357

Nonlinear time-varying compensation for simultaneous performance 1. Introduction

J e f f S. S h a m m a

Departmentof ElectricalEngineering, Universityof Minnesota, Minneapolis,MN 55455, U.S.A. Received 28 April 1990

Abstract: This short note considers the use of nonlinear timevarying compensation for linear time-invariant discrete-time plants. It is shown via counterexample that the problem of simultaneous performance presents fundamental limitations which cannot be overcome by nonlinear time-varying compensation. This result is in contrast to results on simultaneous stabilization which show that limitations due to linear compensation may be removed using nonlinear time-varyingcompensation. As a corollary of these results, the conjecture that achievable disturbance rejection over stable nonlinear timevarying compensation equals that for linear compensation is refuted.

Keywords: Nonlinear

time-varying compensation; simultaneous performance; linear systems; stabilization; disturbance rejection.

Notation

LTI := linear time-invariant. LTV := linear time-varying. N L T V .'= nonlinear time-varying. y2:= (f=

( f ( 0 ) , f ( 1 ) , f ( 2 ) . . . . ):

I]f l] := ( ~ l f(n) lz)l/2< °°} • [Ifllta,bl :=

~

If(n)l 2

.vt=a

H7" JI := sup

NTf II

i~,,~ IIi It S=~o

z := unit right shift operator on t 2 (i.e., time delay).

The use of N L T V compensation for the individual objectives of simultaneous stabilization and disturbance rejection has been studied extensively. These results are summarized briefly as follows. The problem of simultaneous stabilization is to find a single N L T V compensator which stabilizes every plant in a given family of LTI plants. For families of LTI plants characterized by parametric uncertainty, N L T V compensation is superior to LTI compensation. For example, given any finite collection of LTI plants, there always exists a simultaneously stabilizing N L T V compensator (e.g., [10]). In case the plant family is characterized by a single block of dynamic uncertainty, N L T V compensation offers no advantage over LTI compensation (e.g., [6,8,14,16]). For families of LTI plants characterized by both parametric and dynamic uncertainty, N L T V compensation is generally superior to LTI compensation. The most general result along these lines may be found in [12] where necessary and sufficient conditions for simultaneous N L T V stabilization are given for certain general families of LTI plants. Further background and motivation to simultaneous stabilization problems may be found in the survey articles [7,15], the book [2], and references contained therein. The problem of disturbance rejection is to find some compensator which stabilizes a given linear time-invariant feedback control system and also minimizes the m a x i m u m response of certain 'error signals' to possible exogenous disturbances. Contrary to simultaneous stabilization objectives, N L T V compensation offers no advantage over LTI compensation for disturbance rejection. In [4,10] it was shown that in the context of optimal rejection of finite-energy (i.e., ~2) disturbances for an LTI plant, LTV compensation offers no advantages over LTI compensation. That is, LTV compensators cannot do better than LTI compensators in uniformly reducing the energy of the

0167-6911/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

358

J.S. S h a m m a / N L T V c o m p e n s a t i o n f o r s i m u l t a n e o u s p e r f o r m a n c e

resulting error responses to exogenous finiteenergy disturbances. In [9], this result was strengthened to encompass NLTV compensation. The question of LTV compensation for minimizing the maximum response to persistent bounded (i.e., fo~) disturbances was addressed in [16], where again it was shown that LTV compensation offers no advantages over LTI compensation. Further background and motivation to optimal disturbance rejection problems may be found may be found in [1,5,19] and references contained therein. In short, NLTV compensation is generally superior for simultaneous stabilization but offers no advantage for disturbance rejection. This short note addresses the possible advantage of NLTV compensation for the combined objective of simultaneous performance. That is, for a given family of LTI plants, find an NLTV compensator which (1) stabilizes every admissible plant and (2) achieves a prescribed level of disturbance rejection for every admissible plant.

2. Conjectures and eounterexamples In this section, it is shown via counterexample that unlike problems of simultaneous stabilization, the problem of simultaneous performance presents fundamental limitations which cannot be overcome by NLTV compensation. In the discussion that follows, familiarity with the disturbance rejection problem framework and related notions of stabilization, causality, and well-posedness is assumed (cf., [5,7,20]). Let J ( P ) denote some measure of optimal disturbance rejection. That is J ( P ) := inf{ lIT(P, K)I1" K is any LTI

stated as the following optimization: inf sup { ]l T ( P , K ) ] I : K is any NLTV K

P~F

compensator which stabilizes every P ~ ~ }. A lower bound on the achievable simultaneous performance for this family of LTI plants is given by the quantity sup J ( P ) . P~

A reasonable conjecture is that this lower bound may be approached via NLTV compensation. In other words, the achievable simultaneous performance is equal to the worst case individual performance. The intuition behind such a conjecture is taken from the method of proof in NLTV stabilization results. More precisely, in showing NLTV compensation is superior to LTI compensation for simultaneous stabilization, one typically constructs an N L T V compensator which appropriately 'cycles' through a collection of stabilizing LTI compensators. Thus, it is reasonable to believe that such an approach may be possible for simultaneous performance. It is shown via counterexample that this conjecture is not true in general.

Conjecture 2.1. Given any e > 0, there exists an NLTV compensator, K, which stabilizes every P ~ , ~ and sup liT(P, K)II ~ sup J ( P ) + e .

p ~,~-

p ~ oj

The following lemma will prove useful in constructing the counterexample. Essentially, an example is provided for which the lack of stable invertibility is an 'open' property.

stabilizing compensator) where T(P, K ) is a given operator depending on P and K (e.g., T(P, K ) = ( I + P K ) - 1 ) . Let ~denote a finite family of discrete-time LTI plants. Since ~ represents a finite collection of LTI plants, simultaneous stabilization is always possible [10]. That is, there always exists an NLTV compensator which stabilizes every P ~ . The problem of simultaneous performance may be

Lemma 2.1. Let A be any causal finite-gain stable NLTV operator such that I - 2z + A has a causal finite-gain stable inverse. Then II A II > 1. Proof. Let g ~ 2 be given by g = ( 1 , 0 , 0 . . . . ). Then f = ( I - 2z + A) lg satisfies f ( n ) - 2(zf )(n) + ( A f )(n) = g(n), n=0,1,2 .....

J.S. Shamma / NLTV compensation for simultaneous performance

From the proposed validity of Conjecture 2.1, there exists a sequence of causal stable operators ( Qn } such that (1) T ( P a, K , ) = 2z - O, ---, O, (2) the operators I - PaQ, = I - Q, are stably invertible. From Lemma 2.1, the stable invertibility of I - Q, implies [I2z - Q, II-> 1, a contradiction.

or

f(1)

=

f(2)

/.10, / 2f11)

359

(A f ) ( 0 ) (a f ) ( 1 ) (a f)(2)

Thus for any n > 1,

[]

I[ f II LI,.1 > 2 II f II [0..-a/ - II A f II [1,.1 so that [I f ]l t0.,l >- 2 II f ]1[0,,-11 - II A II II f II t0,< and hence

It is noted that the above proposition also provides a counterexample to the following conj ecture. Conjecture 2.2. Let P be a given discrete-time LT1 plant, Then

2 Ilfllt0,<~ 1 + [IA[I Ilfllt0,,-xl.

J ( P ) := inf{ lIT(P, K ) I I : g is any LTI

It follows that IIA II < 1 implies f ~ :2 which contradicts the stable invertibility of ( I - 2 z + A) -~. []

stabilizing compensator } = inf{ l I T ( P , K)I[: K is anystable K NLTV stabilizing compensator).

Proposition 2.1. The family

(Po, where P. = I and Pb = 0, and the disturbance rejection problem given by T ( P , K ) = 2 z - K ( I + P K ) -1 together provide a counterexample to Conjecture

2.1. Proof. By employing the LTI compensators K a = 2 z ( I - 2z) -1 and K b = 2z for the plants P. and Ph, respectively, it follows that J ( P a ) = J ( P b ) = O. Suppose that Conjecture 2.1 is true. Then there exists a sequence of NLTV compensators, ( K . }, which (1) simultaneously stabilize P. and Ph and (2) lead to T(P~, K . ) ~ 0

and

T ( P b, K . ) ---, O.

Since Pb = 0, it follows that the compensators K. also must be stable (note that Ka is unstable). Now all NLTV compensators which stabilize P. are given by [14,17,18,21] { K = Q ( I - p o Q ) - l : Q is any causal stable operator }.

3. Concluding remarks

It has been shown that given a family of LTI plants, the achievable simultaneous performance need not equal the 'worst case' LTI performance. Thus, the objective of simultaneous performance presents fundamental limitations which cannot be overcome by NLTV compensation. It is interesting that this limitation is present even though the family of LTI plants is a finite collection - the situation in which the advantages of NLTV compensation are most significant. Open questions are the computation of the achievable simultaneous performance and the quantification of the degree to which NLTV compensation offers an advantage over LTI compensation for simultaneous performance. An especially interesting case is where the family of plants is characterized by a single block of dynamic uncertainty. In this case simultaneous performance may be given the viewpoint of a 'structured uncertainty' problem [3]. Finally, it is worth noting that via adaptive control, the worst case LTI performance may be achievable in an 'asymptotic' sense(e.g., [11,13]). The example presented in this note further justi-

360

J.S. Shamma / N L T V compensation for simultaneous performance

ties the use of asymptotic measures of performance for adaptive control. [11]

References [1] M.A. Dahleh and J.B. Pearson, Jr., all-Optimal feedback controllers for MIMO discrete-time systems, IEEE Trans. Automat. Control 32 (4) (1987) 314-322. [2] P. Dorato, Ed., Robust Control (IEEE Press, New York, 1987). [3] J.C. Doyle, J.E. Wall and G. Stein, Performance and robustness analysis for structured uncertainty, in: Pro-

[12]

[13]

[14]

ceedings of the 21st IEEE Conference on Decision and Control (1982) 629-636.

[4] A. Feintuch and B.A. Francis, Uniformly optimal control of linear time-varying systems, Systems Control Lett. 5 (1985) 67-71. [5] B.A. Francis, A Course in ,g'°%Optimal Control Theory (Springer-Verlag, New York, 1987). [6] T.T. Georgiou, A.M. Pasoal and P.P. Khargonekar, On the robust stabilizability of uncertain linear time-invariant plants using nonlinear time-varying controllers, Automatica 23 (1987) 617-624. [7] P.P. Khargonekar, Control of uncertain systems using nonlinear feedback, Proceedings of the 1989 International Symposium on Circuits and Systems.

[8] P.P. Khargonekar, T.T. Georgiou and A.M. Pascoal, On the robust stabilization of linear time-invariant plants with unstructured uncertainty, IEEE Trans. Automat. Control 32 (1987) 201-207. [9] P.P. Khargonekar and K.R. Poolla, Uniformly optimal control of linear time-varying plants: Nonlinear timevarying controllers, Systems Control Lett. 5 (1986) 303308. [10] P.P. Khargonekar, K.R. Poolla and A. Tannenbaum,

[15]

[16]

[17]

[18] [19]

[20] [21]

Robust control of linear time-invariant plants by periodic compensation, IEEE Trans. Automat. Control 30 (1985) 1088-1096. J.M. Krause, P.P. Khargonekar and G. Stein, Robust adaptive control: Stability and asymptotic performance, in: Proceedings of the 28th IEEE Conference on Decision and Control (1989) 1019-1024. K. Poolla and S. Cusumano, A novel approach to adaptive robust control, IEEE Trans. Automat. Control (1990, to appear). K. Poolla and J.S. Sharnma, Asymptotic performance through adaptive robust control, 29th IEEE Conference on Decision and Control (1990), submitted. K. Poolla and T. Ting, Nonlinear time-varying controllers for robust stabilization, IEEE Trans. Automat. Control 32 (1987) 195-200. K. Poolla, J.S. Shamma and K.A. Wise, Linear and nonlinear controllers for robust stabilization problems: A survey, in: Proceedings of the 1990 IFAC Conference on Automatic Control, to appear. J.S. Shamma and M.A. Dahleh, Time-varying vs. time-invariant compensation for rejection of persistent disturbances and robust stabilization, IEEE Trans. Automat. Control, to appear. M.S. Verma, Coprime factorizational representations and stability of nonlinear feedback systems, lnternat. J. Control 48 (1988) 897-918. M.Vidyasagar, Control Systems Synthesis: A Factorization Approach (MIT Press, Cambridge, MA, 1985). M. Vidyasagar, Optimal rejection of persistent bounded disturbances, IEEE Trans. Automat. Control 31 (1986) 527-534. J.C. Willems, The Analysis of Feedback Systems (MIT Press, Cambridge, MA, 1971). D.C. Youla, H.A. Jabr and J.J. Bongiorno, Jr., Modern Wiener-Hopf design of optimal controllers: Part If, IEEE Trans. Automat. Control 21 (1976) 319-338.