277
CORRESPOhmENCE
LINEAR TIME-NWRIANT
Exponential Boundedness of Nonlinear and
Time-Varying Feedback Systems
Absiracf-This correspondence considers a system with a linear
time-invariant part and a nonlinearity or time-varying gain in a
feedback loop, and obtains conditions for the exponential boundedness of the signals, utilizing the passiveoperator technique. It
turns out that the transfer function of the linear part shifted in
argumenthas to satisfy thesame conditions as forstability in
order to ensure exponential boundedness.
Fig. 1.
Feedback system.
Fig. 2.
Transformation.
I. INTRODUCTIOS
Studies in the qualitativebehavior of feedback systems have thus
far been madly confined to thedetermination of criteria for a s y m p
t.otic and input-out,put st.ability [I]. In many situations it would be
of interest. to have a bet,ter knowledge of the behavior of signals in
the system, suchas the rateof decay [2]. An att,empt in this direction
has been made iu this correspondence by considering a system consisting of a. linear time-invariant. partand a nonlinear or timevarying gain in cascade in a feedback loop, and deriving conditions
for the L2 boundedness of e<exp ( a t ) , where e, represents signals in
the system.
11.
NATHENATICAL
PRELIN~ARIES
Here deta.iled definitions will be omitted a.s these can be found in
considered t.hroughout the
Zames [I]. L? boundedness willbe
correspondence.
a) X denotes t.he input, space containing funct,ions z. For establishing exponent,ial boundedness, a space Lz* is defined: z E L f , if
x(t) exp ( a t ) E
( a > 0).
b)Lineartransferfunction:
G(s) represents the Laplace t.ransform of t.he linear t.ime-invariant operator G.Let G, and G b denote
the operat.ors corresponding to G(s - a) and G(s a ), respectively. I t is fnrther required that g ( t ) E L l a , i.e., g ( t ) exp ( a t ) must
be in L1.
c) 0perat.or
Let X: Ln. +
be any opera.tor having the form
-Vx(t) = A ' [ x ( t ) ] . The operator N can belong tothe following
classes.
I
I
+
J
Hz
Fig. 3.
Transformation.
A V :
X m if the inequality 0 5 (x - y)[N(z) - -V(y)] <
K ( x - y)* holds for all x and y and A'(O) = 0.
2) S E X O , . if
E X m and X ( -z) = - N ( z ) .
3) X E IYt if X x ( t ) = f ( t ) x ( t ) , where 0 5 f ( t ) < K a n d f ( t ) is a
differentiable function.
1) B E
d ) Let G be the class of operators of t,he type 2:L?, -+ Lze satisfying an equation [a]
m
zz(t) = x Z C C ( t
-~
i=Q
where
i
+
)
L-
z(t -
-
m)
d7
T ) ~ ( T )
- - - and 0
1) zi are real constants, for i = 0, 1,
2) 2 0 > 0
3) z ( ) is a real valued funct.ion on [0,
where G and N are the operators defined in Section 11.
Definition: The system defined by (1) is said t o be exponent.ially
bounded with order a,if el and e~ are in Lf,Le., 11 ei exp ( a t ) 11 < m
and 11 e e x p (at) 11 < m.
The problem is t o find under what conditionsinputs Z I , belonging
~
t o LUare mapped into el and e which are also in LP. Now, for convenience, the system is transformed in steps as shown in Figs. 2 and 3.
The t.ransfonned signa.ls in Fig. 3 are related to the signals in the
original system (1) by
=
T~
< T~ < ,.-
-
and
Theorem 1
Suppose there is an operator 2 in 6: and constants
which
111. STATENENT
OF THE PROBLEM
The system (Fig. 1) is described by thefollowing set of equations:
el
= 51 - y ~ , e2 = 22
yl = Gel,
+ yl
Re [ Z ( j w )
If y # 0, then let
+ -j+][G(jw
- a)
+ l/X]
lim I wG(jw - a) I
=
2
6,
y
2 0,8 > 0 for
for all
0.
yz = N ~ z
Manuscript received March 24, 1969; revisedOctober 21, 1969.
1) zi (for i 2 1) and z ( - ) are everywhere nonpositive,
W.
(3)
278
IEEE TBAWSXCTlONS ON AUTOhL4TIC COSTROL, APRIL
1970
Comments on “On the Optimal Angular Velocity
Control of Asymmetrical Space Vehicles”
Abstract-The purpose of this correspondence is toexplain
the surprisingly simple feedback laws derived by Debs and Athans
in their paper,l and to indicate a broader class of systems that
can be solved by this approach.
then 1; el exp (ai)11 < m and e, exp (at) 11 < m .
Two useful lemmas which can be easily derived are stated first.
I. S-Y
Lemma 1
OF THE RESULTS OF
The state equations of the system are
Let. P be any positive (strongly positive) operator; let T be a
time-varying gain whose derivative exists almost everywhere. Then
rP is positive (strongly positive) if T is posit.ive and nonincreasing
dmost everywhere.
21 = alZ?za
Lemma 2
Let Q be a positive (strongly positive) operator; let r be a timevarying gain. Then rQr is posit.ive (strongly positive).
PToof: Let x E IT,,,. Referring to Fig. 3, let 11f = z(j w )
~ j
+
where conditions 1) and 2) hold. Now, the inputs to the
system are in
L.H I k strongly positive as per (3) and Lemma 1. X * ( j ~ =)
M (j w
a) and hence, MI, also belongs to the class of allowable
multipliers for AT,,,[SI. Hence, H? is positive from Lemma 2. Thus,
from a well-knom theorem of Zames [l], and €2 are in Ls. From (2)
it follows that el and e2 are in L a .
The result for N E N o , can be established similarly using t,he corresponding mult.iplier. These resdts can be improved by extending
the interval of definition of the mult,iplier from [0, m ) to ( - Q , m )
~41.
+
w
= a223X1
53
= afl12s
1 ) H ( s - j3) is strictly positive real (SPR);
2) H ( s ) [ G ( s - a) 1 / K ] is SPR;
3) f I
2Bf(l - f/K).
+
JI
=
+ + X?) + (l/q)(ul2+ US’ +
2
(~(zI’ 5 2 ’
l[
where p
1 dt
(2)
up = -px2,
u3
=
(3)
-qx3.
If the performance index is
JZ =
[
IpCf~(xd+A(m)Cf3(23)1
(4)
where ft(-) and g l t ( - ) , k = 1,2,3, are positive definite functions
of a single variable such that
fk(0)
k = 1,2,3,
gt(0) = 0,
= 0,
Q
> 0.
The control law which minimizes ( 4 ) is
PTooj: Fig. 3 reduces t.0 a linear system G(s - a) f 1;‘K with a
time-varying feedback gain ( f ( t ) ) / ( l - f ( t ) / K ) when N E N t .
The Theorem follows from earlier results [5] recast in a functional
setting. Some recent results of Freedman and Zames [6] can also be
used here. These involve a v e r e e logarkhmic variation criteria on
j ( t ) in contrast t o condition 3).
u1 =
-qh1(21),
h t ( 0 ) = 0, h,t-l(.) exists,
REFERENCES
1
.11. G. Zames. “On the inDut-outout st,ahilit.v of time-xrarrine nonlinear feedback system, pt. I < c o n d i t i ~ n s ~ d e r i ~ . e d uii&ip:s
sin~
of loop gain,
conicit,y,
and positivity
IEEETrans.
Automatic
Control. vol. AC-11, pp. 228-238, April i966.
- “On theinput-output stabllay of time-varyin* nonlinear
feeddack systems,pt. 11: con$tions involving circlesoin the freauencv Dlane and sector nonhnearit.ies.” IEEE Trans. Automatic
I21
I31
> 0,
and
gl;!ut) =
nonlinearities,” S I A M ‘ J . Control, vol. 6, no. 1, pp. 891U8, 1 Y 6 8 .
[5] M . Gruber and J. L. W i e m s , “On a generalization of the circle
crit,erion,” PTOC.4th Ann. Allerton Conf. Circuzt and System Theory,
196% nn
_=. 837--RA!?
161 M . I. Freedman and G. Zames, “Logarithmic variation criteria
for thestability of systemswithtime
varying gains,’’ S I A M J .
Controt, vol. 6, pp. 48‘7-505, August 1968.
h’
hk(2)
n’[(-ut/p)hk-l(-Uk/Q)
x1
#
0, X.
=
1,2,3
ax
- fk(hk-’(--lLk/Q))l.
11. EXPLANATION
OF THE RESULTS
The results of the optimization problem considered in Section I
become obvious if the corresponding optimization problem of a n
equivalent system is considered. Thesystem equivalent to (1)
for the performance indices considered is the simple decoupled
linear system giwn by
x 1 = u1
x 2
- “Stability. conditions for svstems with- monotone and slope
(5)
u3 = -ph3(xa)
hk(~l;)~l;
fk(2d =
H. S. RXYGANATH
Dept. of Elec. Engrg.
Indian Institute of Science
Bangalore 12, India
u2 = -qhr(Zd,
where hl;(- ) are continuous and differentiable scalar valued functions of a single variable such that
M. A . L. THATHACHAR
-_--. --. ---.
[fa2)
> 0, the optimal feedback control law is
u1 = -qx1,
+
resJr&Jed
(1)
+ (l/dcg*(Ul)+ gn(u2) + 93(~”)11dt
The system (1) with 11’ E Nt is exFonentially bounded with order
if there e.nists an H ( s ) such that
[41
x2
+ u1
+ US
+ u3
t hf a2
a3 = 0.
Corresponding to a performance index which is to be minimized,
given by
~
Themem 2
a
DEBSA X D ATEL~SS~
=
u2
53 = ua.
Manuscript received October 9, 1969.
1-4. S. Debs and M. bthans, I E E E Trans. Automatic Control (Short
Papers), vol. AC-14, pp. 80-83, February 1969.