Open access peer-reviewed chapter

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

Written By

Junmin Li, Jinsha Li and Ruirui Duan

Submitted: November 1st, 2016 Reviewed: May 16th, 2017 Published: August 30th, 2017

DOI: 10.5772/intechopen.69777

Chapter metrics overview

1,196 Chapter Downloads

View Full Metrics


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.


  • fuzzy control
  • non-fragile guaranteed cost control
  • delay-dependent
  • linear matrix inequality (LMI)
  • T-S fuzzy bilinear model

1. 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 [18]. 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 be obtained. A fuzzy controller is designed via the method titled ‘parallel distributed compensation (PDC)’ [36], 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,, T-S fuzzy model has been widely studied, and many results have been obtained [18]. 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 [38]. In general, when delay-dependent 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 closed-loop control system [9]. So the problem of non-fragile control has been important issues. Recently, the research of non-fragile control has been paid much attention, and a series of productions have been obtained [1013].

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 dθdt=θv. In this equation, vis the birth rate minus death rate, and θdenotes the population. Obviously, the equation cannot be approximated by a linear model [14].

Most of the existing results focus on the stability analysis and synthesis based on T-S fuzzy model with linear local model. However, when a nonlinear system has of complex nonlinearities, the constructed T-S 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 T-S fuzzy control, so many researchers paid their attentions to the T-S fuzzy models with bilinear rule consequence [1618]. From these papers, it is evident that the T-S 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 T-S 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 discrete-time fuzzy bilinear system. Very recently, a class of nonlinear systems is described by T-S 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 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, d˙(t)<1is required. Refs. [2123] dealt with the uncertain fuzzy systems with time-delay 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, 2432] 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. [3539]. 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. [4042], 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 non-fragile 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, d˙(t)<1and h˙(t)<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 Pbeing 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. The superscript Tmeans matrix transposition. The notion i,j=1sis an abbreviation of i=1sj=1s. Matrices, if the dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.


2. 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:

Plant Rule i:IFϑ1(t) is Fi1 and  and ϑv(t) is FivTHEN x˙(t)=Aix(t)+Adix(td(t))+Biu(t)+Bhiu(th(t))+Nix(t)u(t)+Ndix(td(t))u(th(t))            x(t)=φ(t),   t[τ1,0] , i=1,2,,sE1

where Fijis the fuzzy set, sis the number of fuzzy rules, x(t) ∈ Rnis the state vector, and u(t) ∈ Ris the control input, ϑ1(t), ϑ2(t),…,ϑv(t) are the premise variables. It is assumed that the premise variables do not depend on the input u(t). Ai,Adi,Ni,NdiRn×n,Bi,BhiRn×1denote the system matrices with appropriate dimensions. d(t) is a time-varying differentiable function that satisfies 0 ≤ d(t) ≤ τ1, 0 ≤ h(t) ≤ τ2, where τ1, τ2 are real positive constants as the upper bound of the time-varying delay. It is also assumed that d˙(t)σ1, h˙(t)σ2, and σ1, σ2 are known constants. The initial conditions φ(t), ϕ(t) are continuous functions of t, t[-τ,  0],τ=min(τ1,τ2).

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:



hi(ϑ(t))=ωi(ϑ(t))/i=1sωi(ϑ(t)),ωi(ϑ(t))=j=1vμij(ϑ(t)), μij(ϑ(t)) is the grade of membership of ϑi(t) in Fij. In this paper, it is assumed that ωi(ϑ(t))0,i=1sωi(ϑ(t))>0for all t. Then, we have the following conditions hi(ϑ(t)) ≥ 0, i=1shi(ϑ(t))=1for all t. In the consequent, we use abbreviation hi, hhi, xd(t), ud(t), xh(t), uh(t), to replace hi(ϑ(t)), hi(ϑ(th(t))), x(td(t)), u(td(t)), x(th(t)), u(th(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).


3. Non-fragile guaranteed cost controller design

Extending the design concept in Ref. [17], we give the following non-fragile fuzzy control law:

IF ϑ1(t) is F1i and  and ϑv(t) is FviTHEN u(t)=ρ(Ki+ΔKi)x(t)1+xT(Ki+ΔKi)T(Ki+ΔKi)x=ρsinθi=ρcosθi(Ki+ΔKi)x(t)E3

where ρ> 0 is a scalar to be assigned, and KiRl×nis a local controller gain to be determined. ΔKirepresents the additive controller gain perturbations of the form ΔKi= HiFi(t)Ekiwith Hiand Ekibeing known constant matrices, and Fi(t) the uncertain parameter matrix satisfying FiT(t)Fi(t)I.sinθi=K¯ix(t)1+xTK¯iTK¯ix, cosθi=11+xTK¯iTK¯ix, θi[π2,π2], K¯i=Ki+ΔKi(t)=Ki+HiFi(t)Eki.

The overall fuzzy control law can be represented by


When there exists an input delay h(t), we have that


where sinϕl=K˜lxh(t)1+xhTK˜lTK˜lxh, cosϕl=11+xhTK˜lTK˜lxh, ϕl[π2,π2], K˜l=Kl+ΔKl(th(t))=Kl+HlFl(th(t))Ekl.

So, it is natural and necessary to make an assumption that the functions hiare well defined all t∈ [−τ2, 0], and satisfy the following properties:

hi(ϑ(th(t)))0, for i= 1,2,…,s, and i=1shi(ϑ(th(t)))=1.

By substituting Eq. (5) into Eq. (2), the closed-loop system can be given by



Λij=Ai+ρsinθjNi+ρcosθjBiK¯j, Λdil=Adi+ρsinϕlNdi, Λhil=ρcosϕlBhiK˜l.

Given positive-definite symmetric matrices SRn×nand WR, 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 J0 such that the closed-loop system (6) is asymptotically stable and the closed-loop value of the cost function (7) satisfies JJ0.


4. 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 Mand N with appropriate dimensions such that ε> 0, we have MTN + NT MεMTM+ ε−1 NTN.

Lemma 2[21]: Given constant matrices G, Eand a symmetric constant matrix Sof appropriate dimensions. The inequality S + GFE+ ETFTGT< 0 holds, where F(t) satisfies FT(t) F(t) ≤ I if and only if, for some ε> 0, S + εGGT+ ε−1ETE< 0.

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, τ1 > 0, τ2 > 0, σ1 > 0, σ2 > 0, if there exist matrices P> 0, Q1 > 0, Q2 > 0, R1 > 0, R2 > 0, Ki, i= 1, 2,…, s, X1, X2, X3, X4, Y1, Y2, Y3, Y4, 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,

[Tijl**τ1XTτ1R1*τ2ZT0τ2R2]<0,    i,j,l=1,2,,s  E8


T11,ij=Q1+Q2+X1+X1T+Y1Ai+AiTY1T+S+2ερ2Y1Y1T+4ε1(NiTNi+(BiK¯j)T(BiK¯j))         + Z1T+Z1+ρ2K¯iTWK¯i,T21,i=X1T+X2+Z2+Y2Ai+AdiTY1T,     T31,i=Z3Z1+X3+Y3Ai,T22,ij=(1σ1)Q1X2X2T+Y2Adi+AdiTY2T+2ερ2Y2Y2T+4ε1NdiTNdi,T32,i=X3+Y3AdiZ2T,     T33,il=(1σ2)Q2Z3Z3T+2ερ2Y3Y3T+4ε1(BhiK˜l)TBhiK˜lT41,i=P+X4+Z4+Y4AiY1T,T42,i=X4+Y4AiY2T,T43=Z4Y3T,T44=τ1R1+τ2R2Y4Y4T+2ερ2Y4Y4T.

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 X=[X1TX2TX3TX4T]T, Y=[Y1TY2TY3TY4T]T,Z=[Z1TZ2TZ3TZ4T]T, where XkRn×n, YkRn×n, ZkRn×n, k= 1, 2, 3, 4 will be determined later.

Using the Leibniz-Newton formula and system equation (6), we have the following identical equations:


Then, substituting Eq. (12) into Eq. (11) yields


where η(t)=[xT(t), xdT(t), xhT(t), x˙T(t)]T.

Applying Lemma 1, we have the following inequalities:


Substituting Eq. (13) into Eq. (12) results in

V˙(x(t),t)i,j,l=1shihjhhlηT(t)Tijη(t)td(t)tx˙T(s)R1x˙(s)dsth(t)tx˙T(s)R2x˙(s)ds2ηT(t)Xtd(t)tx˙(s)ds2ηT(t)Zth(t)tx˙(s)ds[xT(t)Sx(t)+uT(t)Wu(t)]i,j,l=1shihjhhlηT(t)(Tijl+τ1XR11XT+τ2ZR21ZT)η(t)td(t)t(ηT(t)X+x˙T(s)R1)R11(ηT(t)X+x˙T(s)R1)Tdsth(t)t(ηT(t)Z+x˙T(s)R2)R21(ηT(t)X+x˙T(s)R2)Tds[xT(t)Sx(t)+uT(t)Wu(t)]       i,j,l=1shihjhhlηT(t)(T˜ijl+τ1XR11XT+τ2ZR21ZT)η(t)[xT(t)Sx(t)+uT(t)Wu(t)]E14


  T˜ijl=[T˜11,ij***T21,iT22,i**T31,iT32,ijT33,il*T41,iT42,iT43T44],  T˜11,ij=T11,ij+ρ2K¯iTcosθiWK¯jcosθjρ2K¯iTWK¯j.E107

In light of the inequality K¯iTWK¯j+K¯jTWK¯iK¯iTWK¯i+K¯jTWK¯j, we have


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 Tproduces


Because of V(x(t),t) ≥ 0 and V˙(x(t),t)<0, thus limTV(x(T),T)=c,where cis a nonnegative constant. Therefore, the following inequality can be obtained:


This completes the proof.

Remark 2:In the derivation of Theorem 1, the free-weighting matrices XkRn×n, YkRn×n, k= 1, 2, 3, 4 are introduced, the purpose of which is to reduce conservatism in the existing delay-dependent 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 Y1, Y2, Y3, Y4 are non-singular, we can define the matrix YiT=λZ, i= 1, 2, 3, 4, Z= P−1,λ> 0.

Pre- and post-multiply (8) and (9) with Θ=diag{Y11,Y21,Y31,Y41,Y41,Y41}and ΘT=diag{Y1T,Y2T,Y3T,Y4T,Y4T,Y4T}, respectively, and letting Q¯1=Y11Q1Y1T, Q¯2=Y11Q2Y1T, R¯k=Y41RkY4T,k=1,2,X¯i=Yi1XiYiT,Z¯i=Yi1ZiYiT, i = 1, 2, 3, 4, we obtain the following inequality (18), which is equivalent to (8):

[T¯11,ij*****T¯21,iT¯22,i****T¯31,iT¯32,iT¯33,il***T¯41,iT¯42,iT¯43T¯44**τ1X1τ1X2τ1X3τ1X4τ1R¯1*τ2Z¯1τ2Z¯2τ2Z¯3τ2Z¯40τ2R¯2]<0,    i,j,l=1,2,,sE18


Τ¯11,ij=Q¯1+Q¯2+X¯1+X¯1T+λAiZ+λZAiT+λ2ZSZ+2ερ2I+4ε1λ2ZNiTNiZ+Z1+Z1T+4ε1λ2(BiK¯jZ)T(BiK¯jZ)+ρ2λ2ZK¯iTWK¯iZ,Τ¯21,i=X¯1T+X¯2+Z¯2+λAiZ+λZAdiT,   Τ¯31,i=Z¯3Z¯1+X¯3+λAiZ,Τ¯41,i=λ2Z+λAiZλZ+X¯4+Z¯4,Τ¯22,i=(1σ1)Q¯1X¯2X¯2T+λAdiZ+λZAdiT+2ερ2I+4ε1λ2ZNdiTNdiZ,Τ¯32,i=X¯3Z¯2+λAdiZλZAdiT, Τ¯42,iX¯4+λAiZλZ, Τ¯33,il=(1σ2)Q¯2Z¯3Z¯3T+4ε1λ2(BhiK˜lZ)TBhiK˜lZ+2ερ2I,Τ¯43=Z¯4λZ, Τ¯44=τ1R1+τ2R2λZλZT+2ερ2I.E2256314

Applying the Schur complement to Eq. (18) results in

Γijl=[Φ11,i**Φ21,ijΦ22*Φ31,il0Φ33]<0,    i,j,l=1,2,,sE19





Obviously, the closed-loop fuzzy system (6) is asymptotically stable, if for some scalars λ> 0, there exist matrices Z>0,Q¯>0,R¯>0and X¯1,X¯2,X¯3,K¯i,i=1,2,..,ssatisfying the inequality (19).

Theorem 2.Consider the system (6) associated with cost function (7). For given scalars ρ> 0, τ1 > 0, τ2 > 0, σ1 > 0, σ2 > 0 and λ> 0, δ> 0, if there exist matricesZ>0,Q¯1>0,R¯1>0,Q¯2>0,R¯2>0andX¯1,X¯2,X¯3,X¯4,Mi, i= 1,2,…,s and scalar ε> 0 satisfying the following LMI (20), the system (6) is asymptotically stable and the control law (5) is a fuzzy non-fragile guaranteed cost control law

[Θ1,ijlΘ2,ijlΘ3]<0,    i,j,l=1,2,,sE20

Moreover, the feedback gains are given by






Proof: At first, we prove that the inequality (20) implies the inequality (19). Applying the Schur complement to Eq. (20) results in


Using Lemma 2 and noting Mi= KiZ, by the condition (21), the following inequality holds:


where ΔK˜i=ΔKi(td(t)).

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 α1, α2, α3, α4, α5, such that Z1α1I,1λ2PQ¯1Pα2I,1λ2PQ¯2Pα3I,1λ2PR¯1Pα4I,1λ2PR¯2Pα5I.

Then, define SQ1=Q¯11,SQ2=Q¯21,SR1=R¯11,SR2=R¯21, 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:

minimize{trace(PZ+SQ1Q¯1+SQ2Q¯2+SR1R¯1+SR2R¯2+α1xT(0)x(0)+α2d(0)0xT(s)x(s)ds+α4τ10θ0x˙T(s)x˙(s)dsdθ+α3h(0)0xT(s)x(s)ds+α5τ20θ0x˙T(s)x˙(s)dsdθ}subject to (20),(23),  ε>0,Q¯1>0,Q¯2>0,R¯1>0,R¯2>0,Z>0,αi>0,i=1,,5E24

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


From the system equation (25), some equilibrium points are tabulated in Table 1. According to these equilibrium points, [xeue], which are also chosen as the desired operating points, [xe ue], we can use the similar modeling method that is described in Ref. [16].

[2.0422 1.2178][2.0422 1.2178]20.307720.3077
[3.6626 2.5443][3.6626 2.5443]77.727277.7272
[5.9543 5.5403][5.9543 5.5403]296.2414296.2414

Table 1.

Data for equilibrium points.

Thus, the system (25) can be represented by

R1: if x1 is about 2.0422then

R2: if x1 is about 3.6626, then         x˙δ(t)=A2xδ(t)+Ad2xdδ(t)+B2uδ(t)+Bd2uhδ(t)+N2xδ(t)uδ(t)+Nd2xdδ(t)uhδ(t)R3: if x1 is about 5.9543,then     x˙δ(t)=A3xδ(t)+Ad3xdδ(t)+B3uδ(t)+Bd3uhδ(t)+N3xδ(t)uδ(t)+Nd3xdδ(t)uhδ(t)E26



The cost function associated with this system is given with S=[1001],W=1. The controller gain perturbation ΔKof the additive form is give with H1 = H2 = H3 = 0.1, Ek1 = [0.05 −0.01], Ek2 = [0.02 0.01], Ek3 = [−0.01 0].

The membership functions of state x1 are shown in Figure 1.

Figure 1.

Membership functions.

Then, solving LMIs (23) and (24) for ρ= 0.45, λ= 1.02 and δ=0.11,τ1=τ2=2,σ1=0,σ2=0gives the following feasible solution:

P=[4.27271.30071.30076.4906],Q1 =[14.1872-1.9381-1.938113.0104],Q2 =[3.10291.28381.28382.0181],R1=[8.3691-1.3053-1.30537.0523],R1=[5.20202.27302.27301.0238],ε=1.8043,K1=[-0.4233 -0.5031],K2=[-0.5961 -0.7049],K1=[-0.4593 -0.3874].E43232623

Figures 24 illustrate the simulation results of applying the non-fragile fuzzy controller to the system (25) with xe=[3.66262.5443]Tand ue=77.7272under initial condition ϕ(t) = [1.2 −1.8]T, t∈ [−2 0]. 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 J0 = 292.0399. The simulation results show that the fuzzy non-fragile guaranteed controller proposed in this paper is effective.

Figure 2.

State responses ofx1(t).

Figure 3.

State responses ofx2(t).

Figure 4.

Control trajectory of system.


6. Conclusions

In this paper, the problem of non-fragile guaranteed cost control for a class of fuzzy time-varying 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.



This work is supported by NSFC Nos. 60974139 and 61573013.


  1. 1. Pang CT, Lur YY. On the stability of Takagi-Sugeno fuzzy systems with time-varying uncertainties. IEEE Transactions on Fuzzy Systems. 2008;16:162–170
  2. 2. Zhou SS, Lam J, Zheng WX. Control design for fuzzy systems based on relaxed non-quadratic stability and H∞ performance conditions. IEEE Transactions on Fuzzy Systems. 2007;15:188–198
  3. 3. Zhou SS, Li T. Robust stabilization for delayed discrete-time fuzzy systems via basis-dependent Lyapunov-Krasovskii function. Fuzzy Sets and Systems. 2005;151:139–153
  4. 4. Gao HJ, Liu X, Lam J. Stability analysis and stabilization for discrete-time fuzzy systems with time-varying delay. IEEE Transactions on Systems, Man, Cybernetics, Part B. 2009;39:306–316
  5. 5. Wu HN, Li HX. New approach to delay-dependent stability analysis and stabilization for continuous-time fuzzy systems with time-varying delay. IEEE Transactions on Fuzzy Systems. 2007;15:482–493
  6. 6. Chen B, Liu XP. Delay-dependent robust H∞ control for T-S fuzzy systems with time delay. IEEE Transactions on Fuzzy Systems. 2005;13:238–249
  7. 7. Chen M, Feng G, Ma H, Chen G. Delay-dependent H∞ filter design for discrete-time fuzzy systems with time-varying delays. IEEE Transactions on Fuzzy Systems. 2009;17:604–616
  8. 8. Zhang J, Xia Y, Tao R. New results on H∞ filtering for fuzzy time-delay systems. IEEE Transactions on Fuzzy Systems. 2009;17:128–137
  9. 9. Keel LH, Bhattacharryya SP. Robust, fragile, or optimal?. IEEE Transactions on Automatic Control. 1997;42:1098–1105
  10. 10. Yang GH, Wang JL, Lin C. H∞ control for linear systems with additive controller gain variations. International Journal of Control. 2000;73:1500–1506
  11. 11. Yang GH, Wang JL. Non-fragile H∞ control for linear systems with multiplicative controller gain variations. Automatica. 2001;37:727–737
  12. 12. Zhang BY, Zhou SS, Li T. A new approach to robust and non-fragile H∞ control for uncertain fuzzy systems. Information Sciences. 2007;177:5118–5133
  13. 13. Yee JS, Yang GH, Wang JL. Non-fragile guaranteed cost control for discrete-time uncertain linear systems. International Journal of Systems Science. 2001;32:845–853
  14. 14. Mohler RR. Bilinear Control Processes. New York, NY: Academic; 1973
  15. 15. Elliott DL. Bilinear Systems in Encyclopedia of Electrical Engineering. New York, NY: Wiley; 1999
  16. 16. Li THS, Tsai SH. T-S fuzzy bilinear model and fuzzy controller design for a class of nonlinear systems. IEEE Transactions on Fuzzy Systems. 2007;15:494–505
  17. 17. Tsai SH, Li THS. Robust fuzzy control of a class of fuzzy bilinear systems with time-delay. Chaos, Solitons and Fractals. 2009;39:2028–2040
  18. 18. Li THS, Tsai SH, et al. Robust H∞ fuzzy control for a class of uncertain discrete fuzzy bilinear systems. IEEE Transactions on Systems, Man, Cybernetics, Part B. 2008;38:510–526
  19. 19. Dong J, Wang Y, Yang G. Control synthesis of continuous-time T-S fuzzy systems with local nonlinear models. IEEE Transactions on Systems, Man, Cybernetics, Part B. 2009;39:1245–1258
  20. 20. Zhang G, Li JM. Non-fragile guaranteed cost control of discrete-time fuzzy bilinear system. Journal of Systems Engineering and Electronics. 2010;21:629–634
  21. 21. Ho DWC, Niu Y. Robust fuzzy design for nonlinear uncertain stochastic systems via sliding-mode control. IEEE Transactions on Fuzzy Systems. 2007;15:350–358
  22. 22. Yang DD, Cai KY. Reliable guaranteed cost sampling control for nonlinear time-delay systems. Mathematics and Computers in Simulation. 2010;80:2005–2018
  23. 23. Chen WH, Guan ZH, Lu XM. Delay-dependent output feedback guaranteed cost control for uncertain time-delay systems. Automatica. 2004;44:1263–1268
  24. 24. Chen B, Lin C, Liu XP, Tong SC. Guaranteed cost control of T–S fuzzy systems with input delay. International Journal of Robust Nonlinear Control. 2008;18:1230–1256
  25. 25. Chen B, Liu XP, Tong SC, Lin C. Observer-based stabilization of T–S fuzzy systems with input delay. IEEE Transactions on Fuzzy Systems. 2008;16:652–663
  26. 26. Chen B, Liu X, Tong S, Lin C. guaranteed cost control of T-S fuzzy systems with state and input delay. Fuzzy Sets and Systems. 2007;158:2251–2267
  27. 27. Du BZ, Lam J, Shu Z. Stabilization for state/input delay systems via static and integral output feedback. Automatica. 2010;46:2000–2007
  28. 28. Kim JH. Delay-dependent robust and non-fragile guaranteed cost control for uncertain singular systems with time-varying state and input delays. International Journal of Control, Automation, and Systems. 2009;7:357–364
  29. 29. Li L, Liu XD. New approach on robust stability for uncertain T–S fuzzy systems with state and input delays. Chaos, Solitons and Fractals.2009;40:2329–2339
  30. 30. Yu KW, Lien CH. Robust H-infinite control for uncertain T–S fuzzy systems with state and input delays. Chaos, Solitons and Fractals. 2008;37:150–156
  31. 31. Yue D, Lam J. Non-fragile guaranteed cost control for uncertain descriptor systems with time-varying state and input delays. Optimal Control Applications and Methods. 2005;26:85–105
  32. 32. Zhang G, Li JM. Non-fragile guaranteed cost control of discrete-time fuzzy bilinear system with time-delay. Journal of Dynamic Systems, Measurement and Control. 2014;136:044502–044504
  33. 33. Yue HY, Li JM. Output-feedback adaptive fuzzy control for a class of nonlinear systems with input delay and unknown control directions. Journal of the Franklin Institute. 2013;350:129–154
  34. 34. Yue HY, Li JM. Adaptive fuzzy tracking control for a class of perturbed nonlinear time-varying delays systems with unknown control direction. International Journal of Uncertainty, Fuzziness and Knowledge-based Systems. 2013;21:497–531
  35. 35. Wang RJ, Lin WW, Wang WJ. Stabilizability of linear quadratic state feedback for uncertain fuzzy time-delay systems. IEEE Transactions on Systems, Man, Cybernetics, Part B. 2004;34:1288–1292
  36. 36. Xia ZL, Li JM, Li JR. Delay-dependent fuzzy static output feedback control for discrete-time fuzzy stochastic systems with distributed time-varying delays. ISA Transaction. 2012;51:702–712
  37. 37. Xia ZL, Li JM. Switching fuzzy filtering for nonlinear stochastic delay systems using piecewise Lyapunov-Krasovskii function. International Journal of Fuzzy Systems. 2012;14:530–539
  38. 38. Li JR, Li JM, Xia ZL. Delay-dependent generalized H2 fuzzy static-output-feedback control for discrete T-S fuzzy bilinear stochastic systems with mixed delays. Journal of Intelligent and Fuzzy Systems. 2013;25:863–880
  39. 39. Xia ZL, Li JM, Li JR. Passivity-based resilient adaptive control for fuzzy stochastic delay systems with Markovian switching. Journal of the Franklin Institute-Engineering and Applied Mathematics. 2014;351:3818–3836
  40. 40. Li JM, Li YT. Robust stability and stabilization of fractional order systems based on uncertain T-S fuzzy model with the fractional order. Journal of Computational and Nonlinear Dynamics. 2013;8:041005
  41. 41. Li YT, Li JM. Stability analysis of fractional order systems based on T-S fuzzy model with the fractional order α: 0<α<1. Nonlinear Dynamics. 2014;78:2909–2919
  42. 42. Li YT, Li JM. Decentralized stabilization of fractional order T-S fuzzy interconnected systems with multiple time delays. Journal of Intelligent and Fuzzy Systems. 2016;30:319–331
  43. 43. Li JM, Yue HY. Adaptive fuzzy tracking control for stochastic nonlinear systems with unknown time-varying delays. Applied Mathematics and Computation. 2015;256:514–528
  44. 44. Yue HY, Yu SQ. Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay. Journal of the Franklin Institute-Engineering and Applied Mathematics. 2016;353:713–734
  45. 45. Duan RR, Li JM, Zhang YN, Yang Y, Chen GP. Stability analysis and H-inf control of discrete T-S fuzzy hyperbolic systems. International Journal of Applied Mathematics and Computer Science. 2016;26:133–145
  46. 46. Wang JX, Li JM. Stability analysis and feedback control of T-S fuzzy hyperbolic delay model for a class of nonlinear systems with time-varying delay. Iranian Journal of Fuzzy Systems. 2016;13:111–134
  47. 47. Li JR, Li JM, Xia ZL. Observer-based fuzzy control design for discrete time T-S fuzzy bilinear systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. 2013;21:435–454

Written By

Junmin Li, Jinsha Li and Ruirui Duan

Submitted: November 1st, 2016 Reviewed: May 16th, 2017 Published: August 30th, 2017