LP APPROACH TO EXPONENTIAL STABILIZATION OF SINGULAR LINEAR POSITIVE TIME-DELAY SYSTEMS VIA MEMORY STATE FEEDBACK

. This paper deals with the exponential stabilization problem by means of memory state feedback controller for linear singular positive systems with delay. By using system decomposition approach, singular systems theory and Lyapunov function method, we obtain new delay-dependent suﬃcient conditions for designing such controllers. The conditions are given in terms of standard linear programming (LP) problems, which can be solved by LP optimal toolbox. A numerical example is given to illustrate the eﬀectiveness of the proposed method.


1.
Introduction. Over the past few decades, singular systems (also referred to as descriptor systems, semi-state systems, implicit systems, differential-algebraic systems, or generalized state-space systems) have attracted much attention due to the comprehensive applications in control theory [4,5,12,19]. The analysis and the controller design for such systems have received considerable attention in the past decades (see e.g. [2,8,20,22]). However, physical systems in the real world involve variables that have nonnegative sign, say, population levels, electrical circuits, power systems, absolute temperature, and so on. Such systems are referred to as positive systems (see e.g. [9,11,25]), where their output states are nonnegative provided the initial conditions are nonnegative. Due to widespread applications, it is necessary to investigate the stability and control problem for positive systems. Example applications include irrigation networks, highway and air-traffic flows, job-balancing in computer clusters, and chemical networks, to name just a few. The states of positive systems are confined within a "cone" located in the positive orthant rather than in the wholes pace, which makes their analysis and synthesis a challenging and interesting job (see e.g. [3,26]). It is worth noting that a lot of well-known results for normal linear systems cannot be simply applied to positive systems, since the states of positive systems are located in the positive orthant rather than in linear spaces. Among the large number of research results obtained for singular linear positive systems, much attention has been devoted to the stability analysis of such systems. A successful application to exponential stability of positive systems has been proposed in [10,13,18,27], however, they did not consider the singularity case.
The problem of observer and control design via LP approach for singular delay systems has been considered in [7,14,17], but the positivity case was not considered. Note that the LP approach may have a numerical advantage versus the LMI approach since the existing LMI softwares cannot handle large size problems and are not numerically stable. It should be noted that due to the singularity of derivative matrix and the non-negativity of variables in positive singular systems, much of the developed theory for such systems is still not up to a quantitative level. This feature makes the analysis and synthesis of singular linear positive systems a challenging task [3,23]. Moreover, stability and control problems of singular positive systems with time delay have been extensively studied due to the fact that the singular positive time-delay system better describes physical systems than state-space systems. By using state-space decomposition approach and linear matrix inequality technique, a solution to stability problem for singular linear positive systems was given in [16,24,27], unfortunately, the conditions are obtained there under the regularity and impulse-free assumptions and the design feedback controller was not considered. To the best of our knowledge, very few results on stability and control for singular linear positive systems with time delay have been reported in the literature.
In this paper, we provide a treatment based on linear programming for positivity and stabilization of singular linear positive systems with delay. The particular property of singular positive systems that we uncover in this paper (and which renders structured memory state feedback control design convex) has consequences for general plants and it is also applicable to pairs of systems and supply rates that can be transformed into the problem studied here via a change of control variables. Our main propose is to design a memory state feedback controller which guarantees the exponential stability of the closed-loop system. The significant contribution of this paper lies in two aspects. First, new delay-dependent sufficient conditions are provided, which determine the regularity, impulse-free and positivity of the system. Second, based on the structural decomposition of system matrices and the singular value decomposition method, we establish new sufficient conditions for designing memory state feedback controllers to stabilizes the closed-loop system. The derived conditions are described in a linear programming form and do not impose any restriction on the matrix dynamics of the governed system, so they can be easier solved by a standard linear programming problem than by the LMI method.
The remainder of this paper is organized as follows. In Section 2, necessary preliminaries and some propositions are provided for the proof of the main result. Section 3 proposes the delay-depending sufficient conditions for designing memory state feedback controllers of exponential stabilization problem. The paper ends with a conclusion and cited references.
Nomenclatures: x ∈ R n is called nonnegative (positive) if all ist entries are nonnegative (positive). x ∈ R n : x = max 1≤i≤n |x i | ; R n 0,+ (R n + ) denotes the space of all nonnegative (positive) vectors in R n ; R m×n denotes the set of all real (m × n) matrices. I n denotes the identity matrix in R n×n . A matrix B ∈ R n×n is called Metzler if all its off diagonal elements are non-negative.

2.
Preliminaries. Consider the following singular linear system with delay where x(t) ∈ R n is the state vector, u(t) ∈ R m is the control vector, A 0 , A 1 ∈ R n×n , B ∈ R n×m , and the matrix E ∈ R n×n is singular and rank E = r < n. The initial condition  Definition 2.3. Given α > 0. The singular unforced system (1) is said to be α−stable if it is regular, impulse-free and there exist a positive number N > 0 such that the solution x(t, ϕ) satisfies One way to solve the stabilization problem of singular systems with delay is to design memoryless controllers (see, e.g. [8,20,22]). The memoryless controllers are the control laws of the form u(t) = Kx(t) or of more general controllers with memory that include, nevertheless, an instantaneous feedback term Although the memoryless controllers in the mentioned papers are easy to implement, it was pointed out in [15], that they tend to be more conservative when the time delay is small. In fact, information on the size of the delay is often available in many processes. Hence, by using this information and employing a feedback of the past control history as well as the current state, we may expect to achieve an improved performance. Therefore, in this paper, as in [13] we will design the memory static feedback controller for the exponential stabilization of singular linear positive systems with delay.
is positive and α−stable.
Since rank E = r < n, it is known ( [5]) that there are two nonsingular matrices P, Q such that P EQ = I r 0 0 0 . Let us denote where Proposition 1. ( [7]) Assume that the system (1) is regular and impulse-free, written in the form (2) with det(A 04 ) = 0. The system (2) is positive if and only if A 04 is Hurwitz,Ã 0 is Metzler andÃ 1 0,B 0.
Applying the memory feedback control where Proposition 2. Assume that Q 0. If the system (2) is α− stabilizable by the feedback control (3), then the system (1) is α− stabilizable by the feedback control Proof. Systems (4) is regular, impulse-free if and only if det(A 04 +B 2 K 2 ) = 0 ( [12]). Setting It is easy to verify that Moreover, noting that det(sI r −Â 0 ) = r k=1 a k s k , a r = 1, and due to the singularity of P and Q, the polynomial det(sE − (A 0 + BKQ −1 )) is not identically zero and which implies that the closed-loop system of (1) is regular and impulse-free. Suppose that system (4) is positive, i.e. y(t) 1) A is Hurwitz.

4) The matrix A is nonsingular and satisfies
Proposition 4. Given a number ν > 0, a matrix Y 0, and X is Metzler and Hurwitz, if for some vector λ ∈ R m + satisfying λ T X + e ν Y ≺ 0, then there exist a vector η ∈ R m + and a number ρ ∈ (0, 1) such that Proof. From (5) and λ 0, Y 0, e ν > 1 it follows that Since X + Y is Metzler, using Proposition 3 and (7), we can find a vector η 1 ∈ R m Moreover, since X is Hurwitz and Metzler, using Proposition 3 again we get −X −1 0. Now, pre-multiplying both sides of equation (8) with the nonsingular matrix −X −1 0, we have which derives by Proposition 3 that there exists η ∈ R m + satisfying

Positivity, regularity and impulse-free For this we rewrite the system (4) in the form
where E = I r 0 0 0 , and Note that the condition a ij β j + b T i k j ≥ 0, i, j = 1, .., n; i = j, is equivalent to a we have which givesÃ 1 +BF 0. Since and using condition (13) we have From (15) and (16) it follows that On the other hand, from αE + A 1 e αh β 0 and (17) it follows that which gives Since A 0 is Metzler matrix, v 1 ∈ R r + , we have (A 03 + B 2 K 1 )v 1 0, and hence from (20), we obtain Remark that the matrix A 04 + B 2 K 2 is Metzler, using Proposition 3 and (21) we obtain that the matrix A 04 + B 2 K 2 is Hurwitz, and det(A 04 + B 2 K 2 ) = 0, which implies that the system (4) is regular, impulse-free. Moreover, since A 1 0, A 0 is Metzler and A 04 + B 2 K 2 is Hurwitz, det(A 04 + B 2 K 2 ) = 0, using Proposition 1 we get that the system (4) is positive.
2. Exponential stability Since system (4) is a linear differential-algebraic equation, we need to prove the exponential stability of each solution y 1 (t), y 2 (t).
Remark 1. Theorem 3.1 gives sufficient conditions for the positivity and stabilization of the system (1). The conditions (12), (13) are described by a system of linear scalar inequalities (a linear programming form) w.r.t. β, k j , f j , j = 1, 2, ..., n, and do not impose any restriction on the matrix dynamics of the governed system, so these conditions can be easier solved by a standard linear programming problem than by the LMI method. Since there exist powerful LP softwares (as Cplex) that can solve efficiently very large size problems, the LP approach is more simple and can have a legitimate numerical advantage in comparison to the LMI method [1,3,13,18,21,26].

Remark 2.
It is worth noting that the stability analysis of linear positive timedelay system has been studied in [10,13], but the results were limited to linear normal regular systems and the method used there can not be applied to singular systems. Also, singular value approach to positivity and stability of system (1) was proposed in [16], however the conditions are obtained under the regularity and impulse-free assumptions and the design feedback controller was not considered.
Remark 3. The following procedure for constructing the memory state feedback controllers can be applied.

5.
Conclusions. In this paper, we have studied problem of exponential stabilization for singular linear positive systems with delay. Based on the system decomposition approach, singular systems theory and Lyapunov function methods, we have provided delay-dependent sufficient conditions for designing memory state feedback controllers of exponential stabilization problem via linear programming. The results on positivity and stability analysis have been obtained in this paper for the systems with constant delay. We have a strong prospect that the results can be extended to the linear singular positive systems with time-varying delays.