# Zero Controllability Criterion for Discrete Positive Systems with Multiple Delays on Both States and Inputs

Authors: Mouhcine Naim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

Zero controllability criterion for positive linear discrete systems with multiple delays in both states and inputs is obtained and proved. An example is given to support our main result.


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## Body

## 1. Introduction and Preliminaries

Throughout this paper, we use the following notation:ℕ is the set of nonnegative integers, ℕ∗ is the set of positive integers, σnm=n,n+1,…,m is the finite subset of ℕ with n≤m, ℝn is the set of real vectors with n components, ℝ+n is the set of vectors in ℝn with nonnegative components, ℝn×m is the set of real matrices of size n×m, In is the identity matrix in ℝn×n, and ℝ+n×m is the set of real matrices with nonnegative entries.Positive systems are a wide class of systems in which state variables are constrained to be positive or at least nonnegative for all time whenever the initial conditions and inputs are nonnegative [1, 2]. The mathematical theory of positive linear systems is based on the theory of nonnegative matrices developed by Perron and Frobenius, see, e.g., [3, 4]. Since positive systems are not defined on linear spaces but on cones, then many concepts of linear systems cannot be directly generalized to linear positive systems without reformulation. One such property is the notion of controllability of linear positive systems.Controllability is one of the fundamental concepts in the mathematical control theory. A positive system is controllable if it is possible to transfer it from an arbitrary nonnegative initial state to an arbitrary nonnegative final state using only certain admissible nonnegative controls. Since late 1980s, controllability of discrete positive linear systems without delays has been a subject of much research [5–9]. In particular, Coxson and Shapiro in [6] showed that the discrete linear positive system is controllable if and only if it is reachable (controllability from zero initial conditions) and zero controllable (controllability to zero final state). The reachability of positive linear discrete systems with multiple delays in both state and control is addressed in [10]. On zero controllability of positive linear discrete systems with delay, the authors of [11] show that the following system with a single state delay(1)xi+1=A0xi+A1xi−1+Bui,i∈ℕ,is zero controllable if and only if the matrix A=A0A1In0 is nilpotent. In this paper, we will extend the result of zero controllability in [11] to the more general case, namely, positive discrete systems with multiple time delays both in state and in input. For this, we consider the general discrete linear time delay systems:(2)xi+1=∑j=0pAjxi−j+∑j=0qBjui−j,i∈ℕ,where xi∈ℝn is the state, ui∈ℝm is the input, Aj∈ℝn×nj∈σ0p, Bj∈ℝn×mj∈σ0q, and p and q are the nonnegative integer maximal values of delays on state and input, respectively. The initial conditions for (2) are given arbitrarily by x−j∈ℝn for j∈σ0p and u−j∈ℝm for j∈σ1q.Definition 1.
(positivity). System (2) is said to be positive if the state xi∈ℝ+n, i∈ℕ, for any initial states x−j∈ℝ+nj∈σ0p, for any initial inputs u−j∈ℝ+mj∈σ1q, and all inputs ui∈ℝ+m, i∈ℕ.Lemma 1.
(see [10]). System (2) is positive if and only if Aj∈ℝ+n×nj∈σ0p and Bj∈ℝ+n×mj∈σ0q.
In all the sequels in this paper, we assume that system (2) is positive.Definition 2.
(zero controllability). System (2) is said to be zero controllable if any initial state sequence x−j∈ℝ+nj∈σ0p and any initial input sequence u−j∈ℝ+mj∈σ1q, there exist a positive integer N and an input sequence ui∈ℝ+mi∈σ0N−1 such that the state of the system is driven from x−j to 0, that is, xN=0.
The paper is organized as follows. In Section2, we give and prove the criterion of the zero controllability of the general system (2) which is the main result of this paper. A numerical example is given in Section 3. Finally, the conclusion is provided in Section 4.


## 2. Main Result

In this section, we give the proof of the main result of this paper, which is accomplished in Theorem1.Lemma 2 (see [12, 13]).
The general solution to (2) is given by(3)xi=Gix0+∑j=1p∑k=1p−j+1Gi−kAk−1+jx−j+∑j=1q∑k=1q−j+1Gi−kBk−1+ju−j+∑j=0i−1∑k=0qGi−1−j−kBkuj,i∈ℕ,where the transition matrix Gi∈ℝn×ni∈ℕ is determined by the recurrence relation(4)Gi=In,fori=0,∑k=0pAkGi−1−k,fori∈ℕ∗,with the assumption(5)Gi=0,fori<0.Lemma 3.
The transition matrixGi also satisfies the following equation:(6)Gi=∑k=0pGi−1−kAk,fori∈ℕ∗.Proof.
See Appendix A.
Then, for anyi∈ℕ, we pose Hi0=Gi, and hence, for all i∈ℕ∗, we pose(7)Hij=∑k=1p−j+1Hi−k0Ak−1+j,j∈σ0p,Lij=∑k=1q−j+1Hi−k0Bk−1+j,j∈σ1q,with(8)Hij=0,forj∈σ1pandi≤0,Lij=0,forj∈σ1qandi≤0.
Moreover, fori∈ℕ, we put(9)Ki=∑k=0qHi−k0Bk,with Ki=0 for i<0.
Clearly by (7) and (9), solution (3) is given by the following new formula:(10)xi=∑j=0pHijx−j+∑j=1qLiju−j+∑j=0i−1Ki−1−juj,i∈ℕ.Theorem 1.
System (2) is zero controllable if and only if the matrix(11)A=A0A1⋯⋯ApB1⋯⋯⋯BqIn0⋯⋯00⋯⋯⋯00⋱⋱⋮⋮⋮⋮⋱⋱⋱⋮⋮⋮0⋯0In00⋯⋯⋯00⋯⋯⋯00⋯⋯⋯0⋮⋮Im⋱⋮⋮⋮0⋱⋱⋮⋮⋮⋮⋱⋱⋱⋮0⋯⋯⋯00⋯0Im0is nilpotent.
We introduce the following useful two lemmas that will aid us in the proof of our main result.Lemma 4.
For alli∈ℕ, we have(12)Hi+1j=Hij+1+Hi0Aj,j∈σ0p−1,Hi+1p=Hi0Ap,(13)Li+1j=Lij+1+Hi0Bj,j∈σ1q−1,Li+1q=Hi0Bq.Proof.
See Appendix B.
Without loss of generality, we assume thatp≥q. Indeed, if p<q, we can set Aj=0 for j∈σp+1q, and then we come back to p=q case.Lemma 5.
For alli≥p, we have(14)Ai=Hi0Hi1⋯HipLi1⋯LiqHi−10Hi−11⋯Hi−1pLi−11⋯Li−1q⋮⋮⋮⋮⋮Hi−p0Hi−p1⋯Hi−ppLi−p1⋯Li−pq0⋯⋯00⋯0⋮⋮⋮⋮0⋯⋯00⋯0.Proof.
See Appendix C.Remark 1.
SinceHp−p0=H00=In, then Ap≠0.
Now, we prove our main result.Proof of Theorem1 (sufficiency).
SinceA is nilpotent, then there exists a positive integer N such that AN+p=0. Hence, by Lemma 5, we have HNi=0 for i∈σ0p and LNj=0 for j∈σ1q. Thus, system (2) is zero controllable.
(Necessity). Since system (2) is zero controllable, there exists a positive integer N such that HNj=0 for i∈σ0p and LNj=0 for j∈σ1q. According to Lemma 4, we get that HN+kj=0 and LN+kj=0 for k∈σ0p. Thus, by Lemma 5, we have AN+p=0. This implies that A is nilpotent. The theorem is proved.Remark 2.
If one diagonal element of the matrixA0 is nonzero, system (2) is nonzero controllable.


## 3. Example

Consider system (2) with p=q=2 and matrices(15)A0=000000100,A1=000000100,A2=000200000,B1=010,B2=002.By calculation, we get that matrixA=A0A1A2B1B2I300000I300000000000I10 is nilpotent with index k=6, that is, Ak−1≠0 and Ak=0. Thus, by Theorem 1, system (2) is zero controllable.


## 4. Conclusion

In this paper, we have investigated the zero controllability of discrete linear positive systems with delays. Necessary and sufficient conditions have been established for the zero controllability discrete linear positive systems with multiple delays in both state variables and input signals. A numerical example is presented to explore the proposed theory.



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