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System stability with combined derivative constraints on time varying gains
Automatica, Vol. 14, pp. 93 94.
Pergamon Press, 1978.
Printed in Great Britain
Correspondence Item System Stability with Combined Derivative Constraints on Time Varying Gains* S. R A M A R A J A N t
and
K . L. P. M I S H R A t
Key Word index System theory; linear systems; time-varying systems; stability constraints tnot in the standard list).
Summary For feedback systems with a time-varying gain in the
multiplier given by tl), a quite different criterion is obtained involving k(t), I~ttj and k'(t) for the stability of the system, namely (for a > b )
feedback path novel conditions are obtained for L2-stability by proving the positivity of certain operators. The proof is presented for the linear time-varying system and the corresponding results for non-linear time-varying systems can be obtained in a similar fashion. An example illustrates the superiority of the new criterion even though it may not be always superior. An additional requirement to the usual ones is that the time-varying gain must he differentiable twice with respect to time.
<(a-b)
2ab-
.
An example is given to illustrate the advantage. The above criterion combining kit), k(t) and ~tt) seems to be new. By stability here we mean Lz(R+)-stability in which square integrable inputs produce square integrable outputs defined on the positive real time. The principle employed to prove stability is Zames'[6] positive operator theory. Briefly, if the open-loop operator can be factored into two positive operators one of which is strongly positive with finite gain then the feedback system is L2input output stable. For detailed definitions and other details regarding the positive operator theory and L2-stability refer to[6, 7].
Introduction BROCKETT AND FORYS[1] obtained (for feedback systems with a linear time-invariant operator having a t ransfer-function G (s)in the forward path and a linear time-varying gain k(t) in the feedback loop as shown in Fig. l ) a multiplier Z(s)=(s+b)/(s+a)
criteria: derivative
(1)
for stability together with a restriction on k(t), namely,
[~ltl/kltl<2 min (a, hi.
121 M I
(,+ +ox )
'?
-f
•.
"l
. . . . . .
l"1
i
I
M2 Fff;. 1. Linear time-varying feedback system. FIG. 2. System after the-introduction of the multiplier. But a notable result was obtained by Gruber and Willems[2] with a very general multiplier Z(s) and the condition on k(t) being
[((t I/k(t) < 2b
System description The feedback system considered is as shown in Fig. 1, as mentioned before, with a linear time-invariant operator having a causal transfer-function G(s) in the forward path and a linear time-varying gain k(t) in the feedback path. It is assumed that
(3)
where Z ( s - b ) is strictly positive-real. (The difference and relation between strict positive-realness of Z(s) and Re Z(j¢~)) > 6 > 0, V(o is well discussed in[3]). Sundareshan and Thathachar[4] obtained using noncausal multipliers (for stability) a condition on k(t)ofthe form
-2a<[qt)/k(tl<2b
L- I G(sla=g(t)ELl ~L~
(i)
k(t)=>c>O, Vt
(ii)
and that the gain
(4)
and difli~rentiable twice with respect to t. It is assumed that the input xlt) and also 2{t)eL 2 and also that all initial conditions are zero without loss of generality[7]. After the introduction of the multiplier Z(s) given by (1) suitably, the system looks as one shown in Fig. 2. Thus there results the operators M 1 and M 2 and later it is shown that M1 is strongly positive with finite gain and M2 positive which implies that the system is L2-stable. Now the main result :
both with upper and lower bounds on ~(t)/k(t). Very recently Venkatesh[5] has obtained some improved integral conditions on ,{(t)/k(t) employing causal multipliers. In this paper for the
*Received 13 December 1976, revised 31 May 1977. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by associate editor B. D. O. Anderson. +The authors are with the Department of Electrical and Electronics Engineering, Regional Engineering College, Tiruchirapalli-620015, India.
Theorem. The system shown in Fig. 1 with assumptions ti) and (ii) is L2-stable if(for a > b) Re ( r i ° + b ! G(jo))> ~ >0, V(o (1¢o + a l 93
(5)
94
Correspondence
Item
Now an example is given which illustrates the advantage of the above theorem.
and
(2ah
(6)
Example Consider a system in which G(s) is such that ZG(s) satisfies condition (5) where
Proof It is easily seen that operator M~ is strongly positive with finite gain under condition (5) of the theorem. Referring to Fig. 2, the input and output of operator Mz are (.~, + by) and (.9k+ y~.+ ayk) respectively, for simplicity y is written for y(t) and k for k(t) and the dot above denotes differentiation with respect to time t. Now the operator M e is positive if (.('+by,
y(ak+[;)+f'k),>_O
Z(s)-ts + l)/(s+30) so that a = 30 and b - 1. Let the gain k(t) be given by
k(t) = 1 0 - e x p [sin wt] so that k(t ) > 0 for all t and is obviously differentiab[e twice with respect to t. For this k(t) Brockett and Forys'[I] criterion assures stability for
for all T > 0 . The l.h.s, of the inequality q-
= ~, ~
-
7'
yZb(ak+fQdt
k~;'2dt + ~
w =< 11.69
([;+a+b k)y)',dt
r~
and the present result for
o
2.24_< w_< 12.48.
and since the first term is non-negative this is
Conclusions >
y2(abk+bfQdt+ o
(l~+a+b k)d(y2/2) o
which after integraiing the last term by parts and assuming that 3'(0 ) = 0 becomes
=
I
I
v2{abk+b~ldt
o
+[]({T)+a+hk(T)]
y2(T) 2
_ f l ) ~ (k'+a+b k)dt.
Combining the first and the last term this is
= [ , ~ { T ) + a + b k ( T J ] Y a l T} + f r y 2 [ a b k + ( b - a ) [~ ](~dt
and is >0 for all ~ 0 under condition 16) of the theorem (for a > b) and hence the positivity of M2. Thus the proof.
Stability results are obtained for linear time-varying systems with combined conditions on k(t), ,(:(t ) and ~:'(t). The appearance of ]~(t) in the stability inequality appears to be new. Whether similar criteria can be obtained with more general and also with noncausal multipliers is a problem to be investigated.
Acknowledgement - T h e authors thank Prof. P. S. Mani Sundaram, Principal, Regional Engineering College, Tiruchirapalli620015, for his encouragement and help. Also sincere thanks are due to the referees for their valuable suggestions. RcteYell('es
[1] R.W. BROCKETTand L. J. FORYS: On the stability of systems containing a time-varying gain. Pro('. 2rot .4llerton Cotl[i Circuit and System 7 heory, Urbana, Illinois i 19641. [2] M. GRUBER and J. L. WILLEMS: On a generalization ol ~.hc circle criterion. Proe. 4th Allerton Con£ Circuit aml Sv.stem 7heory, Urbana, Illinois ( 19661. [3] J. H. TAYLOR: Strictly positive-real functions and the Lefschetz-Kalman Yakubovich (LKY) lemma, lEE1: 7runs. Circuits and Systems, CAS-21,310-311 (19741. [4] M. K. SUNDARESHAN and M. A. L. THATHACHAR: L.~stability of linear time-varying systems conditions involving noncausal multipliers. IEEE Trans. Auto. Comrol AC17,504-510 (1972). [5] Y. V. VENKATESH: Geometric stability criteria for certain non-linear time varying systems. Int. d. Non-Linear Mech. 10,245 252 (1975). [6] G. ZAMES: On the stability of non-linear time-varying feedback systems. Proc. N.E.C. 20, 725~ 730 (1964). [7] Y.S. CHO and K. S. NARENDRA: Stability of non-linear timevarying ieedback systems. Automatica 4,309 325 (1968).
Pergamon Press, 1978.
Printed in Great Britain
Correspondence Item System Stability with Combined Derivative Constraints on Time Varying Gains* S. R A M A R A J A N t
and
K . L. P. M I S H R A t
Key Word index System theory; linear systems; time-varying systems; stability constraints tnot in the standard list).
Summary For feedback systems with a time-varying gain in the
multiplier given by tl), a quite different criterion is obtained involving k(t), I~ttj and k'(t) for the stability of the system, namely (for a > b )
feedback path novel conditions are obtained for L2-stability by proving the positivity of certain operators. The proof is presented for the linear time-varying system and the corresponding results for non-linear time-varying systems can be obtained in a similar fashion. An example illustrates the superiority of the new criterion even though it may not be always superior. An additional requirement to the usual ones is that the time-varying gain must he differentiable twice with respect to time.
<(a-b)
2ab-
.
An example is given to illustrate the advantage. The above criterion combining kit), k(t) and ~tt) seems to be new. By stability here we mean Lz(R+)-stability in which square integrable inputs produce square integrable outputs defined on the positive real time. The principle employed to prove stability is Zames'[6] positive operator theory. Briefly, if the open-loop operator can be factored into two positive operators one of which is strongly positive with finite gain then the feedback system is L2input output stable. For detailed definitions and other details regarding the positive operator theory and L2-stability refer to[6, 7].
Introduction BROCKETT AND FORYS[1] obtained (for feedback systems with a linear time-invariant operator having a t ransfer-function G (s)in the forward path and a linear time-varying gain k(t) in the feedback loop as shown in Fig. l ) a multiplier Z(s)=(s+b)/(s+a)
criteria: derivative
(1)
for stability together with a restriction on k(t), namely,
[~ltl/kltl<2 min (a, hi.
121 M I
(,+ +ox )
'?
-f
•.
"l
. . . . . .
l"1
i
I
M2 Fff;. 1. Linear time-varying feedback system. FIG. 2. System after the-introduction of the multiplier. But a notable result was obtained by Gruber and Willems[2] with a very general multiplier Z(s) and the condition on k(t) being
[((t I/k(t) < 2b
System description The feedback system considered is as shown in Fig. 1, as mentioned before, with a linear time-invariant operator having a causal transfer-function G(s) in the forward path and a linear time-varying gain k(t) in the feedback path. It is assumed that
(3)
where Z ( s - b ) is strictly positive-real. (The difference and relation between strict positive-realness of Z(s) and Re Z(j¢~)) > 6 > 0, V(o is well discussed in[3]). Sundareshan and Thathachar[4] obtained using noncausal multipliers (for stability) a condition on k(t)ofthe form
-2a<[qt)/k(tl<2b
L- I G(sla=g(t)ELl ~L~
(i)
k(t)=>c>O, Vt
(ii)
and that the gain
(4)
and difli~rentiable twice with respect to t. It is assumed that the input xlt) and also 2{t)eL 2 and also that all initial conditions are zero without loss of generality[7]. After the introduction of the multiplier Z(s) given by (1) suitably, the system looks as one shown in Fig. 2. Thus there results the operators M 1 and M 2 and later it is shown that M1 is strongly positive with finite gain and M2 positive which implies that the system is L2-stable. Now the main result :
both with upper and lower bounds on ~(t)/k(t). Very recently Venkatesh[5] has obtained some improved integral conditions on ,{(t)/k(t) employing causal multipliers. In this paper for the
*Received 13 December 1976, revised 31 May 1977. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by associate editor B. D. O. Anderson. +The authors are with the Department of Electrical and Electronics Engineering, Regional Engineering College, Tiruchirapalli-620015, India.
Theorem. The system shown in Fig. 1 with assumptions ti) and (ii) is L2-stable if(for a > b) Re ( r i ° + b ! G(jo))> ~ >0, V(o (1¢o + a l 93
(5)
94
Correspondence
Item
Now an example is given which illustrates the advantage of the above theorem.
and
(2ah
(6)
Example Consider a system in which G(s) is such that ZG(s) satisfies condition (5) where
Proof It is easily seen that operator M~ is strongly positive with finite gain under condition (5) of the theorem. Referring to Fig. 2, the input and output of operator Mz are (.~, + by) and (.9k+ y~.+ ayk) respectively, for simplicity y is written for y(t) and k for k(t) and the dot above denotes differentiation with respect to time t. Now the operator M e is positive if (.('+by,
y(ak+[;)+f'k),>_O
Z(s)-ts + l)/(s+30) so that a = 30 and b - 1. Let the gain k(t) be given by
k(t) = 1 0 - e x p [sin wt] so that k(t ) > 0 for all t and is obviously differentiab[e twice with respect to t. For this k(t) Brockett and Forys'[I] criterion assures stability for
for all T > 0 . The l.h.s, of the inequality q-
= ~, ~
-
7'
yZb(ak+fQdt
k~;'2dt + ~
w =< 11.69
([;+a+b k)y)',dt
r~
and the present result for
o
2.24_< w_< 12.48.
and since the first term is non-negative this is
Conclusions >
y2(abk+bfQdt+ o
(l~+a+b k)d(y2/2) o
which after integraiing the last term by parts and assuming that 3'(0 ) = 0 becomes
=
I
I
v2{abk+b~ldt
o
+[]({T)+a+hk(T)]
y2(T) 2
_ f l ) ~ (k'+a+b k)dt.
Combining the first and the last term this is
= [ , ~ { T ) + a + b k ( T J ] Y a l T} + f r y 2 [ a b k + ( b - a ) [~ ](~dt
and is >0 for all ~ 0 under condition 16) of the theorem (for a > b) and hence the positivity of M2. Thus the proof.
Stability results are obtained for linear time-varying systems with combined conditions on k(t), ,(:(t ) and ~:'(t). The appearance of ]~(t) in the stability inequality appears to be new. Whether similar criteria can be obtained with more general and also with noncausal multipliers is a problem to be investigated.
Acknowledgement - T h e authors thank Prof. P. S. Mani Sundaram, Principal, Regional Engineering College, Tiruchirapalli620015, for his encouragement and help. Also sincere thanks are due to the referees for their valuable suggestions. RcteYell('es
[1] R.W. BROCKETTand L. J. FORYS: On the stability of systems containing a time-varying gain. Pro('. 2rot .4llerton Cotl[i Circuit and System 7 heory, Urbana, Illinois i 19641. [2] M. GRUBER and J. L. WILLEMS: On a generalization ol ~.hc circle criterion. Proe. 4th Allerton Con£ Circuit aml Sv.stem 7heory, Urbana, Illinois ( 19661. [3] J. H. TAYLOR: Strictly positive-real functions and the Lefschetz-Kalman Yakubovich (LKY) lemma, lEE1: 7runs. Circuits and Systems, CAS-21,310-311 (19741. [4] M. K. SUNDARESHAN and M. A. L. THATHACHAR: L.~stability of linear time-varying systems conditions involving noncausal multipliers. IEEE Trans. Auto. Comrol AC17,504-510 (1972). [5] Y. V. VENKATESH: Geometric stability criteria for certain non-linear time varying systems. Int. d. Non-Linear Mech. 10,245 252 (1975). [6] G. ZAMES: On the stability of non-linear time-varying feedback systems. Proc. N.E.C. 20, 725~ 730 (1964). [7] Y.S. CHO and K. S. NARENDRA: Stability of non-linear timevarying ieedback systems. Automatica 4,309 325 (1968).