H_∞ Filtering for a Class of Nonlinear Systems with Interval Time-varying Delay

구간시변 지연을 가지는 비선형시스템의 H_∞ 필터링

Abstract

In this paper, a delay-dependent $H_{\infty}$ filtering problem is investigated for discrete-time delayed nonlinear systems which include a more general sector nonlinear function instead of employing the commonly used Lipschitz-type function. By using the Lyapunov-Krasovskii functional approach, a less conservative sufficient condition is established for the existence of the desired filter, and then, the corresponding solvability condition guarantee the stability of the filter with a prescribed $H_{\infty}$ performance level. Finally, two simulation examples are given to show the effectiveness of the proposed filtering scheme.

1. Introduction

During the past decades, the filter design problem has been widely studied due to its extensive applications in control systems and signal processing. The purpose of the filtering design is to estimate the unavailable state variables of a given system through noisy measurements. There are basically two approaches to the problem: the Kalman filtering approach and the H∞ filtering approach. The Kalman filtering approach is based on the assumption that a linear system model is required and all noise terms and measurements have Gaussian distributions [1]. However, the prior knowledge of noise may not always be precisely known. In order to overcome this problem, H∞ filtering method, which provides both a guaranteed noise attenuation level and robustness against unmodeled dynamics, has been proposed. Compared with the Kalman filtering method, the main advantage of H∞ filtering method is that the noise sources are supposed to be arbitrary signal with bounded energy, and no exact statistics are required to be known. On the other hand, since time delay is commonly encountered in various engineering systems and is frequently a source of instability and poor performance, the problem of H∞ filtering for time delay systems have been received increasing attention in the last decades[2-17].

It is well-known that the main objective of H∞ filtering is to design a suitable filter such that the bound of induced L2 norm of the operator from the noise signals to the filtering error is less than a prescribed level. In order to get less conservative results, that is, obtain a smaller H∞ disturbance attenuation lever 𝛾, various methods are utilized. For example, in [3], a new finite sum inequality was employed to get a sufficient condition for the existence of a suitable filter. In [8], the input-output(IO) approach is used to get the less conservative result than [3]. But some delay terms are neglected for estimating the derivative bound of the constructed Lyapunov functional in the IO approach. Therefore, the results in [8] are conservative to some extent, and there is much room to improve the result in [8]. Moreover, it should be noted that the results in [2-9] are considered for only linear system.

It is well known that nonlinearities exist universally in practical systems, so the H∞ filtering problem for nonlinear dynamical system have been investigated by many researchers [10-17. Xu [15] was concerned with the problem of robust H∞ filtering for a class of discrete-time nonlinear systems with state delay and norm-bounded parameter uncertainty. In [11], a stable full or reduced order filter with the same repeated scalar nonlinearities was designed to guarantee the induced L2 or generalized H∞ performance. In [14], the problem of H∞ filtering for systems with repeated scalar nonlinearities under unreliable communication was investigated. In [15], a robust H∞ filtering problem for a class of discrete-time nonlinear systems was considered. But it should be pointed out that the time delay was not taken into consideration in [11,14,15], and only constant time delay was considered in [10]. To the best of our knowledge, there are few results on the problem of the H∞ filtering for a nonlinear system with time-varying delay.

In this paper, we consider the H∞ filtering problem for a class of discrete-time systems involving sector nonlinearities and interval time-varying delay. Inspired by the work [13], the sector nonlinearities considered in the paper are more general than usual Lipschitz conditions. By using a new Lyapunov functional and Linear matrix inequality technique, delay dependent conditions are obtained for designing a filter with an H∞ disturbance attenuation level 𝛾. When the involved LMIs are feasible, a set of the parameters of a desired filter can be obtained. Two numerical examples are provided to show the usefulness and effectiveness of the proposed design method.

Notation: Throughout the paper, Rn denotes the n-dimensional Euclidean space, Rm × n denotes the set of m by n real matrix. For symmetric matrices X, X ＞ 0 and X＜0, mean that X is a positive/negative definite symmetric matrix, respectively. I and 0 denote the identity matrix and zero matrix with appropriate dimension. ★ represents the elements below the main diagonal of a symmetric matrix. diag… denotes the diagonal matrix. refers to the induced matrix 2-norm. L2 means the space of square integral vector functions on [0, ∞) with norm

 

2. Problem Statements

Consider a discrete-time nonlinear system with time-varying delay and disturbance:

where x(k) ∈ Rn is the state, w(k) ∈ Rq is a disturbance input belongs to L2[0, ∞), y(k) ∈ Rm is the signal to be estimated, f(.) and h(.) are known vector-valued nonlinear functions A,Ad,F,Bw,C,Cd,H1,H2,Dw,L,Ld,Gw are known constant matrices of appropriate dimensions, and d(k) is the time varying delay satisfying

d1≤d(k)≤d2

where d1＞0 and d2＞0 demote the lower and upper bounds of the delay, respectively.

In this paper, without loss of generality, we always assume that f(0) = 0,h(0) = 0 and for vector-valued functions f,h, we assume

where U1,U2,V1,V2 are known real constant matrices, and U2−U1,V2−V1 are positive definite matrices.

Remark 1 Eq. (2) and (3) are the so-called sector-bounded conditions [18], which are more general than the Lipschitz conditions, and have been widely adopted in the literature [19-20]. The reason is that if we use the Lipschitz condition, the matrix U1,U2,V1,V2 are diagonal matrix, it is a special case included in our considered condition.

The objective of this paper is to estimate the system states x(k). In this paper, we consider a full-order linear asymptotically stable filter for system (1) with state-space realization of the form

where is the filter state vector and AFBF,CFDF are appropriately dimensioned filter gains to be determined.

Denote

Then the following error system is obtained

where

The aim of this paper is to design the H∞ filter satisfying that the filtering error system (5) with w(k) = 0 is asymptotically stable and H∞ performance

is guaranteed under zero-initial conditions.

Lemma 1 [5] For any matrix M＞0, integers 𝛾1 and 𝛾2 satisfying 𝛾2＞𝛾1 , and vector function w: N[𝛾1,𝛾2]→Rn such that the sums concerned are well defined, then

Lemma 2 [5] For any matrix , scalars α1(k) ＞0, α2(k) ＞0 satisfying α1(k)+α2(k)=1, vector functions δ1(k) and δ2(k) : N→Rn, the following inequality holds

 

3. Main Results

In this section, first of all, let us give a sufficient condition, which ensure system (1) to be asymptotically stable with H∞ performance level 𝛾. In convenience, we define

The following theorem provides a sufficient condition, which ensures the system (5) to be asymptotically stable with H∞ performance 𝛾.

Theorem 1. For given d2 ＞ d1 ＞ 0, 𝛾 ＞ 0 and matrix U1,U2,V1,V2,W1,W2, the nonlinear filtering error system (5) is asymptotically stable with H∞ performance 𝛾, if there exist positive definite symmetric matrix P,Q1,Q2,R1,R2 and appropriate dimension matrix , satisfying the following LMIs

Proof. Consider the following L-K functional candidate as

where

with the cost function (6).

Calculating the difference of V1(k), V2(k) and V3(k), we have

Since

an upper bound of the difference of V4(k) is obtained from Lemma 2

where

Note that if d(k) = 0 or d(k) = d2 , we have x(k) − x(k−d(k)) = 0 or x(k − d(k)) − x(k−d2) = 0, respectively. Thus Eq.(14) holds based on Lemma 1.

Similar to Eq. (11), the difference of V5(k) is

where

Note that if d(k) = d1 or d(k) = d2, we have x(k−d1)−x(k−d(k)) = 0 or x(k−d(k))−x(k−d2) = 0, respectively. Thus, Eq. (15) holds based on Lemma 1.

It follows readily from Eq. (2) and Eq. (3) that

To establish the H∞ performance for the filtering error system (5), if the difference of V(k) is negative, then z(k) goes to zero as k→∞. Next, assuming zero initial conditions for the filtering error system, the performance index is

If the inequality e(k)Te(k)−𝛾2w(k)Tw(k)+ΔV(k) ＜ 0 holds, then V(k) goes to zero as k→∞.

Combining with Eq. (10)-(18), one can obtain

By Schur complement, inequality (19) is equivalent to

In the inequality (20), the positive-definite matrix P and the filter parameters Af, Bf, Cf, Df, which included in the matrix are unknown. Hence it should be converted to LMI via proper variable substitution method.

Let us define V as

For positive definite matrix P−1 and nonzero matrix V , it follows that

Combined with the Eq. (21), pre and post multiplying the matrix inequality (20) by the matrix diag{I,I,V,I} and diag{I,I,VT,I}, then one can get the following inequality

By simple matrix calculation, it is straightforward to verify that

Now, define a new set of variables as follows

Note that the inequality (8) implies that J＜0 for any nonzero w(k)∈L2 , i.e., the filtering error system has a guaranteed 𝛾 level of disturbance attenuation. This completes the proof.

Remark 2. If F= H1 = H2 =0, the system (1) is reduced to the following linear system:

Combining the same filter system with Eq. (4), the corresponding error system is obtained as following

In convenience, we define

Based on Theorem 1, the following Corollary provides a sufficient condition, which ensures the system (24) to be asymptotically stable with H∞ performance 𝛾.

Corollary 1. For given d2＞d1＞0, 𝛾＞0, the linear filtering error system (24) is asymptotically stable with H∞ performance 𝛾, if there exist positive definite symmetric matrix P,Q1,Q2,R1,R2 and appropriate dimension matrix , satisfying the following LMIs

Remark 3. For any solutions of the LMIs (7}-(8) and LMIs (25)-(26) in Theorem 1 and Corollary 1, respectively, a corresponding filter of the form (4} can be reconstructed from the relations

Remark 4. In [13], the time-varying delay term x(k−d(k)) was not considered for estimating the bound of . In order to obtain a less conservative result, Lemma 2 is applied by using the time-varying delay term x(k−d(k)) in the Eq. (14) and (15). The following examples will be given to demonstrate the effectiveness of this method.

 

4. Numerical Examples

In this section, two examples are given to show the effectiveness of our method on the design of the robust H∞ filter.

Example 1 Consider the following simplified longitudinal flight system [10]:

The measurement signal and signal to be estimated are

It is easy to see that the system described by Eqs. (27)-(31) which is satisfied Eqs. (2) and (3) and has the form (1) with

When d1 = 1,d2 = 5,𝛾 = 0.0670, applying Theorem 1 to above system by utilizing MATLAB(with YALMIP 3.0 an SeDuMi 1.3), the corresponding parameter of the filter gains (Remark 3) are obtained by

Fig. 1 shows that the output z(k) and its estimated output under the initial condition z(k) = [1.5−1 1.5]T , =[−0.5 0 0.5]T respectively. Also, the estimated error e(k) is described in Fig. 2, where the external disturbance w(k) = sin(k)e−10k. From these simulation results, we can see that the disturbance is effectively attenuated by designed H∞ filter for the discrete-time system (1) with nonlinearities and time-varying delay.

Fig. 1z(k) (dashed) and its estimate (solid) in Example 1.

Fig. 2Estimated error e(k) in Example 1.

Furthermore, when d1 =d2 =2, the time delay become constant delay, we can obtain the minimal H∞ performance lever 𝛾=0.0456, which is less conservative than 1.5 derived in [10].

Example 2. Consider the system (23) with the following parameters:

The allowable H∞ performance lever 𝛾 obtained by different methods is depicted in Table 1. When d1 = 1,d2 = 4, we can see that the minimal H∞ performance lever 𝛾 is 3.8447, which is much small than 4.9431. It means that the obtained result in this paper is less conservative than the one derived in [13]. Solving the LMIs (25) and (26) by using the MATLAB(with YALMIP 3.0 an SeDuMi 1.3), the corresponding parameter of the filter gains (Remark 3) are given by

Table 1Comparison of H∞ performance 𝛾

 

5. Conclusions

In this paper, a robust H∞ filtering problem for a class of discrete-time systems with nonlinear sensor and interval time-varying delay has been proposed. Based on the Lyapunov-Krasovskii functional approach, sufficient conditions have been provided for the stability of the filtering error system with a prescribed H∞ performance level. Finally, numerical examples have been given to show the usefulness and effectiveness of the proposed filter design method.