Switched controllers and their applications in bilinear systems

doi:10.1016/S0005-1098(00)00172-2

Automatica

Volume 37, Issue 3, March 2001, Pages 477-481

https://doi.org/10.1016/S0005-1098(00)00172-2 Get rights and content

Abstract

In this paper, switched controllers are designed for a class of nonlinear processes. The nonlinear processes, together with the switched controllers, constitute a type of switched nonlinear systems. An invariant principle is presented for such switched nonlinear systems. The invariant principle and the switched controllers are used to globally stabilize a class of bilinear systems that are not open-loop neutrally stable.

Introduction

Switched systems have numerous applications in control of mechanical systems, automotive industry, aircraft and air traffic control, switching power converters and many other fields. In the last few years, every major control conference has had several regular and invited sessions on switched systems and control. Moreover, almost every major technical control journal has had a special issue on switched and hybrid systems (Liberzon & Morse, 1999).

A switched system arises in two cases. One is when there are abrupt changes in the structures and parameters of the dynamic system, which can be caused by component failures or repairs, variations in the environmental disturbance or subsystems interconnections (Narendra & Balakrishnan, 1997). The other is when a switched controller is used for a continuous system (Savkin, Skafidas & Evans, 1999). The major reason why we chose a switched controller rather than a continuous one is that a switched controller can be applied to achieve better performances. For example, a switched controller can improve transient performance (Narendra & Balakrishnan, 1997). Over the past three decades, there has been much interest in the study of switched systems (Hilhorst, Amerongen, Lohnberg & Tulleken, 1994; Devasia, Paden & Carlo, 1997; Narendra & Balakrishnan, 1997; Liberzon & Morse, 1999; Savkin et al., 1999).

In this paper, we give some basic knowledge of a switched controller for a nonlinear system and present some interesting cases where a switched controller should be used. The nonlinear process, together with the switched controller, constitutes a switched nonlinear system. An invariant principle is presented for such a switched nonlinear system. The invariant principle and the switched controller are used to globally stabilize a bilinear system that is not open-loop neutrally stable. The stabilization of such a bilinear system cannot be solved by any existing method (Quinn, 1980; Chen, 1998; Mohler, 1991).

The paper is composed of four parts. In the following section, some basic knowledge of switched controllers is introduced. The switched controller is used to stabilize a bilinear system in Section 3. Finally, some concluding remarks are given in Section 4.

Section snippets

Switched controllers

Consider a nonlinear system defined on the infinite discrete time points ${0, 1, 2, 3,…}$ $X(k+1)=f (X(k),u(k)),$ where X(k)∈R^r is the state, u(k)∈R is the control input. We shall now introduce some basic definitions.

Definition 1

$ζ(X)=[ζ_{1} (X),…, ζ_{n} (X)]^{T}$ is said to be well defined on a compact set $Ω$ (or R^r) if each $ζ_{i} : Ω$ (or R^r)→[0,1] can be well defined and ∑_i=1ⁿζ_i(X)=1 holds for all $X∈ Ω$ (or R^r).

Definition 2 A switched controller

Suppose that we have a collection of given nonlinear state feedback controllers: $U_{i} (k)=K_{i} (X(k)), 1≤i≤n,$ where $K_{i} (1≤i≤n) : Ω$ (or R^r)→

Problem formulation

Consider the global stabilization of single-input bilinear discrete-time systems $X(k+1)=AX(k)+u(k)NX(k),$ where X(k)∈R^r is the system state vector, A∈R^r×r and N∈R^r×r are constant square matrices and u(k) is a scalar control input with value subject to the constraints $u_{min} ≤u(k)≤u_{max}; (u_{min} <0<u_{max}).$ We suppose that the following assumptions hold.

Assumption 1

There exist an a satisfyingu_min<a<u_maxand a symmetricQ>0 such that $(A+aN)^{T} Q(A+aN)=Q.$

Assumption 2

The onlyX(0)=X₀for which $X_{0}^{T} [(A+aN)^{k}]^{T} N^{T} Q(A+aN)^{k+1} X_{0} =0$ for all k is X₀=0.

Remark 3

From

Conclusion

We have proposed a switched controllers for a nonlinear system. The nonlinear process, together with the switched controller, constitutes a switched nonlinear system. An invariant principle was presented for such a switched nonlinear system. The invariant principle and the switched controller were used to stabilize a bilinear system that is not open-loop neutrally stable.

References (9)

M.S. Chen
Exponential stabilization of a constrained bilinear system
Automatica
(1998)
R.A. Hilhorst et al.
A supervisor control of mode-switch processes
Automatica
(1994)
J.P. Quinn
Stabilization of bilinear systems by quadratic feedback control
Journal of Mathematical Analysis and Application
(1980)
A.V. Savkin et al.
Robust output feedback stabilizability via controller switching
Automatica
(1999)

There are more references available in the full text version of this article.

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^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. Sepulchre under the direction of Editor Paul Van den Hof.

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Technical CommuniqueSwitched controllers and their applications in bilinear systems☆

Abstract

Introduction

Section snippets

Switched controllers

Problem formulation

Conclusion

Automatica

Automatica

Journal of Mathematical Analysis and Application

Automatica

Technical Communique
Switched controllers and their applications in bilinear systems☆