Improved Robust H ∞ Filtering Approach for Nonlinear Systems

and Applied Analysis 3 dynamic equation inferred from formulas (4) and (7) can be described as follows:


Introductions
In recent years, when the external disturbance and the statistical properties of the measurement noise are unknown, using H ∞ 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][2][3][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 design H ∞ 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 fuzzy H ∞ filtering problem.After that, Tseng [9,10] and Tian et al. [11] discussed the design problem of robust H ∞ fuzzy filter for a class of continuous nonlinear systems.Moreover, the aboveobtained results were extended to the fuzzy H ∞ filter or robust H ∞ filter design for nonlinear systems with time delay [12][13][14][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][27][28].Unfortunately, to the best of our knowledge, this idea has not yet been introduced to the design of robust H ∞ filter for the uncertain continuous nonlinear systems.
Taking into account the above-mentioned results, this paper will discuss a new design method of robust H ∞ filter for a class of uncertain nonlinear systems.Firstly, the 2 Abstract and Applied Analysis 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 robust H ∞ 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.

Problem Formulation
Consider a class of uncertain nonlinear systems described by the following T-S fuzzy models.
Plant Rule : where () ∈ R  is the state vector, () ∈ R  is the measured output, () ∈ R  is the signal to be estimated, and () ∈   2 [0, ∞) is the noise signal vector (including process and measurement noises). 0 is the initial state condition of the system, which is considered to be known and, without loss of generality, assumed to be zero.V 1 (), . . ., V  () are the premise variables,   ( = 1, 2, . . ., ) is the fuzzy set, and  is the number of IF-THEN rules.  ,   ,   ,   , and   are known real constant matrices with appropriate dimensions of the th subsystem, respectively.The uncertain time-varying matrices Δ  () and Δ  () represent the parameter uncertainties in the system model and are assumed to be normbounded of the following forms: where   ,   ,   , and   are known constant matrices of appropriate dimensions, which reflect the structural information of uncertainty, and () is an uncertainty matrix function with Lebesgue measurable elements and satisfies  Τ ()  () ≤ .
(3) By using the weighted average method for defuzzification, the uncertain fuzzy dynamic model for the system (1) can be inferred as follows: where in which   (V  ()) is the grade of membership of V  () in the fuzzy set   , while   (V()) is the grade of membership of the th rule.
Filter Rule : where x() ∈ R  is the state vector of filter and ẑ() ∈ R  is an estimate value of the filter output.The matrices   ,   ,  , and   are filter parameters to be determined.Here, it is assumed that the initial condition of filter is x0 = 0. Then the whole fuzzy filter can be expressed as Set the state variable as x() = [ Τ () xΤ ()] Τ and estimated error as z() = () − ẑ().Then the filtering error dynamic equation inferred from formulas ( 4) and ( 7) can be described as follows: where Let  z () be the transfer function from the disturbance input () to the estimation error z().Then the robust H ∞ filter design problem considered in this paper can be described as follows: for a given constant  > 0, find a H ∞ full-order filter in the form of (7) so that the filtering error dynamic system (8) is robustly asymptotically stable and the H ∞ norm of the transfer function  z () is less than the given constant ; that is, ‖ z ()‖ ∞ <  is satisfied.Here, constant  is called a prescribed H ∞ performance level.

Robust 𝐻 ∞ Filter Design
According to the bounded real lemma of continuous-time systems, this section firstly gives a sufficient condition for the existence of robust H ∞ filter for the uncertain fuzzy system (4).That is, for a given constant  > 0, the filtering error system ( 8) is robustly asymptotically stable and satisfies ‖ z ()‖ ∞ < , if there exists a symmetric positive definite matrix  ∈ R 2×2 , such that the following matrix inequality holds: With the aid of the basic idea of [25], an improved robust H ∞ 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 constant  > 0, the filtering error system (8) is robustly asymptotically stable and satisfies ‖ z ()‖ ∞ < , if there exist a symmetric positive definite matrix  ∈ R 2×2 and matrix  ∈ R 2×2 , such that the following matrix inequality holds: Proof.Rewrite the matrix inequality (11) in the following form:, where According to the Projection Theorem [25], inequality ( 12) is equivalent to the following inequality; that is, 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 prescribed H ∞ performance level.
Take into account that inequality ( 11) is a nonlinear matrix inequality on the matrix variables (,   ,   ,   ,   ,  = 1, 2, . . ., ), 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 robust H ∞ filter can be easily achieved by applying the MATLAB LMI toolbox.

Let matrices 𝑉, 𝑉
where  11 ,  11 ,  11 ∈ R × .Then introduce the following nonsingular matrices: Obviously, the equation where Then inequality (17) can be equivalent to the following form: where In the following, by substituting expression (2) of the uncertain matrices Δ  () and Δ  () into the matrix inequality (20), it can be obtained that where According to Lemma 2, the matrix inequality ( 22) holds for all admissible uncertainty matrices () satisfying condition (3), if and only if there exist constants  1 > 0 and  2 > 0, ,  = 1, 2, . . ., , such that the following matrix inequality holds: 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 constant  > 0, the filtering error system (8) is robustly asymptotically stable and satisfies ‖ z ()‖ ∞ < , if there exist constant  1 > 0,  2 > 0, symmetric positive definite matrix P11 , P22 , and matrices P12 ,  11 , , ,   ,   ,   ,   , ,  = 1, 2, . . ., , such that for all admissible uncertainties (3) the following linear matrix inequalities hold: Using the matrix relations of formula (19) and the equivalence of the transfer function, filter parameter matrices are given as follows: Set  =  2 , and an optimization problem about robust H ∞ filter can be described in the following: min  s.t.(25) .
Thus, the obtained filter can be called an optimal robust H ∞ filter of the uncertain fuzzy system (4), and the corresponding optimal disturbance attenuation level is  * = √ .

Numerical Example
In this section, a numerical example will be given to illustrate the effectiveness of robust H ∞ filtering approach developed in the previous section (see Figure 1) [30].
According to the literature [30], Figure 1 can be described by the following state equations: where  1 () = V  () is capacitor voltage and  2 () =   () is inductor current.
Assume that the state variable  1 () satisfies | 1 ()| ≤ 3.In order to simplify the calculation, two fuzzy rules will be used to approximate the nonlinear system (28).
Plant Rule 1: Plant Rule 2: where model parameters are given below: And the fuzzy membership functions corresponding to the above two rules are given in Figure 2. By giving the H ∞ performance level  = 1 and constructing the fuzzy filter (7), by solving linear matrix inequalities (25), the modified filter parameters can be obtained as follows: Assume that the initial state of system is the initial state of filter is x0 = [0 0] Τ , and the uncertain matrix is selected as () = sin().Apply the above-obtained filter to the system (28) for filtering simulation.When the exogenous interference noise is set as () = 0.5 sin(5), 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.Moreover, by solving the optimization problem (27), the minimum disturbance attenuation level is obtained as  * = 5.1 × 10 −7 .By comparison, the result without introducing a free matrix variable is also given as  * = 3.3 × 10 −6 .Thus it can be seen that the system can obtain lower disturbance attenuation level by introducing a free matrix variable.

Conclusions
This paper successfully extends the ideology of literature [25] to robust H ∞ filter design for a class of uncertain nonlinear systems.By introducing a free matrix variable, this paper gives a new systematic design methodology of robust H ∞ 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.

Figure 4 :
Figure 4: Filtering results for random noise interference.