Elsevier

Automatica

Volume 40, Issue 6, June 2004, Pages 1025-1034
Automatica

Brief paper
Matrosov theorem for parameterized families of discrete-time systems

https://doi.org/10.1016/j.automatica.2004.01.016Get rights and content

Abstract

A version of Matrosov's theorem for parameterized discrete-time time-varying systems is presented. The theorem is a discrete-time version of the continuous-time result in Loria et al., 2002 (δ-persistency of excitation: a necessary and sufficient condition for uniform attractivity, 2002, submitted for publication). Our result facilitates controller design for sampled-data nonlinear systems via their approximate discrete-time models. An application of the theorem to establishing uniform asymptotic stability of systems controlled by model reference adaptive controllers designed via approximate discrete-time plant models is presented.

Introduction

The prevalence of computer controlled systems and the fact that the nonlinearities in the plant model can often not be neglected strongly motivate consideration of the class of nonlinear sampled-data models. One of the main difficulties in dealing with this class of models is that it is not clear what is the model one should use when designing the controller. For example, the exact discrete-time model of this class of systems is typically not available for the controller design and one can only use an approximate model for this purpose. However, it was shown in Teel, Nešić, and Kokotović (1999a) and Nešić and Teel (2002) that controllers that stabilize an approximate discrete-time model of the system may destabilize the exact discrete-time model for all sampling periods. Hence, a careful analysis and design are needed if one is using an approximate discrete-time model for controller design.

This has motivated us to present checkable conditions (Teel et al., 1999a; Nešić & Teel, 2002) on the controller, approximate model and the continuous-time plant model that guarantee that if the controller stabilizes the approximate model then it would also stabilize the exact discrete-time model in an appropriate sense (semiglobal-practical) for sufficiently small sampling periods. These results provide a framework for controller design for sampled-data nonlinear systems via their approximate discrete-time models. In particular, Teel et al. (1999a, Theorem 2) gives Lyapunov like conditions on the approximate model to provide such a framework. It is the purpose of this paper to extend the results in Teel et al. (1999a) and Nešić and Teel (2002) in the following way. We prove a version of Matrosov's theorem for establishing uniform asymptotic stability using several Lyapunov like functions that typically have negative semi-definite first difference (in Teel et al. (1999a) and Nešić and Teel (2002) we required a negative definite first difference of the Lyapunov function; moreover in Teel et al. (1999a) we considered only time-invariant systems).

Our results are important for the following reasons. Time-varying systems arise in a range of control applications, such as tracking control, adaptive control or when time-varying controllers are used (such as, stabilization of non-holonomic systems). Matrosov's theorem generalizes, in an appropriate sense, La Salle's theorem to time-varying systems and it is very important in situations when we have a negative semi-definite derivative of the Lyapunov function along the trajectories of the system dynamics (such as, when we use the storage function for the passivity property to establish stability). The classical Matrosov theorem establishes UGAS via two Lyapunov like functions (see Matrosov, 1962; Rouche & Mawhin, 1980, Theorem 5.5; Habets, Rouche, & Laloy, 1977, Theorem 2.5; Paden & Panja, 1988, Appendix). Certain extensions of Matrosov theorem can be found in Panteley, Teel, and Loria (2002) and more recently in Loria, Panteley, Popović, and Teel (2002a) where it was shown how it is possible to combine an arbitrary number of Lyapunov like functions to test UGAS of time-varying continuous-time systems. Our main result extends the main result from Loria et al. (2002a) to parameterized families of time-varying discrete-time systems that naturally arise when an approximate discrete-time model is used to design a controller for a sample-data system. While our proofs follow similar steps as in the continuous-time case (Loria et al., 2002a), they are complicated by the fact that we also require certain uniformity of the stability property with respect to the parameter, which is crucial for sampled-data applications. An application to systems for which a model reference adaptive controller is designed via an approximate discrete-time model is presented (a continuous-time analogue of this result can be found in Loria, Panteley, Popović, & Teel, 2002b).

The paper is organized as follows. First, we present preliminaries and definitions in Section 2. The main result is presented and proved in Section 3. Section 4 explains how our results can be used for controller design of sampled-data nonlinear systems via their approximate discrete-time models. Finally, in Section 5 we show how our result can be used to analyze systems controlled by model reference adaptive controllers that are designed via their approximate discrete-time plant models.

Section snippets

Preliminaries

We denote sets of real and integer numbers, respectively, as R and Z. A function α:R⩾0R⩾0 is said to be of class K (α∈K), if it is continuous, strictly increasing and zero at zero; α∈K if, in addition, it is unbounded. A function β:R⩾0×R⩾0R⩾0 is of class KL if for all t>0, β(·,t)∈K, for all s>0, β(s,·) is decreasing to zero. We denote by |·| the Euclidean norm of vectors. Given arbitrary L⩾0 and T>0 we use the following notation: L,T≔⌊L/T⌋=max{z∈Z:z⩽L/T}. Also, we denote BΔ≔{x:|x|⩽Δ} and H

A type of Matrosov theorem

The following theorem is an adaptation of the Matrosov Theorem given in Loria et al. (2002a) to parameterized discrete-time time-varying systems that naturally arise when one uses an approximate discrete-time model of the sampled-data plant to design a discrete-time controller. A comparison between the statement of this theorem and the statement of Corollary 2, the latter of which is essentially a translation of the result in Loria et al. (2002a) to discrete-time, indicates the complications

Sampled-data systems

In this section we state an extension of Teel et al. (1999a, Theorem 2) that can be used to deal with time-varying systems using Matrosov functions (time-invariant systems using Lyapunov functions were considered in Teel et al. (1999a, Theorem 2)). This result motivates the stability definitions that we use. Consider the class of time-varying systems:ẋ(t)=f(t,x(t),u(t)),where x∈Rnx and u∈Rm are, respectively, the state and control input. Using the assumption of sampler and zero order hold (the

MRAC via approximate models

We consider the problem of sampled data, adaptive tracking control (sometimes called model reference adaptive control or MRAC for nonlinear systems in the form)ξ̇=f(ξ)+g(ξ)[u+h(ξ)θ].The parameter vector θ is unknown. The functions f, g and h are supposed to be locally Lipschitz. We will assume we can find a family of certainty equivalence discrete-time, tracking feedback laws for the Euler approximation of system (29):ξ(k+1)=ξ(k)+Tf(ξ(k))+Tg(ξ(k))[u(k)+h(ξ(k))θ].In other words, we will assume:

Conclusions

We presented a Matrosov theorem for parameterized discrete-time time-varying models that facilitates controller design for sampled-data nonlinear systems via their approximate discrete-time models. Our main theorem is an analogue of the continuous-time result in Loria et al. (2002a) that generalized the classical Matrosov theorem for continuous-time systems. We have also related stability conditions needed on an approximate model to the stability properties of the exact discrete-time model and

Acknowledgements

The first author was supported by the Australian Research Council under the large grants scheme. The second author was supported in part by the AFOSR under grant F49620-00-1-0106 and the NSF under grant ECS-9988813.

Dragan Nešić is currently an Associate Professor and Reader in the Department of Electrical and Electronic Engineering (DEEE) at The University of Melbourne, Australia. He received his BE degree from The University of Belgrade, Yugoslavia in 1990, and his Ph.D. degree from Systems Engineering, RSISE, Australian National University, Canberra, Australia in 1997. In the period of 1997–1999 he held postdoctoral positions at DEEE, The University of Melbourne, Australia; CESAME, Université Catholique

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Dragan Nešić is currently an Associate Professor and Reader in the Department of Electrical and Electronic Engineering (DEEE) at The University of Melbourne, Australia. He received his BE degree from The University of Belgrade, Yugoslavia in 1990, and his Ph.D. degree from Systems Engineering, RSISE, Australian National University, Canberra, Australia in 1997. In the period of 1997–1999 he held postdoctoral positions at DEEE, The University of Melbourne, Australia; CESAME, Université Catholique de Louvain, Louvain la Neuve, Belgium; and ECE, University of California, Santa Barbara, CA, USA. Since February 1999 he has been with The University of Melbourne. His research interests include discrete-time, sampled-data and continuous-time nonlinear control systems, nonlinear systems with disturbances, applications of symbolic computation in control theory and networked control systems. In 2003 Dr. Nešić has been awarded a Humboldt Research Fellowship funded by the Alexander von Humboldt Foundation, Germany. Dr. Nesic has been also awarded a five year (2004–2008) Australian Professorial Fellowship by the Australian Research Council. Dr. Nešić is a Senior Member of IEEE and a member of IEAust. He is an Associate Editor for the journals Systems and Control Letters, Automatica and IEEE Transactions on Automatic Control.

Andrew R. Teel received his A.B. degree in Engineering Sciences from Dartmouth College in Hanover, New Hampshire, in 1987, and his M.S. and Ph.D. degrees in Electrical Engineering from the University of California, Berkeley, in 1989 and 1992, respectively. After receiving his Ph.D., Dr. Teel was a postdoctoral fellow at the Ecole des Mines de Paris in Fontainebleau, France. In September of 1992 he joined the faculty of the Electrical Engineering Department at the University of Minnesota where he was an assistant professor until September of 1997. In 1997, Dr. Teel joined the faculty of the Electrical and Computer Engineering Department at the University of California, Santa Barbara, where he is currently a professor. His research interests include nonlinear dynamical systems and control with application to aerospace and related systems. Professor Teel has received NSF Research Initiation and CAREER Awards, the 1998 IEEE Leon K. Kirchmayer Prize Paper Award, the 1998 George S. Axelby Outstanding Paper Award, and was the recipient of the first SIAM Control and Systems Theory Prize in 1998. He was also the recipient of the 1999 Donald P. Eckman Award and the 2001 O. Hugo Schuck Best Paper Award, both given by the American Automatic Control Council. He is a Fellow of the IEEE.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Alessandro Astolfi under the direction of Editor Hassan Khalil.

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