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

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.


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, say ε, 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 L 2 -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 perturbation ε, when ε is 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.

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: where is the premise variable vector that may depend on states in many cases, μ i (ν(t)) denotes the normalized time-varying fuzzy weighting functions for each rule (i.e., μ i (ν(t)) ≥ 0 and ∑ r i=1 μ i (ν(t)) = 1), ϑ is the number of fuzzy sets, x(t) ∈ n is the state vector, u(t) ∈ m is the input, w(t) ∈ p is the disturbance which belongs to L 2 [0, ∞), y(t) ∈ is the measurement and z(t) ∈ s is the controlled output, the matrices where H j i , j = 1, 2, • • • , 7 are known matrix functions which characterize the structure of the uncertainties.Furthermore, the following inequality holds: for any known positive constant ρ.
Next, let us recall the following definition.
Definition 1. Suppose γ is a given positive number.A system ( 1) is said to have an L 2 -gain less than or equal to γ if with x(0) = 0, where (z(t) − ẑ(t)) is the estimated error output, for all T f ≥ 0 and w(t) ∈ L 2 [0, T f ].

Robust H ∞ fuzzy filter design
Without loss of generality, in this section, we assume that u(t) = 0. Let us recall the system (1) with u(t) = 0 as follows: We are now aiming to design a full order dynamic H ∞ fuzzy filter of the form where x(t) ∈ n is the filter's state vector, ẑ ∈ s is the estimate of z(t), Âij (ε), Bi and Ĉi are parameters of the filter which are to be determined, and μi denotes the normalized time-varying fuzzy weighting functions for each rule (i.e., μi ≥ 0 and ∑ r i=1 μi = 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.

Case I-ν(t) is available for feedback
The premise variable of the fuzzy model ν(t) is available for feedback which implies that μ i is available for feedback.Thus, we can select our filter that depends on μ i as follows: Before presenting our next results, the following lemma is recalled.
Lemma 1.Consider the system (4).Given a prescribed H ∞ performance γ > 0 and a positive constant δ, if there exist matrices , satisfying the following ε-dependent linear matrix inequalities: where with , then the prescribed H ∞ performance γ > 0 is guaranteed.Furthermore, a suitable filter is of the form (6) with where 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 ε is 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 > 0 and a positive constant δ, if there exist matrices X 0 , Y 0 , B 0 i and C 0 i , i = 1, 2, • • • , r, satisfying the following ε-independent linear matrix inequalities: where E = I 0 0 0 , D = 0 0 0 I , , then there exists a sufficiently small ε > 0 such that for ε ∈ (0, ε], the prescribed H ∞ performance γ > 0 is guaranteed.Furthermore, a suitable filter is of the form (6) with where Proof: Suppose the inequalities ( 17)-( 19) hold, then the matrices X 0 and Y 0 are of the following forms: Substituting X 0 and Y 0 into (27), respectively, we have Clearly, X ε = X T ε , and Knowing the fact that the inverse of a symmetric matrix is a symmetric matrix, we learn that Y ε is a symmetric matrix.Using the matrix inversion lemma, we can see that Employing the Schur complement, one can show that there exists a sufficiently small ε such that for ε ∈ (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 Using ( 35) and the Schur complement, it can be shown that there exists a sufficiently small ε > 0 such that for ε ∈ (0, ε], (7) holds.

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 μ i is unavailable for feedback.Hence, we cannot select our filter which depends on μ i .Thus, we select our filter as (5) where μi depends on the premise variable of the filter which is different from μ i .Let us re-express the system (1) in terms of μi , 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 where . ρ is derived by utilizing the concept of vector norm in the basic system control theory and the fact that μ i ≥ 0, μi ≥ 0, ∑ r i=1 μ i = 1 and ∑ r i=1 μi = 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 H ∞ performance γ > 0 and a positive constant δ, if there exist matrices X 0 , Y 0 , B 0 i and C 0 i , i = 1, 2, • • • , r, satisfying the following ε-independent linear matrix inequalities: where E = I 0 0 0 , D = 0 0 0 I , , then there exists a sufficiently small ε > 0 such that for ε ∈ (0, ε], the prescribed H ∞ performance γ > 0 is guaranteed.Furthermore, a suitable filter is of the form (??) with where Proof: It can be shown by employing the same technique used in the proof for Theorem 1.

Example
Consider the tunnel diode circuit shown in Figure 1 where the tunnel diode is characterized by Assuming that the inductance, L, is the parasitic parameter and letting x 2 (t) = i L (t) as the state variables, we have where w(t) is the disturbance noise input, y(t) is the measurement output, z(t) is the state to be estimated and J is the sensor matrix.Note that the variables x 1 (t) and x 2 (t) are treated as the deviation variables (variables deviate from the desired trajectories).The parameters of the circuit are C = 100 mF, R = 10 ± 10% Ω and L = ε H.With these parameters (49) can be rewritten as ẋ1 (t) = −0.1x For the sake of simplicity, we will use as few rules as possible.Assuming that |x 1 (t)| ≤ 3, the nonlinear network system (50) can be approximated by the following TS fuzzy model: Plant Rule 2: where Now, by assuming that F(x(t), t) ≤ ρ = 1 and since the values of R are uncertain but bounded within 10% 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 ε < 0.006 for the fuzzy filter design in Case I and ε < 0.008 for the fuzzy filter design in Case II, the LMIs become ill-conditioned and the Matlab LMI solver yields the error message, "Rank Deficient".Hence, the resulting fuzzy filter is where μ 1 = M 1 (x 1 (t)) and μ 2 = M 2 (x 1 (t)).
Remark 2. The ratios of the filter error energy to the disturbance input noise energy are depicted in Figure 3 when ε = 100 μH.The disturbance input signal, w(t), 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 x 1 (t) and x 2 (t) in Cases I and II.Table I shows the performance index γ with different values of ε in Cases I and II.After 50 seconds, the ratio of the filter error energy to the disturbance input noise energy tends to a constant value which is about 0.02 in Case I and 0.08 in Case II.Thus, in Case I where γ = √ 0.02 = 0.141 and in Case II where γ = √ 0.08 = 0.283, both are less than the prescribed value 0.6.From Table 9.1, the maximum value of ε that guarantees the L 2 -gain of the mapping from the exogenous input noise to the filter error energy being less than 0.6 is 0.30 H, i.e., ε ∈ (0, 0.30] H in Case I, and 0.25 H, i.e., ε ∈ (0, 0.25] H in Case II.

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

Figure 2 .
Figure 2. Membership functions for the two fuzzy set.

Figure 3 .
Figure 3.The ratio of the filter error energy to the disturbance noise energy: D 12 i and D 21 i are of appropriate dimensions, and the matrices ΔA i , ΔB 1 i , ΔB 2 i , ΔC 1 i , ΔC 2 i , ΔD 12 i and ΔD 21 i represent the uncertainties in the system and satisfy the following assumption.
This implies that μ i is available for feedback.Using the LMI optimization algorithm and Theorem 1 with ε = 100 μH, γ = 0.6 and δ = 1, we obtain the following results: Case I-ν(t) are available for feedbackIn this case, x 1 (t) = ν(t) is assumed to be available for feedback; for instance, J = [1 0].

Table 1 .
The histories of the state variables, x 1 (t) and x 2 (t).The performance index γ of the system with different values of ε.