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

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 [1–5] 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 [6–11] 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 [12–14]. The controllability and observability of various hybrid systems have been extensively investigated using different approaches such as geometric analysis [7–9], 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 [7–9] 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, ℝ^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 [18–22]. 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 by𝑥̇(𝑡)=Δ𝑥𝐴(𝑡𝑡_𝑘)𝑥(𝑡)+𝐵(𝑡)𝑢(𝑡),𝑡≠𝑡_𝑘,𝑦(𝑡)=𝐶𝑥=(𝑡)𝑡𝐸𝑥_𝑘(𝑥𝑡)+𝑡_𝑘𝐷+(𝑡𝐹)𝑢_𝑘𝑢(𝑡_𝑘),,₊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, 𝑥∈ℂ^𝑛 is the state vector, 𝑢∈ℂ^𝑚 is the control input, 𝐸_𝑘 and 𝐹_𝑘 are complex 𝑛×𝑛 and 𝑛×𝑚 constant matrices, respectively, 𝑦∈ℂ^𝑝 is the output, 𝐽=[𝑡₀,+∞), Δ𝑥(𝑡_𝑘)=𝑥(𝑡^+𝑘)−𝑥(𝑡^−𝑘), where 𝑥(𝑡^+𝑘)=lim_ℎ→0⁺𝑥(𝑡_𝑘+ℎ), 𝑥(𝑡_𝑘)=𝑥(𝑡^−𝑘)=lim_ℎ→0⁺𝑥(𝑡_𝑘−ℎ) with discontinuity points 𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯,lim_𝑘→∞𝑡_𝑘=∞, which implies that the solution of system (2.1) is left-continuous at 𝑡_𝑘. It should be noticed that the impulsive instants 𝑡_𝑘 can be chosen freely in this paper. We know that 𝑥(𝑡)∶ℝ→ℂ^𝑛, and ℂ^𝑛 is a Banach space on the complex field ℂ. 𝐴(𝑡)∶ℝ→𝔘 where 𝔘=𝔏(ℂ^𝑛,ℂ^𝑛) is the bounded ℂ^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 ℂ. Let 𝐴^∗=𝐴^⊤ be the conjugated transpose of the complex matrix 𝐴. ∏^{1𝑖=𝑘−1}𝐴_𝑖 stands for the matrix product 𝐴_𝑘−1𝐴_𝑘−2⋯𝐴₁.

Corresponding to system (2.1), consider the following complex differential equation:𝑥̇(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)𝑋⁻¹(𝑠), (𝑡,𝑠∈𝐽) 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)=𝑋⁻¹(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∈(𝑡_𝑘−1,𝑡_𝑘], 𝑘=1,2,…, the solution of system (2.1) is given by 𝑥(𝑡)=𝑋𝑡,𝑡_𝑘−1¹_𝑗_=𝑘−1𝐼+𝐸_𝑗𝑋𝑡_𝑗,𝑡_𝑗−1𝑥₀+^𝑘−1_𝑖𝑖=1_{𝑗=𝑘−1}𝐼+𝐸_𝑗𝑋𝑡_𝑗×_{𝑡𝑖𝑡𝑖−1},𝑡_𝑗−1𝑋𝑡_𝑖−1,𝑠𝐵(𝑠)𝑢(𝑠)𝑑𝑠+^𝑘−1_𝑖𝑖=2_{𝑗=𝑘−1}𝐼+𝐸_𝑗𝑋𝑡_𝑗,𝑡_𝑗−1𝐹_𝑖−1𝑢_𝑖−1+𝐹_𝑘−1𝑢_𝑘−1+_{𝑡𝑡𝑘−1}𝑋(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 by𝑥̇(𝑡)=𝑥𝐴𝑥𝑡_+𝑘(𝑡)+𝐵𝑢(𝑡),𝑡≠𝑡_𝑘,𝑦𝑥(𝑡)𝑡==𝐶𝐸𝑥𝑥(𝑡𝑡)_𝑘,₊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 𝑡_𝑓, Lemma 2.1 gives the solution of (2.4) as follows:𝑥𝑡_𝑓=𝑒^𝐴ℎ𝑀¹_{𝑚=𝑀−1}𝐸𝑒^𝐴ℎ𝑚𝑥𝑡₀+^𝑀−2_𝑚₌₁^𝑚+1_{𝑗=𝑀−1}𝐸𝑒^𝐴ℎ𝑗𝐸^𝑡𝑚_𝑡𝑚−1+𝐸_{𝑡𝑀−1𝑡𝑀−2}𝑒^{𝐴(𝑡𝑚−𝑠)}𝐵𝑢(𝑠)𝑑𝑠𝑒^{𝐴(𝑡𝑀−1−𝑠)}𝐵𝑢(𝑠)𝑑𝑠+^𝑡𝑀_𝑡𝑀−1𝑒^{𝐴(𝑡𝑀−𝑠)}𝐵𝑢(𝑠)𝑑𝑠,(2.5) where 𝑡_𝑓=𝑡_𝑀 and ℎ_𝑖=𝑡_𝑖−𝑡_𝑖−1, 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:𝑥̇(𝑡)𝑦=𝑥(𝑡𝐴)𝑡𝑥=(𝐶𝑡)𝑥+(𝑡𝐵)𝑢,₊₀(𝑡),=𝑥₀.(3.1) For the complex constant matrix 𝐵∈ℂ^𝑛×𝑚, denote Im𝐵 as the range of 𝐵 spanned by the columns of 𝐵, that is, ℬ≜Im𝐵={𝑦∣𝑦=𝐵𝑢,∀𝑢∈ℂ^𝑚}. For a given matrix 𝐴∈ℂ^𝑛×𝑛 and a linear space 𝒲⊆ℂ^𝑛, let ⟨𝐴∣𝒲⟩ be the minimal 𝐴-invariant subspace containing 𝒲, that is, ⟨𝐴∣𝒲⟩=∑^{𝑛−1𝑖=0}𝐴^𝑖𝒲. For simplicity, we denote ⟨𝐴∣𝐵⟩=⟨𝐴∣Im𝐵⟩. By [13], for any complex matrices 𝐴 and 𝐵, we have {𝑥∶𝑥=∫^𝑡𝑡₀𝑒^{𝐴(𝑡−𝜏)}𝐵𝑢(𝜏)𝑑𝜏,forsomep.c.v.function𝑢,𝑡>𝑡₀}=⟨𝐴∣𝐵⟩, 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 (𝐵𝐴𝐵⋯𝐴^𝑛−1𝐵)=𝑛 is equivalent to Im(𝐵)+𝐴Im(𝐵)+⋯+𝐴^𝑛−1Im(𝐵)=ℂ^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 𝑥_𝑓 is said to be reachable from zero with fixed final time and fixed impulse times, if given 𝑡₀,𝑡_𝑓>𝑡₀ and a set of impulse times 𝒯, there exists a piecewise continuous input 𝑢(𝑡),𝑡∈[𝑡₀,𝑡_𝑓], such that the system is driven from 𝑥(𝑡₀)=0 to 𝑥(𝑡_𝑓)=𝑥_𝑓. The set of reachable states with fixed final time 𝑡_𝑓 and fixed impulse times 𝒯 is denoted by ℛ_fixed(𝑡₀,𝑡_𝑓,𝒯).

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

From the definitions, we obtain ℛ_free=⋃_𝒯⋃_{𝑡𝑓>𝑡0}ℛ_fixed(𝑡₀,𝑡_𝑓,𝒯). Given an impulse times set 𝒯, by (2.5) and Definition 3.1, ℛ_fixed(𝒯) is given byℛ_fixed=𝑒^𝐴ℎ𝑀¹_{𝑚=𝑀−1}𝐸𝑒^𝐴ℎ𝑚𝑥𝑡₀+^𝑀−2_𝑚₌₁^𝑚+1_{𝑗=𝑀−1}𝐸𝑒^𝐴ℎ𝑗𝐸⟨𝐴∣𝐵⟩+𝐸⟨𝐴∣𝐵⟩+⟨𝐴∣𝐵⟩.(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), ℛ_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 𝐴=⎡⎢⎢⎢⎢⎢⎢⎣0000000010000000⎤⎥⎥⎥⎥⎥⎥⎦,𝐵=⎛⎜⎜⎜⎜⎜⎜⎝1+0002𝑖⎞⎟⎟⎟⎟⎟⎟⎠,𝐸=⎡⎢⎢⎢⎢⎢⎢⎣00011000+𝑖01000000⎤⎥⎥⎥⎥⎥⎥⎦.(3.3) It is clear that ⟨𝐴∣𝐵⟩=Im(𝐵). For the case 𝑡₀<𝑡₁<𝑡_𝑓, ℛ_fixed𝑡₀,𝑡_𝑓,𝒯=⎡⎢⎢⎢⎢⎢⎢⎣(1+𝑖)(1+2𝑖)ℎ¹(1+𝑖)00(1+2𝑖)1+0002𝑖⎤⎥⎥⎥⎥⎥⎥⎦.(3.4) For any even number 𝑘≥2 and 𝑡₀<𝑡₁<⋯<𝑡_𝑘<𝑡_𝑓, it yields that ℛ_fixed𝑡₀,𝑡_𝑓,𝒯=⎡⎢⎢⎢⎢⎢⎢⎣(1+𝑖)^𝑘−10(1+2𝑖)(1+𝑖)0_𝑘−20(1+𝑖)_𝑘(1+2𝑖)1+2𝑖(01+2𝑖)ℎ_𝑘(1+𝑖)_𝑘−200(1+2𝑖)ℎ_𝑘−10⎤⎥⎥⎥⎥⎥⎥⎦.(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<𝑡_𝑓, ℛ_free can be characterized as follows: ℛ_free=_ℎ1,ℎ2>0ℛ_fixed𝑡₀,𝑡_𝑓,𝒯=_ℎ1,ℎ2>0Im⎡⎢⎢⎢⎢⎢⎢⎣(21𝑖+(1𝑖)+0(12𝑖+)ℎ2₂𝑖)(1+02𝑖)1+02𝑖0(1+02𝑖)ℎ₁00⎤⎥⎥⎥⎥⎥⎥⎦.(3.6) For a vector given by 𝑥=[^{𝑎𝑏𝑐𝑑}]^𝑇(𝑎,𝑏,𝑐,𝑑≠0), it can be represented by the linear combination of elements in ℛ_free while it is not included in ℛ_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 ℛ_free is denoted by ℛ_span. A complex impulsive system for which ℛ_span=ℂ^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:𝒲𝒱₀=⟨𝐴∣𝐵⟩,𝒲_𝑚=⟨𝐴∣𝐸𝒲_𝑚−1⟩,𝑚≥1,_𝑚=^𝑚_𝑖=0𝒲_𝑖,𝑚=0,1,2,….(3.7) It is clear that 𝒱₀⊆𝒱₁⊆⋯⊆𝒱_𝑚−1⊆𝒱_𝑚, dim𝒱_𝑚<∞. If there exists an integer 𝑚>0 such that 𝒱_𝑚=𝒱_𝑚+1, by the construction of 𝒱_𝑚, it is easy to verify 𝒱_𝑚=𝒱_𝑚+1=𝒱_𝑚+2=…. This implies that the sequence {𝒱_𝑚,𝑚=1,2,…} converges to 𝒱_𝑛. 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 𝐴∈ℂ^𝑛×𝑛, for almost 𝑇∈ℝ, one has ⟨𝐴∣𝒲⟩=⟨𝑒^𝐴𝑇∣𝒲⟩.

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

Proof. For any 𝑥_𝑓∈ℛ_free, by (3.2) and the property of invariant subspace, we have 𝑥_𝑓∈𝒱_𝑛. Then ℛ_span⊆𝒱_𝑛. Next we prove the reverse inclusion. From Lemma 3.5, there exists ℎ>0 such that sequence (3.7) can be redefined as follows: 𝒲₀=⟨𝐴∣𝐵⟩,𝒲_𝑚=⟨𝑒^𝐴ℎ∣𝐸𝒲_𝑚−1⟩,𝑚≥1.(3.9) Using the property of invariant subspace, (3.9) can be rewritten as 𝒲₀=⟨𝐴∣𝐵⟩,𝒲_𝑚=𝑒^𝐴ℎ⟨𝑒^𝐴ℎ∣𝐸𝒲_𝑚−1⟩,𝑚≥1,(3.10) which implies that 𝒱_𝑛 has the following form: 𝒱_𝑛=⟨𝐴∣𝐵⟩+^𝑛_𝑚₌₁_{𝑙𝑚,…,𝑙2,𝑙1∈{1,2,…,𝑛}}𝑒^𝐴ℎ^𝑙𝑚𝐸⋯𝑒^𝐴ℎ^𝑙1𝐸⟨𝐴∣𝐵⟩.(3.11) Denote an impulse times set to be {𝑙₁ℎ,𝑙₂ℎ,…,𝑙_𝑛ℎ}. It is easy to get that ⟨𝐴∣𝐵⟩+∑^𝑛𝑚=1[𝑒^𝐴ℎ]^𝑙𝑚𝐸⋯[𝑒^𝐴ℎ]^𝑙1𝐸⟨𝐴∣𝐵⟩⊆ℛ_fixed. Hence, we obtain 𝒱_𝑛⊆ℛ_free. Since any element of ℛ_span can be expressed as a linear combination of elements from ℛ_free, we conclude that 𝒱_𝑛⊆ℛ_span. This completes the proof.

Remark 3.7. From Definition 3.4 and Theorem 3.6, it can be found that if 𝒱_𝑛=ℂ^𝑛, system (2.4) is span reachable. For fixed final time and impulse times, if ⟨𝐴∣𝐵⟩=ℂ^𝑛, which implies that ℛ_fixed constitutes the entire space, then we know that rank(𝐵,𝐴𝐵,…,𝐴^𝑛−1𝐵)=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 ⟨𝐴∣𝐵⟩=ℂ^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 ℛ_span for complex linear time-driven impulsive systems. A follow-up question is that whether the span reachable set ℛ_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, ∀𝑡≥𝑡₀.

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 𝐴𝒱⊂𝒱, 𝐸𝒱⊂𝒱. Now, for system (2.4), we relate ℛ_span to the infimal invariant subspace ⟨𝐴,𝐸∣𝐵⟩ containing Im𝐵.

Theorem 3.9. For complex linear impulsive system (2.4), one has ℛ_span=𝒱_𝑛=⟨𝐴,𝐸∣𝐵⟩.(3.12)

Proof. First, we prove that 𝒱_𝑛⊇⟨𝐴,𝐸∣𝐵⟩. From (3.7), it is obvious that Im𝐵⊆𝒲₀⊆𝒱_𝑛, 𝐴𝒲₀=𝐴⟨𝐴∣𝐵⟩⊆⟨𝐴∣𝐵⟩=𝒲₀, 𝐴𝒲_𝑚=𝐴⟨𝐴∣𝐸𝒲_𝑚−1⟩⊆⟨𝐴∣𝐸𝒲_𝑚−1⟩=𝒲_𝑚, 𝑚≥1; 𝐸𝒲_𝑚⊆⟨𝐴∣𝐸𝒲_𝑚⟩=𝒲_𝑚+1⊆𝒱_𝑛, 𝑚≥0. Thus 𝒱_𝑛 is an invariant subspace containing Im𝐵. Since ⟨𝐴,𝐸∣𝐵⟩ is the infimal one, we obtain that 𝒱_𝑛⊇⟨𝐴,𝐸∣𝐵⟩.
Next, we prove that 𝒱_𝑛⊆⟨𝐴,𝐸∣𝐵⟩. Since ⟨𝐴,𝐸∣𝐵⟩ is the infimal invariant subspace containing Im𝐵, we get Im𝐵⊆⟨𝐴,𝐸∣𝐵⟩, 𝐴^𝑖Im𝐵⊆⟨𝐴,𝐸∣𝐵⟩, 𝑖=1,…,𝑛−1. Then 𝒲₀⊆⟨𝐴,𝐸∣𝐵⟩ and 𝐸𝒲₀⊆⟨𝐴,𝐸∣𝐵⟩. By the same reasoning, the fact that 𝐴^𝑖𝐸𝒲₀⊆⟨𝐴,𝐸∣𝐵⟩ implies that 𝒲₁⊂⟨𝐴,𝐸∣𝐵⟩,  𝑖=0,…,𝑛−1. Similarly, for 𝑚>1, 𝒲_𝑚⊆⟨𝐴,𝐸∣𝐵⟩, which means that ∑^𝑛𝑖=0𝒲_𝑖=𝒱_𝑛⊆⟨𝐴,𝐸∣𝐵⟩. 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=⎡⎢⎢⎢⎢⎢⎢⎣0010(1100++𝑖𝑖)00001000⎤⎥⎥⎥⎥⎥⎥⎦.(3.13) Using the construction proposed in (3.7), we have 𝒲₀=[0(1+2𝑖)00]^𝑇, 𝒲₁=⟨𝐴∣𝐸𝒲₀⟩=[000(1+2𝑖)]^𝑇, 𝒲₂=⟨𝐴∣𝐸𝒲₁⟩=[(1+2𝑖)000]^𝑇 and 𝒲₃=⟨𝐴∣𝐸𝒲₂⟩=[00(1+2𝑖)0]^𝑇. It can be easily verified that 𝒱₄ spans the entire complex state space ℂ⁴. 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 𝐶∈ℂ^𝑚×𝑛, let 𝒦 be the kernel of 𝐶, that is, 𝒦≜Ker𝐶={𝑥∈ℂ^𝑛∣𝐶𝑥=0}. Given a matrix 𝐴∈ℂ^𝑛×𝑛 and a linear space ℳ⊆ℂ^𝑛, the largest 𝐴-invariant subspace contained in ℳ is given by ⟨ℳ∣𝐴⟩∶=ℳ∩𝐴⁻¹ℳ∩𝐴⁻²ℳ∩⋯∩𝐴^−𝑛+1ℳ which is the unobservable subspace for complex system (3.1) when ℳ=𝒦, where 𝐴⁻¹ℳ denotes the inverse image of subspace ℳ. Also we have Ker(𝑀𝐴)=𝐴⁻¹Ker(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 𝑥₀∈𝒳 is said to be unobservable on [𝑡₀,𝑡_𝑓] with fixed impulse times, if given 𝑡_𝑓>𝑡₀, impulse times set 𝒯 and 𝑥₀=𝑥(𝑡₀), the output 𝑦(𝑡) is identically equal to zero for all 𝑡∈[𝑡₀,𝑡_𝑓]. The set of unobservable states with finite interval and fixed impulse times is denoted by 𝒬_fixed.

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

By the above definitions, we have 𝒬_free=⋂_𝒯⋂_{𝑡𝑓>𝑡0}𝒬_fixed(𝑡_𝑓,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 𝑥₀∈𝒳, the output 𝑦(𝑡) is given by𝑦(𝑡)=⎧⎪⎨⎪⎩𝐶𝑒^{𝐴(𝑡−𝑡0)}𝐶𝑒𝑥₀,_{𝐴(𝑡−𝑡𝑚−1)1}𝑡∈𝑡₀,𝑡₁,_𝑗=_𝑚−1𝐸𝑒^𝐴ℎ𝑗𝑥₀,𝑡∈𝑡_𝑚−1,𝑡_𝑚,2≤𝑚≤𝑀.(4.1) Denote the following subspace sequences:𝒫𝒪₀=⟨𝒦∣𝐴⟩,𝒪_𝑚=𝐸⁻¹𝒪_𝑚−1∣𝐴,𝑚≥1,_𝑚=^𝑚_𝑖=0𝒪_𝑖,𝑚=0,1,2,….(4.2) Similar to the discussion about 𝒱_𝑚, the sequence {𝒫_𝑚,𝑚=0,1,…} converges to 𝒫_𝑛.

Theorem 4.3. For complex linear impulsive system (2.4), one has 𝒬_free=𝒫_𝑛.

Proof. For an initial state 𝑥₀∈𝒫_𝑛 and a given impulsive times set 𝒯∩(𝑡₀,𝑡_𝑓)={𝑡₀,𝑡₁,…,𝑡_𝑀}, it is obvious that from 𝑥₀∈𝒪₀=Ker(𝐶)∩𝐴⁻¹Ker(𝐶)∩⋯∩𝐴^{−(𝑛−1)}Ker(𝐶), we have 𝐶𝐴^𝑘𝑥₀=0, 𝑘=0,1,…,𝑛−1 which implies that 𝐶𝑒^{𝐴(𝑡−𝑡0)}𝑥₀=0, 𝑡∈(𝑡₀,𝑡₁]. Since 𝑥₀∈𝒪₁=⟨𝐸⁻¹⟨𝒦∣𝐴⟩∣𝐴⟩, the definition of the largest invariant subspace implies that 𝑥₀∈⋂^{𝑛−1𝑘=0}𝐴^−𝑘𝐸⁻¹⟨𝒦∣𝐴⟩. Then 𝐸𝐴^𝑘𝑥₀∈⟨𝒦∣𝐴⟩, 𝑘=0,1,…,𝑛−1. From the property of matrix exponent, it follows that 𝐸𝑒^𝐴ℎ1𝑥₀∈⟨𝒦∣𝐴⟩, which means that 𝐶𝑒^{𝐴(𝑡−𝑡1)}𝐸𝑒^𝐴ℎ1𝑥₀=0,𝑡∈(𝑡₁,𝑡₂]. By the same reasoning, we get 𝐶𝑒^{𝐴(𝑡−𝑡𝑚−1)}∏^{1𝑗=𝑚−1}(𝐸𝑒^𝐴ℎ𝑗)𝑥₀=0, 𝑡∈(𝑡_𝑚−1,𝑡_𝑚], 2≤𝑚≤𝑀. It means that the output 𝑦(𝑡)≡0, 𝑡∈[𝑡₀,𝑡_𝑓]. From Definition 4.2, we conclude that 𝑥₀∈𝒬_free and 𝒫_𝑛⊆𝒬_free.
On the other hand, if 𝑥₀∈𝒬_free, then for any impulse times set 𝒯, 0=𝑦(𝑡)=⎧⎪⎨⎪⎩𝐶𝑒^{𝐴(𝑡−𝑡0)}𝐶𝑒𝑥₀,_{𝐴(𝑡−𝑡𝑚−1)1}𝑡∈𝑡₀,𝑡₁,_𝑗=_𝑚−1𝐸𝑒^𝐴ℎ𝑗𝑥₀,𝑡∈𝑡_𝑚−1,𝑡_𝑚,2≤𝑚≤𝑀.(4.3) The first equation in (4.3) shows that 𝑥₀∈⟨Ker(𝐶)∣𝐴⟩=𝒪₀. If 𝑚=2, (4.3) becomes 𝐶𝑒^{𝐴(𝑡−𝑡1)}𝐸𝑒^𝐴ℎ1𝑥₀=0, 𝑡∈(𝑡₁,𝑡₂], then it follows from the definition of unobservable subspace that 𝑥₀∈Ker𝐶𝑒^{𝐴(𝑡−𝑡1)}𝐸𝑒^𝐴ℎ1==𝐸Ker𝐶𝑒^{𝐴(𝑡−𝑡1)}𝐸∣𝐴=𝐸⁻¹Ker𝐶𝑒^{𝐴(𝑡−𝑡1)}∣𝐴₋₁⟨𝒦∣𝐴⟩∣𝐴.(4.4) Repeating the same process, we obtain 𝑥₀∈𝒪_𝑖, 𝑖∈{0,1,…,𝑛}. This means that 𝑥₀∈𝒫_𝑛 and 𝒬_free⊆𝒫_𝑛. The proof is completed.

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

Theorem 4.4. For complex linear impulsive system (2.4), one has 𝒬_free=𝒫_𝑛=⟨𝒦∣𝐴,𝐸⟩.(4.5)

Proof. First, we prove that 𝒫_𝑛⊇⟨𝒦∣𝐴,𝐸⟩. Given any 𝑥₀∈⟨𝒦∣𝐴,𝐸⟩, since ⟨𝒦∣𝐴,𝐸⟩ is the largest invariant subspace contained in Ker𝐶, we have 𝐴^𝑖𝑥₀∈⟨𝒦∣𝐴,𝐸⟩⊆Ker𝐶, 𝑖=0,…,𝑛−1, which means that 𝑥₀∈𝒪₀=⋂^{𝑛−1𝑖=0}𝐴^−𝑖Ker𝐶. Furthermore, 𝐴^𝑗𝐸𝐴^𝑖𝑥₀∈⟨𝒦∣𝐴,𝐸⟩⊆Ker𝐶, 𝑖,𝑗=0,…,𝑛−1, which means that 𝑥₀∈𝒪₁=⋂^{𝑛−1𝑖=0}𝐴^−𝑖𝐸⁻¹𝒪₀. By the same deduction, 𝑥₀∈𝒪_𝑚 indicates that 𝑥₀∈𝒫_𝑛 by the definition of 𝒫_𝑛, 𝑚>1. Then we have 𝒫_𝑛⊇⟨𝒦∣𝐴,𝐸⟩.
Next, we prove that 𝒫_𝑛⊆⟨𝒦∣𝐴,𝐸⟩. Given any 𝑥₀∈𝒫_𝑛, 𝑥₀∈𝒪₀=⋂^{𝑛−1𝑖=0}𝐴^−𝑖Ker𝐶⊆Ker𝐶, then 𝒫_𝑛⊆Ker𝐶. Moreover, since 𝒪_𝑚 are 𝐴-invariant subspaces, we have 𝐴𝑥₀∈𝐴𝒪_𝑚⊆𝒪_𝑚, 𝑚=0,1,…,𝑛. It is clear that 𝐴𝑥₀∈⋂^𝑛𝑚=0𝒪_𝑚=𝒫_𝑛. On the other hand, the sequence {𝒫_𝑚} converges to 𝒫_𝑛, which implies that 𝑥₀∈𝒫_𝑛=𝒫_𝑛+1 and 𝑥₀∈𝒪_𝑚, 𝑚=1,2,…,𝑛+1. Thus 𝐸𝑥₀∈𝐸𝒪_𝑚=⋂^{𝑛−1𝑖=0}𝐸𝐴^−𝑖𝐸⁻¹𝒪_𝑚−1⊆𝐸𝐸⁻¹𝒪_𝑚−1=𝒪_𝑚−1, 𝑚=1,2,…,𝑛+1. This shows that 𝐸𝑥₀∈𝒪_𝑚, 𝑚=0,1,…,𝑛. In conclusion, we have 𝐸𝑥₀∈𝒫_𝑛 and 𝐴𝑥₀∈𝒫_𝑛. It means that 𝒫_𝑛 is an invariant subspace contained in 𝒦 with respect to matrices 𝐴 and 𝐸. Since ⟨𝒦∣𝐴,𝐸⟩ is the largest one, we conclude that 𝒫_𝑛⊂⟨𝒦∣𝐴,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⟩)^⟂=⟨𝐴^⊤∣𝐶^⊤⟩=ℂ^𝑛, that is, rank(𝑆)=𝑛, which means that system (2.4) is observable, where 𝑆=[𝐶^⊤𝐴^⊤𝐶^⊤⋯(𝐴^𝑛−1)^⊤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=⎡⎢⎢⎢⎢⎢⎢⎣0000000010000000⎤⎥⎥⎥⎥⎥⎥⎦,𝐸=⎡⎢⎢⎢⎢⎢⎢⎣0100001000001100⎤⎥⎥⎥⎥⎥⎥⎦,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(𝐶𝐴^𝑖𝐸𝐴^𝑗)=^{001+𝑖00001+𝑖} and 𝒪₂=^{1+𝑖0001+𝑖0001+𝑖000}. Then we have 𝒪₀∩𝒪₁∩𝒪₂=𝒬_free={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.