335
CORRESPONDENCE
Model Reference
1
and
1
j(Bk - Bk*)
gk
where
Then
+ b16 + b2s
* denotesthe
(4)
complex conjugate.
Yk
(.- 4)
hTR& = -
Bk
where
t
dk
= G(jpJ
1
+=
>0
K -
and
hTg& = 0.
(5)
Further, from (3) and (4) it follows that
Yk
--R~,Af--Ifpdr;=O
Bk
9d
+ I&
= 0.
(6)
In addition, it can be shown that2,3
Re [hTRk(iWI - A)-%]
for all real W,
Fig. 3.
Suggested Liapunov redesign.
THELMPUXOV
FUNCTION
-~
THEPROBLEM
not conclude that the system is asymptotically stable if r(t) is not constant. However,
Consider thesystem
defined bythe
it is stable and should tend to operate near
equations
e = 0, whereas the design using the Hermite
f = Ax-bKu
matrix may not.
RICHARDV . MONOPOLI
u = hTx
(1)
Dept. of Elec. Engrg.
University of Massachusetts
which represents a transfer function
Amherst. Mass.
G(s) = hT(d- A)-%
(2)
Stability of Linear TirneInvariant Systems
INTRODUCTIOS
I t is well known that the stability of a
linear time-invariant system can be detera Liapunovfunction
of the
minedusing
quadratic form.However, forestablishing
the asymptotic stabilityof such a system for
all linear negative feedback gains in a given
range, a Liapunovfunction whichyields
both necessary and sufficient conditions has
not been available so far. This correspondence presents such a function and affirms a
recentconjecture
of Narendra.1The
interest in this problem is due to its being a
limiting case of the Lur'e problem.
Manuscript received December 30, 1966.
1 K. S. h'arendra. 'A conjecture on the existence
of a quadratic Liapunov function for linear time-in?n
variant feedback systems." Pvoc. 4fhAlkrbn Co~rf.
Circuit and System Tlzeory (Umvermty of Illinois.
Urbana). 1966.
with a linear negative feedback gain K.
A is an ( n x n )stable matrix, x, b, and h
are real n vectors, and u is a scalar function
of time.
I t is required tofind a Liapunov function
which yields necessary and sufficient conditions for the asymptotic stabilityof the system for all feedback gains K in (0,
Consider as a possible Liapunov function2.3
v = +xTPx +
+
(8)
p = +xT( ATP+ P A ) x
- KuxT [P b - Bo ATh] - &hTb
+ S k R [ ( h T R(dh T R d x )
+ (hTga)(hTSLAx)]
-
&Kz[(hTRkx)u(hTR&)
k
+ (hTgk~)u(hTg&)].
) be frequencies such
Using (5)
P can
P = 3 x (ATP
I m G(jpk) = 0
and
+ (hT4S)'2j
where P >O.
Thederivative
along thetrajectories
of the system (1) is given by
PRELIMINARIES
* *
y [(hTR-)'
k
a).
Let pk (k = 1,2,
that
+r(jaKu2
K
+ P A ) x - BohTb(Ku)*
- KoxTIPb
,
-!-
be rewritten as,
- BoA'h]
&K[(hrRs)(hTRkAX)
k
Let B k be amatrix
defined by
associatedwith
pk
+ (hTgm)
(hTiJdx)]
2 K. S. Narendra and C. P. Neuman. 'Stability of
a class of differential equations with a single monotone
nonlinearity." J . S I A M on Control, vol. 4. pp. 295308 May 1966.
i M. A. L. Thathachar, Y. D. Srinath, and H. K.
Ramapriyan. "Solution,of a modified Lur'e problem."
submitted for publicatlon In I E E E Trans. on Automatic Confrol.
IEEE TRANS-4CTIONS ON AUTOM4TIC CONTROL,
336
INTERPRETATION
OF THE CRITERION
Adding and subtracting thefollowing terms,
1) K6a2 (1
Brockett and WillemsL have shown that
a necessary and sufficientconditionfor
a
transfer function G(5) to be stable f o r all
linearnegativefeedbackgainsin
(0, R ) is
thatthere should existamultiplier
Z(s)
such that
- $)
takes the form
Z(s) [ W )
iJ= +xT(ATP+ P A ) x
+ f]
is positive real, and
- (s&T~+
(KD
n) ~
P
JIJNE 1967
is the existence of a Liapunov function of
the form
V = xTPx
+ KxTAfx
where P is positive definite, and
A€ = &hhT
+
&[RkThhTRk
Is
+9kThhTgk]
is positive semidefinite.
Thus the Liapunov function is the sum
of a positive-definite quadratic form and a
positive-semidefinite quadratic form
dependinglinearly
on K asconjecturedby
Narendra.1
$1. A. L. THATHACHAR
?VI.D. SRINATH
Dept. of Elec. Engrg.
Indian Inst. of Sci.
Bangalore 12, India
where Xi and pk are frequencies a t which
and
- K6a2 (1 -
$) .
d Arg [ GGu)
du
Setting
f
=
ak=&p~
+ P A ) x - (BohTb+
(KU)~
[
K
(1 - =
K
- K6a2 (1
-
< 0 a t o = hi
3
-
- K U X ~ Pb Bo-ATh - 6h
yk
-4ufkor's Reply6
and using relations (6),
QxT(ATP
-
'I
+=
K
This assumes thatthe
first
nonzero
value of Arg GCjo) is negative.
A slight extension of the result yields the
form of multiplierfor asymptotic stability
as
Kdk)
RxTh]
$) .
THESTABILITY
CRITERION
A direct application of the LIeyer-Kalman-Yakubovich lemma2 gives the criterion for asymptotic stability as
+ 1 (1 - K f dX)h'Rr]
yk
.[id - A ] - ' b 2 0
(12)
for all real a,m-here 6>0.
Use of property (7) yields the criterionin
the form
Re Z Go)G(ju)2 0
(13)
for all real o,where
Z(S) = 6
+ Bos
Since &>_0 and K < X , one canalways
choose 71;and p k to get
where 6>0 and Bo, Ck are any non-negative
is denumbers. If only neutralstability
sired, 620.
where 6>0 can be arbitrarily small.
By expanding (18)in partial fractions,
Z(s) can be put in the same form as (15).
Thus the criterion (13) developed using
the Liapunovfunction (8) is equivalent to
the criterion of Brockett and IVillems and
is hence necessary and sufficient.
If the phase angle of GGo) starts in the
positive direction, one can consider l/(GCjw)
1/ X )in the place of GGu) 1
and proceed as before.
The results can be extended to the case
where G(s) is the ratio of equal-order polynomials. L-sing a well-known transformation: the stability of such a system is equicalent to that of a system having a transfer
function G ( 5 )-G( m ) withnegativefeedbackgains in [0, KT/(l+EG( m))], and hence
the results derived earlier can be applied.
+
+ /x
Thathachar and Srinath are to be congratulated onderiving a simple Liapunov
function for proving the stability of a linear
time-invariant feedback system in the entire
range 0 <K < m (or E ) of the feedback gain.
The proof is based on the fact that a ZLC(S)
function always exists so that G ( s ) Z x ( s )is
positive real if the feedback systemis stable
for all feedback gains
in therange 0 <K < m .
Theauthors'contribution
lies inchoosing
theterms of theLiapunovfunction,particularly the matrices RK and Ig in (8) so
that the desired multiplier ZLC(S)
can be obtained. Since the existence of the multiplier
Z L C ( S )is both necessary and sufficient t o
prove the stability of the feedback system,
the existence of the Liapunov function of
the form derived is both necessary and sufficient for stability in the range 0 <K < m
(or E ) .
On the basis of the results obtained, one
canproceed to determine, for the time-invariant case, aLiapunovfunctionhaving
the general form
V(r)= XTPZ
+
KXTQZ
(19)
where P is a positive-definite matrix and Q
a semidefinite matriv which can beexpressed as
Q=
where Hi is an
1&HihltTHiT
( E Xn)
matrix. The Liapunov
function (Sj derived in the correspondence
can be used for the most restrictive case t o
obtainan LC function Z L C ( S ) ,for which
G(s)Zx(s) is positivereal. Inmost cases,
however,where Z R C ( S ) , Z R L ( Sor) , Z R L C ( S )
multipliersexist,simpler
Liapunov funcCONCLKSION
tions of the general form (19) can be deterIt has been shown that a necessary and
mined. In suchcases, theLiapunov funcsufficient condition forthe asymptotic stabil- tionfor the linear time-invariant case can
ity of a linear time-invariant system for ail
be suitably modified for different nonlinear
linear negative feedback gains K in (0, IC)
situations.2
The form of the 0 matrix is closely re-
CORRESPONDENCE
337
TABLE I
'i
>ft!ltiplier ~ ( s )
GW a s ) P.R.
Q
Nonlinear Gain for which
System Is Stable
1) Q = 0 (Null Matrix)
Konlinear
and
Time-Varying
Gains
The matrix B determined in this manner is
the matrix of a positive definite quadratic
form.
\\-ith B now given by (41, one has
Constant
i
0
f(C.
o<--<m
-1
w2
1
+1
c
f(C)
2) Q = uhhT
O<-<m
(1 + a d
0
(Popor)
Q = ahhT
3)
+
riT = hT(qiI
ZRL or ZRC
tiTiTiT
I
+ A1-l
and thus
Monotonic Increasing Function f(u)
p ( x , t ) = - x'Dx
I
where
f(u)
o<-<=
r
C
-vi
gain functions for which the system is also
stable. Examples of this relationship are indicated in Table I.
I t is this author's opinion that the same
form (19j of the Liapunov function can also
be used to derive conditions forthe stability
of linear time-varying systems b y replacing
IC b y K ( t ) i? (19). In such cases, the time
in
derivative K ( t ) hastobeconstrained
some manner.
K.0 < K
Constant
Gain
:
ZLC
4) Q [in Equation (S)l
D =
<m
[A
+ eP(t)]x
(2)
x = ( a f?)'
P(t) =
6w2
'>
-qs
(-10 00) cos 2t.
In order that V ( x ) be a Liapunov function, those values of E , 6, and w are sought
such that D is positive semidefinite. .A sufficient condition thatthereal,symmetric,
D bepositive semitime-varyingmatrix
definite is that its principal determinants be
strictlygreaterthan
zero. Thiscondition
leads to the following constraints on E , 6,
and w :
1
K. S. SARENDR~
Yale University
Xew Haven, Conn.
The "prime" is used t o denote the transpose
of a vectoror a matrix.Theeigenvalues
XI,? for thetime-invariantpart
of (2) are
easily determined to be
-6
Xl,Z
Explicit Stability Criteria for the
Damped Mathieu Equation
INTRODUCTION
The purpose of this correspondence is t o
establishexplicitstabilitycriteria
for the
Mathieu equation with damping term 6>0:
1-
+ 66.i- + (w2 +
E
cos 2)s = 0.
(1)
Since the time-invariant part
of (1) is uniformly
asymptotically
stable,
it
is thus
knoxvn that for E sufficiently small the perturbed equation will also be stable. An application of Liapunov'sdirectmethodenables one t o obtain a bound on the parameter e within which boundstability is assured. In particular, it will be shown that
under the reasonable assumptions of large w
and small 6, the condition for stability
of (1j
becomes a very simple inequality relationship involving only e and 6, Le., independent
of the frequency w .
LIAPUSOV'S
DIRECTMETHOD
-~PPLICATIOS OF
Equation
(l),
the
damped
hlathieu
equation
with
damping
term
quency U , and small parameter
alent to the system
x1
= x2
x 2
=
-
(w'
+
E
6>0, freE , is equic-
cos 2 t h l - 46x2.
Manuscript received January 3, 1967.
=
+
__16w2
4 6 2 -
= -
€ p 2 )cos?2: - f
cos 2t - 1 < 0.
-2
The aboveinequalitiesare
t when
1 el
from which it is seen thatRe (X1.2] < O
for all 6 >0, w #O. Hence, the time-invariant
asymptotically
part of (2) is uniformly
E sufficiently
stable. This ensures that for
small, theperturbedsystem
will alsobe
stable. I t is now desired t o obtain explicit
stability criteria involving e, 6, and W.
Toward that end, choose as a tentative
(2) thequadratic
Liapunovfunctionfor
form
I:(x! 1) = V(x) = x'Bx
F(x, tj
+ w2 cos 21 > 0
4
where B ' = B = ( b i , j .
leads to
X'[C
This choice of
+
- ~(P'B
V(xj
BP~]X
where
A'B+ BA
=
-C
(3)
with the matrix C restricted to be positive
definite. Then, noting that b12 =bZ1, and
choosing C equal to the identity matrix, (3)
becomes
-2w2b12
b l l - +6bln - w2b?,>
( b l 1 - 36b12 - w2b2?
2(bll - $6b12)
= -
(;
i.
l e ~ C O S 2 t
where
A = (-w?
O
-E
I
I w'+1
One can then write
X =
e
!1 + > c o s 2 t
is a real zero of G(s)
Y).
satisfied for all
<us
( 3
and
B-
< E < ET
(64
where
\$'henever (51 and (6a) are satisfied,
then V ( x ) is positive definite with negative
V(x)
semidefinitetimederivative.Hence,
qualifies asaLiapunovfunctionandthe
system (2) is stable. If all the principal determinants of D were greaterthan some
number -y, Le.,
1
+
_f_
cos 2f > 7
0 2
where y is a positive constant independent
of X and f , then p ( x , t ) would be negative
definite, and the system ( 2 ) would therefore
be asymptotically stable.
SIMPLIFICATION
OF THE STABILITY
CRITERI.4
Consider the case when w is large and 6
relatively small, i.e.,