REALIZATION PROBLEM FOR POSITIVE LINEAR SYSTEMS WITH TIME DELAY

In positive systems inputs, state variables and outputs take only nonnegative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear systems behavior can be found in engineering, management science, economics, social sciences, biology, and medicine, and so forth. Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of positive systems is more complicated and less advanced. An overview of state of the art in positive systems theory is given in the monographs [4, 5]. Recent developments in positive systems theory and some new results are given in [5]. Realizations problem of positive linear systems without time delays has been considered in many papers and books [1, 4, 5]. Explicit solution of equations describing the discrete-time systems with time delay has been given in [2]. Recently, the reachability, controllability, and minimum energy control of positive linear discrete-time systems with time delays have been considered in [3, 6]. In this paper, the realization problem for positive single-input single-output discretetime systems with time delay will be formulated and solved. Necessary and sufficient conditions for the solvability of the realization problem will be established and a procedure for computation of a minimal positive realization of a proper rational function will be presented. To the best knowledge of the author, the realization problem for positive linear systems with time delays has not been considered yet.


Introduction
In positive systems inputs, state variables and outputs take only nonnegative values.Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models.A variety of models having positive linear systems behavior can be found in engineering, management science, economics, social sciences, biology, and medicine, and so forth.
Positive linear systems are defined on cones and not on linear spaces.Therefore, the theory of positive systems is more complicated and less advanced.An overview of state of the art in positive systems theory is given in the monographs [4,5].Recent developments in positive systems theory and some new results are given in [5].Realizations problem of positive linear systems without time delays has been considered in many papers and books [1,4,5].
Explicit solution of equations describing the discrete-time systems with time delay has been given in [2].
Recently, the reachability, controllability, and minimum energy control of positive linear discrete-time systems with time delays have been considered in [3,6].
In this paper, the realization problem for positive single-input single-output discretetime systems with time delay will be formulated and solved.Necessary and sufficient conditions for the solvability of the realization problem will be established and a procedure for computation of a minimal positive realization of a proper rational function will be presented.
To the best knowledge of the author, the realization problem for positive linear systems with time delays has not been considered yet.

Problem formulation
Consider the single-input single-output discrete-time linear system with one time delay where x i ∈ R n , u i ∈ R, y i ∈ R are the state vector, input, and output, respectively, and Initial conditions for (2.1a) are given by Let R n×m + be the set of n × m real matrices with nonnegative entries and R n

(see [3]). The system (2.1) is positive if and only if
3) The transfer function of (2.1) is given by The positive realization problem can be stated as follows.
Given a proper rational function T(z), find a positive realization (2.3) of the rational function T(z).
Necessary and sufficient conditions for the solvability of the problem will be established and a procedure for computation of a positive realization will be presented.

Problem solution
The transfer function (2.4) can be rewritten in the form where and [ Tadeusz Kaczorek 457 From (3.1), we have The strictly proper part of T(z) is given by Therefore, the positive realization problem has been reduced to finding matrices for a given strictly proper rational matrix (3.4).
Lemma 3.1.The strictly proper transfer function (3.4) has the form if and only if detA 1 = 0, where Proof.From the definition of (3.2) of d(z) for z = 0, it follows that a 0 = det A 1 .Note that d(z) = zd (z) if and only if a 0 = 0 and (3.4) can be reduced to (3.6).
Lemma 3.2.If the matrices A 0 and A 1 have the forms Proof.Expansion of the determinant with respect to the first row yields det (3.10) Matrices A 0 and A 1 having the forms (3.8) will be called the matrices in canonical forms.
The following two remarks are in order.
Lemma 3.6.If the matrices A 0 and A 1 have the canonical forms (3.8), then the pair (A 0 ,A 1 ) is cyclic.
Proof.It is well known that ϕ(z) = ψ(z) if and only if the greatest common divisor of all n − 1 degree minors of the polynomial matrix [ Tadeusz Kaczorek 459 we obtain Note that the n − 1 degree minor corresponding to the entry −a 1 z − a 0 of the matrix (3.11) is equal to 1. Therefore, we have ϕ(z) = ψ(z) and by Definition 3.5, the pair (A 0 ,A 1 ) is cyclic. Let where Using (3.2) and (3.12), we obtain Comparison of the coefficients at like powers of z in (3.13) yields where  Note that the matrix A of the dimension (2n − 1) × n 2 has more columns than rows.If the condition (3.16) is satisfied then without loss of generality, we may assume that the matrix A has full row rank equal to 2n − 1 (otherwise, we may eliminate the linearly dependent equations from (3.14)).
Choosing n 2 − 2n + 1 = (n − 1) 2 nonnegative components of the vector x and solving the corresponding matrix equation with nonsingular (2n − 1) × (2n − 1) coefficient matrix, we may compute the desired entries of b and c (that should be nonnegative).Therefore, we have established the following necessary and sufficient conditions for the existence of the solution to the positive realization problem.
Theorem 3.8.The positive realization problem has a solution if and only if the following conditions are satisfied.
(3) The matrix equation (3.14) has a nonnegative solution, x ∈ R n 2 + .If the conditions of Theorem 3.7 are satisfied, then the desired positive realization (2.3) of T(z) can be found by the use of the following procedure.
Step 4 .Find the nonnegative solution x ∈ R n 2 + of (3.14) and the matrices b and c.Remark 3.10.A positive realization computed by the use of Procedure 3.9 is a minimal one.

Example
Find a positive realization (3.5) of the strictly proper function Using Procedure 3.9, we obtain successively the following steps.

Concluding remarks
The realization problem for positive single-input single-output discrete-time systems with one time delay has been formulated and solved.Canonical forms (3.8) of the system matrices A 0 and A 1 have been introduced.It has been shown that the pair (3.8) is cyclic.Necessary and sufficient conditions for the existence of positive minimal realization (2.3) of a proper rational function T(z) have been established.A procedure for computation of a minimal positive realization of proper rational function has been presented Tadeusz Kaczorek 463 and illustrated by an example.The considerations can be extended for the following: (1) single-input single-output discrete-time linear systems with many time delays; (2) multi-input multi-output discrete-time linear systems with one and many time delays.An extension of the considerations for continuous-time linear systems with time delays is also possible.

Remark 3 . 3 .Remark 3 . 4 .Definition 3 . 5 .
The matrices (3.8) have nonnegative entries if and only if the coefficients a k , k = 0,1,...,2n − 1, of the polynomial (3.9) are nonnegative.The dimension n × n of matrices (3.8) is the smallest possible one for(3.4).The pair of matrices (A 0 ,A 1 ) is called cyclic if and only if the characteristic polynomial .15) By Kronecker-Capelli theorem, the matrix equation (3.14) has a solution x if and only if rank A,l = rank A, (3.16) therefore, we have the following theorem.Theorem 3.7.The positive realization problem has a solution only if the condition (3.16) is satisfied.