# Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inputs

Authors: Hong Shi; Guangming Xie; Wenguang Luo
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The controllability issues for discrete-time linear systems with delay in state and control are addressed. By introducing a new concept, the controllability realization index (CRI), the characteristic of controllability is revealed. An easily testable necessary and sufficient condition for the controllability of discrete-time linear systems with state and control delay is established.


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## Body

## 1. Introduction

The concept of controllability, first given by Kalman in the 1960s [1], plays a fundamental role in the modern control theory and has close connections with pole assignment, structure decomposition, quadratic control, and so forth [2, 3]. The various aspects of the controllability of linear systems with delay were considered by several authors [4–11]. The discrete cases have been considered by Klamka [7], Watanabe [8], and Phat [9], but the mathematical conditions given for investigating the controllability are not suitable for real verification and application.In our recent paper [12], a new concept called controllability realization index (CRI) is proposed, which is crucial in determining the controllability of such kind of discrete systems with delays. In that paper, it is proved that the value of CRI is finite for discrete systems with delays, and a general CRI value for planar discrete systems with delays is given. Thus, the judging condition of controllability for planar case is established. In this paper, we will extend our result to the more general case, namely, discrete systems with any order, with time delays both in state and in control.This paper is organized as follows. In Section2, some basic definitions and preliminary results are presented. Section 3 is the main results. An easily testable necessary and sufficient condition for the controllability of discrete-time linear systems with state and control delay is established. A numerical example is given in Section 4. Finally, the conclusion is provided in Section 5.


## 2. Problem Formulation and Preliminaries

In this paper, we consider the discrete-time case the system model is described as follows:(1)x(k+1)=∑i=0p‍Aix(k-i)+∑j=0q‍Bju(k-j),
where x(k)∈ℝn is the state, u(k)∈ℝs is the input, Ai∈ℝn×n and Bj∈ℝn×s are constant matrices, and the positive integers i,j are the lengths of the steps of time delays. The initial states x(-p),x(-p+1),…,x(0) and the initial input u(-q),u(-q+1),…,u(-1) are given arbitrarily.The controllability discussed here refers to the unconstrained controllability or completely controllability.Definition 1 (controllability).
The system (1) is said to be (completely) controllable if, for any initial input u(-q),u(-q+1),…,u(-1), any initial state x(-p),x(-p+1),…,x(0), and any terminal state xf, there exist a positive integer k and, input u(0),…,u(k-1) such that x(k)=xf.Definition 2 (controllability realization index, CRI).
For the system (1), if there exists a positive integer K such that for any initial input u(-q),u(-q+1),…,u(-1), initial state x(-p),x(-p+1),…,x(0), and any terminal state xf, there exists an input u(0),…,u(K-1) such that x(K)=xf, then one calls K the controllability realization index (CRI) of the system (1). Obviously, if exists, such K is not unique, so the one calls the smallest K among them the minimum controllability realization index (MinCRI).Denote byℕ the nonnegative integer set. The matrices A1,…,AN∈ℝn×n are said to be linearly dependent on ℝn×n, if there exist scalars c1,…,cN∈ℝ, not all are zero, such that ∑i=1NciAi=0. In the following statement, span{A1,…,AN} will be used to denote the space constructed by the linear combinations of matrices {A1,…,AN}.


## 3. Main Results

### 3.1. Delay in State

In this section, we first investigate the controllability of the systems only with delay in state(2)x(k+1)=∑i=0p‍Aix(k-i)+Bu(k).Now, we introduce a matrix sequence{Gk}k=0∞⊆ℝn×n as follows:
(3)Gk={Iifk=0∑i=0k-1‍AiGk-1-iifk=1,…,p∑i=0p‍AiGk-1-iifk=p+1,p+2,….Lemma 3.
The general solution of the system (2) is given by
(4)x(k+1)=Ψ(k,x(-p),…,x(0))+∑i=0k‍Gk-iBu(i),k∈ℕ,
where Ψ(k,x(-p),…,x(0)) is the part of the solution with zero input.Proof.
See AppendixA.Lemma 4.
The matrix sequence{Gk}k=0∞ given by (3) satisfies
(5)span{G0,…,Gk,…}=span{G0,…,Gn(p+1)-1}.Proof.
See AppendixB.Theorem 5.
The system (2) is controllable if and only if rank[G0B|⋯|Gn(p+1)-1B]=n.Proof.
(Necessity) If the system is controllable, then we know thatspan{G0,…,Gk,…}=ℝn. Thus, by Lemma 4, we have rank[G0B|⋯|Gn(p+1)-1B]=n.
(Sufficiency) By Lemma3, we have
(6)x(n(p+1))=Ψ(n(p+1)-1,x(-p),…,x(0))+∑i=0n(p+1)-1‍Gn(p+1)-1-iBu(i).
Since rank[G0B|⋯|Gn(p+1)-1B]=n, for any initial state x(-p),…,x(0) and any terminal state x(n(p+1)), we can select appropriate inputs u(0),…,u(n(p+1)-1) such that the equation x(n(p+1))=xf, where x(n(p+1)) is given by the above equation and xf is arbitrary. Thus, the system is controllable.Corollary 6.
n
(
p
+
1
) is a CRI of the system (2).Proof.
It is directly followed from Theorem5.Remark 7.
This work has improved the result in [12]. When the system is second order, that is, n=2, we prove 2p+2 to be a CRI value, which differs from the CRI value 2p+4 in [12]. The difference lies in that the CRI of a system is not unique. For practical applications, obviously the less, the better.


### 3.2. Delays in Both State and Input

Now we investigate the controllability of the system (1). We only consider the case when p=q, for the case p≠q, the discussion is similar (Without loss of generality, we assume that p>q, and let Bj=0,  j=q+1,q+2,…,p, then we come back to the p=q case.).Lemma 8.
The solution of the system (1) can be expressed as
(7)x(k+1)=Φ(k,x(-p),…,x(0),u(-p),…,u(-1))+∑i=0k‍Hk-iu(i),k∈ℕ,
where
(8)Hk={G0B0ifk=0∑i=0k‍Gk-iBiifk=1,…,p-1∑i=0p‍Gk-iBiifk=p,p+1,…;
and Φ(k,x(-p),…,x(0),u(-p),…,u(-1)) is the part of the solution corresponding only to the initial state and initial input.Proof.
The proof is similar to that of Lemma3.Lemma 9.
The matrix sequence{Hk}k=0∞ given by (8) satisfies
(9)span{H0,…,Hk,…}=span{H0,…,Hn(p+1)+p-1}.Proof.
See AppendixC.Theorem 10.
The system (1) is controllable if and only if rank[H0|⋯|Hn(p+1)+p-1]=n.Proof.
The proof is similar to that of Theorem5.Corollary 11.
n
(
p
+
1
)
+
p is a CRI of the system (1).Proof.
It is directly followed from Theorem10.Remark 12.
This corollary provides a complete and verifiable method to testify the controllability of a general discrete-time system with delay in state or in control or both. Approximately the computation work for eachHk is O(pn3), and the entire testing work takes O(p2n4).


## 3.1. Delay in State

In this section, we first investigate the controllability of the systems only with delay in state(2)x(k+1)=∑i=0p‍Aix(k-i)+Bu(k).Now, we introduce a matrix sequence{Gk}k=0∞⊆ℝn×n as follows:
(3)Gk={Iifk=0∑i=0k-1‍AiGk-1-iifk=1,…,p∑i=0p‍AiGk-1-iifk=p+1,p+2,….Lemma 3.
The general solution of the system (2) is given by
(4)x(k+1)=Ψ(k,x(-p),…,x(0))+∑i=0k‍Gk-iBu(i),k∈ℕ,
where Ψ(k,x(-p),…,x(0)) is the part of the solution with zero input.Proof.
See AppendixA.Lemma 4.
The matrix sequence{Gk}k=0∞ given by (3) satisfies
(5)span{G0,…,Gk,…}=span{G0,…,Gn(p+1)-1}.Proof.
See AppendixB.Theorem 5.
The system (2) is controllable if and only if rank[G0B|⋯|Gn(p+1)-1B]=n.Proof.
(Necessity) If the system is controllable, then we know thatspan{G0,…,Gk,…}=ℝn. Thus, by Lemma 4, we have rank[G0B|⋯|Gn(p+1)-1B]=n.
(Sufficiency) By Lemma3, we have
(6)x(n(p+1))=Ψ(n(p+1)-1,x(-p),…,x(0))+∑i=0n(p+1)-1‍Gn(p+1)-1-iBu(i).
Since rank[G0B|⋯|Gn(p+1)-1B]=n, for any initial state x(-p),…,x(0) and any terminal state x(n(p+1)), we can select appropriate inputs u(0),…,u(n(p+1)-1) such that the equation x(n(p+1))=xf, where x(n(p+1)) is given by the above equation and xf is arbitrary. Thus, the system is controllable.Corollary 6.
n
(
p
+
1
) is a CRI of the system (2).Proof.
It is directly followed from Theorem5.Remark 7.
This work has improved the result in [12]. When the system is second order, that is, n=2, we prove 2p+2 to be a CRI value, which differs from the CRI value 2p+4 in [12]. The difference lies in that the CRI of a system is not unique. For practical applications, obviously the less, the better.


## 3.2. Delays in Both State and Input

Now we investigate the controllability of the system (1). We only consider the case when p=q, for the case p≠q, the discussion is similar (Without loss of generality, we assume that p>q, and let Bj=0,  j=q+1,q+2,…,p, then we come back to the p=q case.).Lemma 8.
The solution of the system (1) can be expressed as
(7)x(k+1)=Φ(k,x(-p),…,x(0),u(-p),…,u(-1))+∑i=0k‍Hk-iu(i),k∈ℕ,
where
(8)Hk={G0B0ifk=0∑i=0k‍Gk-iBiifk=1,…,p-1∑i=0p‍Gk-iBiifk=p,p+1,…;
and Φ(k,x(-p),…,x(0),u(-p),…,u(-1)) is the part of the solution corresponding only to the initial state and initial input.Proof.
The proof is similar to that of Lemma3.Lemma 9.
The matrix sequence{Hk}k=0∞ given by (8) satisfies
(9)span{H0,…,Hk,…}=span{H0,…,Hn(p+1)+p-1}.Proof.
See AppendixC.Theorem 10.
The system (1) is controllable if and only if rank[H0|⋯|Hn(p+1)+p-1]=n.Proof.
The proof is similar to that of Theorem5.Corollary 11.
n
(
p
+
1
)
+
p is a CRI of the system (1).Proof.
It is directly followed from Theorem10.Remark 12.
This corollary provides a complete and verifiable method to testify the controllability of a general discrete-time system with delay in state or in control or both. Approximately the computation work for eachHk is O(pn3), and the entire testing work takes O(p2n4).


## 4. Example

In this section, we present a numeric example.Example 13.
Consider the system (1) with p=q=2,n=3and
(10)A0=[100000000],A1=[000010001],A2=[100011001],B0=[100],B1=[010],B2=[001].
By simple calculation, we get
(11)G0=[100010001],G1=[100000000],G2=[100010001],G3=[200011001],G4=[300010001],H0=[100],H1=[100],H2=[100],H3=[210],H4=[411].
Thus, by Theorem 10, the system is controllable.


## 5. Conclusion

In this paper, we have investigated the controllability of discrete-time linear systems with time delays. Necessary and sufficient conditions have been established for discrete-time linear systems with state delay or both state and control delays. The proposed conditions are suitable for real verification and can be efficiently computed.



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