Elsevier

Automatica

Volume 36, Issue 1, January 2000, Pages 45-52
Automatica

Brief Paper
Robust output feedback control of nonlinear singularly perturbed systems

https://doi.org/10.1016/S0005-1098(99)00105-3Get rights and content

Abstract

In this work, we consider nonlinear singularly perturbed systems with time-varying uncertain variables, for which the fast subsystem is asymptotically stable and the slow subsystem is input/output linearizable and possesses input-to-state stable (ISS) inverse dynamics. For these systems, we synthesize a robust output feedback controller that ensures boundedness of the state and enforces robust asymptotic output tracking with attenuation of the effect of the uncertain variables on the output of the closed-loop system. The controller is constructed through combination of a high-gain observer with a robust state feedback controller synthesized via Lyapunov's direct method. The proposed controller enforces the aforementioned properties in the closed-loop system, for initial conditions, uncertainty and rate of change of uncertainty in arbitrarily large compact sets, provided that the singular perturbation parameter is sufficiently small and the observer gain is sufficiently large.

Introduction

Many industrial processes exhibit nonlinear behavior and involve physicochemical phenomena occurring in separate time-scales. It is well established that a direct application of standard control methods to multiple-time-scale processes, without accounting for the presence of time-scale multiplicity, may lead to controller ill-conditioning and/or closed-loop instability. To circumvent these problems, the control of multiple-time-scale processes is usually addressed within the framework of singular perturbations (e.g., Kokotovic, Khalil, & O'Reilly, 1986; Christofides & Daoutidis, 1996).

In addition to nonlinearities and time-scale multiplicity, many industrial processes involve unknown process parameters and external disturbances. Therefore, the problem of designing controllers for nonlinear systems with uncertain variables, that enforce output tracking with attenuation of the effect of the uncertain variables on the output, has received considerable attention. For feedback linearizable nonlinear systems with time-varying uncertain variables that satisfy the so-called matching condition, robust state feedback controllers have been designed via Lyapunov's direct method to solve this problem locally (the reader may refer to Corless (1993) for a review of results in this area). Recently, for a class of nonlinear systems with time-varying uncertain variables that admit a disturbance-strict-feedback form without zero dynamics, a robust output feedback controller was designed in Khalil (1994) that solves this problem for arbitrarily large initial conditions and uncertainty (semi-global result). This result was generalized in Mahmoud and Khalil (1996) to nonlinear systems with asymptotically stable zero dynamics. In Christofides, Teel and Daoutidis, (1996), robust state feedback controllers were synthesized for nonlinear singularly perturbed systems with time-varying uncertain variables.

In this work, we address the problem of synthesizing a robust output feedback controller for nonlinear singularly perturbed systems with uncertain variables, for which the fast subsystem is asymptotically stable and the slow subsystem is input/output linearizable and possesses input-to-state stable inverse dynamics. A dynamic controller is synthesized, through combination of a high-gain observer with a robust state feedback controller synthesized via Lyapunov's direct method, that ensures boundedness of the state and achieves arbitrary degree of asymptotic attenuation of the effect of the uncertain variables on the output of the closed-loop system. The derived controller enforces the requested objectives in the closed-loop system, for initial conditions, uncertainty and rate of change of uncertainty in arbitrarily large compact sets, as long as the singular perturbation parameter is sufficiently small and the observer gain is sufficiently large. A successful application of the proposed control method to a chemical reactor example can be found in Christofides (1998).

Section snippets

Notation

  • |·| denotes the standard Euclidean norm, sgn(·) denotes the sign function, and sat(·) denotes the saturation function, defined assat(s)=amifs≥am,sif−am<s<am,−amifs≤−am,where s∈R and am is a positive real number. For a vector x∈Rn, sat(x)=[sat(x1)sat(x2)sat(xn)]T. O(ε) denotes the standard order of magnitude notation i.e., δ(ε)=O(ε) if there exist positive constants k and c such that |δ(ε)|≤k|ε|,∀|ε|<c. Lfh denotes the Lie derivative of a scalar field h with respect to the vector field f. Lfkh

Preliminaries

We consider single-input single-output nonlinear singularly perturbed systems with uncertain variables of the formẋ=f1(x,θ(t))+Q1(x,θ(t))z+g1(x,θ(t))u,εż=f2(x,θ(t))+Q2(x,θ(t))z+g2(x,θ(t))u,y=h(x),where x∈Rn and z∈Rp denote vectors of state variables, u∈R denotes the manipulated input, θ∈Rq denotes the vector of the uncertain time-varying variables, y∈R denotes the controlled output, and ε is a small positive parameter. f1(x,θ(t)), f2(x,θ(t)), g1(x,θ(t)) and g2(x,θ(t)) are sufficiently smooth

Robust output feedback controller synthesis

Motivated by the assumption of global asymptotic stability of the fast dynamics of the system of Eq. (2), we will synthesize the requisite robust dynamic output feedback controller on the basis of the slow subsystem of Eq. (3). To this end, we will need to impose the following three assumptions on the slow subsystem of Eq. (3). The first assumption is motivated by the requirement of output tracking and states the existence of a coordinate change that renders the system of Eq. (3) partially

Acknowledgements

Financial support in part by UCLA through the SEAS Dean's Fund and the Petroleum Research Fund, administered by the ACS, is gratefully acknowledged.

Panagiotis D. Christofides was born in Athens, Greece, in 1970. He received the Diploma in Chemical Engineering degree, in 1992, from the University of Patras, Greece, the M.S. degrees in Electrical Engineering and Mathematics, in 1995 and 1996, respectively, and the Ph.D. degree in Chemical Engineering, in 1996, all from the University of Minnesota. Since July 1996 he has been an Assistant Professor in the Department of Chemical Engineering at the University of California, Los Angeles. His

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    A direct application of standard control or estimation methods without taking into account time-scale multiplicity to systems with different time scales may lead to ill-conditioning or even the loss of closed-loop stability (Christofides, 2000; Kokotovic, Khalil, & O’Reilly, 1986). The singular perturbation theory is the standard tool for the analysis of systems with time-scale multiplicity (Christofides, 2000; Kokotovic et al., 1986). Within the singular perturbation framework, the original system is typically decomposed into reduced-order subsystems with “fast” and “slow” dynamics.

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Panagiotis D. Christofides was born in Athens, Greece, in 1970. He received the Diploma in Chemical Engineering degree, in 1992, from the University of Patras, Greece, the M.S. degrees in Electrical Engineering and Mathematics, in 1995 and 1996, respectively, and the Ph.D. degree in Chemical Engineering, in 1996, all from the University of Minnesota. Since July 1996 he has been an Assistant Professor in the Department of Chemical Engineering at the University of California, Los Angeles. His theoretical research interests include nonlinear, robust and optimal control, singular perturbations, and model reduction, optimization and control of nonlinear distributed parameter systems, with applications to chemical processes, advanced materials and semiconductor processing, particle technology and fluid flows. Professor Christofides is a recipient of a 1998 National Science Foundation CAREER award, and the 1999 Ted Peterson Student Paper Award of the Computing and Systems Technology Division of the American Institute of Chemical Engineers.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Y. Yamamoto under the direction of Editor R. Tempo.

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