Nonlinear Systems
Third Edition
HASSANK. KHALIL
Department of Electrical and Computer Engineering
Michigan State University
PRENTICE HALL
Upper Saddle River, New Jersey 07458
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Contents
<
,
.
xiii
Preface
1 Introduction
1.1 Nonlinear Models and Nonlinear Phenomena . . . . . . . . . . . . . . .
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Pendulum Equation . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Tunnel-Diode Circuit . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Mass-Spring System . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Negative-Resistance Oscillator . . . . . . . . . . . . . . . . . .
1.2.5 Artificial Neural Network . . . . . . . . . . . . . . . . . . . . .
1.2.6 Adaptive Control . . . . . . . . : . . . . . . . . . . . . . . . .
1.2.7 Common Nonlinearities . . . . . . . . . . . . . . . . . . . . . .
1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
'
1
1
5
5
6
8
11
14
16
18
24
2 Second-Order Systeins
35..
2.1 Qualitative Behavior o f Linear Systems
37
2.2 Multiple Equilibria
4 ~ '.
2.3 Qualitative Behavior Near Equilibrium Points . . . . . . . . . . . . . . . 51
2.4 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5 Numerical Construction o f Phase Portraits
59
2.6 Existence o f Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . 61
2.7 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
..................
..............................
................
3 F'lindamental Properties
87
3.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.2 Continuous Dependence on Initial Conditions
and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3 Differentiability o f Solutions and Sensitivity
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . .102
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
vii
.
.
S
r
continuous, square-integrable functions; that is,
uT(t)u(t)dt < m . TO 1 1 1 ~ sure the size of a signal, we introduce the norm function ( ( ~ ( 1 ,which satisfies tllrce
properties:
The norm of a signal is zero if and only if the signal is identically zero and is
strictly positive otherwise.
gcl~crateall output y that does not bclo~igto LQ. Therefore, H is usually defined
as a mapping from an extended space tz to an extended space Lj, where L r is
defined by
L: = { u 1 u, E L ~ : V T [O,m)}
E
and u, is a truncation of u defined by
Scaling a signal results in a corresponding scaling of the norm; that is, llnull =
aJlu\lfor any positive constant a and every signal u.
+
+
The norm satisfies the triangle inequality ((ul u2((I((ulll Ilu2ll for any
signals ul and uz.
The extended space L r is a linear space that contains the unextended space Lm as
a subset. It allows us to deal with unbounded "ever-growing" signals. For example,
the signal u(t) = t does not belong to the space L,, but its truncation
For the space of piecewise continuous, bounded functions, the norm is defined as
belongs to L,
and the space is denoted by L g . For thc space of piecewise continuous, squareintegrable functions, the norm is defined by
<
and the space is denoted by L F . More generally, the space L r for 1 p < m is
defined as the set of all piecewise continuous functions u : (0,bo) -r Rm such that
for every finite
T.
Hence, u(t) = t belongs to the extended space
.c,e
A mapping H : L F -$ L j is said to be causal if the value of the output (Hu)(t)
at any time t depends only on the Values of the input up to time t. This is equivalent
to
.,
(Hu)r = ( H U T ) ,
Causality is an intrillsic property of dynalnical systems represented by state models.
With the space of input and output signals defined, we can now define inputoutput stability.
Definition 5.1 A mapping H : Lp --, L j is L stable i f there exist a class K:
function a, defined on (0, m ) , and a nonnegative constant P such that
The subscript p in L r refers to the type of pnorm used to define the space, while
the superscript m is the dimension of the signal u. If they are clear from the context,
we may drop one or both of them and refer to the space simply as Lp, Lm, or L.
To distinguish the norm of u as a vector in the space t from the norm of u(t) as a
vector in Rm, we write the first norm as 11 l l ~ . ~
If we think of u E Lm as a "well-behaved" input, the question to ask is whether
the output y will be "well behaved" in the sense that y E Lq, where LQis the same
space as Lm, except that the number of output variables q is, in general, different
from the number of input variables m. A system having the property that any
"well-behaved" input will generate a "well-behaved" output will be defined as a
stable system. However, we cannot define H as a mapping from Lm to LQ,because
we have to deal with systems which are unstable, in that an input u E Lm may
2Note that the norm I(.1) used in tho definition of 11. \IC,, for any p E (1, oo], can be any pnorm
in fin'; thc number p is not necessarily thc samc in the two norms. For example, we may define the
LW spwc with Ilullc, =SUPDO Ilu(t)l/l, II~llc, = SUPr>o Ilu(t)llz~Or IIullc, = " P P ~ ~II4t)llO
Ijowevcr, it is common to define the Lz space with the 2-norm in Rm.
+
II(Hu)rllc 5 c-k (llurllc) P
for all u E LT and T E [0,m ) . It is finite-gain L stable i f there &t
constants y and ,O such that
for all u E L r and
T
(5.1)
nonnegative
E (0, m ) .
The constant ,f3 in (5.1)or (5.2)is called the bias term: It is included in the definition
to allow for systems where H u does not vanish a t 'u = 0.3 When inequality (5.2)
is satisfied, we are usually interested in the smallest possible y for which there is P
such that (5.2) is satisfied. When this value of y is well defined, we will call it the
gain of the system. When inequality (5.2) is satisfied with some y 2 0, we say that
the system has an L gain less than or equal to y.
3See Exercise 5.3 for a diKerent role of thc bias term.
198
CHAPTER 5. INPUT-OUTPUT STABILITY
For causal, L stable systems, it can be shown by a simple argument that
199
5.1. 1: STABILITY
Example 5.2 Consider a single-input-singleoutput system defined by the causal
convolution operator
It
y(t) = o h(t - u)u(u) du
and
IIHullc 5 a (llullc) + PI V E Lm
For causal, finite-gain L stable systems, the foregoing inequality takes the form
IlHullc IrlluTllc + P, v u E Lm
The definition of L, stability is the familiar notion of bounded-input-boundedoutput stability; namely, if the system is L, stable, then for every bounded input
u(t), the output Hu(t) is bounded.
where h(t) = 0 for t
< 0. Suppose h E Lie; that is, for every T E [0,m),
Ilhrll~.
If u E Lxe and
T
=
1-
lhr(u)l da =
1;
l h ( a ) ~da
<
1 t, then
Example 5.1 A memoryless, possibly time-varying, function h : [0, m) x R -,R
can be viewed as an operator H that assigns to every input signal u(t) the output
signal y(t) = h(t,u(t)). We use this simple operator to illustrate the definition of L
stability. Let
C
Consequently,
for some nonnegative constants a, b, and c. Using the fact
we have
+
Ih(u)l I a bclul, V u E R
Hence, H is finite-gain L, stable with 7 = bc and /3 = a. F'urt,hermore, if a = 0.
then for each p E [1,oo) ,
Il~rllc, 5 Ilhrllcl ll~TlIc,r V
E
Thus, for each p E 11,oo], the operator H is finite-gain Lp stable with zero bias and
7 = bc. Let h be a time-varying function that satisfies
for some positive constant a. For each p E [l,oo], the operator H is finite-gain L,
stable with zero bias and 7 = a. Finally, let
Since
2
+
C
'1 !&
I -
;:&
sho~vsthat the system is finite-gain L, stable. The condition Ilhllc, < actually
guarantees finite-gain Lp stability for rnrll p E [l,co].Consider first. the case p = 1.
For t 5 T < m, we have
Reversing the order of integration yields
t>o
H is L, stable with zero bias and a ( r ) = r2. It is not finite-gain L, stable
because the function h(u) = u2 cannot be bounded by a straight line of the form
(h(u)(5 r(u( P, for fill u E R.
A
C
C
This inequality resembles (5.2), hut it is not the same as (5.2) because the constant
-y in (5.2) is required to be indcpentlcnt uf T . While IlhTllcl is finitc for every finite T ,
it inay not be bounded uniformly in r. For example, h(t) = et has l)h,)\c, = (eT -I),
which is finitc for all T E [O,oo) but not uniformly bounded in r. Inequality (5.2)
will be satisfied if h E L1; that is.
Then. the inequality
I1
c
10,~)
Thus,
:
c:
1.d
_
i
#
I
.
I L..-
,.
.
. .
/
200
.
_
_
'
"'
.._
8
2
.
.
-
.
...
.
.
5.2.
.
.
.
-
. : < .
..
L STABILITY OF STATE MODELS
.lt,
\ '
h
a .
.
. -.-. .
< ;. ..
,
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..
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...--.----*-
,
-. .
_.
._"
201
CIIAPTER 5 . INPUT-OUTPUT STABILITY
Consider now the case p E (1, co) and let q E ( 1 , ~ be
) defined by l l p
For t 5 T < co, we have
Example 5.3 Colisider ti sii1gle-i1i~)11t-si1ig1~out~)ut
systelii dcli~icdby 1110lionlill-
+ l / q = 1.
earity
y =tanu
The output y ( t ) is defined only when the input signal is restricted to
Jo
5
(l
lh(t - 41 d o ) llq
(/f lh(t - 41 iu(o)lpd o )
l/P
Thus, the system is not L,
restrict u ( t ) to the set
o
stable in the sense of Definition 5.1. However, if we
( u J5 r
then
l ~ 5l
wherc the second inequality is ol)taiiletl I y npplying Holder's inequality.4 Thus,
(F)
I4
for every u E Lp such that Iu(t)l 5 r for all t 2 0 , where p could be any number
in [1,oo]. Ig the space L,, the requirement Iu(t)l 5 r implies that ~ J U ( ( 5~ ,r,
showing that the foregoing inequality holds only for input signals of small norm.
However, for other Lp spaces with p < oo the instantaneoua bound on (u(t)l does
not necessarily restrict the norm of the input signal. For example, the signal
u ( t )=
~
~
5
~
~
~
~~ ~ u r ~ l I c J ~ L
h
~
~
~
~
l
1
~
In summary, if h l l L , < co,,t.helhfor each p E [ I , co], the causal convolution operator
is finite-gain Lp stable and (5.2) is satisfied with 7 = llhllLl and P = 0.
A
One drawback of ~efiniiion5.1 is the implicit requirement that inequality (5.1)
or inequality (5.2) be satiified for all signals in the input space Lm. This excludes
systems where the input-out.put relation may be defined only for a subset of the
i~lputslx~(:(:.The next cxairiplc c~xplorcst.hc point and motivates the definition of
small-signal L stability that follows tlie example.
.'H61<ler1sinequality states that if f E C,,,.and g E Cp,, where p E ( 1 , ~ and
) l/p
tllcll
for every r E [O,m). (See [14].)
2
and the system will satisfy the inequality
By reversing the order of integration, we obtain
Hence,
A
<-
+ l/q = 1,
4
>0
-.
'
,->
.
which belongs to Lp for each p E [l,oo], satisfi'& the-instantaneous bound lu(t)1 5 r
while its L, norm
can be arbitrarily large.
A
Definition 5.2 A mapping H : L r -, Lz is small-signal L stable (respectively,
small-signal finite-gain L stable) if there is a positive constant r such that inequality
(5.1) [respectively, inequality (5.2)]is satisfied for all u E L r with sup,,^,^, ((u(t)(l5
r.
' '
(
1
f
5.2
f, Stability of State Models
The notion of input-output stability is intuitively appealing. This is probably
why most of us were iiltroduced to dynamical systein stability in tlic framework
of bounded-input-bounded-output stability. Since, in Lyapunov stability, we put a
I
202
CHAPTER 5. INPUT-OUTPUT STABILITY
lot of emphasis 011 studying the stability of equilibriu111poilits and blie asymptotic
behavior of state variables, one may wonder: Whnt can we see about input-outpl~t
stability starting from the formalism of Lyapunov stability? In this section. we
show how Lyapunov stability tools can be used to establish L stability of nonlinear
systems represented by state models.
Consider the system
203
5.2. L STABILITY OF STATE MODELS
Then, for each xo with lxol\ 5 r m , the system (6.3)--(6.4)is snrcrll-siglrul
finite-gain L, stable for each p E [ I , co]. In particular, for each u E Lp, with
supo,,,, ((u(t)ll5 mill{r,, c l c 3 r / ( c ~ c ~ Lthe
) } , output ~ ( tsatisfies
)
(5.11)
IIY~IIL, I
YJI'LLTIIL,
+P
for all 7 E [O, m), with
f 1,
where x E Rn, u E Rm, y E Rq, f : [0,co) x D x D, 4 Rn is piecewise continuous in
t and locally Lipschitz in ( 2 ,u), h : [O, m)x D x D, 4 Rq is piecewise cont.inuous in
t and continuous in (I,u), D c Rn is a domain that contains x = 0, and D, C R'"
is a domain that contains u = 0. For each fixed xo E D, the state model given
)
by (5.3) and (5.4) defines all operator H that assigns to each input signal ~ ( tthe
corresponding output signal y(t). Suppose x = 0 is an equilihritim point of the
unforced system
i = f(t,x,O)
(5.5)
The theme of this section is that if the origin of (5.5) is uniformly asymptotir~lly
stable (or exponentially stable), then, under some assumpt.ions on f and h. the
system (5.3) and (6.4) will be L 3able or small-signal L stable for a certain sigl~ill
space L. We pursue this idea first in the case of exponent.ially stability, and then
for the more general case of uniform asymptotic stability.
Theorem 5.1 Consider the system (5.3)-(5.4) and take r > 0 and r, > 0 smh
that (11x11 i: r r c D and (I(uJ(I r U )c Du.Suppose that
x = 0 iB an eezponentially stable equilibrium point of (6.5), and therc is
Lyapunov hnction V(t,x) that satisfies
if p = m
Furthermore, if the origin is globally exponentially stable and all the assumptions
hold globally (with D = Rn and D, = Rm), then, for each x0 E Rn, the system
0
(5.3)-(5.4) is finite-gain Lp stable for each p E. [I,co].
Proof: The derivative of V along the trajectories of (5.3) satisfies
I -c311xl12
Take \V(t) =
obtain
+ c4LIlxl1 lI'1~1l
d m . Lirhen V ( t , r ( t ) #) 0, use w = ~ / ( 2 f land
) (5.6) to
) 0, it can be verified5 that
When V ( t , x ( t ) =
u
Hence,
for all values of V ( t , x ( t ) ) . By (the comparison) Lemma 3.4, I.V(t) satisfies the
inequality
1
for all ( t ,x ) E [ O , c u ) x D for some positive constank cl, cz, CJ, an.d c4.
f and h satisfy the inequalities
Ilf ( 4
XI
U)-
f ( t ,x,O)Il I
LII'LLII
(5.9)
Using (5.6), we obtain
(5.10)
Ilh(t,214
11 < 771 11x11 + 772 llull
for all ( t ,x, u ) 6 [O, m) x D x D, for some nonnegative constants L. 71, and
112
"~ee
Exercise 5.6.
.
.
,' . . .
204
-:
-
- 2 .
CHAPTER 5. INPUT-OUTPUT STABILITY
It, (:ail I,(! casily verified that
ellsure that ( ( x ( t ) ( 5
( r; hence, z ( t ) stays within the domain of validity of the
n.sr~mpt~ioris.
Using (5.10), we 1 1 i ~
208
5.2. C STABILITY OF STATE AlODELS
Corollary 5.1 Suppose that, i n sonre neighborhood of ( x = 0 , u = 0 ) , the junction
f ( t ,x , u) is continuously diflerentiable, the Jacobian matrices [ 8j / B x ] and [af /au]
are bounded, uniformly i n t , and h ( t ,x, u ) satisfies (5.10). If the origin x = 0 is a n
exponentially stable equilibrium point of (5.5), then there is a conatant ro > 0 such
that for each xo with llxoll < ro, the system (5.3)-(5.4) is small-signal finite-gain
t pstable for each p E [ I ,oo]. Furthermore, i j all the assumptions hold globally and
the origin x = 0 is a globally exponentially stable equilibrium point of (5.5), then for
each t o E Rn,the system (5.3)-(5.4) is finite-gain Lp stable for each p E [l,oo]. 0
For the linear time-invariant system
where
c3
k3 = m,' a = 2c2
= *,
2c1
Set
the global exponential stability condition of Theorem 5.1 is equivalent to the condition that A is Hurwitz. Thus, we have the following result for linear systems:
Suppose now that u E C$ for some p E [ l ,oo].Using the results of Example 5.2, it
can be easily verified that
k2
5
Corollary 5.2 The linear time-invariant system (5.14)-(5.15) is finite-gain Lp stable for each p E [ l ,oo] if A i s Hurwitz. Moreower, (5.11) is satisfied with
a
((M~LT(~c,~
((uT((~p
It is also straightforward to see rhat
where
f
ifp=oo
As for the first term, y l ( t ) , it car1 be verified that
and P is the solution of the Lyapunov equation P A + A ~ =
P -I.
We leave it for the reader to derive the foregoing expressions for 7 and
Thus, by the triangle in equal it.^, (5.11) is satisfied with
When all the assumptions hold globally, there is no need to restrict l(xoll or the
illst,ant,ancousvalues of Ilu(t)ll. Therefore, (5.11) is satisfied for each xo E Rn and
21 E Lpe.
The ~ w of
c (the converse Lynpiinov) Throrrm 4.14 shows t.he existence of a Lya1111nt1vfiirlc*t,ionsatisfying (5.6) tllrc111g11(5.8). Comcc~uc~~tly,
we haw t l ~ efollowing
corol1al.y.
0
P.
Example 5.4 Consider the single-input-single-output, first-order system
The origin of j. = -I - x3 is globally exponentially stable, as can be seen by
the Lyapunov function V ( x ) = x2/2. The function V satisfies (5.6) through (5.8)
globally with cl = c2 = 1/2, cs = cd = 1. The functions f and h satisfy (5.9) and
(5.10) globally with L = 711 = q 2 = 1. Hence, for each xo E R and each p E 11, oo],
the system is finite-gain L,, stable.
A
206
CHAPTER 5. INPUT-OUTPUT STABILITY
Example 5.5 Consider the single-input-singleoutput second-order system
$/
.".
21
f and h satisfy the inequalities
for all (t, x, u) E [O, CO) x D x
nonnegative constant q.
V(x) = x T p x = p11x: f 2 ~ 1 2 5 1 ~+p22x;
2
as a Lyapunov function candidate for the ullforced system:
li = -2plz(x:+ax1
207
STABILITY OF STATE MODELS
(5.20)
Ilh(tj~l~5
) \ la~(11~11)
+ a7(llull) + q
where a is a nonnegative constant. Use
U
&"
=
5.2. L
D,,for some class K functions as to a7, and a
Then, for each xo with llxoll 5 cr;'(al(r)),
L, stable.
th'< system (5.3)-(5.4) is small-signal
0
Proof: The derivative of V along the trajectories of (5.3) satisfies
tanhx1)+2(~ll- P I P - P ~ ~ ) X I X P - ~ ~ Ptanh
P ~ XXIP- 2 ( p ~ ~ - ~ ~ ~ ) x i
+
Choose pll = pl2 p22 to cancel the cross-product term 21x2. Then, taking p22 =
2p12 = 1 makes P positive definite and results in
v = -2: - x; - ax1 tanhxl - 2ax2 t,anhxl
Using the fact that
XI
where 0 < 0 < 1. Set
tanhxl 2 0 for all X I E R, we obtain
ad?.)as ( s u ~ o ~Ilu(t)
t ~ , 11)
=a,'
Thus, for all a < 1, V satisfies (6.6) through (5.8) globally with cl = Xmi,(P),
= Am,(P), ca = 1 - a, and c4 = 21(P((2= 2Am,(P). The functions f and h
satisfy (5.9) and (5.10) globally with L = ql = 1, 112 = 0. Hence, for each $0 E R
and each p E [1,oo], the system is finite-gain L, stable.
A
CP
We turn now to the more general case when the origin of (5.5) is uniforlnlp
asymptotically stable and restrict our attention to the study of L, stability. The
next two theorems give conditions for small-signal L, stability and L, stability,
respectively
Theorem 5.2 Consider the systen~(5.3)-(5.4) and take r
r ) C D. Suppose that
> 0 such that {(Jx(I5
6
and choose r, > 0 small enough that {lluII
SUP,^,^, llu(t)ll 5 ru. 'rhen,
I r,)
C D U and 1-1
By app1yillg Theorem 4.18, we conclude from (4.42) and (4.43) that ((x(t)((satisfies
the inequality
(5.21)
for all 0 I t I s, whcre fi and
Using (5.20), we obtain
-, arc class KC and class K filnctions, respectively.
x = 0 is a uniformly asymptotically stable equilibrium point of (5.5). nnd there
is a Lyapunov function V(t,x) that satisfies
where .we used t.he general property of class K functions
a(a
for all ( t . ~ E) [O. c ~ x)D fov some class K functions crl to a4.
2,
< a i l ( a l ( r ) ) for
=See Exercise 4.35.
+ b) 5 a(2a) + a(2b)
-.. .... . ..
.
5.3.
:>
'
;
Thus,
IIYTI~L,
j
where
yo = a s 0 27
..,
-
-\
.-I
./
.~,)
-..
..i
-
+ a7
5 YO ( I I U T ~ I +
L ~PO
)
and
(5.22)
Po = a~(2P(Il~oll,
0)) +
-
.-'
.
-
I
,/'
,,'
.~,,
..
.
i
.
.. .
.4,
. '
L2 GAIN
"
$
.
d
a
,
..'_.
_
.-.-..-
.,-.,,,----.
-!
209
c
We saw in Example 4.26 tliut the st~itcotluntio~lis input-t o-stat>est.111)lc.T l ~ ootput
function h satisfies (5.23) globally with a l ( r ) = r2, a2(r) = r , and q = 0. Thus,
A
the system is L, stable.
Example 5.7 Consider the singleinputsingleoutput second-order system
The use of (the converse Lyapunov) Theorem 4.16 shows the existence of a
Lyapunov function satisfying (5.16) hhrough (5.18). Consequently, we have the
following corollary:
Corollary 5.3 Suppose that, i n some neighborhood of (x = 0, u = O), the function
f ( t ,x, U) is continuous1y differentiable, the Jacobian matrices [af /ax] and [af /Bu]
are bounded, uniformly i n t , and h(t, x, u) satisfies (5.20). If the unforced system
(5.5) has a uniformly asymptotically, stable equilibrium point at the origin x = 0,
then the system (5.3)-(5.4) is small-signal L, stable.
0
.-.'\
.J
I .
To extend the proof of Theorem 5.2 to show Lm stability, we need to demonstrate that (5.21) holds for any initial state s o E Rn and any bounded input. As we
discussed in Section 4.9, such inequality will not automatically hold when the conclitions of Theorem 5.2 are satisfied globally, even when the origin of (5.5) is globally
uniformly asymptotically stable. However, it will follow from input-testate stability
of the system (5.3), which can be checked using Theorem 4.19. ,
Theorem 5.3 Consider the system (5.3)-(5.4) with D = Rn and D, = Rm. Suppose that
The system (5.3) is input-to-state stable.
where g(t) is continuous and bounded for all t
2 0. Taking V = (x: + x i ) , we have
Using
yields
Thus, V satisfies inequalities (4.48) and (4.49) of Theorem 4.19 globally, with
al (r) = az(r) = r2, W3(x) = -(1 - 0) llxllf, and p(r) = (2r/B)ll3. Hence, the
state equation is input-testate stable. Furthermore, the function h = x l 2 2 satisfies (5.23) globally with a1(r) = f i r , a 2 = 0, and q = 0. Thus, the system ia L,
stable.
A
+
h satisfies the inequality
5.3
for all ( t ,x, u) E [0, oo) x R" x Rm for some class K: functions a l ,
nonnegative constant q .
Then, for each xo E Rn, the system (5.3)-(6.4) is L,
stable.
122,
and a
0
Proof: Input-testate stability shows that an inequality similar to (5.21) holds
for any xo E Rn and any u E Lmf Tlie rest of the proof is the same as that of
Theorem 6.2.
Example 5.6 Consider the single-input-singlooutput first-order system
L2 Gain
L2 stability plays a special role in systems analysis. It is natural to work with squareintegrable signals, which can be viewed as finiteenergy signals.' .In many control
the system is represented as an input-output map, from a disturbance
input u to a controlled output y, which is required to be small. .With L2 input
signals, the control system is designed to make the input-output map finitegain
L2 stable and to minimize the L2 gain. In such problems, it is important not only
to be able to find out that the system is finitegain L2 stable, but also to calculate
the L2 gain or an upper bound on it. In this section, we show how to calculate the
L2 gain for a special class of timeinvariant systems. We start with linear systems.
'If you think of u(t) as current or voltage, then uT(t)u(t)is proportional to the instantaneoue
power of the signal, and its integral over all time is a measure of the energy content of the signal.
the literature on H , control; for example, (201, [54], [61], [go], (1991, or (2191.
.
- ..a
210
CHAPTER 5. INPUT-OUTPUT STABILITY
Theorem 5.5 Consider the time-invariant nonlinear system
Theorem 5.4 Consider the linear time-invariant system
2 = Ax+Bu
y = Cx+Du
where A is ~ u k t z .Let G ( s ) = C ( s I - A)-'B
system is SUPwE~
llG(jw)ll~.~
5.3. L2 GAIN
(6.24)
(5.25)
+D.
Then, the Lz gain oj the
0
Proof: Due to lincnrity, we set x(0) = 0. From Fourier trnnsforln tlleory,'" w c
know that for a causal signal y E L 2 , the Fourier transform Y ( j d ) is given by
where f (rc) is locally Lipschitz, and G ( x ), h ( x ) are continuous over Rn . The matrix
G is n x m and h : Rn -t Rq. Tllc fi~nctions f and h vanish at the origin;
that is, f ( 0 ) = 0 and h(0) = 0 . Let y be a positive number and suppose there is
positil~esenridefinite function V ( x ) that satisfies the
a continuowly diffei~ntiable,
inequality
def DV
( V ,f G , h 7 ) = f
(
1
av
x )- G ( x ) G T ( )
27 a x
( ) + 51h T ( x ) h ( x ) 5 0 (5.28)
for all x E Rn. Then, for each xo E Rn, the system (5.26)-(5.27) is finite-gain L2
0
stable and its L2 gain is less than or equal to 7 .
and
Y ( j w )= G ( j w ) U ( j w )
Proof: By completing the squares. we have
Using Parseval's theorem," we can write
Substituting (5.28) yields
I
which shows that the Lz gain is less than or cqual to SUPwE~
(IG(jw)1J2.Showing
1JG(jw)II2is dolie by a contrudictioii arguilieilt
that the Lz gain is equal to S\IPwE~
that is given in Appelidis C.10.
Hence,
Sote that. tlie left-liand side of (5.29) is the derivative of V along the trajectories
of the system (5.2G). Integrating (5.20) yields
The case of linear time-inwriant systems is cxccptional ill that we can actually
find the exact Lz gain. In more general cases. like the case of the next theorem. we
can only find an upper bound on the Lz gain.
BThis is the induced 2-norm of the complex matrix G ( j w ) , which is equal to
Am,[GT(-jw)G(jw)] = umsx[G(jw)]. Thls quantity is known as the H, norm of G ( j w ) ,
$en G ( j w ) is viewed r an element of the Hardy apace H,. (See [GI].)
'Osee
>
where x ( t ) is the solution of (5.26) for a given u E LZe. Using V ( x ) 0, we obtain
(53).
'LPamval'stheorem (531 states that for a caus~lsignal y E Ls,
Taking the square roots and using the inequality
numbers a and b, we obtain
+d
ll~l'llc?5 711~~rllc2
d
m 5 a + b for nonnegative!
m
(6.30)
The idea of t.he preceding example is generalized in the next one.
•
which completes the proof.
Example 6.9 Consider the nonlinear system (5.26)-(5.27), with m = q, and sup-
Inequality (5.28) is knawn as the Hamilton-Jacobi inequality (or the HamiltonJacobi equation when is replaced by =). The search for a function V(x) that satisfies (5.28) requires basically the solution of a partial differential equation, which
might be difficult to solve. If we succeed in finding V(x), we obtain a finite-gain
L2 stability result, which, unlike Theorem 5.1, does not require the origin of the
unforced system to be exponent,ially stable. This point is illustrated by the next
..
example.
pose there is a continuously differentiable positive semidefinite function W(x) that
satisfies12
<
awX
(
)
<
-khT&)h(x),
To satisfy (5.28), we need to choose a > 0 and -y
where a and k are positive constants. The unforced system is a special case of
the class of syst.ems treatad in Esnlnplc 4.9. In that example, ivc used the energylike Lyapunov function V(x) = a r f / 4 xi12 to show that the origin is globally
asymptotically stable. Using V(x) = a(axfl4 x;/2) with a > 0 as a candidate
for the solution of the Hamilton-Jacohi inequality (5.28), it can be shown that
(
To satisfy (5.28), we need to choose a
a2 1
+272 1)xi
+
Example 5.10 Consider the nonlinear system (5.26)-(5.27), with m = q, and suppose there is a continuously differentiable positive semidefinite function W(x) that
satisfies13
,
> 0 and 7 > 0 such that
d2
1
-ab+-+-<O
27'
2-
aw
aw
-G(x)
ax
-ax
f(~)
(5.31)
By simple algebraic manipulati~n,we can rewrite this inequality as
\7
'
5 0
(5.34)
= hT(x)
(5.35)
for all x E Rn.The output feedback control
a2
2- 2ak - 1
u=-ky+v,
Since we are interested in the smallest possible 7,we choose a to minimize the
right-hand side of the preceding i~aquality. The minimum value l/k2 is achieved
at a = 1/k. Thus, choosing 7 = l/k, we conclude that the system is finite-gain
L2 stable and its Lz gain is less than or equal to Ilk. We note that. the conditions
of Theorem 5.1 are not satisfied in t.his example because the origin of the unforced
system 'is not exponentially stable. Linearization at the origin yields the matrix
I
which is not Hurwitz.
> 0 such that
This inequality is the same as inequality (5.31) of Example 5.8. By repeating the
argument used there, it can be shown that the system is finite-gain L2 stable and
A
its Lz gain is legs than or equal to l l k .
+
Z(V, f.,G, h, y) = -ak
(5.32)
for all x E Rn.Using V(x) = aW(x) with a > 0 as a candidate for the solution of
the Hamilton-Jacobi inequality (5.28), it can be shown that
Example 5.8 Consider the single-input-single-output system
+
k>O
A
k>O
results in the closed-loop system
= f(x)
y
aw + G(X)V dl(
- ~G(X)GT(X)(z)
= h(x) = GT(x)
(z)
+
fC(x) G(X)V
-
''A system satisfying (5.32) and (5.33) will be defined in the next chapter as an output strictly
passive system.
13A system satisfying (5.34) and (5.35) will be defined in the next chapter as a passive system.
We will come back to this example in Section 6.5 and look at it as a feedback connection of two
passive systems.
214
CHAPTER 5. INPUT-OUTPUT STABILITY
5.3.
Lz GAIN
It can bc easily verified that. for tlic closed-loop systcni, I.V(x) sntisfics (5.32) and
(5.33) of the previous example. Hence, the input-output map from v to y is finitegain L2 stable and its L2 gain is less than or equal to ilk. This shows, in essence,
that the L2 gain can be made arbitrarily small by choosing the feedback gaiu k
sufficiently large.
A
Lemma 5.1 Suppose tlre assuttlplio7rs of Tlreorem 5.5 an: sulis/icrl on u rlo71rr~i7r
D c R" that contains the origin, f ( x ) is continuously diferentiable, and x = 0 is
an asymptotically stable equilibrium point of x = f ( x ) . Then, there is kl > 0 such
Example 5.11 Consider the linear time-invariant syst.em
Proof: Take r > 0 such that {llxll 5 r ) c D: By (the converse Lyapunov) Theorem 4.16, thcrc exist ro > 0 and a coiitinuously differentiable Lyapunov function
W ( x ) that sat.isfics
a1(lIxll) 5 I.V(x) 5 a2(11x11)
5 = Ax+Bu
y = Cx
that for each xo with Ilxol) <- kl, the sy8tem (5.26)-(5.27) is small-signal finite-gain
0
L2 stable with L2 gain less than or equal to 7 .
Suppose there is a positive semidefinite solution P of the Riccati equation
c.
for all J J x )<) ro: for some class K functions a1 to as. The derivative of W along
the trajectories of (5.26) satisfies
4
for some 7 > 0. Taking V ( x ) = ( 1 / 2 ) x T P x and using the expression [ 8 V / a z ] =
x T P , it can be easily seen that V ( x ) satisfies the Hamilton-Jacobi equation
Hence, the system is finite-gain L2 stable and it,s L2 gain is less than or equd to
7. This result gives an alternative method for computing an upper bound on the
L2 gain, as opposed to the frequency-domain calculation of Theorem 5.4. I t is
interesting to note that the existence of a positive semidefinite solution of (5.3G) is
a necessary and sufficient co~iditionfor the L2 gain to be less than or equd to ?.I4
A
where k is an upper bound on )/6L1'/8x(J.L is a Lipschitz constant of f with respect
to u , and 0 < 0 < 1. Similar to the proof of Theorem 5.2, we can apply Theorem 4.18
to show that there exist a class KC function 0, a class K function 70,and positive
constants kl and k2 such that, for any initial state xo with llxoll 5 kl and any input
u ( t ) with supost,, JJu(t)ll5 k2, the solution x ( t ) satisfies
In Theorem 5.5, we assumed that the assumptions hold globally. It is clear from
the proof of the theorem that if the assumptions hold only on a finite domain D,
we will still arrive a t inequality (5.30) as long as the solution of (5.26) stays ill D.
+ 70(ozwT
II~(~)II)
I&
j<
'C
I I X ( ~ ) I5
I P(IIXOII.~)
for all 0 I t 5 T . Thus, by choosing k1 a i d k2 small enough, we can be sure that
((x(t)(I5 I. for all 0 5 t 5 7.The lemnia follows then from Corollary 5.4.
Corollary 5.4 Suppose the assumptions of Theorem 5.5 are s a t w e d on n domnin.
D C Rn that contains the origin. Then, for any x0 E D and any u E Clefor which
the solution of (5.26) satisfies x ( t ) E D for all t E 10, T ] , we have
To apply Lemma 5.1, we need to check asymptotic stability of the origin of x = f ( x ) .
This task can be done by using linearization or searching for a Lyapunov function.
The next lemma shows that, under certain conditions, we can use the same function
V that satisfies the Hamilton-Jacobi illequality (5.28) as a Lyapunov function for
showing asyniptotic stability.
+ @TJ
I I ~ r l l ~I z7 l l ~ r l l ~ Z
Lemma 5.2 Suppose the assumptions of Theorem 5.5 are satisfied on a domain
D C Rn that contains the origin, f ( x ) is continuously diflerentiable, and no solution
of x = f ( x ) can stay identically in S = { x E D I h ( x ) = 0) other than the trivial
solution x ( t ) r 0. Then, the ortgin of x = f ( x ) is asymptotically stable and there is
Ensuring that the solutioil x ( t ) of (5.26) remains in some neighborhood of the origin, when both ))xo((and supoltlrl)u(t)ll are sufficiently small, follows from asymp
totic stability of the origin of k = f ( x ) . This fact is used to show small-signal L2
stability in the next lemma.
' L<
:a:
3
;d:
'I L
kl > 0 such that for each xo with ))xO1l5 kl: the system (5.26)-(5.27) is small-signal
finite-gain L2 stable u~tthL2 gain less than or equal to 7.
'"SW 1541 for tlrc proof of ~ r ~ c ~ i t y .
C
-
.-
-.
.
.. -
.- .
.
.....
/
.......
.
.......
.k.
.
- .
- -- --.-.
/
....
- ..
.,
, .
216
,
.
.
.
.I..-.
CHAPTER 5. INPUT-OUTPUT STABILITY
Proof: Take u(t) 3 0. By (5.28), we have
5 T ) c D. We will show that V(x) is positive
Take T > 0 such that B, = {JJxJJ
definite in B,. Toward that end, let +(t; x) be the solution of i = f (x) that starts
at d(0;x) = x E B,. By existence and uniqueness of solutions (Theorem 3.1) and
continuous dependence of the solution on initial states (Theorem 3.4), there exists
6 > 0 such that for each x E B, the solution d(t; x) stays in D for all t E [O, 61.
Integrating (5.37) over [0, r] for r < 6, we obtain
07
1'
+ 0 su~cllthat V(E) = 0.
Ilh(d(t; s))ll; dt = 0, V r E [O,61
+
The foregoing inequality
217
Repeating the argument used in Example 5.8, it can be easily seen that by choosing
a = 7 = 1/k, inequality (5.28) is satisfied for all x E R2. Since the conditions
of Theorem 5.5 are not satisfied globally, we investigate small-signal finite-gain
stabiiity by using Lemma 5.1. We need to show that the origin of the unforced
system is asymptotically stable. This can be shown by linearization at the origin,
which results in a Hurwitz matrix. Alternatively, we can apply Lemma 5.2, whose
conditions are satisfied in the domain D = {(xl(< d),
because
Thus, we conclude that the system is small-signal finite-gain L2 stable and its Lz
gain is less than or equal to 1/k.
A
5.4
Using V(+(r;x)) 2 0, we obtain
Suppose now that there i s 5
implies that
5.1. FEEDBACK SI'STEAfS: TIIE SAfALL-GAIN THEOREAl
Feedback Systems: The Small-Gain Theorem
The formalism of input-output stability is particularly useful in studying stabiiity
of interconnected systems, since the gain of a system allows us t o track how the
norm of a signal increases or decreases a s it passes through the system. This is
particularly so for the feedback connection of Figure 5.1. Heri, we have two systems
H1 : C r C,9 tnd Hz :Lz -* L,". Suppose both systems are finite-gain L stable;15
that is,
h(+(t; 2)) r 0. V t E (0.4
Since during this interval the solr~tionstays in S and, by assumption, the only
solution that can stay identically in S is the trivial solution, we conclude that
$(t;x) r 0 + x = 0. Thus, V(x) is positive definite in B,. Using V(s) as a
Lyapunov function candidate for x = f (x), we conclude from (5.37) and LaSalle's
invariance principle (Corollary 4.1) that the origin of x = f (x) is asymptotically
0
stable. Application of Lemma 5.1 completes the proof.
Example 5.12 As a variation on tke theme of Examples 5.8 and 5.9, consider the
system
where a, k > 0. The function V ( x ) = a [a (2712 - xf/12) + xi/2], with cr > 0,
is positive semidefinite in the set {Ixll 5 6).
Using V(x) as a candidate for the
solrrtion of t,he Hamilton-Jncobi inrtl~~alit,y
(5.28), it can be shown that .
Suppose further that the feedback system is well defined in the sense that for.everv
pair of inputs u1 E L r and u2 E Lz, there exist unique outputs el, y2 E C r and
e2, yl E Cf.16 Define
The question of interest is whether the feedback connection, when viewed
.- m
-n
mapping from the input u to the output e or a mapping from input u to the output
y, is finite-gain C stable.17 It is not hard to see (Exercise 5.21) that the mapping
from u to e is finite-gain C stable if and only if the mapping from u to y is finite-gain
L stable. Therefore, we can simply say that the feedback connection is finite-gain
'=In this section, we present a version of the cla~sicalsmall-gain theorem that applies to finitegain t stability. For more general versions which apply to C stability, eee [93] and (1231.
'"ufficient conditions for existence and uniqueness of solutions are available in the iiterature.
The most common approach uses the contraction mapping principle. (See, for example, (53,
Theorem IIL3.11.) A more recent approach that makes use of existence and uniquenw of the
solution of state equations can be found in (931.
"See Exercise 5.20 for an explanation of why we have to consider both Inputs and outputs in
studying the stabiiity of the f e e d b d connection.
.
CHAPTER 5. INPUT-OUTPUT STABILITY
5.4. FEEDBACK SYSTEAIS: THE SMALL-GAIN TIfEOREM
219
The feedback connection of Figure 5.1 provides a convenient setup for studying
robustness issues in dynamical systcms. Quite often, dynamical systems suhjcct
to model uncertainties can be represented in the form of a feedback connection
with HI, say, as a stable nominal system and Hz as a stable perturbation. Then,
the requirement 7172 < 1 is satisfied whenever 7 2 is small enough. Therefore,
the small-gain thcorem provides a conceptual framework for understanding many
of tlie robustness results that arise in the study of dynamical systems, especially
when feedback is used. Many of the robustness results that we can derive by using
Lyapunov stability techniques can be interpreted as special cases of the small-gain
theorem..
Example 5.13 Consider the feedback connection of Figure 5.1. Let H1 be a linear
time-invariant system with a Hurwitz square transfer function matrix G(s)= C(sIA)-'B. Let Hz be a memoryless function e2 = $(t, y2) that satisfies
Figure 5.1: Feedback connection.
L stable if either mapping is finite-gain L stable. The following theorem, known as
the small-gain theorem, gives a sufficient condition for finite-gain L stability of tlie
feedback connection.
Theorem 5.6 Under the preceding assumptions, the feedback connection is finitegain L stable ifrlr2< 10
Proof: Assuming existence of tlie solution, we can write
elr = ulr
- ( H ~ Q ) ~ ,e 2 =~ U ~ +T ( H l e d r
fiom Theorem 5.4, we know that HI is finite-gain L2 stable and its L2 gain is given
by
71 = SUP llG(j4112
WER
We have seen'in Example 5.1 that Hz is finitegain L2 stable and its L2 gain is less
than or equal to 72. Assuming the feedback connection is well defined, we conclude
by the small-gain theorem that it will be finite-gain L2 stable if 7172 < 1.
A
Example 5.14 Consider the system
Then,
1
llelrll~5 (Ilulrllc + r z l l ~ 2 ~ l+
l cP2 + r ~ P 1 )
1 - 7172
,
(5.40)
for all r E [0,oo). Similarly,
1
l(e2~llcI (11wtTllc+ n ~ ~ ~+PI
l r+ ~ 3~ 2 c)
1 - 7172
whcre f is a smooth function of its argumcnts, A is a Hurwitz matrix, -CA-'B = I ,
is a small positive parameter, and dl, dz arc disturbaiice signals. The liiiear part
of this modrl represents actuator dynamics that are, typically, much faster t,haii thc
plant dynamics represented here by the nonlinear equation S = f . The disturbance
signals dl and d2 enter the system at the input of the plant and the input of the
actuator, respectively. Suppose the disturbance signals dl and d2 belong to a signal
space L, where L could be any L, space, and the control goal is to attenuate the
effect of this disturbance on the state x. This goal can be met if feedback control can
be designed so that the closed-loop input-output map from (dl, d2) to x is finitegain L stable and the L gain is less than some given tolerance 6 > 0. To simplify
the design problem, it is common to neglect the actuator dynamics by setting E = 0
and substituting v = -CA-I B(u + d2) = u + d2 in the plant equation to obtain the
reduced-order model
x = f(t,rr,u f d)
-.
Since 7172 < 1,
'*
(5.41)
for all r E [O,oo). The proof is competed by noting that Ilellc 5 Ilelllc + llezllc,
which follows from the triangle inequality.
.......
-..
.
....
.
.
.-:.
.-
,.-
CIqAPTER 5. INPUT-OUTPUT STABILITY
where d = dl + d 2 . Assuming that the state variables are available for measurement,
we use this model to design a state feedback control law u = y ( t , x ) to meet the
design objective. Suppose we have succeeded in designing a smooth state feedback
control u = ~ ( xt ), such that
I I ~ L5 7
~ ~+dfi ~ ~ c
(5.42)
for some y < 6. Will the control meet the design objective when applied to the
actual system with the actuator dynamics included? This is a question of robustness
of the controller with respect to the unmodeled actuator dynamics.18 When the
control is applied to the actual systeln, the closed-loop equation is given by
.,
.......
.- . . . . . -.. .....
r
...,
.. ....
+ A-'B(?(t, + d?(t)]
:1
7
=-
+
- '.
.
I
.
.
..&
.
*
I
..\_....
f ( t .x , $ 4
5)
-..
.
.
.-..
I1
I ~ Y ~ I6L ^~zlkzllr.+ P 2
U ~ y f l l e z l l r .+
where
+d ( t )+ C q )
(5.43)
2~k~=(Q)IIA-1Bl1211~l12,
P2
Amin(&)
1,
, ~ 2
/-
p , l c l 1 2 1 1 ~ L~ ,o=~( Q
l l)
xmin(Q)
ifp=w
and Q is the solution of the Lyapunov equation QA+ATQ = -I.lg Thw, ssuming
the feedback connection is well defined, we conclude from the small-gain theorem
that the input-output map from u to e is f, stable. &om (5.40), we have
f ( t ,x , r ( t ,5 ) + e l )
ar
a7
Yi .= ? = ~ + ~ f ( t , s , ~ ( t , x ) + e ~ )
Using
llxllr. 5 ~llelllr.+ P
which follows from (5.42), and the definition of u1 and u2, we obtain
H2 defined by
I t is interesting to note that the righehand side of (5.44) approaches
and
~1
= dl
+ d? = d ,
u2 = d2
In this representation, the system H I is t,he nominal reduced-order closed-loop system, while Hz represents the effect of the u~lmodeleddynamics. Setting E = 0
'"n Example 11.14, we investigate a sinrilar robustness problem, in the context of stabilization,
using singular perturbation theory.
..
221
+
7f=
+ or
-.-
+ $ f ( t l x j 7 ( t j x ) + e l ) i c i l l ~ l l+c2llelll
X)
- f ( t ,x , y ( t , x ) + d ( t ) + C q )
at ax
It is not difficult to see that the closed-loop system can be represented in the form
of Figure 5.1 with Hl defined by
S =
+
b
for all ( t X,
~ e l ) , where ci and c2 are nonnegative constants. Using (5.42) and (5.43),
it can be shown that
IlYlll~5 ~ l l e l l +
l ~PI
where
71 = c17 C2, Pi = CIP
Since H2 is a linear time-invariant system and A is Hurwitz, we apply Corollary 5.2
to show that H2 is finitegain tpstable for any p E [ I ,w ] and
€4 = AV + E A - ~ B [+
? d2(t)]
where
.
opens the loop and the overall closed-loop system reduces to the nominal one. ~~t
us assume that the feedback function 7 ( t , x )satisfies the inequality
brings the closed-loop system into the form
j. =
'
5.4. FEEDBACK SYSTEMS: THE SAlALL-GAIN THEOREAl
Let us assume that d 2 ( t ) is differentiable and d2 E L. The change of variables
q =z
*!,
...
as E -+ 0 , which shows that for sufficiently small E the upper bound on the L gain
of the map from d to x , under the actual closed-loop system, will be close to the
corresponding quantity under the nominal closed-loop system.
A
''P of Corollary 5.2 is taken as EQ SO that (fQ)(A/f)
+ (A/f)=(fQ)
= -I.
'.
222
5.5
CHAPTER 5. INPUT-OUTPUT STABILITY
5.5.
P.
Exercises
5.1 Show that the series connection of two L stable (respectively, finite-gain
stable) systems is L stable (respectively, finite-gain L stable).
L
5.2 Show that the parallel connection of two L stable (respectively, finitegain
stable) systems is L stable (respectively, finite-gain L stable).
L
EXERCISES
5.3 Consider a system defined by the memoryless function y = u'l3.
(a) Show that the system is L, stable with zero bias.
(b) For any positive constant a, show that the system is finite-gain L, stable with
7 = a and 0 = (l/a)'I2.
(a) on-off with hysteresis
(b) On-off with dead zone and hysteresis
(c) Compare the two statements.
94
5.4 Consider a system defined by the memoryless function by y = h(u), where
h : Rm + R* is globally Lipschitz. Investigate L, stability for each p E 11, co]when
(1) h(0) = 0.
?-
(2) h(0) # 0.
6.5 For each of the relay charactcristices sllown in Figure 5.2, investigatc
stability.
L, aid
"
L2
.
5.6 Verify that D+W(t) satisfies (5.12) when V(t, x(t)) = 0.
Hint: Using Exercise 3.24, show that V(t h,x(t h)) 5 ~ 4 h * L ~ 1 / ~ ) h) ~~/(2h ) , ~t
where o(h)/h + 0 as h + 0. Then, use the fact that q 2 2cl.
+
+
+
5.7 Suppose the assumptions of Theorem 5.1 are satisfied, except (5.10), which is
replaced by
Ilh(t,x,~)II 7 1 ~ 1 1 ~ 1 1 +v2llull+ 1731 713 > 0
Show that the system is small-signal finite-gain L, stable (or finite-gain L, stable,
if the assumptions hold globally) and find the constants 7 and 0 in (5.11).
<
I
(c) Ideal on-off
((1) 011-off with dead zone
Figure 5.2: Relay characteristics
5.11 For each of the following systenas. illvestigate L, and finite-gain L, stability:
= -XI
+ x:x2,
+22,
(1)
il
(2)
~1 = -XI
(3)
51 = (xl
5.9 Derive a result similar t,o Corollary 5.2 for linear time-varying systems.
(4)
5 1 = - ~ 1 - ~ ~ + ~ ~
f 2 ,= X I
5.10 For each of tlic following syst.clns, investigate L, and finitegain L, stability:
(5)
~1 = -XI
(6)
51 = x2,
(7)
51 = -XI
,
5.8 Suppose the nss~lmptionsof Theorem 5.1 are satisfied, except (5.10), which i
replaced by (5.20). Sliow that the systelrl is small-signal L, stable (or L, stab
if the assumptions hold globally).
= -(l + u)x3 - x5
y = X+U
= -(I+U)X~
y = x
(2)
x = -x/(l + x 2 ) + u
= r/(l+s2)
(4)
.
-..r
I..,
-I.
,"
,
-
.,,
. s .
,,
,
.
.
+x:x2,
y=x1
x2 = -x; - x2 + u ,
Y = $2
2
-1
=2
-x;+u2,
x2 = 21 - 2 2
jr
jr
(3)
+ u)(llx11; - I),
x2 = -x; - x 2 + u I
.
-
.
a
". , .:
.+
= -x - 5 3 + x 2 u
9 = xsinu
, . ,.
,
.
.
..---
.
- : :
. .. .
'
.
. ..--
- 12,
i 2
= -2;
~2
= XI
+ u,
- 2 2 + U,
- xg + u,
y = Xl
y=x1(x2+u1)
y=x1+u
y = x*
~ ( t =) xi (t - 2')
I
3
.
I.-.,
1' I)
>
-1
---
' ...
.
- . ..
-. .
..
.
-- --
- . . .. .
. ..:. .
.
...
CIIAPTER 5. INPUT-OUTPUT STABILITY
224
. .. .
:"
.
..
,
.
... ......
. - . . % . -,..
L. .4..,
5.5. EXEItCISES
"
;
.
.
$
.
d
,
,.
7
. , .
..
.-.-,L,...
.
226
5.17 (1771) Consider tlie systcln
5.12 Consider the system
2 = f (x)
where h is continuously differentiable, h(0) = 0, and zh(t) > az2 for all z E R, for
some a > 0. Show that the system is finite-gain Lp stable for each p E [l,m].
+ G(x)u,
y
= h(x) + J(x)u
where f , G, h, and J are smooth functions of x. Suppose there is a positive constant
7 such that 721
- JT(x)J(x) > 0 and
5.13 ([192]) Consider the time-invariant system
j. = f(x, u),
y = h(x, 21)
V x. Show that the system is finite-gain L2 stable with L2 gain less than or equal
whcrc f is locally Lipschitz, h is continuous, f (0,O) = 0, and h(0,O) = 0. Suppose
there is a continuously differentiable, positive definite, radially unbounded function
V(x) such that
(x,u)
a x (2, u) 5 -IV(x) $(,)I
to 7.
Hint: Set
where W(x) is continuous, positive definite, and radially unbounded, $(u) is continuous, and $(O) = 0. Show that the system is Lm stable.
and show that the following inequality holds V u
+
Ef
.
.
5.14 Let H(s) be a Hunvitz strictly proper transfer function, and h ( t )= L-'{H(s)}
be the corresponding impulse response function. Show that
5.15 For each of the following systems, show that the system is finite-gain (or
small-signal finite-gain) C2 stable and find an upper bound on the L2 gain:
5.18 ([199])a Consider the system
where u is a control input and w is a disturbance input. The functions f , G, K,
and h are smooth, and f(0) = 0, h(0) = 0. Let 7 > 0. Suppose there is a smooth
posit.ive semidefinite function V(x) that sat.isfies
V x. Show that with the feedback control u = -GT(x)(dV/ax)T the closed-loop
map from u to
5.16 Consider the system
where a is locally Lipschitz, u(0) = 0, and zu(z) 2 0 for all z E R.
(a) Is the system finite-gain L,
stable?
(b) Is it finite-gain L2 stable? If ycs, find an upper bound on the L2 gain.
[ :]
is finite-gain L2 stable with
4 gain less than or equd to 7.
5.19 ([200]) The purpose of this exercise is to show that the 122 gain of a linear
time-invariant system of the form (5.24)-(5.25), with a Hurwitz matrix A, is the
same, whether the space of functions is defined on R+ = [0, m ) or on the whole
real line R = (-m, m). Let L2 be the space of square integrable functions on R+
with the norm ( I ~ l l : ~ =
uT(t)u(t) dt and L ~ be
R the space of square integrable
m uT(t)u(t) dt. Let 72 and 72R be the
functions on R with the norm llu((inR= J-m
L2 gains on L2 and LZR,respectively. Since L2 is a subset of LZR,it is clear that
% 5 %R. We will show that 72 = YZR by showing that, for every E > 0, there is a
signal u E LZ such that y E L2 and JJyllr,2 (1 - E ) Y ~ R ~ ~ u ~ ~ L ~ .
Jr
.
226
CIfAPTER 8. INPUT-OUTPUT STABlLl'l'Y
(a) Given E
> 0 show that we can always choose 0 < 6 < 1 such that
(b) Show that we can always select u E tzR
and time tl < oo such that
= ~ ( t+uz(t),
)
where u1 vanishes for t < tl and u2 vanishes for t > tl.
Let pl(t) be the output corresponding to the input ul(t). Show that
( c ) Let u(t)
(d) For d ' t
> 0, define u(t) and p(t) by u(t) = ul(t + tl) and y(t) = yl(t + tl).
Chapter 6
Passivity
Show that both u and y belong to L2, y(t) is the output corresponding to
and I ~ u I I c ~ 2 (1 - ~ ) " 1 2 ~ ~ 1 ~ 1 1 ~ ~ .
5.20 Consider the feedback connection of Figure 5.1, where Hl and Hz are linear
timeinvariant systems represented by the transfer functions Hl (s) = ( s - l ) / ( s + 1)
and Hz(s) = 1/(3 - 1). Find the closed-loop transfer functions from (u1,u2) t~
(pi, p2) and from ( ~ 1uz)
, to (el, el). Use these transfer functions to discuss why
we have to considcr both inputs (u1,uz) and both outputs (el, e2) (or (plyB))in
studying the stability of the feedback connection.
6.21 Consider the feedback connection of Figure 5.1. Show that the mapping from
( ~ 1uz)
, to (pl,yz) is finitegain C stable if and only if the mapping from (ul, a 2 )to
(el, ez) is h i t e g a i n t stable.
6.22 Let dz(t) = asinwt in Example 5.14, where a and w are positive constants.
(a) Show that, for sufficientlysmall E , the state of the closed-loop system is uni-
formly bounded.
(b) Investigate the effect of increcrsi~lgw.
5.23 Consider the feedback connection of Figure 5.1, where H1 and Hz are given
by
HI :
{
il =
-XI
i2 = -xt
Yl
=
+
2.2
- x2 + el
22
and Hz :
{ ::==
-x:
e2
(1/2)x;
Let uz = 0, u = u1 be the input, and y = yl be the output.
(a) Using x = 1x1, xz,
x3IT
as the state vector; find the state model of the systsem.
(b) Is the syst.cm t2st,able?
passivity provides us with a useful t.001 for the analysis of nonlinear syst.eins!whic2.i
relates nicely to Lyapunov and C2 stal~ility.We start in Scctioll6.l I)y dcfiliing passivity of memorylcss ~lo~llil~caritics.
\V(!cxtcnd the definition t . tlynamiwl
~
systc~ns,
represented by state models, in Scctio~i6.2. In both cases, we use clcctrical 1iet.works
to motivate the definitions. In Section 6.3, we study positive real and stxictly positive real t.ra11sfer fullctions ant1 sho~vthat they represent passive and strictly passi~~e
systems, respectively. The coil~~ectioi~
bct~veenpassivity ant1 both Lyapuno~~
and
C2 stability is estahlislled in Section G.4. These four sections prepare us to address
the main results of the c11al)tcr. 11~11101y.
t,l~epassivity theorems of Sect.io116.5. The
main passivity theorem stat.es that the (negative) feedback connection. of two passive
systems is passive. Under additiollal obser~.abilityconditions, t.lie feedback connection is also asymptotically stable. The passivity theorems of Section 6.5 and the
importiult gc~lcri~li./.i~t.i~
small-gain theom111 of Sect.io115.4 l)rovidc a co~~ccptually
of the fact that the feedback connectioll of two stable linear systems will be stable
if t.he loop gain is less than 'one or the loop phase is less than 180 degrees. The
~ dphase of a transfer function comes from the
connection between passivity a ~ thc
frequency-domain cl~aractcrizfitio~l
of positive real transfer funct,ions, give11 i11 Scction 6.3. There we kl~owthat. tllc pllt~scof u positive real transfer funct-io11c~1111ot.
exceed 90 degree. Hence, t.he lool) phase cannot exceed 180 degrees. If one of tile
two transfer functions is strictly positive real, the loop phase will be strictly less
than 180 degrees. Section 6.5 discusses also loop transformations, which allow us,
in certain cases, to transform the feetlback connection of two s y s t e ~ that
~ s may l~ot,
be passive into t.he feedback connection of t.wa passive systems! hence ext,endil~gt , l ~
utility of tlic passivity thcorc~lis.
. CHAPTER 6. PASSIVITY
228
(a)
Figure 6.1: (a) A passive resistor; (b) u-y characteristic lies in the first-third quadrant.
6.1
--
G.1. MEA,IORYLESS FUNCTIONS
9'19
I
(b)
(c)
Figure 6.2: (a) and (b) are examples of nonlinear passive resistor characterirtiu; (c) is
an example of a nonpassive resistor.
Memoryless Functions
Our goal in this section is to define passivity of the memoryless function y = h(t,u),
where h : [O,co) x RP + RP. We uso electric networks to motivate the definition.
Figure 6.l(a) shows a one-port resistive element with voltage u and current y. We
view this element as a syst*emwith input, u and output y. The resistive element
that is, if uy 2 0 for all
is passive if the inflow of power is always n~nnegat~ive;
points (u, y) on its u-y characteristic. Geometrically, this means that the u-y curve
must lie in the first and third,quadrants, as shown in Figure 6.l(b). The simplest
such resistive element is a linear resistor that obeys Ohm's law u = Ry or y = Gu,
where R is the resistance and G = l / R is the conductance. For positive resistallce,
the u-y characteristic is a straight line of slope G and the prod~lctuy = Gy2 is
always nonnegative. In fact, it is always positive except at the origi'n point (0,O).
Nonlinear passive resistive elements have nonlinear u-y curves lying in the first and
third quadrants; examples are shown in Figures 6.2(a) and (b). Notice that the
tunnel-diode characteristic of Figure 6.2(b) is still passive even though the curve
has negative slope in some region. As all example of an element that is not passive,
Figure 6.2(c) shows the. u-y characteristic of a negative resistance that was used
in Section 1.2.4 to construct the. negative resistance oscillator. Such characteristic
can be only realized using active devices such as the twin tunnel-diode circuit of
Figure 1.7. For a multipart network where u and y are vectors, the power flow
into the network is the inner product uTy = Cf,,uiyi = Cy=luihi(u). The network
is passive if uTy 2 0 for all u. This concept of passivity is now abstracted and
nssigned to any function y = h(t,u) irrespective of its physical origin. We think of
uTy as the power flow into the system and say that the system is passive if uTy 2 0
for all u. For the scalar case, the graph of the input-output relation must lie in the
first and third quadrants. We also say that the graph belongs to the sector [O,a],
where zero and infinity are the slopes of the boundaries of the first-third quadrant
region. The graphical representation is valid even whea h is time varying. In this
case, the u-y curve will be changing with time, but will always belong to the sector
10, oo]. For a vector function, we can give a graphical representation in the special
case when h(t, u) is decoupled in t.11~
sc~~sc:
t.h;~t.h i ( t ,u) dc~)c~lcls
ollly on ui; that is,
L
1
hp(ttu,)
In this case, the graph of each component belongs to the sector [ O , 4 In the general
case, such graphical representation is not possible, but we will continue to use the
sector terminology by saying that h belongs to the sector [O, cm]if uTh(t, u ) 1 0 for
all (t,u).
An extreme case of passivity happens when uTy = 0. In this case, we say that
the system is lossless: An example of a Iossless system is the ideal transformer
shown in Figure 6.3. Here y Su, where
-
The matrix S is skew-symmetric; that is, S
(1/2)uT(S ST)u =o.
+
+ ST = 0. Hence, uTy = uTSu =
'
Figure 6.3: Ideal transformer
Consider now a function h satisfying uTy 2 uTv(u) for some fuoetion ~ ( u ) .
Wllcn uTp(U) > 0 for all u # 0,h is called input strictly passive because passivity
is strict in the sense that uTY = 0 only if u = 0. Equivalently, in the malar case,
230
CHAPTER 6. PASSIVITY
6.1. IIIEI\IORYLESS FUNCTIONS
231
Figure 6.4: A graphical representation of uTy 2 euTu for (a) E > 0 (excess of passivity); (b) E < 0 (shortage of passivity): (c) removal of excess or shortage of passivity by
input-feedforward operation.
Figure 6.8: A graphical representation of r r T y 2 6yTy for (a) 6 > 0 (excess of passivity); (b) 6 < 0 (shortage of passivity); (c) removal of excess or shortage of passivity by
output-feedback operation.
the u y graph does not touch the u-axis, except at the origin. The term uTP(u)
represents the LLexcess"of passivity. On the other hand, if uTp(u) is negative for
some values of u, then the function h is not necessarily passive. The term uTp(u)
represents the "shortage" of passivity. Excess and shortage of passivity are more
transparent when h is scalar and ~ ( u =) EU. In this case, h belongs to the sector
[E,m), shown in Figure 6.4, with excess of passivity when E > 0 and shortage of
passivity when E < 0. Excess or shortage of passivity can be removed by the inputfeedforward operation shown in Figure 6.4(c). With the new output defined as
8 = y - p(u), we have
Definition 6.1 The system y = h(t:u) is
passive if uTy 2 0.
lossless if uTy = 0.
input-feedforward passive if uTy 2 uTrp(u) for some function rp.
input strictly passive if uTY 2 uTrp(u) und uTp(u) > 0, V u # 0.
output-feedback passive if uTy 2 yTp(y) for some function p.
output strictly passive i j u T y 2 yTp(y) and yTp(y) > 0, V y # 0.
>
Thus, any function satisfying uTy uTQ(u) can be transformed into a function that
belongs to the sector [0,m] via input feedforward. Such a function is called inputfeedforwad passive. On the other hand, suppose uTy yTp(y) for some function
p(y). Similar to the foregoing case, there is excess of passivity when yTp(y) > 0 for
all y # 0, and shortage of passivity when yTp(y) is negative for some values of y.
A graphical representation of the scalar case with p(y) = dy is shown in Figure 6.5.
There is "excess" of passivity when 6 > 0 and shortage of passivity when 6 < 0.
Excess or shortage of passivity can be removed by the output-feedback operation
shown in Figure 6.5(c). With the new input defined as ii = u - p(y), we have
>
Hence, any function satisfying uTy 2 yTp(y) can be transformed into a function
that belongs to the sector [0, m] via output feedback. Such a function is called
output-feedback passive. When yTp(y) > 0 for all y # 0, the function is called
output strictly passive because passivity is strict in the sense that uTy = 0 only if
y = 0. Equivalently, in the scalar case, the u y graph does not touch the y-axis,
except at the origin. For convenience, we summnrize the various notions of passivit.~
in tlic nest dchition.
C
In all cases, the inequality sl~ouldhold for ull (t, u).
Consider next a scalar function y = ll(t, u), which satisfies the in equal it,^
for all (t, u), where a and are real num1)crs with p 2 a. The graph of this function
belongs to a sector whose boundaries are the lines y = a u and y = pu. We say
that h belongs to the sector [a,P]. Figure 6.6 shows the sector [a,P] for /3 > 0
and different signs of a. If strict inequality is satisfied on either side of (6.2), we
say that h belongs to a sector (a,P], [a,P). or (a,P), with obvious implications.
Comparing the sectors of Figure 6.6 with thosc of Figures 6.4 alid 6.8 shows that a
function in the sector (a,P] combines input-feedforward passivity with output strict
passivity since the sector [a,
P] is the intersection of the sectors [a?
m] and [O,p].
Indeed, we can show that such a function can be transformed into a function that
belongs to the sector [O!x]by a sequence of input-feedforward and output-feedback
operations. Before we do that. we extend the sector definition to the vector case.
Toward that end, note that (6.2) is equivalent to
.
,
,
,.. . .
.
.
____ _ . _.-__
' < .
.;
:
9, ,.
,
b
6.2. STATE AlODELS
CHAPTER 6. PASSIVITY
4
..
T
. ".. .- " ..~ . . .
..
-,,.:,*
,
.
. ..;; ,. ,"...~ 3,a.
.. ..n&a-.-"""--,
'
233
Figure 6.7: A function in the sector [Kl, Kz], where K = K2 - K1 = K T > 0, can be
transformed into a function in the sector 10,w] by input feedforward followed by output
feedback.
Figure 6.6: The sector [alp]for ,f3 > 0 and (a) a > 0; (b) a < 0.
for all (t,u). For the vector case. let us consider first a function h(t,u) that is
decoupled as in (6.1). Suppose each component h, satisfies the sector condition
(6.2) with constants ai and 0, > a,.Taking
it can be easily seen that
4
.
The sector 10, m ] corresponds to passivity. The sector IKl, w] corresponds to inputfeedforward passivity with p(u) = Klu. The sector [0, K2] with K2 = (l/d)I > 0
corresponds to output strict passivity with p(y) = 6y. We leave it to the reader
(Exercise 6.1) to verify that a function in the sector IK1, Kz] can be transformed into
a function in the sector (0,m ] by input feedforward followed by output feedback, as
shown in Figure 6.7.
for all (t,u). Note that EC; = K2 - K1 is a positive definite symmetric (diagonal)
matrix. Inequality (6.4) may hold for more general vector functions. For example,
suppose h(t, u) satisfies the inequality
for all (t, u). Taking K1 = L - TIand K2 = L
+ 71, we can write
4
Once. again, I<= K2 - KIis a positive definite symmetric (diagonal) matrix. We
use inequality (6.4) with a positive definite symmetric matrix K = K2 K1 as a
-
definition of the sector [K1, K2] in the vector case. The next definition summaries
the sector tcrminology.
Definition 6.2 A memoyless ji~nctionh : [O,m) x RP
the sector
4
fl is said to belong to
In all cases, the inequality should hold for all (t,u). I' in any case the inequality is
strict, we write the sector as (0, w ) , (Kl, m ) , (0, K2), or (K1, K 2 ) . In the scalar
P), or ( a , P) to indicate .that one o r both sides of (6.2) is
case, we mite (a, 01, [a,
satisfied as a strict inequality.
4
6.2
State Models
Let us now define passivity for a dynamical system represented by the state model
where j : Rn x RP --t Rn is locally Lipschitz, h : Rn x RP --t RP is continuous,
j(0,O) = 0, and h(0,O) = 0. The system has the same number of inputs and
outputs. The following RLC circuit motivates the definition.
Example 6.1 The RLC circuit of Figure 6.8 features a voltage source connected
[O,K2] with Kz= KT > 0 ij hT(t, u)[h(t,u) - K ~ u 5] 0.
to an RLC network wit,h linear inductor and capacitor and nonlinear resistors. The
nonlinear resistors 1 and 3 are represented by their v-i characteristics il = h l ( y )
and i3 = h3(v3), while resistor 2 is represented by its i-v characteristic v2 = hz(i2).
CIIAPTER (i. PASSIVITY
62. STATE I\IODEL.S
We take the voltage u as the input and the current y as the output. The product
uy is tbe power Row i1lt.o the netww.ork. Takillg the c~~rrellt
51 t l ~ r ~ u gtllc
l l ili(ltl(:t.or
and the voltage xz across the capacitor as the state variables, we call write the state
model as
The new feature of an RLC network over a resistive network is the presence of the
energy-storing elements L and C. The system is passive if the energy absorbed by
the network over any period of time [0,t ] is greater than or equal to the increase in
the energy stored in the network over the same period; that is,
(6.8)
+
where V(x) = (1/2)Lx:
(1/2)Cxg is the energy stored in network. If (6.8) holds
with strict inequality, then the difference between the absorbed energy and the
increase in the stored energy must be the energy dissipated in the resistors. Since
(6.8) must hold for every t 1 0, the illstantaneous power inequality
Figure 6.8: RLC circuit of Example 6.1.
Tlle term uhl (u) could represent excess or shortage of passivity. If uhl ( u ) > 0
for all u 0. there is excess of passivity since the energy absorbed over [0, t]
will be greater than the increase ill the stored energy, unless the input u(t) is
identically zero. This is a case of input strict passivity. On the other hand,
if uhl(u) is negative for some values of u, there is shortage of passivity. As
we saw with memoryless fullctions. this type of excess or shortage of passivity
can be removed by input fectlforwartl shown in Figure 6.4(c).
+
Case 3: If hl = 0 and
must hold for all t; that is, the power Row into the network must be greater than or
equal to the rate of change of the energy stored in the network. We can investigate
inequality (6.9) by calculating the derivative of V along the trajectories of the
system. We have
h3 E
Excess or shortage of passivity of h2 results in the same pr0pcrt.y for the
network. Once again, as with ~nomorylessfunctions, this type of excess or
shortage of passivity call be rr.1~1ovrtlby output feedback, as ill Figure G.5(c).
When yh2(y) > 0 for all y # 0. wvc liave output strict passivity because the
energy absorbed over (0: t] wvill be greater than the increase ill the stored
energy, unless the output y ( t ) is identically zero.
Case 4: If hl E (0,m ] ?h2 E (0: x ) .and
Thus,
+
try = l'~ t~hl(tl)+ .rlhz(.~l)t .r2h3(.r?)
Lf hi. hz. and ha me pi~.ssi\~.
uu 2 i'nnd the system is passive. Other possibilities
(UC
illlstratd by follr different spcrinl rnscs of the network.
im d there is no e n e r g dissipation iu the
tht- s p t t ~ ins l l ~ s l c s .
k s e 1: If hl = h: = h3 = 0. tiy =
lwt\\vrk:
lh~ris.
Ic
[O?x].
h3
'I
E (0, m ) ,
whcrc x1h2(x1)+ ~ ~ 1 1 : ~is( ;Ix j)osil.iv~
~)
d~fillitc1111i(:t,io11
01 .r. ?'his is ;I c:ilS(!
of state strict passivity because thc crlcrgy absorbed over [O:t] wvill bc grcatcr
than the increase in the stored cilcrgy, unless the state x(t) is identically
zero. A system having this property is called state strictly passive or. simply
strictly passice. Clearly there is 110 counterpart for state strict pasivity ill
memoryless functions since there is no state.
1
.
'
-- -. .. .
.
.
.
-
.-
- - ...
.
.
.
-:.
.....
CHAPTER 6. PASSIVITY
6.3. POSITIVE REAL TRANSFER FUNCTIONS
237
Definition 6.3 The system (6.6)-(6.7) is said to be passive if there exists a contin-
uously differentiable positive semidefinite function V ( x ) (called the storage function)
such that
av
(6.10)
uTY > v = -f ( x ,u ) , V ( I , U ) E Rn x RP
ax
Moreover, it is said to be
input-feedforward passive if u T y 2 v
.
+ uTcp(u) for
I
Figure 6.9: Example 6.2
some function cp.
v + uTcp(u) and u T q ( u ) > 0, v # 0
ontput-feedback passive if uT?j _> v + yTP(y) for some function p.
i n p t strictly passive i f u T y _>
output strictly passive i f u T y > v
strictly passive if u T y 2 v
+ yTp(y) and yTp(y) > 0 , V y # 0.
+ $ ( x ) for some positive definite function $.
In all cases, the inequalitu sho?clrl liold l o r all ( I , u ) .
Figure 6.10: Example 6.3
Definition 6.3 reads almost the same as Definition 6.1 for memoryless functions,
except for the presence of a storage function V ( x ) . If we adopt the convention
that V ( x ) = 0 for a memoryless function, Definition 6.3 can be used for both state
models arid memoryless functioils.
Example 6.2 The integrator of Figure 6.9(a), represented by
Example 6.3 The cascade connection of an integrator and a passive memoryless
function, s h o h jn Figure 6.10(a), is represented by
'
Passivity of h guarantees that
h ( u ) d u 2 0 for all x. With V ( x ) = h ( o ) d u
as the storage function, we have = h ( x ) x = yu. Hence, the system is lossleap.
Suppose now the integrator is replaced by the transfer hnction l / ( a s 1 ) with
a > 0, as shown in Figure 6.lO(b). The system can be represented by the state
model
ax=-x+u
Y =h(x)
With V ( x ) = a l ; h ( u ) du as the storage function; we have
v
is a lossless system since, with V ( z ) = (1/2)x2 as the storage function, we have
u y = V . When a memoryless function is connected in parallel with the integrator,
as shown in Figure 6.9(b), the system is represented b y
'\
Clearly, the system is input-feedforward passive since the parallel path h ( u ) can
be cancelled by feedforward from the input. With V ( x ) = ( 1 / 2 ) x 2 as the storage
function, we have u y = ~ + u h ( u ) If. h E [O, m], the system is passive. If u h ( u ) > 0
for all u # 0 , the system is input strictly passive. When the loop is closed around an
integrator with a memoryless funct.ion, as in Figure 6.9(c), the system is represented
by
x = -h(z)+u,
y= x
Plainly, the system is output-feedback passive since the feedback path can be callcelled by a feedback from the output. With V ( x ) = ( 1 / 2 ) x 2 as the storage function,
we have u y = v yh(y). If h E [O,oo],tllc systen~is passive. If yh(y) > 0 for all
A
y # 0, the system is output strictly passive.
+
I
Hence, the system is passive. When x h ( x ) > 0 for all x
passive.
6.3
+
# 0, the system is strictly
A
Positive Real Transfer Functions
Definition 6.4 A p x p proper mtional tmnsfer function m a t h G ( s ) is called
positive real if
poles of all elements of G ( s ) are i n Re[s]5 0,
B
238
CHAPTER 6. PASSIVITY
for all real w for which jw is not a pole of any element of G ( s ) , the matrix
G(jw) GT(-jw) is positive semidefinite, and
+
any pure imaginaly pole jw of any element of G(s) is a simple pole and the
residue matrix lim,,jw(s - jw)G(s) is positive semidefinite Hermitian.
The transfer function G(s) is called strictly positive real ' if G(s- E ) is positiz~ereal
for some E > 0.
When p = 1, the second condition of Definition 6.4 reduces to Re[G(jw:)]2
0, V w E R, which holds when the Nyquist plot of of G(jw) lies in the closed righthalf complex plane. This is a condition that can be satisfied only if the relative
degree of the transfer function is zero or one.2
The next lemma gives an equivalent characterization of strictly positive real
transfer functions.
23!)
6.3. POSITIC'E REAL TR;\NSFER FUh1C,'T1O~Y.S
It is not strictly positivc real siilcc: l / ( s - E ) lias a pole in Re[s]> 0 ror iuly E > 0.
Thc t.ra~~sfcr
h111ctio11
G ( s )= l / ( s + n ) wit.11 (1 > 0 is positive rcal, si11c:c il, I~tls11o
poles in Re(s]2 0
Since this is so for cvrry a > 0. \\T SCP tli;rt ror ally E E (0,a) tlic transfer f~~nctioii
G ( s - E ) = 1 / ( s+ a - E ) will I J positivc
~
rcal. Hence, G ( s )= l / ( s + a) is strictly
positive real. The sallic conclusioli (.;11i I)(\ clra~viifrom Lemma 6.1 by noting that
w2a
liin w2Re[G(jw)]
= lim -= a > 0
d-z ~2 +a2
w-32
The transfer functiori
1
sZ+s+l
is not positive real because it,srelativc tlcgrce is two. We can see it also I J calculatirig
~
C ( s )=
Lemma 6.1 Let G ( s ) be a p x p proper rational transfer function matrix, and
suppose det [G(s)+ GT(-s)j is not identically zero.3 Then, G ( s ) is strictly positive
real if and only if
Consider t.he 2 x 2 transfer functioii iilatrix
G(s) is Hurwitz; that is, poles of all elements of G ( s ) have negative real parts,
G(jw)+ GT(-jw) is positive definite for all w E R, and
+
I
either G ( w ) G T ( w ) is positive definite or it is positive semidefinite and
lirn,,,
w2M T [ G ( j w ) + G T ( - j w ) ]is~positive definite for any px (p-q) fullrank mat* M such that h f T [ G ( ~ ) + G T ( ~=) ]0,Mwhere q = rank[G(w)+
GT(m)l.
\\lc cannot apply Leinma 6.1 because tlat[G(s)+ GT(-s)] G 0 V s. I-Iowcva, G ( s )
is strictly posit.ive real. as call be seen by checking the conditions of Definition 6.4.
Note that.! for E < 1?the poles of the elcinents of G(s - E ) are in Re[s]< 0 and
Proof: See Appendix C.11.
>
is positive semidefillite for all w E R. Si~nilarly.it can be seen that tlie 2 x 2 transfer
+
If G ( w ) G T ( w ) = 0, we can take ll1 = I. In the scalar case ( p = 1). the
frequency-domain conditioii of the lemma reduces to Re[G(jw)]> 0 for all w E R
and either G ( m ) > 0 or G ( w )= 0 and lirn,,,
w2Re[G(jw)]> 0.
Example 6.4 The transfer function G ( s ) = l / s is positive real since it has no
poles in Reis] > 0, has n simple pole at s = 0 whose residue is 1, and .
'The definition of strictly poaitive real transfer functions is not uniform in the literature. (See
[206] for various definitions nnd the relntionsl~ipbetween them.)
h he relatiw degree of a rational transfer function G(a) = n(a)/d(a) is deg d deg n. For a
proper trnnrfcr ft~nction.thc relnti\.e dcgrce is a nonnegntive integer.
-
;'Kqt~i\nln~fl~.
C(r) + (:I.(-$)
IIW n n n r ~ ~ ~r n ~l ~11kclr.cVrt ltc lic.l<Iof rnfioiinl f~iiirtioiis of 8 .
+
is strictly posit,ive ~ e a l .This tinic, ho~vcver,det[G(s) G ~ ( - s ) is] not idelltically
zero, and we can apply Leinlna G.1 to arrive at the same conclusioii by noting that
G ( w ) G T ( w )is positive dcfiiiitc arid
+
is positive definite for all w E R. Finally. the 2 x 2 transfer function matrix
-
-
- - - -...-
CHAPTER 6. PASSIVITY
6.4. Lz AND LYAPUNOV STABILITY
231
Proof: Suppose there exist P = pT > 0, L, 1Y,and E > 0 that satisfy (6.14)
through (6.16). Set p = ~ / and
2 recall that G ( s - 11) = C ( s I - pI A)-IB + D.
From (6.14),we have
-
has
I
-1
It can be verified that
It follows from Lemma 6.2 that G(s - p) is positive real. Hence, G ( s ) is strictly
positive real. On the other hand, suppose G ( s )is strictly positive real. There exists
p > 0 such that G(s - p) is positive real. It follows from Lemma 6.2 that there
are matrices P = PT > 0, L, and W ,which satisfy (6.15) through (6.17). Setting
E = 2p shows that P, L, IY, and E satisfy (6.14) through (6.16).
0
Lemma 6.4 The linear time-iwvariant minimal realization
.d
,i
+
G(jw) ~ ~ ( - j =d )
-
is positive definite for all w E R. Taking AIT = [ 0 1
1, it can be verified that
lim w 2 ~ ' [ G ' ( j w+
) ~ ~ ( - j w )=]4~ l
x = Ax+Bu
y = Cx+Du
w d m
.>
Consequently, by Lemma 6.1, we collclude that G ( s )is strictly positive real.
A
with G ( s )= C ( s l - A)-'B
Passivity properties of positivc real transfer functions can be shown by using the
next two lcmmas, which are known, rcspcctivcly, as thc positive real lemma and the
Kalman-Yakubovich-Popov lemma. The lemmas give algebraic characterization of
positive real and strictly positive real transfer functions.
Lemma 6.2 (Positive Real) Let G(s) = C(sI - A)-'B + D be a p x p transfer
function matrix where ( A ,B ) is controllabh and ( A ,C ) is obsennble. Then. G(s)
is positive real if and only if thew exist mntrins P = PT > 0, L, m d IV S U thnt
~
+ D is
passive if G ( s ) i s positive real;
strictly passive if G(s) is strictly positive real.
".
0
Proof: Apply Lemmas 6.2 and 6.3, respectively, and use V ( s )= ( l / 2 ) x T P x8~ the
storage function.
av
uTg - z ( A x
+ Bu)
+
+
U ~ C X+ ; u ~ ( D
+ D ~ ) U- ; X ~ ( P +A A ~ P ) -X X ~ P B U
u T ( B T p+ W T L ) x+ $ u T 1 l r T ~ ~ u
+ $ X ~ L ~ L+X ; E X ~ P X - X ~ P B U
;(Lx + W U ) ~ ( L X+ W u )+ ;ExTPx 2 ;txTPz
= uT(Cx Du) - ~ P ( A XB u )
=
=
=
Proof: See Appendix C.12.
In the case of Lemma 6.2, E = 0,and we conclude that the system is passive, while in
the case of Lemma 6.3, E > 0, and we conclude that the system is strictly pwsive.
Lemma 6.3 (Kalman-Yakubovich-Popov) Let G(s)= C(sl-A)-lB+D be a
p x p transferhnction mat*, where ( A ,B ) is controllable and (A,C ) is observable.
Then, G(s) is strictly positive real if and only ij there exist matrices P = PT > 0,
L , and IY,arid a positive constartt E sricli tllat
6.4
L2and Lyapunov Stability
In this section, we study Lp and Lyapunov stability of passive systems of the form
-
ri. =
f
u)
(21
wl~eref : R" x RJ' Rn is locally Lipschitz, h : Rn x RJ'
f (0,O) = 0, and h(0,O)= 0.
-
(6.18)
.
- ~ ,
RJ' is continuous,
CHAPTER 6. PASSIVITY
243
6.4. L2 AA'D LY.4PUSOi' STABILIT\'
> +
L e m m a 8.5 If the system (6.18)-(6.19) is output strictly passive with u T y v
6yTy, for some 6 > 0 , then it is finite-gain Lz stable and its L2 gain is less than or
equal to 1/6.
Definition 6.5 The systenr (6.18) (6.13) is said to be zeiu-state olr.sci.~~c~blc
if iro
. ~ o l ~ ~ t iofo nx = f (1.0) can. stay idci~.Linlll!lin S = { x E Rn I h ( x ,0 ) = O ) , olhei,
thnn. the trivial solrltioit. x ( t ) = 0.
ProoE The derivative of the storage function V ( x ) satisfies
C
6.
. C,
L e m m a 6.7 Considei the systcn~ (6.18)-(6.19). The origin of x = f(x,O) is
asymptotically stuble if the systeirl is
IC
strirtly pnrrsirlc or.
t
output strictly passive and zeiv-state observable.
f-
integrating both sides over [0,T ] yields
Thus,
where we used the facts that V ( x )
numbers a and b.
/-
.
.. .,
i
Furtheirnore, if the storage function is i,adially unbounded, the origin will be globally
asymptotically sfable.
0
L
Proof: Suppose thr.system is st.rictly passive and let V ( x ) he its storage function.
Then. with u = 0. I/ satisfies thc inrqr~alityV
--$J(x),where * ( x ) is positive
definite. \Ve can use this incquality to show that V ( x ) is positive definite. In
particular, for any x E Rn.the equation i = f ( x ,0 ) has a solution o ( t :x ) . starting
ffom x at t = O arid defined on sotne ir~terval[0,6]. Integrating the inequality
V 5 -v(x) yields
m5
a
+ b for nonl~egative
V(.(T. x ) ) - V ( x )
L e m m a 6.8 If the system (6.18)-(6.19) is passive with a positive definite storage
function V ( x ) , then the origin of x = f ( x , 0 ) is stable.
.
..
> 0 and d
..
8
<
1
I I V T I I L5
~ J ~ ~ u T l l ~ 2
+
.
,
Proof: Take V as a Lyapunov function candidate for x = f ( x ,0). Then v 5 0.
Using V ( Q ~ (xT) .)
lr
$ ( 4 ( t ;x ) ) dt, V
T
L
E [ O dl
L
i,
> 0. we obtain
1'
$,>?;
T? show asymptotic stability of the origin of j. = f ( x ,0 ) , we nccd to either sl~ow
that V is negative definite or apply the invariance principle. I n the next lemma, we
apply the invariance principle by considering a case where V = 0 when v = 0 and
then require the additional property that
<-
~ ( r2 )
k
b
il(m(t;z))dt
Suppose now that there is f # O such that V ( Z ) = 0. The foregoing inequa1it.y
implies
'1
&
!'
1
,
"T
for all solutions of (6.18) when u = 0. Equivalently, no solutions of x = f ( x , 0 )
can stay identically in S = { x E Rn I h ( x ,0 ) = 0 } , other than the trivial solution
x ( t ) I 0. The property (6.20) can be interpreted as an observability condition.
.Recall that for the linear systenl
x=Ax,
y=Cx
observability is equivalent to
C.
For eRsy reference. we define (6.20) ns m obsrrvnhility property of the system.
C
which c0ntradict.s t,he claim that f # 0. Thus! V ( x )> 0 for all x # 0. This qualifies
V ( x )as a Lyapunov function candidate. and since V ( X ) 5 - $ ( I ) , we conclude that
the origin is asymptotically stable.
Suppose now the system.is output strictly passive and let V ( x ) be its storage
function. Then, with u = 0 , V satisfies the inequality V 5 -yTp(y), where yTp(y) >
0 for all y # 0. By repeating the precetlii~gargument, we can use the inequality to
show that V ( x ) is posit.ive definite. In part.icular, for any x E Rn:we have
V(x)
>
1'
l ~ ~ ( br() f,~:) p ( h ( m ( xt ;) , 0 ) ) dt
- . - .. --- . . -..
'
.
- --.......- .-. .. ...--
.-
.
/
-
:. . -
.
....
CHAPTER 6. PASSIVITY
244
Suppose now that there is
implies
r#
...
.
..
-.--..-.----.-
- . - ... . . . , ;:L
'T."
a
.
.
.
.
"
.$..,'
$
A
.
d
- --.
a
-.-,.,,
*
. . ~.~"..;,-&~&2.,L.A..
>,-,
6.5. PASSIVITY THEOREblS
245
0 s~ichthat V ( i ) = 0. The foregoing inequality
LT
hT(m(t; l),O)p(h(#(t; I ) , 0)) dt = 0, V 7 E lO,6]
*
h(#(t; 5 ) .0)
0
which, due to zero-state observability, implies
where a and k are positive constants. Consider also the positive definite, radially
unbounded fi~nctionV(x) = (1/4)ax:
(1/2)xi as a storage function candidate.
The derivative v is given by
+
Hence, V(x) > 0 for all x # 0. This qualifies V(x) as a Lyapunov function can-yTp(y) and y(t) = 0 + x(t) = 0, we conclude by
didate, and since V(x)
the invariance principle that the "rigill is asymptotically stahle. Finally, if V(x) is
radially unbounded, we can infer global asymptotic stability from Theorem 4.2 and
Corollary 4.2, respectively.
<
Therefore. the system is output strict.1~pnssive wit.11 p ( y ) = by. It follows froin
Lemrna 6.5 that the systeln is firiite-gain L2stable with L2gain less than or equal
to Ilk. h,loreover, when u = 0,
Example 6.5 Consider the pinput-poutput system4
.@:
4~
'
j:.
lfi:
where f is locally Lipschitz, G and h are continuous, f(0) = 0, and h(0) = 0.
Suppose there is a continuously differentiable positive semidefinite function V(t)
such that
av
av
%f (x) 0, %G(x) = hT(x)
<
Then,
BV
uTY- -[f
Ox
BV
(x)
av f (x) - hT (x)u = - %f (2) 2 0
+ G(x)u] = uTh(x) - Bx
Hence, the system is zero-state observable. It follows from Lemma 6.7 that the
origin of the unforced system is globally asymptotically stable.
A
6.5
Feedback Systems: Passivity Theorems
Consider the feedback connection of Figure 6.11 where each of the feedback components HIand H2is either a timeinvariant dynamical system represented by the
state model
which shows that the system is passive. If V(x) is positive definite, we can conclude
that the origin of x = f (2)is stable. If we have the stronger condition
av f (x) < -khT(x)h(x), $x)jj-,av
ax
or a (possibly time-varying) memoryless function represented by
= hT(x)
for some k > 0, then
and the system is output strictly passive with p ( ~ =
) ky. It follows from Lemma 6.5
that the system is finitegain L2stable and its L2gain is less than or equal to Ilk.
If, in addition, the system is zero-state observable, then the origin of x = f(x) is
asymptotically stable. Furthermore, if V(x) is radially unbounded, the origin will
A
he globally asymptotically stable.
"Lz stability OF this system was studied in Examples 5.9 and 5.10.
We are interested in using passivity properties of the feedback components HIand
Hz to analyze stability of the feedback connection. We will study both L2 and
Lyapunov stability. We require the feedback connection to have a well-defined state
model. When both components HIand Hz are dynamical systems, the closed-loop
state model takes the form
j. =
f (x, U)
y = h(x,u)
'La and Lynpunov stability of this system were studied in Examples 5.8 and 4.9.
(6.24)
(6.25)
CHAPTER 6. PASSIVITY
6.5. PASSIVITY THEOREbfS
The feedback coi~nectionwill haw a \rcll-defined state model if the cquatiorls
have a unique solution (el, ez) for every (xl, t , ~1,212).This will be always the casc
when hl is independent of el. The case when both components are memoryless
functions is less important and follows trivially as a special case when the statc x
does not exist. In this case. the feedback connection is represented by y = h(t,u).
Figure 6.11: Feedback connection.
The starting point of our analysis is the following fundamental property:
T h e o r e m 6.1 The feedback connection of two passive systems is passive.
Proof: Let Vl(xl) and fi(x2) be t.he storage functions for H1 and Hz, respectively.
If either component is a memoryless function, take = 0. Then,
We assume that f is locally Lipschitz, h is continuous, f (0,O) = 0, and h(0, 0) = 0.
It can be easily verified that the feedback connectioli will have a well-defined state
model if the equations
e: yi 2 v i
From t.he feedback connection of Figme 6.11, we see that
Hence:
have a unique solution (el, en) for every (21, x2, ~1,212).The properties f (0,O) = 0
and h(0,O) = 0 follow from fi(O,O) = 0 and hi(0,O) = 0. It is also easy to see
that (6.26) and (6.27) will always have a unique solution if hl is illdependent of el
or h2 is independent.of e2. In this case, the functions f and h of tlic closetf-loop
state model inherit smoothnessproperties of the functions fi and hi of the feedback
components. In particular, if fi and hi are locally Lipschitz, so are f and h. For
linear systems, requiring hi to be independent of ei is equivalent to requiring t,he
transfer function of Hi to be strictly proper.G
When one component, Hl say, is a dynamical system, while the other one is a
memoryless function, the closed-loop stat,e model takes the form
Taking V(x) = 6 ( x l ) + Vz(x2)as the storage function for the feedback connection,
na obtain
.uTy 2 v
Using Theorem 6.1 and the results of the previous section on stability properties
of passive systems, we can arrive at some straightforward conclusions on stability
of the feedback connection. We start with t2stability. The next lemma is an
immediate consequence of Lemr~ia6.5.
Lemma 6.8 The feedback connection of two output strictly pass'ive sy.~temswith
where
is finite-gain tzstable and its Lz gain is less than or equal to 1/ min{gl, 62).
We assume that f is piecewise continuous in t and locally Lipschitz in (x, u), h is
piecewise continuous in t and cont.inuous in (x, u), f (t, 0,O) = 0, and h(t, 0,O) = 0.
::5,
:.,
I
.,,
,,
j.
:
1,
k
*
I
"Tl
"
rslstrnr~of soll~lionsfor (6.26) n~lcl(6.27) is ~rors~~cul
fr~rtl~rr
in Esrrrisr 6.12.
Proof: With V = Vl
+ l5 and b =nlin{bl, 621, we have
.
.
/
-:.
.
.-
CHAPTER 6. PASSIVITY
21 8
w--. . . . .
.-L-
_ . . .
.
-.. . . . . . . . .
.....
..
--
_
__
:...
$
.,
"
$
-
d
',
.
.
.
.
.
-<.
I
.
.
---.
.,
-,.
-.-I ,
...
,.. -
,
.
-.
- . .
-..-,
..
210
6.5. PASSIVITY THEORE!vlS
+
where k2 = b2 2ac. Iiltegratiilg over (0, T ] , using V(r) 2 0, and taking tlle square
roots, we arrive at
k
Reading t,he proof of Lemma 6.5 shows that we use the inequality
5 ;ll~711Ll
11~7Il~l
+
JmmE
which completes the proof of the theorem.
to arrive at the inequality
1
v 5 -uTu
26
-
6
pTY
(6.33)
which is then used to show finite-gain C2 stability. In Lemma 6.8, we establish
(6.32) for the feedback connection, which then leads to (6.33). However, even if
(6.32) does liot hold for the feedl~i~ck
connection, we may still be able to show an
inequality of the form (6.33). This idea is used in the next theorem to prove a more
general result that includes Lemina 6.8 as a special case.
Theorem 6.2 Consider the feedback connection of Figure.6.11 and suppose each
feedback component satisfies the in,equulity
eTyi 2
fi + +;eTei + biYTyi,
for i = 1,2
(6.34)
for some storage function K(xi). Then, the closed-loop map from u to y is finite
gain C2 stable if
(6.35)
€1
62 > 0 and €2 61 > 0
0
+
0
Theorem 6.2 reduces to Lemma 6.8 when (6.34) is satisfied with €1 = €2 = 0,
61 > 0, and 62 > 0. However, condition (6.35) is satisfied in several other cases.
For example, it is satisfied when both H1 and Hz are input strictly passive with
eTyi 2 fi+~iu'ui for some ei > 0. It is also satisfied when one component (HI say)
.
is passive, while the other component satisfies (6.34) with positive €2 and 6 ~ What
is more interesting is that (6.35) can be satisfied even when some of the constants
~i and 6i are negative. For example, a negative €1 can be compensated for by a
6 ~ .This is a case where shortage of passivity (at the input side) of HI
is compensated for by excess of passivity (at the output side) of Hz. Similarly, a
negative J2 can be compensated for by a positive €1. This is a case where shortage
of passivity (at the output side) of H2 is compensated for by excess of passivity (at
the input side) of HI.
Example 6.7 Consider the feedback connection of
+
Proof: Adding inequalities (6.34) for i = 1,2 and using
where k > 0 and ei, yi E RP. Suppose there is a positive definite function VI fx)
such that
av1 (x),< 01
zf
ah
-G(x)
ax
= hT(x), V
Both components are passive. Moreover, Hz satisfies
we obtain
E Rn
Thus, (6.34) is satisfied with €1 = 61 = 0, €2 = yk, and 62 = (1 - y)/k. This shows
that (6.35) is satisfied, and we conclude that the closed-loop map from u to y is
finite-gain C2 stable.
A
where
Example 6.8 Consider the feedback connection of
+
+
and V(r) = VI(XI) Vz(x2). Let a = inin{e2 bl,
c = JJn'1))z
1 0. Then
+d2) > 0, b = ((Nllz2 0, and
31 = xz
~2 = -ax? -U(XZ)+ el
Y1 = 5 2
and
H2 : yz = kez
where o E [-a, co], a > 0, a > 0, and k > 0. If o was in the sector (0, co], we could
have shown that HI is passive with the storage function Vi (x) = (a/4)xf (1/2)x;.
For u E [-a, co], we have
+
'
250
CHAPTER 6. PASSIVITY
Hence, (6.34) is satisfied for H1 with €1 = 0 ~ i i d61 = -a. Since
(6.34) is satisfied for HZ with €2 = yk and 62 = (1 - y)/k. If k > a, we can clloose
y such that yk > a. Then, €1 62 > 0 and 9 61 > 0. We conclude that the
closed-loop map from u to y is finite-gain t2stable.
A
+
+
Let us turn now to studying Lyapunov stability of thc feedback conllectioll. \Ire
are interested in studying stability and asymptotic stability of the origin of the
closed-loop system when the input u = 0. Stability of the origin follows trivially
from Theorem 6.1 and Lemma 6.6. Therefore, we focus our attention on st,udying
asymptotic stability. The next theorem is an im1nediat.econsequence of Theorem 6.1
and Lemma 6.7.
Theorem 6.3 Consider the feedback connection of ttrro time-in11ntian.t dyna~n.ica1
systems of the f o m (6.21)-(6.22). The origin of the closed-loop system (6.24) (when
u = 0) is asymptotically stable if
both feedback com.ponents are strictly passiz~e,
251
6.5. PASSIVITY THEOREMS
and v = 0 implies $1 = 0 and yz = 0. Note that y2(t) a 0 + el(t) = 0, which
together with xl(t) = 0 imply that !jl(t) = 0. Hence, e2(t) = 0 and zero-state
observability of Hz shows that y2(t) 0 + xz(t) G 0. Thus. the origin is asymptotically stable. Finally. if Vl(xl) arid 15(z2) are radially unbounded, so is V(x),
and we can conclude global asymptotic stabilify.
The proof uses a simple idea, namely, that the sum of the storage functions
for the feedback components is usctl as a Lyapunov function candidate for the
feedback connection. Beyond this sirnple itlea, the rest of the proof is straightforyard Lyapunov analysis. In fact. tlie analysis is restrjctive because to show that
V = Vl Vz 5 0, we insist that both Vl 5 0 and V2 < 0. Clearly, this is not
necessary. One term. vl say, could be positive over some region as long as the sum
v 5 0 over the same region. This is again a manifestation of the idea that shortage
of passivity of one component can be compensated for by excess of passivity of the
other component. This idea is exploitrd in Examples 6.10 and 6.11, whilc Exarnplc
6.9 is a straightforward application of Theorem 6.3.
+
Example 6.9 Consider the feedback connection of
both feedback cony)onents are output strictly passive and zero-state obsenmble,
or
one component is strictly passiire and the other one is oi~tputstrictly pa~si~re
and zero-state observable.
firthemore, if the storage bnction for each componen,t is radially unbounded. the
origin is globally asymptotically stable.
0
Proof: Let Vl(x1) and Vz(x2) be the storage functions for H1 and Hz, respectively.
As in the proof of Lemma 6.7, we can show that Vl(x1) and Vz(x2) are positive
definite functions. Take V(z) = Vl(z1) V2(x2) as a Lyapupov function candidate
for the closed-loop system. In the first case, the derivative V satisfies
+
since u = 0. Hence, the origin is asymptotically stable. 111 the second case,
where a, b, and li are positivc coilstiuit.~.Using Vl = (a/4)x:
storage function for H1, we obt.ain
+ (1/2)2:
as the
Hence, H1 is output strictly passive. Besides, with el = 0, we have
which shows that H I is zero-state observable. Using V2 = (b/2)x:
storage function for Hz, we obtain
+ (1/2)x:
as the
% = bx3x4 - b 5 3 ~ -4 .r: + x4e2 = -y24 + yzez
where y'~i(vi) > 0 for all yi # 0. Here v is only negative semidefinite and v = 0 +
y = 0. To apply the invariance principle, we need to show that y(t) n 0 + x(t) E 0.
Note that y2(t)
0 + el(t) m 0. Then, zero-state observability of HI sho~vs
that yl(t) E 0 =+ xl(t) E 0. Similarly, yl(t) E 0 + ez(t) = 0 and zerostate observability of H2 shows that y2(t) E 0 =+ x2(t) r 0. Thus. the origin is
asymptotically stable. 111the third case (with H1 as the strictly passive component).
=
Therefore, Hz is output strictly passivc. Rloreover, with ez = 0: we have
y2(t) = 0
H
x4(t)
0 .=. x3(t) I 0
which shows that Hz is zero-state observable. Thus, by the second case of Theorem 6.3 and the fact that Vl and 14 arc radially unbounded, we conclude that the
A
origin is globally asymptotically stahlr.
.
.
. ..
.
. . ' ...
--
--
--.-.
.
.
- ..
-
...,
CHAPTER 6. PASSIVITY
252
-.
.<
--
-.. .! ,... >
"
:
.*,A, .
T
I
3
,
-
.'
.
,
. -.
'.
. .-.--.I="L;
.
.
: .-A
6.5. PASSIVITY THEOREAlS
:',**;',A <sic:,/
253
Example 6.10 Reconsider the feedback coililection of the previous example, hut
change the output of Hl to yl = x2 + el. F+om the expression
we can conclude that H1 is passive, but we cannot conclude strict passivity or
output strict passivity. Therefore, we cannot apply Theorem 6.3. Using
as a Lyapunov function candidate for the closed-loop system, we obtain
Moreover, v = 0 implies that
12
L - - - - - - - J
= xa = 0 and
Figure 6.12: Example 6.11.
Thus, by the invariance principle and the Fact that V is radially unbounded, we
conclude that the origin isglobally asymptotically stable.
Example 6.11 Reconsider the system
from Examples 4.8 and 4.9, where hl and h2 are locally Lipschitz and belong to
the sector (0, oo). The system can be viewed as the state model of the feedback
connection of Figure 6.12, where H1 consists of a negative feedback loop around the
integrator x2 with hz in the feedback path, and Hz consists of a cascade connection
of the integrator xl with hl. We saw in Example 6.2 that H1 is output strictly
passive with the storage function 13 = (1/2)x: and, in Example 6.3, that H2 is
lossless with the storage function V2 = 'S: hl (u) du. We cannot apply Theorem 6.3
because Hz is neither strictly passive nor output strictly passive. However, using
hl(u) du (1'/2)xi as a Lyapunov function candidate, we can
V = Vl -I-IT2=
proceed to investigate asymptotic stability of the origin. This is already done in
Examples 4.8 and 4.9, where it is shown that the origin is asymptotically stable and
will be globally asymptotically stable if S l hl(z) dz + m as (yl + m. We will
not repeat the analysis of these two examples here, but let us note that if hl(y) and
h2(y) belong to the sector (0, m) only for y E (-a, a), then the Lyapunov analysis
can be limited to some region around t.he origin, leading t o a local asymptotic
stability conclusion, as in Example 4.8. This shows that passivity remains useful as
a tool for Lyapunov analysis evcii wlic!li it l~oltlso~ilyon a finite region, rather than
the whole space.
'A
stoZ'
+
When the feedback connection has a dynamical system as one component and
a memoryless function as the other component, we can perform Lyapunov analysis
by using the'i;torage function of the dynamical system as a Lyapunov function
candidate. It is important, however, t o distinguish between time-invariant and
time-tarying memoryless functions, for in the latter case the closed-loop system
will he nonaritonomo~lsand we cannot apply the invariance principle as we did in
the proof of Theorem 6.3. We treat these two cases separately in the next two
theorems.
T h e o r e m 6.4 Consider the feedback connection of a strictly passive, time-invariant,
dynamical system of the form (6.21)-(6.22) with a passive (possibly time-varying)
memotyless function of the form (6.23). Then, the origin of the closed-loop system
(6.28) (when ri = 0) is uniformly asymptotically stable. firthemnore, the storage
function for the dynamical system is mdially unbounded, the origin will be globailg
uniformly asymptotically stable.
0
Proof: As in the proof of Lemma 6.7, it can be shown that Vl(zl) is positive
definite. Its derivative is given by
8%
fi = -&1
flki,ei)
2 eTyl- h ( z 1 ) = -eTy2
- qjl(zl) 5 -qj1(zl)
The conclusion follows from Theorem 4.9.
T h e o r e m 6.5 Consider the feedback connection of a time-invariant dylamieal system H1 of the form (6.21)-(6.22) with a time-invariant memoryless function Hz of
;?$>>:g?q
I
i
CHAPTER 6. PASSIVITY
theform (6.23). Suppose that Hl is rem-state obsertrable and has a positive definite
stomge function, which satisfies
eTyi L
li,+ yTpl(y1)
and that HZ satisfies
eTyz 1 eTvz(ez)
(G.37)
Then, the o ~ g i nof the closed-loop system (6.28) (when u = 0) is asymptotically
stable if
vT[Pi(v) vz(v)] > 0, V v # 0
(6.38)
Furthermom, if Vl is mdially unbounded, the odgin will be globally asymptotically
stable.
0
Proof: Use K(x1) as a Lyapunov function candidate, t.o obtain
+
= -e?yz - YTPI(YI) 5 -[YTv2(~1)+ Y ? P ~ ( Y ~ ) ]
Inequality (6.38) shows that I$
0 and fi = 0 9 yl = 0. Noting that yl(t)
0
ez(t) 0
el(t) s 0, we see that zeroatate observability of HI implies that
xi (t) 0. The conclusion follows from the invariance principle.
0
*
*
<
Example 6.12 Consider the feedback connection of a strictly positive real transfer
function and a passive time-varying memoryless function. From Lemina 6.4. wc
know that the dynamical system is strictly passive with a positive definite storage
function of the form V(x) = (1/2)xTPx. J+om Theorem 6.4, we conclude that the
origin of the closed-loop system is globally uniformly asymptotically stable. This is
a version of the circle criterion of Section 7.1.
A
E x a m p l e 6.13 Consider the feedback connection of
where a E (0,m) and ei, yi E R*. Suppose there is a radially unbounded positive
definite function K(x) such that
av1f (x) 5 0
ax
av1
ax
-G(x)
= hT(x),
V x E Rn
' and HI is zero-state observable. Both components are passive. bIoreover. Hz
satisfies
T
e2 YZ = era(e2)
Thus, (6.36) is snt,isfirtl wit,ll pi = 0, ailtl (6.37) is sat.isfictl wit,h cpz = a. Sillce
a E (O,oo), (6.39) is sntisfictl. It follows from Tlieorcin 6.5 that the origin of tlie
closed-loop system is globnlly asymptoticnlly stnblr.
A
6.5. PASSIVITY THEOREAIS
We conclude this scction by prcsci~tiiigloop transformations, which extend the
utility of passivity theorems. Starting with a feedback connection in which one of the
two feedback components is not passivc or does not satisfy a condition that is needed
in one of the theorems. we may be alde to reconfigure the feedback connection
into an equivalent connection that has thc desired properties. We illustrate the
process first for loop transformations that use constant gains. Suppose H I is a
time-invariant dyliamical system, while Hz is a (possibly timevarying) memoryless
functioii that belongs to the sector [IC1,K2], where K = Kz - Ki is a positive
definite symmetric matrix. We saw in Section 6.1 that a function in the sector
[K1. K2] can he transformed into a function inthe sector [O, CO]by input feedforward
followed by output feedback, as shown in Figure 6.7. Input feedforward on Hz
can be nullified by output feedback on Hi. a~ shown in Figure 6.13(b), resulting
in an equivalent feedback connection, as far as asymptotic stability of the origin
is concerned. Similarly. premultiplying the modified Hz by K-' can be nullified
by postmultiplying the modified Hi by I<, as shown in Figure 6.13(c). Finally,
output feedback oil the component in the feedback path can be nullified by input
feedforward on the component in the forward path, as shoyn in Fi_gure 6.13(d].
The reconfigured feedback connection has two components H1 and HZ, where Hz
is a memoryless function that belongs to the sector [O,CO]. We can now apply
Theorem 6.4 or 6.5 if H1 satisfies the conditions of the respective theorem.
~ x a m ~ l e ' 6 . 1Consider
4
the feedback coiiiicction of
f 1 = x2
j.2
= 4x1)
Y1
= 52
+ bx2 + e l
and
Hz : yz = a(ez)
where a E [a,P], 12 E [al, CO],b > O! 01 > 0! and k = P - a > 0. Applying the loop
trnnsfoririatioii of Figure G.13(d) (witli IC1 = a aud K2 = P) results ill tlie feedback
connection of
fi:
[
= 12
= h
-
2
+
and
& : h=b(h)
where 5 E 10,CO]and a = a - b. If a > b. it can he shown (Exercise 6.4) that fil
is strictly passive with a storage funct.iori of the form Vl = k
h(s) ds xTPx,
where P = P' > 0. Thus, we conclude from Theorem 6.4 that thc origiii of trhc
A
feedback connectiou is globally asymptotically stable.
St'
+
Next. we consider loop transformations with dynamic multipliers, as shown in
Figure 6.14. Premultiplying Hz by a transfer function W(s) can by nullified by
postmultiplying HI by W-'(s), providctl thc inverse exists. For example, wlicrl Hz
is a passive, time-invariant, memoryless function h, we saw in Example 6.3 that
premultiplying h by the transfer fu~~ction
l/(as + 1) results in a strictly passive
-
-
,.
d,.
/
.- --
.
.. ..
. CHAPTER 6. PASSIVITY
.
-?
7
"
- ...-. . . . .
-.:...,
-
-.
.
"..-.-.------
.
,
<
.
>,L.*.....
d
.
+
\r.
..&-
-
..
...
n
. .
-.-'
'
_..
.-
257
6.5. PASSIVITY THEOREMS
Figure 6.14: Loop transformation with dynamic multipliers.
+
dynamical system. If postmultiplying H1 by (as 1 ) results in a strictly passive
system or an output strictly passive system that is zero-state observable, we can
employ Theorem 6.3 to conclude asymptotic stability of the origin. This idea is
illustrated in the,,next two examples for cases where H1 is linear and nonlinear,
respectively.
Example 6.15 Let H1be a linear time-invariant system represented by the state
model
x=Ax+Bel, y , = C x
+ +
Its transfer function 1/(s2 s 1 ) has relative _degreetwo; hence, it is not positive
real. Postmultiplying H Iby (as+ 1 ) results in HI,
which can be represented by the
state model
x = A x + B e l , , $1
=ex
+
Figure 6.13: Loop transformation with constant gains. A memorylew function Hz in
the sector [K1,1c2]is transformed into a memoryless function H2in the sector (0,CO].
where C = C aCA = [ 1 a
the condition
1. Its transfer function (as+ l ) / ( s 2+ s + 1 ) satisfies
it a 2 1. Thus, choosing a 2 1, we can apply Lemmas 6.3 and 6.4 to conclude that
Hl is strictly passive with the storage function ( l / 2 ) x T P x where P satisfies the
equations
PA A ~ =
P - L ~ L- EP, P B = CT
+
258
(7IIAPTER 6. PASSIVITY
6.6. EXERCISES
for some L and E > 0. On the otlier lialid. let H2 be given by y2 = A(e2). where
h E [0, w]. We saw in Example 6.3 tliat prelni~ltiplyiligh by tlie tralisfer fiinctioli
l / ( a s + l ) results in a strictly passive system with the storage function a Joe' h(s) ds.
Application of Theorem 6.3 shows that the origin of the transformed feedback tollnection of Figure 6.14(b) (with zero input) is asymptot.ically stable with tlie Lyepunov function V = (1/2)xTPx a
h(s) ds. Notice, hotvevrr. tliat tllr transformed feedback connectioll of Figure 6.14(b) has a state model of dimension three,
while the original feedback connection has a state model of dimellsioli two; so more
work is needed to establish asymptotic stability of the origin of the original fecclback
connection. The extra work can be alleviatecl if n7e use the trarisformed feedback
connection only to come up with the Lyapunov function V and tlien proceed to
calculate the derivative of V with respect to tlie original feedback collnectioll. Sricli
derivative is given by
connect ion)
e2
1' = (1/4)b x j
+ (l/2)xTpx +
h(s) ds
as a Lyapuliov function caiididate for tlic original feedback connection (whcn u = 0)
I
yields
+ Jbe2
which is negative definite. Since V is positive definite and radially unbounded, we
A
conclude tliat tlie origin is glol~allynsympt,otically stable.
6.6
Exercises
6.1 Verify thaea function in the sector [K1,K2]can be transformed into a function
in the sector [O! x ] by input feedforward followed by output feedback, as shown in
Figure 6.7.
!I
which shows that the origin is asymptotically stable. In fact, silice I' is radially
unbounded, we conclude tllnt the origin is globally asymptotically stable.
A
where a a11d I; are positive colistallts iuld h E [O, k]. Show that the system is pacsive
with V(x) = a h(u) du as the storage fuliction.
St
Example 6.16 Consider the feedback connection of
+
where b > 0,- k > 0, and h E [0, w]. Postmultiplying H1 by (as 1) resu1t.s
in a system H1 represented by the same state equation but with a new output
fil =-XI ax2. Using Vl = (1/4)bxi (1/2)xTpx as a storage functioli caiididntc
for HI, we obtain
+
6.2 Consider the system
+
\vhrre 0 < a < a and h E (0. m]. Sliotv that the system is strictly passive.
Hint: Usc \'(I) of Example 4.5 as thc storage function.
i
6.4 Consider t,he system
+
Taking pll = k, pl2 = p22 = l 9a = 1, and assuming that k
r
> 1, we obtain
wliicli shows that H] is strictly passive. On the other hand. prcniultiplying h by
thr trnrisfer fiilictioli l/(s + 1) results ill n strictly passive systrin with tlie stnr~ ~ IIIIN.~
g r io11
/I(x) (I,*. l'si~igIIiv stonigv hnic-tion (or tlis 1r o t ~ s i o ~ ~fi~>tll):ttsk
~i~~cl
and a, > 0. Let V(x) = k S t ' h(s) ds xTPx,
where a > 0: k > O? h E [al,
where pi1 = uplz, p22 = k/2. and 0 < pi2 < min{2al, ak/2). Using V(x) as a
storage function, show that the systein is strictly passive.
6.5 Consider t.he system represented by the block diagram of Figure 6.15, where
E RJ', A.1 and K are positive definite symmetric matrices, h E [O,K], and
h T ( u ) N do 2 0 for all x. Sho~vthat the system is output strictly passim.
u?y
st
. ,-
. - .
,
-.
.
....
.
.,
:
'
.
'
, *.,
---*
.- .
,
.*
...
.---
I
,
..._.......
t
Figure (3.15: Exercise 6.5
+
~
,is lossless.
u ~ ]
~
+
(c) Show that, when v = 0, the origin w = 0 is globally asymptotically stable.
+
+
6.8 Consider equations (6.14) through (6.16) and suppose that (D + DT) is nonsingular. Show that P satisfies the Riccati equation
6.13 Consider equations (6.26)-(6.27) and (6.30)-(6.31), and suppose hl =.hl (xl),
independent of el. Show, in each case, that the equations have a unique solution
(e1,ez).
Bo = B ( D + DT)-IBT, and Co =
6.14 Consider the feedback connection of Figure 6.11 with
351
+
.>
(b) Let u = -Kw v, where K is a positive definite symmetric matrix. Show that
the map from v to w is finite-gain t 2 stable.
+
6.7 Show that the transfer function (bos + bl)/(s2 a l s + a2) is strictly positive
real if and only if all coefficients are positive and bl < ulbo.
6.10 Consider the equations of motion of an m-link robot, described in Exercise 1.4. Assume that P(q) i s a positive definite function of q and g(q) = 0 has an
isolated root at q = 0.
..
6.12 Consider the feedback system of Figure 6.11 where HI and Hz have the state
models
xi = fi(xi) G i ( ~ i ) e i , Pi = hi(zi) Ji(Xi)ei
for i = 1,2. Show that the feedback system has a well-defined state model if the
matrix I J2(x2)JI(XI) is nonsingular for all xl and x2.
6.6 Show that the parallel connection of two passive (respectively, input strictly
passive, output strictly passive, strictly passive) dynamical systems is passive ( r e
spectively, input strictly passive, output strictly passive, strictly passive).
6.9 Show that if a system is input strictly passive, with cp(u) = eu, and finite-gain
C2 stable, then there is a storage function V and positive constants e l and 61 such
that
_ ? _
261
(a) Sliow tliat tlie Jnltp from u = [ul, uz,u3IT to w = [dl,~
+ B ( D + DT)-'C,
'
6.6. EXERCISES
CHAPTER 6. PASSI\'ITY
where A. = - ( ~ / 2 ) 1 - A
-cT(D + D ~ ) - ~ c .
..;
-
.
.
C
= x2
-21
QI1
,
4if
(a) Using the total energy V = 1z q' T A l ( q ) ~ P(q) as a storage function, show that
the map from u to q is passive.
=
- h r ( n ) +el
and H2 :
{
=
Y2
22
-23
+ e2
= h2(~3)
where hl and h2 are locally Lipschitz functions, which satisfy hl E ( 0 , 4 , h2 E
(0, oo],and (hz(z)(2 ( z ( / ( l z2) for aI1 Z.
+
(a) Show that the feedback connection is passive.
(b) Show that the origin of the unforced system is globally asymptotically stable.
6.15 Repeat the previous exercise for
+
(b) With u = -Kdq v, where Kd is a positive diagonal constant matrix, show
t,hat the map from v to Q is 04tput
strictly passive.
\
(c)
Show that u = -Kdq, where Kd is a positive diagonal constant matrix, makes
the origin ( q = 0, q = 0) asyn~ptoticallystable. Under what additional
conditions will it be globally asyfnptotically stable?
6.11 ([151]) Euler equations for a rotating rigid spacecraft are given by
where w l to w3 are the components of the angnlar velocity vector along the principal
axes, ul to ug are the torque inputs ilppliccl about the principal axes, and J1to J3
are the principal moments of inertia.
2
6.16 ([78]) Consider the feedback system of Figure 6.11, where H1 and H2 are
passive dynamical systems of the form (6.21)-(6.22). Suppose the feedback connection has a well-defined state model and the series connection HI(-Hz), with input
e2 and output yl, is zero-state observable. Show that the origin is asymptotically
stable if Hz is input strictly passive or HI is output strictly passive.
6.17 ([78]) Consider the feedback system of Figure 6.11, where Hl and Hz are
passive dynamical systems of the form (6.21)-(6.22). Suppose the feedback cannection has a well-defined state model and the series connection H2H1,with input
el and output y2, is zero-state observable. Show that the origin is asymptotically
stable if HI is input strictly passive or Hz is output strictly passive.
.:
CHAPTER ti. PASSIVITY
6.18 ([78]) As a generalization of the concept of passivity, a dynalnical system of
the form (6.6)-(6.7) is said to be dissipative with respect to a supply rate U J ( U ,y)
if there is a positive definite storage function l f ( x ) such that v 5 w. Consitler the
feedback system of Figure 6.11 where H1 and H2 are zero-state observable, dynalnical systems of the form (6.21)-(6.22). Suppose each of HI and H2 is dissipative with
storage function K ( . r ; ) nlid supply rate I U ~ ( U ;p, i ) = $Qiyi
IITR~u~,
where Qi and Ri are real symmetric matrices and Si is a real matrix. Show tliat
the origin is stable (respectively, asymptotically stable) if the matrix
+
+
Chapter 7
Frequency Domain Analysis of
Feedback Systems
is negative semidefinite (respectively, negative definite) for some a > 0.
6.19 Consider the feedback connection of two time-invariant dynamical systems
of the form (6.21)-(6.22). Suppose both feedback cornpolleiits are zero-state observable and there exist positive definite storage functions which satisfv
eTyi 1
~+
eTcpi(ei)
+ YrPi(31i), for i = 1,2
Show that the origin of the closed-loop system (6.24) wheu u = 0 is asymptotically
stable if
vT[p1(v) + ~ 2 ( v )>
] 0 and vTb2(v) - pi(-v)]
> 0, V v # 0
Under what additional conditions will the origin be globally asymptotically stable?
F
1
1
Many nonlinear physical systems can be represented as a feedback connection of a
linear dynamical system and a nonlinear element, as shown in Figure 7.1. The process of representing a system in this form depends on the particular system involved.
For instance, in the case in which a control system's only nonlinearity is in the form
of a relay or actuator/sensor nonlinearity, there is no difficulty in representing the
system in the feedback form of Figurc 7.1. In other cases, the representation may
be less obvious. We assume that the csteriial input r = 0 and study the behavior of
the unforced system. What is unique about this chapter is the use of the frequency
response of the linear system, which builds on classical control tools like the Nyquist
plot and the Nyquist criterion. In Section 7.1, we study absolute stability. The system is said to absolutely stable if it has a globally uniformly asymptotically stable
equilibrium point at the origin for all nonlinearities in a given sector. The circle and
Popov criteria give frequency-domain sufficient conditions for absolute stability in
Figure 7.1: Feedback connection.
t
........
CIMPTER 7. I*'EEDBACh' SYSTEAJS
the forin of strict positive realness of certain transfer functions. In the single-inputsingle-output case, both criteria can be applied graphically. In Section 7.2, we use
tlic tlt!st:ril)irig function method to study the existence of periodic solutions for a
single-input-single-output system. We derive frequency-domain conditions, which
can be applied graphically, to predict. the existence or absence of oscillations and
estimate the frequency and arnplit.ude of oscillation when there is one.
Consider the feedback connection of Figure 7.1. We assume that the external input
r = 0 ant1 study the behavior of tlie unforced system, represented by
. . .
--
.
7
-......
. - --- ----$
r
-------A-
'
0
.
4
.
4 .
...
- -.
+..-...266
7.1. ABSOLllTE STABILITY
Definition 7.1 Consider the system (7.1)-(7.3), ,where II, satisfies a sector condition per Definition 6.2. The system is absobtelg stable if the origin is globally
uniformly asymptotically stable for any nonlinearity in the given sector. It is absolutely stable with a finite domain if the origin is uniformly asymptotically stable.
We will investigate asymptotic stability of the origin by using Lyapunov analysis.
A Lyapunov function candidate can be chosen by using the passivity tools of the.
previous chapter. In particular, if the closed-loop system can be represented as
a feedback connection of two passive systems, then tlie sum of the two storage
functions can be used as a Lyapunov function candidate for the closed-loop system.
The llse of loop transformations allows us to rover various sectors and Lyapunov
function candidates, leading to the circle and Popov criteria.
7.1.1
Circle Criterion
T h e o r e m 7.1 The system (7.1)-(7.3) is absolutely stable if
whcre .r E R",U ,y E RP, (A, B) is coiit,rollablc, (A, C) is observable, and $J :
[O, w) x R" --IRP is a mel~~oryless,
possibly time-varyillg, i~oi~lincurity,
wliich is
piecewise coiitinuous in t and locally Lipschitz in y. We assunie that the feedback
connection has a well-defined state inodel, which is the case when
$J E
[Icl,m] and G(s)[I + KlG(s)]-' is strictly positive real, or
$ E [K1,K2], with K = Kz - K1 = K~
is st~z'ctlypositive real.
> 0, and [ I + ~ z G ( s ) ] [ l +KIG(S)]-'
If the sector condition is satisfied only on a set Y C RP,then the foregoing conditions
has a unique solution u for every (t, x) in the domain of interest. THis is always the
case when D = 0. The transfer funct,ion matrix of the linear system
is square and proper. The controllability and observability assumptions ensure
that {A, B, C, D) is a minimal realization of G(s). From linear system theory, we
know that for any rational proper G(s), a minimal realization always exists. The
nonlinearity 1/, is required to satisfy a sector condition per Definition 6.2. The sector
condition may be satisfied globally, that is, for all y E RP,or satisfied only for y € Y ,
a subset of Rp, whose interior is connected and contains the origin.
For all nonlinearities satisfying the sector condition, the origin r = 0 is an
equilibrium point of the system (7.1)-(7.3). The problem of interest. here is to
study the stability of the origin, not for a given nonlinearity, but rather for a class
of nonlinearities that satisfy a given sector condition. If we succeed in showing that
t.lle origin is uniformly asymptot,ically stable for all nonlinearities in the sector, the
system is said to be absolutely stahlc. The problem was originally formulated by
Lure and is sometimes called Lure's.proble7n. Tkaditionally, absolute stability has
been definer1 for the case when the origin is globally uniformly asymptotically stable.
To keep up t.his tradition, we will use the phrase "absolute stability" when the sector
coritlition is satisfied globally anrl t11c origiii is globally uniformly asymptotically
stable. Ot,hermise,we will use the phrase '.ahsolute stability with a finite domain."
ensure that the system is absolutely stable with a finite domain.
0
We refer to this theoremas the multivariable cimle criterion, although the reason
for using this name will not be clear until we specialize it to the scalar case. A necessary condition for equation (7.4) to have a unique solution u for every 11, E [Kl, m]
or $JE [Kl, Kz] is the nonsingularity of the matrix ( I K1D). This can be seen by
taking $J= K l y in (7.4). Therefore, the transfer function [I+KlG(s)]-I is proper;
+
Proof of T h e o r e m 7.1: We prove the theorem first for the sector [0, m] and recover the other cases by loop transformations. If 11, € (0, m] and G(s) is strictly positive real, we have a feedback connection of two passive systems. From Lemma 6.4,
we know that the storage function for the linear dynarnical system is V(x) =
(1/2)xTPx, where P = PT > 0 satisfies the Kalman-Yakubovich-Popov equations
and
E
> 0. Using V(x) as a Lyapunov function candidat,e, we obtain
.,
CIfAPTER 7. FEEDBACK S)-STEAIS
?
Figure 7.2:
-
-
-
267
7.1. ABSOLUTE STABILITY
r - - - - - - - - - - - - - - -
-------
I
11 E [I<l, oo] is transformed to 4E [0, oo] via a loop transformation.
Figure 7.3: 11, E [I<1,Kz] is transformed to
,
4 E [0, oo] via a loop transformation.
Using (7.6) and (7.7) yields
where urn,[.] denotes the maximum singular value of a complex matrix. The constant 71 is finite, since G(s) is Hurwit.~.Suppose ?I, satisfies the inequality
Using (7.8) and the fact that uTDu = ;uT(D
+ DT)u, we obttlill
then it belongs to the sector [I<1,If2] with K1 =
Theorem 7.1, we need to show that
-721
and K2 = y21. To apply
Since yTIL(t, y) 10, we have
v 5 - ;€xTpx
which shows that the origin is globnlly cxpo~icntinllystablc. If sntisfics thc sccutor
condition only for y E Y , the foregoing analysis will be valid in some neighborhood
of the origin, showing that the origin is exponentially stable. Tlle case JI E [ICl,m ]
can be transformed to a casc where the nonlinearity beloligs to (0, oo] via tlie loop
transformation of Figure 7.2. Hence, the systeln is absolutely stable if G(s)[I +
I<lG(s)]-' is strictly positive real. The case II, E [I<l,I<2] call be transforlncd
to a case where the 1ionlinearit.y beloilgs to [O,oo] via the loop t,rai~sformatio~l
of
Figure 7.3. Hence, the system is absolutely stable if
$J
is strictlv positive real. LVc not(>t.l~;l.ttlrt.[Z(s) + ZT(-s)] is not itlcnt.ically zcro
because Z ( m ) = I. LVe apply Lclllina G.1. Since G(s) is Hurwitz, Z(s) will be
Hurwitz if [I- -y2G(s)]-I is Hurwitz. Noting that1
we see that if 71-72< 1, the plot of rlet[I --y2G(jw)] will not go t,hrough nor encircle
the origin. Hence, by the multivariablc Nyquist criterion,' [I-y2G(s)]-1 is Hurwitz;
consequently, Z(s) is Hurwitz. Next. we show that
:&,
+
I + I<G(s)[I+ I<iG(s)]-' = [I I<~G(s)]
[I+ IC1G(s)]-l
is strictly positive real.
0
Example 7.1 Consider the system (7.1)-(7.3) and suppose G(s) is Hurwitz and
strictly proper. Let
%
= SUP gm,[G(jw)l=
UF R
sup IlG(jw)112
wER
.id-L
'The rollott4ng properties of singular values of a complex matrix are used:
det G # 0 o o,I~[G] > 0
amaxIG-'] = l l a m i n [ q , if a m ~ . [ G l
a m i n [ l + G ] 2 1 - amax[Gl
o m a x [ G G~ 2 ]
-
>0
< ~rnax[Gl]~rnax[G~]
1,
!lp
ii
2See (33. pp. 160-161) lor a statement of tlie nulltivariable Nyquist criterion.
f
:A
-
,
.v:'d$&
,
,.. .
. ._
-..
...,
- 2'!..
.
____
. <
*..
*
-
.
..I .
',
. .
,.. * .
-_- . "
.-.- ....
-: ,
. ,..
260
7.1. ABSOLUTE STABlI,lTY
Tlle left.-hand side of this inequality is give11 by
Hence, Z(jw) + ZT(-jw) is positive definite for all w if and only if
Now, for y1y2
< 1, we h a w
Figure 7.4: Graphical representation of the circle criterion.
+
Finally, Z ( w ) + ZT(co) = 21. Thus, all the conditions of Lemma 6.1 are satisfied
and we co~lcludethat Z(s) is strictly positive real and the system is absolutely stable
the loop around
if 7172 < 1. This is a robustness rcsult, which shows that closi~~g
a Hurwitz transfer function with a nonlinearity satisfying (7.9). with a sufficiently
A
small y2, does not destroy t.he stabilit,y of the system.3
-(l/a)
j O , respectively, as shown in Figure 7.4. The real part of the ratio of two
complex numbers is positive when the angle difference between the two numbers
is less than n/2; that is, the angle (81 - 82) in Figure 7.4 is less than n/2. If we
define D(a,P) to be the closed disk in the coinplex plane whose diameter is the
line segment'connecting the points -(l/a)
j O and -(l/P)
j0, then it is simple
to see that the angle (81 - 82) is less than n/2 when q is outside the disk D(a, P).
Since (7.11) is required to hold for all w, all points on the Nyquist plot of G(jw)
must be strictly outside the disk D(a,P). On the other hand, Z(s) is Hurwitz if
G(s)/[l -iaG(s)] is Hurwitz. The Nyquist criterion states that G(s)/[l rrG(s)]
is Hurwitz if and only if the Nyquist plot of G(jw) does not intersect the point
-(l/a)+jO and encircles it exactly m times in the counterclockwise direction, where
m is the number of poles of G(s) in the open right-half complex plane.4 Therefore,
the conditions of Theorem 7.1 are satisfied if the Nyquist plot of G(jw) does not
enter the disk D ( a , P ) and encircles it m times in the counterclockwise direction.
Consider, next, the case when 0 > 0 and a = 0. For this case, Theorem 7.1 requires
1 flG(s) to be strictly positive real. This is the case if G(s) is Hurwitz and
+
In the scalar case p = 1, the collditions of Theorem 7.1 call be verified graphically
by exainil~ingthe Nyquist plot of G(jsr). For ?I, E [a, PI, with > a, the system is
absolutely stable if the scalar transfrr function
+
+
is strictly positive real. To verify that Z(s) is strictly positive real, we can use
Lemma 6.1 which states that Z(s) is strictly posit,ive real if it is Hurwitz and
+
i
Re[l
To relate condition (7.10) to the Nyquist plot of G(jw), we have to distinguish
between three different cases, depending on the sign of a. Consider first the case
when p > a > 0. In this case, condition (7.10) can be rewritten as
which is equivalent to the graphical condition that t.he Nyquist plot of G(jw) lies
to the right of the vertical line defined by Re[s] = -1/P. Finally, consider the case
+
< 1 can he clcrivcrl also lro~nthe srr~all-gaintheorem. (See Example 5.13.)
V w E [-co, w]
The latter condition can be rewritten as
For a point q on the Nyquist plot nf G(jw). tlir two col~iplrxnr~mbers(l/P)+G(jw)
ant1 (l/tu) + G(jw) can bc rcprc~sciitc~tl
by tllc lir~csconncrting q to -(l/P) j0 ant1
J'rhe inequality 7172
+ PG(jw)] > 0,
6
'when G(s) has poles on the imaginary axis, the Nyquist path is indented in the right-half
plane, as usual.
270
C W P T E R 7. FEEDBACK SI'STEAIS
7.1. ABSOLUTE STABILITY
when a < 0 < 0. In this case, conditioii (7.10) is cquivalcnt to
where the inequality sign is reversed because. as we go from (7.10) to (7.12), ;ve
multiply by alp. which is iiow iiegative. Repeati~igprevious arguments, it can be
easily seen that for (7.12) to hold, the Nyquist plot of G(jw) must lie inside the
disk D(a,D). Consequently, the Nyquist plot cannot encircle the point - ( l / a ) +
jO. Therefore, from the Nyquist criterion, we see that G(s) must be Hurwvitz for
G(s)/[l aG(s)] to be so. The stability criteria for the three cases are summarized
in the following theorem, which is known as the circle criterion.
+
Theorem 7.2 Consider a scalar system of the form (7.1)-(7.3), where { A :B:C:D}
is a minimal realization of G(s) and $JE [a, 01. Then, the svstem is nbsoltrtelg stable
if one of the following conditions is satisfied, as appropriate:
I. If 0 < a < p, the Nyquist plot of G(jw) does not enter the disk D(a, 3) and
encircles it m times in the counterclockwise direction, where rn is the i~.ulnber
of poles of G(s) with positive real parts.
2. I f 0 = a < p, G(s) is Hunuitz and the ~ ~ ~ uploti sofiG(jw) lies to the right
of the vertical line defined by Re[s] = -lip.
< 0 < p , G(s) is Htrnuitz and the Nyquist plot of G(jw) lies in the
interior of the disk D(a,P).
9. If a
If the sector condition is satisfied on$ on an interval [a,b], then the f0regoin.g conditions ensure that the system is absolutely stable with a finite domaiii.
0
The circle criterion allows us to investigate absolute stability by using only
the Nyquist plot of G(jw). This is important because the Nyquist plot can be
determined directly from experimental data. Given the Nyquist plot of G(j&), we
can determine permissible sectors for which the system is absolutely stable. T l ~ c
next two exrllnples il11ist.rnte tlic usc of tlic circlc criterion.
Example 7.2 Let
The Nyquist plot of G(jw) is show11 ill Figure 7.5. Since G(s) is Hurwitz. we can
allow a to be negative and apply the third case of the circle criterion. So, we need
to determine a disk D ( a , P ) that encloses the Nyquist plot. Clearly, the choice of
the disk is not uniqm. Suppose wc decide to locatmetile center of tlie disk at t l ~ c
origin of the complex plaiic. This means that wwet will work with a disk D(-72,32).
R ~ C ~tllc
C rndills ( l / ~ ?>)0 is to br cliosrii. Tllr Nyquist plot wwpil br i ~ ~ s i d
this
r
t
Figure 7.5: Nyquist plot for Example 7.2.
disk if IG(jw)l < 1/72. In particular, if we set 71 = SUP,,=R IG(jw)I, then 7 2 must
be chosen to satisfy 71% < 1. This is the same condition we found in Example 7.1.
It is not hard to see that IG(jw)( is maximum at w = 0 and 71 = 4. Thus, 7 2 must
be less than 0.25. Hence, we can concludc that the system is absolutely stable for all
nonlinearities in the sector [-0.25+~,0.25-€1, where E > 0 can be arbitrarily small.
Inspectioh of the Nyquist plot and the disk D(-0.25,0.25) in Figure 7.5 suggests
that the choice t o locate the center at the origin may not be the best one. By locating
the center at another point, we might be able to obtain a disk that encloses the
Nyquist plot more tightly. For example, let us locate the center at the point 1.5 jO.
The maximum distance from this point to the Nyquist plot is 2.834. Hence, choosing
the radius of the disk to be 2.9 ensures that the Nyquist plot is inside the disk
D(-1/4.4,1/1.1), and we can conclude that the system is absolutcly stable for all
nonlinearities in the sector [-0.227,0.714]. Comparing this sector with the previous
one (see Figure 7.6) shows that by giving in a little bit on the lower bound of the
sector, we achieve a significant improvement in the upper bound. Clearly, there is
still room for optimizing the choice of the center of the disk, but we will not pursue
it. The point we wanted to show is that the graphical representation used in the
circle criterion gives us a closer look at tlie problem, compared wit11 tlie use of norm
inequalities as in Example 7.1, which allows us to obtain less conservative estimates
of the sector. Another direction we can pursue in applying the circle criterion is to
restrict a to zero and apply the second case of the circle criterion. The Nyquist plot
lies to the right of the vertical line Re[s] = -0.857. Hence, we can conclude that
the system is absolutely stable for all nonlinearities in the sector [O, 1.1661. This
sector is sketched in Figure 7,6, together with the previous two sectors. It gives the
best estimate of p , which is achieved at the expense of limiting the nonlinearity to
be a first-quadrant-third-quadrant nonlinearity. To appreciate how this flexibility
in usiiig tlie circle criterion could be useful in applications, let us suppose that we
are interested in studying the stability of the system of Figure 7.7, which includes a
liiniter or saturation noillinearity (a typical nonlinearity in feedback co~itrolsystems
+
. _. .
.
- . .
.-
CHAPTER 7. FEEDBACK SYSTEMS
.
.
.
".:.
,_ "__I___.._.___
..
.a>.
.
\
'
"-- ..--
. ..
-"--- ti--
8
..-
b - ' .
.
.-
,
...
"
,, ,
.
273
7.1. ABSOLUTE STABILITY
Figure 7.6: Sectors for Example 7.2.
Figure 7.8: Nyquist plot for Example 7.3.
due to constraints on physical variables). The saturation nonlinearity belongs to a
sector [O,1]. Therefore, it is included in the sector [O, 1.1661, but not in the sector
(-0.25,0.25) or [-0.227,0.714]. Thus, based on the application of the second case
of the circle criterion, we can conclude that the feedback system of Figure 7.7 has
A
a globally asymptotically stable equilibrium point at the origin.
choose a and p to locate the disk D(a,P) inside the left lobe. Let us locate the
jO, about halfway between the two ends of
center of the disk a t the point -3.2
the lobe on the real axis. The minimum distance from this center to the Nyquist
plot is 0.1688. Hence, choosing the radius to be 0.168, we conclude that the system
is absolutely stable for all nonlinearitiea in the sector [0.2969,0.3298].
A
+
In ~ x a r n ~ l $7.1
A through 7.3, we have considered cases where the sector condition
is satisfied globally. In the next example, the sector condition is satisfied only on a
finite interval.
Example 7.4 Consider the feedback connection of Figure 7.1, where the linear
system is represented by the transfer function
G(s) =
Figure 7.7:. Feedback connection with saturation nonlinearity.
',
Example 7.3 Let
a
This transfer function is not Hurwitz, since it has a pole in the open right-half plane.
So, we must restrict a to be positive and apply the first case of the circle criterion.
The Nyquist plot of G(jw) is shown in Figure 7.8. From the circle criterion, we know
that the Nyquist plot must encircle the disk D(a,P) once in the counterclockwise
direction. Inspection of the Nyquist plot shows that a disk can be encircled by the
Nyquist plot only if it is totally inside one of the two lobes formed by the Nyquist
plot in the left-half plane. A disk inside the right lobe is encircled once in the
clockwise direction. Hence, it does not satisfy the circle criterion. A disk inside
the left lobe is encircled once in the counterclockwise direction. Thus, we need to
s+2
(8
+ 1)(8 - 1)
and the nonlinear element is $(y) = sat(y). The nonlinearity belongs globally to
the sector (0, I]. However, since G(s) is not Hurwitz, we must apply the first case
of the circle criterion, which requires the sector condition to hold with a positive
a. Thus, we cannot conclude absolute stability by using the circle ~ r i t e r i o n .The
~
best we can hope for is to show absolute stability with a finite domain. Figure 7.9
shows that on the interval 1-a,a], the nonlinearity @ belongs to the sector [alp]
with a = l / a and p = 1. Since G(s) has a pole with positive real part, the Nyquist
plot of G(jw), shown in Figure 7.10, must encircle the disk D ( a , l ) once in the
counterclockwise direction. It can be verified, analyticauy, that condition (7.10) is
satisfied for a > 0.5359. Thus, choosing a = 0.55, the sector condition is satisfied on
the interval [-1.818,1.818] and the disk D(0.55.1) is encircled once by t h e ' N y q ~ t
plot in the counterclockwise direction. From the first case of the circle criterion, we
=1n fact, the origin is not globally asymptotically stable because the system has three equilib
rium points.
CHAPTER 7. FEEDBACK SYSTEMS
7.1. ABSOLUTE STABILITY
Figure 7.9: Sector for Example 7.4.
Figure 7.11: Region of attraction for Example 7.4.
where
A=
[
-i55],
B=
[
1, C=
[
0.9 0.45
1,
and D = l
The matrix P is the solution of equations (7.6) through (7.8). It can be verified
that6
E
.
= 0.02, P =
[
0'4946 0'4834
0.4834 1.0774
], L
=
[
0.i946
-0.4436
],
and bV = &
Figure 7.10: Nyquist plot for Example 7.4.
satisfy (7.6) through (7.8). Thus, V(x) = xTPx is a Lyapunov function for the
system. We estimate the region of attraction by
conclude that the system is absolutely stable with a finite domain. We can also use
a quadratic Lyapunov function V(x) = xTPx to estimate the region of attraction.
Consider the state model
V(x) = 0.3445 to ensure that 0, is contained in the set
where c < min~lyl=1.818)
{Jyl< 1.818). Taking c = 0.34 gives the estimate shown in Figure 7.11.
A
7.1.2
Popov Criterion
Consider a special case of the system (7.1)-(7.3), given by
'The loop transformation of Figure 7.3 is given by
where x E Rn, u, y E RP, (A, B) is controllable, ( A ,C) is observable, and : R * R
is a locally Lipschitz memoryless nonlinearity that belongs to the sector [0,ki]. In
this special case, the transfer function G(s) = C ( s I - A)-'B is strictly proper
$J~
Thus, the transformed linear system is given by
I
6The value of E is chosen such that C(s - ~ / 2 )is positive real and [(~/2)1
+A] is Hurwitz, where
D. Then, P is calculated by solving a Riccati equation, as described in
Exercise 6.8.
G(s) = C(sI - A)-'B
+
.
-
1
.
.
..,
CHAPTER 7. FEEDBACK SYSTEBIS
7.1. ABSOLUTE STABILITY
'
277
The condition (1 +Xl;y,) # 0 inlplies that (A,C)is observable; hence, tlle realizatioll
{A, B,C, D) is minimal. If A4 ( I sr)G(s) is strictly positive real, we c m apply
the ICalman-Yakubovich-Popov lemma to collclude that there are matrices P =
PT > 0, L,and Wl and a positive constant E that satisfy
+ +
and V = (1/2)xTPx is a-storage function for f i l . One the other hand, it can be verified (Exercise 6.2) that H2is passive with the storage function
yi $" A(@)do.
Thus, the storage function for the transformed feedback connection of Figure 7.12
is
i
We use V as a Lyapunov function candidate for the original feedback connection
(7.13)-(7.15). The derivative V is given by
.,3
xf=l
f
Figure 7.12: Loop transformation.
and $J is time invariant and decoupled; that is, $Ji(y) = $Ji(yi). Since D = 0, the
feedback connection has a well-defined state model. The following theorem, known
as the multivariable Popov criterion, is proved .by using a (Lure-type) Lyapunov
function of the form V = (1/2)xTPx C y i J ' l(li(u) du, which is motivated by
the application of a loop transformation that transforms the system (7.13)-(7.15)
into the feedback connection of two passive dynamical systems.
i
i
+
Using (7.16) and (7.17) yields
Theorem 7.3 The system (7.13)-(7.15) is absolutely stable ifl jor 1 5 a 5 p,
qiE [O, ki],O < ki 5 CO, and the~eedsts a constant ~i 2 0, urith (1 h y i ) # 0
for every eigenualue Xk of A, such that M + ( I sr)G(s) is stn'ctly positive real,
where r = diag(yl,. . ,-yp) and M = diag(l/kl,. .., Ilkp). If the sector conditiorc
qiE [O, ki] is satisfied only on a'set Y c RP, then the foregoing conditions ensure
0
that the system is absolutely stable w.th a finite domain.
+
+
Proof: The loop tr_ansformationof Figure 7.12 results in a feedback connection of
H~ and &, where HI is a linear system whose transfer function is
Substituting u = -$J(v) and using (7.18), we obtain
v = - IEX~PX
- ~ ( L +x W U ) ~ ( L X+ Wu) - ( ~ ( y ) ~-[ yM$(y)] 5 - $ C X ~ P X
which shows that the origin is globally asymptotically stable. If $J satisfies the sector
condition only for y E Y , the foregoing analysis will be valid in some neighborhood
of the origin, showing that the origin is asymptotically stable.
+ +
. .
+
Thus, M ( I + sr)G(s) :can be realized by the state model {A, B,C, 271, when:
A = A, B = B, C = C + rCA, nl~dD = hf + r C B . Let Xk be an eigenvalue of A
ant1 ur; be the associated eigenvector. Then
(C
+ rCA)vh = (C+ rCXk)Uk = (I+ Xpr)Cvk
For M (I sr)G(s) to be strictly positive real, G(s) must be Hurwitz. As
we have done in the circle criterion, this restriction on G(r) may be removed by
performing a loop transformation that replaw C(s) by G(s)P K,G(s)J-l. We
will not repeat this idea in general, but will illustrate it by an example. In the scalar
case p = 1, we can test the strict positive realneaa of Z(s) = (I/*) + (1 + s?)G(s)
graphically. By Lemma 6.1, Z(s) is strictly positive real if G(8) is Hunsitz and
+
i
i
CHAPTER 7. FEEDBACK SYSTEhlS
Figure 7.13: Popov plot.
Figure 7.14: Popov plot for Example 7.5.
where G(jw) = Re[G(jw)] jIm[G(jw)]. If we plot Re[G(jw)] versus wIm[G(jw)]
with w as a parameter, then condition (7.19) is satisfied if the plot lies to the
right of the l i e that intercepts the point -(l/k)
j O with a slope 117. (See
Figure 7.13.) Such a plot is known as a Popov plot, in contrast to a Nyquist plot,
which is a plot of Re[G(jw)] versus Im[G(jw)j. If condition (7.19) is satisfied only
for w E (-co, w ) , while the left-hand side approaches zero as w tends to oc, then
we need to analytically verify that
Assume that h belongs to a sector [a,P], where P > a. Then, $J belongs to the
sector [0, k], where k = /3 - a. Condition (7.19) takes the form
+
+
1
a - ~ ' + ~ ~
> 0, V w E 1-03, co]
+
k ( a - w ~+w2
) ~
For all 6iiite positive values of a and k, this inequality is satisfied by choosing 7 > 1.
Even at k = m, the foregoing inequality is satisfied for all w E (-co, oo) and
lim
u-00
This case arises when k = w and G(s) has relative degree two.
With 7 = 0, condition (7.19) reduces to the circle criterion condition Re[G(jw)] >
-1/k, which shows that, for the system (7.13)-(7.15), the conditions of the Popov
criterion are weaker than those of the circle criterion. In other words, with 7 > 0,
absolute stability can be established under less stringent conditions.
Example 7.5 Consider the second-order system
'
279
7.1. ABSOLUTE STABILITY
This system would fit the form (7.13)-(7.15) if we took $J = h, but the m a t r ~A
would not be Hurwitz. Adding and subtracting the term a y to the right-hand side
of the second state equation, where a > 0, and defining $J(y) = h(y) ay, the
system takes the form (7.13)-(7.15), with
w2(a-J+7w2)
( a - w2)2 + w2
>O
Hence, the system is absolutely stable for all nonlinearities h in the sector [a,co],
where a can be arbitrarily small. Figure 7.14 shows the Popov plot of G(jw) for
a = 1. The plot is drawn only for w 2 0, since Re[G(jw)] and wIm[G(jw)] are
even functions of w. The Popov plot asymptotically approaches the line through
the origin of unity slope from the right side. Therefore, it lies to the right of any
line of slope less than one that intersects the real axis at the origin and approaches
it asymptotically as w tends to co. To see the advantage of having 7 > 0, let us
take 7 = 0 and apply the circle critcriol~.Fkom the second case of Theorem 7.2,
the system is absolutely stable if the Nyquist plot of G(jw) lies to the right of the
vertical line defined by Re[s] = -l/k. Since a portion of the Nyquist plot lies in
the left-half plane, k cannot be arbitrarily large. The maximum permissible value
of k can be determined analytically from the condition
-
A=[-:
B=[:],
and C = [ 1 O ]
which yields k < l f 2 f i . Thus, using the circle criterion, we can only conclude that
the system is absolutely stable for all nonlinearities h in the sector [a,l+a+2fi-E],
A
where a > 0 and E > 0 can be arbitrarily small.
_
.
.
/
280
.-..
-
.-
,
-..
7.2.
,
_
._-
. ._._._......
.
.
a
<
.
98.
. ..'.
.
7.- ....
d "
r
'
t
'
.$. ,
:,
. ..
..
,-,&.A
281
THE DESCRIBING FUNCT~ONMETHOD
CHAPTER 7. FEEDBACK SYSTEMS
The Describing Function Method
7.2
Consider a single-input-singleoutput nonlinear system represeoted by the feedback
wnnection of Figure 7.1, where G(s) is a strictly proper, rational t r d r function
and 4 is a time-invariant, memoryless nonlinearity. We assume that the externd
input T = 0 and study the existence of periodic solutions. A periodic solution
satisfies y(t 27r/w) = y(t) for all t, where w is the frequency of oscillation. We
will use a general method for finding periodic solutions, known as the method of
h a n o n i c balance. The idea of the method is to represent a periodic solution by a
hurier series and seek a frequency w and a set of Fourier coefficients that satish,
the system's equation. Suppose y(t) is periodic and let
+
Using tlle ortllogondity of t l ~ efu~~ctions
exp(jl;wt) for different values of I*., we find
that the Fourier coefficients must satisfy
G(jkw)ck
(7.20)
for all integers k. Because G(jkw) = G(-jksr), ar, = &,k, and ck = 5-k, we need
only look at (7.20) for k 2 0. Equation (7.20) is an infinitedimensional equation,
which we can hardly solve. We need to find a finite-dimensional approximation of
(7.20). Noting that the transfer function G(s) is strictly proper, that is, G(jw) -,0
as w -,oo, it is reasonable to assume that there is an integer q > 0 such that for
all k > q, (G(jkw)(is small enough to replace G(jltw) (and consequently ak) by 0.
This approximation reduces (7.20) t o a finite-dimensional problem
G(jkw)&+Bk=O,
be its Fourier series, where a t are complex coefficients: a* = I - k and j = fl.
Since $(.) is a time-invariant nonlinearity, $(g(t)) is periodic with the same f r e
quency w and can be written as
m
Il(y(t)) =
ckex~(jkwt)
k=-m
is a function of all ails. For y(t) t o be a solution
where cadi complex co&cient
of the feedback system, it must satisfy the differential equation
d(~)ll(t)+ n(p)$(y(t)) = 0
where p is the differential operator P(.) = d(.)/dt and n(s) and d(s) are the numerator and denominator polynomials of G(s). Because
we have
M
d(p)
M
ak exp(jk*t) =
k=-w
d(jkw)ai exp(jkut)
k=-m
and
k=-w
k=-m
Substituting these expressions back into the differential equation yields
w
[d(jkw)ab + n(jkw)ck]exp(jkwt) = 0
k=-m
'A bar over a complex variable denbtes its complex conjugate.
k = 0 , 1 , 2 ,...,q
(7.21)
where the Fourier coefficients are written with a hat accent to emphasize that a
solution of (7.21) is only an approximation t o the solution of (7.20). In essence, we
can proceed to solve (7.21). However, the complexity of the problem will grow with
q and, for a large q, the finite-dimensional problem (7.21) might still be difficult
to solve. The simplest problem results if we can choose q = 1. This, of course,
requires the transfer function G(s) to have sharp "low-pass filtering'' characteristics
to allow us to approximate G(jkw) by 0 for all k > 1. Even though we know G(s),
we cannot judge whether this is a good approximation, since we do not know the
frequency of oscillation w. Nevertheless, the classical describing function method
makes this approximation and sets Bk = 0 for k > 1 to reduce the problem t o one
of solving the two equations
G(0)?o(Borci,)+cio = 0
(7.22)
G(jw)E1(cio,B1) B1 = 0
(7.23)
Notice that (7.22) and (7.23) define one real equation (7.22) and one complex equation (7.23) in two real unknowns, w and Bo, and a complex unknown B1. When
expressed as real quantities, they define three equations in four unknowns. This
is expected because the time origin is arbitrary for an autonomous system, so if
(B0,dl) satisfies the equation, then (Bo, BleJe) will give another solution for arbitrary real .'6 To take care of this nonuniqueness, we take the first harmonic of y(t)
t o be asinwt, with a 2 0;that is, we choose the time origin such that the phase of
the first harmonic is zero. Using
a
a
a sin wt = [exp(jwt) exp(-jut)] + bl = 2j
we rewrite (7.22) and (7.23) as
+
3
--
+ ak = 0
-
.
989
CHAPTER 7. FEEDBACK SYSTEMS
283
7.2. THE DESCRIBING FUNCTION METHOD
Since (7.24) does not depend on w, it may be solved for 60 as a function of a. Note
is an odd function, that is,
that if
$(a)
then 60 = & = 0 is a solution of (7.24) because
For convenience, let us restrict our attention to nonlinearities with odd symmetry
and take do = & = 0. Then, we can rewrite (7.25) es
describing function of the nonlinearity 4. It is obtained by applying a sinusoidal
signal asinwt at the input of the nonlinearity and by calculating the ratio of the
Fourier coefficient of the first harmonic at the output to a. It can be thought of as
an "equivalent gain" of a linear time-invariant element whose response to a sinwt is
@(a)asinwt. This equivalent gain concept (sometimes called equivalent linearization) can be applied to more general time-varying nonlinearities or nonlinearities
with memory, like hysteresis and b a c k l a ~ h .In
~ that general context, the describing function might be complex and dependent on both a and w. We will only
deal with describing fuilctions of odd, time-invariant, memoryless nonlinearities for
which @(a) is real, dependent only on a, and given by the expression
@(a) =
The coefficient &(0, a/2j) is the complex Fourier coefficient of the first harmonic at
the output of the nonlinearity when its input is the sinusoidal signal asinwt. It is
given by
&&
2a/w
PI (0, a/2j)
=
$(a sin wt) eip(- jwt) dt
2n/w
rg
re
i
."**
s:.
J,%
'*I
B
.?'
?&
.>>
=
E1
W(a sin wt) cos wt - j$(a sin wt) sin wt] dt
The first term under the integral sign is an odd function, while the second term
is an even function. Therefore, the integration of the first term over one complete
cycle is zero, and the integral simplifies to
,?<
.,<.:
,*.
38
$
.*,
61(0, a/'&) = - J ;
r;i
:"P
;*.
'*>'A,
6""
A&
$(a i n 0) sins do
(7.30)
which is obtained from (7.27) by changing the integration variable from t to 0 = wt.
The describing function method states that if (7.29) has a solution (a,, w,), then
there is "probably" a periodic solution of the system with frequency and amplitude
(at the input of the nonlinearity) close to w, and a,. Conversely, if (7.29) has no
solutions, then the system "probably" does not have a periodic solution. More
analysis is needed to replace the word "probably" with "certainly" and to quantify
the phrase "close to w, and a," when there is a periodic solution. We will postpone
these investigations until a later point in the section. For now, we would like to
look more closely at the calculation of the describing function and the question of
solving the harmonic balance equation (7.29). The next three examples illustrate
the calculation of the describing function for odd nonlinearities.
Example 7.6 Consider the signum nonlinearity $(y) = sgn(y). The describing
function is giver1 by
$(a sin wt) sin wt dt
Define a function @(a) by
.,
...,
&;<
Example 7.7 Considcr the piecewise-linear function of Figurc 7.15,'If s sinusoidal
input to this nonlinearity has amplitude a 6, the nonlinearity will act as a linear
gain. The output will be a sirlusoid with amplitude sla. Hence, the describing
function is Q(a) = s l , independent of a. When a > 6, we divide the integral on
the right-hand side of (7.30) into pieces, with each piece corresponding to a linear
portion of $(.). Furthermore, using tlie odd symmetry of the output waveform, we
simplify the integration to
<
'*>
so that (7.26) can be rewritten as
Since we are not interested in a solution with a = 0, we can solve (7.28) completely
by finding all solutions of
G(jw)@(a) 1 = 0
(7.29)
Equation (7.29) is known as the first-order harmonic balance equation, or simply
the harmonic balance equation. The function @(a) defined by (7.27) is called the
+
@(a) =
2
na
/'
0
$(a sin 0) sin 0 do =
*
- . , . -* - 4. , ,
7.2. T H E DESCRIBING FUNCTION M E T H O D
A
.,-
.
r
- L _
I.
285
~ ( sine)
a
Figure 7.16: Describing function for the piecewise-linear function of Figure 7.15.
its describing function is given by
Example 7.8 Consider an odd nonlinearity that satisfies the sector condition
Figure 7.15: Piecewise-linear function.
<
all2 Y*(Y) I Py2
for all y E R. The describing function l ( a ) satisfies the lower bound
~ ( a =)
$1'
*(a sin 0) sin 0 do 2 -L r s i n 2 0 do = or
and the upper bound
a
)=
$"
s i n )s i n 0
Therefore,
or 5 l ( a )
5 P,
Va
20
A
Thus,
l(a) =
4
K'~
lin-l
(:)+
Since the describing function l ( a ) is real, (7.29) can be rewritten as
+ s2
A sketch of the describing function is shown in Figure 7.16. By selecting specific
values for 6 and the slopes sl and sz, we can obtain the describing function of severd
common nonlinearities. For example, the saturation nonlinearity is a special case of
the piecewise-linear function of Figure 7.15 with 6 = 1, sl = 1, and s 2 = 0. Hence,
q'
6
R
,
+
+
{Re[G(jw)] jIm[G(jw)]) @(a) 1 = 0
This equation is equivalent to the two real equations
--I
287
7.2. THE DESCRIBING FUNCTION IIIETHOD
Because (7.32) is independent of a , we can solve it first for w to determine the poc
sible frequencies of oscillation. For each solution w, we solve (7.31) for a. Note that
the possible frequencies of oscillation are determined solely by the transfer function
G(s); they are independent of the nonlinearity $I(.). The nonlinearity determines
the corresponding value of a, that is, the possible amplitude of oscillation., This prccedure can be carried out analytically for low-order transfer functions, as illustrated
by the next examples.
Example 7.9 Let
and consider two nonlinearities: the signum nonlinearity and the saturation nonlinearity. By simple manipulation, we can write G(jw) as
Equation (7.32) has a unique positive root w = 2 f i . Evaluating Re[G(jw)] at
w = 2 f i and substituting it in (7.31), we obtain Q(a) = 0.8. For the saturation
nonlinearity, the describing fulictioll is give11 in Example 7.7, arid lP(a) = 0.8 has
the unique solution a = 1.455. Therefore, we expect that the nonlinear system
formed of G(s) and the saturation nonlinearity will oscillate with frequency close
to 2 f i and amplitude (at the input of the nonlinearity) close to 1.455. For the
dead-zone nonlinearity, the describing function q ( a ) is less than 0.8 for all a. Thus,
Q(a) = 0.8 has no solutions, and we expect that the nonlinear system formed of
G(s) and the dead-zone nonlinearity will not have sustained oscillations. In this
particular example, we can confirm the no oscillation conjecture by showing that
the system is absolutely stable for a class of sector nonlinearities, which includes
the given dead-zone nonlinearity. It can be easily checked that
Therefore, from the circle criterion (Theorem 7.2), we know that the system is
absolutely stable for a sector [O,P] with 0 < 0.8. The given dead-zone nonlinearity belongs to this sector. Consequently, the origin of the state space is globally
A
asymptotically stable and the system cannot have a sustained oscillation.,
Equation (7.32) takes the form
Example 7.11 Consider Raleigh's equation
which has one positive root w = ,fi. Note that for each positive root of (7.32),
there is a negative root ofequal magnitude. We only consider the positive roots.
Note also that a root a t w = 0 would be of no interest because it would not give rise
to a nontrivial periodic solution. Evaluating Re[G(jw)] at w = & and substituting
it in (7.31), we obtain Q(a) = 6. All this information has.been gathered without
specifying the nonlinearity $(.). Consider now the signum nonlinearity. We found
in Example 7.6 that Q(a) = 4/na. Therefore, Q(a) = 6 has a unique solution
a = 2/3?r. Now we can say that the nonlinear system formed of G(s) and the signum
nonlinearity will "probably" oscillate with frequency close to fi and amplitude
(at the input of the nonlinearity) close to 2/3?r. Consider next the saturation
1 for d l a. Therefore, Q(a) = 6
nonlinearity. We found in Example 7.7 that *(a)
has no solutions. Therefore, we expect that the nonlinear system formed of G(s)
and the saturation nonlinearity will not have a sustained oscillation.
A
<
Example 7.10 Let
and consider two nonliiearities: the saturation nonlinearity and a dead-zone nonlinearity that is a special case of the piecewise-linear function of Example 7.7 with
sl = 0, s 2 = 0.5, and 6 = 1. We can write G(jw) as
where E is a positive constant. To study existence of periodic solutions, we represent
the equation in the feedback form of Figure 7.1. Let u = -i3/3 and rewrite the
system's equation as
The first equation defines a linear systeni. Taking y = i to be its output, its transfer
function is
ES
G(s) =
s2 - ES 1
+
The second equation defines a nonlinearity $(Y) = y3/3. The two equations together
represent the system in the feedback form of Figure 7.1. The describing function of
$(y) = y3/3 is given by
@(a) =
&
The function G(jw) can be written as
sin o ) sin
~ 0 dB = f a 2
. -.
..- .
.
. .-
/
288
CHAPTER 7.
FEEDBACK&TE~?S
" <@
d
The equation Im[G(jw)] = 0 yields ~ ( -1w 2 ) = 0; hence, there is a unique positive
solution w = 1. Then,
1 @(a)Re[G(j)]= 0 =+ a = 2
+
Therefore, we expect that Raleigh's equation has a periodic solution of frequency
near 1 rad/sec and that the amplitude of oscillation in t is near 2.
n
For higher-order transfer functions, solving the harmonic balance equation (7.29)
analytically might be very complicated. Of course, we can always resort to numerical
methods for solving (7.29). However, the power of the describing function method is
not in solving (7.29) analytically or numerically; rather, it is the graphical solution
of (7.20) that made the method popular. Equation (7.20) can be rewritten as
Figure 7.17: Nyquist plot for Example 7.12.
A nonlinearity $(-) with these features belongs to a sector [a,A. Hence, from
Example 7.8, its describing function satisfies a 5 Q(a) P for all a 2 0. It should
be noted, however, that the slope restriction is not the same as the sector condition.
A nonlinearity may satisfy the foregoing slope restriction with bounds a and 0, and
could belong to a sector [a,fl with different bounds B and P.1° We emphmize that
in the forth$ominp analysis, we should use the slope bounds a and P, not the sector
boundaries 6 and j.
<
Equation (7.33) suggests that we call solve (7.29) by plotting the Nyquist plot of
G(jw) for w > 0 and the locus of -1/P(a) for a 2 0. Intersections of these loci give
the solutions of (7.29). Since @(a)is real for odd nonlinearities, the locus of -1/8(a)
in the complex plane will be confined to the real axis. Equation (7.34) suggests a
similar procedure by plotting the inverse Nyquist plot of G(jw) (that is, the locus in
the complex plane of l/G(jw) as w varies) and the locus of -@(a). The important
role of Nyquist plots in classical control theory made this graphical implementation
of the describing function method a popular tool with control engineers as they
faced nonlinearities.
Example 7.12 Consider again the transfer function G(s) of Example 7.9. The
Nyquist plot of G(jw) is shown in Figure 7.17. It intersects the real awis a t (-1/6,0).
For odd nonlinearities, (7.29) will have a solution if the locus of -l/Q(a) on the
A
real axis includes this point of intersection.
Let us turn now to the question of justifying the describing function method
Being an approximate method for solving the infinitedimensional equation (7.20),
the describing function method .can be justified by providing estimates of the error
caused by the approximation. In the interest of simplicity, u7e will pursue this
analysis only for nonlinearities with the following two feature^:^
Odd nonlinearity, that is, $(y) = -+(-y),
V y ;f. 0.
Single-valued nonlinearity with a slope between a and P; that is,
for all real numbers yl and 32 > yl.
%ee [24], [129],and
[I891 for descriljirrg function theory for more general nonlinearities
i
d
#'1I
.
I
(
I
Example 7.13 Consider the piecewiselinear odd nonlinearity
2
'=
{
2'1
4 -Yl
y-2,
f010lyl2
for2_<2'_<3
for y 1 3
shown in Figure 7.18. The nonlinearity satisfies the slope restriction
as well as the sector condition
In the forthcoming analysis, we should take a = -1 and P = 1.
A
We will restrict our attention to the question of the exk.tenm of half-wave symmetric
periodic solutions;ll that is, periodic solutions that only have odd harmonics. This
is a reasonable restriction in view of the odd symmetry of .31. The Fourier coefficients
loverify that [ti, C [a,
61.
"This restriction is made only for convenience. See I1281 for a more general analysis that d o e
not make this assumption.
i
CHAPTER 7. FEEDBACK SYSTEMS
kr:
7.2. THE DESCRIBING FUNCTION METHOD
Figure 7.18: Nonlinearity of Example 7.13.
Figure 7.19: Finding p(w).
f~
2'
%?
-r
T s b ~
of the odd harmonics of a periodic solution y(t) satkfy (7.20) for k = 1,3,5,. ...
The basic idea of the error analysis is to split the periodic solution y(t) into a first
harmonic yl(t) and higher harmonics yh(t). We choose the time origin such that
the phase of the first harmonic is zero; that is, yl(t) = asinwt. Thus,
&%
U"
y(t) = a sin wt
lb"l
,y
~
+ yh(t)
Using this representation, the Fourier coefficients of the first harmonic of y(t) and
*(v(t)) are
a
a1 = 2j
cl =
flrh
$(asinut
iT
+ yh(t)) exp(-jut)
dt
Fkom (7.20), with k = 1, we have
6Q = @(a)- @*(a,yh)
When yh = 0, @*(a,O)= ,@(a). Thus, 6Q = 0 and (7.36) reduces to the harmonic
balance equation
Therefore, the harmonic balance equation (7.37) is an approximate version of the
exact equation (7.36). The error term 6Q cannot be found exactly, but its size can
often be estimated. Our next step is to find an upper bound on 6%~.To that end,
let us define two functions p(w) and a(w). Start by drawing the locus of l/G(jw)
in the complex plane. On the same graph paper, draw a (critical) circle with the
interval [-P, -a]on the real axis as a diameter. Notice that the locus of -@(a) lies
inside this circle on the real axis, since a 5 Q(a) 0. Now consider an w such that
the points on the locus 1/G corresponding to k u (k > 1 and odd) lie outside the
critical circle, as shown in Figure 7.19. The distance from any one of these points
to the center of the critical circle is
<
G(jw)cl
+ a1 = 0
Introducing the functiol;
@*(a,yh) = - = j-2w
a1
we can rewrite the equation as
where
J"'~$(aSinwt + yh(t))exp(-jut)
dt
0
1
-+ **(a, yh) = 0
G(jw)
Adding Q(a) to both sides of (7.35), we can rewrite it as
1
+@(a)= 6Q
C(jw)
Define
(7.35)
in,
p(w) = k > l ; k
odd
lT+-l
1
G(jkw)
Note that we have defined p(w) only for w at which l/G(jkw) lies outside the critical
circle for all k = 3,5,. . .; that is, for w in the set
(7.36)
.... .. ..--
.
..--. .
--""",%
$-
- . ..
"
_.
..
-7.2.. . TIIE, DESCRIBING
. 2. ., ,
FUNCTION AfETIIOD
'
. ..
.-..
CHAPTER 7. FEEDBACK SYSTEMS
-..
..
,
,
,
. ., ,
,
,
.
'
4
I
..
5
.e-
.. .
i
,
.-.- -..'.
.
.
c
On any connected subset R' of R, define
The positive quantity u(w) is an upper bound on the error term 69, as stated in
the next lemma.
Lemma 7.1 Under the stated assumptions,
Figure 7.20: A complete intersection.
Proof: See Appendix C.13.
The proof of Lemma 7.1 is based on writing an equation for yh(t) in the form
yh = T(gh) and showing that T(.)is a contraction mapping. This allows us to
calculate the upper bound of (7.40), which is then used to calculate the upper
bound of (7.41) on the error term. The slope restrictions on the nonlinearity are
is a contraction mapping.
used in showing that T(.)
Using the bound of (7.41) in (7.36), we see that a necessary condition for the
existence of a half-wave symmetric periodic solution with w E R' is
$J
Geometricdly, this condition states that the point -@(a) must be contained in a
circle with a center at l/G(jw) and radius o(w). For each w E R' C a, we can
draw such an error circle. The envelope of all error circles over the connected set
fl' forms an uncertainty band. The reason for choosing a subset of R is that, as w
approaches the boundary of R, the error circles become arbitrarily large and cease
to give any useful information. The subset R' should be chosen with the objective
of drawing a narrow band. If G(jw) llas sharp low-pass filtering characteristics, the
uncertainty band can be quite narrow over 0'.Note that p(w) is a measure of the
low-pass filtering characteristics of G(jw); for the smaller (G(jkw)l for k > 1, the
larger p(w), as seen from (7.38). A large p(w) results in a small radius u(w) for the
error circle, as seen from (7.39).
We are going to look a t intersections of the uncertainty band with the locus
of -@(a). If no part of the band intersects the -@(a) locus, then clearly (7.36)
has no solution with w E 0'. If t11c barlcl intersects the locus completely, as in
Figure 7.20, then we expect that there is a solution. This is indeed true, provided
we exclude some degenerate cases. Actually, we can find error bounds by examining
the intersection. Let a1 and a2 be the amplitudes corresponding to the intersections
of the boundary of the uncertainty band with the -@(a) locus. Let wl and w2 be
the frequencies corresponding to the error circles of radii u(wl) and u(wz), which
are tangent to the -@(a) locus on either side of it. Define a rectangle r in the
(w, a ) plane by
...
~={(w,a)(w1<w<wz,al<a<a2)
The rectangle r contains the point (w,,a,) for which the loci of 1/G and -!?
intersect, that is, the solution of the harmonic balance equation (7.37). It turns out
that if certain regularity conditions hold, then it is possible to show that (7.36) has
a solution in the closure of J?. These regularity conditions are
A complete intersection between the uncertainty band and the -!?(a) locus can
now be precisely defined as taking place when the l/G(jw) locus intersects the
-rk(a) locus and a finite set r can be defined, as shown, such that (w,, a,) is the
unique intersection point in r and the regularity conditions hold.
Finally, notice that a t high frequencies for which all harmonics (including the
first) have the corresponding l/G(jw) points outside the critical circle, we do not
need to draw the uncertainty band. Therefore, we defme a set
and take the smallest frequency in fi as the largest. frequency in R', then decrease
w until the error circles become uncomfortably large.
The next theorem on the justification of the describing function method is the
main result of this section.
CHAPTER 7. FEEDBACK SYSTEMS
7.2. THE DESCRIBING FUNCTION METHOD
290
Theorem 7.4 Consider the feedback connection of Figure 7.1, where the nonlineority
is memoryless, time invariant, odd, and single valued with slopes between
a and 0. Dmw the loci of l/G(jw) and -9(a) in the complez plane and construct
the critical circle and the band of uncertainty as described earlier. Then,
*(a)
$:$*
1.*
$
&
4
,
sv
%y<T
the system has no half-wave symmetric periodic solutions with fundamental
frequency w E 6.
1-+
i
t
$t *
the system has no half-wave symmetric periodic solutions with bndamental
frequency w E R' if the comsponding error circle does not intersect the -9(a)
locua.
>
&6$
*&'
k
for each complete intersection defining a set in the (w, a) plane, there is at
least one half-wave symmetric periodic solution
?&
Y
y(t) = a sin wt
+ yh(t)
with (w, a) in f and yh(t) satisfies the bound of (7.40).
Proof: See Appendix C.14.
>*<,
Note that the theorem gives a sufficient condition for oscillation and a sufficient condition for nonoscillation. Between the two conditions, there is an area of
ambiguity where we cannot reach conclusions of oscillation or nonoscillation.
,-
+
*\,,
Example 7.14 Consider again
,
*
together with the saturation nonlinearity. We have seen in Example 7.10 that the
harmonic balance equation has a unique solution w. = 2 f i = 2.83 and a, = 1.455.
The saturation nonlinearity satisfies the slope restrictions with a = 0 and ,8 = 1.
Therefore, the critical circle is centered at -0.5 and its radius is 0.5. The function
l/G(jw) is given by
&*
T5.
.*
Hence, the locus of l/G(jw) lies on the lime Rels] = -0.8, as shown in Figure 7.21.
The radius of the error circle u(w) has been calculated for eight frequencies starting
with w = 2.65 and ending with w = 3.0, with uniform increments of 0.05. The
centers of the error circles are spread on the line Re[s] = -0.8 inside the critical
circle. The d u e of u(w) st w = 2.65 is 0.0388 and at w = 3.0 is 0.0321, with
monotonic change between the two extremes. In all cases, the closest harmonic to
the critical circle is the third harmonic, so that the infimum in (7.38) is achieved at
k = 3. The boundaries of the uncertainty band are almost vertical. The intersection
of the uncertainty b m d with the -@(a) 1oc11s correspond to the points a1 = 1.377
p
9
;
;
Q
+*
*,
-
t-
*\
:"
i
0
4.
-
v
,
Figure 7.21: Uncertainty band for Example 7.14.
and a2 = 1.539. The error circle corresponding to w = 2.85 is almost tangent to the
real axis from the lower side, so we take w2 = 2.85. The error circle corresponding
to w = 2.8 is the closest circle to be tangent to the real axis from the upper side.
This means that wl > 2.8. Trying w = 2.81, we have obtained a circle that is almost
tangent to the real axis. Therefore, we define..the set r as
r = {(w, a) 1 2.81 < w < 2.85, 1.377 < a < 1.539)
There is only one intersection point in I?. U'e need to check the regularity conditions.
The derivative
is different from zero at a = 1.455, and
Thus, by Theorem 7.4, we conclude that the system indeed has a periodic solution.
Moreover, we conclude that the frequency of oscillation w belongs to the interval
7.3. EXERCISES
297
Figure 7.22: Inverse Nyquist plot and critical circle for Example 7.15.
7.2 Consider the feedback connection of Figure 7.1 with G(s) = 2s/(s2
[2.81,2.85], and the amplitude of the first harmonic at the input of the nonlinearity
belongs to the interval [1.377,1.539]. F'roin the bound of (7.40), we know also.that
the higher harmonic component yh(t) satisfies
!
X
~ " y 2 ( t )dt 5 0.0123, Q (w, a) E
r
+ s + 1).
(a) Show that the system is absolutely stable for nonllnearities in the sector [O, 11.
(b) Show that the system has no limit cycles when $(y) = sat(y).
7.3 Consider the system
0.
which shows that the waveform of the oscillating signal at the nonlinearity input is
A
fairly close to its first harmonic asin wt.
where h is a smooth function satisfying
Example 7.15 Reconsider Example 7.9 with
and the saturation nonlinearity. The noillinearity satisfies the slope restriction with
a = 0 and p = 1. The inverse N y q ~ist plot of G(jw), shown in Figure 7.22, lies
outside the critical circle for all w > Hence, fl = (0, m), and we conclude that
A
there is no oscillation.
3
J h ( y ) l l c,
for a1 c v c a2 and
- a2 c y c -a1
(a) Show that the origin is the unique equilibrium point.
(b) Show, using the circle criterion, that the origin is globally asymptotically stable.
7.4 ([201]j Consider the system
7.3
Exercises
7.1 Using the circle criterion, study absolute stability for each of the scalar transfer
functions given next. In each case, find a sector [a,P] for which the system is
absolutely stable.
S
(1)
G(s).=
I
(2)
1
G(s) = (S +2)(s +3)
s2-S+1
where p, a, q, and w are positive constants. Represent the system in the form of
Figure 7.1 with $(t, y) = qy coswt and use the circle criterion to derive conditions
on p, a, q, and w, which ensure that the origin is exponentially stable.
+
7.5 Consider the linear time-varying system x = [A B E ( t ) q x , where A ia
Hurwitz, JIE(t)lJzI 1, Q t 2 0, and s ~ p , , ~ u , , ~ ( j w I - A)-'B] c 1. Show that
the origin is uniformly asymptotically stable.
298
CHAPTER 7. FEEDBACK SYSTEMS
9"
ki
+
7.6 Consider the system x = Ax Bu and let u = -Fx be a stabilizing state
feedback control; that is, the matrix (A - BF) is Hurwitz. Suppose that, due to
physical limitations, we have to use a limiter to limit the value of ui to (ui(t)(<_ L.
The closed-loop system can be represented by x = Ax - B L sat(Fx/L), where
sat(v) is a vector whose ith component is the saturation function.
(a) Show that the system can be represented in the form of Figure 7.1 with G(s) =
F(sI - A BF)-lB and $(y) = L sat(y/L) - y.
+
(b) Derive a condition for asymptotic stability of the origin using the multiwiable
circle criterion.
zu*
7.3. EXERCISES
(a) Show that the system can be represented as the feedback connection of Figure 7.1 with G(6) = [H(s) + li]/s.
(b) Using the version of the Popov criterion in Exercise 7.8, find a lower bound
kc
such that the system is absolutely stable for all k > kc.
i6
(c) Apply the result to the case
Figure 7.23: Exercise 7.9.
and estimate the region of attraction.
7.10 For each odd nonlinearity $(y) on the following list, verify the given expree
sion of the describing function Q(a):
7.7 Repeat Exercise 7.1 using the Popov criterion.
7.8 In this exercise, we derive a version of thepopov criterion for a scalar transfer
function G(s) with all poles in the open left-half plane, except for a simple pole on
the imaginary axis having a positive residue. The system is represented by
E = Az
- B$(y),
6 = -$(y),
and
y = Cz
+ dv
&.
(I)
$(y)
= y5;
k
(2)
$(y)
= y3(yl;
(3)
$(y)
:
Figure 7.2.L(a);
(4)
$(y)
:
Figure 7.24(b)0,
where d > 0, A is Hurwitz, (A, B) is controllable, (A,C) is observable, and $
belongs to a sector (0, k]. Let V ( z ,v) = zTPz a(y - C Z ) ~ b J
: $(u) du, whrre
~=~~>0,a>o,andb>0.
+
+
(a) Show that V is positive definite and radially unbounded.
(b) Show that
.r
v satisfies the inequality
v 5 Z ~ ( P A+ A ~ P ) Z- 2zT(PB - w)$(y) - r$2(Y)
+
where w = adCT (1/2)bATCT and 7 = (2adlk)
chosen such that 7 2 0.
-k + Re[(l + jwg)G(jw)] > 0,
V w E R, where g =
b
7.9 ([85]) The feedback system of Figure 7.23 represents a control system where
H(s) is the (scalar) transfer function of the plant and the inner loop models the
actuator. Let H(s) = ( s 6)/(s 2)(s 3) and suppose k 2 0 and $ belongs to a
sector (O,@],where p could be arbitrarily large, but finite.
+
+
+
(5)
+ b(d + CB). Assume b is
(c) Show that the system is absolutely stable if
1
=
t
f
$(y)
:
*(a)
=
{
*(a) = 5a4/8
*(a) = 32a3/15a
*(a) =
4A
faa
for a I A
( ~ S / r a ) [l ( ~ / a ) ~ ] ' / ~ , for a 2 A
Figure 7.24(c)0,
k[l - N(a/A)I,
k[N(a/B) - N(a/A)],
for a 5 A
for A i a I B
for a 2 B
where
7.11 Using the describing function method, investigate the existence of periodic
solutions and the possible frequency and amplitude of oscillation in the feedback
connection of Figure 7.1 for each of the following cases:
jfp''=
i(y)l
7:
7
(a)
(b)
G(s) =
ABY
(2) G ( ~ =
) (1 - s)/s(s
1
(8
+ 1 ) y s + 2)2 '
where b > 0.
(c)
{"
sgn(y)
-1SbyS1
blyl > 1
origin of the closed-loop system is globally asymptotically stable.
a
( b ) Using the Popov criterion, find the largest b for which we can confirm that the
origin of the closed-loop system is glibally asymptotically stable.
+ 1) alld rl/(u) = y5.
+ 1) and $J is the nonlinearity of Exercise 7.10, part
*(Y) =
i
(a) Using the circle criterion, find the largest b for which we can confirm that the
Figure 7.24: Exercise 7.10.
(1) G(S) = (1 - s)/s(s
7.14 Coasiii~rtlic fixdbnrk colulcct i o of
~ I:igl~ns
~
7.1 wit11
.
(c) Using the describing function method, find the smallest b for which the system
(51,
will oscillate and estimate the frequency of oscillation.
A = 1, B = 312, and k = 2.
(4) G(s) = (s
+ 6)/s(s + 2)(s + 3) and $J(y) = sgn(y).
a
.I
- s + 1) ant1 $(y) = y5.
( 5 ) G(S) = s/(s2
+
10
+
(6) G(s) = 5(s 0.25)/s2(s 2)2 and $ is the nonlinearity of Exercise 7.10, part
(3), with A = 1 and k = 2.
(7) G(s) = 5(s + 0.25)/s2(s + 2)2 and $ is the nonlinearity of Exercise 7.10, part
(4), with A = 1 and B = 1.
(8) G(s) = 5(9
a
(5), with A = 1, B = 312, and k = 2.
1
+ 1)3 and $(Y) = sgn(y).
(10) G(s) = l / ( s
+
.
and $(y) = sat(y).
7.12 Apply the describing function method to study the existence of periodic
solutions in the negative resistance oscillator of Section 1.2.4 with h(v) = -u+v3 v5/5 and E = 1. For each possibleperiodic solution, estimate the frequency and
amplitude of oscillation. Using computer simulation, determine how accurate the
describing function results are.
7.13 Consider the feedback connection of Figure 7.1 with G(s) = 2bs/(s2 - bs+ 1)
and $(y) = sat(y). Using the describing function method, show that for sufficiently
small b > 0 the system has a periodic solution. Confirm your conclusion by applying
Tl~eorem7.4 and estimate the frequency and amplitude of oscillation.
i
i. find the frequency interval [wl ,wz] and the amplitude interval [ a ~az];
,
ii. use Lemma 7.1 to find an upper bound on the energy content of the higherorder har~nonicsand express it as a percentage of the energy content of
,
the first harmonic; and
iii. simulate the system and compare the simulation results with the foregoing
analytical results.
$ : :
(e) Repeat part (dJ for b = 30.
+ 0.25)/s2(s + 2)2 and $ is the nonlinearity of Exercise 7.10, part
(9) G(s) = l / ( s
:
(d) For b = 10, study the existence of periodic solutions by using Theorem 7.4. For
each oscillation, if any,
+ 1)6 and $(Y) = sgn(y).
(3) ~ ( s =) l / ( s
i
7.15 Repeat parts (a) to (c) of the previous exercise for G(s) = 1 0 / ( ~ + 1 ) ~ ( 8 + 2 ) .
.,
*
,
.
+
7.16 Consider the feedback connection of Figure 7.1, where G(s) = l/(s
and $(y) is the piecewise-linear function of Figure 7.15 with d = l/k, sl = k, and
s 2 = 0.
(a) Using the describing function method, investigate the existence of periodic
solutions and the possible frequency and amplitude of oscillation when k = 10.
(b) Continuing with k = 10, apply Theorem 7.4. For each oscillation, if any, find
the frequency interval [wl, w2] and the amplitude interval [al, a2].
(c) What is the largest slope k > 0 for which Theorem 7.4 ensures that there is no
oscillation?
7.17 For each of the following cases, apply Theorem 7.4 to study the existence of
periodic solutions in the feedback connection of Figure 7.1. For each oscillation, if
any, find the frequency interval [wl,w2] and the amplitude interval [allaz].
*,
~
302
(1) G(a) = 2(s
CHAPTER 7. FEEDBACK SYSTEMS
- l)/s3(s + 1) and $(y) = sat(y).
+ 0.85 + 8) and $(y) = (112) sin y.
(3) G(s) = -a/($ + 0.85 + 8) and $(y) is the nonlinearity of Example 7.13.
(4) G(5) = -24/s2(s + 1)' and $(y) is an odd nonlinearity defined by
y3 + 1112,
forO<yll
(2) G(s) = -s/(s2
$(I/) =
2~ - 112,
Chapter 8
for y 2 1
Advanced Stability Analysis
In Chapter 4, we gave the basic concepts and tools of Lyapunov stability. In this
chapter, we examine some of these concepts more closely and present a number of
extensions and refinements.
We saw in Chapt.er 4 how to use linearization to study stability of equilibrium
points of autonomous systems. We saw also that linearization fails when the Jacobian matrix, etaluated at the equilibrium point, has some eigenvalues with zero
real parts and no eigenvalues with posit.ive real parts. In Section 8.1, we introduce
the center manifold theorem and use it to study stability of equilibrium points of
autonomous systems in the critical case when linearization fails.
The concept of the region of attraction of an asymptotically stable equilibrium
point was introduced in Section 4.1. In Section 8.2, we elaborate further on that
concept and present some ideas for providing estimates of this region.
LaSalle's invariance principle for autonomous systems is very useful in applications. For a general nonautonomous system, there is no invariance principle in the
same form that tvas presented in Theorem 4.4. There are, however, theorems which
capture some features of the invariance principle. Two such theorems are given in
Section 8.3. The first theorem shows convergence of the trajectory to a set, while
the second one shows uniform asymptotic stability of the origin.
Finally, in Section 8.4, we introduce 11oLio11sof stability of periodic solutions and
invariant sets.
8.1
The Center Manifold Theorem
15
Consider the autonomous system
x =f(x)
(8.1)
where f : D 4 Rn is continuously differentiable and D c Rn is a domain that
contains the origin x = 0. Suppose that the origin is an equilibrium point of (8.1).
..
..
.
- .-
.
.. ..
,
,
- . - --.
304
-
.*. .-.-. . -. -.
.
:/
-:.
-
.L)
CHAPTER 8. ADVANCED STABILITY ANALYSIS
8.1. THE CENTER MANIFOLD THEOREM
305
Theorem 4.7 states that if the linearization of f at the origin, that is, the matrix
1
is twice continuously differentiable and
has all eigenvalues with negative real parts, then the origin is asymptotically stable;
if it has some eigenvalues with positive real parts, then the origin is unstable. If
A has some eigenvalues with zero real parts with the rest of the eigenvalues having
negative real parts, then linearization fails to determine the stability properties of
the origin. In this section, we take a closer look into the case when linearization
fails. Failure of linearization leaves us with the task of analyzing the nth-order
nonlinear system (8.1) in order to determine stability of the origin. The interesting
finding that we are going to present in the next few pages is that stability properties
of the origin can be determined by airalyzing a lower order nonlinear system - a
system whone order is exactly equal to the number of eigenvalues of A with zero
real parts. This will follow as an application of the center manifold theory.'
A k-dimensional manifold in Rn (1 5 k < n) has a rigorous mathematical
definiti~n.~
For our purpose here, it is sufficient to think of a k-dimensional manifold
as the solution of the equation
V(X) = 0
Since our interest is in the case when linearization fails, assume that A has k eigenvalues with zero r e d parts and m = n - k eigenvalues with negative real parts.
We can always find a similarity transformation T that transforms A into a block
diagonal matrix, that is,
where all eigenvalues of A1 have zero real parts and all eigenvalues of A2 have
negative real parts. Clearly, A1 i6 k x k and A2 is m x m. The change of variables
I
transforms (8.1) into the form
where 7 : Rn -, Rn-k is sufficiently smooth (that is, sufficiently many times continuously differentiable). For example, the unit circle
is a one-dimensional manifold in R2. Similarly, the unit sphere
is an (n - 1)-dimensional manifold in Rn. A manifold {g(x) = 0) is said to be an
invariant manifold for (8.1) i f ,
q(x(0)) = 0 & v(x(t))
= 0,
where gl and g2 inherit properties of
differentiable and
j. In particular, they are twice continuously
for i = 1,2. If z = h(y) is an inmiant manifold for (8.2)-(8.3) and h is smooth,
then it is called a center manifold if
V t E [0, tl) C R
where (0,tl) is any time interval over which the solution x ( t ) is defined.
Suppose now that f(x) is twice cont,inuously differentiable. Equation (8.1) can
be represented as
Theorem 8.1 If gl and g2 are twice continuously dgerentiable and satCb (8.4),
a l l eigenvalue8 of A1 have zero realparts, and all eigenwaluea.o&A2.have negative ml;
parts, then there exist a constant 6 > 0 and a wntinuowly diflerentiable functio;
h(y), defined for all llyll < 6, n u h that z = h(y) C a center manifad for (8.2)-(8.3).
0
'The center manifold theory has several applications to dynamical systems. It is presented here
only insofar as it relates to determining the stability of the origin. For a broader viewpoint of the
ti~&ry,the reader may consult 1341.
2See, for example, (711.
Proof: See Appendix C.15.
If the initial state of the system (8.2)-(8.3) lies in the center manifold; that is,
z(0) = h(y(O)), then the solution ( y ( t ) , z ( t ) ) will lie in the manifold for all t 2 0;
i ?
CHAPTER 8. ADVANCED STABILITY ANALYSIS
$&'
P.
that is, z(t) = h(y(t)). In this case, the motion of the system in the center manifold
is described by the kth-order differential equation
.
.
9 = A ~ +Ygl(y, h(y))
(8.51
. ,
which we refer to as the reduced system. If z(0) # h(y(O)), then the difference
~ ( t-) h(y(t)) represents the deviation of the trajectory from the center manifold at
any time t. The change of variables
$3
transforms (8.2)-(8.3) into
8.1. THE CENTER AIANIFOLD THEOREAI
for i = 1,2. Consequently, in the domain
Nl and N2 satisfy
IINi(ylw)l12 I killwll, i = 192
where the positive constants kl and k2 can be made arbitrarily small by choosing
p small enough. These inequalities, together with the fact that A2 is Hurwitz,
suggest that the stability properties of the origin are determined by the reduced
system (8.5). The next theorem, known as the reduction principle, confirms this
conjecture.
Substituting these identities into (8.7) results in
in a neighborhood of the origin, wlicre a3 and a4 are class
other hand, since A2 is Hurwitz, tlic Lyapunov equation
as
+
= Alar + gl(y, h ( d ) + Nib, w)
= A2w+ N2(y,w)
(8.9)
(8.10)
where
Nl(~1.w)= gl(y,w + h b ) ) - g1(y, h(y))
and
N2(2/1w) = g2(v, w + h(y))
,
,
6
PA2
+A
K
,C
/&-
1k.
/ (.
c
L
1
c
functions. On the
&
~ =
P -I
has a unique positive definite solution P. Consider
"(Y, w) = V(y)
+
as a Lyapunov function candidatevor the full system (8.9) and (8.10). The dcrivative of v along the trajectories of the system is given by
ah
- g2(y, h(y)) - -(y)
N i b , w)
ay
It is not difficult to verify that Nl and N2 are twice continuously differentiable, and
;c
'Q
Proof: The change of coordinates from (y, I ) to (y, w) does not change the stability
properties of the origin (Exercise 4.26); therefore, we can work with the system (8.9)
and (8.10). If the origin of the reduced system (8.5) is unstable, then, by invariance,
the origin of (8.9) and (8.10) is unstable. This is so because for any solution y(t)
of (8.5), there is a corresponding solution (y(t),O) of (8.9) and (8.10). Suppose
now that the origin of the reduced system (8.5) is asymptotically stable. By (the
converse Lyapunov) Theorem 4.16. there is a continuously differentiable function
V(y) that is positive definite and satisfies the inequalities
Since the equation must be satisfied by any solution that lies in the center manifold, we conclude that the function h(y) must satisfy the partial differential equation
(8.8). Adding and subtracting gl(y, h(y)) to the right-hand side of (8.6), and subtracting (8.8) from (8.7), we can rewrite the equation in the transformed coordinates
-,
C
c-
T h e o r e m 8.2 Under the &;&itions
of Theorem 8.1, if the origin y = 0 of the
reduced system (8.5) is asymptotically stable (respectively, unstable) then the origin
ofthe full system (8.2) and (8.3) is also asymptotically stable (respectively, unstable).
0
In the new coordinates, the center manifold is w = 0. The motion in the manifold
is characterized by
w ( t ) s O =+ w ( t ) e O
I
C
C
3The function u ( y , w ) is continuously differentiable everywhere around the origin, except on
the manifold w = 0. Both u ( y , w ) and ir(y,w) are defined and continuous around the origin. It
can be easily seen that the statement of Theorem 4.1 is still valid.
QP
C
b
b
&
bc
.sWB
fb
I
i
e
..
.
.. -
.
.
308
--- . -. /
.
.- . .
.
.
-. .
--
..-......, -
.-
.
.-..
.... .. .
,
..
. .
-
..
...-..
.,
.
-
-.
. .'
..
*r,
"
-T;
<
.
b
d
.
.
- . . ... . ..
..
r
..
-
--. -.- .
,
. -.
@..
8.1. THE CENTER AIANIFOLD THEOREAI
CHAPTER 8. ADVANCED STABILITY ANALYSIS
. . .-,
,?
.
.-..
309
To use Theorem 8.2, we 11c4 to find the ccllter ~llw~ilold
:= h(y). The f u ~ ~ c t i o ~ ~
h is a solution of the partial differential equation
def Bh
N ( h b ) ) = ~ ( 9 [ A) l p gi(& h ( ~ ) ) l A2h(y)
-h ( p ,h ( y ) ) = 0
+
max
:.
. :
with boundary conditions
,
h ( 0 ) = 0;
Bh
( 0 )= 0
BY
This equation for h cannot be solved exactly in most cases (to do so would imply
that a solution of the full system (8.2)-(8.3) has been found), but its solution can
be approximated arbitrarily closely as a Taylor series in y.
Since k1 and kz can be made arbitrarily small by restricthg the domain around the
origin to be sufficiently small, we can choose them small enough to ensure that
Hence,
(8.11)
T h e o r e m 8.3 If a continuously diflerentiable function d ( y ) with 4(O) = 0 and
[O@/ay](O)= 0 mn be found such that N ( 4 ( y ) ) = O(IIyIJP)for some p > 1, then for
suficiently small J ( y ( J
h ( v ) 4 ( v ) = 0(lll/llP)
and the reduced system can be represented as
-
1
4 % ~5)-a3(llvl12) - 4 & J F l l w 1 1 2
B = Ala, + g i b , 4 ( y ) )+ O ( I I Y I I'~1 ~
which shows that L(y, w ) is negative definite. Consequently, the origin of the full
system (8.9)-(8.10) is asymptotically stable.
Proof: See Appendix C.15.
We leave it to the reader (Exercises 8.1 and 8.2) to extend the proof of Theorem 8.2 to prove the next two corollaries.
The order of magnitude notation O(.) will be formally introduced in Chapter 10.
(Definition 10.1). For our purpose here, it is enough to think of f ( y ) = O ( J J y p )
as a shorthand notation for (If (y)l( 5 kJJy(lPfor sufficiently small I J y J JLet
. us now:;
illustrate the application of the center manifold theorem by examples. In the first.
two examples, we will make use of the observation that for a scalar state equation
of the form
Corollary 8.1 Under the assumptions of Theorem 8.1, if the origin y = 0 of the
reduced system (8.5) is stable and there is a continuously differentiable Lyapunov
function V ( y ) such that
I
where p is a positive integer, the origin is asymptotically itable if p L - o d d b d
a < 0. It is unstable if p is odd and a > 0, or p is even and a # 0.5
in some nezghborhood of y = 0, then the origin of the full system (8.2)-(8.3) is
stable.
0
\
Corollary 8.2 Under the assumptions of Theorem 8.1, the origin of the reduced
system (8.5) is asymptotically stable i f and only i f the origin of the full system
0
(8.2)-(8.3) is asymptotically sttzble.
4The existence of the Lyapunov function V ( y )cannot be inferred from a converse Lyapunov
theorem. The converse Lyapunov theorem for stability [72, 107) guarantees the existence of a
Lyapunov function V ( t ,y ) whose derivative satisfies ~ ( yt ) , 0. In general, this function cannot be
mule indepcn[l(!nt o f t. (See (72,page 2281.) I<vc:r~t l ~ o l ~ gwe
b can choose V ( t ,y) to be continuously
differentiable in its arguments, it cannot be gllarantccd that the partial derivatives aV/Byi, 8V/8t
will be uniformly bounded in a neighborhood of the origin for all t 0. (See (107, page 531.)
<
>
Example 8.1 Consider the system
=
x2 =
x1
.
22
-22
+ ax: + bx1x2
where o # 0. The system has a unique equilibrium point a t the origin. The
linearization a t the origin results in the matrix
1
CHAPTER 8. ADVANCED STABILITY ANALYSIS
"!
which has .eigenvalues a t 0,and -1. Let M be.a,&atrix +hose col-s
ei&nvectors .of A;jthat,is; : .i
c<7.e,z:
..?.
8.1. THE CENTER AlANIFOLD THEOREhl
are the "
Example 8.2 Consider the system
~..?
I.!
:
-:'.... ....
=.M;~?
Then,
and3:t,+ii.,T
I
which is already represented in the ( y ,z ) coordinates. The center manifold equation
(8.11) with the boundary condition (8.12) is
The change of variables
We start by trying 4 ( y ) = 0. The reduced system is
a)
'7):
3 -
"
j,
.
puts the system into the form
Clearly, we cannot reach any conclusion about the stability of the origin. Therefore,
we substitute h(y) =h& +O(JyI3)into the center manifold equation and calculate
hz, by matching coefficients of y2, t o obtain h2 = a. The reduced system is"
#~k3
The center manifold equation (8.11) with the bo&dary condition (8.12) becomes
+
"'Ath ( y ) =
+
.
+.
+
.............
,.:-.
. .-,.,
hi#
hay3..+. and substitute' this series in the center manifold
ustionto find t h e u n k n o p coefficients h2,ks,... by matching coefficients of like
powers in y (since the equation holds as an identity in y). We do not know in
a d W c e howmany t e r m of the series we need. Westart'with the simplest np-.
' ' ' p d m a t i o n h(y)"= O. :We substitute h(y) = O(JyI2)into the reduced system and
study stability of its origin. If the stability properties of the origin can be determined, we & don$. Otherwise, we calculate the coefficient h2, substitute h(y) =
O(bP), and study st,ability of the origin. If it c t g o t be resolved, we proceka to the approximation h(y) %. h2y2 hi$, and so on. Let us start with the
approximation h(y) * 0. The reduced system is
+
dJ
,':
>'A
?
,.
.&,
pp.:i.,:,
,, :
p$*;:;.
&A
.X
p+'
.$
t$
,
..: ?,:.
+ O((yI4)/
which has the exact solution h(y) = 0. The reduced system y = 0 has a stable origin
with V ( y )= y2 as a Lyapunov functio~~.
Tlicrefore, by Corollary 8.1, we conclude
A
that the origin of the full system is stnhle if a = 0
f
Example 8.3 Consider the system (8.2)-(8.3)with
ai = all2 + O ( I V I ~ )
Notice that an O(IY12)error in h(y) results in an O(lyI3)error in the right-hand side
of the reduced system. This is a consequence of the fact that the function gl(y, z ) ,
which appears on the right-hand side of the reduced system (8.5) as gl(y, h ( y ) ) ,
has a partial derivative with respect to z that vanishes at the origin. Clearly, this
observation is rrlso valid for higher order approximations; that is, an error of order
O(ly(k)in h(y) results in an error of order O(JylkC1)
in gl(y, h(y)),for k 2 2. The
term ay2 is the dominant term on the right-hand side of the reduced system. For
a # 0, the origin of t b reduced system isunstable. Consequently, by ~ i e o r e m8.2,
tile origin of the full system is unstable.
A
y = ay3
Therefore, the origin is asymptotically stable if a < 0 and unstable if a > 0. Consequently, by Theorem 8.2, we conclude that the origin of the full system is asymptotically stable if a < 0 and unstable if a > 0. If a = 0, the center manifold equation
(8.11) with the boundary condition (8.12) reduces to
N ( h ( v ) ) = h'(ar)[a(e
- b(yh(y)+ h2(y))l+ h(y)
a(y h ( ~ )* )b(yh(y)
~
h2(y))= 0 , , h(0) = hr(0)= 0
+.
Y =0(ly13)
4
1
6)
It can be verified that ?(y) = results in N ( ( ( y ) )= 0 ( [ ~ Y I I ;and
)
Using V ( y )= (yi + y;)/2 as a Lyapunov function candidate, we obtain
v = -Y: - Y: + y T o ( I l ~ l l : ) r -Il~lli+ kllvll;
=The error on the right-hand side of the reduced system is actually 0((y15) since, if we wrlte
h(y) = hzy2 + hay3 + .... we will find that 113 = 0.
-...
-... -.-.....
.
- .........
:
.
.-
...
CHAPTER 8. ADVANCED STABILITY ANALYSIS
ill sonie ileighborhood of the origin where I; > 0. Hence,
I - ;llYJI;,
for
<
. ." . : .
.........................
-
"'
I
. - , .. . . . .
.
....-
...
sf,
4
,
'".':'---.
.
.-",.
^,.
....
313
8.2. REGION OF ATTRACTION
Estimate
anra@ of the region
.
.
toe tc
1
lI~Jl2
which shows that the origin of the reduced system is asymptotically stable. ConseA
quently, the origin of the full system is asymptotically stable.
Notice that in the preceding example it is not enough to study the system
Region of attraction
tVe have to find a Lyapunov function that confirms asymptotic stability of the origin
for all perturbations of the order 0 (lly11;). The importance of this observation is
illustrated by the next example.
Figure 8.1: Critical clearance time.
1
Example 8.4 Consider the previous example, but change A1 to
With d(y) = 0, the reduced system can be represented as
Without the perturbation term 0 (lly114), the origin of this system is asymptotically
~ O V V(y) to show asymptotic stability
stable.' If you try to find a L ~ R ~ Ufunction
in the presence of the perturbation term, you will not succeed. In fact, it can
be verified that the center manifold equation (8.11) with the boundary condition
(8.12) has the exact solution h(y),= y;, so that the reduced system is given by the
equation
-
A
whose origin is unstable.'
8.2
Region of Attraction
,D-k sf
@*f~hh+
o * ~ a c ~ ~
Quite often, it is not sufficient to determine that a given system has an asymp
totically stable equilibrium point, Rather, it is important to find the region of
att,raction of that point, or at least an estimate of it. To appreciate the importance
of determining the region of attraction, let us run a scenario of events that could
?See Exercise 4.56.
BSee Exercise 4.13.
happen in the operation of a nonlinear system. Suppose that a nonlinear system
has an asymptotically stable equilibrium point, which is denoted by x,, in Suppose
the system is operating at steady state a t xpr. Then, at time to a fault' that ch*iges
the structure of the system takes place, for example, a short circuit in an electrical
network. Syppose the faulted system does not have equilibrium points a t xpr or
in its neighborhood. The trajectory of the system will be driven away from xpr.
Suppose further that the fault is cleared at time tl and the postfault system has
an asymptotically stable equilibrium point at xps, where either xpa = xp, or xps
is sufficiently close to xpr so that steady-state operation at x,, is still acceptable.
At time tl the state o f t h e system, say, x(tl), could be far from the postfault equilibrium xp8. Whether or not the system will return to steady-state operation at
x,, depends on whether x(tl) belongs to the region of attraction of xps, as determined by the postfault system equation. A crucial factor in determining how far
x(tl) could be from xp8 is the time it takes the operators of the system to remove
the fault, that is, the time difference (tl - to). If ( t l - to) is very short, then, by
continuity of the solution with respect to t , it is very likely that x(tl) will be in the
region of attraction of xp,. However, operators need time to detect the fault and fix
it. How much time they have is a critical question. In planning such a system, it is
valuable to give operators a "critical clearance time," say t,, such that they have to
clear the fault within this time; that is, (tl - to) must be less than t,. If we know
the region of attraction of xps, we can find t, by Figure 8.1. integrating the faulted
system equation starting from the prefault equilibrium xpr until the trajectory hits
the boundary of the region of attraction. The time it takes the trajectory to reach
the boundary can be taken as the critical clearance time because if the fault is
cleared before that time the state x(tl) will be within the region of attraction. Of
course, we are assuming that xpr belongs to the region of attraction of xp,, which is
reasonable. If the actual region of attraction is not known, and an estimate t,, oft,
is obtained by using an estimate of the region of attraction, then t, < t,, since the
. - * *"%-*
5
&$#<
'
:
314
CHAPTER 8. ADVANCED STABILITY ANALYSIS
boundary of the estimate of the region of attraction will be inside the actual boundary of the region. (See Figure 8.1.) Thii scenario shows an example where finding
the region of attraction is needed in planning the operation of a nonlinear system.
It also shows the importance of finding estimates of the region of attraction that
are not too conservative. A very conservative estimate of the region of attraction
would result in t,, that is too small to be useful. Let us conclude this motivating
discussion by saying that the scenario of events described here is not hypothetical.
It is the essence of the transient stability problem in power system^.^
Let the origin z = 0 be an asymptotically stable equilibrium point for the
nonlinear system . ,
x = f (z)
(8.13)
where f : D + - R y i s locally Lipschit2 and D c Rn is a domain containing the
origin.) Let $(t;z) be the solution of (8.13) that starts at initial state z at time
t = 0. he region of attraction of the origin, denoted by RA, is defined by,
8.2. REGION OF ATTRACTION
315
:
;s;;j
R;={ZE
c
-
.%a
?x
%
Figure 8.2: Phase portrait for Example 8.5.
D 1 q5(t;z) isdefined'dt 2 0 a n d q5(t;z)+0ast+oo)
Some properties of the region of attraction are stated in the next lemma, whose
proof is given in Appendix
C.16.
-".
~ e l h m a8.1 If z = 0 an asynptotically stable equilibrium point for (8.13), then
itg region of attmction RA i s an open, connected, invariant set. Moreover, the
boundary of RA b f o n e d by tmjectories.
0
origin is a stable focus; hence, it is asymptotically stable. This can be confirmed by
linearization, since
Lemma 8.1 suggests that one way to determine the region of attraction is to
characterize those trajectories that lie on the boundary of RA. There are some
methods that approach the problem from this viewpoint, but they use geometric
notions from the theory of dynamical systems that are not introduced in this book.
Therefore, we will not describe this class of methods.1° We may, however, get a
flavor of these geometric methods in the case of second-order systems (n = 2) by
employing phase portraits. Examples 8.5 and 8.6 show typical cases in the state
plane. In the first example, the boundary of the region of attraction is a limit cycle,
while in the second one the boundary is formed of stable trajectories of saddle points.
Example 8.7 shows a rather pathological case where the boundary is a closed curve
of equilibrium points.
has eigenvalues at -112 f jfi/2. Clearly, the region of attraction is bounded
because trajectories starting outside the limit cycle cannot cross it to reach the
origin. Because there are no other equilibrium points, the boundary of RA must be
the limit cycle. Inspection of the phase portrait shows that indeed all trajectories
A
starting inside the limit cycle spiral toward the origin.
k
#A',
e
I .
.'
.
$be
?+-Yt
&>
'>
s*
<2
('mt ,
'
P'c
ew
r
4
Example 8.6 Consider the second-ordcr system
Example 8.5 The second-order system
is a Van der Pol equation in reverse time, that is, with t replaced by -t. The system
has one equilibrium point at the origin and one unstable limit cycle, as determined
from the phase portrait shown in Figure 8.2. The phase portrait shows that the
Osee (1701 for an Introduction to the transient stability'problem in power systems.
l0Exnrnplcs of these ~ncthociscnn bo founcl in (361ant1 (216).
This system has three isolated equilibrium points at (0,0), (fi, o), and (-8.0).
The phase portrait of the system is shown in F i g m 8.3. The phase portrait shows
that the origin is a stable focus, and the other two equilibria are saddle points.
Thus, the origin is asymptotically stable and the other equilibria are unstable; a
fact that can be confirmed by linearization. From the phase portrait, we can also
. . see,
. that the stable trajectories of the saddle points form two separatrices that are
A
the boundaries of the region of attraction. The region is unbounded.
'?
-.y-
....................
>
................
-
-.........
,
.........
CIIAP'l'ER 8. ADVANCED STABILITY ANALYSIS
:$l(j
7.
.-..
.-
...
/
I
317
8.2. REGION OF ATrRAC'170N
find a Lyapunov function V(x) that is positive definite in D and G(x) is negative
definite in D or negative eemidefinite, but no solution can stay-identically in the
set {V(x)= 0) except for the zero solution x = 0, then the origin is asymptotiof R A . This
cally stable. One may jump to the conclusionthatc o n j e c t u r e j s n o t as illustrated by the next example.
t
Example 8.8 Consider again the system of Example 8.6:
x1 = x2
= -21
x2
+ i x f - 22
2
t
!
This system is a special case of that of Example 4.5 with
h(xl)= xl
Example 8.7 The system
,
=
and
has an isolated equilibrium point at. the origin and a continuum of equilibrium points
on the unit circle; that is, every point on the unit circle is an equilibrium point.
Clearly, RA must be confined to the interior of the unit circle. The trajectories of
the system are the radii of the unit circle. This can be seen by transforming the
system into polar coordinates. The change of variables
yields
L
fi
C
I
6
I
a:..
B
p = -p(l - p2), 0 = 0
All trajectories starting with p < 1 approach the origin as t
is the interior of the unit circle.
c,
(
and a = l
Therefore, a Lyapunov function is given by
Figure 8.3: Phase portrait for Example 8.6.
I-,
- 1x3
Ifly0c~&
\
-+
co. Therefore, RA
A
Lyapunov's method can be used to find the region of attraction RA or an estimate of it. The basic tool for finding the boundary of RA is Zubov's theorem,
which is given in Exercise 8.10. The theorem, however, has the character of an
existence theorem and requires the solution of a partial differential equation. Via
much simpler procedures, we can find estimates of RA by using Lyapunov's method.
By an estimate of RA, we mean a set R C RA such that every trajectory starting
in R approaches the origin as t + co. For the rest of this section, we will discuss
some aspects of estimating RA. Let 11s start by showing that the domain D of Theorem 4.1 (or Corollary 4.1) is not an estimate of R A We have seen in Theorem 4.1
and Corollary 4.1 that if D is a domain that contains the origin, and if we can
ix; -
~~4
12 1
+ iX
2 12+Bx$
~ ( z=)- 1x2(11x2)
2 1
3 1
ix;
Defining a domain D by
it can be easily seen that V(x) > 0 and V(x)< 0 in D - (0). Inspecting the phase
portrait in Figure 8.3 shows that D is not a subset of RA.
A
In view of this example, it is not difficult to see why D of Theorem 4.1 or
Corollary 4.1 is not an estimate of RA. Even though a trajectory starting in D
will move from one Lyapunov surface V(x) = cl to an inner Lyapunov surface
V(x) = q ,with c2 < c1, there is no guarantee that the trajectory will remain
)
forever in D. Once the trajectory leaves D, there is no guarantee that ~ ( xwill
be negative. Consequently, the whole argument about V(x)decreasing to zero &
f
a
s This problem does not arise when RA is estimated by a compact positively
invariant subset of D; that is, a compact set R c Dxsuch that every trajectory
starting in R stays in R for all future time. Theorem 4.4 shows that R is a subset
of RA.The simplest such estimate is the setu
R c = { x E Rn I V(X)1 C)
"The set {V(z) 5 c} may have more than one component, but there can be only one bounded
component in D, and that is the component we work with. For exam~le.if VIrl = xa/(l
z4) and D =
{[*I2
{CI < I ) ,
m}.
the set {V(x) S 1/41 hm two mmponentu
We work with {lzl
5
-1.
. -, --
{hI5
,-,
\/z--J
+
i d
318
CHAPTER 8. ADVANCED STABILITY ANALYSIS
..~~...
:.. .-.
when,^^ is b
xTp,and
. . ...
D
7..i_.--..l
~ d e and
d containtid iid. For a quadratic Lyapunov function V ( x ) =
{l/xl/z< r } , we can ensure that
c D by choosing
c < min x T p x = X ~ ~ , , ( P ) ~ '
CL-r
319
a certain nrighhorhood of the origin. Because our interest here is in estimating
r
min xTPx =
(bTzI=i
< c) will be a subset of D = {IbTxl
< ri, i = 1,...PI. if we
ri"
bTP-1 bi
c < min
I<~_<P
The simplicity of estimating the region of attraction by 0,= { x T p x _< c) has
increased significance .in.rie.y;of.the linearization results of Section 4.3. There, we
saw that if the J&obihmatrix i
A=
for P. The unique solutiol~is the positive definite matrix
The quadratic function V ( x ) = xTPx is a Lyapunov function for the system in
For D = {(bTx(< T ) , where b E Rn,12
Therefore, { x T P x
choose
8.2. REGION OF ATTRACTION
..
the region of attraction, we need to dcter~ninea domain D about the origin where
V ( x ) is negative definite and a constant c > 0 such that R, = { V ( x ) 5 c ) is a
subset of D. \Ire are interested in the largest set R, that we can determine, that
is, the largest value for the constant c. Notice that we do not have to worry about
dieckiiig positive definiteiless of V ( 2 ) ill D hecatrse V ( x ) is positive definite for all
x. hloreovcr, V ( x ) is radially unbounded: hence R, is bounded for any c > 0. The
derivative of V ( x ) along the trajectories of the system is given by
The right-hand side of V ( X ) is writtcn as the sum of two terms. The first term,
-IIxII$, is the contribution of the linear part A x , while the second term is the
contribution of the nonlinear tern1 g(x) = f ( x ) - Ax. Since
31
a x .=o
is Hurwitz, then we can always find a quadratic Lyapunov function V ( x )= xTPx
by solving the Lyapunov equation p A + A T P = -Q for any positive definite matrix
Q. Putting the pieces together, we see that wheneverA is Hunvitz, we can atimate
the region of attmctta of the ora'gin. This is illustrated by the next example.
we know that there is an open hall D = {z E RZ I llxl12 < r ) such that V ( x ) is
negative definite in D.Once we find such a hall, we can find 52, C D by choosing
E x a m p l e 8.9 The second-order system
Thus, to cnlar~ethe estimate of the rrgion of tit,trrrction,we need to find t.he largest
ball on which V(z) is negative definite. \Ve have
k1 =
k2 =
-22
21
(4
+
(2:
- 1)x2
was treated in Example 8.5. There, we saw that the origin is asymptotically stable
8f
~=bl,~=[!
'
I:]
is Hurwitz. A Lyapunov function for the system can be found by tsking Q = I and
solving the Lyapunov equation
12~ollowing1122, Section 10.31, the Lagrangian arrsociated with the conatrained optimization
problem is L(z, A)
xTPx + X[(bTz)2 r2). The first-order necessary conditions are 2Px +
2X(bTz)b= 0 and (bTr)l ? = 0. It cnn he vcrlficd tl~ntthe solr~tio~~s
X = -1/(bTP-'b) and
z = *r~-'b/(b~'P-'b)yicld the rt~lnlmdmluc ~ ~ / ( b ' ~ P - l b ) .
m
-
--.,..
..
-
. ..
,,
,
.
'-'
2
I
we get
v
PA+ A ~ =
P -I
-
where we used J x l J5 llxllz, lrlxzl 5 11x11;/2, and )xl - 2x21 5 fillxllz. Thus, V ( X )
is negative definite on a ball D of radius give11 by r Z = 2/& = 0.8944. In this
second-order example, a less conservative estimate of fi, can be found by searching
for the ball D in polar coordinates. Taking
, *; . .,.-,-. - .
.
' " . ..'
.+-
-
- . .-
-.
>
"
. .- .
= -p2
< -p2
<_ -p2
-
5
+ p4cos2 6 sin e(2 sin 8 - cos 8 )
+ p4 1 cos2 9 sin 9 ) .I2 sin 6' - cos 81
+ p4 x 0.3849 x 2.2361
1
-p2 + 0.8~1~" 0, for pZ < 0.861
...............
.
.
/
:i20
,
-
.
.-
C:IIA PTER R. ADVANCED S T A B ~ L ~ TANALYSIS
Y
_ . ..
.
...r
4.
.
q
, ,. .
'i .
,
.
-<
:.
_.
._.
. . . .
321
8.2. REGION OF ATTRACTION
-5
Figure 8.4: (a) Contours of V(X) = 0 (dashed), V(x) = 0.8 (dash-dot), and V(x) =
2.25 (solid) for Example 8.9; (b) comparison of the region of attraction with its estimate.
1
. . . .
5
0
Figure 8.5: Estimates of the region of attraction for Example 8.10.
as a Lyapunov function candidate.13 The derivative V ( X ) is given by
Using this last equation, together with Xmi,(P) 2 0.69, we choose
The set R, with c = 0.8 is an estimate of the region of attraction. A less conservative
(that is, larger) estimate can be obtained by plotting contours 'of V(x) = 0 and
V(x) = c for increying values of c until we determine the largest c for which
V(x) = c will be in {V(x) < 0). This is shown in Figure 8.4(a) where c is determined
to be c = 2.25. Figure 8.4(b) compares this estimate with the region of attraction
n
whose boundary is a limit cycle.
Estimating the region of attraction by R, = {V(x) 5 c) is simple, but usudly
conservative. According to LaSa e's theorem (Theorem 4.4), we can work with
any compact set R c D provided we ca;n show that R is positively invariant. It
typically requires investigating the vector field at the boundary of 0 to ensure that
trajectories starting in fi cannot leave it. The next example illustrates this idea.
'I
Example 8.10 Consider the system
where h : R -, R is a locally Lipschitz function that satisfies
Considcr the quadratic function
Therefore, V(X) is negative definite in the set
and we can conclude that the origin is asymptotically stable. ib estimate RA, let
us start by an estimate of,the form fl, = (V(x) 5 c ) . The largest c > 0 for which
R, C G is given by
c=
-
1
min xTPx =
bTp-lb = I
121+221=1
where bT = [l 11. Hence, R, with c = 1 is an estimate of RA. (See Figure 8.5.)
*Ipthis,example, wegan obtain a better estimate of RA by not restricting ourselves
to estimates of the form R,. A key point in the development is to observe that
trajectories inside G cannot leave through certain segments of the boundary Ixl +
221 = 1. This can be seen by examining the vector field at the boundary or by the
following analysis: Let
u=x1
+x2
1 3 ~ h i Lyapunov
s
function candidate can be derived by using the variable gradient method or
by applying the circle criterion and the Kal~nan-Yakubovich-Popov lemma.
i
22
CHAPTER 8. ADVANCED STABlLlTY ANALYSlS
such that the boundary of G is given by a = 1 and u = -1. The derivative of u2
along the trajectories of the system is given by
d
where E is the set of all points in 0 where V(X)= 0. In the case of lionautoi~ol~iouv
systems, it. may not even be clcar how to define a set E , since V ( t , x )is a fu~lctiol~
of both t and x. The situatio~lwill bc siinplcr if it can be shown t h d
;7ja2 = 2u(.i1+ .i2) = 20x2 - 8a2 - 2ah(a) 5 20x2 - 8u2, V In( 5 1
On the boundary a = 1,
f
d
dt
-a2<2x2-850,
~2214
This implies that when the trajectory is at any point on the segment of the boundary
u = 1 for which xz 5 4, it cannot move outside the set G, because at such point u2
is nonincreasing. Similarly, on the boundary a = -1,
Hence, the trajectory cannot leave the set G through the segment of the boundary
a = -1 for wbkh x2 2 -4. This information can'be used to form a closed, bounded,
positively invariant set R that satisfies the conditions of Theorem 4.4. Using the
two segments of the boundary of G just identified to define the boundtuy of R, we
now need two other segments t o close the set. These segments should have the
property that trajectories cannot leave the set through them. We can take them
as segments of a Lyapunov surface. Let cl be such that the Lyapunov surface
V(x) = c1 intersects the boundary XI x2 = 1 at 2 2 = 4, that is, at the point
(-3,4). (See Figure 8.5.) Let cz be such that the Lyapunov surface V(x) = c2
intersects the boundary XI +xz = -1 a t 3 2 = -4, that is, a t the point (3, -4). The
required Lyapunov surface is defined by V(x) = min{cl, c2). The constants cl and
cz are given by
for, then, a set E may be dcfincd as tlir set of points where W ( x )= 0. We may
expect that the trajectory of the systeiri appyoaclies E as t tends to w . This is,
basically, the statement of the next theoreln. Before we state the theorem, we state
a lemma that will be used in the proof of the t.heorem. The lemma is interesting in
its own sake and is known as Barhalat 's 1em.m.a.
Lemma 8.2 Let d : R -+ R be a unijomly continuow function on [0,w ) . Suppose
that limt,, Si 4(7) d~ exists and is finite. Then,
Proof: If it is not true. then there is a positive constant kl surh that for every
T > 0. we can find Tl T with 14(&)J2 kl. Since $ ( t ) is uniformly continuous,
there is ipositive constant k2 such tllrit I@(t+ T) - qh(t)J < k1/2 for all t 2 0 and
all 0 5 T 5 k2. Hence,
>
+
C1
= v(x)/,1=-3,2,=4 = 10,
d(t1
= 10
c2 = V(X)I,~=~
Therefore, we take c = 10 and d e h e the set R by
= {x E R2 1 V(x) 5 10 and )xl
%,
EnR E B
4
m
L
R
T
'
f-
..,.
.,
..
.1.--.-A.
*.
,,
.
....,
.
)
I
I
.
:
.
.
.
.
..-2-..---.-
'
-
-
. .
.
.-l
fr,b
TI
;...--....
.
-_..
.
+ kz.
Q
Theorem 8.4 Let D c Rn he a dornain containing x = 0 and suppose f (t, x) is
piecewise continuous in t and locally Lipscllitz in x, uniformly irl t , on [0,m ) x D.
Furthermore, suppose f(t, 0) is 1~r~ifu17r~ly
bounded for all t 2 0. Let S : (0, oo) x
r ) -,R be a continuously differantiable function such that
11'1(x) 5 V(t,x) 5 I.Y?(x)
In the case of autonomous systems, LaSalle's invariance theorem (Theorem 4.4)
sl~owsthnt tho t.rnjcct,onrof t.lie system approaches the largest invariant set iii E,
--
= Jl'lrk2 id(t)\ dt >
fi
k
8.3 Invariance-like Theorems
dtI
where the equality holds, since 4(t) rctains t,he same sign for TI 5 t 5 TI
Thus, I#J(T)d7 cannot convergc 10 it fil~itclimit as 1 -+ ca7 a contradiction.
+ xzl < 1)
This set is closed, bounded, and positively invariant. Moreover, V(X) is negative
definite in 52, since R C G. Thus, all the conditions of Theorem 4.4 are satisfied and
we can conclude that all trajectories starting in R approach the origin as t -+ m ;
that is, R C RA.
A
'
4"
Therefore?
.
'
:r24
.
/
-
-
...
('11~\1W"1 H. ADVANCED S'1'AOILj'j'Y ANALI'SIS
V t 2 0, V x E D , where lV1(x)nnd 1lJ2(x)are contin.uol~spositive definite functions
and IY(x) is a continuous positive semidefinite function on D. Choose r > 0 such
that Br c D and let p < minll,ll=r1.K ( x ) . Then, all solutions of x = f ( t ,X ) vlth
x(to)E { x E Br ( W 2 ( x )5 p ) are bounded and satisfy
Moreover, if all the assumptions hold globally and Wl ( x ) is radially unbounded, the
0
statement is tme for all x(to)E R n .
Proof: Similar to the proof of Theoreln 4.8, it can be shown that
Therefore, limt,,
J:~W ( X ( T )dr) exists and is finite. Since x ( t ) is bounded, i ( t )=
f ( t ,x ( t ) )is bounded, uniformly in t. for all t 2 to. Hence, x(t) is uniformly continuous in t on [to,co). Consequently, 1V(x(t))is uniformly continuous in t on [to,M )
because 1V(x) is uniformly continuous in x on the compact set,Br. Therefore, by
Lemma 8.2, we conclude that l V ( x ( t ) )-, 0 as t -, M . If all the assumptio~ls110ld
globally and W l ( x )is radially uribounded, then for any x(to), we can choose p so
large that x(t0) E { x E Rn ( I Y ~ ( x<_) p } .
-+
+
[t,t 6].15
t
Theorem 8.5 Let D C Rn be a domain containing x = 0 and suppose f ( t , x ) b
piecewise continuous in t and locally Lipschitt in x for all t 2 0 and x E D. Let
x = 0 be an equilibrium point for x = f ( t , x ) at t = 0. Let V : 10, oo) x D R be a
continuously differentiable function such that
-+
since ~ ( t , x5) 0. Hence, (Ix(t)((< T for all t 2 to. Because V ( t , x ( t ) )is m o m
tonically o on increasing and boilndrtl from below by zcro, it converges as t -+ M .
Now,
The limit W ( x ( t ) )-+ 0 iml)lics that x(t) approaches E as t
in E allowed us to arrive at Corolla~y4.1, where asyniptotic stability of the origill
is established by showing that the set E does not contain an entire trajectory of
the system, other than the trivial solution. For a general nonautono~noussystem,
there is no extension of Corollary 4.1 that would show uniform asymptotic stability.
However, the next theorem shows'that it is possible to conclude uniform asymptotic
stability if, in addition to V ( t ,x ) I 0, we can show that V decreases over the interval
co, whcrc
Therefore, the positive limit set of z ( t ) is a subset of E. The mere knowledge that
x(t) approaches E is much weakcr tl1a11 thc invariance principle for autonomous
syst,ems, which states that x ( t ) approaches the largest invariant set in E. The
stronger conclusion in the case of autonomous systems is a consequence of the
property of autonomous systems stated in Lemma 4.1, namely the positive limit
set is ,an invariant set. There are somc special classes of nonautonomous systems
where positive limit sets have some sort of an invaria~iceproperty. I.' However, for a
general nonautonomous system, the positive limit sets are not invariant. The fact
t,liat, in the case of autonomous systems. x ( t ) approrrches the largest invariant S C ~
l.li.'xir~~l~~ias
arc pcriotlic systcms, irlrrlcrsL-~rc!rioclicsyslcrs, and asymptotically a~~tonomolls
fiy*
lclos. Scc (154, Chapter Bj for invariance Lhcorcn>sfor tiles@classes of systems. See, a\=, (1361for
a tlilicrent.gcneralization bf the invariance principle.
bVl(.)
+
I
:
I. V ( t , x )5 LV2(x)
+
V ( t 6,q5(t 6;t ,2 ) )- V ( t ,x ) 5 -XV(t, x ) , 0 < X < 1''
V t 2 0, V x E D , for some d > 0, where W l ( x ) and IV2(x) are continuous positive
definite functions on D and d ( ~ ; t , xis) the solution of the system that starts at
( t , x ) . Then,. the origin is uniformly asymptotically stable. 0 all the assumptions
hold globally and W l ( x ) is radially unbounded, then the origin is globally uniformly
asymptotically stable. If
then the origin is exponentially stable.
0
Proof: Choose r > 0 such that B, E D. Similar to the proof of Theorem 4.8, it
can be shown that
where p
< minll,ll=, ~ V I ( Xbecause
),
~
( tx ), 1 0. Now, for all t
2 to, we have
+
V ( t 6,x(t+ 6 ) )5 V ( t ,4 t ) )- XV(t,x(t))= (1 - X)V(t,x(t))
i
hloreover, since ~ ( xt ), < 0,
V ( Tx, ( T ) )< V ( t , x ( t ) ) ,V f E [t,t
151t L sllown in [I] that the condition
can be shown if
+ 61
v _< 0 can be dropped and uniform aqymptotic stability
-
V(t + 6,b(t + 6;t,z)) ~'(L,z)5 -r(Ilz(J)
for some class K function
Y
......
lGThercis no loss of generality in assu~ningthat X < 1, for if the inequality is satisfied with
X I 1, then i L is satisfied for any positive < 1, since -A1 V 5 -XV. Notice, however, that this
inequality co~lldnot be satisfied with A > 1, since V(t,z) > 0, V z # 0.
>
I
t
CHAPTER 8. ADVANCED STABILITY ANALYSIS
+
For any t 2 to, let N be the smallest positive integer such that t 5 to N6. Divide
the interval [to,to+ ( N - 1)6]into ( N - 1) equal subintervals of length 6 each. Tllen,
+
327
8.3. INVARlANCELlKE THEOREAIS
The solution of the linear system is given by 4(r;t , x ) = @ ( T , t ) x , whcrc ~I)(T,t ) is
the state transition matrix. Therefore,
V ( t , x ( t ) ) ' < V(to+ ( N - l ) b , ~ ( t o ( N - 1)6))
5 (1 X)V(to + ( N 2)6,x(t0 ( N - 2)6))
-
-
+
5 (1 - X ) ( N - ' ) ~ ( tx(t0))
o,
1
(1 - 4
5 -(I--
~ ) ( ~ - ' 0 ) ' ~ v (x(to))
to,
where
Suppose there is a positive constarit k < cz such that
where
1
1
b = -In6 (1 - A)
Taking
u(r,9 ) =
r
m)e-b8
it can be easily seen that u(r,s) is a class K& function and V ( t ,x(t))satisfies
V ( t l ~ ( t5
) )4V(to1x(to))1t - to),
Example 8.11 Consider the linear time-"arying system
>
where A(t) is continuous for all t 0. Suppose there is a continuously differentiable,
symmetric matrix P ( t ) that satisfies
v t >0
as well as the matrix differential equation
+
V ( t +6,d(t
+
-P(t) = ~ ( t ) ~ (~t ~) ( t ) ~c (T (t t ) c ( t )
where C(t) is continuous in t. The derivative of the quadratic function
Thus, all the assumptions of Theorem 8.5 are satisfied globally with
and we conclude that the origin is globally exponentially stable. Readers familiar
with li~iearsystem theory will recogiliiic that the matrix IV(t,t 6 ) is the observability Gramian of the pair ( A ( t )C
, ( t ) )and that the inequality LV(t,t 6 ) 2 kI
is implied by uniform observabi1it.y of ( A ( t )C
, ( t ) ) .Comparing this example with
Example 4.21 shows that Theorem 8.5 allows us to replace the positive definiteness requirement on the matrix Q ( t )of (4.28) by the weaker requirement Q ( t )=
A
CT(t)C(t),where the pair ( A ( t )Cbt))
,
is uniformly observable.
+
.
3
Example 8.12 In Section 1.2.6, we saw that the closed-loop equation of a model
reference adaptive control system, with plant 9p = apyp+ kpu and reference model
aim = amym k,r, is given I)y
+
dl
along the trajectories of the system is
P(t,x) = -x=cT(t)c(t)x 5 0
+
Theorems 8.4 and 8.5 and their application to linear systems, as in Example 8.11,
are cstcrisively used in the analysis of nrlaptivc control syst,ems.17 As an example,
we analyze the adaptive control system of Section 1.2.6.
V ( t ,x) = x T p ( t ) x
.
+ 6 ; t , x ) )- V ( t : x )5 - k l ( ~ ( <( $- -c2kV ( t , x )
v V(to1x(t0))E [O,p]
&om this point on, the rest of the proof is identical to that of Theorem 4.9. The
proof of the statements on global uniform asymptotic stability and exponential
stability are the same as the proofs of Theorems 4.9 and 4.10.
0 < c1I I P(t) < czI,
then
$2
':See,
= -reor(t)
= -woko ym(t)l
for cxan~ple,1871 and 11681.
+
.+ . ..
.
.
.,,,-. ..
. --
/
...
,
.
. ..
.-
-
- -... .. . .,
. ..-,
CHAPTER8. ADVANCED STABILITY ANALYSIS
328
where > 0 is the adaptation gain, e, = yp - ym is the output error, and $1 and 4 2
are the parameter errors. It was assumed that kp > 0 and, of course, the reference
model must have a, < 0. Furthermore, we assume that ~ ( t is) piecewise continuous
and bounded. Using
-.
--
am
-,.(ti
-7yp(t)
kPr(t) kPyp(t)
0
0
0
0
]
.
-..
,
."
.li...
"
<
- : -%
.
, ,.
.
.,.
.
329
I
tlie systmii i = rl(t).r. Olicc again, nsing \ ' LU11 L ~ ~ I ~ I II111ictio11
I ~ O Y ct~iitlid~ke,
\\v
obtain
P=a" e:
= -xTCTCx,
where C =
kP
,/?[
1 0 0
]
From Example 8.12, we see that the origin will be uiliformly asymptotically stable if
the pair (A(t), C ) is uniformly observable. Since uniform observability of (A(t),C)
is equivalent to uniform observability of (A(t) - I((t)C, C) for any piecewise continuous, bounded matrix K(t),lg we take
By applying Theorem 8.4, we conclude that for any c > 0 and for. all initial states in
the set {V 5 c}, all state variables are bounded for all t 2 to and limtdm eo(t) = 0.
This shows that the output of the plant yp tracks the desired output y,, but
says nothi~igaboi~tthe convergence of the parameter errors 41 and 42 to zero.
In fact, they may not converge to zero. For example, if T and ym are nonzero
constant signals, the closed-loop system will have an equilibrium subspace {e, =
0, d2 = (am/km)41)l which shows clearly that, in general, dl and $2 do not
converge to zero. -Toderive conditions under which
and
will converge to
zero, we apply Theorem 8.5. This will yield conditions under which the origin
(c, = 0, dl = 0, d2 = 0) is uniformly asymptotically stable. Since we have already
shown that all state variables are bounded, we can represent the closed-loop system
as the linear time-varying system
[
a
8.4. STABILITY OF PERIODIC SOLUTIONS
as a Lyapunov function candidate, we obtain
I=
"
,
.
wher~ x =
x.,
[2 ]
~ ( t =)
1.
-am
[ am
1
-~as(t) - ~ ~ a ( t )
to simplify the pair to
By investigating observability of this pair for a given reference signal, we can determine whether the conditions of Theore1118.5 are sntisfied. For example, if r is
t pair is not observable.
a nonzero cdnstant signal, it can be easily SCCII t l ~ t ~the
This is not surprising, since we have already seen that in this case the origin is not
uniformly asymptotically stable. On the other hnnd. if ~ ( t =) asinwt with positive
) ya,(t) = ad4sin(wt 6), where M and 6 are
a and w, we have ~,,(t) = ~ ( t and
determined by the transfer function of the reference model. It can be verified t.hat
the pair is uniformly observable; hence the origin (e, = 0, $1 = 0, $2 = 0) is
uniformly asymptotically stable and the parameter errors &(t) and &(t) converge
to zero as t tends to infinity.20
+
Suppose the reference signal r ( t )has a steady-state value r,,(t); that is, limt,,[T(t)1-,,(t)] = 0. Then, lirnt,,[y,(t)
- y,,(t)] = 0, where y,,(t) is the steady-state response of the reference model. These limits, together with limt,,, e,(t) = 0, show
that the linear system can'be represented by
8.4
where
In Chapter 4, we developed an extensive theory for the stability of equilibrium
points. In this section, we consider the corresponding problem for periodic solutions.
If u(t) is a periodic solution of the system
am
I
kp~ss(t) hyhS(t)
0O
,
and
Stability of Periodic Solutions
what can we say about other solutions that start arbitrarily close to u(t)? Will
they remain in some neighborhood of u(t) for all t? Will they eventually approach
+
9.6.
A
lim B(t) = 0
t-m
If we can show that the origin of x = A(t)x is uniformly asymptotically stable, we
cikn IISC t,hc property limr,,
B(t) = 0 to show that t,he origin of x = [A(t) B(t)]x
is uniformly asymptotically stable.18 Therefore, we concentrate our attention on
LWSooExample
~
lgSee (87, Lemma 4.8.11.
2oFor this example, the reference r(t) = asinwt is said Lo be persistently exciting, while a
constant reference is not persistently exciting. 'Ib read more about persistence of excitation, see
151, (151,(871,[139],[lea],or Section 13.4 of the second edition of this book.
,
CHAPTER 8. ADVANCED STABILITY ANALYSIS
? Such stability properties of the periodic solution u(t) can be characterized
h o m these espressions, we see that the systcm has a periodic solution
investigated in the sense of Lyapunov. Let
Sl (t) = cost?
y=s-u(t)
+$
,
:$4%
&%$.-
$&
qd-2
$j
s
p<fd>
&
>
e
2
2 g
z
-LGG
%<:
$i.
y"T
p9
53,
Z.
*J+
z s
$
6 = f ( t , +~ u ( t ) )- f (t,u(t))
;
Pk
-
(8.15)
The behavior of solutions of (8.14) near u(t) is equivalent to the behavior of solutions of (8.15) near y = 0. Therefore, we can characterize stability properties of
u(t) from those of the equilibrium y = 0. In particular, we say that the periodic
solution u(t) is uniformly asymptotically stable if y = 0 is a uniformly asymptotically stable equilibrium point for the system (8.15). Similar statements can be
made for other stability properties, like uniform stability. Thus, investigating the
stability of u(t) has been reduced to studying the stability of an equilibrium point of
a nonautonomous system, which we studied in Chapter 4. We shall find this notion
' of uniform asymptotic stability of periodic solutions in the sense of Lyapunov to be
useful when we study nonautonomous systems dependent on small parameters in
Chapter 10. The notion, however, is too restrictive when we analyze periodic solutions of autonomous systems. The next example illustrates the restrictive nature of
this notion.
+ [ ~ ( tsin) B(t) - sin t]' -+ 0 as t --,
for sufficiently small [ro cosBo - 11' + [ro sin^^]'. Because r ( t ) -, 1 as t -,m, we
[r(t) cos e(t) - cost]'
1-2:
11-cos(B(t)-t)l-,O
which clearly is not satisfied when ro
tonically increasing function of t.
+-28)
X;
3 ] - z 2 1 +I1 - x(~ - 2 8 )
1-2:
X;
X:
# 1: since (B(t) - t) is an ever-gro\ving monoA
where f : D -t R" is a continuously diffcrerltiable map from a doinain D C Rn into
R". Let A l C D be a closed invariant set of (8.1G).Define an E-neigliborllood of M
by
U, = {x E Rn I tlist(r, A1) < E )
represented in the polar coordinates
21=rcosB,
as t - t c a
The point illustrated by this example is true in general. In particular, a nontrivial
periodic solution of an autonomous system can never be asymptotically stable in
the sense of LYapunov.''
The stability-like propcrties of the periodic orbit of Example 8.13 can be captured by extending the notion of stability ill the sense of Lyapunov from stability
of an equilibrium point to stability of all illvuriallt set. Co~lsidcrthe autonomous
system
j. = f ( r )
'I
-2i)
* z = ~ ~ [ (+
3]+21b+(1-x;-x;) 'I
Z:
SC)
nlust have
le 8.13 Consider the second-order system
. L 1 = R [ (
S2(t)= sin t
'
The corresponding periodic orbit is the unit. circle r = 1. All nearby solutioils
spiral t.oward this periodic orbit as t
cc. This spiralling is clearly the kind of
"asymptotically stable" behavior we expect. to see with a periodic orbit. In fact,
the periodic orbit has been known classically as a stable limit cycle. However,
tlie periodic solution f ( t ) is not uniformly asymptotically stable in the sense of
Lyapunov. Recall that for the solutioll t o be ulliformly asymptotically stable, we
must have
at the origin y = 0 becomes an equilibrium point for the nonautonomous
e
331
8.4. STABILITY OF PERIODIC SOLUTIOXS
x2=rsine
where dist(2,ll.i) is the minimum distance from z to a point in ill; tliat is,
,
)=1+(1-,.')'
r
dist(x, AI) =
The solution starting at (ro, 00) L given by
1
1-r$
=
+
1 4t (1 r;)
6(t) =
60'
+ t + 4 ln [l + 4 t ( l -
_I~.....
.
-
3...
.
......,
.
..
I
"'
r;)']
;
.
> 0, there is 6 > 0 such that.
x(0)~U~~x(t)~
VU
t 2, 0,
2'See (72, Theorem 81.11 for a proof of this statement.
t
a
1. - 1/11
Definition 8.1 The closed invariant set 111 of (8.16) is
stable if, for each E
,,--;-**
,'
illf
YEAI
-... .
. \ . -.*-
.--
-
. .-..,
.
.
. ,-.
-
_
. . .
.,
.
1 '
"-..-
.
.
."
.J
-
'7
'
.
.
/
L
.)
J
332
......
............
- I
.
..,
( ~ ~ J , \ P ' ~8.E RADVANCED STABILITY ANALYSIS
aslJmptotically stable if it is stable and 6 can be chosen such that
x(0) E Ua + t-m
lim dist(x(t), M) = 0
This definition reduces to Definition 4.1 when M is an equilibrium point. Lyapunov
stability theory for equilibrium points, as presented in Chapter 4, can be extended
to invariant sets.22 For example, by repeating the proof of Theorem 4.1, it is
not hard to see that if there is a function V(x), which is zero on A4 and positive
in some neighborhood D of MI excluding M itself, and if !he derivative V(X) =
[OV/ax]f ( x ) 5 0 in D, then M is stable. Furthermore, if V(x) is negative in D,
excluding hi, then M is asymptotically st,able.
Stability and asymptotic stability of invariant sets are interesting concepts in
their own sake. We will apply them here to the specific case when the invariant set
A.I is the closed orbit associated with a periodic solution. Let u(t) be a nontrivial
periodic solution of the autonomous syst.em (8.16) with pcriod T I and let y be the
closed orbit defined by
The periodic orbit y is the image of u(t) in the state space. It is an invariant
set whose stability properties are characterized by Definition 8.1. It is common,
especially for second-order systems, to refer to asymptotically stable periodic orbits
as stable limit cycles
Example 8.14 The harmonic oscillator
-
'.
..................
I
.
:.-
..
.*
..--
"
'(.
q
.
........
.
2
a
.
,
. . .
,
8.4. STABILIT'I' 01.' PBRIOI)ICS O L L ~ ~ I O N S
:rq
of
docs not approach 7, as t -, m, 110 lrlatter IIOIV sn~alld is. Stability of the
periodic orbit { r = c) can be also shown by the Lyapunov function
-
-
V(x) = (r2 c ~ =)(x:~ + x i c ~ ) ~
whose derivative along the trajectories of the system is
A
Example 8.15 Consider the system of Example 8.13. It has an isolated periodic
orbit
Y = { x E R ~l r = l ) , w h e r e r =
For x $ y, we have
dist(x, 7) = YET
inf Ilx - vll2 = YET
illf J(xl
m
X:+X
- y1)2 + (x2 - 2/2)2= Ir
- 11
Recalling that
-
it can be easily seen that the E-6 requirement for stability is satisfied and
Hence, the periodic orbit is asymptotically stable. The same conclusion can be
arrived at using the Lyapunov function
whose derivative along the trajectories of the system is
has a continuum of periodic orbits, which are concentric circles with a center at
the origin. Any one of these perio ic orbits is stable. Consider, for example, the
periodic orbit yc defined by
\
4
y c = { x ~ R 1I r = c > O } ,
,
wherer=
Js
~ ( x =) 4(r2 - 1 ) r i = -4(r2
- I)' < 0, for r # 1
Having defined the stability properties of periodic orbits, we can now define the
stability properties of periodic solutions.
The U, neighborhood of y, is,defined by the annular region
Definition 8.2 A nontrivial periodic solution u ( t ) of (8.16) is
orbitally stable if the closed orbit y generated by u(t) is stable.
This annular region itself is an invariant set. Thus, given E > 0, we can take 6 = E
and see that any solution starting in the U6 neighborhood at t = 0 will remain in
the U, neighborhood for all t 2 0.' Hence, the periodic orbit yc is stable. However,
it is not asymptotically stable, because a solution starting in a Ua neighborhood
asymptotically orbitally stable if the closed orbit y generated by u(t) ia asymptotically stable.
Notice that different terminology is used depending on whether we are talking about
t.he periodic solution or the corresponding periodic orbit. In Example 8.14 we say
that the unit circle is an asymptotically stable periodic orbit, but we my that the
periodic solution (cos t, sin t) is orbitally asymptotically stable.
%ee, for example, I2131 and [221]for comprehensive coverage and (1181 for some interesting
results on converse Lyapunov theorems.
335
8.5. EXERCISES
CHAPTER 8. ADVANCED STABILITY ANALYSIS
Investigate stability of the origin by rlsilig the center manifold theorem for each of
the following cases:
( 1 ) a+c>O.
( 3 ) a + c = 0 and cd + bc2 < 0.
( 5 ) a c = cd bc2 = 0.
+
.1 are satisfied in a case where gl(y,0 ) =
0, g2(1,0) = 0, and A1 = 0. Show that the origin of the full system is stable.
+
= ax?
+
k l = ax1x2 - x;,
the full system.
ms,investigate stability of the origin by
= ax21 - x &
i2 = -xz+x:+
kl
.
,
I,
*:
<,..
,
.
?
::F*:
.
0 < V ( x )< 1 , V x E G - {O)
11~11
As x approaches the boundary of G. 01. in case of unbounded G
lim V ( X )= I .
t
a#o
21x2
= x;x2
= -x! - 1 2
x,
(4)
2'2)+ x i
51 = xz
(7) x~ = -12
a x f / ( l + x;)
a#O
C
+
t
00,
-+
av
ax
+ - 1)
= -xi
xf(x1 x2
= xg(xl+ 1 2 1 )
-
8.11 ( [ 7 2 ] ) Consider the second-order system
+ +
+ bxlxi,
x2 = -12
XI
. ..
.
..
\
...
-.
,. ,..,.,
.
,,.-.,-...*
*-.-Ip-. : .
= - h 1 ( ~ 1+) g 2 ( ~ 2 ) ?
.t2 = -gi(xl)
where
+ CX? + dxtx2
B
.
hl(0) = 0 , r h l ( r ) > 0 V -a1
-. .
.
"..:I-"-.
---
.,.
--
(8.17)
Show that x = 0 is asymptotically stable and G is the region of attract.ion.
8.7 ( [ 3 4 ] )Consider the system
xl = xlxz +ax:
For x E G, V ( x )satisfies the partial differential equation
-f ( x ) = -h(z)[l- V ( X ) ]
xl = -2x1-3a2+x3+xz
( 8 ) x~ = X I x: 1 2
x3 = X:
+
,"
+ bxlx2 + cx:
h is continuous and positive definite on Rn.
-12 + X l X 3
(6) dl
5,
.;
52 = - 1 2
V is continuously differentiable and positive definite in G and satisfies
t
..
+,xlx2 - I ;
8.10 (Zubov's Theorem) Consider the system (8.13) and let G C Rn be a domain containi'ng the origin. Suppose there exist two functions V : G -, R and
h : Rn 4 R with the following properties:
totically (but not exponentially) stable equilib
,XI,) = (0,O) is an asymptotically stable
-
+ X:
Investigate stability of the origili by using the center manifold theorem for all p a sible values of the real coilstants a, b, and c.
(a) Show that if x, = 0 is an exponentialljr stable equilibrium point of 5, =
fa(Xa9 O), then (x,, xb) = (0,O) is an exponentially stable equilibrium point of
+ 12x3
- (x: +
.t2= -.r2
8.9 ( [ 8 8 ] )Consider the system
where dim(x,) = n l , dim(xb) = nz, Ab is Hurwitz, fa and fb are cont,inuously
dserentiable, (8f b / & b ] ( ~ , 0) = 0, and fb(x,, 0 ) = 0 in a neighborhood of x, = 0.
21
-53
+ r:x2,
1nvestigat.estability of t,he origin by using the center manifold theorem for all possible values of the real parameter a.
50 =' fa(&, xb)
kb = Abxb f b ( ~ a , ~ b )
51 =
( 3 ) 52 =
x3 =
+ bc2 > 0.
8.8 ([34]) Consider the system
8.1 with a = 0. Show that the origin is stable.
(1
+
( 2 ) a c < 0.
( 4 ) a + c = 0 and cd
.
,
.
-
'U...
-
.-
-
.
-
< z <bl
.
-.
.
,
.
. ,
1
--
.
.
.
-
..- .
.
JO
for some positive constants ai, bi (ai ,= w or bi = w is allowed). Apply Zubov's
theorem to show that the region of attraction is {X E R2 1 - ai < xi < bi).
Hint: Take h(x) = gl(xl)hl(xl) and seek a solution of the partial differential
equation (8.17) in the form V(x) = 1 - IVl(xl)W2(x2). Note that, with this choice
of h, V(x) is only negative semidefinite; apply LaSallels invariance principle.
8.12 Find the region of attraction of the system
Figure 8.6: Exercise 8.17.
Hint: Use the previous exercise.
8 . 1:)
1,181.
$2
\I(! 1111
c!vtbryt.ri{ic!ct.ory ill
o11(!11,l)osil.i\-c!lyi~~viuirrl~b
sot cont.nining the origin. Suppose
R approi~clicst.11~origin as t -+ m. Show that R is connected.
8.14 Consider a second-order systeln x = f (x) with asymptotically stable origin.
Let V(x) = x: + x;, and D = {x E R2 I lxZl < 1, lxl - 2 2 1 < 1). SIII)I)OW'
[8V/a.-c] f (x) is negative definite in D. Estimate the region of attraction.
(a) Show that the origin is the unique equilibrium point.
(b) Show, using linearization, that the origin is asymptotically stable.
+ 2 2 . S ~ ~ OtlliltW U b < -1Ul (u(2 1.
(d) Lct V(x) = x: + 0.52; + 1 - cosxl. Sliow tliat,
(c) L c ~U = 21
8.15 Consider the system
(a) Using V(x) = 51:
+ 2 x 1 +~2x22.
~ show that the origin is asymptotically stable.
(b) Lct,
S = {XE
n2 1 V(X)I5 ) n {XE R2 ( 1x215 1)
Show that S is an estimate of the region of attraction.
8.16 Show that the origin of
is positively invariant and trajectories in Ad, approach the origin as t
4
(e) Show t,hat the origin is globally asymptotically stable.
8.19 Consider the synchronous generator model described in Exercise 1.8. Take
the state variables and parameters as in parts (a) and (b) of the exercise. Moreover,
take T = 6.6 sec, A.I = 0.0147 (per unit power) x sec2/rad, and D/Ai = 4 sec-I.
<
(a) Find all equilibrium points in the region -n 5 x l
n , and determine the
stability properties of each equilibrium by using linearization.
is asymptotically stable and estimate the region of attraction.
8.17 Consider a second-order system x = f (x), together wit.h a Lyapunov function
V(x). Suppose that V (x) < 0 for all x: +xi 2 a2. The sketch, given in Figure 8.6,
shows four different directions of the vector field at a point on the circle x f + x i = aZ.
Which of these directions are possible and which are not? Justify your answer.
(b) Estimate the region of attraction of each asymptotically stable equilibrium.
8.20 ([113]) Consider the system
x l = 52,
x2 = -51
- g(t)z2
where g(t) is continuously differentiable and 0 < kl 5 g(t) 5 k2 for all t 2 0.
8.18 Consider the system
x l = x2,
oo.
.i.? = - . ~ 2 - sin X I - 2 sat(xl + x2)
(a) Show that the origin is exponentially stable.
(b) LVould (a) be h u e if g(t) were not bounded? Consider g(t) = 2
+ exp(t).
CHAPTER 8. ADVANCED STABILITY ANALYSIS
'
;; >8.21Consider the system
x l = x2,
x2 = - sinxi
- g(t)x2
where g(t) is continuously differentiable and 0 < kl 5 g(t) 5 k2 for all t 2 0. Show
that the origin is exponentially stable.
Hint: Use the previous exercise.
9 '
Chapter 9
: ~ b ( h M IC&:~N
\f
(M
~ I
8.22 Consider the system
where a(t) = sin t + sin 2t. Show that the origin is exponentially stable.
*s
, k.,'
$5
&&
Stability of Perturbed Systems
8.23 Consider the single-input-singleoutput nonlinear system
Consider the system
x = f(t,x) +g(t.x)
fo, fl, and go are known smooth functions of x, defined for all x E Rn,
while 8' E RP is a vector of unknown constant parameters.. The function go(x) is
bounded away from zero; that is, Jgo(x)l2 ko > 0, for all x E Rn. We assume that
all state variables can be measured. It is desired to design a state feedback adaptive
controller such that x l asymptotically tracks a desired reference signal ~ ( t )where
,
T and its derivatives up to T(") are continuous and bounded for all t 2 0.
' where
..., enIT, show that e satisfies the equation
d = Ae + B [fo(x) + (19')~ (x) + go(x)u - T(")]
(a) '@king ei = x1-r('-l)
fl
By?
,
p
:
'r
and e = [el,
where (A, B) is a controllable pair.
(b) Design K such that A - B K is Hurwitz and let P be the positive definite
solution of the Lyapunov equation P(A B K ) (A - B K ) ~ P= - I . Using
the Lyapunov function candidate V = eTpe + &TI'-1&,
where & = 9 - O* and
I' is a symmetric positive definite matrix, show that the adaptive controller
-
+
(9.1)
where f : [0, x) x D -+ Rn and g : [0,m) x D -+ Rn are piecewise continuous in
t aiid locally Lipschitz in x on [0,oo) x D, aiid D c Rn is a domain that contains
the origin x = 0. \Ire think of this system as a perturbation of the nominal system
Tlic perturbatioi~term g(t, x) could result from inodcling errors, aging, or uncertainties and disturbances, which exist in any realistic problem. In a typical situation,
nrc do not know g(f.x). hut we know some information about it, like knowing an
upper bound on Ilg(t.x)I/. Here, we represent the perturbation as an additive term
on the right-hand side of the state equation. Uncertainties that do not change the
system's order can always be represented in this form. For if the perturbed righthand side is some function f (t, x), then by adding and subtracting f (t, x), we can
rewrite the right-liaiid side as
and define
@
H
ensures that all state variables are bounded and limt-+aoe(t) = 0.
.
$9
1:
(4 Let
"t)
=
..
[
-Bf'(R)
OPXP
],
c = [ In
Onrp
1
where R = [T, . , T ( ~ - ' ) ] ~Show
.
that if (A(~),c) is uniformly observable,
then the pnrruneter error & converges to zero as t -+ 00.
@d
rCo*r
a:
g(t:x) = J(t, 5) - f (t, X)
Suppose the nominal system (9.2) has a'uniformly asymptotically stable equilibrium
poilit a t the origin, what can we say about the stability behavior of the pertutbed
system (9.1)? A natural approach to address this question is to use a Lyapunov
function for the nominal system as a Lyapunov function candidate for the perturbed
system. This is what we have done in the analysis of the linearization appioach in
Sections 4.3 and 4.6. The new element here is that the perturbation term could be
more general than the perturbation term in the case of linearization. The conclusions we can arrive a t depend critically on ivliether the perturbation term vanishes
)I
'1
"i
5.c
f
jA
!1
i
5]
5'
5
3
% ,
tlie stability of tlic origill IIS all equilibrium poillt for tlie perturbed
to iii~~stigate
system (9.1). The derivative of V along the trajectories of (9.1) is given by
at the origin. If g(t, 0) = 0, tlie perturbed system (9.2) has an equilibrium point
at the origin. In this case, we analyze the stability behavior of the origin as an
equilibrium point of the perturbed system. If g(t,O) # 0, the origin will not be
an equilibrium point of the perturbed system. In this case, we study ultimate
boundedness of the solutions of the perturbed system.
The cases of vanishing and nonvanishing perturbations are treated in Sections 9.1
and 9.2, respectively. In Section 9.3, we restrict our attention to the case when tlie
nominal system has an exponentially stable equilibrium point at the origin and use
the comparison lemma to derive some sharper results on the asymptotic behavior of
the solution of the perturbed system. In Section 9.4, we give a result that establishes
continuity of the solution of the state equation on the infinite-time interval.
The last two sections deal with interconnected systems and slowly varying systems, respectively. In both cases, stability analysis is simplified by viewing the
system as a perturbation of a simpler system. In the case of interconnected systems, the analysis is simplified by decomposing the system into smaller isolated
~ubsystems,while in the case of slowly varying systems, a nonautonornous system
with slowly varying inputs is approximated by an autonomous system where the
slowly varying inputs are treated as constant parameters.
c *uuz;J
*
0
9.1 vakshing Perturbation
( c k d ~ o s ~ ba\r
av av
+
1 1
BV
-
5~
3
(
er6b;l&J
)
-
+
-c311xl12
+ q.rllx~12
If 'Y is small enough to satisfy the bound
--
&>qcwiH,GI)
with the case g(t, 0) = Oi Suppose X = 0 is an exponentially
~~t us
librium point of the nominal system (9.2), and let V(t,x) be a,Lyapunov
that satisfies
~111x11~
5 v(tt1.31 < ~211x11~
f
at
The first two terms on the right-hand side constitute the derivative of V(t,x) along
the trajectories of the nominal system, which is negative definite and satisfies (9.4).
The third term, [bV/bx]g, is the effect of the perturbation. Since we do not have
complete knowledge of g, we cannot judge whether this term helps or hurts the
cause of making V(t,x) negative definite. With the growth bound (9.6) 8s our
only information on g, the best we can do is worst case analysis where [bV/8x]g is
bounded by a nonnegative term. Using (9.4) through (9.6), we obtain
equi-
then
w,x) I-(c3 - .r~4)11~II~,(c3 - 7 ~ 4 >) 0
Therefore, by Theorem 4.10, we conclude the next lemma.
Lemma 9.1 Let x = 0 be an exponentially stable equilibrium point of the nominal
(9.3)
(9.4)
~11~11
$
for all (t,x) E [O, m) x D for some positive constants cl, c2, c3, and cd. The
existence of a Lyapunov function satisfying (9.3) through (9.5) is guaranteed by
Theorem 4.14, under some.additional assumptions. Suppose the perturbation term
g(t, x) satisfies the linear growth bound
where 7 is a nonnegative constant,. This bound is natural in view of thq assumptions on g(t,x). In fact, any function g(t,x) that vanishes at the origin and is locally
Lipschitz in x, uniformly in t for all t 2 0, in a bounded neighborhood ofthe origin
satisfies (9.6) over that neighborhood.' IVe use V as a Lyapunov function candidate
'Note, liowever, that the linear growl11 b o l ~ ~ l(9.6)
d beco~nesrestrictive when required to hold
globally, because that would require g to be globally Lipschitz in I.
:
,
:
system (9.2). Let V(t, X) be a Lyapunovfunction of the nominal system that eat&&
(9.3) t h m ~ g h(9.5) in [O, m) x D. Suppose the perturbation t e r n g(tlx) 8&&fi@
(9.6) and (9.7). Then, the origin b an clponentially stable equilibrium point of
the perturbed system (9.1). Moreover, if all the assumptions hold globally,. then the
origin is globally exponentially stable.
0
This lemma is conceptually important because it shows that exponential stability
of the origin is robust with respect to a class of perturbations that satisfy (9.6) and
(9.7). TOassert this robustness property, we do not have to know V(t, x) explicitly.
It is just enough to know that the origin is an exponentially stable equilibrium
of the nominal system. Sometimes, we may be able to show that the origin is
exponentially stable without actually finding a Lyapunov function that satisfies
(9.3) through (9.5).' Irrespective of the method we use to show exponential stability
of the origin, we can assert the existence of V(t,x) satisfying (9.3) through (9.5) by
application of Theorem 4.14 (provided the Jacobian matrix [af/ax] is bounded).
However, if we do not know the Lyapunov function V(t, x) we cannot calculate the
2Tl~isis the case, for example, when exponential stability of the origin is shown using T h e
rem 8.5.
342
CHAPTER 9. STABILITY OF PERTURBED S17STEh4S
343
9.1. VANISHII\'G PERTURBATION
The eigenvalues of A are -1 f j L 6 . Hence, A is Hurwitz. The solution of the
Lyapunov equatior~
P A A'"P = -1
bound of (9.7). Consequently, our robustness conclusion becomes a qualitative one
where we say that the origin is exponentially stable for all perturbatiolls satisfying
+
is given by
wit.h sufficiently small 7. On t.he other hand? if we know V(t, x)? we can calculate
the bound of 19.71.
, ,. which is an additional ~ i e c eof informntio~i. We ssl~oultlbe
caroful not t.o o\~erernphasizethe usefulness of such bounds because they could bc
conservative for a given perturbation g(t, x). The conservatism is a consequence of
the worst case analysis we have adopted from the beginning.
As we saw in Example 9.1?the Lyapurlov function V(x) = xTPx satisfies inequalities
(9.3) through (9.5) with ca = 1 and
Example 9.1 Consider the system
The perturbation tcrin g(x) satisfies
where A is Hurwitz and 11g(t,~)11~
I r11x112 for a11 t 2 0 and a11 x E R". Let
Q = QT > 0 and solve the Lyapunov equation
<
for all 1x21 k2. At this point ill the analysis, we do not know a bound on x2(t),
alt.hough we know that x2(t) will hc bounded whenever the trajectory x(t) is confined to a compact set. We keep kz undetermilled and proceed with the analysis.
Using V(x) as a Lyapunov funct.ioi1 cantlidate for the perturbed syst.em, we obtain
for P. From Theorem 4.6, we know that there is a unique solution P = PT > 0.
The quadratic Lyapunov function V(x) = xTPx satisfies (9.3) through (9.5). In
particular,
An~ilP+l/~/l;
-(WY
&nnx(P)llxII~
LC negative tlcfiliitc! if
)
Hence, ~ ( x will
. .
"-:*.
The derivative of V(x) along the trajectories of t.he perturbed system satisfies
<
~ ( x ) -Xrn~n(Q)Ilxllf + 2~rnex(P)7ll~ll~
Hence, the origin is globally exponeritially stable if 7 < Xrnin(Q)/2Xrn,,(P). Since
this bound depends on the choice of Q, one may u-onder how to choose Q to maximize the ratio XmIn(Q)/Xrn,(P). 'It turns out bhat this ratio is maximized with the,
A
choice Q = I (Exercise 9.1).
Example 9.2 Consider the second-order system
where the constant /3 2 0 is unknown. We view the system as a perturbed system
of the form (9.1) with
.
:...!
\
'
....
. :.
.
:
.
;. .
.
. . ...... .. .. .. .<;..::*~'. fl
1
. . .
<3.02Gk;
To estimate the bound k2, let 52, = { x E R2 I V(x) 5 c). For any positive constant
c: the set 52, is closed and bounded. The boundary of 52, is the Lyapunov surface
The largest value of Ix21 on t.he surface V ( r ) = c can be determined by differentiating
the surface equat.ioxl partially with respcct to xl. This results in
Therefore, the ext,reme values of 1 2 are obtained at the interscctioxl of tllc lilic
xl = -22112 with the Lyapunov surface. Simple calculations show that the largest
value of xif on the Lyapunov surface is 9Gc/29. Thus, all points inside 52, satisfy
t.he bound
96c
1x2( kZ, wllere kg = 29
<
Therefore, if
.
,
ir(r) will be ~legatisedefinite in R, and wc call coliclude that the origin x = 0
is exponentially stable with R, as an estimate of the region of attraction. The
inequality /3 < O.l/c shows a tradeoff between the estimate of the region of attraction
and the estimate of the upper bound on P. The smaller the upper bound on 0 , the
larger the estimate of the region of attraction. This tradeoff is not artificial; it does
exist in this example. The change of variables
.
,..
. .___l_ll._...^_.
'
'
5
--1
: L .!!A,, * , .
9.1. VANISHING PERTURBATlON
.,..
_
-
I
.."
.- -
f
. ...
345
When the origin of the iloluinal systeiu (9.2) is uiliforluly asymptotically stable,
but not exponentially stable, the stability analysis of the perturbed system is more
involved. Suppose the nominal system has a positive definite, decrescent Lyapunov
function V(t, x) that satisfies
for all (t, x) E [0, oo) x D, where W3(x) is positive definite and continuous. The
derivative of V along the trajectories of (9.1) is given by
transforms the state equation into
Our task now is to show that
which was shown in Example 8.5 to have a bounded region of attraction surrounded
by an instable limit cycle. When transformed into the x-coordinates, the region
of attraction will expand with decreasing and shrink with increasing P. Finally,
let us use this example to illustrate our remarks on the conservative nature of the
bound of (9.7). Using this bound, we came up with the inequality0 < 1/3.026k;,
which allows the perturbation term g(t, x) to be any second-order vector that satisfies 11g(t,~)11~
< Pk2211~11~.This class of perturbations is more general than the
perturbation we have in this specific problem. We have a structured perturbation in
the sense that the first component of g is always zero, while our analysis allowed
for an unstructured perturbation where the vector g could change in all directions.
Such disregard of the structure of the perturbation will, in general, lead to conservative bounds. Suppose we repeat t e analysis, this time taking into consideration
the structure of the perturbation. In tead of using the general bound of (9.7), we
calculate the derivative of V(t, x ) along the trajectories of the perturbed system to
obtain
'a
I
V(X) = -11x11;
= -11x11;
,.
t: i-
3
k"'
i
;
CJ
b-
+ 2xTPg(x)
+ 2Px5 ( Q x ~ x+~Ax;)
5 -llxll; + 2Px; ( ~ l l x l l ;+ &llxll;)
5 -11~1122 + ~ ~ ~ ; 1 1 ~ 1 1 2 2
Hence, V(X) is negative definit.e for B < 4/3k;. Using, again, the fact that for all
x E R,, (x2I25 ki = 96~129,we arrive at the bound < 0.4/c, which is four times
A
the bound we obtained by using (9.7).
for all (t,x) E (0,oo) x D, a task that cannot be done by putting a simple order
of magnitude bound on IJg(t,x)JI,as we have done in the exponential stability case.
The growth bound on ((g(t,x)JJwill depend on the nature of the Lyapunov function
of the nominal system. One class of Lyapunov functions for which the analysis
is almost as simple as in exponential stability ip the case when V(t,x) is positive
definite, decrescent, and satisfies
..
2
. .-
av + -f
sv
a t . BX
(t,2) < - ~ 3 4 ~ ( 2 )
for all (t, x) E (0,oo) x D for some positive constants c3 and c4, where 4 : Rn -r R
is positive definite and continuous. A Lyapunov function satisfying (9.8) and (9.9)
is usually called a quadratic-type Lyapunov function. It is clear that a Lyapunov
function satisfying (9.3) through (9.5) is quadratic type, but a quadratic-type Lyapunov function may exist even when the origin is not exponentially stable. We
will illustrate this point shortly by an example. If the nominal system (9.2) has a
quadratic-type Lyapunov function V(t, x), then its derivative along the trajectories
of (9.1) satisfies
i.(t, 2) I- ~ 3 4 ~ ( 2+) c44(x)llg(t, x)ll
Suppose now that the perturbation term satisfies the bound
I
346
CHAPTER 9. STADILITY OF PERTURBED SlSTEAIS
347
9.2. NONVANlSHlNG PERTURBATIOK
Then,
,-
Lemma 9.2 Let s = 0 be an exponen.tially stlblble equilibrium point of the nominal
system (9.2). Let V ( t ,x ) be a L?japl~n,olr
fim.ction of the nonrinal system that sata.$es
(9.3) through (9.5) in (0, oo) x D, where D:= { x E R" 1 ))I))
< r } . Suppose the
perturbation t e r n g(t, x) satisfies
.
~ ( xt ), 5 -(c3 - c47)d2(x)
which slio\vs that ~ ( .r)
t , is negative dcfiiiite.
/-----.
Example 9.3 Consider the scalar system
The nominal system
1)
> 0 , all x E D. and some positive constant 0 < 1. Then, for all Ilx(to)ll <
GT,
the solution x ( t ) of the pertui.bed system (9.1) satisJcs
for all t
= -z3
has a globally asymptotically stable equilibriuin point at tlie origin. but., as we s a w
in Example 4.23, the origin is riot exponeiit,i~llystable. Tlius, there is no Lyapuiiov
function that satisfies (9.3) tlirough (9.5). The Lyapunov function V(x) = z4
satisfies (9.8) and (9.9), wit11 4 ( x ) = lrI3! c3 = 4, and cd = 4. Suppose tllc
perturbation term g ( t , x ) satisfies the bound Jg(t,x)l 5 y1xI3 for all x, with 7 < 1.
Then, the derivative of V along the trajectories of the perturbed system satisfies
Hence, the origin is a globally uniformly asymptotically stable equilibrium point of
the perturbed system.
A
In contrast to the case of exponential stability, it is important to iiotice tllot. a
nominal system with uniformly asyinptotically stable, but not expolientially stable,
origin is not robust to smooth perturbations with arbitrarily small lii~cargro\vtll
bounds of tlie form of (9.G). This point is illustrated by t.he next e ~ a m p l e . ~
and
6
1
for some finite T , where
.
0
I
B
1
Proof: X'e use V ( t .t) as a Lyapunov function candidate for the perturbed system
(9.1). The derivative of V ( t , x ) along the trajectories of (9.1) satisfies
Example 9.4 Consider the s c n l ~ system
r
of tlie previous example with perrurbation g = yx where 7 > 0; that is,
It can be easily seen, via linearization, that for any 7 > 0 the origin is unstable, no
matter how small r is.
A
19.2
Nonvanishing Perturbation
Let us turn now to the more general case wheu we do not know that g(t,0 ) = 0.
The origin x = 0 may not be an equilibrium point of the perturbed system (9.1).
We can no longer study stability of the origin as an equilibrium point, nor should
we expect the solution of the perturbed system to approacli the origin as t -i oc.
The best we can hope for is that r ( t )will be ulti~nat~ely
bounded by a sillall bound.
if the perturbation term g ( t , x ) is small in some sense. We start with the case when
the origin of the nominal system (9,2) is exponentially stable.
3 ~ walso,
, Excrciar 9.7.
Applying Theorem 4.18 and Exercise 4.51 completes the proof.
N0t.e that the ultimate bound b in Lemma 9.2 is proportional to t.he upper
bound on the perturbation 6. Once again, tliis result can be viewed as a robustllcss
property of iiominal systems having espol~efially stable equilibria at the origin,
because it shows that arbitrarily small (uniformly bounded) pert.url)i~tionswill not
result in large steady-state deviations from the origin.
Example 9.5 Consider the second-order system
where 0 2 0 is unkno~vnand d ( t ) is a uniformly bounded disturbance that satisfies
Id(t)l 5 6 for all t 2 0. This is the same system we studied in Example 9.2, except
.-
.
- --
-
.
-
--
-..-.
CHAPTER 9. STABILITY OF PERTURBED SYSTEAIS
.
/
348
:
r-.
-.--. .
'
, ,
..
,..
.
"
'
"
$
.
.
_.,..I....
....., ; L..!..., . * + . .
9.2. NONVANISIIING PERTURBATION
. -.
. . . .,
-- . .
,
*\._:
:-
..--
.,
;
-.
< , . .
:"&-
.-q
"..," ..,<.:1
3119
*aI
for the additional perturbatiol~terin d(t). Again, the system can be viewed as
a ~erturbationof a nominal linear system that has a Lyapunov function V(x) =
X?PX, where
in [O,oo) x D, where D = {x E Rn ( 11x11 < r) and ai(.), i = 1,2,3,4, are class K
functions. Suppose the perturbation term g(t, x) satisfies the uniform bound
We use If(x) as a Lyapunov function candidate for the perturbed system, but
we treat the two perturbation terms pxi and d(t) differently, since the first term
vanishes at the origin while the second one does not. Calculating the derivative of
V(x) along the trajectories of the perturbed system, we obtain
and
where we have used the inequality
+
12x1 5x21 5 11x112and k2 is an upper bound on 1x21. Suppose
Then,
v t xIX
I
m 6
for all t 2 0 , all x E D , and some positive constant 0 < 1. Then, for all 11x(to)ll <
a;'(al(r)), the solution x(t) of the perturbed system (9.1) satisfies
0 I 4(1 - ~)/3k;, where 0 < C < 1.
+I I I I I - 1 -I
Ilx(t)ll I P@), v t 2 to + T
for some class KL function p and some finite T , where p is a class K function of
d defined by
p(6) = a;'
l
(a2
(a;'
836
l v lIxII2 2 P = -
(y
)))
0
where 0 < 0 < 1. As we saw ill Example 9.2, 1x2I2 is bounded on Cl, by 96~129.
Thus, if 0 5 0.4(1- C)/c and d is so small that p2X,,(p)
< c, then B, c Cl, and
all trajectories starting inside Cl, remain for all future time in R,. Furthermore,
the conditions of Theorem 4.18 are satisfied in Cl,. Therefore, the solutions of the
perturbed system are uniformly ulti~natelybounded by
Proof: We use V(t, x) as a Lyapunov function candidate for the perturbed system
(9.1). The derivative of V(t, x) along the trajectories of (9.1) satisfies
In the more general case when the origin x = 0 is a uniformly asymptotically
stable equilibrium point of &thenominal system (9.2), rather than exponentially
stable, the analysis of the perturbed system proceeds in a similar manner.
Applying Theorem 4.18 completes the proof.
L e m m a 9.3' Let x = 0 be a unifo~mlyasymptotically stable equilibrium point of the
nominal system (9.2). Let V(t,x) be a Lyapunov function of the nominal system
that satisfies the inequalities4
al(llxll) I v ( t ! x ) 5 az(llxll)
(9.11)
"he existence of a Lyapunov function satisfying these inequalities (on a bounded domain) is
guaranteed by Theorem 4.16 under somc additional assumptions.
This lemma is similar to the one we arrived at in the special case of exponential stability. However, there is an important feature of our analysis in the case of
exponential stability, which has no counterpart in the more general case of uniform
asymptotic stability. In the case of exponential stability, d is required to satisfy
(9.10). The right-hand side of (9.10) approaches oo as r -+ oo. Therefore, if the
assumptions hold globally, we can conclude that for all uniformly bounded disturbances, the solution of the perturbed system will be uniformly bounded. This is the
case because, for any 6, we can choose r large enough to satisfy (9.10). In the case
1
i
350
CHAPTER 9. STXBILITI' OF PERTURBED SI'TEAlS
of uniform asymptotic stability, 6 is required to satisfy (9.14). Iilspcctio~lof (9.14)
shows that, without further informati011 about the c l ~ s sK functions, we cannot
say anything about the limit of the right-hand side as r -+ oo. Thus, we cannot
conclude that uniformly bounded pertorbations of n nominal system 1vit.h a uniformly asymptotically stable equilibrium at the origin will have bounded solutions
irrespective of the size of the perturbation. Of course the fact that we cannot show
it, does not mean it is not true. It turns out., however, that such a st.ntenient is
not true. It is possible to construct examples (Exercise 9.13) where the origin is
globally uniformly asymptoticellp stable, but a bounded perturbation could drive
the solution of the perturbed system to infinity.
b.3
Comparison ~ e t h o d ]
Consider the pert,urbed system (0.1). Let V(t, x) be a Lyapunov filnct.ion for t.he
nominnl system (9.2) and suppose the tlerivative of V along the t,rajectories of (9.1)
satisfies the differential inequality
E@
9.3. COAIPARISON METHOD
I
Using (9.3), we can find an upper boulltl on
v as
w
t
dt
I
To obtain a linear differential inequality. we take W(t) =
fact lif = ~ / 2 f l .when V # 0. to obtain
and use the
When \/ = 0. it can be showllS that D+IV(t) 5 c46(t)/2fi. Hence, D+\V(t)
satisfies (9.17) for all values of V. By the comparison lemrna, W(t) satisfies tllc
inequality
\.\f(t)
< 4(t.to)lV(to)+
& 1;
!
where the transitioll function d(ttto) is given by
1
Using (9.3) in (9.18), we obtain
I
Suppose now that 7(t) satisfies t,lie contlit.ion
4(t17)6(7)dr
By (the comparison) Lemma 3.4,
where y(t) is the solution of t.he differential equat.ion
This approacli is particularly useful when the differential ir~cqualit~y
is linear, that is,
when h(t, V) = a(t)V b(t). for then we can write down a closed-form expression
for the solution of the first-order linear differential equation of y. Arriving at. a
linear differential inequality is possible when the origin of the nominal system (9.2)
is exponentially stable.
Let V(t,x) be a Lyapullor function of the nornilla1 system (0.2) that satisfies
(9.3) through (9.5) for all (t..r) E [ O , x ) x D, whcrc D = {T E Rn 1 llrll < r ) .
Suppose the perturbation term g(t, x) satisfies the bound
+
where y : R
R is ilorlneg~t,iveant1 contiriuous for nll t 2 O? and 6 : R -+ R is
nonnegative, continuous, and bounded for all t 2 0. The derivative of V along the
trajectories of (9.1) satisfies
-+
.V(t,x)
-av
av
-fax
at
+
=
av-
( t , ~+) zg(t,x)i
t
1 7 ( 7 ) d r 5 ~ (-t to)
I
for some nonnegati\re const.ants E anrl q. where
I
Defining the constants a and p by
t
and using (9.20) and (9.21) in (9.19). we obtain
"~ee
Exercise 9.14.
+n
(9.20)
7
72
7
3
-
.- --
---
+---
. .
/
CHAM'ER 9. STABILITY OF PERTURBED SYSTEAlS
352
.
For this bound to be valid, we must ensure that IJx(t)lJ< r for all t
thatG
9.3. COhlPARISON METHOD
> to. Noting
Lemma 9.5
1. If
then (9.20) is satisfied with E = 0 and q = k .
3
we see that the condition I(x(t)ll < r will be satisfied if
I
> 0 , there is q = q ( ~>) 0 such that (9.20) is satisfied.
3. If there are constants A > 0 , T 2 0 , and €1 > 0 such that
then for any E
and
2clar
sup 6 ( t ) < -,>to
c4 P
For easy refel.ence, we summarize our findings in the next lemma.
then (9.20) is satisfied with
Lemma 9.4 Let x = 0 be an ezporientially stable equilibrium point of the nominal
E
+
= E I and q = E I A JoTy(t) dt.
I
0
of the nominal svstem that satisfies
system (9.3). Let V ( t ,x ) be a Lynp~lnor~frmction
(9.3) through (9.5) i n [0,ao) x D , where D = { x E Rn I llxl12 < T ) . Suppose
the perturbation term g ( t , x ) satisfies (9.15), where y ( t ) satisfies (9.20) and (9.21).
Then, provided x ( t o ) satisfies (9.24) and sup,,,,, 6 ( t ) satisfies (9.25), the solution of
the perturbed system (9.1) satisfies (9.23). bh>hermore, if all the assumptions hold
0
globally, then (9.23) is satisfied for any .r(to) and any bounded 6 ( t ) .
Proof: The first case is obvious. To prove the second case, note that, because
limt+rn y ( t ) = 0 , for any E > 0, there is TI = T I ( € )> 0 such that y ( t ) < E for all
t 2 TI. Let q =
y ( t ) dt. If to T I ,then
Specializing the foregoing lemma to the casc of vanishing perturbations; that is,
when 6 ( t ) E-0, we obtain the following result:
If t
Corollary 9.1 Let x = 0 be an ezponentially stable equilibrium point of the nominal
system (9.2). Let V ( t , x ) be a Lyapunolrfunction of the nominal system that satisfies
(9.3) through (9.5) i n [O, ao) x D . Suppose the perturbation term g(t, x ) satisfies
If to < T I
11g(t,4
11 1 r(t)llxll
where y ( t ) sntisfies (9.20) and (9.21). Then, the origin is an ezponentially stable
equilibrium point of the perturbed system (9.1). Moreover, if all the assumptions
0
Fold globally, then the origin is globnlly ezponentially stable.
If y ( t ) = y = constant, then Corollary 9.1 requires y to satisfy the bound
y < c1c3/c2c4,which has no advnntage over the bound y < ca/c4 required by
Lemma 9.1, since (cl/c2) 1. In fact, whenever ( c l / c 2 ) < 1, the current bound
will be more conservative (that is, smaller) than the bound required by Lemma 9.1.
The advantage of Corollary 9.1 is seen ill the case when the integral of y ( t ) satisfies
conditions (9.20) and (9.21), even when sup,,,,, y ( t ) is not small enough to satisfy
sup,,,,,
- y ( t ) < c3/c4. Three such cases arc giGn in the next lemma.
>
J,T'
'6.
l o t y ( T )d7
<
lo
t
E
d7 = ~ (-tto)
< T I ,then
< t, then
In the last case, if t 5 T , then
rt
rT
<
+
w e use the fact that the function ue-''I
b ( l - e - I " ) , witli positive a, b, and a, relaxes
monoton~callyfrom its initial value a to its filial value b. Hence, it is bounded by the maximum
of the two ~ ~ l i n ~ l ~ c r s .
For t
1 tl 2 T , let N be the integer for which ( N - l ) A 5 t - tl 5 N A . Then,
354
355
9.4. CONTINUITY OF SOLUTIONS
CHAPTER 9. STABILITY OF PERTURBED SYSTEAB
Lemma 9.6 Suppose the conditions of Lemma 9.4 are satisfied, and let ~ ( tdenote
)
the solution of the perturbed system (9.1).
This inequality is used next with ti = to when t 2 t o 2 T ?and with t l = T when
_< T I t. If t _> to 2 T. then
to
1.
If
e-4'-')6(r) d~ 5 P, V t 2 to
while if to 5 T j t , then
a, tho^ .r(t) is uniformly ultintutely bounded wit11
for soirlr positive coilstal~t
the ~rltimatebound
b = - c4pP
-2~~e
where 0 E (O? 1 ) is an arbiti'ary constant.
In the first case of the foregoing lemma, condition (9.20) is satisfied wit11 E = 0 ,
while in the second case. it is satisfied with arbitrarily small E. Therefore. in both
cases, condition (9.21) is always satisfied and the origin of the pertllrhed system
(9.2) is exponentially stable. The third case of the lemma sets a bound on a moving
average of 7 ( t ) as t becomes sufficiently Inrge. The o4gin of the perturb~dsystem
(9.2) will be exponentially stable if this bound is sufficiently small.
Example 9.6 Consider the'linear system
1!
then x ( t ) is uniformly ultimately bounded with the ultimate bound
b e -c4pbm ,
2nc1e :
!
r
where 8 E (0.1)is an arbitrary conatnnt.
I
lim 6(1) = 0 ,
i-u
n l ~ c wr l ( t ) nnd B ( t ) are c~olitinuousand A ( t ) is bo~ultlctlon (0,cc). Suppose thc
origin is all exponelltially stable equilibrium point of the nominal system
I
i
From Theorem 4.12, we know that there is a quadratic Lyapunov function V ( t .x ) =
x T P ( t ) x that satisfies (9.3) through (9.5) globally. The perturbation ten11 S ( t ) r
satisfies the inequality
IlB(t)xll IlB(t)ll Ilxll
Since ( ( B ( t ) (4
( 0 as t -, w, we conclude from Corollary 9.1 and the second case
of Lemma 9.5 that the origill is a glohnllp expol~cliti~lly
stablc eqllilibriun point of
the perturbed system.
A
M
(Exercise 9.16) and
In the case of nonvanishing perturbations, that is, when b ( t ) $ 0. the next
lemma states a number of conclusions concerning the asymptotic behavior of x ( t )
w t f --.,
V the conditions
of Lemma 9.4 are s~rtiqficd
hold for any initial state x(t0).
then the foregoing statente?its
0
,
aiid li111~,~~ ( 1=) wrn: thcn limt,,
s
S
r
lim ~ ( t=) 0
t-m
Proof: All three cases follow easily from inequality (9.23). In the 'first two cases,
we use the property t,hat if u ( t ) = ~ ( t+)a with a > 0 and liml,, w ( t ) = 0 , then
u ( l ) is ullill~atclybounded by.a/8 for auy pusitiw 0 < 1. This is su because there
is a finite t i ~ n cT sucl~that ( w ( t ) (5 a(1 - B)/Q for all t 2 T.In tho last tw! cases,
w
tho property that if u ( t ) = ee\p(-a(t - T ) ) w ( T ) d~ where ul(t) is bounded
and
Similar conclusiorn can be drawn when :S IIB(t)ll dt <
11B(t)l12dt < w (Exercise 9.16).
~~I('II
u ( t ) - v!;/n.7
0
9.4 Continuity of Solutions on the Infinite Interval
!I
t
I
In Section 3.2, we studied continuous dependence of the solution of the state equation on init.ia1 states and parameters. h particular, in Theorem 3.1, we examined
the nominal system
j.= f ( t , x )
(9.26)
7Sec (33. Theorem 3.3.2.333.
. . .
,
.
-.
-.. .
--..A
'
.d*-.,
'
2
d
f
.
*
9.4. CONTINUITY OF SOLUTIONS
357
i
and t.he pert.urbed system
xi.=f ( t , x )+ g ( t , x )
(9.27)
under the assumption that 11g(t,x)ll 5 b in the domain of interest. Using the
Gronwall-Bellman inequality, we found that if y(t) and z ( t ) are well-defined solutions of the nominal and perturbed systems, respectively, then
-
'. I
-.,
..A
L-,
J
r',,
i
t.,
where L is a Lipschitz constant for f . This bound is valid only on compact time
intervals, since the exponential term exp[L(t - to)]grows unbounded as t -+ cm. In
fact, the bound is useful only on an interval [to,t l ] where t l is reasonably small, for
if t l is large, the bound will be too large to be of any use. This is not surprising,
because in Section 3.2, we did not impose any stability conditions on the system.
In this section, we use Lemma 9.4 to calculate a bound on the error between the
solutions of (9.26) and (9.27) that is valid uniformly in t for all t 2 to.
Theorem 9.1 Let D 1 c Rn be domain that contains the origin and suppose
f ( t ,x ) and its first partial det-ivntives with respect to x are continuous, bounded,
and Lipschitz in x , uniformly in t , for all ( t ,x ) E [0,cm) x Do, for every compact set Do C D;
g(t, x ) is piecewise continuous i n t , locally Lipschitz i n x , and
of the set 0. When tlie ~lonli~lal
systeln (9.26) is autonomous, the function V is
provided by (the converse Lyapunov) Theorem 4.17 and the set S-2 can be any compact subset of the region of attraction. Exponential stability is used only locdy
when the error z ( t ) - y(t) is sufficiently small.
if
1
<
This shows that v is negative on V ( t ,x ) = c; hence the set { V ( t ,x ) 5 c} is positively
invariant. Therefore, for all t ( t 0 ) E { W ~ ( X ) c), the solution z ( t ) of (9.27) is
>0
uniformly bounded. Since S-2 is, in the interior of { W 2 ( x ) c), there ie
such that at01 E { W z ( x ) c) +enever y(t0) E 0 and (Iz(to)- y(to)))5 pl. I t is
also clear that for y(t0) E a, y(t) is uniformly bounded and y(t)
0 as t
oo,
uniformly in to. The error e(t) = z ( t ) - y(t) satisfies the equation
<
<
<
+ g(t, z ) .-
where
- -
f (t,y) = f (t,e ) + A ( t , e) + g(t, z )
(9.31)
-
A ( t , e ) = f ( t ,YO) + e ) - f ( 4y ( t ) ) f ( t ,e )
We analyze the error equation (9.31) over the ball {Ilell I r ) c D. Equation (9.31)
can be viewed as a perturbation of the system
Let y(t) and z ( t ) denote solutions of the nominal system (9.26) and the perturbed
system (9.27), respectively. Tlien, for each compact set R C {lVz(x) 5 pc, 0 <
p < 11, there emkt positive constants p, 7,q, p, and k , independent of 6, such that
if y(to) E R, 6 < q, and (Iz(to)- y(to)(l < p , the solutions y(t) and z ( t ) will be
uniformly bounded for all t .? to 2 0 and
whose origin is exponentially stable. By Theorem 4.14, there exists a Lyapunov
function V ( t ,e) that satisfies (9.3) through (9.5) for llell < ro < r. By the mean
value theorem, the error term Ai can be written as
While the origin is exponentially st,able, the Lyapunov function V is required to
satisfy the conditions of uniform tlsyrnptotic stability, rather than (the more stringent) co~lclitionsof exponential stsbi1it.y. This provides less conservative estimates
!i
for aU, x , {Wl(x).-s,.~}.,
~
.where kl is an upper bound on BV/ax over { W l ( x )
c}. Let k2 > 0 be the minimum of W3(x) omr the compact set A = { W l ( x ) 5
c and W2( x ) 2 c). Then
there is a Lyapunov fut~ction 17(t,x) that satisfies the conditions of Theorem 4.9 for the nominal system (9.26) for ( t ,x ) E [O,oc) x D and {lVl ( x ) 5 c}
is a compact subset of D.
\
!
Proof of Theorem 9.1: The derivative of If along the trajectories of the perturbed system (9.27) satisfies
k = 5 - Y = f (t!2 )
.the origin x = 0 is an exponentially stable equilibrium point of the nominal
sl~stem(9.26);
!
':.i
.
'
.
,.
where 0 < A, < 1. Since the Jacobian matrix [ a f / a x ]is Lipschitz in x, uniformly
in t, the perturbation term ( A g) satisfies
+
IlA(t1e) + g(t,t)ll 5 L1llelI2 + L2llell ll~(t)ll+6
I
:
358
-
CHAPTER 9. STABILITY OF PERTURBED Sl'STEAlS
where y(t)
-+
0 as t
-+
x. uniformly in to. Consequently.
IlA(t, e) + g(t, 2)ll I {Llrl + Lzll~(t)ll)llell + 6
for all llell
I r~ < ro.
This inequality takes the forin (9.15) with
+
~ ( t =) {LIT] L2Ily(t)(l}
and
6(t) r 6
€1
Given any > 0, there is Tl > 0 such that Ily(t)ll 5 c1 for a11 t 2 to+Tl. Therefore,
(9.20) is satisfied with
lo
1
'(7) d~ 5 (€1 + L I ~ I )-(to)
~ + T I maxLzllu(t)Il
tall
9.5. INTERCONNECTED Sl'STEAIS
each isolat.ed subsystem. Suppose this scnrcl~has' been successful and tliat, for cacli
subsystem, we have a positivc dcti~iitc!tlccrcsccnt Lyapunov function V,(t, xi) whosc
derivative along the trajectories of the isolated subsystem (9.33) is negative definite.
The function
m
\ / ( t , r ) = xi= dI i ~ , ( t , x i ) , di > 0
is a composite Lyapzrnov function for the collection of the m isolated subsystems
for all values of the positive constants d,. Viewing the interconncctcd system (9.32)
as a perturbation of the isolated subsystems (9.33), it is reasonable to try V(t,x)
as a Lyapunov function candidate for (9.32). The derivative of V(t,x) along the
trajectories of (9.32) is given by
By taking €1 and rl small enough, we can satisfy (9.21). Thus, all the assumpt.ions
of Lemma 9.4 are satisfied and (9.30) follows from (9.23).
9.5
Interconnected Systems
bpbd*
~1st-
When we analyze the stability of a nonlinear dynainical system, the complexity
of the analysis grows rapidly as the order of the system increases. This situat.ion
motivates us to look for ways to simplify the analysis. If the system can be modeled
as an interconnection of lower order subsystems, then we ]nay pursue the st.abi1it.y
analysis in two steps. 111 the first step, we tlccornl~oset l ~ esyst.cin illto srrlnllcr
isolnted siibsystenls by igl~oril~g
interconnections, and analyze the st.ability'of each
from the first step with
subsystem. In the second step, we combine our co~~clusions
information about the int.erconnections to draw conclusions about the st.abi1it.y of
the interconnected system. In this section, we illustrate how this idea can he utilized
in searching for Lyapunov functions for interconnected systems.
Consider the interconnected system
+
where xi E Rn',n l ... + n, = n, nntl x = [r:?. .. ,I:] T . Suppose fi aild gi
are smooth enough to ensure local existence ant1 uniqueness of the solution for all
initial conditions in a domain of interest, and that
fi(t,O) = 0,
gi(t,O) = 0,
The first term on the right-hand side is negative definite by virtue of the fact that
C
& is a Lyapunov function for the ith isolated subsystem, but the second term is, in
IL
general, indefinite. The situation is si~nilarto our earlier investigation of perturbed
systems in Section 9.1. Therefore, 1ve inay approach the problem by performing
worst cask analysis where the term [al/,/a.r,Ig, is bounded by a nonnegative upper
bound. Let us illustrate the idea hy using quadratic-type Lyapunov functions,
introduced in Section 9.1. Suppose that. for i = 1,2,. .. , m , &(t, r,) satisfies
for all t 2 0 and 11x11 < r for so~ncpositive constants ai and Pi, whcre di : Rni + R
are positive definite and continuous. Furthermore, suppose that tlie interconncctiorl
terms gi(t, x) satisfy the hound
m
<
JIgi(t,x)II j=
x T1i j d j ( x j )
(9.3G)
Vi
so that the origin x = 0 is an equilibrium point of the system. Ignoring the interconnection terms gi, the system decomposes into m isolated subsystems:
with ench one having an equilibrium point at its origin xi = 0. i r e start. by searching
i1syi111)toticstnbility of tllr origiii for
for Lynp~lllovfu~lctionst Ililt cstnl)lisl~u11ifo1.111
for all t 2 0 and JJxJJ
< r for some nonnegative constants yij. Then? the derivative
of V(t,x) = CEl di&(t,xi) along t . 1 ~t.rajectories of the interconnected system
(9.32) satisfies the inequality
L
L
C
c:
. . . . . . . . . . . . .
--
.'
.
3GO
..
.
.... -. ..
.
.
. . . . . .- :. ...
-
.......
STD)d
where
d = [ ~ l l . . . , ~ m ] T l D = d i a g ( d l , . . .,dm)
and S is an m x m matrix whoseelements are defined by
If there is a positive diagonal matrix D such that
then ~ ( x)
t , is negative definite. since d(x) = 0 if and only if x = 0; recall that
di(xi) is a ~~ositive
definite fuiictioii of xi. Tlius, a si~ficientcolldition for u~iiforiil
asymptotic stability of the origin as an equilibrium point of the interconllected system is the existence of a positive diagonal matrix D such that DS+STD is positive
definite. The matrix S is spccial in that its off-diagonal elements are nonpositive.
The next lemma applies to this class of matrices.
Lemma 9.7 There exists a positive diagonal matriz D such that D S + STD is
positive definite if and only if S is an M-matrix; that is, the leading principal
minors of S are positive:
$.
,
-----
d . .
I
I
. . . . . . . . .
361
Theorem 9.2 Coi~sider.the systeiu (9.3'2) urid si~pposeUheia a~ positiue definite
decrescent Lyapunovfvnctions K ( t , x i ) that satisfy (9.34) and (9.35) and that gi(tlz)
satisfies (9.36) for all t 2 0 and llxll < r . Suppose the matrix S defined by (9.37) is
an kl-matrix. Then, the origin is uniformly asymptotically stable. Moreover, if all
the assumptions hold globally and K ( t , x i ) are radially unbounded, it will be globally
uniformly asymptotically stable.
0
The right-hand side is a quadratic form in d l , . .. , d m , which we rewrite as
- ~$(Ds+
:
.;L..!*,.,.,
9.6. INTERCONNECTED SYSTEhIS
CHAPTER 9. STABILITY OF PERTURBED SYSTEhfS
~ ( xt ), 5
.
.. ....,,
r>r
I
Example 9.7 Consider the second-order system
i
The system can be represented in the form (9.32) with
t
fi(x1) = - 2 1 ,
g 1 ( ~=
) -1.5~:x,3,
fz(x2) = - x i ,
3
and g2(x)= 0 . 5 ~ : ~ ;
The first isolated subsystem X I = -XI has a Lyapunov function K ( x l ) = x:/2,
which satisfies
sv1
-f1(x1)
= -x: = -ald;(xl)
ax1
where ct1 = 1 and dl(x1) = 1x11. The second isolated subsystem x2 = -xi has a
Lyapunov function h ( x 2 ) = ~ 4 1 4which
,
satisfies
i
I
.A
1
1
where cr2 = 1 and h ( x 2 ) = )x2I3. The Lyapunov functions satisfy (9.35) with
= 1. The interconnection term g l ( x ) satisfies the inequality
P1 =
Proof: See [57].
The.Af-matrix condition can be interpreted as a requirement that the diagonal
elements of S be "larger as a whole" than the off-diagonal elements. It can be seen
(Exercise 9.22) that diagonally dominant matrices with nonpositive off-diagonal
elements are M-matrices. The diagonal elements of S are measures of the "degree of
stability" for the isolated subsystems in the sense that the constant cti gives a lower
bound on the rate of decrease of the Lyapunov function with respect to &(xi).
The off-diagonal elements of S reprtxent the "strength of the interconnections" in
the sense that they give an upper hound on g i ( t , x ) with respect to $ j ( z j ) for j =
1,....m. Thus, the M-matrix coiidit~ionsays that if the degrees of stability for the
isolatc(l subsystems are larger ns n ~rrllolethan the strength of the interconnections,
then the interconnected system has a uniformly asymptotically stable equilibrium at
the o7Qin. We summarize our conclusion in the next theorem.
)
the inequality
for all 1x11 5 cl. The interconnection term g ~ ( xsatisfies
<
Igz(X)I = 0 . 5 ~ : ~ ; O . ~ C I & I
for all 1x11
(XI)
:
< cl and 1x21 5 ~ 2 Thus,
.
if we restrict our attention to the set
we can conclude that the interconnection terms satisfy (9.36) with
711 = OI 712 = 1 . 5 ~ : ~721 = 0 . 5 ~ ~ ~ 2and
2~
The matrix
rz2= 0
s
362
C H A P T E R 9. STABILIT't' OF P E R T U R B E D S Y S T E M S
363
9.5. I N T E R C O N N E C T E D S Y S T E h l S
is an A{-matrix if 0.75~:~; < 1. This will bc the case, for csamplc, when cl =
cz = 1. Thus, the origin is asymptotically stable. If we are interested in est.imnting
the region of attraction? we need t.0 k n o ~the
~~
col~lpositeLyapu~lovfu~lctionI/ =
dlV; dzV;; that is. nre need to ~ I I O I Va positive diagonal matrix D such t,hat
DS STD > 0. Taking cl = c2 = 1: we have
+
+
which is positive definite for 1 < d2/dl < 9. Siiice there is 110 loss of generality in
multiplying a Lyapunov function by a positive const.ant., we take dl = 1 and write
the composite Lyapunov function as
An estimate of the region of attraction is given by
Figure 9.1: The sector nonlinearity qi(xi) of Example 9.8.
<
where c
min {1/2, d2/4} to ensure that R, is il~siclcthe rectangle ( . E ~ ( 5 1.
Noting that the surface V(x) = c iiltcrsccts the rl-asis and the x2-axis at 6
and (4c/d2)'I4, respectively. we max~mizethese distances by choosing d2 = 2 and
c = 0.5.
A
..
Example 9.8 The lnr~tl~eillatical
nlotlol of NI nrtificinl neural rletwork was pro
sentcd in Section 1.2.5, ~ I I its
I ~ sbability properties were analyzed in Example 4.11
by using LaSalle's invariance principle. A key assumpt,ion in Example 4.11 is the
symmetry requirement Tij = T j i , which allows us t,o represent. the right,-hand side
of the state equation as tihe gradient of n scnlnr function. Let us relax this rrquirement and allow Tij# Tji.
We will analyze the stability properties of the net\vork
by viewing it as an intercoliiiect~ionof subsystems; each subsystem corresponds to
one neuron. We find it convenient here to work with t,he voltages.at t,l~eamplifier
inputs ui. The equations of motion are
To analyze the stability properties of a given equilibrium point u*, wesl~iftit to
the origin. Let xi = ui - ul. Then:
where
vi(ri) = gi(ai
+ u;) - g i ( ~ ; )
Assume that qi(.) satisfies the sector condition
u2kil _< uqj(u) 5 u2ki2, for u E I-~i,ri]
for i = 1,2,. . . ,n, where gi(.) are sigmoid functions, Ii are constant current inputs,
Ri > 0, and Ci > 0. We assume that the system has a finite number of isolated
equilibrium point,s. Each equilibrium point u' satisfies the equation
where k,] and ki2 are positive c o ~ s t a ~ l t Figure
s.
9.1 shows that such co~ldit~io~l
is
indeed satisfied when gi(ui) .= (2l/n,/T) tan-' (X~ui/2Vj\,),X > 0. tVe can recast
this system in'the form (9.32) with
Using
,
.. _ .
.
-
.
364
'
.
~
.
.
-
. . ..-. ,..,
.
. .
. . - ... .-.
_,LC.
CIII\PTE~~.
!I. STABILITY OF PERTURBED SYSTEMS
365
9.6. SLO\VI,Y VARYING SI'STEhlS
?
as a Lyapunov function candidate for the ith isolated subsystem, we obtain
If T,i 5 0, then
where c 5 0.5mini{diCir?) to ensure that R, is inside the set lxil -< ri. This analysis is repeated for each asymptotically stable equilibrium point. The conclusions
we could arrive at in this example are more conservative compared with the conclusions we arrived a t using LaSalle's invariance principle. First, the interconnection
coefficients Tij must be restricted to satisfy the M-matrix condition. Second, we
obtain only local estimates of the regions of attractions for the isolated equilibrium
points. The union of these estimates does not cover the whole domain of inte'rest.
On the other hand, we do not have to assume that Tij = Tji.
A
9.6
which is negative definite. If Tii > 0, then
-
I
Slowly Varying Systems
The system
s = f XI^)
and
In this casc, we assume that Tj~l;?~
< l/Ri, so that thc dcrivativc of V, is llcgativc
definite. To simplify the notation, let
Then, K(xi) satisfies (9.34) and (9.35) on the interval [-ri:ri] with
where cri is positive by assumption. The interconnection term gi(x) satisfies the
ineaualitv
Thus, gi(r) satisfies (9.36) with yii = 0 and yij = kj21Tijl/Ci for i # j. Now wc
can form the matrix S as
6i
+ l/R,,
-ITii152
where x E Rn and u(t) E r C Rm for all t > 0 is considered to be slowly varying if
u(t) is continuously differentiable and Ilu(t)(1 is "sufficiently" small. The components
of u(t) could be input variables or time-varying parameters. In the analysis of
(9.38), one usually treats u as a "frozen" parameter and assumes that for each
fixed u = a E r , the .frozen system has an isolated equilibrium point defined by
x = h(cr):.. If a property of a: = h(a) is uniform in h, then it i s reasonable tb
expect that the slowly varying system (9.38) will possess a similar property. The
underlying characteristic of such systems is that the motion caused by changes
of initial conditions is much faster than that caused by inputs or timevarying
parameters. In this section, we will see how Lyapunov stability can be used to
analyze slowly varying systems.
Suppose f (2, u) is locally Lipschitz on Rn x r, and for every u E r the equation
i
,
r
9
I
I
i
1
i
O=f(x,u)
has a continuously differentiable isolated root x = h(u); that is,
0 = f ( h ( ~ )U)
,
for i = j
for i # j
To analyze the stability properties of the frozen equilibrium point x = h(a), we
shift it to the origin via the change of variables z = x - h(a) to obtain the equation
The equilibrium point u* is asyinptotically stable if S is an Al-matrix. We may
estimate the region of attraction by t-he set.
Now we search for a Lyapunov function to show that z = 0 is asymptotically stable.
Since g(z, a ) depends on the parameter a , a Lyapunov function for the system may
!
StiU
CHAPTER 9. STABILITY OF PERTURBED SI'STEAlS
depend, in general. on a. Suppose we cnn fincl a Lyapullov fuilction V ( n ,a ) that
satisfies the conditions
dl'
z g ( z 7 a )5
-~311~11~
(9.42)
in which a1 and
pl
are defii~edI)!.
then z ( t ) sat.isfies the inequality
for all z E D = { z
RR 1 IJzIJ< r ) and a E r. where c,, i = 1,2.. .. , 5 are
positive constants independent of a. Inequalities (9.41) and (9.42) state the usual
requirements that V be positive definite and dccrescent nnd has a negative definite
derivative along the trajectories of the system (9.40). Furthermore. they show that
the origin z = 0 is exponentially stable. The special requirement here is that
these inequalities hold uniformly in a. Inequalities (9.43) and (9.44) are needed
to handle the perturbations of (9.40). which will result from the fact that u ( t ) is
not constant, but a time-varying function. \Vit,h V ( z ,u ) as a Lyapunov function
candidate, the analysis of (9.38) proceeds as follows: Tlie change of variables z =
z - h ( u ) transforms (9.38) into the form
Depending upoil tlie assuinpt ioiis for (1 i 11, several conclusiol~scall be drawn from
the foregoing inequality. Sonle of these coilclusions are stated in the next theorem.
Theorem 9.3 Consider the system (9.45). Suppose that [ah/8u] satisfies (9.39),
(lu(t)([5 E for all t 2 0. and there is a Lyapunou function V ( z , u ) that satisfies
(9.41) through (9.44). If
r
cic~
E < - X
czc;
r
+ cq L/c5
then for all J ( t ( 0I ( ) < r a , the sohriiails of (9.45) are unifolnlly 6ol~n.dcdfor 011
t 2 0 and unifoinlly ultiinately Do~~nded
6y
wlicre the effect,of the time variat.ion of u appears as a perturbation of the frozen
system (9.40). The derivative of V ( Z , U )a1011g the trajectories (9.45) is given by
where 6' E ( 0 ,1) is an arbitrary constnnt. If, in addition, u ( t ) -+ 0 as t + m, then
t ( t ) -t 0 as t -, m. Finally, if h ( u ) = 0 for all u E r and E < c3/c5. then z = 0
is an ezponentially stable equili6r.ilrn1 point of (9.45). Eguiualen.tly, x = 0 is an
0
exponentially stable egtrilibri~tmpoint of (9.38).
Proof: Since Ilu(t)ll 5 E < cIc3/c;c5.ineq~lality(9.46) is sat,isfied wit.h E I = E and
Setting
7 ( t )= ~ l l u ( t ) l land d(t) = LIld(t)ll
we can rewrite the last in equal it..^ as
which takes the form of inequality (9.16) of Section 9.3. Therefore, by applying the
comparison lemma, as in Sect,ion 9.3. it can be shown that, if 4 ( t ) satisfies
Using the give11 upper bound on
E,
we have
.......
368
.
-
.
..'
.
...-
.
...
. . . . .
.....
.-
.....
CHAPTER 9. STABILITY OF PERTURBED W S ~ E M s
Hence, the inequality suptyo ll~(t)ll< 2 c i a l ~ / c d Lis satisfied and
obtain
(9'47), we
.................
.
T
A
.
..%
:, .
:,. _
-.-.-.*
369
where T = h1 (2k2)/27. Silllilar t o the prod 01 Tlicorelu 4.14, it can be &own.
that V ( z ,a ) satisfia (9.41) through (9.43) with cl = [l - exp ( - ~ L ~ T ) ] /c2~=L ~ ,
k2[l- exp(-27T)V27, CJ = 112, and CA = 2k{l - exp [-(7 L l ) T ] } / ( 7 L ~ ) .
T o ahow that V ( z l a )satisfies (9.44), note that the sensitivity fvnction &(ti z , a )
satisfies the sensitivity equation
-
B
=4
;
4
47
+(
-
4.(0; z , a ) = o
4 )
from which we obtain
I
1
1
< 1'
1
I
5
1
5
which shows that z = 0 will be exponentially stable if E < c3/c5.
Theorem 9.3 requires the existence o f a Lyapunov function V ( z ,a ) for the frozen
system (9.40), which satisfies inequalities (9.41) through (9.44). Lemma 9.8 shows
that such Lyapunov function will exist, under some mild smoothness requirements,
if the equilibrium point z = 0 o f the frozen system is exponentially stable uniformly
in a . This is done by deriving a converse Lyapunov function for the system, as in
the converse Lyapunov theorems o f Section 4.7.
.-
.., ......
,
.....
9.6. SLOWLY VARYING S1'Sl'/?hIS
4
After a finite time, the exponentially decaying term will be less than (1 -%)b, which
shows that z(t) will be ultimately bounded by b. I f , in addition, C(t) --+ 0 as t -+ oo,
then it is clear from (9.47) that z ( t ) -+ 0 as t -+ oo.. I f h ( u ) = 0 for all u E r, we
can take L = 0. Consequently, the upper bound on V simplifies to
"
.A,.
,. . . . . . .. , ---
----
..........
. . . . . . ,.. ,....,, . .A..
Llll4.(r;~,a)li2 dr +
/
1
~ 2 l l 4 ( r ; z , a ) l 1dr
2
0
I'
+
Lil14a(~;'z,
a1112 dr +
~ l l l + ( r i 2, a1112dr
L2ke-7T dr~lzIl2
L2k
-11~112
7
Use o f the Gronwall-Bellman inequality yields
.
114.(t;2 , 4 1 1 25 * 1 1 z 1 1 2 e ~ l t
7
Hence,
Lemma 9.8 Consider the system (9.40) and suppose g(z,a) is continuously differentiable and the Jacobian matrices [8g/Bz] and [Bg/bo]satisfy
for all ( z ,a ) E D x I?, where D = { z E Rn 1 1 1 ~ 1 1 < r } . Let k , 7 , and ro be positive
constants with ro < r / k , and define'go = { z E Rn I llzll < ro}. Assume that the
trajectories of the system satisfy , \
IIz(t)ll I k l l z ( ~ ) l l e - ~ ' Q, 4 0 ) E Do,
E
r, t 2 0
which completes the proof of the lemma.
o
Then, there 2s a function V : Do x I' -, R that satisfies (9.41) through (9.44).
Moreover, if all the assumptions hold globally (in z), then v ( z , a ) is defined and
0
satisfies (9.41) through (9.44) on Rn x r.
when the frozen System (9.40) is linear, a Lyapunov function s a t k h g (9.41)
through (9.44) can be explicitly determined by solving a parameterized Lyapunov
equation. This fact is stated in the next lemma.
Proof: Owing t o the equivalence of norms, it is sufficient to Prove the lemma
for the norm. Let d(t;z , a ) be the solution of (9.40) that starts at (092); that
is, 4(0;z , a ) = *. T h e notation emphasizes the dependence o f the Solution on the
parameter a. Let
Lemma 9.9 Consider the system i = A ( a ) z , whe,hen o E I. and A ( ~is)
O u s l ~diffe~ntiable.~uPPosethe elements of A and their jrst padial derivcrthes
with respect to a are uniformly bounded; that is,
V ( Z ,Q ) =
l
T
mT(t;2 , a)m(t;2 , a ) dt
IIA(a)II2 5 c, A
)
2
< bi,
Q a E I', Q 1
< i < rn
370
CHAPTER 9. STABILITY OF PERTURBED SYSTEMS
Suppose further that A(a) is Hurwitr uniformly in a; that is,
Rc[X(A(a))] 5 -0
< 0,
Applying the Gro~~\vall-Bcll~r~a~~
inrc111;llityyicltls
11 cxp\f ( A + B)]\\ 5 k ( . ~ \ ) r ~ - ( ~ ' - k ~ " ) ~ 'VB t~ ~2) 0' ,
V aEr
Then, the Lyapunov equation
Hcrice. tllcre csists n positive coi~stal~t
7 < +3 n~rtla neighborhood N(A) of A sucl~
that if C E N(A). t h w
has a unique positive definite solution P ( a ) for every a E I?. Moreo.ver, P ( a ) is
continuously differentiable and satisfies
for all ( z , a ) E Rn x I?, where cl, c2, and pi are positive constants independent of
a. Consequently, V(z,a) =
(9.42) through (9.44) in the 2-nonn
with cs = 1, cg = 2c2, and
0
Proof: The uniform Hurwitz property of A(a) implies that the exponential matrix
exp [tA(a)] satisfies
11 exp [tA(a)]ll ,< k ( ~ ) e - O ~ V, t L 0, V a E
where p > 0 is independent of a, but k(A) > 0 depends on a. For the exponentially
decaying bound to hold uniformly in a, we need to use the property that IIA(a)ll
is bounded. The set of matrices satisfying Re[X(A(a))] _< -a and JJA(a)JJ
,< c is
a compact set, which we denote by S. Let A and B be any two elements of S.
Consider8
exp[t(A
+ B)] = exp[tA] +
1'
exp[(t - r)A]Bexp[r(A
+ B)] d r
Using the exponentially decaying bound on exp[tA], we get
Siurr S is rolnpact. it is roi.crec1 by a fil~itr1111n1l,crof these neighborlloods. Therecnll fincl a positivc cons~iuitk i ~ ~ ( l v l ) ( * ~of~ (nl eS lU~Ct~ Ithat.
fore.
Considcr no\\- the Lyapunov cquatio~~
(9.48). Existence of a unique positdivedefinite
soh~tionfor every a E I? follows from Theorein 4.6, Moreover, the proof of that
theorem shows that
Shce A(a) is continuously differentia1,le. su is P(o). LVe have
zTp(n)z 5
1"
k2
k2e-27'1(;11i dt = -));
27)I;
*
cz =
k2
27
Let y(t) = el"(")z. Then. y = A(a)gl
-yT(t)3i(t) = -yT(t)~(a)y(t) I I A ( ~ ) I I ~ Y ~I(cyT(t)y(t)
~)u(~)
and
Z ~ P ( ) Z=
bx
yT(t)y(t) dt 2
&-
;vT(t)i(t)
dt
Multiply through by ePt,
Diffcrerltiatc P(a)A(a)i-AT(a)P(a)= -I p;1ltially wit.hrespect to iu~yco~nponent
ai of a, and denote the derivative of P ( a ) by PJ(cr). Then,
+
'This matrix identity follows by writing i= (A B)z a s i= Az f Bz and viewing Bz as an
input term. Substituting x(t) = exp[t(A$ B)]zointo the input term yields
P1(a)A(a)+ A ~ ( ~ ) P ' ( O
= )-{P(a)A1(n)
Thus, P1(a) is given by
Slnm this c'xprmion
Ilolds for nil $0 E R". we nrrive nt the nlntrix identity.
+[ ~ ' ( a ) ] ~ ~ ( a ) }
.
.
. ... ,...
,
' .,.
- ,.
.
.
...
-.
,
9, STABILITY OF PERTURBED SYSTEAIS
372
~t follows that
IIP1(n)l\25
lm
-
k2
b.k4
k2e-2Tt2-b, dt =
27
2y2
.-
..
.
a
t
t
b,k4
* r =2-Y2
4
which completes the proof of the lemma.
be noted that the set r in emm ma 9.9 is not n e c e s s ~ icompact.
l~
When
the boundedness of A ( a ) and its partial derivatives follow from the
assumption that A ( a ) is continuously differentiable.
.. .
..
.
,. . .... . . ;. ~ . .3.
:.L
9.7. EXERCISES
,
:
r
<
.
.., >.*..
373
+
where 6 > 0.
x =A(~t)x
.J
6
( 1 + 6)
-Ivil, V lvil
.y'
.*
< L ( l + 6)
(b) Let P be the solution of
-1
P ( A- B F )
+ ( A- B F ) ~ P= -I
Show that the derivative of V ( x )= xTPx along the trajectories of the closedloop system will be negative definite over the region ( ( F X ) ~L(1+
~ 6 ) , V i,
provided 6/(1+ 6) < 1/(211PBllz IIFIIz).
<
(c) Show,that the origin is asymptotically stable and discuss how you would esti(I
e
where cs is an upper bound on / / P ' ( ( Y ) (Therefore,
(~.
for all E < l/c5, the origin
x = 0 is an exponentially stable equilibrium point of 5 = A ( ~ t ) x .
A
mate the region of attraction.
(d) Apply the result obtained in part (c) to the case
A=[,&
1 8
Exercises
i],B = [ ; ] , ' F = [ ~
2 1 , and 1 - 1
and estimate the region of attraction.
a
9.3 Consider the system
9.1 ([lso])Consider the Lyapunov equation p A + A T P = -8, where Q = QT >
0 and A is Hurwitz. Let p(Q) = ~ r n i n ( ~ ) / ~ m a x ( P ) .
/*
5 = f ( t ,X )
(b) Let Q = QT > 0 have x,~,,(Q) = 1. Show that ~ ( 12)P(Q).
Q = QT > 0.
(b), let PI and 9 be the solutions of the Lyapunov equation for Q = I
Hint: In
and Q = Q, respectively. Show that '
m
- p2 =
exp(ATt)(I- Q )e x p ( ~ tdt) 5 0
+ Bu,
y = C x , and u = -g(t, y)
4
t
where f ( t ,0) = 0, g(t,0 ) = 0, and I(g(t,y) 11 I 711y(I for all t L 0. Suppose that the
origin of 5 = f ( t ,x ) is globally exponentially stable and let V ( t ,x ) be a Lyapunov
function that satisfies (9.3) through (9.5) globally. Find a bound 7' on 7 such that
the origin of the given system is globally exponentially stable for 7 < 7'.
( a ) show that p ( k ~=) p(Q) for any positive constant k.
(c) Show that p ( I ) > p(Q),v
.
3".
( a ) Show that
Iht(v)I I
where E > 0. When e is s~fficient~ly
small, we can treat this system as a slowly
varying system. It is in the form of (9.38) with u = et and = [O,m).For all
u E r, the origin x = 0 is an equilibrium point. Hence, this is a special case where
h ( u ) = 0. Suppose Re[A(A(a))]< -0 < 0, and A ( a ) and A1(a)are uniformly
bounded for all a E I?. Then, the solution of the Lyapunov equation (0.48) holds
the properties stated in Lemnla 9.9. Using V ( x , u ) = xTP(u)x as a Lyapunov
function candidate for x = A ( u ) z , we obtain
1
.
9.2 Co~isiderthe system S = Ax Bu ulid Ict 11 = -Fr be a stl\bilizilg stntc
feedback control; that is, the matrix (A - B F ) is Hurwitz. Suppose that, due to
physical limitations, we have to use a limiter to limit the value of ui to Iui(t)l 5 L. .
The closed-loop system can be represented by x = Ax - B L sat(Fx/L), where
sat(v) is a vector whose ith component is the saturation function. By adding and
subtracting the term BFx, we can rewrite the closed-loop state equation as j.=
( A- BF)X - ~ h ( F x )where
,
h(v) = L sat(u/L) - v. Thus, the effect of the limiter
can be viewed as a perturbation of the nominal system without the limiter.
Example 9.9 Consider the system
p,
..
.
I+,
r is
9.7
\
.
,..
b
k
9.4 Consider the perturbed system
I
x = Ax+ B[u + g ( t , x ) ]
where g ( t , x ) is continuously differentiable and satisfies Ilg(t,x)(12I k11~11~,
vt2
0, V x E B , for some r > 0. Let P = PT > 0 be the solution of the Riccati equation
P A + A ~ P + Q- P B B ~ P + ~ ( =YOP
I
374
CHAPTER 9. STABILITY OF PERTURBED SYSTEMS
>
where Q
k?1 and a
perturbed system.
I
> 0.
375
9.7. EXERCISES
Show' that u = -BTPx stabilizes the origin of the
9.5 ([loll) Consider the perturbed system
x=Ax+Bu+Dg(t,y),
I
3
where g(t, y) is continuously differentiable and satisfies Ilg(t, y)(J25 k(lyl12,V t
( ( ~ ( ( 27 for Some T > 0. Suppcise the equ,@tion
<
V
<
y=Cx
> 0,
wlicrc j(x) and g(x) arc colitinuously differentiable and Jlg(x)JJ yJIxJlfor all
llxll < r . Suppose the origin of the llolninal system j. = f(x) is asymptotically stable
and t.l,ere is a Lyapunov functioll V(x) that satisfies inequalit.ies (9.11) t.hrough
(9.13) for all IJ.rll < 1'. Lct = {If(.) 5 c)! with c < crl(r).
(a) Sl~owthat there is a positive col~siant. y' such that, for y < y*. the solutions of
tlie perturl>ed syste111 startillg ill stay in for all t 2 0 and are ultilnately
.
bounded by a class K fullct ioil of y.
Q = QT > 0, E > 0, and 0 < y < I l k has a positive definite solution
P = pT> 0. Show that u = - ( 1 / 2 ~ ) B ~ P stabilizes
x
the origin of the perturbed
where
system.
9.6 Consider the system
where a > 0, 0, y, and w
(b) Suppose the no~ninalsystcin liils t11c additional property that A = [6j/6x](0)
is Hurwitz. Show that tllere is 7; such that, for y < y;. the solutiolls of the
perturbed system starting in fl converge to the origin as t
w.
-
(c) \\'auld (b) hold if A was i ~ o Hurwitz?
t
Consider
> 0 are constants.
(a) By viewing this system as a perturbation of the linear system
Hint: For the exiumplc of part (c). ~lsc!
If (x) = .rf
show that the origin of the perturbed system is exponentially stable with
(11x112 5 r ) included in the region of attraction, provided 1/31 and ( y ( are
sufficiently small. Find upper bounds on 1/31 and 171 in terms of r.
+
(b) Using V(X) = sf x i as a Lyapunov function candidate for the perturbed
system, show that the origin is globally exponentially stable when /3 5 0 and
exponentially stable with {JJxJJz
<
included in the region of attraction
when
0.
+ i.ri + ixg + 11x3
to show that the origin of x = jS() is nspmpt.otically stable and t.hcn apply Thcorem 4.16 t.o obtain a Lyapunov function that satisfies (9.11) through (9.13).
9.9 Collsidcr t,llc system
m)
(c) Compare the results of (a) and (b) and comment on the conservative nature of
the result of (a).
9.7 Consider the perturbed system
s = f ( 4 + 9(x)
Suppose the origin of the nominal system x = f(x) is asymptotically (but not
exponentially) stable. Show that, for any y > 0, there is a function g(x) satisfying
()g(x)ll rllxll in some neighborhood of the origin such that the origin of the
perturbed system is unstable.
(a) With y = 0, sl~owthat the origill is globally asymptotically stable. Is it exponentially stable?
(b) With 0 < y 5 112: show that. the origin is unstable and the solutions of the
system are globally ultimately bounded by an ultimate bound that is a class
K function of y.
9.10 ([19]) Consider tlie system
:
<
where a? b > a, c! and y arc positive consbants and q(t) is a continuous function.
(a) With q(t)
(c) Sliow t.lit\t tlic riglit-llcui(1 sick ol' (!l.lil) sppronc.11rswro ns 1%+ CQ.
= 0, use
V ( x )= (b
+ +
( d ) Consider the perturbed system i = - x / ( l x 2 ) 6, where 6 is a positive
constant. Show that whenever 6 > 1/2, the solution x ( t ) escapes to w for
any initial state x(0).
+ ic2) x i + ~ ~ 1 +xi
x 2 + 2 4 1 - cosxl)
to show that the origin is globally exponentially stable.
( b ) Study the stability of the system when q ( t ) # 0 and (q(t)l5 k for all t 2 0.
f
9.11 Consider the system
( a ) Let b = 0. Show that the origin is globally asymptotically stable. Is it exponentially stable?
(b) Let b > 0. Show that the origin is exponentially stable for b < min{l,az).
b > 0.
1
!
:i
(a) Show that inequalities (9.11) through (9.13) are satisfied globally with
(b) Verify that these functions belong to class K,.
I)
v ( t ) dt 5 \l(b - a )
[
>
G ( t ) dt, V u ( t ) 0
which follows from the Cauchy-Schwartz illcquality.
.
9.17 Consider the system j. = A ( t ) x where A ( t ) is continuous. Suppose limt,,
A exists and A is Hurwitz. Show that the origin is exponentially stable.
A(t) =
9.18 Repeat part(b) of Exercise 9.10 when q(t) is bounded and q ( t ) + 0 as t -,w.
9.19 Consider the system x = f ( t ,x ) , where 11 f ( t ,x ) - f (0,x)112 5 -y(t)llxll2 for all
t 2 0 , x E R 2 , ~ ( t+) 0 as t + 00,
and a, P, w, R are positive constants. Show that the origin is globally exponentially
stable.
Hint: In part (d), note that the Jacobian matrix of the nominal system is not
globally bounded.
+ x Z )and V ( x )= x4.
0
I
( d ) Discuss the results of parts '(a) through (c) in view of the robustness results of
Section 9.1, and show that when b = 0 the origin is not globally exponentially
stable.
9.13 Consider the scalar system x = - x / ( l
+
9.16 Consider the linear system of Example 9.6, but change the assumption on
B ( t ) to
llB(t)112dt < w. Show that the origin is exponentially stable.
Hint: Use the inequality
1
9.12 ([S]) Consider the system
+
jp
(b) With b # 0, show that the origin is exponentially stable for sufficiently small
lbl, but not globally asymptotically stable, no matter how small Jbl is.
(c) Discuss the results of parts (a) and (b) in view of the robustness results of
Section 9.1, and show that when 6 = 0 the origin is not globally exponentially
stable.
b
9.14 Verify that D + W ( t ) satisfies (9.17) when V = 0.
Hint: Show that V ( t h , x ( t h ) ) 5 0.5c4h2(lg(t,0)(12 h o(h), where o ( h ) / h
as h + 0. Then, use the fact that
2 1.
+
4
9.15 Consider the linear system of Example 9.6, but change the assumption on
B ( t ) to
IIB(t)ll dt < w. Show that the origin is exponentially stable.
( a ) With b = 0 , show that the origin is exponentially stable and globally asymp
totically stable.
(c) Show that the origin is not .globally asymptotically stable for any
a
i1
1
f
+
+
+
9.20 Consider the system x = f ( x ) G ( x ) u w ( t ) , where I(w(t))lz5 a c e".
Suppose there exist a symmetric positive definite matrix P , a positive semidefinite
function 1Y(x),and positive constants 7 arid u such that
2 x T pf ( x )
+ -yxTpx + W ( x ) - 2uxTPG(x)GT( X ) P X
5 0, V x E Rn
Show that with u = -uGT(x)Px, the trajectories of the closed-loop system are
uniformly ultimately bounded by 2akX,,(P)/.yXm1,(P),
for some k > 1.
i
378
CHAPTER 9. STABILI??' OF PERTURBED SYSTEMS
9.21 Consider the perturbed system (9.1). Suppose there is a Lyapunov function V ( t , x )that satisfies (9.11) through (9.13),and the perturbation term satisfies
lJg(t,x)l( 6 ( t ) , V t 2 0, V x E D. Show that for any E > 0 and A > 0 , thcrc exist
q > 0 and p > 0 such that whenever ( l / A )J,'+*6 ( ~d~
) < q, every solution of the
perturbed system with I(x(to))l< p will satisfy Ilx(t)ll < E , V t to.
(This result is known as total stability in the presence of perturbation that is
<
>
+
379
9.7. EXERCISES
\\,liere e i j is a I)ii~;~ry
variilblc tli;~tt:~kcsthe value 1 when t,he jth s~ll)syst,c!intu:t.s
on the ith subsystcin aiid the valuc 1) otherwise. The origin of tlic iiitcrco~i~rnct.c!tl
systcln is said to I)c coniicctivcly ;rsyli11)tol.ic;~lly
stable if it is ;~s.vliil,I.ol,i(:i~lly
sl,id)l(:
for a11 iiitercuiiiicctioli pattcriis. tlii~tis, for all possible valucs of the binary varia1)les
eij. Supposc that. all the assuinptioiis of Theorem 9.2 are satisfied, with (9.36)t-J i i i g
the foiln
m
the time interval with sampling points at to i A
for i = 0 , 1 , 2 , .. ., and show that W(to i A ) satisfies the difference inequality
+
Sllow that the origin is ronncctively asyniptot ically stable.
<
9.22 Let A be an n x n matrix with aij 0 for all i # j and aii > CjZi(aij!, i =
1,2,. . ,n. Show that A is an M-matrix.
Hint: Show that Cj",, aij > 0 for i = 1 , . ,n , and use mathematical induction to
show that all the leading principal minors are positive.
.
9.28 ((491) The output y(t) of the liliear system
..
is required to track a reference iiiput
I..
Consider the integral controller
9.23 Suppose the conditions of Theorem 9.3 are satisfied with
Show that the origin is exponentially stable.
have assurned that the state x can be measured and the matrices Fi and
where
F2 can be designed suc11 that the inatrix
9.24 ([132]) Study the stability of the origin of the system
is Hurwitz.
by using composite Lyapunov analysis.
9.25 Study the stability of the origin of the system
( a ) Show that. if r = constant, then y(f)
--t
r as t
-+
oo.
( b ) St.ucly the trackil~gproperties olt.11~
syst,e111when r ( t ) is a slowly varying input.
9.29 ( [ 8 6 ] )The output y(t) of the nonlinear system
by using composite Lyapunov analysis.
9.26 Consider the linear interconnected system
where, for each i, xi is an ni-dimensional vector and Aii is a Hurwitz matrix. Study
the stability of the origin by usilig composite Lyapunov analysis.
9.27 ( [ 1 7 5 ] )Complex interconnected systems could be subject to structural perturbations that cause groups of subsystems to be connected or disconnected from
each other during operation. Such structural perturbations can be represented a s
is required to track a reference inp11t r. Consider the int.egra1 colltroller
where we have assumed that the state x can be measured, tlw f\mction y can be
designed such that t l ~ ecloscd-lool) syst.c!111
has an exponentially stable equilibrium point ( 2 ,z), and the funct,ions f: h? and y
are twice continuously differentiable in tlleir i argument.^.
( a ) Show that. if r = constant. a i d tlie initial state ( ~ ( 0~ () 0~) is
) sufficiently close
to ( f , f ) . then y(t) --t r as t --t x.
..
/
.
. :. ..
,'?
CHAPTER 9. STABILITY OF PERTURBED SYSTEhfS
(b) Study the tracking properties of tlib systelll wllen ~ ( t is) a slowly varying input.
9.30 ([86]) Consider the tracking problem of Exercise 9.29, but assume that we
can only measure y = h(x). Consider the observer-based integral controller
i1=f(n1u)+G(r)b-h(a)],
&=T-y,
and u = ~ ( z ~ , z z , ~ )
Suppose 7 and G can be designed such that the closed-loop system has an exponentially stable equilibrium point (2,i l , i2). Study the tracking properties of the
system when
(2) ~ ( t is) slowly varying.
(1) T = constant.
2
9.31 Consider the linear system x = A(t)x where I(A(t)\\ k and the eigendues
of A(t) satisfy Re[A(t)] 5 -o for all t 2 0. Suppose that
/ ( ~ ( t ) ldtl ~ p. Show
that the origin of x = A(t)x is exponentially stable.
so
Chapter 10
Perturbation Theory and Averaging
<
Exact closed-form analytic solutions of nonlinear differential equations are possible
only for a limited number of special classes of differential equations. In general,
we have to resort to approximate solutions. There are two distinct categories of
approximation methods that engineers and scientists should have a t their disposal as
they analyze nonlinear systems: (1) numerical solution methods and (2) asymptotic
methods. In this and the next chapter, we introduce the reader to some asynlpt,otic
methods for'the analysis of nonlinear differential equations.'
Suppose we are given the state equation
where E is a "small" scalar parameter, and, under certain conditions, the equation
has an exact solution d(t, E ) . Equations of this type are encountered in many applications. The goal of an asymptotic method is to obtain an approximate solution
Z(t, E) such that the approximation error x ( t , E ) - Z(t, E) is small, in some norm,
for small IEI and the approximate solution 5(t, E) is expressed in terms of equations
simpler than the original equation. The practical significance of asymptotic methods is in revealing underlying structural properties possessed by the original state
equation for small I E ~ . We will see, in Section 10.1, examples where asymptotic
methods reveal a weak coupling structure among isolated subsystems or the structure of a weakly nonlinear system. More important, asymptotic methods reveal
multiple-t.imoscale structures inherent in many practical problems. Quite often,
the solution of the state equation exhibits the phenolnenon that some variables
move in time faster than other variables, leading ta the classification of variables
as "slow" and "fast." Both the averaging method of this chapter and the singular
perturbation method of the next chapter deal with the interaction of slow and fast
variables.
'N~~rnericsl
solution methods are not stutlied in illis textbook on !Ile premise lhat most studcnts
arc introdllcctl Lo t1le111in elel~lentarydiflerentialequation courses and they get their in-depth study
of the subject in numerical analysis courses.
382
CHAPTER 10. PERTURBATION AND AVERAGING
Section 10.1 presents the classical perturbation method of seeking an approximate solution as a finite Taylor expansion of the exact solution. The asymptotic
validity of the approximation is established in Section 10.1 on finite time intervals
and in Section 10.2 on .the infinite-time interval. Section 10.3 examines an autonomous system under the iduence of a weak periodic perturbation. While the
results of the f i s t three sections are interesting in their own sake, they provide the
technical basis for the averaging method. In Section 10.4, we introducc the averaging method in its simplest form, which is sometimes called "periodic averaging"
since the right-hand side function is periodic in time. Section 10.5 gives an application of the averaging method to the study of periodic solutions of weakly nonlinear
second-order systems. Finally, we present a more general form of the averaging
method in Section 10.6.
10.1
The Perturbation Method
"?p",
353
10.1. THE PERTURBATION AIETHOD
When the approxilnatioll crror s;ltislics the bound of (10.4), we say that the error
is of order O ( E ) and write
. K ( / . E ) - .Ucr(t)
= O(E)
.,
This crder of rnngilitude notatiou will I,e used frequently in this chapter and the
next one. It is defined nest.
Definition 10.1 61( E ) = O ( & ( E ) ) if there exist positive constan.ts k and c such that
Example 10.1
$?
,\@>
eQ
Consider the system
E"
= O ( E ~for
) all n
.
1 m?since
I E J"
=J
<
E ~ " ' J E ( ~ -/ E~l r n ,
v I€( < 1
+
~ ~ l ( 0 . 5E ) = O ( E ~ since
):
k = f(t,X,€)
where f : [to,t l ] x D x [ - E ~€01
, -' Rn is "sufficiintly smooth" in its arguments over
a domain D C Rn.The required smoothness conditions will be spelled out as we
proceed. Suppose we want to solve the state equation (10.1) for a given initial state
ik
.
i + 2~ = O ( 1 ) . since
11 + 2 ~ 1< 1 + 2a, V
where, for more generality, we allow the initial state to depend "smoothly" on
The solution of (10.1) and (10.2) depends on the parameter E , a point that we
emphasize by writing the solution as x ( t , E ) . The goal of the perturbation method is
to exploit the "smallness" of the perturbation parameter E to construct approximate
solutions that are valid for sufficiently small lal. The simplest approximation results
by setting E = 0 in (10.1) and (10.2) to obtain the nominal or unperturbed problem
E.
where qo = q(0). Suppose this problem has a unique solution x o ( t ) defined on
[to,tl] and x O ( t ) E D for all t E [ t O l t l ] . Suppose further that f is continuous in
( t , x , ~ and
) locally Lipschitz in ( x , E ) , uniformly in t , and q is locally Lipschitz in E
for ( t , x , E ) in [to,t l ] x D x [-Eo, E O ] . The closeness of the solutions of the perturbrd
and unperturbed problems follows from continuity of solutions with respect to initial
states and parameters. In particular, Theorem 3.5 shows that there is a positive
, problem of (10.1) and (10.2) has
constant €1 5 EO such that for all JEI 5 ~ 1 the
a unique solution x ( t , E ) defined on [to,t l ] . Furthermore, Theorem 3.4 shows that
there is a positive constant k such that
f
I'
JE(
<a
exp(-U/E) \vith positive a all115 is O ( c n )for ally positivc irltcger n, since
+
5
\\:hat can we say about the ~lulllrricalvalue of the approsilnatiun error z ( t , E ) ~ ( tfor
) a given iiurnerical value OI.Ewheli the error is O ( E ) ' ! Uiifortullately! we
callnot trailslate tile O ( E ) ortlcr u l ~llil~l~itudc
stillc~~lcllt
ill1 o ;I 11111ilc:ri~;l1
1)01111(1
on the prror. Knowing that the error is O ( E ) means t,hat it.s norin is less than
~ ( E for
J some positive constalit k tliat. is independent of E. However, we do not
know the value of k? which rniglit>be1: 20, or any positive
The fact that
k is independent of E guar.antees that the bound k l ~ ldecreases rnoiiotoliically a.
J E decreases.
~
Therefore, for sufficiellt.ly small IE(,the error will be small. hlore
precisely, given any tolerance 6: we kuow that the norm of the rrror will be less
- -
21t should be noted. however, that in a well-formulated perturbation problen~where variables
parameter, one should
are normalized to have dimensionless slntc variables, time, and perturbatio~~
expect the numerical value of k not t o bc nll~chlarger than one. See Example 10.4 for further
discussion of normalization, or consult 198) and [141] for more examples.
than 6 for all (EI < 6lk. If this ralige is too sillall to cover the numerical values of
interest for E, we then need to extend the range of validity by obtaining a higher
order approximation. An O(.c2) approximation will meet the same 6 tolerance for
all J E ~< $&, an O ( E ~approximation
)
will do it for all ~ E I < (6/k3)'13, and so
on. Although the constants k, k2,k3,. . . are not necessarily equal, these intervals
are increasing in length, since the tolerance 6 is typically much smaller than one.
Another way to look at higher order approximations is to see that, for a given
)
"sufficiently small1' value of E, an O(E") error will be smaller than an O ( E ~error
for n > m, since
l/(n-m)
Q
4
as an identity in E. Helice, coetficicnts of likc po\\.vrs of E ~ilustbe equal. hlatching
those coefficients, we can derive the equations that must be satisfied by xo, xl, and
so on. Before we do that, we have to generate the coefficients of the Taylor series
of h(t, E). The zeroth-order term h,-,(t) is given by
ho(t) = f(tI~0(t)lO)
Consequently, matching coefficients of EO in (10.6), we determine that xo(t) satisfies
which, not surprisingly, is the unperturbed problem (10.3). The first-order term
hl(t) is given by
Higher-order approximations for sol~~tions
of (10.1) and (10.2) can be obtained
in a straightforward manner, provided the functions f and q are sufficiently smooth.
Suppose f and q have continuous partial derivatives with respect to (x, E) up to order
) [to,tl] x D x [ - E ~ , E ~TO
) obtain a higher order approximation of
N for ( t , x , ~ E
x(t, E), we construct a finite Taylor series
N-1
x ( t , ~=
)
C x k ( t ) ~+~E ~ & ( ~ , E )
(10.5)
k=O
Two things need to be done here. First, we need to calculate the terms XO,XI,
. . . , XN-~;in the process of doing that, it will be shown that these terms are well
defined. Second, we need to show that the remainder term R, is well defined and
bounded on [to,tl], which will establish that
xk(t)~
is ~an O(cN) (Nth-order)
approximation of x(t, E). By Taylor's theoremI3 the smoothness requirement on the
)
the existence of a finite Taylor series for q ( ~ )that
; is,
initial state q ( ~ guarantees
Matching coefficients of
~1
af
= -ax
PI
E
in (10.6), we find that xl(t) satisfies
xo(t), 0)
Xl
+ $(tI
xo(t), o), ,
x1(t0) = q1
Define
A ) = tax I 0
0)
af
g1(t, xo(t)) = -(t,
a&
xo(t), 0)
and rewrite the equation for zl as
Therefore,
Xk(tO)= qk, k = 0,1,2,. .., N - 1
Substituting (10.5) into (10.1) yields
where the coefficients of the Taylor series of h(t, E) are functions of the c0efficient.sof
the Taylor series of x(t, E). Since (10.6) holds for all sufficiently small E, it must hold
3See [lo,Theorem 5-14].
4
This linear equation has a unique solution defined on [to,tl].
The process can be continued to derive the equations satisfied by x2, xa, and
so on. This, however, will involve higher order differentials o f f with respect to x,
which makes the notation cumbersome. There is no point in writing the equations
in a general form. Once the idea is clear, we can generate the equations for the
specific problem of interest. Nevertheless, to set the pattern that these equations
take, we will, a t the risk of boring some readers, derive the equation for x2. The
second-order coefficient in the Taylor series of h(t,E) is given by
386
10.1. THE PERTURBATION i\fE?'HOL)
CHAPTER 10. PERTURBATION AND AVERAGING
Example 10.2 Consider the Van drr Pol state equation
Now,
xl(0) = 7 1 ( ~ )
.fl = n . ~ .
=
~2
f
-Xi
~ ( - 1. T ~ ) x z ,~ ~ ( =
0 )7 p 2 ( E )
Suppose lie want to construct a finite Taylor series with N = 3. Let
To simplify the notation, let
Xi
= X,IJ
+ E X 1 ] + E2x,2
E3&,, i = 1,2
and
and continue to calculate the second derivative of h with respect to E:
.f-
qi = 7i0 + E l l i l + E27i2 + E ~ R ? i~=, 2
Substituting the series for xl alirl 2-2 into the state equation results in
.! &.
;&.
x
Thus,
hlatching coefficic~itsof
+
h2(t)= A ( t ) x ~ ( t ) g2(t,xo(t)r XI(^))
EO:
(a20
+
+
E X Z ~ ~
~
+E
~
'id-
~ 2 R 2~ ~ )
!&
~ v eobt,ai~i
;c.
.
.
r t..
where
which is the unperturbed probleln at z = 0. hlatching coefficicnts of
E.
we obtain
Matching coefficients of E' in (10.6) yields
In summary, the Taylor series coefficients so, X I , ... , X N - 1 are obtained by
solving the equations
ko = f ( t ,xo, O ) , xo(to) = 770
(10.7)
5k = A(t)zk + gk(t,x0(t)1
...
xk-l(t)),
xk(t0) = 771;
(10.8)
for k = 1,2, ,N- 1, where A(t) is the Jacobian [ a f l a x ]evaluated at x =
xo(t) and E = 0, and the term gk(trxO(t)!xl(t), . ,xk-l(t)) is a polynomial in
X I , . . ,xk-1 with coefficients depending continuously on t and xo(t). The assump
tion that xO(t)is defined on [to,t l ] implies that A(t) is defined on the same interval;
hence, the linear equations (10.8) have unique solutions defined on [to,tl]. Let
us now illustrate the calculation of the Taylor series coefficients by a second-order
csmplc.
.
..
while matching coefficients of
x22 = -xl2
f
E'
results ill
(1 - x:o)x2l - 2 ~ 1 0 ~ 1 1 ~ 2~ 02 ~
2 ( 0 ) = 722
The lat.ter two sets of equations are in t.he form of (10.8) for k = 1:2.
Having calculated t.he terms xo,
XI:
.. . , X N - 1 ,
our task now is to show t,hat
E ) . Consider the approxi-
~ rx ks( t ) ~~% indeed an O ( E ~approximation
)
of x(t,
mation error
N-1
A
.-.--
-. -
. -.-.-.
.
/
CIIAPTER 10. PERTURBATION A N D ' A M C I N G
388
'
Differentiating both sides of (10.9) with respect to t and substituting for the derivatives of x and xk from (10.1), (10.7), and (10.8), it can be shown that e satisfies the
equation
(10.10)
c = A(t)e+ pl(t,e,€) pz(t,E), e(to) = E ~ R ? ( & )
+
Tlren, tlrei-e exists E* > 0 stir11 thrrt V It) < e*, tltt, 1)tnblctti !)ituS11 I I (10.1)
~
ui~ti
(10.2) lras a ui~iqzresol~rtioilx(t,e), defiiied on [to, tl], wlrich satisfies
i
9
where
When we approximate x(t, E ) by xo(t), we need not know the value of the parameter E, which could be an unknown parameter that represents deviations of the
system's parameters from their nominal values. When we use a higher order a p
proximation
x k ( t ) ~for~ N 2 2, we need to know the value of E to construct
the series, even though we do not need it to calculate the terms X I , x2, and so on. If
we have to know E to construct the Taylor series approximation, we must then compare the computational effort needed to approximate the solution via a Taylor series
with the effort needed to calculate the exact solution. The exact solution x(t, E ) can
be obtained by solving the nonlinear state equation (10.1), while the approximate
solution is obtained by solving the nonlinear state equation (10.7) and a number of
linear state equations (10.8), depending on the order of the approximation. Since,
in both cases, we have to solve a nonlinear state equation of order n, we must ask
ourselves, What do we gain by solving (10.7) instead of (10.1)? One situation where
the Taylor series approximation will be clearly preferable is the case when t i e solution is sought for several values of E. In the Taylor series approximation, equations
(10.7) and (10.8) will be solved only once; then, different Taylor expansions will be
constructed for different values of E. Aside from this special (repeated values of E )
case, we find the Taylor series approximation to be effective when
z,"=ol
By assumption, xo(t) is bounded and belongs to D for all t E [to,tl]. Hence, there
exist X 1 0 and el > 0 such that for all llell <- X and
<- €1, the functions xo(t),
xk(t)rk, and e +
~ ~ ( t belong
) & ~to a compact subset of D. It can be
-.
.
easily verified that
\&I1.
z;=o1
( l ~ l ( t , e z ,-~ )PI(^, el,E)II 5 k~llez- ell1
Ilpz(t,&)I\-< h!€IN
I
(10.12)
(10.13)
, some positive coytant. kl and k2.
for all t E [to,tl], el,e2 E BA,E E [ - ~ l , & l ]for
Equation (10.10) can be viewed as a perturbation of
(10.14)
Co = A(t)eo pl(t, eo, E), eo(t0) = 0
+
r the unperturbed state equation (10.7) is consitlerilbly simpler than the
E-dependent state equation (10.1),, and
which has the unique solution eo(t,s) = 0 for t E [to,tl]. Applying Theorem 3.5
shows that (10.10) has a unique solution defined on [to,tl] for sufficiently small 161.
Applying Theorem 3.4 shows that
We summarize our conclusion in the next theorem.
Theorem 10.1 Suppose
f and its partial derivatives vnth respect to (x, E ) up to order N are continuow
in (t, 2, E) for (t, x,E) E [to, tll x D x [-€0, €01;
-T,
and its derivatives up to order N are continuorn for E E [-EO,EO];
the nominal problem given in (10.3) has a unique solution xo(t) defined on
[to,tl] and xo(t) E D for all t E [to,ti].
is reasonably small that an "acceptable" approximation can be achieved
with a few terms in the series.
E
.$
In most engineering applications of the perturbation method, adequate approximac
tions are achieved with N = 2 or 3, and the process of setting 6 = 0 simplifies
the state equation considerably. In the next two examples, we look at two typical
cases where setting E = 0 reduces the complexity of the state equation. In the
first example, we consider again the Van der Pol equation of Example 10.2, which
represents a wide class of "weakly nonlinear systems" that become linear at E = 0.
To construct a Taylor series approximation, we only solve linear equations. In the
second example, we look at a system formed of interconnected subsystems with
''weak" or E-coupling. At E = 0, the system decomposes into lower order decoupled
subsystems. To construct a Taylor series approximation, we always solve lower order decoupled equations as opposed to solving the original higher order equation
(10.1).
CHAPTER 10. PERTURBATION AND AVERAGING
10.1. THE PERTURUAI'ION i\IEl'HULJ
oJ1
Figure 10.2: Electric circuit of Example 10.4.
Figure 10.1: Example 10.3 at E = 0.i: (a) xl(t, E) (solid) and xlo(t) (dashed); (b)
xl(t, E) - xlO(t) (solid) and xl(t, E) - xlo(t) - EXII (t) (dashed).
whose, solution is
Example 10.3 Suppose we want to solve the Van der Pol equation
+
By Theorem 10.1, the second-order approximation xo(t) &xl(t)is O($) close to
the exact solution for sufficiently sniall E . To compaie the approximate solution
wit.h thc cract one at E = 0.1, ~ v ecnlculatc
over the time interval [0, T ] . We start by setting E = 0 t,o obtain the linear unperturbed equation
xl0 '= 5201 510(0) ' 1
k2"
whose solution is
= -510,
x20(0) = 0
-
xlo(t) = cost, xZ0(t)= sint
Clearly, all the assumptions of Theorem 10.1 are satisfied, and we conclude that the
approximation error x(t, E ) - xo(t) is O(E). Calculating x(t, E) numerically a t three
different values of E and using
.
.
1
which shows a reduction in the approximation error by almost an order of magnitude. Figure lO.l(b) shows the approxilnation errors in the first coinpollent of thc
state vector for the first-order approximation x0 and the second-order approximaA
tion s o €21 at E = 0.1.
+
Example 10.4 The circuit shown in Figure 10.2 contains nonlinear resistors whose
I-V characteristics are given by i = @(v). The differential equations for the voltages
across the capacitors are
Eo = a<t<_rr
max Ilx(t, €1 - xo(t)ll2
'
as a measure of the approximation error, we find that Eo = 0.0112, 0.0589, and
0.1192 for E = 0.01, 0.05, and 0.1, respectively. These numbers show that the error
is bounded by 1.26 for E 0.1. Figure lO.l(a) shows the exact and approximate
trajectories of the first component of the state vector when E = 0.1. Suppose we
want to improve the approximation a t E = 0.1. Rom Example 10.2, we know t,hat
x l l and xzl satisfy the equation
<
The circuit has two similar RC sections connected through' the resistor R,. When
R, is "relatively large," the colinection between the two sections becomes "weak."
In particular, when R, = oo,the connection is open circuit and the two sections are
decoupled from each other. This circuit lends itself to an E-coupling representation
where the coupling between the t,wo sections may be parameterized by a small
.
-
- 20.2.
p;traiiic!t.cr E. At. first glance, it appcars t.hat. a reasonable choice of E is E = l/Rc.
Indeed, with this choice, the couplil~gterins in the foregoing equations will be
multiplied by E. However, such achoice makes E dependent on the absolute value of
a physical parameter whose value, no rnatter how small or large, has no significance
by itself without considering the values of other physical parameters in the system.
In a well-formulated perturbation problem, t.he parameter E would be chosen as a
ratio betwcci~physical parameters that rcflects the "true smallness" of E in a relative
sense. To choose E this way, we usually start by choosing the state variables or the
time variable (or both) as dimensionless quantities. In our circuit, the clear choice
of st,at.e variables is vl and v2. Instead of working with vl and v2, we scale them in
such a way that the typical extreme values of the scaled variables would be close to
ztl. Duo to thc weak coupling betwrnn tlie two identical sections, it is reasonable
to use the same scaling factor a for both state variables. Define the state variables
as xl = v l / a and x2 = v2/a. Taking a dimensionless time T = t/RC and writing
dxldr = i-,we obtain the state equation
- ---.--"-.
- --.5
6,
a
.
- ',---' .-.
I
-
P E R ? ~ B A T ~ O 6N~ i ' l r INIZINI'I'E
~
INTERVAL
0.05
0
0.5
1
Time
1.5
393
I
2
Figure 10.3: Exact solution (solid), first-order approximation (dashed), and secondorder approximation (dash-dot) for Example 10.4 at E = 0.3.
where hl(.)is the derivative of h(.). Figure 10.3 shows the exact solution as well as
the first-order and second-order approximations for E = 0.3.
A
It appears now that a reasonable choice of E is RIR,. Suppose that R = 1.5 x lo3
0, E = 1.2 V, and the nonlinear resistors are tunnel diodcs with
Take a = 1 and rewrite the state equation as
where h(v) = 1.5 x lo3 x $(TI).
Suppose we want to solve this equation for the
initial state
)
x1 (0) As0.15; ~ ~ (=00.6
X
Setting E = 0, we obtain the decoupl d equations
kl =. 1.2 - z l - h ( r l ) , ~ ~ ( 0= ) 0.15
x2 = .1.2-x2-h(x2)?
x~(O) = 0.6
which are solved independdntly of each other. Let xlo(t) and xzo(t) be the sohtions. According to Theorem 10.1, tliey provide an O(E)approximation of the exact
)
we set up the
solution for sufficiently small E. To obtain an O ( E ~approximation,
equations for 211 and x21 as
A serious limitation of Theorem 10.1 is that the O ( E ~error
)
bound is valid
only on finite (order O(1)) time intervals [to, tl]. It does not hold on intervals like
[to,T/~]nor on the infinite-time interval [to,oo). The reason is that the constant k
in the bound kJ€JNdepends on tl in such a way that it grows unbounded as t l increases. In particular, since the constant k results from application of Theorem 3.4,
it has a component of the form exp(Lt1). In the next section, we will see how to
employ stability conditions to extend Theorem 10.1 to the infinite interval. In the
lack of such stability canditions, the approximation may not be valid for large t,
even though it is valid on O(1) time intervals. Figure 10.4 shows the exact m d
approximate solutions for the Van der Pol equation of Example 10.3, at E = 0.1,
over a large time interval. For large t, the error xl(t, E ) - xlo(t) is no longer O(E).
b o r e seriously, the error x l (t, E) - xlo(t) - ~ x 1(t)
1 grows unbounded, which is a
consequence of the term t cos t in 311(t).
10.2 Perturbation on the Infinite Interval
The perturbation result of Theorem 10.1 can be extended to the infinite time interval
[to,oo) under some additional stability conditions. In the next theorem, we require
the nominal system (10.7) to have an exponentially stable equilibrium point at the
origin and use a Lyapunov function to estimate its region of attraction. There is no
loss of generality in taking the equilibrium point a t the origin, since m y equilibrium
point can be shifted to the origin by a change of variables.
Theorem 10.2 Let D C Rn be a domain that contains the origin and suppose
CHAPTER 10. PERTURBATION AND AVERAGING
10.2. PERTURBATION ON THE INFINITE IN W R V A L
395
such that for all [el < .ell x ( t , ~is) uniformly bounded and x ( t , ~-) xo(t) is O ( E ) ,
uniformly in t , for all t 2 to. It is also clear that for qo E R, xo(t) is uniforn~ly
bounded and limr,, xo(t) = 0. Col~sitlcrthe linear equations (10.8). Wc know
from bounded-input-bounded-output. stability (Theorem 5.1) that the solution of
(10.8) will be uniformly bounded if the origin of i.=A(t)z is exponentially stable
and the input term gk is bounded. The input gk is a polynomial in X I , . . . , xk-1
with coefficients depending on t arid xo(t). The dependence on t comes through
the partial derivatives of f , which are hounded on compact subsets of D. Since
xo(t) is bounded, the polynon~ialcoefficients are'bounded for all t 2 to. Hence,
boundedness of gk will follow from boundedness of xi, . . . , xk-1. The matrix A ( t )
is given by
Figure 10.4: Exact solution (solid), first-order approximation (dash-dot), and secondorder approximation (dashed) for the Van der Pol equation over a large time interval.
f and its partial derivatives with respect to ( 2 ,E ) up to order N are continzlous
and bounded for ( t , x , ~ E) [O,oo)x Do x [ - E O , E O ] , for every compact set
Do c D; if N = 1, f/ax](t,x, E ) is Lipschit2 in (1,E ) , unifonnly in t;
where xo(t) is the solut,ion of the nominal system (10.7). It turns out that exponential st.ability of the origin as an equilibrium for (10.7) ensures that the origin of
i = A(t)z will be exponentially stable for every solution xo(t)that starts in the set
R. To see this point, let
af
[a
17 and its derivatives up to order N are continuous for
E
E [-.so, €01;
and write
the origin is an exponentially stable equilibiizrm point of the nominal system
(10.7);
there is a Lyapunov function V ( t , x ) that satisfies the conditions of Theorem 4.9 for the nominal system (10.7) for ( t ,x ) E [O, oo) x D and { l V l ( x )5 c )
is a compact subset of D.
Then, for each compact set R c {1.V2(x)5 pc, 0 < p < I ) , there is a positizie
constant E* such that for all to 2 0, 770 E R, and (€1 < E * , eqlrations (10.1) and
(10.2) have a unique solution x(t,E ) , unifonnly bounded on [to,oo), an,d
where O ( E ~holds
) uniformly in t for all t 2 to.
Ao(t) = z ( t , O,0)
0
If the nominal system (10.7) is autonomous, the set R in Theorem 10.2 can
be any compnct subset of the regio~iof at.tract,ion of the origin. This is a co~isequence of (the converse Lyapunov) Theorem 4.17. since the Lyapunov function V ( x )
provided by the theorem has the property that any compact subset of the region
of attraction can be included in the interior of a compact set. of the form { V ( x )5 c).
Proof of Theorem 10.2: Application of Tlirorriii 9.1 sllows that tlirrr is 51 > 0
,
+
efAo(t) + B ( t )
A(t) = Ao(t) [ A ( t )- Ao(t)]
so that the linear system i = A(t)z can be viewed as a linear perturbation of
3i = Ao(t)y. Since (af/ax](t,x,O)is Lipschitz in x, uniformly in t ,
On the other hand. by exponential stnbility of the origin of (10.7) and Theorem 4.15,
we know that the origin of the linear system a = Ao(t)y is exponentially stable.
xo(t) = 0 to show that the
Therefore, similar to Example 9.6. we can use limt,,
origin of the linear system i = A(t)z is exponentially stable.
xo(t),O), wc sac that yl is
Since Ilxo(t)()is boulided alid gl(t,t o ( t ) )= [Df/aE](t,
bounded for all t 2 to. Hence, by Theorem 5.1, we conclude that x l ( t ) is bounded.
By a simple induction argument, we can see that x2(t), . . . , ~ ~ - ~are( bounded.
t )
So far, we have verified that the exact solution x(t,E ) and the approximate
solution ~ f ! :xk(t)Ekare uniformly bounded on [to,oo) for sufficiently small IEI.
All that remains now is to analyze the approximation error e = x - C~G'
xk(t)~~.
The error analysis is quite similar to what we have done in Sectioli 10.1. The error
satisfies (10.10), where pl and p l satisfy (10.11), (10.13) and