1. Introduction
In recent years, TS (TakagiSugeno) modelbased fuzzy control has attracted wide attention, essentially because the fuzzy model is an effective and flexible tool for the control of nonlinear systems [1–8]. Through the application of sector nonlinearity approach, local approximation in fuzzy partition spaces or other different approximation methods, TS 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 be obtained. A fuzzy controller is designed via the method titled ‘parallel distributed compensation (PDC)’ [3–6], the main idea of which is that for each linear subsystem, the corresponding linear controller is carried out. Finally, the overall nonlinear controller is obtained via fuzzy blending of each individual linear controller. Based on the above content,， TS fuzzy model has been widely studied, and many results have been obtained [1–8]. In practical applications, time delay often occurs in many dynamic systems such as biological systems, network systems, etc. It is shown that the existence of delays usually becomes the source of instability and deteriorating performance of systems [3–8]. In general, when delaydependent results were calculated, the emergence of the inner product between two vectors often makes the process of calculation more complicated. In order to avoid it, some model transformations were utilized in many papers, unfortunately, which will arouse the generation of an inequality, resulting in possible conservatism. On the other hand, due to the influence of many factors such as finite word length, truncation errors in numerical computation and electronic component parameter change, the parameters of the controller in a certain degree will change, which lead to imprecision in controller implementation. In this case, some small perturbations of the controllers’ coefficients will make the designed controllers sensitive, even worse, destabilize the closedloop control system [9]. So the problem of nonfragile control has been important issues. Recently, the research of nonfragile control has been paid much attention, and a series of productions have been obtained [10–13].
As we know, bilinear models have been widely used in many physical systems, biotechnology, socioeconomics and dynamical processes in other engineering fields [14, 15]. Bilinear model is a special nonlinear model, the nonlinear part of which consists of the bilinear function of the state and input. Compared with a linear model, the bilinear models have two main advantages. One is that the bilinear model can better approximate a nonlinear system. Another is that because of nonlinearity of it, many real physical processes may be appropriately modeled as bilinear systems. A famous example of a bilinear system is the population of biological species, which can be showed by
Most of the existing results focus on the stability analysis and synthesis based on TS fuzzy model with linear local model. However, when a nonlinear system has of complex nonlinearities, the constructed TS model will consist of a number of fuzzy local models. This will lead to very heavy computational burden. According to the advantages of bilinear systems and TS fuzzy control, so many researchers paid their attentions to the TS fuzzy models with bilinear rule consequence [16–18]. From these papers, it is evident that the TS fuzzy bilinear model may be suitable for some classes of nonlinear plants. In Ref. [16], a nonlinear system was transformed into a bilinear model via Taylor’s series expansion, and the stability of TS fuzzy bilinear model was studied. Moreover, the result was stretched into the complex fuzzy system with state time delay [17]. Ref. [18] presented robust stabilization for a class of discretetime fuzzy bilinear system. Very recently, a class of nonlinear systems is described by TS fuzzy models with nonlinear local models in Ref. [19], and in this paper, the scholars put forward a new fuzzy control scheme with local nonlinear feedbacks, the advantage of which over the existing methods is that a fewer fuzzy rules and less computational burden. The nonfragile guaranteed cost controller was designed for a class of TS discretetime fuzzy bilinear systems in Ref. [20]. However, in Refs. [19, 20], the timedelay 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 timedelay,
So far, the problem of nonfragile guaranteed cost control for fuzzy system with local bilinear model with different timevarying state and input delays has not been discussed.
In this paper, the problem of delaydependent nonfragile guaranteed cost control is studied for the fuzzy timevarying delay systems with local bilinear model and different state and input delays. Based on the PDC scheme, new delaydependent stabilization conditions for the closedloop 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 crossproduct 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 timevarying delay in both state and input; (2) some freeweighting matrices are introduced in the derivation process, where the constraint of the derivatives of timedelay,
The paper is organized as follows. Section 2 introduces the fuzzy delay system with local bilinear model, and nonfragile controller law for such system is designed based on the parallel distributed compensation approach in Section 3. Results of nonfragile 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 semidefinite). In symmetric block matrices, the asterisk (*) refers to a term that is induced by symmetry, and diag{….} denotes a blockdiagonal matrix. The superscript T means matrix transposition. The notion
2. System description and assumptions
In this section, we introduce the TS fuzzy timedelay system with local bilinear model. The ith rule of the fuzzy system is represented by the following form:
(1) 
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).
Remark 1: The fuzzy system with timevarying state and input delays will be investigated in this paper, which is different from the system in Ref. [17]. In Ref. [17], only state timevarying delay is considered. And also, here, we assume that the derivative of timevarying delay is less than or equal to a known constant that may be greater than 1; the assumption on timevarying 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:
where
The objective of this paper is to design a statefeedback nonfragile guaranteed cost control law for the fuzzy system (2).
3. Nonfragile guaranteed cost controller design
Extending the design concept in Ref. [17], we give the following nonfragile fuzzy control law:
(3) 
where ρ > 0 is a scalar to be assigned, and K_{i} ∈ R^{l×n} 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 satisfying
The overall fuzzy control law can be represented by
When there exists an input delay h(t), we have that
where
So, it is natural and necessary to make an assumption that the functions h_{i} are well defined all t ∈ [−τ_{2}, 0], and satisfy the following properties:
By substituting Eq. (5) into Eq. (2), the closedloop system can be given by
where
Given positivedefinite symmetric matrices S ∈ R^{n×n} and W ∈ R, we take the cost function
Definition 1. The fuzzy nonfragile control law u(t) is said to be nonfragile 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 closedloop value of the cost function (7) satisfies J ≤ J_{0}.
4. Analysis of stability for the closedloop 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 M^{T} N + N^{T} M ≤ ε M^{T} M + ε^{−1} N^{T} N.
Lemma 2 [21]: Given constant matrices G, E and a symmetric constant matrix S of appropriate dimensions. The inequality S + GFE + E^{T}F^{T}G^{T} < 0 holds, where F(t) satisfies F^{T}(t) F(t) ≤ I if and only if, for some ε > 0, S + εGG^{T} + ε^{−1}E^{T} E < 0.
The following theorem gives the sufficient conditions for the existence of the nonfragile 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, τ_{1} > 0, τ_{2} > 0, σ1 > 0, σ_{2} > 0, if there exist matrices P > 0, Q_{1} > 0, Q2 > 0, R_{1} > 0, R_{2} > 0, K_{i}, i = 1, 2,…, s, X_{1}, X_{2}, X_{3}, X_{4}, Y_{1}, Y_{2}, Y_{3}, Y_{4}, and scalar ε > 0 satisfying the inequalities (8), the system (6) is asymptotically stable and the control law (5) is a fuzzy nonfragile guaranteed cost control law, moreover,
where
Proof: Take a Lyapunov function candidate as
(9) 
The time derivatives of V(x(t),t), along the trajectory of the system (6), are given by
(10) 
Define the freeweighting matrices as
Using the LeibnizNewton formula and system equation (6), we have the following identical equations:
(11) 
Then, substituting Eq. (12) into Eq. (11) yields
(12) 
where
Applying Lemma 1, we have the following inequalities:
(13) 
Substituting Eq. (13) into Eq. (12) results in
(14) 
where
In light of the inequality
Applying the Schur complement to Eq. (8) yields
Therefore, it follows from Eq. (15) that
which implies that the system (6) is asymptotically stable.
Integrating Eq. (16) from 0 to T produces
Because of V (x(t),t) ≥ 0 and
(17) 
This completes the proof.
Remark 2: In the derivation of Theorem 1, the freeweighting matrices X_{k} ∈ R^{n×n}, Y_{k} ∈ R^{n×n}, k = 1, 2, 3, 4 are introduced, the purpose of which is to reduce conservatism in the existing delaydependent stabilization conditions, see Ref. [17].
In the following section, we shall turn the conditions given in Theorem 1 into linear matrix inequalities (LMIs). Under the assumptions that Y_{1}, Y_{2}, Y_{3}, Y_{4} are nonsingular, we can define the matrix
Pre and postmultiply (8) and (9) with
(18) 
where
Applying the Schur complement to Eq. (18) results in
where
With
Obviously, the closedloop fuzzy system (6) is asymptotically stable, if for some scalars λ > 0, there exist matrices
Theorem 2. Consider the system (6) associated with cost function (7). For given scalars ρ > 0, τ_{1} > 0, τ_{2} > 0, σ_{1} > 0, σ_{2} > 0 and λ > 0, δ > 0, if there exist matrices
Moreover, the feedback gains are given by
and
where
Proof: At first, we prove that the inequality (20) implies the inequality (19). Applying the Schur complement to Eq. (20) results in
(21) 
Using Lemma 2 and noting M_{i} = K_{i}Z, by the condition (21), the following inequality holds:
(22) 
where
Therefore, it follows from Theorem 1 that the system (6) is asymptotically stable and the control law (5) is a fuzzy nonfragile 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
Then, define
(23) 
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:
(24) 
Using the following cone complement linearization (CCL) algorithm [24] can iteratively solve the minimization problem (24). □
5. Simulation examples
In this section, the proposed approach is applied to the Van de Vusse system to verify its effectiveness.
Example: Consider the dynamics of an isothermal continuous stirred tank reactor for the Van de Vusse
(25) 
From the system equation (25), some equilibrium points are tabulated in Table 1. According to these equilibrium points, [x_{e} u_{e}], which are also chosen as the desired operating points,

 u_{e}  u_{de} 

[2.0422 1.2178]  [2.0422 1.2178]  20.3077  20.3077 
[3.6626 2.5443]  [3.6626 2.5443]  77.7272  77.7272 
[5.9543 5.5403]  [5.9543 5.5403]  296.2414  296.2414 
Thus, the system (25) can be represented by
(26) 
where
The cost function associated with this system is given with
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
Figures 2–4 illustrate the simulation results of applying the nonfragile fuzzy controller to the system (25) with
6. Conclusions
In this paper, the problem of nonfragile 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 delaydependent asymptotically stability of the closedloop TS fuzzy local bilinear system have been obtained. Moreover, the designed fuzzy controller has guaranteed the cost functionbound constraint. Finally, the effectiveness of the developed approach has been demonstrated by the simulation example. The robust nonfragile guaranteed cost control and robust nonfragile Hinfinite control based on fuzzy bilinear model will be further investigated in the future work.