Robust and Passive Constrained Fuzzy Control for Discrete Fuzzy Systems with Multiplicative Noises and Interval Time Delay

The passive fuzzy control for discrete-time uncertain Takagi-Sugeno (T-S) fuzzy models with multiplicative noises and time delay is investigated subject to robust asymptotical stability. Applying Jensen’s inequality and free-weighting matrix technique, less conservative sufficient conditions are derived via choosing Lyapunov function to analyze and synthesize the robust asymptotical stability and passivity of closed-loop system. The derived conditions are not strictly linear matrix inequality (LMI) problems, thus the cone complementarity technique is employed to propose a suboptimal technique to solve the proposed nonstrictly LMI problems. An algorithm is developed in this paper to design the fuzzy controller which can be accomplished by state-feedback scheme or output-feedback scheme. Finally, numerical examples are provided to demonstrate the feasibility and applicability of the proposed fuzzy controller design technique.


Introduction
The delay-dependent stability analysis and synthesis of T-S fuzzy model with time delay has been extensively discussed in [1][2][3][4].The time delay is an inherent and unavoidable effect on many practical dynamic nonlinear systems.In literature, the stability conditions of T-S fuzzy model with time delay were derived in terms of LMI problem [5] that can be solved by convex optimization technique.Furthermore, many relaxed techniques have been proposed to extend the maximum allowable delay range.For example, the piecewise Lyapunov function technique, fuzzy Lyapunov function approach, and free-weighting matrix approach have been used in [6][7][8][9][10] to reduce the conservatism of stability and stabilization problems for T-S fuzzy model with time delay.Although the conservatism of concerned problems can be decreased by such approaches, too many variables are needed to be found for satisfying their stability conditions.It is known that the complications of control synthesis and computational demands are increasing when the number of free variables is increased.Hence, the less conservative stability criteria with few free variables for dealing with T-S fuzzy model with time delay are worth to be discussed and investigated.
As well known, the performance requirement is the most important issue in the stability analysis and synthesis of control systems.For attenuating the effect of the external disturbance on systems, many efforts [11][12][13][14][15][16][17] proposed useful techniques such that the attenuation performance of system can be achieved.From [17], it can be found that the dissipativity and its particular case of passivity can be defined as  ∞ performance constraint, positive real performance constraint, strictly input passive performance constraint, strictly output passive performance constraint, and strictly vary passive performance constraint by setting different power supply function [18].Based on the power supply function, the passivity theory proposes a general and elastic tool for dealing with the effect of disturbance on the systems.On the other hand, uncertainty is often an existing phenomenon which is caused by modeling errors and internal perturbations.Generally, the parameter uncertainties of system are considered as norm-bounded time-varying function [1,2,12].Considering external disturbances and uncertainties, the robust stability and passivity become important performances of control systems.
Recently, the stochastic systems have received much attention based on the stochastic modeling approach [19].Therefore, many efforts have been devoted to expand the stability criteria [20,21] from deterministic systems to stochastic ones.Applying the fuzzy modeling approach, the nonlinear stochastic systems can be approximated by blending linear stochastic subsystems with corresponding membership functions.In literature [22][23][24][25][26][27], the nonlinear stochastic systems were represented by T-S fuzzy models in which the consequent part is structured by Itô stochastic differential equations.Since the consequent part of stochastic fuzzy mode belongs to linear stochastic systems, the Itô formula can also be employed to analyze the stability of stochastic T-S fuzzy systems.And then, the parallel distributed compensation (PDC) technique [17] was employed to design the fuzzy controller such that stability of nonlinear stochastic system is achieved.In case of continuous-time T-S fuzzy model, the delay-dependent stability and stabilization problems were studied in [23], robust fuzzy controller problems were discussed in [24][25][26], and robust fuzzy filtering design problem was addressed in [27,28].However, only few efforts [29][30][31][32] have been proposed for solving the stability and stabilization problems of discrete-time stochastic nonlinear systems.
From the above motivations, the fuzzy controller design of discrete uncertain T-S fuzzy model with multiplicative noise and time delay is investigated in this paper subject to passivity and robust asymptotical stability.The time delay effect is concerned as an interval time-varying delay [9] in this paper.Based on the discrete type Jensen inequality [33] and free-weighting matrix technique, the less conservative sufficient conditions are derived via Lyapunov function to achieve the robust asymptotical stability.In addition, the passivity theory is applied to discuss the external disturbance effect on the system.While deriving the conditions, none of the model transportation is used to avoid the potential conservatism of stability criteria in time delay systems.Since the proposed sufficient conditions belong to nonstrictly LMI problems, an algorithm based on cone complementarity technique [34] is developed in this paper.With the proposed algorithm, the feasible solutions of the conditions and allowable maximum upper bound of interval time-varying delay can be found by LMI technique.The main contributions of this paper can be summarized as follows.(1) Achieving passivity performance constraint, a robust fuzzy controller is developed in this paper for discrete uncertain T-S fuzzy model with multiplicative noise and time delay.(2) Comparing previous researches, the proposed fuzzy control method provides less conservatism because it can find bigger allowable maximum upper bound of time delay and its less desired unknown variables reduce the mathematical complexity.At last, two numerical examples are employed to demonstrate the effectiveness and application of the proposed design method.
Notation.The following notations are applied throughout this paper.The tr(A) denotes the trace of matrix A. The I is identity matrix with appropriate dimension.The diag{⋅ ⋅ ⋅ } means block-diagonal matrix.The * denotes the transposed elements of matrices for symmetric position.The {(⋅)} denotes the expected value of function (⋅).Moreover, let (Ω, F, {F  } ≥0 , P) be a complete probability space with filtration {F  } ≥0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous).

System Description and Problem Statement
Applying the fuzzy modeling approach, the nonlinear stochastic systems can be represented by the T-S fuzzy model with multiplicative noise.Hence, the uncertain T-S fuzzy model with interval time-varying delay and multiplicative noise can be structured as follows: where   () = (A  + ΔA  )  () + (A  + ΔA  )  ( −  ()) Besides, ∑  =1 ℎ  (()) = 1, ℎ  (()) ≥ 0 is the grade of membership function, () is the set of premise variables,  is the number of fuzzy rules, and and () are state vector, state delay vector, controller input vector, controlled output vector, measurable output vector, external disturbance input vector, and initial condition, respectively.In addition, the () denotes standard scalar discrete Wiener process (Brownian motion) [19] on (Ω, F, P) with {()} = 0 and { 2 ()} = 1.The time-varying delay () is a positive integer and satisfies  min ≤ () ≤  max .Here,  min and  max are known lower and upper bounds of delay, respectively.Moreover, ΔA  , ΔA  , ΔB  , ΔE  , Δ A  , Δ A  , Δ B  , and Δ E  are defined as follows: where Substituting ( 4) into (1a), the closed-loop uncertain T-S fuzzy model with interval time-varying delay and multiplicative noise can be inferred as follows: where For deriving the stability criteria of this paper, the following definitions and lemmas are necessary to be introduced.Based on the energy concept, the passivity theory provides a useful tool to discuss the effect of external disturbance for achieving attenuation performance.Here, the passivity property is introduced in the following definition.
Definition 1 (see [18]).If there exist constant matrices S 1 , S 2 , and S 3 for satisfying the following inequality, then the closed-loop system ( 5) is called passive with the disturbance V() and controlled output () for all terminal time   > 0: Via the well-known mathematical definition of power supply function [18], the passivity theory includes several performance constraints with setting matrices S 1 , S 2 , and S 3 .In this paper, the generalized power supply function ( 7) is proposed to be the constraint index.Besides, for illustrating the concerned stability concept clearly, the following definition is introduced.
For analyzing the uncertainties of systems, the following lemma is proposed to convert the uncertain matrices into deterministic matrices.
Lemma 3 (see [26]).Given real compatible dimension matrices A, H, and R for any matrix X > 0,  > 0, Δ with Δ  Δ ≤ I, one can find the following results: where In this paper, the following discrete type Jensen inequality is employed to derive the less conservative sufficient conditions.
Lemma 4 (see [33]).For any compatible constant matrices Q = Q  > 0, scalars  min > 0 and  max > 0 satisfying  min <  max and vector function  : [ min ,  min + 1, . . .,  max ] → R   such that the following sums are well defined, it holds that From the above definitions and lemmas, the sufficient conditions are derived in the following section for guaranteeing the robust asymptotical stability and passivity of closedloop system (5).

Stabilization Criteria and Robust Fuzzy Controller Design
In this section, the stability criteria for closed-loop system (5) are derived with both of state-feedback control scheme and output-feedback control scheme.The sufficient conditions derived in this paper are nonstrictly LMI problems.In order to solve the proposed nonstrictly LMI problems, an algorithm is also developed in this section.
In Theorem 5, condition (10) simultaneously includes variables P 1 , P −1 1 , P 5 , and P −1 5 such that ( 10) is not a strictly LMI problem.For applying the LMI technique, let us introduce two new variables, that is, X 1 and X 5 , such that and use X 1 and X 5 to substitute P −1 1 and P −1 5 in condition (10), respectively.Based on the cone complementarity technique [34], the following nonlinear minimization problem is proposed instead of the original nonconvex condition (10): Minimize tr (P 1 X 1 + P 5 X 5 ) Subject to (10) (P −1 1 and P −1 5 are replaced by X 1 and X 5 , resp.) , Although the above minimization problem gives suboptimal solutions for original problem (10), it is much easier to solve (35) than the original nonconvex problem.In order to find the feasible solutions of (35), the following algorithm is proposed.
Step 2. Solve the following LMI problem: Minimize tr (P 1 X  1 + X 1 P  1 + P 5 X  5 + X 5 P  5 ) Subject to (10) (P −1 1 and P −1 5 are replaced by X 1 and X 5 , resp.) , Step 3. Substitute the feasible solutions obtained from Step 2 into (10).If condition ( 10) is satisfied, then go back to Step 2 after increasing the  max until ( 10) is not satisfied with specified  max .In this case, the feasible solutions are obtained and the algorithm can be stopped.Otherwise, go to the next step.
Remark 7. In order to apply the LMI technique, Algorithm 6 is a useful tool to find the feasible solutions of conditions of Theorem 5.In Algorithm 6, the number of desired unknown variables in fuzzy controller design process is 2 2 +  + 8.
From Theorem 3 of [9], one can find that the number of desired unknown variables is 72 3 + 7 2 + 11 + 6. Obviously, the number of desired unknown variables of the proposed method is less than that developed in [9].
Remark 8.In [9], the quadratic transformation inequality "W + W + P ≥ WP −1 W" is often applied for converting the bilinear matrix inequalities into linear matrix inequalities.
During the transformation process, the conservatisms arise to find the solutions of sufficient conditions of Theorems 2 and 3 in [9].Oppositely, the similar bilinear matrix inequalities are solved via the cone complement technique in this paper.
Applying the cone complement technique, the bilinear matrix inequalities are converted into nonstrictly linear matrix inequalities that can be solved by a suboptimal algorithm, that is, Algorithm 6.
Remark 9.In Theorem 5, the free-weighting matrix technique is applied to reduce the conservatism of considered fuzzy controller design problems.By applying the freeweighting matrix technique, more free matrices are added to reduce the conservatism of derived sufficient conditions.However, adding free matrices also increases the computational complexity.In order to balance the incompatible case, it is recommended to use the free-weighting matrix technique as less as possible.
Theorem 5 provides the sufficient conditions ( 10) to design state-feedback fuzzy controller for guaranteeing robust asymptotical stability and passivity of closed-loop system (5) in mean square.In the following, with the few modifications, Theorem 5 can also be applied to find the outputfeedback gains for structuring the fuzzy controller.Based on (1c), the output-feedback fuzzy controller can also be structured via PDC technique such as Introducing (37) into (1a), the closed-loop system can be obtained as follows: where Theorem 10.Given performance parameters S 1 , S 2 ≥ 0, and S 3 and values  min > 0 and  max > 0, the closed-loop system (38) is robustly asymptotically stable and passive in the sense of mean square, if there exist positive definite matrices P 1 > 0, P 2 > 0, P 3 > 0, P 4 > 0 and P 5 > 0, any matrices N 1 , N 2 and N 3 , scalars   > 0 and   > 0, and outputfeedback gains K  such that where With the same Lyapunov function (13), the proof of Theorem 10 can be obtained with similar procedure of proof of Theorem 5. Hence, the proof of Theorem 10 is omitted here.Although the feasible solutions condition (39) of Theorem 10 cannot be directly obtained by using LMI technique, one can also apply Algorithm 6 by substituting sufficient condition (39) for (10).And then, the feasible solutions can be obtained for satisfying condition (39) and hence the modified algorithm is omitted here.
In the following section, the two numerical examples are provided to apply the proposed fuzzy controller design technique in this paper.

Numerical Examples
In this section, two numerical examples apply the proposed fuzzy controller design method in this paper.In the first example, the less conservatism of stability criteria in this paper can be shown and demonstrated.On the other hand, in Example 2, both of Theorems 5 and 10 are applied to design the state-feedback fuzzy controller and outputfeedback fuzzy controller, respectively.
And the membership function of (43a), (43b), and (43c) is chosen as ĥ1 = (1 − 2 1 ())/2 and ĥ2 = 1 − ĥ1 .For comparing the proposed method with that developed in [9], the passivity performance is chosen as  ∞ performance constraint by setting S 1 ≜ 0, S 2 ≜ I, and S 3 ≜ − 2 I.For finding maximum allowable  max , let us study different cases with  min = 2,  min = 5, and  min = 10.From Table 1, one can find that the allowed upper bound of delay  max controlled by the proposed design method is bigger than that of [9].It means that the proposed design method can provide bigger maximum delay bound than the approach developed in [9].Besides, the smaller  ∞ performance level  can be found by using the proposed design method.It should be noted that the stability criterion of this paper possesses less conservatism than that proposed in [9].
Next, we apply the proposed design techniques to find both of state-feedback fuzzy controller and output-feedback fuzzy controller for nonlinear delay Hénon system.
In the following, both design methods in Theorems 5 and 10 will be applied to design the PDC-based fuzzy controller in the terms of ( 4) and (37), respectively, such that closed-loop system is robustly asymptotically stable and passive in mean square.
Case A: State-Feedback.Through Theorem 5 and Algorithm 6 of this paper, the following feasible solutions can be found with the range of time-varying delay between  max = 5 and  min = 1 and chosen passivity (7) From PDC concept, the fuzzy controller can be established with sublinear state-feedback gains in (46) and the membership function, such as With (47), the responses of system (44a), (44b), (44c), (44d), and (44e) by adding the uncertainties and stochastic behaviors are stated in Figure 1 with initial condition (0) = [1 0]  .And the time delay effect is chosen by random block in Simulink of MATLAB and bounded as 1 ≤ () ≤ 5 in this case.Based on the simulation results in this case, the attenuation performance can be checked by the following equation: Obviously, because the radio value of ( 48) is smaller than one, the chosen passivity inequality can be satisfied with the simulation results of this case.Hence, from (48) and Figure 1, the robust asymptotical stability and passivity of system (45a), (45b), and (45c) by adding the uncertainties and stochastic behaviors can be achieved by design fuzzy controller (47).Next, the case of output-feedback controller design for the same considered system will be shown.
Case B: Output-Feedback.Considering the system (45a), (45b), and (45c) with  max = 2 and  min = 1, the following feasible solutions are found to satisfying the Theorem 10 by using modified Algorithm 6: Since the value of ( 51) is smaller than one, the passivity of system can be achieved by the fuzzy controller (50).From (51) and Figure 2, (45a), (45b), and (45c) with design fuzzy controller are robustly asymptotically stable and passive in the mean square.

Conclusion
The fuzzy controller design problems of discrete uncertain T-S fuzzy systems with interval time-varying delay and multiplicative noise were discussed and investigated in this paper.With the free-weighting matrices technique and discrete type Jensen inequality, the less conservative stability criterion was derived by applying Lyapunov function.Although the derived conditions were not strictly convex problems, the cone complementarity method provided a suboptimal technique to solve it with the LMI technique.Through the proposed design method, the PDC-based fuzzy controller can be established by both of state-feedback and output-feedback schemes for guaranteeing the robust asymptotical stability and passivity constraints.Finally, two numerical examples have been provided to demonstrate the effectiveness and usefulness of the proposed design methods.