# On Homogeneous Parameter-Dependent Quadratic Lyapunov Function for RobustH∞ Filtering Design in Switched Linear Discrete-Time Systems with Polytopic Uncertainties

Authors: Guochen Pang; Kanjian Zhang
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

This paper is concerned with the problem of robustH∞ filter design for switched linear discrete-time systems with polytopic uncertainties. The condition of being robustly asymptotically stable for uncertain switched system and less conservative H∞ noise-attenuation level bounds are obtained by homogeneous parameter-dependent quadratic Lyapunov function. Moreover, a more feasible and effective method against the variations of uncertain parameter robust switched linear filter is designed under the given arbitrary switching signal. Lastly, simulation results are used to illustrate the effectiveness of our method.


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## Body

## 1. Introduction

Switched systems are a class of hybrid systems that consist of a finite number of subsystems and a logical rule orchestrating switching between the subsystems. Since this class of systems has numerous applications in the control of mechanical systems, the automotive industry, aircraft and air traffic control, switching power converters, and many other fields, the problems of stability analysis and control design for switched systems have received wide attention during the past two decades [1–15]. Reference [2] proposed the H∞ weight learning law for switched Hopfield neural networks with time-delay under parametric uncertainty. Reference [8] dealt with the delay-dependent exponentially convergent state estimation problem for delayed switched neural networks. Some criteria for exponential stability and asymptotic stability of a class of nonlinear hybrid impulsive and switching systems have been established using switched Lyapunov functions in [9]. Reference [14] investigated the problem of designing a switching compensator for a plant switching amongst a family of given configurations.On the other hand, within robust control theory scheme, theH∞ noise-attenuation level is an important index for the influence of external disturbance on system stability [16–19]. However, the uncertainties which generally exist in many practical plants and environments may result in significant changes in robust H∞ noise-attenuation level. In order to suppress the conservativeness, many new methods have been considered. Among these methods, homogeneous polynomial parameter-dependent quadratic Lyapunov function is one of the most effective methods. The main feature of these functions is that they are quadratic Lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form. It is firstly introduced to study robust stability of polynomial systems in [20]. Most results have been presented in [21–25]. In [19], homogeneous parameter-dependent quadratic Lyapunov functions were used to establish tightness in robust H∞ analysis. Reference [21] presented some general results concerning the existence of homogeneous polynomial solutions to parameter-dependent linear matrix inequalities whose coefficients are continuous functions of parameters lying in the unit simplex. Reference [23] investigated the problems of checking robust stability and evaluating robust H2 performance of uncertain continuous-time linear systems with time-invariant parameters lying in polytropic domains. Reference [25] introduced Gram-tight forms, that is, forms whose minimum coincides with the lower bound provided by LMI optimizations based on SOS (sum of squares of polynomials) relaxations.Thus, in order to suppress the influence of uncertainties on the system’s robustH∞ control, the homogeneous parameter-dependent quadratic Lyapunov functions are used to design robust H∞ filter in this paper. We first consider the system’s anti-interference of external disturbance. Along this direction, the robust H∞ filters for switched linear discrete-time systems are designed. Lastly, through the comparison, we have the fact that homogeneous parameter-dependent quadratic Lyapunov functions can suppress conservativeness which the uncertainties bring.The rest of this paper is organised as follows. We state the problem formulation in Section2. The main results are presented in Section 3. Section 4 illustrates the obtained result by numerical examples, which is followed by the conclusion in Section 5.Notation.
R
n denotes the n-dimension Euclidean space and Rn×m is the real matrices with dimension n×m; R0n means Rn/{0}; the notation X≥Y (resp., X>Y), where X and Y are symmetric matrices, represents the fact that the matrix X-Y is positive semidefinite (resp., positive definite); AT denotes the transposed matrix of A; sq(x) represents (x12,…,xn2) with x∈Rn; he(X) means X+XT with X∈Rn×n; x⊗y denotes the Kronecker product of vectors x and y. ∥·∥ denotes Euclidean norm for vector or the spectral norm of matrices.


## 2. Problem Statement

Consider a class of uncertain switched linear discrete-time systems which were given in [1]:
(1)xk+1=Aiλxk+Biλωk,yk=Ciλxk+Diλωk,zk=Hiλxk+Liλωk,
where x(k)∈Rn is state vector, ω(k)∈Rl is disturbance input which belongs to l2[0,+∞), y(k) is the measurement output, z(k) is objective signal to be attenuated, and i is switching rule, which takes its value in the finite set Π∶={1,…,N}. As in [1], the switching signal i is unknown a priori, but its instantaneous value is available in real time. As an arbitrary discrete time k, the switching signal i is dependent on k or x(k) or both or other switching rules. λ is an uncertain parameter vector supposed to satisfy λ∈Λ={λj≥0,∑j=1s‍=1}. The vector λ represents the time-invariant parametric uncertainty which affects linearly the system dynamics. The vector λ can take any value in Λ, but it is known to be constant in time.The matrices of each subsystem have appropriate dimensions and are assumed to belong to a given convex-bounded polyhedral domain described bys vertices in the ith subsystem, that is,
(2)AiλBiλCiλDiλHiλLiλ∈Γi,
where
(3)Γi≔∑j=1s‍λjAijBijCijDijHijLij;λj≥0∑j=1s‍λjAiλBiλCiλDi(λ)HiλLiλ∑j=1s‍aaλj=∑j=1s‍λjAijBijCijDijHijLij;∑j=1s‍aaλjλj≥0∑j=1s‍λj=1.Hence, we are interested in designing an estimator or filter of the form(4)xfk+1=Afixfk+Bfiyk,zfk=Cfixfk+Dfiyk,
where xf(k)∈Rn, zf(k)∈Rq; Aij, Bij, Cij and Dij are the parameterized filter matrices to be determined. The filter with the above structure may be called switched linear filter, in which the switching signal i is also assumed unknown a priori but available in real-time and homogeneous with the switching signal in system (1).Augmenting the model of (1) to include the state of filter (4), we obtain the filtering error system:
(5)xek+1=Aeiλxek+Beiλωk,ek=Ceiλxek+Deiλωk,
where
(6)xek=xkxfkT,ek=zk-zfk,Aei(λ)=Ai(λ)0BfiCi(λ)Afi,Bei(λ)=Bi(λ)BfiDi(λ)T,Cei(λ)=Hi(λ)-DfiCi(λ)-Cfi,Dei(λ)=Li(λ)-DfiDi(λ).Based on [1], the robust H∞ filtering problem addressed in this paper can be formulated as follows: finding a prescribed level of noise attention γ>0 and determining a robust switched linear filter (4) such that the filtering error system is robustly asymptotically stable and
(7)e2<γ2ω2.This problem has received a great deal of attention. In order to get a better level of noise attentionγ>0, we will use homogeneous polynomial functions which have demonstrated nonconservative result for serval problems. In this paper, we extend these methods to design robust switched linear filter.The following preliminaries are given, which are essential for later developments. We firstly recall the homogeneous polynomial function from Chesi et al. [26].Definition 1.
The functionh:Rn→R is a form of degree d in n scalar variables if
(8)hx=∑q∈Ln,s‍aqxq,
where Ln,s={q∈Nn:∑i=1n‍qi=s} and x∈Rn and aq∈R is coefficient of the monomial xq.The set of forms of degrees in n scalar variables is defined as
(9)Ξn,s={h:Rn⟶R:(8)holds}.Definition 2.
The functionf:Rn→R is a polynomial of degree less than or equal to s, in n scalar variables, if
(10)fx=∑i=0s‍hix,
where x∈Rn and hi∈Ξn,i, i=1,…,s.Definition 3.
Consider the vectorx∈Rn, x=[x1,…,xn]T. The power transformation of degree m is a nonlinear change of coordinates that forms a new vector xm of all integer powered monomials of degree m that can be made from the original x vector:
(11)xjm=cjx1mj1x2mj2⋯xnmjn,mji∈1,…,n,∑i=1n‍mji=m,j=1,…,d(n,m),d(n,m)=(n+m-1)!(n-1)!m!.
Usually we takecj=1; then, with m>0,
(12)x1x2⋮xnm=x1x1,x2,x3,…,xnm-1Tx2x2,x3,…,xnm-1T⋮xnxnm-1T;
otherwise
(13)x1x2⋮xnm=1.
For example, n=2, m=2, ⇒d(n,m)=3, and
(14)x=x1x2⟹x2=x12x1x2x22.Definition 4.
The functionM:Rn→Rr×r is a homogeneous parameter-dependent matrix of degree m in n scalar variables if
(15)Mi,j∈Ξn,m,∀i,j=1,…,r.
We denote the set ofr×r homogeneous parameter-dependent matrices of degree m in n scalar variables as
(16)Ξn,m,r#={M:Rn⟶Rr×r:(15)holds}
and the set of symmetric matrix forms as
(17)Ξn,m,r={M∈Ξn,m,r#:M(x)=MT(x)∀x∈Rn}.Definition 5.
LetM∈Ξn,2m,r and H∈Srd(n,m) such that
(18)Mx=ΦH,xm,r,
where Φ(H,xm,r)=(xm⊗Ir)TH(xm⊗Ir). Then (18) is called a square matricial representation (SMR) of M(x) with respect to xm⊗Ir. Moreover, H is called a SMR matrix of M(x) with respect to xm⊗Ir.Lemma 6 (Chesi et al. [26]).
LetM∈Ξn,2m,r. Then H={H∈Srd(n,m):  (18)  holds} is an affine space. Moreover,
(19)HM=H+L:H∈Srdn,msatisfies18,L∈Ln,m,r,
where Ln,m,r is linear space:
(20)Ln,m,r={L∈Srd(n,m):Φ(L,xm,r)=0r×r∀x∈Rn}
whose dimension is given by
(21)ωn,m,r=r2dn,mrdn,m+1-(r+1)d(n,2m).Lemma 7 (Chesi et al. [26]).
LetM∈Ξn,s,r. Then
(22)M(x)>0,∀x∈γq=x∈Rq:xi≥0,∑i=1q‍xi=1
holds if and only if
(23)Msqx>0,∀x∈R0n.Lemma 8 (Boyd et al. [27]).
The linear matrix inequality(24)S=S11S12S12TS22<0,
where S11=S11T and S22=S22T, is equivalent to
(25)S11<0,S22-S12TS11-1S12<0.


## 3. Main Result

In this section, the sufficient condition for existence of robustH∞ filter for uncertain switched systems is formulated. For this purpose, we firstly consider the anti-interference of system (1) for disturbance.Theorem 9.
For a given scalarγ>0. Consider system (1); if there exist matrices Pij>0 and matrices Rij such that
(26)Ψ11(Pi¯j,Rij)0Ψ13(Rij,Aij)Ψ14(Rij,Bij)*-IcΨ23(Cij)Ψ24(Dij)**Ψ33(Pij)0***-γ2Ic+L11L12L13L14*L22L23L24**L33L34***L44<0i,i¯∈∏‍,j∈[1,s]
with
(27)ΦPi¯Ri(λm+1)=∑j=1s‍λjPi¯λm-Riλm-RiTλm;ΦRiAi(λm+1)=Ri(λm)Ai(λ);ΦRiBi(λm+1)=Ri(λm)Bi(λ);ΦCi(λm+1)=∑j=1s‍λjmCi(λ);ΦDi(λm+1)=∑j=1s‍λjmDi(λ);ΦI(λm+1)=∑j=1s‍λjm+1I;ΦPiλm+1=∑j=1s‍λjPi(λm);ΦPi¯Risqλm+1=ϖTλm+1Ψ11(Pi¯j,Rij)ϖ(λm+1);ΦRiAi(sq(λm+1))=ϖT(λm+1)Ψ13(Rij,Aij)ϖ(λm+1);ΦRiBisqλm+1=ϖTλm+1Ψ14Rij,Bijϖλm+1;ΦCisqλm+1=ϖTλm+1Ψ23(Cij)ϖ(λm+1);ΦDi(sq(λm+1))=ϖTλm+1Ψ24Dijϖλm+1;ΦPisqλm+1=ϖTλm+1Ψ33Pijϖλm+1;ΦIsqλm+1=ϖTλm+1Icϖλm+1;ϖ(λm+1)=λm+1⊗I;L11⋯L44∈Ls,m,n,
where Ψ(·) is a matrix which is made up of Aij, Bij, Cij, Dij, Pij, Rij and the set Ls,m,n is defined in Lemma 6, then system (1) is robustly asymptotically stable with H∞ performance γ for any switching signal.Proof.
Consider the following homogeneous parameter-dependent quadratic Lyapunov function:(28)V(k,x(k))=xT(k)Pi(λm)x(k).
Then, along the trajectory of system (1), we have
(29)ΔV=Vk+1,xk+1-Vk,xk=xTkAiTλPi¯λmAiλ-Piλmx(k)+2xTkAiTλPi¯λmBiλω(k)+ωTkBiTλPi¯λmBiλω(k).
Wheni=i¯, the switched system is described by the ith mode. When i≠i¯, it represents the switched system being at the switching times from mode i¯ to mode i.
Before and after multiplying inequality (26) by {λm+1⊗I}T and {λm+1⊗I}, we have(30)ΦPi¯Risqλm+10ΦRiAisqλm+1ΦRiBisqλm+1*-ΦIsqλm+1ΦCisqλm+1ΦDisqλm+1**ΦPisqλm+10***-γ2ΦIsqλm+1<0.From Lemma 7, one has(31)ΦPi¯Ri(λm+1)0ΦRiAi(λm+1)ΦRiBi(λm+1)*-ΦI(λm+1)ΦCi(λm+1)ΦDi(λm+1)**ΦPi(λm+1)0***-γ2ΦI(λm+1)<0which implies(32)Pi¯(λm)-Ri(λm)-RiT(λm)0Ri(λm)Ai(λ)Ri(λm)Bi(λ)*-ICi(λ)Di(λ)**Pi(λm)0***-γ2I<0.
Under the disturbanceω(k)=0, we get
(33)ΔV=xTkAiTλPi¯λmAiλ-Piλmxk.
If
(34)AiTλPi¯λmAiλ-Piλm<0,∀i,i¯∈Π,
then ΔV<0. It implies system (1) is robust asymptotic stable.
In terms of Lemma8, condition (34) is equivalent to
(35)-Pi¯(λm)Pi¯(λm)Ai(λ)*Pi(λm)<0.
Since (26), it follows that
(36)Pi¯(λm)-Ri(λm)-RiT(λm)<0
which implies the matrices Ri(λm) are nonsingular for each i. Then, we have
(37)Pi¯λm-RiλmPi¯-1λmPi¯λm-RiλmT≥0
which is equivalent to
(38)Pi¯λm-Riλm-RiTλm≥-RiλmPi¯-1λmRiTλm.
Hence, it can be readily established that (32) is equivalent to(39)-Ri(λm)Pi¯-1(λm)RiT(λm)0Ri(λm)Ai(λ)*-ICi(λ)**-Pi(λm)***Ri(λm)Bi(λ)Di(λ)0-γ2I<0.Before and after multiplying the above inequality by diag⁡{-Ri-T(λm)Pi¯(λm),I,I,I}, we obtain
(40)-Pi¯(λm)0Pi¯(λm)Ai(λ)Pi¯(λm)Bi(λ)*-ICi(λ)Di(λ)**-Pi(λm)0***-γ2I<0,
which implies that (35) holds. Then, the stability of system (1) can be deduced.
On the other hand, let(41)J=∑k=0∞‍yTkyk-γ2ωTkωk
be performance index to establish the H∞ performance of system (1). When the initial condition x(0)=0, we have Vk,xk|k=0=0 which implies
(42)J<∑k=0∞‍yTkyk-γ2ωTkωk+ΔV=∑k=0∞‍xTkωTkΘ11Θ12Θ12TΘ22xkωk,
where
(43)Θ11=AiTλPi¯λmAiλ-Piλm+CiTλCiλ;Θ12=AiTλPi¯λmBiλ+CiTλDiλ;Θ22=-γ2I+BiTλPi¯λmBiλ+DiTλDiλ.
In terms of Lemma 8, we have J<0 which means that y2<γ2ω2. Then, the proof is completed.Remark 10.
Within robustlyH∞ performance scheme, the basic idea is to get an upper bound of H∞ noise-attenuation level. Since model uncertainties may result in significant changes in H∞ noise-attenuation level, we should effectively reduce this effect. For this purpose, the homogeneous parameter-dependent quadratic Lyapunov function is exploited in Theorem 9.Remark 11.
In Theorem9, Ψ(·) is a matrix which is made up of Aij, Bij, Cij, Dij, Pij, Rij. For example, when n=2 and m=1, condition (26) can be written as
(44)Ψ11(Pi¯j,Rij)=Pi¯1-Ri1T00-Ri1*-Ri1-Ri1T0-Ri2-Ri2T+Pi¯1+Pi¯2**Pi¯2-Ri2-Ri2T,Ψ13(Rij,Aij)=Ri1Ai100*Ri1Ai2+Ri2Ai10**Ri2Ai2,Ψ14(Rij,Bij)=Ri1Bi100*Ri1Bi2+Ri2Bi10**Ri2Bi2,Ψ23(Cij)=Ci100*Ci1+Ci20**Ci2,Ψ24(Cij)=Di100*Di1+Di20**Di2,Ψ33(Pij)=-Pi100*-Pi1-Pi20**-Pi2.Remark 12.
About the calculation of matricesL11⋯L44, the following algorithm is presented in [26]: (1)
choosexm and x2m as in Definition 3 where x∈Rn(2)
setA=0d(n,2m)d(r,2)×3 and b=0 and define the variable α∈Rω(n,m,r)(3)
fori=1,…,d(n,m) and j=1,…,r(4)
setc=r(i-1)+j(5)
fork=1,…,d(n,m) and l=max⁡{1,j+r(i-k)},…,r(6)
setd=r(k-1)+l and f=ind((xm)i(xm)k,x2m)(7)
setg=ind(yjyl,y2) and a=fdr,2+g and Aa,1=Aa,1+1(8)
ifAa,1=1(9)
setAa,2=c and Aa,3=d(10)
else(11)
setb=b+1 and G=0rd(n,m)×rd(n,m) and Gc,d=1(12)
seth=Aa,2 and p=Aa,3 and Gh,p=Gh,p-1(13)
setL=L+αbG(14)
endif(15)
endfor(16)
endfor(17)
setL=0.5he(L).
Next, the condition for robustH∞ filter is formulated. For convenience, we define n=2, m=1, and s=2.Theorem 13.
Given a constantγ>0, if there exist matrices P2ij, yi, ζij, A¯fi, B¯fi, C¯fi, D¯fi, xij and positive definite matrices P1ij,P3ij and scalar εi such that
(45)Λ11Λ120Λ14Λ15Λ16*Λ220Λ24Λ25Λ26**Λ33Λ34Λ35Λ36***Λ44Λ450****Λ550*****Λ66+L11L12L13L14L15L16*L22L23L24L25L26**L33L34L35L36***L44L45L46****L55L56*****L66<0
with
(46)Λ11=P1i¯1-xi100-xi1T0-xi1-xi1T0-xi2-xi2T+P1i¯1+P1i¯200P1i¯2-xi2-xi2T,Λ12=P2i1-εiyi00-ζi10P2i1+P2i20-ζi1-ζi2-2εiyi00P2i1-εiyi-ζi2,Λ14=xi1Ai100+εiB¯fiCi10xi1Ai2+εiB¯fiCi10+xi2Ai1+εiB¯fiCi200xi2Ai2+εiB¯fiCi2,Λ15=εiA¯fi0002εiA¯fi000εiA¯fi,Λ16=xi1Bi100+εiB¯fiDi10xi1Bi2+εiB¯fiDi10+xi2Bi1+εiB¯fiDi200xi2Ai2+εiB¯fiDi2,Λ22=P3i1-yi00-yiT0P3i1+P3i20-2yi-2yiT00P3i2-yi-yiT,Λ24=ζi1Ai100+B¯fiCi10ζi1Ai2+B¯fiCi10+ζi2Ai1+B¯fiCi200ζi2Ai2+B¯fiCi2,Λ25=A¯fi0002A¯fi000A¯fi,Λ26=ζi1Bi100+B¯fiDi10ζi1Bi2+B¯fiDi10+ζi2Bi1+B¯fiDi200ζi2Ai2+B¯fiDi2,Λ33=-I1×1000-2I1×1000-I1×1,Λ34=Hi1-D¯fiCi1000Hi1-D¯fiCi10+Hi2-D¯fiCi200Hi2-D¯fiCi2,Λ35=-C¯fi000-2C¯fi000-C¯fi,Λ36=Li1-D¯fiDi1000Li1-D¯fiDi10+Li2-D¯fiDi200Li2-D¯fiDi2,Λ44=-P1i1000-P1i1-P1i2000-P1i2,Λ45=-P2i1000-P2i1-P2i2000-P2i2,Λ55=-P3i1000-P3i1-P3i2000-P3i2,Λ66=-γ2I1×1000-2γ2I1×1000-γ2I1×1,L11⋯L66∈L2,2,2,
then there exists a robust switched linear filter in form of (4) such that, for all admissible uncertainties, the filter error system (5) is robustly asymptotically stable and performance index holds for any nonzero ω∈l2[0,∞), where Afi=yi-1A¯fi, Bfi=yi-1B¯fi, Cfi=C¯fi, and Dfi=D¯fi.Proof.
Let(47)Piλ=P1iλP2iλ*P3iλ,Riλ=xiλεiyiζiλyi,
where
(48)P1isqλ=λ⊗ITP1i100P1i2λ⊗I,P2isqλ=λ⊗ITP2i100P2i2λ⊗I,P3isqλ=λ⊗ITP3i100P3i2λ⊗I,xisqλ=λ⊗ITxi100xi2λ⊗I,ζisqλ=λ⊗ITζi100ζi2λ⊗I.
From Theorem 9, the filter error system is robustly asymptotically stable with a prescribed H∞ noise-attenuation level bound γ if the following matrix inequality holds: (49)P1iλ-xiλP2iλ-εiyi0xiλAiλεiA¯fixiλBiλ-xiλT-ζiλ+εiB¯fiCiλ+εiB¯fiDiλ*P3i-yi0ζiλAiλA¯fiζiλBiλ-yiT+B¯fiCiλ+B¯fiDiλ**-IHiλC¯fiLiλ-D¯fiCiλ-D¯fiDiλ***-P1iλ-P2iλ0****-P3iλ0*****-γ2I<0.
By (48) and Lemma 7, one can obtain that inequality (45) is equivalent to (49). Thus, if (45) holds, the filter error system is robustly asymptotically stable with an H∞ noise-attenuation level bound γ>0. Then, the proof is completed.


## 4. Examples

The following example exhibits the effectiveness and applicability of the proposed method for robustH∞ filtering problems with polytopic uncertainties.Consider the following uncertain discrete-time switched linear system (1) consisting of two uncertain subsystems which is given in [1]. There are two groups of vertex matrices in subsystem 1:
(50)A11=ρ0.820.10-0.060.77,B11=ρ00.1,C11=ρ10,D11=ρ,H11=ρ10,L11=0,A12=ρ0.820.10-0.06-0.75,B12=ρ0-0.1,C12=ρ10.2,D12=0.8ρ,H12=ρ10,L12=0
and two groups of vertex matrices in subsystem 2:
(51)A21=ρ0.820.06-0.100.77,B21=ρ0.10,C21=ρ0-1,D21=-ρ,H21=ρ10,L21=0,A22=ρ0.820.06-0.10-0.75,B22=ρ-0.10,C22=ρ0.2-1,D22=-0.8ρ,H22=ρ10,L22=0.
Moreover, we define the disturbance:
(52)ω(k)=0.001e-0.003ksin(0.002πk).Firstly, consider the problem of being robustly asymptotically stable with anH∞ noise-attenuation for the uncertain switched system which was given in Theorem 9. The different minimum H∞ noise-attenuation level bounds γ can be obtained by different methods. Table 1 lists the different calculation results. From Table 1, it can be clearly seen that the results which are gotten by homogeneous parameter-dependent quadratic Lyapunov functions are better.Table 1
Different minimumγ for uncertain switched system.
ρ
1
1.1
1.2
[1]
1.2799
1.6985
6.2048
Theorem9
1.2799
1.6984
4.0988In addition, for givenρ=1.2 and initial condition x(0)=[0.1,-0.3]T, Figures 1 and 2 show system (1) is robustly asymptotically stable with an H∞ noise-attenuation level bound γ=4.0988.Figure 1
The states responses corresponding to uncertain parametersλ1=0.4 and λ2=0.6.Figure 2
H
∞ noise-attenuation level bound γ=4.0988.Next, we consider the problem of robustH∞ filtering. In order to get H∞ noise-attenuation level bound γ, we define ε1=ε2=1. By solving the corresponding convex optimization problems in Theorem 13, we obtain the minimum H∞ noise-attenuation level bounds γ. Table 2 lists our results and [1]’s results. Moreover, when ρ=1.2, the admissible filter parameters can be obtained according to Theorem 13 as
(53)Af1=0.87700.0505-0.73340.3356,Bf1=-0.0852-0.1077,Cf1=-1.20570.0112,Af2=1.14570.6743-1.2492-0.8883,Bf1=-0.25930.5929,Cf2=-0.8695-0.1525,Df1=-0.0073,Df2=-0.0045.
By comparison, using homogeneous parameter-dependent quadratic Lyapunov function for the existence of a robust switched linear filter in Theorem 13 is better.Table 2
Different minimumγ for robust switched linear filter.
ρ
1
1.1
1.2
[1]
0.5375
1.1414
4.4493
Theorem13
0.5148
1.0826
4.2892By givingH∞ noise-attenuation level bound γ=4.2892 and initial condition xe(0)=[0.1-0.300]T, Figure 3 shows the filtering error system is robustly asymptotically stable and Figure 4 shows the error response of the resulting filtering error system by applying above filter. It is clear that the method in Theorem 13 is feasible and effective against the variations of uncertain parameter.Figure 3
The states responses corresponding to uncertain parametersλ1=0.4 and λ2=0.6.Figure 4
Filtering error response corresponding to uncertain parametersλ1=0.4 and λ2=0.6.


## 5. Conclusions

In this paper, the problems of being robustly asymptotically stable with anH∞ noise-attenuation level bounds γ and switched linear filter design for uncertain switched linear system are studied by homogeneous parameter-dependent quadratic Lyapunov functions. By using this method, the less conservative H∞ noise-attenuation level bounds are obtained. Moreover, we also get a more feasible and effective method against the variations of uncertain parameter under the given arbitrary switching signal. Numerical examples illustrate the effectiveness of our method.



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*Source: 271765-2015-02-15.xml*