Delay-Dependent Stability Analysis of Uncertain Fuzzy Systems with State and Input Delays under Imperfect Premise Matching

This paper discusses the stability and stabilization problem for uncertain T-S fuzzy systemswith time-varying state and input delays. A new augmented Lyapunov function with an additional triple-integral term and different membership functions of the fuzzy models and fuzzy controllers are introduced to derive the stability criterion, which is less conservative than the existing results. Moreover, a new flexibility design method is also provided. Some numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed method.


Introduction
Since the Takagi-Sugeno (T-S) [1] fuzzy systems were firstly proposed in 1985, their stability analysis has received considerable research attention [2,3].However, most of the existing results are only for the T-S fuzzy systems free of time delay.Actually, time delay often occurs in many practical systems, such as [4,5].It has been shown that the existence of time delay usually becomes the source of instability and deteriorated performance of systems.Therefore, study of the time delay is important in both theory and practice [6].The first stability analysis work on the T-S fuzzy systems with time delay is done in [7,8] by using the Lyapunov-Razumikhin functional approach.If some uncertainties exist in a T-S fuzzy time delay system, they may also significantly affect the system performances and even cause unstable system outputs.Therefore, the issue of the stability for the uncertain T-S fuzzy time delay systems has been widely explored [9][10][11].In the literature, two basic approaches have been utilized, that is, delay-independent approach [12] and delay-dependent approach.The latter makes use of the information on the length of the delays, and it is less conservative than the former one.A lot of stability analysis results have been reported based on the delay-dependent approach [13,14].
During the recent years, some research work for different types of the delays of the T-S fuzzy systems has been published, such as constant delay [15,16], bounded time delay [17], time varying delay [18], and interval time varying delay [19,20].However, all these results are only for the T-S fuzzy systems with state delays.Thus, they may be invalid when applied to the systems with input delays.As we know, input delays extensively exist in industrial processes and can cause instability or serious performance deterioration.In fact, in modern industrial systems, sensors, controllers, and plants are often connected together, and the sampled data and controller signals are transmitted through networks.In view of this, the input delays should be taken into consideration for robust controller design.Intensive results on the stabilization for the T-S fuzzy systems with state and input delays are reported in [21][22][23][24][25][26][27][28].For example, the work in [21] is based on the linear systems with input delays.In [22,24], the authors only consider the fuzzy systems with constant input delays, and their results are conservative.In [23], the authors study the fuzzy systems with both the state and input delays.Unfortunately, the results are obtained without any uncertainty, and the state delay is assumed to be equal to the input delay.In [26], the uncertainty has been considered in the analysis, but the state delay is also assumed Mathematical Problems in Engineering to be the same as the input delay.Some interesting results for the uncertain fuzzy systems with state and input delays have been obtained in [25,27,28], most of which introduce some Lyapunov-Krasovskii functions containing integral terms, for example, ∫  −   ()(), and double-integral terms, ∫ 0 − ∫  + ẋ  () ẋ () .Several triple-integral terms are used in the Lyapunov function [29] to yield less conservative results for the fuzzy systems with state delay.A large portion of robust controller design topics have been investigated on the basis of the Parallel Distribution Compensation (PDC) design technique, where the fuzzy controller shares the same premise membership functions as those of the T-S fuzzy time delay model [7-11, 15-20, 22-28].As a matter of fact, if the membership functions in the premise of the fuzzy rules of the fuzzy controllers are allowed to be designed arbitrarily, we can even achieve better design flexibility.For instance, a fuzzy controller not sharing the same premise rules as those of the T-S fuzzy model referred to as imperfect premise matching is employed to control the nonlinear plants [30,31], and [12] extends the available results to the T-S fuzzy time-delay systems with only the state delay.
In this paper, an augmented Lyapunov-Krasovskii function that contains a triple-integral term is introduced to investigate the stability and stabilization problem for uncertain T-S fuzzy systems with the state and input delays under the imperfect premise matching, in which the fuzzy time delay model and fuzzy controller are with different premise.Some less conservative delay-dependent stability and robust stability conditions are obtained by two integral inequalities and a parameterized model transformation method.Moreover, different from the general PDC design technique, a new design approach of robust stable controllers is proposed.Two simulation examples are further given to illustrate that the proposed design methods are less conservative and more flexible.

System Description and Preliminaries
Let  be the number of the fuzzy rules describing the time delay nonlinear plant.The th rule can be represented as follows. If where    is a fuzzy term of rule  corresponding to the function   (()),  = 1, 2, . . ., ;  = 1, 2, . . ., . () ∈   is the system state vector, and () ∈   is the input vector.The matrices  1 ,  2 , and   ,  = 1, 2, . . ., , are of appropriate dimensions.The initial condition () is a continuous vectorvalued function.The delays  1 () and  2 () are time varying, and satisfy where ℎ  are constants representing the upper bound of the delay. is a positive constant.Δ 1 , Δ 2 , and Δ  denote the uncertainties in the system, and they are the form of where   ,  1 ,  2 , and   are known constant matrices of appropriate dimensions, and   () is unknown matrix function satisfying    ()  () ≤ . is an appropriately dimensioned identity matrix.Hence, the overall fuzzy model can be formulized as follows: ( (()) is the normalized grade of membership function that is a nonlinear function of ().    (  (())) is the grade of the membership corresponding to the fuzzy term of    .Different from the popular PDC design technique, the following fuzzy control law under imperfect premise matching is employed to deal with the problem of stabilization via state feedback.Under the imperfect premise matching, the th rule of the fuzzy controller is defined as follows. If where    denotes the fuzzy set. is a positive integer, and   ∈  × is the feedback gain of rule .The state feedback fuzzy control law is represented by where where Lemma 1 (see [32] (Schur complement)).Given constant matrices Ω 1 , Ω 2 , and Ω 3 , where Lemma 2 (see [33]).Let Q = Q  , D, E, and K() satisfies K  ()K() ≤ I; the following inequality holds if and only if the following inequality holds for any smaller  > 0: Lemma 3 (see [34]).For any constant matrix Σ = Σ  > 0 and a scalar  > 0 such that the following integrations are well defined, we have

Main Results
In this section, some new delay-dependent criteria for the T-S fuzzy systems with state and input delays are proposed by introducing a novel Lyapunov function with an additional triple-integral term under the imperfect premise matching.Moreover, a robust stabilization criterion is also investigated.
Since V(()) < 0, the fuzzy control system with the time varying state and input delays ( 15) is asymptotically stable with the state feedback control law (17).

Robust Stability of Uncertain Fuzzy Systems.
We also examine the design of a robust stable controller for the uncertain system (4) under the imperfect premise matching.Consider the following uncertain fuzzy time varying delay control system: Based on the above results of Theorem 4, a robust stabilization criterion for the T-S fuzzy systems with time varying state and input delays is investigated.The following result can be obtained.
Remark 7. The augmented Lyapunov function with an additional triple-integral term is introduced in analyzing the stability problem for the T-S fuzzy systems with time varying state and input delays.In addition, two integral inequalities are used to derive Theorem 4, and less free-weighting matrices are introduced, which lead to get less conservative results.It can be observed that some existing results are the special cases of this paper.For example, the method in [25] is from (18) with  12 = 0,  22 = 0,  12 =  22 = 0,  12 =  22 = 0, and  1 =  2 = 0 in Theorem 4. The results in [22] are based on system (33) with  2 () = 0 and (18) with  2 = 0,  12 =  22 = 0,  1 =  2 = 0, and  1 =  2 = 0. Remark 8. Different from the general PDC technique, the design method under the imperfect premise matching is much more flexible, because the membership functions of the fuzzy controllers do not need to be chosen the same as those of the fuzzy time delay models.Instead, they can be designed arbitrarily.Thus, the design flexibility is significantly enhanced.On the other hand, some simple membership functions of the fuzzy controllers might be employed, which can reduce the implementation cost.

Numerical Examples
In this section, two numerical examples are given to illustrate the conservativeness and effectiveness of the proposed methods.The first example compares our techniques with the existing ones in the literature for stability analysis, which shows that Theorem 4 in this paper is less conservative than the other results.The second example is used to illustrate the advantage of the robust stability conditions of Theorem 5 and demonstrate how to design robust fuzzy controllers by using our approach.
Example 11.Consider the well-known truck-trailer system, which can be described by the following T-S fuzzy system: (1) Let  = 1, system (47) is the same as the fuzzy system of [22], which implies that our system is more general.Table 2 gives the maximum input delay value of ℎ 2 , for which the stabilization is guaranteed by Theorem 5 as well as the state feedback gains, when  1 = 0.75,  2 = 0.95  2 = 0.1,  3 = 0.2,  4 = 0.7,  5 = 0.1,  6 = 0.4,  7 = 0.8, and  8 = 2.It is clearly visible that the results derived in this paper are better than those from [22,28].
(2) In the case of  ̸ = 1, the state and input delays of system (47) are all exist.The approach proposed in [22] can not be used to get the maximum of state delay ℎ 1 , as the system in [22] only contains the input delay.However, the method in [28] and Theorem 5 can be used.Here, let  = 0.7,  1 = 0.75,  2 = 0.95,  2 = 0.1,  3 = 0.2,  4 = 0.7,  5 = 0.1,  6 = 0.4,  7 = 0.8, and  8 = 3, and Table 3 provides the maximum upper bounds of ℎ 1 and ℎ 2 and the state feedback gains for which the robust stabilization is guaranteed by Theorem 5.
By Theorem 5, we can conclude that the uncertain fuzzy model (47) is robust stable based on the following fuzzy control law: With the PDC design technique, it is required that the fuzzy controller must share the same fuzzy membership functions as those of the fuzzy time delay model.Under such  a condition, the implementation cost of the fuzzy controller is high by employing these complex membership functions.However, from Theorem 5, the membership functions do not need to be chosen the same as  1 ( 1 ()) and  2 ( 1 ()).Thus, we can select some simple membership functions instead, such as Remark 12.In Example 11, our method offers less conservative results in the sense of allowing longer time delay and obtaining smaller feedback control gains.Moreover, compared with [22][23][24][25][26][27][28], where the fuzzy controller must have the same fuzzy membership functions as those of the fuzzy time delay model, the above membership functions of our fuzzy controller are much simper, which can considerably lower the implementation cost of the fuzzy controller and enhance the design flexibility.
Remark 13.If the function of () in  1 ( 1 ()) is unknown, on the basis of the PDC design technique [7-11, 15-20, 22-28], the fuzzy controller cannot be implemented as () is not available.However, using the proposed design method, we can select some simper and well-known membership functions that are different from those of the fuzzy time delay models, for example,  1 ( 1 ()) and  2 ( 1 ()), so as to realize the fuzzy controller.In other words, the proposed design approach can be applied even for the fuzzy time delay models with uncertain grades of membership.

Conclusions
In this paper, we study the robust stability and stabilization problems for general nonlinear fuzzy systems with time varying state and input delays.A less conservative delaydependent robust stability criterion has been obtained by introducing a novel augmented Lyapunov function with an additional triple-integral term.Moreover, a new design approach different from the traditional PDC design technique is proposed, which can significantly improve the design flexibility and reduce the implementation cost of the fuzzy controller by arbitrarily selecting simple fuzzy membership functions.Two numerical examples are used to illustrate the conservativeness and effectiveness of our proposed methods.