# Output Reachable Set Analysis for Periodic Positive Systems

Authors: Ilyas Sitli; Fouad Lahmidi; Abdelwahed Namir; Mouhcine Naim
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

In recent decades, periodic systems, whose parameters contain periodic properties, have received a lot of attention. The 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., [2–10].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. This 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. The 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 [18–23]. The 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.This 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.The organization of this paper is the following form. In Section2, 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. The notations used in this work are ℤ+ set of nonnegative integers, ℤ+0 set of positive integers,ℝ set of real numbers, ℝn set of n-dimensional real vectors, ℝm×n set of m×n real matrices, ℝ+n positive orthant of ℝn,ℝ¯+n closed positive orthant of ℝn, AT transpose of A, 1 vector 1,1,…,1T, I identity matrix, x≥0x>0 every component of the vector x is nonnegative (positive), A≥0A>0 every entry of matrix A is nonnegative (positive), and σsk=s,s+1,…,k is the finite subset of ℤ+ with s≤k.


## 2. Preliminaries

Consider the discrete-time periodic linear system described by(1)xt+1=Atxt+Btϑt,t∈ℤ+,x0∈ℝn,(2)yt=Ctxt,t∈ℤ+,where xt∈ℝn is the state vector, ϑt∈ℝm is the exogenous input signal, yt∈ℝr is the output vector, and At,Bt, and Ct are real matrices with appropriate dimensions and we assume that there exists T∈ℤ+0 such that, for all t∈ℤ+, we have At=At+T, Bt=Bt+T, and Ct=Ct+T. We note x≔xtt≥0, y≔ytt≥0, and ϑ≔ϑtt≥0.Definition 1.
Systems (1)-(2) are said to be positive if, for any initial condition x0∈ℝ+n and for any exogenous input ϑt∈ℝ+m and t∈ℤ+, we have xt∈ℝ+n and yt∈ℝ+r, for all t∈ℤ+.Remark 1.
Systems (1)-(2) are positive if At≥0,Bt≥0,andCt≥0, for all t∈σ0T−1.In the rest of this paper, we assume that systems (1)-(2) are positive.The results presented in this paper are divided into two cases according to the following norms:ϑ1,1=∑t=0∞ϑt1 and ϑ∞,1=supt≥0ϑt1, where ϑt1=∑i=0mϑt,i, for ϑt=ϑt,1,ϑt,2,…,ϑt,mT∈ℝm.Case 1:ϑ∈Σ1,1+≔ϑ|ϑ1,1≤1andϑt≥0,for allt∈ℤ+.The output reachable set in this case is defined as follows:(3)ℜ1,1+≔y|there isϑ∈Σ1,1+such thatyandϑsatisfy1−2withx0=0.Case 2:ϑ∈Σ∞,1+≔ϑ|ϑ∞,1≤1andϑt≥0,for allt∈ℤ+.The output reachable set in this case is defined as follows:(4)ℜ∞,1+≔y|there isϑ∈Σ∞,1+such thatyandϑsatisfy1−2withx0=0.The output reachable set in this paper will be bounded by hyperpyramids of the form(5)ℑξ≔ζ∈ℝ¯+r|ξTζ≤1,whereξ∈ℝ+r.


## 3. 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.


### 3.1. Estimation of Output Reachable Sets

#### 3.1.1. Case 1

In this case, we consider systems (1)-(2) under zero initial conditions and exogenous input ϑ∈Σ1,1+.Lemma 1.
Letφt:ℝ+r⟶ℝ+,t≥0, be a set of functions such that(i)
φ00=0.(ii)
Ify∈ℜ1,1+ and ϑ∈Σ1,1+ is the corresponding input, we have(6)φt+1yt+1−φtyt≤1Tϑt,∀t∈ℤ+.
Then, we obtainφtyt≤1, ∀y∈ℜ1,1+and∀t∈ℤ+.Proof.
Lety∈ℜ1,1+ and ϑ∈Σ1,1+ be the corresponding input. Then, ∀t∈ℤ+0, and we have(7)φtyt−φt−1yt−1≤1Tϑt−1=ϑt−11,⋮φ1y1−φ0y0≤1Tϑ0=ϑ01.
So, we obtain(8)φtyt−φ0y0≤∑s=0t−11Tϑs≤ϑ1,1≤1.
Sinceφ0y0=φ00=0, then we get φtyt≤1. The proof is completed.According to this lemma, we get the following bounding of the output reachable set.Theorem 1.
Consider systems (1)-(2), and assume that there exist ξl∈ℝ+r, l∈σ0T, with ξT=ξ0, such that, for all l∈σ0T−1,(9)AlTCl+1Tξl+1−ClTξl≤0,BlTCl+1Tξl+1≤1.
Then,ℜ1,1+⊂∪l=0T−1ℑξl (i.e., the output reachable set is bounded by the union of a set of hyperpyramids).Proof.
Letξtt≥0 be the sequence defined by ξt=ξt+T, ∀t∈ℤ+, and let, for t≥0,(10)φt:ℝ+r⟶ℝ+,z⟼ξtTz.
Lety∈ℜ1,1+ and ϑ∈Σ1,1+ be the corresponding input. For any t≥0, there is l∈σ0T−1 such that t≡lT. Then, we obtain(11)φt+1yt+1−φtyt=ξt+1TCt+1At−ξtTCtxt+ξt+1TCt+1Btϑt=ξl+1TCl+1Al−ξlTClxt+ξl+1TCl+1Blϑt≤1Tϑt.
According to Lemma1, we get φtyt≤1. And, the periodicity of ξt implies that yt∈∪l=0T−1ℑξl. So, ℜ1,1+⊂∪l=0T−1ℑξl. The proof is completed.IfT=1, then systems (1)-(2) reduce to the linear positive time-invariant system given by(12)xt+1=Axt+Bϑt,t∈ℤ+,x0∈ℝn.(13)yt=Cxt,t∈ℤ+.From Theorem1, we can deduce the following corollary to determine an estimation of the output reachable set of systems (12)-(13).Corollary 1.
Consider systems (12)-(13), and assume that there exists ξ∈ℝ+r such that the following condition holds:(14)ATCTξ−CTξ≤0,BTCTξ≤1.
Then,ℜ1,1+⊂ℑξ.


#### 3.1.2. Case 2

In this case, we consider systems (1)-(2) under zero initial conditions and exogenous input ϑ∈Σ∞,1+.Lemma 2.
Letφt:ℝ+r⟶ℝ+,t∈ℤ+, be a set of functions such that(i)
φ00=0.(ii)
There existθt∈0,1,t∈ℤ+, such that if y∈ℜ∞,1+ and ϑ∈Σ∞,1+ is the corresponding input, then(15)φt+1yt+1−θtφtyt≤1−θt1Tϑt,∀t∈ℤ+.
Then, we obtainφtyt≤1,∀t∈ℤ+.Proof.
Lety∈ℜ∞,1+ and ϑ∈Σ∞,1+ be the corresponding input, and we have(16)φt+1yt+1−θtφtyt≤1−θt1Tϑt≤1−θtϑ∞,1≤1−θt.
Then,φt+1yt+1−1≤θtφtyt−1.
For any timet∈ℤ+0, we have(17)φtyt−1≤θt−1φt−1yt−1−1≤θt−1θt−2φt−2yt−2−1⋮≤θt−1⋯θ0φ0y0−1.
Sinceφ0y0=φ00=0≤1, it follows that φtyt≤1. The proof is completed.Using this lemma, we can derive the following bounding of the output reachable set.Theorem 2.
Consider systems (1)-(2), and assume that there exist ξl∈ℝ+r, l∈σ0T, with ξT=ξ0 and θl∈0,1, l∈σ0T−1, such that, for all l∈σ0T−1, we have(18)AlTCl+1Tξl+1−θlClTξl≤0,BlTCl+1Tξl+1≤1−θl1.
Then,ℜ∞,1+⊂∪l=0T−1ℑξl.Proof.
We construct two sequencesξtt≥0 and θtt≥0 by posing ξt=ξt+T and θt=θt+T,∀t∈ℤ+. We consider the family of functions:(19)φt:ℝ+r⟶ℝ+,z⟼ξtTz.
Lety∈ℜ∞,1+ and ϑ∈Σ∞,1+ be the corresponding input. For all t∈ℤ+, there is l∈σ0T−1 such that t≡lT. Then, we obtain(20)φt+1yt+1−θtφtyt=ξt+1TCt+1At−θtξtTCtxt+ξt+1TCt+1Btϑt=ξl+1TCl+1Al−θlξlTClxt+ξl+1TCl+1Blϑt≤1−θl1Tϑt=1−θt1Tϑt.
So, Lemma2 implies that φtyt≤1, and from the periodicity of ξt, we obtain yt∈∪l=0T−1ℑξl. So, ℜ∞,1+⊂∪l=0T−1ℑξl. The proof is completed.IfT=1, we can deduce the following corollary for the estimation of the output reachable set of the linear positive time-invariant systems (12)-(13).Corollary 2.
Consider systems (12)-(13), and assume that there exists ξ∈ℝ+r such that the following condition holds:(21)ATCTξ−θCTξ≤0,BTCTξ≤1−θ1.
Then,ℜ∞,1+⊂ℑξ.Remark 2.
In Theorem1 (Theorem 2), the norms of the exogenous inputs are no greater than 1. We can deduce the estimation of the output reachable set if the norms of the exogenous inputs are no greater than a scalar ϑ^>0. If ϑ1,1≤ϑ^ϑ∞,1≤ϑ^, then the output reachable set ℜ1,1+ℜ∞,1+ can be bounded by the union of a set of hyperpyramids ∪l=0T−1ℑξl/ϑ^. Indeed, if we assume that ϑ1,1≤ϑ^ϑ∞,1≤ϑ^, then we obtain ϑ/ϑ^1,1≤1ϑ/ϑ^∞,1≤1. Systems (1)-(2) can be rewritten in the following form:(22)xt+1ϑ^=Atxtϑ^+Btϑtϑ^,ytϑ^=Ctxtϑ^.
So, according to Theorem1 (Theorem 2), yt/ϑ^ can be bounded by ∪l=0T−1ℑξl. Then, yt can be bounded by ∪l=0T−1ℑξl/ϑ^.The hypervolume of the hyperpyramid (5) is equal to 1/r!∏i=1r1/ξi (Carter and Champanerkar [30]). The volume of the bounding hyperpyramids considered in the two cases can be minimised by solving the following optimization problem:(23)min−∑m=1r∑l=0T−1logξl,m,which subjects to the conditions in Theorems 1 or 2, with ξl=ξl,1,ξl,2,…,ξl,rT, l∈σ0T−1.To solve optimization problem (23) which subjects to the conditions in Theorem 2, we adopt genetic algorithm (GA). For more details about GA, see [31, 32].


### 3.2. Examples

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


#### 3.2.1. Case 1

Consider systems (1)-(2) with period T=2, and(24)A0,B0,C0=0.10.60.20.4,0.50.3,0.30.70.10.1,A1,B1,C1=0.50.20.40.2,0.20.1,0.20.60.30.5.By solving problem (23), we obtain(25)ξ0=1.23214.1090,ξ1=1.25492.1621.The bounding hyperpyramids are shown in Figure1 with exogenous inputs defined by ϑt=1−m×mt,m=0,0.1,…,0.9.Figure 1
Estimation of output reachable set withϑ∈Σ1,1+.


#### 3.2.2. Case 2

Consider systems (1)-(2) with periodT=2 and(26)A0,B0,C0=0.50.20.60.4,0.50.3,0.20.50.10.2,A1,B1,C1=0.70.10.30.2,0.10.4,0.10.10.40.6.The optimal values of problem (23) are(27)θ0=0.854,θ1=0.935,ξ0=0.1710.317,ξ1=0.5310.182.The bounding hyperpyramids are shown in Figure2 with exogenous inputs defined by aϑt=randt, bϑt=0.7, and cϑt=sint.Figure 2
Estimation of the output reachable set withϑ∈Σ∞,1+.
(a)(b)(c)


## 3.1. Estimation of Output Reachable Sets

### 3.1.1. Case 1

In this case, we consider systems (1)-(2) under zero initial conditions and exogenous input ϑ∈Σ1,1+.Lemma 1.
Letφt:ℝ+r⟶ℝ+,t≥0, be a set of functions such that(i)
φ00=0.(ii)
Ify∈ℜ1,1+ and ϑ∈Σ1,1+ is the corresponding input, we have(6)φt+1yt+1−φtyt≤1Tϑt,∀t∈ℤ+.
Then, we obtainφtyt≤1, ∀y∈ℜ1,1+and∀t∈ℤ+.Proof.
Lety∈ℜ1,1+ and ϑ∈Σ1,1+ be the corresponding input. Then, ∀t∈ℤ+0, and we have(7)φtyt−φt−1yt−1≤1Tϑt−1=ϑt−11,⋮φ1y1−φ0y0≤1Tϑ0=ϑ01.
So, we obtain(8)φtyt−φ0y0≤∑s=0t−11Tϑs≤ϑ1,1≤1.
Sinceφ0y0=φ00=0, then we get φtyt≤1. The proof is completed.According to this lemma, we get the following bounding of the output reachable set.Theorem 1.
Consider systems (1)-(2), and assume that there exist ξl∈ℝ+r, l∈σ0T, with ξT=ξ0, such that, for all l∈σ0T−1,(9)AlTCl+1Tξl+1−ClTξl≤0,BlTCl+1Tξl+1≤1.
Then,ℜ1,1+⊂∪l=0T−1ℑξl (i.e., the output reachable set is bounded by the union of a set of hyperpyramids).Proof.
Letξtt≥0 be the sequence defined by ξt=ξt+T, ∀t∈ℤ+, and let, for t≥0,(10)φt:ℝ+r⟶ℝ+,z⟼ξtTz.
Lety∈ℜ1,1+ and ϑ∈Σ1,1+ be the corresponding input. For any t≥0, there is l∈σ0T−1 such that t≡lT. Then, we obtain(11)φt+1yt+1−φtyt=ξt+1TCt+1At−ξtTCtxt+ξt+1TCt+1Btϑt=ξl+1TCl+1Al−ξlTClxt+ξl+1TCl+1Blϑt≤1Tϑt.
According to Lemma1, we get φtyt≤1. And, the periodicity of ξt implies that yt∈∪l=0T−1ℑξl. So, ℜ1,1+⊂∪l=0T−1ℑξl. The proof is completed.IfT=1, then systems (1)-(2) reduce to the linear positive time-invariant system given by(12)xt+1=Axt+Bϑt,t∈ℤ+,x0∈ℝn.(13)yt=Cxt,t∈ℤ+.From Theorem1, we can deduce the following corollary to determine an estimation of the output reachable set of systems (12)-(13).Corollary 1.
Consider systems (12)-(13), and assume that there exists ξ∈ℝ+r such that the following condition holds:(14)ATCTξ−CTξ≤0,BTCTξ≤1.
Then,ℜ1,1+⊂ℑξ.


### 3.1.2. Case 2

In this case, we consider systems (1)-(2) under zero initial conditions and exogenous input ϑ∈Σ∞,1+.Lemma 2.
Letφt:ℝ+r⟶ℝ+,t∈ℤ+, be a set of functions such that(i)
φ00=0.(ii)
There existθt∈0,1,t∈ℤ+, such that if y∈ℜ∞,1+ and ϑ∈Σ∞,1+ is the corresponding input, then(15)φt+1yt+1−θtφtyt≤1−θt1Tϑt,∀t∈ℤ+.
Then, we obtainφtyt≤1,∀t∈ℤ+.Proof.
Lety∈ℜ∞,1+ and ϑ∈Σ∞,1+ be the corresponding input, and we have(16)φt+1yt+1−θtφtyt≤1−θt1Tϑt≤1−θtϑ∞,1≤1−θt.
Then,φt+1yt+1−1≤θtφtyt−1.
For any timet∈ℤ+0, we have(17)φtyt−1≤θt−1φt−1yt−1−1≤θt−1θt−2φt−2yt−2−1⋮≤θt−1⋯θ0φ0y0−1.
Sinceφ0y0=φ00=0≤1, it follows that φtyt≤1. The proof is completed.Using this lemma, we can derive the following bounding of the output reachable set.Theorem 2.
Consider systems (1)-(2), and assume that there exist ξl∈ℝ+r, l∈σ0T, with ξT=ξ0 and θl∈0,1, l∈σ0T−1, such that, for all l∈σ0T−1, we have(18)AlTCl+1Tξl+1−θlClTξl≤0,BlTCl+1Tξl+1≤1−θl1.
Then,ℜ∞,1+⊂∪l=0T−1ℑξl.Proof.
We construct two sequencesξtt≥0 and θtt≥0 by posing ξt=ξt+T and θt=θt+T,∀t∈ℤ+. We consider the family of functions:(19)φt:ℝ+r⟶ℝ+,z⟼ξtTz.
Lety∈ℜ∞,1+ and ϑ∈Σ∞,1+ be the corresponding input. For all t∈ℤ+, there is l∈σ0T−1 such that t≡lT. Then, we obtain(20)φt+1yt+1−θtφtyt=ξt+1TCt+1At−θtξtTCtxt+ξt+1TCt+1Btϑt=ξl+1TCl+1Al−θlξlTClxt+ξl+1TCl+1Blϑt≤1−θl1Tϑt=1−θt1Tϑt.
So, Lemma2 implies that φtyt≤1, and from the periodicity of ξt, we obtain yt∈∪l=0T−1ℑξl. So, ℜ∞,1+⊂∪l=0T−1ℑξl. The proof is completed.IfT=1, we can deduce the following corollary for the estimation of the output reachable set of the linear positive time-invariant systems (12)-(13).Corollary 2.
Consider systems (12)-(13), and assume that there exists ξ∈ℝ+r such that the following condition holds:(21)ATCTξ−θCTξ≤0,BTCTξ≤1−θ1.
Then,ℜ∞,1+⊂ℑξ.Remark 2.
In Theorem1 (Theorem 2), the norms of the exogenous inputs are no greater than 1. We can deduce the estimation of the output reachable set if the norms of the exogenous inputs are no greater than a scalar ϑ^>0. If ϑ1,1≤ϑ^ϑ∞,1≤ϑ^, then the output reachable set ℜ1,1+ℜ∞,1+ can be bounded by the union of a set of hyperpyramids ∪l=0T−1ℑξl/ϑ^. Indeed, if we assume that ϑ1,1≤ϑ^ϑ∞,1≤ϑ^, then we obtain ϑ/ϑ^1,1≤1ϑ/ϑ^∞,1≤1. Systems (1)-(2) can be rewritten in the following form:(22)xt+1ϑ^=Atxtϑ^+Btϑtϑ^,ytϑ^=Ctxtϑ^.
So, according to Theorem1 (Theorem 2), yt/ϑ^ can be bounded by ∪l=0T−1ℑξl. Then, yt can be bounded by ∪l=0T−1ℑξl/ϑ^.The hypervolume of the hyperpyramid (5) is equal to 1/r!∏i=1r1/ξi (Carter and Champanerkar [30]). The volume of the bounding hyperpyramids considered in the two cases can be minimised by solving the following optimization problem:(23)min−∑m=1r∑l=0T−1logξl,m,which subjects to the conditions in Theorems 1 or 2, with ξl=ξl,1,ξl,2,…,ξl,rT, l∈σ0T−1.To solve optimization problem (23) which subjects to the conditions in Theorem 2, we adopt genetic algorithm (GA). For more details about GA, see [31, 32].


## 3.1.1. Case 1

In this case, we consider systems (1)-(2) under zero initial conditions and exogenous input ϑ∈Σ1,1+.Lemma 1.
Letφt:ℝ+r⟶ℝ+,t≥0, be a set of functions such that(i)
φ00=0.(ii)
Ify∈ℜ1,1+ and ϑ∈Σ1,1+ is the corresponding input, we have(6)φt+1yt+1−φtyt≤1Tϑt,∀t∈ℤ+.
Then, we obtainφtyt≤1, ∀y∈ℜ1,1+and∀t∈ℤ+.Proof.
Lety∈ℜ1,1+ and ϑ∈Σ1,1+ be the corresponding input. Then, ∀t∈ℤ+0, and we have(7)φtyt−φt−1yt−1≤1Tϑt−1=ϑt−11,⋮φ1y1−φ0y0≤1Tϑ0=ϑ01.
So, we obtain(8)φtyt−φ0y0≤∑s=0t−11Tϑs≤ϑ1,1≤1.
Sinceφ0y0=φ00=0, then we get φtyt≤1. The proof is completed.According to this lemma, we get the following bounding of the output reachable set.Theorem 1.
Consider systems (1)-(2), and assume that there exist ξl∈ℝ+r, l∈σ0T, with ξT=ξ0, such that, for all l∈σ0T−1,(9)AlTCl+1Tξl+1−ClTξl≤0,BlTCl+1Tξl+1≤1.
Then,ℜ1,1+⊂∪l=0T−1ℑξl (i.e., the output reachable set is bounded by the union of a set of hyperpyramids).Proof.
Letξtt≥0 be the sequence defined by ξt=ξt+T, ∀t∈ℤ+, and let, for t≥0,(10)φt:ℝ+r⟶ℝ+,z⟼ξtTz.
Lety∈ℜ1,1+ and ϑ∈Σ1,1+ be the corresponding input. For any t≥0, there is l∈σ0T−1 such that t≡lT. Then, we obtain(11)φt+1yt+1−φtyt=ξt+1TCt+1At−ξtTCtxt+ξt+1TCt+1Btϑt=ξl+1TCl+1Al−ξlTClxt+ξl+1TCl+1Blϑt≤1Tϑt.
According to Lemma1, we get φtyt≤1. And, the periodicity of ξt implies that yt∈∪l=0T−1ℑξl. So, ℜ1,1+⊂∪l=0T−1ℑξl. The proof is completed.IfT=1, then systems (1)-(2) reduce to the linear positive time-invariant system given by(12)xt+1=Axt+Bϑt,t∈ℤ+,x0∈ℝn.(13)yt=Cxt,t∈ℤ+.From Theorem1, we can deduce the following corollary to determine an estimation of the output reachable set of systems (12)-(13).Corollary 1.
Consider systems (12)-(13), and assume that there exists ξ∈ℝ+r such that the following condition holds:(14)ATCTξ−CTξ≤0,BTCTξ≤1.
Then,ℜ1,1+⊂ℑξ.


## 3.1.2. Case 2

In this case, we consider systems (1)-(2) under zero initial conditions and exogenous input ϑ∈Σ∞,1+.Lemma 2.
Letφt:ℝ+r⟶ℝ+,t∈ℤ+, be a set of functions such that(i)
φ00=0.(ii)
There existθt∈0,1,t∈ℤ+, such that if y∈ℜ∞,1+ and ϑ∈Σ∞,1+ is the corresponding input, then(15)φt+1yt+1−θtφtyt≤1−θt1Tϑt,∀t∈ℤ+.
Then, we obtainφtyt≤1,∀t∈ℤ+.Proof.
Lety∈ℜ∞,1+ and ϑ∈Σ∞,1+ be the corresponding input, and we have(16)φt+1yt+1−θtφtyt≤1−θt1Tϑt≤1−θtϑ∞,1≤1−θt.
Then,φt+1yt+1−1≤θtφtyt−1.
For any timet∈ℤ+0, we have(17)φtyt−1≤θt−1φt−1yt−1−1≤θt−1θt−2φt−2yt−2−1⋮≤θt−1⋯θ0φ0y0−1.
Sinceφ0y0=φ00=0≤1, it follows that φtyt≤1. The proof is completed.Using this lemma, we can derive the following bounding of the output reachable set.Theorem 2.
Consider systems (1)-(2), and assume that there exist ξl∈ℝ+r, l∈σ0T, with ξT=ξ0 and θl∈0,1, l∈σ0T−1, such that, for all l∈σ0T−1, we have(18)AlTCl+1Tξl+1−θlClTξl≤0,BlTCl+1Tξl+1≤1−θl1.
Then,ℜ∞,1+⊂∪l=0T−1ℑξl.Proof.
We construct two sequencesξtt≥0 and θtt≥0 by posing ξt=ξt+T and θt=θt+T,∀t∈ℤ+. We consider the family of functions:(19)φt:ℝ+r⟶ℝ+,z⟼ξtTz.
Lety∈ℜ∞,1+ and ϑ∈Σ∞,1+ be the corresponding input. For all t∈ℤ+, there is l∈σ0T−1 such that t≡lT. Then, we obtain(20)φt+1yt+1−θtφtyt=ξt+1TCt+1At−θtξtTCtxt+ξt+1TCt+1Btϑt=ξl+1TCl+1Al−θlξlTClxt+ξl+1TCl+1Blϑt≤1−θl1Tϑt=1−θt1Tϑt.
So, Lemma2 implies that φtyt≤1, and from the periodicity of ξt, we obtain yt∈∪l=0T−1ℑξl. So, ℜ∞,1+⊂∪l=0T−1ℑξl. The proof is completed.IfT=1, we can deduce the following corollary for the estimation of the output reachable set of the linear positive time-invariant systems (12)-(13).Corollary 2.
Consider systems (12)-(13), and assume that there exists ξ∈ℝ+r such that the following condition holds:(21)ATCTξ−θCTξ≤0,BTCTξ≤1−θ1.
Then,ℜ∞,1+⊂ℑξ.Remark 2.
In Theorem1 (Theorem 2), the norms of the exogenous inputs are no greater than 1. We can deduce the estimation of the output reachable set if the norms of the exogenous inputs are no greater than a scalar ϑ^>0. If ϑ1,1≤ϑ^ϑ∞,1≤ϑ^, then the output reachable set ℜ1,1+ℜ∞,1+ can be bounded by the union of a set of hyperpyramids ∪l=0T−1ℑξl/ϑ^. Indeed, if we assume that ϑ1,1≤ϑ^ϑ∞,1≤ϑ^, then we obtain ϑ/ϑ^1,1≤1ϑ/ϑ^∞,1≤1. Systems (1)-(2) can be rewritten in the following form:(22)xt+1ϑ^=Atxtϑ^+Btϑtϑ^,ytϑ^=Ctxtϑ^.
So, according to Theorem1 (Theorem 2), yt/ϑ^ can be bounded by ∪l=0T−1ℑξl. Then, yt can be bounded by ∪l=0T−1ℑξl/ϑ^.The hypervolume of the hyperpyramid (5) is equal to 1/r!∏i=1r1/ξi (Carter and Champanerkar [30]). The volume of the bounding hyperpyramids considered in the two cases can be minimised by solving the following optimization problem:(23)min−∑m=1r∑l=0T−1logξl,m,which subjects to the conditions in Theorems 1 or 2, with ξl=ξl,1,ξl,2,…,ξl,rT, l∈σ0T−1.To solve optimization problem (23) which subjects to the conditions in Theorem 2, we adopt genetic algorithm (GA). For more details about GA, see [31, 32].


## 3.2. Examples

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


### 3.2.1. Case 1

Consider systems (1)-(2) with period T=2, and(24)A0,B0,C0=0.10.60.20.4,0.50.3,0.30.70.10.1,A1,B1,C1=0.50.20.40.2,0.20.1,0.20.60.30.5.By solving problem (23), we obtain(25)ξ0=1.23214.1090,ξ1=1.25492.1621.The bounding hyperpyramids are shown in Figure1 with exogenous inputs defined by ϑt=1−m×mt,m=0,0.1,…,0.9.Figure 1
Estimation of output reachable set withϑ∈Σ1,1+.


### 3.2.2. Case 2

Consider systems (1)-(2) with periodT=2 and(26)A0,B0,C0=0.50.20.60.4,0.50.3,0.20.50.10.2,A1,B1,C1=0.70.10.30.2,0.10.4,0.10.10.40.6.The optimal values of problem (23) are(27)θ0=0.854,θ1=0.935,ξ0=0.1710.317,ξ1=0.5310.182.The bounding hyperpyramids are shown in Figure2 with exogenous inputs defined by aϑt=randt, bϑt=0.7, and cϑt=sint.Figure 2
Estimation of the output reachable set withϑ∈Σ∞,1+.
(a)(b)(c)


## 3.2.1. Case 1

Consider systems (1)-(2) with period T=2, and(24)A0,B0,C0=0.10.60.20.4,0.50.3,0.30.70.10.1,A1,B1,C1=0.50.20.40.2,0.20.1,0.20.60.30.5.By solving problem (23), we obtain(25)ξ0=1.23214.1090,ξ1=1.25492.1621.The bounding hyperpyramids are shown in Figure1 with exogenous inputs defined by ϑt=1−m×mt,m=0,0.1,…,0.9.Figure 1
Estimation of output reachable set withϑ∈Σ1,1+.


## 3.2.2. Case 2

Consider systems (1)-(2) with periodT=2 and(26)A0,B0,C0=0.50.20.60.4,0.50.3,0.20.50.10.2,A1,B1,C1=0.70.10.30.2,0.10.4,0.10.10.40.6.The optimal values of problem (23) are(27)θ0=0.854,θ1=0.935,ξ0=0.1710.317,ξ1=0.5310.182.The bounding hyperpyramids are shown in Figure2 with exogenous inputs defined by aϑt=randt, bϑt=0.7, and cϑt=sint.Figure 2
Estimation of the output reachable set withϑ∈Σ∞,1+.
(a)(b)(c)


## 4. Conclusion

In this work, we have studied the estimation of the output reachable set for positive periodic systems. Results (Theorem1 and Theorem 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.



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