Robust H ∞ Filtering for Uncertain Discrete-Time Fuzzy Stochastic Systems with Sensor Nonlinearities and Time-Varying Delay

1 College of Computer and Information, Hohai University, Changzhou 213022, China 2 Changzhou Key Laboratory of Sensor Networks and Environmental Sensing, Changzhou 213022, China 3 Jiangsu Key Laboratory of Power Transmission and Distribution Equipment Technology, Changzhou 213022, China 4 School of Mathematical Sciences, Anhui University, Hefei 230601, China 5 Department of Mathematics and Physics, Hohai University, Changzhou 213022, China


Introduction
Fuzzy systems in the Takagi-Sugeno T-S model can represent a lot of complex nonlinear systems in 1-3 .By using a T-S fuzzy plant model, one can describe a nonlinear system as a weighted sum of some simple linear subsystems.This fuzzy model is described by a family of fuzzy if-then rules that represent local linear input/output relations of the system.The overall fuzzy model of the system is achieved by smoothly blending these local linear models together through membership functions.Consequently, stability, control, and filtering problems for T-S fuzzy systems have attracted considerable attention, and many important results have been reported in 4-9 .symmetric matrices X and Y , the notation X > Y resp., X ≥ Y means that the X − Y is positive definite resp., positive semidefinite .* denotes a block that is readily inferred by symmetry.E{•} stands for the mathematical expectation operator with respect to the given probability measure P.

Problem Description
Consider a class of uncertain discrete-time stochastic systems that can be approximated by the following time-delay T-S fuzzy stochastic model with r plant rules.
Plant rule i: if θ 1 k is η i 1 and . . .and θ p k is η i p , then where η i j is the fuzzy set.θ t θ 1 t , θ 2 t , . . ., θ p t T is the premise variable vector, x k ∈ R n is the state vector, y k ∈ R q is the measurable output vector, z k ∈ R r is the state combination to be estimated, w k is a real scalar process on a probability space Ω, F, P relative to an increasing family F k k∈N of σ-algebra F k ⊂ F generated by w k k∈N , and N is the set of natural numbers.The stochastic process {w k } is independent, which is assumed to satisfy where the stochastic variables w 0 , w 1 , w 2 , . . .are assumed to be mutually independent.The exogenous disturbance signal v k ∈ R p is assumed to belong to L e2 0 ∞ , R p , and is F k−1 measurable for all k ∈ N, where L e2 0 ∞ , R p denotes the space of k-dimensional nonanticipatory square summable stochastic process f • f k k∈N on N with respect to

2.3
The time-varying delay τ k satisfies where τ 1 and τ 2 are known positive integers representing the minimum and maximum delays, respectively.In addition, A i , A di , B i , E i , E di , G i , C i , C di , D i , and L i are known real constant matrices, ΔA i k , ΔA di k , ΔB i k , ΔE i k , ΔE di k , ΔG i k , ΔC di k , and ΔD i k are unknown matrices representing time-varying parameter uncertainties, and the admissible uncertainties are assumed to be modeled in the form where The defuzzified output of the T-S fuzzy system 2.1 is inferred as follows: where h i θ k μ i θ k / r j 1 h j θ k , and μ i θ k p j 1 η i j θ j k , in which η i j θ j k is the grade of membership of η j k in η i j .According to the theory of fuzzy sets, we have μ i θ k ≥ 0, i 1, 2, . . ., r and r i 1 h i θ k > 0. Therefore, it is implied that h i θ k ≥ 0, i 1, 2, . . ., r and r i 1 h i θ k 1.For some given diagonal matrices K 1 ≥ 0 and K 2 ≥ 0 with K 2 > K 1 , the nonlinear vector functions φ ∈ K 1 , K 2 , is assumed to represent sensor nonlinearities and satisfies the following sensor condition 29 : Further, the nonlinear function φ ξ can be decomposed into a linear and a nonlinear part as follows: where the nonlinearity φ s ξ belongs to the set Φ s given by We consider the following fuzzy filter for the estimation of z k : where x k ∈ R n , z k ∈ R r , and the matrices A fi and B fi i 1, 2, . . ., r are to be determined.
Remark 2.1.Similar to 27, 29 , the sensor nonlinearities satisfying 2.10 are also considered in this paper.Here, there exists the nonlinear function φ ξ in the system 2.1 , which is called sensor nonlinearities.It is noted that in the previous filter, the matrix L i is assumed to be constant in order to avoid more verbosely mathematical derivation.
Defining T , and augmenting the model 2.9 to include the states of the filter 2.13 , we obtain the following filtering error system: where

2.16
The H ∞ filtering problem to be addressed in this paper can be formulated as follows.Given discrete-time fuzzy stochastic systems 2.9 , a prescribed level of noise attenuation γ > 0, and any φ ∈ K 1 K 2 , find a suitable filter in the form of 2.13 such that the following requirements are satisfied.
1 The filtering error system 2.14 -2.15 with v k 0 is said to be robustly stochastic stable if there exists a scalar c > 0 such that for all admissible uncertainties satisfying 2.5 -2.8 where x k denotes the solution of stochastic systems with initial state x 0 .
2 For the given disturbance attenuation level γ > 0 and under zero initial conditions for all v k ∈ L e2 0 ∞ , R p , the performance index γ satisfies the following inequality:

2.18
Before concluding this section, we introduce the following lemmas, which will be used in the derivation of our main results in the next section.Lemma 2.2 Gu et al. 33 .Given constant matrices Γ 1 , Γ 2 , and Γ 3 with appropriate dimensions, where if and only if Lemma 2.3 see 34 .For given matrices H, E, and F t with F T t F t ≤ I and scalar ε > 0, the following inequality holds:

Moreover, if the previous condition is satisfied, an acceptable state-space realization of the H ∞ filter is given by
Proof.We first establish the condition of robustly stochastic stability for the filtering error system 2.14 -2.15 .It can be shown that LMI 3.1 implies Define a matrix P > 0 by where X > 0 and Y > 0 satisfy the solvability of 3.1 .
Then, it can be shown from 3.3 and 3.5 that LMI 3.4 can be rewritten as where which is equivalent to the following inequality:

3.9
On the other hand, it is noted that the following equality holds:

3.10
By Lemmas 2.2 and 2.3, from 3.8 -3.10 , we have where Then there exists a small scalar α > 0 such that Consider the Lyapunov-Krasovskii functional candidate as follows: where e T l H T QHe l .

3.15
Calculating the difference of V k along the filtering error system 2.14 with v k 0 and taking the mathematical expectation, we have

3.16
Noting 2.2 and 2.12 , we have

3.17
By some calculations, we have 18 e T i H T QHe i .

3.19
Since Combining 3.18 -3.21 , we have Combining 3.17 and 3.22 yields where and Γ is given in 3.11 .Thus, it follows from 3.13 and 3.23 that

3.25
Hence, by summing up both sides of 3.25 from 0 to N for any integer N > 1, we have which yields where c 1/α λ max P .Taking N → ∞, it is shown from 2.17 and 3.27 that the filtering error system 2.14 is robustly stochastic stable for v k 0. Next, we will show that the filtering error system 2.14 -2.15 satisfies To this end, define with any integer N > 0. Then for any nonzero v k , we have where

3.31
It can be shown that if there exist real matrices X > 0, Y > 0, Q > 0, any matrices A Fj , B Fj , scalars ε 1i > 0, ε 2i > 0 and ε 3i > 0 for i, j 1, 2, . . ., r satisfying LMI 3.1 .Since v k ∈ L e2 0 ∞ , R p / 0 and by Lemma 2.2, it is implied that Γ ij < 0, and, thus, J N < 0. That is, || z k || e2 < γ||v k || e2 .This completes the proof.Remark 3.2.When τ 1 and τ 2 are given, matrix inequality 3.1 is linear matrix inequalities in matrix variables 1, 2, . . ., r, which can be efficiently solved by the developed interior point algorithm 33 .Meanwhile, it is esay to find the minimal attenuation level γ.Remark 3.3.Theorem 3.1 is suggested to the H ∞ filter design for uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities and time-varying delay.This approach is called fuzzy-rule-dependent approach, which is applicable to the case that the information of the premise variable θ i k i 1, 2, . . ., r is available; for example, see 24 .For the special case that the premise variables are unavailable, the fuzzy-rule-independent approach as in 24, 29 is adopted, which may result in lager conservativeness due to lack of the information of the premise variables.
In the following, we will present the fuzzy-rule-dependent H ∞ filter design for uncertain discrete-time fuzzy stochastic system with time-varying delay, which are less conservative than Theorem 3.1.Theorem 3.4.Consider the uncertain discrete-time fuzzy stochastic systems in 2.1 .A filter of form 2.13 and constants τ 1 and τ 2 , the filtering error system 2.14 -2.15 is robustly stochastic stable with performance γ, if there exist real matrices X > 0, Y > 0, Q > 0, any matrices A Fi , B Fi , scalars ε 1i > 0, ε 2i > 0 and ε 3i > 0 for i, j 1, 2, . . ., r, i < j, such that the following LMIs are satisfied: where

3.35
Our elaborate estimation J N is negative definite if and only if Γ ii < 0 and Γ ij Γ ji < 0. By Lemmas 2.2 and 2.3, we can obtain that 3.32 is equivalent to Γ ii < 0, and 3.33 is equivalent to Γ ij Γ ji < 0. Thus, we can conclude that the LMIs 3.32 and 3.33 can guarantee J N < 0. That is, || z k || e2 < γ||v k || e2 .This completes the proof.
Theorem 3.5.Consider the uncertain discrete-time fuzzy stochastic systems in 2.1 .A filter of form 2.13 and constants τ 1 and τ 2 , the filtering error system 2.14 -2.15 is robustly stochastic stable with performance γ, if there exist real matrices X > 0, Y > 0, Q > 0, any matrices A Fi , B Fi , scalars ε 1i > 0, ε 2i > 0 and ε 3i > 0 for i, j 1, 2, . . ., r, i < j, such that the following LMIs are satisfied: Proof.This theorem can be proved by employing the same techniques as in the proof of Theorem 3.4; hence, the detailed procedure is omitted here.
Remark 3.6.It is noted that the convexifying procedure proposed in this paper is based on the relaxation inequality 3.36 , and extensions of the current derivations based on the more powerful relaxation techniques presented in 8, 24, 30 are straightforward.In this way, the design conservatism can be further reduced but the computation complexity will also increase.By considering the information on the premises 8, 24, 30 , Theorems 3.4 and 3.5 relaxed the conservatism of the previous works 29 by representing the interactions among the fuzzy subsystems in a matrix and solving it in a numerical manner.For comparison, if there is no time-varying delay exit in 2.1 , we consider the following discrete-time stochastic systems 29 :

3.39
Remark 3.7.This class of uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities has been considered in 29 ; a new different Lyapunov functional is then employed to deal with systems with sensor nonlinearities.Then from Theorems 3.1, 3.4, and 3.5, we can get the following corollaries, which are less conservative than Theorem 3.1 in 29 .
Corollary 3.8.Consider the uncertain discrete-time fuzzy stochastic systems in 3.39 .A filter of form 2.13 and constants τ 1 and τ 2 , the filtering error system is robustly stochastic stable with performance γ, if there exist real matrices X > 0, Y > 0, any matrices A Fj , B Fj , scalars ε 1i > 0 and ε 3i > 0 for i, j 1, 2, . . ., r such that the following LMI is satisfied: where

3.41
Moreover, if the previous condition is satisfied, an acceptable state-space realization of the H ∞ filter is given by LMI 3.3 .
Corollary 3.9.Consider the uncertain discrete-time fuzzy stochastic systems in 3.39 .Give a filter of form 2.13 and constants τ 1 and τ 2 , the filtering error system is robustly stochastic stable with performance γ, if there exist real matrices X > 0, Y > 0, any matrices A Fj , B Fj , scalars ε 1i > 0 and ε 3i > 0 for i, j 1, 2, . . ., r such that the following LMIs are satisfied: 13 and constants τ 1 and τ 2 , the filtering error system is robustly stochastic stable with performance γ, if there exist real matrices X > 0, Y > 0, any matrices A Fj , B Fj , scalars ε 1i > 0 and ε 3i > 0 for i, j 1, 2, . . ., r such that the following LMIs are satisfied: In the sequel, special results for the case of linear sensor, that is, φ Cx k Cx k , may be obtained from Theorems 3.1, 3.4, and 3.5.

3.45
Moreover, if the previous condition is satisfied, an acceptable state-space realization of the H ∞ filter is given by LMI 3.3 .

Numerical Example
In this section, two simulation examples are given to illustrate the effectiveness and benefits of the proposed approach.
Example 4.1.Consider the system 3.39 with parameters as follows:  By Theorem 3.1, the minimum achievable noise attenuation level is given by γ min 0.3474 and we can obtain the corresponding filter parameters as follows:   The relaxation technique is adopted in this paper, so Theorems 3.4 and 3.5 relaxed the conservatism more than Theorem 3.1.There has been a decrease in the minimum achievable noise attenuation level γ min .
With the initial conditions x t and x t are 2 − 1.5 T and 2 − 1.5 T , respectively, for an appropriate initial interval.For given τ 1 1, τ 2 5 with γ min 0.2563, according to Theorem 3.5, we apply the filter parameters mentioned previously to the system 2.1 and obtain the simulation results as in Figures 1-3. Figure 1 shows the state response x k under the initial condition.Figure 2 shows the estimation of filter x k .Figure 3 shows error response x k .From these simulation results, we can see that the designed H ∞ filter can stabilize the system 2.1 with sensor nonlinearities and time-varying delay.

Conclusions
In this paper the robust filtering problem for a class of uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities and time-varying delay has been developed.A new type of Lyapunov-Krasovskii functional has been constructed to derive some sufficient conditions for the filters in terms of LMIs, which guarantees a prescribed H ∞ performance index for the filtering error system.A numerical example has shown the usefulness and effectiveness of the proposed filter design method.

2 Figure 1 :
Figure 1: The state response x k .

Figure 2 :
Figure 2: The estimation of filter x k .

Figure 3 :
Figure 3: The error response x k .
Consider the uncertain discrete-time fuzzy stochastic systems in 3.39 .A filter of form 2.

Table 1 :
Minimum index γ for different methods.Σ 11 , Σ 22 , Σ 33 , Σ 44 , Σ 37 , Σ 47 , Σ 711 , Σ 712 , and Σ 17 are defined in Corollary 3.12, Theorem 3.1, and Corollary 3.11, respectively.Moreover, if the previous conditions are satisfied, an acceptable statespace realization of the H ∞ filter is given by LMI 3.3 .Remark 3.14.In the previous results, there still exists conservativeness; for these, we can also use the delay-dependent stability results of discrete-time systems in 13-16 to derive the less conservative theorems and corollaries for filtering problem of uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities and time-varying delay.
By Theorem 3.4, the minimum achievable noise attenuation level is given by γ min 0.3419 and we can obtain the corresponding filter parameters as follows: By Theorem 3.5, the minimum achievable noise attenuation level is given by γ min 0.2563 and we can obtain the corresponding filter parameters as follows: