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Mathematics » "Modern Fuzzy Control Systems and Its Applications", book edited by S. Ramakrishnan, ISBN 978-953-51-3390-2, Print ISBN 978-953-51-3389-6, Published: August 30, 2017 under CC BY 3.0 license. © The Author(s).

Chapter 2

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

By Junmin Li, Jinsha Li and Ruirui Duan
DOI: 10.5772/intechopen.69777

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Overview

Membership functions.
Figure 1. Membership functions.
State responses of x1(t).
Figure 2. State responses of x1(t).
State responses of x2(t).
Figure 3. State responses of x2(t).
Control trajectory of system.
Figure 4. Control trajectory of system.

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

Junmin Li, Jinsha Li and Ruirui Duan
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Abstract

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.

Keywords: 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, v is the birth rate minus death rate, and θ denotes 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)<1 is 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)<1 and 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 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. The superscript T means matrix transposition. The notion i,j=1s is 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,,s
(1)

where Fij is the fuzzy set, s is the number of fuzzy rules, x(t) ∈ Rn is the state vector, and u(t) ∈ R is 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×1 denote 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:

x˙(t)=i=1shi(ϑ(t))[Aix(t)+Adix(td(t))+Biu(t)+Bhiu(th(t))+Nix(t)u(t)+Ndix(td(t))u(th(t))]
(2)

where

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))>0 for all t. Then, we have the following conditions hi(ϑ(t)) ≥ 0, i=1shi(ϑ(t))=1 for 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)
(3)

where ρ > 0 is a scalar to be assigned, and KiRl×n is a local controller gain to be determined. ΔKi represents the additive controller gain perturbations of the form ΔKi = HiFi(t)Eki with Hi and Eki being 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

u(t)=i=1shiρK¯ix(t)1+xTK¯iTK¯ix=i=1shiρsinθi=i=1shiρcosθiK¯ix(t)
(4)

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

uh(t)=l=1shhlρsinϕl=l=1shhlρcosϕlK˜lxh(t)
(5)

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 hi are 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

x˙(t)=i,j,l=1shihjhhl(Λijx(t)+Λdijxd(t)+Λhilxh(t))
(6)

where

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

Given positive-definite symmetric matrices SRn×n and WR, we take the cost function

J=0[xT(t)Sx(t)+uT(t)Wu(t)]dt
(7)

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 M and N with appropriate dimensions such that ε > 0, we have MT N + NT Mε MT M + ε−1 NT N.

Lemma 2 [21]: Given constant matrices G, E and a symmetric constant matrix S of 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−1ET E < 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,

JxT(0)Px(0)+d(0)0xT(s)Q1x(s)ds+τ10θ0x˙T(s)R1x˙(s)dsdθ+h(0)0xT(s)Q2x(s)ds+τ20θ0x˙T(s)R2x˙(s)dsdθ=J0
[Tijl**τ1XTτ1R1*τ2ZT0τ2R2]<0,    i,j,l=1,2,,s  
(8)

where Tijl=[T11,ij***T21,iT22,i**T31,iT32,ijT33,il*T41,iT42,iT43T44],  

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

V(x(t),t)=xT(t)Px(t)+td(t)txT(s)Q1x(s)ds+τ10t+θtx˙T(s)R1x˙(s)dsdθ+th(t)txT(s)Q2x(s)ds+τ20t+θtx˙T(s)R2x˙(s)dsdθ
(9)

The time derivatives of V(x(t),t), along the trajectory of the system (6), are given by

V˙(x(t),t)=2xT(t)Px˙(t)+xT(t)(Q1+Q2)x(t)(1d˙(t))xdT(t)Q1xd(t)+x˙T(t)(τ1R1+τ2R2)x˙(t)tτ1tx˙T(s)R1x˙(s)ds(1h˙(t))xhT(t)Q2xh(t)tτ2tx˙T(s)R2x˙(s)ds
(10)

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:

[xT(t)X1+xdT(t)X2+xhT(t)X3+x˙T(t)X4][x(t)xd(t)td(t)tx˙(s)ds]0,[xT(t)Z1+xdT(t)Z2+xhT(t)Z3+x˙T(t)Z4][x(t)xh(t)th(t)tx˙(s)ds]0,i,j=1shihjhl[xT(t)Y1+xdT(t)Y2+xhT(t)Y4+x˙T(t)Y4][Λijx(t)+Λdilxd(t)+Λhilxh(t)x˙(t)]0
(11)

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

V˙(x(t),t)=2xT(t)Px˙(t)+xT(t)(Q1+Q2)x(t)+x˙T(t)(τ1R1+τ2R2)x˙(t)(1d˙(t))xdT(t)Q1xd(t)(1h˙(t))xhT(t)Q2xh(t)tτ1tx˙T(s)R1x˙(s)ds+2ηT(t)X[x(t)xd(t)td(t)tx˙(s)ds]tτ2tx˙T(s)R2x˙(s)ds+2ηT(t)Z[x(t)xh(t)th(t)tx˙(s)ds]+2ηT(t)Yi,j,l=1shihjhhl[Λijx(t)+Λdilxd(t)+Λhilxh(t)x˙(t)]2xT(t)Px˙(t)+xT(t)(Q1+Q2)x(t)+x˙T(t)(τ1R1+τ2R2)x˙(t)(1σ1)xdT(t)Q1xd(t)(1σ2)xhT(t)Q2xh(t)td(t)tx˙T(s)R1(s)x˙(s)ds+2ηT(t)X[x(t)xd(t)td(t)tx˙(s)ds]th(t)tx˙T(s)R2(s)x˙(s)ds+2ηT(t)Z[x(t)xh(t)th(t)tx˙(s)ds]+2ηT(t)Yi,j,l=1shihjhhl[Λijx(t)+Λdilxd(t)+Λhilxh(t)x˙(t)]+xT(t)Sx(t)+i,j=1shihjρ2xT(t)K¯iTcosθiWK¯jcosθjx(t)[xT(t)Sx(t)+uT(t)Wu(t)]
(12)

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

Applying Lemma 1, we have the following inequalities:

2xT(t)Y1Λijx(t)2xT(t)Y1Aix(t)+ερ2xT(t)Y1Y1Tx(t)+ε1xT(t)(NiTNi+(BiK¯j)T(BiK¯j))x(t),2xT(t)Y1Λdilxd(t)2xT(t)Y1Adixd(t)+ερ2sin2φlxT(t)Y1Y1Tx(t)+ε1xdT(t)NdiTNdixd(t),2xT(t)Y1Λhilxh(t)ερ2cos2φlxT(t)Y1Y1Tx(t)+ε1xhT(t)(BhiK˜l)T(BhiK˜l)xh(t),2xdT(t)Y2Λijx(t)2xdT(t)Y2Aix(t)+ερ2xdT(t)Y2Y2Txd(t)+ε1xT(t)(NiTNi+(BiK¯j)T(BiK¯j))x(t),2xT(t)Y1Λhilxh(t)ερ2cos2φlxT(t)Y1Y1Tx(t)+ε1xhT(t)(BhiK˜l)T(BhiK˜l)xh(t),2xdT(t)Y2Λijx(t)2xdT(t)Y2Aix(t)+ερ2xdT(t)Y2Y2Txd(t)+ε1xT(t)(NiTNi+(BiK¯j)T(BiK¯j))x(t),2xdT(t)Y2Λdilxd(t)2xdT(t)Y2Adixd(t)+ερ2sin2φlxdT(t)Y2Y2Txd(t)+ε1xdT(t)NdiTNdixd(t),2xdT(t)Y2Λhilxh(t)ερ2cos2φlxdT(t)Y2Y2Txd(t)+ε1xhT(t)(BhiK˜l)T(BhiK˜l)xh(t),2xhT(t)Y3Λijx(t)2xhT(t)Y3Aix(t)+ερ2xhT(t)Y3Y3Txh(t)+ε1xT(t)(NiTNi+(BiK¯j)T(BiK¯j))x(t),2xhT(t)Y3Λdilxd(t)2xdT(t)Y3Adixd(t)+ερ2sin2φlxhT(t)Y3Y3Txh(t)+ε1xdT(t)NdiTNdixd(t),2xhT(t)Y3Λhilxh(t)ερ2cos2φlxhT(t)Y3Y3Txh(t)+ε1xhT(t)(BhiK˜l)T(BhiK˜l)xh(t),2x˙T(t)Y4Λijx(t)2x˙T(t)Y4Aix(t)+ερ2x˙T(t)Y4Y4Tx˙(t)+ε1xT(t)(NiTNi+(BiK¯j)T(BiK¯j))x(t),2x˙T(t)Y4Λdilxd(t)2x˙T(t)Y4Adixd(t)+ερ2x˙T(t)Y4Y4Tx˙(t)+ε1xdT(t)NdiTNdixd(t),2x˙T(t)Y4Λhilxh(t)ερ2cos2φlx˙T(t)Y4Y4Tx˙(t)+ε1xhT(t)(BhiK˜l)T(BhiK˜l)xh(t)
(13)

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)]
(14)

where

  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.

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

V˙(x(t),t)i,j,l=1shihjhhlηT(t)(Τijl+τ1XR11XT+τ2ZR21ZT)η(t)[xT(t)Sx(t)+uT(t)Wu(t)]
(15)

Applying the Schur complement to Eq. (8) yields

Τii++τ1XR11XT+τ2ZR21ZT<0,Τij+Τji+2τ1XR11XT+2τ2ZR21ZT<0.

Therefore, it follows from Eq. (15) that

V˙(x(t),t)[xT(t)Sx(t)+uT(t)Wu(t)]<0
(16)

which implies that the system (6) is asymptotically stable.

Integrating Eq. (16) from 0 to T produces

0T[xT(t)Sx(t)+uT(t)Wu(t)]dtV(x(T),T)+V(x(0),0)<V(x(0),0)

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

JxT(0)Px(0)+d(0)0xT(s)Q1x(s)ds+τ10θ0x˙T(s)R1x˙(s)dsdθ+h(0)0xT(s)Q2x(s)ds+τ20θ0x˙T(s)R2x˙(s)dsdθ=J0
(17)

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,,s
(18)

where

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.

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

Γijl=[Φ11,i**Φ21,ijΦ22*Φ31,il0Φ33]<0,    i,j,l=1,2,,s
(19)

where

Φ11,i=[T¯¯11,i*****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]

With

Τ¯¯11,i=Q¯1+Q¯2+X¯1+X¯1T+λAiZ+λZAiT+Z1+Z1T+2ερ2I,Τ¯¯22,i=(1σ1)Q¯1X¯2X¯2T+λAdiZ+λZAdiT+2ερ2I,Τ¯¯33=(1σ2)Q¯2Z¯3Z¯3T+2ερ2I.
Φ21,ij=[λZ000λNiZ000λBiK¯jZ000ρλK¯jZ000]Φ31,il=[0λBhiK˜lZ0000λNdiZ0]Φ22=[S10000ε4I0000ε4I0000W1]Φ33=[ε4I00ε4I]

Obviously, the closed-loop fuzzy system (6) is asymptotically stable, if for some scalars λ > 0, there exist matrices Z>0,Q¯>0,R¯>0 and X¯1,X¯2,X¯3,K¯i,i=1,2,..,s satisfying 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 matrices Z>0,Q¯1>0,R¯1>0,Q¯2>0,R¯2>0 and X¯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,,s
(20)

Moreover, the feedback gains are given by

Ki=MiZ1,i=1,2,,s

and

JxT(0)Px(0)+d(0)0xT(s)Q1x(s)ds+τ10θ0x˙T(s)R1x˙(s)dsdθ+h(0)0xT(s)Q2x(s)ds+τ20θ0x˙T(s)R2x˙(s)dsdθ=J0

where

Θ2,ijl=[λEkjZ00000000000λEkiZ0000000000000λEklZ00000000000000000(BiHj)T000000000000ρHiT0000000000000(BhiHl)T],Θ3=diag{δI,δI,δI,δ1I,δ1I,δ1I},
Θ1,ijl=[Τ¯¯11,i***********Τ¯21,iΤ¯¯22,i**********Τ¯31,iΤ¯32,iΤ¯33*********Τ¯41,iΤ¯42,iΤ¯43,iΤ¯44********τ1X¯1τ1X¯2τ1X¯3τ1X¯4τ1R¯1*******τ2Z1τ2Z2τ2Z3τ2Z40τ2R¯21******λZ00000S1*****λNiZ000000ε4I****λBiMj0000000ε4I***ρλMi00000000W1**0λNdiZ00000000ε4I*00λBhiMl00000000ε4I]

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

Φ1,ijl+δ[000000000000000000000000BiHj000ρHi000000BhiHl][00000000(BiHj)T000000000000ρHiT0000000000000(BhiHi)T]
+δ1[(λEkjZ)T(λEkiZ)T000000(λEklZ)T000000000000000000000000000][λEkjZ00000000000λEkiZ0000000000000λEklZ000000000]<0
(21)

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

Φ1,ijl+[0***********00**********000*********0000********00000*******000000******0000000*****00000000****λBiΔK¯jZ00000000***ρλΔK¯iZ000000000**00000000000*00λBhiΔK˜lZ000000000]<0
(22)

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

JxT(0)Px(0)+d(0)0xT(s)Q1x(s)ds+τ10θ0x˙T(s)R1x˙(s)dsdθ+h(0)0xT(s)Q2x(s)ds+τ20θ0x˙T(s)R2x˙(s)dsdθ=J0

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:

[α1IIIZ]0,[α2I1λP1λPSQ1]0,[α3I1λP1λPSQ2]0,[α4I1λP1λPSR1]0,[α5I1λP1λPSR2]0,[ZIIZ]0,[SQ1IIQ¯1]0,[SQ2IIQ¯2]0,[SR1IIR¯1]0,[SR2IIR¯2]0,
(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:

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,,5
(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

x˙1=50x110x13+u(10x1)+u(th)+u(th)(0.5x1(td)+0.2x2(td))+5x2(td)x˙2=50x1100x2u(th)+u(th)(0.3x1(td)0.2x2(td))+10x2(td)5x1(td)
(25)

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

xeT xdeT ueude
[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 2.0422 then

x˙δ(t)=A1xδ(t)+Ad1xdδ(t)+B1uδ(t)+Bh1udδ(t)+N1xδ(t)uδ(t)+Nd1xdδ(t)uhδ(t)
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)
(26)

where

A1=[75.23837.794650100],A2=[98.300511.731550100],A3=[122.12288.857750100],N1=N2=N3=[1001];B1=B2=B3=[100];Ad1=Ad2=Ad3=[05105],Nd1=Nd2=Nd3=[0.50.20.30.2],Bh1=Bh2=Bh3=[10],xδ=x(t)xe',uδ=u(t)ue',xdδ=x(td)xde',uhδ=u(td)uhe'.

The cost function associated with this system is given with S=[1001],W=1 . The controller gain perturbation ΔK of 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.

media/F1.png

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=0 gives 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].

Figures 24 illustrate the simulation results of applying the non-fragile fuzzy controller to the system (25) with xe=[3.66262.5443]T and ue=77.7272 under 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.

media/F2.png

Figure 2.

State responses of x1(t).

media/F3.png

Figure 3.

State responses of x2(t).

media/F4.png

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.

Acknowledgements

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

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