# Improved RobustH∞ Filtering Approach for Nonlinear Systems

Authors: Jun Chen; Haiqiao Sun
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

An improved design approach of robustH∞ filter for a class of nonlinear systems described by the Takagi-Sugeno (T-S) fuzzy model is considered. By introducing a free matrix variable, a new sufficient condition for the existence of robust H∞ filter is derived. This condition guarantees that the filtering error system is robustly asymptotically stable and a prescribed H∞ performance is satisfied for all admissible uncertainties. Particularly, the solution of filter parameters which are independent of the Lyapunov matrix can be transformed into a feasibility problem in terms of linear matrix inequalities (LMIs). Finally, a numerical example illustrates that the proposed filter design procedure is effective.


---

## Body

## 1. Introductions

In recent years, when the external disturbance and the statistical properties of the measurement noise are unknown, usingH∞ filtering approach to estimate the states of a linear system becomes one of the focuses on the estimated theoretical research, and some useful research results [1–4] are obtained. However, how to design an effective filter for nonlinear systems is still a very difficult problem. Over the past two decades, there has been a rapidly growing interest in fuzzy control of nonlinear systems. In particular, the fuzzy model proposed by Takagi and Sugeno [5] receives a great deal of attention. And it indicates that this type of fuzzy model has a good approximation performance for the complex nonlinear systems, so some scholars attempt to apply this fuzzy model to designH∞ filter for nonlinear systems. Feng et al. [6] were prior scholars to study the filter for nonlinear systems by using T-S fuzzy model and linear matrix inequality (LMI) techniques. For a class of discrete nonlinear dynamic systems, Tseng and Chen [7] and Pan et al. [8] studied a fuzzyH∞ filtering problem. After that, Tseng [9, 10] and Tian et al. [11] discussed the design problem of robustH∞ fuzzy filter for a class of continuous nonlinear systems. Moreover, the above-obtained results were extended to the fuzzyH∞ filter or robustH∞ filter design for nonlinear systems with time delay [12–15]. In addition,H∞ filtering approach is also applied to Markovian jump systems [16], nonlinear interconnected systems [17], chaotic systems [18], and networked nonlinear systems [19] for the discrete-time case and stochastic systems [20] and singular systems [21] for the continuous-time case. Nevertheless, in the above-mentioned results, the solving process of filter parameters is related to the Lyapunov matrix, which will more or less bring some conservative to the results. The reason is that most of the existence conditions of filter are sufficient conditions; if the Lyapunov matrix cannot be found, then the filter parameters which maybe exist cannot be constructed. For this reason, de Oliveira et al. [22] proposed a novel filter design method by introducing free matrices to the framework of the quadratic Lyapunov function. By means of decoupling the relations between the Lyapunov matrix and the system matrix, the conservative of the results will be reduced. But due to the restrictions of LMI characteristics, this method can only be applied to the discrete systems [17, 23, 24]. Lately, Apkarian et al. [25] extended this idea to the linear continuous systems with the aid of Projection Theorem. And this idea has been used in other fields [26–28]. Unfortunately, to the best of our knowledge, this idea has not yet been introduced to the design of robustH∞ filter for the uncertain continuous nonlinear systems.Taking into account the above-mentioned results, this paper will discuss a new design method of robustH∞ filter for a class of uncertain nonlinear systems. Firstly, the T-S fuzzy model is employed to represent the nonlinear systems. Then, on the basis of the bounded real lemma of continuous systems, a new criterion for the existence of the improved robustH∞ filter is obtained via introducing a free matrix variable. Based on this criterion, the solution of the filter parameters independent of the Lyapunov matrix can be obtained. Combined with the linear matrix inequality techniques, the filter design problem can be transformed into a feasibility problem of a set of linear matrix inequalities. Finally, a simulation example will be given to verify the validity of the proposed method.


## 2. Problem Formulation

Consider a class of uncertain nonlinear systems described by the following T-S fuzzy models.Plant Rulei:
(1)IFv1(t)isMi1and⋯andvp(t)isMipTHENx˙(t)=(Ai+ΔAi(t))x(t)+Biw(t),iiiiiiiiiiiy(t)=(Ci+ΔCi(t))x(t)+Diw(t),iiiiiiiiiiiz(t)=Lix(t),iiiiiiiiiiiiix0=x(0),i=1,2,…,r,
wherex(t)∈ℜnis the state vector,y(t)∈ℜm is the measured output,z(t)∈ℜl is the signal to be estimated, and w(t)∈L2q[0,∞) is the noise signal vector (including process and measurement noises).x0is the initial state condition of the system, which is considered to be known and, without loss of generality, assumed to be zero.v1(t),…,vp(t) are the premise variables, Mij(j=1,2,…,p) is the fuzzy set, and r is the number of IF-THEN rules.Ai, Bi, Ci, Di, and Liare known real constant matrices with appropriate dimensions of the ith subsystem, respectively. The uncertain time-varying matrices ΔAi(t) andΔCi(t)represent the parameter uncertainties in the system model and are assumed to be norm-bounded of the following forms:
(2)ΔAi(t)=EaiF(t)Hai,ΔCi(t)=EciF(t)Hci,iiiiiiiiiiiii=1,2,…,r,
where Eai, Eci, Hai, andHciare known constant matrices of appropriate dimensions, which reflect the structural information of uncertainty, andF(t)is an uncertainty matrix function with Lebesgue measurable elements and satisfies
(3)FΤ(t)F(t)≤I.By using the weighted average method for defuzzification, the uncertain fuzzy dynamic model for the system (1) can be inferred as follows:
(4)x˙(t)=∑i=1rhi(v(t))[(Ai+ΔAi(t))x(t)+Biw(t)],y(t)=∑i=1rhi(v(t))[(Ci+ΔCi(t))x(t)+Diw(t)],z(t)=∑i=1rhi(v(t))Lix(t),
where v(t)=[v1(t),v2(t),…,vp(t)]Τ, and
(5)hi(v(t))=μi(v(t))∑i=1rμi(v(t)),μi(v(t))=∏j=1pMij(vj(t)),iiiiiiiiiiiiiii=1,2,…,r,
in which Mij(vj(t)) is the grade of membership ofvj(t)in the fuzzy setMij, whileμi(v(t))is the grade of membership of the ith rule.In general, it is assumed thatμi(v(t))≥0, i=1,2,…,r,and∑i=1rμi(v(t))>0. Therefore, it is easy to obtain that hi(v(t))≥0, i=1,2,…,r, and∑i=1rhi(v(t))=1.Based on the T-S fuzzy models (1), the full-order filter is constructed as follows.Filter Rulei:
(6)IFv1(t)isMi1and⋯andvp(t)isMipTHENx^˙(t)=Afix^(t)+Bfiy(t),iiiiiiiiiz^(t)=Cfix^(t)+Dfiy(t),i=1,2,…,r,
wherex^(t)∈ℜnis the state vector of filter and z^(t)∈ℜl is an estimate value of the filter output. The matrices Afi, Bfi,Cfi, and Dfi are filter parameters to be determined. Here, it is assumed that the initial condition of filter is x^0=0. Then the whole fuzzy filter can be expressed as
(7)x^˙(t)=∑i=1rhi(v(t))[Afix^(t)+Bfiy(t)],z^(t)=∑i=1rhi(v(t))[Cfix^(t)+Dfiy(t)].Set the state variable asx~(t)=[xΤ(t)x^Τ(t)]Τ and estimated error asz~(t)=z(t)-z^(t). Then the filtering error dynamic equation inferred from formulas (4) and (7) can be described as follows:
(8)x~˙(t)=∑i=1r∑j=1rhi(v(t))hj(v(t))iiiiiiiii×[(A~ij+ΔA~ij(t))x~(t)+B~ijw(t)],z~(t)=∑i=1r∑j=1rhi(v(t))hj(v(t))iiiiiiiii×[(C~ij+ΔC~ij(t))x~(t)+D~ijw(t)],
where
(9)iiiiiiA~ij=[Aj0BfiCjAfi],B~ij=[BjBfiDj],iiiiiiC~ij=[Lj-DfiCj-Cfi],D~ij=-DfiDj,ΔA~ij(t)=[ΔAj(t)0BfiΔCj(t)0],ΔC~ij(t)=[-DfiΔCj(t)0].LetHz~w(s)be the transfer function from the disturbance inputw(t)to the estimation errorz~(t). Then the robustH∞ filter design problem considered in this paper can be described as follows: for a given constantγ>0, find aH∞ full-order filter in the form of (7) so that the filtering error dynamic system (8) is robustly asymptotically stable and theH∞ norm of the transfer functionHz~w(s)is less than the given constantγ; that is, ∥Hz~w(s)∥∞<γis satisfied. Here, constantγis called a prescribedH∞ performance level.For brevity, the functionshi(v(t))will be replaced byhiin the subsequence, andΔAi(t),ΔCi(t),ΔA~ij(t),ΔC~ij(t)will be replaced byΔAi,ΔCi,ΔA~ij,ΔC~ij.


## 3. RobustH∞ Filter Design

According to the bounded real lemma of continuous-time systems, this section firstly gives a sufficient condition for the existence of robustH∞ filter for the uncertain fuzzy system (4). That is, for a given constantγ>0, the filtering error system (8) is robustly asymptotically stable and satisfies∥Hz~w(s)∥∞<γ, if there exists a symmetric positive definite matrixP∈ℜ2n×2n, such that the following matrix inequality holds:(10)∑i=1r∑j=1rhihj[(A~ij+ΔA~ij)ΤP+P(A~ij+ΔA~ij)PB~ij(C~ij+ΔC~ij)ΤB~ijΤP-γ2ID~ijΤC~ij+ΔC~ijD~ij-I]<0.With the aid of the basic idea of [25], an improved robustH∞ filter design method is obtained by introducing a free matrix variable, in which the Lyapunov function matrix and the filtering error system matrix are separated. Then the filter parameters to be determined can be solved independently of the Lyapunov function matrix. This kind of processing method can reduce the conservatism of the results.Theorem 1.
For a given constantγ>0, the filtering error system (8) is robustly asymptotically stable and satisfies∥Hz~w(s)∥∞<γ, if there exist a symmetric positive definite matrixP∈ℜ2n×2nand matrixV∈ℜ2n×2n, such that the following matrix inequality holds:(11)∑i=1r∑j=1rhihj[-(V+VΤ)VΤ(A~ij+ΔA~ij)+PVΤB~ij0VΤ*-P0(C~ij+ΔC~ij)Τ0**-γ2ID~ijΤ0***-I0****-P]<0.Proof.
Rewrite the matrix inequality (11) in the following form:,
(12)∑i=1r∑j=1rhihj[Φ+M~ΤVΤN~+N~ΤVM~]<0,
where
(13)Φ=[0P000*-P0(C~ij+ΔC~ij)Τ0**-γ2ID~ijΤ0***-I0****-P],M~=[I0000],N~=[-IA~ij+ΔA~ijB~ij0I].According to the Projection Theorem [25], inequality (12) is equivalent to the following inequality; that is, (14)∑i=1r∑j=1rhihj[P(A~ij+ΔA~ij)+(A~ij+ΔA~ij)ΤP-PPB~ij(C~ij+ΔC~ij)ΤPB~ijΤP-γ2ID~ijΤ0C~ij+ΔC~ijD~ij-I0P00-P]<0.Applying the Schur complement, it is easy to know that inequality (10) can be deduced from the above inequality. That is to say, inequality (11) is a sufficient condition of the establishment of inequality (10), which can guarantee the filtering error system (8) is robustly asymptotically stable and satisfies the prescribedH∞ performance level.Take into account that inequality (11) is a nonlinear matrix inequality on the matrix variables (P, Afi, Bfi, Cfi, Dfi, i=1,2,…,r), so it is very difficult to solve these variables directly. In this end, the variable substitution method will be utilized in the following derivation to transform inequality (11) into the form of linear matrix inequalities. Then the parameters of robustH∞ filter can be easily achieved by applying the MATLAB LMI toolbox.Lemma 2 (see [29]).
Given matricesY, H, and E of appropriate dimensions, where Y is symmetric, then the inequalityY+HFE+EΤFΤHΤ<0holds for all F satisfyingFΤF≤I, if and only if there exists a constantε>0such that the equalityY+εHHΤ+ε-1EΤE<0holds.
Let matricesV, V-1, andPbe partitioned as follows:
(15)V=[V11V12V21V22],V-1=[W11W12W21W22],P=[P11P12P12ΤP22],
whereV11,W11,P11∈ℜn×n.
Then introduce the following nonsingular matrices:(16)Π1=[V11IV210],Π2=[IW110W21].
Obviously, the equationVΠ2=Π1holds.
DenoteT1=diag⁡{Π2,Π2,I,I,Π2}, P-=[P-11P-12P-12ΤP-22]=Π2ΤPΠ2. Let inequality (11) be pre- and postmultiplied byT1ΤandT1, respectively, and substitute the expression of the matrix variablesA~ij,B~ij,C~ij, andD~ij. The following matrix inequality can be obtained:(17)∑i=1r∑j=1rhihj[Ξ11Ξ12Ξ13Ξ14Ξ150V11ΤΞ18*Ξ22Ξ23Ξ24Bj0IW11**-P-11-P-120Ξ3600***-P-220Ξ4600****-γ2I-DjΤDfiΤ00*****-I00******-P-11-P-12*******-P-22]<0,where
(18)Ξ11=-V11-V11Τ,Ξ12=-I-V11ΤW11-V21ΤW21,Ξ13=P-11+V11Τ(Aj+ΔAj)+V21ΤBfi(Cj+ΔCj),Ξ14=P-12+V11Τ(Aj+ΔAj)W11iiiiiiiii+V21ΤBfi(Cj+ΔCj)W11+V21ΤAfiW21,Ξ15=V11ΤBj+V21ΤBfiDj,Ξ18=V11ΤW11+V21ΤW21,Ξ22=-W11-W11Τ,Ξ23=P-12Τ+Aj+ΔAj,Ξ24=P-22+(Aj+ΔAj)W11,Ξ36=LjΤ-(Cj+ΔCj)ΤDfiΤ,Ξ46=W11ΤLjΤ-W11Τ(Cj+ΔCj)ΤDfiΤ-W21ΤCfiΤ.Moreover, denoteT2=diag⁡{I,W11-1,I,W11-1,I,I,I,W11-1}. Similarly, multiply inequality (17) byT2Τon the left and byT2on the right. At the same time, let
(19)P~=[P~11P~12P~12ΤP~22]=[I00W11-1]Τ[P-11P-12P-12ΤP-22][I00W11-1],Q=W11-1,R=V21ΤW21Q,Xi=V21ΤAfiW21Q,Yi=V21ΤBfi,Zi=CfiW21Q.
Then inequality (17) can be equivalent to the following form:(20)∑i=1r∑j=1rhihj[Ξ11Ξ~12Ξ~13Ξ~14Ξ~150V11ΤV11Τ+R*Ξ~22Ξ~23Ξ~24QΤBj0QΤQΤ**-P~11-P~120Ξ3600***-P~220Ξ~4600****-γ2I-DjΤDfiΤ00*****-I00******-P~11-P~12*******-P~22]<0,where
(21)Ξ~12=-Q-V11Τ-R,Ξ~13=P~11+V11Τ(Aj+ΔAj)+Yi(Cj+ΔCj),Ξ~14=P~12+V11Τ(Aj+ΔAj)+Yi(Cj+ΔCj)+Xi,Ξ~15=V11ΤBj+YiDj,Ξ~22=-Q-QΤΞ~23=P~12Τ+QΤ(Aj+ΔAj),Ξ~24=P~22+QΤ(Aj+ΔAj),Ξ~46=LjΤ-(Cj+ΔCj)ΤDfiΤ-ZiΤ.In the following, by substituting expression (2) of the uncertain matricesΔAi(t)andΔCi(t)into the matrix inequality (20), it can be obtained that
(22)∑i=1r∑j=1rhihj[Ξ^ij+E1jF(t)H1j+H1jΤFΤ(t)E1jΤiiiiiiiiiiiiiiiii+E2ijF(t)H2j+H2jΤFΤ(t)E2ijΤ]<0,
where
(23)Ξ^ij=[Ξ11Ξ~12Ξ^13Ξ^14Ξ~150V11ΤV11Τ+R*Ξ~22Ξ^23Ξ^24QΤBj0QΤQΤ**-P~11-P~120Ξ3600***-P~220Ξ^4600****-γ2I-DjΤDfiΤ00*****-I00******-P~11-P~12*******-P~22],Ξ^13=P~11+V11ΤAj+YiCj,Ξ^23=P~12Τ+QΤAj,Ξ~14=P~12+V11ΤAj+YiCj+Xi,Ξ^24=P~22+QΤAj,Ξ^46=LjΤ-CjΤDfiΤ-ZiΤ,E1jΤ=[EajΤV11EajΤQ000000],H1j=[00HajHaj0000],E2ijΤ=[EcjΤYiΤ0000-EcjΤDfiΤ00],H2j=[00HcjHcj0000].According to Lemma2, the matrix inequality (22) holds for all admissible uncertainty matrices F(t)satisfying condition (3), if and only if there exist constantsε1ij>0andε2ij>0,i,j=1,2,…,r, such that the following matrix inequality holds:
(24)∑i=1r∑j=1rhihj[Ξ^ij+ε1ijH1jΤH1j+ε1ij-1E1jE1jΤiiiiiiiiiiiiiiii+ε2ijH2jΤH2j+ε2ij-1E2ijE2ijΤ]=∑i=1rhi2[Ξ^ii+ε1iiH1iΤH1i+ε1ii-1E1iE1iΤiiiiiiiiiiiiiiii+ε2iiH2iΤH2i+ε2ii-1E2iiE2iiΤ]+∑i=1r∑i<jrhihj[Ξ^ij+ε1ijH1jΤH1j+ε1ij-1E1jE1jΤiiiiiiiiiiiiiiiiiiiiiiiiii+ε2ijH2jΤH2j+ε2ij-1E2ijE2ijΤ+Ξ^jiiiiiiiiiiiiiiiiiiiiiiiiiii+ε1jiH1iΤH1i+ε1ji-1E1iE1iΤiiiiiiiiiiiiiiiiiiiiiiiiii+ε2jiH2iΤH2i+ε2ji-1E2jiE2jiΤ]<0.Applying the Schur complement lemma to the above matrix inequality, the following conclusion can be reached from the above deduction.Theorem 3.
For a given constantγ>0, the filtering error system (8) is robustly asymptotically stable and satisfies∥Hz~w(s)∥∞<γ, if there exist constantε1ij>0,ε2ij>0, symmetric positive definite matrixP~11,P~22, and matrices P~12,V11,Q,R,Xi,Yi,Zi,Dfi,i,j=1,2,…,r, such that for all admissible uncertainties (3) the following linear matrix inequalities hold:(25)[Ξ^ii+ε1iiH1iΤH1i+ε2iiH2iΤH2iE1iE2iiE1iΤ-ε1iiI0E2iiΤ0-ε2iiI]<0,i=1,2,…,r,[(Ξ^ij+Ξ^ji+ε1ijH1jΤH1j+ε1jiH1iΤH1i+ε2ijH2jΤH2j+ε2jiH2iΤH2i)E1jE2ijE1iE2jiE1jΤ-ε1ijI000E2ijΤ0-ε2ijI00E1iΤ00-ε1jiI0E2jiΤ000-ε2jiI]<0,i<j.Using the matrix relations of formula (19) and the equivalence of the transfer function, filter parameter matrices are given as follows:
(26)Afi=R-1Xi,Bfi=R-1Yi,Cfi=Zi,Dfi=Dfi,iiiiiiiiiiiiiiiiiiiiiiii=1,2,…,r.Setρ=γ2, and an optimization problem about robustH∞ filter can be described in the following:
(27)min⁡ρs.t.(25).
Thus, the obtained filter can be called an optimal robustH∞ filter of the uncertain fuzzy system (4), and the corresponding optimal disturbance attenuation level isγ*=ρ.


## 4. Numerical Example

In this section, a numerical example will be given to illustrate the effectiveness of robustH∞ filtering approach developed in the previous section (see Figure 1) [30].Figure 1
Tunnel diode circuit system.According to the literature [30], Figure 1 can be described by the following state equations:
(28)x˙1(t)=-0.1x1(t)-0.5x13(t)+50x2(t),x˙2(t)=-x1(t)-10x2(t)+w(t),y(t)=x1(t)+w(t),z(t)=x1(t),
wherex1(t)=vC(t)is capacitor voltage andx2(t)=iL(t)is inductor current.Assume that the state variablex1(t)satisfies|x1(t)|≤3. In order to simplify the calculation, two fuzzy rules will be used to approximate the nonlinear system (28).Plant Rule 1:(29)IFx1(t)isM1(x1(t)),THENx˙(t)=(A1+ΔA1(t))x(t)+B1w(t).THENy(t)=(C1+ΔC1(t))x(t)+D1w(t),THENz(t)=L1x(t).Plant Rule 2:(30)IFx1(t)isM2(x1(t)),THENx˙(t)=(A2+ΔA2(t))x(t)+B2w(t)THENy(t)=(C2+ΔC2(t))x(t)+D2w(t),THENz(t)=L2x(t),
where model parameters are given below:
(31)A1=[-0.150-1-10],A2=[-4.650-1-10]B1=B2=[01],C1=C2=[10],D1=D2=1,L1=L2=[10],Ea1=Ea2=[00.1],Ha1=Ha2=[0.10.1],Ec1=Ec2=1,Hc1=Hc2=[-0.10.1].And the fuzzy membership functions corresponding to the above two rules are given in Figure2.Figure 2
Fuzzy membership functions.By giving theH∞ performance levelγ=1and constructing the fuzzy filter (7), by solving linear matrix inequalities (25), the modified filter parameters can be obtained as follows:
(32)Af1=[-1.678041.9421-1.8564-9.1733],Af2=[-5.081743.5742-1.8467-9.1707],Bf1=[1.07850.9305],Bf2=[-0.10950.9196],Cf1=[0.6950-0.3013],Cf2=[0.7422-0.3745],Df1=0.2516,Df2=0.2566.Assume that the initial state of system isx0=[-10]Τ, the initial state of filter isx^0=[00]Τ, and the uncertain matrix is selected asF(t)=sin(t). Apply the above-obtained filter to the system (28) for filtering simulation. When the exogenous interference noise is set asw(t)=0.5sin(5t), the simulation results are shown in Figure 3, in which the blue dotted line indicates the case without introducing a free matrix variable, while the red dotted line represents the case with introducing a free matrix variable. Similarly, Figure 4 shows the filtering results when the noise is a random noise with zero mean and variance of 0.01. Obviously, from the simulation results, it can be seen that the filtering results with introducing a free matrix variable are better than those of not introducing, and the former makes the system have a higher error estimation accuracy.Figure 3
Filtering results forw(t)=0.5sin(5t).Figure 4
Filtering results for random noise interference.Moreover, by solving the optimization problem (27), the minimum disturbance attenuation level is obtained asγ*=5.1×10-7. By comparison, the result without introducing a free matrix variable is also given asγ*=3.3×10-6. Thus it can be seen that the system can obtain lower disturbance attenuation level by introducing a free matrix variable.


## 5. Conclusions

This paper successfully extends the ideology of literature [25] to robustH∞ filter design for a class of uncertain nonlinear systems. By introducing a free matrix variable, this paper gives a new systematic design methodology of robustH∞ filter. In particular, the filter parameters can be designed independent of the Lyapunov matrix. This method can decouple between the Lyapunov matrix and the system matrix, so it can reduce the conservatism of the system to a certain extent. The solution of filter can be converted into a standard LMI problem. From the simulation results, it can be seen that the improved filter has the lower conservatism and the higher estimation accuracy, which is useful for engineering application.



---

*Source: 613026-2014-04-27.xml*