The stabilization problem for discrete time-varying linear systems

doi:10.1016/j.sysconle.2008.05.003

Systems & Control Letters

Volume 57, Issue 11, November 2008, Pages 936-939

https://doi.org/10.1016/j.sysconle.2008.05.003 Get rights and content

Abstract

In this paper, we consider the strong stabilization problem for a time-varying linear system which admits a coprime factorization. We also study the simultaneous and robust stabilization problems for this class of systems.

Introduction

The stabilization problem for linear systems has been studied within various frameworks. For a discussion of the origins of the problem, we refer to [1]. It is shown in [1], [2] that every internally stabilizable causal linear time-varying system admits doubly coprime factorizations and there is a Youla–Kuc˘era-like parametrization of all stabilizing controllers which is conceptually similar to the classical result for LTI systems. However, internal stabilizability is generally not equivalent to the existence of doubly coprime factorizations (see [3], [4], [5], [6], [7], [8]). In this context it is natural to ask whether among these controllers there exist stable systems. One deep result in this area up to this point is that of Treil [9]. For single-input/single-output LTI distributed systems on the unit circle, every stabilizable linear system (the underlying field is the complex numbers) is strongly stabilizable, that is, stabilizable by a stable controller. It was recently shown by Quadrat [3] that this result can be extended to every MIMO plant defined by a transfer matrix $P$ with entries from an algebra of stable range one. Thus the issue is resolved for MIMO LTI systems.

Within the framework of nest algebras (i.e., see the book [10]), the purpose of the paper is to consider the problem of stabilizing a family of plants (resp., robust stabilizability) by means of a stable controller. We give some results to these problems for certain classes of systems. The simultaneous and robust stabilization problems are well studied in the literature [10] but we generally do not ask the controller to be stable. The good reference where the simultaneous stabilization problem with a stable controller was briefly studied is Quadrat [3], using the concept of $n$ -fold ring.

This paper is organized as follows. In Section 2, we give some definitions, notations and auxiliary propositions. In Section 3, we obtain parametrizations of the stable controllers for a linear time-varying system which admits a coprime factorization. Sections 4 Application to simultaneous stabilization, 5 Robust stability are devoted to applications of the obtained results to the simultaneous stabilization and robust stability problems. The paper ends with acknowledgement and cited references.

Section snippets

Preliminaries

We recall some basic concepts (see [10]) that will be useful in this paper. First, we introduce the definitions about complete nest and nest algebra.

Let $H$ be a separable complex Hilbert space. $L (H)$ denotes the set of all bounded linear operators on $H$ . For $T \in L (H)$ , the range $Ran(T)$ of $T$ is ${T x : x \in H}$ . $T$ is an orthogonal projection if $T$ is idempotent $(T = T^{2})$ and self-adjoint.

Definition 2.1

A family $N$ of closed subspaces of the Hilbert space $H$ is a complete nest if

1.
${0}, H \in N$ .
2.
For $N_{1}, N_{2} \in N$ , either $N_{1} \subseteq N_{2}$ or $N_{2} \subseteq N_{1}$ .
3.
If $N_{α}$ is

Strong stabilization

Practicing engineers are reluctant to use unstable compensators for the purpose of stabilization. This gives us a motivation for considering whether or not there exists a stable compensator for a given plant $L$ that admits a left or right coprime factorization. Also, the knowledge of when a plant is or is not stabilizable with a stable controller is useful for another problem, namely, the simultaneous stabilization problem, which studies the stabilizability of several plants by means of the same

Application to simultaneous stabilization

We now turn to the problem of simultaneous stabilization. Given $L_{0}, L_{1}, \dots, L_{n} \in S$ , when does there exist $C \in S$ for which ${L_{0}, C}$ , ${L_{1}, C}, \dots, {L_{n}, C}$ are stable? We first see the case of two systems.

Theorem 4.1

Suppose that $L_{1}$ and $L_{2}$ admit left coprime factorizations $B_{1}^{- 1} A_{1}$ and $B_{2}^{- 1} A_{2}$ respectively, with $A_{i}, B_{i}, X_{i}$ and $Y_{i} \in S$ that satisfy the Bezout equation $A_{i} X_{i} + B_{i} Y_{i} = I (i = 1, 2)$ . If $L_{1}$ and $L_{2}$ are strongly stabilizable, then there exists a controller $C \in S$ which simultaneously stabilizes $L_{1}$ and $L_{2}$ if and only if there exists $T \in S$

Robust stability

Consider the usual feedback system with plant $L$ and controller $C$ , both in $L$ . Suppose $L$ is not fixed but belongs to some set $B$ . The robust stability problem is to find, if one exists, a controller $C$ that achieves internal stability for every $L \in B$ . Similarly to Theorem 8.1.1 of [10], we obtain the theorem about robust stability with a stable controller.

Theorem 5.1

Suppose that $L \in L$ admits a left coprime factorization $B^{- 1} A$ , $A X + B Y = I$ , where $A, B, X$ and $Y \in S$ . Let $C \in S$ be a stable controller of $L$ admitting a right

Acknowledgement

The suggestions given by the referee for the improvement of the original paper are gratefully acknowledged.

References (11)

A. Quadrat
On a generalization of the Youla-Kuc˘era parametrization. Part I: The fractional ideal approach to SISO systems
Systems & Control Lett.
(2003)
S. Treil
The stable rank of the algebra $H^{\infty}$ equals 1
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On strong stabilization for linear time-varying systems
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Control System Synthesis: A Factorization Approach
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Stabilizability and existence of system representations for discrete-time time-varying systems
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^☆: This research is supported by NSFC, Item Number: 10671028.

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The stabilization problem for discrete time-varying linear systems☆

Abstract

Introduction

Section snippets

Preliminaries

Strong stabilization

Application to simultaneous stabilization

Robust stability

Acknowledgement

Systems & Control Lett.

J. Funct. Anal.

Systems & Control Letters.

Control System Synthesis: A Factorization Approach

Stabilizability and existence of system representations for discrete-time time-varying systems

SIAM J. Control. Optim.