Elsevier

Automatica

Volume 48, Issue 6, June 2012, Pages 1057-1068
Automatica

Analysis for a class of singularly perturbed hybrid systems via averaging

https://doi.org/10.1016/j.automatica.2012.03.013Get rights and content

Abstract

A class of singularly perturbed hybrid dynamical systems is analyzed. The fast states are restricted to a compact set a priori. The continuous-time boundary layer dynamics produce solutions that are assumed to generate a well-defined average vector field for the slow dynamics. This average, the projection of the jump map in the direction of the slow states, and flow and jump sets from the original dynamics define the reduced, or average, hybrid dynamical system. Assumptions about the average system lead to conclusions about the original, higher-dimensional system. For example, forward pre-completeness for the average system leads to a result on closeness of solutions between the original and average system on compact time domains. In addition, global asymptotic stability for the average system implies semiglobal, practical asymptotic stability for the original system. We give examples to illustrate the averaging concept and to relate it to classical singular perturbation results as well as to other singular perturbation results that have appeared recently for hybrid systems. We also use an example to show that our results can be used as an analysis tool to design hybrid feedbacks for continuous-time plants implemented by fast but continuous actuators.

Introduction

Systems exhibiting two time scale behavior (or multiple time scale behavior), with fast and slow dynamical variables, arise in all areas of science and engineering; properties of such systems can be analyzed using singular perturbation methods. For continuous-time systems, the celebrated Levinson–Tikhonov approach to analyzing singularly perturbed systems relates the dynamical properties of the perturbed system to the properties of two auxiliary systems, fast (boundary layer) and slow (quasi-steady state) systems. The main singular perturbation results can be classified into two main categories: closeness of solutions of the original perturbed system to solutions of its approximation on compact or infinite time intervals (Khalil, 2002, Teel et al., 2003); and stability results of the original system that are based on appropriate stability properties of the slow and the fast systems (Balachandra and Sethna, 1975, Tikhonov et al., 1985). Analogs of Levinson–Tikhonov theorem are established for differential inclusions, such as Dontchev et al., 1996, Veliov, 1997 on finite time intervals and Watbled (2005) on infinite time intervals with the assumptions that the boundary layer system converges to a Lipschitz set-valued map and global asymptotic stability of the reduced system. There are also results on singular perturbations for discrete-time systems (Grammel, 1999, Litkuhi and Khalil, 1985).

The averaging method was developed for continuous-time systems, discrete-time systems and differential inclusions (Bitmead and Johnson, 1987, Donchev and Grammel, 2005, Sanders and Verhulst, 1985, Wang and Nešić, 2010), but there exist only a few results for special classes of hybrid systems. For example, results on averaging of switched systems and dither systems to approximate time-varying hybrid systems by non-hybrid systems were considered in Iannelli, Johansson, Jonsson, and Vasca (2006), Porfiri, Roberson, and Stilwell (2008) and Wang and Nešić (2010). Recently, asymptotic stability for a class of time-varying hybrid systems via averaging has been considered in Teel and Nešić (2010), where states change continuously in a set in the state space and change instantaneously in another set in the state space.

Combining averaging and singular perturbation techniques, the results in Balachandra and Sethna (1975) consider continuous-time systems when the boundary layer system (obtained in the singular perturbation approach by setting the derivative of slow state variables to zero) is time-varying and possesses a time-varying integral manifold on which the derivative of slow state variables can be averaged. The results can be applied to adaptive control systems (Riedle & Kokotovic, 1986) and extremum seeking control systems (Tan, Nešić, & Mareels, 2006). The averaging method is also helpful in considering the singular perturbation problem when the boundary layer system is not time varying. Instead of insisting that trajectories of the boundary layer system converge to an equilibrium manifold, as in the classical singular perturbation theory, a set is used to replace the equilibrium manifold. In particular, trajectories of the boundary layer system are assumed to converge to a family of limit cycles parameterized by slow state variables, which then can be used to average the derivative of slow state variables. This idea can be found in the optimal control results in Gaitsgory (1992), the work of Artstein, 1999, Artstein, 2002, the work of Grammel, 1996, Grammel, 1997 and more recently in a unified framework for studying robustness to slowly-varying parameters, rapidly-varying signals and generalized singular perturbations in Teel et al. (2003). Artstein considered limiting behavior of the slow dynamics and statistical limit behavior of the fast dynamics for differential equations and differential inclusions respectively in Artstein, 1999, Artstein, 2002, by replacing the equilibria of the boundary layer system with an invariant probability measure. With the assumption that finite time average of the slow dynamics converges to a continuous limit set, Grammel, 1996, Grammel, 1997 constructed a limit differential inclusion for the slow motion. Singular perturbation theory based on averaging leads to a reduced order system, where fast motions appear implicitly and only their average influence on slow motions is considered. In general, this approach requires the assumption that large time scale behavior of trajectories of the fast dynamics is in some sense independent of its initial values (Grammel, 1999), or properties guaranteed by a unique invariant measure (Artstein, 1999, Artstein, 2002, Grammel, 1997) or some stability properties (Dontchev et al., 1996, Teel et al., 2003, Veliov, 1997, Watbled, 2005).

Hybrid dynamical systems are considered in the present paper, which naturally arise in a range of engineering applications including power electronics, robotics, manufacturing, automated highway systems, air traffic management systems, chemical process, and so on (Engell et al., 2000, Livadas et al., 2000, Song et al., 2000). Moreover, even for a continuous-time plant, the capabilities of nonlinear feedback control can be enhanced by using a hybrid controller (Goebel et al., 2009, Mayhew et al., 2007, Prieur, 2001). For hybrid dynamical systems, singular perturbation results appear in Sanfelice and Teel (2011), Sanfelice, Teel, Goebel, and Prieur (2006). Robustness to measurement noise and unmodeled dynamics of stability in hybrid systems was considered in Sanfelice et al. (2006) with the assumption that the boundary layer system converges to a quasi-steady state equilibrium manifold. Sanfelice and Teel (2011) presents results on stability of hybrid control systems singularly perturbed by fast but continuous actuators, where a more general set-valued mapping used to approximate the limiting behavior of the boundary layer system replaces the equilibrium manifold in the classical singular perturbation theory. It shows that if a hybrid control system has a compact set that is globally asymptotically stable when the actuator dynamics are omitted, or equivalently, are infinitely fast, then the same compact set is semi-globally practically asymptotically stable in the singular perturbation parameter.

We combine the averaging and the singular perturbation techniques to consider both closeness of solutions of a hybrid system with solutions of its average and stability properties of the actual system based on stability of its average in this paper. We show that each solution of the slow dynamics of the singularly perturbed hybrid system can be made arbitrarily close on compact time domains to some solution of its average system when the average system is forward complete. We also show that a compact set is semi-globally practically asymptotically stable for the actual hybrid system if it is globally asymptotically stable for the average system. Compared to hybrid singular perturbation results in Sanfelice and Teel (2011), our results give sharper conclusions in some cases and an example is used to illustrate this claim.

The paper is organized as follows. In Section 2, some basic definitions under the hybrid system framework are reviewed. We introduce a class of singularly perturbed hybrid systems in Section 3. The main results are given in Section 4. The proofs of main results are listed in Section 5 and Section 6 contains the conclusions.

Section snippets

Preliminaries

The singularly perturbed hybrid systems that we consider are based on two time scales, (τ,j) and (t,j) with τ=εt, with the notations x=dxdτ, ẋ=dxdt. Z0={0,1,2,}. B is the closed unit ball in an Euclidean space, the dimension of which should be clear from the context. A set-valued mapping M:RnRn is outer semi-continuous at xRn if for all sequences xix and yiM(xi) such that yiy we have yM(x), and M is outer semi-continuous (OSC) if it is outer semi-continuous at each xRn. A set-valued

Singularly perturbed hybrid systems

Consider a class of singularly perturbed hybrid systems with the time variables (τ,j): Hεx=f(x,z,ε)z=1εψ(x,z,ε)},(x,z)C×Ψ,(x,z)+G(x,z),(x,z)D×Ψ, where xRn, zRm, C,DRn, ΨRm, f:C×Ψ×R0Rn, ψ:C×Ψ×R0Rm, G:Rn×RmRn×Rm, and ε>0 is a small parameter reflecting that the flow dynamic of z are much faster than x. Let f0(x,z)f(x,z,0) and ψ0(x,z)ψ(x,z,0). We assume that system Hε satisfies the following conditions.

Assumption 1

The sets C and D are closed and the set Ψ is compact. G is outer semi-continuous

Main results

First in Theorem 1 we present results on closeness of the slow solutions x of the singularly perturbed system Hε to the solutions of its average system Hav on compact time domains, under the assumption that the system Hav is forward pre-complete from a given compact set.

Theorem 1

Suppose that the singularly perturbed system Hε in (3) satisfies Assumption 1, Assumption 2 and that its average system Hav defined in (1), (16), (17) is forward pre-complete from a compact set K0Rn . Then, for each ρ>0 and any

Proofs

The proofs of Theorem 1, Theorem 2 are given in this section. Applying the coordinate transformation technique, we show in Section 5.2 that for any compact set, solutions of the actual hybrid system included in this set are actually the solutions of its average system under small perturbations when the parameter ε is sufficiently small. Then, with the preliminary results on properties of general hybrid systems listed in Section 5.1, we show Theorem 1, Theorem 2 in Sections 5.3 Proof of, 5.4

Conclusions

We considered a class of hybrid dynamical systems with the singular perturbations theory and the averaging method. We showed that if there exists a well defined average for the actual perturbed hybrid system, the slow solutions of the actual system on compact time domains are arbitrarily close to the solution of the average system that approximates the slow dynamics of the actual system for arbitrarily small values of the singular perturbation parameter. We also showed that the global

Wei Wang received her B.E. and M.E. degrees in Mechanical Engineering from Qingdao University of Science and Technology in 1994 and 2002, respectively. She also worked for her Ph.D. degree from 2008 to 2011 in the Department of Electrical and Electronic Engineering (DEEE) at the University of Melbourne, Australia. She is currently a postdoctoral fellow in DEEE at the University of Melbourne, Australia. Her research interests include nonlinear control systems, hybrid systems and networked

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    Wei Wang received her B.E. and M.E. degrees in Mechanical Engineering from Qingdao University of Science and Technology in 1994 and 2002, respectively. She also worked for her Ph.D. degree from 2008 to 2011 in the Department of Electrical and Electronic Engineering (DEEE) at the University of Melbourne, Australia. She is currently a postdoctoral fellow in DEEE at the University of Melbourne, Australia. Her research interests include nonlinear control systems, hybrid systems and networked control systems.

    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., he was a postdoctoral fellow at the Ecole des Mines de Paris in Fontainebleau, France. In September 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, he joined the faculty of the Electrical and Computer Engineering Department at the University of California, Santa Barbara, where he is currently a professor.

    Dragan Neišić is a Professor in the Department of Electrical and Electronic Engineering (DEEE) at The University of Melbourne, Australia. He received his B.E. degree in Mechanical Engineering from The University of Belgrade, Yugoslavia in 1990, and his Ph.D. degree from Systems Engineering, RSISE, Australian National University, Canberra, Australia in 1997. Since February 1999 he has been with The University of Melbourne. His research interests include networked control systems, discrete-time, sampled-data and continuous-time nonlinear control systems, input-to-state stability, extremum seeking control, applications of symbolic computation in control theory, hybrid control systems, and so on. He was awarded a Humboldt Research Fellowship (2003) by the Alexander von Humboldt Foundation, an Australian Professorial Fellowship (2004–2009) and Future Fellowship (2010–2014) by the Australian Research Council. He is a Fellow of IEEE and a Fellow of IEAust. He is currently a Distinguished Lecturer of CSS, IEEE (2008–2010). He served as an Associate Editor for the journals Automatica, IEEE Transactions on Automatic Control, Systems and Control Letters and European Journal of Control.

    Supported by the Australian Research Council under the Discovery Project and Future Fellow program, AFOSR (Grant FA9550-09-1-0203) and NSF (Grants ECCS-0925637 and CNS-0720842). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo.

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