# Controllability and Observability Criteria for Linear Piecewise Constant Impulsive Systems

Authors: Hong Shi; Guangming Xie
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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. Section2 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(2.1)x˙(t)=Akx(t)+Bku(t),t∈[tk-1,tk),x(tk+)=Ekx(tk-)+Fku(tk),y(t)=Ckx(t)+Dku(t),t∈[tk-1,tk),x(t0+)=x0,t0≥0,
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)∈ℝn is the state vector, and u(t)∈ℝp the input vector, y(t)∈ℝq the output vector; x(t+):=limh→0+x(t+h), x(t-):=limh→0-x(t-h), and the discontinuity points are
(2.2)t1<t2<⋯<tk<⋯,lim⁡k→∞tk=∞,
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 anyt∈(tk-1,tk],k=1,2,…, the general solution of the system (2.1) is given by
(2.3)x(t)=exp⁡⁡[Ak(t-tk-1)]{∏i=k-11‍Eiexp⁡⁡(Aihi)x(t0)x(t)=exp⁡⁡[Ak(t-tk-1)]+∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡⁡(Ajhj)(Ei∫ti-1ti‍exp⁡⁡[Ai(ti-s)]Biu(s)ds+Fiu(ti))+Ek-1∫tk-2tk-1‍exp⁡⁡[Ak-1(tk-1-s)]Bk-1u(s)ds+Fk-1u(tk-1)∏i=k-11‍}x(t)=+∫tk-1t‍exp⁡⁡[Ak(t-s)]Bku(s)ds,
where hk=tk-tk-1, k=1,2,….Proof.
Fort∈(t0,t1], we have
(2.4)x(t)=exp⁡[A1(t-t0)]x(t0)+∫t0t‍exp⁡[A1(t-s)]B1u(s)ds.
For t=t1+, we have
(2.5)x(t1+)=E1(exp⁡(A1h1)x(t0)+∫t0t1‍exp⁡[A1(t1-s)]B1u(s)ds)+F1u(t1).
Similarly, for t∈(ti-1,ti], i=2,3,…,k, we have
(2.6)x(t)=exp⁡[Ai(t-ti-1)]x(ti-1+)+∫ti-1t‍exp⁡[Ai(t-s)]Biu(s)ds.
And, for t=ti+,i=2,3,…,k, we have
(2.7)x(ti+)=Ei(exp⁡(Aihi)x(ti-1)+∫ti-1ti‍exp⁡[Ai(ti-s)]Biu(s)ds)+Fiu(ti).
Thus, by (2.4), (2.5), (2.6), and (2.7), it is easy to verify (2.3).Iftf∈(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∈(tk-1,tk],k=2,3,….Now, we give some mathematical preliminaries as the basic tools in the following discussion.Given matricesA∈ℝn×n and B∈ℝn×p, denote ℐm(B) as the range of B, that is, ℐm(B)={y∣y=Bx,forallx∈ℝn×n}, and denote 〈A∣B〉 as the minimal invariant subspace of A on ℐm(B), that is, 〈A∣B〉=ℐm(B)+ℐm(AB)+⋯+ℐm(An-1B). Given a linear subspace 𝒲⊆ℝn, denote 𝒲⊥ as the orthogonal complement of 𝒲, that is, 𝒲⊥={x∣xT𝒲=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 matricesA,E∈ℝn×n,B,F∈ℝn×p, for any 0≤t0<tf<+∞, one has
(2.8){x∣x=E∫t0tf‍exp⁡⁡[A(tf-s)]Bu(s)ds+Fu(tf),∀piecewisecontinuousu}=E〈A∣B〉+Im(F).Proof.
See AppendixA.Lemma 2.3.
Given two matricesA∈ℝn×n,C∈ℝq×n, two scalars t0<tf, and a vector x∈ℝn, the following two statements are equivalent:(a)
Cexp[A(t-t0)]x=0,t∈[t0,tf],(b)
xT〈AT∣CT〉=0.Proof.
See AppendixB.Lemma 2.4.
Given a matrixA∈ℝn×n and a linear subspace 𝒲⊆ℝn, the following two statements are equivalent:(a)
ℐm(A)⊆𝒲,(b)
ATW⊥=0.Proof.
See AppendixC.


## 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 xf∈ℝn, there exists a piecewise continuous input u(t):[t0,tf]→ℝp 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 ℛ[t0,tf].Theorem 3.2.
For the system (2.1), the reachable set on [t0,tf], where tf∈(tk-1,tk], is given by
(3.1)R[t0,tf]=exp⁡⁡[Ak(tf-tk-1)]∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡⁡(Ajhj)(Ei〈Ai∣Bi〉+Im(Fi))+exp⁡[Ak(tf-tk-1)](Ek-1〈Ak-1∣Bk-1〉+Im(Fk-1))+〈Ak∣Bk〉.Proof.
By Lemma2.1, letting x(t0)=0, we have
(3.2)x(t)=exp⁡[Ak(t-tk-1)]{∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡(Ajhj)x(t)=exp⁡[Ak(t-tk-1)]×(Ei∫ti-1ti‍exp⁡[Ai(ti-s)]Biu(s)ds+Fiu(ti))x(t)=exp⁡[Ak(t-tk-1)]∑i=1k-2‍∏j=k-1i+1‍+Ek-1∫tk-2tk-1‍exp⁡⁡[Ak-1(tk-1-s)]Bk-1u(s)ds+Fk-1u(tk-1)}x(t)=+∫tk-1t‍exp⁡[Ak(t-s)]Bku(s)ds.
It follows that
(3.3)R[t0,tf]={∑i=1k-2‍∏j=k-1i+1‍x∣x=exp⁡[Ak(tf-tk-1)]×{∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡(Ajhj)(Ei∫ti-1ti‍exp⁡[Ai(ti-s)]Biu(s)ds+Fiu(ti))+Ek-1∫tk-2tk-1‍exp⁡[Ak-1(tk-1-s)]Bk-1u(s)ds+Fk-1u(tk-1)∑i=1k-2‍∏j=k-1i+1‍}+∫tk-1tf‍exp⁡[Ak(tf-s)]Bku(s)ds,∀piecewisecontinuousu∑i=1k-2‍∏j=k-1i+1‍}=exp⁡[Ak(tf-tk-1)](∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡(Ajhj)=exp⁡[Ak(tf-tk-1)]×{x∣x=Ei∫ti-1ti‍exp⁡[Ai(ti-s)]Biu(s)ds+Fiu(ti),=exp⁡[Ak(tf-tk-1)]×∀piecewisecontinuousu∫ti-1ti‍}=exp⁡[Ak(tf-tk-1)]×+{x∣x=Ek-1∫tk-2tk-1‍exp⁡[Ak-1(tk-1-s)]Bk-1u(s)ds+Fk-1u(tk-1)=exp⁡[Ak(tf-tk-1)]×∀piecewisecontinuousu∫ti-1ti‍}∑i=1k-2‍∏j=k-1i+1‍)+{x∣x=∫tk-1tf‍exp⁡[Ak(tf-s)]Bku(s)ds,∀piecewisecontinuousu}.
By Lemma2.2, we get
(3.4)R[t0,tf]=exp⁡⁡[Ak(tf-tk-1)](∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡⁡(Ajhj)(Ei〈Ai∣Bi〉+Im(Fi))R[t0,tf]=exp⁡⁡[Ak(tf-tk-1)]+Ek-1〈Ak-1∣Bk-1〉+Im(Fk-1)∑i=1k-2‍∏j=k-1i+1‍)R[t0,tf]=+〈Ak∣Bk〉.
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∈(tk-1,tk], if and only if
(3.5)∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡⁡(Ajhj)(Ei〈Ai∣Bi〉+Im(Fi))+Ek-1〈Ak-1∣Bk-1〉+Im(Fk-1)+〈Ak∣Bk〉=Rn.Proof.
Since(3.6)R[t0,tf]=exp⁡[Ak(tf-tk-1)]R[t0,tf]=×(∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡(Ajhj)(Ei〈Ai∣Bi〉+Im(Fi))+Ek-1〈Ak-1∣Bk-1〉+Im(Fk-1))R[t0,tf]=+〈Ak∣Bk〉R[t0,tf]=exp⁡[Ak(tf-tk-1)](∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡(Ajhj)(Ei〈Ai∣Bi〉+Im(Fi))∑i=1k-2‍∏j=k-1i+1‍+Ek-1〈Ak-1∣Bk-1〉+Im(Fk-1)+〈Ak∣Bk〉)
and the matrix exp[Ak(tf-tk-1)] is nonsingular, the proof directly follows from Theorem 3.2.Remark 3.4.
Theorem3.3 is a geometric type condition. By simple transformation, we can get an algebraic type condition. In fact, for i=1,2,…, denote
(3.7)Qi=[Bi,AiBi,…,Ain-1Bi],
for i=1,2,…,k-2, denote
(3.8)Hi=[∏j=k-1i+1‍Ejexp⁡⁡(Ajhj)EiQi,∏j=k-1i+1‍Ejexp⁡⁡(Ajhj)Fi],Hk-1=[Ek-1Qk-1,Fk-1],
and, finally, denote
(3.9)Q[t0,tf]=[H1,H2,…,Hk-1,Qk].
Then, it is easy to verify that
(3.10)exp⁡⁡[Ak(tf-tk-1)]Im(Q[t0,tf])=R[t0,tf].
Thus, we get the following algebraic type criterion.Corollary 3.5.
The system (2.1) is reachable on [t0,tf], where tf∈(tk-1,tk], if and only if
(3.11)rank⁡(Q[t0,tf])=n.


## 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 x0∈ℝn, there exists a piecewise continuous input u(t):[t0,tf]→ℝp 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,…,k-1, then the controllable set on [t0,tf], where tf∈(tk-1,tk], satisfies
(4.1)(exp⁡⁡[Ak(t-tk-1)]∏i=k-11‍Eiexp⁡⁡(Aihi))C[t0,tf]⊆R[t0,tf].Proof.
By Lemma2.1, letting x(tf)=0, we have
(4.2)0=exp⁡[Ak(tf-tk-1)]∏i=k-11‍Eiexp⁡(Aihi)x(t0)exp⁡[Ak(tf-tk-1)]×{∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡(Ajhj)×(Ei∫ti-1ti‍exp⁡[Ai(ti-s)]Biu(s)ds+Fiu(ti))+Ek-1∫tk-2tk-1‍exp⁡[Ak-1(tk-1-s)]Bk-1u(s)ds+Fk-1u(tk-1)∑i=1k-2∏j=k-1i+1‍}+∫tk-1tf‍exp⁡[Ak(tf-s)]Bku(s)ds.
It is equivalent to
(4.3)-exp⁡[Ak(tf-tk-1)]∏i=k-11‍Eiexp⁡(Aihi)x(t0)=exp⁡[Ak(tf-tk-1)]{∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡(Ajhj)×(Ei∫ti-1ti‍exp⁡[Ai(ti-s)]Biu(s)ds+Fiu(ti))+Ek-1∫tk-2tk-1‍exp⁡[Ak-1(tk-1-s)]Bk-1u(s)ds+Fk-1u(tk-1)∑i=1k-2∏j=k-1i+1‍}+∫tk-1tf‍exp⁡[Ak(tf-s)]Bku(s)ds.
This implies that
(4.4)(exp⁡[Ak(tf-tk-1)]∏i=k-11‍Eiexp⁡⁡(Aihi))x(t0)∈R[t0,tf].
Hence,
(4.5)(exp⁡[Ak(tf-tk-1)]∏i=k-11‍Eiexp⁡(Aihi))C[t0,tf]⊆R[t0,tf].Based on Theorem4.2, 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∈(tk-1,tk], if and only if
(4.6)Im(∏i=k-11‍Eiexp⁡(Aihi))⊆∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡(Ajhj)(Ei〈Ai∣Bi〉+Im(Fi))+Ek-1〈Ak-1∣Bk-1〉+Im(Fk-1)+〈Ak∣Bk〉.Proof.
First, it is easy to prove that (4.6) is equivalent to
(4.7)Im(exp⁡[Ak(tf-tk-1)]∏i=k-11‍Eiexp⁡(Aihi))⊆R[t0,tf].
Necessity: since the system is controllable, we have(4.8)C[t0,tf]=Rn.
Then, by Theorem 4.2, we get
(4.9)R[t0,tf]⊇(exp⁡[Ak(tf-tk-1)]∏i=k-11‍Eiexp⁡(Aihi))Rn=Im(exp⁡[Ak(tf-tk-1)]∏i=k-11‍Eiexp⁡⁡(Aihi)).
Sufficiency: suppose that (4.7) holds. For any x∈ℝn, we have
(4.10)(exp⁡⁡[Ak(tf-tk-1)]∏i=k-11‍Eiexp⁡⁡(Aihi))x∈R[t0,tf].
This implies that there exists a piecewise continuous function u(t),t∈[t0,tf], such that
(4.11)0=exp⁡[Ak(tf-tk-1)]∏i=k-11‍Eiexp⁡(Aihi)x×exp⁡[Ak(tf-tk-1)]{∑i=1k-2‍∏j=k-1i+1‍Ejexp⁡(Ajhj)×exp⁡[Ak(tf-tk-1)]×(Ei∫ti-1ti‍exp⁡[Ai(ti-s)]Biu(s)ds+Fiu(ti))×exp⁡[Ak(tf-tk-1)]+Ek-1∫tk-2tk-1‍exp⁡[Ak-1(tk-1-s)]Bk-1u(s)ds+Fk-1u(tk-1)∑i=1k-2‍∏j=k-1i+1‍}+∫tk-1tf‍exp⁡[Ak(tf-s)]Bku(s)ds.
Then, we know that x∈𝒞[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,…,k-1, then the following statements are equivalent:(a)
the system is reachable,(b)
the system is controllable,(c)
∑i=1k-2∏j=k-1i+1Ejexp(Ajhj)(Ei〈Ai∣Bi〉+ℐm(Fi))+Ek-1〈Ak-1∣Bk-1〉+ℐm(Fk-1)+〈Ak∣Bk〉=ℝn.Proof.
SinceEi is nonsingular, i=1,2,…,k-1, we have that
(4.12)exp⁡[Ak(t-tk-1)]∏i=k-11‍Eiexp⁡(Aihi)
is nonsingular. It follows that
(4.13)(exp⁡[Ak(t-tk-1)]∏i=k-11‍Eiexp⁡(Aihi))C[t0,tf]=R[t0,tf].
It is easy to see that 𝒞[t0,tf]=ℝn⇔ℛ[t0,tf]=ℝn.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 x0∈ℝn 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 xf∈ℝn 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:
(5.1)x˙(t)=Akx(t),t∈[tk-1,tk),x(tk+)=Ekx(tk-),y(t)=Ckx(t),t∈[tk-1,tk),x(t0+)=x0,t0≥0.
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
(5.2)y(t)={C1exp⁡⁡[A1(t-t0)]x(t0),t∈(t0,t1],Ciexp⁡⁡[Ai(t-ti-1)]∏j=i-11‍Ejexp⁡(Ajhj)x(t0),t∈(ti-1,ti],i=2,…,k.Theorem 5.3.
The system (5.1) is observable on [t0,tf], where tf∈(tk-1,tk], if and only if
(5.3)∑i=k2‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉+〈A1T∣C1T〉=Rn.Proof.
We prove the complementary proposition of Theorem5.3, that is, the system (5.1) is not observable on [t0,tf], where tf∈(tk-1,tk], if and only if
(5.4)∑i=k2‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉+〈A1T∣C1T〉⊊Rn.
Necessity: if the system (5.1) is not observable on [t0,tf], where tf∈(tk-1,tk], then there exists x0∈ℝn, nonzero, such that y(t)≡0,t∈[t0,tf]. This means that
(5.5)C1exp⁡[A1(t-t0)]x0=0,t∈(t0,t1],Ciexp⁡[Ai(t-ti-1)]∏j=i-11‍Ejexp⁡(Ajhj)x0=0,t∈(ti-1,ti],i=2,…,k-1,Ckexp⁡[Ak(t-tk-1)]∏j=k-11‍Ejexp⁡(Ajhj)x0=0,t∈(tk-1,tf].
By Lemma 2.3, we get
(5.6)x0T〈A1T∣C1T〉=0,x0T∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉=0,i=2,…,k.
It follows that
(5.7)x0T(〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉)=0.
Then, we know that
(5.8)x0∉〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉.
This implies (5.4).
Sufficiency: on the contrary, if (5.4) holds, there exists x0∈ℝn, nonzero, such that
(5.9)x0T(〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉)=0.
It follows that
(5.10)x0T〈A1T∣C1T〉=0,x0T∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉=0,i=2,…,k.
By Lemma 2.3, we get
(5.11)C1exp⁡[A1(t-t0)]x0=0,t∈(t0,t1],Ciexp⁡⁡[Ai(t-ti-1)]∏j=i-11‍Ejexp⁡(Ajhj)x0=0,t∈(ti-1,ti],i=2,…,k-1,Ckexp⁡[Ak(t-tk-1)]∏j=k-11‍Ejexp⁡(Ajhj)x0=0,t∈(tk-1,tf].
This means that y(t)≡0,t∈[t0,tf]. Thus, the system (5.1) is not observable.Remark 5.4.
Theorem5.3 is a geometric type condition. By simple transformation, we can get an algebraic type condition. In fact, for i=1,2,…, denote
(5.12)Oi=[CiT,AiTCiT,…,(AiT)n-1CiT],
for i=2,…,k, denote
(5.13)Gi=∏j=1i-1‍exp⁡(AjThj)EjTOi,
and, finally, denote
(5.14)O[t0,tf]=[O1,G2,…,Gk].
Then, it is easy to verify that
(5.15)Im(O[t0,tf])=〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉.
Thus, we get the following algebraic type criterion.Corollary 5.5.
The system (5.1) is observable on [t0,tf], where tf∈(tk-1,tk], if and only if
(5.16)rank⁡(O[t0,tf])=n.Next, we establish a criterion for determinability.Theorem 5.6.
The system (5.1) is determinable on [t0,tf], where tf∈(tk-1,tk], if and only if
(5.17)Im(∏j=1k-1‍exp⁡(AjThj)EjT)⊆∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉+〈A1T∣C1T〉.Proof.
First, by Lemma2.4, we know that (5.17) is equivalent to
(5.18)∏j=k-11‍Ejexp⁡(Ajhj)(∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉+〈A1T∣C1T〉)⊥=0.
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∈(tk-1,tk], if and only if
(5.19)∏j=k-11‍Ejexp⁡(Ajhj)(∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉+〈A1T∣C1T〉)⊥≠0.
Necessity: if the system (5.1) is not determinable on [t0,tf], where tf∈(tk-1,tk], then there exists a terminal xf∈ℝn, nonzero, such that y(t)=0,t∈[t0,tf]. Then, there exists a nonzero x0∈ℝn as the initial state such that the system is driven from x(t0)=x0 to x(tf)=xf, that is, xf=exp[Ak(tf-tk-1)]∏j=k-11‍Ejexp(Ajhj)x0. This means that
(5.20)C1exp⁡[A1(t-t0)]x0=0,t∈(t0,t1],Ciexp⁡[Ai(t-ti-1)]∏j=i-11‍Ejexp⁡(Ajhj)x0=0,t∈(ti-1,ti],i=2,…,k-1,Ckexp⁡[Ak(t-tk-1)]∏j=k-11‍Ejexp⁡(Ajhj)x0=0,t∈(tk-1,tf].
By Lemma 2.3, we get
(5.21)x0T〈A1T∣C1T〉=0,x0T∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉=0,i=2,…,k.
It follows that
(5.22)x0T(〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉)=0.
This implies that
(5.23)x0∈(〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉)⊥.
Since exp[Ak(tk-1-tf)]xf=∏j=k-11‍Ejexp(Ajhj)x0, we know that
(5.24)exp⁡[Ak(tk-1-tf)]xf∈∏j=k-11‍Ejexp⁡(Ajhj)(〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉)⊥.
It implies that
(5.25)∏j=k-11‍Ejexp⁡(Ajhj)(〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉)⊥≠0.
Hence, (5.19) holds.
Sufficiency: on the contrary, if (5.19) holds, then we know that
(5.26)∏j=k-11‍Ejexp⁡(Ajhj)(〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡⁡(AjThj)EjT〈AiT∣CiT〉)⊥≠0.
Then, there exists a nonzero xf satisfying
(5.27)exp⁡[Ak(tk-1-tf)]xf∈∏j=k-11‍Ejexp⁡(Ajhj)(〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉)⊥
such that there exists a nonzero x0 satisfying
(5.28)exp⁡[Ak(tk-1-tf)]xf=x0,x0T(〈A1T∣C1T〉+∑i=2k‍∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉)=0.
It follows that
(5.29)x0T〈A1T∣C1T〉=0,x0T∏j=1i-1‍exp⁡(AjThj)EjT〈AiT∣CiT〉=0,i=2,…,k.
By Lemma 2.3, we get
(5.30)C1exp⁡[A1(t-t0)]x0=0,t∈(t0,t1],Ciexp⁡[Ai(t-ti-1)]∏j=i-11‍Ejexp⁡(Ajhj)x0=0,t∈(ti-1,ti],i=2,…,k-1,Ckexp⁡[Ak(t-tk-1)]∏j=k-11‍Ejexp⁡(Ajhj)x0=0,t∈(tk-1,tf].
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,…,k-1, then the following statements are equivalent:(a)
the system is observable,(b)
the system is determinable,(c)
∑i=k2∏j=1i-1exp(AjThj)EjT〈AiT∣CiT〉+〈A1T∣C1T〉=ℝn.Proof.
IfEi is nonsingular, i=1,2,…,k-1, then we know that ∏j=k-11‍Ejexp(Ajhj) 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(6.1)A1=[000000000],B1=[100],C1=[010],D1=0,E1=[100010000],F1=[010],A2=[000000000],B2=[100],C2=[100],D2=0,E2=[110100000],F2=[110],A3=[000000000],B3=[010],C3=[100],D3=0,E3=[000100000],F3=[010],
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(6.2)E2exp⁡(A2)(E1〈A1∣B1〉+Im(F1))+E2〈A2∣B2〉+Im(F2)+〈A3∣B3〉=span⁡{[100],[010]}.
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=[x10x20x30]T, we have
(6.3)x(tf)=[**0].
This fact shows that the system is indeed not reachable.Next, we consider the controllability. By a simple calculation, we have(6.4)Im(E2exp⁡⁡(A2)E1exp⁡⁡(A1))=span⁡{[100],[010]}.
It is easy to see that
(6.5)Im(E2exp⁡⁡(A2)E1exp⁡⁡(A1))⊆E2exp⁡⁡(A2)(E1〈A1∣B1〉+Im(F1))+E2〈A2∣B2〉+Im(F2)+〈A3∣B3〉.By Theorem4.3, the system should be controllable. In fact, we can take the piecewise constant input
(6.6)u(t)={c1,t∈(0,1],0,t∈(1,2],c3,t∈(2,3].
Then, for any nonzero initial state x0=[x10x20x30]T, we have
(6.7)x(tf)=[x10+0.5c1x20+1.5c1+(2-2tf+0.5tf2)c30].
Obviously, if c1=-2x10, c3=(-x20-1.5c1)/(2-2tf+0.5tf2), then x(tf)=0. This fact shows that the system is indeed controllable.Next, we consider the observability. By a simple calculation, we have(6.8)〈A1T∣C1T〉+exp⁡(A1T)E1T〈A2T∣C2T〉+exp⁡(A1T)E1Texp⁡(A2T)E2T〈A3T∣C3T〉=span⁡{[100],[010]}.
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
(6.9)y(t)≡0,t∈[0,tf).
This fact shows that the system is indeed not observable.Finally, we consider the determinability. By a simple calculation, we have(6.10)E2exp⁡⁡(A2)E1exp⁡⁡(A1)×(〈A1T∣C1T〉+exp⁡⁡(A1T)E1T〈A2T∣C2T〉+exp⁡⁡(A1T)E1Texp⁡⁡(A2T)E2T〈A3T∣C3T〉)⊥=[100]span⁡{[001]}=0.
It follows that
(6.11)E2exp⁡(A2)E1exp⁡(A1)×(〈A1T∣C1T〉+exp⁡(A1T)E1T〈A2T∣C2T〉+exp⁡(A1T)E1Texp⁡(A2T)E2T〈A3T∣C3T〉)⊥=0.
By Theorem 5.6, the system should be determinable. In fact, for any nonzero terminal state xf=[x1fx2fx3f]T, there must exist a nonzero initial state x0=[x10x20x30]T such that
(6.12)exp⁡[A3(2-tf)]xf=E2exp⁡(A2)E1exp⁡(A1)x0.
It follows that
(6.13)(2-tf)xf=[110100000]x0.
This means that x3f=x30=0 and |x10|+|x20|≠0. It is easy to verify that, for any initial state x0 satisfying |x10|+|x20|≠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(6.14)A1=[000000000],B1=[100],C1=[010],D1=0,E1=[100010001],F1=[010],A2=[000000000],B2=[010],C2=[100],D2=0,E2=[100010001],F2=[110],A3=[000000000],B3=[001],C3=[100],D3=0,E3=[100110001],F3=[010],
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(6.15)E2exp⁡(A2)(E1〈A1∣B1〉+Im(F1))+E2〈A2∣B2〉+Im(F2)+〈A3∣B3〉=span⁡{[100],[010],[001]}=R3.
By Theorem 3.3, the system should be reachable. In fact, we take the piecewise constant input
(6.16)u(t)={c1,t∈(0,1],c2,t∈(1,2],c3,t∈(2,3].
Then, letting x(0)=0, for any nonzero terminal state x(3)=[x1fx2fx3f]T, we have
(6.17)[x1fx2fx3f]=[110120001][c1c2c3].
Obviously, we can select suitable c1,c2, and c3 such that xf is any state in ℝ3. This fact shows that the system is indeed reachable.Next, by Theorem4.3, the system should be reachable. In fact, we take the piecewise constant input
(6.18)u(t)={c1,t∈(0,1],c2,t∈(1,2],c3,t∈(2,3].
Then, for any nonzero initial state x(0)=[x10x20x30]T, letting x(3)=0, we have
(6.19)0=[x10x20x30]+[110120001][c1c2c3].
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 Corollary4.4, 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.



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*Source: 182040-2012-09-02.xml*