Controllability and Observability Criteria for Linear Piecewise Constant Impulsive Systems

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 𝑥̇(𝑡)=𝑥𝐴𝑡_𝑘+𝑘𝑥(𝑡)+𝐵_𝑘𝑢(𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑦(𝑡)=𝐶_𝑘=𝐸_𝑘𝑥𝑡^−𝑘+𝐹_𝑘𝑢𝑡_𝑘,𝑥𝑥(𝑡𝑡)₊₀+𝐷_𝑘𝑢(𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,=𝑥₀,𝑡₀≥0,(2.1) where 𝑘=1,2,…,𝐴_𝑘,𝐵_𝑘,𝐶_𝑘,𝐷_𝑘,𝐸_𝑘, and 𝐹_𝑘 are the known 𝑛×𝑛, 𝑛×𝑝, 𝑝×𝑛, 𝑞×𝑝, 𝑛×𝑛, and 𝑛×𝑝 constant matrices; 𝑥(𝑡)∈ℝ^𝑛 is the state vector, and 𝑢(𝑡)∈ℝ^𝑝 the input vector, 𝑦(𝑡)∈ℝ^𝑞 the output vector; 𝑥(𝑡⁺)∶=lim_ℎ→0⁺𝑥(𝑡+ℎ), 𝑥(𝑡⁻)∶=lim_ℎ→0⁻𝑥(𝑡−ℎ), and the discontinuity points are 𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯,lim_𝑘→∞𝑡_𝑘=∞,(2.2) where 𝑡₀<𝑡₁ and 𝑥(𝑡^−𝑘)=𝑥(𝑡_𝑘), which implies that the solution of (2.1) is left-continuous at 𝑡_𝑘.

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

Lemma 2.1. For any 𝑡∈(𝑡_𝑘−1,𝑡_𝑘],𝑘=1,2,…, the general solution of the system (2.1) is given by 𝑥(𝑡)=exp𝐴_𝑘𝑡−𝑡_𝑘−1¹_𝑖_=𝑘−1+_𝑘−2𝐸_𝑖exp𝐴_𝑖ℎ_𝑖𝑥𝑡₀_{𝑖=1𝑖+1𝑗=𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐸_𝑖^𝑡𝑖_𝑡𝑖−1+𝐸_𝑘−1exp𝐴_𝑖𝑡_𝑖−𝑠𝐵_𝑖𝑢(𝑠)𝑑𝑠+𝐹_𝑖𝑢𝑡_𝑖^𝑡𝑘−1_𝑡𝑘−2+_{𝑡𝑡𝑘−1}exp𝐴_𝑘−1𝑡_𝑘−1−𝑠𝐵_𝑘−1𝑢(𝑠)𝑑𝑠+𝐹_𝑘−1𝑢𝑡_𝑘−1exp𝐴_𝑘(𝑡−𝑠)𝐵_𝑘𝑢(𝑠)𝑑𝑠,(2.3) where ℎ_𝑘=𝑡_𝑘−𝑡_𝑘−1, 𝑘=1,2,….

Proof. For 𝑡∈(𝑡₀,𝑡₁], we have 𝑥(𝑡)=exp𝐴₁𝑡−𝑡₀𝑥𝑡₀+^𝑡𝑡₀exp𝐴₁(𝑡−𝑠)𝐵₁𝑢(𝑠)𝑑𝑠.(2.4) For 𝑡=𝑡⁺¹, we have 𝑥𝑡⁺¹=𝐸₁exp𝐴¹ℎ₁𝑥𝑡₀+^𝑡1_𝑡0exp𝐴₁𝑡₁−𝑠𝐵₁𝑢(𝑠)𝑑𝑠+𝐹₁𝑢𝑡₁.(2.5) Similarly, for 𝑡∈(𝑡_𝑖−1,𝑡_𝑖], 𝑖=2,3,…,𝑘, we have 𝑥(𝑡)=exp𝐴_𝑖𝑡−𝑡_𝑖−1𝑥𝑡^+𝑖−1+^𝑡𝑡_𝑖−1exp𝐴_𝑖(𝑡−𝑠)𝐵_𝑖𝑢(𝑠)𝑑𝑠.(2.6) And, for 𝑡=𝑡^+𝑖,𝑖=2,3,…,𝑘, we have 𝑥𝑡^+𝑖=𝐸_𝑖exp𝐴^𝑖ℎ_𝑖𝑥𝑡_𝑖−1+^𝑡𝑖_𝑡𝑖−1exp𝐴_𝑖𝑡_𝑖−𝑠𝐵_𝑖𝑢(𝑠)𝑑𝑠+𝐹_𝑖𝑢𝑡_𝑖.(2.7) Thus, by (2.4), (2.5), (2.6), and (2.7), it is easy to verify (2.3).

If 𝑡_𝑓∈(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 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘],𝑘=2,3,….

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

Given matrices 𝐴∈ℝ^𝑛×𝑛 and 𝐵∈ℝ^𝑛×𝑝, denote ℐ𝑚(𝐵) as the range of 𝐵, that is, ℐ𝑚(𝐵)={𝑦∣𝑦=𝐵𝑥,forall𝑥∈ℝ^𝑛×𝑛}, and denote ⟨𝐴∣𝐵⟩ as the minimal invariant subspace of 𝐴 on ℐ𝑚(𝐵), that is, ⟨𝐴∣𝐵⟩=ℐ𝑚(𝐵)+ℐ𝑚(𝐴𝐵)+⋯+ℐ𝑚(𝐴^𝑛−1𝐵). Given a linear subspace 𝒲⊆ℝ^𝑛, denote 𝒲_⟂ as the orthogonal complement of 𝒲, that is, 𝒲_⟂={𝑥∣𝑥^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,𝐸∈ℝ^𝑛×𝑛,𝐵,𝐹∈ℝ^𝑛×𝑝, for any 0≤𝑡₀<𝑡_𝑓<+∞, one has 𝑥∣𝑥=𝐸^𝑡𝑓_𝑡0=𝐸⟨𝐴∣𝐵⟩ex+pℐ𝐴𝑚(𝑡𝐹_𝑓)−.𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑢𝑡_𝑓,∀piecewisecontinuous𝑢(2.8)

Proof. See Appendix A.

Lemma 2.3. Given two matrices 𝐴∈ℝ^𝑛×𝑛,𝐶∈ℝ^𝑞×𝑛, two scalars 𝑡₀<𝑡_𝑓, and a vector 𝑥∈ℝ^𝑛, the following two statements are equivalent:(a)𝐶exp[𝐴(𝑡−𝑡₀)]𝑥=0,𝑡∈[𝑡₀,𝑡_𝑓],(b)𝑥^𝑇⟨𝐴^𝑇∣𝐶^𝑇⟩=0.

Proof. See Appendix B.

Lemma 2.4. Given a matrix 𝐴∈ℝ^𝑛×𝑛 and a linear subspace 𝒲⊆ℝ^𝑛, the following two statements are equivalent:(a)ℐ𝑚(𝐴)⊆𝒲,(b)𝐴^𝑇𝑊_⟂=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,𝑡_𝑓](𝑡₀<𝑡_𝑓) if, for any terminal state 𝑥_𝑓∈ℝ^𝑛, there exists a piecewise continuous input 𝑢(𝑡)∶[𝑡₀,𝑡_𝑓]→ℝ^𝑝 such that the system (2.1) is driven from 𝑥(𝑡₀)=0 to 𝑥(𝑡_𝑓)=𝑥_𝑓. Moreover, the set of all the reachable states on [𝑡₀,𝑡_𝑓] is said to be the reachable set on [𝑡₀,𝑡_𝑓], denoted as ℛ[𝑡₀,𝑡_𝑓].

Theorem 3.2. For the system (2.1), the reachable set on [𝑡₀,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], is given by ℛ𝑡₀,𝑡_𝑓=exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1^𝑘−2_{𝑖=1𝑖+1𝑗=𝑘−1}+exp𝐴_𝑘𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐸_𝑖⟨𝐴_𝑖∣𝐵_𝑖⟩+ℐ𝑚𝐹_𝑖𝑡_𝑓−𝑡_𝑘−1𝐸_𝑘−1⟨𝐴_𝑘−1∣𝐵_𝑘−1⟩+ℐ𝑚𝐹_𝑘−1+⟨𝐴_𝑘∣𝐵_𝑘⟩.(3.1)

Proof. By Lemma 2.1, letting 𝑥(𝑡₀)=0, we have 𝑥(𝑡)=exp𝐴_𝑘𝑡−𝑡_𝑘−1^𝑘−2_{𝑖=1𝑖+1𝑗=𝑘−1}×𝐸_𝑖𝐸_𝑗exp𝐴_𝑗ℎ_𝑗^𝑡𝑖_𝑡𝑖−1+𝐸_𝑘−1exp𝐴_𝑖𝑡_𝑖−𝑠𝐵_𝑖𝑢(𝑠)𝑑𝑠+𝐹_𝑖𝑢𝑡_𝑖^𝑡𝑘−1_𝑡𝑘−2+_{𝑡𝑡𝑘−1}exp𝐴_𝑘−1𝑡_𝑘−1−𝑠𝐵_𝑘−1𝑢(𝑠)𝑑𝑠+𝐹_𝑘−1𝑢𝑡_𝑘−1exp𝐴_𝑘(𝑡−𝑠)𝐵_𝑘𝑢(𝑠)𝑑𝑠.(3.2) It follows that ℛ𝑡₀,𝑡_𝑓=×𝑥∣𝑥=exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1_𝑘−2_{𝑖=1𝑖+1𝑗=𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐸_𝑖^𝑡𝑖_𝑡𝑖−1+𝐸_𝑘−1exp𝐴_𝑖𝑡_𝑖−𝑠𝐵_𝑖𝑢(𝑠)𝑑𝑠+𝐹_𝑖𝑢𝑡_𝑖^𝑡𝑘−1_𝑡𝑘−2+_{𝑡𝑓𝑡𝑘−1}exp𝐴_𝑘−1𝑡_𝑘−1−𝑠𝐵_𝑘−1𝑢(𝑠)𝑑𝑠+𝐹_𝑘−1𝑢𝑡_𝑘−1=exp𝐴_𝑘exp𝐴_𝑘𝑡_𝑓−𝑠𝐵_𝑘𝑢(𝑠)𝑑𝑠,∀piecewisecontinuous𝑢𝑡_𝑓−𝑡_𝑘−1×^𝑘−2𝑥∣_𝑖=1𝑖+1𝑥_{𝑗=𝑘−1}𝐸=_𝑗e𝐸x_𝑖p𝐴_𝑗ℎ_𝑗^𝑡𝑖_𝑡𝑖−1+∀p𝑥ie∣c𝑥ew=is𝐸eco_𝑘−1enxtipnu𝐴ou_𝑖s𝑡𝑢_𝑖−𝑠𝐵_𝑖𝑢(𝑠)𝑑𝑠+𝐹_𝑖𝑢𝑡_𝑖,^𝑡𝑘−1_𝑡𝑘−2+𝑥∣𝑥=_{𝑡𝑓𝑡𝑘−1}∀piecewisecontinueoxups𝑢𝐴_𝑘−1𝑡_𝑘−1−𝑠𝐵_𝑘−1𝑢(𝑠)𝑑𝑠+𝐹_𝑘−1𝑢𝑡_𝑘−1exp𝐴_𝑘𝑡_𝑓−𝑠𝐵_𝑘𝑢(𝑠)𝑑𝑠,∀piecewisecontinuous𝑢.(3.3)
By Lemma 2.2, we get ℛ𝑡₀,𝑡_𝑓=exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1^𝑘−2_{𝑖=1𝑖+1𝑗=𝑘−1}+𝐸_𝑘−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐸_𝑖⟨𝐴_𝑖∣𝐵_𝑖⟩+ℐ𝑚𝐹_𝑖+⟨𝐴_𝑘⟨𝐴_𝑘−1∣𝐵_𝑘−1⟩+ℐ𝑚𝐹_𝑘−1∣𝐵_𝑘⟩.(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,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], if and only if ^𝑘−2_{𝑖=1𝑖+1𝑗=𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐸_𝑖⟨𝐴_𝑖∣𝐵_𝑖⟩+ℐ𝑚𝐹_𝑖+𝐸_𝑘−1⟨𝐴_𝑘−1∣𝐵_𝑘−1⟩+ℐ𝑚𝐹_𝑘−1+⟨𝐴_𝑘∣𝐵_𝑘⟩=ℝ(^𝑛3..5)

Proof. Since ℛ𝑡₀,𝑡_𝑓=×exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1_𝑘−2_{𝑖=1𝑖+1𝑗=𝑘−1}+⟨𝐴_𝑘𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐸_𝑖⟨𝐴_𝑖∣𝐵_𝑖⟩+ℐ𝑚𝐹_𝑖+𝐸_𝑘−1⟨𝐴_𝑘−1∣𝐵_𝑘−1⟩+ℐ𝑚𝐹_𝑘−1=exp𝐴∣_𝑘𝐵_𝑘⟩𝑡_𝑓−𝑡_𝑘−1+𝐸^𝑘−2_{𝑖=1𝑖+1𝑗=k−1𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐸_𝑖⟨𝐴_𝑖∣𝐵_𝑖⟩+ℐ𝑚𝐹_𝑖⟨𝐴_𝑘−1∣𝐵_𝑘−1⟩+ℐ𝑚𝐹_𝑘−1+⟨𝐴_𝑘∣𝐵_𝑘⟩(3.6) and the matrix exp[𝐴_𝑘(𝑡_𝑓−𝑡_𝑘−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 𝑄_𝑖=𝐵_𝑖,𝐴_𝑖𝐵_𝑖,…,𝐴^𝑖𝑛−1𝐵_𝑖,(3.7) for 𝑖=1,2,…,𝑘−2, denote 𝐻_𝑖=^𝑖+1_𝑗_=𝑘−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐸_𝑖Q_𝑖,^𝑖+1_𝑗_=𝑘−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐹_𝑖𝐻_𝑘−1,=𝐸_𝑘−1𝑄_𝑘−1,𝐹_𝑘−1,(3.8) and, finally, denote 𝑄_{[𝑡0,𝑡𝑓]}=𝐻₁,𝐻₂,…,𝐻_𝑘−1,𝑄_𝑘.(3.9) Then, it is easy to verify that exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1ℐ𝑚𝑄_{[𝑡0,𝑡𝑓]}=ℛ𝑡⁰,𝑡_𝑓.(3.10) Thus, we get the following algebraic type criterion.

Corollary 3.5. The system (2.1) is reachable on [𝑡₀,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], if and only if rank𝑄_{[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,𝑡_𝑓](𝑡₀<𝑡_𝑓) if, for any initial state 𝑥₀∈ℝ^𝑛, there exists a piecewise continuous input 𝑢(𝑡)∶[𝑡₀,𝑡_𝑓]→ℝ^𝑝 such that the system (2.1) is driven from 𝑥(𝑡₀)=𝑥₀ to 𝑥(𝑡_𝑓)=0. Moreover, the set of all the controllable states on [𝑡₀,𝑡_𝑓] is said to be the controllable set on [𝑡₀,𝑡_𝑓], denoted as 𝒞[𝑡₀,𝑡_𝑓].

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

Theorem 4.2. For the system (2.1), if 𝐸_𝑖 is nonsingular, for 𝑖=1,…,𝑘−1, then the controllable set on [𝑡₀,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], satisfies exp𝐴^𝑘𝑡−𝑡_𝑘−1¹_{𝑖=𝑘−1}𝐸_𝑖exp𝐴_𝑖ℎ_𝑖𝒞𝑡₀,𝑡_𝑓⊆ℛ𝑡₀,𝑡_𝑓.(4.1)

Proof. By Lemma 2.1, letting 𝑥(𝑡_𝑓)=0, we have 0=exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1¹_𝑖_=𝑘−1×𝐸_𝑖exp𝐴_𝑖ℎ_𝑖𝑥𝑡₀exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1_𝑘−2_{𝑖=1𝑖+1𝑗=𝑘−1}×𝐸_𝑖𝐸_𝑗exp𝐴_𝑗ℎ_𝑗^𝑡𝑖_𝑡𝑖−1+𝐸_𝑘−1exp𝐴_𝑖𝑡_𝑖−𝑠𝐵_𝑖𝑢(𝑠)𝑑𝑠+𝐹_𝑖𝑢𝑡_𝑖^𝑡𝑘−1_𝑡𝑘−2+_{𝑡𝑓𝑡𝑘−1}exp𝐴_𝑘−1𝑡_𝑘−1−𝑠𝐵_𝑘−1𝑢(𝑠)𝑑𝑠+𝐹_𝑘−1𝑢𝑡_𝑘−1exp𝐴_𝑘𝑡_𝑓−𝑠𝐵_𝑘𝑢(𝑠)𝑑𝑠.(4.2) It is equivalent to −exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1¹_𝑖_=𝑘−1=exp𝐴_𝑘𝐸_𝑖exp𝐴_𝑖ℎ_𝑖𝑥𝑡₀𝑡_𝑓−𝑡_𝑘−1×𝐸^𝑘−2_𝑖_{𝑖=1𝑖+1𝑗=𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗^𝑡𝑖_𝑡𝑖−1+𝐸_𝑘−1exp𝐴_𝑖𝑡_𝑖−𝑠𝐵_𝑖𝑢(𝑠)𝑑𝑠+𝐹_𝑖𝑢𝑡_𝑖^𝑡𝑘−1_𝑡𝑘−2+_{𝑡𝑓𝑡𝑘−1}exp𝐴_𝑘−1𝑡_𝑘−1−𝑠𝐵_𝑘−1𝑢(𝑠)𝑑𝑠+𝐹_𝑘−1𝑢𝑡_𝑘−1exp𝐴_𝑘𝑡_𝑓−𝑠𝐵_𝑘𝑢(𝑠)𝑑𝑠.(4.3) This implies that exp𝐴^𝑘𝑡_𝑓−𝑡_𝑘−1¹_{𝑖=𝑘−1}𝐸_𝑖exp𝐴_𝑖ℎ_𝑖𝑥𝑡₀∈ℛ𝑡₀,𝑡_𝑓.(4.4) Hence, exp𝐴^𝑘𝑡_𝑓−𝑡_𝑘−1¹_{𝑖=𝑘−1}𝐸_𝑖exp𝐴_𝑖ℎ_𝑖𝒞𝑡₀,𝑡_𝑓⊆ℛ𝑡₀,𝑡_𝑓.(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,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], if and only if ℐ𝑚¹_𝑖_=𝑘−1⊆_𝑘−2𝐸_𝑖exp𝐴_𝑖ℎ_𝑖_{𝑖=1𝑖+1𝑗=𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐸_𝑖⟨𝐴_𝑖∣𝐵_𝑖⟩+ℐ𝑚𝐹_𝑖+𝐸_𝑘−1⟨𝐴_𝑘−1∣𝐵_𝑘−1⟩+ℐ𝑚𝐹_𝑘−1+⟨𝐴_𝑘∣𝐵_𝑘(⟩4..6)

Proof. First, it is easy to prove that (4.6) is equivalent to ℐ𝑚exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1¹_𝑖_=𝑘−1𝐸_𝑖exp𝐴_𝑖ℎ_𝑖⊆ℛ𝑡₀,𝑡_𝑓.(4.7)
Necessity: since the system is controllable, we have 𝒞𝑡₀,𝑡_𝑓=ℝ^𝑛.(4.8) Then, by Theorem 4.2, we get ℛ𝑡₀,𝑡_𝑓⊇exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1¹_𝑖_=𝑘−1𝐸_𝑖exp𝐴_𝑖ℎ_𝑖ℝ^𝑛=ℐ𝑚exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1¹_𝑖_=𝑘−1𝐸_𝑖exp𝐴_𝑖ℎ_𝑖.(4.9)
Sufficiency: suppose that (4.7) holds. For any 𝑥∈ℝ^𝑛, we have exp𝐴^𝑘𝑡_𝑓−𝑡_𝑘−1¹_{𝑖=𝑘−1}𝐸_𝑖exp𝐴_𝑖ℎ_𝑖𝑥∈ℛ𝑡₀,𝑡_𝑓.(4.10) This implies that there exists a piecewise continuous function 𝑢(𝑡),𝑡∈[𝑡₀,𝑡_𝑓], such that 0=exp𝐴_𝑘𝑡_𝑓−𝑡_𝑘−1¹_𝑖_=𝑘−1×exp𝐴_𝑘𝐸_𝑖exp𝐴_𝑖ℎ_𝑖𝑥𝑡_𝑓−𝑡_𝑘−1×^𝑘−2𝐸_{𝑖=1𝑖+1𝑗=𝑘−1𝑖}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗^𝑡𝑖_𝑡𝑖−1+𝐸_𝑘−1exp𝐴_𝑖𝑡_𝑖−𝑠𝐵_𝑖𝑢(𝑠)𝑑𝑠+𝐹_𝑖𝑢𝑡_𝑖^𝑡𝑘−1_𝑡𝑘−2+_{𝑡𝑓𝑡𝑘−1}exp𝐴_𝑘−1𝑡_𝑘−1−𝑠𝐵_𝑘−1𝑢(𝑠)𝑑𝑠+𝐹_𝑘−1𝑢𝑡_𝑘−1exp𝐴_𝑘𝑡_𝑓−𝑠𝐵_𝑘𝑢(𝑠)𝑑𝑠.(4.11) Then, we know that 𝑥∈𝒞[𝑡₀,𝑡_𝑓]. 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 𝐸_𝑖 is nonsingular, 𝑖=1,2,…,𝑘−1, then the following statements are equivalent:(a)the system is reachable,(b)the system is controllable,(c)∑^{𝑘−2𝑖=1}∏^{𝑖+1𝑗=𝑘−1}𝐸_𝑗exp(𝐴_𝑗ℎ_𝑗)(𝐸_𝑖⟨𝐴_𝑖∣𝐵_𝑖⟩+ℐ𝑚(𝐹_𝑖))+𝐸_𝑘−1⟨𝐴_𝑘−1∣𝐵_𝑘−1⟩+ℐ𝑚(𝐹_𝑘−1)+⟨𝐴_𝑘∣𝐵_𝑘⟩=ℝ^𝑛.

Proof. Since 𝐸_𝑖 is nonsingular, 𝑖=1,2,…,𝑘−1, we have that exp𝐴_𝑘𝑡−𝑡_𝑘−1¹_𝑖_=𝑘−1𝐸_𝑖exp𝐴_𝑖ℎ_𝑖(4.12) is nonsingular. It follows that exp𝐴^𝑘𝑡−𝑡_𝑘−1¹_{𝑖=𝑘−1}𝐸_𝑖exp𝐴_𝑖ℎ_𝑖𝒞𝑡₀,𝑡_𝑓=ℛ𝑡₀,𝑡_𝑓.(4.13) It is easy to see that 𝒞[𝑡₀,𝑡_𝑓]=ℝ^𝑛⇔ℛ[𝑡₀,𝑡_𝑓]=ℝ^𝑛.

Remark 4.5. For system (2.1), assume that 𝐴_𝑖=𝐴,𝐵_𝑖=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 𝐸_𝑖=𝐼,𝐹_𝑖=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,𝑡_𝑓] (𝑡₀<𝑡_𝑓) if any initial state 𝑥₀∈ℝ^𝑛 can be uniquely determined by the corresponding system input 𝑢(𝑡) and the system output 𝑦(𝑡), for 𝑡∈[𝑡₀,𝑡_𝑓].

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

In order to investigate observability and determinability for the system (2.1), we first investigate those of the following zero input system: 𝑥̇(𝑡)=𝑥𝐴_𝑘𝑡_+𝑘𝑥(𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑦(𝑡)=𝐶_𝑘=𝐸_𝑘𝑥𝑡^−𝑘,𝑥𝑡₊₀𝑥(𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,=𝑥₀,𝑡₀≥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 𝑦(𝑡)=⎧⎪⎨⎪⎩𝐶¹𝐶_𝑖exp𝐴₁𝑡−𝑡₀𝑥𝑡₀,𝑡∈𝑡₀,𝑡₁,exp𝐴_𝑖𝑡−𝑡_𝑖−1¹_𝑗_=𝑖−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝑥𝑡₀,𝑡∈𝑡_𝑖−1,𝑡_𝑖,𝑖=2,…,𝑘.(5.2)

Theorem 5.3. The system (5.1) is observable on [𝑡₀,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], if and only if ²_{𝑖=𝑘𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖+𝐴^𝑇1∣𝐶^𝑇1=ℝ^𝑛.(5.3)

Proof. We prove the complementary proposition of Theorem 5.3, that is, the system (5.1) is not observable on [𝑡₀,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], if and only if ²_{𝑖=𝑘𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖+𝐴^𝑇1∣𝐶^𝑇1⊊ℝ^𝑛.(5.4)
Necessity: if the system (5.1) is not observable on [𝑡₀,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], then there exists 𝑥₀∈ℝ^𝑛, nonzero, such that 𝑦(𝑡)≡0,𝑡∈[𝑡₀,𝑡_𝑓]. This means that 𝐶_𝑖𝐶₁exp𝐴₁𝑡−𝑡₀𝑥₀=0,𝑡∈𝑡₀,𝑡₁,exp𝐴_𝑖𝑡−𝑡_𝑖−1¹_𝑗_=𝑖−1𝐶𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝑥₀=0,𝑡∈𝑡_𝑖−1,𝑡_𝑖,𝑖=2,…,𝑘−1,_𝑘exp𝐴_𝑘𝑡−𝑡_𝑘−1¹_𝑗_=𝑘−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝑥₀=0,𝑡∈𝑡_𝑘−1,𝑡_𝑓.(5.5) By Lemma 2.3, we get 𝑥𝑥^𝑇0𝐴^𝑇1∣𝐶^𝑇1=0,_{𝑇0𝑖−1}_𝑗=1exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖=0,𝑖=2,…,𝑘.(5.6) It follows that 𝑥^𝑇0𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖=0.(5.7) Then, we know that 𝑥₀∉𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖.(5.8) This implies (5.4).
Sufficiency: on the contrary, if (5.4) holds, there exists 𝑥₀∈ℝ^𝑛, nonzero, such that 𝑥^𝑇0𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖=0.(5.9) It follows that 𝑥𝑥^𝑇0𝐴^𝑇1∣𝐶^𝑇1=0,_{𝑇0𝑖−1}_𝑗=1exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖=0,𝑖=2,…,𝑘.(5.10) By Lemma 2.3, we get 𝐶_𝑖𝐶₁exp𝐴₁𝑡−𝑡₀𝑥₀=0,𝑡∈𝑡₀,𝑡₁,exp𝐴_𝑖𝑡−𝑡_𝑖−1¹_𝑗_=𝑖−1𝐶𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝑥₀=0,𝑡∈𝑡_𝑖−1,𝑡_𝑖,𝑖=2,…,𝑘−1,_𝑘exp𝐴_𝑘𝑡−𝑡_𝑘−1¹_𝑗_=𝑘−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝑥₀=0,𝑡∈𝑡_𝑘−1,𝑡_𝑓.(5.11) This means that 𝑦(𝑡)≡0,𝑡∈[𝑡₀,𝑡_𝑓]. 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 𝑂_𝑖=𝐶^𝑇𝑖,𝐴^𝑇_i𝐶^𝑇𝑖,…,𝐴^𝑇𝑖^𝑛−1𝐶^𝑇𝑖,(5.12) for 𝑖=2,…,𝑘, denote 𝐺_𝑖=^𝑖−1_𝑗=1exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝑂_𝑖,(5.13) and, finally, denote 𝑂_{[𝑡0,𝑡𝑓]}=𝑂₁,𝐺₂,…,𝐺_𝑘.(5.14) Then, it is easy to verify that ℐ𝑚𝑂_{[𝑡0,𝑡𝑓]}=𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖.(5.15) Thus, we get the following algebraic type criterion.

Corollary 5.5. The system (5.1) is observable on [𝑡₀,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], if and only if rank𝑂_{[𝑡0,𝑡𝑓]}=𝑛.(5.16)

Next, we establish a criterion for determinability.

Theorem 5.6. The system (5.1) is determinable on [𝑡₀,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], if and only if ℐ𝑚^𝑘−1_𝑗=1exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗⊆^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖+𝐴^𝑇1∣𝐶^𝑇1.(5.17)

Proof. First, by Lemma 2.4, we know that (5.17) is equivalent to ¹_{𝑗=𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖+𝐴^𝑇1∣𝐶^𝑇1_⟂=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,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], if and only if ¹_{𝑗=𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖+𝐴^𝑇1∣𝐶^𝑇1_⟂≠0.(5.19) Necessity: if the system (5.1) is not determinable on [𝑡₀,𝑡_𝑓], where 𝑡_𝑓∈(𝑡_𝑘−1,𝑡_𝑘], then there exists a terminal 𝑥_𝑓∈ℝ^𝑛, nonzero, such that 𝑦(𝑡)=0,𝑡∈[𝑡₀,𝑡_𝑓]. Then, there exists a nonzero 𝑥₀∈ℝ^𝑛 as the initial state such that the system is driven from 𝑥(𝑡₀)=𝑥₀ to 𝑥(𝑡_𝑓)=𝑥_𝑓, that is, 𝑥_𝑓=exp[𝐴_𝑘(𝑡_𝑓−𝑡_𝑘−1)]∏^{1𝑗=𝑘−1}𝐸_𝑗exp(𝐴_𝑗ℎ_𝑗)𝑥₀. This means that 𝐶_𝑖𝐶₁exp𝐴₁𝑡−𝑡₀𝑥₀=0,𝑡∈𝑡₀,𝑡₁,exp𝐴_𝑖𝑡−𝑡_𝑖−1¹_𝑗_=𝑖−1𝐶𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝑥₀=0,𝑡∈𝑡_𝑖−1,𝑡_𝑖,𝑖=2,…,𝑘−1,_𝑘exp𝐴_𝑘𝑡−𝑡_𝑘−1¹_𝑗_=𝑘−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝑥₀=0,𝑡∈𝑡_𝑘−1,𝑡_𝑓.(5.20) By Lemma 2.3, we get 𝑥𝑥^𝑇0𝐴^𝑇1∣𝐶^𝑇1=0,_{𝑇0𝑖−1}_𝑗=1exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖=0,𝑖=2,…,𝑘.(5.21) It follows that 𝑥^𝑇0𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖=0.(5.22) This implies that 𝑥₀∈𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖_⟂.(5.23) Since exp[𝐴_𝑘(𝑡_𝑘−1−𝑡_𝑓)]𝑥_𝑓=∏^{1𝑗=𝑘−1}𝐸_𝑗exp(𝐴_𝑗ℎ_𝑗)𝑥₀, we know that exp𝐴_𝑘𝑡_𝑘−1−𝑡_𝑓𝑥_𝑓∈¹_𝑗_=𝑘−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖_⟂.(5.24) It implies that ¹_{𝑗=𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖_⟂≠0.(5.25) Hence, (5.19) holds.
Sufficiency: on the contrary, if (5.19) holds, then we know that ¹_{𝑗=𝑘−1}𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖_⟂≠0.(5.26) Then, there exists a nonzero 𝑥_𝑓 satisfying exp𝐴_𝑘𝑡_𝑘−1−𝑡_𝑓𝑥_𝑓∈¹_𝑗_=𝑘−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖_⟂(5.27) such that there exists a nonzero 𝑥₀ satisfying 𝑥_𝑇0exp𝐴_𝑘𝑡_𝑘−1−𝑡_𝑓𝑥_𝑓=𝑥₀,𝐴^𝑇1∣𝐶^𝑇1+^𝑘_{𝑖=2𝑖−1𝑗=1}exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖=0.(5.28) It follows that 𝑥𝑥^𝑇0𝐴^𝑇1∣𝐶^𝑇1=0,_{𝑇0𝑖−1}_𝑗=1exp𝐴^𝑇𝑗ℎ_𝑗𝐸^𝑇𝑗𝐴^𝑇𝑖∣𝐶^𝑇𝑖=0,𝑖=2,…,𝑘.(5.29) By Lemma 2.3, we get 𝐶_𝑖𝐶₁exp𝐴₁𝑡−𝑡₀𝑥₀=0,𝑡∈𝑡₀,𝑡₁,exp𝐴_𝑖𝑡−𝑡_𝑖−1¹_𝑗_=𝑖−1𝐶𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝑥₀=0,𝑡∈𝑡_𝑖−1,𝑡_𝑖,𝑖=2,…,𝑘−1,_𝑘exp𝐴_𝑘𝑡−𝑡_𝑘−1¹_𝑗_=𝑘−1𝐸_𝑗exp𝐴_𝑗ℎ_𝑗𝑥₀=0,𝑡∈𝑡_𝑘−1,𝑡_𝑓.(5.30) This means that 𝑦(𝑡)≡0, 𝑡∈[𝑡₀,𝑡_𝑓]. Thus, we find a nonterminal nonzero state 𝑥_𝑓 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 𝐸_𝑖 is nonsingular, 𝑖=1,2,…,𝑘−1, then the following statements are equivalent:(a)the system is observable,(b)the system is determinable,(c)∑^2𝑖=𝑘∏^{𝑖−1𝑗=1}exp(𝐴^𝑇𝑗ℎ_𝑗)𝐸^𝑇𝑗⟨𝐴^𝑇𝑖∣𝐶^𝑇𝑖⟩+⟨𝐴^𝑇1∣𝐶^𝑇1⟩=ℝ^𝑛.

Proof. If 𝐸_𝑖 is nonsingular, 𝑖=1,2,…,𝑘−1, then we know that ∏^{1𝑗=𝑘−1}𝐸_𝑗exp(𝐴_𝑗ℎ_𝑗) is nonsingular. Hence, we get (5.3) and (5.17) are equivalent.

Remark 5.8. For system (2.1), assume that 𝐴_𝑖=𝐴,𝐵_𝑖=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 𝐸_𝑖=𝐼,𝐹_𝑖=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=𝐷⎡⎢⎢⎢⎢⎣000000000⎤⎥⎥⎥⎥⎦,𝐵₁=⎡⎢⎢⎢⎢⎣100⎤⎥⎥⎥⎥⎦,𝐶₁=010₁,𝐴₂=0,𝐸₁=⎡⎢⎢⎢⎢⎣100010000⎤⎥⎥⎥⎥⎦,𝐹₁=⎡⎢⎢⎢⎢⎣010⎤⎥⎥⎥⎥⎦,=𝐷⎡⎢⎢⎢⎢⎣000000000⎤⎥⎥⎥⎥⎦,𝐵₂=⎡⎢⎢⎢⎢⎣100⎤⎥⎥⎥⎥⎦,𝐶₂=100₂,𝐴₃=0,𝐸₂=⎡⎢⎢⎢⎢⎣110100000⎤⎥⎥⎥⎥⎦,𝐹₂=⎡⎢⎢⎢⎢⎣110⎤⎥⎥⎥⎥⎦,=𝐷⎡⎢⎢⎢⎢⎣000000000⎤⎥⎥⎥⎥⎦,𝐵₃=⎡⎢⎢⎢⎢⎣010⎤⎥⎥⎥⎥⎦,𝐶₃=100₃,=0,𝐸₃=⎡⎢⎢⎢⎢⎣010000000⎤⎥⎥⎥⎥⎦,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,𝑡_𝑓], where 𝑡_𝑓∈(2,3], of the system in Example 6.1.

First, we consider the reachability. By a simple calculation, we have 𝐸₂ex=psp𝐴an₂⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎢⎢⎣𝐸100₁⎤⎥⎥⎥⎥⎦⟨,𝐴⎡⎢⎢⎢⎢⎣010₁⎤⎥⎥⎥⎥⎦∣⎫⎪⎪⎬⎪⎪⎭𝐵.₁⟩+ℐ𝑚𝐹₁+𝐸₂⟨𝐴₂∣𝐵₂⟩+ℐ𝑚𝐹₂+⟨𝐴₃∣𝐵₃⟩(6.2) By Theorem 3.3, the system should not be reachable. In fact, for any piecewise continuous input 𝑢(𝑡),𝑡∈[0,𝑡_𝑓], and any nonzero initial state 𝑥₀=[𝑥⁰¹𝑥⁰²𝑥⁰³]^𝑇, we have 𝑥𝑡_𝑓=⎡⎢⎢⎢⎢⎣∗∗0⎤⎥⎥⎥⎥⎦.(6.3) This fact shows that the system is indeed not reachable.

Next, we consider the controllability. By a simple calculation, we have ℐ𝑚𝐸₂exp𝐴₂𝐸₁exp𝐴₁=span⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎢⎢⎣100⎤⎥⎥⎥⎥⎦,⎡⎢⎢⎢⎢⎣010⎤⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭.(6.4) It is easy to see that ℐ𝑚⊆𝐸𝐸₂e₂xp𝐴₂𝐸₁exp𝐴₁exp𝐴₂𝐸₁⟨𝐴₁∣𝐵₁⟩+ℐ𝑚𝐹₁+𝐸₂⟨𝐴₂∣𝐵₂⟩+ℐ𝑚𝐹₂+⟨𝐴₃∣𝐵₃⟩.(6.5)

By Theorem 4.3, the system should be controllable. In fact, we can take the piecewise constant input 𝑢(𝑡)=⎧⎪⎨⎪⎩𝑐¹𝑐0,,𝑡𝑡∈∈((01,,12₃]],,,𝑡∈(2,3].(6.6) Then, for any nonzero initial state 𝑥₀=[𝑥⁰¹𝑥⁰²𝑥⁰³]^𝑇, we have 𝑥𝑡_𝑓=⎡⎢⎢⎢⎢⎣𝑥⁰¹+0.5𝑐₁𝑥⁰²+1.5𝑐₁+2−2𝑡_𝑓+0.5𝑡^2𝑓𝑐³0⎤⎥⎥⎥⎥⎦.(6.7) Obviously, if 𝑐₁=−2𝑥⁰¹, 𝑐₃=(−𝑥⁰²−1.5𝑐₁)/(2−2𝑡_𝑓+0.5𝑡^2𝑓), then 𝑥(𝑡_𝑓)=0. This fact shows that the system is indeed controllable.

Next, we consider the observability. By a simple calculation, we have 𝐴^𝑇1=∣𝐶sp^𝑇1an⎧⎪⎪⎨⎪⎪⎩+⎡⎢⎢⎢⎢⎣e100x⎤⎥⎥⎥⎥⎦p,⎡⎢⎢⎢⎢⎣010𝐴⎤⎥⎥⎥⎥⎦^𝑇1⎫⎪⎪⎬⎪⎪⎭.𝐸^𝑇1𝐴^𝑇2∣𝐶^𝑇2+exp𝐴^𝑇1𝐸^𝑇1exp𝐴^𝑇2𝐸^𝑇2𝐴^𝑇3∣𝐶^𝑇3(6.8) By Theorem 5.3, the system should not be observable. In fact, for any piecewise continuous input 𝑢(𝑡),𝑡∈[0,𝑡_𝑓], and nonzero initial state 𝑥₀=[001]^𝑇, we have 𝑦(𝑡)≡0,𝑡∈0,𝑡_𝑓.(6.9) This fact shows that the system is indeed not observable.

Finally, we consider the determinability. By a simple calculation, we have 𝐸₂exp×𝐴₂𝐸𝐴_1𝑇1exp𝐴₁∣𝐶^𝑇1+exp𝐴^𝑇1𝐸^𝑇1𝐴^𝑇2∣𝐶^𝑇2+exp𝐴^𝑇1𝐸^𝑇1exp𝐴^𝑇2𝐸^𝑇2𝐴^𝑇3∣𝐶^𝑇3_⟂=⎡⎢⎢⎢⎢⎣100⎤⎥⎥⎥⎥⎦span⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎢⎢⎣001⎤⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭=0.(6.10) It follows that 𝐸₂ex×p𝐴𝐴_2𝑇1𝐸₁exp𝐴₁∣𝐶^𝑇1+exp𝐴^𝑇1𝐸^𝑇1𝐴^𝑇2∣𝐶^𝑇2+exp𝐴^𝑇1𝐸^𝑇1exp𝐴^𝑇2𝐸^𝑇2𝐴^𝑇3∣𝐶^𝑇3_⟂=0.(6.11) By Theorem 5.6, the system should be determinable. In fact, for any nonzero terminal state 𝑥_𝑓=[𝑥^𝑓1𝑥^𝑓2𝑥^𝑓3]^𝑇, there must exist a nonzero initial state 𝑥₀=[𝑥⁰¹𝑥⁰²𝑥⁰³]^𝑇 such that exp𝐴₃2−𝑡_𝑓𝑥_𝑓=𝐸₂exp𝐴₂𝐸₁exp𝐴₁𝑥₀.(6.12) It follows that 2−𝑡_𝑓𝑥_𝑓=⎡⎢⎢⎢⎢⎣110100000⎤⎥⎥⎥⎥⎦𝑥₀.(6.13) This means that 𝑥^𝑓3=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,𝑡_𝑓). This fact shows that the system is indeed determinable.

Example 6.2. Consider a 3-dimensional linear piecewise constant impulsive system with 𝐴₁=𝐷⎡⎢⎢⎢⎢⎣000000000⎤⎥⎥⎥⎥⎦,𝐵₁=⎡⎢⎢⎢⎢⎣100⎤⎥⎥⎥⎥⎦,𝐶₁=010₁,𝐴₂=0,𝐸₁=⎡⎢⎢⎢⎢⎣100010001⎤⎥⎥⎥⎥⎦,𝐹₁=⎡⎢⎢⎢⎢⎣010⎤⎥⎥⎥⎥⎦,=𝐷⎡⎢⎢⎢⎢⎣000000000⎤⎥⎥⎥⎥⎦,𝐵₂=⎡⎢⎢⎢⎢⎣010⎤⎥⎥⎥⎥⎦,𝐶₂=100₂,𝐴₃=0,𝐸₂=⎡⎢⎢⎢⎢⎣100010001⎤⎥⎥⎥⎥⎦,𝐹₂=⎡⎢⎢⎢⎢⎣110⎤⎥⎥⎥⎥⎦,=𝐷⎡⎢⎢⎢⎢⎣000000000⎤⎥⎥⎥⎥⎦,𝐵₃=⎡⎢⎢⎢⎢⎣001⎤⎥⎥⎥⎥⎦,𝐶₃=100₃,=0,𝐸₃=⎡⎢⎢⎢⎢⎣110010001⎤⎥⎥⎥⎥⎦,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,𝑡_𝑓], where 𝑡_𝑓∈(2,3], of the system in Example 6.2.

First, we consider reachability. By a simple calculation, we have 𝐸₂=exsppan𝐴⎧⎪⎪⎨⎪⎪⎩₂⎡⎢⎢⎢⎢⎣100𝐸⎤⎥⎥⎥⎥⎦,₁⎡⎢⎢⎢⎢⎣⟨010𝐴⎤⎥⎥⎥⎥⎦₁,∣⎡⎢⎢⎢⎢⎣001𝐵⎤⎥⎥⎥⎥⎦₁⎫⎪⎪⎬⎪⎪⎭⟩=+ℝℐ₃𝑚𝐹₁+𝐸₂⟨𝐴₂∣𝐵₂⟩+ℐ𝑚𝐹₂+⟨𝐴₃∣𝐵₃⟩.(6.15) By Theorem 3.3, the system should be reachable. In fact, we take the piecewise constant input 𝑢(𝑡)=⎧⎪⎨⎪⎩𝑐¹𝑐,𝑡∈(0,1₂],𝑐,𝑡∈(1,2₃],,𝑡∈(2,3].(6.16) Then, letting 𝑥(0)=0, for any nonzero terminal state 𝑥(3)=[𝑥^𝑓1𝑥^𝑓2𝑥^𝑓3]^𝑇, we have ⎡⎢⎢⎢⎢⎣𝑥^𝑓1𝑥^𝑓2𝑥^𝑓3⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣110120001⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑐₁𝑐₂𝑐₃⎤⎥⎥⎥⎥⎦.(6.17) Obviously, we can select suitable 𝑐₁,𝑐₂, and 𝑐₃ such that 𝑥_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 𝑢(𝑡)=⎧⎪⎨⎪⎩𝑐¹𝑐,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 0=⎡⎢⎢⎢⎢⎣𝑥𝑥⁰¹⁰²𝑥⁰³⎤⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎣110120001⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑐₁𝑐₂𝑐₃⎤⎥⎥⎥⎥⎦.(6.19) Obviously, we can select suitable 𝑐₁,𝑐₂, and 𝑐₃ such that 𝑥₀ 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 𝑥∣𝑥=^𝑡𝑓_𝑡0=⟨𝐴∣𝐵⟩ex.p𝐴𝑡_𝑓−𝑠𝐵𝑢(𝑠)𝑑𝑠,∀piecewisecontinuous𝑢(A.1) Thus, it is easy to see that 𝑥∣𝑥=𝐸^𝑡𝑓_𝑡0⊆𝐸⟨𝐴∣𝐵⟩ex+pℐ𝐴𝑚(𝑡𝐹_𝑓)−.𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑢𝑡_𝑓,∀piecewisecontinuous𝑢(A.2) Moreover, we have 𝑥∣𝑥=^𝑡𝐸_𝑡0=⟨𝐴∣𝐵⟩e,xp𝐴𝑡_𝐸−𝑠𝐵𝑢(𝑠)𝑑𝑠,∀piecewisecontinuous𝑢(A.3) where 𝑡_𝐸=(𝑡₀+𝑡_𝑓)/2. Then, for any 𝑥∈𝐸⟨𝐴∣𝐵⟩+ℐ𝑚(𝐹), there exist a piecewise continuous function 𝑢(𝑡), 𝑡∈[𝑡₀,𝑡_𝐸], and 𝑦∈ℝ^𝑛 such that 𝑥=𝐸^𝑡𝐸_𝑡0exp𝐴𝑡_𝐸−𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑦.(A.4) Then, we can take 𝑣(𝑡)=⎧⎪⎨⎪⎩𝑢(𝑡),𝑡∈𝑡⁰0,𝑡∈𝑡_𝐸,𝑡_𝐸,𝑦,𝑡=𝑡_𝑓,𝑡_𝑓,,(A.5) such that 𝑥=𝐸^𝑡𝑓_𝑡0exp𝐴𝑡_𝑓−𝑠𝐵𝑣(𝑠)𝑑𝑠+𝐹𝑣𝑡_𝑓.(A.6) This implies that 𝑥∈𝑥∣𝑥=𝐸^𝑡𝑓_𝑡0exp𝐴𝑡_𝑓−𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑢𝑡_𝑓,∀piecewisecontinuous𝑢.(A.7) It follows that 𝑥∣𝑥=𝐸^𝑡𝑓_𝑡0⊇𝐸⟨𝐴∣𝐵⟩ex+pℐ𝐴𝑚(𝑡𝐹_𝑓)−.𝑠𝐵𝑢(𝑠)𝑑𝑠+𝐹𝑢𝑡_𝑓,∀piecewisecontinuous𝑢(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(𝑡−𝑡₀)]𝑥=0,𝑡∈[𝑡₀,𝑡_𝑓], we get 𝐶𝑥=0. Then, for 𝑖=1,…,𝑛−1, calculating the 𝑖th derivative of 𝐶exp[𝐴(𝑡−𝑡₀)]𝑥 with respect to 𝑡 at 𝑡=𝑡₀, we get 𝐶𝐴^𝑖𝑥=0.(B.1) Thus, we know that 𝑥^𝑇𝐶^𝑇,𝐶^𝑇𝐴^𝑇,…,𝐶^𝑇𝐴^𝑇^𝑛−1=0.(B.2) Hence, 𝑥^𝑇⟨𝐴^𝑇∣𝐶^𝑇⟩=0.

((a)⇐(b)) If 𝑥^𝑇⟨𝐴^𝑇∣𝐶^𝑇⟩=0, it follows that 𝑥^𝑇[𝐶^𝑇,𝐶^𝑇𝐴^𝑇,…,𝐶^𝑇(𝐴^𝑇)^𝑛−1]=0. That is 𝐶𝐴^𝑖𝑥=0,𝑖=0,1,…,𝑛−1.(B.3) Then, it is easy to prove that 𝐶exp[𝐴(𝑡−𝑡₀)]𝑥=0,𝑡∈[𝑡₀,𝑡_𝑓].

C. Proof of Lemma 2.4

Given a matrix 𝐴∈ℝ^𝑛×𝑛 and a linear subspace 𝒲⊆ℝ^𝑛, the following two statements are equivalent:(a)ℐ𝑚(𝐴)⊆𝒲,(b)𝐴^𝑇𝑊_⟂=0.

((a)⇒(b)) Assume that ℐ𝑚(𝐴)⊆𝒲 ((b)⇒(a)). It is equivalent to ℐ𝑚(𝐴)⋂𝒲_⟂=0. It follows that, for any 𝑥∈𝒲_⟂,𝑥^𝑇𝐴=0. That is, 𝐴^𝑇𝑥=0. This implies that 𝐴^𝑇𝑊_⟂=0.

((b)⇒(a)) Assume that 𝐴^𝑇𝑊_⟂=0. It follows that, for any 𝑥∈𝒲_⟂, 𝐴^𝑇𝑥=0. That is, 𝑥^𝑇𝐴=0. This implies that ℐ𝑚(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.