On the Output Controllability of Positive Discrete Linear Delay Systems

Necessary and sufficient conditions for output reachability and null output controllability of positive linear discrete systems with delays in state, input, and output are established. It is also shown that output reachability and null output controllability together imply output controllability


Introduction
The research devoted to controllability was started by Kalman in the 1960s [1] and refers to linear dynamical systems.Controllability is one of the fundamental concepts in the modern mathematical control theory ( [2][3][4],. ..) and continually appears as a necessary condition for the existence of solutions to many control problems, for example, stabilization of unstable system by feedback and optimal control.Basically a system is controllable if it is possible to transfer it around its entire configuration space using only certain admissible controls.There exist many definitions of controllability that depends on the framework or the class of models applied.The following are examples of variations of controllability notions which have been introduced in the control literature: asymptotic controllability [5], relative controllability [6], constrained controllability [7], complete controllability [8], approximate controllability [9], small controllability [10], output controllability [11,12], and so on.
In most engineering applications, it is needed to direct the output toward some desired value.In fact, having control over the output of the system has a significant importance if not more than the states.For example, the control of a multilink cable-driven manipulator, where the task is typically defined in terms of end effector pose, rather than the joint positions and velocities which can define the system's state [13], also, controlling the output of fixed-speed wind turbines in the electrical network, which can directly affect the behavior of power systems [14].Output controllability is a property of the impulse response matrix of a linear invariant-time system which reflects the dominant ability of an external input to move the output from any initial condition to any final condition in a finite time [2].In general, the output controllability means that the system's output can be directed regardless of its state [15].The necessary and sufficient criterion for output controllability of linear time-invariant systems is addressed in, for example, [12].
Positive systems are a wide class of systems in which state variables and outputs are constrained to be positive, or at least nonnegative for all time whenever the initial conditions and inputs are nonnegative.Since the state variables and outputs of many real-world processes represent quantities that may not have meaning unless they are nonnegative because they measure concentrations, numbers, populations, and so on, positive systems arise frequently in mathematical modeling of engineering problems, management sciences, economics, social sciences, chemistry, biology, ecology, pharmacology, medicine, and so forth.
An excellent survey of positive systems with an emphasis on their applications in the areas of management and social sciences is given by Luenberger in [16].The more recent monographs by Farina and Rinaldi in [17] and Kaczorek in [18] are devoted entirely to positive linear systems and some of their applications.Since positive systems are confined within a cone located in the positive orthant rather than in the whole space [19,20], their analysis and synthesis are more complicated and more challenging.
The state controllability of positive linear discrete systems is largely studied by several authors since late 1980s [21][22][23][24][25][26], the problem of controllability of linear positive discrete systems with delays in state or control was discussed in [27].The problem of output reachability of positive linear discrete systems is addressed in [28].The output reachability of positive discrete linear systems with state delay has been studied in [29].
In this paper we examine the issue of output reachability, null output controllability, and output controllability for positive linear systems with multiple delays in state, input, and output.These concepts are equivalent for unconstrained systems.The output reachability of discrete positive linear systems are characterized and proven by a simple algebraic proof.The criteria for the null output controllability will be established.We show that these properties are not equivalent for positive systems.In addition we prove that the positive system is output controllable only if it is output reachable and null output controllable.
The structure of the paper is as follows.In the next section some mathematical preliminaries of positive linear discrete systems with delays are presented.We investigate the output reachability and null output controllability of positive linear discrete systems with delays in state, input, and output, respectively, in Sections 3 and 4. In Section 5, necessary and sufficient conditions for the output controllability of positive delay systems are provided.Numerical examples will be presented in Section 6.

Preliminaries
First we introduce some notations.N is the set of nonnegative integers, N + the set of positive integers,    = {,  + 1, . . ., } the finite subset of N with  ≤ , R  the set of real vectors with  components, and R  + the set of vectors in R  with nonnegative components; that is, where  denotes the transpose, R × the set of real matrices of order  ×  (R  = R ×1 ),   the identity matrix in R × , and  −1 the inverse of  ∈ R × .
In this work, we consider the discrete linear delay system with the output equation where   ∈ R  is the system state,   ∈ R  is the input (or control),   ∈ R  ,   ∈ R × ( ∈   0 ) are the matrices of the state,   ∈ R × ( ∈   0 ) are the matrices of the input,   ∈ R × ( ∈   0 ) are the matrices of the output and   ∈ R × ( ∈  V 0 ) are the matrices of the feedthrough (or feedforward), and ,  and V, and  are the nonnegative integer maximal values of delays on state, input, and output, respectively.
Definition 1.The system modeled by ( 2) and ( 3) is said to be positive if the state   ∈ R  + and the output ) and for any initial inputs ) and all inputs The mathematical theory of positive linear systems is based on the theory of nonnegative matrix developed by Perron and Frobenius (see [16,30]).Definition 2. A matrix  = (  ) in R × is said to be nonnegative and denoted by  ∈ R × + , if all of its elements are nonnegative; that is, Indeed, suppose one of the elements of ,   , is negative.Then, for the nonnegative vector  = (0, . . ., 0, 1, 0, . . ., 0)  ∈ R  + with the one in the th component, the th component of  would be   , which is negative.It is also easy to verify the converse.
The following proposition provides a necessary and sufficient conditions for positivity of system (2) and (3).

Proposition 4. System (2) and (3) is positive if and only if
Proof.
In all the sequel, we assume that system (2) and ( 3) is positive.
In the next proposition, we will present the explicit solution of system (2).

Proposition 5. The general solution to (2) is given by
where the transition matrix   ∈ R × ( ∈ N) is determined by the recurrence relation with the assumption Proof.The proof is given in [31].
Clearly by (15), (16), and ( 17), the solution of ( 2) is given by the following new formula: In the following and without loss of generality, we assume that  = V.Indeed, for example, if  > V we can set   = 0 for  ∈   V+1 .
Now, we introduce the matrices sequence as follows: For 0 ≤  < , the output equation ( 3) can be rewritten as Hence with where Abstract and Applied Analysis 5 For  ≥ , we have Then, we get the linear algebraic equation with The following lemmas will be needed in the sequel.
For  ∈   1 , we have For  ≥  + 1, we have Thus, ( 28) is satisfied in step  + 1.Hence, (28) holds for any  ∈ N + .Lemma 7.For all  ∈ N, we have Abstract and Applied Analysis Proof.For  = 0, we have Let  ∈ N + .For  = 0, we have then by Lemma 6, we get For  ∈ And for  = , we have Similarly, we prove that (32) holds.
Lemma 8. We have And for all  ∈ N, we have Proof.Let  < .For  = 0, we have for  ∈  −1 1 , we have and, for  = , we have For  ≥ , with  ∈  −1 0 , we have and, for  = , we have Similarly, we prove that (40) holds.

Output Reachability
In this section we will present necessary and sufficient conditions for output reachability of system (2) and (3).By generalization of definition given in [29] we obtain the following definitions.
Definition 9.The system modeled by ( 2) and ( 3) is said to be output reachable in  ∈ N + steps if, for any nonnegative final output   ∈ R  + , there exists a nonnegative input sequence   ∈ R  + ,  ∈  −1 0 , which steers the output of the system from  − = 0,  ∈   0 to   , with  − = 0 for  ∈   1 ; that is,   =  −1 .
Definition 10.The system modeled by ( 2) and ( 3) is said to be output reachable if there exists a positive integer  ∈ N + such that the system is output reachable in  steps.Now, we present a class of nonnegative matrices, called the monomial matrices [18,30].The utility of such a matrix will be highlighted in the study of the output reachability of positive linear systems.
A vector V ∈ R  + with exactly one of its components being nonzero and all the others being zero is called monomial vector or -monomial if the nonzero component is in the th position.
Definition 11.A square matrix  ∈ R × + is said to be monomial if it contains  linearly independent monomial columns.
An important property of monomial matrices is given by the following result.
The characterization of the output reachability is given by the following proposition.
Proposition 13.The system modeled by ( 2) and ( 3) is output reachable if and only if, for some  ∈ N + , the output reachability matrix R  includes a monomial submatrix of order  ×  ( ≤ ).

Proof.
Sufficiency.Let   ∈ R  + be the final output to be reached.From (21) or (25), we have With x0 = 0, this gives The matrix R  includes a monomial submatrix of order  × , and without loss of generality, we can assume that . Hence, by Lemma 12, we have  −1 1 ∈ R × + .Thus, for we get that is, system (2) and ( 3) is output reachable.
Necessity.Assume that system (2) and ( 3) is output reachable for some  ∈ N + .Thus, for every  ∈ R  + there exists an input   ∈ R  + such that with R  = (  ) ∈  1 ,∈  (54) The same reasoning gives the existence of a 1-monomial column or another null column of R  .Since the columns of R  are not all null, then R  has at least one 1-monomial column.
The same reasoning for  =   ,  ∈   2 , leads to the existence of a -monomial column.Hence by Definition 11, the matrix R  contains a monomial submatrix of order  × .The proposition is proved.Remark 14.If system (2) and ( 3) is output reachable and then the nonnegative input   0 ∈ R  + which steers the output of the system from  − = 0,  ∈   0 , to any desired nonnegative final output   ∈ R  + , with  − = 0 for  ∈   1 , can be computed by the formula

Null Output Controllability
By generalization of definition given in [11] the precise definitions of the null output controllability of system (2) and (3) are given as follows.
Definition 15.The system modeled by ( 2) and ( 3) is said to be null output controllable in  ∈ N + steps if, for any