Elsevier

Automatica

Volume 45, Issue 1, January 2009, Pages 225-229
Automatica

Brief paper
On partitioned controllability of switched linear systems

https://doi.org/10.1016/j.automatica.2008.06.009Get rights and content

Abstract

When a switched linear system is not completely controllable, the controllability subspace is not enough to describe the controllability of the system over whole state space. In this case the state space can be divided into two or three control-invariant sub-manifolds, which form a control-related partition of the state space. This paper investigates when each component is a controllable sub-manifold. First, we consider when a sub-manifold is controllable for no control input case. Then the results are used to produce a necessary and sufficient condition assuring the controllability of the partitioned control-invariant sub-manifolds of a class of switched linear systems. An example is given to demonstrate the effectiveness of the results.

Introduction

A switched linear system is a hybrid system which consists of several linear subsystems and a rule that orchestrates the switching among them. There are many studies on the controllability of switched linear systems. For instance, studies for low-order switched linear systems have been presented in Loparo, Aslanis, and IIajek (1987) and Xu and Antsaklis (1999). Some sufficient conditions and necessary conditions for controllability were presented in Ezzine and Haddad (1989) and Szigeti (1992) for switched linear systems under the assumption that the switching sequence is fixed. The complexity of stability and controllability of hybrid systems was addressed in Blondel and Tsitsiklis (1999) and Hu, Zhang, and Deng (2004). Sun and Zheng (2001), Sun, Ge, and Lee (2002), and Sun and Ge (2005) investigated the controllability and reachability issues for switched linear systems in detail.

Consider a switched linear systemẋ(t)=Aσ(t)x(t)+Bσ(t)u(t),x(t)Rn,u(t)Rm, where σ:[0,)Λ={1,2,,N} is a piece-wise constant, right continuous mapping, called switching signal. As a particular case when there is no control input we have ẋ(t)=Aσ(t)x(t),x(t)Rn, which is called a switched linear system without control.

The reachable set of x0, denoted by R(x0), is defined as: yR(x0), if there exist u, σ and T>0, such that y=φ(u,σ,x0,T). (Correspondingly, for system (2), y=φ(σ,x0,T).)

Here φ(u,σ,x0,t) is the trajectory of system (1) with initial point x(0)=x0, control u(t) and switching signal σ(t). Similarly, we use φ(σ,x0,t) to denote the trajectory of system (2).

For system (1) we define a subspace as C=<A1,,AN|B1,,BN>, which is the smallest subspace containing Bi and Ai invariant. The main result about the controllability of system (1) is the following:

Theorem 1 Sun et al. (2002)

For system(1), the largest reachable set from the origin isR(0)=C. Moreover, for any two pointsx,yC,xR(y).

System(1)is completely controllable, if and only if,dim(C)=n.

We call C the controllable subspace of system (1). It is clear that the controllable subspace for system (2) is C={0}.

Definition 2

A sub-manifold URn is called a controllable sub-manifold if for any two points x,yU, xR(y).

From Theorem 1 one sees easily that the controllable subspace C is a controllable sub-manifold. Moreover, it is the largest subspace, which is also a controllable sub-manifold.

Definition 3

A sub-manifold URn is called a control invariant sub-manifold if for any two points xU and yUc, xR(y), and yR(x).

Note that if U is a control invariant sub-manifold, then so is its complement Uc. We also have (with mild revision).

Proposition 4 Cheng, Lin, and Wang (2006)

Cis a control invariant sub-manifold.

Assume the controllable subspace, C, of system (1) is not the whole space. Then C becomes a zero measure set. To describe the controllability of the system over whole state space, we are interested in finding (non-subspace type of) controllable sub-manifolds in Cc. For block diagonal systems or symmetric systems the problem has been discussed in Cheng et al. (2006). This paper investigates the same problem for more general cases. Moreover, the procedure for designing controls and switching laws is also provided.

Section snippets

Controllability of switched linear systems without control

Consider system (2). It is obvious that {0} and Rn{0} are control-invariant. So we ask when Rn{0} is a controllable sub-manifold?

Before giving a useful sufficient condition, we need some preliminaries.

Definition 5

A point x00 is called an interior point of system (2), if 0 is an interior point of the convex cone generated by {Aλx0|λΛ}.

The geometric meaning of the interior points is obvious, but we need a clear algebraic description for verification. We briefly cite some well known results as follows (

Controllability of switched linear systems

Consider system (1). Denote Cλ=<Aλ|Bλ>,λΛ. Assume the controllable subspace of system (1), C, is composed by the controllable subspaces of the switching modes. That is,

A1C=C1C2CN. Then system (1) can be expressed as {żi1=j=1NAσ(t)ijzj1+Aσ(t)i(N+1)z2+Bσ(t)iui,i=1,2,,N,ż2=Aσ(t)(N+1)(N+1)z2, where zi1 corresponds to Ci respectively. An immediate consequence is

Lemma 9

Assumption A1 assures that(Aiii,Bii),i=1,,N, are controllable.

For system (7), we have the following result:

Theorem 10

Consider system(7).

An illustrative example

The proof of Theorem 10 is constructive, so it can be used to construct the control. In the following example, a detailed design process of the control is depicted.

Example 11

Consider the following system with n=3, m=1, Λ={1,2}: ẋ=Aσ(t)x+Bσ(t)u where A1=(112011001),B1=(100);A2=(101112001),B2=(010). Denote the controllable subspace of system (9) by C, the controllable subspace of every mode of system (9) by C1, C2 respectively. Then C=span{(100),(010)}={xR3|x3=0},C1=span{(100)},C2=span{(010)}.

Conclusion

This paper considered when control-invariant sub-manifolds of switched linear systems are controllable. The main controllability results of the paper consisted of two parts. First, the controllability via switching law was investigated, a sufficient condition was obtained. Then in the case that the controllable subspace is partitioned by the controllable subspaces of switching models, a necessary and sufficient condition for Cc being a controllable sub-manifold was obtained. The proof provided

Acknowledgements

This work is supported by NNSF of China under Grant 60674022, 60736022, 60221301 and Grant SIC07010201.

The authors would like to acknowledge the anonymous reviewers and the Associate Editor for their accurate reading and useful suggestions.

Yupeng Qiao received her bachelor degree and master degree in Science from the Harbin Institute of Technology in 2003 and 2005, respectively, and Ph.D. from Institute of Systems Science, AMSS, CAS in 2008. Her research interests include control of nonlinear and hybrid systems.

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Yupeng Qiao received her bachelor degree and master degree in Science from the Harbin Institute of Technology in 2003 and 2005, respectively, and Ph.D. from Institute of Systems Science, AMSS, CAS in 2008. Her research interests include control of nonlinear and hybrid systems.

Daizhan Cheng graduated from Tsinghua University in 1970, received M.S. from Graduate School, Chinese Academy of Sciences in 1981, Ph.D. from Washington University, St. Louis, in 1985. Since 1990, he has been a professor with Institute of Systems Science, AMSS, CAS. His research interests include nonlinear systems, hybrid systems and numerical method for system control. He is an IEEE Fellow and IFAC Fellow.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Maria Elena Valcher under the direction of Editor Roberto Tempo.

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