International Journal of Mathematics and Mathematical Sciences

Volume 2021, Issue 1 4665468

Research Article

Open Access

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

Mouhcine Naim,

Corresponding Author

Mouhcine Naim

[email protected]

orcid.org/0000-0002-6130-3633

Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P. O. Box 7955 Sidi Othman, Casablanca, Morocco uh2c.ac.ma

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Mouhcine Naim,

Corresponding Author

Mouhcine Naim

[email protected]

orcid.org/0000-0002-6130-3633

Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, P. O. Box 7955 Sidi Othman, Casablanca, Morocco uh2c.ac.ma

Search for more papers by this author

First published: 12 May 2021

https://doi.org/10.1155/2021/4665468

Citations: 1

Academic Editor: Niansheng Tang

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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.

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, is the finite subset of ℕ with n ≤ m, ℝⁿ is the set of real vectors with n components, is the set of vectors in ℝⁿ with nonnegative components, ℝ^n×m is the set of real matrices of size n × m, I_n is the identity matrix in ℝ^n×n, and 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)

is zero controllable if and only if the matrix

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)

where x_i ∈ ℝⁿ is the state, u_i ∈ ℝ^m is the input,

, 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

and u_−j ∈ ℝ^m for

Definition 1. (positivity). System (2) is said to be positive if the state , i ∈ ℕ, for any initial states , for any initial inputs , and all inputs , i ∈ ℕ.

Lemma 1. (see [10]). System (2) is positive if and only if and .

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 and any initial input sequence , there exist a positive integer N and an input sequence such that the state of the system is driven from x_−j to 0, that is, x_N = 0.

The paper is organized as follows. In Section 2, 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 Theorem 1.

Lemma 2 (see [12], [13].)The general solution to (2) is given by

(3)

where the transition matrix G_i ∈ ℝ^n×n(i ∈ ℕ) is determined by the recurrence relation

(4)

with the assumption

(5)

Lemma 3. The transition matrix G_i also satisfies the following equation:

(6)

Proof. See Appendix A.

Then, for any i ∈ ℕ, we pose , and hence, for all i ∈ ℕ^∗, we pose

(7)

with

(8)

Moreover, for i ∈ ℕ, we put

(9)

with K_i = 0 for i < 0.

Clearly by (7) and (9), solution (3) is given by the following new formula:

(10)

Theorem 1. System (2) is zero controllable if and only if the matrix

(11)

is nilpotent.

We introduce the following useful two lemmas that will aid us in the proof of our main result.

Lemma 4. For all i ∈ ℕ, we have

(12)

(13)

Proof. See Appendix B.

Without loss of generality, we assume that p ≥ q. Indeed, if p < q, we can set A_j = 0 for , and then we come back to p = q case.

Lemma 5. For all i ≥ p, we have

(14)

Proof. See Appendix C.

Remark 1. Since , then A^p ≠ 0.

Now, we prove our main result.

Proof of Theorem 1 (sufficiency). Since A is nilpotent, then there exists a positive integer N such that A^N+p = 0. Hence, by Lemma 5, we have for and for . Thus, system (2) is zero controllable.

(Necessity). Since system (2) is zero controllable, there exists a positive integer N such that for and for . According to Lemma 4, we get that and for . Thus, by Lemma 5, we have A^N+p = 0. This implies that A is nilpotent. The theorem is proved.

Remark 2. If one diagonal element of the matrix A₀ is nonzero, system (2) is nonzero controllable.

3. Example

Consider system (2) with p = q = 2 and matrices

(15)

By calculation, we get that matrix is nilpotent with index k = 6, that is, A^k−1 ≠ 0 and A^k = 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.

Conflicts of Interest

The author declares no conflicts of interest.

Appendix

A. Proof of Lemma 3

Proof. First, for i = 1, we have and (6) holds. Secondly, suppose that (6) holds for . We prove that it holds for k = i + 1.

For , we have

(A.1)

For i ≥ p + 1, we have

(A.2)

Thus, (6) is satisfied in step i + 1. Hence, (6) holds for any i ∈ ℕ^∗.

B. Proof of Lemma 4

Proof. Let i ∈ ℕ. For , we have

(B.1)

and, for j = p, we have

(B.2)

Similarly, we prove that (13) holds.

C. Proof of Lemma 5

Proof. We introduce a new state variable for i ∈ ℕ by

(C.1)

It is easy to verify that

(C.2)

where A is defined in (11) and

(C.3)

Let u_i = 0 for i ∈ ℕ. Then, the solution of system (C.2) is given by

(C.4)

On the other hand, from (10), for all i ≥ p, we have

(C.5)

Hence, by identification between (C.4) and (C.5), we get that (14) holds.

Open Research

Data Availability

No data were used to support this study.

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

Abstract

1. Introduction and Preliminaries

2. Main Result

3. Example

4. Conclusion

Conflicts of Interest

Appendix

A. Proof of Lemma 3

B. Proof of Lemma 4

C. Proof of Lemma 5

Open Research

Data Availability

References

Citing Literature

References

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

Abstract

1. Introduction and Preliminaries

2. Main Result

3. Example

4. Conclusion

Conflicts of Interest

Appendix

A. Proof of Lemma 3

B. Proof of Lemma 4

C. Proof of Lemma 5

Open Research

Data Availability

References

Citing Literature

References

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