Nonlinear Analysis: Hybrid Systems 9 (2013) 9–17

Contents lists available at SciVerse ScienceDirect

Nonlinear Analysis: Hybrid Systems

journal homepage: www.elsevier.com/locate/nahs

Stability, L1-gain and control synthesis for positive switchedsystems with time-varying delayMei Xiang, Zhengrong Xiang ∗

School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

a r t i c l e i n f o

Article history:Received 11 June 2012Accepted 7 January 2013

Keywords:Exponential stabilityWeighted L1-gainController designPositive switched systemsTime-varying delay

a b s t r a c t

Exponential stability, L1-gain performance and controller design problems for a classof positive switched systems with time-varying delay are investigated in this paper.First, by constructing an appropriate co-positive type Lyapunov–Krasovskii functional,sufficient conditions for the exponential stability are developed by using the averagedwell time approach. Then, the weighted L1-gain performance is investigated for thesystem considered. Based on the results obtained, an effective method is proposed forthe construction of a stabilizing feedback controller with L1-gain property. All the resultsare formulated as a set of linear matrix inequalities (LMIs) and therefore can be easilyimplemented. Finally, the theoretical results obtained are demonstrated by a numericalexample.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

A switched system is a type of hybrid dynamical system that consists of a number of subsystems and a switching signal,which defines a specific subsystem being activated during a certain interval. As a special class of hybrid system, manydynamical systems can bemodeled as switched systems [1–3]. Recently, the importance of positive switched systems,whosestates and outputs are non-negative whenever the initial conditions and inputs are non-negative, has been highlightedand investigated by many researchers due to their broad applications in communication systems [4,5], the viral mutationdynamics under drug treatment [6], formation flying [7], and system theory [8–12]. A positive switched system means aswitched system in which each subsystem is itself a positive system. It should be noted that, although switched systemshave been studied in much recent control engineering and mathematics literature, there are still many open questionsrelating to positive switched systems.

It is well known that the reaction of real-world systems to exogenous signals is always not instantaneous, and isaffected by certain time delays, for example long-distance transportation systems, hydraulic pressure systems, networkcontrol systems, and so on. Time delay is frequently a source of instability: it often causes undesirable performance infeedback systems such as chaos [13,14], and it even causes a system to become out of control. Although many resultshave been reported for time-delay systems [15–18], only recently have positive switched systems with time delay beeninvestigated [19,20].

It should be noted that many previous results on positive switched system mainly focus on stability and stabilization[21,22]. It is well known that the traditional Lyapunov–Krasovskii functional may give conservative stability conditions forpositive systems, as it fails to take account of the fact that the trajectories are naturally constrained to the positive orthant, so,when the stability of positive systems is considered, it is natural to apply a linear co-positive Lyapunov function. In addition,positive switched systems with disturbances are commonly found in practice. And due to the non-negative property, it

∗ Corresponding author. Tel.: +86 13951012297; fax: +86 25 84313809.E-mail addresses: xiangzr@mail.njust.edu.cn, xiangzr@tom.com (Z. Xiang).

1751-570X/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.doi:10.1016/j.nahs.2013.01.001

10 M. Xiang, Z. Xiang / Nonlinear Analysis: Hybrid Systems 9 (2013) 9–17

would be natural to evaluate the size of positive systems via the L1-gain (i.e., the L1-induced norm) in terms of the input andoutput signals. Thus, stability and L1-gain analysis problems have become interesting issues for disturbed positive switchedsystems. Some results on L1-gain analysis have been reported for positive systems [23,24], and positive switched systems[25,26]. Unfortunately, little attempt has beenmade to investigate the issue of L1-gain analysis for positive switched systemswith time delay, which motivates the present research.

The main contributions of this paper are three-fold: (1) by constructing an appropriate co-positive typeLyapunov–Krasovskii functional, sufficient conditions for the exponential stability are proposed by using the averagedwell time approach; (2) weighted L1-gain performance analysis for positive switched systems with time-varying delayis performed for the first time; (3) the desired controller is proposed under which exponential stability of a closed-loopsystem with weighted L1-gain performance is obtained.

The remainder of this paper is organized as follows. In Section 2, the system formulation and some necessary lemmas aregiven. Section 3 is devoted to deriving the results on stability, L1-gain analysis and controller design. An example is providedto illustrate the feasibility of the obtained results in Section 4. Concluding remarks are given in Section 5.Notation: In this paper, A≻0 (≺0) means that all entries of matrix A are non-negative (non-positive); A ≻ 0 (≺ 0) meansthat all entries of A are positive (negative); A ≻ B (A≻B)means that A− B ≻ 0 (A− B≻0); AT means the transpose of matrixA. R (R+) is the set of all real (positive real) numbers; Rn (Rn

+) is n-dimensional real (positive) vector space; Rn×k is the set

of all real matrices of dimension (n × k); Z+ refers to the set of all positive integers. ∥x∥ =n

k=1 |xk|, where xk is the kthelement of x ∈ Rn.

2. Problem statements and preliminaries

Consider the following switched linear systems with time-varying delay:x(t) = Aσ(t)x(t)+ Adσ(t)x(t − d(t))+ Bσ(t)w(t),xθ = φ(θ), θ ∈ [−τ , 0] ,z(t) = Cσ(t)x(t)+ Dσ(t)w(t),

(1)

where x(t) ∈ Rn and z(t) ∈ Rq denote the state and controlled output, respectively; w(t) ∈ Rw is the disturbance input,which belongs to L1 [0,∞); σ(t) : [0,∞) → M = {1, 2, . . . ,m} is the switching signal, with m being the number ofsubsystems,which depends on time t or state x(t); Ap, Adp, Bp, Cp andDp are constantmatriceswith appropriate dimensions,and p denotes the pth subsystem; φ(θ) is the initial condition on [−τ , 0], τ > 0; t0 is the initial time, and tq denotes theqth switching instant; d(t)denotes the time-varying delay satisfying 0 ≤ d(t) ≤ τ , d(t) ≤ d for known constants τ and d.

Without loss of generality, we let t0 = 0 in this paper.

Definition 1. System (1) is said to be positive if, for any initial condition φ(θ)≻0, θ ∈ [−τ , 0] , w(t)≻0 and any switchingsignal σ(t), the corresponding trajectory x(t)≻0 and z(t)≻0 holds for all t ≥ 0.

Definition 2 ([27]). A is called a Metzler matrix if the off-diagonal entries of matrix A are non-negative.

The following lemma is a direct extension of Lemma 3 in [20] and Proposition 1 in [19].

Lemma 1. System (1) is positive if and only if Ap, ∀p ∈ M,are Metzler matrices, and Adp≻0, Bp≻0, Cp≻0, Dp≻0, p ∈ M.

Definition 3 ([28]). The switched system (1) withw(t) = 0 is said to be exponentially stable under σ(t) if the solution x(t)satisfies

∥x(t)∥ ≤ αxt0 e−β(t−t0), ∀t0 ≥ 0, t ≥ t0,

for constants α > 0, β > 0, wherext0 = sup−τ≤δ≤0 ∥xδ∥.

Definition 4 ([29]). For any switching signal σ(t) and any T2 > T1 ≥ 0, let Nσ (T1, T2) denote the number of switchings ofσ(t) over the interval (T1, T2). For given Ta > 0,N0 ≥ 0, if the inequality

Nσ (T1, T2) ≤ N0 +T2 − T1

Taholds, then the positive constant Ta is called an average dwell time and N0 is called a chattering bound.

As commonly used in the literature, for convenience, we choose N0 = 0 in this paper.

Definition 5. For λ > 0 and γ > 0, system (1) is said to have weighted L1-gain if, under zero initial condition φ(θ) =

0, θ ∈ [−τ , 0], there exists switching signal σ(t). Thus the following conditions hold.

(a) System (1) is exponentially stable whenw(t) = 0.

M. Xiang, Z. Xiang / Nonlinear Analysis: Hybrid Systems 9 (2013) 9–17 11

(b) System (1) satisfies∞

t0e−λ(t−t0) ∥z(t)∥ dt ≤ γ

∞

t0

∥w(t)∥ dt (2)

whenw(t) = 0.

Remark 1. In Definition 5, γ is the system’s suppression to exogenous disturbance. The smaller γ is, the better theperformance is.

The aim of this paper is to find a class of switching signals σ(t) under which system (1) is exponentially stable andhas weighted L1-gain. Furthermore, the control synthesis problem will be investigated for the following positive switchedsystem with time-varying delay.x(t) = Aσ(t)x(t)+ Adσ(t)x(t − d(t))+ Bσ(t)w(t)+ Eσ(t)u(t),

xθ = φ(θ), θ ∈ [−τ , 0] ,z(t) = Cσ(t)x(t)+ Dσ(t)w(t).

(3)

3. Main results

This section will focus on the problem of stability analysis and control synthesis for positive switched systems withtime-varying delay.

3.1. Stability analysis

First, we consider the following non-switched delay positive system:x(t) = Ax(t)+ Adx(t − d(t)),xθ = φ(θ), θ ∈ [−τ , 0] , (4)

where A and Ad are constant matrices.Choose the co-positive type Lyapunov–Krasovskii functional candidate for system (4) as follows:

V (t, x(t)) = V1(t, x(t))+ V2(t, x(t))+ V3(t, x(t)) (5)

where

V1(t, x(t)) = xT (t)v,

V2(t, x(t)) =

t

t−d(t)eλ(−t+s)xT (s)υds,

V3(t, x(t)) =

0

−τ

t

t+θeλ(−t+s)xT (s)ϑdsdθ,

and v, υ, ϑ ∈ Rn+.

For the sake of simplicity, V (t, x(t)) is written as V (t) in this paper.

Lemma 2. Given positive constant λ, if there exist v, υ, ϑ ∈ Rn+and ς ∈ Rn such that

Ψ = diagψ1, ψ2, . . . , ψn, ψ

′

1, ψ′

2, . . . , ψ′

n

≤ 0 (6)

Π = diagπ1, π2, . . . , πn, π

′

1, π′

2, . . . , π′

n

≤ 0, (7)

where

ψr = aTr v + λv + υr + τϑr + ςr , ψ ′

r = aTdrv − (1 − d)e−λτυr − ςr ,

πr = −aTr ς − e−λτϑr , π ′

r = −aTdrς, r ∈ N = {1, 2, . . . , n} ,

ar (adr) represents the rth column vector of matrix A (Ad), and v = [v1, v2, . . . , vn]T , υ = [υ1, υ2, . . . , υn]T , ϑ =

[ϑ1, ϑ2, . . . , ϑn]T , ς = [ς1, ς2, . . . , ςn]T , then along the trajectory of system (4), we have

V (t) ≤ e−λ(t−t0)V (t0).

12 M. Xiang, Z. Xiang / Nonlinear Analysis: Hybrid Systems 9 (2013) 9–17

Proof. Along the trajectory of system (4) with co-positive type Lyapunov–Krasovskii functional (5), we have

V1(t) = xT (t)v = xT (t)ATv + xT (t − d(t))ATdv

V2(t) = −λ

t

t−d(t)eλ(−t+s)xT (s)υds + xT (t)υ − (1 − d(t))e−λd(t)xT (t − d(t))υ

≤ −λ

t

t−d(t)eλ(−t+s)xT (s)υds + xT (t)υ − (1 − d)e−λτ xT (t − d(t))υ

V3(t) = −λ

0

−τ

t

t+θeλ(−t+s)xT (s)ϑdsdθ + τxT (t)ϑ −

0

−τ

eλθxT (t + θ)ϑdθ

≤ −λ

0

−τ

t

t+θeλ(−t+s)xT (s)ϑdsdθ + τxT (t)ϑ −

t

t−d(t)e−λτ xT (s)ϑds

V (t)+ λV (t) ≤ xT (t)(ATv + λv + υ + τϑ)+ xT (t − d(t))(ATdv − (1 − d)e−λτυ)−

t

t−d(t)e−λτ xT (s)ϑds. (8)

Using Leibniz–Newton formula, one can obtain t

t−d(t)x(s)ds = x(t)− x(t − d(t)) (9)

and t

t−d(t)x(s)ds =

t

t−d(t)(Ax(s)+ Adx(s − d(s)))ds. (10)

Further, from (9) and (10), the following is obtained for any vector ς ∈ Rn:x(t)− x(t − d(t))−

t

t−d(t)(Ax(s)+ Adx(s − d(s)))ds

T

ς = 0. (11)

Combining (8) and (11) leads to

V (t)+ λV (t) ≤ xT (t)(ATv + λv + υ + τϑ + ς)+ xT (t − d(t))(ATdv − (1 − d)e−λτυ − ς)

−

t

t−d(t)

x(s)

x(s − d(s))

T ATς

ATdς

+

e−λτϑ

0

ds. (12)

By (6) and (7), ∀r ∈ N ,

ATv + λv + υ + τϑ + ς≺0 (13)

ATdv − (1 − d)e−λτυ − ς≺0 (14)

−ATς − e−λτϑ≺0 (15)

−ATdς≺0. (16)

From (12)–(16), one obtains

V (t) ≤ −λV (t).

Then, along the trajectory of system (4), we have

V (t) ≤ e−λ(t−t0)V (t0).

The proof is completed. �

Now we are in a position to provide sufficient conditions of exponential stability for the following positive switchedsystem:

x(t) = Aσ(t)x(t)+ Adσ(t)x(t − d(t)),xθ = φ(θ), θ ∈ [−τ , 0] , (17)

where Ap and Adp are constant matrices for any p ∈ M = {1, 2, . . . ,m}.

M. Xiang, Z. Xiang / Nonlinear Analysis: Hybrid Systems 9 (2013) 9–17 13

Theorem 1. Given positive constant λ, if there exist vp, υp, ϑp ∈ Rn+and ςp ∈ Rn, such that, ∀(p, k) ∈ M × M,

Ψp = diagψp1, ψp2, . . . , ψpn, ψ

′

p1, ψ′

p2, . . . , ψ′

pn

≤ 0 (18)

Πpk = diagπpk1, πpk2, . . . , πpkn, π

′

pk1, π′

pk2, . . . , π′

pkn

≤ 0, (19)

where

ψpr = aTprvp + λvpr + υpr + τϑpr + ςpr , ψ ′

r = aTdprvp − (1 − d)e−λτυpr − ςpr ,

πpkr = −aTkrςp − e−λτϑpr , π ′

pkr = −aTdkrςp, r ∈ N = {1, 2, . . . , n} ,

apradpr

represents the rth column vector of matrix Ap

Adp

; and vp =

vp1, vp2, . . . , vpn

T, υp =

υp1, υp2, . . . , υpn

T, ϑp =

ϑp1, ϑp2, . . . , ϑpnT, ςp =

ςp1, ςp2, . . . , ςpn

T , then system (17) is exponentially stable for any switching signal σ(t) withaverage dwell time

Ta > T ∗

a =lnµλ. (20)

Moreover, the state decay of system (17) is given by

∥x(t)∥ ≤ (ε2/ε1 + (ε3/ε1)τ + (ε4/ε1)τ2)e−

λ−

lnµTa

(t−t0)

xt0 , (21)

where ε1 = min(r,p)∈N×Mvpr

, ε2 = max(r,p)∈N×M

vpr

, ε3 = max(r,p)∈N×M

υpr

, ε4 = max(r,p)∈N×M

ϑpr

and µ ≥ 1

satisfies

vi≺µvj, υi≺µυj, ϑi≺µϑj, ∀i, j ∈ M. (22)

Proof. Choose the following multiple co-positive type Lyapunov–Krasovskii functional for system (17):

Vσ(t)(t) = xT (t)vσ(t) + t

t−d(t)eλ(−t+s)xT (s)υσ(t)ds +

0

−τ

t

t+θeλ(−t+s)xT (s)ϑσ(t)dsdθ, (23)

where vp, υp, ϑp ∈ Rn+, ∀p ∈ M .

According to (18), (19), (22) and Lemma 2, we can easily obtain

Vi(tk) ≤ µVj(t−k ), ∀i, j ∈ M, k = 1, 2, . . . . (24)

When t ∈ [tk, tk+1), from (24), we have

Vσ(t)(t) ≤ e−λ(t−tk)Vσ(tk)(tk). (25)

Therefore, it follows from (25) and the relation k = Nσ (t0, t) ≤t−t0Ta

that

Vσ(t)(t) ≤ e−λ(t−tk)µVσ(t−k )(t−

k )

≤ · · · ≤ e−λ(t−t0)µkVσ(t0)(t0)

≤ e−

λ−

lnµTa

(t−t0)Vσ(t0)(t0). (26)

Considering the definition of Vσ(t)(t), and denoting ε1 = min(r,p)∈N×Mvpr

, ε2 = max(r,p)∈N×M

vpr

, ε3 = max(r,p)∈N×M

υprand ε4 = max(r,p)∈N×M

ϑpr

, yields that

Vσ(t)(t) ≥ ε1 ∥x(t)∥ (27)

Vσ(t0)(t0) ≤ ε2 ∥x(t0)∥ + (ε3 + ε4τ)

t0

t0−τ∥x(s)∥ ds. (28)

Combining (26)–(28), we obtain

∥x(t)∥ ≤1ε1

e−

λ−

lnµTa

(t−t0)

ε2 ∥x(t0)∥ + (ε3e−λτ

+ ε4τe−λτ )

t0

t0−τ∥x(s)∥ ds

≤ (ε2/ε1 + (ε3/ε1)τ + (ε4/ε1)τ

2)e−

λ−

lnµTa

(t−t0) sup

−τ≤δ≤0∥xδ∥ . (29)

14 M. Xiang, Z. Xiang / Nonlinear Analysis: Hybrid Systems 9 (2013) 9–17

Thus, by denoting α = ε2/ε1 + (ε3/ε1)τ + (ε4/ε1)τ2, β = λ −

lnµTa

, it can be seen from (29) that ∥x(t)∥ ≤

αxt0 e−β(t−t0),∀t0 ≥ 0, t ≥ t0, where

xt0 = sup−τ≤δ≤0 ∥xδ∥. Finally, we can conclude that, if (18), (19) and (22)hold, system (17) is exponentially stable for any switching signal with average dwell time (20).

This completes the proof. �

Remark 2. When µ = 1 in (22), which leads to vi = µvj, υi = µυj, ϑi = µϑj, ∀i, j ∈ M , and T ∗a = 0, then system (17)

possesses a common co-positive type Lyapunov–Krasovskii functional, and switching signals can be arbitrary.

3.2. L1-gain analysis

The following result establishes sufficient conditions of exponential stability with weighted L1-gain property for system(1).

Theorem 2. For given positive constants λ and γ , if there exist vp, υp, ϑp ∈ Rn+and ςp ∈ Rn, such that, ∀(p, k) ∈ M × M,

Ψp = diagψp1, ψp2, . . . , ψpn, ψ

′

p1, ψ′

p2, . . . , ψ′

pn, ψ′′

p1, ψ′′

p2, . . . , ψ′′

pn

≤ 0 (30)

Πpk = diagπpk1, πpk2, . . . , πpkn, π

′

pk1, π′

pk2, . . . , π′

pkn, π′′

pk1, π′′

pk2, . . . , π′′

pkn

≤ 0, (31)

where

ψpr = aTprvp + λvpr + υpr + τϑpr + ςpr +cpr , ψ ′

pr = aTdprvp − (1 − d)e−λτυpr − ςpr ,

ψ ′′

pr = bTprvp +dpr − γ ,

πpkr = −aTkrςp − e−λτϑpr , π ′

pkr = −aTdkrςp, π ′′

pkr = −bTkrςp, r ∈ N = {1, 2, . . . , n} ,

apradpr , bpr , cpr , dpr

represents the rth column vector of matrix Ap

Adp, Bp, Cp,Dp

; and vp =

vp1, vp2, . . . , vpn

T, υp =

υp1, υp2, . . . , υpnT, ϑp =

ϑp1, ϑp2, . . . , ϑpn

T, ςp =

ςp1, ςp2, . . . , ςpn

T , then system (1) is exponentially stable withweighted L1-gain performance for any switching signal σ(t), and the average dwell time satisfies

Ta > T ∗

a =lnµλ, (32)

where µ ≥ 1 satisfies

vi≺µvj, υi≺µυj, ϑi≺µϑj, ∀i, j ∈ M. (33)

Proof. By Theorem 1, the exponential stability of system (1) with w(t) = 0 is ensured if (30) and (31) hold. To show theweighted L1-gain, we choose the Lyapunov functional (23). From (33), we have

Vi(tk) ≤ µVj(t−k ), ∀i, j ∈ M, k = 1, 2, . . . . (34)

For any t ∈ [tk, tk+1), noticing (30) and (31), we have

V (t) ≤ e−λ(t−tk)V (tk)−

t

tke−λ(t−s)Λ(s)ds, (35)

whereΛ(s) = ∥z(s)∥ − γ ∥w(s)∥. Combining (34) and (35) leads to

V (t) ≤ µe−λ(t−tk)V (t−k )−

t

tke−λ(t−s)Λ(s)ds

≤ µkV (t0)e−λt− µk

t1

t0e−λ(t−s)Λ(s)ds − µk−1

t2

t1e−λ(t−s)Λ(s)ds − · · · −

t

tke−λ(t−s)Λ(s)ds

= e−λ(t−t0)+Nσ (t0,t) lnµV (t0)−

t

t0e−λ(t−s)+Nσ (s,t) lnµΛ(s)ds. (36)

Under zero initial condition, (36) gives

0 ≤ −

t

t0e−λ(t−s)+Nσ (s,t) lnµΛ(s)ds. (37)

M. Xiang, Z. Xiang / Nonlinear Analysis: Hybrid Systems 9 (2013) 9–17 15

Multiplying both sides of (37) by e−Nσ (t0,t) lnµ yields t

t0e−λ(t−s)−Nσ (t0,s) lnµ ∥z(s)∥ ds ≤ γ

t

t0e−λ(t−s)−Nσ (t0,s) lnµ ∥w(s)∥ ds. (38)

Noticing that Nσ (t0, s) ≤ (s− t0)/Ta and Ta > T ∗a = lnµ/λ, we have Nσ (t0, s) lnµ ≤ λ(s− t0). Thus, it follows from (38)

that tt0e−λ(t−s)−λ(s−t0) ∥z(s)∥ ds ≤ γ

tt0e−λ(t−s) ∥w(s)∥ ds. Integrating both sides of this inequality from t = t0 to ∞ leads

to inequality (2).This completes the proof. �

Remark 3. When µ = 1 in Theorem 2, integrating both sides of inequality (38) from t = t0 to ∞ leads to∞

t0

∥z(s)∥ ds ≤ γ

∞

t0

∥w(s)∥ ds,

which gives the standard L1-gain performance.

3.3. Control synthesis

Inwhat follows, for positive switched system (3),we design a state feedback controller tomake the corresponding closed-loop system exponentially stable with a weighted L1-gain. By Theorem 2, this problem reduces to finding u(t) = Kσ(t)x(t)such thatx(t) = (Aσ(t) + Eσ(t)Kσ(t))x(t)+ Adσ(t)x(t − d(t))+ Bσ(t)w(t),

xθ = φ(θ), θ ∈ [−τ , 0] ,z(t) = Cσ(t)x(t)+ Dσ(t)w(t),

(39)

is exponentially stable, and satisfies the L1-gain performance under zero initial condition.

Theorem 3. Consider system (39), for given positive scalars λ and γ . If Ap + EpKp are Metzler matrices and there existvp, υp, ϑp ∈ Rn

+and ςp, gp, hpk ∈ Rn, such that, ∀(p, k) ∈ M × M,

Ψp = diagψp1, ψp2, . . . , ψpn, ψ

′

p1, ψ′

p2, . . . , ψ′

pn, ψ′′

p1, ψ′′

p2, . . . , ψ′′

pn

≤ 0 (40)

Πpk = diagπpk1, πpk2, . . . , πpkn, π

′

pk1, π′

pk2, . . . , π′

pkn, π′′

pk1, π′′

pk2, . . . , π′′

pkn

≤ 0, (41)

where

ψpr = aTprvp + λvpr + gpr + υpr + τϑpr + ςpr +cpr , ψ ′

pr = aTdprvp − (1 − d)e−λτυpr − ςpr ,

ψ ′′

pr = bTprvp +dpr − γ ,

πpkr = −aTkrςp − hpkr − e−λτϑpr , π ′

pkr = −aTdkrςp, π ′′

pkr = −bTkrςp, r ∈ N = {1, 2, . . . , n} ,

apradpr , bpr , cpr , dpr

represents the rth column vector of matrix Ap

Adp, Bp, Cp,Dp

; vp =

vp1, vp2, . . . , vpn

T, υp =

υp1,

υp2, . . . , υpnT, ϑp =

ϑp1, ϑp2, . . . , ϑpn

T, ςp =

ςp1, ςp2, . . . , ςpn

T, gp = K T

p ETp vp, hpk = K T

k ETk ςp, then system (3) is

positive and exponentially stable with L1-gain performance. Moreover, the feedback controller is

u(t) = Kσ(t)x(t), (42)

with average dwell time

Ta > T ∗

a =lnµλ, (43)

where µ ≥ 1 satisfies

vi≺µvj, υi≺µυj, ϑi≺µϑj, ∀i, j ∈ M. (44)

Proof. Letting gp = K Tp E

Tp vp, hpk = K T

k ETk ςp, we readily obtain Theorem 3.

This completes the proof. �

Based on Theorem 3, we now present an effective algorithm for constructing a feedback controller which exponentiallystabilizes positive switched system (3).

Algorithm. Step (1) Input the matrices Ap, Adp, Bp, Cp,Dp and Ep.Step (2) Solve the linear matrix inequality (40) to obtain vp, υp, ϑp, ςp, gp.

16 M. Xiang, Z. Xiang / Nonlinear Analysis: Hybrid Systems 9 (2013) 9–17

Fig. 1. Switching signal.

Step (3) Compute the gain matrices Kp by gp = K Tp E

Tp vp, then substitute them in (41), by adjusting the parameter λ such

that (41) holds.Step (4) Construct the feedback controller u(t) = Kσ(t)x(t), where Kp, p ∈ M , are the gain matrices.

4. Numerical example

Consider positive switched system (3) with two subsystems described by

A1 =

−3 81 −3

, Ad1 =

0.1 0.10.0 0.0

, B1 =

0.10.1

,

C1 =0.2 0.3

, D1 = [0.4] , E1 =

0.20.1

,

A2 =

−2 49 −5

, Ad2 =

0.1 0.10 0

, B2 =

0.20.1

,

C2 =0.1 0.2

, D2 = [0.2] , E2 =

0.10.3

.

Let d(t) = 0.5 |sin(0.2t)|, and take λ = 0.1, γ = 0.6. Solving the matrix inequalities in Theorem 3 gives rise to

v1 =

0.75960.7596

, v2 =

0.70710.7729

, υ1 =

0.83860.8386

, υ2 =

0.83860.8386

,

ϑ1 =

0.83860.8386

, ϑ2 =

0.83860.8386

, ς1 =

0.19660.1966

, ς2 =

0.19140.1914

,

and the state feedback gain matrices

K1 =−4.1691 −27.9412

, K2 =

−26.7708 −5.3850

.

Then, according to (44), we can get µ = 1.0742; then T ∗a = 0.716. The simulation results are shown in Figs. 1 and 2, where

the initial condition x(0) =2 3

T . Fig. 1 shows the switching signal with average dwell time Ta = 0.8; the state responsesof the corresponding closed-loop system are shown in Fig. 2.

5. Conclusions

In this paper we have investigated the problem of stability analysis and L1-gain performance for a class of positiveswitched systemswith time-varying delay. By constructing an appropriate co-positive type Lyapunov–Krasovskii functionaland using the average dwell time approach, sufficient conditions for the exponential stability and the weighted L1-gainperformance have been proposed. An effectivemethod has been proposed for the construction of a state feedback controllerwhich not only exponentially stabilizes the positive switched system but also ensures satisfactory L1-gain performance.Finally, an example was provided to show the effectiveness and applicability of the proposed results.

M. Xiang, Z. Xiang / Nonlinear Analysis: Hybrid Systems 9 (2013) 9–17 17

Fig. 2. State responses of the system.

Acknowledgment

Thisworkwas supported by theNational Natural Science Foundation of China under Grant Nos. 60974027 and 61273120.

References

[1] R. Goebel, R.G. Sanfelice, A.R. Teel, Hybrid dynamical systems, IEEE Control Syst. Mag. 29 (2) (2009) 28–93.[2] D. Liberzon, Switching in Systems and Control, Springer Verlag, Boston, MA, 2003.[3] Z. Li, Y. Soh, C. Wen, Switched and Impulsive Systems: Analysis, Design and Applications, Springer-Verlag, Berlin, Germany, 2005.[4] R. Shorten, F. Wirth, D. Leith, A positive systems model of TCP-like congestion control: asymptotic results, IEEE/ACM Trans. Netw. 14 (3) (2006)

616–629.[5] R. Shorten, D. Leith, J. Foy, R. Kilduff, Towards an analysis and design framework for congestion control in communication networks, in: Proc. the 12th

Yale Workshop on Adaptive and Learning Systems, 2003.[6] E. Hernandez-Varga, R. Middleton, P. Colaneri, F. Blanchini, Discrete-time control for switched positive systems with application to mitigating viral

escape, Internat. J. Robust Nonlinear Control 21 (10) (2011) 1093–1111.[7] A. Jadbabaie, J. Lin, A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Control 48 (6)

(2003) 988–1001.[8] T. Kaczorek, The choice of the forms of Lyapunov functions for a positive 2D Roesser model, Int. J. Appl. Math. Comput. Sci. 17 (4) (2007) 471–475.[9] T. Kaczorek, A realization problem for positive continuous-time systemswith reduced numbers of delays, Int. J. Appl. Math. Comput. Sci. 16 (3) (2006)

325–331.[10] L. Benvenuti, A. Santis, L. Farina, Positive Systems, in: Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, Germany, 2003.[11] M. Rami, F. Tadeo, A. Benzaouia, Control of constrained positive discrete systems, in: Proc. American Control Conference, New York, USA, 2007,

pp. 5851–5856.[12] M. Rami, F. Tadeo, Positive observation problem for linear discrete positive systems, in: Proc. the 45th IEEE Conference on Decision and Control, San

Diego, USA, 2006, pp. 4729–4733.[13] A. Ucar, A prototype model for chaos studies, Internat. J. Engrg. Sci. 40 (3) (2002) 251–258.[14] A. Ucar, On the chaotic behavior of a prototype delayed dynamical system, Chaos Solitons Fractals 16 (2) (2003) 187–194.[15] H.R. Karimi, H. Gao, New delay-dependent exponential H∞ synchronization for uncertain neural networks with mixed time delays, IEEE Trans. Syst.

Man Cybern. 40 (1) (2010) 173–185.[16] M.S. Mahmoud, P. Shi, Robust stability, stabilization and H∞control of time-delay systems with Markovian jump parameters, Internat. J. Robust

Nonlinear Control 13 (8) (2003) 755–784.[17] Z. Xiang, R. Wang, Non-fragile observer design for nonlinear switched systems with time delay, Int. J. Intell. Comput. Cybern. 2 (1) (2009) 175–189.[18] Z. Xiang, R. Wang, Robust control for uncertain switched non-linear systems with time delay under asynchronous switching, IET Control Theory Appl.

3 (8) (2009) 1041–1050.[19] X. Zhao, L. Zhang, P. Shi, Stability of a class of switched positive linear time-delay systems, Internat. J. Robust Nonlinear Control (2012).

http://dx.doi.org/10.1002/rnc.2777.[20] X. Liu, C. Dang, Stability analysis of positive switched linear systems with delays, IEEE Trans. Automat. Control 56 (7) (2011) 1684–1690.[21] E. Fornasini, M. Valcher, Stability and stabilizability of special classes of discrete-time positive switched systems, in: Proc. American Control

Conference, San Francisco, USA, July 2011, pp. 2619–2624.[22] F. Najson, State-feedback stabilizability, optimality, and convexity in switched positive linear systems, in: Proc. of American Control Conference, San

Francisco, USA, Jul. 2011, pp. 2625–2632.[23] Y. Ebihara, D. Peaucelle, D. Arzelier, L1 gain analysis of linear positive systems and its application, in: Proc. 50th IEEE Conference on Decision and

Control and European Control Conference, Orlando, FL, USA, December 2011, pp. 4029–4034.[24] X. Chen, J. Lam, P. Li, Z. Shu, Controller synthesis for positive systems under L1-induced performance, in: Proc. of 24th Chinese Control and Decision

Conference, May 2012, pp. 92–97.[25] A. Zappavigna, P. Colaneri, J. Geromel, R. Middleton, Stabilization of continuous-time switched linear positive systems, in: Proc. of American Control

Conference, Marriott Waterfront, Baltimore, MD, USA, June 2010, pp. 3275–3280.[26] A. Zappavigna, P. Colaneri, J. Geromel, R. Middleton, Dwell time analysis for continuous-time switched linear positive systems, in: Proc. of American

Control Conference, Marriott Waterfront, Baltimore, MD, USA, June 2010, pp. 6256–6261.[27] O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE Trans. Automat. Control 52

(7) (2007) 1346–1349.[28] X. Sun, W. Wang, G. Liu, J. Zhao, Stability analysis for linear switched systems with time-varying delay, IEEE Trans. Syst. Man Cybern. 38 (2) (2008)

528–533.[29] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell-time, in: Proc. of the 38th IEEE Conference on Decision and Control, 1999,

pp. 2655–2660.