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New Results on Robust H∞ Filter for Uncertain Fuzzy Descriptor Systems

Written By

Wudhichai Assawinchaichote

Submitted: 06 December 2011 Published: 27 September 2012

DOI: 10.5772/48569

From the Edited Volume

Fuzzy Controllers - Recent Advances in Theory and Applications

Edited by Sohail Iqbal, Nora Boumella and Juan Carlos Figueroa Garcia

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1. Introduction

The problem of filter design for descriptor systems system has been intensively studied by a number of researchers for the past three decades; see Ref.[1]-[6]. This is due not only to theoretical interest but also to the relevance of this topic in control engineering applications. Descriptor systems or so called singularly perturbed systems are dynamical systems with multiple time-scales. Descriptor systems often occur naturally due to the presence of small “parasitic” parameter, typically small time constants, masses, etc.

The main purpose of the singular perturbation approach to analysis and design is the alleviation of high dimensionality and ill-conditioning resulting from the interaction of slow and fast dynamics modes. The separation of states into slow and fast ones is a nontrivial modelling task demanding insight and ingenuity on the part of the analyst. In state space, such systems are commonly modelled using the mathematical framework of singular perturbations, with a small parameter, sayε, determining the degree of separation between the “slow” and “fast” modes of the system.

In the last few years, many researchers have studied the H filter design for a general class of linear descriptor systems. In Ref.[3], the authors have investigated the decomposition solution of H filter gain for singularly perturbed systems. The reduced-order H optimal filtering for system with slow and fast modes has been considered in Ref.[4]. Although many researchers have studied linear descriptor systems for many years, the H filtering design for nonlinear descriptor systems remains as an open research area. This is because, in general, nonlinear singularly perturbed systems can not be easily separated into slow and fast subsystems.

Fuzzy system theory enables us to utilize qualitative, linguistic information about a highly complex nonlinear system to construct a mathematical model for it. Recent studies show that a fuzzy linear model can be used to approximate global behaviors of a highly complex nonlinear system; see for example, Ref.[7]-[19]. In this fuzzy linear model, local dynamics in different state space regions are represented by local linear systems. The overall model of the system is obtained by “blending" these linear models through nonlinear fuzzy membership functions. Unlike conventional modelling where a single model is used to describe the global behaviour of a system, the fuzzy modelling is essentially a multi-model approach in which simple sub-models (linear models) are combined to describe the global behaviour of the system.

What we intend to do in this paper is to design a robust H filter for a class of nonlinear descriptor systems with nonlinear on both fast and slow variables. First, we approximate this class of nonlinear descriptor systems by a Takagi-Sugeno fuzzy model. Then based on an LMI approach, we develop an H filter such that the L2-gain from an exogenous input to an estimate error is less or equal to a prescribed value. To alleviate the ill-conditioning resulting from the interaction of slow and fast dynamic modes, solutions to the problem are given in terms of linear matrix inequalities which are independent of the singular perturbationε, when εis sufficiently small. The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard nonlinear descriptor systems.

This paper is organized as follows. In Section 2, system descriptions and definitions are presented. In Section 3, based on an LMI approach, we develop a technique for designing a robust H filter for the system described in section 2. The validity of this approach is demonstrated by an example from a literature in Section 4. Finally in Section 5, conclusions are given.


2. System descriptions

In this section, we generalize the TS fuzzy system to represent a TS fuzzy descriptor system with parametric uncertainties. As in Ref.[19], we examine a TS fuzzy descriptor system with parametric uncertainties as follows:


whereEε=I00εI, ε>0is the singular perturbation parameter, ν(t)=[ν1(t)    νϑ(t)]is the premise variable vector that may depend on states in many cases, μi(ν(t))denotes the normalized time-varying fuzzy weighting functions for each rule (i.e., μi(ν(t))0andi=1rμi(ν(t))=1), ϑis the number of fuzzy sets, x(t)Rnis the state vector, u(t)Rmis the input, w(t)Rpis the disturbance which belongs toL2[0,), y(t)Rlis the measurement and z(t)Rsis the controlled output, the matrices Ai,B1i,B2i,C1i,C2i,D12i and D21i are of appropriate dimensions, and the matricesΔAi, ΔB1i, ΔB2i, ΔC1i, ΔC2i, ΔD12iand ΔD21irepresent the uncertainties in the system and satisfy the following assumption.

Assumption 1

ΔAi=F(x(t),t)H1i,    ΔB1i=F(x(t),t)H2i,    ΔB2i=F(x(t),t)H3i,E2
ΔC1i=F(x(t),t)H4i,    ΔC2i=F(x(t),t)H5i,    ΔD12i=F(x(t),t)H6iE3
and     ΔD21i=F(x(t),t)H7iE4

whereHji, j=1,2,,7are known matrix functions which characterize the structure of the uncertainties. Furthermore, the following inequality holds:


for any known positive constantρ.

Next, let us recall the following definition.

Definition 1 Suppose γis a given positive number. A system (1) is said to have an L2-gain less than or equal to γif

withx(0)=0, where (z(t)-z^(t)) is the estimated error output, for all Tf0 andw(t)L2[0,Tf].

3. Robust H fuzzy filter design

Without loss of generality, in this section, we assume thatu(t)=0. Let us recall the system (1) with u(t)=0as follows:


We are now aiming to design a full order dynamic H fuzzy filter of the form


where x^(t)Rn is the filter’s state vector, z^Rsis the estimate ofz(t), A^ij(ε), B^iand C^i are parameters of the filter which are to be determined, and μ^i denotes the normalized time-varying fuzzy weighting functions for each rule (i.e., μ^i0andi=1rμ^i=1), such that the inequality (3) holds. Clearly, in real control problems, all of the premise variables are not necessarily measurable. In this section, we then consider the designing of the robust H fuzzy filter into two cases as follows.

3.1. Case I–ν(t)is available for feedback

The premise variable of the fuzzy model ν(t)is available for feedback which implies that μi is available for feedback. Thus, we can select our filter that depends on μi as follows:


Before presenting our next results, the following lemma is recalled.

Lemma 1 Consider the system (4). Given a prescribed H performance γ>0and a positive constantδ, if there exist matricesXε=XεT, Yε=YεT, Bi(ε)andCi(ε), i=1,2,,r, satisfying the following ε-dependent linear matrix inequalities:

Ψ11ii(ε)<0,    i=1,2,,rE13
Ψ22ii(ε)<0,    i=1,2,,rE14
Ψ11ij(ε)+Ψ11ji(ε)<0,    i<jrE15
Ψ22ij(ε)+Ψ22ji(ε)<0,    i<jrE16




 and     λ=1+ρ2i=1rj=1r[H2iTH2j+H7iTH7j]12,E23

then the prescribed H performance γ>0is guaranteed. Furthermore, a suitable filter is of the form (6) with




Proof. It can be shown by employing the same technique used in Ref.[18]-[19].

Remark 1 The LMIs given in Lemma 1 may become ill-conditioned when εis sufficiently small, which is always the case for the descriptor systems. In general, these ill-conditioned LMIs are very difficult to solve. Thus, to alleviate these ill-conditioned LMIs, we have the following ε-independent well-posed LMI-based sufficient conditions for the uncertain fuzzy descriptor systems to obtain the prescribed H performance.

Theorem 1 Consider the system (4). Given a prescribed H performance γ>0and a positive constantδ, if there exist matricesX0, Y0, B0iandC0i, i=1,2,,r, satisfying the following ε-independent linear matrix inequalities:

EX0T=X0E,    X0TD=DX0,    X0E+DX0>0E28
EY0T=Y0E,    Y0TD=DY0,    Y0E+DY0>0E29
Ψ11ii<0,    i=1,2,,rE30
Ψ22ii<0,    i=1,2,,rE31
Ψ11ij+Ψ11ji<0,    i<jrE32
Ψ22ij+Ψ22ji<0,    i<jrE33





 and     λ=1+ρ2i=1rj=1r[H2iTH2j+H7iTH7j]12,E41

then there exists a sufficiently small ε^>0 such that forε(0,ε^], the prescribed H performance γ>0is guaranteed. Furthermore, a suitable filter is of the form (6) with



Xε={X0+εX~}Eε     and     Yε-1={Y0-1+εNε}EεE45

with X~=D(X0T-X0) andNε=D((Y0-1)T-Y0-1).

Proof. Suppose the inequalities (17)-(19) hold, then the matrices X0 and Y0 are of the following forms:

X0=X1X20X3     and     Y0=Y1Y20Y3E46
withX1=X1T>0, X3=X3T>0, Y1=Y1T>0andY3=Y3T>0. Substituting X0 and Y0 into (27), respectively, we have

Clearly, Xε=XεT, andYε-1=(Yε-1)T. Knowing the fact that the inverse of a symmetric matrix is a symmetric matrix, we learn that Yε is a symmetric matrix. Using the matrix inversion lemma, we can see that


whereY~=Y0Nε(I+εY0Nε)-1Y0. Employing the Schur complement, one can show that there exists a sufficiently small ε^ such that forε(0,ε^], (8)-(9) holds.

Now, we need to show that


By the Schur complement, it is equivalent to showing that


Substituting (28) and (29) into the left hand side of (32), we get


The Schur complement of (17) is


According to (34), we learn that

X1-Y1-1>0         and         X3-Y3-1>0.E54

Using (35) and the Schur complement, it can be shown that there exists a sufficiently small ε^>0 such that forε(0,ε^], (7) holds.

Next, employing (28), (29) and (30), the controller’s matrices given in (16) can be re-expressed as follows:


Substituting (28), (29), (30) and (36) into (14) and (15), and pre-post multiplying byEε00I, we, respectively, obtain

Ψ11ij+ψ11ij     and     Ψ22ij+ψ22ijE56

where the ε-independent linear matrices Ψ11ij and Ψ22ij are defined in (24) and (25), respectively and the ε-dependent linear matrices are


Note that the ε-dependent linear matrices tend to zero when εapproaches zero.

Employing (20)-(22) and knowing the fact that for any given negative definite matrixW, there exists an ε>0such thatW+εI<0, one can show that there exists a sufficiently small ε^>0 such that forε(0,ε^], (10)-(13) hold. Since (7)-(13) hold, using Lemma 1, the inequality (3) holds.

3.2. Case II–ν(t)is unavailable for feedback

The fuzzy filter is assumed to be the same as the premise variables of the fuzzy system model. This actually means that the premise variables of fuzzy system model are assumed to be measurable. However, in general, it is extremely difficult to derive an accurate fuzzy system model by imposing that all premise variables are measurable. In this subsection, we do not impose that condition, we choose the premise variables of the filter to be different from the premise variables of fuzzy system model of the plant. In here, the premise variables of the filter are selected to be the estimated premise variables of the plant. In the other words, the premise variable of the fuzzy model ν(t)is unavailable for feedback which implies μi is unavailable for feedback. Hence, we cannot select our filter which depends onμi. Thus, we select our filter as (5) where μ^i depends on the premise variable of the filter which is different fromμi. Let us re-express the system (1) in terms ofμ^i, thus the plant’s premise variable becomes the same as the filter’s premise variable. By doing so, the result given in the previous case can then be applied here. Note that it can be done by using the same technique as in subsection. After some manipulation, we get



ΔA-i=F-(x(t),x^(t),t)H-1i,    ΔB-1i=F-(x(t),x^(t),t)H-2i,    ΔB-2i=F-(x(t),x^(t),t)H-3i,E60
ΔC-1i=F-(x(t),x^(t),t)H-4i,    ΔC-2i=F-(x(t),x^(t),t)H-5i,    ΔD-12i=F-(x(t),x^(t),t)H-6iE61
 and     ΔD-21i=F-(x(t),x^(t),t)H-7iE62


H-1i=H1iT  A1T  ArT  H11TH1rTT,    H-2i=H2iT  B11TB1rT  H21TH2rTT,E63
H-3i=H3iT  B21TB2rT  H31TH3rTT,    H-4i=H4iT  C11TC1rT  H41TH4rTT,E64
H-5i=H5iT  C21TC2rT  H51TH5rTT,    H-6i=H6iT  D121TD12rT  H61TH6rTTE65
H-7i=H7iT  D211T    D21rT  H71TH7rTT     and E66
F-(x(t),x^(t),t)=[F(x(t),t)    (μ1-μ^1)        (μr-μ^r)    F(x(t),t)(μ1-μ^1)    . Note that F(x(t),t)(μr-μ^r)] whereF-(x(t),x^(t),t)ρ-. ρ-={3ρ2+2}12is derived by utilizing the concept of vector norm in the basic system control theory and the fact thatρ-, μi0, μ^i0andi=1rμi=1.

Note that the above technique is basically employed in order to obtain the plant’s premise variable to be the same as the filter’s premise variable; e.g. [17]. Now, the premise variable of the system is the same as the premise variable of the filter, thus we can apply the result given in Case I. By applying the same technique used in Case I, we have the following theorem.

Theorem 2 Consider the system (4). Given a prescribed i=1rμ^i=1 performance Hand a positive constantγ>0, if there exist matricesδ, X0, Y0andB0i, C0i, satisfying the following i=1,2,,r-independent linear matrix inequalities:

X0E+DX0IIY0E+DY0    >    0E68
EX0T=X0E,    X0TD=DX0,    X0E+DX0    >    0E69
EY0T=Y0E,    Y0TD=DY0,    Y0E+DY0    >    0E70
Ψ11ii<0,    i=1,2,,rE71
Ψ22ii<0,    i=1,2,,rE72
Ψ11ij+Ψ11ji<0,    i<jrE73

whereΨ22ij+Ψ22ji<0,    i<jr,





then there exists a sufficiently small  and     λ-=1+ρ-2i=1rj=1r[H-2iTH-2j+H-7iTH-7j]12, such that forε^>0, the prescribed ε(0,ε^] performance His guaranteed. Furthermore, a suitable filter is of the form (??) with




with Xε={X0+εX~}Eε     and     Yε-1={Y0-1+εNε}Eε andX~=D(X0T-X0).

Proof. It can be shown by employing the same technique used in the proof for Theorem 1.


4. Example

Consider the tunnel diode circuit shown in Figure 1 where the tunnel diode is characterized by


Figure 1.

Tunnel diode circuit.

Assuming that the inductance, iD(t)=0.01vD(t)+0.05vD3(t)., is the parasitic parameter and letting L and x1(t)=vC(t) as the state variables, we have


where Cx˙1(t)=-0.01x1(t)-0.05x13(t)+x2(t)Lx˙2(t)=-x1(t)-Rx2(t)+0.1w2(t)y(t)=Jx(t)+0.1w1(t)z(t)=x1(t)x2(t)is the disturbance noise input, w(t)is the measurement output, y(t)is the state to be estimated and z(t)is the sensor matrix. Note that the variables J and x1(t) are treated as the deviation variables (variables deviate from the desired trajectories). The parameters of the circuit arex2(t), C=100  mFandR=10±10%  Ω. With these parameters (49) can be rewritten as

L=ε  HE88

For the sake of simplicity, we will use as few rules as possible. Assuming thatx˙1(t)=-0.1x1(t)+0.5x13(t)+10x2(t)εx˙2(t)=-x1(t)-(10+ΔR)x2(t)+0.1w2(t)y(t)=Jx(t)+0.1w1(t)z(t)=x1(t)x2(t)., the nonlinear network system (50) can be approximated by the following TS fuzzy model:

Plant Rule 1: IF |x1(t)|3 is x1(t) THEN

Eεx˙(t)=[A1+ΔA1]x(t)+B11w(t),    x(0)=0,E90

Plant Rule 2: IF y(t)=C21x(t)+D211w(t). is x1(t) THEN

Eεx˙(t)=[A2+ΔA2]x(t)+B12w(t),    x(0)=0,E93


x(t)=[x1T(t)    x2T(t)]TE95

w(t)=[w1T(t)    w2T(t)]TE96
A1=-0.110-1-1,    A2=-4.610-1-1,    B11=B12=0000.1,E97
C1=1001,    C21=C22=J,    D21=0.10,E98

Now, by assuming that ΔA1=F(x(t),t)H11,    ΔA2=F(x(t),t)H12     and     Eε=100ε. and since the values of F(x(t),t)ρ=1are uncertain but bounded within R of their nominal values given in (49), we have


Note that the plot of the membership function Rules 1 and 2 is the same as in Figure 2. By employing the results given in Lemma 1 and the Matlab LMI solver, it is easy to realize that H11=H12=0001.for the fuzzy filter design in Case I and ε<0.006for the fuzzy filter design in Case II, the LMIs become ill-conditioned and the Matlab LMI solver yields the error message, “Rank Deficient".

Figure 2.

Membership functions for the two fuzzy set.

Case I-ε<0.008 are available for feedback

In this case, ν(t)is assumed to be available for feedback; for instance,x1(t)=ν(t). This implies that J=[10] is available for feedback. Using the LMI optimization algorithm and Theorem 1 with μiH, ε=100  μandγ=0.6, we obtain the following results:


Hence, the resulting fuzzy filter is

A^21ε=-0.0928-0.3138-34.7355-3.8964, A^22ε=-0.0928-0.3138-34.7355-3.8964, B^1=1.58353.2008, B^2=1.25673.8766, C^1=-1.7640-0.8190, C^2=4.5977-0.8190.E101



Case II: μ1=M1(x1(t))     and     μ2=M2(x1(t)).are unavailable for feedback

In this case, ν(t)is assumed to be unavailable for feedback; for instance,x1(t)=ν(t). This implies that J=[01] is unavailable for feedback. Using the LMI optimization algorithm and Theorem 2 with μiH, ε=100  μandγ=0.6, we obtain the following results:


The resulting fuzzy filter is

A^21ε=-2.3549-0.3748-32.4539-3.9044,  A^22ε=-2.3549-0.3748-32.4539-3.9044,  B^1=-0.30533.9938, B^2=-0.37345.1443, C^1=4.3913-0.1406, C^2=1.9832-0.1406.E105



Figure 3.

The ratio of the filter error energy to the disturbance noise energy:μ^1=M1(x^1(t))     and     μ^2=M2(x^1(t))..

The performance index 0Tf(z(t)-z^(t))T(z(t)-z^(t))dt0TfwT(t)w(t)dt
γFuzzy Filter in Case IFuzzy Filter in Case II

Table 1.

The performance index >0.6of the system with different values ofγ.

Figure 4.

The histories of the state variables, εandx1(t).

Remark 2 The ratios of the filter error energy to the disturbance input noise energy are depicted in Figure 3 when x2(t)H. The disturbance input signal, ε=100  μ, which was used during the simulation is the rectangular signal (magnitude 0.9 and frequency 0.5 Hz). Figures 4(a) - 4(b), respectively, show the responses of w(t) and x1(t) in Cases I and II. Table I shows the performance index x2(t)with different values of γin Cases I and II. After ε seconds, the ratio of the filter error energy to the disturbance input noise energy tends to a constant value which is about 50 in Case I and 0.02 in Case II. Thus, in Case I where 0.08and in Case II whereγ=0.02=0.141, both are less than the prescribed valueγ=0.08=0.283. From Table 9.1, the maximum value of 0.6that guarantees the ε-gain of the mapping from the exogenous input noise to the filter error energy being less than L2 is 0.6 H, i.e., 0.30H in Case I, and ε(0,0.30] H, i.e., 0.25H in Case II.


5. Conclusion

The problem of designing a robust ε(0,0.25] fuzzy H-independent filter for a TS fuzzy descriptor system with parametric uncertainties has been considered. Sufficient conditions for the existence of the robust ε fuzzy filter have been derived in terms of a family of H-independent LMIs. A numerical simulation example has been also presented to illustrate the theory development.


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Written By

Wudhichai Assawinchaichote

Submitted: 06 December 2011 Published: 27 September 2012