Output Reachable Set Analysis for Periodic Positive Systems

In this study, we use the union of the bounding hyperpyramids to estimate the output reachable set for periodic positive systems under two classes of exogenous disturbances. Optimization algorithms are used to obtain the smallest bounding hyperpyramids possible. Finally, numerical examples are given to verify the theoretical results.


Introduction
In recent decades, periodic systems, whose parameters contain periodic properties, have received a lot of attention. e study of periodic systems is motivated by the fact that many real-world practical systems possess periodic characteristics and can thus be described as periodic systems. For example, a pendulum system is a periodic system with cyclic behavior [1]. In the literature, a lot of important results on linear periodic systems have been reported, see, e.g., [2][3][4][5][6][7][8][9][10].
On the contrary, several dynamical systems indicate that the state and output variables are forced to be positive, or at least nonnegative, at all times when initial conditions and inputs are nonnegative. is type of system is known in the literature as the positive system. Positive systems have many fields of application: chemistry, biology, sociology, and economics. Many important properties and applications of positive systems can be found in the works of Luenberger [11], Farina and Rinaldi [12], and Kaczorek [13].
Recently, dynamical systems with both periodic and positive properties have attracted the interest of many researchers. e stability and stabilization problems of discrete-time periodic positive systems were studied by Bougatef et al. [14] and Ait Rami and Napp [15]. For periodic positive systems with time delays, the stability problem was addressed in [16]. See also [17], for more periodic systems' research results.
Reachability, as a fundamental concept in control theory, has received a lot of attention. Numerous authors have investigated the reachability of positive systems for both discrete and continuous systems [18][19][20][21][22][23]. e set containing all system outputs that are reachable from the origin under a prescribed set of inputs is called the output reachable set. Many researchers have focused on characterizing the outputs' reachable sets for dynamical systems, but when the input signal is constrained, transferring the output of the system to an arbitrary desired output from the origin is generally difficult, so the popular technique in the literature is to determine a region as small as possible to bind the output reachable set. A usual strategy is to estimate the output reachable set by a few ellipsoids that can be determined by solving linear matrix inequality (LMI). Reachable set estimation problem has been studied for time-delay systems [24,25], singular systems [26], periodic systems [27], positive systems [28], switched positive systems [29], etc.
is paper aims to solve the output reachable set estimation problem for periodic positive systems under two possible classes of nonnegative exogenous disturbances based on two norms. e organization of this paper is the following form. In Section 2, the formulation of the problem is given and the positivity of the considered system is defined. In Section 3, results on the estimation of the output reachable set are given under two classes of exogenous disturbances, and the optimization techniques are used to obtain the smallest possible hyperpyramids which bound the output reachable set. Some examples are considered to verify the theoretical results.
Notations. e notations used in this work are Z + set of nonnegative integers, Z 0 + set of positive integers, R set of real numbers, R n set of n-dimensional real vectors, R m×n set of m × n real matrices, R n + positive orthant of R n , R n + closed positive orthant of R n , A T transpose of A, 1 vector

Preliminaries
Consider the discrete-time periodic linear system described by where x t ∈ R n is the state vector, ϑ t ∈ R m is the exogenous input signal, y t ∈ R r is the output vector, and A t , B t , and C t are real matrices with appropriate dimensions and we assume that there exists T ∈ Z 0 + such that, for all t ∈ Z + , we Definition 1. Systems (1)-(2) are said to be positive if, for any initial condition x 0 ∈ R n + and for any exogenous input ϑ t ∈ R m + and t ∈ Z + , we have x t ∈ R n + and y t ∈ R r + , for all t ∈ Z + .
In the rest of this paper, we assume that systems (1)-(2) are positive.
e results presented in this paper are divided into two cases according to the following norms: e output reachable set in this case is defined as follows: Case 2: ϑ ∈ Σ + ∞,1 ≔ ϑ|‖ϑ‖ ∞,1 ≤ 1 and ϑ t ≥ 0, for all t ∈ Z + }. e output reachable set in this case is defined as follows: e output reachable set in this paper will be bounded by hyperpyramids of the form

Main Results
In this section, we will estimate the output reachable set of systems (1)-(2) by hyperpyramids for the two cases mentioned above. For that, we will use optimization techniques to obtain the smallest possible hyperpyramids.

□
According to this lemma, we get the following bounding of the output reachable set. Theorem 1. Consider systems (1)- (2), and assume that there

□
If T � 1, then systems (1)-(2) reduce to the linear positive time-invariant system given by From eorem 1, we can deduce the following corollary to determine an estimation of the output reachable set of systems (12)-(13). (13), and assume that there exists ξ ∈ R r + such that the following condition holds:

□
If T � 1, we can deduce the following corollary for the estimation of the output reachable set of the linear positive time-invariant systems (12)-(13). (12)- (13), and assume that there exists ξ ∈ R r + such that the following condition holds:
To solve optimization problem (23) which subjects to the conditions in eorem 2, we adopt genetic algorithm (GA). For more details about GA, see [31,32].

Examples.
In this section, we will give examples for the two cases.

Conclusion
In this work, we have studied the estimation of the output reachable set for positive periodic systems. Results ( eorem 1 and eorem 2) have been found for two exogenous disturbance classes, and optimization techniques have been used to minimize the volume of bounding hyperpyramids. Numerical examples have been used to verify the theoretical results.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. International Journal of Mathematics and Mathematical Sciences 5