# Input-Output Finite-Time Control of Positive Switched Systems with Time-Varying and Distributed Delays

Authors: Leipo Liu; Xiangyang Cao; Zhumu Fu; Shuzhong Song
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

The problem of input-output finite-time control of positive switched systems with time-varying and distributed delays is considered in this paper. Firstly, the definition of input-output finite-time stability is extended to positive switched systems with time-varying and distributed delays, and the proof of the positivity of such systems is also given. Then, by constructing multiple linear copositive Lyapunov functions and using the mode-dependent average dwell time (MDADT) approach, a state feedback controller is designed, and sufficient conditions are derived to guarantee that the corresponding closed-loop system is input-output finite-time stable (IO-FTS). Such conditions can be easily solved by linear programming. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Positive switched systems are a class of dynamics whose state and output are nonnegative whenever the initial conditions and inputs are nonnegative. In the last decades, the research of positive switched systems is a hot topic due to their many practical applications in communication networks [1], the viral mutation dynamics under drug treatment [2], formation flying [3], systems theory [4, 5], and so forth. The stability analysis and controller synthesis of positive switched systems have been highlighted by many researchers [6–10]. However, most of the results mentioned above discussed the asymptotic stability (Lyapunov stability).Contrary to asymptotic stability, some researchers focus on finite-time stability (FTS) of positive switched systems: that is, given a bound on the initial condition, the system state does not exceed a certain threshold during a specified time interval. Some works dealing with analysis and controller design of FTS control problems have been published [11–16]. Recently, the definition of IO-FTS is proposed by [17]; this definition is that a system is said to be IO-FTS if, given a class of norm bounded input signals over a specified time interval 0,T, the outputs of the system are also norm bounded over 0,T. IO-FTS involves signals defined over a finite-time interval and does not necessarily require the inputs and outputs to belong to the same class, and IO-FTS constraints permit specifying quantitative bounds on the controlled variables to be fulfilled during the transient response. This definition of IO-FTS is fully consistent with the definition of FTS. Some related results have been obtained in [18–25]. These results about IO-FTS are mainly involved in impulsive systems [18], linear systems [19–22], stochastic systems [23], Markovian jump systems [24], and impulsive switched linear systems [25]. It is worth noting that the results mentioned above are mainly concerned with nonpositive systems. Huang et al. [26] extended IO-FTS to a class of discrete-time positive switched systems with state delays via average dwell time (ADT), but the time-varying delay and distributed delay were not considered. As we know, multiple time delays, such as time-varying delays and distributed delays, usually occur in many practical systems and may result in system performance deterioration, even instability. Therefore, they must be taken into account in analyzing and implementing any controller scheme. Moreover, MDADT approach allows that every subsystem has its own ADT to make the individual properties of each subsystem unneglected, which is more applicable and less conservative compared with ADT. Then, it is necessary to investigate IO-FTS analysis of positive switched systems with time-varying and distributed delays via MDADT approach. However, this problem is still open.Motivated by the aforementioned factors, the problem of IO-FTS of positive switched systems with time-varying and distributed delays is considered. The main contributions of this paper are as follows: (1) the positivity of positive switched systems with time-varying and distributed delays is proved; (2) the definition of IO-FTS is for the first time extended to positive switched systems with time-varying and distributed delays; (3) by using multiple copositive type Lyapunov function and MDADT approach, a state feedback controller is designed and sufficient conditions for IO-FTS of the corresponding closed-loop system are given. Such conditions can be easily solved by linear programming. The rest of this paper is organized as follows. In Section 2, problem statements and necessary lemmas are given. Main results are given in Section 3. A numerical example is provided in Section 4. Section 5 concludes this paper.Notations. The representation A≻0 (⪰0,≺0,⪯0) means that aij>0 (≥0,<0,≤0), which is applicable to a vector. A≻BA⪰B means that A-B≻0A-B⪰0. R+n is the n-dimensional nonnegative (positive) vector space. Rn×n denotes the space of n×n matrices with real entries. AT denotes the transpose of matrix A. Let x∈Rn; 1-norm x is defined by x=∑k=1n‍|xk|. L1,0,T denotes the space of absolute integrable vector-valued functions on the interval 0,T; that is, st∈L1,0,T if ∫0Tstdt<∞ holds. L∞,0,T denotes the space of the uniformly bounded vector-valued functions on the interval 0,T; that is, st∈L∞,0,T if maxt∈0,Twt<∞ holds. Matrices are assumed to have compatible dimensions for calculating if their dimensions are not explicitly stated.


## 2. Preliminaries and Problem Statements

Consider the following positive switched systems with time-varying and distributed delays:(1)x˙t=Aσtxt+Adσtxt-dt+Abσt∫t-τ2txsds+Gσtut+Bσtwtyt=Cσtxtxθ=φθ,θ∈-τ1,0,where xt∈Rn is the system state and ut∈Rm and yt∈Rs represent the control input and output. σt:0,∞→M_={1,2,…,M} is the system switching signal, where M is the number of subsystems; ∀p∈M_, Ap, Adp, Abp, Bp, Cp, and Gp are constant matrices with appropriate dimensions, dt≥0 denotes time-varying delay, which satisfies d˙t≤h<1, and τ2 is the distributed delay, where τ2 and h are known positive constants. φθ is the initial condition on -τ1,0, τ1=max⁡dt,τ2. wt∈Rl is the exogenous disturbance and defined as(2)W1=w·∈L1,0,T:∫0Twtdt≤dwith a known scalar d>0.Next, we will present some definitions and lemmas for the positive switched system (1) to our further study.Definition 1 (see [14]).
System (1) is said to be positive if for any switching signals σt, any initial conditions φθ⪰0, θ∈-τ1,0, and any disturbance input wt⪰0, the corresponding trajectory satisfies xt⪰0 and yt⪰0 for all t⪰0.Definition 2 (see [14]).
A is called a Metzler matrix if the off-diagonal entries of matrix A are nonnegative.Definition 3 (see [16]).
For any switching signalσt and any t2≥t1≥0, let Nσpt1,t2 denote the switching numbers where the pth subsystem is activated over the interval t1,t2 and Tpt1,t2 denote the total running time of the pth subsystem over the interval t1,t2. If there exist N0p≥0 and Tαp>0 such that(3)Nσpt1,t2≤N0p+Tpt1,t2Tαp,∀t2≥t1≥0,∀p∈M_then Tαp and N0p are called MDADT and mode-dependent chattering bounds, respectively. Generally, we choose N0p=0.Lemma 4 (see [16]).
A matrixA∈Rn×n is a Metzler matrix if and only if there exists a positive constant η such that A+ηIn⪰0.Lemma 5 (see [9]).
LetA∈Rn×n. Then eAt⪰0, ∀t≥0, if and only if A is a Metzler matrix.Lemma 6.
System (1) is positive if and only if Ap, ∀p∈M_, are Metzler matrices and ∀p∈M_, Adp⪰0, Abp⪰0, Bp⪰0, Cp⪰0, and Gp⪰0.Proof.
Sufficiency. For any T>0, t0=0, let t1,t2,…,tNσt0,T denote the switching instants on the interval t0,T. From system (1), ∀t∈t0,t1, we have(4)exp⁡-Aσt0tx˙t-Aσt0xt=exp⁡-Aσt0tAdσt0xt-dt+Abσt0∫t0-τ2t0xsds+Gσt0ut+Bσt0wtyt=Cσt0xt.Integrating both sides of the first equation of the above equations from t0 to t, it yields (5)exp⁡-Aσt0txt=∫t0texp⁡-Aσt0sAdσt0xs-ds+Abσt0∫t0-τ2t0xsds+Gσt0us+Bσt0wsds+xt0.It leads to (6)xt=∫t0texp⁡Aσt0t-sAdσt0xs-ds+Abσt0∫t0-τ2t0xsds+Gσt0us+Bσt0wsds+exp⁡Aσt0tφ0.By Lemma 5, if Aσt0 is a Metzler matrix, Adσt0⪰0, Abσt0⪰0, Bσt0⪰0, Cσt0⪰0, and Gσt0⪰0, then it is easy to obtain that xt⪰0, yt⪰0, ∀t∈t0,t1, and xt1=xt1-⪰0. Similar to the above process, ∀t∈t1,t2, it follows that (7)xt=∫t1texp⁡Aσt1t-sAdσt1xs-ds+Abσt1∫t1-τ2t1xsds+Gσt1us+Bσt1wsds+exp⁡Aσt1tφt1⪰0yt=Cσt1xt⪰0and xt2=xt2-⪰0. Recursively, if Ai, ∀i∈M_, are Metzler matrices, Adi⪰0, Abi⪰0, Bi⪰0, Ci⪰0, and Gi⪰0, then one has that xT⪰0, yT⪰0, ∀t>0.
Necessary. Conversely, suppose that there exists an element aiqg<0, i∈M_, q≠g, where aiqg is in the qth row and gth column of Ai. From system (1), we can obtain (8)x˙qt=∑k=1,k≠q,k≠gnaiqkxkt+aiqqxqt+aiqgxgt+∑k=1nadiqkxkt-dt+∑k=1nabiqk∫t-τ2txksds+∑k=1nbiqkwkt+∑k=1ngiqkukt,where xkt, wkt, and ukt represent the kth elements of xt, wt, and ut, respectively. Then, if xgt≠0, it is obvious that x˙qt<0 is possible whenever xqt=0, which means that xqt+<0. It follows that system (1) is not positive.
Then, suppose thatAdi has an element adiqg<0, i∈M_; we obtain (9)x˙qt=∑k=1,k≠qnaiqkxkt+aiqqxqt+∑k=1,k≠gnadiqkxkt-dt+adiqgxgt-dt+∑k=1nabiqk∫t-τ2txksds+∑k=1nbiqkwkt+∑k=1ngiqkukt.It can be obtained that xqt+<0 is possible whenever xqt=0. It shows that system (1) is not positive.
In the same way, ifAbi has an element abiqg<0, Bi has an element biqg<0, or Gi has an element giqg<0, i∈M_, then system (1) is exactly not positive.
Finally, assume that there exists an elementCiqg<0, i∈M_, q≠g, where Ciqg is in the qth row and gth column of Ci. According to system (1), (10)yqt=∑k=1,k≠q,k≠gnciqkxkt+ciqqxqt+ciqgxgt.It is easy to obtain that yqt<0 is possible whenever xgt>0; it means that system (1) is not positive.
From the above, system (1) is positive under any switching signals if and only if Ai are Metzler matrices, Adi⪰0, Abi⪰0, Bi⪰0, Ci⪰0, and Gi⪰0, ∀i∈M_.
The proof is completed.Next, we will give the definitions of input-output finite-time stability for the positive switched system (1).Definition 7 (IO-FTS).
Consider zero initial conditionx0=0, for a given time constant Tf, disturbances signals W1 defined by (2), and a vector δ≻0; system (1) is said to be IO-FTS with respect to δ,Tf,d,σt, if(11)wt∈W1⟹yTδ≤1,∀t∈0,Tf.If the disturbance wt satisfies wt∈W2, W2 is defined as (12)W2=w·∈L∞,0,T:maxt∈0,Twtdt≤d;then we give Definition 8.Definition 8 (IO-FTS).
Consider zero initial conditionx0=0, for a given time constant Tf, disturbances signals W2, and a vector δ≻0; system (1) is said to be IO-FTS with respect to δ,Tf,d,σt, if (13)wt∈W2⟹yTδ≤1,∀t∈0,Tf.Remark 9.
[26] gives two definitions of discrete positive switched systems for two classes of exogenous disturbances, respectively. Similarly, for continuous positive switched systems, two classes of exogenous disturbances for IO-FTS problem can be also proposed, that is, norm bounded integrable signals wtwt∈L1,0,T,∫0Twtdt≤d and the uniformly bounded signals wtwt∈L∞,0,T,maxt∈0,T⁡wt≤d. Because of the similar process, in this paper, we only focus on the former.The aim of this paper is to design a state feedback controllerut and find a class of switching signals σt for the positive switched system (1) such that the corresponding closed-loop system is IO-FTS.


## 3. Main Results

### 3.1. IO-FTS Analysis

In this subsection, we will focus on the problem of IO-FTS of positive switched system (1) with ut≡0. The following theorem gives sufficient conditions of IO-FTS of system (1) via the MDADT approach.Theorem 10.
Consider system (1) with ut≡0. Given positive constants Tf, λp, p∈M_, and γ and a vector δ≻0, if there exist positive vectors νp, ρp, ϱp, p∈M_, the following inequalities hold:(14)Ψ=diagψp1,ψp2,…,ψpn,ψp1˙,ψp2˙,…,ψpn˙,ψp1¨,ψp2¨,…,ψpn¨≤0(15)bprTνp<γ(16)γd<e-λpTf(17)cprTδ≺νp(18)νp⪯μpνq,ρp⪯μpρq,ϱp⪯μpϱq,where ψpr=adprTνpr-λpνpr+ρpr+τ1ϱpr, ψpr˙=adprTνpr-1-hρpr, ψpr¨=abprTνpr-ϱpr, apradpr,abpr,bpr,cpr represents the rth column vector of the matrix ApAdp,Abp,Bp,Cp, νp=νp1,νp2,…,νpn, ρp=ρp1,ρp2,…,ρpn, ϱp=ϱp1,ϱp2,…,ϱpn, and νpr, ρpr, and ϱpr represent the rth elements of the vectors νp, ρp, and ϱp, respectively. μpμp≥1 satisfies (18); then system (1) is IO-FTS for any switching signal σt with the MDADT(19)Tα>Tα∗=Tfln⁡μpln⁡e-λpTf-ln⁡γd.Proof.
Construct the multiple copositive type Lyapunov-Krasovskii functional for system (1) as follows:(20)Vσtt=Vσtt,xt=xTtνp+∫t-dtteλpt-sxTsρpds+∫-τ10∫t+θteλpt-sxTsϱpdsdθ,where νp, ρp, and ϱp∈R+n, ∀p∈M_.
Along the trajectory of system (1) with ut≡0, we have(21)V˙σtt=xTtApTνp+xTt-dtAdpTνp+∫t-τ2txTsAbpTνpds+wTtBpTνp+λp∫t-dtteλpt-sxTsρpds+xTsρp-1-d˙teλpdtxTt-dtρp+λp∫-τ10∫t+θteλpt-sxTsϱpdsdθ+τ1xTtϱp-∫-τ10e-λpθxTt+θϱpdθ≤xTtApTνp+xTt-dtAdpTνp+∫t-τ2txTsAbpTνpds+wTtBpTνp+λp∫t-dtteλpt-sxTsρpds+xTsρp-1-heλpdtxTt-dtρp+λp∫-τ10∫t+θteλpt-sxTsϱpdsdθ+τ1xTtϱp-∫t-τ1txTsϱpds.Combining (20) and (21) yields(22)V˙σtt-λpVσtt≤xTtApTνp-λpνp+ρp+τ1ϱp+xTt-dtAdpT-1-hρp+∫t-τ2txTsAbpTνpds-∫t-τ1txTsϱpds+wTtBpTνp.Noting that τ1=max⁡dt,τ2, we have(23)∫t-τ2txTsAbpTνpds-∫t-τ1txTsϱpds≤∫t-τ1txTsAbpTνpds-∫t-τ1txTsϱpds=∫t-τ1txTsAbpTνp-ϱpds.From (22) and (23), we obtain(24)V˙σtt-λpVσtt≤xTtApTνp-λpνp+ρp+τ1ϱp+wTtBpTνp+xTt-dtAdpT-1-hρp+∫t-τ1txTsAbpTνp-ϱpds.From (14), (15), and (24), we have(25)V˙σtt-λpVσtt≤wTtBpTνp≤γwTt.Integrating both sides of (25) during the period tk,t for t∈tk,tk+1 leads to(26)Vσtt≤eλσtkt-tkVσtktk+γ∫tkteλσtkt-swTsds.On the other hand, from (18) and (20), one can easily obtain(27)Vσtktk≤μσtkVσtk-tk-.Let N be the switching number of σt over 0,Tf, and denote t1,…,tk,…,tN as the switching instants over the interval 0,Tf. Combining with (2), (26), and (27), we have(28)Vσtt≤eλσtkt-tkVσtktk+γ∫tkteλσtkt-swTsds≤μσtkeλσtkt-tkVσtk-tk-+γ∫tkteλσtkt-swTsds≤μσtkeλσtkt-tkeλσtk-1tk-tk-1Vσtk-1tk-1+γ∫tk-1tkeλσtk-1tk-swTsds+γ∫tkteλσtkt-swTsds≤⋯≤∏i=1kμσtieλσtkt-tke∑i=1k-1λσtiti-ti-1Vσ00+∏i=1kμσtiγ∫0t1eλσ0t1-swTsds+∏i=2kμσtiγ∫t1t2eλσt1t2-swTsds+⋯+γ∫tkteλσtkt-swTsds≤∏p=1MμpNσp0,te∑p=1MλpTp0,tVσ00+γ∫0t∏p=1MμpNσp0,te∑p=1MλpTps,twTsds.According to (3) and choosing λ=maxp∈M_⁡λp+ln⁡μp/Tαp, for t∈0,Tf, one can obtain from (28)(29)Vσtt≤∏p=1MμpTp0,t/Tαpe∑p=1MλpTp0,tVσ00+γd=e∑p=1Mλp+ln⁡μp/TαpTp0,tVσ00+γd≤eλTfVσ00+γd.Under the zero initial condition x0=0, noting the definition of Vσtt and (29), we have(30)Vσtt≥xTtνp,Vσtt≤eλTfγd.From (3) and (30), one has(31)xTtνp≤eλTfγd.From (17) and (31), we have(32)yTtδ=xTtCσtTδ<xTtνp≤eλTfγd.From (19) and (32), we obtain(33)yTtδ<1.Thus, the proof is completed.


### 3.2. IO-FTS Controller Design

In this subsection, the state feedback controllerut=Kσtxt will be designed. For system (1), under the controller ut, the corresponding closed-loop system is given by(34)x˙t=Aσt+GσtKσtxt+Adσtxt-dt+Abσt∫t-τ2txsds+Bσtwtyt=Cσtxtxθ=φθ,θ∈-τ1,0.According to Lemma 4, to guarantee the positivity of system (34), Ap+GpKp should be Metzler matrices, ∀p∈M_. Theorem 11 gives some sufficient conditions to guarantee that the closed-loop system (34) is IO-FTS.Theorem 11.
Consider the positive switched system (34). Given positive constants Tf, λp, and γ and a vector δ≻0, if there exists a set of positive vectors νp, ρp, ϱp, gp, p∈M_, (15)–(17) and the following conditions hold:(35)Ap+GpKpareMetzlermatrices.(36)Φ=diag⁡ϕp1,ϕp2,…,ϕpn,ϕp1˙,ϕp2˙,…,ϕpn˙,ϕp1¨,ϕp2¨,…,ϕpn¨≤0,where ϕpr=adprTνpr-λpνpr+ρpr+τ1ϱpr+gpr, ϕpr˙=adprTνpr-1-hρpr, ϕpr¨=abprTνpr-ϱpr, gp=KpTGpTνp, apradpr,abpr,bpr,cpr represents the rth column vector of the matrix ApAdp,Abp,Bp,Cp, gpr represents the rth elements of vector gp, νp=νp1,νp2,…,νpn, ρp=ρp1,ρp2,…,ρpn, ϱp=ϱp1,ϱp2,…,ϱpn, and νpr, ρpr, and ϱpr represent the rth elements of the vectors νp, ρp, and ϱp, respectively. μpμp≥1 satisfies (18); then the closed-loop system (34) is IO-FTS for any switching signal σt with MDADT (19).Proof.
According to Lemma4, we get that Ap+GpKp is a Metzler matrix for each p∈M_. Replacing Ap in (14) with Ap+GpKp, then letting gp=KpTGpTνp, similar to Theorem 10, we can get (36) and the resulting closed-loop system (34) is IO-FTS with the MDADT (19).
The proof is completed.Next, an algorithm is presented to obtain the feedback gain matricesKp, p∈M_.Step 1.
By adjusting the parametersλp and solving (15)–(18) and (36) via linear programming, positive vectors νp, ρp, ϱp, and gp can be obtained.Step 2.
Substitutingνp and gp into gp=KpTGpTνp, Kp can be obtained.Step 3.
The gainKp is substituted into Ap+GpKp. If Ap+GpKp are Metzler matrices, then Kp are admissible. Otherwise, return to Step 1.


## 3.1. IO-FTS Analysis

In this subsection, we will focus on the problem of IO-FTS of positive switched system (1) with ut≡0. The following theorem gives sufficient conditions of IO-FTS of system (1) via the MDADT approach.Theorem 10.
Consider system (1) with ut≡0. Given positive constants Tf, λp, p∈M_, and γ and a vector δ≻0, if there exist positive vectors νp, ρp, ϱp, p∈M_, the following inequalities hold:(14)Ψ=diagψp1,ψp2,…,ψpn,ψp1˙,ψp2˙,…,ψpn˙,ψp1¨,ψp2¨,…,ψpn¨≤0(15)bprTνp<γ(16)γd<e-λpTf(17)cprTδ≺νp(18)νp⪯μpνq,ρp⪯μpρq,ϱp⪯μpϱq,where ψpr=adprTνpr-λpνpr+ρpr+τ1ϱpr, ψpr˙=adprTνpr-1-hρpr, ψpr¨=abprTνpr-ϱpr, apradpr,abpr,bpr,cpr represents the rth column vector of the matrix ApAdp,Abp,Bp,Cp, νp=νp1,νp2,…,νpn, ρp=ρp1,ρp2,…,ρpn, ϱp=ϱp1,ϱp2,…,ϱpn, and νpr, ρpr, and ϱpr represent the rth elements of the vectors νp, ρp, and ϱp, respectively. μpμp≥1 satisfies (18); then system (1) is IO-FTS for any switching signal σt with the MDADT(19)Tα>Tα∗=Tfln⁡μpln⁡e-λpTf-ln⁡γd.Proof.
Construct the multiple copositive type Lyapunov-Krasovskii functional for system (1) as follows:(20)Vσtt=Vσtt,xt=xTtνp+∫t-dtteλpt-sxTsρpds+∫-τ10∫t+θteλpt-sxTsϱpdsdθ,where νp, ρp, and ϱp∈R+n, ∀p∈M_.
Along the trajectory of system (1) with ut≡0, we have(21)V˙σtt=xTtApTνp+xTt-dtAdpTνp+∫t-τ2txTsAbpTνpds+wTtBpTνp+λp∫t-dtteλpt-sxTsρpds+xTsρp-1-d˙teλpdtxTt-dtρp+λp∫-τ10∫t+θteλpt-sxTsϱpdsdθ+τ1xTtϱp-∫-τ10e-λpθxTt+θϱpdθ≤xTtApTνp+xTt-dtAdpTνp+∫t-τ2txTsAbpTνpds+wTtBpTνp+λp∫t-dtteλpt-sxTsρpds+xTsρp-1-heλpdtxTt-dtρp+λp∫-τ10∫t+θteλpt-sxTsϱpdsdθ+τ1xTtϱp-∫t-τ1txTsϱpds.Combining (20) and (21) yields(22)V˙σtt-λpVσtt≤xTtApTνp-λpνp+ρp+τ1ϱp+xTt-dtAdpT-1-hρp+∫t-τ2txTsAbpTνpds-∫t-τ1txTsϱpds+wTtBpTνp.Noting that τ1=max⁡dt,τ2, we have(23)∫t-τ2txTsAbpTνpds-∫t-τ1txTsϱpds≤∫t-τ1txTsAbpTνpds-∫t-τ1txTsϱpds=∫t-τ1txTsAbpTνp-ϱpds.From (22) and (23), we obtain(24)V˙σtt-λpVσtt≤xTtApTνp-λpνp+ρp+τ1ϱp+wTtBpTνp+xTt-dtAdpT-1-hρp+∫t-τ1txTsAbpTνp-ϱpds.From (14), (15), and (24), we have(25)V˙σtt-λpVσtt≤wTtBpTνp≤γwTt.Integrating both sides of (25) during the period tk,t for t∈tk,tk+1 leads to(26)Vσtt≤eλσtkt-tkVσtktk+γ∫tkteλσtkt-swTsds.On the other hand, from (18) and (20), one can easily obtain(27)Vσtktk≤μσtkVσtk-tk-.Let N be the switching number of σt over 0,Tf, and denote t1,…,tk,…,tN as the switching instants over the interval 0,Tf. Combining with (2), (26), and (27), we have(28)Vσtt≤eλσtkt-tkVσtktk+γ∫tkteλσtkt-swTsds≤μσtkeλσtkt-tkVσtk-tk-+γ∫tkteλσtkt-swTsds≤μσtkeλσtkt-tkeλσtk-1tk-tk-1Vσtk-1tk-1+γ∫tk-1tkeλσtk-1tk-swTsds+γ∫tkteλσtkt-swTsds≤⋯≤∏i=1kμσtieλσtkt-tke∑i=1k-1λσtiti-ti-1Vσ00+∏i=1kμσtiγ∫0t1eλσ0t1-swTsds+∏i=2kμσtiγ∫t1t2eλσt1t2-swTsds+⋯+γ∫tkteλσtkt-swTsds≤∏p=1MμpNσp0,te∑p=1MλpTp0,tVσ00+γ∫0t∏p=1MμpNσp0,te∑p=1MλpTps,twTsds.According to (3) and choosing λ=maxp∈M_⁡λp+ln⁡μp/Tαp, for t∈0,Tf, one can obtain from (28)(29)Vσtt≤∏p=1MμpTp0,t/Tαpe∑p=1MλpTp0,tVσ00+γd=e∑p=1Mλp+ln⁡μp/TαpTp0,tVσ00+γd≤eλTfVσ00+γd.Under the zero initial condition x0=0, noting the definition of Vσtt and (29), we have(30)Vσtt≥xTtνp,Vσtt≤eλTfγd.From (3) and (30), one has(31)xTtνp≤eλTfγd.From (17) and (31), we have(32)yTtδ=xTtCσtTδ<xTtνp≤eλTfγd.From (19) and (32), we obtain(33)yTtδ<1.Thus, the proof is completed.


## 3.2. IO-FTS Controller Design

In this subsection, the state feedback controllerut=Kσtxt will be designed. For system (1), under the controller ut, the corresponding closed-loop system is given by(34)x˙t=Aσt+GσtKσtxt+Adσtxt-dt+Abσt∫t-τ2txsds+Bσtwtyt=Cσtxtxθ=φθ,θ∈-τ1,0.According to Lemma 4, to guarantee the positivity of system (34), Ap+GpKp should be Metzler matrices, ∀p∈M_. Theorem 11 gives some sufficient conditions to guarantee that the closed-loop system (34) is IO-FTS.Theorem 11.
Consider the positive switched system (34). Given positive constants Tf, λp, and γ and a vector δ≻0, if there exists a set of positive vectors νp, ρp, ϱp, gp, p∈M_, (15)–(17) and the following conditions hold:(35)Ap+GpKpareMetzlermatrices.(36)Φ=diag⁡ϕp1,ϕp2,…,ϕpn,ϕp1˙,ϕp2˙,…,ϕpn˙,ϕp1¨,ϕp2¨,…,ϕpn¨≤0,where ϕpr=adprTνpr-λpνpr+ρpr+τ1ϱpr+gpr, ϕpr˙=adprTνpr-1-hρpr, ϕpr¨=abprTνpr-ϱpr, gp=KpTGpTνp, apradpr,abpr,bpr,cpr represents the rth column vector of the matrix ApAdp,Abp,Bp,Cp, gpr represents the rth elements of vector gp, νp=νp1,νp2,…,νpn, ρp=ρp1,ρp2,…,ρpn, ϱp=ϱp1,ϱp2,…,ϱpn, and νpr, ρpr, and ϱpr represent the rth elements of the vectors νp, ρp, and ϱp, respectively. μpμp≥1 satisfies (18); then the closed-loop system (34) is IO-FTS for any switching signal σt with MDADT (19).Proof.
According to Lemma4, we get that Ap+GpKp is a Metzler matrix for each p∈M_. Replacing Ap in (14) with Ap+GpKp, then letting gp=KpTGpTνp, similar to Theorem 10, we can get (36) and the resulting closed-loop system (34) is IO-FTS with the MDADT (19).
The proof is completed.Next, an algorithm is presented to obtain the feedback gain matricesKp, p∈M_.Step 1.
By adjusting the parametersλp and solving (15)–(18) and (36) via linear programming, positive vectors νp, ρp, ϱp, and gp can be obtained.Step 2.
Substitutingνp and gp into gp=KpTGpTνp, Kp can be obtained.Step 3.
The gainKp is substituted into Ap+GpKp. If Ap+GpKp are Metzler matrices, then Kp are admissible. Otherwise, return to Step 1.


## 4. Numerical Example

Consider the positive switched system (1) with the parameters as follows: (37)A1=-4121-3212-3.5,Ad1=0.10.20.20.20.10.20.10.10.2,Ab1=0.10.20.30.30.10.10.20.10.1,G1=0.10.20.3,A2=-5223-5212-3,Ad2=0.20.20.10.20.10.10.10.20.1,Ab2=0.20.10.30.10.20.10.10.10.2,G2=0.10.20.3,B1=0.10.20.2,B2=0.20.10.2,C1=0.20.20.20.20.30.2,C2=0.20.20.20.30.20.2.Let dt=0.1+0.1sin⁡t, then we get τ2=0.1,τ1=max⁡dt,τ2=0.2,h=0.1. Choose γ=5,λ1=0.18,λ2=0.22,μ1=1.1,μ2=1.21,wt=0.05e-0.5t,d=0.01,δ=1.732.28T. Solving the inequalities in Theorem 11 by linear programming, we have(38)ν1=6.78339.493110.2540,ρ1=4.40164.63416.0117,ϱ1=9.90605.55487.1065,g1=-1.2804-2.9308-2.3817,ν2=8.16238.660611.9049,ρ2=4.35765.53095.7083,ϱ2=9.96936.12827.5314,g2=-1.8343-2.4662-2.7991.By gp=KpTGpTνp, we get (39)K1=-1.8877-1.5437-0.7742,K2=-2.2474-1.4238-0.7837.It is easy to verify that Ap+GpKp are Metzler matrices. Then, from (19), we can obtain Tα1∗=0.7971, Tα2∗=2.3956. Choose Tα1=0.8>Tα1∗ and Tα2=2.4>Tα2∗. The simulation results are shown in Figures 1–3, where x0=000T. The state trajectories of the closed-loop system with MDADT are shown in Figure 1. The switching signal σt with MDADT is depicted in Figure 2. Figure 3 plots the evolution of yTtδ of system (1). From Figure 3, we easily know that yTtδ<1, which implies that the resulting closed-loop system (1) is IO-FTS.Figure 1
State trajectories of closed-loop system (1).Figure 2
Switching signal of system (1) with MDADT.Figure 3
The evolution ofyTtδ of system (1).


## 5. Conclusions

In this paper, the problem of IO-FTS of positive switched systems with time-varying and distributed delays is investigated. The definition of IO-FTS of continuous positive switched systems is proposed. Then, by constructing multiple linear copositive Lyapunov functions, a state feedback controller is designed. Based on the MDADT approach, some sufficient conditions are obtained to guarantee that the closed-loop system is IO-FTS. Such sufficient conditions can be solved by linear programming. Our further work will focus on the IO-FTS of positive switched nonlinear systems with multiple delays.



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*Source: 4896764-2017-03-06.xml*