Controllability Analysis of Linear Discrete Time Systems with Time Delay in State

Abstract

The controllability issues for linear discrete-time systems with delay in state are addressed. By introducing a new concept, the minimum controllability realization index (MinCRI), the characteristic of controllability is revealed. It is proved that the MinCRI of a system with state delay exists and is finite. Based on this result, a necessary and sufficient condition for the controllability of discrete-time linear systems with state delay is established.

1. Introduction

The concept of controllability of dynamical systems was first proposed by Kalman in 1960s [1]. Since then, controllability of dynamical systems has been studied by many authors in various contexts [2–11], because controllability turns out to be a fundamental concept in modern control theory and has dose connections with pole assignment, structure decomposition, quadratic control, and so forth [12, 13]. On the other hand, time delay phenomena are very common in practical systems, for instance, in economic, biological, and physiological systems. Studying the time delay phenomena in control systems has become an important topic in control theory. Chyung studied the controllability for linear time-invariant systems with constant time delay in the control functions. Simple algebraic-type necessary and sufficient criteria are established [3, 4]. However, once the time delay appears in state, the problem becomes much more complex. There are some preliminary results in [5, 6]. However, all these results are not suitable for verification and application.

In this paper, we consider the discrete-time case, the system model described as follows:𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐷𝑥(𝑘−ℎ)+𝐵𝑢(𝑘), (1.1) where 𝑥(𝑘)∈ℝ^𝑛 is the state, 𝑢(𝑘)∈ℝ^𝑝 is the input, 𝐴, 𝐷∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑝 are constant matrices, and positive integer ℎ is the length of the steps of time delay. The initial states 𝑥(−ℎ),𝑥(−ℎ+1),…,𝑥(0) are given arbitrarily.

The controllability discussed here refers to the unconstrained controllability, or say completely controllability.

Definition 1.1 (controllability). System (1.1) is said to be (completely) controllable, if for any initial state 𝑥(−ℎ),𝑥(−ℎ+1),…,𝑥(0) and any final state 𝑥_𝑓, there exist a positive integer 𝑘 and input 𝑢(0),…,𝑢(𝑘−1) such that 𝑥(𝑘)=𝑥_𝑓.

Definition 1.2 (controllability realization index, (CRI)). For system (1.1), if there exists a positive integer 𝐾 such that for any initial state 𝑥(−ℎ),𝑥(1−ℎ),…,𝑥(0) and any terminal state 𝑥_𝑓, there exist input 𝑢(0),…,𝑢(𝐾−1) such that 𝑥(𝐾)=𝑥_𝑓; then one calls 𝐾 the controllability realization index (CRI) of system (1.1). Obviously, if exists, such 𝐾 is not unique, so one calls the smallest 𝐾 among them the minimum controllability realization index (MinCRI).

In this paper, our main concern is the following. Can the controllability of system (1.1) be realized completely in finite steps, or say can the MinCRI of the system be finite?

Here we demonstrate that the answer to this question is yes. And in the case of planar systems, the exact value of the controllability realization index is obtained, and we will prove it is independent of the choices of 𝐴 and 𝐷.

This paper is organized as follows. In Section 2 some concepts are introduced, which will be used in the later discussion. Section 3 contains the main results. Finally, the conclusion is provided in Section 4.

2. Preliminaries

Denote by ℕ the nonnegative integer set, ℝ the real number set, respectively. The matrices 𝐴₁,…,𝐴_𝑁∈ℝ^𝑛×𝑛 are said to be linearly dependent in ℝ^𝑛×𝑛, if there exist scalars 𝑐₁,…,𝑐_𝑁∈ℝ, not all zero, such that ∑^𝑁_𝑖=1𝑐_𝑖𝐴_𝑖=0. In what follows, span{𝐴₁,…,𝐴_𝑁} will be used to denote the linear subspace constructed by the linear combinations of matrices {𝐴₁,…,𝐴_𝑁}.

Definition 2.1 (symmetrical subspace). Given matrices 𝐴,𝐷∈ℝ^𝑛×𝑛, one defines a subset of ℝ^𝑛×𝑛 as follows: 𝑃[𝐴,𝐷]∶=𝑓(𝐴,𝐷)∣𝑓(𝐴,𝐷)=^𝑀_𝑚=1𝑐_𝑚𝐴^𝑖1𝑚,𝐷^𝑗1𝑚,⋯𝐴^{𝑖𝑚𝑚,𝑛}𝐷^{𝑗𝑚𝑚,𝑛},𝑛_𝑚∈ℕ,𝑖_𝑚,1,𝑗_𝑚,1,…,𝑖_{𝑚,𝑛𝑚},𝑗_{𝑚,𝑛𝑚}∈ℕ,0≤𝑖_𝑚,1,𝑗_𝑚,1,…,𝑖_{𝑚,𝑛𝑚},𝑗_{𝑚,𝑛𝑚}𝑐≤𝑛,_𝑚∈ℝ,𝑚=1,…,𝑀;𝑀∈ℕ. (2.1) Then a one-to-one mapping Θ∶𝑃[𝐴,𝐷]→𝑃[𝐴,𝐷] is constructed as follows:

give any 𝑓(𝐴,𝐷)∈𝑃[𝐴,𝐷], supposing 𝑓(𝐴,𝐷)=∑^𝑀_𝑚=1𝑐_𝑚𝐴^𝑖1𝑚,𝐷^𝑗1𝑚,,…,𝐴^{𝑖𝑚𝑚,𝑛}𝐷^{𝑗𝑚𝑚,𝑛}, one defines Θ𝑓(𝐴,𝐷)∶=^𝑀_𝑚=1𝑐_𝑚𝐴^𝑖1𝑚,𝐷^𝑗1𝑚,⋯𝐴^{𝑖𝑚𝑚,𝑛}𝐷^{𝑗𝑚𝑚,𝑛}+𝐷^{𝑗𝑚𝑚,𝑛}𝐴^{𝑖𝑚𝑚,𝑛}⋯𝐷^𝑗1𝑚,𝐴^𝑖1𝑚,. (2.2) Then we get a subspace of 𝑃[𝐴,𝐷] by 𝑃_Sym[𝐴,𝐷]∶=Θ𝑓(𝐴,𝐷)∣∀𝑓(𝐴,𝐷)∈𝑃[𝐴,𝐷]. (2.3) We call 𝑃_Sym[𝐴,𝐷] the symmetrical subspace of 𝑃[𝐴,𝐷].

By Definition 2.1, it is easy to verify that 𝐼,𝐴,𝐷,𝐴𝐷+𝐷𝐴,𝐴𝐷𝐴,𝐷𝐴𝐷∈𝑃_Sym[𝐴,𝐷], where 𝐼 is the identity matrix in ℝ^𝑛×𝑛.

Now, we introduce a matrix sequence {𝐺_𝑘}^∞_𝑘=0∈ℝ^𝑛×𝑛 as follows: 𝐺_𝑘=𝐴^𝑘𝐺,if𝑘=0,1,…,ℎ,_𝑘−1𝐴+𝐺_{𝑘−1−ℎ}𝐷,if𝑘=ℎ+1,ℎ+2,… (2.4)

Then the solution of system (1.1) can be expressed as 𝑥(𝑘+1)=Ψ_𝑘(𝑥(0),…,𝑥(−ℎ))+^𝑘_𝑖=0𝐺_𝑘−𝑖𝐵𝑢(𝑖),𝑘>0, (2.5) where Ψ_𝑘(𝑥(0),…,𝑥(−ℎ)) is the part of the solution with zero input.

Lemma 2.2. Given the matrix sequence {𝐺_𝑘}^∞_𝑘=0 in (2.3), the following statements hold:(a) there are no similar terms between 𝐺_𝑘𝐴 and 𝐺_𝑘−ℎ𝐷;(b) for any 𝐺_𝑘, it can be expressed as 𝐺_𝑘=_{()∀𝑓𝐴,𝐷∈𝑄𝑘[]𝐴,𝐷}𝑓(𝐴,𝐷), (2.6) where 𝑄_𝑘[𝐴,𝐷] is a subset of ℝ^𝑛×𝑛 defined as follows: 𝑄_𝑘[𝐴,𝐷]=𝑓(𝐴,𝐷)∣𝑓(𝐴,𝐷)=𝐴^𝑖1𝐷^𝑗1⋯𝐴^𝑖𝑘𝐷^𝑗𝑘,∀𝑖₁,𝑗₁,…,𝑖_𝑘,𝑗_𝑘∈ℕ,^𝑘_𝑚=1𝑖_𝑚+(ℎ+1)^𝑘_𝑚=1𝑗_𝑚=𝑘; (2.7)(c) for all 𝑘∈ℕ,𝐺_𝑘∈𝑃_Sym[𝐴,𝐷].

Proof. Statement (a) is nearly self-evident because any term from 𝐺_𝑘𝐴 is ended by 𝐴, whereas any term from 𝐺_𝑘−ℎ𝐷 is ended by 𝐷.
To prove the result of statement (b), mathematical induction is invoked.(I)The first ℎ+1 terms of {𝐺_𝑘}^∞_𝑘=0 are 𝐼,𝐴,…,𝐴^ℎ, and it is obvious that 𝑄_𝑘[𝐴,𝐷]={𝐴^𝑘}, for 𝑘=0,1,…,ℎ. Thus, for 𝑘=0,1,…,ℎ, (2.5) holds.(II)Assume that, for 𝑙=𝑘,𝑘+1,…,𝑘+ℎ, 𝐺_𝑙 is expressed in form (2.5). We will prove that 𝐺_𝑘+ℎ+1 can be expressed in form (2.5) as well.
First, we prove that, for any 𝑘, we have 𝑄_𝑘+1[𝐴,𝐷]=𝑄_𝑘[𝐴,𝐷]𝐴𝑄_𝑘−ℎ[𝐴,𝐷]𝐷, (2.8) where the sets 𝑄_𝑘[𝐴,𝐷]𝐴, 𝑄_𝑘−ℎ[𝐴,𝐷]𝐷 are defined as 𝑄_𝑘[𝐴,𝐷]𝐴=𝑓(𝐴,𝐷)𝐴∣𝑓(𝐴,𝐷)∈𝑄_𝑘[,𝑄𝐴,𝐷]_𝑘−ℎ[𝐴,𝐷]𝐷=𝑓(𝐴,𝐷)𝐷∣𝑓(𝐴,𝐷)∈𝑄_𝑘−ℎ[𝐴,𝐷]. (2.9) By the assumption, we have 𝑄_𝑘[]𝐴=𝑓()()𝐴,𝐷𝐴,𝐷𝐴∣𝑓𝐴,𝐷∈𝑄_𝑘[=𝑓()()𝐴,𝐷]𝐴,𝐷𝐴∣𝑓𝐴,𝐷=𝐴^𝑖1𝐷^𝑗1⋯𝐴^𝑖𝑘𝐷^𝑗𝑘,𝑖₁,𝑗₁,…,𝑖_𝑘,𝑗_𝑘∈ℕ,_𝑘𝑚=1𝑖_𝑚+(ℎ+1)^𝑘_𝑚=1𝑗_𝑚,=𝑔()()=𝑘𝐴,𝐷∣𝑔𝐴,𝐷=𝐴^𝑖1𝐷^𝑗1⋯𝐴^𝑖𝑘+1𝐷^𝑗𝑘+1,𝑖₁,𝑗₁,…,𝑖_𝑘,𝑗_𝑘𝑖∈ℕ,_𝑘+1=1,𝑗_𝑘+1=0,^𝑘+1_𝑚=1𝑖_𝑚+(ℎ+1)^𝑘+1_𝑚=1𝑗_𝑚=𝑘+1⊆𝑄_𝑘+1[],𝑄𝐴,𝐷_𝑘−ℎ[]=𝑓()()𝐴,𝐷𝐷,𝐴,𝐷𝐷∣𝑓𝐴,𝐷∈𝑄_𝑘−ℎ[=𝑓()()𝐴,𝐷]𝐴,𝐷𝐷∣𝑓𝐴,𝐷=𝐴^𝑖1𝐷^𝑗1⋯𝐴^{𝑖𝑘−ℎ}𝐷^{𝑗𝑘−ℎ},𝑖₁,𝑗₁,…,𝑖_𝑘−ℎ,𝑗_𝑘−ℎ∈ℕ,_{𝑘−ℎ𝑚=1}𝑖_𝑚+(ℎ+1)^𝑘−ℎ_𝑚=1𝑗_𝑚=𝑔()()=𝑘−ℎ𝐴,𝐷∣𝑔𝐴,𝐷=𝐴^𝑖1𝐷^𝑗1⋯𝐴^𝑖𝑘+1𝐷^𝑗𝑘+1,𝑖₁,𝑗₁,…,𝑖_𝑘−ℎ,𝑗_𝑘−ℎ𝑖∈ℕ,_𝑘−ℎ+1=⋯=𝑖_𝑘+1=0,𝑗_𝑘−ℎ+1=⋯=𝑗_𝑘=0,𝑗_𝑘+1=1,_𝑘+1𝑚=1𝑖_𝑚+(ℎ+1)^𝑘+1_𝑚=1𝑗_𝑚=𝑘+1⊆𝑄_𝑘+1[𝐴,𝐷]. (2.10) Thus, we have 𝑄_𝑘[𝐴,𝐷]𝐴⋃𝑄_𝑘−ℎ[𝐴,𝐷]𝐷⊆𝑄_𝑘+1[𝐴,𝐷].
On the contrary, given any 𝑓(𝐴,𝐷)∈𝑄_𝑘+1[𝐴,𝐷], suppose that 𝑓(𝐴,𝐷)=𝐴^𝑖1𝐷^𝑖1⋯𝐴^𝑖𝑘+1𝐷^𝑗𝑘+1, (2.11) where 𝑖₁,𝑗₁,…,𝑖_𝑘+1,𝑗_𝑘+1∈ℕ, ∑^𝑘+1_𝑚=1𝑖_𝑚+(ℎ+1)∑^𝑘+1_𝑚=1𝑗_𝑚=𝑘+1.
There are two cases: (i) the final term of 𝑓(𝐴,𝐷) is 𝐴^𝑖𝑚, 𝑖_𝑚>0, 1≤𝑚≤𝑘+1, then there exists a matrix polynomial 𝑔(𝐴,𝐷) such that 𝑓(𝐴,𝐷)=𝑔(𝐴,𝐷)𝐴, then it is easy to verify that 𝑔(𝐴,𝐷)∈𝑄_𝑘[𝐴,𝐷], and it follows that 𝑓(𝐴,𝐷)∈𝑄_𝑘[𝐴,𝐷]𝐴; (ii) the final term of 𝑓(𝐴,𝐷) is 𝐷^𝑗𝑚, 𝑗_𝑚>0, 1≤𝑚≤𝑘+1, then there exists a matrix polynomial 𝑔(𝐴,𝐷) such that 𝑓(𝐴,𝐷)=𝑔(𝐴,𝐷)𝐷, then it is easy to verify that 𝑔(𝐴,𝐷)∈𝑄_𝑘−ℎ[𝐴,𝐷], and it follows that 𝑓(𝐴,𝐷)∈𝑄_𝑘−ℎ[𝐴,𝐷]𝐷.
Thus, we have 𝑄_𝑘[𝐴,𝐷]𝐴⋃𝑄_𝑘−ℎ[𝐴,𝐷]𝐷⊇𝑄_𝑘+1[𝐴,𝐷]. Hence, we know that (2.7) holds.
Secondly, since 𝐺_𝑘+ℎ+1=𝐺_𝑘+ℎ𝐴+𝐺_𝑘𝐷, we have 𝐺_𝑘+1+ℎ=_{𝑓()𝐴,𝐷∈𝑄𝑘+ℎ[]𝐴,𝐷}𝑓(𝐴,𝐷)𝐴+_{𝑓()𝐴,𝐷∈𝑄𝑘[]𝐴,𝐷}𝑓()𝐷=𝐴,𝐷_{𝑔()𝐴,𝐷∈𝑄𝑘+ℎ[]𝐴⋃𝑄𝐴,𝐷𝑘[]𝐷𝐴,𝐷}𝑔(𝐴,𝐷)=_{𝑔()𝐴,𝐷∈𝑄𝑘+ℎ+1[]𝐴,𝐷}𝑔(𝐴,𝐷). (2.12) Thus, (2.5) holds for 𝑘+ℎ+1. Hence, statement (b) holds.
As for statement (c), it can be deduced naturally from statement (b). Note that if 𝐴^𝑖1𝐷^𝑗1⋯𝐴^𝑖𝑘𝐷^𝑗𝑘∈𝑄_𝑘[𝐴,𝐷], then 𝐴^𝑖1𝐷^𝑗1⋯𝐴^𝑖𝑘𝐷^𝑗𝑘∈𝑄_𝑘[𝐴,𝐷] as well. So they must appear in 𝐺_𝑘. By Definition 2.1, 𝐺_𝑘∈𝑃_Sym[𝐴,𝐷].

3. Main Results

First, we consider the general case.

Theorem 3.1. System (1.1) is controllable if and only if rank[𝐺₀𝐵,𝐺₁𝐵,…,𝐺_𝑘𝐵,…]=𝑛.

Proof. From (2.4) and Definition 1.1, this theorem holds naturally.

Theorem 3.2. For system (1.1), there exists 𝐾∈ℕ such that rank𝐺₀𝐵,𝐺₁𝐵,…,𝐺_𝑘𝐵,…=rank𝐺₀𝐵,𝐺₁𝐵,…,𝐺_𝐾𝐵. (3.1)

Proof. It is obvious that rank[𝐺₀𝐵,𝐺₁𝐵,…,𝐺_𝐾𝐵]=dim(span{𝐺₀𝐵,𝐺₁𝐵,…,𝐺_𝐾𝐵}). Consider an auxiliary scalar sequence {𝑑_𝑘}^∞_𝑘=0, where 𝑑_𝑘=dim(Span{𝐺₀𝐵,𝐺₁𝐵,…,𝐺_𝑘𝐵}). By definition, 𝑑_𝑘≤𝑑_𝑘+1≤𝑛² for any 𝑘. Hence, there exists a constant 𝑑^∗ such that, lim_𝑘→∞𝑑_𝑘=𝑑^∗. Then for 𝜀=0.5, there exists 𝐾>0 such that for all 𝑘>𝐾, we have 𝑑^∗−0.5<𝑑_𝑘<𝑑^∗+0.5. It implies that 𝑑_𝑘=𝑑^∗.

Remark 3.3. From Theorem 3.2, it is clear that the controllability of the system (1.1) can be realized completely within first 𝐾 steps during the evolving process, which means 𝐾 is just a CRI of system (1.1), so the existence of MinCRI of linear discrete-time delay systems is ensured and the value of MinCRI is finite. Moreover, 𝐾 is dependent on 𝐴, 𝐷, and ℎ.

Now we consider the second-order case, that is, 𝑛=2.

Theorem 3.4. If 𝑛=2, then 2ℎ+4 is a CRI of system (1.1).

Before giving the proof of Theorem 3.4, we first give some lemmas.

Lemma 3.5. Given 𝐴,𝐷∈ℝ^2×2, for any 𝑘∈ℕ, one has (𝐴𝐷)^𝑘𝐴,(𝐷𝐴)^𝑘𝐷,(𝐴𝐷)^𝑘+(𝐷𝐴)^𝑘∈span{𝐼,𝐴,𝐷,𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷}.

Proof. According to the different cases, we formulate the proof into two cases.
(a) 𝐼,𝐴,𝐷 are linearly dependent. Without loss of generality, suppose that 𝐴=𝜆₁𝐼+𝜆₂𝐷. (3.2) Then we have (𝐴𝐷)^𝑘𝐴=𝜆₁𝐼+𝜆₂𝐷𝐷^𝑘𝜆₁𝐼+𝜆₂𝐷(𝐷𝐴)^𝑘𝐷=𝐷𝜆₁𝐼+𝜆₂𝐷^𝑘𝐷(𝐴𝐷)^𝑘+(𝐷𝐴)^𝑘=𝐷𝜆₁𝐼+𝜆₂𝐷𝐷^𝑘+𝐷𝜆₁𝐼+𝜆₂𝐷^𝑘. (3.3) It is obvious that (𝐴𝐷)^𝑘𝐴,(𝐷𝐴)^𝑘𝐷,(𝐴𝐷)^𝑘+(𝐷𝐴)^𝑘∈span{𝐼,𝐷}.
(b) 𝐼,𝐴,𝐷 are linearly independent. If span{𝐼,𝐴,𝐷,𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷}=ℝ^2×2, the proof is completed. Otherwise, suppose that 𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷∈span{𝐼,𝐴,𝐷}. We use mathematical induction.(i)For 𝑘=1, by the assumption, (𝐴𝐷)¹𝐴,(𝐷𝐴)¹𝐷,𝐴𝐷+𝐷𝐴∈span{𝐼,𝐴,𝐷}.(ii)Suppose that for 𝑘, (𝐴𝐷)^𝑘𝐴,(𝐷𝐴)^𝑘𝐷∈span{𝐼,𝐴,𝐷}. Assume (𝐴𝐷)^𝑘𝐴=𝜆₁+𝜆₂𝐴+𝜆₃𝐷(𝐷𝐴)^𝑘𝐷=𝜆₄+𝜆₅𝐴+𝜆₆𝐷. (3.4)For 𝑘+1, we have (𝐴𝐷)^𝑘+1𝐴=𝐴(𝐷𝐴)^𝑘𝐷𝐴=𝐴𝜆₄+𝜆₅𝐴+𝜆₆𝐷𝐴=𝜆₄𝐴²+𝜆₅𝐴³+𝜆₆()𝐴𝐷𝐴𝐷𝐴^𝑘+1𝐷=𝐷𝜆₁+𝜆₂𝐴+𝜆₃𝐷𝐷=𝜆₁𝐷²+𝜆₂𝐷𝐴𝐷+𝜆₃𝐷³(𝐴𝐷)^𝑘+1+(𝐷𝐴)^𝑘+1=(𝐴𝐷)^𝑘−1𝐴𝐷𝐴𝐷+(𝐷𝐴)^𝑘−1=()𝐷𝐴𝐷𝐴𝐴𝐷^𝑘−1𝐴𝜆₄+𝜆₅𝐴+𝜆₆𝐷+(𝐷𝐴)^𝑘−1𝜆₄+𝜆₅𝐴+𝜆₆𝐷𝐴=𝜆₆(𝐴𝐷)^𝑘+𝜆₆(𝐷𝐴)^𝑘+(𝐴𝐷)^𝑘−1𝐴𝜆₄+𝜆₅𝐴+(𝐷𝐴)^𝑘−1𝜆₄+𝜆₅𝐴𝐴. (3.5) It is obvious that (𝐴𝐷)^𝑘+1𝐴,(𝐷𝐴)^𝑘+1𝐷,(𝐴𝐷)^𝑘+1+(𝐷𝐴)^𝑘+1∈span{𝐼,𝐴,𝐷}.

Lemma 3.6. Given 𝐴,𝐷∈ℝ^2×2, one has span{𝐼,𝐴,𝐷,𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷}=𝑃_Sym[𝐴,𝐷].

Proof. First, it is obvious that span{𝐼,𝐴,𝐷,𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷}⊆𝑃_Sym[𝐴,𝐷]. (3.6) Now we prove that span{𝐼,𝐴,𝐷,𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷}⊇𝑃_Sym[𝐴,𝐷].
For any 𝑓(𝐴,𝐷)∈𝑃_Sym[𝐴,𝐷], assume that 𝑓(𝐴,𝐷)=^𝑀_𝑚=1𝑐_𝑚𝐴^𝑖1𝑚,𝐷^𝑗1𝑚,⋯𝐴^{𝑖𝑚𝑚,𝑛}𝐷^{𝑗𝑚𝑚,𝑛}+𝐷^{𝑗𝑚𝑚,𝑛}𝐴^{𝑖𝑚𝑚,𝑛}⋯𝐷^𝑗1𝑚,𝐴^𝑖1𝑚,, (3.7) where 𝑛_𝑚∈ℕ,𝑖_𝑚,1,𝑗_𝑚,1,…,𝑖_{𝑚,𝑛𝑚},𝑗_{𝑚,𝑛𝑚}∈{0,1},𝑐_𝑚∈ℝ,𝑚=1,…,𝑀; 𝑀∈ℕ.
It is easy to see that if we could prove that each 𝐴^𝑖1𝑚,𝐷^𝑗1𝑚,⋯𝐴^{𝑖𝑚𝑚,𝑛}𝐷^{𝑗𝑚𝑚,𝑛}+𝐷^{𝑗𝑚𝑚,𝑛}𝐴^{𝑖𝑚𝑚,𝑛}⋯𝐷^𝑗1𝑚,𝐴^𝑖1𝑚,∈span{𝐼,𝐴,𝐷,𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷}, (3.8) then we would have proved that 𝑓(𝐴,𝐷)∈span{𝐼,𝐴,𝐷,𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷}. (3.9) It is easy to verify that 𝐴^𝑖1𝑚,𝐷^𝑗1𝑚,⋯𝐴^{𝑖𝑚𝑚,𝑛}𝐷^{𝑗𝑚𝑚,𝑛}+𝐷^{𝑗𝑚𝑚,𝑛}𝐴^{𝑖𝑚𝑚,𝑛}⋯𝐷^𝑗1𝑚, (3.10) can only be rewritten in three different forms:(a)(𝐴𝐷)^𝑘𝐴, where 𝑘∈ℕ;(b)(𝐷𝐴)^𝑘𝐷, where 𝑘∈ℕ;(c)(𝐴𝐷)^𝑘+(𝐷𝐴)^𝑘, where 𝑘∈ℕ.
By Lemma 3.5, we know that (3.8) holds. Thus, we have 𝑓(𝐴,𝐷)∈span{𝐼,𝐴,𝐷,𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷}. (3.11) It follows that 𝑃_Sym[𝐴,𝐷]⊆span{𝐼,𝐴,𝐷,𝐷𝐴+𝐴𝐷,𝐴𝐷𝐴,𝐷𝐴𝐷}. (3.12)

Now we are in a position to prove Theorem 3.4.

Proof of Theorem 3.4. By the definition of CRI, we only need to prove that span𝐺₀𝐵,𝐺₁𝐵,…,𝐺_𝑘𝐵,…=span𝐺₀𝐵,𝐺₁𝐵,…,𝐺_2ℎ+3𝐵. (3.13) By Lemmas 2.2 and 3.6, we have span𝐺₀,𝐺₁,…,𝐺_𝑘,…⊆span{𝐼,𝐴,𝐷,𝐴𝐷+𝐷𝐴,𝐴𝐷𝐴,𝐷𝐴𝐷}. (3.14) Now we prove that span𝐺₀,𝐺₁,…,𝐺_2ℎ+3=span{𝐼,𝐴,𝐷,𝐴𝐷+𝐷𝐴,𝐴𝐷𝐴,𝐷𝐴𝐷}. (3.15) By the definition of 𝐺_𝑘, we have 𝐺_𝑘=𝐴^𝑘, 𝑘=0,1,…,ℎ, 𝐺_ℎ+1=𝐴^ℎ+1+𝐷, 𝐺_ℎ+2=𝐴^ℎ+2+𝐷𝐴+𝐴𝐷, 𝐺_ℎ+3=𝐴^ℎ+3+𝐷𝐴²+𝐴𝐷𝐴+𝐴²𝐷. Note that 𝐷𝐴²+𝐴²𝐷 can be linearly expressed by 𝐼,𝐴,𝐷,𝐴𝐷+𝐷𝐴; then, we have span𝐺₀,𝐺₁,…,𝐺_ℎ+3=span{𝐼,𝐴,𝐷,𝐴𝐷+𝐷𝐴,𝐴𝐷𝐴}. (3.16) Consider the term 𝐷𝐴𝐷; it first appears in 𝐺_2ℎ+3. It is easy to verify that all other terms in 𝐺_2ℎ+3 can be linearly expressed by 𝐼,𝐴,𝐷,𝐴𝐷+𝐷𝐴,𝐴𝐷𝐴. Thus, we know that (3.14) holds. This implies that (3.8) holds.

Without proof, we can get the following corollaries directly.

Corollary 3.7. Assuming  𝑛=2, system (1.1) is controllable if and only if rank[𝐵,𝐴𝐵,𝐷𝐵,(𝐴𝐷+𝐷𝐴)𝐵,𝐴𝐷𝐴𝐵,𝐷𝐴𝐷𝐵]=2. (3.17)

Corollary 3.8. Assuming 𝑛=2, if system (1.1) is controllable, then the controllability can be realized in 2ℎ+4 steps; that is, one can select appropriate 𝑢(0),𝑢(1),…,𝑢(2ℎ+3) such that system (1.1) can be driven from any initial state to any terminal state.

Remark 3.9. Note that the parameter ℎ does not appear in Corollary 3.7. This implies that the specific number of the delay steps has nothing to do with the controllability of the system.

Remark 3.10. Lemma 3.6 plays a key role in the proof of Theorem 3.4. When 𝑛>2, it is difficult to build up a similar result like Lemma 3.6 for 𝑛>2. Thus the CRI problem for more general case is still open.

4. Examples

In this section, we present some numerical examples to illustrate the validity of our theoretic results.

Example 4.1. Consider the system (1.1) with 𝑛=2, ℎ=2, and 10𝐴=0000,𝐷=0010,𝐵=. (4.1) By simple calculation, we have rank[𝐵,𝐴𝐵,𝐷𝐵,(𝐴𝐷+𝐷𝐴)𝐵,𝐴𝐷𝐴𝐵,𝐷𝐴𝐷𝐵]=2. By Corollary 3.7, the system should be controllable. In fact, we have 𝑥(4)=Φ(𝑥(0),𝑥(−1),𝑥(−2))+𝐼,𝐴,𝐴²𝑢(3)𝑢(2)𝑢(1)𝑢(0),𝐷. (4.2) Leting 𝑢(2)=𝑢(1)=0, we can select suitable 𝑢(3) and 𝑢(0) such that 𝑥(4) be any state in ℝ². Thus, the system is controllable indeed.

Example 4.2. Consider the system (1.1) with 𝑛=2, ℎ=2, and 10𝐴=1001,𝐷=𝐴,𝐵=. (4.3) By simple calculation, we have rank[𝐵,𝐴𝐵,𝐷𝐵,(𝐴𝐷+𝐷𝐴)𝐵,𝐴𝐷𝐴𝐵,𝐷𝐴𝐷𝐵]=1. By Corollary 3.7, the system should not be controllable. In fact, it is easy to see that the second element of the state 𝑥(𝑘) is not affected by any input 𝑢(𝑘). Thus, the system is not controllable indeed.

5. Conclusion

This paper discussed the controllability of linear discrete-time systems with delay in state. After introducing a new concept called MinCRI to describe the controllability feature of delay systems, we proved the existence and finiteness of MinCRI. Based on this, a necessary and sufficient condition for the controllability of linear delay systems has been derived.

Acknowledgments

The authors are grateful to the Associate Editor and the reviewers for their helpful and valuable comments and suggestions for improving this paper. This work is supported by the National Natural Science Foundation (NNSF) of China (60774089, 10972003, and 60736022). This work is supported by N2008-07 (08010702014) from Beijing Institute of Petrochemical Technology and supported by the Opening Project of Guangxi Key Laboratory of Automobile Components and Vehicle Technology(NO.2012KFZD03).