# L

Authors: Leipo Liu; Xiangyang Cao; Bo Fan; Zhumu Fu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, the problem ofL1 input-output finite-time control of positive switched nonlinear systems with time-varying and distributed delays is investigated. Nonlinear functions considered in this paper are located in a sector field. Firstly, the proof of the positivity of switched positive nonlinear systems with time-varying and distributed delays is given, and the concept of L1 input-output finite-time stability (L1 IO-FTS) is firstly introduced. Then, by constructing multiple co-positive-type nonlinear Lyapunov functions and using the average dwell time (ADT) approach, a state feedback controller is designed and sufficient conditions are derived to guarantee the corresponding closed-loop system is L1 IO-FTS. Such conditions can be easily solved by linear programming. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

In the past decades, positive switched systems have been paid much attention due to its broad applications in many areas such as communication systems [1], biological systems [2], and systems theory [3–6]. Many meaningful results have been presented in the literatures [7–11], and the references therein. Most of related results are mainly concerned with positive switched linear systems. In practice, many systems are nonlinear, such as physical systems, chemical systems, and transport systems. Although the study of dynamic behavior of positive switched nonlinear systems is not an easy work, because it is difficult to define the positivity of a nonlinear system, and few effective control techniques with respect to positive switched nonlinear systems are proposed. There are already some available results about positive switched nonlinear systems [12–16].However, the literatures mentioned above mainly focus on asymptotic stability, but in many practical applications, one is more interested in the behavior of the systems within a finite-time interval. Reference [17] firstly defined the concept of finite-time stability (FTS) for linear deterministic systems. For positive switched nonlinear systems, only [18] considered the guaranteed cost finite-time control problem. Recently, the definition of input-output finite-time stability (IO-FTS) is proposed in [19], which is fully consistent with the definition of FTS. IO-FTS means that given a class of norm bounded input signals over a specified time interval 0,T, the outputs of the system do not exceed an assigned threshold during such time intervals, and it does not necessarily require the inputs and outputs to belong to the same class. It is noteworthy that the concept of IO-FTS is different from that of input-output stability, the later deals with the behavior of a system within an infinite time interval. As we know, only a few results considered the IO-FTS problem of positive switched linear systems [20–22]. Reference [20] extended IO-FTS to positive switched linear systems with state delays, but the time-varying delay and distributed delay were not considered. References [21, 22] discussed the input-output finite-time control problem of positive switched systems and uncertain positive impulsive switched systems, respectively. However, for positive switched nonlinear systems, the IO-FTS problem has not been studied.Moreover, multiple time delays have great effect on the systems, such as oscillation and instability. On the other hand, compared withH∞ performance, L1 gain performance is more suitable to describe the performances of the systems with positive properties [23–25]. When taking multiple time delays and L1 gain performance into account, the problem of choosing an appropriate nonlinear copositive Lyapunov function and analyzing the L1 IO-FTS of the positive switched nonlinear systems will be more complex and challenging, which motivates our study.In this paper, we are interested in the problem ofL1 IO-FTS for a class of positive switched nonlinear systems with time-varying and distributed delays via ADT. The proposed system model is more general such that the systems dealt with in [6, 13, 17, 21–23] can be regarded as special forms. The main contributions of this paper are (1) the proof of the positivity of positive switched nonlinear systems with time-varying and distributed delays is given, (2) integrating L1 performance index into IO-FTS, the concept of L1 IO-FTS is for the first time extended to positive switched nonlinear systems with time-varying and distributed delays, and (3) a state feedback controller is designed and sufficient conditions for L1 IO-FTS of the corresponding closed-loop system are given. Such conditions can be easily solved by linear programming. The rest of this paper is organized as follows. In Section 2, problem statements and necessary lemmas are given. Main results are given in Section 3. A numerical example is provided in Section 4. Section 5 is the conclusion.Notations.
The representationA≻0 ≽0,≺0,≼0 means that aij>0 ≥0,<0,≤0, which is applicable to a vector. A≻B A≽B means that A−B≻0 A−B≽0. The symbols R, Rn, and Rn×n denote the set of real numbers, the space of the vectors of n-tuples of real numbers, the space of n×n matrices with real numbers, respectively. R+n is the n-dimensional nonnegative (positive) vector space. 1l denotes the l-dimensional vector 1,…,1T. AT denotes the transpose of matrix A. Let x∈Rn, 1-norm x is defined by x=∑k=1nxk. L1,0,T denotes the space of absolute integrable vector-valued functions on the interval 0,T; that is, st∈L1,0,T if ∫0Tstdt<∞ holds. L∞,0,T denotes the space of the uniformly bounded vector-valued functions on the interval 0,T; that is, st∈L∞,0,T if maxt∈0,Twt<∞ holds. Matrices are assumed to have compatible dimensions for calculating if their dimensions are not explicitly stated.


## 2. Preliminaries and Problem Statements

Consider a class of positive switched nonlinear systems with time-varying and distributed delays:(1)x˙t=Aσtfxt+Adσtfxt−dt+Abσtf∫t−τ2txsds+Gσtut+Bσtwt,yt=Cσtgxt,xθ=φθ,θ∈−τ1,0,where xt∈Rn is the system state, ut∈Rm and yt∈Rs represent the control input and output, respectively. fx=f1x1,f2x2,…,fnxnT∈Rn, gx=g1x1,g2x2,…,gnxnT∈Rn. The function σt:0,∞ is the system switching signal, and it is continuous from the right everywhere for a switching sequence, 0≤t0≤t1≤,…,N denotes the number of subsystems. t0=0 is the initial time, tk denotes the kth switching instant. ∀p∈N¯, Ap, Adp, Abp, Bp, Cp, and Gp are constant matrices with appropriate dimensions. The differentiable function dt≥0 represents time-varying delay, which satisfies d˙t≤h<1, τ2 is the distributed delays, where τ2 and h are known positive constants. φθ is the initial condition on −τ1,0, τ1=maxdt,τ2. wt∈Rl is the exogenous disturbance and is defined as(2)W1=w⋅∈L1,0,T:∫0Twtdt≤ε,with a known scalar ε>0.Next, we will present some definitions and lemmas for the positive switched nonlinear systems (1).Definition 1.
System (1) is said to be positive if for any switching signals σt, any initial conditions φθ≽0, θ∈−τ1,0, wt≽0, the corresponding trajectory satisfies xt≽0 and yt≽0 for all t≽0.Definition 2 (see [20]).
A is called a Metzler matrix if the off-diagonal entries of matrix A are nonnegative.Definition 3 (see [20]).
For anyT2>T1≥0, let NσT1,T2 denote the switching number of σt, over the interval T1,T2, for given Tα>0 and N0≥0, if the inequality(3)NσT1,T2≤N0+T2−T1Tαholds, then the positive constant Tα is called an ADT, and N0 is called a chattering bound. Generally speaking, N0=0 is chosen in this paper.Assumption 1.
The nonlinear functionsfx and gx lie in sector fields satisfying(4)m1xi2≤fixixi≤m2xi2,(5)m3xi2≤gixixi≤m4xi2,for xi∈R and i=1,2,…,n, where 0<m1≤m2,0<m3≤m4, and fi0=0, gi0=0.Remark 1.
The system model (1) is a more general form. If Abp=0, then system (1) is turned into the system in [13]. If m1=m2=m3=m4=1 (it means fixi=xi, gixi=xi), then system (1) is transformed into the systems in [21, 22]. Moreover, if m1=m2=m3=m4=1 and Abp=Adp=0, then system (1) will be turned into the systems in [6, 10, 17, 23].Remark 2.
Assumption 1 is a necessary assumption, because it takes full advantage of the characteristics of nonnegative states of positive switched nonlinear systems.Lemma 1 (see [25]).
Let A∈Rn×n be a Metzler matrix, then there exists a vector v≻0 such that Av≺0.Lemma 2.
Under Assumption 1. System (1) is positive if and only if Ap, ∀p∈N¯, are Metzler matrices and ∀p∈N¯, Adp≽0, Abp≽0, Bp≽0, Cp≽0, and Gp≽0.Proof (Necessity).
Firstly, letφθ=0 on θ∈−τ,0 and wt=0. Then, x˙0=Gpu0 for some p∈N¯. By definition 1, x˙0≽0 for every u0≽0, which means that Gp≽0 for each p∈N¯. In the same way, we can get Bp≽0.Secondly, we prove thatAp,∀p∈N¯, are Metzler matrices. Let ut=0 and wt=0. Suppose there exists an element apik<0,i≠k. From the system (1), we have(6)x˙it=∑j=1,j≠i,j≠knapijfjxjt+apiifixit+apikfkxkt+∑j=1nadpijfjxjt−dt+∑j=1nabpijfj∫t−τ2txjsds.We can see thatxit<0 is possible if fixkt=0, fkxkt≠0, and apik takes a small value enough. Thus, xit+<0, which brings a contradiction with the positivity of the system (1). So, Ap, ∀p∈N¯, are Metzler matrices.Thirdly, we proveAdp≽0 by the same way. Suppose there exists an element adpik<0, then we have(7)x˙it=∑j=1napijfjxjt+∑j=1,j≠knadpijfjxjt−dt+adpikfkxkt−dt+∑j=1nabpijfj∫t−τ2txjsds.It is easy to get thatx˙it<0 is possible if adpik takes a small value enough. Furthermore, xit+<0, which brings a contradiction with the positivity of the system (1). So, Adp≽0.Finally, suppose there exists an elementabpik<0, similar to the above process, we have(8)x˙it=∑j=1napijfjxjt+∑j=1nadpijfjxjt−dt+∑j=1,j≠knabpijfj∫t−τ2txjsds+abpikfk∫t−τ2txksds.In the same way, it is possible thatx˙it<0 if abpik takes a small value enough. It shows that system (1) is not positive. Based on these four points addressed above, the necessity is obtained.


### 2.1. Sufficiency

Letφt≽0. To prove that xt≽0 for all t≥0, it is sufficient to ensure that the vector x˙t does not point towards the outside of R+n whenever xt is on the boundary of R+n. This is equivalent to verifying that the components of the vector x˙t corresponding to the zero components of xt≽0 are nonnegative. Denoting by Ω the set of indices of such components, that is, xit=0 for i∈Ω. Then, for some p∈N¯, we have(9)x˙it=∑j∉Ωapijfjxjt+∑j=1nadpijfjxjt−dt+∑j=1nabpijfj∫t−τ2txjsds+gpijujt+bpijwjt,i∈Ω,where apij, adpij, abpij, gpij, bpij are the ith row jth column element of Ap, Adp, Abp, Gp, Bp, respectively. From the condition (4), it follows from φθ≽0 that fjxjt≥0 for t∈−τ1,Tf. According to system (1), apij≥0 for i≠j, adpij≥0, abpij≥0, gpij≥0, bpij≥0. So, we have x˙it≽0 for ut∈R+m and wt≽0. This means xt≽0 for φt≽0. From (5), we have gxt≽0 for xt≽0. Combining this with Cp≽0, it yields yt≽0.That reveals the system (1) is positive under any switching signals if and only if Ai are Metzler matrices, Adp≽0, Abp≽0, Bp≽0, Cp≽0, and Gp≽0, ∀i∈N¯.From the above, we can conclude the system (1) is positive.Remark 3.
Lemma 2 proves the positivity of a new class of switched nonlinear systems, which plays a key role in our latter work. From Assumption 1, iffx≽0 and gx≽0 for x≽0, then Lemma 2 still holds. That is to say, the system (1) is positive under any switching signals if and only if it consists of a family of positive nonlinear subsystems.Next, we will give the definitions of IO-FTS andL1 IO-FTS for the positive switched nonlinear system (1).Definition 4.
(IO-FTS) Consider zero initial condition (x0=0), for a given time constant Tf, disturbances signals W1 defined by (2) and a vector δ≻0, the system (1) is said to be IO-FTS with respect to (δ, Tf, ε, σt), if(10)wt∈W1⟹yTδ≤1,∀t∈0,Tf.
If the disturbancewt satisfies wt∈W2, where W2 is defined as(11)W2=w⋅∈L∞,0,T:maxt∈0,Twt≤ε,then, we give Definition 5.Definition 5.
(IO-FTS) Consider zero initial condition (x0=0), for a given time constant Tf, disturbances signals W2, and a vector δ≻0. System (1) is said to be IO-FTS with respect to (δ, Tf, ε, σt), if(12)wt∈W2⟹yTδ≤1∀t∈0,Tf.Remark 4.
In Definition 4 and 5, two classes of exogenous disturbances are norm bounded integrable signalsW1 and the uniformly bounded signals W2, respectively. Because of the similarity of the proof process, we only focus on the former in this paper.Definition 6.
(L1 IO-FTS) Under the zero initial condition (φθ=0), for a given time constant Tf, the system (1) is said to be L1 IO-FTS with respect to (δ, Tf, ε, σt), if the following conditions are satisfied: (1) system (3) is IO-FTS with respect to (δ, Tf, ε, σt); (2) the output yt satisfies(13)∫0Tfe−βtytdt≤η∫0Tfwtdt,where β>0, η>0, and wt satisfies (2).Remark 5.
In Definition 6,L1-gain performance index provides a more useful description for the disturbance attenuation performance of positive switched systems. Integrating L1-gain performance index into IO-FTS, the concept of L1 IO-FTS is more suitable to describe the system behavior during a specified time interval.The aim of this paper is to design a state feedback controllerut and find a class of switching signals σt for positive switched nonlinear system (1) such that the corresponding closed-loop system is L1 IO-FTS.


## 2.1. Sufficiency

Letφt≽0. To prove that xt≽0 for all t≥0, it is sufficient to ensure that the vector x˙t does not point towards the outside of R+n whenever xt is on the boundary of R+n. This is equivalent to verifying that the components of the vector x˙t corresponding to the zero components of xt≽0 are nonnegative. Denoting by Ω the set of indices of such components, that is, xit=0 for i∈Ω. Then, for some p∈N¯, we have(9)x˙it=∑j∉Ωapijfjxjt+∑j=1nadpijfjxjt−dt+∑j=1nabpijfj∫t−τ2txjsds+gpijujt+bpijwjt,i∈Ω,where apij, adpij, abpij, gpij, bpij are the ith row jth column element of Ap, Adp, Abp, Gp, Bp, respectively. From the condition (4), it follows from φθ≽0 that fjxjt≥0 for t∈−τ1,Tf. According to system (1), apij≥0 for i≠j, adpij≥0, abpij≥0, gpij≥0, bpij≥0. So, we have x˙it≽0 for ut∈R+m and wt≽0. This means xt≽0 for φt≽0. From (5), we have gxt≽0 for xt≽0. Combining this with Cp≽0, it yields yt≽0.That reveals the system (1) is positive under any switching signals if and only if Ai are Metzler matrices, Adp≽0, Abp≽0, Bp≽0, Cp≽0, and Gp≽0, ∀i∈N¯.From the above, we can conclude the system (1) is positive.Remark 3.
Lemma 2 proves the positivity of a new class of switched nonlinear systems, which plays a key role in our latter work. From Assumption 1, iffx≽0 and gx≽0 for x≽0, then Lemma 2 still holds. That is to say, the system (1) is positive under any switching signals if and only if it consists of a family of positive nonlinear subsystems.Next, we will give the definitions of IO-FTS andL1 IO-FTS for the positive switched nonlinear system (1).Definition 4.
(IO-FTS) Consider zero initial condition (x0=0), for a given time constant Tf, disturbances signals W1 defined by (2) and a vector δ≻0, the system (1) is said to be IO-FTS with respect to (δ, Tf, ε, σt), if(10)wt∈W1⟹yTδ≤1,∀t∈0,Tf.
If the disturbancewt satisfies wt∈W2, where W2 is defined as(11)W2=w⋅∈L∞,0,T:maxt∈0,Twt≤ε,then, we give Definition 5.Definition 5.
(IO-FTS) Consider zero initial condition (x0=0), for a given time constant Tf, disturbances signals W2, and a vector δ≻0. System (1) is said to be IO-FTS with respect to (δ, Tf, ε, σt), if(12)wt∈W2⟹yTδ≤1∀t∈0,Tf.Remark 4.
In Definition 4 and 5, two classes of exogenous disturbances are norm bounded integrable signalsW1 and the uniformly bounded signals W2, respectively. Because of the similarity of the proof process, we only focus on the former in this paper.Definition 6.
(L1 IO-FTS) Under the zero initial condition (φθ=0), for a given time constant Tf, the system (1) is said to be L1 IO-FTS with respect to (δ, Tf, ε, σt), if the following conditions are satisfied: (1) system (3) is IO-FTS with respect to (δ, Tf, ε, σt); (2) the output yt satisfies(13)∫0Tfe−βtytdt≤η∫0Tfwtdt,where β>0, η>0, and wt satisfies (2).Remark 5.
In Definition 6,L1-gain performance index provides a more useful description for the disturbance attenuation performance of positive switched systems. Integrating L1-gain performance index into IO-FTS, the concept of L1 IO-FTS is more suitable to describe the system behavior during a specified time interval.The aim of this paper is to design a state feedback controllerut and find a class of switching signals σt for positive switched nonlinear system (1) such that the corresponding closed-loop system is L1 IO-FTS.


## 3. Main Results

### 3.1. IO-FTS Analysis

Consider the system (1) with ut=0, the system is described by(14)x˙t=Aσtfxt+Adσtfxt−dt+Abσtf∫t−τ2txsds+Bσtwt,yt=Cσtgxt,xθ=φθθ∈−τ1,0.In this subsection, we concern with the IO-FTS analysis of positive switched nonlinear system (14). The following theorem gives some sufficient conditions of IO-FTS for system (14) via the ADT technique.Theorem 1.
Consider the system (14), for given constants Tf, γ, λp, p∈N¯, and a vectors δ≻0, if there exist positive vectors νp∈Rn, ρp∈Rn, ϱp∈Rn, p∈N¯, such that the following inequalities hold:(15)ApTνp+ρp+τ1ϱp−λpm2νp≺0,(16)AdpTνp−1−hρp≺0,(17)m2AbpTνp−m1ϱp≺0,(18)εγ<e−λTf,(19)m4CpTδ≺νp,(20)νp≼μνq,ρp≼μρq,ϱp≼μϱq.(21)BpTνp≤γ1l,where νp=νp1,νp2,…,νpnT, ρp=ρp1,ρp2,…,ρpnT, ϱp=ϱp1,ϱp2,…,ϱpnT, λ=maxp∈N¯λp, μμ≥1 satisfies (20), then the system (14) is IO-FTS for switching signal σt with the ADT scheme:(22)Tα>Tα∗=Tflnμlne−λTf−lnεγ.Proof.
Construct the multiple co-positive-type nonlinear Lyapunov functional for the system (14):(23)Vσtt=Vσtt,xt,the form of each Vpt,xt,∀p∈N¯ can be given by(24)Vpt=Vpt,xt=xTtνp+∫t−dtteλpt−sfTxsρpds+∫−τ10∫t+θteλpt−sfTxsϱpdsdθ,where νp, ρp, and ϱp∈R+n, ∀p∈N¯.
Along the trajectory of system (14), we have(25)V˙pt=fTxtApTνp+fTxt−dtAdpTνp+fT∫t−τ2txsdsAbpTνp+λp∫t−dtteλpt−sfTxsρpds−1−d˙teλpdtfTxt−dtρp+λp∫−τ10∫t+θteλpt−sfTxsϱpdsdθ+τ1fTxtϱp−∫−τ10e−λpθfTxt+θϱpdθ+wTtBpTνp+fTxtρp.
Consider (4) and xt≽0, m1xt≼ft≼m2xt can be obtained, then (25) is transformed into(26)V˙pt=λpVpt+fTxtApTνp+ρp+τ1ϱp−λpxTtνp+fTxt−dtAdpTνp−1−d˙tρp+fT∫t−τ2txsdsAbpTνp−∫−τ10e−λpθfTxt+θϱpdθ+wTtBpTνp≤λpVσtt+fTxtApTνp+ρp+τ1ϱp−λpm2νp+fTxt−dtAdpTνp−1−hρp+m2∫t−τ2txTsdsAbpTνp−∫t−τ1tfTxsϱpds+wTtBpTνp≤λpVσtt+fTxtApTνp+ρp+τ1ϱp−λpm2νp+fTxt−dtAdpTνp−1−hρp+m2∫t−τ2txTsdsAbpTνp−m1∫t−τ1txTsϱpds+wTtBpTνp.
Noting thatτ1=maxdt,τ2, we have(27)m2∫t−τ2txTsdsAbpTνp−m1∫t−τ1txTsϱpds≤m2∫t−τ1txTsdsAbpTνp−m1∫t−τ1txTsϱpds=∫t−τ1txTsm2AbpTνp−m1ϱpds.
According to (27), (26) can be rewritten as(28)V˙pt−λpVpt≤fTxtApTνp+ρp+τ1ϱp−λpm2νp+fTxt−dtAdpTνp−1−hρp+∫t−τ1txTsm2AbpTνp−m1ϱpds+wTtBpTνp.
From (15)–(17) and (21), we get(29)V˙pt−λpVpt≤wTtBpTνp≤γwt.
Integrating both sides of (29) during the period tk,t for t∈tk,tk+1 leads to(30)Vσtt≤eλσtkt−tkVσtktk+∫tkteλσtkt−sγwsds.
From (20) and (24), we get(31)Vσtktk≤μVσtk−tk−.
Denotet1,…,tN, as the switching instants over the interval 0,Tf. From (2), (30), and (31), we have(32)Vσtt≤eλσtkt−tkVσtktk+∫tkteλσtkt−sγwsds≤μeλt−tkVσtk−tk−+∫tkteλt−sγwsds≤μeλt−tkeλtk−tk−1Vσtk−1tk−1+∫tk−1tkeλtk−sγwsds+∫tkteλt−sγwsds≤μ2eλt−tkeλtk−tk−1Vσtk−1−tk−1−+∫tk−1tkeλtk−sγwsds+∫tkteλt−sγwsds≤⋯≤μkeλtVσ00+μk∫0t1eλt−sγwsds+μk−1∫t1t2eλt−sγwsds+⋯+∫tkteλt−sγwsds≤μNeλTfVσ00+∫0tμNσts,teλt−sγwsds≤μNeλTfVσ00+μN∫0TfeλTfγwsds≤μNeλTfVσ00+εγ.
From (24), (32) and the zero initial condition x0=0, we have(33)Vσtt≥xTtνp,(34)Vσtt≤μNeλTfεγ.
From (3), (33) and (34), one has(35)xTtνp≤eλ+lnμ/TαTfεγ.
According to (22) and (35), we can easily obtain(36)xTtνp<1.
By (5) and (19), we get(37)yTtδ=gTxtCpTδ≤m4xTtCσtTδ<xTtνp.
From (36) and (37), we obtain(38)yTtδ<1.
Thus, the proof is completed.


### 3.2.L1 IO-FTS Performance Analysis

In this section, we will consider the problem ofL1 IO-FTS for system (14).Theorem 2.
Consider the system (14), for given constants Tf, λp, p∈N¯, η, and a vector δ≻0, if there exist positive vectors νp∈Rn, ρp∈Rn, ϱp∈Rn, p∈N¯, such that (16), (17), (19), (20), and the following inequalities hold,(39)ApTνp+ρp+τ1ϱp−λpm2νp+m4m1Cp≺0,(40)BpTνp<η1l,(41)εη<e−λTf,where νp=νp1,νp2,…,νpnT2, ρp=ρp1,ρp2,…,ρpnT, ϱp=ϱp1,ϱp2,…,ϱpnT, λ=maxp∈N¯λp, μμ≥1 satisfies (20), then the system (14) is L1 IO-FTS for any switching signal σt with the ADT scheme(42)Tα>Tα∗=maxTflnμlne−λTf−lnεη,lnμη.Proof.
(39) can be easily derived from (15). Let η>γ in Theorem 1, from (16), (17), (19), (20), (39), (40), and (41), we can obtain that the system (14) is IO-FTS with respect to (δ, Tf, ε, σt).Choosing the multiple co-positive-type Lyapunov functional (24). Similar to the proof process of Theorem 1, from (16), (17), and (39), we have(43)V˙pt−λpVpt+yt−ηwt<0.Denoting∇s=ηws−ys and integrating both sides of (43) from tk to t for t∈tk,tk+1, it gives rise to(44)Vσtt≤eλσtkt−tkVσtktk+∫tkteλσtkt−s∇sds.Similar to the proof process of (32), for any t∈0,Tf, we can obtain(45)Vσtt≤μNσt0,teλtVσ00+∫0teλt−sμNσts,t∇sds.Considering the zero initial condition, we have(46)∫0teλt−sμNσts,tysds<η∫0teλt−sμNσts,twsds.Multiplying both sides of (46) by μ−Nσt0,t leads to(47)∫0teλt−sμ−Nσt0,sysds<η∫0teλt−sμ−Nσt0,swsds.Noting thatμ>1, Nσt0,s≤s/t, and Tα>lnμ/λ, we obtain that 0<Nσt0,s≤s/t≤λs/lnμ; that is, e−λs≤μ−Nσt0,s<1. Then (47) can be turned into(48)∫0teλt−se−λsysds<η∫0teλt−swsds.Lett=Tf, then multiplying both sides of (48) by e−λTf yields(49)∫0Tfe−2λsysds<η∫0Tfwsds.Denotingβ=2λ, (49) can be rewritten as(50)∫0Tfe−βsysds<η∫0Tfwsds.Thus, the proof is completed.


### 3.3.L1 IO-FTS Controller Design

In this subsection, we consider the design ofL1 IO-FTS controller. Considering system (1), under the controller ut=Kσtfxt, the corresponding closed-loop system is given by(51)x˙t=Aσt+GσtKσtfxt+Adσtfxt−dt+Abσtf∫t−τ2txsds+Bσtwt,yt=Cσtgxt,xθ=φθ,θ∈−τ1,0.By Lemma 2, to guarantee the positivity of system (51), Ap+GpKp should be Metzler matrices.Theorem 3.
Consider the system (51), for given constants Tf, λp, p∈N¯, and a vector δ≻0, if there exist positive vectors νp∈Rn, ρp∈Rn, ϱp∈Rn, ℏp, p∈N¯, such that (16), (17), (19), (20), (40)–(42), and the following inequalities hold:(52)Ap+GpKpareMetzlermatrices,(53)ApTνp+ρp+τ1ϱp−λpm2νp+m4m1Cp+ℏp≺0,where ℏp=KpTGpTνp, νp=νp1,νp2,…,νpnT, ρp=ρp1,ρp2,…,ρpnT, and ϱp=ϱp1,ϱp2,…,ϱpnT; then, under the ADT scheme (42), the resulting closed-loop system (51) is L1 IO-FTS with respect to (δ, Tf, ε, σt), where μμ≥1 satisfies (20).Proof.
ReplacingAp in (39) with Ap+GpKp, then letting ℏp=KpTGpTνp, similar to Theorem 2, we can get (53); then, the resulting closed-loop system (51) is L1 IO-FTS with ADT scheme (42).The proof is completed.Next, an algorithm is presented to obtain the feedback gain matricesKp, p∈S¯.Algorithm 1
(1)
Step 1. By adjusting the parametersλp and solving (16), (17), (19), (20), (40)–(42), (52), and (53) via linear programming, positive vectors νp, ρp, ϱp, and ℏp can be obtained.
(2)
Step 2. Substitutingνp and hp into ℏp=KpTGpTνp, Kp can be obtained. According to (42), Tα can also be obtained.
(3)
Step 3. If the gainKp satisfies (52), then, Kp is admissible. Otherwise, return to Step 1.Remark 6.
There is not a systemic method to adjust these parameters; the selection is generally by experience.


## 3.1. IO-FTS Analysis

Consider the system (1) with ut=0, the system is described by(14)x˙t=Aσtfxt+Adσtfxt−dt+Abσtf∫t−τ2txsds+Bσtwt,yt=Cσtgxt,xθ=φθθ∈−τ1,0.In this subsection, we concern with the IO-FTS analysis of positive switched nonlinear system (14). The following theorem gives some sufficient conditions of IO-FTS for system (14) via the ADT technique.Theorem 1.
Consider the system (14), for given constants Tf, γ, λp, p∈N¯, and a vectors δ≻0, if there exist positive vectors νp∈Rn, ρp∈Rn, ϱp∈Rn, p∈N¯, such that the following inequalities hold:(15)ApTνp+ρp+τ1ϱp−λpm2νp≺0,(16)AdpTνp−1−hρp≺0,(17)m2AbpTνp−m1ϱp≺0,(18)εγ<e−λTf,(19)m4CpTδ≺νp,(20)νp≼μνq,ρp≼μρq,ϱp≼μϱq.(21)BpTνp≤γ1l,where νp=νp1,νp2,…,νpnT, ρp=ρp1,ρp2,…,ρpnT, ϱp=ϱp1,ϱp2,…,ϱpnT, λ=maxp∈N¯λp, μμ≥1 satisfies (20), then the system (14) is IO-FTS for switching signal σt with the ADT scheme:(22)Tα>Tα∗=Tflnμlne−λTf−lnεγ.Proof.
Construct the multiple co-positive-type nonlinear Lyapunov functional for the system (14):(23)Vσtt=Vσtt,xt,the form of each Vpt,xt,∀p∈N¯ can be given by(24)Vpt=Vpt,xt=xTtνp+∫t−dtteλpt−sfTxsρpds+∫−τ10∫t+θteλpt−sfTxsϱpdsdθ,where νp, ρp, and ϱp∈R+n, ∀p∈N¯.
Along the trajectory of system (14), we have(25)V˙pt=fTxtApTνp+fTxt−dtAdpTνp+fT∫t−τ2txsdsAbpTνp+λp∫t−dtteλpt−sfTxsρpds−1−d˙teλpdtfTxt−dtρp+λp∫−τ10∫t+θteλpt−sfTxsϱpdsdθ+τ1fTxtϱp−∫−τ10e−λpθfTxt+θϱpdθ+wTtBpTνp+fTxtρp.
Consider (4) and xt≽0, m1xt≼ft≼m2xt can be obtained, then (25) is transformed into(26)V˙pt=λpVpt+fTxtApTνp+ρp+τ1ϱp−λpxTtνp+fTxt−dtAdpTνp−1−d˙tρp+fT∫t−τ2txsdsAbpTνp−∫−τ10e−λpθfTxt+θϱpdθ+wTtBpTνp≤λpVσtt+fTxtApTνp+ρp+τ1ϱp−λpm2νp+fTxt−dtAdpTνp−1−hρp+m2∫t−τ2txTsdsAbpTνp−∫t−τ1tfTxsϱpds+wTtBpTνp≤λpVσtt+fTxtApTνp+ρp+τ1ϱp−λpm2νp+fTxt−dtAdpTνp−1−hρp+m2∫t−τ2txTsdsAbpTνp−m1∫t−τ1txTsϱpds+wTtBpTνp.
Noting thatτ1=maxdt,τ2, we have(27)m2∫t−τ2txTsdsAbpTνp−m1∫t−τ1txTsϱpds≤m2∫t−τ1txTsdsAbpTνp−m1∫t−τ1txTsϱpds=∫t−τ1txTsm2AbpTνp−m1ϱpds.
According to (27), (26) can be rewritten as(28)V˙pt−λpVpt≤fTxtApTνp+ρp+τ1ϱp−λpm2νp+fTxt−dtAdpTνp−1−hρp+∫t−τ1txTsm2AbpTνp−m1ϱpds+wTtBpTνp.
From (15)–(17) and (21), we get(29)V˙pt−λpVpt≤wTtBpTνp≤γwt.
Integrating both sides of (29) during the period tk,t for t∈tk,tk+1 leads to(30)Vσtt≤eλσtkt−tkVσtktk+∫tkteλσtkt−sγwsds.
From (20) and (24), we get(31)Vσtktk≤μVσtk−tk−.
Denotet1,…,tN, as the switching instants over the interval 0,Tf. From (2), (30), and (31), we have(32)Vσtt≤eλσtkt−tkVσtktk+∫tkteλσtkt−sγwsds≤μeλt−tkVσtk−tk−+∫tkteλt−sγwsds≤μeλt−tkeλtk−tk−1Vσtk−1tk−1+∫tk−1tkeλtk−sγwsds+∫tkteλt−sγwsds≤μ2eλt−tkeλtk−tk−1Vσtk−1−tk−1−+∫tk−1tkeλtk−sγwsds+∫tkteλt−sγwsds≤⋯≤μkeλtVσ00+μk∫0t1eλt−sγwsds+μk−1∫t1t2eλt−sγwsds+⋯+∫tkteλt−sγwsds≤μNeλTfVσ00+∫0tμNσts,teλt−sγwsds≤μNeλTfVσ00+μN∫0TfeλTfγwsds≤μNeλTfVσ00+εγ.
From (24), (32) and the zero initial condition x0=0, we have(33)Vσtt≥xTtνp,(34)Vσtt≤μNeλTfεγ.
From (3), (33) and (34), one has(35)xTtνp≤eλ+lnμ/TαTfεγ.
According to (22) and (35), we can easily obtain(36)xTtνp<1.
By (5) and (19), we get(37)yTtδ=gTxtCpTδ≤m4xTtCσtTδ<xTtνp.
From (36) and (37), we obtain(38)yTtδ<1.
Thus, the proof is completed.


## 3.2.L1 IO-FTS Performance Analysis

In this section, we will consider the problem ofL1 IO-FTS for system (14).Theorem 2.
Consider the system (14), for given constants Tf, λp, p∈N¯, η, and a vector δ≻0, if there exist positive vectors νp∈Rn, ρp∈Rn, ϱp∈Rn, p∈N¯, such that (16), (17), (19), (20), and the following inequalities hold,(39)ApTνp+ρp+τ1ϱp−λpm2νp+m4m1Cp≺0,(40)BpTνp<η1l,(41)εη<e−λTf,where νp=νp1,νp2,…,νpnT2, ρp=ρp1,ρp2,…,ρpnT, ϱp=ϱp1,ϱp2,…,ϱpnT, λ=maxp∈N¯λp, μμ≥1 satisfies (20), then the system (14) is L1 IO-FTS for any switching signal σt with the ADT scheme(42)Tα>Tα∗=maxTflnμlne−λTf−lnεη,lnμη.Proof.
(39) can be easily derived from (15). Let η>γ in Theorem 1, from (16), (17), (19), (20), (39), (40), and (41), we can obtain that the system (14) is IO-FTS with respect to (δ, Tf, ε, σt).Choosing the multiple co-positive-type Lyapunov functional (24). Similar to the proof process of Theorem 1, from (16), (17), and (39), we have(43)V˙pt−λpVpt+yt−ηwt<0.Denoting∇s=ηws−ys and integrating both sides of (43) from tk to t for t∈tk,tk+1, it gives rise to(44)Vσtt≤eλσtkt−tkVσtktk+∫tkteλσtkt−s∇sds.Similar to the proof process of (32), for any t∈0,Tf, we can obtain(45)Vσtt≤μNσt0,teλtVσ00+∫0teλt−sμNσts,t∇sds.Considering the zero initial condition, we have(46)∫0teλt−sμNσts,tysds<η∫0teλt−sμNσts,twsds.Multiplying both sides of (46) by μ−Nσt0,t leads to(47)∫0teλt−sμ−Nσt0,sysds<η∫0teλt−sμ−Nσt0,swsds.Noting thatμ>1, Nσt0,s≤s/t, and Tα>lnμ/λ, we obtain that 0<Nσt0,s≤s/t≤λs/lnμ; that is, e−λs≤μ−Nσt0,s<1. Then (47) can be turned into(48)∫0teλt−se−λsysds<η∫0teλt−swsds.Lett=Tf, then multiplying both sides of (48) by e−λTf yields(49)∫0Tfe−2λsysds<η∫0Tfwsds.Denotingβ=2λ, (49) can be rewritten as(50)∫0Tfe−βsysds<η∫0Tfwsds.Thus, the proof is completed.


## 3.3.L1 IO-FTS Controller Design

In this subsection, we consider the design ofL1 IO-FTS controller. Considering system (1), under the controller ut=Kσtfxt, the corresponding closed-loop system is given by(51)x˙t=Aσt+GσtKσtfxt+Adσtfxt−dt+Abσtf∫t−τ2txsds+Bσtwt,yt=Cσtgxt,xθ=φθ,θ∈−τ1,0.By Lemma 2, to guarantee the positivity of system (51), Ap+GpKp should be Metzler matrices.Theorem 3.
Consider the system (51), for given constants Tf, λp, p∈N¯, and a vector δ≻0, if there exist positive vectors νp∈Rn, ρp∈Rn, ϱp∈Rn, ℏp, p∈N¯, such that (16), (17), (19), (20), (40)–(42), and the following inequalities hold:(52)Ap+GpKpareMetzlermatrices,(53)ApTνp+ρp+τ1ϱp−λpm2νp+m4m1Cp+ℏp≺0,where ℏp=KpTGpTνp, νp=νp1,νp2,…,νpnT, ρp=ρp1,ρp2,…,ρpnT, and ϱp=ϱp1,ϱp2,…,ϱpnT; then, under the ADT scheme (42), the resulting closed-loop system (51) is L1 IO-FTS with respect to (δ, Tf, ε, σt), where μμ≥1 satisfies (20).Proof.
ReplacingAp in (39) with Ap+GpKp, then letting ℏp=KpTGpTνp, similar to Theorem 2, we can get (53); then, the resulting closed-loop system (51) is L1 IO-FTS with ADT scheme (42).The proof is completed.Next, an algorithm is presented to obtain the feedback gain matricesKp, p∈S¯.Algorithm 1
(1)
Step 1. By adjusting the parametersλp and solving (16), (17), (19), (20), (40)–(42), (52), and (53) via linear programming, positive vectors νp, ρp, ϱp, and ℏp can be obtained.
(2)
Step 2. Substitutingνp and hp into ℏp=KpTGpTνp, Kp can be obtained. According to (42), Tα can also be obtained.
(3)
Step 3. If the gainKp satisfies (52), then, Kp is admissible. Otherwise, return to Step 1.Remark 6.
There is not a systemic method to adjust these parameters; the selection is generally by experience.


## 4. Numerical Examples

Example 1.
Consider system (1) with the parameters as follows:(54)A1=−1.60.30.2−1.2,Ad1=0.060.020.080.06,Ab1=0.020.010.010.02,G1=0.40.2,A2=−1.10.10.2−0.8,Ad2=0.010.050.030.04,Ab2=0.010.020.020.01,G2=0.20.3,B1=0.040.05,B2=0.020.03,C1=0.030.010.040.04,C2=0.030.040.020.02,where fixit=xit+xit/xi2t+2, gixit=xit; then, we get m1=1, m2=2, m3=m4=1. Choosing dt=0.1sint+0.2, wt=0.05e−0.5t, τ2=0.1, τ1=maxdt,τ2=0.3, h=0.1, λ1=λ2=0.5, η=0.5, ε=0.01, μ=1.5, δ=1.251.75T. Solving the inequalities in Theorem 3 by linear programming, we get(55)ν1=5.44934.4656,ρ1=2.33190.5649,ϱ1=15.937612.7097,ν2=7.36466.2864,ρ2=2.28850.6289,ϱ2=15.386113.9692,ℏ1=0.74440.2401,ℏ2=0.71910.5509.
Byℏp=KpTGpTνp, we obtain(56)K1=0.34150.2688,K2=0.48990.2921.
It is easy to verify thatAp+GpKp are Metzler matrices. Then, from (42), we can obtain Tα∗=0.8105. Choosing Tα=0.82>Tα∗. The simulation results are shown in Figures 1–3, where the x0=0,0T. Figure 1 shows the state trajectories of the closed-loop system (1). The switching signal σt is depicted in Figure 2. Figure 3 plots the evolution of yTtδ of system (1). From the simulation results, we know yTtδ<1, which implies the effectiveness of our proposed method.Figure 1
State trajectories of closed-loop system (1).Figure 2
Switching signal of system (1).Figure 3
Evolution ofyTtδ of system (1).Example 2.
A practical price dynamics model is considered as an uncertain positive switched system with time-varying delays in [3]. In an economic system, affected by the national macroeconomic regulation and control policy, the relation between supply and demand always presents nonlinear behavior. So, it is more suitable to describe the price dynamic model by positive switched nonlinear systems [18]. Consider the parameters as follows:(57)A1=0.40.20.3−2.5,Ad1=0.10.30.20.2,G1=0.30.4,A2=−1.50.30.20.2,Ad2=0.20.10.30.1,G2=0.20.3,B1=0.30.2,B2=0.10.2,C1=0.20.30.10.1,C2=0.30.20.20.1.where fixit=xit+xit/xit+2, gixit=xit; then, we get m1=1, m2=2, m3=m4=1. Choosing dt=0.1cost+0.2, wt=0.05e−0.5t, τ1=0.3, h=0.2, λ1=λ2=0.5, η=0.5, ε=0.01, μ=1.3, δ=12T. Solving the inequalities in Theorem 3 by linear programming, we get(58)ν1=0.89270.9982,ρ1=1.22180.2905,ϱ1=0.83980.1101,ν2=1.39571.3527,ρ2=0.96710.2873,ϱ2=0.77780.1292,ℏ1=0.12020.5457,ℏ2=0.01720.0814.
Byℏp=KpTGpTνp, we obtain(59)K1=0.17910.8060,K2=0.02460.1143.
It is easy to verify thatAp+GpKp are Metzler matrices. Then, from (42), we can obtain Tα∗=0.7483. Choosing Tα=0.75>Tα∗. The simulation results are shown in Figures 4–6, where the x0=0,0T. Figure 4 shows the state trajectories of the closed-loop system (1). The switching signal σt is depicted in Figure 5. Figure 6 plots the evolution of yTtδ of system (1). From the simulation results, we know yTtδ<1, which implies the effectiveness of our proposed method.Figure 4
State trajectories of closed-loop system (1).Figure 5
Switching signal of system (1).Figure 6
Evolution ofyTtδ of system (1).


## 5. Conclusions

This paper has investigated the problem ofL1 IO-FTS for positive switched nonlinear systems with time-varying and distributed delays. Firstly, the concept of L1 IO-FTS of positive switched nonlinear systems is firstly proposed. Then, by constructing nonlinear Lyapunov–Krasovskii functions, a state feedback controller is designed. Based on the ADT approach, some sufficient conditions are obtained to guarantee that the closed-loop system is L1 IO-FTS.



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