Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 182040 | https://doi.org/10.1155/2012/182040

Hong Shi, Guangming Xie, "Controllability and Observability Criteria for Linear Piecewise Constant Impulsive Systems", Journal of Applied Mathematics, vol. 2012, Article ID 182040, 24 pages, 2012. https://doi.org/10.1155/2012/182040

Controllability and Observability Criteria for Linear Piecewise Constant Impulsive Systems

Academic Editor: Junjie Wei
Received14 May 2012
Accepted22 Jul 2012
Published02 Sep 2012

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 [1–15].

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 [17–19]. 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 Μ‡π‘₯(𝑑)=π΄π‘˜π‘₯(𝑑)+π΅π‘˜π‘’ξ€Ίπ‘‘(𝑑),π‘‘βˆˆπ‘˜βˆ’1,π‘‘π‘˜ξ€Έ,π‘₯𝑑+π‘˜ξ€Έ=πΈπ‘˜π‘₯ξ€·π‘‘βˆ’π‘˜ξ€Έ+πΉπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ,𝑦(𝑑)=πΆπ‘˜π‘₯(𝑑)+π·π‘˜ξ€Ίπ‘‘π‘’(𝑑),π‘‘βˆˆπ‘˜βˆ’1,π‘‘π‘˜ξ€Έ,π‘₯𝑑+0ξ€Έ=π‘₯0,𝑑0β‰₯0,(2.1) where π‘˜=1,2,…,π΄π‘˜,π΅π‘˜,πΆπ‘˜,π·π‘˜,πΈπ‘˜, and πΉπ‘˜ are the known 𝑛×𝑛, 𝑛×𝑝, 𝑝×𝑛, π‘žΓ—π‘, 𝑛×𝑛, and 𝑛×𝑝 constant matrices; π‘₯(𝑑)βˆˆβ„π‘› is the state vector, and 𝑒(𝑑)βˆˆβ„π‘ the input vector, 𝑦(𝑑)βˆˆβ„π‘ž the output vector; π‘₯(𝑑+)∢=limβ„Žβ†’0+π‘₯(𝑑+β„Ž), π‘₯(π‘‘βˆ’)∢=limβ„Žβ†’0βˆ’π‘₯(π‘‘βˆ’β„Ž), and the discontinuity points are 𝑑1<𝑑2<β‹―<π‘‘π‘˜<β‹―,limπ‘˜β†’βˆžπ‘‘π‘˜=∞,(2.2) where 𝑑0<𝑑1 and π‘₯(π‘‘βˆ’π‘˜)=π‘₯(π‘‘π‘˜), which implies that the solution of (2.1) is left-continuous at π‘‘π‘˜.

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

Lemma 2.1. For any π‘‘βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜],π‘˜=1,2,…, the general solution of the system (2.1) is given by 𝐴π‘₯(𝑑)=expπ‘˜ξ€·π‘‘βˆ’π‘‘π‘˜βˆ’1ξƒ―ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έπ‘₯𝑑0ξ€Έ+π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έξ‚΅πΈπ‘–ξ€œπ‘‘π‘–π‘‘π‘–βˆ’1𝐴expπ‘–ξ€·π‘‘π‘–π΅βˆ’π‘ ξ€Έξ€»π‘–π‘’(𝑠)𝑑𝑠+𝐹𝑖𝑒𝑑𝑖+πΈπ‘˜βˆ’1ξ€œπ‘‘π‘˜βˆ’1π‘‘π‘˜βˆ’2𝐴expπ‘˜βˆ’1ξ€·π‘‘π‘˜βˆ’1π΅βˆ’π‘ ξ€Έξ€»π‘˜βˆ’1𝑒(𝑠)𝑑𝑠+πΉπ‘˜βˆ’1π‘’ξ€·π‘‘π‘˜βˆ’1ξ€Έξƒ°+ξ€œπ‘‘π‘‘π‘˜βˆ’1𝐴expπ‘˜ξ€»π΅(π‘‘βˆ’π‘ )π‘˜π‘’(𝑠)𝑑𝑠,(2.3) where β„Žπ‘˜=π‘‘π‘˜βˆ’π‘‘π‘˜βˆ’1, π‘˜=1,2,….

Proof. For π‘‘βˆˆ(𝑑0,𝑑1], we have 𝐴π‘₯(𝑑)=exp1ξ€·π‘‘βˆ’π‘‘0π‘₯𝑑0ξ€Έ+ξ€œπ‘‘π‘‘0𝐴exp1(ξ€»π΅π‘‘βˆ’π‘ )1𝑒(𝑠)𝑑𝑠.(2.4) For 𝑑=𝑑+1, we have π‘₯𝑑+1ξ€Έ=𝐸1𝐴exp1β„Ž1ξ€Έπ‘₯𝑑0ξ€Έ+ξ€œπ‘‘1𝑑0𝐴exp1𝑑1π΅βˆ’π‘ ξ€Έξ€»1𝑒(𝑠)𝑑𝑠+𝐹1𝑒𝑑1ξ€Έ.(2.5) Similarly, for π‘‘βˆˆ(π‘‘π‘–βˆ’1,𝑑𝑖], 𝑖=2,3,…,π‘˜, we have 𝐴π‘₯(𝑑)=expπ‘–ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1π‘₯𝑑+π‘–βˆ’1ξ€Έ+ξ€œπ‘‘π‘‘π‘–βˆ’1𝐴exp𝑖(ξ€»π΅π‘‘βˆ’π‘ )𝑖𝑒(𝑠)𝑑𝑠.(2.6) And, for 𝑑=𝑑+𝑖,𝑖=2,3,…,π‘˜, we have π‘₯𝑑+𝑖=𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έπ‘₯ξ€·π‘‘π‘–βˆ’1ξ€Έ+ξ€œπ‘‘π‘–π‘‘π‘–βˆ’1𝐴expπ‘–ξ€·π‘‘π‘–π΅βˆ’π‘ ξ€Έξ€»π‘–ξ‚Άπ‘’(𝑠)𝑑𝑠+𝐹𝑖𝑒𝑑𝑖.(2.7) Thus, by (2.4), (2.5), (2.6), and (2.7), it is easy to verify (2.3).

If π‘‘π‘“βˆˆ(𝑑0,𝑑1], 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 π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜],π‘˜=2,3,….

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

Given matrices π΄βˆˆβ„π‘›Γ—π‘› and π΅βˆˆβ„π‘›Γ—π‘, denote β„π‘š(𝐡) as the range of 𝐡, that is, β„π‘š(𝐡)={π‘¦βˆ£π‘¦=𝐡π‘₯,forallπ‘₯βˆˆβ„π‘›Γ—π‘›}, and denote ⟨𝐴∣𝐡⟩ as the minimal invariant subspace of 𝐴 on β„π‘š(𝐡), that is, ⟨𝐴∣𝐡⟩=β„π‘š(𝐡)+β„π‘š(𝐴𝐡)+β‹―+β„π‘š(π΄π‘›βˆ’1𝐡). Given a linear subspace π’²βŠ†β„π‘›, denote π’²βŸ‚ as the orthogonal complement of 𝒲, that is, π’²βŸ‚={π‘₯∣π‘₯𝑇𝒲=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,πΈβˆˆβ„π‘›Γ—π‘›,𝐡,πΉβˆˆβ„π‘›Γ—π‘, for any 0≀𝑑0<𝑑𝑓<+∞, one has ξ‚»ξ€œπ‘₯∣π‘₯=𝐸𝑑𝑓𝑑0𝐴𝑑expπ‘“ξ€·π‘‘βˆ’π‘ ξ€Έξ€»π΅π‘’(𝑠)𝑑𝑠+𝐹𝑒𝑓,βˆ€piecewisecontinuous𝑒=𝐸⟨𝐴∣𝐡⟩+β„π‘š(𝐹).(2.8)

Proof. See Appendix A.

Lemma 2.3. Given two matrices π΄βˆˆβ„π‘›Γ—π‘›,πΆβˆˆβ„π‘žΓ—π‘›, two scalars 𝑑0<𝑑𝑓, and a vector π‘₯βˆˆβ„π‘›, the following two statements are equivalent:(a)𝐢exp[𝐴(π‘‘βˆ’π‘‘0)]π‘₯=0,π‘‘βˆˆ[𝑑0,𝑑𝑓],(b)π‘₯π‘‡βŸ¨π΄π‘‡βˆ£πΆπ‘‡βŸ©=0.

Proof. See Appendix B.

Lemma 2.4. Given a matrix π΄βˆˆβ„π‘›Γ—π‘› and a linear subspace π’²βŠ†β„π‘›, the following two statements are equivalent:(a)β„π‘š(𝐴)βŠ†π’²,(b)π΄π‘‡π‘ŠβŸ‚=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 [𝑑0,𝑑𝑓](𝑑0<𝑑𝑓) if, for any terminal state π‘₯π‘“βˆˆβ„π‘›, there exists a piecewise continuous input 𝑒(𝑑)∢[𝑑0,𝑑𝑓]→ℝ𝑝 such that the system (2.1) is driven from π‘₯(𝑑0)=0 to π‘₯(𝑑𝑓)=π‘₯𝑓. Moreover, the set of all the reachable states on [𝑑0,𝑑𝑓] is said to be the reachable set on [𝑑0,𝑑𝑓], denoted as β„›[𝑑0,𝑑𝑓].

Theorem 3.2. For the system (2.1), the reachable set on [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], is given by ℛ𝑑0,𝑑𝑓𝐴=expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—πΈξ€Έξ€·π‘–βŸ¨π΄π‘–βˆ£π΅π‘–ξ€·πΉβŸ©+β„π‘šπ‘–ξ€Ίπ΄ξ€Έξ€Έ+expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1πΈξ€Έξ€»ξ€·π‘˜βˆ’1βŸ¨π΄π‘˜βˆ’1βˆ£π΅π‘˜βˆ’1ξ€·πΉβŸ©+β„π‘šπ‘˜βˆ’1ξ€Έξ€Έ+βŸ¨π΄π‘˜βˆ£π΅π‘˜βŸ©.(3.1)

Proof. By Lemma 2.1, letting π‘₯(𝑑0)=0, we have 𝐴π‘₯(𝑑)=expπ‘˜ξ€·π‘‘βˆ’π‘‘π‘˜βˆ’1ξƒ―ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€ΈΓ—ξ‚΅πΈπ‘–ξ€œπ‘‘π‘–π‘‘π‘–βˆ’1𝐴expπ‘–ξ€·π‘‘π‘–π΅βˆ’π‘ ξ€Έξ€»π‘–π‘’(𝑠)𝑑𝑠+𝐹𝑖𝑒𝑑𝑖+πΈπ‘˜βˆ’1ξ€œπ‘‘π‘˜βˆ’1π‘‘π‘˜βˆ’2𝐴expπ‘˜βˆ’1ξ€·π‘‘π‘˜βˆ’1π΅βˆ’π‘ ξ€Έξ€»π‘˜βˆ’1𝑒(𝑠)𝑑𝑠+πΉπ‘˜βˆ’1π‘’ξ€·π‘‘π‘˜βˆ’1ξ€Έξƒ°+ξ€œπ‘‘π‘‘π‘˜βˆ’1𝐴expπ‘˜ξ€»π΅(π‘‘βˆ’π‘ )π‘˜π‘’(𝑠)𝑑𝑠.(3.2) It follows that ℛ𝑑0,𝑑𝑓=𝐴π‘₯∣π‘₯=expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1Γ—ξƒ―ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έξ‚΅πΈπ‘–ξ€œπ‘‘π‘–π‘‘π‘–βˆ’1𝐴expπ‘–ξ€·π‘‘π‘–π΅βˆ’π‘ ξ€Έξ€»π‘–π‘’(𝑠)𝑑𝑠+𝐹𝑖𝑒𝑑𝑖+πΈπ‘˜βˆ’1ξ€œπ‘‘π‘˜βˆ’1π‘‘π‘˜βˆ’2𝐴expπ‘˜βˆ’1ξ€·π‘‘π‘˜βˆ’1π΅βˆ’π‘ ξ€Έξ€»π‘˜βˆ’1𝑒(𝑠)𝑑𝑠+πΉπ‘˜βˆ’1π‘’ξ€·π‘‘π‘˜βˆ’1ξ€Έξƒ°+ξ€œπ‘‘π‘“π‘‘π‘˜βˆ’1𝐴expπ‘˜ξ€·π‘‘π‘“π΅βˆ’π‘ ξ€Έξ€»π‘˜π‘’ξƒ°ξ€Ίπ΄(𝑠)𝑑𝑠,βˆ€piecewisecontinuous𝑒=expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξƒ©ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€ΈΓ—ξ‚»π‘₯∣π‘₯=πΈπ‘–ξ€œπ‘‘π‘–π‘‘π‘–βˆ’1𝐴expπ‘–ξ€·π‘‘π‘–π΅βˆ’π‘ ξ€Έξ€»π‘–π‘’(𝑠)𝑑𝑠+𝐹𝑖𝑒𝑑𝑖,ξ‚Ό+ξ‚»βˆ€piecewisecontinuous𝑒π‘₯∣π‘₯=πΈπ‘˜βˆ’1ξ€œπ‘‘π‘˜βˆ’1π‘‘π‘˜βˆ’2𝐴expπ‘˜βˆ’1ξ€·π‘‘π‘˜βˆ’1π΅βˆ’π‘ ξ€Έξ€»π‘˜βˆ’1𝑒(𝑠)𝑑𝑠+πΉπ‘˜βˆ’1π‘’ξ€·π‘‘π‘˜βˆ’1ξ€Έξ‚Όξƒͺ+ξ‚»ξ€œβˆ€piecewisecontinuous𝑒π‘₯∣π‘₯=π‘‘π‘“π‘‘π‘˜βˆ’1𝐴expπ‘˜ξ€·π‘‘π‘“π΅βˆ’π‘ ξ€Έξ€»π‘˜π‘’ξ‚Ό.(𝑠)𝑑𝑠,βˆ€piecewisecontinuous𝑒(3.3)
By Lemma 2.2, we get ℛ𝑑0,𝑑𝑓𝐴=expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξƒ©ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—πΈξ€Έξ€·π‘–βŸ¨π΄π‘–βˆ£π΅π‘–ξ€·πΉβŸ©+β„π‘šπ‘–ξ€Έξ€Έ+πΈπ‘˜βˆ’1βŸ¨π΄π‘˜βˆ’1βˆ£π΅π‘˜βˆ’1ξ€·πΉβŸ©+β„π‘šπ‘˜βˆ’1ξ€Έξƒͺ+βŸ¨π΄π‘˜βˆ£π΅π‘˜βŸ©.(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 [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], if and only if π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—πΈξ€Έξ€·π‘–βŸ¨π΄π‘–βˆ£π΅π‘–ξ€·πΉβŸ©+β„π‘šπ‘–ξ€Έξ€Έ+πΈπ‘˜βˆ’1βŸ¨π΄π‘˜βˆ’1βˆ£π΅π‘˜βˆ’1ξ€·πΉβŸ©+β„π‘šπ‘˜βˆ’1ξ€Έ+βŸ¨π΄π‘˜βˆ£π΅π‘˜βŸ©=ℝ𝑛.(3.5)

Proof. Since ℛ𝑑0,𝑑𝑓𝐴=expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1Γ—ξƒ©ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—πΈξ€Έξ€·π‘–βŸ¨π΄π‘–βˆ£π΅π‘–ξ€·πΉβŸ©+β„π‘šπ‘–ξ€Έξ€Έ+πΈπ‘˜βˆ’1βŸ¨π΄π‘˜βˆ’1βˆ£π΅π‘˜βˆ’1ξ€·πΉβŸ©+β„π‘šπ‘˜βˆ’1ξ€Έξƒͺ+βŸ¨π΄π‘˜βˆ£π΅π‘˜βŸ©ξ€Ίπ΄=expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξƒ©ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=kβˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—πΈξ€Έξ€·π‘–βŸ¨π΄π‘–βˆ£π΅π‘–ξ€·πΉβŸ©+β„π‘šπ‘–ξ€Έξ€Έ+πΈπ‘˜βˆ’1βŸ¨π΄π‘˜βˆ’1βˆ£π΅π‘˜βˆ’1ξ€·πΉβŸ©+β„π‘šπ‘˜βˆ’1ξ€Έ+βŸ¨π΄π‘˜βˆ£π΅π‘˜βŸ©ξƒͺ(3.6) and the matrix exp[π΄π‘˜(π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1)] 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=𝐡𝑖,𝐴𝑖𝐡𝑖,…,π΄π‘–π‘›βˆ’1𝐡𝑖,(3.7) for 𝑖=1,2,…,π‘˜βˆ’2, denote 𝐻𝑖=𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€ΈπΈπ‘–Q𝑖,𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€ΈπΉπ‘–ξƒ­,π»π‘˜βˆ’1=ξ€ΊπΈπ‘˜βˆ’1π‘„π‘˜βˆ’1,πΉπ‘˜βˆ’1ξ€»,(3.8) and, finally, denote 𝑄[𝑑0,𝑑𝑓]=𝐻1,𝐻2,…,π»π‘˜βˆ’1,π‘„π‘˜ξ€».(3.9) Then, it is easy to verify that 𝐴expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ‚€π‘„ξ€Έξ€»β„π‘š[𝑑0,𝑑𝑓]𝑑=β„›0,𝑑𝑓.(3.10) Thus, we get the following algebraic type criterion.

Corollary 3.5. The system (2.1) is reachable on [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], if and only if 𝑄rank[𝑑0,𝑑𝑓]=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 [𝑑0,𝑑𝑓](𝑑0<𝑑𝑓) if, for any initial state π‘₯0βˆˆβ„π‘›, there exists a piecewise continuous input 𝑒(𝑑)∢[𝑑0,𝑑𝑓]→ℝ𝑝 such that the system (2.1) is driven from π‘₯(𝑑0)=π‘₯0 to π‘₯(𝑑𝑓)=0. Moreover, the set of all the controllable states on [𝑑0,𝑑𝑓] is said to be the controllable set on [𝑑0,𝑑𝑓], denoted as π’ž[𝑑0,𝑑𝑓].

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,…,π‘˜βˆ’1, then the controllable set on [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], satisfies 𝐴expπ‘˜ξ€·π‘‘βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έξƒͺπ’žξ€Ίπ‘‘0,π‘‘π‘“ξ€»ξ€Ίπ‘‘βŠ†β„›0,𝑑𝑓.(4.1)

Proof. By Lemma 2.1, letting π‘₯(𝑑𝑓)=0, we have 𝐴0=expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έπ‘₯𝑑0𝐴expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1Γ—ξƒ―ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€ΈΓ—ξ‚΅πΈπ‘–ξ€œπ‘‘π‘–π‘‘π‘–βˆ’1𝐴expπ‘–ξ€·π‘‘π‘–π΅βˆ’π‘ ξ€Έξ€»π‘–π‘’(𝑠)𝑑𝑠+𝐹𝑖𝑒𝑑𝑖+πΈπ‘˜βˆ’1ξ€œπ‘‘π‘˜βˆ’1π‘‘π‘˜βˆ’2𝐴expπ‘˜βˆ’1ξ€·π‘‘π‘˜βˆ’1π΅βˆ’π‘ ξ€Έξ€»π‘˜βˆ’1𝑒(𝑠)𝑑𝑠+πΉπ‘˜βˆ’1π‘’ξ€·π‘‘π‘˜βˆ’1ξ€Έξƒ°+ξ€œπ‘‘π‘“π‘‘π‘˜βˆ’1𝐴expπ‘˜ξ€·π‘‘π‘“π΅βˆ’π‘ ξ€Έξ€»π‘˜π‘’(𝑠)𝑑𝑠.(4.2) It is equivalent to ξ€Ίπ΄βˆ’expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έπ‘₯𝑑0𝐴=expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξƒ―ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€ΈΓ—ξ‚΅πΈπ‘–ξ€œπ‘‘π‘–π‘‘π‘–βˆ’1𝐴expπ‘–ξ€·π‘‘π‘–π΅βˆ’π‘ ξ€Έξ€»π‘–π‘’(𝑠)𝑑𝑠+𝐹𝑖𝑒𝑑𝑖+πΈπ‘˜βˆ’1ξ€œπ‘‘π‘˜βˆ’1π‘‘π‘˜βˆ’2𝐴expπ‘˜βˆ’1ξ€·π‘‘π‘˜βˆ’1π΅βˆ’π‘ ξ€Έξ€»π‘˜βˆ’1𝑒(𝑠)𝑑𝑠+πΉπ‘˜βˆ’1π‘’ξ€·π‘‘π‘˜βˆ’1ξ€Έξƒ°+ξ€œπ‘‘π‘“π‘‘π‘˜βˆ’1𝐴expπ‘˜ξ€·π‘‘π‘“π΅βˆ’π‘ ξ€Έξ€»π‘˜π‘’(𝑠)𝑑𝑠.(4.3) This implies that 𝐴expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έξƒͺπ‘₯𝑑0ξ€Έξ€Ίπ‘‘βˆˆβ„›0,𝑑𝑓.(4.4) Hence, 𝐴expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έξƒͺπ’žξ€Ίπ‘‘0,π‘‘π‘“ξ€»ξ€Ίπ‘‘βŠ†β„›0,𝑑𝑓.(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 [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], if and only if ξƒ©β„π‘š1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€ΈξƒͺβŠ†π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—πΈξ€Έξ€·π‘–βŸ¨π΄π‘–βˆ£π΅π‘–ξ€·πΉβŸ©+β„π‘šπ‘–ξ€Έξ€Έ+πΈπ‘˜βˆ’1βŸ¨π΄π‘˜βˆ’1βˆ£π΅π‘˜βˆ’1ξ€·πΉβŸ©+β„π‘šπ‘˜βˆ’1ξ€Έ+βŸ¨π΄π‘˜βˆ£π΅π‘˜βŸ©.(4.6)

Proof. First, it is easy to prove that (4.6) is equivalent to ξƒ©ξ€Ίπ΄β„π‘šexpπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έξƒͺξ€Ίπ‘‘βŠ†β„›0,𝑑𝑓.(4.7)
Necessity: since the system is controllable, we have π’žξ€Ίπ‘‘0,𝑑𝑓=ℝ𝑛.(4.8) Then, by Theorem 4.2, we get ℛ𝑑0,π‘‘π‘“ξ€»βŠ‡ξƒ©ξ€Ίπ΄expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έξƒͺℝ𝑛𝐴=β„π‘šexpπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έξƒͺ.(4.9)
Sufficiency: suppose that (4.7) holds. For any π‘₯βˆˆβ„π‘›, we have 𝐴expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έξƒͺ𝑑π‘₯βˆˆβ„›0,𝑑𝑓.(4.10) This implies that there exists a piecewise continuous function 𝑒(𝑑),π‘‘βˆˆ[𝑑0,𝑑𝑓], such that 𝐴0=expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έπ‘₯𝐴×expπ‘˜ξ€·π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1ξƒ―ξ€Έξ€»π‘˜βˆ’2𝑖=1𝑖+1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€ΈΓ—ξ‚΅πΈπ‘–ξ€œπ‘‘π‘–π‘‘π‘–βˆ’1𝐴expπ‘–ξ€·π‘‘π‘–π΅βˆ’π‘ ξ€Έξ€»π‘–π‘’(𝑠)𝑑𝑠+𝐹𝑖𝑒𝑑𝑖+πΈπ‘˜βˆ’1ξ€œπ‘‘π‘˜βˆ’1π‘‘π‘˜βˆ’2𝐴expπ‘˜βˆ’1ξ€·π‘‘π‘˜βˆ’1π΅βˆ’π‘ ξ€Έξ€»π‘˜βˆ’1𝑒(𝑠)𝑑𝑠+πΉπ‘˜βˆ’1π‘’ξ€·π‘‘π‘˜βˆ’1ξ€Έξƒ°+ξ€œπ‘‘π‘“π‘‘π‘˜βˆ’1𝐴expπ‘˜ξ€·π‘‘π‘“π΅βˆ’π‘ ξ€Έξ€»π‘˜π‘’(𝑠)𝑑𝑠.(4.11) Then, we know that π‘₯βˆˆπ’ž[𝑑0,𝑑𝑓]. 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,…,π‘˜βˆ’1, then the following statements are equivalent:(a)the system is reachable,(b)the system is controllable,(c)βˆ‘π‘˜βˆ’2𝑖=1βˆπ‘–+1𝑗=π‘˜βˆ’1𝐸𝑗exp(π΄π‘—β„Žπ‘—)(πΈπ‘–βŸ¨π΄π‘–βˆ£π΅π‘–βŸ©+β„π‘š(𝐹𝑖))+πΈπ‘˜βˆ’1βŸ¨π΄π‘˜βˆ’1βˆ£π΅π‘˜βˆ’1⟩+β„π‘š(πΉπ‘˜βˆ’1)+βŸ¨π΄π‘˜βˆ£π΅π‘˜βŸ©=ℝ𝑛.

Proof. Since 𝐸𝑖 is nonsingular, 𝑖=1,2,…,π‘˜βˆ’1, we have that 𝐴expπ‘˜ξ€·π‘‘βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έ(4.12) is nonsingular. It follows that 𝐴expπ‘˜ξ€·π‘‘βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑖=π‘˜βˆ’1𝐸𝑖𝐴expπ‘–β„Žπ‘–ξ€Έξƒͺπ’žξ€Ίπ‘‘0,𝑑𝑓𝑑=β„›0,𝑑𝑓.(4.13) It is easy to see that π’ž[𝑑0,𝑑𝑓]=ℝ𝑛⇔ℛ[𝑑0,𝑑𝑓]=ℝ𝑛.

Remark 4.5. For system (2.1), assume that 𝐴𝑖=𝐴,𝐡𝑖=𝐡,𝑖=1,…,π‘˜. 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,…,π‘˜. 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 [𝑑0,𝑑𝑓] (𝑑0<𝑑𝑓) if any initial state π‘₯0βˆˆβ„π‘› can be uniquely determined by the corresponding system input 𝑒(𝑑) and the system output 𝑦(𝑑), for π‘‘βˆˆ[𝑑0,𝑑𝑓].

Definition 5.2 (determinability). The system (2.1) is said to be (completely) determinable on [𝑑0,𝑑𝑓] (𝑑0<𝑑𝑓) if any terminal state π‘₯π‘“βˆˆβ„π‘› can be uniquely determined by the corresponding system input 𝑒(𝑑) and the system output 𝑦(𝑑), for π‘‘βˆˆ[𝑑0,𝑑𝑓].

In order to investigate observability and determinability for the system (2.1), we first investigate those of the following zero input system: Μ‡π‘₯(𝑑)=π΄π‘˜π‘₯𝑑(𝑑),π‘‘βˆˆπ‘˜βˆ’1,π‘‘π‘˜ξ€Έ,π‘₯𝑑+π‘˜ξ€Έ=πΈπ‘˜π‘₯ξ€·π‘‘βˆ’π‘˜ξ€Έ,𝑦(𝑑)=πΆπ‘˜ξ€Ίπ‘‘π‘₯(𝑑),π‘‘βˆˆπ‘˜βˆ’1,π‘‘π‘˜ξ€Έ,π‘₯𝑑+0ξ€Έ=π‘₯0,𝑑0β‰₯0.(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 ⎧βŽͺ⎨βŽͺβŽ©πΆπ‘¦(𝑑)=1𝐴exp1ξ€·π‘‘βˆ’π‘‘0π‘₯𝑑0𝑑,π‘‘βˆˆ0,𝑑1ξ€»,𝐢𝑖𝐴expπ‘–ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έξ€»1𝑗=π‘–βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έπ‘₯𝑑0𝑑,π‘‘βˆˆπ‘–βˆ’1,𝑑𝑖,𝑖=2,…,π‘˜.(5.2)

Theorem 5.3. The system (5.1) is observable on [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], if and only if 2𝑖=π‘˜π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬+𝐴𝑇1βˆ£πΆπ‘‡1=ℝ𝑛.(5.3)

Proof. We prove the complementary proposition of Theorem 5.3, that is, the system (5.1) is not observable on [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], if and only if 2𝑖=π‘˜π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬+𝐴𝑇1βˆ£πΆπ‘‡1ξ¬βŠŠβ„π‘›.(5.4)
Necessity: if the system (5.1) is not observable on [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], then there exists π‘₯0βˆˆβ„π‘›, nonzero, such that 𝑦(𝑑)≑0,π‘‘βˆˆ[𝑑0,𝑑𝑓]. This means that 𝐢1𝐴exp1ξ€·π‘‘βˆ’π‘‘0π‘₯ξ€Έξ€»0𝑑=0,π‘‘βˆˆ0,𝑑1ξ€»,𝐢𝑖𝐴expπ‘–ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έξ€»1𝑗=π‘–βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έπ‘₯0𝑑=0,π‘‘βˆˆπ‘–βˆ’1,𝑑𝑖𝐢,𝑖=2,…,π‘˜βˆ’1,π‘˜ξ€Ίπ΄expπ‘˜ξ€·π‘‘βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έπ‘₯0𝑑=0,π‘‘βˆˆπ‘˜βˆ’1,𝑑𝑓.(5.5) By Lemma 2.3, we get π‘₯𝑇0𝐴𝑇1βˆ£πΆπ‘‡1π‘₯=0,𝑇0π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬=0,𝑖=2,…,π‘˜.(5.6) It follows that π‘₯𝑇0𝐴𝑇1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬ξƒͺ=0.(5.7) Then, we know that π‘₯0βˆ‰ξ«π΄π‘‡1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬.(5.8) This implies (5.4).
Sufficiency: on the contrary, if (5.4) holds, there exists π‘₯0βˆˆβ„π‘›, nonzero, such that π‘₯𝑇0𝐴𝑇1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬ξƒͺ=0.(5.9) It follows that π‘₯𝑇0𝐴𝑇1βˆ£πΆπ‘‡1π‘₯=0,𝑇0π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬=0,𝑖=2,…,π‘˜.(5.10) By Lemma 2.3, we get 𝐢1𝐴exp1ξ€·π‘‘βˆ’π‘‘0π‘₯ξ€Έξ€»0𝑑=0,π‘‘βˆˆ0,𝑑1ξ€»,𝐢𝑖𝐴expπ‘–ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έξ€»1𝑗=π‘–βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έπ‘₯0𝑑=0,π‘‘βˆˆπ‘–βˆ’1,𝑑𝑖𝐢,𝑖=2,…,π‘˜βˆ’1,π‘˜ξ€Ίπ΄expπ‘˜ξ€·π‘‘βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έπ‘₯0𝑑=0,π‘‘βˆˆπ‘˜βˆ’1,𝑑𝑓.(5.11) This means that 𝑦(𝑑)≑0,π‘‘βˆˆ[𝑑0,𝑑𝑓]. 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=𝐢𝑇𝑖,𝐴𝑇i𝐢𝑇𝑖𝐴,…,π‘‡π‘–ξ€Έπ‘›βˆ’1𝐢𝑇𝑖,(5.12) for 𝑖=2,…,π‘˜, denote 𝐺𝑖=π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—π‘‚π‘–,(5.13) and, finally, denote 𝑂[𝑑0,𝑑𝑓]=𝑂1,𝐺2,…,πΊπ‘˜ξ€».(5.14) Then, it is easy to verify that ξ‚€π‘‚β„π‘š[𝑑0,𝑑𝑓]=𝐴𝑇1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬.(5.15) Thus, we get the following algebraic type criterion.

Corollary 5.5. The system (5.1) is observable on [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], if and only if 𝑂rank[𝑑0,𝑑𝑓]=𝑛.(5.16)

Next, we establish a criterion for determinability.

Theorem 5.6. The system (5.1) is determinable on [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], if and only if ξƒ©β„π‘šπ‘˜βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξƒͺβŠ†π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬+𝐴𝑇1βˆ£πΆπ‘‡1.(5.17)

Proof. First, by Lemma 2.4, we know that (5.17) is equivalent to 1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έξƒ©π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬+𝐴𝑇1βˆ£πΆπ‘‡1ξƒͺβŸ‚=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 [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], if and only if 1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έξƒ©π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬+𝐴𝑇1βˆ£πΆπ‘‡1ξƒͺβŸ‚β‰ 0.(5.19) Necessity: if the system (5.1) is not determinable on [𝑑0,𝑑𝑓], where π‘‘π‘“βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜], then there exists a terminal π‘₯π‘“βˆˆβ„π‘›, nonzero, such that 𝑦(𝑑)=0,π‘‘βˆˆ[𝑑0,𝑑𝑓]. Then, there exists a nonzero π‘₯0βˆˆβ„π‘› as the initial state such that the system is driven from π‘₯(𝑑0)=π‘₯0 to π‘₯(𝑑𝑓)=π‘₯𝑓, that is, π‘₯𝑓=exp[π΄π‘˜(π‘‘π‘“βˆ’π‘‘π‘˜βˆ’1∏)]1𝑗=π‘˜βˆ’1𝐸𝑗exp(π΄π‘—β„Žπ‘—)π‘₯0. This means that 𝐢1𝐴exp1ξ€·π‘‘βˆ’π‘‘0π‘₯ξ€Έξ€»0𝑑=0,π‘‘βˆˆ0,𝑑1ξ€»,𝐢𝑖𝐴expπ‘–ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έξ€»1𝑗=π‘–βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έπ‘₯0𝑑=0,π‘‘βˆˆπ‘–βˆ’1,𝑑𝑖𝐢,𝑖=2,…,π‘˜βˆ’1,π‘˜ξ€Ίπ΄expπ‘˜ξ€·π‘‘βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έπ‘₯0𝑑=0,π‘‘βˆˆπ‘˜βˆ’1,𝑑𝑓.(5.20) By Lemma 2.3, we get π‘₯𝑇0𝐴𝑇1βˆ£πΆπ‘‡1π‘₯=0,𝑇0π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬=0,𝑖=2,…,π‘˜.(5.21) It follows that π‘₯𝑇0𝐴𝑇1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬ξƒͺ=0.(5.22) This implies that π‘₯0βˆˆξƒ©ξ«π΄π‘‡1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬ξƒͺβŸ‚.(5.23) Since exp[π΄π‘˜(π‘‘π‘˜βˆ’1βˆ’π‘‘π‘“)]π‘₯𝑓=∏1𝑗=π‘˜βˆ’1𝐸𝑗exp(π΄π‘—β„Žπ‘—)π‘₯0, we know that 𝐴expπ‘˜ξ€·π‘‘π‘˜βˆ’1βˆ’π‘‘π‘“π‘₯ξ€Έξ€»π‘“βˆˆ1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έξƒ©ξ«π΄π‘‡1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬ξƒͺβŸ‚.(5.24) It implies that 1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έξƒ©ξ«π΄π‘‡1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬ξƒͺβŸ‚β‰ 0.(5.25) Hence, (5.19) holds.
Sufficiency: on the contrary, if (5.19) holds, then we know that 1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έξƒ©ξ«π΄π‘‡1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬ξƒͺβŸ‚β‰ 0.(5.26) Then, there exists a nonzero π‘₯𝑓 satisfying 𝐴expπ‘˜ξ€·π‘‘π‘˜βˆ’1βˆ’π‘‘π‘“π‘₯ξ€Έξ€»π‘“βˆˆ1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έξƒ©ξ«π΄π‘‡1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬ξƒͺβŸ‚(5.27) such that there exists a nonzero π‘₯0 satisfying 𝐴expπ‘˜ξ€·π‘‘π‘˜βˆ’1βˆ’π‘‘π‘“π‘₯𝑓=π‘₯0,π‘₯𝑇0𝐴𝑇1βˆ£πΆπ‘‡1+π‘˜ξ“π‘–=2π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬ξƒͺ=0.(5.28) It follows that π‘₯𝑇0𝐴𝑇1βˆ£πΆπ‘‡1π‘₯=0,𝑇0π‘–βˆ’1𝑗=1𝐴expπ‘‡π‘—β„Žπ‘—ξ€ΈπΈπ‘‡π‘—ξ«π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–ξ¬=0,𝑖=2,…,π‘˜.(5.29) By Lemma 2.3, we get 𝐢1𝐴exp1ξ€·π‘‘βˆ’π‘‘0π‘₯ξ€Έξ€»0𝑑=0,π‘‘βˆˆ0,𝑑1ξ€»,𝐢𝑖𝐴expπ‘–ξ€·π‘‘βˆ’π‘‘π‘–βˆ’1ξ€Έξ€»1𝑗=π‘–βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έπ‘₯0𝑑=0,π‘‘βˆˆπ‘–βˆ’1,𝑑𝑖𝐢,𝑖=2,…,π‘˜βˆ’1,π‘˜ξ€Ίπ΄expπ‘˜ξ€·π‘‘βˆ’π‘‘π‘˜βˆ’1ξ€Έξ€»1𝑗=π‘˜βˆ’1𝐸𝑗𝐴expπ‘—β„Žπ‘—ξ€Έπ‘₯0𝑑=0,π‘‘βˆˆπ‘˜βˆ’1,𝑑𝑓.(5.30) This means that 𝑦(𝑑)≑0, π‘‘βˆˆ[𝑑0,𝑑𝑓]. Thus, we find a nonterminal nonzero state π‘₯𝑓 such that the output 𝑦(𝑑) 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,…,π‘˜βˆ’1, then the following statements are equivalent:(a)the system is observable,(b)the system is determinable,(c)βˆ‘2𝑖=π‘˜βˆπ‘–βˆ’1𝑗=1exp(π΄π‘‡π‘—β„Žπ‘—)πΈπ‘‡π‘—βŸ¨π΄π‘‡π‘–βˆ£πΆπ‘‡π‘–βŸ©+βŸ¨π΄π‘‡1βˆ£πΆπ‘‡1⟩=ℝ𝑛.

Proof. If 𝐸𝑖 is nonsingular, 𝑖=1,2,…,π‘˜βˆ’1, then we know that ∏1𝑗=π‘˜βˆ’1𝐸𝑗exp(π΄π‘—β„Žπ‘—) is nonsingular. Hence, we get (5.3) and (5.17) are equivalent.

Remark 5.8. For system (2.1), assume that 𝐴𝑖=𝐴,𝐡𝑖=𝐡,𝑖=1,…,π‘˜. 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,…,π‘˜. 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=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦000000000,𝐡1=⎑⎒⎒⎒⎒⎣100⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐢1=,𝐷0101=0,𝐸1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦100010000,𝐹1=⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐴2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦000000000,𝐡2=⎑⎒⎒⎒⎒⎣100⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐢2=,𝐷1002=0,𝐸2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦110100000,𝐹2=⎑⎒⎒⎒⎒⎣110⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐴3=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦000000000,𝐡3=⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐢3=,𝐷1003=0,𝐸3=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦000100000,𝐹3=⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,(6.1) where 𝑑0=0, 𝑑1=1, 𝑑2=2, and 𝑑3=3.

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

First, we consider the reachability. By a simple calculation, we have 𝐸2𝐴exp2𝐸1⟨𝐴1∣𝐡1ξ€·πΉβŸ©+β„π‘š1ξ€Έξ€Έ+𝐸2⟨𝐴2∣𝐡2ξ€·πΉβŸ©+β„π‘š2ξ€Έ+⟨𝐴3∣𝐡3⟩⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩⎑⎒⎒⎒⎒⎣100⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭.=span(6.2) By Theorem 3.3, the system should not be reachable. In fact, for any piecewise continuous input 𝑒(𝑑),π‘‘βˆˆ[0,𝑑𝑓], and any nonzero initial state π‘₯0=[π‘₯01π‘₯02π‘₯03]𝑇, we have π‘₯𝑑𝑓=βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£βˆ—βˆ—0⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(6.3) This fact shows that the system is indeed not reachable.

Next, we consider the controllability. By a simple calculation, we have ξ€·πΈβ„π‘š2𝐴exp2𝐸1𝐴exp1⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩⎑⎒⎒⎒⎒⎣100⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎫βŽͺβŽͺ⎬βŽͺβŽͺβŽ­ξ€Έξ€Έ=span.(6.4) It is easy to see that ξ€·πΈβ„π‘š2𝐴exp2𝐸1𝐴exp1ξ€Έξ€ΈβŠ†πΈ2𝐴exp2𝐸1⟨𝐴1∣𝐡1ξ€·πΉβŸ©+β„π‘š1ξ€Έξ€Έ+𝐸2⟨𝐴2∣𝐡2ξ€·πΉβŸ©+β„π‘š2ξ€Έ+⟨𝐴3∣𝐡3⟩.(6.5)

By Theorem 4.3, the system should be controllable. In fact, we can take the piecewise constant input ⎧βŽͺ⎨βŽͺβŽ©π‘π‘’(𝑑)=1],],𝑐,π‘‘βˆˆ(0,10,π‘‘βˆˆ(1,23].,π‘‘βˆˆ(2,3(6.6) Then, for any nonzero initial state π‘₯0=[π‘₯01π‘₯02π‘₯03]𝑇, we have π‘₯𝑑𝑓=⎑⎒⎒⎒⎒⎣π‘₯01+0.5𝑐1π‘₯02+1.5𝑐1+ξ‚€2βˆ’2𝑑𝑓+0.5𝑑2𝑓𝑐30⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(6.7) Obviously, if 𝑐1=βˆ’2π‘₯01, 𝑐3=(βˆ’π‘₯02βˆ’1.5𝑐1)/(2βˆ’2𝑑𝑓+0.5𝑑2𝑓), then π‘₯(𝑑𝑓)=0. This fact shows that the system is indeed controllable.

Next, we consider the observability. By a simple calculation, we have 𝐴𝑇1βˆ£πΆπ‘‡1𝐴+exp𝑇1𝐸𝑇1𝐴𝑇2βˆ£πΆπ‘‡2𝐴+exp𝑇1𝐸𝑇1𝐴exp𝑇2𝐸𝑇2𝐴𝑇3βˆ£πΆπ‘‡3⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩⎑⎒⎒⎒⎒⎣100⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭.=span(6.8) By Theorem 5.3, the system should not be observable. In fact, for any piecewise continuous input 𝑒(𝑑),π‘‘βˆˆ[0,𝑑𝑓], and nonzero initial state π‘₯0=[001]𝑇, we have 𝑦(𝑑)≑0,π‘‘βˆˆ0,𝑑𝑓.(6.9) This fact shows that the system is indeed not observable.

Finally, we consider the determinability. By a simple calculation, we have 𝐸2𝐴exp2𝐸1𝐴exp1×𝐴𝑇1βˆ£πΆπ‘‡1𝐴+exp𝑇1𝐸𝑇1𝐴𝑇2βˆ£πΆπ‘‡2𝐴+exp𝑇1𝐸𝑇1𝐴exp𝑇2𝐸𝑇2𝐴𝑇3βˆ£πΆπ‘‡3ξ¬ξ€ΈβŸ‚=⎑⎒⎒⎒⎒⎣100⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩⎑⎒⎒⎒⎒⎣001⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭span=0.(6.10) It follows that 𝐸2𝐴exp2𝐸1𝐴exp1×𝐴𝑇1βˆ£πΆπ‘‡1𝐴+exp𝑇1𝐸𝑇1𝐴𝑇2βˆ£πΆπ‘‡2𝐴+exp𝑇1𝐸𝑇1𝐴exp𝑇2𝐸𝑇2𝐴𝑇3βˆ£πΆπ‘‡3ξ¬ξ€ΈβŸ‚=0.(6.11) By Theorem 5.6, the system should be determinable. In fact, for any nonzero terminal state π‘₯𝑓=[π‘₯𝑓1π‘₯𝑓2π‘₯𝑓3]𝑇, there must exist a nonzero initial state π‘₯0=[π‘₯01π‘₯02π‘₯03]𝑇 such that 𝐴exp3ξ€·2βˆ’π‘‘π‘“π‘₯𝑓=𝐸2𝐴exp2𝐸1𝐴exp1ξ€Έπ‘₯0.(6.12) It follows that ξ€·2βˆ’π‘‘π‘“ξ€Έπ‘₯𝑓=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦π‘₯1101000000.(6.13) This means that π‘₯𝑓3=π‘₯03=0 and |π‘₯01|+|π‘₯02|β‰ 0. It is easy to verify that, for any initial state π‘₯0 satisfying |π‘₯01|+|π‘₯02|β‰ 0, we have 𝑦(𝑑)β‰’0, π‘‘βˆˆ(0,𝑑𝑓). This fact shows that the system is indeed determinable.

Example 6.2. Consider a 3-dimensional linear piecewise constant impulsive system with 𝐴1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦000000000,𝐡1=⎑⎒⎒⎒⎒⎣100⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐢1=,𝐷0101=0,𝐸1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦100010001,𝐹1=⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐴2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦000000000,𝐡2=⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐢2=,𝐷1002=0,𝐸2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦100010001,𝐹2=⎑⎒⎒⎒⎒⎣110⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐴3=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦000000000,𝐡3=⎑⎒⎒⎒⎒⎣001⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐢3=,𝐷1003=0,𝐸3=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦100110001,𝐹3=⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,(6.14) where 𝑑0=0, 𝑑1=1, 𝑑2=2, and 𝑑3=3.

Now, we try to use our criteria to investigate the reachability and controllability on [0,𝑑𝑓], where π‘‘π‘“βˆˆ(2,3], of the system in Example 6.2.

First, we consider reachability. By a simple calculation, we have 𝐸2𝐴exp2𝐸1⟨𝐴1∣𝐡1ξ€·πΉβŸ©+β„π‘š1ξ€Έξ€Έ+𝐸2⟨𝐴2∣𝐡2ξ€·πΉβŸ©+β„π‘š2ξ€Έ+⟨𝐴3∣𝐡3⟩⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩⎑⎒⎒⎒⎒⎣100⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,⎑⎒⎒⎒⎒⎣010⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,⎑⎒⎒⎒⎒⎣001⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦⎫βŽͺβŽͺ⎬βŽͺβŽͺ⎭=span=ℝ3.(6.15) By Theorem 3.3, the system should be reachable. In fact, we take the piecewise constant input ⎧βŽͺ⎨βŽͺβŽ©π‘π‘’(𝑑)=1],𝑐,π‘‘βˆˆ(0,12],𝑐,π‘‘βˆˆ(1,23].,π‘‘βˆˆ(2,3(6.16) Then, letting π‘₯(0)=0, for any nonzero terminal state π‘₯(3)=[π‘₯𝑓1π‘₯𝑓2π‘₯𝑓3]𝑇, we have ⎑⎒⎒⎒⎒⎣π‘₯𝑓1π‘₯𝑓2π‘₯𝑓3⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘1101200011𝑐2𝑐3⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(6.17) Obviously, we can select suitable 𝑐1,𝑐2, and 𝑐3 such that π‘₯𝑓 is any state in ℝ3. 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 ⎧βŽͺ⎨βŽͺβŽ©π‘π‘’(𝑑)=1],𝑐,π‘‘βˆˆ(0,12],𝑐,π‘‘βˆˆ(1,23].,π‘‘βˆˆ(2,3(6.18) Then, for any nonzero initial state π‘₯(0)=[π‘₯01π‘₯02π‘₯03]𝑇, letting π‘₯(3)=0, we have ⎑⎒⎒⎒⎒⎣π‘₯0=01π‘₯02π‘₯03⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦+⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π‘1101200011𝑐2𝑐3⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.(6.19) Obviously, we can select suitable 𝑐1,𝑐2, and 𝑐3 such that π‘₯0 is any state in ℝ3. 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 ξ‚»ξ€œπ‘₯∣π‘₯=𝑑𝑓𝑑0𝐴𝑑expπ‘“ξ‚Όβˆ’π‘ ξ€Έξ€»π΅π‘’(𝑠)𝑑𝑠,βˆ€piecewisecontinuous𝑒=⟨𝐴∣𝐡⟩.(A.1) Thus, it is easy to see that ξ‚»ξ€œπ‘₯∣π‘₯=𝐸𝑑𝑓𝑑0𝐴𝑑expπ‘“ξ€·π‘‘βˆ’π‘ ξ€Έξ€»π΅π‘’(𝑠)𝑑𝑠+𝐹𝑒𝑓,βˆ€piecewisecontinuousπ‘’βŠ†πΈβŸ¨π΄βˆ£π΅βŸ©+β„π‘š(𝐹).(A.2) Moreover, we have ξ‚»ξ€œπ‘₯∣π‘₯=𝑑𝐸𝑑0𝐴𝑑expπΈξ‚Όβˆ’π‘ ξ€Έξ€»π΅π‘’(𝑠)𝑑𝑠,βˆ€piecewisecontinuous𝑒=⟨𝐴∣𝐡⟩,(A.3) where 𝑑𝐸=(𝑑0+𝑑𝑓)/2. Then, for any π‘₯∈𝐸⟨𝐴∣𝐡⟩+β„π‘š(𝐹), there exist a piecewise continuous function 𝑒(𝑑), π‘‘βˆˆ[𝑑0,𝑑𝐸], and π‘¦βˆˆβ„π‘› such that ξ€œπ‘₯=𝐸𝑑𝐸𝑑0𝐴𝑑expπΈβˆ’π‘ ξ€Έξ€»π΅π‘’(𝑠)𝑑𝑠+𝐹𝑦.(A.4) Then, we can take ⎧βŽͺ⎨βŽͺβŽ©π‘’ξ€Ίπ‘‘π‘£(𝑑)=(𝑑),π‘‘βˆˆ0,𝑑𝐸,𝑑0,π‘‘βˆˆπΈ,𝑑𝑓,𝑦,𝑑=𝑑𝑓,(A.5) such that ξ€œπ‘₯=𝐸𝑑𝑓𝑑0𝐴𝑑expπ‘“ξ€·π‘‘βˆ’π‘ ξ€Έξ€»π΅π‘£(𝑠)𝑑𝑠+𝐹𝑣𝑓.(A.6) This implies that ξ‚»ξ€œπ‘₯∈π‘₯∣π‘₯=𝐸𝑑𝑓𝑑0𝐴𝑑expπ‘“ξ€·π‘‘βˆ’π‘ ξ€Έξ€»π΅π‘’(𝑠)𝑑𝑠+𝐹𝑒𝑓.,βˆ€piecewisecontinuous𝑒(A.7) It follows that ξ‚»ξ€œπ‘₯∣π‘₯=𝐸𝑑𝑓𝑑0𝐴𝑑expπ‘“ξ€·π‘‘βˆ’π‘ ξ€Έξ€»π΅π‘’(𝑠)𝑑𝑠+𝐹𝑒𝑓,βˆ€piecewisecontinuousπ‘’βŠ‡πΈβŸ¨π΄βˆ£π΅βŸ©+β„π‘š(𝐹).(A.8) By (A.2) and (A.8), we know that (2.8) holds.

B. Proof of Lemma 2.3

((a)β‡’(b)) If 𝐢exp[𝐴(π‘‘βˆ’π‘‘0)]π‘₯=0,π‘‘βˆˆ[𝑑0,𝑑𝑓], we get 𝐢π‘₯=0. Then, for 𝑖=1,…,π‘›βˆ’1, calculating the 𝑖th derivative of 𝐢exp[𝐴(π‘‘βˆ’π‘‘0)]π‘₯ with respect to 𝑑 at 𝑑=𝑑0, we get 𝐢𝐴𝑖π‘₯=0.(B.1) Thus, we know that π‘₯𝑇𝐢𝑇,𝐢𝑇𝐴𝑇,…,πΆπ‘‡ξ€·π΄π‘‡ξ€Έπ‘›βˆ’1ξ‚„=0.(B.2) Hence, π‘₯π‘‡βŸ¨π΄π‘‡βˆ£πΆπ‘‡βŸ©=0.

((a)⇐(b)) If π‘₯π‘‡βŸ¨π΄π‘‡βˆ£πΆ