# RobustH∞ Control for the Nonlinear Cascade Systems with Passive and Nonpassive Subsystems

Authors: Hongpeng Zhao; Xingtao Wang
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

In this paper,H∞ control for the uncertain switched nonlinear cascade systems with passive and nonpassive subsystems is investigated. Based on the average dwell time method, for any given passivity rate, average dwell time, and disturbance attenuation level, the feedback controllers of the subsystems by predetermined constants are designed to solve the exponential stability and L2-gain problems of H∞ control for switched nonlinear cascade systems. Two examples are provided to demonstrate the effectiveness of the proposed design method.


---

## Body

## 1. Introduction

With the development of scientific computing technology, the research onH∞ control problems of nonlinear systems has been greatly promoted, and the results of nonlinear control problems continue to emerge [1, 2]. However, these methods usually bring a difficulty that needs to solve the Hamilton–Jacobi equation.The passivity, from the electrical network, becomes an extremely useful property in switched systems, and many results about the passivity of switched systems have been published [3–11]. Storage functions that characterize passivity can be used as Lyapunov functions to analyze stabilization problems [3]. And the passivity is closely related to the robust stability of systems under certain negative feedback disturbances [6]. Recently, the storage function method has been found to ensure a top limit of the minimum dwell time to keep the passivity of linear systems [5]. For switched nonlinear systems, stability was inferred from the passivity described by using multiple storage functions [10]. The necessary and sufficient conditions were obtained for the local passivity of discrete-time switched nonlinear systems which consisted of passive and nonpassive modes, and the passivity of the affine system was studied [9]. Using multiple barrier storage functions, sufficient conditions were derived for guaranteeing the regional passivity of the switched systems [8]. And literature [7] considered the stability of switched nonlinear systems with feedback incrementally passive subsystems via the average dwell time method.With the systems becoming more complex in actual problems, the robustness caused by external disturbance becomes a source of trouble, and there are a few achievements on passivity ofH∞ control problem of switched systems [12–15]. The stability of two types of passive H∞ control for discrete-time linear switched systems was considered by multiple storage functions [12, 13]. And combining the piecewise Lyapunov function and the average dwell time method, the literature [15] investigated the disturbance of time-controlled switched systems consisting of several linear time-invariant subsystems. The H∞ control of uncertain switched nonlinear systems with passivity was researched, and this research avoided solving the Hamilton–Jacobi inequality problem [14].Complex nonlinear systems can be transformed into cascaded systems through certain conditions, and the stability of the cascaded system is studied by the stability and cascade properties of the subsystems [16]. This not only reduces the complexity of the controller but also reduces the difficulty of stability analysis [17–20]. A natural question is how to study the stability of switching nonlinear cascade robust H∞ control systems through passivity, and in this paper, we will work to solve these problems.In this paper, based on the method of average dwell time, the robustH∞ control problem for a class of passive uncertain cascade switched systems with passiveness is considered. For passive subsystems and nonpassive subsystems, we design controllers and apply the multiple storage functions method to solve the stability and L2-gain of the nonlinear uncertain cascade switched system under the given conditions. Finally, two numerical simulations are illustrated to support our analytical results. Compared with the method of existing nonlinear cascade switched systems’ H∞ control problem, the advantages are that we adopt the parametric equation method to avoid the Lyapunov function construction and the Hamilton–Jacobi equation solution, which reduces the computational difficulty.Notions:ℝn is the n-dimensional real Euclidean space; T denotes the matrix transposition; λminQ1,Q2 means the smallest eigenvalue of the matrices Q1 and Q2, and λmaxQ1,Q2 is the largest; ⋅ is the Euclidean norm of vector; LfVx stands for LfVx=∂Vx/∂xTfx, where fx,Vx∈C1ℝn,ℝ; and gt∈L20,+∞ means ∫0∞gt2dt<∞.


## 2. Problem Statement and Preliminaries

Consider the uncertain switched nonlinear cascade system with the form(1)z˙=pσtz,x+qσtz,xω,x˙=fσtz,x+Δfσtz,x+gσtz,xuσt+cσtz,xω,y=hσtz,x+dσtz,xω,where z∈ℝn−m,x∈ℝm, and X=zT,xT, ω∈ℝ is the disturbance input and ω∈L20,+∞, uσ∈ℝm is the control input, and y∈ℝp is the output, defining the right continuous function σt:ℝ+⟶l¯=1,2,…,l is the switching law. For ∀i∈l¯, pi⋅,⋅, fi⋅,⋅, gi⋅,⋅, and hi⋅,⋅ are smooth functions of appropriate dimensions, qi⋅,⋅, ci⋅,⋅, and di⋅,⋅ are bounded and smooth functions of appropriate dimensions, and Δfi⋅,⋅ is uncertain nonlinear functions of appropriate dimensions. Especially, pi0,0=0, fi0,0=0, and hi0,0=0. In the ideal state, the subsystem switching signal σt is defined on the following switching sequence:(2)∑=z0T,x0TT;t0,σt0,t1,σt1,t2,σt2,…,tk,σtk,k∈l¯,where t0 and z0T,x0TT are initial time and initial state, respectively, tk,σtk means that the ikth subsystem is activated at t∈tk,tk+1. Without loss of generality, we assume t0=0. In order to better understand the switching between subsystems, the block diagram of the switched system (1) is shown in Figure 1.Figure 1
The block diagram of the switched system.Assumption 1.
(see [21]). For z∈ℝn−m,x∈ℝm, the constants a1,a2,a3>0, and μ≥1, there exist positive definite functions Wiz,x∈C1,i∈I, such that the conditions(3)a1zt+xt2≤Wiz,x≤a2zt+xt2,(4)∂Wiz,x∂x≤a3x,∂Wiz,x∂z≤a3z,(5)Wiz,x≤μWjz,x,μ>0,i,j=1,…,N,hold.
For the subsystems of the switched system (1), we classify them into two groups: i∈Ip⊂l¯ represents that the ith closed-loop subsystem is passive; i∈In⊂l¯−Ip represents nonpassive. Then, Ip and In satisfy Assumption 2.Assumption 2.
Fori∈Ip,(6)LfiVix≤0,LpiUiz≤0,LgiVix=∂Vix∂xgi=hiTz,x.
Fori∈In, there exists a constant λ>0 satisfying(7)LfiVix+LpiUiz≤λWiz,x,where Uσ and Vσ are smooth functions of appropriate dimensions.Assumption 3.
(see [6]). For uncertain function Δfiz,x, it satisfies the bound Δfiz,x≤ζtz+x, ∀i∈I, where ζt is a nonnegative function and satisfies ∫t0tζτdτ≤κt−t0+η for nonnegative constants κ and η.Definition 1.
(see [22]). Let Nστ,t represent the number of switchings of σt in the interval τ,t for any switching signal σt and 0<τ<t. If(8)Nστ,t≤N0+t−ττaholds for N0,τa>0. The constant τa is called average dwell time, and N0 is the chatter bound. Without loss of generality, we choose N0=0.
The notion of average dwell time is often used for identifying switching signals which have certain desirable properties.Definition 2.
(see [23]). For any 0≤T1<T2, let TpT1,T2 denote the total time when the passive subsystems are active on T1,T2. Then, the passivity rate of the switched system is recorded as rpT1,T2=TpT1,T2/T2−T1. Clearly, 0<rpT1,T2≤1.
In this paper, we will study the following robustH∞ control problem for system (1). For any constant γ>0, define the control laws of each subsystems ui=uiz,x and ui0,0=0,i=1,…,l. Under the switching signal σt, system (1) has the following properties [6, 19]:(i)
The closed-loop system (1) with wt≡0 is globally robustly exponentially stable for all admissible uncertainties.(ii)
The closed-loop system (1) has a weighted L2-gain level γ for some real-valued function with βz,x and β0,0=0, that is, there exist a constant λ>0 and ωt∈L20,∞, such that(9)∫0∞e−λsyTsysds≤γ2∫0∞ωTsωsds+βz0,x0,holds.Definition 3.
In the nonlinear system,(10)z˙=pσtz,x,x˙=fσtz,x,y=hσtz,x,for degree λ¯, it is exponentially small-time norm-observable if there exist positive constants δ>0 and c>0, such that when yt+s≤δ holds for t≥t0 and 0<s≤τ, τ>0, zt+τ+xt+τ≤ce−λ¯τzt+xt is established.Remark 1.
The small-time norm-observability has been proposed for ensuring the asymptotical stability of switched systems [24]. In this paper, the exponential small-time norm-observability with degree λ¯ is exponential form, and it is used to research global robust exponential stability of system (1).Remark 2.
A method is given to verify that system (10) is exponentially small-time norm-observable. Assume that there exist positive constants δ and λ¯ and positive definite matrices Q1 and Q2, such that the following condition is satisfied:(11)2zTQ1pz,x+2xTQ2fz,x+δ+2λ¯−yz,xzTQ1z+xTQ2x≤0.
Let(12)Wt=zTtQ1zt+xTtQ2xt,l1=λminQ1,Q2,l2=λmaxQ1,Q2.
We can get(13)l1z+x2≤zTQ1z+xTQ2x≤l2z+x2.
From (11), the time derivative of Wt along the trajectory of system (10) is(14)dWtdt=2zTQ1pz,x+2xTQ2fz,x≤yz,x−δ−2λ¯Wt.
Whenyz,x≤δ holds for t∈t∗,t∗+τ with length τ, we obtain(15)dWtdt≤−2λ¯Wt,t∈t∗,t∗+τ.
By (13) and (15), using the differential inequality theory, we obtain(16)Wt≤e−2λ¯t−t∗Wt∗.
Hence,(17)l1zt+xt2≤l2e−2λ¯t−t∗zt∗+xt∗2,t∈t∗,t∗+τ,which means(18)zt+xt≤ce−λ¯t−t∗zt∗+xt∗,c=l2l1,t∈t∗,t∗+τ.
According to Definition3, system (10) is exponentially small-time norm-observable.Lemma 1.
If system (10) is exponentially small-time norm-observable with degree λ¯, for any k≥0, it has(19)zt+τ+xt+τ2≤c1e−2λ¯−kτzt+xt2−∫tt+τe−2λ¯−kt+τ−θyzθ,xθ2dθ,where c1=k0+c2, t≥t0, and τ>0.Proof.
If system (10) is exponentially small-time norm-observable with degree λ¯, there exists a constant k0>0, such that(20)yzs,xs2≤k0τe−2λ¯τzt+xt2,∀s∈t,t+τ,holds. By (20), we have(21)yzs,xs2≤k0τe−2λ¯−ks−tzt+xt2,∀s∈t,t+τ,namely,(22)τe2sλ¯−kyzs,xs2≤k0e2tλ¯−kzt+xt2,∀s∈t,t+τ.
Apply the integral mean value theorem to the above formula, and there exists a constants0, and t≤s0≤t+τ, such as(23)∫tt+τe2λ¯−kθyzθ,xθ2dθ=τe2λ¯−ks0yzs0,xs02≤k0e2tλ¯−kzt+xt2.
Then,(24)−k0e−2τλ¯−kzt+xt2≤−∫tt+τe−2λ¯−kt+τ−θyzθ,xθ2dθ.
Because system (10) is exponentially small-time norm-observable, if yt+s≤δ can be given with t0≤t,0<τ,and0<s≤τ, we obtain(25)zt+xt≤ce−λ¯τzt∗+xt∗,t∈t∗,t∗+τ,c=l2l1,which means(26)zt+τ+xt+τ≤ce−λ¯τzt+xt.
Then,(27)zt+τ+xt+τ2≤c2e−2λ¯τzt+xt2.
Then, the sum of (24) and (27) is(28)zt+τ+xt+τ2≤c1e−2λ¯−kτzt+xt2−∫tt+τe−2λ¯−kt+τ−θyzθ,xθ2dθ,where c1=k0+c2.


## 3. Main Results

In this section, we will discuss system (1) in two parts. Part I: when ω≡0, we will analyze the globally robustly exponentially stable of system (1) for all admissible uncertainties. Part II: when ω≠0, the weighted L2-gain level will be researched.


### 3.1. Part I: The Stability Analysis ofω≡0

Theorem 1.
Under the conditions of Assumptions1 and 2, let the positive constants τa and r be any given average dwell time and passivity rate, respectively. For all admissible uncertainties, system (1) with ui=0 is assumed to be exponentially small-time norm-observable with the positive constants λ¯,c, and δ satisfying λ¯≥1/2λ∗−a3κ/a1, ξ=ea3η/a1, and c≤a1/a2ξ, where(29)λ∗=λ1r+λr+lnμξrτa+a3κra1−λ,for a constant λ>0. Design the controllers(30)uix=−kiWiz,x,τa,rLgiVixT,i∈Ip,0,i∈In,where(31)kiWiz,x,τa,r=λ∗LgiVix2−1Wiz,x,LgiVix>δ,0,LgiVix≤δ.Then, the switched system (1) with w≡0 is globally robustly exponentially stable under any switching signals with the average dwell time τa and passivity rate rpT1,T2≥r.Proof.
Let(32)Wσzt,xt=Uσzt+Vσxt,where Uσ and Vσ are smooth functions of appropriate dimensions.
Fori∈Ip, we make the set Si=t:LgiVixt≤δ. Then, we divide the proof into two cases: one is Si=∅, and the other is Si≠∅.Case 1:.
Si=∅.
Assume that theith subsystem is active. For ω≡0, the time derivative of Wiz,x along the trajectory of the switched system (1) is(33)∂Wizt,xt∂t=∂Uiz∂zpi+∂Vix∂xfi+Δfi+giui.
Substituting controller (30) into (33), from (3), (4), and (7), for i∈Ip, we obtain that(34)∂Wizt,xt∂t≤LpiUiz+LfiVix+a3a1Wiz,xζt−∂Vix∂xgiλ∗Wiz,xLgiVixTLgiVix2≤a3a1Wiz,xζt−λ∗Wiz,x=−λ∗−a3a1ζtWiz,x,where ∂Vix/∂xΔfi≤a3Δfix≤a3Δfix+z≤a3ζtx+z2≤a3/a1Wiz,xζt.
Similarly, fori∈In, it follows from (3), (4), and (6) that(35)∂Wizt,xt∂t≤λWiz,x+a3a1Wiz,xζt=λ+a3a1ζtWiz,x.
Fort∈tk,tk+1, we apply the integral of (34) and (35) that(36)Wikzt,xt≤ϕ˜ikt,tkWikztk,xtk,t∈tk,tk+1,where ϕ˜ikt,tk=e−λ∗t−tk+a3/a1∫tktζτdτ,ik∈Ipeλt−tk+a3/a1∫tktζτdτ,ik∈In., Define(37)ϕikt,tk≔ξe−a∗t−tk,ik∈Ip,ξeat−tk,ik∈In,where ϕikt,tk:=ξe−a∗t−tk,ik∈Ip,ξeat−tk,ik∈In. From Assumption 3, then(38)Wikzt,xt≤ϕikt,tkWikztk,xtk,t∈tk,tk+1.
Choose the piecewise function:(39)Wzt,xt=Wikzt,xt,where Wzt0xt0=Wi0zt0,xt0.
On the contrary,ϕikt,τϕik−1τ,s=ξϕik−1t,s, for t∈tk,tk+1, and we obtain(40)Wzt,xt=Wikzt,xt≤ϕikt,tkWikztk,xtk≤ϕikt,tkμWik−1ztk,xtk≤ϕikt,tkμϕik−1tk,tk−1Wik−1ztk−1,xtk−1…≤ϕikt,tk,…,ϕi0t1,t0Wi0zt0,xt0=ξkϕt,t0μkWi0zt0,xt0=ξeN0−t−t0/τalnμξ−a∗Tpt0,t+aTnt0,tWzt0,xt0.
Fromλ∗=λ1/r+λ/r+lnμξ/rτa+a3κ/ra1−λ, we have λ1=λ∗r−λ−lnμξ/τa−a3κ/a1+λr; then,(41)N0+t−t0τalnμξ−a∗Tpt0,t+aTnt0,t=N0lnμξ−λ1t−t0.
Taking (41) into (40), we obtain(42)Wzt,xt≤ξN0+1μN0e−λ1t−t0Wzt0,xt0.
From (3), we get that(43)a1zt+xt2≤ξN0+1μN0a2e−λ1t−t0zt0+xt02,which means(44)zt+xt≤a2a1ξN0+1μN0e−λ1/2t−t0zt0+xt0.Case 2:.
Si≠∅.
In the case, fori∈Ip, we suppose that t:LgiVixt≤δ=ti1,ti1′∪ti2,ti2′∪⋯⊂t0,t.
From Definition3, we have zt+τ+xt+τ≤ce−λ¯τzt+xt. Then,(45)ztik′+xtik′≤ce−λ¯tik′−tikztik+xtik≤a1ξa2e−a∗/2tik′−tikztik+xtik.
From (3), we obtain(46)Wikztik′,xtik′≤ξe−a∗tik′−tikWikztik,xtik.
Similar to Case 1, we can achieve(47)zt+xt≤a2a1ξN0+1μN0e−λ1/2t−t0zt0+xt0.
Therefore, the closed-loop switched system (1) is exponential stability for all admissible uncertainties. This completes the proof.


### 3.2. Part II: TheL2-Gain Analysis of ω≠0

In this section, we investigate theH∞ performance analysis of system (1) by the L2-gain γ.Theorem 2.
Assume the positive constantsτ, r, and γ are average dwell time, passivity rate, and disturbance attenuation level, respectively. Uσ, Vσ, and Wσ still satisfy (3)–(7). For all admissible uncertainties and disturbance inputs, assume the passive subsystems are exponentially small-time norm-observability with positive constants λ¯, c, δ, k0, and k. All of these satisfy k=λ¯−1/2λ∗−a3κ/a1−a32ρ2/4a1γ2 and a2c2≤a1−k0ea3η/a1. Then, we design the controllers:(48)uix=−kiWiz,x,τa,r,γLgiVixT,i∈Ip,−LgiVixT,i∈In,where(49)kiWiz,x,τa,r,γ=λ∗Wiz,x+yiyiTLgiVix2,LgiVix>δ,0,LgiVix≤δ,and λ∗ is given by Theorem 1 with λ1=λ2+a32ρ2/4a1γ2 and λ2>0. Then, the switched system (1) achieves a weighted L2-gain from ω to y for all admissible uncertainties.Proof.
On the basis of Theorem1, the time derivative of Wiz,x along the trajectory of switched system (1) is(50)∂Wizt,xt∂t=∂Uiz∂zpi+qiω+∂Vix∂xfi+Δfi+giui+ciω.
Wheni∈Ip, Si=∅, similarly the proof of Theorem 1, we have(51)∂Wizt,xt∂t≤−λ∗−a3ξa1Wiz,x+∂Uiz∂zqiω+∂Vix∂xciω−yiTyi≤−λ∗−a3a1ζtWiz,x+a3zqi+a3xciω−yiTyi≤−λ∗−a3a1ζtWiz,x+a32ρ24γ2z+x2+γ2ω2−yiTyi≤−λ∗−a3a1ζt−a32ρ24a1γ2Wiz,x+γ2ωTtωt−yiTyi,where ρ=maxqi,ci.
Wheni∈Ip,Si≠∅, on the interval tik,tik′, we have(52)ztik′+‖xtik′2≤c1e−2λ¯−ktik′−tikztik+xtik2−ξa2∫tiktik′e−2λ¯−ktik′−θyiTyidθ.
From (3), we obtain(53)Wikztik′,xtik′≤c2a2+ξk0e−2λ¯−ktik′−tikztik+xtik2−∫tiktik′ξe−2λ¯−ktik′−θyikTyikdθ≤c2a2+ξk0a1e−2λ¯−ktik′−tikWikztikxtik−∫tiktik′ξe−2λ¯−ktik′−θyikTyikdθ.
Due tok=λ¯−1/2λ∗−a3κ/a1−a32ρ2/4a1γ2, we get λ¯−k=1/2λ∗−a3κ/a1−a32ρ2/4a1γ2=1/2a∗−a32ρ2/4a1γ2. Substituting λ¯−k into (53), we know(54)Wikztik′,xtik′≤c2a2+ξk0a1e−a∗−a32ρ2/4a1γ2tik′−tikWikztik,xtik+∫tiktik′ξe−a∗−a32ρ2/4a1γ2tik′−θγ2ω2θ−yikTyikdθ.
Fora2c2≤a1−k0ea3η/a1=a1−k0ξ=a1ξ−k0ξ, which means c2a2+ξk0/a1≤ξ; then,(55)Wikztik′,xtik′≤ξe−a∗−a32ρ2/4a1γ2tik′−tikWikztik,xtik+∫tiktik′ξe−a∗−a32ρ2/4a1γ2tik′−θγ2ω2θ−yikTyikdθ.
Similarly, wheni∈In,(56)∂Wizt,xt∂t≤λ+a3ξa1Wiz,x+∂Uiz∂zqiω+∂Vix∂xciω−yiTyi≤λ+a3a1ζt+a32ρ24a1γ2Wiz,x+γ2ωTtωt−yiTyi.
Leta˜∗=a∗−a32ρ2/4a1γ2,a˜=a+a32ρ2/4a1γ2,andΓit=γ2ωTtωt−yiTyi, where a∗=λ∗−a3κ/a1,a=λ+a3κ/a1,andξ=ea3η/a1∫t0tζτdτ≤κt−t0+η. For (51), (55), and (56), the differential equation theory and the constant variable formula are used, respectively. When i∈Ip and Si=∅,(57)Wikzt,xt≤e∫tkt−λ∗−a3/a1ζt−a32ρ2/4a1γ2dsWikztk,xtk+∫tktΓikτe−∫tkτ−λ∗−a3/a1ζs−a32ρ2/4a1γ2dsdτ≤ξe−a˜∗t−tkWikztk,xtk+∫tkte−λ∗−a3/a1ζt−a32ρ2/4a1γ2t−τ+a3/a1ηΓikτdτ≤ξe−a˜∗t−tkWikztk,xtk+∫tktξe−a˜∗t−τΓikτdτ.
Wheni∈IpandSi≠∅,(58)Wikzt,xtk≤ξe−a˜∗t−tkWiktk+∫tktξe−a˜∗t−τΓikτdτ.
And wheni∈In,(59)Wikzt,xt≤e∫tktλ+a3/a1ζt+a32ρ2/4a1γ2dsWikztk,xtk+∫tktΓikτe−∫tkτλ+a3/a1ζt+a32ρ2/4a1γ2dsdτ≤ξea˜t−tkWikztk,xtk+∫tkteλ+a3/a1ζt+a32ρ2/4a1γ2t−τ+a3/a1ηΓikτdτ≤ξea˜t−tkWikztk,xtk+∫tktξea˜t−τΓikτdτ,where(60)e∫tkt−λ∗−a3/a1ζt−a32ρ2/4a1γ2ds∫tktΓikτe−∫tkτ−λ∗−a3/a1ζs−a32ρ2/4a1γ2dsdτ=∫tktΓikτe∫τt−λ∗−a3/a1ζs−a32ρ2/4a1γ2dsdτ≤∫tktΓikτe−λ∗−a3κ/a1−a32ρ2/4a1γ2t−τ+a3/a1ηdτ=∫tktξe−a˜∗t−τΓikτdτ.
Combining (57)–(59), we obtain(61)Wikzt,xt≤ψt,tkWikztk,xtk+∫tktψikt,τΓikτdτ,t∈tk,tk+1,where ψikt,τ=ξe−a˜∗t−τ,ik∈Ipξea˜t−τ,ik∈In. Define the piecewise function Wzt,xt=Wikzt,xt and t∈tk,tk+1. When the time t satisfies t0<t1<⋯<tn<t<tn+1<⋯, by the property ψikt,τψik−1τ,s=ξψik−1t,s, i∈I, and from (61), we have(62)Wt=Wikzt,xt≤ψikt,tkWikztk,xtk+∫tktψikt,τΓikτdτ≤ψikt,tkμWik−1ztk,xtk+∫tktψikt,τΓikτdτ≤ψikt,tkμ≤ψik−1tk,tk−1Wik−1ztk−1,xtk−1+∫tk−1tkψik−1tk,τΓik−1τdτ+∫tktψikt,τΓikτdτ…≤ψikt,tk,…,ψi0t1,t0Wi0zt0,xt0μk+ψikt,tk,…,ψi1t2,t1μk∫t0t1ψi0t1,τΓi0τdτ+⋯+ψikt,tkμ∫tk−1tkψik−1tk,τΓik−1τdτ+∫tktψikt,τΓikτdτ=ψikt,tk,…,ψi0t1,t0Wi0zt0,xt0μk+∑n=1k∫tn−1tnξk−n+1μk−n+1ψin−1t,τΓin−1τdτ+∫tktψikt,τΓikτdτ≤ξk+1μke−a˜∗Tpt0,t+a˜Tnt0,tWzt0,xt0+ξ∫t0tμξNστ,te−a˜∗Tpτ,t+a˜Tnτ,tΓτdτ.
Due toλ1=λ2+a32ρ2/4a1γ2, we have −a˜∗Tpt0,,t+a˜Tnt0,t=−λ2−lnμξ/τat−t0.
Then,(63)0≤ξkμke−λ2−lnμξ/τat−t0Wzt0,xt0+∫t0tμξNστ,te−λ2−lnμξ/τat−τΓτdτ,which means(64)0≤e−λ2−lnμξ/τat−t0+Nδt,t0lnμξWzt0,xt0+∫t0te−λ2−lnμξ/τat−τ+Nδt,τlnμξΓτdτ.
We multiply both sides of the above formula bye−Nσt0,tlnμξ:(65)0≤e−λ2−lnμξ/τat−t0Wzt0,xt0+∫t0te−λ2−lnμξ/τat−τ+Nδt0,τlnμξΓτdτ.
Obviously,−Nδt0,τ≤0,ξ>1,μ>1, then e−Nδt0,τlnμξ≤1, and putting Γτ, Nδt0,τ=τ−t0/τa into (65), we obtain(66)∫t0te−λ2−lnμξ/τat−τ−lnμξ/τaτ−t0yTτyτdτ≤γ2∫t0te−λ2−lnμξ/τat−τωTτωτdτ+e−λ2−lnμξ/τat−t0Wzt0,xt0.
For the trivial case ofμ=1 and ξ=1, we obtain(67)∫t0∞yTτyτdτ≤γ2∫t0∞ωTτωτdτ+Wzt0,xt0.
Next, we consider the nontrivial case ofμ>1. Rearranging the double-integral area leads(68)∫t0+∞e−lnμξ/τaτ−t0yTτyτdτ≤γ2∫t0+∞ωTτωτdτ+Wzt0,xt0.
Hence, the switched system (1) achieves a weighted L2-gain from ω to y for all admissible uncertainties. This completes the proof.Remark 3.
Under zero initial condition, we haveW0,0=0, and from (68), we can get the weighted L2-gain level γ2=∫0+∞e−lnμξ/τaτ−t0yTτyτdτ/∫0+∞ωTτωτdτ. The smaller the weighted L2-gain level γ is, the better the performance of robust H∞ control of system (1) is [25, 26].Remark 4.
The system in [19] is similar to system (1) in this paper, and it needs to satisfy these conditions in [19]: (i) γ2−γ32−γd2>0; (ii) for unbounded positive definite differentiable functions Vix1,i=1,…,N, constants γ1>0 and λ0>0, such that ∂Vi/∂x1f1,ix1,0+1/4γ12∂Vi/∂x1cix1,0ciTx1,0∂ViT/∂x1+hiTx1,0hix1,0+λ0Vi≤0 holds. In our paper, we just need γ>0, and the positive definite differentiable functions Wi do not need to satisfy condition (ii). So, this paper gets less conservative.


## 3.1. Part I: The Stability Analysis ofω≡0

Theorem 1.
Under the conditions of Assumptions1 and 2, let the positive constants τa and r be any given average dwell time and passivity rate, respectively. For all admissible uncertainties, system (1) with ui=0 is assumed to be exponentially small-time norm-observable with the positive constants λ¯,c, and δ satisfying λ¯≥1/2λ∗−a3κ/a1, ξ=ea3η/a1, and c≤a1/a2ξ, where(29)λ∗=λ1r+λr+lnμξrτa+a3κra1−λ,for a constant λ>0. Design the controllers(30)uix=−kiWiz,x,τa,rLgiVixT,i∈Ip,0,i∈In,where(31)kiWiz,x,τa,r=λ∗LgiVix2−1Wiz,x,LgiVix>δ,0,LgiVix≤δ.Then, the switched system (1) with w≡0 is globally robustly exponentially stable under any switching signals with the average dwell time τa and passivity rate rpT1,T2≥r.Proof.
Let(32)Wσzt,xt=Uσzt+Vσxt,where Uσ and Vσ are smooth functions of appropriate dimensions.
Fori∈Ip, we make the set Si=t:LgiVixt≤δ. Then, we divide the proof into two cases: one is Si=∅, and the other is Si≠∅.Case 1:.
Si=∅.
Assume that theith subsystem is active. For ω≡0, the time derivative of Wiz,x along the trajectory of the switched system (1) is(33)∂Wizt,xt∂t=∂Uiz∂zpi+∂Vix∂xfi+Δfi+giui.
Substituting controller (30) into (33), from (3), (4), and (7), for i∈Ip, we obtain that(34)∂Wizt,xt∂t≤LpiUiz+LfiVix+a3a1Wiz,xζt−∂Vix∂xgiλ∗Wiz,xLgiVixTLgiVix2≤a3a1Wiz,xζt−λ∗Wiz,x=−λ∗−a3a1ζtWiz,x,where ∂Vix/∂xΔfi≤a3Δfix≤a3Δfix+z≤a3ζtx+z2≤a3/a1Wiz,xζt.
Similarly, fori∈In, it follows from (3), (4), and (6) that(35)∂Wizt,xt∂t≤λWiz,x+a3a1Wiz,xζt=λ+a3a1ζtWiz,x.
Fort∈tk,tk+1, we apply the integral of (34) and (35) that(36)Wikzt,xt≤ϕ˜ikt,tkWikztk,xtk,t∈tk,tk+1,where ϕ˜ikt,tk=e−λ∗t−tk+a3/a1∫tktζτdτ,ik∈Ipeλt−tk+a3/a1∫tktζτdτ,ik∈In., Define(37)ϕikt,tk≔ξe−a∗t−tk,ik∈Ip,ξeat−tk,ik∈In,where ϕikt,tk:=ξe−a∗t−tk,ik∈Ip,ξeat−tk,ik∈In. From Assumption 3, then(38)Wikzt,xt≤ϕikt,tkWikztk,xtk,t∈tk,tk+1.
Choose the piecewise function:(39)Wzt,xt=Wikzt,xt,where Wzt0xt0=Wi0zt0,xt0.
On the contrary,ϕikt,τϕik−1τ,s=ξϕik−1t,s, for t∈tk,tk+1, and we obtain(40)Wzt,xt=Wikzt,xt≤ϕikt,tkWikztk,xtk≤ϕikt,tkμWik−1ztk,xtk≤ϕikt,tkμϕik−1tk,tk−1Wik−1ztk−1,xtk−1…≤ϕikt,tk,…,ϕi0t1,t0Wi0zt0,xt0=ξkϕt,t0μkWi0zt0,xt0=ξeN0−t−t0/τalnμξ−a∗Tpt0,t+aTnt0,tWzt0,xt0.
Fromλ∗=λ1/r+λ/r+lnμξ/rτa+a3κ/ra1−λ, we have λ1=λ∗r−λ−lnμξ/τa−a3κ/a1+λr; then,(41)N0+t−t0τalnμξ−a∗Tpt0,t+aTnt0,t=N0lnμξ−λ1t−t0.
Taking (41) into (40), we obtain(42)Wzt,xt≤ξN0+1μN0e−λ1t−t0Wzt0,xt0.
From (3), we get that(43)a1zt+xt2≤ξN0+1μN0a2e−λ1t−t0zt0+xt02,which means(44)zt+xt≤a2a1ξN0+1μN0e−λ1/2t−t0zt0+xt0.Case 2:.
Si≠∅.
In the case, fori∈Ip, we suppose that t:LgiVixt≤δ=ti1,ti1′∪ti2,ti2′∪⋯⊂t0,t.
From Definition3, we have zt+τ+xt+τ≤ce−λ¯τzt+xt. Then,(45)ztik′+xtik′≤ce−λ¯tik′−tikztik+xtik≤a1ξa2e−a∗/2tik′−tikztik+xtik.
From (3), we obtain(46)Wikztik′,xtik′≤ξe−a∗tik′−tikWikztik,xtik.
Similar to Case 1, we can achieve(47)zt+xt≤a2a1ξN0+1μN0e−λ1/2t−t0zt0+xt0.
Therefore, the closed-loop switched system (1) is exponential stability for all admissible uncertainties. This completes the proof.


## 3.2. Part II: TheL2-Gain Analysis of ω≠0

In this section, we investigate theH∞ performance analysis of system (1) by the L2-gain γ.Theorem 2.
Assume the positive constantsτ, r, and γ are average dwell time, passivity rate, and disturbance attenuation level, respectively. Uσ, Vσ, and Wσ still satisfy (3)–(7). For all admissible uncertainties and disturbance inputs, assume the passive subsystems are exponentially small-time norm-observability with positive constants λ¯, c, δ, k0, and k. All of these satisfy k=λ¯−1/2λ∗−a3κ/a1−a32ρ2/4a1γ2 and a2c2≤a1−k0ea3η/a1. Then, we design the controllers:(48)uix=−kiWiz,x,τa,r,γLgiVixT,i∈Ip,−LgiVixT,i∈In,where(49)kiWiz,x,τa,r,γ=λ∗Wiz,x+yiyiTLgiVix2,LgiVix>δ,0,LgiVix≤δ,and λ∗ is given by Theorem 1 with λ1=λ2+a32ρ2/4a1γ2 and λ2>0. Then, the switched system (1) achieves a weighted L2-gain from ω to y for all admissible uncertainties.Proof.
On the basis of Theorem1, the time derivative of Wiz,x along the trajectory of switched system (1) is(50)∂Wizt,xt∂t=∂Uiz∂zpi+qiω+∂Vix∂xfi+Δfi+giui+ciω.
Wheni∈Ip, Si=∅, similarly the proof of Theorem 1, we have(51)∂Wizt,xt∂t≤−λ∗−a3ξa1Wiz,x+∂Uiz∂zqiω+∂Vix∂xciω−yiTyi≤−λ∗−a3a1ζtWiz,x+a3zqi+a3xciω−yiTyi≤−λ∗−a3a1ζtWiz,x+a32ρ24γ2z+x2+γ2ω2−yiTyi≤−λ∗−a3a1ζt−a32ρ24a1γ2Wiz,x+γ2ωTtωt−yiTyi,where ρ=maxqi,ci.
Wheni∈Ip,Si≠∅, on the interval tik,tik′, we have(52)ztik′+‖xtik′2≤c1e−2λ¯−ktik′−tikztik+xtik2−ξa2∫tiktik′e−2λ¯−ktik′−θyiTyidθ.
From (3), we obtain(53)Wikztik′,xtik′≤c2a2+ξk0e−2λ¯−ktik′−tikztik+xtik2−∫tiktik′ξe−2λ¯−ktik′−θyikTyikdθ≤c2a2+ξk0a1e−2λ¯−ktik′−tikWikztikxtik−∫tiktik′ξe−2λ¯−ktik′−θyikTyikdθ.
Due tok=λ¯−1/2λ∗−a3κ/a1−a32ρ2/4a1γ2, we get λ¯−k=1/2λ∗−a3κ/a1−a32ρ2/4a1γ2=1/2a∗−a32ρ2/4a1γ2. Substituting λ¯−k into (53), we know(54)Wikztik′,xtik′≤c2a2+ξk0a1e−a∗−a32ρ2/4a1γ2tik′−tikWikztik,xtik+∫tiktik′ξe−a∗−a32ρ2/4a1γ2tik′−θγ2ω2θ−yikTyikdθ.
Fora2c2≤a1−k0ea3η/a1=a1−k0ξ=a1ξ−k0ξ, which means c2a2+ξk0/a1≤ξ; then,(55)Wikztik′,xtik′≤ξe−a∗−a32ρ2/4a1γ2tik′−tikWikztik,xtik+∫tiktik′ξe−a∗−a32ρ2/4a1γ2tik′−θγ2ω2θ−yikTyikdθ.
Similarly, wheni∈In,(56)∂Wizt,xt∂t≤λ+a3ξa1Wiz,x+∂Uiz∂zqiω+∂Vix∂xciω−yiTyi≤λ+a3a1ζt+a32ρ24a1γ2Wiz,x+γ2ωTtωt−yiTyi.
Leta˜∗=a∗−a32ρ2/4a1γ2,a˜=a+a32ρ2/4a1γ2,andΓit=γ2ωTtωt−yiTyi, where a∗=λ∗−a3κ/a1,a=λ+a3κ/a1,andξ=ea3η/a1∫t0tζτdτ≤κt−t0+η. For (51), (55), and (56), the differential equation theory and the constant variable formula are used, respectively. When i∈Ip and Si=∅,(57)Wikzt,xt≤e∫tkt−λ∗−a3/a1ζt−a32ρ2/4a1γ2dsWikztk,xtk+∫tktΓikτe−∫tkτ−λ∗−a3/a1ζs−a32ρ2/4a1γ2dsdτ≤ξe−a˜∗t−tkWikztk,xtk+∫tkte−λ∗−a3/a1ζt−a32ρ2/4a1γ2t−τ+a3/a1ηΓikτdτ≤ξe−a˜∗t−tkWikztk,xtk+∫tktξe−a˜∗t−τΓikτdτ.
Wheni∈IpandSi≠∅,(58)Wikzt,xtk≤ξe−a˜∗t−tkWiktk+∫tktξe−a˜∗t−τΓikτdτ.
And wheni∈In,(59)Wikzt,xt≤e∫tktλ+a3/a1ζt+a32ρ2/4a1γ2dsWikztk,xtk+∫tktΓikτe−∫tkτλ+a3/a1ζt+a32ρ2/4a1γ2dsdτ≤ξea˜t−tkWikztk,xtk+∫tkteλ+a3/a1ζt+a32ρ2/4a1γ2t−τ+a3/a1ηΓikτdτ≤ξea˜t−tkWikztk,xtk+∫tktξea˜t−τΓikτdτ,where(60)e∫tkt−λ∗−a3/a1ζt−a32ρ2/4a1γ2ds∫tktΓikτe−∫tkτ−λ∗−a3/a1ζs−a32ρ2/4a1γ2dsdτ=∫tktΓikτe∫τt−λ∗−a3/a1ζs−a32ρ2/4a1γ2dsdτ≤∫tktΓikτe−λ∗−a3κ/a1−a32ρ2/4a1γ2t−τ+a3/a1ηdτ=∫tktξe−a˜∗t−τΓikτdτ.
Combining (57)–(59), we obtain(61)Wikzt,xt≤ψt,tkWikztk,xtk+∫tktψikt,τΓikτdτ,t∈tk,tk+1,where ψikt,τ=ξe−a˜∗t−τ,ik∈Ipξea˜t−τ,ik∈In. Define the piecewise function Wzt,xt=Wikzt,xt and t∈tk,tk+1. When the time t satisfies t0<t1<⋯<tn<t<tn+1<⋯, by the property ψikt,τψik−1τ,s=ξψik−1t,s, i∈I, and from (61), we have(62)Wt=Wikzt,xt≤ψikt,tkWikztk,xtk+∫tktψikt,τΓikτdτ≤ψikt,tkμWik−1ztk,xtk+∫tktψikt,τΓikτdτ≤ψikt,tkμ≤ψik−1tk,tk−1Wik−1ztk−1,xtk−1+∫tk−1tkψik−1tk,τΓik−1τdτ+∫tktψikt,τΓikτdτ…≤ψikt,tk,…,ψi0t1,t0Wi0zt0,xt0μk+ψikt,tk,…,ψi1t2,t1μk∫t0t1ψi0t1,τΓi0τdτ+⋯+ψikt,tkμ∫tk−1tkψik−1tk,τΓik−1τdτ+∫tktψikt,τΓikτdτ=ψikt,tk,…,ψi0t1,t0Wi0zt0,xt0μk+∑n=1k∫tn−1tnξk−n+1μk−n+1ψin−1t,τΓin−1τdτ+∫tktψikt,τΓikτdτ≤ξk+1μke−a˜∗Tpt0,t+a˜Tnt0,tWzt0,xt0+ξ∫t0tμξNστ,te−a˜∗Tpτ,t+a˜Tnτ,tΓτdτ.
Due toλ1=λ2+a32ρ2/4a1γ2, we have −a˜∗Tpt0,,t+a˜Tnt0,t=−λ2−lnμξ/τat−t0.
Then,(63)0≤ξkμke−λ2−lnμξ/τat−t0Wzt0,xt0+∫t0tμξNστ,te−λ2−lnμξ/τat−τΓτdτ,which means(64)0≤e−λ2−lnμξ/τat−t0+Nδt,t0lnμξWzt0,xt0+∫t0te−λ2−lnμξ/τat−τ+Nδt,τlnμξΓτdτ.
We multiply both sides of the above formula bye−Nσt0,tlnμξ:(65)0≤e−λ2−lnμξ/τat−t0Wzt0,xt0+∫t0te−λ2−lnμξ/τat−τ+Nδt0,τlnμξΓτdτ.
Obviously,−Nδt0,τ≤0,ξ>1,μ>1, then e−Nδt0,τlnμξ≤1, and putting Γτ, Nδt0,τ=τ−t0/τa into (65), we obtain(66)∫t0te−λ2−lnμξ/τat−τ−lnμξ/τaτ−t0yTτyτdτ≤γ2∫t0te−λ2−lnμξ/τat−τωTτωτdτ+e−λ2−lnμξ/τat−t0Wzt0,xt0.
For the trivial case ofμ=1 and ξ=1, we obtain(67)∫t0∞yTτyτdτ≤γ2∫t0∞ωTτωτdτ+Wzt0,xt0.
Next, we consider the nontrivial case ofμ>1. Rearranging the double-integral area leads(68)∫t0+∞e−lnμξ/τaτ−t0yTτyτdτ≤γ2∫t0+∞ωTτωτdτ+Wzt0,xt0.
Hence, the switched system (1) achieves a weighted L2-gain from ω to y for all admissible uncertainties. This completes the proof.Remark 3.
Under zero initial condition, we haveW0,0=0, and from (68), we can get the weighted L2-gain level γ2=∫0+∞e−lnμξ/τaτ−t0yTτyτdτ/∫0+∞ωTτωτdτ. The smaller the weighted L2-gain level γ is, the better the performance of robust H∞ control of system (1) is [25, 26].Remark 4.
The system in [19] is similar to system (1) in this paper, and it needs to satisfy these conditions in [19]: (i) γ2−γ32−γd2>0; (ii) for unbounded positive definite differentiable functions Vix1,i=1,…,N, constants γ1>0 and λ0>0, such that ∂Vi/∂x1f1,ix1,0+1/4γ12∂Vi/∂x1cix1,0ciTx1,0∂ViT/∂x1+hiTx1,0hix1,0+λ0Vi≤0 holds. In our paper, we just need γ>0, and the positive definite differentiable functions Wi do not need to satisfy condition (ii). So, this paper gets less conservative.


## 4. Numerical Example

In this section, we give two examples to demonstrate the effectiveness of the proposed method.Example 1.
Consider a switched continuous stirred tank reactor system with two modes feed stream [27, 28]:(69)ξ˙1=FσtVξ1,in,σt−ξ1+Kσtφσtξ1,ξ2−dt,ξ˙2=FσtVξ2,in,σt−ξ2−ΔHσtξ1,ξ2φσtξ1,ξ2+γξ2c−ξ2+dt.
In this paper, we ignore the influence of temperature on reaction speed, and only consider the disturbancedt on the concentration. And the physical meaning of the parameters in system (69) can be found in [28].
The control objective is to make the temperature to some constant referenceξ1∗andξ2∗. And uk∗=γξ2c∗ is a steady-state control corresponding to the temperature set points ξ1∗andξ2∗. Let z=ξ1−ξ1∗,x=ξ2−ξ2∗,uk=γξ2c−ξ2c∗. System (69) can be expressed in the form with equilibrium point at the origin:(70)z˙=f1σz,x+dt,x˙=f2σz,x+Δf2σz,x+uσ+dt,where f1σ=Fσ/Vξ1,in,σ−ξ1∗−z+Kσφσz+ξ1∗,x+ξ2∗,f2σ=Fσ/Vξ2,in,σ−ξ2∗−x+γξ2c∗−ξ2∗−x and Δf2σ=−ΔHσz+ξ1∗,x+ξ2∗φσz+ξ1∗,x+ξ2∗.
Then, let the steady-state pointξ2∗=ξ2c∗=0K, ξ1∗=1mol/L, and parameters for the simulation F1=4L/s,γ=K1=1, ξ2in1=ξ2in2=0K,V=1L,ξ1in1=ξ1in2=1mol/L,ΔH1=ΔH2=−0.5,F2=1L/s,K2=2,φ1=ze−x,φ2=z+xe−2x,and dt=ωt. And defining the output y=2x, we get two subsystems as follows:(71)z˙=−4z+ze−x+ωt,x˙=−5x+0.5ze−x+u1+ωt,y=2x,(72)z˙=−z+2z+xe−2x+ωt,x˙=−2x+0.5z+xe−2x+u2+ωt,y=2x,where ui,i=1,2, are controllers and θi,i=1,2, are unknown constants.
It is not difficult to know thatΔf1t,x≤0.5z and Δf2t,x≤0.5z+x. So, ∫t0tζτdτ≤∫t0t0.5dτ≤κt−t0+η, where κ=1andη=0.5.
LetW1=z2+0.5x2 and W2=z2+x2. For system (71),(73)Lp1U1z=∂U1∂zp1=2z×−z+ze−x=−2z2+z2e−x≤0,Lf1V1x=∂V1∂xf1=2x×−5x=−10x2≤0,Lg1V1x=2x.
For system (72),(74)Lp2U2z=2z×−z+2z+xe−2x=−2z2+4zz+xe−2x,Lf2V2x=2x×−x=−4x2,Lg2V2x=2x.
Then,(75)Lp2U2z+Lf2V2x=−2z2+4zz+xe−2x−4x2≤2z2+4zx−4x2≤3z2≤3W2z,x.
It is easy to verify that system (71) is passive and system (72) is nonpassive.
A simple calculation shows thata1=0.5, a2=2, a3=2, μ=2, ρ=1, and λ=2. In addition, we acquire that c=3 and ξ=ea3η/a1. Let the average dwell time τa=2, the passivity rate r=1, the disturbance attenuation level γ=2, and λ2=1.2. Then, λ1=λ2+a32ρ2/4a1γ2=1.7000, λ∗=λ1/r+λ/r+lnμξ/rτa+a3κ/ra1−λ=7.0466, λ¯=10λ∗−a3κ/a1=30.4657, and λ¯−cκ−1/2λ∗−a3κ/a1−a32/4a1γ2=26.9424>0. According to Theorem 1, when ωt≡0, we construct the controllers(76)ui=−14.0932z2+0.5x2x2x2,i∈Ip,2x>0.1,0,i∈Ip,2x≤0.1,0,i∈In.
Figure2 gives the control input of systems (71) and (72) and the switching signal σt. Figure 3 is the simulation result with the initial states z1=2 and x1=1.
Whenωt=e−t, we construct the controllers(77)n∗ui=−14.0932z2+0.5x2−8x2x2x2,i∈Ip,2x>0.1,0,i∈Ip,2x≤0.1,−2x,i∈In.
From Remark3, we define the function(78)γ2s=∫0se−lnμξ/τaτ−t0yTτyτdτ∫0sωTτωτdτ.
In this example, we can getγ2s=∫0se−26.9315yTτyτdτ/∫0sωTτωτdτ. Figure 4 gives the L2-gain level with the initial states z1=0 and x1=0. And we can easily see the L2-gain less than γ=2.Figure 2
The control inputu and the switching signal σt.Figure 3
The system state response.Figure 4
TheL2-gain level γ2.Example 2.
Consider the uncertain switched nonlinear cascade systems with two systems:(79)z1.=−3z1−z2x1,z2.=−z2+5z1x1,x1.=−x1−z1x2+θ1x1−u1,x2.=−2x2+0.3z1x1+θ1x2−3u1,y=−6x1−6x2,(80)z1.=z1−0.2z2x1,z2.=z2+2z1x1,x1.=x1+z1x2+θ2x1+0.1u2+sinx12ω,x2.=x2−z1x1+θ2x2−0.1u2+sinx22ω,y=2x1+2x2+ω,where θ1=−0.48 and θ2=−0.06 are generated constants by random numbers.
LetW1=0.5z12+0.1z22+3x12+x22, and W2=z12+0.1z22+x12+x22. For system (79),(81)Lp1U1z=z10.2z2−3z1−z2x1−z2+5z1x1=−3z12−0.2x22≤0,Lf1V1x=6x12x2−x1−z1x2−2x2+0.3z1x1=−6x12−4x22≤0,Lg1V1x=−6x1−6x2.
For system (80),(82)Lp2U2z=2z10.2z2z1−0.2z2x1z2+2z1x1=2z12+0.2z22,Lf2V2x=2x12x2x1+z1x2x2−z1x1=2x12+2x22,Lg2V2x=0.2x1−0.2x2.
Then,(83)Lp2U2z+Lf2V2x=2z12+0.2z22+2x12+2x22≤2W2z,x.
It is easy to see that system (79) is passive and system (80) is nonpassive. A simple calculation shows that a1=0.1,a2=3,a3=2,μ=10, κ=0, η=0.5, ρ=1, and λ=2. In addition, we acquire that c=20 and ξ=ea3η/a1. Let the average dwell time τa=2, the passivity rate r=6, the disturbance attenuation level γ=1, and λ2=1.2. Hence, λ1=λ2+a32ρ2/4a1γ2=3.7000, λ∗=λ1/r+λ/r+lnμξ/rτa+a3κ/ra1−λ=0.3919, λ¯=10λ∗−a3κ/a1=3.9188, and λ¯−cκ−1/2λ∗−a3κ/a1−a32/4a1γ2=4.0354>0.
According to Theorem1, we construct the controllers; when ωt≡0,(84)ui=0.39190.5z12+0.1z22+3x12+x22x1+x26x1+x22,i∈Ip,6x1+x2>0.3,0,i∈Ip,6x1+x2≤0.3,0,i∈In.
Figure5 shows the control input of systems (79) and (80) and the switching signal σt. Figure 6 is the simulation result with the initial states z11=1, z21=2, x11=−1, and x21=−1.
Whenωt=e−te−t, we construct the control functions:(85)ui=0.39190.5z12+0.1z22+3x12+x22+y1y1Tx1+x26x1+x22i∈Ip,6x1+x2>0.3,0,i∈Ip,6x1+x2≤0.3,−0.2x1−0.2x2,i∈In.
In this example, we can getγ2s=∫0se−103.8155yTτyτdτ/∫0sωTτωτdτ. Figure 7 is the L2-gain level with the initial states z11=0, z21=0, x11=0, and x21=0. And we can easily see the L2-gain less than γ=1.Figure 5
The control inputu and the switching signal σt.Figure 6
The system state response.Figure 7
TheL2-gain level γ2.


## 5. Conclusion

Based on the method of average dwell time, we give sufficient conditions to ensure the solvability of the problem avoiding the Lyapunov function construction by the storage functions and reducing the computational complexity of the solution. For any switching signal, the system can achieve stability and have the weightedL2-gain property under the action of the feedback controller designed by the given passivity rate, average dwell time, and interference attenuation level. The proposed scheme supplements the research methods of robust H∞ control for the nonlinear cascade systems. In the future, we will extend the results of this paper to global stabilization of switched stochastic nonlinear robust H∞ control systems.



---

*Source: 8866775-2021-01-16.xml*