# Guaranteed Cost Finite-Time Control for Positive Switched Linear Systems with Time-Varying Delays

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

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## Abstract

This paper considers the guaranteed cost finite-time control for positive switched linear systems with time-varying delays. The definition of guaranteed cost finite-time boundedness is firstly given. Then, by using the mode-dependent average dwell time approach, a static output feedback law and a state feedback control law are constructed, respectively, and sufficient conditions are obtained to guarantee that the closed-loop system is guaranteed cost finite-time boundedness. Such conditions can be easily solved by linear programming. Finally, an example is given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Positive switched systems are a class of hybrid systems consisting of a family of positive subsystems and a switching law that specifies which subsystem will be activated along the system trajectory at each instant of time. In recent years, many remarkable results about the problems of stability analysis and controller synthesis of positive switched systems have been presented [1–8]. Among these existing results, the average dwell time (ADT) approach and mode-dependent average dwell time (MDADT) technique are widely used; the latter allows that every subsystem has its own average dwell time to make the individual properties of each subsystem unneglected, which is more applicable and less conservative compared with ADT technique [9].In fact, many studies focus on the asymptotic stability in the area of positive switched systems, which reflects the asymptotic behavior of the system in an infinite time interval. However, in many practical applications, one is more interested in what happens over a finite-time interval. Peter [10] has firstly defined finite-time stability (FTS) for linear deterministic systems, which means that, given a bound on the initial condition, the system state does not exceed a certain threshold during a specified time interval. Then Amato et al. [11] extended this definition to finite-time boundedness (FTB) when they dealt with the behavior of the state in the presence of external disturbances. Then many related results have been presented [12–17]. Recently, [18] first extends the concept of FTS to positive switched systems and gives some FTS conditions of positive switched systems. In [19], the problem of finite-time L1 control for a class of positive switched linear systems with time-varying delays is considered and the concept of finite-time L1 boundedness is also proposed. In [20], the problem of output feedback L1 finite-time control for switched positive systems with time-varying delays is investigated via MDADT approach. On the other hand, when controlling a real plant, it is also desirable to design a control system which not only is stable but also guarantees an adequate level of performance. One approach to this problem is the guaranteed cost control. It has the advantage of providing an upper bound on a given system performance index and thus the system performance degradation incurred by the uncertainties or time delays is guaranteed to be less than this bound [21]. So it is necessary to study the design problem of guaranteed cost finite-time controller. There are some results about this problem; see [22–29] and references therein. These results mainly focus on nonpositive systems, involved in fuzzy systems [22, 27], continuous-time nonlinear systems [23], interconnected systems [24], Markov switching systems [25], stochastic systems [26, 28], neural networks [29], and so on. However, to the best of our knowledge, there are few results available on guaranteed cost finite-time control for positive switched linear systems with time-varying delays, which motivates our present study.In this paper, the problem of the finite-time boundedness with guaranteed cost control for positive switched linear systems with time-varying delays is considered. Firstly, the definition of guaranteed cost finite-time boundedness is given, which is extended to positive switched linear systems with time-varying delays for the first time. Secondly, by using the MDADT approach and copositive Lyapunov-Krasovskii functional method, a static output feedback and a state feedback are designed, respectively, and sufficient conditions are obtained to guarantee that the closed-loop system is guaranteed cost finite-time boundedness (GCFTB). Such conditions can be easily solved by linear programming.Notations. Throughout this paper, A⪰0 (≻0, ⪯0, ≺0) means that all entries of matrix A are nonnegative (positive, nonpositive, negative). R+n is the n-dimensional nonnegative (positive) vector space, the n-dimensional Euclidean space is denoted by Rn and Rn×n represents the space of n×n matrices with real entries, and AT is the transpose of a matrix A. x=∑k=1nxk, where xk is the kth element of x∈Rn; if not explicitly stated, matrices are assumed to have compatible dimensions.


## 2. Problem Statements and Preliminaries

Consider the following positive switched linear systems with time-varying delays:(1)x˙t=Aσtxt+Adσtxt-dt+Gσtut+Bσtwt,xθ=φθ,θ∈-τ,0,yt=Cσtxt,where x(t)∈Rn,  u(t)∈Rm, and y(t)∈Rz represent the system state, control input, and control output, respectively. w(t)∈Rl is the disturbance input, which satisfies(2)∃d>0:∫0Tfwtdt<d.σ(t):[0,∞)→S_={1,2,…,S} is the switching signal, where S is the number of subsystems; ∀p∈S_,  Ap,  Adp,  Bp,  Cp, and Gp are constant matrices with appropriate dimensions, p denotes pth subsystems, and tq denotes the qth switching instant; φ(θ) is the initial condition on [-τ,0],  τ>0, and  d(t) denotes the time-varying delay satisfying 0≤d(t)≤τ,   d˙(t)≤h<1, where τ and h are known positive constants.Then we will present some definitions and lemmas for the following positive switched systems:(3)x˙t=Aσtxt+Adσtxt-dt+Bσtwt,xθ=φθ,θ∈-τ,0,yt=Cσtxt.The following definitions and lemmas are necessary for our further study.Definition 1 (see [19]).
System (3) is said to be positive if, for any initial conditions φ(θ)⪰0,  θ∈[-τ,0], disturbance input w(t)⪰0, and any switching signals σ(t), the corresponding trajectory x(t)⪰0 holds for all t⪰0.Definition 2 (see [19]).
A is called a Metzler matrix if the off-diagonal entries of matrix A are nonnegative.Lemma 3 (see [6]).
A matrixA is a Metzler matrix if and only if there exists a positive constant η such that A+ηIn⪰0.Lemma 4 (see [19]).
System (3) is positive if and only if Ap,  ∀p∈S_, are Metzler matrices and ∀p∈S_,  Adp⪰0,  Bp⪰0,  Gp⪰0, and Cp⪰0.Definition 5 (see [20]).
For any switching signalσ(t) and any t2≥t1≥0, let Nσp(t1,t2) denote the switching numbers that the pth subsystem is activated over the interval [t1,t2) and Tp(t1,t2) denote the total running time of the pth subsystem over the interval [t1,t2). If there exist N0p≥0,  ταp>0,  ∀p∈S_, such that (4)Nσpt1,t2≤N0p+Tpt1,t2ταp,∀t2≥t1≥0.Then ταp and N0p are called MDADT and mode-dependent chatter bounds, respectively. Generally, we choose N0p=0.Definition 6 (FTS).
For a given time constantTf and vectors δ≻ε≻0, positive switched system (3) with w(t)≡0 is said to be finite-time stable with respect to (δ,ε,Tf,σ(t)), if(5)sup-τ≤t≤0⁡xTtδ≤1⟹xTtε<1,∀t∈0,Tf.If the above condition is satisfied for any switching signals σ(t), then system (3) is said to be uniformly finite-time stable with respect to (δ,ε,Tf).Definition 7 (FTB).
For a given time constantTf and vectors δ≻ε≻0, positive switched system (3) is said to be finite-time boundedness with respect to (δ,ε,Tf,d,σ(t)), where w(t) satisfies (2), if (5) applies.Now we give some definitions about GCFTB for the positive switched systems (1).Definition 8.
Define the cost function of positive switched system (1) as follows: (6)J=∫0TfxTtR1+uTtR2dt,where R1≻0 and R2≻0 are two given vectors.Remark 9.
It should be noted that the proposed cost function is different from the existing literatures [21–24]; it is for the first time introduced in positive switched systems. This definition provides a more useful description, because it takes full advantage of the characteristics of nonnegative states of positive switched systems.Definition 10 (GCFTB).
For a given time constantTf and vectors δ≻ε≻0, consider positive switched system (1) and cost function (6); if there exist a feedback control law u(t) and a positive scalar J∗ such that the closed-loop system is FTB with respect to (δ,ε,Tf,d,σ(t)) and the cost function satisfies J≤J∗, then the closed-loop system is called GCFTB, where J∗ is a guaranteed cost value and u(t) is a guaranteed cost finite-time controller.The aim of this paper is to design a static output feedback controlleru(t)=Kσ(t)y(t), a state feedback controller u(t)=K-σ(t)x(t), and a class of switching signals σ(t) for positive switched system (1) such that the corresponding closed-loop system is GCFTB.


## 3. Main Results

### 3.1. GCFTB Analysis

In this subsection, we will focus on the problem of GCFTB for positive switched system (3). The following theorem gives sufficient conditions of GCFTB for system (3) with the MDADT.Theorem 11.
Consider the positive switched system (3). For given constants Tf,  λp,  γ and vectors δ≻ε≻0 and R1≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, the following inequalities hold: (7)Ψp=diagψp1,ψp2,…,ψpn,ψ′p1,ψ′p2,…,ψ′pn≤0,(8)ξ1ε≺νp≺ξ2δ,υp≺ξ3δ,ϑp≺ξ4δ,(9)ξ2bprTδ<γ,(10)ξ2+τeλτξ3+τ2eλτξ4+γd<ξ1e-κTf,where(11)ψpr=aprT-λpνpr+υpr+τϑpr+R1,ψ′pr=adprTνp-1-hνprλ=maxp∈S_λp,r∈n_=1,2,…,n.apr(adpr) represents the rth column vector of the matrix Ap(Adp),  νp=[νp1,νp2,…,νpn]T,  υp=[υp1,υp2,…,υpn]T, and ϑp=[ϑp1,ϑp2,…,ϑpn]T, and  νpr,  υpr, and ϑpr represent the rth elements of the vectors νp,  υp, and ϑp, respectively; then under the following MDADT scheme (12)ταp>ταp∗=maxTfln⁡μpln⁡ξ1e-λpTf-ln⁡ξ2+τeλpτξ3+τ2eλpτξ4+γd,ln⁡μpλ,system (3) is GCFTB with respect to (δ,ε,Tf,d,σ(t)), where μp  (μp≥1) satisfies (13)νp⪯μpνq,υp⪯μpυq,ϑp⪯μpϑq,∀p,q∈S_.And the guaranteed cost value of system (3) is given by (14)J=∫0TfxTsR1ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Proof.
Construct the following copositive Lyapunov-Krasovskii functional for system (3): (15)Vσtt,xt=xTtνp+∫t-dtteλpt-sxTsυpds+∫-τ0∫t+θteλpt-sxTsϑpdsdθ,where νp,  υp, and ϑp∈R+n,  ∀p∈S_.
For the sake of simplicity,Vσ(t)(t,x(t)) is written as Vσ(t)(t) in this paper.
Along the trajectory of system (3), we have(16)V˙σtt=xTtApTνp+xTt-dtAdpTνp+wTtBpTνp+λp∫t-dtteλpt-sxTsυpds+xTsυp-1-d˙teλpdtxTt-dtυp+λp∫-τ0∫t+θteλpt-sxTsϑpdsdθ+τxTtϑp-∫-τ0e-λpθxTt+θϑpdθ≤xTtApTνp+xTt-dtAdpTνp+wTtBpTνp+λp∫t-dtteλpt-sxTsυpds+xTsυp-1-hxTt-dtυp+λp∫-τ0∫t+θteλpt-sxTsϑpdsdθ+τxTtϑp-∫t-dttxTsϑpds.Combining (8), (9), (15), and (16) leads to(17)V˙σtt-λpVσtt+xTtR1=xTtApTνp-λpxTtνp+xTsυp+τxTtϑp+xTt-dtAdpTνp-1-hxTt-dtυp+wTtξ2BpTδ-∫t-dttxTsϑpds+xTtR1≤xTtApTνp-λpνp+υp+τϑp+R1+xTt-dtAdpTνp-1-hυp+γwt.Substituting (7) into (17) yields (18)V˙σtt-λpVσtt+xTtR1-γwt≤0.It implies (19)V˙σtt-λpVσtt-γwt≤0.Integrating both sides of (19) during the period [tk,t) for t∈[tk,tk+1) leads to (20)Vσtt≤eλσtkt-tkVσtktk+γ∫tkteλσtkt-swTsds.For any Tf>0, let N be the switching number of σ(t) over [0,Tf), and denote t1,t2,…,tk,…,tN as the switching instants over the interval [0,Tf). Then, for t∈[0,Tf),  Vσ(tk)(tk)≤μσ(tk)Vσ(tk-)(tk-) is easily obtained from (13) and (15). Combining with (19), we have (21)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=1kuσtieλσtkt-tke∑i=1k-1λσtiti-ti-1Vσ00+∏i=1kuσtiγ∫0t1eλσ0t1-swTsds+∏i=2kuσtiγ∫t1t2eλσt1t2-swTsds+⋯+γ∫tkteλσtkt-swTsds≤∏p=1SupNσp0,te∑p=1SλpTp0,tVσ00+γ∫0t∏p=1SupNσp0,te∑p=1SλpTps,twTsds.According to the definition of MDADT, one can obtain from (21)(22)Vσtt≤∏p=1SupTp0,t/ταpe∑p=1SλpTp0,tVσ00+γd=e∑p=1Sλp+ln⁡μp/ταpTp0,tVσ00+γd.Choose κ=maxp∈S_λp+ln⁡μp/ταp. Therefore, for t∈[0,Tf)(23)Vσtt≤eκTfVσ00+γd.Noting the definition of Vσ(t)(t) and (8), we have (24)Vσtt≥ξ1xTtε,(25)Vσ00≤ξ2xT0δ+τeλτξ3sup-τ≤θ≤0xTθδ+τ2eλτξ4sup-τ≤θ≤0xTθδ≤ξ2+τeλτξ3+τ2eλτξ4sup-τ≤θ≤0xTθδ≤ξ2+τeλτξ3+τ2eλτξ4.Combining (21)–(25), we obtain (26)xTtε≤1ξ1eκTfξ2+τeλτξ3+τ2eλτξ4+γd.Substituting (10) into (26), one has (27)xTtε<1.From Definition 7, we conclude that system (3) is FTB with respect to (δ,ε,Tf,d,σ(t)).
Next, we will give the guaranteed cost value of system (3).
According to (17), denoting ∇(t)=γw(t)-xT(t)R1 and integrating both sides of (18) from tk to t for t∈[tk,tk+1], it gives rise to (28)Vσtt≤eλpt-tkVσtktk+∫tkteλpt-s∇sds.Similar to the proof process of (21), for any t∈[0,Tf], we can obtain (29)Vσtt≤∏p=1SupNσp0,te∑p=1SλTp0,tVσ00+∫0t∏p=1SupNσps,te∑p=1SλTps,t∇sds,where λ=maxi∈s_λp. From (29), we can get (30)∫0t∏p=1SupNσps,te∑p=1SλTps,txTsR1ds≤∏p=1SupNσp0,te∑p=1SλTp0,tVσ00+γ∫0t∏p=1SupNσps,te∑p=1SλTps,twsds.Multiplying both sides of (30) by ∏p=1Sμ-Nσp(0,t) leads to (31)∫0t∏p=1Sup-Nσp0,seλt-sxTsR1ds≤eλTfVσ00+γ∫0t∏p=1Sup-Nσp0,seλt-swsds.Noting that Nσp(0,s)≤Tp(0,s)/ταp and ταp>ln⁡μp/λ, we obtain that 0<Nσp(0,s)≤Tp(0,s)/ταp≤λs/ln⁡μp; that is, e-λs≤μp-Np(0,s)<1.
Then (31) can be turned into(32)∫0te-λseλt-sxTsR1ds≤∫0tμ-Nσp0,seλt-sxTsR1ds≤eλTfVσ00+γ∫0teλt-swsds.Let t=Tf; then multiplying both sides of (32) by e-λTf leads to (33)∫0Tfe-2λsxTsR1ds≤Vσ00+γ∫0Tfe-λswsds≤Vσ00+γ∫0Tfwsds.Substituting (2) into (33) yields (34)e-2λTf∫0TfxTsR1ds≤Vσ00+γdwhich can be rewritten as (35)∫0TfxTsR1ds≤e2λTfVσ00+γd.Substituting (25) into (35), the guaranteed cost value of system (3) is given by (36)J=∫0TfxTsR1ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Therefore, according to Definition 10, we can conclude that system (3) is GCFTB with respect to (δ,ε,Tf,d,σ(t)). Thus, the proof is completed.Remark 12.
In Theorem11, if μp≡1 in (12), then one can obtain ταp∗=0, which means that the switching signal can be arbitrary.When the exogenous disturbancew(t)=0, the result on guaranteed cost finite-time stability can be obtained as follows.Corollary 13.
Consider switched positive system (3) with w(t)=0. For given constants Tf,  λ, and vectors δ≻ε≻0,  R1≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, such that (7), (8), and the following inequality holds(37)ξ2+τeλτξ3+τ2eλτξ4<ξ1e-κTfthen under the following MDADT scheme(38)τap>τap∗=Tfln⁡μpln⁡ξ1e-λpTf-ln⁡ξ2+τeλτξ3+τ2eλτξ4system (3) is guaranteed cost finite-time stable to (δ,ε,Tf,d,σ(t)), where μp≥1 satisfies (13); the corresponding guaranteed cost value is given by (39)J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4.


### 3.2. GCFTB Controller Design

In this section, we are concerned with the GCFTB controller design of positive switched system (1). The static output feedback control law and state feedback control law will be designed to ensure the close-loop system (1) is GCFTB, respectively.


#### 3.2.1. Static Output Feedback Controller Design

Consider system (1); under the controller u(t)=Kσ(t)y(t), the corresponding closed-loop system is given by(40)x˙t=Aσt+GσtKσtCσtxt+Adσtxt-dt+Bσtwt,xθ=φθ,θ∈-τ,0,yt=Cσtxt.By Lemma3, to guarantee the positivity of system (40), Ap+GpKpCp should be Metzler matrices, ∀p∈S_. Theorem 14 gives some sufficient conditions to guarantee that the closed-loop system (1) is GCFTB.Theorem 14.
Consider the positive switched system (1). For given constants Tf,  λp,  γ and vectors δ≻ε≻0,  R1≻0, and R2≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, (8)–(10) and the following conditions hold: (41)Ap+GpKpCpareMetzlermatrices,(42)Ψ-p=diagψ-p1,ψ-p2,…,ψ-pn,ψ-′p1,ψ-′p2,…,ψ-′pn≤0,where(43)ψ-pr=aprT-λpνpr+υpr+τϑpr+R1+gpr,ψ-′pr=adprTνp-1-hνprλ=maxp∈S_λp,r∈n_=1,2,…,n.gp=CpTKpTGpTνp+R2, where gpr represents the rth elements of vector gp,  apr(adpr) represents the rth column vector of the matrix Ap(Adp), and νp=[νp1,νp2,…,νpn]T,  υp=[υp1,υp2,…,υpn]T, and ϑp=[ϑp1,ϑp2,…,ϑpn]T,  νpr,  υpr, and ϑpr represent the rth elements of the vectors νp,  υp, and ϑp, respectively, and μp  (μp≥1) satisfies (13); then under the MDADT scheme (12), the resulting closed-loop system (40) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value of system (40) is given by (44)J=∫0TfxTsR1+xTsCσtTKσtTR2ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Proof.
By Lemma3, we know that Ap+GpKpCp is a Metzler matrix for each p∈S_. According to Lemma 4, system (40) is positive if Adp,  Bp,  Gp, and Cp,  p∈S_ are all nonnegative. Replacing Ap in (7) with Ap+GpKpCp and letting gp=CpTKpTGpTνp+R2, similar to Theorem 11, we easily obtain that the resulting closed-loop system (40) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value is given by (44).
The proof is completed.


#### 3.2.2. State Feedback Controller Design

Consider system (1); under the controller u(t)=K-σ(t)x(t), the corresponding closed-loop system is given by (45)x˙t=Aσt+GσtK-σtxt+Adσtxt-dt+Bσtwt,xθ=φθ,θ∈-τ,0,yt=Cσtxt.By Lemma3, to guarantee the positivity of system (45), Ap+GpK-p should be Metzler matrices, ∀p∈S_. The following Theorem 15 gives some sufficient conditions to guarantee that closed-loop system (1) is GCFTB.Theorem 15.
Consider the positive switched system (1). For given constants Tf,  λp,  γ and vectors δ≻ε≻0,  R1≻0, and R2≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, (8)–(10) and the following conditions hold:(46)Ap+GpK-pareMetzlermatrices,(47)Ψ-p=diagψ-p1,ψ-p2,…,ψ-pn,ψ-′p1,ψ-′p2,…,ψ-′pn≤0,where(48)ψ-pr=aprT-λpνpr+υpr+τϑpr+R1+gpr,ψ-′pr=adprTνp-1-hνpr,λ=maxp∈S_λp,r∈n_=1,2,…,n.gp=K-pTGpTνp+R2, where  gpr represents the rth elements of vector gp,  apr(adpr) represents the rth column vector of the matrix Ap(Adp), and νp=[νp1,νp2,…,νpn]T,  υp=[υp1,υp2,…,υpn]T, and ϑp=[ϑp1,ϑp2,…,ϑpn]T,  νpr,  υpr, and ϑpr represent the rth elements of the vectors νp,  υp, and ϑp, respectively, and μp  (μp≥1) satisfies (13); then under the MDADT scheme (12), the resulting closed-loop system (45) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value of system (45) is given by (49)J=∫0TfxTsR1+xTsK-σsTR2ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Proof.
By Lemma3, we know that Ap+GpK-p is a Metzler matrix for each p∈S_. According to Lemma 4, system (45) is positive if Adp,  Bp,  Gp, and Cp are all nonnegative. Replacing Ap in (7) with Ap+GpK-p and letting gp=K-pTGpTνp+R2, similar to Theorem 11, we easily obtain that the resulting closed-loop system (45) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value is given by (49).
The proof is completed.Next, a procedure is presented to obtain the feedback gain matricesKp(K-p),  p∈S_.Step 1.
By adjusting the parametersλp and solving (8)–(10), (13), (42), and (47) via linear programming, positive vectors νp,  υp,  ϑp, and gp can be obtained.Step 2.
Substitutingνp and gp into gp=CpTKpTGpTνp+R2  (or gp=K-pTGpTνp+R2),  Kp(K-p) can be obtained.Step 3.
The gainKp(K-p) is substituted into Ap+GpKpCp  (or Ap+GpK-p). If Ap+GpKpCp  (or Ap+GpK-p) are Metzler matrices, then Kp(K-p) are admissible. Otherwise, return to Step 1.


## 3.1. GCFTB Analysis

In this subsection, we will focus on the problem of GCFTB for positive switched system (3). The following theorem gives sufficient conditions of GCFTB for system (3) with the MDADT.Theorem 11.
Consider the positive switched system (3). For given constants Tf,  λp,  γ and vectors δ≻ε≻0 and R1≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, the following inequalities hold: (7)Ψp=diagψp1,ψp2,…,ψpn,ψ′p1,ψ′p2,…,ψ′pn≤0,(8)ξ1ε≺νp≺ξ2δ,υp≺ξ3δ,ϑp≺ξ4δ,(9)ξ2bprTδ<γ,(10)ξ2+τeλτξ3+τ2eλτξ4+γd<ξ1e-κTf,where(11)ψpr=aprT-λpνpr+υpr+τϑpr+R1,ψ′pr=adprTνp-1-hνprλ=maxp∈S_λp,r∈n_=1,2,…,n.apr(adpr) represents the rth column vector of the matrix Ap(Adp),  νp=[νp1,νp2,…,νpn]T,  υp=[υp1,υp2,…,υpn]T, and ϑp=[ϑp1,ϑp2,…,ϑpn]T, and  νpr,  υpr, and ϑpr represent the rth elements of the vectors νp,  υp, and ϑp, respectively; then under the following MDADT scheme (12)ταp>ταp∗=maxTfln⁡μpln⁡ξ1e-λpTf-ln⁡ξ2+τeλpτξ3+τ2eλpτξ4+γd,ln⁡μpλ,system (3) is GCFTB with respect to (δ,ε,Tf,d,σ(t)), where μp  (μp≥1) satisfies (13)νp⪯μpνq,υp⪯μpυq,ϑp⪯μpϑq,∀p,q∈S_.And the guaranteed cost value of system (3) is given by (14)J=∫0TfxTsR1ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Proof.
Construct the following copositive Lyapunov-Krasovskii functional for system (3): (15)Vσtt,xt=xTtνp+∫t-dtteλpt-sxTsυpds+∫-τ0∫t+θteλpt-sxTsϑpdsdθ,where νp,  υp, and ϑp∈R+n,  ∀p∈S_.
For the sake of simplicity,Vσ(t)(t,x(t)) is written as Vσ(t)(t) in this paper.
Along the trajectory of system (3), we have(16)V˙σtt=xTtApTνp+xTt-dtAdpTνp+wTtBpTνp+λp∫t-dtteλpt-sxTsυpds+xTsυp-1-d˙teλpdtxTt-dtυp+λp∫-τ0∫t+θteλpt-sxTsϑpdsdθ+τxTtϑp-∫-τ0e-λpθxTt+θϑpdθ≤xTtApTνp+xTt-dtAdpTνp+wTtBpTνp+λp∫t-dtteλpt-sxTsυpds+xTsυp-1-hxTt-dtυp+λp∫-τ0∫t+θteλpt-sxTsϑpdsdθ+τxTtϑp-∫t-dttxTsϑpds.Combining (8), (9), (15), and (16) leads to(17)V˙σtt-λpVσtt+xTtR1=xTtApTνp-λpxTtνp+xTsυp+τxTtϑp+xTt-dtAdpTνp-1-hxTt-dtυp+wTtξ2BpTδ-∫t-dttxTsϑpds+xTtR1≤xTtApTνp-λpνp+υp+τϑp+R1+xTt-dtAdpTνp-1-hυp+γwt.Substituting (7) into (17) yields (18)V˙σtt-λpVσtt+xTtR1-γwt≤0.It implies (19)V˙σtt-λpVσtt-γwt≤0.Integrating both sides of (19) during the period [tk,t) for t∈[tk,tk+1) leads to (20)Vσtt≤eλσtkt-tkVσtktk+γ∫tkteλσtkt-swTsds.For any Tf>0, let N be the switching number of σ(t) over [0,Tf), and denote t1,t2,…,tk,…,tN as the switching instants over the interval [0,Tf). Then, for t∈[0,Tf),  Vσ(tk)(tk)≤μσ(tk)Vσ(tk-)(tk-) is easily obtained from (13) and (15). Combining with (19), we have (21)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=1kuσtieλσtkt-tke∑i=1k-1λσtiti-ti-1Vσ00+∏i=1kuσtiγ∫0t1eλσ0t1-swTsds+∏i=2kuσtiγ∫t1t2eλσt1t2-swTsds+⋯+γ∫tkteλσtkt-swTsds≤∏p=1SupNσp0,te∑p=1SλpTp0,tVσ00+γ∫0t∏p=1SupNσp0,te∑p=1SλpTps,twTsds.According to the definition of MDADT, one can obtain from (21)(22)Vσtt≤∏p=1SupTp0,t/ταpe∑p=1SλpTp0,tVσ00+γd=e∑p=1Sλp+ln⁡μp/ταpTp0,tVσ00+γd.Choose κ=maxp∈S_λp+ln⁡μp/ταp. Therefore, for t∈[0,Tf)(23)Vσtt≤eκTfVσ00+γd.Noting the definition of Vσ(t)(t) and (8), we have (24)Vσtt≥ξ1xTtε,(25)Vσ00≤ξ2xT0δ+τeλτξ3sup-τ≤θ≤0xTθδ+τ2eλτξ4sup-τ≤θ≤0xTθδ≤ξ2+τeλτξ3+τ2eλτξ4sup-τ≤θ≤0xTθδ≤ξ2+τeλτξ3+τ2eλτξ4.Combining (21)–(25), we obtain (26)xTtε≤1ξ1eκTfξ2+τeλτξ3+τ2eλτξ4+γd.Substituting (10) into (26), one has (27)xTtε<1.From Definition 7, we conclude that system (3) is FTB with respect to (δ,ε,Tf,d,σ(t)).
Next, we will give the guaranteed cost value of system (3).
According to (17), denoting ∇(t)=γw(t)-xT(t)R1 and integrating both sides of (18) from tk to t for t∈[tk,tk+1], it gives rise to (28)Vσtt≤eλpt-tkVσtktk+∫tkteλpt-s∇sds.Similar to the proof process of (21), for any t∈[0,Tf], we can obtain (29)Vσtt≤∏p=1SupNσp0,te∑p=1SλTp0,tVσ00+∫0t∏p=1SupNσps,te∑p=1SλTps,t∇sds,where λ=maxi∈s_λp. From (29), we can get (30)∫0t∏p=1SupNσps,te∑p=1SλTps,txTsR1ds≤∏p=1SupNσp0,te∑p=1SλTp0,tVσ00+γ∫0t∏p=1SupNσps,te∑p=1SλTps,twsds.Multiplying both sides of (30) by ∏p=1Sμ-Nσp(0,t) leads to (31)∫0t∏p=1Sup-Nσp0,seλt-sxTsR1ds≤eλTfVσ00+γ∫0t∏p=1Sup-Nσp0,seλt-swsds.Noting that Nσp(0,s)≤Tp(0,s)/ταp and ταp>ln⁡μp/λ, we obtain that 0<Nσp(0,s)≤Tp(0,s)/ταp≤λs/ln⁡μp; that is, e-λs≤μp-Np(0,s)<1.
Then (31) can be turned into(32)∫0te-λseλt-sxTsR1ds≤∫0tμ-Nσp0,seλt-sxTsR1ds≤eλTfVσ00+γ∫0teλt-swsds.Let t=Tf; then multiplying both sides of (32) by e-λTf leads to (33)∫0Tfe-2λsxTsR1ds≤Vσ00+γ∫0Tfe-λswsds≤Vσ00+γ∫0Tfwsds.Substituting (2) into (33) yields (34)e-2λTf∫0TfxTsR1ds≤Vσ00+γdwhich can be rewritten as (35)∫0TfxTsR1ds≤e2λTfVσ00+γd.Substituting (25) into (35), the guaranteed cost value of system (3) is given by (36)J=∫0TfxTsR1ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Therefore, according to Definition 10, we can conclude that system (3) is GCFTB with respect to (δ,ε,Tf,d,σ(t)). Thus, the proof is completed.Remark 12.
In Theorem11, if μp≡1 in (12), then one can obtain ταp∗=0, which means that the switching signal can be arbitrary.When the exogenous disturbancew(t)=0, the result on guaranteed cost finite-time stability can be obtained as follows.Corollary 13.
Consider switched positive system (3) with w(t)=0. For given constants Tf,  λ, and vectors δ≻ε≻0,  R1≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, such that (7), (8), and the following inequality holds(37)ξ2+τeλτξ3+τ2eλτξ4<ξ1e-κTfthen under the following MDADT scheme(38)τap>τap∗=Tfln⁡μpln⁡ξ1e-λpTf-ln⁡ξ2+τeλτξ3+τ2eλτξ4system (3) is guaranteed cost finite-time stable to (δ,ε,Tf,d,σ(t)), where μp≥1 satisfies (13); the corresponding guaranteed cost value is given by (39)J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4.


## 3.2. GCFTB Controller Design

In this section, we are concerned with the GCFTB controller design of positive switched system (1). The static output feedback control law and state feedback control law will be designed to ensure the close-loop system (1) is GCFTB, respectively.


### 3.2.1. Static Output Feedback Controller Design

Consider system (1); under the controller u(t)=Kσ(t)y(t), the corresponding closed-loop system is given by(40)x˙t=Aσt+GσtKσtCσtxt+Adσtxt-dt+Bσtwt,xθ=φθ,θ∈-τ,0,yt=Cσtxt.By Lemma3, to guarantee the positivity of system (40), Ap+GpKpCp should be Metzler matrices, ∀p∈S_. Theorem 14 gives some sufficient conditions to guarantee that the closed-loop system (1) is GCFTB.Theorem 14.
Consider the positive switched system (1). For given constants Tf,  λp,  γ and vectors δ≻ε≻0,  R1≻0, and R2≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, (8)–(10) and the following conditions hold: (41)Ap+GpKpCpareMetzlermatrices,(42)Ψ-p=diagψ-p1,ψ-p2,…,ψ-pn,ψ-′p1,ψ-′p2,…,ψ-′pn≤0,where(43)ψ-pr=aprT-λpνpr+υpr+τϑpr+R1+gpr,ψ-′pr=adprTνp-1-hνprλ=maxp∈S_λp,r∈n_=1,2,…,n.gp=CpTKpTGpTνp+R2, where gpr represents the rth elements of vector gp,  apr(adpr) represents the rth column vector of the matrix Ap(Adp), and νp=[νp1,νp2,…,νpn]T,  υp=[υp1,υp2,…,υpn]T, and ϑp=[ϑp1,ϑp2,…,ϑpn]T,  νpr,  υpr, and ϑpr represent the rth elements of the vectors νp,  υp, and ϑp, respectively, and μp  (μp≥1) satisfies (13); then under the MDADT scheme (12), the resulting closed-loop system (40) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value of system (40) is given by (44)J=∫0TfxTsR1+xTsCσtTKσtTR2ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Proof.
By Lemma3, we know that Ap+GpKpCp is a Metzler matrix for each p∈S_. According to Lemma 4, system (40) is positive if Adp,  Bp,  Gp, and Cp,  p∈S_ are all nonnegative. Replacing Ap in (7) with Ap+GpKpCp and letting gp=CpTKpTGpTνp+R2, similar to Theorem 11, we easily obtain that the resulting closed-loop system (40) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value is given by (44).
The proof is completed.


### 3.2.2. State Feedback Controller Design

Consider system (1); under the controller u(t)=K-σ(t)x(t), the corresponding closed-loop system is given by (45)x˙t=Aσt+GσtK-σtxt+Adσtxt-dt+Bσtwt,xθ=φθ,θ∈-τ,0,yt=Cσtxt.By Lemma3, to guarantee the positivity of system (45), Ap+GpK-p should be Metzler matrices, ∀p∈S_. The following Theorem 15 gives some sufficient conditions to guarantee that closed-loop system (1) is GCFTB.Theorem 15.
Consider the positive switched system (1). For given constants Tf,  λp,  γ and vectors δ≻ε≻0,  R1≻0, and R2≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, (8)–(10) and the following conditions hold:(46)Ap+GpK-pareMetzlermatrices,(47)Ψ-p=diagψ-p1,ψ-p2,…,ψ-pn,ψ-′p1,ψ-′p2,…,ψ-′pn≤0,where(48)ψ-pr=aprT-λpνpr+υpr+τϑpr+R1+gpr,ψ-′pr=adprTνp-1-hνpr,λ=maxp∈S_λp,r∈n_=1,2,…,n.gp=K-pTGpTνp+R2, where  gpr represents the rth elements of vector gp,  apr(adpr) represents the rth column vector of the matrix Ap(Adp), and νp=[νp1,νp2,…,νpn]T,  υp=[υp1,υp2,…,υpn]T, and ϑp=[ϑp1,ϑp2,…,ϑpn]T,  νpr,  υpr, and ϑpr represent the rth elements of the vectors νp,  υp, and ϑp, respectively, and μp  (μp≥1) satisfies (13); then under the MDADT scheme (12), the resulting closed-loop system (45) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value of system (45) is given by (49)J=∫0TfxTsR1+xTsK-σsTR2ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Proof.
By Lemma3, we know that Ap+GpK-p is a Metzler matrix for each p∈S_. According to Lemma 4, system (45) is positive if Adp,  Bp,  Gp, and Cp are all nonnegative. Replacing Ap in (7) with Ap+GpK-p and letting gp=K-pTGpTνp+R2, similar to Theorem 11, we easily obtain that the resulting closed-loop system (45) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value is given by (49).
The proof is completed.Next, a procedure is presented to obtain the feedback gain matricesKp(K-p),  p∈S_.Step 1.
By adjusting the parametersλp and solving (8)–(10), (13), (42), and (47) via linear programming, positive vectors νp,  υp,  ϑp, and gp can be obtained.Step 2.
Substitutingνp and gp into gp=CpTKpTGpTνp+R2  (or gp=K-pTGpTνp+R2),  Kp(K-p) can be obtained.Step 3.
The gainKp(K-p) is substituted into Ap+GpKpCp  (or Ap+GpK-p). If Ap+GpKpCp  (or Ap+GpK-p) are Metzler matrices, then Kp(K-p) are admissible. Otherwise, return to Step 1.


## 3.2.1. Static Output Feedback Controller Design

Consider system (1); under the controller u(t)=Kσ(t)y(t), the corresponding closed-loop system is given by(40)x˙t=Aσt+GσtKσtCσtxt+Adσtxt-dt+Bσtwt,xθ=φθ,θ∈-τ,0,yt=Cσtxt.By Lemma3, to guarantee the positivity of system (40), Ap+GpKpCp should be Metzler matrices, ∀p∈S_. Theorem 14 gives some sufficient conditions to guarantee that the closed-loop system (1) is GCFTB.Theorem 14.
Consider the positive switched system (1). For given constants Tf,  λp,  γ and vectors δ≻ε≻0,  R1≻0, and R2≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, (8)–(10) and the following conditions hold: (41)Ap+GpKpCpareMetzlermatrices,(42)Ψ-p=diagψ-p1,ψ-p2,…,ψ-pn,ψ-′p1,ψ-′p2,…,ψ-′pn≤0,where(43)ψ-pr=aprT-λpνpr+υpr+τϑpr+R1+gpr,ψ-′pr=adprTνp-1-hνprλ=maxp∈S_λp,r∈n_=1,2,…,n.gp=CpTKpTGpTνp+R2, where gpr represents the rth elements of vector gp,  apr(adpr) represents the rth column vector of the matrix Ap(Adp), and νp=[νp1,νp2,…,νpn]T,  υp=[υp1,υp2,…,υpn]T, and ϑp=[ϑp1,ϑp2,…,ϑpn]T,  νpr,  υpr, and ϑpr represent the rth elements of the vectors νp,  υp, and ϑp, respectively, and μp  (μp≥1) satisfies (13); then under the MDADT scheme (12), the resulting closed-loop system (40) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value of system (40) is given by (44)J=∫0TfxTsR1+xTsCσtTKσtTR2ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Proof.
By Lemma3, we know that Ap+GpKpCp is a Metzler matrix for each p∈S_. According to Lemma 4, system (40) is positive if Adp,  Bp,  Gp, and Cp,  p∈S_ are all nonnegative. Replacing Ap in (7) with Ap+GpKpCp and letting gp=CpTKpTGpTνp+R2, similar to Theorem 11, we easily obtain that the resulting closed-loop system (40) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value is given by (44).
The proof is completed.


## 3.2.2. State Feedback Controller Design

Consider system (1); under the controller u(t)=K-σ(t)x(t), the corresponding closed-loop system is given by (45)x˙t=Aσt+GσtK-σtxt+Adσtxt-dt+Bσtwt,xθ=φθ,θ∈-τ,0,yt=Cσtxt.By Lemma3, to guarantee the positivity of system (45), Ap+GpK-p should be Metzler matrices, ∀p∈S_. The following Theorem 15 gives some sufficient conditions to guarantee that closed-loop system (1) is GCFTB.Theorem 15.
Consider the positive switched system (1). For given constants Tf,  λp,  γ and vectors δ≻ε≻0,  R1≻0, and R2≻0, if there exist a set of positive vectors νp,  υp,  ϑp,  p∈S_, and positive constants ξ1,  ξ2,  ξ3, and ξ4, (8)–(10) and the following conditions hold:(46)Ap+GpK-pareMetzlermatrices,(47)Ψ-p=diagψ-p1,ψ-p2,…,ψ-pn,ψ-′p1,ψ-′p2,…,ψ-′pn≤0,where(48)ψ-pr=aprT-λpνpr+υpr+τϑpr+R1+gpr,ψ-′pr=adprTνp-1-hνpr,λ=maxp∈S_λp,r∈n_=1,2,…,n.gp=K-pTGpTνp+R2, where  gpr represents the rth elements of vector gp,  apr(adpr) represents the rth column vector of the matrix Ap(Adp), and νp=[νp1,νp2,…,νpn]T,  υp=[υp1,υp2,…,υpn]T, and ϑp=[ϑp1,ϑp2,…,ϑpn]T,  νpr,  υpr, and ϑpr represent the rth elements of the vectors νp,  υp, and ϑp, respectively, and μp  (μp≥1) satisfies (13); then under the MDADT scheme (12), the resulting closed-loop system (45) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value of system (45) is given by (49)J=∫0TfxTsR1+xTsK-σsTR2ds≤J∗=e2λTfξ2+τeλτξ3+τ2eλτξ4+γd.Proof.
By Lemma3, we know that Ap+GpK-p is a Metzler matrix for each p∈S_. According to Lemma 4, system (45) is positive if Adp,  Bp,  Gp, and Cp are all nonnegative. Replacing Ap in (7) with Ap+GpK-p and letting gp=K-pTGpTνp+R2, similar to Theorem 11, we easily obtain that the resulting closed-loop system (45) is GCFTB with respect to (δ,ε,Tf,d,σ(t)) and the guaranteed cost value is given by (49).
The proof is completed.Next, a procedure is presented to obtain the feedback gain matricesKp(K-p),  p∈S_.Step 1.
By adjusting the parametersλp and solving (8)–(10), (13), (42), and (47) via linear programming, positive vectors νp,  υp,  ϑp, and gp can be obtained.Step 2.
Substitutingνp and gp into gp=CpTKpTGpTνp+R2  (or gp=K-pTGpTνp+R2),  Kp(K-p) can be obtained.Step 3.
The gainKp(K-p) is substituted into Ap+GpKpCp  (or Ap+GpK-p). If Ap+GpKpCp  (or Ap+GpK-p) are Metzler matrices, then Kp(K-p) are admissible. Otherwise, return to Step 1.


## 4. Numerical Example

In this section, a numerical example in [6] is provided to show the effectiveness of the proposed approach. Consider positive switched system (1) with the parameters as follows: (50)A1=-3.50.10.3-3,Ad1=0.20.50.80.6,B1=0.20.10.10.3,G1=12,C1=0.10.20.40.1,A2=-50.10.1-2.5,Ad2=0.60.30.80.5,B2=0.30.10.20.1,G2=21,C2=0.20.50.30.1,R1=0.20.3,R2=0.30.2,ε=0.10.1,δ=12.Assuming that d(t)=0.2+0.2sin⁡(t), we can get τ=0.4 and h=0.2.Let(51)wt=f1tf2t,f1t=f2t=1+sintt3≤t≤10,0others.Choosing the parameters Tf=10,  λ1=0.2,  λ2=0.14,  γ=1, and  μ1=μ2=1.4 and solving the inequalities in Theorem 14 by linear programming, we get(52)ν1=1.25902.1813,υ1=1.80963.1278,ϑ1=1.48481.9703,ν2=1.18252.2998,υ2=1.81913.0665,ϑ2=1.49451.9646,g1=0.71490.7630,g2=0.84740.7229. (I) By gp=CpTKpTGpTνp+R2, we obtain the static output feedback controller:(53)K1=0.19000.4910,K2=0.26500.1050. (II) By gp=K-pTGpTνp+R2, we obtain the state feedback controller:(54)K-1=0.28810.1998,K-2=0.12750.4291.It is easy to verify that (41) and (46) are satisfied. Then, according to (12), we get τα1∗=1.6824 and  τα2∗=2.4034.The simulation results are shown in Figures1–5; the initial conditions of system (1) are x(0)=0.50.2T and x(θ)=0,  θ∈[-τ,0), which satisfy the condition sup-τ≤t≤0{xT(t)δ}≤1. Figure 1 depicts the switching signal with MDADT. Under the static output feedback control law, Figures 2 and 3 are obtained to show the state trajectories and the evolution of xT(t)ε of the closed-loop system (1). Under the state feedback control law, Figures 4 and 5 depict the state trajectories and the evolution of xT(t)ε of the closed-loop system (1). The cost value is J∗=50.9407, which can be obtained by (44) or (49).Figure 1
Switching signal of system (1).Figure 2
State trajectories of the closed-loop system (1) under static output feedback law.Figure 3
The evolution ofxT(t)ε of system (1) under static output feedback law.Figure 4
State trajectories of the closed-loop system (1) under state feedback law.Figure 5
The evolution ofxT(t)ε of system (1) under state feedback law.Remark 16.
Reference [6] considered the problem of exponential L1 output tracking control for positive switched linear systems with ADT, which focuses on the asymptotic stability and the exponential L1 performance index. There are some differences between [6] and this paper. Firstly, the finite-time stability in this paper and the asymptotic stability in [6] are two absolutely different concepts. Secondly, the performance indexes are also different. Finally, this paper uses MDADT approach instead of ADT method in [6]. So we do not compare the simulation results.


## 5. Conclusions

This paper has dealt with the problem of guaranteed cost finite-time control for positive switched systems with tine-varying delays. A novel guaranteed cost performance index is introduced. Based on the MDADT approach, some sufficient conditions and two kinds of controllers are obtained to guarantee that the closed-loop system is GCFTB. Finally, a numerical example is provided to show the effectiveness of the proposed method.



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*Source: 7051658-2017-02-19.xml*