Output Monomial Reachability and Zero Output Controllability of Positive Switched Systems

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.


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][8][9][10][11][12][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. ey have the property that all descriptive variables can only take positive values, or at least nonnegative values. ese 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. e 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]. e 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].
is 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 Section 3. In Section 4, necessary and sufficient conditions for the output monomial reachability are provided. e zero output controllability problem is explored in Section 5.

Preliminaries
e symbols Z + and R + denote the sets of nonnegative integers and nonnegative real numbers, respectively. R n is the n-dimensional Euclidean space and R n + is the set of all n-dimensional nonnegative real vectors. In addition, R n×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 A T for the transpose of the matrix A and I n the n × n identity matrix. For any i, j ∈ Z + , with Let A � a 1 , a 2 , . . . , a r , 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 � a i 1 a i 2 , . . . , a i l where a i j ∈ A, l ∈ Z + \ 0 { }. e length of the word ω is the number of letters it is composed of, written as |ω| � l.
e set of all words over the alphabet is a free monoid A * for concatenation, whose neutral element is the empty word denoted by ε � ∅. Clearly, for any word ω ∈ A * , εω � ωε � ω and |ε| � 0.
Let A 1 , A 2 , . . . , A r be a set of n × n matrices and i 1 , i 2 , . . . , i l ∈ [1, r]. If ω is a word in A * , we set if ω � ε, Next, we introduce a class of nonnegative matrices, namely, the monomial matrices.

Definition 1.
A nonnegative vector v 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 matrix E 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 where x t ∈ R n is the state vector, u t ∈ R m is the control input, y t ∈ R p is the output vector, and δ: is a switching sequence. Given a control u t , t ∈ [0, k − 1], and a switching sequence δ(t), t ∈ [0, k − 1], the solution of system (2), with the initial condition x 0 , at time k, can be expressed as [25] where Definition 3. e discrete system (2) is called positive if for any switching sequence δ, any initial condition x 0 ∈ R n + , and for any input u t ∈ R m + , t ∈ Z + , the state x t ∈ R n + and the output y t ∈ R p + for all t ∈ Z + . Proposition 1.
e discrete system (2) is positive if and only if, for all i ∈ Ω, then equations (3) and (4) imply that for all x 0 ∈ R n + and

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.
It is clearly seen that when x 0 � 0, the output can be written as where 2 International Journal of Mathematics and Mathematical Sciences R k (δ) is called the output reachability matrix associated to the switching sequence δ. Definition 5. e set of all nonnegative linear combinations of the columns of a matrix A ∈ R p×n is called polyhedral convex cone, namely, Polyhedral convex cones play an important role in the output reachability of positive systems since the set of all reachable outputs in k steps is a polyhedral cone belonging to the nonnegative orthant.
Cone(R k (δ)) is a polyhedral cone generated by the columns of the output reachability matrix R k (δ) associated to the switching sequence δ of length k + 1. e 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.

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. e positive switched system (2) with x 0 � 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 u 0 , u 1 , . . . , u k−1 such that with e i being the ith canonical vector of R p .

Lemma 1.
If B ∈ R m×n + and V ∈ R n + are such that BV is an imonomial vector, then B includes an i-monomial column.
which implies that for all j ∈ Since AB contains an i-monomial column, then there exists j ∈ [1, m] such that Ab j is an i-monomial vector. Applying Lemma 1, it yields that the matrix A has an imonomial vector. e proposition below contains a necessary and sufficient condition for output monomial reachability using the output reachable matrix R N associated with all possible switching sequences.

Proposition 2. e 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
International Journal of Mathematics and Mathematical Sciences 3 includes an p × p monomial submatrix.
Proof. Assume that for all i ∈ [1, p], there exist k ≥ 1, a switching sequence δ: [0, k] ⟶ Ω, and nonnegative control inputs u 0 , u 1 , . . . , u k−1 such that y k � e i . is implies that the following equality and pose δ(k) � l i and δ(l − 1) � j i . en, Conversely, let i ∈ [1, p]; then, R N includes an i-monomial column, which implies that there exist Let k � |ω i | + 1 and pose . We get from (5) that erefore, the system is output monomially reachable. □ Remark 1. In the case of single output systems (p � 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: For the two subsystems, we have x 0 � 0, y 1 � C i B i u 0 � 0, and i ∈ [1,2]. So, neither one is output reachable in one step. But R 1 � C 1 B 1 C 1 B 2 C 2 B 1 C 2 B 2 � 0 1 1 0 , and hence the positive system (2) is output reachable in one step. Indeed, let δ(0) � 1 and δ(1) � 2; then, for all y d ∈ R + , for u 0 � y d we get Example 3. Consider the positive system switching among the following subsystems: We have So, the two subsystems are not output monomially reachable in one step. But Hence, the positive system (2) is output monomially reachable in one step. Indeed, for any Also, for any y d � 0 β , let δ(0) � δ(1) � 1 and u 0 � β.
On the other hand, it is clearly seen that this system is not reachable in one step because the vector 1 1 can never be reached in one step.

Corollary 2.
If the positive switched system (2) is output monomially reachable, then the matrix C 1 C 2 · · · C r has an p × p monomial submatrix.
Proof. Suppose the system is output monomially reachable. us, for all i ∈ [1, p], there exist ω i ∈ A * , j i , l i ∈ Ω such that C l i ω i (A 1 , A 2 , . . . , A r )B j i has an i-monomial column. Applying Corollary 1, it yields that the matrix C l i has an imonomial column.
Hence, the matrix C 1 C 2 · · · C r has an p × p monomial submatrix.

Zero Output Controllability
To present our main results for zero output controllability, we introduce the following definition.
□ Example 4. Consider the positive switched system composed of two subsystems with the following matrices: By choosing j � 2 and ω � a 1 a 2 ∈ A * , we get erefore, positive switched system (2) is zero output controllable. Also, the positive switched system (2) is zero output controllable, since there exists a word ω � a 2 a 1 such that (ω(A 1 , A 2 )) 4 � 0, that is, ω(A 1 , A 2 ) is nilpotent.

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 discretetime 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 (Proposition 2) and the zero output controllability of discrete-time positive switched systems (Proposition 3). ese conditions were then applied to numerical examples to illustrate their application and to support the theoretical results. e results discussed here will be of great value for our future work that will treat another class of positive systems.

Data Availability
No data were used to support this study.