Elsevier

Automatica

Volume 39, Issue 3, March 2003, Pages 455-462
Automatica

Brief Paper
Low-gain integral control of continuous-time linear systems subject to input and output nonlinearities

https://doi.org/10.1016/S0005-1098(02)00238-8Get rights and content

Abstract

Continuous-time low-gain integral control strategies are presented for tracking of constant reference signals for finite-dimensional, continuous-time, asymptotically stable, single-input single-output, linear systems subject to a globally Lipschitz and non-decreasing input nonlinearity and a locally Lipschitz, non-decreasing and affinely sector-bounded output nonlinearity. Both non-adaptive (but possibly time varying) and adaptive integrator gains are considered. In particular, it is shown that applying error feedback using an integral controller ensures asymptotic tracking of constant reference signals, provided that (a) the steady-state gain of the linear part of the plant is positive, (b) the positive integrator gain is ultimately sufficiently small and (c) the reference value is feasible in a very natural sense. The classes of actuator and sensor nonlinearities under consideration contain standard nonlinearities important in control engineering such as saturation and deadzone.

Introduction

The synthesis of low-gain integral and proportional-plus-integral controllers for (uncertain) stable plants has received considerable attention in the last 20 years. The following principle is well known (see, for example, Davison 1976; Lunze, 1988): closing the loop around an asymptotically stable, finite-dimensional, continuous-time, single-input, single-output linear plant Σ, with transfer function G, compensated by a pure integral controller with gain k, will result in a stable closed-loop system which achieves asymptotic tracking of arbitrary constant reference signals, provided that |k| is sufficiently small and kG(0)>0. Therefore, if a plant is known to be asymptotically stable and if the sign of G(0) is known (this information can be obtained from plant step response data), then the problem of tracking by low-gain integral control reduces to that of tuning the gain parameter k. The problem of tuning the integrator gain adaptively has been addressed in a number of papers, see Cook (1992) and Miller and Davison 1989, Miller and Davison 1993 (with input constraints treated in Miller and Davison 1989, Miller and Davison 1993). Recently, Logemann et al. have developed tuning regulator results for infinite-dimensional systems with input nonlinearities (Logemann & Ryan, 2000; Logemann, Ryan, & Townley, 1998).

In this paper, we present results which show that the above principle remains true if the plant to be controlled is a stable, finite-dimensional single-input, single-output, linear system subject to an input and/or output nonlinearity (see Fig. 1). Precisely, we prove that, if G(0)>0 and if the constant reference signal r is feasible in an entirely natural sense, then there exists a number k>0 such that, for all non-decreasing, piecewise continuously differentiable, globally Lipschitz input nonlinearities ϕ and all non-decreasing, piecewise continuously differentiable, locally Lipschitz and affinely sector-bounded output nonlinearities ψ the following holds: for all positive, bounded and continuous integrator gains k(·) (thus in particular for positive constant gains), the output y(t) of the closed-loop system converges to r as t→∞, provided that limsupt→∞k(t)<k and k is not of class L1 (under some additional assumptions on the nonlinearities, results concerning the rate of convergence are derived). When compared with Logemann and Ryan (2000) and Logemann et al. (1998), the novelty in this paper is not only the inclusion of output nonlinearities, but also a different Lyapunov analysis which, for finite-dimensional systems, is more natural and powerful than the (infinite-dimensional) approaches developed in Logemann and Ryan (2000) and Logemann et al. (1998).

Finally, in Section 3.2, we show that one consequence of the above principle is that the following simple adaptation law (introduced in Logemann and Ryan (2000)) k(t)=1/l(t), l̇(t)=|r−y(t)| with l(0)=l0>0, produces an integrator gain k so that the output y(t) of the closed-loop system converges to r as t→∞.

Section snippets

Problem formulation

The problem of tracking constant reference signals r∈R will be addressed in the context of a class of finite-dimensional (state space RN) single-input (u(t)∈R), single-output (y(t)∈R), continuous-time (time domain R+≔[0,∞)), real linear systems Σ=(A,B,C,D) having a nonlinearity in the input and output channel:ẋ=Ax+Bϕ(u),x(0)=x0RN,w=Cx+Dϕ(u),y=ψ(w).

Integral control

Let Σ=(A,B,C,D)∈S and (ϕ,ψ)∈N. To achieve the objective of tracking feasible reference values r∈R, we will investigate integral control actionu(t)=u0+0tk(τ)[r−ψ(Cx(τ)+Dϕ(u(τ))]dτ=u0+0tk(τ)[r−y(τ)]dτwith control gain k (possibly constant) which is either prescribed or determined adaptively.

Thomas Fliegner was born in Dermbach (Thüringen), Germany, on January 17, 1962. He received the Diploma degree from the University of Jena, Germany, in 1987, and the Ph.D. degree from the University of Twente, The Netherlands, in 1995, both in mathematics. He was a Research Fellow in the Department of Engineering at the University of Cambridge, UK, from 1996 to 1998, and in the Department of Mathematical Sciences, University of Bath, UK, from 1998 to 2001. Since 2001 he has been with the

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Thomas Fliegner was born in Dermbach (Thüringen), Germany, on January 17, 1962. He received the Diploma degree from the University of Jena, Germany, in 1987, and the Ph.D. degree from the University of Twente, The Netherlands, in 1995, both in mathematics. He was a Research Fellow in the Department of Engineering at the University of Cambridge, UK, from 1996 to 1998, and in the Department of Mathematical Sciences, University of Bath, UK, from 1998 to 2001. Since 2001 he has been with the Department of Sciences and Liberal Arts at the International University of Germany where he is currently a Lecturer in Mathematics. His research interests are in nonlinear control theory.

Hartmut Logemann was born in Varel (Niedersachsen), Germany, on January 22, 1956. He received the Diploma degree from the University of Oldenburg, Germany, in 1981, and the Ph.D. degree from the University of Bremen, Germany, in 1986, both in mathematics. From 1986 to 1988 he was a Research Fellow in the Department of Mathematics at the University of Strathclyde, Glasgow, UK. From 1988–1993 he was with the Institute for Dynamical Systems at the University of Bremen. Since 1993 he has been with the Department of Mathematical Sciences at the University of Bath, UK, where he is currently a Professor of Mathematics. His research interests include adaptive control, infinite-dimensional systems theory, robust control and sampled-data systems. He has served as an Associate Editor of Automatica, SIAM Journal on Control and Optimization and Systems and Control Letters.

Eugene Ryan received the B.E. Degree from the National University of Ireland (University College, Cork) and the Degree of Ph.D. from the University of Cambridge, UK. He is currently Professor of Mathematics at the University of Bath, UK. His research interests include dynamical systems, stability, nonsmooth feedback, and adaptive control.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Andrew R. Teel under the direction of Editor Hassan Khalil. This work was supported by the UK Engineering & Physical Sciences Research Council (Grant Ref: GR/L78086).

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