Non-Fragile Guaranteed Cost Control of Nonlinear Systems with Different State and Input Delays Based on T-S Fuzzy Local Bilinear Models

This paper focuses on the non-fragile guaranteed cost control problem for a class of Takagi- Sugeno (T-S) fuzzy time-varying delay systems with local bilinear models and different state and input delays. A non-fragile guaranteed cost state-feedback controller is designed such that the closed-loop T-S fuzzy local bilinear control system is delay-dependent asymptotically stable, and the closed-loop fuzzy system performance is constrained to a certain upper bound when the additive controller gain perturbations exist. By employing the linear matrix inequality (LMI) technique, sufficient conditions are established for the existence of desired non-fragile guaranteed cost controllers. The simulation examples show that the proposed approach is effective and feasible.


Introduction
In recent years, T-S (Takagi-Sugeno) model-based fuzzy control has attracted wide attention, essentially because the fuzzy model is an effective and flexible tool for the control of nonlinear systems [1][2][3][4][5][6][7][8]. Through the application of sector nonlinearity approach, local approximation in fuzzy partition spaces or other different approximation methods, T-S fuzzy models will be used to approximate or exactly represent a nonlinear system in a compact set of state variables. The merit of the model is that the consequent part of a fuzzy rule is a linear dynamic subsystem, which makes it possible to apply the classical and mature linear systems theory to nonlinear systems. Further, by using the fuzzy inference method, the overall fuzzy model will existing methods is that a fewer fuzzy rules and less computational burden. The non-fragile guaranteed cost controller was designed for a class of T-S discrete-time fuzzy bilinear systems in Ref. [20]. However, in Refs. [19,20], the time-delay effects on the system is not considered. Ref. [17] is only considered the fuzzy system with the delay in the state and the derivatives of time-delay, _ dðtÞ < 1 is required. Refs. [21][22][23] dealt with the uncertain fuzzy systems with timedelay in different ways. It should be pointed out that all the aforementioned works did not take into account the effect of the control input delays on the systems. The results therein are not applicable to systems with input delay. Recently, some controller design approaches have been presented for systems with input delay, see [2,3,4,18,[24][25][26][27][28][29][30][31][32] for fuzzy T-S systems and [8,15,33,34] for non-fuzzy systems and the references therein. All of these results are required to know the exact delay values in the implementation. T-S fuzzy stochastic systems with state time-vary or distributed delays were studied in Refs. [35][36][37][38][39]. The researches of fractional order T-S fuzzy systems on robust stability, stability analysis about "0 < α < 1", and decentralized stabilization in multiple time delays were presented in Refs. [40][41][42], respectively. For different delay types, the corresponding adaptive fuzzy controls for nonlinear systems were proposed in Refs. [33,43,44]. In Refs. [45,46], to achieve small control amplitude, a new T-S fuzzy hyperbolic model was developed, moreover, Ref. [46] considered the input delay of the novel model. In Ref. [25,47], the problems of observer-based fuzzy control design for T-S fuzzy systems were concerned.
So far, the problem of non-fragile guaranteed cost control for fuzzy system with local bilinear model with different time-varying state and input delays has not been discussed.
In this paper, the problem of delay-dependent non-fragile guaranteed cost control is studied for the fuzzy time-varying delay systems with local bilinear model and different state and input delays. Based on the PDC scheme, new delay-dependent stabilization conditions for the closed-loop fuzzy systems are derived. No model transformation is involved in the derivation. The merit of the proposed conditions lies in its reduced conservatism, which is achieved by circumventing the utilization of some bounding inequalities for the cross-product between two vectors as in Ref. [17]. The three main contributions of this paper are the following: (1) a nonfragile guaranteed cost controller is presented for the fuzzy system with time-varying delay in both state and input; (2) some free-weighting matrices are introduced in the derivation process, where the constraint of the derivatives of time-delay, _ dðtÞ < 1 and _ hðtÞ < 1, is eliminated; and (3) the delay-dependent stability conditions for the fuzzy system are described by LMIs. Finally, simulation examples are given to illustrate the effectiveness of the obtained results.
The paper is organized as follows. Section 2 introduces the fuzzy delay system with local bilinear model, and non-fragile controller law for such system is designed based on the parallel distributed compensation approach in Section 3. Results of non-fragile guaranteed cost control are given in Section 4. Two simulation examples are used to illustrate the effectiveness of the proposed method in Section 5, which is followed by conclusions in Section 6.
Notation: Throughout this paper, the notation P > 0(P ≥ 0) stands for P being real symmetric and positive definite (or positive semi-definite). In symmetric block matrices, the asterisk (*) refers to a term that is induced by symmetry, and diag{….} denotes a block-diagonal matrix.

System description and assumptions
In this section, we introduce the T-S fuzzy time-delay system with local bilinear model. The ith rule of the fuzzy system is represented by the following form: where F ij is the fuzzy set, s is the number of fuzzy rules, x(t) ∈ R n is the state vector, and u(t) ∈ R is the control input, ϑ 1 (t), ϑ 2 (t),…,ϑ v (t) are the premise variables. It is assumed that the premise variables do not depend on the input u(t). A i , A di , N i , N di ∈ R nÂn , B i , B hi ∈ R nÂ1 denote the system matrices with appropriate dimensions. d(t) is a time-varying differentiable function that satisfies 0 ≤ d(t) ≤ τ 1 , 0 ≤ h(t) ≤ τ 2 , where τ 1 , τ 2 are real positive constants as the upper bound of the timevarying delay. It is also assumed that _ dðtÞ ≤ σ 1 , _ hðtÞ ≤ σ 2 , and σ 1 , σ 2 are known constants. The initial conditions φ(t), ϕ(t) are continuous functions of t, t ∈ ½Àτ, 0, τ ¼ minðτ 1 , τ 2 Þ.
Remark 1: The fuzzy system with time-varying state and input delays will be investigated in this paper, which is different from the system in Ref. [17]. In Ref. [17], only state time-varying delay is considered. And also, here, we assume that the derivative of time-varying delay is less than or equal to a known constant that may be greater than 1; the assumption on time-varying delay in Ref. [17] is relaxed.
By using singleton fuzzifier, product inferred and weighted defuzzifier, the fuzzy system can be expressed by the following globe model: is the grade of membership of ϑ i (t) in F ij . In this paper, it is assumed that ω i ðϑðtÞÞ ≥ 0, X s i¼1 ω i ðϑðtÞÞ > 0 for all t. Then, we have the following conditions h i (ϑ(t)) ≥ 0, X s i¼1 h i ðϑðtÞÞ ¼ 1 for all t. In the consequent, we use abbreviation h i , h hi , x d (t), u d (t), x h (t), u h (t), to replace h i (ϑ(t)), h i (ϑ(t À h(t))), x(t À d(t)), u(t À d (t)), x(t À h(t)), u(t À h(t)), respectively, for convenience. The objective of this paper is to design a state-feedback non-fragile guaranteed cost control law for the fuzzy system (2).

Non-fragile guaranteed cost controller design
Extending the design concept in Ref. [17], we give the following non-fragile fuzzy control law: where ρ > 0 is a scalar to be assigned, and K i ∈ R lÂn is a local controller gain to be determined. ΔK i represents the additive controller gain perturbations of the form ΔK i = H i F i (t)E ki with H i and E ki being known constant matrices, and F i (t) the uncertain parameter matrix The overall fuzzy control law can be represented by When there exists an input delay h(t), we have that where sin ϕ l ¼K l xhðtÞ So, it is natural and necessary to make an assumption that the functions h i are well defined all t ∈ [Àτ 2 , 0], and satisfy the following properties: h i ðϑðt À hðtÞÞÞ ≥ 0, for i = 1,2,…,s, and P s i¼1 h i ðϑðt À hðtÞÞÞ ¼ 1.
By substituting Eq. (5) into Eq. (2), the closed-loop system can be given by where Given positive-definite symmetric matrices S ∈ R nÂn and W ∈ R, we take the cost function Definition 1. The fuzzy non-fragile control law u(t) is said to be non-fragile guaranteed cost if for the system (2), there exist control laws (4) and (5) and a scalar J 0 such that the closedloop system (6) is asymptotically stable and the closed-loop value of the cost function (7) satisfies J ≤ J 0 .

Analysis of stability for the closed-loop system
Firstly, the following lemmas are presented which will be used in the paper.
Lemma 1 [20]: Given any matrices M and N with appropriate dimensions such that ε > 0, we have Lemma 2 [21]: Given constant matrices G, E and a symmetric constant matrix S of appropriate dimensions. The inequality S + GFE The following theorem gives the sufficient conditions for the existence of the non-fragile guaranteed cost controller for system (6) with additive controller gain perturbations.
Theorem 1. Consider system (6) associated with cost function (7). For given scalars ρ > 0, , and scalar ε > 0 satisfying the inequalities (8), the system (6) is asymptotically stable and the control law (5) is a fuzzy non-fragile guaranteed cost control law, moreover, where T ijl ¼ Proof: Take a Lyapunov function candidate as The time derivatives of V(x(t),t), along the trajectory of the system (6), are given by Define the free-weighting matrices as where X k ∈ R nÂn , Y k ∈ R nÂn , Z k ∈ R nÂn , k = 1, 2, 3, 4 will be determined later.

Remark 2:
In the derivation of Theorem 1, the free-weighting matrices X k ∈ R nÂn , Y k ∈ R nÂn , k = 1, 2, 3, 4 are introduced, the purpose of which is to reduce conservatism in the existing delaydependent stabilization conditions, see Ref. [17].
Therefore, it follows from Theorem 1 that the system (6) is asymptotically stable and the control law (5) is a fuzzy non-fragile guaranteed cost control law. Thus, we complete the proof.

Now consider the cost bound of
Similar to Ref. [23], we supposed that there exist positive scalars α 1 , α 2 , α 3 , α 4 , α 5 , such that Z À1 ≤ α 1 I, 1 Then, define S Q1 ¼Q 1 À1 , S Q2 ¼Q 2 À1 , S R1 ¼R 1 À1 , S R2 ¼R 2 À1 , by Schur complement lemma, we have the following inequalities: Using the idea of the cone complement linear algorithm in Ref. [24], we can obtain the solution of the minimization problem of upper bound of the value of the cost function as follows: Using the following cone complement linearization (CCL) algorithm [24] can iteratively solve the minimization problem (24). □

Simulation examples
In this section, the proposed approach is applied to the Van de Vusse system to verify its effectiveness.
The cost function associated with this system is given with S ¼  The membership functions of state x 1 are shown in Figure 1.
Then, solving LMIs (23) and (24) for ρ = 0.45, λ = 1.02 and δ ¼ 0:11, τ 1 ¼ τ 2 ¼ 2, σ 1 ¼ 0, σ 2 ¼ 0 gives the following feasible solution:  . It can be seen that with the fuzzy control law, the closed-loop system is asymptotically stable and an upper bound of the guaranteed cost is J 0 = 292.0399. The simulation results show that the fuzzy non-fragile guaranteed controller proposed in this paper is effective.

Conclusions
In this paper, the problem of non-fragile guaranteed cost control for a class of fuzzy timevarying delay systems with local bilinear models has been explored. By utilizing the Lyapunov stability theory and LMI technique, sufficient conditions for the delay-dependent asymptotically stability of the closed-loop T-S fuzzy local bilinear system have been obtained. Moreover, the designed fuzzy controller has guaranteed the cost function-bound constraint. Finally, the effectiveness of the developed approach has been demonstrated by the simulation example. The robust non-fragile guaranteed cost control and robust non-fragile H-infinite control based on fuzzy bilinear model will be further investigated in the future work.