# Output Monomial Reachability and Zero Output Controllability of Positive Switched Systems

Authors: Maryam Riouali; Fouad Lahmidi; Abdelwahed Namir
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

In this paper, we present a sufficient condition for the output reachability of discrete-time positive switched systems. Besides, necessary and sufficient conditions for output monomial reachability and zero output controllability are provided. Further, some examples are shown.


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## Body

## 1. Introduction

In recent years, engineers and applied mathematicians have been interested in the study and analysis of switched systems, which represent an important class of hybrid dynamical systems. A switched system is the association of a finite set of differential or difference subsystems and a switching law that indicates at each instant the active system. Switched systems are of a great interest, since they are very convenient in the mathematical modelling of several systems such as network control systems, near-space vehicle control systems [1], biological systems [2], dc/dc convertors, oscillators [3], chaos generators [4], and so on. Based on previous research, many mathematical problems have been posed and investigated such as stability and stabilizability properties [5, 6]. Recent studies examined other issues such as reachability and controllability [7–13]. It is important to note that Babiarz [14] provided important results on the output controllability for standard switched systems.It should be noted that positive systems are of great importance in practice as they appear naturally in various fields of science and technology. They have the property that all descriptive variables can only take positive values, or at least nonnegative values. These systems can be found in economics [15], biology, stochastic processes (Markov chains or hidden Markov models) [16], chemical processing [17], communication science and information [18], etc. The theory of positive systems is more complicated than the one of the standard systems because positive linear systems are defined on cones and not on linear spaces [19]. As a result, some known properties of linear systems cannot be applied for positive systems (for more details, see [20]).Combing the characteristics of general switched systems and positive systems, results are obtained on positive switched systems [21]. The strong interest that this type of system has recently raised is due to its strong presence in the most important areas. As an example, in the field of biology and pharmacokinetics, they are used to describe the dynamics of the viral mutation under drug treatment [22]. It is also applied in HIV treatment modelling [21], formation flying [23], and communication systems [18]. Many problems have been examined concerning positive switched systems, such as stability and stabilizability [24] as well as structural properties, like reachability, controllability, and observability [25, 26].This paper which deals with the output reachability and controllability problem of discrete-time positive switched systems is organized as follows. After some preliminaries in the following section, we provide necessary conditions for the output reachability in Section3. In Section 4, necessary and sufficient conditions for the output monomial reachability are provided. The zero output controllability problem is explored in Section 5.


## 2. Preliminaries

The symbolsℤ+ and ℝ+ denote the sets of nonnegative integers and nonnegative real numbers, respectively. ℝn is the n-dimensional Euclidean space and ℝ+n is the set of all n-dimensional nonnegative real vectors. In addition, ℝn×m represents the space of n×m matrices with real entries and ℝ+n×m represents the set of all n×m matrices with nonnegative entries. If E∈ℝ+n×m, we say that E is nonnegative and write E≥0. We write AT for the transpose of the matrix A and In the n×n identity matrix. For any i,j∈ℤ+, with i≤j, we set i,j = i,i+1,…,j.LetA=a1,a2,…,ar,r∈ℤ+\0, be an alphabet whose elements are called letters. A word over the alphabet A is a finite sequence of elements of A; it will be denoted by ω=ai1ai2,…,ail where aij∈A, l∈ℤ+\0. The length of the word ω is the number of letters it is composed of, written as ω=l.The set of all words over the alphabet is a free monoidA∗ for concatenation, whose neutral element is the empty word denoted by ε=∅. Clearly, for any word ω∈A∗, εω=ωε=ω and ε=0.LetA1,A2,…,Ar be a set of n×n matrices and i1,i2,…,il∈1,r. If ω is a word in A∗, we set(1)ωA1,A2,…,Ar=In,ifω=ε,Ai1Ai2,…,Ail,ifω=ai1ai2,…,ail.Next, we introduce a class of nonnegative matrices, namely, the monomial matrices.Definition 1.
A nonnegative vectorv is said to be monomial if it contains precisely one nonzero entry. We will call it an i-monomial vector if the nonzero component is in the ith position.Definition 2.
A nonnegative matrixE is a monomial matrix if it has only one nonzero entry in every row and every column.
In this paper, we consider a discrete-time switched system described by the difference state equation(2)xt+1=Aδtxt+Bδtut,t∈ℤ+,yt=Cδtxt,where xt∈ℝn is the state vector, ut∈ℝm is the control input, yt∈ℝp is the output vector, and δ:ℤ+⟶Ω=1,r is a switching sequence.
Given a controlut, t∈0,k−1, and a switching sequence δt, t∈0,k−1, the solution of system (2), with the initial condition x0, at time k, can be expressed as [25](3)xk=ℋδk,0x0+∑t=0k−1ℋδk,t+1Bδtut,where(4)ℋδt,l=In,t=l≥0,Aδt−1Aδt−2⋯Aδlt>l.Definition 3.
The discrete system (2) is called positive if for any switching sequence δ, any initial condition x0∈ℝ+n, and for any input ut∈ℝ+m, t∈ℤ+, the state xt∈ℝ+n and the output yt∈ℝ+p for all t∈ℤ+.Proposition 1.
The discrete system (2) is positive if and only if, for each i∈Ω, Ai∈ℝ+n×n, Bi∈ℝ+n×m, and Ci∈ℝ+p×n.Proof.
IfAi∈ℝ+n×n, Bi∈ℝ+n×m, and Ci∈ℝ+p×n for all i∈Ω, then equations (3) and (4) imply that for all x0∈ℝ+n and ut∈ℝ+m,t∈ℤ+ we have xt∈ℝ+n and yt∈ℝ+p for all t∈ℤ+.
Conversely, assume that the positive switched system (1) is positive. Let δt=i, i∈Ω, and ut=0 for all t∈ℤ+. Then, from (2), for t=0, we obtain x1=Aix0∈ℝ+n and y0=Cix0∈ℝ+p. Thus, Ai∈ℝ+n×n and Ci∈ℝ+p×n, since x0 may be arbitrary. Now, assuming that x0=0, then from (2), for t=0, we have x1=Biu0∈ℝ+n. It follows that Bi∈ℝ+n×m, since u0 may be arbitrary.


## 3. Output Reachability of Switched Positive Systems

In the main result of this section, we provide a sufficient condition for the output reachability of system (2). Before giving our result, some definitions concerning the output reachability of positive switched systems should be cited.Definition 4.
An outputyd∈ℝ+p is said to be reachable in N steps if there exists a switching sequence δt, t∈0,N, and inputs ut∈ℝ+m for t∈0,N−1 that steer the output of system (2) from x0=0 to yd, namely, yd=yN. System (2) is called output reachable if every nonnegative output yd∈ℝ+p is reachable in some step N.
It is clearly seen that whenx0=0, the output can be written as(5)yk=ℛkδuk,where(6)ℛkδ=Cδkℋδk,1Bδ0Cδkℋδk,2Bδ1⋯CδkBδk−1,uk=u0u1⋮uk−1.
ℛkδ is called the output reachability matrix associated to the switching sequence δ.Definition 5.
The set of all nonnegative linear combinations of the columns of a matrixA∈ℝp×n is called polyhedral convex cone, namely,(7)ConeA=y∈ℝp:y=Ax,x∈ℝ+n.
Polyhedral convex cones play an important role in the output reachability of positive systems since the set of all reachable outputs ink steps is a polyhedral cone belonging to the nonnegative orthant.
Coneℛkδ is a polyhedral cone generated by the columns of the output reachability matrix ℛkδ associated to the switching sequence δ of length k+1. The length of the switching sequence δ is the cardinality of the discrete-time interval 0,k and it is denoted, for short, by means of the notation δ=k+1.
Clearly, the positive switched system (2) is output reachable if there exist switching sequences δ1,δ2,…,δh of lengths k1+1,k2+1,…,kh+1, respectively, h≥1, such that(8)∪t=1hConeℛktδt=ℝ+p.Example 1.
Consider positive switched system (2) with Ω=1,2 and the following matrices:(9)A1=1111,A2=1100,B1=0110,B2=1001,C1=1101,C2=1011.
Define(10)δ10=2,δ11=2,δ20=1,δ21=1.
We get(11)Coneℛ1δ1∪Coneℛ1δ2=Cone1011∪Cone1110=ℝ+2.
Therefore, the system is output reachable.


## 4. Output Monomial Reachability of Switched Positive Systems

We study in this section the concept of output monomial reachability and provide necessary and sufficient conditions for this property. First, we recall the following definition and give some preliminary results.Definition 6.
The positive switched system (2) with x0=0 is said to be output monomially reachable if, for all i∈1,p, there exist k≥1, a switching sequence δ:0,k⟶Ω, and nonnegative control inputs u0,u1,…,uk−1 such that(12)yk=ei,with ei being the ith canonical vector of ℝp.Lemma 1.
IfB∈ℝ+m×n and V∈ℝ+n are such that BV is an i-monomial vector, then B includes an i-monomial column.Proof.
LetB=b1b2…bn∈ℝ+m×n, with bl=b1l,b2l,…,bmlT being the vector columns of B and V=v1,v2,…,vnT∈ℝ+n. If BV is i-monomial, then(13)BV=v1b1+v2b2+⋯+vnbn=0,0,…,α,0,…,0T,withα>0,which implies that for all j∈1,n, we have vjbj=0,0,…,vjbij,0,…,0T and there exists some j∈1,n such that vjbj≠0. Therefore, there exists j∈1,n such that bj=0,0,…,bij,0,…,0T≠0. Hence, B includes an i-monomial column.Corollary 1.
LetA∈ℝ+p×n and B∈ℝ+n×m. If AB includes an i-monomial column, then A has an i-monomial column.Proof.
Letb1,b2,…,bm be the vector columns of B; then,(14)AB=Ab1Ab2…Abm
SinceAB contains an i-monomial column, then there exists j∈1,m such that Abj is an i-monomial vector. Applying Lemma 1, it yields that the matrix A has an i-monomial vector.
The proposition below contains a necessary and sufficient condition for output monomial reachability using the output reachable matrixℛN associated with all possible switching sequences.Proposition 2.
The positive switched system (2) is output monomially reachable if and only if there exists some positive integer N such that the output reachability matrix in N steps(15)ℛN=C1ωA1,…,ArB1⋯C1ωA1,…,ArBrC2ωA1,…,ArB1⋯CrωA1,…,ArBrω∈A∗,0≤ω≤N−1includes an p×p monomial submatrix.Proof.
Assume that for alli∈1,p, there exist k≥1, a switching sequence δ:0,k⟶Ω, and nonnegative control inputs u0,u1,…,uk−1 such that yk=ei. This implies that the following equality ∑t=0k−1Cδkℋδk,t+1Bδtut=eiholds.
Then, there existsl∈1,k such that CδkAδk−1⋯AδlBδl−1ul−1 is an i-monomial vector. By Lemma 1, CδkAδk−1⋯AδlBδl−1 includes an i-monomial column. Let(16)ωi=ε,ifk=l,aδk−1aδk−2⋯aδl,ifl≤k−1,and pose δk=li and δl−1=ji. Then, CliωiA1,A2,…,ArBji includes an i-monomial column. For N=max1≤i≤pωi+1, we have ℛN includes an p×p monomial submatrix.
Conversely, leti∈1,p; then, ℛN includes an i-monomial column, which implies that there exist ωi∈A∗, 0≤ωi≤N−1, and li,ji∈Ω such that CliωiA1,A2,…,ArBji contains an i-monomial column.
Letk=ωi+1 and pose(17)ωi=ε,ifωi=0,aik−1aik−2,…,ai1,ifωi≥1.
Letδ:0,k⟶Ω satisfying δ0=ji, δk=li and δh=ih, where 1≤h≤k−1. Then,(18)CliωiA1,A2,…,ArBji=CδkAδk−1⋯Aδ1Bδ0.
SetM=CδkAδk−1⋯Aδ1Bδ0. Then, there exists j∈1,m such that Mij>0 and Mlj=0, for l∈1,p\i.
Letu0=0,…,0,1/Mij,0,…,0T∈ℝ+m and ul=0∈ℝ+m, for all j∈1,k−1.
We get from (5) that(19)yk=CδkAδk−1⋯Aδ1Bδ0u0=Mu0=ei.
Therefore, the system is output monomially reachable.Remark 1.
In the case of single output systemsp=1, the Proposition 2 gives in fact a characterization of the output reachability of system (2).
Let us now consider some examples.Example 2.
Consider positive switched system (2) consisting of two subsystems with the following matrices:(20)A1,B1,C1=1010,10,01,A2,B2,C2=0101,01,10.
For the two subsystems, we havex0=0, y1=CiBiu0=0, and i∈1,2. So, neither one is output reachable in one step. But ℛ1=C1B1C1B2C2B1C2B2=0110, and hence the positive system (2) is output reachable in one step. Indeed, let δ0=1 and δ1=2; then, for all yd∈ℝ+, for u0=yd we get y1=C2B1u0=yd.Example 3.
Consider the positive system switching among the following subsystems:(21)A1,B1,C1=1010,10,0010,A2,B2,C2=0101,01,0100.
We have(22)C1B1=01,C2B2=10.
So, the two subsystems are not output monomially reachable in one step. But(23)R1=C1B1C1B2C2B1C2B2=00011000.
Hence, the positive system (2) is output monomially reachable in one step. Indeed, for any yd=α0, let δ0=δ1=2 and u0=α. Then, y1=C2B2u0=α0=yd.
Also, for anyyd=0β, let δ0=δ1=1 and u0=β. Then, y1=C1B1u0=0β=yd.
On the other hand, it is clearly seen that this system is not reachable in one step because the vector11 can never be reached in one step.Corollary 2.
If the positive switched system (2) is output monomially reachable, then the matrix C1C2⋯Cr has an p×p monomial submatrix.Proof.
Suppose the system is output monomially reachable. Thus, for alli∈1,p, there exist ωi∈A∗, ji,li∈Ω such that CliωiA1,A2,…,ArBji has an i-monomial column. Applying Corollary 1, it yields that the matrix Cli has an i-monomial column.
Hence, the matrixC1C2⋯Cr has an p×p monomial submatrix.


## 5. Zero Output Controllability

To present our main results for zero output controllability, we introduce the following definition.Definition 7.
The positive switched system (2) is said to be zero output controllable if, for all x0∈ℝ+n, there exist k≥1, a switching sequence δ:0,k⟶Ω, and nonnegative control input ut∈ℝ+m, t∈0,k−1 such that(24)yk=0.Proposition 3.
The positive switched system (2) is zero output controllable if and only if there exist ω∈A∗ and j∈1,r such that(25)CjωA1,A2,…,Ar=0.Proof.
If the system is zero output controllable, then in particular, forx0=1,1,…,1T, there exist k≥1, a switching sequence δ:0,k⟶Ω, and ut∈ℝ+m, t∈0,k−1 such that yk=0.
It follows that(26)CδkAδk−1⋯Aδ0x0=0.
Then,(27)CδkAδk−1⋯Aδ0=0.
Letω=aδk−1⋯aδ0 and j=δk; then, CjωA1,A2,…,Ar=0.
Conversely, letk=ω with ω=aik−1⋯ai0, δl=il, l∈0,k−1, δk=j, and u0=⋯=uk−1=0. Then, for each x0∈ℝ+n, we have(28)yk=CδkAδk−1⋯Aδ0x0=CjωA1,A2,…,Arx0=0.
Ifω=0, then Cj=0, and for δ0=δ1=j and u0=0, we obtain y1=0, for all x0∈ℝ+n, which completes the proof.Corollary 3.
The positive switched system (2) is zero output controllable if there exists ω∈A∗ such that ωA1,A2,…,Ar is nilpotent.Proof.
Assume that there existsω=ai1⋯aik such that ωA1,A2,…,ArN=Ai1⋯AikAi1⋯Aik⋯⋯Ai1⋯Aik=0.
Letω˜=ai1⋯aikai1⋯aik⋯⋯ai1⋯aik. Then, ω˜A1,A2,…,Ar=0. According to Proposition 3, system (2) is zero output controllable.Example 4.
Consider the positive switched system composed of two subsystems with the following matrices:(29)A1,B1,C1=0100100000100001,1000,10001000,A2,B2,C2=1011001100010000,0100,00010001.
By choosingj=2 and ω=a1a2∈A∗, we get C2ωA1,A2=C2A1A2=0. Therefore, positive switched system (2) is zero output controllable. Also, the positive switched system (2) is zero output controllable, since there exists a word ω=a2a1 such that ωA1,A24=0, that is, ωA1,A2 is nilpotent.


## 6. Conclusions

In this paper, we have addressed a number of issues related to the output reachability, output monomial reachability, and the zero output controllability properties of discrete-time positive switched systems. By means of certain concepts borrowed from the algebra of noncommutative polynomials, we have been able to establish the necessary and sufficient conditions guaranteeing the output monomial reachability (Proposition2) and the zero output controllability of discrete-time positive switched systems (Proposition 3). These conditions were then applied to numerical examples to illustrate their application and to support the theoretical results. The results discussed here will be of great value for our future work that will treat another class of positive systems.



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