# On the Output Controllability of Positive Discrete Linear Delay Systems

Authors: Mouhcine Naim; Fouad Lahmidi; Abdelwahed Namir; Mostafa Rachik
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

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.


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## Body

## 1. 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 ([2–4],…) 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 [21–26], 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 Sections3 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.


## 2. Preliminaries

First we introduce some notations.N is the set of nonnegative integers, N+ the set of positive integers, σsk=s,s+1,…,k the finite subset of N with s≤k, Rn the set of real vectors with n components, and R+n the set of vectors in Rn with nonnegative components; that is, (1)R+n=x=x1,x2,…,xnT∈Rn:xi≥0,i∈σ1n,where T denotes the transpose, Rn×m the set of real matrices of order n×mRn=Rn×1, In the identity matrix in Rn×n, and A-1 the inverse of A∈Rn×n.In this work, we consider the discrete linear delay system(2)xi+1=∑j=0pAjxi-j+∑j=0qBjui-j,i∈N,u-j∈Rmforj∈σ1q,x-j∈Rnforj∈σ0p, with the output equation(3)yi=∑j=0lCjxi-j+∑j=0vDjui-j,i∈N,withl≤p,v≤q, where xi∈Rn is the system state, ui∈Rm is the input (or control), yi∈Rr,Aj∈Rn×nj∈σ0p are the matrices of the state, Bj∈Rn×mj∈σ0q are the matrices of the input, Cj∈Rr×nj∈σ0l are the matrices of the output and Dj∈Rr×mj∈σ0v are the matrices of the feedthrough (or feedforward), and p, q and v, and l 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 xi∈R+n and the output yi∈R+r,i∈N, for any initial states x-j∈R+n(j∈σ0p) and for any initial inputs u-j∈R+m(j∈σ1q) and all inputs ui∈R+m,i∈N.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 matrixA=aij in Rn×m is said to be nonnegative and denoted by A∈R+n×m, if all of its elements are nonnegative; that is, aij≥0 for all i∈σ1n,j∈σ1m.Remark 3.
A
∈
R
+
n
×
m if and only if Ax∈R+n for all x∈R+m. Indeed, suppose one of the elements of A,aij, is negative. Then, for the nonnegative vector x=0,…,0,1,0,…,0T∈R+m with the one in the jth component, the ith component of Ax would be aij, 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 (4)Aj∈R+n×nj∈σ0p,Bj∈R+n×mj∈σ0q,(5)Cj∈R+r×nj∈σ0l,Dj∈R+r×mj∈σ0v.Proof.
Sufficiency. If the condition (4) is satisfied, then(6)x1=∑j=0pAjx-j+∑j=0qBju-j∈R+n,since x-j∈R+nj∈σ0p and u-j∈R+mj∈σ0q. Assume that xk∈R+n for k∈σ1i. From (2) we have(7)xi+1=∑j=0pAjxi-j+∑j=0qBjui-j∈R+n,since (4) holds and xi-j∈R+nj∈σ0p, u-j∈R+mj∈σ1q, and ui∈R+m,i∈N. Hence xi∈R+n for any i∈N. Consequently, if condition (5) is satisfied, we get that yi∈R+r for every i∈N since x-j∈R+nj∈σ0l,  u-j∈R+m(j∈σ1v), and ui∈R+m,i∈N.
Necessity. Assuming that system (2) and (3) is positive, let u-j=0 for j∈σ0q. Then from (2) and (3), for i=0, we have(8)x1=∑j=0pAjx-j=A-x-0∈R+n,y0=∑j=0lCjx-j=C-x-1∈R+r,with(9)A-=A0A1⋯Ap∈Rn×np+1,C-=C0C1⋯Cl∈Rr×nl+1,x-0=x0,x-1,…,x-pT∈Rnp+1,x-1=x0,x-1,…,x-lT∈Rnl+1.Hence by Remark 3, we have A-∈R+n×np+1; that is, Aj∈R+n×nj∈σ0p and C-∈R+r×nl+1; that is, Cj∈R+r×nj∈σ0l since x-0∈R+np+1 and x-1∈R+nl+1 are arbitrary. Now, assume that x-j=0 for j∈σ0p, and for i=0, we obtain(10)x1=∑j=0qBju-j=B-u-0∈R+n,y0=∑j=0vDju-j=D-u-1∈R+r,with(11)B-=B0B1⋯Bq∈Rn×mq+1,D-=D0D1⋯Dv∈Rr×mv+1,u-0=u0,u-1,…,u-qT∈Rmq+1,u-1=u0,u-1,…,u-vT∈Rmv+1,which implies that B-∈R+n×mq+1; that is, Bj∈R+n×mj∈σ0q and D-∈R+r×mv+1, that is, Dj∈R+r×mj∈σ0v since u-0∈R+mq+1 and u-1∈R+mv+1 are arbitrary. This completes the 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(12)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(13)Gi=Infori=0,∑k=0pAkGi-1-kfori∈N+,with the assumption(14)Gi=0fori<0.Proof.
The proof is given in [31].We poseHi0=Gi, and then(15)Hi0=Infori=0,∑k=0pAkHi-1-k0fori∈N+,0fori<0,and, for all i∈N+, we pose(16)Hij=∑k=1p-j+1Hi-k0Ak-1+j,j∈σ1p,Lij=∑k=1q-j+1Hi-k0Bk-1+j,j∈σ1q, with Hij=Lij=0 for i≤0.Moreover, fori∈N, we pose(17)Ki=∑k=0qHi-k0Bk, with Ki=0, for i<0.Clearly by (15), (16), and (17), the solution of (2) is given by the following new formula:(18)xi=Hi0x0+∑j=1pHijx-j+∑j=1qLiju-j+∑j=0i-1Ki-1-juj,i∈N.In the following and without loss of generality, we assume thatl=v. Indeed, for example, if l>v we can set Dj=0 for j∈σv+1l.Now, we introduce the matrices sequence as follows:(19)Hij=∑k=0lCkHi-kj,j∈σ0p,i∈N,Lij=∑k=0lCkLi-kj,j∈σ1q,i∈N,Ki=∑k=0lCkKi-k,i∈N,K-i=Ki+Di+1,i∈σ0l-1.For0≤i<l, the output equation (3) can be rewritten as(20)yi=∑k=0iCkxi-k+∑k=i+1lCkxi-k+∑k=0iDkui-k+∑k=i+1lDkui-k=∑k=0iCkHi-k0x0+∑k=0iCk∑j=1pHi-kjx-j+∑k=0iCk∑j=1qLi-kju-j+∑k=i+1lCkxi-k+∑k=i+1lDkui-k+∑k=0iCk∑j=0i-k-1Ki-k-1-juj+∑k=0iDkui-k=∑k=0lCkHi-k0x0+∑k=0lCk∑j=1pHi-kjx-j+∑k=0lCk∑j=1qLi-kju-j+∑k=i+1lCkxi-k+∑k=i+1lDkui-k+∑k=0i-1Ck∑j=0i-k-1Ki-k-1-juj+∑k=0iDkui-k=∑k=0lCkHi-k0x0+∑j=1p∑k=0lCkHi-kjx-j+∑j=1q∑k=0lCkLi-kju-j+∑j=i+1lCjxi-j+∑j=i+1lDjui-j+∑j=0i-1∑k=0i-j-1CkKi-j-1-kuj+∑j=0i-1Di-juj+D0ui=Hi0x0+∑j=1pHijx-j+∑j=1qLiju-j+∑j=1l-iCi+jx-j+∑j=1l-iDi+ju-j+∑j=0i-1Ki-j-1+Di-juj+D0ui=Hi0x0+∑j=1l-iHij+Ci+jx-j+∑j=l-i+1pHijx-j+∑j=1l-iLij+Di+ju-j+∑j=l-i+1qLiju-j+∑j=0i-1K-i-j-1uj+D0ui.Hence(21)yi=Qi+1x~0+Ri+1u0i+1, with(22)Qi+1=Mi+1Oi+1∈R+r×np+1+mq,where(23)Mi+1=Hi0Hi1+Ci+1Hi2+Ci+2⋯Hil-i+ClHil-i+1⋯Hip∈R+r×np+1,Oi+1=Li1+Di+1⋯Lil-i+DlLil-i+1⋯Liq∈R+r×mq,x~0=x0x-1⋮x-pu-1⋮u-q∈R+np+1+mq,Ri+1=K-i-1K-i-2⋯K-1K-0D0∈R+r×i+1m,u0i+1=u0u1⋮ui∈R+i+1m.Fori≥l, we have(24)yi=∑k=0lCkHi-k0x0+∑k=0lCk∑j=1pHi-kjx-j+∑k=0lCk∑j=1qLi-kju-j+∑k=0lCk∑j=0i-k-1Ki-k-1-juj+∑k=0lDkui-k=∑k=0lCkHi-k0x0+∑j=1p∑k=0lCkHi-kjx-j+∑j=1q∑k=0lCkLi-kju-j+∑j=0i-l-1∑k=0lCkKi-j-1-kuj+∑j=i-li-1∑k=0i-j-1CkKi-j-1-kuj+∑j=0lDjui-j=∑k=0lCkHi-k0x0+∑j=1p∑k=0lCkHi-kjx-j+∑j=1q∑k=0lCkLi-kju-j+∑j=0i-1∑k=0lCkKi-j-1-kuj+∑j=0lDjui-j=Hi0x0+∑j=1pHijx-j+∑j=1qLiju-j+∑j=0i-1Ki-j-1uj+∑j=0lDjui-j=Hi0x0+∑j=1pHijx-j+∑j=1qLiju-j+∑j=0i-l-1Ki-j-1uj+∑j=i-li-1Ki-j-1+Di-juj+D0ui=Hi0x0+∑j=1pHijx-j+∑j=1pLiju-j+∑j=0i-l-1Ki-j-1uj+∑j=i-li-1K-i-j-1uj+D0ui.Then, we get the linear algebraic equation(25)yi=Qi+1x~0+Ri+1u0i+1,with(26)Qi+1=Hi0Hi1⋯HipLi1⋯Liq∈R+r×np+1+mq,(27)Ri+1=Ki-1Ki-2⋯KlK-l-1⋯K-0D0∈R+r×i+1m.The following lemmas will be needed in the sequel.Lemma 6.
For anyi∈N+, we have(28)Hi0=∑k=0pHi-1-k0Ak.Proof.
First, fori=1, we have H10=A0=∑k=0pH-k0Ak and (28) holds. Secondly, suppose that (28) holds for k∈σ1i. We prove that it holds for k=i+1.
Fori∈σ1p, we have(29)Hi+10=∑k=0iAkHi-k0=∑k=0i-1AkHi-k0+Ai=∑k=0i-1Ak∑j=0pHi-k-1-j0Aj+Ai=∑k=0i-1Ak∑j=0i-k-1Hi-k-1-j0Aj+Ai=∑j=0i-1∑k=0i-j-1AkHi-j-1-k0Aj+Ai=∑j=0i-1Hi-j0Aj+Ai=∑j=0iHi-j0Aj=∑j=0pHi-j0Aj.
Fori≥p+1, we have(30)Hi+10=∑k=0pAkHi-k0=∑k=0pAk∑j=0pHi-k-1-j0Aj=∑j=0p∑k=0pAkHi-j-1-k0Aj=∑j=0pHi-j0Aj.
Thus, (28) is satisfied in step i+1. Hence, (28) holds for any i∈N+.Lemma 7.
For alli∈N, we have(31)Hi+1j=Hij+1+Hi0Aj,j∈σ0p-1,Hi+1p=Hi0Ap,(32)Li+1j=Lij+1+Hi0Bj,j∈σ1q-1,Li+1q=Hi0Bq.Proof.
Fori=0, we have(33)H0j+1+H00Aj=Aj=H1j,j∈σ0p-1,H00Ap=Ap=H1p.Let i∈N+. For j=0, we have(34)Hi1+Hi0A0=∑k=1pHi-k0Ak+Hi0A0=∑k=0pHi-k0Ak;then by Lemma 6, we get (35)Hi+10=Hi1+Hi0A0.For j∈σ1p-1, we have(36)Hi+1j-Hij+1=∑k=1p-j+1Hi+1-k0Ak-1+j-∑k=1p-jHi-k0Ak+j=∑k=0p-jHi-k0Ak+j-∑k=1p-jHi-k0Ak+j=Hi0Aj.And for j=p, we have(37)Hi+1p=∑k=11Hi+1-k0Ak-1+p=Hi0Ap.
Similarly, we prove that (32) holds.Lemma 8.
We have(38)Hi+10=Hi1+Hi0A0+Ci+1,i<l,Hi+10=Hi1+Hi0A0,i≥l.
And for alli∈N, we have(39)Hi+1j=Hij+1+Hi0Aj,j∈σ1p-1,Hi+1p=Hi0Ap,(40)Li+1j=Lij+1+Hi0Bj,j∈σ1q-1,Li+1q=Hi0Bq.Proof.
Leti<l. For j=0, we have(41)Hi+10=∑k=0lCkHi+1-k0=∑k=0i+1CkHi+1-k0=∑k=0iCkHi+1-k0+Ci+1=∑k=0iCkHi-k1+Hi-k0A0+Ci+1=∑k=0iCkHi-k1+∑k=0iCkHi-k0A0+Ci+1=Hi1+Hi0A0+Ci+1,for j∈σ1p-1, we have(42)Hi+1j=∑k=0lCkHi+1-kj=∑k=0iCkHi+1-kj=∑k=0iCkHi-kj+1+Hi-k0Aj=Hij+1+Hi0Aj,and, for j=p, we have(43)Hi+1p=∑k=0iCkHi+1-kp=∑k=0iCkHi-k0Ap=Hi0Ap.
Fori≥l, with j∈σ0p-1, we have(44)Hi+1j=∑k=0lCkHi+1-kj=∑k=0lCkHi-kj+1+Hi-k0Aj=Hij+1+Hi0Aj,and, for j=p, we have(45)Hi+1p=∑k=0lCkHi+1-kp=∑k=0lCkHi-k0Ap=Hi0Ap.
Similarly, we prove that (40) holds.


## 3. 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∈N+ steps if, for any nonnegative final output yf∈R+r, there exists a nonnegative input sequence ui∈R+m,i∈σ0N-1, which steers the output of the system from x-j=0,j∈σ0p to yf, with u-j=0 for j∈σ1p; that is, yf=yN-1.Definition 10.
The system modeled by (2) and (3) is said to be output reachable if there exists a positive integer N∈N+ such that the system is output reachable in N 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 vectorv∈R+n with exactly one of its components being nonzero and all the others being zero is called monomial vector or i-monomial if the nonzero component is in the ith position.Definition 11.
A square matrixA∈R+n×n is said to be monomial if it contains n linearly independent monomial columns.An important property of monomial matrices is given by the following result.Lemma 12 (see [18]).
LetA∈R+n×n. Then A-1 exists and is nonnegative if and only if A is a monomial matrix. Furthermore, A-1 is also a monomial matrix.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∈N+, the output reachability matrix RN includes a monomial submatrix of order r×r(r≤Nm).Proof.
Sufficiency. Let yf∈R+r be the final output to be reached. From (21) or (25), we have (46)yN-1=QNx~0+RNu0N.With x~0=0, this gives(47)yN-1=RNu0N.
The matrixRN includes a monomial submatrix of order r×r, and without loss of generality, we can assume that (48)RN=R1R2such that R1∈R+r×r is a monomial matrix and R2∈R+r×Nm-r. Hence, by Lemma 12, we have R1-1∈R+r×r. Thus, for (49)u0N=R1-1yf0∈R+Nm,we get(50)yN-1=R1R2R1-1yf0=yf;that is, system (2) and (3) is output reachable.
Necessity. Assume that system (2) and (3) is output reachable for some N∈N+. Thus, for every z∈R+r there exists an input uN∈R+Nm such that (51)z=RNuN,with RN=riji∈σ1r,j∈σ1Nm and uN=ujj∈σ1Nm. In particular, for z=e1, with e1 being the first column of Ir, we have(52)∑j=1Nmr1juj=1,and for i∈σ2r, we have(53)∑j=1Nmrijuj=0.
So by (52), there exists k∈σ1Nm such that uk≠0, and consequently by equation (53) we have rik=0 for all i∈σ2r. Hence, if r1k≠0, then the kth column of RN is monomial. If r1k=0, then the kth column of RN is null, which implies that(54)∑j=1Nmr1juj=1,j≠k,∑j=1Nmrijuj=0,j≠k,i∈σ2r.
The same reasoning gives the existence of a1-monomial column or another null column of RN. Since the columns of RN are not all null, then RN has at least one 1-monomial column.
The same reasoning forz=ei, i∈σ2r, leads to the existence of a i-monomial column. Hence by Definition 11, the matrix RN contains a monomial submatrix of order r×r. The proposition is proved.Remark 14.
If system (2) and (3) is output reachable and(55)RNTRNRNT-1∈R+Nm×r,then the nonnegative input u0N∈R+Nm which steers the output of the system from x-j=0,j∈σ0p, to any desired nonnegative final output yf∈R+r, with u-j=0 for j∈σ1q, can be computed by the formula (56)u0N=RNTRNRNT-1yf.


## 4. 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∈N+ steps if, for any nonnegative initial state sequence x-j∈R+n(j∈σ0p) and any nonnegative initial input sequence u-j∈R+m(j∈σ1q), there exists a nonnegative input sequence ui∈R+m,i∈σ0N-1, which steers the output of the system from x-j to zero; that is, yN-1=0.Definition 16.
The system modeled by (2) and (3) is said to be null output controllable if there exists a positive integer N∈N+ such that the system is null output controllable in N steps.The characterization of the null output controllability is given by the following proposition.Proposition 17.
The system modeled by (2) and (3) is null output controllable if and only if, for some N∈N+, the null output controllability matrix QN is null.Proof.
Sufficiency. From (21) or (25), at the step i=N-1, we have(57)yN-1=QNx~0+RNu0N;since QN=0, then, for u0N=0, we have yN-1=0; that is, system (2) and (3) is null output controllable.
Necessity. If system (2) and (3) is null output controllable, then, for some N∈N+, there exists an input u0N∈R+Nm such that(58)QNx~0+RNu0N=0.
SinceRNu0N∈R+r and QNx~0∈R+r, then QNx~0=0, which ensures that QN=0 because x~0∈R+np+1+mq by Definition 15, is arbitrary. This finishes the proof.System (2) and (3) describes the evolution of the state and output of a system in the nonnegative orthant with delays in the state, input, and output. However, we can rewrite this system in such a way that these delays disappear from the state equation. Let xii∈N be the solution of (2) and define a new state variable x~i∈R+np+1+mq for i∈N by(59)x~i=xixi-1⋮xi-pui-1⋮ui-q.It is readily verified that the statex~i satisfies(60)x~i+1=Ax~i+Bui,i∈N,x~0∈R+np+1+mq, and the output yi satisfies (61)yi=Cx~i+D0ui,i∈N,where(62)A=A0A1⋯⋯ApB1⋯⋯⋯BqIn0⋯⋯00⋯⋯⋯00⋱⋱⋮⋮⋮⋮⋱⋱⋱⋮⋮⋮0⋯0In00⋯⋯⋯00⋯⋯⋯00⋯⋯⋯0⋮⋮Im⋱⋮⋮⋮0⋱⋱⋮⋮⋮⋮⋱⋱⋱⋮0⋯⋯⋯00⋯0Im0,B=B00⋮0Im0⋮0∈R+np+1+mq×m,C=C11C12,where(63)C11=C0C1⋯Cl0⋯0∈R+r×np+1,C12=D1⋯Dl0⋯0∈R+r×mq.Then we have the following result.Proposition 18.
The system modeled by (2) and (3) is null output controllable if and only if there exists N∈N+ such that CAN-1=0. In particular, if A is nilpotent, then system (2) and (3) is null output controllable.Proof.
Sufficiency. The general solution of (60) is given by (64)x~i=Aix~0+∑j=0i-1Ai-j-1Buj,i∈N.
Forui=0,i∈σ0N-1, we have x~N-1=AN-1x~0, this implies that (65)yN-1=Cx~N-1+D0uN-1=CAN-1x~0+D0uN-1=0;since CAN-1=0. Hence system (2) and (3) is null output controllable.
Necessity. System (2) and (3) is null output controllable, according to Proposition 17, QN=0 for some N∈N+. For u0N=0, we have(66)yN-1=QNx~0+RNu0N=0∀x~0∈R+np+1+mq.
On the other hand, we haveyN-1=CAN-1x~0=0; then CAN-1=0 since x~0 is arbitrary. This completes the proof.In the remainder of this section and without loss of generality, we assume thatp≥q. Indeed, if p<q we can set Aj=0 for j∈σp+1q.Lemma 19.
For alli≥p, we have(67)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.
Letui=0 for i∈N. Then, according to (64), we have(68)x~i=Aix~0,i∈N. On the other hand, from (18), for all i≥p we have(69)x~i=xixi-1⋮xi-p0⋮0=Hi0Hi1⋯HipLi1⋯LiqHi-10Hi-11⋯Hi-1pLi-11⋯Li-1q⋮⋮⋮⋮⋮Hi-p0Hi-p1⋯Hi-ppLi-p1⋯Li-pq0⋯⋯00⋯0⋮⋮⋮⋮0⋯⋯00⋯0x0x-1⋮x-pu-1⋮u-q.
Hence by identification between (68) and (69), we get that (67) holds.Proposition 20.
If, for somes∈σ0l,Cs is injective, that is, rankCs=n, then system (2) and (3) is null output controllable implying that A is a nilpotent matrix.Proof.
System (2) and (3) is null output controllable; then by Proposition 17, for some N∈N+, we have QN=0. If N≤l, then HN-1j=0,j∈σ0p, Ck=0,k∈σNl and LN-1k=0,k∈σ1q, Dj=0,j∈σNl. Then s∈σ0N-1 and CsHN-1-sj=0,CsLN-1-sk=0. Since Cs is injective, then CsTCs is invertible, which implies that HN-1-sj=0 and LN-1-sk=0. By Lemma 7, for i∈σ0p we get HN-1-s+ij=0 and LN-1-s+ik=0. According to Lemma 19, we have AN-1-s+p=0, that is, A is nilpotent. Similarly, we prove that A is nilpotent if N≥1+l. This finishes the proof.


## 5. Output Controllability

By generalization of definition given in [11] we shall formulate the fundamental definitions for output controllability of system (2) and (3) as follows.Definition 21.
The system modeled by (2) and (3) is said to be output controllable in N∈N+ steps if for any nonnegative initial state sequence x-j∈R+nj∈σ0p and any nonnegative initial input sequence u-j∈R+mj∈σ1q, there exists a nonnegative input sequence ui∈R+m,i∈σ0N-1, which steers the output of the system from x-j to any desired nonnegative final output yf∈R+r, i.e., yN-1=yf.Definition 22.
The system modeled by (2) and (3) is said to be output controllable if there exists a positive integer N∈N+ such that the system is output controllable in N steps.The characterization of the output controllability is given by the following proposition.Proposition 23.
The system modeled by (2) and (3) is output controllable if and only if it is output reachable and null output controllable.Proof.
Necessity. It is evident.
Sufficiency. Since system (2) and (3) is output reachable, then, according to Proposition 13, RN1 for some N1∈N+ includes a monomial submatrix of order r×r. On the other hand, system (2) and (3) is null output controllable; hence, according to Proposition 17, QN2=0 for some N2∈N+. Then, for N=max⁡N1,N2, the matrix (70)RN=R~RN1contains a monomial submatrix of order r×r, with R~∈R+r×N-N1m. Hence, by proof of Proposition 13, for any yf∈R+r, there exists a nonnegative input u0N∈R+Nm such that (71)yf=RNu0N.
And by Lemma8, we have QN=0. Then for every x~0∈R+np+1+mq we get that(72)yN-1=QNx~0+RNu0N=yf;that is, system (2) and (3) is output controllable. The proposition is proved.


## 6. Numerical Examples

Example 1 (output reachability).
Suppose that we are given system (2) and (3) with p=q=l=2 and matrices(73)A0=000000001,A1=000010000,A2=000010110,B0=010,B1=100,B2=010,C0=100001,C1=001000,C2=000001,D0=10,D1=00,D2=10.The conditions of Proposition 13 are satisfied because the output reachability matrix in five steps(74)R5=K3K2K-1K-0D0=0020110000contains a monomial submatrix of order 2×2.
By simple calculation, we get(75)R5TR5R5T-1=01000.40000.20∈R+5×2.Then the nonnegative input sequence that permitted to transfer the output from the zero initial conditions to the final output yf=10.5T according to (56) is (76)u05=R5TR5R5T-1yf=0.500.400.2T.Table 1 gives the values of the output at each step. We see that the final output has been reached within a number of steps of the input data sequence greater than n+1=4.Table 1
Values of the outputs in the transfer steps.
N
1
2
3
4
5
yN-1
0.5
0
0.5
0
1.4
0
0
0
1
0.5This comes up to be a particularity of discrete delay systems. This is not satisfied in the case of discrete systems without delay where the steps to reach the final outputyf are always less than or equal to n+1. This results from the Cayley-Hamilton theorem.The next two examples study, respectively, the conditions of the null output controllability and output controllability.Example 2 (null output controllability).
Consider the system modeled by (2) and (3) with matrices(77)A0=000000100,A1=000000100,A2=000200000,B1=010,B2=002,C0=300001,C1=100100,C2=200000.System (2) and (3) is null output controllable because the null output controllability matrix in four steps(78)Q4=H30H31H32L31L32is null.
System (2), (3) in this example is null output controllable for any Ck∈R+r×3k∈σ02 because the matrix A is nilpotent with index k=6; that is, Ak-1≠0 and Ak=0.Example 3 (output controllability).
Consider the system modeled by (2) and (3) with matrices(79)A0=000100100,A1=000000110,A2=000000101,B0=001000,B1=001200,B2=000013,C0=000100,C1=100000,C2=100100,D0=1403,D1=1101,D2=0211.
System (2) and (3) is output reachable because the output reachability matrix in tree steps(80)R3=K-1K-0D0=021114110103contains a monomial submatrix of order 2×2.
The conditions of Proposition17 are satisfied because the null output controllability matrix in four steps(81)Q4=H30H31H32L31L32is null, so by proof of Proposition 23, the system is output controllable in four steps.


## 7. Conclusion

The output controllability of positive discrete linear systems with delays in state, control, and output has been considered. Necessary and sufficient conditions for the positivity of discrete systems have been established (Proposition4). Criteria for output reachability (Proposition 13) and null output controllability (Proposition 17) of the positive discrete systems have been also proved. It has been shown that output reachability and null output controllability together imply output controllability (Proposition 23). Numerical examples were given to illustrate the results.We think that the techniques used in this paper can be useful to investigate the output reachability, null output controllability, and output controllability problems for different positive dynamical systems such as switched systems, fractional systems with different orders, and fractional switched systems.



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