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Journal of Applied Mathematics
Volume 2012, Article ID 182040, 24 pages
http://dx.doi.org/10.1155/2012/182040
Research Article

Controllability and Observability Criteria for Linear Piecewise Constant Impulsive Systems

1Mathematics and Physics Department, Beijing Institute of Petrochemical Technology, Beijing 102617, China
2Center for Systems and Control, LTCS, and Department of Industrial Engineering and Management, Peking University, Beijing 100871, China
3School of Electrical and Electronic Engineering, East China Jiaotong University, Nanchang 330013, China

Received 14 May 2012; Accepted 22 July 2012

Academic Editor: Junjie Wei

Copyright © 2012 Hong Shi and Guangming Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Impulsive differential systems are an important class of mathematical models for many practical systems in physics, chemistry, biology, engineering, and information science that exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. This paper studies the controllability and observability of linear piecewise constant impulsive systems. Necessary and sufficient criteria for reachability and controllability are established, respectively. It is proved that the reachability is equivalent to the controllability under some mild conditions. Then, necessary and sufficient criteria for observability and determinability of such systems are established, respectively. It is also proved that the observability is equivalent to the determinability under some mild conditions. Our criteria are of the geometric type, and they can be transformed into algebraic type conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.

1. Introduction

In recent years, there has been increasing interest in the analysis and synthesis of impulsive systems, or impulsive control systems, due to their significance both in theory and in applications [115].

Different from another type of systems associated with the impulses, that is, the singular systems or the descriptor systems, impulsive control systems are described by impulsive ordinary differential equations. Many real systems in physics, chemistry, biology, engineering, and information science exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the continuous dynamical processes. This kind of impulsive behaviors can be modelled by impulsive systems.

Controllability and observability of impulsive control systems have been studied by a number of papers [4, 6, 12, 13, 15, 16]. Leela et al. [4] investigated the controllability of a class of time-invariant impulsive systems with the assumption that the impulses of impulsive control are regulated at discontinuous points. Lakshmikantham and Deo [12] improved Leela et al.’s [4] results. Then, George et al. [13] extended the results to the linear impulsive systems with time-varying coefficients and nonlinear perturbations. Benzaid and Sznaier [6] studied the null controllability of the linear impulsive systems with the control impulses only acting at the discontinuous points. Guan et al. [15] investigated the controllability and observability of linear time-varying impulsive systems. Sufficient and necessary conditions for controllability and observability are established and their applications to time-invariant impulsive control systems are also discussed. Xie and Wang [16] investigated controllability and observability of a simple class of impulsive systems. Necessary and sufficient conditions are obtained.

Controllability and observability are the two most fundamental concepts in modern control theory [1719]. They have close connections to pole assignment, structural decomposition, quadratic optimal control and observer design, and so forth. In this paper, we aim to derive necessary and sufficient criteria for controllability and observability of linear piecewise constant impulsive control systems. We first investigate the reachability of such systems and a geometric type necessary and sufficient condition is established. Then, we investigate the controllability and an equivalent condition is established as well. Moreover, it is shown that the controllability is not equivalent to reachability for such systems in general case but is equivalent under some extra conditions. Next, we investigate the observability and determinability of such systems, and get similar results as the controllability and reachability case.

This paper is organized as follows. Section 2 formulates the problem and presents the preliminary results. Sections 3 and 4 investigate reachability and controllability, respectively. Observability and determinability are investigated in Section 5. Section 6 contains a numerical example. Finally, we provide the conclusion in Section 7.

2. Preliminaries

Consider the piecewise linear impulsive system given by ̇𝑥(𝑡)=𝐴𝑘𝑥(𝑡)+𝐵𝑘𝑢𝑡(𝑡),𝑡𝑘1,𝑡𝑘,𝑥𝑡+𝑘=𝐸𝑘𝑥𝑡𝑘+𝐹𝑘𝑢𝑡𝑘,𝑦(𝑡)=𝐶𝑘𝑥(𝑡)+𝐷𝑘𝑡𝑢(𝑡),𝑡𝑘1,𝑡𝑘,𝑥𝑡+0=𝑥0,𝑡00,(2.1) where 𝑘=1,2,,𝐴𝑘,𝐵𝑘,𝐶𝑘,𝐷𝑘,𝐸𝑘, and 𝐹𝑘 are the known 𝑛×𝑛, 𝑛×𝑝, 𝑝×𝑛, 𝑞×𝑝, 𝑛×𝑛, and 𝑛×𝑝 constant matrices; 𝑥(𝑡)𝑛 is the state vector, and 𝑢(𝑡)𝑝 the input vector, 𝑦(𝑡)𝑞 the output vector; 𝑥(𝑡+)=lim0+𝑥(𝑡+), 𝑥(𝑡)=lim0𝑥(𝑡), and the discontinuity points are 𝑡1<𝑡2<<𝑡𝑘<,lim𝑘𝑡𝑘=,(2.2) where 𝑡0<𝑡1 and 𝑥(𝑡𝑘)=𝑥(𝑡𝑘), which implies that the solution of (2.1) is left-continuous at 𝑡𝑘.

First, we consider the solution of the system (2.1).

Lemma 2.1. For any 𝑡(𝑡𝑘1,𝑡𝑘],𝑘=1,2,, the general solution of the system (2.1) is given by 𝐴𝑥(𝑡)=exp𝑘𝑡𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝑥𝑡0+𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐸𝑖𝑡𝑖𝑡𝑖1𝐴exp𝑖𝑡𝑖𝐵𝑠𝑖𝑢(𝑠)𝑑𝑠+𝐹𝑖𝑢𝑡𝑖+𝐸𝑘1𝑡𝑘1𝑡𝑘2𝐴exp𝑘1𝑡𝑘1𝐵𝑠𝑘1𝑢(𝑠)𝑑𝑠+𝐹𝑘1𝑢𝑡𝑘1+𝑡𝑡𝑘1𝐴exp𝑘𝐵(𝑡𝑠)𝑘𝑢(𝑠)𝑑𝑠,(2.3) where 𝑘=𝑡𝑘𝑡𝑘1, 𝑘=1,2,.

Proof. For 𝑡(𝑡0,𝑡1], we have 𝐴𝑥(𝑡)=exp1𝑡𝑡0𝑥𝑡0+𝑡𝑡0𝐴exp1(𝐵𝑡𝑠)1𝑢(𝑠)𝑑𝑠.(2.4) For 𝑡=𝑡+1, we have 𝑥𝑡+1=𝐸1𝐴exp11𝑥𝑡0+𝑡1𝑡0𝐴exp1𝑡1𝐵𝑠1𝑢(𝑠)𝑑𝑠+𝐹1𝑢𝑡1.(2.5) Similarly, for 𝑡(𝑡𝑖1,𝑡𝑖], 𝑖=2,3,,𝑘, we have 𝐴𝑥(𝑡)=exp𝑖𝑡𝑡𝑖1𝑥𝑡+𝑖1+𝑡𝑡𝑖1𝐴exp𝑖(𝐵𝑡𝑠)𝑖𝑢(𝑠)𝑑𝑠.(2.6) And, for 𝑡=𝑡+𝑖,𝑖=2,3,,𝑘, we have 𝑥𝑡+𝑖=𝐸𝑖𝐴exp𝑖𝑖𝑥𝑡𝑖1+𝑡𝑖𝑡𝑖1𝐴exp𝑖𝑡𝑖𝐵𝑠𝑖𝑢(𝑠)𝑑𝑠+𝐹𝑖𝑢𝑡𝑖.(2.7) Thus, by (2.4), (2.5), (2.6), and (2.7), it is easy to verify (2.3).

If 𝑡𝑓(𝑡0,𝑡1], then we are just concerned with a linear time-invariant system. Controllability and observability criteria can be found in standard text books [18, 19]. Thus, in the remainder of the paper, we will only be concerned with the case 𝑡𝑓(𝑡𝑘1,𝑡𝑘],𝑘=2,3,.

Now, we give some mathematical preliminaries as the basic tools in the following discussion.

Given matrices 𝐴𝑛×𝑛 and 𝐵𝑛×𝑝, denote 𝑚(𝐵) as the range of 𝐵, that is, 𝑚(𝐵)={𝑦𝑦=𝐵𝑥,forall𝑥𝑛×𝑛}, and denote 𝐴𝐵 as the minimal invariant subspace of 𝐴 on 𝑚(𝐵), that is, 𝐴𝐵=𝑚(𝐵)+𝑚(𝐴𝐵)++𝑚(𝐴𝑛1𝐵). Given a linear subspace 𝒲𝑛, denote 𝒲 as the orthogonal complement of 𝒲, that is, 𝒲={𝑥𝑥𝑇𝒲=0}.

The following lemma is a generalization of Theorem  7.8.1 in [17], which is the starting point for deriving the criteria of reachability and controllability.

Lemma 2.2. Given matrices 𝐴,𝐸𝑛×𝑛,𝐵,𝐹𝑛×𝑝, for any 0𝑡0<𝑡𝑓<+, one has 𝑥𝑥=𝐸𝑡𝑓𝑡0𝐴𝑡exp𝑓𝑡𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑢𝑓,piecewisecontinuous𝑢=𝐸𝐴𝐵+𝑚(𝐹).(2.8)

Proof. See Appendix A.

Lemma 2.3. Given two matrices 𝐴𝑛×𝑛,𝐶𝑞×𝑛, two scalars 𝑡0<𝑡𝑓, and a vector 𝑥𝑛, the following two statements are equivalent:(a)𝐶exp[𝐴(𝑡𝑡0)]𝑥=0,𝑡[𝑡0,𝑡𝑓],(b)𝑥𝑇𝐴𝑇𝐶𝑇=0.

Proof. See Appendix B.

Lemma 2.4. Given a matrix 𝐴𝑛×𝑛 and a linear subspace 𝒲𝑛, the following two statements are equivalent:(a)𝑚(𝐴)𝒲,(b)𝐴𝑇𝑊=0.

Proof. See Appendix C.

3. Reachability

In this section, we first investigate the reachability of system (2.1).

Definition 3.1 (reachability). The system (2.1) is said to be (completely) reachable on [𝑡0,𝑡𝑓](𝑡0<𝑡𝑓) if, for any terminal state 𝑥𝑓𝑛, there exists a piecewise continuous input 𝑢(𝑡)[𝑡0,𝑡𝑓]𝑝 such that the system (2.1) is driven from 𝑥(𝑡0)=0 to 𝑥(𝑡𝑓)=𝑥𝑓. Moreover, the set of all the reachable states on [𝑡0,𝑡𝑓] is said to be the reachable set on [𝑡0,𝑡𝑓], denoted as [𝑡0,𝑡𝑓].

Theorem 3.2. For the system (2.1), the reachable set on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], is given by 𝑡0,𝑡𝑓𝐴=exp𝑘𝑡𝑓𝑡𝑘1𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐸𝑖𝐴𝑖𝐵𝑖𝐹+𝑚𝑖𝐴+exp𝑘𝑡𝑓𝑡𝑘1𝐸𝑘1𝐴𝑘1𝐵𝑘1𝐹+𝑚𝑘1+𝐴𝑘𝐵𝑘.(3.1)

Proof. By Lemma 2.1, letting 𝑥(𝑡0)=0, we have 𝐴𝑥(𝑡)=exp𝑘𝑡𝑡𝑘1𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗×𝐸𝑖𝑡𝑖𝑡𝑖1𝐴exp𝑖𝑡𝑖𝐵𝑠𝑖𝑢(𝑠)𝑑𝑠+𝐹𝑖𝑢𝑡𝑖+𝐸𝑘1𝑡𝑘1𝑡𝑘2𝐴exp𝑘1𝑡𝑘1𝐵𝑠𝑘1𝑢(𝑠)𝑑𝑠+𝐹𝑘1𝑢𝑡𝑘1+𝑡𝑡𝑘1𝐴exp𝑘𝐵(𝑡𝑠)𝑘𝑢(𝑠)𝑑𝑠.(3.2) It follows that 𝑡0,𝑡𝑓=𝐴𝑥𝑥=exp𝑘𝑡𝑓𝑡𝑘1×𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐸𝑖𝑡𝑖𝑡𝑖1𝐴exp𝑖𝑡𝑖𝐵𝑠𝑖𝑢(𝑠)𝑑𝑠+𝐹𝑖𝑢𝑡𝑖+𝐸𝑘1𝑡𝑘1𝑡𝑘2𝐴exp𝑘1𝑡𝑘1𝐵𝑠𝑘1𝑢(𝑠)𝑑𝑠+𝐹𝑘1𝑢𝑡𝑘1+𝑡𝑓𝑡𝑘1𝐴exp𝑘𝑡𝑓𝐵𝑠𝑘𝑢𝐴(𝑠)𝑑𝑠,piecewisecontinuous𝑢=exp𝑘𝑡𝑓𝑡𝑘1𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗×𝑥𝑥=𝐸𝑖𝑡𝑖𝑡𝑖1𝐴exp𝑖𝑡𝑖𝐵𝑠𝑖𝑢(𝑠)𝑑𝑠+𝐹𝑖𝑢𝑡𝑖,+piecewisecontinuous𝑢𝑥𝑥=𝐸𝑘1𝑡𝑘1𝑡𝑘2𝐴exp𝑘1𝑡𝑘1𝐵𝑠𝑘1𝑢(𝑠)𝑑𝑠+𝐹𝑘1𝑢𝑡𝑘1+piecewisecontinuous𝑢𝑥𝑥=𝑡𝑓𝑡𝑘1𝐴exp𝑘𝑡𝑓𝐵𝑠𝑘𝑢.(𝑠)𝑑𝑠,piecewisecontinuous𝑢(3.3)
By Lemma 2.2, we get 𝑡0,𝑡𝑓𝐴=exp𝑘𝑡𝑓𝑡𝑘1𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐸𝑖𝐴𝑖𝐵𝑖𝐹+𝑚𝑖+𝐸𝑘1𝐴𝑘1𝐵𝑘1𝐹+𝑚𝑘1+𝐴𝑘𝐵𝑘.(3.4) This is just (3.1).

Since we have obtained the geometric form of the reachable set, we can establish a geometric type criterion as follows.

Theorem 3.3. The system (2.1) is reachable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], if and only if 𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐸𝑖𝐴𝑖𝐵𝑖𝐹+𝑚𝑖+𝐸𝑘1𝐴𝑘1𝐵𝑘1𝐹+𝑚𝑘1+𝐴𝑘𝐵𝑘=𝑛.(3.5)

Proof. Since 𝑡0,𝑡𝑓𝐴=exp𝑘𝑡𝑓𝑡𝑘1×𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐸𝑖𝐴𝑖𝐵𝑖𝐹+𝑚𝑖+𝐸𝑘1𝐴𝑘1𝐵𝑘1𝐹+𝑚𝑘1+𝐴𝑘𝐵𝑘𝐴=exp𝑘𝑡𝑓𝑡𝑘1𝑘2𝑖=1𝑖+1𝑗=k1𝐸𝑗𝐴exp𝑗𝑗𝐸𝑖𝐴𝑖𝐵𝑖𝐹+𝑚𝑖+𝐸𝑘1𝐴𝑘1𝐵𝑘1𝐹+𝑚𝑘1+𝐴𝑘𝐵𝑘(3.6) and the matrix exp[𝐴𝑘(𝑡𝑓𝑡𝑘1)] is nonsingular, the proof directly follows from Theorem 3.2.

Remark 3.4. Theorem 3.3 is a geometric type condition. By simple transformation, we can get an algebraic type condition. In fact, for 𝑖=1,2,, denote 𝑄𝑖=𝐵𝑖,𝐴𝑖𝐵𝑖,,𝐴𝑖𝑛1𝐵𝑖,(3.7) for 𝑖=1,2,,𝑘2, denote 𝐻𝑖=𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐸𝑖Q𝑖,𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐹𝑖,𝐻𝑘1=𝐸𝑘1𝑄𝑘1,𝐹𝑘1,(3.8) and, finally, denote 𝑄[𝑡0,𝑡𝑓]=𝐻1,𝐻2,,𝐻𝑘1,𝑄𝑘.(3.9) Then, it is easy to verify that 𝐴exp𝑘𝑡𝑓𝑡𝑘1𝑄𝑚[𝑡0,𝑡𝑓]𝑡=0,𝑡𝑓.(3.10) Thus, we get the following algebraic type criterion.

Corollary 3.5. The system (2.1) is reachable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], if and only if 𝑄rank[𝑡0,𝑡𝑓]=𝑛.(3.11)

4. Controllability

In this section, we investigate the controllability of system (2.1).

Definition 4.1 (controllability). The system (2.1) is said to be (completely) controllable on [𝑡0,𝑡𝑓](𝑡0<𝑡𝑓) if, for any initial state 𝑥0𝑛, there exists a piecewise continuous input 𝑢(𝑡)[𝑡0,𝑡𝑓]𝑝 such that the system (2.1) is driven from 𝑥(𝑡0)=𝑥0 to 𝑥(𝑡𝑓)=0. Moreover, the set of all the controllable states on [𝑡0,𝑡𝑓] is said to be the controllable set on [𝑡0,𝑡𝑓], denoted as 𝒞[𝑡0,𝑡𝑓].

First, we show the relationship between the controllable set and the reachable set.

Theorem 4.2. For the system (2.1), if 𝐸𝑖 is nonsingular, for 𝑖=1,,𝑘1, then the controllable set on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], satisfies 𝐴exp𝑘𝑡𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝒞𝑡0,𝑡𝑓𝑡0,𝑡𝑓.(4.1)

Proof. By Lemma 2.1, letting 𝑥(𝑡𝑓)=0, we have 𝐴0=exp𝑘𝑡𝑓𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝑥𝑡0𝐴exp𝑘𝑡𝑓𝑡𝑘1×𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗×𝐸𝑖𝑡𝑖𝑡𝑖1𝐴exp𝑖𝑡𝑖𝐵𝑠𝑖𝑢(𝑠)𝑑𝑠+𝐹𝑖𝑢𝑡𝑖+𝐸𝑘1𝑡𝑘1𝑡𝑘2𝐴exp𝑘1𝑡𝑘1𝐵𝑠𝑘1𝑢(𝑠)𝑑𝑠+𝐹𝑘1𝑢𝑡𝑘1+𝑡𝑓𝑡𝑘1𝐴exp𝑘𝑡𝑓𝐵𝑠𝑘𝑢(𝑠)𝑑𝑠.(4.2) It is equivalent to 𝐴exp𝑘𝑡𝑓𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝑥𝑡0𝐴=exp𝑘𝑡𝑓𝑡𝑘1𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗×𝐸𝑖𝑡𝑖𝑡𝑖1𝐴exp𝑖𝑡𝑖𝐵𝑠𝑖𝑢(𝑠)𝑑𝑠+𝐹𝑖𝑢𝑡𝑖+𝐸𝑘1𝑡𝑘1𝑡𝑘2𝐴exp𝑘1𝑡𝑘1𝐵𝑠𝑘1𝑢(𝑠)𝑑𝑠+𝐹𝑘1𝑢𝑡𝑘1+𝑡𝑓𝑡𝑘1𝐴exp𝑘𝑡𝑓𝐵𝑠𝑘𝑢(𝑠)𝑑𝑠.(4.3) This implies that 𝐴exp𝑘𝑡𝑓𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝑥𝑡0𝑡0,𝑡𝑓.(4.4) Hence, 𝐴exp𝑘𝑡𝑓𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝒞𝑡0,𝑡𝑓𝑡0,𝑡𝑓.(4.5)

Based on Theorem 4.2, we can establish a criterion for controllability of the system (2.1) as follows.

Theorem 4.3. The system (2.1) is controllable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], if and only if 𝑚1𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐸𝑖𝐴𝑖𝐵𝑖𝐹+𝑚𝑖+𝐸𝑘1𝐴𝑘1𝐵𝑘1𝐹+𝑚𝑘1+𝐴𝑘𝐵𝑘.(4.6)

Proof. First, it is easy to prove that (4.6) is equivalent to 𝐴𝑚exp𝑘𝑡𝑓𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝑡0,𝑡𝑓.(4.7)
Necessity: since the system is controllable, we have 𝒞𝑡0,𝑡𝑓=𝑛.(4.8) Then, by Theorem 4.2, we get 𝑡0,𝑡𝑓𝐴exp𝑘𝑡𝑓𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝑛𝐴=𝑚exp𝑘𝑡𝑓𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖.(4.9)
Sufficiency: suppose that (4.7) holds. For any 𝑥𝑛, we have 𝐴exp𝑘𝑡𝑓𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝑡𝑥0,𝑡𝑓.(4.10) This implies that there exists a piecewise continuous function 𝑢(𝑡),𝑡[𝑡0,𝑡𝑓], such that 𝐴0=exp𝑘𝑡𝑓𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝑥𝐴×exp𝑘𝑡𝑓𝑡𝑘1𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗×𝐸𝑖𝑡𝑖𝑡𝑖1𝐴exp𝑖𝑡𝑖𝐵𝑠𝑖𝑢(𝑠)𝑑𝑠+𝐹𝑖𝑢𝑡𝑖+𝐸𝑘1𝑡𝑘1𝑡𝑘2𝐴exp𝑘1𝑡𝑘1𝐵𝑠𝑘1𝑢(𝑠)𝑑𝑠+𝐹𝑘1𝑢𝑡𝑘1+𝑡𝑓𝑡𝑘1𝐴exp𝑘𝑡𝑓𝐵𝑠𝑘𝑢(𝑠)𝑑𝑠.(4.11) Then, we know that 𝑥𝒞[𝑡0,𝑡𝑓]. Hence, the system (2.1) is controllable.

In the general case, for system (2.1), controllability is not equivalent to reachability. But under some mild conditions, we can show that they are equivalent.

Corollary 4.4. For the system (2.1), if 𝐸𝑖 is nonsingular, 𝑖=1,2,,𝑘1, then the following statements are equivalent:(a)the system is reachable,(b)the system is controllable,(c)𝑘2𝑖=1𝑖+1𝑗=𝑘1𝐸𝑗exp(𝐴𝑗𝑗)(𝐸𝑖𝐴𝑖𝐵𝑖+𝑚(𝐹𝑖))+𝐸𝑘1𝐴𝑘1𝐵𝑘1+𝑚(𝐹𝑘1)+𝐴𝑘𝐵𝑘=𝑛.

Proof. Since 𝐸𝑖 is nonsingular, 𝑖=1,2,,𝑘1, we have that 𝐴exp𝑘𝑡𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖(4.12) is nonsingular. It follows that 𝐴exp𝑘𝑡𝑡𝑘11𝑖=𝑘1𝐸𝑖𝐴exp𝑖𝑖𝒞𝑡0,𝑡𝑓𝑡=0,𝑡𝑓.(4.13) It is easy to see that 𝒞[𝑡0,𝑡𝑓]=𝑛[𝑡0,𝑡𝑓]=𝑛.

Remark 4.5. For system (2.1), assume that 𝐴𝑖=𝐴,𝐵𝑖=𝐵,𝑖=1,,𝑘. Then, it is easy to see that Theorem 4.2 concludes the results of Theorem 3.4 in [15].

Remark 4.6. For system (2.1), assume that 𝐸𝑖=𝐼,𝐹𝑖=0,𝑖=1,,𝑘. Then, it is easy to see that Theorem  5 in [20] is a special case of Corollary 4.4.

5. Observability and Determinability

In the above analysis, reference is made to reachability and controllability only. It should be noticed that the observability and determinability counterparts can be addressed dualistically. In this section, we outline the relevant concepts and the corresponding criteria.

Definition 5.1 (observability). The system (2.1) is said to be (completely) observable on [𝑡0,𝑡𝑓] (𝑡0<𝑡𝑓) if any initial state 𝑥0𝑛 can be uniquely determined by the corresponding system input 𝑢(𝑡) and the system output 𝑦(𝑡), for 𝑡[𝑡0,𝑡𝑓].

Definition 5.2 (determinability). The system (2.1) is said to be (completely) determinable on [𝑡0,𝑡𝑓] (𝑡0<𝑡𝑓) if any terminal state 𝑥𝑓𝑛 can be uniquely determined by the corresponding system input 𝑢(𝑡) and the system output 𝑦(𝑡), for 𝑡[𝑡0,𝑡𝑓].

In order to investigate observability and determinability for the system (2.1), we first investigate those of the following zero input system: ̇𝑥(𝑡)=𝐴𝑘𝑥𝑡(𝑡),𝑡𝑘1,𝑡𝑘,𝑥𝑡+𝑘=𝐸𝑘𝑥𝑡𝑘,𝑦(𝑡)=𝐶𝑘𝑡𝑥(𝑡),𝑡𝑘1,𝑡𝑘,𝑥𝑡+0=𝑥0,𝑡00.(5.1) It is obvious that observability and determinability of the system (2.1) are equivalent to those of the system (5.1), respectively.

For the system (5.1), by Lemma 2.1, the output is given by 𝐶𝑦(𝑡)=1𝐴exp1𝑡𝑡0𝑥𝑡0𝑡,𝑡0,𝑡1,𝐶𝑖𝐴exp𝑖𝑡𝑡𝑖11𝑗=𝑖1𝐸𝑗𝐴exp𝑗𝑗𝑥𝑡0𝑡,𝑡𝑖1,𝑡𝑖,𝑖=2,,𝑘.(5.2)

Theorem 5.3. The system (5.1) is observable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], if and only if 2𝑖=𝑘𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖+𝐴𝑇1𝐶𝑇1=𝑛.(5.3)

Proof. We prove the complementary proposition of Theorem 5.3, that is, the system (5.1) is not observable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], if and only if 2𝑖=𝑘𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖+𝐴𝑇1𝐶𝑇1𝑛.(5.4)
Necessity: if the system (5.1) is not observable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], then there exists 𝑥0𝑛, nonzero, such that 𝑦(𝑡)0,𝑡[𝑡0,𝑡𝑓]. This means that 𝐶1𝐴exp1𝑡𝑡0𝑥0𝑡=0,𝑡0,𝑡1,𝐶𝑖𝐴exp𝑖𝑡𝑡𝑖11𝑗=𝑖1𝐸𝑗𝐴exp𝑗𝑗𝑥0𝑡=0,𝑡𝑖1,𝑡𝑖𝐶,𝑖=2,,𝑘1,𝑘𝐴exp𝑘𝑡𝑡𝑘11𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝑥0𝑡=0,𝑡𝑘1,𝑡𝑓.(5.5) By Lemma 2.3, we get 𝑥𝑇0𝐴𝑇1𝐶𝑇1𝑥=0,𝑇0𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖=0,𝑖=2,,𝑘.(5.6) It follows that 𝑥𝑇0𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖=0.(5.7) Then, we know that 𝑥0𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖.(5.8) This implies (5.4).
Sufficiency: on the contrary, if (5.4) holds, there exists 𝑥0𝑛, nonzero, such that 𝑥𝑇0𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖=0.(5.9) It follows that 𝑥𝑇0𝐴𝑇1𝐶𝑇1𝑥=0,𝑇0𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖=0,𝑖=2,,𝑘.(5.10) By Lemma 2.3, we get 𝐶1𝐴exp1𝑡𝑡0𝑥0𝑡=0,𝑡0,𝑡1,𝐶𝑖𝐴exp𝑖𝑡𝑡𝑖11𝑗=𝑖1𝐸𝑗𝐴exp𝑗𝑗𝑥0𝑡=0,𝑡𝑖1,𝑡𝑖𝐶,𝑖=2,,𝑘1,𝑘𝐴exp𝑘𝑡𝑡𝑘11𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝑥0𝑡=0,𝑡𝑘1,𝑡𝑓.(5.11) This means that 𝑦(𝑡)0,𝑡[𝑡0,𝑡𝑓]. Thus, the system (5.1) is not observable.

Remark 5.4. Theorem 5.3 is a geometric type condition. By simple transformation, we can get an algebraic type condition. In fact, for 𝑖=1,2,, denote 𝑂𝑖=𝐶𝑇𝑖,𝐴𝑇i𝐶𝑇𝑖𝐴,,𝑇𝑖𝑛1𝐶𝑇𝑖,(5.12) for 𝑖=2,,𝑘, denote 𝐺𝑖=𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝑂𝑖,(5.13) and, finally, denote 𝑂[𝑡0,𝑡𝑓]=𝑂1,𝐺2,,𝐺𝑘.(5.14) Then, it is easy to verify that 𝑂𝑚[𝑡0,𝑡𝑓]=𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖.(5.15) Thus, we get the following algebraic type criterion.

Corollary 5.5. The system (5.1) is observable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], if and only if 𝑂rank[𝑡0,𝑡𝑓]=𝑛.(5.16)

Next, we establish a criterion for determinability.

Theorem 5.6. The system (5.1) is determinable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], if and only if 𝑚𝑘1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖+𝐴𝑇1𝐶𝑇1.(5.17)

Proof. First, by Lemma 2.4, we know that (5.17) is equivalent to 1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖+𝐴𝑇1𝐶𝑇1=0.(5.18) Similar to the proof of Theorem 5.3, we prove the complementary proposition of Theorem 5.6, that is, the system (5.1) is not determinable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], if and only if 1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖+𝐴𝑇1𝐶𝑇10.(5.19) Necessity: if the system (5.1) is not determinable on [𝑡0,𝑡𝑓], where 𝑡𝑓(𝑡𝑘1,𝑡𝑘], then there exists a terminal 𝑥𝑓𝑛, nonzero, such that 𝑦(𝑡)=0,𝑡[𝑡0,𝑡𝑓]. Then, there exists a nonzero 𝑥0𝑛 as the initial state such that the system is driven from 𝑥(𝑡0)=𝑥0 to 𝑥(𝑡𝑓)=𝑥𝑓, that is, 𝑥𝑓=exp[𝐴𝑘(𝑡𝑓𝑡𝑘1)]1𝑗=𝑘1𝐸𝑗exp(𝐴𝑗𝑗)𝑥0. This means that 𝐶1𝐴exp1𝑡𝑡0𝑥0𝑡=0,𝑡0,𝑡1,𝐶𝑖𝐴exp𝑖𝑡𝑡𝑖11𝑗=𝑖1𝐸𝑗𝐴exp𝑗𝑗𝑥0𝑡=0,𝑡𝑖1,𝑡𝑖𝐶,𝑖=2,,𝑘1,𝑘𝐴exp𝑘𝑡𝑡𝑘11𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝑥0𝑡=0,𝑡𝑘1,𝑡𝑓.(5.20) By Lemma 2.3, we get 𝑥𝑇0𝐴𝑇1𝐶𝑇1𝑥=0,𝑇0𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖=0,𝑖=2,,𝑘.(5.21) It follows that 𝑥𝑇0𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖=0.(5.22) This implies that 𝑥0𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖.(5.23) Since exp[𝐴𝑘(𝑡𝑘1𝑡𝑓)]𝑥𝑓=1𝑗=𝑘1𝐸𝑗exp(𝐴𝑗𝑗)𝑥0, we know that 𝐴exp𝑘𝑡𝑘1𝑡𝑓𝑥𝑓1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖.(5.24) It implies that 1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖0.(5.25) Hence, (5.19) holds.
Sufficiency: on the contrary, if (5.19) holds, then we know that 1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖0.(5.26) Then, there exists a nonzero 𝑥𝑓 satisfying 𝐴exp𝑘𝑡𝑘1𝑡𝑓𝑥𝑓1𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖(5.27) such that there exists a nonzero 𝑥0 satisfying 𝐴exp𝑘𝑡𝑘1𝑡𝑓𝑥𝑓=𝑥0,𝑥𝑇0𝐴𝑇1𝐶𝑇1+𝑘𝑖=2𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖=0.(5.28) It follows that 𝑥𝑇0𝐴𝑇1𝐶𝑇1𝑥=0,𝑇0𝑖1𝑗=1𝐴exp𝑇𝑗𝑗𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖=0,𝑖=2,,𝑘.(5.29) By Lemma 2.3, we get 𝐶1𝐴exp1𝑡𝑡0𝑥0𝑡=0,𝑡0,𝑡1,𝐶𝑖𝐴exp𝑖𝑡𝑡𝑖11𝑗=𝑖1𝐸𝑗𝐴exp𝑗𝑗𝑥0𝑡=0,𝑡𝑖1,𝑡𝑖𝐶,𝑖=2,,𝑘1,𝑘𝐴exp𝑘𝑡𝑡𝑘11𝑗=𝑘1𝐸𝑗𝐴exp𝑗𝑗𝑥0𝑡=0,𝑡𝑘1,𝑡𝑓.(5.30) This means that 𝑦(𝑡)0, 𝑡[𝑡0,𝑡𝑓]. Thus, we find a nonterminal nonzero state 𝑥𝑓 such that the output 𝑦(𝑡) remains zero. Hence, the system (5.1) is not determinable.

Similar to the controllability and reachability case, under some simple condition, we can show that for the system (5.1), observability is equivalent to determinability.

Corollary 5.7. For the system (5.1), if 𝐸𝑖 is nonsingular, 𝑖=1,2,,𝑘1, then the following statements are equivalent:(a)the system is observable,(b)the system is determinable,(c)2𝑖=𝑘𝑖1𝑗=1exp(𝐴𝑇𝑗𝑗)𝐸𝑇𝑗𝐴𝑇𝑖𝐶𝑇𝑖+𝐴𝑇1𝐶𝑇1=𝑛.

Proof. If 𝐸𝑖 is nonsingular, 𝑖=1,2,,𝑘1, then we know that 1𝑗=𝑘1𝐸𝑗exp(𝐴𝑗𝑗) is nonsingular. Hence, we get (5.3) and (5.17) are equivalent.

Remark 5.8. For system (2.1), assume that 𝐴𝑖=𝐴,𝐵𝑖=𝐵,𝑖=1,,𝑘. Then, it is easy to see that Theorem 4.3 concludes the results of Theorem  4.2 in [15].

Remark 5.9. For system (2.1), assume that 𝐸𝑖=𝐼,𝐹𝑖=0,𝑖=1,,𝑘. Then, it is easy to see that Theorem  2 in [20] is a special case of Corollary 5.7.

6. Examples

In this section, we give two numerical examples to illustrate how to utilize our criteria.

Example 6.1. Consider a 3-dimensional linear piecewise constant impulsive system with 𝐴1=000000000,𝐵1=100,𝐶1=,𝐷0101=0,𝐸1=100010000,𝐹1=010,𝐴2=000000000,𝐵2=100,𝐶2=,𝐷1002=0,𝐸2=110100000,𝐹2=110,𝐴3=000000000,𝐵3=010,𝐶3=,𝐷1003=0,𝐸3=000100000,𝐹3=010,(6.1) where 𝑡0=0, 𝑡1=1, 𝑡2=2, and 𝑡3=3.

Now, we try to use our criteria to investigate the reachability, controllability, observability, and determinability on [0,𝑡𝑓], where 𝑡𝑓(2,3], of the system in Example 6.1.

First, we consider the reachability. By a simple calculation, we have 𝐸2𝐴exp2𝐸1𝐴1𝐵1𝐹+𝑚1+𝐸2𝐴2𝐵2𝐹+𝑚2+𝐴3𝐵3100,010.=span(6.2) By Theorem 3.3, the system should not be reachable. In fact, for any piecewise continuous input 𝑢(𝑡),𝑡[0,𝑡𝑓], and any nonzero initial state 𝑥0=[𝑥01𝑥02𝑥03]𝑇, we have 𝑥𝑡𝑓=0.(6.3) This fact shows that the system is indeed not reachable.

Next, we consider the controllability. By a simple calculation, we have 𝐸𝑚2𝐴exp2𝐸1𝐴exp1100,010=span.(6.4) It is easy to see that 𝐸𝑚2𝐴exp2𝐸1𝐴exp1𝐸2𝐴exp2𝐸1𝐴1𝐵1𝐹+𝑚1+𝐸2𝐴2𝐵2𝐹+𝑚2+𝐴3𝐵3.(6.5)

By Theorem 4.3, the system should be controllable. In fact, we can take the piecewise constant input 𝑐𝑢(𝑡)=1],],𝑐,𝑡(0,10,𝑡(1,23].,𝑡(2,3(6.6) Then, for any nonzero initial state 𝑥0=[𝑥01𝑥02𝑥03]𝑇, we have 𝑥𝑡𝑓=𝑥01+0.5𝑐1𝑥02+1.5𝑐1+22𝑡𝑓+0.5𝑡2𝑓𝑐30.(6.7) Obviously, if 𝑐1=2𝑥01, 𝑐3=(𝑥021.5𝑐1)/(22𝑡𝑓+0.5𝑡2𝑓), then 𝑥(𝑡𝑓)=0. This fact shows that the system is indeed controllable.

Next, we consider the observability. By a simple calculation, we have 𝐴𝑇1𝐶𝑇1𝐴+exp𝑇1𝐸𝑇1𝐴𝑇2𝐶𝑇2𝐴+exp𝑇1𝐸𝑇1𝐴exp𝑇2𝐸𝑇2𝐴𝑇3𝐶𝑇3100,010.=span(6.8) By Theorem 5.3, the system should not be observable. In fact, for any piecewise continuous input 𝑢(𝑡),𝑡[0,𝑡𝑓], and nonzero initial state 𝑥0=[001]𝑇, we have 𝑦(𝑡)0,𝑡0,𝑡𝑓.(6.9) This fact shows that the system is indeed not observable.

Finally, we consider the determinability. By a simple calculation, we have 𝐸2𝐴exp2𝐸1𝐴exp1×𝐴𝑇1𝐶𝑇1𝐴+exp𝑇1𝐸𝑇1𝐴𝑇2𝐶𝑇2𝐴+exp𝑇1𝐸𝑇1𝐴exp𝑇2𝐸𝑇2𝐴𝑇3𝐶𝑇3=100001span=0.(6.10) It follows that 𝐸2𝐴exp2𝐸1𝐴exp1×𝐴𝑇1𝐶𝑇1𝐴+exp𝑇1𝐸𝑇1𝐴𝑇2𝐶𝑇2𝐴+exp𝑇1𝐸𝑇1𝐴exp𝑇2𝐸𝑇2𝐴𝑇3𝐶𝑇3=0.(6.11) By Theorem 5.6, the system should be determinable. In fact, for any nonzero terminal state 𝑥𝑓=[𝑥𝑓1𝑥𝑓2𝑥𝑓3]𝑇, there must exist a nonzero initial state 𝑥0=[𝑥01𝑥02𝑥03]𝑇 such that 𝐴exp32𝑡𝑓𝑥𝑓=𝐸2𝐴exp2𝐸1𝐴exp1𝑥0.(6.12) It follows that 2𝑡𝑓𝑥𝑓=𝑥1101000000.(6.13) This means that 𝑥𝑓3=𝑥03=0 and |𝑥01|+|𝑥02|0. It is easy to verify that, for any initial state 𝑥0 satisfying |𝑥01|+|𝑥02|0, we have 𝑦(𝑡)0, 𝑡(0,𝑡𝑓). This fact shows that the system is indeed determinable.

Example 6.2. Consider a 3-dimensional linear piecewise constant impulsive system with 𝐴1=000000000,𝐵1=100,𝐶1=,𝐷0101=0,𝐸1=100010001,𝐹1=010,𝐴2=000000000,𝐵2=010,𝐶2=,𝐷1002=0,𝐸2=100010001,𝐹2=110,𝐴3=000000000,𝐵3=001,𝐶3=,𝐷1003=0,𝐸3=100110001,𝐹3=010,(6.14) where 𝑡0=0, 𝑡1=1, 𝑡2=2, and 𝑡3=3.

Now, we try to use our criteria to investigate the reachability and controllability on [0,𝑡𝑓], where 𝑡𝑓(2,3], of the system in Example 6.2.

First, we consider reachability. By a simple calculation, we have 𝐸2𝐴exp2𝐸1𝐴1𝐵1𝐹+𝑚1+𝐸2𝐴2𝐵2𝐹+𝑚2+𝐴3𝐵3100,010,001=span=3.(6.15) By Theorem 3.3, the system should be reachable. In fact, we take the piecewise constant input 𝑐𝑢(𝑡)=1],𝑐,𝑡(0,12],𝑐,𝑡(1,23].,𝑡(2,3(6.16) Then, letting 𝑥(0)=0, for any nonzero terminal state 𝑥(3)=[𝑥𝑓1𝑥𝑓2𝑥𝑓3]𝑇, we have 𝑥𝑓1𝑥𝑓2𝑥𝑓3=𝑐1101200011𝑐2𝑐3.(6.17) Obviously, we can select suitable 𝑐1,𝑐2, and 𝑐3 such that 𝑥𝑓 is any state in 3. This fact shows that the system is indeed reachable.

Next, by Theorem 4.3, the system should be reachable. In fact, we take the piecewise constant input 𝑐𝑢(𝑡)=1],𝑐,𝑡(0,12],𝑐,𝑡(1,23].,𝑡(2,3(6.18) Then, for any nonzero initial state 𝑥(0)=[𝑥01𝑥02𝑥03]𝑇, letting 𝑥(3)=0, we have 𝑥0=01𝑥02𝑥03+𝑐1101200011𝑐2𝑐3.(6.19) Obviously, we can select suitable 𝑐1,𝑐2, and 𝑐3 such that 𝑥0 is any state in 3. This fact shows that the system is indeed controllable.

Finally, according to the conclusion in Corollary 4.4, since the matrices 𝐸1, 𝐸2, and 𝐸3 in Example 6.1 are singular, we know that the reachability might not be equivalent to the controllability in this example. However, the reachability should be equivalent to the controllability in Example 6.2 since the matrices 𝐸1,𝐸2, and 𝐸3 in this example are nonsingular. From the above analysis, all these statements are correct indeed.

7. Conclusion

This paper has studied the controllability and observability of linear piecewise constant impulsive systems. Necessary and sufficient criteria for reachability and controllability have been established, respectively. Moreover, it has been proved that the reachability is equivalent to the controllability under some mild conditions. Then, necessary and sufficient criteria for the observability and determinability of such systems have been established, respectively. It has been also proved that the observability is equivalent to the determinability under some mild conditions. Our criteria are of the geometric type, and they can be transformed into algebraic type conveniently. Finally, a numerical example has been given to illustrate the utility of our criteria.

Appendices

A. Proof of Lemma 2.2

By Theorem  7.8.1 in [17], we have 𝑥𝑥=𝑡𝑓𝑡0𝐴𝑡exp𝑓𝑠𝐵𝑢(𝑠)𝑑𝑠,piecewisecontinuous𝑢=𝐴𝐵.(A.1) Thus, it is easy to see that 𝑥𝑥=𝐸𝑡𝑓𝑡0𝐴𝑡exp𝑓𝑡𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑢𝑓,piecewisecontinuous𝑢𝐸𝐴𝐵+𝑚(𝐹).(A.2) Moreover, we have 𝑥𝑥=𝑡𝐸𝑡0𝐴𝑡exp𝐸𝑠𝐵𝑢(𝑠)𝑑𝑠,piecewisecontinuous𝑢=𝐴𝐵,(A.3) where 𝑡𝐸=(𝑡0+𝑡𝑓)/2. Then, for any 𝑥𝐸𝐴𝐵+𝑚(𝐹), there exist a piecewise continuous function 𝑢(𝑡), 𝑡[𝑡0,𝑡𝐸], and 𝑦𝑛 such that 𝑥=𝐸𝑡𝐸𝑡0𝐴𝑡exp𝐸𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑦.(A.4) Then, we can take 𝑢𝑡𝑣(𝑡)=(𝑡),𝑡0,𝑡𝐸,𝑡0,𝑡𝐸,𝑡𝑓,𝑦,𝑡=𝑡𝑓,(A.5) such that 𝑥=𝐸𝑡𝑓𝑡0𝐴𝑡exp𝑓𝑡𝑠𝐵𝑣(𝑠)𝑑𝑠+𝐹𝑣𝑓.(A.6) This implies that 𝑥𝑥𝑥=𝐸𝑡𝑓𝑡0𝐴𝑡exp𝑓𝑡𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑢𝑓.,piecewisecontinuous𝑢(A.7) It follows that 𝑥𝑥=𝐸𝑡𝑓𝑡0𝐴𝑡exp𝑓𝑡𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑢𝑓,piecewisecontinuous𝑢𝐸𝐴𝐵+𝑚(𝐹).(A.8) By (A.2) and (A.8), we know that (2.8) holds.

B. Proof of Lemma 2.3

((a)(b)) If 𝐶exp[𝐴(𝑡𝑡0)]𝑥=0,𝑡[𝑡0,𝑡𝑓], we get 𝐶𝑥=0. Then, for 𝑖=1,,𝑛1, calculating the 𝑖th derivative of 𝐶exp[𝐴(𝑡𝑡0)]𝑥 with respect to 𝑡 at 𝑡=𝑡0, we get 𝐶𝐴𝑖𝑥=0.(B.1) Thus, we know that 𝑥𝑇𝐶𝑇,𝐶𝑇𝐴𝑇,,𝐶𝑇𝐴𝑇𝑛1=0.(B.2) Hence, 𝑥𝑇𝐴𝑇𝐶𝑇=0.

((a)(b)) If 𝑥𝑇𝐴𝑇𝐶𝑇=0, it follows that 𝑥𝑇[𝐶𝑇,𝐶𝑇𝐴𝑇,,𝐶𝑇(𝐴𝑇)𝑛1]=0. That is 𝐶𝐴𝑖𝑥=0,𝑖=0,1,,𝑛1.(B.3) Then, it is easy to prove that 𝐶exp[𝐴(𝑡𝑡0)]𝑥=0,𝑡[𝑡0,𝑡𝑓].

C. Proof of Lemma 2.4

Given a matrix 𝐴𝑛×𝑛 and a linear subspace 𝒲𝑛, the following two statements are equivalent:(a)𝑚(𝐴)𝒲,(b)𝐴𝑇𝑊=0.

((a)(b)) Assume that 𝑚(𝐴)𝒲 ((b)(a)). It is equivalent to 𝒲𝑚(𝐴)=0. It follows that, for any 𝑥𝒲,𝑥𝑇𝐴=0. That is, 𝐴𝑇𝑥=0. This implies that 𝐴𝑇𝑊=0.

((b)(a)) Assume that 𝐴𝑇𝑊=0. It follows that, for any 𝑥𝒲, 𝐴𝑇𝑥=0. That is, 𝑥𝑇𝐴=0. This implies that 𝑚(𝐴)𝒲.

Acknowledgments

The authors are grateful to Professor Xinghuo Yu, the Associate Editor, and the reviewers for their helpful and valuable comments and suggestions for improving this paper. This work is supported by National Natural Science Foundation (NNSF) of China (60774089, 10972003, and 60736022). This work is also supported by N2008-07 (08010702014) from Beijing Institute of Petrochemical Technology.

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