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Journal of Applied Mathematics
Volume 2012, Article ID 876120, 12 pages
http://dx.doi.org/10.1155/2012/876120
Research Article

Geometric Analysis of Reachability and Observability for Impulsive Systems on Complex Field

Shouwei Zhao,1,2,3 Jitao Sun,2 and Hai Lin3,4

1College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201620, China
2Department of Mathematics, Tongji University, Shanghai 200092, China
3Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576
4Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA

Received 21 October 2011; Revised 8 December 2011; Accepted 8 December 2011

Academic Editor: Nazim I. Mahmudov

Copyright © 2012 Shouwei Zhao et al. 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

Nowadays, quantum systems have become one of the focuses of the ongoing research and they are typical complex systems, whose state variables are defined on the complex field. In this paper, the issue of reachability and observability is addressed for a class of linear impulsive systems on complex field, for simplicity, complex linear impulsive systems. This kind of time-driven impulsive systems allows free impulsive instants, which leads to the limitation of using traditional definitions of reachability and observability directly. New notations about the span reachable set and unobservable set are proposed. Sufficient and necessary conditions for span reachability and observability of such systems are established. Moreover, the explicit characterization of span reachable set and unobservable set is presented by geometric analysis. It is pointed out that the geometric conditions are equivalent to the algebraic ones in known results for special cases. Numerical examples are also presented to show the effectiveness of the proposed methods.

1. Introduction

Recent years have witnessed growing interest in investigating the control theory of hybrid systems and most progress has been made in the stability and stabilization of hybrid systems, see [15] and the references therein. Impulsive dynamical systems are an important class of hybrid systems which exhibit continuous evolutions described by ordinary differential equations and abrupt changes at some instants or impulses. Examples of these systems include evolution processes, optimal control models in economics, stimulated neural networks, frequency-modulated systems, and some motions of missiles or aircrafts. In view of both theoretical and practical significance, much attention has been paid on the analysis and synthesis of impulsive systems, or impulsive control systems, see [611] and the references therein.

Closely related to the pole assignment, structural decomposition, quadratic optimal control and observer design, the controllability, reachability, and observability play a significant role in the control theory and engineering [1214]. The controllability and observability of various hybrid systems have been extensively investigated using different approaches such as geometric analysis [79], algebraic characterization [10, 11], functional analysis [15, 16], and differential geometric method [3]. Particularly, research efforts have been made on the controllability and observability for impulsive systems. By proposing algebraic rank conditions, the state controllability and observability of linear time-varying impulsive systems were investigated in [10, 11]. For impulsive functional differential systems, the controllability is considered with the help of fixed-point theorems [15, 16]. References [79] presented the geometric analysis of reachability, controllability and observability for (switched) impulsive systems. Geometric analysis is effective in providing easily verifiable conditions for the controllability and observability based on the explicit characterization of controllable and observable sets in terms of invariant sets of systems. Hence, it provides an effective and simple method to investigate the fundamental properties of hybrid systems.

However, in the above-mentioned works, the state space of the considered systems is always 𝑛-dimensional real vector space, that is, 𝑛, except few reports on the issue of controllability for complex systems [10, 17]. Nowadays, control of complex systems, especially quantum systems, has attracted considerable attention [1822]. It should be noticed that quantum system models are typical complex dynamical systems whose states evolve in Banach (Hilbert) space on the field of complex number, which are much more complicated than real systems. In view of this, complex dynamics systems have many potential applications ranging from science to engineering. Therefore, it is important and necessary to study the control theory of a special class of complex dynamical systems, complex linear impulsive systems. This motivates us to consider the reachability and observability of complex linear impulsive systems by geometric analysis. The impulsive system considered in the current paper has uncertainty in the impulsive instants which can be regarded as time-driven impulsive systems. This kind of more general systems exists in many practical applications [7]. Due to the novel properties of reachable set and unobservable set for this kind of systems, traditional geometric analysis may be limited to characterize them. Hence, new concepts on the reachable set and unobservable set are introduced. Based on these definitions, we generalize the geometric analysis approach for reachability and observability to complex linear impulsive systems. Specifically, sufficient and necessary criteria for reachability and observability are derived, and explicit characterization of reachable set and unobservable set is proposed consequently. Moreover, it is proved that the span reachable set and unobservable set with free impulsive times are invariant subspaces of the complex impulsive system.

The rest of this paper is organized as follows. In Section 2, the complex linear impulsive systems to be dealt with are formulated and the solution expression for such systems is presented. In Sections 3 and 4, based on the geometric characterization of reachable set and unobservable set for complex linear impulsive systems, sufficient and necessary conditions for state reachability and state observability of complex linear impulsive systems are derived, respectively. Moreover, examples are discussed to illustrate the effectiveness of the proposed methods. Finally, some conclusions are drawn in Section 5.

2. Preliminaries

Consider the complex linear time-varying impulsive system described bẏ𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑢(𝑡),𝑡𝑡𝑘,𝑡Δ𝑥𝑘=𝐸𝑘𝑥𝑡𝑘+𝐹𝑘𝑢𝑘,𝑥𝑡𝑦(𝑡)=𝐶(𝑡)𝑥(𝑡)+𝐷(𝑡)𝑢(𝑡),+0=𝑥0,(2.1) where 𝑘=1,2,,𝐴(𝑡),𝐵(𝑡),𝐶(𝑡), and 𝐷(𝑡) are known 𝑛×𝑛, 𝑛×𝑚, 𝑝×𝑛, and 𝑝×𝑚 continuous-time complex-valued matrices, 𝑥𝑛 is the state vector, 𝑢𝑚 is the control input, 𝐸𝑘 and 𝐹𝑘 are complex 𝑛×𝑛 and 𝑛×𝑚 constant matrices, respectively, 𝑦𝑝 is the output, 𝐽=[𝑡0,+), Δ𝑥(𝑡𝑘)=𝑥(𝑡+𝑘)𝑥(𝑡𝑘), where 𝑥(𝑡+𝑘)=lim0+𝑥(𝑡𝑘+), 𝑥(𝑡𝑘)=𝑥(𝑡𝑘)=lim0+𝑥(𝑡𝑘) with discontinuity points 𝑡0<𝑡1<𝑡2<<𝑡𝑘<,lim𝑘𝑡𝑘=, which implies that the solution of system (2.1) is left-continuous at 𝑡𝑘. It should be noticed that the impulsive instants 𝑡𝑘 can be chosen freely in this paper. We know that 𝑥(𝑡)𝑛, and 𝑛 is a Banach space on the complex field . 𝐴(𝑡)𝔘 where 𝔘=𝔏(𝑛,𝑛) is the bounded 𝑛-linear continuous map. Hence complex impulsive system (2.1) is a special differential-difference equation in Banach space defined on the complex number field . Let 𝐴=𝐴 be the conjugated transpose of the complex matrix 𝐴. 1𝑖=𝑘1𝐴𝑖 stands for the matrix product 𝐴𝑘1𝐴𝑘2𝐴1.

Corresponding to system (2.1), consider the following complex differential equation:̇𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡).(2.2) Suppose that 𝑋(𝑡) is the fundamental solution matrix of system (2.2). Then 𝑋(𝑡,𝑠)=𝑋(𝑡)𝑋1(𝑠), (𝑡,𝑠𝐽) is the transition matrix associated with the matrix 𝐴(𝑡). It is clear that 𝑋(𝑡,𝑡)=𝐼, 𝑋(𝑡,𝜏)𝑋(𝜏,𝑠)=𝑋(𝑡,𝑠) and 𝑋(𝑡,𝑠)=𝑋1(𝑠,𝑡). Now we present the solution expression of complex impulsive system (2.1) which was proved in [10] using ordinary differential equations theory in the complex field.

Lemma 2.1 (see [10]). For 𝑡(𝑡𝑘1,𝑡𝑘], 𝑘=1,2,, the solution of system (2.1) is given by 𝑥(𝑡)=𝑋𝑡,𝑡𝑘11𝑗=𝑘1𝐼+𝐸𝑗𝑋𝑡𝑗,𝑡𝑗1𝑥0+𝑘1𝑖𝑖=1𝑗=𝑘1𝐼+𝐸𝑗𝑋𝑡𝑗,𝑡𝑗1×𝑡𝑖𝑡𝑖1𝑋𝑡𝑖1,𝑠𝐵(𝑠)𝑢(𝑠)𝑑𝑠+𝑘1𝑖𝑖=2𝑗=𝑘1𝐼+𝐸𝑗𝑋𝑡𝑗,𝑡𝑗1𝐹𝑖1𝑢𝑖1+𝐹𝑘1𝑢𝑘1+𝑡𝑡𝑘1𝑋(𝑡,𝑠)𝐵(𝑠)𝑢(𝑠)𝑑𝑠.(2.3)

In the remainder of this paper, we focus our attention on the reachability of time-invariant version of system (2.1) with respect to the continuous-time input 𝑢(𝑡) and observability with respect to the continuous-time output 𝑦(𝑡). The complex linear impulsive system is given bẏ𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑡𝑡𝑘,𝑥𝑡+𝑘𝑡=𝐸𝑥𝑘,𝑥𝑡𝑦(𝑡)=𝐶𝑥(𝑡),+0=𝑥0,(2.4) where 𝐴,𝐵,𝐶,𝐸 are known 𝑛×𝑛, 𝑛×𝑚, 𝑝×𝑛, 𝑛×𝑛 constant complex matrices. Let 𝒯={𝑡1,𝑡2,.} be a countable set of impulse times.

Remark 2.2. System (2.4) is a class of more general linear impulsive systems in the complex field with time-driven impulsive behavior. The system parameter matrices are all complex matrices. It should be noticed that the impulse times could be chosen freely, allowing for a richer interaction between the continuous-time dynamics and the impulsive effects. Hence, with inherent uncertainties, system (2.4) has interesting features in reachability and observability different from that of common impulsive systems. This motivates our current work.

Given an initial time 𝑡0 and final time 𝑡𝑓, Lemma 2.1 gives the solution of (2.4) as follows:𝑥𝑡𝑓=𝑒𝐴𝑀1𝑚=𝑀1𝐸𝑒𝐴𝑚𝑥𝑡0+𝑀2𝑚=1𝑚+1𝑗=𝑀1𝐸𝑒𝐴𝑗𝐸𝑡𝑚𝑡𝑚1𝑒𝐴(𝑡𝑚𝑠)𝐵𝑢(𝑠)𝑑𝑠+𝐸𝑡𝑀1𝑡𝑀2𝑒𝐴(𝑡𝑀1𝑠)+𝐵𝑢(𝑠)𝑑𝑠𝑡𝑀𝑡𝑀1𝑒𝐴(𝑡𝑀𝑠)𝐵𝑢(𝑠)𝑑𝑠,(2.5) where 𝑡𝑓=𝑡𝑀 and 𝑖=𝑡𝑖𝑡𝑖1, 𝑖=1,2,,𝑀. In the subsequent, we proceed to investigate the reachability and observability criteria of complex linear impulsive system (2.4).

3. Geometric Analysis of Reachability

In this section, the main purpose is to characterize the geometric properties of reachability of complex linear impulsive system (2.4) and establish the equivalence between algebraic criteria in known results [10] and the geometric ones obtained here. To discuss the geometric property of reachable set for complex impulsive system (2.4), we first introduce the concept of invariant subspace of complex linear systems.

Consider the following complex linear system:𝑦𝑥𝑡̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),(𝑡)=𝐶𝑥(𝑡),+0=𝑥0.(3.1) For the complex constant matrix 𝐵𝑛×𝑚, denote Im𝐵 as the range of 𝐵 spanned by the columns of 𝐵, that is, Im𝐵={𝑦𝑦=𝐵𝑢,𝑢𝑚}. For a given matrix 𝐴𝑛×𝑛 and a linear space 𝒲𝑛, let 𝐴𝒲 be the minimal 𝐴-invariant subspace containing 𝒲, that is, 𝐴𝒲=𝑛1𝑖=0𝐴𝑖𝒲. For simplicity, we denote 𝐴𝐵=𝐴Im𝐵. By [13], for any complex matrices 𝐴 and 𝐵, we have {𝑥𝑥=𝑡𝑡0𝑒𝐴(𝑡𝜏)𝐵𝑢(𝜏)𝑑𝜏,forsomep.c.v.function𝑢,𝑡>𝑡0}=𝐴𝐵, which is the reachable subspace of complex linear system (3.1). Moreover, the equivalence between the algebraic condition for controllability and the geometric one is given as follows. rank (𝐵𝐴𝐵𝐴𝑛1𝐵)=𝑛 is equivalent to Im(𝐵)+𝐴Im(𝐵)++𝐴𝑛1Im(𝐵)=𝑛.

In view of the special structure of the system considered here, definitions about the reachability are introduced first. The state space for complex impulsive system (2.4) is denoted by 𝒳.

Definition 3.1 (Reachable set with fixed final time and fixed impulse times). For complex linear impulsive system (2.4), a nonzero state 𝑥𝑓 is said to be reachable from zero with fixed final time and fixed impulse times, if given 𝑡0,𝑡𝑓>𝑡0 and a set of impulse times 𝒯, there exists a piecewise continuous input 𝑢(𝑡),𝑡[𝑡0,𝑡𝑓], such that the system is driven from 𝑥(𝑡0)=0 to 𝑥(𝑡𝑓)=𝑥𝑓. The set of reachable states with fixed final time 𝑡𝑓 and fixed impulse times 𝒯 is denoted by xed(𝑡0,𝑡𝑓,𝒯).

Definition 3.2 (Reachable set with free final time and free impulse times). For system (2.4), a nonzero state 𝑥𝑓 is said to be reachable from zero with free final time and free impulse times, if given 𝑡0, there exists 𝑡𝑓>𝑡0, a set of impulse times 𝒯 and a piecewise continuous input 𝑢(𝑡),𝑡[𝑡0,𝑡𝑓], such that the system is driven from 𝑥(𝑡0)=0 to 𝑥(𝑡𝑓)=𝑥𝑓. The set of reachable states with free final time 𝑡𝑓 and free impulse times 𝒯 is denoted by free.

From the definitions, we obtain free=𝒯𝑡𝑓>𝑡0xed(𝑡0,𝑡𝑓,𝒯). Given an impulse times set 𝒯, by (2.5) and Definition 3.1, xed(𝒯) is given byxed=𝑒𝐴𝑀1𝑚=𝑀1𝐸𝑒𝐴𝑚𝑥𝑡0+𝑀2𝑚=1𝑚+1𝑗=𝑀1𝐸𝑒𝐴𝑗𝐸𝐴𝐵+𝐸𝐴𝐵+𝐴𝐵.(3.2) In [7], it was pointed out that for real impulsive systems, the reachable set does not necessarily constitute a subspace. Thus, for complex linear impulsive system (2.4), free may be a subset instead of subspace of the state space. This fact will be clarified in the following example.

Example 3.3. Consider complex linear impulsive system (2.4) with 000𝐴=0010000000000000,𝐵=1+2𝑖,𝐸=0000001001+𝑖001000.(3.3) It is clear that 𝐴𝐵=Im(𝐵). For the case 𝑡0<𝑡1<𝑡𝑓, xed𝑡0,𝑡𝑓=,𝒯(1+𝑖)(1+2𝑖)1001+2𝑖(1+𝑖)(1+2𝑖)000.(3.4) For any even number 𝑘2 and 𝑡0<𝑡1<<𝑡𝑘<𝑡𝑓, it yields that xed𝑡0,𝑡𝑓=,𝒯(1+𝑖)𝑘10((1+2𝑖)001+𝑖)𝑘2(1+2𝑖)1+2𝑖(1+𝑖)𝑘(1+2𝑖)𝑘000(1+𝑖)𝑘2(1+2𝑖)𝑘10.(3.5) Therefore, when the final time and the impulse times are fixed, the system can reach at most a three-dimensional complex subspace of the state space. It follows that when only two impulse times are required, that is, 0<𝑡1<𝑡2<𝑡𝑓, free can be characterized as follows: free=1,2>0xed𝑡0,𝑡𝑓=,𝒯1,2>00(Im(1+𝑖)(1+2𝑖)001+2𝑖)1+2𝑖2𝑖(1+2𝑖)2000(1+2𝑖)10.(3.6) For a vector given by [𝑥=𝑎𝑏𝑐𝑑]𝑇(𝑎,𝑏,𝑐,𝑑0), it can be represented by the linear combination of elements in free while it is not included in free. It should be noticed that the subspace spanned by the reachable set with free impulse times is the entire complex state space.

Example 3.3 motivates us to present a new concept, span reachability, for complex impulsive system (2.4).

Definition 3.4 (Span Reachability). For complex impulsive system (2.4), the subspace spanned by the elements of free is denoted by span. A complex impulsive system for which span=𝑛 is said to be span reachable.

In the following, the explicit construction of span reachable set is proposed and its property is discussed. Denote the following subspaces sequences:𝒲0=𝐴𝐵,𝒲𝑚=𝐴𝐸𝒲𝑚1𝒱,𝑚1,𝑚=𝑚𝑖=0𝒲𝑖,𝑚=0,1,2,.(3.7) It is clear that 𝒱0𝒱1𝒱𝑚1𝒱𝑚, dim𝒱𝑚<. If there exists an integer 𝑚>0 such that 𝒱𝑚=𝒱𝑚+1, by the construction of 𝒱𝑚, it is easy to verify 𝒱𝑚=𝒱𝑚+1=𝒱𝑚+2=. This implies that the sequence {𝒱𝑚,𝑚=1,2,} converges to 𝒱𝑛. For the proof of the main results, a Lemma is presented first. The proof is similar to that of Lemma  2 in [12]. Thus, we omit it here.

Lemma 3.5. Given a complex matrix 𝐴𝑛×𝑛, for almost 𝑇, one has 𝐴𝒲=𝑒𝐴𝑇𝒲.

Theorem 3.6. For complex linear impulsive system (2.4), one has span=𝒱𝑛.(3.8)

Proof. For any 𝑥𝑓free, by (3.2) and the property of invariant subspace, we have 𝑥𝑓𝒱𝑛. Then span𝒱𝑛. Next we prove the reverse inclusion. From Lemma 3.5, there exists >0 such that sequence (3.7) can be redefined as follows: 𝒲0=𝐴𝐵,𝒲𝑚=𝑒𝐴𝐸𝒲𝑚1,𝑚1.(3.9) Using the property of invariant subspace, (3.9) can be rewritten as 𝒲0=𝐴𝐵,𝒲𝑚=𝑒𝐴𝑒𝐴𝐸𝒲𝑚1,𝑚1,(3.10) which implies that 𝒱𝑛 has the following form: 𝒱𝑛=𝐴𝐵+𝑛𝑚=1𝑙𝑚,,𝑙2,𝑙1{1,2,,𝑛}𝑒𝐴𝑙𝑚𝑒𝐸𝐴𝑙1𝐸𝐴𝐵.(3.11) Denote an impulse times set to be {𝑙1,𝑙2,,𝑙𝑛}. It is easy to get that 𝐴𝐵+𝑛𝑚=1[𝑒𝐴]𝑙𝑚𝐸[𝑒𝐴]𝑙1𝐸𝐴𝐵xed. Hence, we obtain 𝒱𝑛free. Since any element of span can be expressed as a linear combination of elements from free, we conclude that 𝒱𝑛span. This completes the proof.

Remark 3.7. From Definition 3.4 and Theorem 3.6, it can be found that if 𝒱𝑛=𝑛, system (2.4) is span reachable. For fixed final time and impulse times, if 𝐴𝐵=𝑛, which implies that xed constitutes the entire space, then we know that rank(𝐵,𝐴𝐵,,𝐴𝑛1𝐵)=𝑛. From Theorem  3 in [10], the above condition indicates that system (2.4) is controllable. Hence, when reduced to linear systems, the algebraic condition (3.11) in [10] and the geometric criterion 𝐴𝐵=𝑛 are equivalent in checking the reachability and controllability of system (2.4). When reduced to complex linear impulsive systems with fixed impulse times and 𝐴𝐸=𝐸𝐴, simple computation follows that the conditions for reachability and controllability are equivalent. While in this paper, we consider a more general system with time-driven impulses, and a new concept, span reachability is introduced. Hence, the derived conditions in this paper and the known literature [10] cannot be compared directly.

The concept of invariant subspace is fundamental to a geometric analysis of linear time-invariant systems. The invariance facilitates the investigation of system control problems such as disturbance decoupling, output stabilization, output regulation, and structure stability. Hence, we develop the invariant subspace characterization of the span reachable set span for complex linear time-driven impulsive systems. A follow-up question is that whether the span reachable set span is an invariant subspace of system (2.4). The invariant subspace of complex impulsive systems (2.4) is defined as follows.

Definition 3.8. For complex impulsive system (2.4) with 𝑢(𝑡)0, 𝒱 is an invariant subspace if for any initial time 𝑡0 and any set of impulse times 𝒯, 𝑥(0)𝒱 implies 𝑥(𝑡)𝒱, 𝑡𝑡0.

Generalizing Lemma  4.2 in [7] to the complex case, we conclude that for complex linear impulsive systems, 𝒱 is an invariant subspace if and only if 𝐴𝒱𝒱, 𝐸𝒱𝒱. Now, for system (2.4), we relate span to the infimal invariant subspace 𝐴,𝐸𝐵 containing Im𝐵.

Theorem 3.9. For complex linear impulsive system (2.4), one has span=𝒱𝑛=𝐴,𝐸𝐵.(3.12)

Proof. First, we prove that 𝒱𝑛𝐴,𝐸𝐵. From (3.7), it is obvious that Im𝐵𝒲0𝒱𝑛, 𝐴𝒲0=𝐴𝐴𝐵𝐴𝐵=𝒲0, 𝐴𝒲𝑚=𝐴𝐴𝐸𝒲𝑚1𝐴𝐸𝒲𝑚1=𝒲𝑚, 𝑚1; 𝐸𝒲𝑚𝐴𝐸𝒲𝑚=𝒲𝑚+1𝒱𝑛, 𝑚0. Thus 𝒱𝑛 is an invariant subspace containing Im𝐵. Since 𝐴,𝐸𝐵 is the infimal one, we obtain that 𝒱𝑛𝐴,𝐸𝐵.
Next, we prove that 𝒱𝑛𝐴,𝐸𝐵. Since 𝐴,𝐸𝐵 is the infimal invariant subspace containing Im𝐵, we get Im𝐵𝐴,𝐸𝐵, 𝐴𝑖Im𝐵𝐴,𝐸𝐵, 𝑖=1,,𝑛1. Then 𝒲0𝐴,𝐸𝐵 and 𝐸𝒲0𝐴,𝐸𝐵. By the same reasoning, the fact that 𝐴𝑖𝐸𝒲0𝐴,𝐸𝐵 implies that 𝒲1𝐴,𝐸𝐵,  𝑖=0,,𝑛1. Similarly, for 𝑚>1, 𝒲𝑚𝐴,𝐸𝐵, which means that 𝑛𝑖=0𝒲𝑖=𝒱𝑛𝐴,𝐸𝐵. The proof is completed.

Example 3.10. Consider complex linear impulsive system (2.4) with the same coefficient matrices as that in Example 3.3. Now we modify the matrix 𝐸 as follows 𝐸=0001000011+𝑖000(1+𝑖)00.(3.13) Using the construction proposed in (3.7), we have 𝒲0=[0(1+2𝑖)00]𝑇, 𝒲1=𝐴𝐸𝒲0=[000(1+2𝑖)]𝑇, 𝒲2=𝐴𝐸𝒲1=[(1+2𝑖)000]𝑇 and 𝒲3=𝐴𝐸𝒲2=[00(1+2𝑖)0]𝑇. It can be easily verified that 𝒱4 spans the entire complex state space 4. Hence, system (2.4) with the above matrices is span reachable. Moreover, form this example, we can find that the explicit construction (3.7) helps us to derive the span reachable set easily.

4. Geometric Characterization of Observability

In this section, we present the geometric characterization of the unobservability of complex linear impulsive system (2.4). For convenience, the unobservable set of complex linear systems and its geometric property are introduced first. For a matrix 𝐶𝑚×𝑛, let 𝒦 be the kernel of 𝐶, that is, 𝒦Ker𝐶={𝑥𝑛𝐶𝑥=0}. Given a matrix 𝐴𝑛×𝑛 and a linear space 𝑛, the largest 𝐴-invariant subspace contained in is given by 𝐴=𝐴1𝐴2𝐴𝑛+1 which is the unobservable subspace for complex system (3.1) when =𝒦, where 𝐴1 denotes the inverse image of subspace . Also we have Ker(𝑀𝐴)=𝐴1Ker(𝑀) [13]. We introduce the following definitions of unobservability.

Definition 4.1 (Unobservable set with finite intervals and fixed impulse times). For complex impulsive system (2.4), a state 𝑥0𝒳 is said to be unobservable on [𝑡0,𝑡𝑓] with fixed impulse times, if given 𝑡𝑓>𝑡0, impulse times set 𝒯 and 𝑥0=𝑥(𝑡0), the output 𝑦(𝑡) is identically equal to zero for all 𝑡[𝑡0,𝑡𝑓]. The set of unobservable states with finite interval and fixed impulse times is denoted by 𝒬xed.

Definition 4.2 (Unobservable set with free impulse times). For complex impulsive system (2.4), a state 𝑥0𝒳 is said to be unobservable on [𝑡0,𝑡𝑓] with free impulse time, if given 𝑡0, 𝑥0=𝑥(𝑡0) yields a response 𝑦(𝑡) that is identically equal to zero for all 𝑡𝑡0 and all impulse times sets 𝒯. The set of these unobservable states is denoted by 𝒬free. System (2.4) is observable if 𝒬free={0}.

By the above definitions, we have 𝒬free=𝒯𝑡𝑓>𝑡0𝒬xed(𝑡𝑓,𝒯).

It is easy to see from Definitions 4.1 and 4.2, the observability of complex linear impulsive system (2.4) is equivalent to that of zero-input complex impulsive system. In this way, given an impulse times set 𝒯 and 𝑥0𝒳, the output 𝑦(𝑡) is given by𝑦(𝑡)=𝐶𝑒𝐴(𝑡𝑡0)𝑥0𝑡,𝑡0,𝑡1,𝐶𝑒𝐴(𝑡𝑡𝑚1)1𝑗=𝑚1𝐸𝑒𝐴𝑗𝑥0𝑡,𝑡𝑚1,𝑡𝑚,2𝑚𝑀.(4.1) Denote the following subspace sequences:𝒪0=𝒦𝐴,𝒪𝑚=𝐸1𝒪𝑚1𝒫𝐴,𝑚1,𝑚=𝑚𝑖=0𝒪𝑖,𝑚=0,1,2,.(4.2) Similar to the discussion about 𝒱𝑚, the sequence {𝒫𝑚,𝑚=0,1,} converges to 𝒫𝑛.

Theorem 4.3. For complex linear impulsive system (2.4), one has 𝒬free=𝒫𝑛.

Proof. For an initial state 𝑥0𝒫𝑛 and a given impulsive times set 𝒯(𝑡0,𝑡𝑓)={𝑡0,𝑡1,,𝑡𝑀}, it is obvious that from 𝑥0𝒪0=Ker(𝐶)𝐴1Ker(𝐶)𝐴(𝑛1)Ker(𝐶), we have 𝐶𝐴𝑘𝑥0=0, 𝑘=0,1,,𝑛1 which implies that 𝐶𝑒𝐴(𝑡𝑡0)𝑥0=0, 𝑡(𝑡0,𝑡1]. Since 𝑥0𝒪1=𝐸1𝒦𝐴𝐴, the definition of the largest invariant subspace implies that 𝑥0𝑛1𝑘=0𝐴𝑘𝐸1𝒦𝐴. Then 𝐸𝐴𝑘𝑥0𝒦𝐴, 𝑘=0,1,,𝑛1. From the property of matrix exponent, it follows that 𝐸𝑒𝐴1𝑥0𝒦𝐴, which means that 𝐶𝑒𝐴(𝑡𝑡1)𝐸𝑒𝐴1𝑥0=0,𝑡(𝑡1,𝑡2]. By the same reasoning, we get 𝐶𝑒𝐴(𝑡𝑡𝑚1)1𝑗=𝑚1(𝐸𝑒𝐴𝑗)𝑥0=0, 𝑡(𝑡𝑚1,𝑡𝑚], 2𝑚𝑀. It means that the output 𝑦(𝑡)0, 𝑡[𝑡0,𝑡𝑓]. From Definition 4.2, we conclude that 𝑥0𝒬free and 𝒫𝑛𝒬free.
On the other hand, if 𝑥0𝒬free, then for any impulse times set 𝒯, 0=𝑦(𝑡)=𝐶𝑒𝐴(𝑡𝑡0)𝑥0𝑡,𝑡0,𝑡1,𝐶𝑒𝐴(𝑡𝑡𝑚1)1𝑗=𝑚1𝐸𝑒𝐴𝑗𝑥0𝑡,𝑡𝑚1,𝑡𝑚,2𝑚𝑀.(4.3) The first equation in (4.3) shows that 𝑥0Ker(𝐶)𝐴=𝒪0. If 𝑚=2, (4.3) becomes 𝐶𝑒𝐴(𝑡𝑡1)𝐸𝑒𝐴1𝑥0=0, 𝑡(𝑡1,𝑡2], then it follows from the definition of unobservable subspace that 𝑥0Ker𝐶𝑒𝐴(𝑡𝑡1)𝐸𝑒𝐴1=Ker𝐶𝑒𝐴(𝑡𝑡1)𝐸=𝐸𝐴1Ker𝐶𝑒𝐴(𝑡𝑡1)=𝐸𝐴1.𝒦𝐴𝐴(4.4) Repeating the same process, we obtain 𝑥0𝒪𝑖, 𝑖{0,1,,𝑛}. This means that 𝑥0𝒫𝑛 and 𝒬free𝒫𝑛. The proof is completed.

From Definition 4.2 and Theorem 4.3, we can see that if 𝒫𝑛={0}, system (2.4) is observable. Similarly, we aim to show the invariance of the unobservable set with free impulse times 𝒬free. Denote the supremal invariant subspace of system (2.4) contained in 𝒦 to be 𝒦𝐴,𝐸.

Theorem 4.4. For complex linear impulsive system (2.4), one has 𝒬free=𝒫𝑛=𝒦𝐴,𝐸.(4.5)

Proof. First, we prove that 𝒫𝑛𝒦𝐴,𝐸. Given any 𝑥0𝒦𝐴,𝐸, since 𝒦𝐴,𝐸 is the largest invariant subspace contained in Ker𝐶, we have 𝐴𝑖𝑥0𝒦𝐴,𝐸Ker𝐶, 𝑖=0,,𝑛1, which means that 𝑥0𝒪0=𝑛1𝑖=0𝐴𝑖Ker𝐶. Furthermore, 𝐴𝑗𝐸𝐴𝑖𝑥0𝒦𝐴,𝐸Ker𝐶, 𝑖,𝑗=0,,𝑛1, which means that 𝑥0𝒪1=𝑛1𝑖=0𝐴𝑖𝐸1𝒪0. By the same deduction, 𝑥0𝒪𝑚 indicates that 𝑥0𝒫𝑛 by the definition of 𝒫𝑛, 𝑚>1. Then we have 𝒫𝑛𝒦𝐴,𝐸.
Next, we prove that 𝒫𝑛𝒦𝐴,𝐸. Given any 𝑥0𝒫𝑛, 𝑥0𝒪0=𝑛1𝑖=0𝐴𝑖Ker𝐶Ker𝐶, then 𝒫𝑛Ker𝐶. Moreover, since 𝒪𝑚 are 𝐴-invariant subspaces, we have 𝐴𝑥0𝐴𝒪𝑚𝒪𝑚, 𝑚=0,1,,𝑛. It is clear that 𝐴𝑥0𝑛𝑚=0𝒪𝑚=𝒫𝑛. On the other hand, the sequence {𝒫𝑚} converges to 𝒫𝑛, which implies that 𝑥0𝒫𝑛=𝒫𝑛+1 and 𝑥0𝒪𝑚, 𝑚=1,2,,𝑛+1. Thus 𝐸𝑥0𝐸𝒪𝑚=𝑛1𝑖=0𝐸𝐴𝑖𝐸1𝒪𝑚1𝐸𝐸1𝒪𝑚1=𝒪𝑚1, 𝑚=1,2,,𝑛+1. This shows that 𝐸𝑥0𝒪𝑚, 𝑚=0,1,,𝑛. In conclusion, we have 𝐸𝑥0𝒫𝑛 and 𝐴𝑥0𝒫𝑛. It means that 𝒫𝑛 is an invariant subspace contained in 𝒦 with respect to matrices 𝐴 and 𝐸. Since 𝒦𝐴,𝐸 is the largest one, we conclude that 𝒫𝑛𝒦𝐴,𝐸. This completes the proof.

Remark 4.5. When the systems in this paper and [10] are reduced to complex linear systems, if Ker(𝐶)𝐴={0}, from [13], we know that (Ker(𝐶)𝐴)=𝐴𝐶=𝑛, that is, rank(𝑆)=𝑛, which means that system (2.4) is observable, where 𝑆=[𝐶𝐴𝐶(𝐴𝑛1)𝐶].
Thus the geometric condition is equivalent to the algebraic one in Theorem  5(i) in [10] for the observability of system (2.4). When reduced to complex linear impulsive systems with 𝐴𝐸=𝐸𝐴, simple computation follows that the algebraic condition in [10] and the geometric one here for the observability are equivalent.

Example 4.6. Consider complex linear impulsive system (2.4) with 𝐴=0010000000000000,𝐸=0001100101000000,𝐶=0100.(4.6) It is easy to get that 𝒪0=ker(𝐶)=1+𝑖0000001+𝑖0001+𝑖. Simple computations from (4.2) yield that 𝒪1=3𝑖,𝑗=1ker(𝐶𝐴𝑖𝐸𝐴𝑗)=001+𝑖00001+𝑖 and 𝒪2=1+𝑖0001+𝑖0001+𝑖000. Then we have 𝒪0𝒪1𝒪2=𝒬free={0}, which means that the system is observable.

5. Conclusion

In this paper, the reachability and observability have been investigated for a class of time-driven complex linear impulsive systems which allow free impulsive times. It has been shown that traditional geometric approach may be not sufficient to study the reachability and observability for such systems. Hence, a new geometric analysis method is developed. New concepts of the reachability and observability have been introduced. Sufficient and necessary conditions for the span reachability and observability of such systems have been established. Moreover, geometric properties of span reachable set and unobservable set have been studied. The equivalence between the algebraic conditions in known results [10] and the geometric ones obtained here has been established. Numerical examples have been provided to show the explicit construction of the reachable subspace and unobservable subspace and easily-verifiable conditions for the reachability and observability of complex linear impulsive systems.

Acknowledgments

The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the paper. Project supported by NNSF of China under Grants 60874027, 60114039, the Fundamental Research Funds for the Central Universities of China, TDSI (TDSI/08-004/1A), TL@NUS (TL/CG/2009/1), and Shanghai Municipal Education Commission Research Funding under Grants gjd10009 and A-3500-11-10.

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