# Exponential Feedback Passivity of Switched Polynomial Nonlinear Systems

Authors: Huawei Zhu; Xiaorong Hou
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The exponential feedback passivity problem of switched polynomial nonlinear systems is studied. To obtain this aim, a method of parameterization of controller is presented and parameter solution algorithm is described. Then, the addressed method is utilized to solve the robust stabilization for a class of switched polynomial nonlinear systems with parameter uncertainties and external disturbance. This result extends the previous results on exponential feedback passivity from the case of nonlinear systems to switched nonlinear systems. A numerical example is given to demonstrate the effectiveness of the proposed result.


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## Body

## 1. Introduction

Switched polynomial nonlinear systems (SPNSs) [1–3] are a class of dynamical systems consisting of nonlinear subsystems with polynomial structure and a switching law, which defines a specific subsystem being activated during a certain interval of time. Due to the theoretical development as well as practical applications, analysis and synthesis of switched systems have recently gained considerable attention. Many tools, such as single Lyapunov function [4], multiple Lyapunov functions [5], and average dwell time [6], have been proved effective in solving the problem of stability and stabilization for switched nonlinear systems. In addition, in practical control systems, the parameter uncertainty and external disturbance are common, which implies that it is meaningful to study the robust stabilization of switched polynomial nonlinear system with parameter uncertainties and external disturbance [6–8].On the other hand, as a particular case of dissipativity, passivity was introduced in [9] and later generalized in [10]. In virtue of the property of keeping a system internal stability, passivity theory is extensively used as an effective tool for the stability analysis and synthesis of nonlinear systems [11–22]. Recently, the exponential passivity problem for nonlinear systems has attracted the attention of many scholars. In [23], the concept of exponential passivity was introduced. In this result, it needs to satisfy that there exists an r-continuously differentiable storage function Vs(·) with supply rate r(u,y)=uTy meeting some conditions. Using exponentially weighted system storage functions with appropriate exponentially weighted supply rates, Chellaboina and Haddad [24] introduced the notion of exponential dissipativity, which was more general than the notion of exponential passivity introduced in [23]. In [25], a delay-dependent exponential passivity condition for memristive neural networks was proposed in terms of linear matrix inequalities. By constructing a proper Lyapunov-Krasovskii functional, utilizing free-weighting matrix method and some stochastic analysis techniques, some delay-dependent sufficient conditions that ensure the exponential passivity of stochastic neural networks were deduced in [26]. In [27], by constructing an appropriate Lyapunov-Krasovskii functional and using the Wirtinger-based integral inequality to estimate its derivative, a delay-range-dependent and delay-rate-dependent criterion was presented to ensure the augmented filtering dynamic system to be robustly exponentially passive with an expected dissipation. The exponential passivity of some systems can be ensured with the conditions of [25–27]. Still and all, whether these conditions are true depend on the structure of the systems themselves. In addition, the nonlinear functions of the systems of [25–27] should satisfy the so-called Lipschitz condition, which restrict the application scope of these results. In [23], some feedback controllers were designed to obtain exponential passivity for a class of affine systems possessing globally defined normal form and factorisable high-frequency gain. In [28], a continuous output feedback controller was designed to solve the exponentially stability problem for a class of uncertain systems that contain a nominal part which is affine in the control and an uncertain part which is norm bounded by a known function. However, the results of [23, 28] are only suitable for some special systems. Based on the above discussions, we known that the related research on the exponential feedback passivity of nonlinear systems is still rather limited. Especially to the author’s knowledge, there exist no results on the feedback exponential passivity of switched nonlinear systems so far, which is the motivation of the research of this paper.In this paper, the exponential feedback passivity problem and the exponential passivity property are investigated for a class of SPNSs. The focus is on designing parameterized controller and solving out the feasible interval solution of the parameter. First, a concept ofCr-parameterized feedback exponentially passive of switched systems is introduced. Second, some parameterized symmetric matrix inequality conditions are proposed to ensure the exponential passivity of the SPNSs. Then, a parameter solution algorithm is formulated to solve the matrix inequalities. At last, the exponential passivity property is utilized to solve the robust stabilization problem of the SPNSs with parameter uncertainties and external disturbance.The remainder of this paper is organized as follows. Section2 is the problem formulation. The main contributions of this paper are then given in Section 3, in which a method of parameterization of controller is produced and a parameter solution algorithm is provided. In Section 4, a numerical example to support the results of this paper is given, which is followed by the conclusion in Section 5.Notations. The notations throughout this paper are quite standard. We use P>0 to denote a positive define matrix P. λM(P) (λm(P)) means the largest (smallest) eigenvalue of matrix P. Let P=(∑j=1n∑i=1nP(i,j)2)1/2. Lf(x)V(x), Lg(x)V(x) mean ∂V(x)/∂xf(x), ∂V(x)/∂xg(x), respectively. I denotes identity matrix with appropriate dimension. Rn denotes the n dimensional Euclidean space. Let x∈R+n=x∈Rn,x>0. x=(xTx)1/2 refers to the Euclidean vector norm. L2[0,+∞) is the space of square integrable vector over [0,+∞). “PT” denotes the transpose of the matrix P. “P-T” denotes the inverse of the matrix PT. Matrices, if not explicitly stated, are assumed to have compatible dimensions.


## 2. Problem Formulation and Preliminaries

Consider following uncertain switched polynomial nonlinear system (SPNS) with the form(1)x˙=fσtx+Δfσtt,x+gσtxuσt+gσt1xwt,f0=0y=hσtx,where x∈Rn, y∈Rp, u∈Rp, and w∈Rq represent, respectively, the state variables, output, control, and external disturbance and w(t)∈L2[0,+∞). σ(t):[0,∞)→M=1,2,…,m∗, called switching signal, is a piecewise right continuous function. m∗ is a finite integer in the nonnegative integer set. ∑={(tk,ik)∣ik∈M;k=0,1,…;0=t0≤t1≤⋯} means the switching sequence corresponding to the switching signal σ(t). σ(t)=i,(i∈M) means the ith subsystem is active. For σ(t)=i, fi(x) and hi(x) are nonlinear polynomial functions, Δfi(x) express the parameter uncertainty, and gi(x) and gi1(x) are polynomial matrices. Assume that the states of system (1) do not jump at switching instants.Remark 1.
For the SPNSs, all the terms of the matrices are polynomial functions about system states. The requirement of possessing polynomial structure is not very restrictive for system (1) since any continuous functions can be approximated by polynomials of sufficient large order.In the following subsections, some definitions and lemma to be used in this paper are provided.Definition 2 (see [29]).
For a switching signalσ(t) and any T>T0≥0, If Nσ(T,T0)≤N0+T-T0/τα, τα>0, N0≥0, τα and N0 are called, respectively, the average dwell time and the jitter bound, where Nσ(T,T0) is the switching numbers of σ(t) over the interval [T0,T).
N 0 refers to the initial number of jitter. Without loss of generality, we make the value N0=0.Definition 3.
Considering the following switched system:(2)x˙=fσtx+gσtxuσt,y=hσtx,if, for ∀i,j∈M, any given average dwell time τα, there exist positive constant α⌢i, μ≥1, positive constants a1,a2, positive definite continuous function Vi(x) called storage function with Vi(0)=0, and parameterized controller ui(x,km)=υi(x,km)+riTνi(x) in which υi(x,km) is the part to parameterize the controller, νi(x)=-hi(x) is the new input, ri is a weighting matrix with full column rank, km is the parameter with serial number m, and there exists feasible interval solution for km, such that, ∀t≥0(i)(3)Lfi+giuiVix≤νiThi-α⌢iVix,(ii)(4)hix=rigiTx∂Vix∂x,(iii)(5)a1x2≤Vix≤a2x2,(iv)(6)Vix≤μVjx,(v)(7)α⌢i>ln⁡μταhold. Then, under the parameterized controller ui(x,km) and switching law with average dwell time τα, the closed-loop system (2) is said to be Cr-parameterized feedback exponentially passive.Remark 4.
In view of the switching law with average dwell timeτα and its related switching sequence ∑, letting α^=mini∈M⁡α⌢i, for t∈[tk,tk+1),k=0,1,…, by (3) and (6), similar to the process of (43), it can be obtained that (8)Vikxt≤μkexp⁡-α^t-t0Vi0xt0+∫tktexp-α^t-τνikTτyikτdτ+∑n=1k∫tn-1tnμk+1-nexp-α^t-τνin-1Tτyin-1τdτ≤exp⁡-α^t-t0+Nσt0,tln⁡μVi0xt0+∫t0texp⁡-α^t-τ+Nστ,tln⁡μνTτyτdτ.By Definition2, it is obtained that Nστ,tln⁡μ≤t-τln⁡μ/τα, which implies that -α^(t-τ)+Nστ,tln⁡μ≤(ln(μ)/τα-α^)(t-τ). Therefore, (9)Vikxt≤exp-α^t-t0+Nσt0,tln⁡μVi0xt0+12exp⁡ln⁡μτα-α^t∫t0t2exp⁡α^-ln⁡μτατνTτyτdτ.In view of Vik(x(t))≥0, by (7), when x(t0)=0, it gives ∫t0teετr^(ν(τ),y(τ))dτ≥0, where ε=α^-ln(μ)/τα>0, r^(ν(τ),y(τ))=2νT(τ)y(τ). Then, in view of Definitions 2.1 and 3.4 of [24], we know that system (2) is exponentially passive. Thus, the notion of Cr-parameterized feedback exponential passivity defined in Definition 3 in this paper is consistent with the notion of exponential passivity in [24].Remark 5.
The systems rendered to beCr-exponentially passive by a parameterized feedback controller are called Cr-parameterized feedback exponentially passive systems or said to be feedback equivalent to Cr-parameterized exponentially passive systems. In Section 3, the Cr-parameterized feedback exponentially passive property will be used to solve the robust stabilization problem for the SPNSs with parameter uncertainties and external disturbance.Lemma 6 (see [30]).
If a vector functionf(x) with f(0)=0 has continuous nth order partial derivatives, f(x) can be expressed as(10)fx=a1xx1+⋯+anxxn=Axx,where A(x)=[a1(x),…,an(x)], ai(x) are vector functions, i=1,…,n, x=[x1,…,xn]T∈Rn.According to Lemma6, system (1) can be transformed into(11)x˙=Aσtxx+Δfσtt,x+gσtxuσt+gσt1xwt,where Aσ(t)(x)=[aσ(t)1(x),…,aσ(t)n(x)]∈Rn×n is called structural matrix.Fori∈M, choose positive definite symmetric matrix Pi. In view of Pi>0, there must exist matrices Ai1(x) such that Ai1(x)=Ai(x)Pi-1 where Pi-1 are the inverse of the matrices Pi. Then, Ai(x) can be expressed as Ai(x)=Ai1(x)Pi and system (1) can be transformed into the following equivalent representation:(12)x˙=Jσtx-RσtxPσtx+Δfσtt,x+gσtxuσt+gσt1xwt,where Ji(x)=1/2(Ai(x)Pi-1-Pi-TAiT(x)) is skew-symmetric and Ri(x)=-1/2(Ai(x)Pi-1+Pi-TAiT(x)) is symmetric.Remark 7.
WhenRi(x)≥0, the ith subsystem described by (12) is called dissipated Hamiltonian system and Ri(x) is the so-called dissipative matrix [11].


## 3. Main Results

In this section, the exponential feedback passivity of SPNSs is addressed with the method of parameterization of controller. In what follows, they will be accomplished in two subsections.


### 3.1. Exponential Feedback Passivity

Theorem 8.
Under the switching law with the given average dwell timeτα, the closed-loop system (1) without parameter uncertainties and external disturbance is globally exponentially stable, if it is said to be feedback equivalent to Cr-parameterized exponentially passive system as Definition 3.Proof.
Choose the following piecewise Lyapunov function:(13)Vxt=Vσtxt.
Then, taking no regard of parameter uncertainties and lettingw=0, the differential of Vi(x) along the trajectory of the close-loop system (1) is obtained (14)V˙ix=∂Vix∂xx˙=∂Vix∂xfix+gixui.
According to the switching law with average dwell timeτα, the switching sequence ∑=(tk,ik) is achieved. For t∈[tk,tk+1),k=0,1,…, noting that νi(x)=-hi(x), by (3), (4), it is obtained that(15)V˙ikx≤-αikVikx-hikThik≤-α^Vikx,where α^=mini∈M⁡αi.
Integrating both sides of (15) over t∈[tk,tk+1),∀k=0,1,2,… leads to(16)Vikxt≤exp⁡-α^t-tkVikxtk.
Then, according to the switching sequence∑ and Definition 2, by (6), it follows from (16) that(17)Vikxt≤μkexp⁡-α^t-t0Vi0xt0≤exp⁡N0+t-t0ταln⁡μ-α^t-t0Vi0xt0.
On the other hand,(18)N0+t-t0ταln⁡μ-α^t-t0=N0ln⁡μ+ln⁡μτa-α^t-t0=N0ln⁡μ-α^0t-t0,where(19)α^0=α^-ln⁡μτα.
Substituting (18) into (17) yields(20)Vikxt≤μN0exp⁡-α^0t-t0Vi0xt0.
According to (7) and noting that α^=mini∈M⁡αi, it gives α^0>0. Furthermore, by (5), we have Vik(x(t))≥a1x(t)2,Vi0(x(t0))≤a2x(t0)2. Then, it follows from (20) that (21)xt≤a2a1μN0exp⁡-12α^0t-t0xt0which implies the globally exponential stability of the closed-loop system (1) without parameter uncertainties and external disturbance. This completes the proof.Theorem 9.
Considering system (1) without parameter uncertainty and external disturbance, for any given average dwell time Tα, i∈M, if there exist positive definite symmetric matrices Pi, positive definite functions Vi(x)=1/2xTPix, symmetrical polynomial parameter matrices Ki(x,km), and positive constants α~i, system (1) can be transformed into the form of (12) and, for ∀x∈R+n, (22)Gi=∂Vix∂xRix+gixKix,kmgiTx∂ViTx∂x-α~iVix>0,(23)hix=rixgiTxPix(24)α~i>ln⁡μTαhold.
Then, under the switching law with average dwell timeTα and following parameterized controller(25)uikm,x=-Kix,km+riTxrixgiTxPix=υix,km+riTνix,system (1) is said to be feedback equivalent to Cr-parameterized exponentially passive systems.Proof.
Leta1=1/2mini∈M⁡λm(Pi), a2=1/2maxi∈M⁡λM(Pi), a3=maxi∈M⁡Pi, μ=maxk,l∈M⁡λM(Pk)/λm(Pl). Obviously, (5) and (6) hold.
Considering theith subsystem, according to the conditions fi(0)=0 and Lemma 6, system (1) can be transformed into the form of (12).
Choose the storage functions for each subsystem(26)Vixt=12xTPix≥0,i∈M.
In view of (22), noting that νi(x)=-hi(x), the differential of Vi(x) along the trajectory of the system (1) is obtained (27)V˙ixt=-∂Vix∂xRix∂ViTx∂x-gixυix,km+riTνix=-∂Vix∂xRix+gixKix,kmgiTx∂ViTx∂x+νiTxhix≤-α~iVix+νiTxhixwhich means (3) holds.
By (23) and (24), we know that (4) and (7) hold. According to Definition 3, the closed-loop system (1) without parameter uncertainty and external disturbance is said to be Cr-parameterized exponentially passive. This completes the proof.Remark 10.
In order to render that (3) holds, condition (22) should be satisfied. However, the major difficulty is to solve out the interval value of km such that (22) holds over the entire system state space. According to Remark 1, expression Gi in the left side of condition (22) is polynomial function. By taking advantage of this feature, a symbolic computation based algorithm is produced to solve the interval value of km and α~i in Section 3.2.Remark 11.
The aims of the parameterization of controller include two aspects. Firstly, compensate forRi and render it to be dissipative matrix: that is, Ri(x)+gi(x)Ki(x,km)giT(x)≥0. Secondly, render that the storage function Vi(x) has decay rate α~i along each subsystem. In Theorem 13 and Remark 14, we will know that the robust stabilization problem of system (1) can also be solved if the values of α~i are big enough. As a matter of fact, by adjusting the values of km contained in the parameterized controller (25), the values of α~i can be improved according to their feasible interval solutions.In the following subsection, theCr-parameterized feedback exponential passivity property of system (1) is utilized to solve its robust stabilization problem. Firstly, introduce an assumption as follows.Assumption 12 (see [22]).
Considering the uncertain terms of system (1), there exists nonnegative function ς(t), which satisfies ∫t0tς(τ)dτ≤κ(t-t0)+η for some nonnegative constants κ, η, such that Δfi(t,x)≤ς(t)x,∀i∈M hold for all x.
The uncertain termΔfi(t,x) described by Assumption 12 is usual in nonlinear system [22].
In this paper, the robust stabilization problem of system (1) is summarized as follows: for a given disturbance attenuation level γ>0, given average dwell time τα, parameterized controllers ui(km,x),i∈M in which km are the parameters with serial number m and there exist feasible interval solutions for km, such that
(a) the closed-loop system (1) is said to be globally robustly exponentially stable when w=0,
(b) the closed-loop system (1) has L2 gain from w to y for all admissible uncertainties. That is to say, there exists a constant λ~≥0 and a real-valued function v~(x) with v~(0)=0, such that(28)∫t0texp⁡-λ~t-t0yTtytdτ≤∫t0tγ2wTwdτ+v~xt0holds.Theorem 13.
Suppose the conditions of Theorem9 are satisfied and the uncertain terms of system (1) satisfy Assumption 12. For any given disturbance attenuation level γ, if there exist positive constants αi,i∈M, a3 such that (29), (30), and (31) hold,(29)α~iVix-14γ2∂Vix∂xgi1xgi1Tx∂ViTx∂x>αiVix,(30)∂Vix∂x≤a3x,(31)αi-ln⁡μexp⁡a3η/a1τα-a3a1κ>0,where α~i, a1, μ, and Vi(x) are the same as Theorem 9 and τα is a reassigned average dwell time. Then, the robust stabilization problem is solvable for system (1) under the parameterized controller (25) and switching law with average dwell time τα.Proof.
Choose the storage functionsVi(x(t))=1/2xTPix≥0,i∈M as (26). By the switching law with average dwell time τα, the switching sequence ∑ is achieved. According to Theorem 9, the system (1) without parameter uncertainty and external disturbance is said to be Cr-parameterized exponentially passive. Then, by Definition 3 and (29), for t∈[tk,tk+1),k=0,1,…, it is obtained that(32)V˙ikxt≤-αikVikx-hikTxhikx-14γ2∂Vikx∂xgik1xgik1Tx∂VikTx∂x+∂Vikx∂xΔfikt,x+∂Vikx∂xgik1xw.
In addition, combining with Assumption12, (5) and (30) lead to(33)∂Vikx∂xΔfikt,x≤∂Vikx∂xςtx≤a3xςtx≤a3a1ςtVikx.
Whenw≡0, it is obtained by substituting (33) into (32) that(34)V˙ikxt≤-αik-a3a1ςtVikx-hikTxhikx≤-αik-a3a1ςtVikx≤-α-a3a1ςtVikx,where =mini∈M⁡αi.
Letξ=exp(a3η/a1), α~=α-a3/a1κ, Φik(t,τ)=ξexp⁡-α~t-τ. Integrating both sides of (34) over t∈[tk,tk+1),∀k=0,1,2,… leads to (35)Vikxt≤exp-αt-tk+a3a1∫tktςτdτVikxtk≤exp-αt-tk+a3a1κt-tk+a3a1ηVikxtk=Φikt,tkVikxtk.
Obviously, according to the structural characteristic ofΦik(t,tk), switching sequence ∑, Definition 2, (6), over the time interval [t0,t], it follows from (35) that(36)Vxt≤Φikt,tkμΦik-1tik,tik-1⋯μΦi0ti1,ti0Vxt0≤μkξk+1exp⁡-α~t-t0Vi0xt0≤ξexp⁡N0+t-t0ταln⁡μξ-α~t-t0Vi0xt0.
Besides, similar to (18), considering Definition 2, it is obtained that(37)N0+t-t0ταln⁡μξ-α~t-t0=N0ln⁡μξ+ln⁡μξτa-α~t-t0=N0ln⁡μξ-α-0t-t0,where(38)α-0=α-ln⁡μξτα-a3a1κ.
Substituting (37) into (36) yields(39)Vikxt≤μN0ξN0+1exp⁡-α-0t-t0Vi0xt0.
Noting thatα=mini∈M⁡αi, due to (31), it gives α-0>0. Furthermore, by (5), we have V(x(t))≥α1x(t)2, V(x(t0))≤α2x(t0)2. Then, it follows from (39) that(40)xt≤a2a1μN0ξN0+1exp⁡-12α-0t-t0xt0which implies the globally exponential stability of the closed-loop system.
Next, theL2 gain disturbance attenuation problem of system (1) from w to y is studied. With the similar procedure of (34), for t∈[tk,tk+1),k=0,1,…, by (32) and (33), it gives(41)V˙ikxt≤-αik-a3a1ςtVikx-hikTxhikx-14γ2xTPikTgik1xgik1TxPikx+∂Vikx∂xgik1xw≤-αik-a3a1ςtVikx-hikTxhikx+γ2w2≤-α-a3a1ςtVikx-hikTxhikx+γ2w2.
LetQik(t)=γ2w2(t)w(t)-hikT(x(t))hik(x(t)). Then, integrating both sides of (41) over t∈[tk,tk+1),∀k=0,1,2,… yields(42)Vikxt≤Φikxt,tkVikxtk+∫tktΦik-t,τQikτdτ.
On the other hand, in view of the switching sequence∑={(tk,ik)∣ik∈M},∀k=1,2,…, for n∈1,…,k, t∈[tk,tk+1), τ∈[tn-1,tn), by Definition 2, it is obtained that Nσ(τ,t)=Nσ(n-1,t)=k+1-n. Then, noting that Vik(x(t))≥0, over the time interval [t0,t], it is obtained that(43)0≤Vikxt≤μkξk+1exp⁡-α~t-t0Vi0xt0+∫tktΦik-t,τQikτdτ+ξ∑n=1k∫tn-1tnμξk+1-nexp⁡-α~t-τQin-1τdτ≤ξexp⁡-α~t-t0+Nσt0,tln⁡μξVi0xt0+ξ∫t0texp⁡-α~t-τ+Nστ,tlnμξQτdτ,where Q(t)=γ2w2(t)w(t)-yT(x(t))y(x(t)).
Noticing thatξ>0, multiplying ξ-1exp-Nσt0,tln⁡μξ with the two sides of (43) leads to (44)0≤exp⁡-α~t-t0Vi0xt0+∫t0texp⁡-α~t-τ+Nστ,t-Nσt0,tlnμξQτdτ=exp⁡-α~t-t0Vi0xt0+∫t0texp-α~t-τ-τ-t0ταlnμξQτdτ=exp⁡-α~+ln⁡μξταt-t0Vi0xt0+∫t0texp-α~+ln⁡μξταt-τ-τ-t0ταlnμξQτdτ,which implies that(45)∫t0texp-α~+ln⁡μξταt-τ-τ-t0ταlnμξyTτyτdτ≤exp-α~+ln⁡μξταt-t0Vi0xt0+∫t0texp⁡-α~+ln⁡μξταt-τexp⁡-τ-t0ταln⁡μξγ2wTτwτdτ≤exp-α~+ln⁡μξταt-t0Vi0xt0+∫t0texp-α~+ln⁡μξταt-τγ2wTτwτdτ.
Due to the positive properties ofa1,a3,η, it gives ξ>1. In addition, by μ>1, α~>0, and τα>0, it is easy to know that α~+ln(μξ)/τα>0. Then, for t→∞, it follows from (45) that(46)∫t0+∞exp⁡-ln⁡μξτατ-t0yTτyτ-γ2wTτwτdτ≤Vt0which means that the switched system achieves the L2 gain level γ from w to y. This completes the proof.Remark 14.
Aside from the two aims described in Remark11, the procedure of parameterization of controller has another effect that compensates some nonlinear actions for the external disturbance terms gi1(x)w. Hence, if the external disturbance structure matrices gi1(x) contain state variables, the intervals of the parameters should be solved again by combining conditions (22) and (29). Namely, synthesize conditions (22) and (29) and obtain a new condition(47)Gi=∂Vix∂xRix+gixKix,kmgiTx-14γ2gi1xgi1Tx∂ViTx∂x-αiVix≥0.
Solve conditions (31) and (47) with the PS algorithm described in Section 3.2 and obtain the interval values of the parameters km and αi. Then the controller satisfying the requirements can be obtained as the same way of Theorem 9.
In addition, it is easy to know that whether conditions (29) and (31) hold depends on the values of α~i. Considering this property, we have the following three ways to confirm the values of the parameters when the external disturbance structure matrices gi1(x) do not contain state variables.
(1) If (29) and (31) are still established with the parameter values which are obtained according to Theorem 9, the controllers obtained in Theorem 9 are still effective in Theorem 13.
(2) If (29) or (31) is not established with the parameter values which is obtained according to Theorem 9, reselect the parameter values according to the interval value of the parameter (see Remark 10) obtained in Theorem 9.
(3) Resolve the interval values of the parameters according to (31) and (47) and reselect the parameter values. This way is always effective.Remark 15.
By the interval value of the parameterkm, several specific values of km can be chosen and several controllers are achieved. Then, the optimized controller can be got by comparing with the performance of system (1) under the above obtained different controllers. Considering that the dissipation of the system is closely related to some performances of the system [11], it is reasonable to optimize the system with this way (see Remark 11). In Section 4, the disturbance attenuation capability of the system is improved with this method.


### 3.2. Parameter Solution Algorithm

In order to obtain the interval values of the parameters restricted by (22), (24) or (31), (47), a symbolic computation based parameter solution (PS) algorithm is produced in this subsection. First, design the parameter matrices and obtain the parameterized controllers; secondly, according to conditions (22), (24) or (31), (47), obtain equivalent inequality set; thirdly, decompose in the parameter space and search out the required cell. At last, choose several values for each km and obtain the optimized controller. In this subsection, we will just consider the case of (31) and (47). The controller for the case of (22) and (24) can be obtained with the same way, which will not be considered. The algorithm now proceeds as follows.PS AlgorithmStep 1 (design the parameter matrices).
For each subsystem, design a symmetrical polynomial parameter matrixKi(x,km) with the terms Kinrnc=∑s=0r~θmkm∏j~=1nxj~sj~, where θm is the coefficient of the item, km is the parameter, m is the serial number of the parameter, s is the serial number of the monomial with maximum number r~, nr is the row number and nc is the column number for the term in matrix Ki(x,km), sj~ is a nonnegative integer, and j~ is the serial number of the state variable with the maximum number n.Step 2 (obtain inequality conditions).
(2.1)
Obtain the values of parameters: choose some symmetric positive definite matricesPi such that (23) holds. Then, let a1=1/2mini∈M⁡λm(Pi), a2=1/2maxi∈M⁡λM(Pi), a3=maxi∈M⁡Pi=maxi∈M⁡(∑j=1n∑i=1nP(i,j)2)1/2, μ=maxk,l∈M⁡λM(Pk)/λm(Pl). Obtain κ, η according to Assumption 12. (2.2)
Obtain inequality conditions: with the given average dwell timeτα and the obtained parameter values, achieve inequality (31), and express it with the form of fi0>0. Substituting Ki(x,km) into (47) leads to the inequality conditions Gi>0.Step 3 (simplify the inequality conditions).
(3.1)
Considering the polynomial expressionsGi(x,km,αi) for the ith subsystem, set the coefficient of the terms, whose degree subjects to the system state x are odd to be zero, and obtain a group of equalities. Transform the equalities to express one parameter with other parameters as follows: kms=f(αi,kmj1,kmj2,…),∀ms,mj1,mj2∈m,ms≠mj1,ms≠mj2. Then, substitute f(αi,kmj1,kmj2,…) with kms in Ki. (3.2)
Choose some terms in the polynomial expressionsGi(x,km,αi) and set their coefficients to be zero. Then, a group of equalities are obtained. Transform the equalities to express one parameter with other parameters as the forms kmr=f(αi,kmk1,kmk2,…),∀mr,mk1,mk2∈m,mr≠mk1,mr≠mk2. Substitute f(αi,kmk1,kmk2,…) with kmr of Ki. Repeat this process several times until only one parameter km retained in each parameter matrix Ki. (3.3)
Obtain a new polynomial inequalityGi>0 with the new Ki(x,ki) and rewrite it into quadratic form. Then the positive of the quadratic form is equivalent to the positive definiteness of quadratic matrix P^i(ki,αi). (3.4)
According to the positive definiteness of the quadratic matrixP^i(ki,αi), we know that all the order principal minors of the quadratic form matrix are positive that are fir>0,r∈N+. Besides, we know that fi0>0. In this way, a polynomial inequality constraint condition set Ci, which is an equivalent condition of (31) and (47), is obtained. (3.5)
Repeat (3.1)–(3.4) and obtain the polynomial inequality constraint condition setCi for each subsystem.Step 4 (solve the interval values of the parameters).
(4.1)
Regard every inequality inCi as implicit function, such as fir(ki,αi)=0 and fi0(αi)=0, and draw them in the ki-αi plane. Then, the ki,αi parameter space is decomposed into many cells. (4.2)
Search out the required regionRCi by verifying any one point in each cell to all the constraint conditions in Ci. (4.3)
Repeat (4.1)-(4.2) for each subsystem and obtain the required regionRCi,i∈M for each subsystem.Step 5 (obtain controllers).
(5.1)
For every subsystem, achieve the parameterized controller by substituting the newKi(x,ki) obtained from Step (3.2) into (25). (5.2)
According to the required regionsRCi obtained in Step (4.2), choose several values for each ki and substitute them into the parameterized controller. Then, several group controllers are obtained. (5.3)
Simulate system (1) with every group controllers obtained in Step (5.2), respectively. Compare the simulating results and find out the controller possessing better capability.Remark 16.
In Step (3.3),Gi is updated with the new Ki(x,ki). Then, the new Gi is transformed into quadratic form. If infeasible result is got, redesign the parameter matrix Ki(x,km) again and rerun the algorithm.Remark 17.
By Step (4.2), for each subsystem, the parameter space is decomposed into cells. According to the Cylindrical Algebraic Decomposition algorithm [31], all the points containing in each cell have the same characteristics. Then, in Step (4.2), at least one required region RCi satisfying the requirements of Ci is obtained by verifying any one point in each cell. If infeasible result is got in Step (4.2), redesign the corresponding parameter matrix Ki(x,km) and rerun this algorithm for the corresponding subsystem.


## 3.1. Exponential Feedback Passivity

Theorem 8.
Under the switching law with the given average dwell timeτα, the closed-loop system (1) without parameter uncertainties and external disturbance is globally exponentially stable, if it is said to be feedback equivalent to Cr-parameterized exponentially passive system as Definition 3.Proof.
Choose the following piecewise Lyapunov function:(13)Vxt=Vσtxt.
Then, taking no regard of parameter uncertainties and lettingw=0, the differential of Vi(x) along the trajectory of the close-loop system (1) is obtained (14)V˙ix=∂Vix∂xx˙=∂Vix∂xfix+gixui.
According to the switching law with average dwell timeτα, the switching sequence ∑=(tk,ik) is achieved. For t∈[tk,tk+1),k=0,1,…, noting that νi(x)=-hi(x), by (3), (4), it is obtained that(15)V˙ikx≤-αikVikx-hikThik≤-α^Vikx,where α^=mini∈M⁡αi.
Integrating both sides of (15) over t∈[tk,tk+1),∀k=0,1,2,… leads to(16)Vikxt≤exp⁡-α^t-tkVikxtk.
Then, according to the switching sequence∑ and Definition 2, by (6), it follows from (16) that(17)Vikxt≤μkexp⁡-α^t-t0Vi0xt0≤exp⁡N0+t-t0ταln⁡μ-α^t-t0Vi0xt0.
On the other hand,(18)N0+t-t0ταln⁡μ-α^t-t0=N0ln⁡μ+ln⁡μτa-α^t-t0=N0ln⁡μ-α^0t-t0,where(19)α^0=α^-ln⁡μτα.
Substituting (18) into (17) yields(20)Vikxt≤μN0exp⁡-α^0t-t0Vi0xt0.
According to (7) and noting that α^=mini∈M⁡αi, it gives α^0>0. Furthermore, by (5), we have Vik(x(t))≥a1x(t)2,Vi0(x(t0))≤a2x(t0)2. Then, it follows from (20) that (21)xt≤a2a1μN0exp⁡-12α^0t-t0xt0which implies the globally exponential stability of the closed-loop system (1) without parameter uncertainties and external disturbance. This completes the proof.Theorem 9.
Considering system (1) without parameter uncertainty and external disturbance, for any given average dwell time Tα, i∈M, if there exist positive definite symmetric matrices Pi, positive definite functions Vi(x)=1/2xTPix, symmetrical polynomial parameter matrices Ki(x,km), and positive constants α~i, system (1) can be transformed into the form of (12) and, for ∀x∈R+n, (22)Gi=∂Vix∂xRix+gixKix,kmgiTx∂ViTx∂x-α~iVix>0,(23)hix=rixgiTxPix(24)α~i>ln⁡μTαhold.
Then, under the switching law with average dwell timeTα and following parameterized controller(25)uikm,x=-Kix,km+riTxrixgiTxPix=υix,km+riTνix,system (1) is said to be feedback equivalent to Cr-parameterized exponentially passive systems.Proof.
Leta1=1/2mini∈M⁡λm(Pi), a2=1/2maxi∈M⁡λM(Pi), a3=maxi∈M⁡Pi, μ=maxk,l∈M⁡λM(Pk)/λm(Pl). Obviously, (5) and (6) hold.
Considering theith subsystem, according to the conditions fi(0)=0 and Lemma 6, system (1) can be transformed into the form of (12).
Choose the storage functions for each subsystem(26)Vixt=12xTPix≥0,i∈M.
In view of (22), noting that νi(x)=-hi(x), the differential of Vi(x) along the trajectory of the system (1) is obtained (27)V˙ixt=-∂Vix∂xRix∂ViTx∂x-gixυix,km+riTνix=-∂Vix∂xRix+gixKix,kmgiTx∂ViTx∂x+νiTxhix≤-α~iVix+νiTxhixwhich means (3) holds.
By (23) and (24), we know that (4) and (7) hold. According to Definition 3, the closed-loop system (1) without parameter uncertainty and external disturbance is said to be Cr-parameterized exponentially passive. This completes the proof.Remark 10.
In order to render that (3) holds, condition (22) should be satisfied. However, the major difficulty is to solve out the interval value of km such that (22) holds over the entire system state space. According to Remark 1, expression Gi in the left side of condition (22) is polynomial function. By taking advantage of this feature, a symbolic computation based algorithm is produced to solve the interval value of km and α~i in Section 3.2.Remark 11.
The aims of the parameterization of controller include two aspects. Firstly, compensate forRi and render it to be dissipative matrix: that is, Ri(x)+gi(x)Ki(x,km)giT(x)≥0. Secondly, render that the storage function Vi(x) has decay rate α~i along each subsystem. In Theorem 13 and Remark 14, we will know that the robust stabilization problem of system (1) can also be solved if the values of α~i are big enough. As a matter of fact, by adjusting the values of km contained in the parameterized controller (25), the values of α~i can be improved according to their feasible interval solutions.In the following subsection, theCr-parameterized feedback exponential passivity property of system (1) is utilized to solve its robust stabilization problem. Firstly, introduce an assumption as follows.Assumption 12 (see [22]).
Considering the uncertain terms of system (1), there exists nonnegative function ς(t), which satisfies ∫t0tς(τ)dτ≤κ(t-t0)+η for some nonnegative constants κ, η, such that Δfi(t,x)≤ς(t)x,∀i∈M hold for all x.
The uncertain termΔfi(t,x) described by Assumption 12 is usual in nonlinear system [22].
In this paper, the robust stabilization problem of system (1) is summarized as follows: for a given disturbance attenuation level γ>0, given average dwell time τα, parameterized controllers ui(km,x),i∈M in which km are the parameters with serial number m and there exist feasible interval solutions for km, such that
(a) the closed-loop system (1) is said to be globally robustly exponentially stable when w=0,
(b) the closed-loop system (1) has L2 gain from w to y for all admissible uncertainties. That is to say, there exists a constant λ~≥0 and a real-valued function v~(x) with v~(0)=0, such that(28)∫t0texp⁡-λ~t-t0yTtytdτ≤∫t0tγ2wTwdτ+v~xt0holds.Theorem 13.
Suppose the conditions of Theorem9 are satisfied and the uncertain terms of system (1) satisfy Assumption 12. For any given disturbance attenuation level γ, if there exist positive constants αi,i∈M, a3 such that (29), (30), and (31) hold,(29)α~iVix-14γ2∂Vix∂xgi1xgi1Tx∂ViTx∂x>αiVix,(30)∂Vix∂x≤a3x,(31)αi-ln⁡μexp⁡a3η/a1τα-a3a1κ>0,where α~i, a1, μ, and Vi(x) are the same as Theorem 9 and τα is a reassigned average dwell time. Then, the robust stabilization problem is solvable for system (1) under the parameterized controller (25) and switching law with average dwell time τα.Proof.
Choose the storage functionsVi(x(t))=1/2xTPix≥0,i∈M as (26). By the switching law with average dwell time τα, the switching sequence ∑ is achieved. According to Theorem 9, the system (1) without parameter uncertainty and external disturbance is said to be Cr-parameterized exponentially passive. Then, by Definition 3 and (29), for t∈[tk,tk+1),k=0,1,…, it is obtained that(32)V˙ikxt≤-αikVikx-hikTxhikx-14γ2∂Vikx∂xgik1xgik1Tx∂VikTx∂x+∂Vikx∂xΔfikt,x+∂Vikx∂xgik1xw.
In addition, combining with Assumption12, (5) and (30) lead to(33)∂Vikx∂xΔfikt,x≤∂Vikx∂xςtx≤a3xςtx≤a3a1ςtVikx.
Whenw≡0, it is obtained by substituting (33) into (32) that(34)V˙ikxt≤-αik-a3a1ςtVikx-hikTxhikx≤-αik-a3a1ςtVikx≤-α-a3a1ςtVikx,where =mini∈M⁡αi.
Letξ=exp(a3η/a1), α~=α-a3/a1κ, Φik(t,τ)=ξexp⁡-α~t-τ. Integrating both sides of (34) over t∈[tk,tk+1),∀k=0,1,2,… leads to (35)Vikxt≤exp-αt-tk+a3a1∫tktςτdτVikxtk≤exp-αt-tk+a3a1κt-tk+a3a1ηVikxtk=Φikt,tkVikxtk.
Obviously, according to the structural characteristic ofΦik(t,tk), switching sequence ∑, Definition 2, (6), over the time interval [t0,t], it follows from (35) that(36)Vxt≤Φikt,tkμΦik-1tik,tik-1⋯μΦi0ti1,ti0Vxt0≤μkξk+1exp⁡-α~t-t0Vi0xt0≤ξexp⁡N0+t-t0ταln⁡μξ-α~t-t0Vi0xt0.
Besides, similar to (18), considering Definition 2, it is obtained that(37)N0+t-t0ταln⁡μξ-α~t-t0=N0ln⁡μξ+ln⁡μξτa-α~t-t0=N0ln⁡μξ-α-0t-t0,where(38)α-0=α-ln⁡μξτα-a3a1κ.
Substituting (37) into (36) yields(39)Vikxt≤μN0ξN0+1exp⁡-α-0t-t0Vi0xt0.
Noting thatα=mini∈M⁡αi, due to (31), it gives α-0>0. Furthermore, by (5), we have V(x(t))≥α1x(t)2, V(x(t0))≤α2x(t0)2. Then, it follows from (39) that(40)xt≤a2a1μN0ξN0+1exp⁡-12α-0t-t0xt0which implies the globally exponential stability of the closed-loop system.
Next, theL2 gain disturbance attenuation problem of system (1) from w to y is studied. With the similar procedure of (34), for t∈[tk,tk+1),k=0,1,…, by (32) and (33), it gives(41)V˙ikxt≤-αik-a3a1ςtVikx-hikTxhikx-14γ2xTPikTgik1xgik1TxPikx+∂Vikx∂xgik1xw≤-αik-a3a1ςtVikx-hikTxhikx+γ2w2≤-α-a3a1ςtVikx-hikTxhikx+γ2w2.
LetQik(t)=γ2w2(t)w(t)-hikT(x(t))hik(x(t)). Then, integrating both sides of (41) over t∈[tk,tk+1),∀k=0,1,2,… yields(42)Vikxt≤Φikxt,tkVikxtk+∫tktΦik-t,τQikτdτ.
On the other hand, in view of the switching sequence∑={(tk,ik)∣ik∈M},∀k=1,2,…, for n∈1,…,k, t∈[tk,tk+1), τ∈[tn-1,tn), by Definition 2, it is obtained that Nσ(τ,t)=Nσ(n-1,t)=k+1-n. Then, noting that Vik(x(t))≥0, over the time interval [t0,t], it is obtained that(43)0≤Vikxt≤μkξk+1exp⁡-α~t-t0Vi0xt0+∫tktΦik-t,τQikτdτ+ξ∑n=1k∫tn-1tnμξk+1-nexp⁡-α~t-τQin-1τdτ≤ξexp⁡-α~t-t0+Nσt0,tln⁡μξVi0xt0+ξ∫t0texp⁡-α~t-τ+Nστ,tlnμξQτdτ,where Q(t)=γ2w2(t)w(t)-yT(x(t))y(x(t)).
Noticing thatξ>0, multiplying ξ-1exp-Nσt0,tln⁡μξ with the two sides of (43) leads to (44)0≤exp⁡-α~t-t0Vi0xt0+∫t0texp⁡-α~t-τ+Nστ,t-Nσt0,tlnμξQτdτ=exp⁡-α~t-t0Vi0xt0+∫t0texp-α~t-τ-τ-t0ταlnμξQτdτ=exp⁡-α~+ln⁡μξταt-t0Vi0xt0+∫t0texp-α~+ln⁡μξταt-τ-τ-t0ταlnμξQτdτ,which implies that(45)∫t0texp-α~+ln⁡μξταt-τ-τ-t0ταlnμξyTτyτdτ≤exp-α~+ln⁡μξταt-t0Vi0xt0+∫t0texp⁡-α~+ln⁡μξταt-τexp⁡-τ-t0ταln⁡μξγ2wTτwτdτ≤exp-α~+ln⁡μξταt-t0Vi0xt0+∫t0texp-α~+ln⁡μξταt-τγ2wTτwτdτ.
Due to the positive properties ofa1,a3,η, it gives ξ>1. In addition, by μ>1, α~>0, and τα>0, it is easy to know that α~+ln(μξ)/τα>0. Then, for t→∞, it follows from (45) that(46)∫t0+∞exp⁡-ln⁡μξτατ-t0yTτyτ-γ2wTτwτdτ≤Vt0which means that the switched system achieves the L2 gain level γ from w to y. This completes the proof.Remark 14.
Aside from the two aims described in Remark11, the procedure of parameterization of controller has another effect that compensates some nonlinear actions for the external disturbance terms gi1(x)w. Hence, if the external disturbance structure matrices gi1(x) contain state variables, the intervals of the parameters should be solved again by combining conditions (22) and (29). Namely, synthesize conditions (22) and (29) and obtain a new condition(47)Gi=∂Vix∂xRix+gixKix,kmgiTx-14γ2gi1xgi1Tx∂ViTx∂x-αiVix≥0.
Solve conditions (31) and (47) with the PS algorithm described in Section 3.2 and obtain the interval values of the parameters km and αi. Then the controller satisfying the requirements can be obtained as the same way of Theorem 9.
In addition, it is easy to know that whether conditions (29) and (31) hold depends on the values of α~i. Considering this property, we have the following three ways to confirm the values of the parameters when the external disturbance structure matrices gi1(x) do not contain state variables.
(1) If (29) and (31) are still established with the parameter values which are obtained according to Theorem 9, the controllers obtained in Theorem 9 are still effective in Theorem 13.
(2) If (29) or (31) is not established with the parameter values which is obtained according to Theorem 9, reselect the parameter values according to the interval value of the parameter (see Remark 10) obtained in Theorem 9.
(3) Resolve the interval values of the parameters according to (31) and (47) and reselect the parameter values. This way is always effective.Remark 15.
By the interval value of the parameterkm, several specific values of km can be chosen and several controllers are achieved. Then, the optimized controller can be got by comparing with the performance of system (1) under the above obtained different controllers. Considering that the dissipation of the system is closely related to some performances of the system [11], it is reasonable to optimize the system with this way (see Remark 11). In Section 4, the disturbance attenuation capability of the system is improved with this method.


## 3.2. Parameter Solution Algorithm

In order to obtain the interval values of the parameters restricted by (22), (24) or (31), (47), a symbolic computation based parameter solution (PS) algorithm is produced in this subsection. First, design the parameter matrices and obtain the parameterized controllers; secondly, according to conditions (22), (24) or (31), (47), obtain equivalent inequality set; thirdly, decompose in the parameter space and search out the required cell. At last, choose several values for each km and obtain the optimized controller. In this subsection, we will just consider the case of (31) and (47). The controller for the case of (22) and (24) can be obtained with the same way, which will not be considered. The algorithm now proceeds as follows.PS AlgorithmStep 1 (design the parameter matrices).
For each subsystem, design a symmetrical polynomial parameter matrixKi(x,km) with the terms Kinrnc=∑s=0r~θmkm∏j~=1nxj~sj~, where θm is the coefficient of the item, km is the parameter, m is the serial number of the parameter, s is the serial number of the monomial with maximum number r~, nr is the row number and nc is the column number for the term in matrix Ki(x,km), sj~ is a nonnegative integer, and j~ is the serial number of the state variable with the maximum number n.Step 2 (obtain inequality conditions).
(2.1)
Obtain the values of parameters: choose some symmetric positive definite matricesPi such that (23) holds. Then, let a1=1/2mini∈M⁡λm(Pi), a2=1/2maxi∈M⁡λM(Pi), a3=maxi∈M⁡Pi=maxi∈M⁡(∑j=1n∑i=1nP(i,j)2)1/2, μ=maxk,l∈M⁡λM(Pk)/λm(Pl). Obtain κ, η according to Assumption 12. (2.2)
Obtain inequality conditions: with the given average dwell timeτα and the obtained parameter values, achieve inequality (31), and express it with the form of fi0>0. Substituting Ki(x,km) into (47) leads to the inequality conditions Gi>0.Step 3 (simplify the inequality conditions).
(3.1)
Considering the polynomial expressionsGi(x,km,αi) for the ith subsystem, set the coefficient of the terms, whose degree subjects to the system state x are odd to be zero, and obtain a group of equalities. Transform the equalities to express one parameter with other parameters as follows: kms=f(αi,kmj1,kmj2,…),∀ms,mj1,mj2∈m,ms≠mj1,ms≠mj2. Then, substitute f(αi,kmj1,kmj2,…) with kms in Ki. (3.2)
Choose some terms in the polynomial expressionsGi(x,km,αi) and set their coefficients to be zero. Then, a group of equalities are obtained. Transform the equalities to express one parameter with other parameters as the forms kmr=f(αi,kmk1,kmk2,…),∀mr,mk1,mk2∈m,mr≠mk1,mr≠mk2. Substitute f(αi,kmk1,kmk2,…) with kmr of Ki. Repeat this process several times until only one parameter km retained in each parameter matrix Ki. (3.3)
Obtain a new polynomial inequalityGi>0 with the new Ki(x,ki) and rewrite it into quadratic form. Then the positive of the quadratic form is equivalent to the positive definiteness of quadratic matrix P^i(ki,αi). (3.4)
According to the positive definiteness of the quadratic matrixP^i(ki,αi), we know that all the order principal minors of the quadratic form matrix are positive that are fir>0,r∈N+. Besides, we know that fi0>0. In this way, a polynomial inequality constraint condition set Ci, which is an equivalent condition of (31) and (47), is obtained. (3.5)
Repeat (3.1)–(3.4) and obtain the polynomial inequality constraint condition setCi for each subsystem.Step 4 (solve the interval values of the parameters).
(4.1)
Regard every inequality inCi as implicit function, such as fir(ki,αi)=0 and fi0(αi)=0, and draw them in the ki-αi plane. Then, the ki,αi parameter space is decomposed into many cells. (4.2)
Search out the required regionRCi by verifying any one point in each cell to all the constraint conditions in Ci. (4.3)
Repeat (4.1)-(4.2) for each subsystem and obtain the required regionRCi,i∈M for each subsystem.Step 5 (obtain controllers).
(5.1)
For every subsystem, achieve the parameterized controller by substituting the newKi(x,ki) obtained from Step (3.2) into (25). (5.2)
According to the required regionsRCi obtained in Step (4.2), choose several values for each ki and substitute them into the parameterized controller. Then, several group controllers are obtained. (5.3)
Simulate system (1) with every group controllers obtained in Step (5.2), respectively. Compare the simulating results and find out the controller possessing better capability.Remark 16.
In Step (3.3),Gi is updated with the new Ki(x,ki). Then, the new Gi is transformed into quadratic form. If infeasible result is got, redesign the parameter matrix Ki(x,km) again and rerun the algorithm.Remark 17.
By Step (4.2), for each subsystem, the parameter space is decomposed into cells. According to the Cylindrical Algebraic Decomposition algorithm [31], all the points containing in each cell have the same characteristics. Then, in Step (4.2), at least one required region RCi satisfying the requirements of Ci is obtained by verifying any one point in each cell. If infeasible result is got in Step (4.2), redesign the corresponding parameter matrix Ki(x,km) and rerun this algorithm for the corresponding subsystem.


## 4. Example and Simulations

In order to verify the effectiveness of the method produced in this paper for the uncertain SPNSs with external disturbance, an example is simulated.Example 18.
Consider the SPNS consisting of two subsystems with following parameters:(48)f1x=-8x2-2x2x3-4.5x12x1-6x22x1x2-4.5x3,f2x=1.5x1+6x2-3x1+2x22x3+x2-2x23+x3,Δf1x=0θ1e-tx10,Δf2x=0θ2x1θ3x2,g1x=020.5211011,g2x=013312021,h1x=8x22x1+4x2+x30.5x1+4x2+x3,h2x=6x2x1+2x2+4x33x1+4x2+2x3,g11x=diag1,1,1,g21x=diag1,1,1,τα=1,γ=6,where θ1∈[-1,1],θ2∈[-0.1,0.1],θ3∈[-0.2,0.2].


### 4.1. Solve Example18 with the Method of This Paper

LetM={1,2}, P1=diag(1,4,1), and P2=diag(1,2,2). Design the following piecewise Lyapunov function: (49)Vx=Vσtx=12xTPσtx.Considering the parameter uncertainty, we can getΔf1(t,x)≤θ1e-tx, Δf2(t,x)≤θx,θ∈maxθ2,θ3. Thus, ∫t0tς(τ)dτ≤∫t0t(θ+θ1e-t)dτ≤κ(t-t0)+η, where κ=0.2,η=1.To be sure,i∈M=1,2, which is the same in hereinafter. It is easy to verify that (23) holds with ri=diag(1,1,1). With the way of Step 2 of the PS algorithm, it gives a1=1/2, a2=2, a3=18, μ=4, which render that (5), (6), and (30) hold.In the following subsection, the parameterized controllers will be designed and the interval value of their parameters will be solved with the PS algorithm described in Section3.2.ByP1 and P2, each subsystem can be transformed into the form of (12), where(50)J1x=0-2-2x22002x200,R1x=920003200092,J2x=030-30x220-x220,R2x=-32000-12000-12.By substituting the values of all the parameters into (31), the conditions f10(α1)>0 (see (54)) and f20(α2)>0 (see (55)) are obtained.By Step1 of the PS algorithm, define the following parameter matrix: (51)Kix,km=diagki,ki,ki.Then, by (25) and (51), the parameterized controllers for the subsystems of Example 18 are obtained as(52)u1k1,x=-8k1+1x2-k1+12x1+4x2+x3-k1+10.5x1+4x2+x3,u2k2,x=-6k2+1x2-k2+1x1+2x2+4x3-k2+13x1+4x2+2x3.Hence, by the values ofγ, R1(x), R2(x), and (51), with the way of Steps 2 and 3, we have(53)P^1=x22+174k1+647144-12α110k152k1∗96k1+2159-2α18k1∗∗2k1+647144-12α1,P^2=10k2-217144-12α214k210k2∗56k2-7336-α216k2∗∗20k2-7336-α2,where “∗” means the symmetric terms in the symmetric matrix P^i.According to the symmetric positive definiteness ofP^i, the constraint conditions fis>0,s∈{1,2,3} are obtained with the way of Step  (3.4) of the PS algorithm.(54)
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1
2
α
1
3
-
445
2
k
1
2
α
1
+
121
4
α
1
2
k
1
+
36229
18
k
1
2
-
161899
288
k
1
α
1
+
359
24
α
1
2
>
0
,
(55)
C
2
=
f
20
=
α
2
-
11.57
>
0
f
21
=
10
k
2
-
217
144
-
1
2
α
2
>
0
f
22
=
364
k
2
2
-
314
3
k
2
-
38
k
2
α
2
+
15841
5184
+
121
48
α
2
+
1
2
α
2
2
>
0
f
23
=
3600
k
2
3
-
20186
9
k
2
2
-
896
k
2
2
α
2
+
354269
1296
k
2
+
8357
36
k
2
α
2
+
48
k
2
α
2
2
-
1156393
186624
-
10585
1296
α
2
-
509
144
α
2
2
-
1
2
α
2
3
>
0
.Decompose theα1-k1 plane into cells by drawing all the curves of f1j=0,j∈0,1,2,3 in the plane and search out the required cell RC1 with the method of Step (4.2) of the PS algorithm. The required cell RC1 marked by gray background is surrounded by the curves of f10=0 and f13=0 in Figure 1. Following the above similar procedures, obtain the RC2 that is partly surrounded by the curves of f20=0 and f23=0 in Figure 2. Then, considering RCi and (54) and (55), we know that k1∈[3.61,k~1],k~1≥15 and k2∈[2.54,k~2],k~2≥15. Choose(56)k1=5k2=5.Figure 1
System constraint curves ink1-α1 plane.Figure 2
System constraint curves ink2-α2 plane.Byτα=1, for i,j∈M, design the following switching law:(57)σt=1,t∈2k+0.2,2k+1.2,k∈N2,t∈2k+1.2,2k+2.2ort∈0,0.2,k∈N.Obtain a group of controllers by substituting (56) into (52).Suppose initial statesx(t0)=-41-7T and external disturbance w=0. Let θ1=cos⁡(t), θ2=0.1sin⁡t, and θ3=0.2sin⁡t+1. Simulate Example 18 under switching law (57) and parameterized controller (52) with parameter (56). Figure 3 (dotted lines) shows the state responses and Figure 4 (dotted lines) shows the partial enlargement of the state responses.Figure 3
State response of case 1 (dotted lines) and the method of [22] (solid lines).Figure 4
Partial enlargement of the state response.Whenx(t0)=000T, suppose external disturbance w=e-te-te-tT is inserted into Example 18. To clearly see the disturbance attenuation capabilities of the controllers designed with the method of this paper, define the function(58)zt=∫t0texp⁡-ln⁡μξ/τατ-t0yTτyτdτ∫t0twTτwτdτto describe the L2 disturbance attenuation gain level of the system from w(t) to y(t). Simulate the system with the switching law (57) and the parameterized controller (52) with parameter (56). The simulation results are showed by Figure 5 (dotted line). In order to verify the tuning characteristic of the parameter containing in the parameterized controller, choose(59)k1=10k2=10,(60)k1=15k2=15,respectively, and identify them as cases 2 and 3, respectively. Meanwhile, identify the previous case as case 1. Simulate the example of case 2 and case 3, respectively, again. The curves of z(t) are showed, respectively, with dot and dash line (case 2) and dash line (case 3) in Figure 5.Figure 5
The curves ofz(t) with different parameter values.Summary 1.
According to Figure5, we know that the disturbance attenuation capability of the parameterized controller is more and more bigger as the values of the parameters increase, which means that the optimized controller can be obtained by choosing larger values for the parameter. The simulation result is in excellent agreement with Remark 15. In other words, when choosing the parameter values according to RCi, the three aims of the parameterized controller described in Remarks 11 and 14 are achieved. When choosing larger parameter values according to RCi for the parameterized controller (52), Example 18 has better L2 gain performance from w to y.


### 4.2. Solve Example18 with the Method of [22]

In order to verify the effectiveness of the method of Theorem13, Example 18 is simulated with the method of Theorem  2 of [22]. At first, it will be verified that the method of [22] is suitable for Example 18.As with the method of this paper, choose the piecewise Lyapunov functionV(x)=Vσ(t)(x) as (49), where V1(x)=0.5x1+2x2+0.5x3 and V1(x)=0.5x1+x2+x3. It is easy to verify that the 1th subsystem is inherently passive and the 2th subsystem is inherently nonpassive with Lf2V2(x)≤3V2(x). Besides, considering g11x=diag(1,1,1), g21x=diag(1,1,1), we know that Example 18 has the structure of system (1) of [22].On the other hand, byVi(x), it is easy to obtain the values of the parameters that is a1=0.5,a2=2,a3=18,μ=4,κ=0.2,η=1,λ=3 and verify that conditions (12)–(16) for Theorem  2 of [22] hold. Noticing that V˙1(x)=∂V1(x)/∂xf1(x)≤xT-4.5000-24000-4.5x≤-9V1(x), by (12) of [22], it gives that x(t+τ)≤2e-9/2τx(t), which means c=2 and λ-=9/2. For δ=0.5, τ=0.0006, it is easy to verify that the 1th subsystem is exponentially small-time norm-observable with degree λ--cκ=4.1. Then, it is easy to verify that a2c2≤a1-k0exp⁡a3η/a1 and h1(x(t+s))2≤k0/τexp⁡-2λ-τx(t)2 for any s∈[0,0.0006], where k0=0.01 and t satisfies h1(x(t+s))≤0.5 for all s∈[0,0.0006]. Let the average dwell time τa=1 and passivity rate r=0.5, the disturbance attenuation level γ=6 and λ2=0.05. Then, λ1=λ2+a32/4a1γ2=0.3, λ∗=λ1/r+λ/r+ln⁡μ/rτa+a3κ/ra1-λ=9.7667 and k=λ--cκ-1/2(λ∗-a3κ/a1-a32/4a1γ2)=0.1902>0. By the obtained parameters, design controller (61)(61)u1x=4.8348x12+4x22+x32Lx+1-8x2-2x1-4x2-x3-0.5x1-4x2-x3Lx>0.250Lx≤0.25,u2x=-6x2-x1-2x2-4x3-3x1-4x2-2x3,where L(x)=64x22+(2x1+4x2+x3)2+(0.5x1+4x2+x3)2.Then, according to Theorem2 of [22], under controller (61) and switching law (57), the system of Example 18 is globally robustly exponentially stable when w≡0 and has L2 gain disturbance attenuation level γ from w to y.Summary 2.
Comparing between (52) and (61), we know that the controllers designed with the method of this paper is simpler than that designed with the method of [22].
Letθ1=cos(t), θ2=0.1sin⁡t, θ3=0.2sin⁡t+1. With the same initial states x(t0)=-41-7T, simulate Example 18 under controller (61) and switching law (57). Figure 3 (solid lines) shows the state responses and Figure 4 (solid lines) shows the partial enlargement of the state responses.Summary 3.
According to Figures3 and 4, the system controlled with the method of this paper became stable for no more than 1.5 s (see the dotted lines in Figures 3 and 4), while the system controlled with the method of [22] became stable for over 2.5 s (see the solid lines in Figures 3 and 4). Overall, the controller designed using Theorem 13 of this paper possesses a better robustness compared with the existing approach [22].
Whenx(t0)=000T, suppose external disturbance w=e-te-te-tT is inserted into Example 18 under controller (61) and switching law (57). Example 18 is simulated and the disturbance attenuation L2 gain from w to y is showed by Figure 6 (solid line).Figure 6
The curves ofz(t) with the method of this paper in case 1 and [22].Summary 4.
With the same external disturbance andL2 gain disturbance attenuation level γ, Figure 6 illustrates that the controllers designed using Theorem 13 of this paper provide better disturbance attenuation capability compared with the controller using the existing approach described in [22].


## 4.1. Solve Example18 with the Method of This Paper

LetM={1,2}, P1=diag(1,4,1), and P2=diag(1,2,2). Design the following piecewise Lyapunov function: (49)Vx=Vσtx=12xTPσtx.Considering the parameter uncertainty, we can getΔf1(t,x)≤θ1e-tx, Δf2(t,x)≤θx,θ∈maxθ2,θ3. Thus, ∫t0tς(τ)dτ≤∫t0t(θ+θ1e-t)dτ≤κ(t-t0)+η, where κ=0.2,η=1.To be sure,i∈M=1,2, which is the same in hereinafter. It is easy to verify that (23) holds with ri=diag(1,1,1). With the way of Step 2 of the PS algorithm, it gives a1=1/2, a2=2, a3=18, μ=4, which render that (5), (6), and (30) hold.In the following subsection, the parameterized controllers will be designed and the interval value of their parameters will be solved with the PS algorithm described in Section3.2.ByP1 and P2, each subsystem can be transformed into the form of (12), where(50)J1x=0-2-2x22002x200,R1x=920003200092,J2x=030-30x220-x220,R2x=-32000-12000-12.By substituting the values of all the parameters into (31), the conditions f10(α1)>0 (see (54)) and f20(α2)>0 (see (55)) are obtained.By Step1 of the PS algorithm, define the following parameter matrix: (51)Kix,km=diagki,ki,ki.Then, by (25) and (51), the parameterized controllers for the subsystems of Example 18 are obtained as(52)u1k1,x=-8k1+1x2-k1+12x1+4x2+x3-k1+10.5x1+4x2+x3,u2k2,x=-6k2+1x2-k2+1x1+2x2+4x3-k2+13x1+4x2+2x3.Hence, by the values ofγ, R1(x), R2(x), and (51), with the way of Steps 2 and 3, we have(53)P^1=x22+174k1+647144-12α110k152k1∗96k1+2159-2α18k1∗∗2k1+647144-12α1,P^2=10k2-217144-12α214k210k2∗56k2-7336-α216k2∗∗20k2-7336-α2,where “∗” means the symmetric terms in the symmetric matrix P^i.According to the symmetric positive definiteness ofP^i, the constraint conditions fis>0,s∈{1,2,3} are obtained with the way of Step  (3.4) of the PS algorithm.(54)
C
1
=
f
10
=
α
1
-
11.57
>
0
f
11
=
647
144
k
1
+
17
4
-
1
2
α
1
>
0
f
12
=
19183
36
k
1
-
113
2
k
1
α
1
+
139105
1296
+
308
k
1
2
+
α
1
2
-
1507
72
α
1
>
0
f
13
=
-
170161
1152
α
1
+
13524241
5184
k
1
+
90000935
186624
+
144
k
1
3
-
1
2
α
1
3
-
445
2
k
1
2
α
1
+
121
4
α
1
2
k
1
+
36229
18
k
1
2
-
161899
288
k
1
α
1
+
359
24
α
1
2
>
0
,
(55)
C
2
=
f
20
=
α
2
-
11.57
>
0
f
21
=
10
k
2
-
217
144
-
1
2
α
2
>
0
f
22
=
364
k
2
2
-
314
3
k
2
-
38
k
2
α
2
+
15841
5184
+
121
48
α
2
+
1
2
α
2
2
>
0
f
23
=
3600
k
2
3
-
20186
9
k
2
2
-
896
k
2
2
α
2
+
354269
1296
k
2
+
8357
36
k
2
α
2
+
48
k
2
α
2
2
-
1156393
186624
-
10585
1296
α
2
-
509
144
α
2
2
-
1
2
α
2
3
>
0
.Decompose theα1-k1 plane into cells by drawing all the curves of f1j=0,j∈0,1,2,3 in the plane and search out the required cell RC1 with the method of Step (4.2) of the PS algorithm. The required cell RC1 marked by gray background is surrounded by the curves of f10=0 and f13=0 in Figure 1. Following the above similar procedures, obtain the RC2 that is partly surrounded by the curves of f20=0 and f23=0 in Figure 2. Then, considering RCi and (54) and (55), we know that k1∈[3.61,k~1],k~1≥15 and k2∈[2.54,k~2],k~2≥15. Choose(56)k1=5k2=5.Figure 1
System constraint curves ink1-α1 plane.Figure 2
System constraint curves ink2-α2 plane.Byτα=1, for i,j∈M, design the following switching law:(57)σt=1,t∈2k+0.2,2k+1.2,k∈N2,t∈2k+1.2,2k+2.2ort∈0,0.2,k∈N.Obtain a group of controllers by substituting (56) into (52).Suppose initial statesx(t0)=-41-7T and external disturbance w=0. Let θ1=cos⁡(t), θ2=0.1sin⁡t, and θ3=0.2sin⁡t+1. Simulate Example 18 under switching law (57) and parameterized controller (52) with parameter (56). Figure 3 (dotted lines) shows the state responses and Figure 4 (dotted lines) shows the partial enlargement of the state responses.Figure 3
State response of case 1 (dotted lines) and the method of [22] (solid lines).Figure 4
Partial enlargement of the state response.Whenx(t0)=000T, suppose external disturbance w=e-te-te-tT is inserted into Example 18. To clearly see the disturbance attenuation capabilities of the controllers designed with the method of this paper, define the function(58)zt=∫t0texp⁡-ln⁡μξ/τατ-t0yTτyτdτ∫t0twTτwτdτto describe the L2 disturbance attenuation gain level of the system from w(t) to y(t). Simulate the system with the switching law (57) and the parameterized controller (52) with parameter (56). The simulation results are showed by Figure 5 (dotted line). In order to verify the tuning characteristic of the parameter containing in the parameterized controller, choose(59)k1=10k2=10,(60)k1=15k2=15,respectively, and identify them as cases 2 and 3, respectively. Meanwhile, identify the previous case as case 1. Simulate the example of case 2 and case 3, respectively, again. The curves of z(t) are showed, respectively, with dot and dash line (case 2) and dash line (case 3) in Figure 5.Figure 5
The curves ofz(t) with different parameter values.Summary 1.
According to Figure5, we know that the disturbance attenuation capability of the parameterized controller is more and more bigger as the values of the parameters increase, which means that the optimized controller can be obtained by choosing larger values for the parameter. The simulation result is in excellent agreement with Remark 15. In other words, when choosing the parameter values according to RCi, the three aims of the parameterized controller described in Remarks 11 and 14 are achieved. When choosing larger parameter values according to RCi for the parameterized controller (52), Example 18 has better L2 gain performance from w to y.


## 4.2. Solve Example18 with the Method of [22]

In order to verify the effectiveness of the method of Theorem13, Example 18 is simulated with the method of Theorem  2 of [22]. At first, it will be verified that the method of [22] is suitable for Example 18.As with the method of this paper, choose the piecewise Lyapunov functionV(x)=Vσ(t)(x) as (49), where V1(x)=0.5x1+2x2+0.5x3 and V1(x)=0.5x1+x2+x3. It is easy to verify that the 1th subsystem is inherently passive and the 2th subsystem is inherently nonpassive with Lf2V2(x)≤3V2(x). Besides, considering g11x=diag(1,1,1), g21x=diag(1,1,1), we know that Example 18 has the structure of system (1) of [22].On the other hand, byVi(x), it is easy to obtain the values of the parameters that is a1=0.5,a2=2,a3=18,μ=4,κ=0.2,η=1,λ=3 and verify that conditions (12)–(16) for Theorem  2 of [22] hold. Noticing that V˙1(x)=∂V1(x)/∂xf1(x)≤xT-4.5000-24000-4.5x≤-9V1(x), by (12) of [22], it gives that x(t+τ)≤2e-9/2τx(t), which means c=2 and λ-=9/2. For δ=0.5, τ=0.0006, it is easy to verify that the 1th subsystem is exponentially small-time norm-observable with degree λ--cκ=4.1. Then, it is easy to verify that a2c2≤a1-k0exp⁡a3η/a1 and h1(x(t+s))2≤k0/τexp⁡-2λ-τx(t)2 for any s∈[0,0.0006], where k0=0.01 and t satisfies h1(x(t+s))≤0.5 for all s∈[0,0.0006]. Let the average dwell time τa=1 and passivity rate r=0.5, the disturbance attenuation level γ=6 and λ2=0.05. Then, λ1=λ2+a32/4a1γ2=0.3, λ∗=λ1/r+λ/r+ln⁡μ/rτa+a3κ/ra1-λ=9.7667 and k=λ--cκ-1/2(λ∗-a3κ/a1-a32/4a1γ2)=0.1902>0. By the obtained parameters, design controller (61)(61)u1x=4.8348x12+4x22+x32Lx+1-8x2-2x1-4x2-x3-0.5x1-4x2-x3Lx>0.250Lx≤0.25,u2x=-6x2-x1-2x2-4x3-3x1-4x2-2x3,where L(x)=64x22+(2x1+4x2+x3)2+(0.5x1+4x2+x3)2.Then, according to Theorem2 of [22], under controller (61) and switching law (57), the system of Example 18 is globally robustly exponentially stable when w≡0 and has L2 gain disturbance attenuation level γ from w to y.Summary 2.
Comparing between (52) and (61), we know that the controllers designed with the method of this paper is simpler than that designed with the method of [22].
Letθ1=cos(t), θ2=0.1sin⁡t, θ3=0.2sin⁡t+1. With the same initial states x(t0)=-41-7T, simulate Example 18 under controller (61) and switching law (57). Figure 3 (solid lines) shows the state responses and Figure 4 (solid lines) shows the partial enlargement of the state responses.Summary 3.
According to Figures3 and 4, the system controlled with the method of this paper became stable for no more than 1.5 s (see the dotted lines in Figures 3 and 4), while the system controlled with the method of [22] became stable for over 2.5 s (see the solid lines in Figures 3 and 4). Overall, the controller designed using Theorem 13 of this paper possesses a better robustness compared with the existing approach [22].
Whenx(t0)=000T, suppose external disturbance w=e-te-te-tT is inserted into Example 18 under controller (61) and switching law (57). Example 18 is simulated and the disturbance attenuation L2 gain from w to y is showed by Figure 6 (solid line).Figure 6
The curves ofz(t) with the method of this paper in case 1 and [22].Summary 4.
With the same external disturbance andL2 gain disturbance attenuation level γ, Figure 6 illustrates that the controllers designed using Theorem 13 of this paper provide better disturbance attenuation capability compared with the controller using the existing approach described in [22].


## 5. Conclusions

In this paper, the exponential feedback passivity of switched polynomial nonlinear systems is presented. Comparing with the existing results of [25–27], the nonlinear functions of the systems do not need to satisfy the so-called Lipschitz condition. In addition, the performance of the SPNSs can be improved by adjusting the value of the parameter contained in the parameterized controller according to the obtained feasible interval solution of the parameters.



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*Source: 6283875-2018-05-13.xml*