# Stability andl1-Gain Control of Positive Switched Systems with Time-Varying Delays via Delta Operator Approach

Authors: Shuo Li; Zhengrong Xiang; Hamid Reza Karimi
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper investigates the problems of stability andl1-gain controller design for positive switched systems with time-varying delays via delta operator approach. The purpose is to design a switching signal and a state feedback controller such that the resulting closed-loop system is exponentially stable with  l1-gain performance. Based on the average dwell time approach, a sufficient condition for the existence of an  l1-gain controller for the considered system is established by constructing an appropriate copositive type Lyapunov-Krasovskii functional in delta domain. Moreover, the obtained conditions can unify some previously suggested relevant methods in the literature of both continuous- and discrete-time systems into the delta operator framework. Finally, a numerical example is presented to explicitly demonstrate the effectiveness and feasibility of the proposed method.


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## Body

## 1. Introduction

Positive systems mean that their states and outputs are nonnegative whenever the initial conditions and inputs are nonnegative [1, 2]. A positive switched system consists of a family of positive subsystems and a switching signal, coordinating the operation of various subsystems to specify when and how the switching takes place among the subsystems. Recently, due to the broad applications in communication systems [3, 4], formation flying [5], viral mutation dynamics under drug treatment [2], and systems theories [6–10], positive systems have been highlighted and investigated by many researchers [11–14]. It has been shown that a linear copositive Lyapunov functional is powerful for the analysis and synthesis of positive systems [15–17].The delta operator, a novel method with good finite word length performance under fast sampling rates, has drawn considerable interest in the past three decades. As we know, the standard shift operator was mostly adopted in the study of control theories for discrete-time systems. However, the dynamic response of a discrete system does not converge smoothly to its continuous counterpart when the sampling period tends to zero; namely, data are taken at high sampling rates. Until Goodwin et al. proposed a delta operator method in [18] to take the place of the traditional shift operator, the above problem is avoided. It was shown that delta operator requires smaller word length when implemented in fixed-point digital control processors than shift operator does [19]. The delta operator model can be regarded as a useful approach to deal with discrete-time systems under high sampling rates through the analysis methods of continuous-time systems [20–23]. Based on significant early investigations such as [24–26] studying the basic properties and performance of delta operator model, numerical properties and practical applications of delta operator model have been extensively investigated [27–29]. The delta operator is defined by
(1)δx(t)={dx(t)dt,T=0(x(t+T)-x(t))T,T≠0,
whereTis a sampling period. WhenT→0, the delta operator model will approach the continuous system before discretization and reflect a quasicontinuous performance.In real engineering, time delays are involved in many fields, such as mechanics, medicine, chemistry, biology, physics, economics, engineering, and control theory [30–33]. The existence of time delay may give rise to the deterioration of system performance and instability. Many results have been reported for time-delay systems [34–39].In addition, exogenous disturbances are commonly unavoidable in practical process, and the output will be inevitably affected by the disturbance in a system. Because of the peculiar nonnegative property of positive systems, thel1-gain (or L1-gain) index [39] can characterize the disturbance rejection property, by means of which we can limit the effect of disturbance in a prescribed level. Some results onl1-gain (or L1-gain) analysis and control for positive systems have been reported in the literature [39, 40]. However, few results on the issue ofl1-gain performance for positive switched systems via delta operator approach are proposed, which motivates the current research.In this paper, we focus our attention on investigating the stability andl1-gain controller design for positive switched systems with time-varying delays via delta operator approach. The main contributions of this paper are fourfold.(1)The positive switched systems via delta operator approach are investigated for the first time.(2)By applying the average dwell time approach, sufficient conditions of exponential stability for positive switched delta operator systems are derived. Moreover, the results obtained can be applied to both continuous-time systems and discrete-time systems.(3)l1-gain performance analysis of the underlying system is developed.(4)A state feedback controller design scheme is proposed such that the corresponding closed-loop system is exponentially stable with anl1-gain performance.The remainder of the paper is as follows. The problem formulation and some necessary lemmas are provided in Section2. In Section 3, the issues of stability,l1-gain performance analysis, and control of the underlying system are developed. A numerical example is presented to demonstrate the feasibility of the obtained results in Section 4. In Section 5, concluding remarks are given.Notations.  A⪰0  (⪯,≻,≺) means that all entries of matrixAare nonnegative (nonpositive, positive, and negative);A≻B(A⪰B)means thatA-B≻0(A-B⪰0);ATmeans the transpose of matrix A;R(R+)is the set of all real (positive real) numbers;Rn(R+n)is ann-dimensional real (positive real) vector space;Rm×nis the set of allm×n-dimensional real matrices;Z+refers to the set of all positive integers; the vector1-norm is denoted by∥x∥=∑k=1n|xk|, wherexkis thekth element ofx∈Rn;1n∈Rndenotes a column vector withnrows containing only1entry;l1[k0,∞)is the space of absolute summable sequence on[k0,∞); that is, we sayz:[k0,∞)→Rpis inl1[k0,∞)if∑k=k0∞∥z(k)∥<∞.


## 2. Problem Formulation

Consider the following switched delta operator system with time-varying delays:(2)δx(k)=Aσ(k)x(k)+Adσ(k)x(k-dk)+Dσ(k)w(k),x(k0+θ)=φ(θ),θ=-d-,-d-+1,…,0,z(k)=Cσ(k)x(k)+Eσ(k)w(k),
wherex(k)∈Rndenotes the state;z(k)∈Rlis the controlled output; andw(k)∈Rwis the disturbance input, which belongs tol1[k0,∞).kmeans the timet=kTandT>0is the sampling period;k0is the initial time.σ(k):[k0,∞)→m_={1,2,…,m}is the switching signal withmrepresenting the number of subsystems.Ap,Adp,Cp,Dp,andEpare constant matrices with appropriate dimensions.dkdenotes the time-varying discrete delay which satisfies0≤d_≤dk≤d-for known integersd_andd-;{φ(θ),θ=-d-,-d-+1,…,0}is a given discrete vector-valued initial condition. The switch is assumed to only occur at the sampling time in this paper.Remark 1.
To illustrate the main advantage of delta operator systems directly, we consider a typical continuous system without time delays as follows:(3)x˙(t)=Ax(t)+Dw(t),z(t)=Cx(t)+Ew(t).
Using the traditional shift operator approach to discretize the system, the following discrete form inz-domain can be obtained(k=0,1,2,…):
(4)x((k+1)T)=Azx(kT)+Dzw(kT),z(kT)=Czx(kT)+Ezw(kT),
whereAz=eAT,Dz=(∫0TeAtdt)D,Cz=C,andEz=E. WhenT→0,limT→0Az=IandlimT→0Dz=0. The movement of the system poles towards stable boundary makes the system defective with the increase in the sampling rates. However, by utilizing the delta operator approach, we can obtain the following system expressed in delta domain:
(5)δx(kT)=Aδx(kT)+Dδw(kT),z(kT)=Cδx(kT)+Eδw(kT),
whereAδ=(Az-I)/T,Dδ=Dz/T,Cδ=C,andEδ=E. WhenT→0,limT→0Aδ=AandlimT→0Dz=D. It can be seen that the system matrices are the same as those of the original continuous system, alleviating the problems encountered with fast sampling.Remark 2.
Since a delta operator system can be regarded as a quasicontinuous system whenT→0, the termδx(k)can be utilized likex˙(t)in normal continuous-time systems.Definition 3.
System (2) is said to be positive if, for any initial conditionsφ(θ)⪰0,θ=-d-,-d-+1,…,0, any inputsw(k)⪰0, and any switching signalsσ(k), the corresponding trajectoriesx(k)⪰0andz(k)⪰0hold for allk≥k0.Remark 4.
Definition3 follows the general positivity definition of a positive system, which means that the state and output are nonnegative whenever the initial condition and input are nonnegative [1, 2].Lemma 5.
System (2) is positive if and only if(I+TAp)⪰0,Adp⪰0, Dp⪰0,Cp⪰0, andEp⪰0, for all p∈m_.Proof.
From the definition of delta operatorδ, the discrete form of system (2) can be obtained as follows:
(6)x(k+1)=(I+TAσ(k))x(k)+TAdσ(k)x(k-dk)+TDσ(k)w(k),x(k0+θ)=φ(θ),θ=-d-,-d-+1,…,0,z(k)=Cσ(k)x(k)+Eσ(k)w(k).
Combining Lemma2in [41] and Lemma1in [42], one can obtain the remaining proof easily.Remark 6.
WhenT→0, system (2) degenerates to a general continuous-time positive switched system as follows:
(7)x˙(t)=Aσ(t)x(t)+Adσ(t)x(t-d(t))+Dσ(t)w(t),x(t0+θ)=φ(θ),θ∈[-d2,0],z(t)=Cσ(t)x(t)+Eσ(t)w(t),
whered(t)denotes the time-varying delay which is everywhere time differentiable and satisfies0≤d1≤d(t)≤d2 andd˙(t)≤dd<1for known constantsd1,d2, anddd. Then according to [39], system (7) is positive if and only ifApare Metzler matrices, andAdp⪰0,Cp⪰0,Dp⪰0,andEp⪰0, for all p∈m_.Remark 7.
In the light of Lemma2.1of [43], it is clear that thepth subsystem in system (2) is positive if and only if (I+TAp)⪰0, Αdp⪰0, Dp⪰0, Cp⪰0, and Ep⪰0, for all p∈m_. Thus we can have an equivalent expression of Lemma 5: system (2) is positive under any switching signals if and only if it consists of a family of positive subsystems.Definition 8 (see [44]).
System (2) withw(k)=0is said to be exponentially stable underσ(k)if, for constantsα>0 andβ>0, the solutionx(k)satisfies
(8)∥x(k)∥≤α∥x(k0)∥ce-β(k-k0),∀k≥k0,
where∥x(k0)∥c=sup-d-≤θ≤0∥x(k0+θ)∥.Definition 9 (see [45]).
For any switching signalσ(k) and anyk2>k1≥0, letNσ(k1,k2)denote the number of switches ofσ(k)over the interval[k1,k2). For givenτa>0andN0≥0, if the inequality
(9)Nσ(k1,k2)≤N0+k2-k1τa
holds, then the positive constantτais called an average dwell time andN0is called a chattering bound.
Without loss of generality, one choosesN0=0in this paper.Definition 10.
For0<α<1/Tandγ>0, system (2) is said to have a prescribedl1-gain performance levelγif there exists a switching signalσ(k)such that the following conditions are satisfied:(a)
system (2) is exponentially stable whenw(k)=0;(b)
under zero initial condition, that is,φ(θ)=0,θ=-d-,-d-+1,…,0, system (2) satisfies
(10)∑k=k0∞(1-Tα)(k-k0)∥z(k)∥≤γ∑k=k0∞∥w(k)∥,iiiiiiiiiiii∀w(k)∈l1[k0,∞),w(k)≠0.Remark 11.
In Definition10, as proposed in [39],l1-gain performance indexγcharacterizes system's suppression to exogenous disturbances. The smaller the value ofγis, the better the performance of the system is, that is, the lesser the effect of the disturbance input on the control output is.
The purposes of this paper are(1)to find a class of switching signalsσ(k)under which system (2) is exponentially stable and possesses anl1-gain performance and(2)to determine a class of switching signals and a state feedback controlleru(k)=Kσ(k)x(k)for the following positive switched delta operator system with time-varying delays:
(11)δx(k)=Aσ(k)x(k)+Adσ(k)x(k-dk)+Bσ(k)u(k)+Dσ(k)w(k),x(k0+θ)=φ(θ),θ=-d-,-d-+1,…,0,z(k)=Cσ(k)x(k)+Eσ(k)w(k)
such that the resulting closed-loop system is exponentially stable with an  l1-gain performance.


## 3. Main Results

This section will focus on the problems of stability analysis andl1-gain controller design for positive switched delta operator systems with time-varying delays.


### 3.1. Stability Analysis

First, we consider the following switched positive delta operator system:(12)δx(k)=Aσ(k)x(k)+Adσ(k)x(k-dk),x(k0+θ)=φ(θ),θ=-d-,-d-+1,…,0,
whereI+TAp⪰0,Adp⪰0forp∈m_, anddkis defined the same as system (2).Sufficient conditions of exponential stability of system (12) are provided in the following theorem.Theorem 12.
Given a positive constant0<α<1/T, if there exist νp,υp,ϑp∈R+n, such that, for all p∈m_,
(13)ApTνp+ανp+(1-Tα)(d--d_+1)υp+(1-Tα)ϑp⪯0,AdpTνp-(1-Tα)d-+1υp⪯0,
whereνp=[νp1,νp2,…,νpn]T,υp=[υp1,υp2,…,υpn]T, andϑp=[ϑp1,ϑp2,…,ϑpn]T, then system (12) is exponentially stable for any switching signalsσ(k)with average dwell time
(14)τa>τa*=-lnμln(1-Tα),
whereμ≥1satisfies
(15)νp⪯μνq,υp⪯μυq,ϑp⪯μϑq,∀p,q∈m_.
Furthermore, the state decay of system (12) is given by
(16)∥x(k)∥≤ab(k-k0)∥x(k0)∥c,
where
(17)a=ε2ε1+Tε3d-ε1+0.5Tε3(d-+d_-1)(d--d_)ε1+Tε4d-ε1,b=μ1/τa(1-Tα),ε1=min(r,p)∈n_×m_{νpr},ε2=max(r,p)∈n_×m_{νpr},ε3=max(r,p)∈n_×m_{υpr},ε4=max(r,p)∈n_×m_{ϑpr},∥x(k0)∥c=sup-d-≤θ≤0∥x(k0+θ)∥,n_={1,2,…,n}.Proof.
Choose the following piecewise copositive type Lyapunov functional for thepth subsystem in system (12):
(18)Vp(k,x(k))=Vp1(k,x(k))+Vp2(k,x(k))+Vp3(k,x(k))+Vp4(k,x(k)),
where
(19)Vp1(k,x(k))=xT(k)νp,Vp2(k,x(k))=T∑s=k-dkk-1(1-Tα)k-sxT(s)υp,Vp3(k,x(k))=T∑l=-d-+1-d_∑s=k+lk-1(1-Tα)k-sxT(s)υp,Vp4(k,x(k))=T∑s=k-d-k-1(1-Tα)k-sxT(s)ϑp,∀p∈m_.
For simplicity,Vp(k,x(k))is written asVp(k)(correspondingly,V(k,x(k))is written asV(k)) in the later section of the paper.
The Lyapunov function in delta domain has the following form:(20)δVp1(k,x(k))=δ(xT(k)νp)=(δxT(k))νp=xT(k)ApTνp+xT(k-dk)AdpTνp,δVp2(k,x(k))=1T[Vp2(k+1)-Vp2(k)]=1T[T∑s=k+1-dk+1(k+1)-1(1-Tα)k+1-sxT(s)υp-T∑s=k-dkk-1(1-Tα)k-sxT(s)υp]≤-Tα∑s=k-dkk-1(1-Tα)k-sxT(s)υp+(1-Tα)xT(k)υp-(1-Tα)d-+1xT(k-dk)υp+∑s=k+1-d-k-d_(1-Tα)k+1-sxT(s)υp,δVp3(k,x(k))=1T[Vp3(k+1)-Vp3(k)]=1T[T∑l=-d-+1-d_∑s=k+1+l(k+1)-1(1-Tα)k+1-sxT(s)υp-T∑l=-d-+1-d_∑s=k+lk-1(1-Tα)k-sxT(s)υp]=-Tα∑l=-d-+1-d_∑s=k+lk-1(1-Tα)k-sxT(s)υp+(1-Tα)(d--d_)xT(k)υp-∑s=k+1-d-k-d_(1-Tα)k+1-sxT(s)υp,δVp4(k,x(k))=1T[Vp4(k+1)-Vp4(k)]=1T[T∑s=k+1-d-(k+1)-1(1-Tα)k+1-sxT(s)ϑp-T∑s=k-d-k-1(1-Tα)k-sxT(s)ϑp]=-Tα∑s=k-d-k-1(1-Tα)k-sxT(s)ϑp+(1-Tα)xT(k)ϑp-(1-Tα)d-+1xT(k-d-)ϑp.
According to (20), we have
(21)δVp(k,x(k))+αVp(k,x(k))≤xT(k)ApTνp+xT(k-dk)AdpTνp+αxT(k)νp+(1-Tα)xT(k)υp-(1-Tα)d-+1xT(k-dk)υp+∑s=k+1-d-k-d_(1-Tα)k+1-sxT(s)υp+(1-Tα)(d--d_)xT(k)υp-∑s=k+1-d-k-d_(1-Tα)k+1-sxT(s)υp+(1-Tα)xT(k)ϑp-(1-Tα)d-+1xT(k-d-)ϑp≤xT(k)[ApTνp+ανp+(1-Tα)×(d--d_+1)υp+(1-Tα)ϑp]+xT(k-dk)[AdpTνp-(1-Tα)d-+1υp].
From (13), we obtain
(22)δVp(k)+αVp(k)≤0⟹δVp(k)=Vp(k+1)-Vp(k)T≤-αVp(k)⟹Vp(k+1)≤Vp(k)-TαVp(k)⟹Vp(k+1)≤(1-Tα)Vp(k).
Letk1<⋯<kgdenote the switching instants ofσ(k)over the interval[k0,k). Consider the following piecewise Lyapunov functional candidate for system (12):
(23)V(k)=Vσ(k)(k).
From (15) and (18), we obtain
(24)Vσ(ki)(k)≤μVσ(ki-)(k),i=1,2,…,g.
Then, it follows from (22), (24), and the relationNσ(k0,k)≤(k-k0)/τathat, for[ki,ki+1),
(25)Vσ(k)(k)=Vσ(ki)(k)≤(1-Tα)(k-ki)Vσ(ki)(ki)≤μ(1-Tα)(k-ki)Vσ(ki-)(ki-)≤μ(1-Tα)(k-ki)+(ki-ki-1)Vσ(ki-1)(ki-1)=μ(1-Tα)(k-ki-1)Vσ(ki-1)(ki-1)=μNσ(ki-1,k)(1-Tα)(k-ki-1)Vσ(ki-1)(ki-1)≤⋯≤μNσ(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)≤μ(k-k0)/τa(1-Tα)(k-k0)Vσ(k0)(k0)≤(μ1/τa(1-Tα))(k-k0)Vσ(k0)(k0).
Considering the definition ofVσ(k)(k),ε1,ε2,ε3,andε4in Theorem 12, it yields that
(26)Vσ(k)(k)≥ε1∥x(k)∥,Vσ(k0)(k0)≤(ε2+Tε3d-+0.5Tε3(d-+d_-1)×(d--d_)+Tε4d-)sup-d-≤θ≤0∥x(k0+θ)∥.
Combining (25)-(26), we obtain
(27)∥x(k)∥≤(ε2ε1+Tε3d-ε1+0.5Tε3(d-+d_-1)(d--d_)ε1+Tε4d-ε1)×(μ1/τa(1-Tα))(k-k0)∥x(k0)∥c,
where∥x(k0)∥c=sup-d-≤θ≤0∥x(k0+θ)∥.
Therefore, according to Definition8, system (12) is exponentially stable for any switching signalsσ(k)with average dwell time (14).
This completes the proof.Remark 13.
Whenμ=1in (15), which leads toνp=νq,υp=υq,ϑp=ϑq, for all p,q∈m_, andτa*=0by (14), system (12) possesses a common copositive type Lyapunov-Krasovskii functional, and the switching signal can be arbitrary.
Whendk=0, system (12) can be represented by
(28)δx(k)=A~σ(k)x(k),
whereA~p=Ap+AdpsatisfiesI+TA~p⪰0, for all p∈m_. Then we have the following corollary.Corollary 14.
Given a positive constant0<α<1/T, if there existνp∈R+n, such that, for all p∈m_,
(29)A~pTνp+ανp⪯0,
then system (28) is exponentially stable for any switching signalsσ(k)with average dwell time (14), whereμ≥1satisfies
(30)νp⪯μνq,∀p,q∈m_.
When the sampling periodT→0, system (12) becomes a continuous-time system as follows:
(31)x˙(t)=Aσ(t)x(t)+Adσ(t)x(t-d(t)),x(t0+θ)=φ(θ),θ∈[-d2,0],
whereApare Metzler matrices and Adp⪰0, for all p∈m_. d(t)denotes the time-varying delay which satisfies0≤d1≤d(t)≤d2 andd˙(t)≤dd<1for known constantsd1,d2,anddd.We can obtain sufficient conditions of exponential stability of system (31) by Theorem 12.Corollary 15.
Given a positive constantα, if there exist νp,υp,ϑp∈R+n, such that, for all p∈m_,
(32)ApTνp+ανp+(d2-d1+1)υp+ϑp⪯0,AdpTνp-(1-dd)e-αd2υp⪯0,
then system (31) is exponentially stable for any switching signalsσ(t)with average dwell time
(33)τa>τa*=lnμα,
whereμ≥1satisfies (15).
LetA-σ(k)=Aσ(k)+I. When the sampling periodT=1, system (12) becomes a discrete-time system as follows:
(34)x(k+1)=A-σ(k)x(k)+Adσ(k)x(k-dk),x(k0+θ)=φ(θ),θ=-d-,-d-+1,…,0,
whereA-p⪰0andAdp⪰0, for all p∈m_. One can obtain sufficient conditions of exponential stability of system (34) by Theorem 12.Corollary 16.
Given a positive constant0<α<1, if there existνp,υp,ϑp∈R+n, such that, for all p∈m_,
(35)A-pTνp+ανp+(1-α)(d--d_+1)υp+(1-α)ϑp⪯0,AdpTνp-(1-α)d-+1υp⪯0,
then system (34) is exponentially stable for any switching signalsσ(k)with average dwell time
(36)τa>τa*=-lnμln(1-α),
whereμ≥1satisfies (15).


### 3.2.l1-Gain Analysis

The following theorem establishes sufficient conditions of exponential stability withl1-gain property for system (2).Theorem 17.
For given positive constants0<α<1/Tandγ, if there exist νp,υp,ϑp∈R+n, such that, for all p∈m_,
(37)ApTνp+ανp+(1-Tα)(d--d_+1)υp+(1-Tα)ϑp+c~p⪯0,AdpTνp-(1-Tα)d-+1υp⪯0,(38)DpTνp+e~p-γ1w⪯0,
where1w=[1,1,…,1︸w]T,c~p=[∥cp1∥,∥cp2∥,…,∥cpn∥]T,cpjrepresents thejth column of matrixCp,j∈n_={1,2,…,n}, e~p=[∥ep1∥,∥ep2∥,…,∥epw∥]T, and epjrepresents thejth column of matrixEp,j∈w_={1,2,…,w}, then system (2) is exponentially stable with anl1-gain performance for any switching signalsσ(k)with average dwell time (14), whereμ≥1satisfies (15).Proof.
By Theorem12, the exponential stability of system (2) withw(k)=0is ensured if (37) holds. To show the weightedl1-gain performance, we choose the Lyapunov functional (18). From (15), we have
(39)Vσ(ki)(ki)≤μVσ(ki-)(ki-),∀i=1,2,….
For anyk∈[ki,ki+1), noticing (37)-(38), we have
(40)V(k)≤(1-Tα)(k-ki)Vσ(ki)(ki)-∑s=kik(1-Tα)(k-s)Λ(s),
whereΛ(s)=∥z(s)∥-γ∥w(s)∥.
Combining (39) and (40) leads to
(41)V(k)≤μ(1-Tα)(k-ki)Vσ(ki-)(ki-)-∑s=kik(1-Tα)(k-s)Λ(s)≤μ(1-Tα)(k-ki-1)Vσ(ki-1)(ki-1)-μ∑s=ki-1ki(1-Tα)(k-s)Λ(s)-∑s=kik(1-Tα)(k-s)Λ(s)=μNσ(ki-1,k)(1-Tα)(k-ki-1)Vσ(ki-1)(ki-1)-μNσ(ki-1,k)∑s=ki-1ki(1-Tα)(k-s)Λ(s)-∑s=kik(1-Tα)(k-s)Λ(s)≤⋯≤μNσ(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)-μNσ(k0,k)∑s=k0k1(1-Tα)(k-s)Λ(s)-μNσ(k1,k)∑s=k1k2(1-Tα)(k-s)Λ(s)-⋯-∑s=kik(1-Tα)(k-s)Λ(s)=μNσ(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)-∑s=k0kμNσ(s,k)(1-Tα)(k-s)Λ(s).
Under the zero initial condition, we obtain from (41) that
(42)0≤-∑s=k0kμNσ(s,k)(1-Tα)(k-s)Λ(s);
namely,
(43)∑s=k0kμNσ(s,k)(1-Tα)(k-s)∥z(s)∥≤γ∑s=k0kμNσ(s,k)(1-Tα)(k-s)∥w(s)∥.
Multiplying both sides of (43) byμ-Nσ(k0,k)yields
(44)∑s=k0kμ-Nσ(k0,s)(1-Tα)(k-s)∥z(s)∥≤γ∑s=k0kμ-Nσ(k0,s)(1-Tα)(k-s)∥w(s)∥.
Noticing thatNσ(k)(k0,s)≤(s-k0)/τa, we have
(45)μ-Nσ(k0,s)≥(1-Tα)(s-k0).
Combining (44) and (45) leads to
(46)∑s=k0k(1-Tα)(s-k0)(1-Tα)(k-s)∥z(s)∥≤γ∑s=k0k(1-Tα)(k-s)∥w(s)∥.
Summing both sides of (46) fromk=k0to∞leads to
(47)∑k=k0∞(1-Tα)(k-k0)∥z(k)∥≤γ∑k=k0∞∥w(k)∥.
From Definition 10, it can be concluded that system (2) is exponentially stable with a prescribedl1-gain performance levelγ.
This completes the proof.Remark 18.
Whenμ=1in Theorem 17, summing both sides of (44) fromk=k0to∞leads to
(48)∑k=k0∞∥z(k)∥≤γ∑k=k0∞∥w(k)∥
which gives the standardl1-gain performance.


### 3.3. Controller Design

In this section, we are interested in designing a state feedback controlleru(k)=Kσ(k)x(k)for positive switched system (11) such that the corresponding closed-loop system
(49)δx(k)=(Aσ(k)+Bσ(k)Kσ(k))x(k)+Adσ(k)x(k-dk)+Dσ(k)w(k),x(k0+θ)=φ(θ),θ=-d-,-d-+1,…,0,z(k)=Cσ(k)x(k)+Eσ(k)w(k)
is exponentially stable with an l1-gain performance.Theorem 19.
Considering system (11), for given positive scalars0<α<1/Tandγ, if there exist νp,υp,ϑp∈R+n andgp∈Rn, such that, for all p∈m_,
(50)ApTνp+gp+ανp+(1-Tα)(d--d_+1)υp+(1-Tα)ϑp+c~p⪯0,(51)AdpTνp-(1-Tα)d-+1υp⪯0,(52)DpTνp+e~p-γ1w⪯0,(53)I+T(Ap+BpKp)⪰0,
wherec~pande~phave been defined in Theorem 17 andgp=KpTBpTνp, then the corresponding closed-loop system (49) is positive and exponentially stable with a prescribedl1-gain performance levelγfor any switching signalsσ(k)with average dwell time (14), whereμ≥1satisfies (15).Proof.
Denotegp=KpTBpTνp. Following the proof line of Theorem 17, one can exactly obtain Theorem 19. It is omitted here.
This completes the proof.Consider the controller design of the following positive switched delta operator system without time delay:(54)δx(k)=A~σ(k)x(k)+Bσ(k)u(k)+Dσ(k)w(k),z(k)=Cσ(k)x(k)+Eσ(k)w(k),
whereA~σ(k)=Aσ(k)+Adσ(k). Then we directly have the following corollary.Corollary 20.
Considering system (54), for given positive scalars0<α<1/Tandγ, if there existνp∈R+nandgp∈Rn, such that, for all p∈m_,
(55)A~pTνp+ανp+gp+c~p⪯0,DpTνp+e~p-γ1w⪯0,I+T(A~p+BpKp)⪰0,
wherec~pande~phave been defined in Theorem 17 andgp=KpTBpTνp, then the corresponding closed-loop system is positive and exponentially stable with a prescribedl1-gain performance levelγfor any switching signalsσ(k)with average dwell time (14), whereμ≥1satisfies (30).
Based on Theorem19, one  is now in a position to present an effective algorithm for constructing the desired controller.Algorithm 21.
Consider the following.Step 1.
Input the matricesAp,Adp,Bp,Cp,Dp, and Ep.Step 2.
Choose the parameters0<α<1/Tandγ>0. By solving (50)–(52), one can obtain the solutions ofνp,υp,ϑp, and gp.Step 3.
By the equationgp=KpTBpTνpwith the obtainedgpandνp, one can get the gain matricesKp.Step 4.
Check condition (53) in Theorem 19. If it holds, go to Step 5; otherwise, adjust the parameterαand return to Step 2.Step 5.
Construct the feedback controlleru(k)=Kσ(k)x(k), whereKp,p∈m_, are the gain matrices.


## 3.1. Stability Analysis

First, we consider the following switched positive delta operator system:(12)δx(k)=Aσ(k)x(k)+Adσ(k)x(k-dk),x(k0+θ)=φ(θ),θ=-d-,-d-+1,…,0,
whereI+TAp⪰0,Adp⪰0forp∈m_, anddkis defined the same as system (2).Sufficient conditions of exponential stability of system (12) are provided in the following theorem.Theorem 12.
Given a positive constant0<α<1/T, if there exist νp,υp,ϑp∈R+n, such that, for all p∈m_,
(13)ApTνp+ανp+(1-Tα)(d--d_+1)υp+(1-Tα)ϑp⪯0,AdpTνp-(1-Tα)d-+1υp⪯0,
whereνp=[νp1,νp2,…,νpn]T,υp=[υp1,υp2,…,υpn]T, andϑp=[ϑp1,ϑp2,…,ϑpn]T, then system (12) is exponentially stable for any switching signalsσ(k)with average dwell time
(14)τa>τa*=-lnμln(1-Tα),
whereμ≥1satisfies
(15)νp⪯μνq,υp⪯μυq,ϑp⪯μϑq,∀p,q∈m_.
Furthermore, the state decay of system (12) is given by
(16)∥x(k)∥≤ab(k-k0)∥x(k0)∥c,
where
(17)a=ε2ε1+Tε3d-ε1+0.5Tε3(d-+d_-1)(d--d_)ε1+Tε4d-ε1,b=μ1/τa(1-Tα),ε1=min(r,p)∈n_×m_{νpr},ε2=max(r,p)∈n_×m_{νpr},ε3=max(r,p)∈n_×m_{υpr},ε4=max(r,p)∈n_×m_{ϑpr},∥x(k0)∥c=sup-d-≤θ≤0∥x(k0+θ)∥,n_={1,2,…,n}.Proof.
Choose the following piecewise copositive type Lyapunov functional for thepth subsystem in system (12):
(18)Vp(k,x(k))=Vp1(k,x(k))+Vp2(k,x(k))+Vp3(k,x(k))+Vp4(k,x(k)),
where
(19)Vp1(k,x(k))=xT(k)νp,Vp2(k,x(k))=T∑s=k-dkk-1(1-Tα)k-sxT(s)υp,Vp3(k,x(k))=T∑l=-d-+1-d_∑s=k+lk-1(1-Tα)k-sxT(s)υp,Vp4(k,x(k))=T∑s=k-d-k-1(1-Tα)k-sxT(s)ϑp,∀p∈m_.
For simplicity,Vp(k,x(k))is written asVp(k)(correspondingly,V(k,x(k))is written asV(k)) in the later section of the paper.
The Lyapunov function in delta domain has the following form:(20)δVp1(k,x(k))=δ(xT(k)νp)=(δxT(k))νp=xT(k)ApTνp+xT(k-dk)AdpTνp,δVp2(k,x(k))=1T[Vp2(k+1)-Vp2(k)]=1T[T∑s=k+1-dk+1(k+1)-1(1-Tα)k+1-sxT(s)υp-T∑s=k-dkk-1(1-Tα)k-sxT(s)υp]≤-Tα∑s=k-dkk-1(1-Tα)k-sxT(s)υp+(1-Tα)xT(k)υp-(1-Tα)d-+1xT(k-dk)υp+∑s=k+1-d-k-d_(1-Tα)k+1-sxT(s)υp,δVp3(k,x(k))=1T[Vp3(k+1)-Vp3(k)]=1T[T∑l=-d-+1-d_∑s=k+1+l(k+1)-1(1-Tα)k+1-sxT(s)υp-T∑l=-d-+1-d_∑s=k+lk-1(1-Tα)k-sxT(s)υp]=-Tα∑l=-d-+1-d_∑s=k+lk-1(1-Tα)k-sxT(s)υp+(1-Tα)(d--d_)xT(k)υp-∑s=k+1-d-k-d_(1-Tα)k+1-sxT(s)υp,δVp4(k,x(k))=1T[Vp4(k+1)-Vp4(k)]=1T[T∑s=k+1-d-(k+1)-1(1-Tα)k+1-sxT(s)ϑp-T∑s=k-d-k-1(1-Tα)k-sxT(s)ϑp]=-Tα∑s=k-d-k-1(1-Tα)k-sxT(s)ϑp+(1-Tα)xT(k)ϑp-(1-Tα)d-+1xT(k-d-)ϑp.
According to (20), we have
(21)δVp(k,x(k))+αVp(k,x(k))≤xT(k)ApTνp+xT(k-dk)AdpTνp+αxT(k)νp+(1-Tα)xT(k)υp-(1-Tα)d-+1xT(k-dk)υp+∑s=k+1-d-k-d_(1-Tα)k+1-sxT(s)υp+(1-Tα)(d--d_)xT(k)υp-∑s=k+1-d-k-d_(1-Tα)k+1-sxT(s)υp+(1-Tα)xT(k)ϑp-(1-Tα)d-+1xT(k-d-)ϑp≤xT(k)[ApTνp+ανp+(1-Tα)×(d--d_+1)υp+(1-Tα)ϑp]+xT(k-dk)[AdpTνp-(1-Tα)d-+1υp].
From (13), we obtain
(22)δVp(k)+αVp(k)≤0⟹δVp(k)=Vp(k+1)-Vp(k)T≤-αVp(k)⟹Vp(k+1)≤Vp(k)-TαVp(k)⟹Vp(k+1)≤(1-Tα)Vp(k).
Letk1<⋯<kgdenote the switching instants ofσ(k)over the interval[k0,k). Consider the following piecewise Lyapunov functional candidate for system (12):
(23)V(k)=Vσ(k)(k).
From (15) and (18), we obtain
(24)Vσ(ki)(k)≤μVσ(ki-)(k),i=1,2,…,g.
Then, it follows from (22), (24), and the relationNσ(k0,k)≤(k-k0)/τathat, for[ki,ki+1),
(25)Vσ(k)(k)=Vσ(ki)(k)≤(1-Tα)(k-ki)Vσ(ki)(ki)≤μ(1-Tα)(k-ki)Vσ(ki-)(ki-)≤μ(1-Tα)(k-ki)+(ki-ki-1)Vσ(ki-1)(ki-1)=μ(1-Tα)(k-ki-1)Vσ(ki-1)(ki-1)=μNσ(ki-1,k)(1-Tα)(k-ki-1)Vσ(ki-1)(ki-1)≤⋯≤μNσ(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)≤μ(k-k0)/τa(1-Tα)(k-k0)Vσ(k0)(k0)≤(μ1/τa(1-Tα))(k-k0)Vσ(k0)(k0).
Considering the definition ofVσ(k)(k),ε1,ε2,ε3,andε4in Theorem 12, it yields that
(26)Vσ(k)(k)≥ε1∥x(k)∥,Vσ(k0)(k0)≤(ε2+Tε3d-+0.5Tε3(d-+d_-1)×(d--d_)+Tε4d-)sup-d-≤θ≤0∥x(k0+θ)∥.
Combining (25)-(26), we obtain
(27)∥x(k)∥≤(ε2ε1+Tε3d-ε1+0.5Tε3(d-+d_-1)(d--d_)ε1+Tε4d-ε1)×(μ1/τa(1-Tα))(k-k0)∥x(k0)∥c,
where∥x(k0)∥c=sup-d-≤θ≤0∥x(k0+θ)∥.
Therefore, according to Definition8, system (12) is exponentially stable for any switching signalsσ(k)with average dwell time (14).
This completes the proof.Remark 13.
Whenμ=1in (15), which leads toνp=νq,υp=υq,ϑp=ϑq, for all p,q∈m_, andτa*=0by (14), system (12) possesses a common copositive type Lyapunov-Krasovskii functional, and the switching signal can be arbitrary.
Whendk=0, system (12) can be represented by
(28)δx(k)=A~σ(k)x(k),
whereA~p=Ap+AdpsatisfiesI+TA~p⪰0, for all p∈m_. Then we have the following corollary.Corollary 14.
Given a positive constant0<α<1/T, if there existνp∈R+n, such that, for all p∈m_,
(29)A~pTνp+ανp⪯0,
then system (28) is exponentially stable for any switching signalsσ(k)with average dwell time (14), whereμ≥1satisfies
(30)νp⪯μνq,∀p,q∈m_.
When the sampling periodT→0, system (12) becomes a continuous-time system as follows:
(31)x˙(t)=Aσ(t)x(t)+Adσ(t)x(t-d(t)),x(t0+θ)=φ(θ),θ∈[-d2,0],
whereApare Metzler matrices and Adp⪰0, for all p∈m_. d(t)denotes the time-varying delay which satisfies0≤d1≤d(t)≤d2 andd˙(t)≤dd<1for known constantsd1,d2,anddd.We can obtain sufficient conditions of exponential stability of system (31) by Theorem 12.Corollary 15.
Given a positive constantα, if there exist νp,υp,ϑp∈R+n, such that, for all p∈m_,
(32)ApTνp+ανp+(d2-d1+1)υp+ϑp⪯0,AdpTνp-(1-dd)e-αd2υp⪯0,
then system (31) is exponentially stable for any switching signalsσ(t)with average dwell time
(33)τa>τa*=lnμα,
whereμ≥1satisfies (15).
LetA-σ(k)=Aσ(k)+I. When the sampling periodT=1, system (12) becomes a discrete-time system as follows:
(34)x(k+1)=A-σ(k)x(k)+Adσ(k)x(k-dk),x(k0+θ)=φ(θ),θ=-d-,-d-+1,…,0,
whereA-p⪰0andAdp⪰0, for all p∈m_. One can obtain sufficient conditions of exponential stability of system (34) by Theorem 12.Corollary 16.
Given a positive constant0<α<1, if there existνp,υp,ϑp∈R+n, such that, for all p∈m_,
(35)A-pTνp+ανp+(1-α)(d--d_+1)υp+(1-α)ϑp⪯0,AdpTνp-(1-α)d-+1υp⪯0,
then system (34) is exponentially stable for any switching signalsσ(k)with average dwell time
(36)τa>τa*=-lnμln(1-α),
whereμ≥1satisfies (15).


## 3.2.l1-Gain Analysis

The following theorem establishes sufficient conditions of exponential stability withl1-gain property for system (2).Theorem 17.
For given positive constants0<α<1/Tandγ, if there exist νp,υp,ϑp∈R+n, such that, for all p∈m_,
(37)ApTνp+ανp+(1-Tα)(d--d_+1)υp+(1-Tα)ϑp+c~p⪯0,AdpTνp-(1-Tα)d-+1υp⪯0,(38)DpTνp+e~p-γ1w⪯0,
where1w=[1,1,…,1︸w]T,c~p=[∥cp1∥,∥cp2∥,…,∥cpn∥]T,cpjrepresents thejth column of matrixCp,j∈n_={1,2,…,n}, e~p=[∥ep1∥,∥ep2∥,…,∥epw∥]T, and epjrepresents thejth column of matrixEp,j∈w_={1,2,…,w}, then system (2) is exponentially stable with anl1-gain performance for any switching signalsσ(k)with average dwell time (14), whereμ≥1satisfies (15).Proof.
By Theorem12, the exponential stability of system (2) withw(k)=0is ensured if (37) holds. To show the weightedl1-gain performance, we choose the Lyapunov functional (18). From (15), we have
(39)Vσ(ki)(ki)≤μVσ(ki-)(ki-),∀i=1,2,….
For anyk∈[ki,ki+1), noticing (37)-(38), we have
(40)V(k)≤(1-Tα)(k-ki)Vσ(ki)(ki)-∑s=kik(1-Tα)(k-s)Λ(s),
whereΛ(s)=∥z(s)∥-γ∥w(s)∥.
Combining (39) and (40) leads to
(41)V(k)≤μ(1-Tα)(k-ki)Vσ(ki-)(ki-)-∑s=kik(1-Tα)(k-s)Λ(s)≤μ(1-Tα)(k-ki-1)Vσ(ki-1)(ki-1)-μ∑s=ki-1ki(1-Tα)(k-s)Λ(s)-∑s=kik(1-Tα)(k-s)Λ(s)=μNσ(ki-1,k)(1-Tα)(k-ki-1)Vσ(ki-1)(ki-1)-μNσ(ki-1,k)∑s=ki-1ki(1-Tα)(k-s)Λ(s)-∑s=kik(1-Tα)(k-s)Λ(s)≤⋯≤μNσ(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)-μNσ(k0,k)∑s=k0k1(1-Tα)(k-s)Λ(s)-μNσ(k1,k)∑s=k1k2(1-Tα)(k-s)Λ(s)-⋯-∑s=kik(1-Tα)(k-s)Λ(s)=μNσ(k0,k)(1-Tα)(k-k0)Vσ(k0)(k0)-∑s=k0kμNσ(s,k)(1-Tα)(k-s)Λ(s).
Under the zero initial condition, we obtain from (41) that
(42)0≤-∑s=k0kμNσ(s,k)(1-Tα)(k-s)Λ(s);
namely,
(43)∑s=k0kμNσ(s,k)(1-Tα)(k-s)∥z(s)∥≤γ∑s=k0kμNσ(s,k)(1-Tα)(k-s)∥w(s)∥.
Multiplying both sides of (43) byμ-Nσ(k0,k)yields
(44)∑s=k0kμ-Nσ(k0,s)(1-Tα)(k-s)∥z(s)∥≤γ∑s=k0kμ-Nσ(k0,s)(1-Tα)(k-s)∥w(s)∥.
Noticing thatNσ(k)(k0,s)≤(s-k0)/τa, we have
(45)μ-Nσ(k0,s)≥(1-Tα)(s-k0).
Combining (44) and (45) leads to
(46)∑s=k0k(1-Tα)(s-k0)(1-Tα)(k-s)∥z(s)∥≤γ∑s=k0k(1-Tα)(k-s)∥w(s)∥.
Summing both sides of (46) fromk=k0to∞leads to
(47)∑k=k0∞(1-Tα)(k-k0)∥z(k)∥≤γ∑k=k0∞∥w(k)∥.
From Definition 10, it can be concluded that system (2) is exponentially stable with a prescribedl1-gain performance levelγ.
This completes the proof.Remark 18.
Whenμ=1in Theorem 17, summing both sides of (44) fromk=k0to∞leads to
(48)∑k=k0∞∥z(k)∥≤γ∑k=k0∞∥w(k)∥
which gives the standardl1-gain performance.


## 3.3. Controller Design

In this section, we are interested in designing a state feedback controlleru(k)=Kσ(k)x(k)for positive switched system (11) such that the corresponding closed-loop system
(49)δx(k)=(Aσ(k)+Bσ(k)Kσ(k))x(k)+Adσ(k)x(k-dk)+Dσ(k)w(k),x(k0+θ)=φ(θ),θ=-d-,-d-+1,…,0,z(k)=Cσ(k)x(k)+Eσ(k)w(k)
is exponentially stable with an l1-gain performance.Theorem 19.
Considering system (11), for given positive scalars0<α<1/Tandγ, if there exist νp,υp,ϑp∈R+n andgp∈Rn, such that, for all p∈m_,
(50)ApTνp+gp+ανp+(1-Tα)(d--d_+1)υp+(1-Tα)ϑp+c~p⪯0,(51)AdpTνp-(1-Tα)d-+1υp⪯0,(52)DpTνp+e~p-γ1w⪯0,(53)I+T(Ap+BpKp)⪰0,
wherec~pande~phave been defined in Theorem 17 andgp=KpTBpTνp, then the corresponding closed-loop system (49) is positive and exponentially stable with a prescribedl1-gain performance levelγfor any switching signalsσ(k)with average dwell time (14), whereμ≥1satisfies (15).Proof.
Denotegp=KpTBpTνp. Following the proof line of Theorem 17, one can exactly obtain Theorem 19. It is omitted here.
This completes the proof.Consider the controller design of the following positive switched delta operator system without time delay:(54)δx(k)=A~σ(k)x(k)+Bσ(k)u(k)+Dσ(k)w(k),z(k)=Cσ(k)x(k)+Eσ(k)w(k),
whereA~σ(k)=Aσ(k)+Adσ(k). Then we directly have the following corollary.Corollary 20.
Considering system (54), for given positive scalars0<α<1/Tandγ, if there existνp∈R+nandgp∈Rn, such that, for all p∈m_,
(55)A~pTνp+ανp+gp+c~p⪯0,DpTνp+e~p-γ1w⪯0,I+T(A~p+BpKp)⪰0,
wherec~pande~phave been defined in Theorem 17 andgp=KpTBpTνp, then the corresponding closed-loop system is positive and exponentially stable with a prescribedl1-gain performance levelγfor any switching signalsσ(k)with average dwell time (14), whereμ≥1satisfies (30).
Based on Theorem19, one  is now in a position to present an effective algorithm for constructing the desired controller.Algorithm 21.
Consider the following.Step 1.
Input the matricesAp,Adp,Bp,Cp,Dp, and Ep.Step 2.
Choose the parameters0<α<1/Tandγ>0. By solving (50)–(52), one can obtain the solutions ofνp,υp,ϑp, and gp.Step 3.
By the equationgp=KpTBpTνpwith the obtainedgpandνp, one can get the gain matricesKp.Step 4.
Check condition (53) in Theorem 19. If it holds, go to Step 5; otherwise, adjust the parameterαand return to Step 2.Step 5.
Construct the feedback controlleru(k)=Kσ(k)x(k), whereKp,p∈m_, are the gain matrices.


## 4. Numerical Example

Consider positive switched delta operator system (11) consisting of two subsystems described by the following.Subsystem 1:(56)A1=[1.84.51.5-2.8],Ad1=[0.50.00.10.0],B1=[0.40.1],C1=[0.10.2],D1=[0.10.2],E1=[0.1],Subsystem 2:(57)A2=[5.24.53.8-1.8],Ad2=[0.50.00.10.0],B2=[0.50.2],C2=[0.20.3],D2=[0.20.1],E2=[0.2],
andd-=2,d_=0,α=1.3,γ=2, and T=0.25. Then, by solving (50)–(52) in Theorem 19, we can obtain the following solutions:
(58)ν1=[9.64722.5911],υ1=[17.49161.9008],ϑ1=[0.73121.4427],ν2=[6.96941.8534],υ2=[12.68661.3511],ϑ2=[0.37271.0628],
and the state feedback gain matrices can be obtained as follows:
(59)K1=[-14.2263-10.7904],K2=[-18.2172-8.8405].Obviously, condition (53) is satisfied.According to (15), we haveμ=2.0193. Then from (14), we getτa>τa*=1.7880. Choosingτa=2, the simulation results are shown in Figures 1 and 2, where the initial conditions arex(0)=[0.20.3]Tand x(k)=[00]T,k=-2,-1, and the exogenous disturbance input isw(k)=0.05e-0.5kwhich belongs tol1[0,∞). The switching signal with average dwell timeτa=2is shown in Figure 1 and the state responses of the corresponding closed-loop system are given in Figure 2. From the simulation results, it can been seen that the closed-loop system is exponentially stable with a prescribedl1-gain performance levelγ=2.Figure 1
Switching signal.Figure 2
State responses of the closed-loop system.


## 5. Conclusions

In this paper, the stability andl1-gain controller design problems for positive switched systems with time-varying delays via delta operator approach have been investigated. By constructing a copositive type Lyapunov-Krasovskii functional and using the average dwell time approach, we proposed sufficient conditions of exponential stability andl1-gain performance for the considered system. The desired state feedbackl1-gain controller was designed such that the corresponding closed-loop system is exponentially stable and satisfies anl1-gain performance. Finally, a numerical example was presented to demonstrate the feasibility of the obtained results. In our future work, we will study the robust stabilization problem of positive switched systems with uncertainties and time-varying delays via delta operator approach.



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