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.
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 , biological systems , dc/dc convertors, oscillators , chaos generators , 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  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 , biology, stochastic processes (Markov chains or hidden Markov models) , chemical processing , communication science and information , 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 . As a result, some known properties of linear systems cannot be applied for positive systems (for more details, see ).
Combing the characteristics of general switched systems and positive systems, results are obtained on positive switched systems . 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 . It is also applied in HIV treatment modelling , formation flying , and communication systems . Many problems have been examined concerning positive switched systems, such as stability and stabilizability  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 Section 3. In Section 4, necessary and sufficient conditions for the output monomial reachability are provided. The zero output controllability problem is explored in Section 5.
The symbols and denote the sets of nonnegative integers and nonnegative real numbers, respectively. is the n-dimensional Euclidean space and is the set of all n-dimensional nonnegative real vectors. In addition, represents the space of matrices with real entries and represents the set of all matrices with nonnegative entries. If , we say that is nonnegative and write . We write for the transpose of the matrix and the identity matrix. For any , with , we set = .
Let , be an alphabet whose elements are called letters. A word over the alphabet is a finite sequence of elements of ; it will be denoted by where , . The length of the word is the number of letters it is composed of, written as .
The set of all words over the alphabet is a free monoid for concatenation, whose neutral element is the empty word denoted by . Clearly, for any word , and .
Let be a set of matrices and . If is a word in , we set
Next, we introduce a class of nonnegative matrices, namely, the monomial matrices.
Definition 1. A nonnegative vector is said to be monomial if it contains precisely one nonzero entry. We will call it an -monomial vector if the nonzero component is in the position.
Definition 2. A nonnegative matrix 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 equationwhere is the state vector, is the control input, is the output vector, and is a switching sequence.
Given a control , , and a switching sequence , , the solution of system (2), with the initial condition , at time , can be expressed as where
Definition 3. The discrete system (2) is called positive if for any switching sequence , any initial condition , and for any input , , the state and the output for all .
Proposition 1. The discrete system (2) is positive if and only if, for each , , , and .
Proof. If , , and for all , then equations (3) and (4) imply that for all and we have and for all .
Conversely, assume that the positive switched system (1) is positive. Let , , and for all . Then, from (2), for , we obtain and . Thus, and , since may be arbitrary. Now, assuming that , then from (2), for , we have . It follows that , since 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 output is said to be reachable in steps if there exists a switching sequence , , and inputs for that steer the output of system (2) from to , namely, . System (2) is called output reachable if every nonnegative output is reachable in some step .
It is clearly seen that when , the output can be written aswhere is called the output reachability matrix associated to the switching sequence .
Definition 5. The set of all nonnegative linear combinations of the columns of a matrix 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 steps is a polyhedral cone belonging to the nonnegative orthant.
is a polyhedral cone generated by the columns of the output reachability matrix associated to the switching sequence of length . The length of the switching sequence is the cardinality of the discrete-time interval and it is denoted, for short, by means of the notation .
Clearly, the positive switched system (2) is output reachable if there exist switching sequences of lengths , respectively, , such that
Example 1. Consider positive switched system (2) with and the following matrices:DefineWe getTherefore, 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 is said to be output monomially reachable if, for all , there exist , a switching sequence , and nonnegative control inputs such thatwith being the ith canonical vector of .
Lemma 1. If and are such that is an i-monomial vector, then includes an i-monomial column.
Proof. Let , with being the vector columns of and . If is i-monomial, thenwhich implies that for all , we have and there exists some such that . Therefore, there exists such that . Hence, includes an i-monomial column.
Corollary 1. Let and . If includes an i-monomial column, then has an i-monomial column.
Proof. Let be the vector columns of ; then,Since contains an i-monomial column, then there exists such that is an i-monomial vector. Applying Lemma 1, it yields that the matrix has an i-monomial vector.
The proposition below contains a necessary and sufficient condition for output monomial reachability using the output reachable matrix 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 stepsincludes an monomial submatrix.
Proof. Assume that for all , there exist , a switching sequence , and nonnegative control inputs such that . This implies that the following equality
Then, there exists such that is an -monomial vector. By Lemma 1, includes an i-monomial column. Letand pose and . Then, includes an i-monomial column. For , we have includes an monomial submatrix.
Conversely, let ; then, includes an i-monomial column, which implies that there exist , , and such that contains an i-monomial column.
Let and poseLet satisfying , and , where . Then,Set . Then, there exists such that and , for .
Let and , for all .
We get from (5) thatTherefore, the system is output monomially reachable.
Remark 1. In the case of single output systems , 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 , , and . So, neither one is output reachable in one step. But , and hence the positive system (2) is output reachable in one step. Indeed, let and ; then, for all , for we get .
Example 3. Consider the positive system switching among the following subsystems:We haveSo, the two subsystems are not output monomially reachable in one step. ButHence, the positive system (2) is output monomially reachable in one step. Indeed, for any , let and . Then, .
Also, for any , let and . Then, .
On the other hand, it is clearly seen that this system is not reachable in one step because the vector can never be reached in one step.
Corollary 2. If the positive switched system (2) is output monomially reachable, then the matrix has an monomial submatrix.
Proof. Suppose the system is output monomially reachable. Thus, for all , there exist , such that has an i-monomial column. Applying Corollary 1, it yields that the matrix has an i-monomial column.
Hence, the matrix has an 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 , there exist , a switching sequence , and nonnegative control input , such that
Proposition 3. The positive switched system (2) is zero output controllable if and only if there exist and such that
Proof. If the system is zero output controllable, then in particular, for , there exist , a switching sequence , and , such that .
It follows thatThen,Let and ; then, .
Conversely, let with , , , , and . Then, for each , we haveIf , then , and for and , we obtain , for all , which completes the proof.
Corollary 3. The positive switched system (2) is zero output controllable if there exists such that is nilpotent.
Proof. Assume that there exists such that
Let . Then, . 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:By choosing and , we get . Therefore, positive switched system (2) is zero output controllable. Also, the positive switched system (2) is zero output controllable, since there exists a word such that , that is, is nilpotent.
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 (Proposition 2) 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.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.