Robust Control for Fuzzy Nonlinear Uncertain Systems with Discrete and Distributed Time Delays

Robust Control for Fuzzy Nonlinear Uncertain Systems with Discrete and Distributed Time Delays

Rathinasamy Sakthivel^a, Ponnusamy Vadivel^b, Kalidass Mathiyalagan^c,d, and Ju H. Park^d

aDepartment of Mathematics, Sungkyunkwan University, Suwon-440 746, South Korea

bDepartment of Mathematics, Kongu Engineering college, Erode-638 052, India

cInstitute of Cyber-System and Control, Zhejiang University, Yuquan Campus, Hangzhou 310027, PR China

dNonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-dong, Kyongsan 712 – 749, Republic of Korea

Reprint requests to R. S.; E-mail:krsakthivel@yahoo.com

Z. Naturforsch.69a, 569 – 580 (2014) / DOI: 10.5560/ZNA.2014-0050

Received July 19, 2013 / revised June 22, 2014 / published online August 27, 2014

This paper addresses the problem of stability and stabilization issue for a class of fuzzy nonlinear uncertain systems with discrete and distributed time delays. By utilizing a new Lyapunov–Krasovskii functional together with free weighting matrix approach, a new set of delay-dependent sufficient con- ditions are derived which makes the closed loop system robustly asymptotically stable. In particular, the parameter uncertainties are assumed to be norm bounded. Further, a state feedback controller is proposed to guarantee the robust stabilization for uncertain systems and subsequently the controller is constructed in terms of the solution to a set of linear matrix inequalities (LMI). The derived conditions are expressed in the form of linear matrix inequalities which can be efficiently solved via standard LMI toolbox. Further, two numerical examples are provided to demonstrate the effectiveness and less conservatism of the obtained results.

Key words:Fuzzy Nonlinear Systems; Robust Control; Delay Fractioning Technique; Linear Matrix Inequality; Lyapunov–Krasovskii Functional.

1. Introduction

Fuzzy systems in the form of the Takagi–Sugeno (TS) model [1] have recently attracted great interest due to its applications in various fields of science and engineering. The TS fuzzy model gives an effec- tive method to represent complex nonlinear systems by fuzzy sets and fuzzy reasoning. One of the most- popular application areas of fuzzy-logic systems is modelling and control of complex dynamical systems.

Many issues related to the stability analysis and control synthesis of TS fuzzy systems due to its simpler form in the defuzzification have been reported in [2–7] and references therein. It is well known that the stability is- sue is one of the most important problems in analysis and design of control systems. In particular, asymptot- ical stability is one of the main properties of dynami- cal systems, which is a crucial feature in the design of dynamical systems. More precisely, the linear matrix

inequality (LMI) approach has been widely applied to discuss stability analysis for TS fuzzy systems [8–11].

It is well known that time delays are unavoidable in hardware implementations. The existence of time de- lays may lead to oscillation, divergence, and even in- stability in dynamical systems. Therefore, the stabil- ity of time delayed systems has become a topic of the- oretical and practical importance [12–15]. Moreover, uncertainties are inevitable in dynamical systems be- cause of the existence of modelling errors and exter- nal disturbance. Also, the parameter uncertainties are significant while modelling the system and should be taken into account. Therefore, many interesting results are presented on the stability analysis in the presence of parameter uncertainties [16–20].

In recent years, there has been increasing interest in studying stabilization of dynamical system by choos- ing proper controller design [21–23]. In particular, the main issue in stabilization of dynamical system is how

© 2014 Verlag der Zeitschrift für Naturforschung, Tübingen·http://znaturforsch.com

570 R. Sakthivel et al.·Robust Control for Fuzzy Nonlinear Uncertain Systems to reduce the possible conservatism by applying so

many new ideas and techniques. More precisely, an im- portant index for checking the conservatism of stability criteria is to find the maximum delay bound. Moreover the idea of delay-fractioning or partitioning becomes an increasing interest of many researchers due to yield of its less conservative results while studying the dy- namical behaviours via LMI approach when the par- titioning number goes thinner [10]. Kwon et al. [24]

derived a novel delay-dependent stability criteria in terms of LMIs for uncertain dynamic systems with time-varying delays by using a new augmented Lya- punov functional with convex optimization algorithms.

Senthilkumar and Balasubramaniam [9] discussed ro- bustH_∞ control for nonlinear uncertain stochastic TS fuzzy systems with time delays. Very recently, Feng and Lam [25] studied the problem of robust delay- dependent stability analysis and stabilization for dis- tributed delay systems with linear fractional uncertain- ties by introducing an integral partitioning technique.

To the best of authors’ knowledge, the stability and stabilization of fuzzy nonlinear systems with discrete and distributed time delays is not yet fully investigated and still there is room for further research, and this motivates our research. By constructing a new type of Lyapunov–Krasovskii functional (LKF) based on some advanced ideas like delay fractioning technique and using the matrix inequality technique, a new set of sufficient conditions are derived to ensure the robust stability and stabilization of the considered fuzzy sys- tems. The obtained conditions are formulated in terms of LMIs, which can be easily solved numerically by using the MATLAB LMI toolbox. Finally, we provide two numerical examples to demonstrate the effective- ness and less conservatism of the proposed results.

2. Problem Formulation and Preliminaries

In this section, we start by introducing some nota- tions, definitions, and basic results that will be used throughout the paper. The superscripts ‘T’ and ‘(−1)’

stand for matrix transposition and matrix inverse, re- spectively. Rⁿ denotes the n-dimensional Euclidean space.P>0 means thatPis real symmetric and pos- itive definite. Inand 0_nrepresentn-dimensional iden- tity matrix and zero matrix, respectively. In symmetric block matrices or long matrix expressions, we use an asterisk (∗) to represent a term that is induced by sym- metry, and sym(A) is defined asA+A^T; diag{.} stands

for a block-diagonal matrix.k·krefers to the Euclidean vector norm.

Consider the following uncertain TS fuzzy systems with discrete and distributed time delays and nonlinear perturbations together with theith rule as follows:

Plant rule i: IF

θ1(t)isM₁ⁱ and . . . and

θp(t)isMⁱ_p THEN

˙

x(t) =A_i(t)x(t) +A_di(t)x(t−τ) +A_hi(t)

· Z _t

t−h

x(s)ds+B_i(t)u(t) +A_{f i}f_i(x(t),x(t−τ),t), x(t) =φ(t), t∈[−¯τ,0], τ¯=max(τ,h), (1) where θ₁(t), . . . ,θ_p(t) is the premise variable vec- tor;M_qⁱ(i=1,2, . . . ,r, q=1,2, . . . ,p)are fuzzy sets, and r is the number of IF-THEN rules; x(t) is the state vector, and u(t) is the control input vector;

τ and h represent the discrete and the distributed time delay, respectively; φ(t) is the initial condi- tion defined on[−¯τ,0];A_i(t) =A_i+∆A_i(t), A_di(t) = A_di+∆A_di(t), A_hi(t) = A_hi+∆A_hi(t), and B_i(t) = Bi+∆Bi(t) in which Ai, A_di, A_hi, Bi are known constant matrices with appropriate dimensions and

∆A_i(t),∆A_di(t),∆A_hi(t),∆B_i(t) denote the parameter uncertainties in the system and are assumed to satisfy the following assumption [9,26]:

Assumption 1. The parameter uncertainties are de- fined as

∆Ai∆A_di∆A_hi∆Bi

=

H_iF_i(t)E_1i H_iF_i(t)E_2i H_iF_i(t)E_3i H_iF_i(t)E_4i , whereH_i, E_1i, E_2i, E_3i, andE_4iare known real con- stant matrices, and Fi(t) denotes an unknown time- varying matrix function satisfyingF_i^T(t)F_i(t)≤I; here Iis the identity matrix.

Assumption 2. There exist known real constant ma- trices G_1i, G_2i∈R^n×n such that the nonlinear func- tion f_i(·,·,·)∈Rⁿsatisfies the following bounded con- dition:

|fi(x(t),x(t−τ),t)| ≤G1i|x(t)|+G2i|x(t−τ)|, i=1,2. . . ,r.

By inferring from the fuzzy models, the final output of fuzzy system is inferred as

˙ x(t) =

r

∑

i=1

η_i(θ(t))

"

A_i(t)x(t) +A_di(t)x(t−τ) +A_hi(t)

· Z t

t−h

x(s)ds+B_i(t)u(t) +A_fif_i(x(t),x(t−τ),t)

# , (2) where ηi(θ(t)) is the normalized membership func- tion of the inferred fuzzy setω_i(θ(t)). ie.,η_i(θ(t)) =

ω_i(θ(t))

∑^r_i=1ω_i(θ(t)), where ωi(θ(t)) =∏_q=1^p M_qⁱ(θ_q(t)), here M_qⁱ(θ_q(t)) is the grade of the membership function of θ_q(t) in M_qⁱ. According to the theory of fuzzy sets, it is obvious that ωi(θ(t))≥0, i=1,2, . . . ,r,

∑^r_i=1ω_i(θ(t))>0 for any θ(t). Therefore, it implies thatηi(θ(t))≥0,∑^r_i=1ηi(θ(t)) =1 for anyθ(t).

Based on the parallel distributed compensation strat- egy, the fuzzy state-feedback controller obeys the fol- lowing rule:

Control rule i: IF

θ₁(t)isM₁ⁱ and . . . and

θ_p(t)isMⁱ_p THEN

u(t) =K_ix(t), i=1,2. . . ,r,

wherex(t)∈Rⁿis the input of the controller,u(t)∈R^m is the output of the controller, andK_i is the gain ma- trix of the state feedback controller. Thus, the overall fuzzy state feedback controller can be represented by the input-output form

u(t) =

r

∑

i=1

η_i(θ(t))K_ix(t). (3) Then under the control law (3), the overall closed-loop system can be written as

˙ x(t) =

r i=1

∑

ηi(θ(t))

r k=1

∑

ηk(θ(t))

"

A_i(t) +B_i(t)K_k

x(t) +A_di(t)x(t−τ) (4) +A_hi(t)

Z t

t−hx(s)ds+A_fif_i(x(t),x(t−τ),t)

# .

The following lemmas will be used in the main proof:

Lemma 1. [7] Given constant matricesΩ₁,Ω₂, and Ω3with appropriate dimensions, whereΩ₁^T=Ω1and Ω₂^T=Ω2>0,thenΩ1+Ω₃^TΩ₂⁻¹Ω3<0if and only if

Ω₁ Ω₃^T

∗ −Ω₂

<0 or

−Ω₂ Ω3

∗ Ω1

<0. Lemma 2. [25] For any constant matrix Z>0 and a scalar τ>0, if there exists a Lebesgue measurable

function x(·):[t−τ, t]→Rⁿ, then the following in- equalities hold:

− Z t

t−τ

x^T(s)Zx(s)ds≤ −1 τ

_{Z t}

t−τ

x^T(s)ds

·Z Z _t

t−τ

x(s)ds

,

− Z 0

−τ Z t

t+θ

x^T(s)Zx(s)dsdθ≤

− 2 τ²

_{Z 0}

−τ Z t

t+θ

x(s)dsdθ T

Z _{Z 0}

−τ Z t

t+θ

x(s)dsdθ

. Lemma 3. [7] Assume that Ω,Mi, and E are real matrices with appropriate dimensions, and F_i(t) is a matrix function satisfying F_i^T(t)F_i(t) ≤I. Then, Ω+M_iF_i(t)E_i+ [M_iF_i(t)E_i]^T<0 holds if and only if there exist a scalarε>0satisfyingΩ+ε⁻¹M_iM_i^T+ εE_i^TEi<0.

3. Main Results

In this section, we present our main results. First, we derive the sufficient condition for robust stability of TS fuzzy system of the form

˙ x(t) =

r i=1

∑

ηi(θ(t))

"

A_i(t)x(t) +Ad_i(t)x(t−τ)

+Ahi(t) Z _t

t−h

x(s)ds

#

. (5)

More precisely, by constructing a novel LKF together with free-weighting matrix technique, a delay depen- dent condition is derived to make system (5) robustly asymptotically stable. Then the results obtained for system (5) can be extended to study robust stabiliza- tion for the nonlinear uncertain TS fuzzy system (4).

Further, an LMI based algorithm will be developed to design a suitable feedback controller for the uncertain TS fuzzy system (4) such that the resulting closed loop system is asymptotically stable.

Theorem 1. Assume that Assumption1 holds. For given integers m,l≥1, the closed loop system given in (5) is robustly asymptotically stable, if there exist n×n matrices R1j>0, R2j>0, Zj>0, j=1,2,3n×3n matrixP¯>0, ln×ln matrices Q₁>0, Q₂>0, any appropriately dimensioned matrices N1∈R^ln×ln, N2∈ R^n×n, N₃∈R^mn×mn, N₄∈R^n×n, and L₁, L₂, M₁, M₂, and scalarε_1i>0, such that the following LMIs hold

572 R. Sakthivel et al.·Robust Control for Fuzzy Nonlinear Uncertain Systems for i=1,2, . . . ,r:

Ψ=













 Ω

qτ lL₁

qh

mL₂ W_N^T

1NH_i ε_1iE_i^T

∗ −M₁ 0 0 0

∗ ∗ −M₂ 0 0

∗ ∗ ∗ −ε_1iI 0

∗ ∗ ∗ ∗ −ε_1iI















<0, (6)

M_j<R_j2, j=1,2, (7)

where

Ω =W_Q^TQW_Q+W_R^T

1R₁W_R1+W_R^T

2R₂W_R2+W_Z^TZW_Z +sym W_P^T₁PW¯ _P2+LW_L+W_N^T

1NW_N2 , W_P1=





I_n 0_n,n(l+m+4)

0_n,n(l+m+2) In 0_n,2n

0_n,n(l+1) I_n 0_n,n(m+3)



,

W_P2=





0_n,n(l+m+4) In

I_n −I_n 0_n,n(l+m+3)

I_n 0_n,n(l+m+2) −I_n 0_n



,

P¯=





P₁₁ P₁₂ P₁₃

∗ P₂₂ P₂₃

∗ ∗ P₃₃



,

W_Q=









I_n,ln 0_n,n(m+5)

0_n I_n,ln 0_n,n(m+4)

0_n,n(l+1) I_n,mn 0_n,4n 0_n,n(l+2) In,mn 0n,3n







 ,

N=







 N₁ N2

N₃ N₄







 ,

W_R1=









 qτ

lI_n 0_n,n(l+m+4)

0_n,n(l+m+2) ql

τI_n 0_n,2n 0_n,n(l+m+4) q

τ lI_n









 ,

W_R2=







 qh

mI_n 0_n,n(l+m+4)

0_n,n(l+m+3) p_m

hIn 0_n

0_n,n(l+m+4) qh

mI_n







 ,

W_Z=











0_n,n(l+m+4) ^√1

2 τ lI_n 0_n,n(l+m+4) ^√1

2 h mI_n

I_n 0_n,n(l+m+1) −√

2^l

τI_n 0_n,2n I_n 0_n,nl −√

2^m_hI_n 0_n,n(m+3)









 ,

W_N1=









I_n,ln 0_n,n(m+5)

0_n,ln In 0_n,n(m+4)

0_n,n(l+1) In,mn 0n,4n

0_n,n(l+m+4) I_n







 ,

Q=diag{Q₁, −Q₁, Q₂,−Q₂}, R₁=diag{R₁₁,−R₁₁,R₁₂}, R₂=diag{R₂₁,−R₂₁,R₁₂},

Z=diag{Z₁,Z₂, −Z₁,−Z₂,}, L= [L₁ L₂], W_L=

I_n −I_n 0_n,n(l+m+3)

I_n 0_n,n(l+m+2) −I_n 0_n

, W_N2=

A_i 0_n,n(l−1) A_di

m

z }| {

A_hi. . .A_hi 03n −I_n

, E_i=

E_1i 0_n,(l−1)n E_2i

m

z }| { E_3i, . . . ,E_3i 0_n,4n

.

Proof. In order to prove the stability result, we con- sider the following Lyapunov–Krasovskii functional for model (5) based on the delay fractioning idea:

V(t,x(t),x_t(θ)) =

3

∑

j=1

Vj(t,x(t),xt(θ)), (8) where

V₁(t,x(t),x_t(θ)) =ζ₁^T(t)





P₁₁ P₁₂ P₁₃

∗ P₂₂ P₂₃

∗ ∗ P₃₃



ζ₁(t) +

Z t t−^τ

l

γ₁^T(s)Q₁γ₁(s)ds+ Z t

t−^h

m

γ₂^T(s)Q₂γ₂(s)ds, V₂(t,x(t),x_t(θ)) =

Z 0

−^τ_l Z t

t+θ

x^T(s)R₁₁x(s)dsdθ +

Z ₀

−^τ_l Z _t

t+θ

˙

x^T(s)R₁₂x(s)˙ dsdθ +

Z ₀

−_m^h Z _t

t+θ

x^T(s)R₂₁x(s)dsdθ +

Z ₀

−_m^h Z _t

t+θ

x˙^T(s)R₂₂x(s)dsdθ˙ , V3(t,x(t),xt(θ)) =

Z ₀

−^τ_l Z ₀

θ Z _t

t+λ

x˙^T(s)Z₁x(s)˙ dsdλdθ +

Z 0

−_m^h Z 0

θ Z t

t+λ

˙

x^T(s)Z₂x(s)˙ dsdλdθ, wherex_t(θ) =x(t+θ),∀θ∈[τ,¯ 0], ζ₁(t) =

x^T(t)

Z t t−^τ

l

x^T(s)ds Z t

t−_m^hx^T(s)ds T

,

γ₁(t) =

x^T(t)x^T t−τ

l

. . .x^T

t−(l−1)τ l

T

∈R^ln,

andγ2(t) = _{Z t}

t−_m^h

x^T(s)ds . . .

. . .

Z t−^(m−2)h_m t−^(m−1)h_m

x^T(s)ds

Z t−^(m−1)h_m t−h

x^T(s)ds

#T

∈R^mn,

herel,m≥1 are integers.

By evaluating the derivative ofV(t,x(t))along the trajectories of system (5), we obtain

V˙1(t,x(t),xt(θ)) =2

x^T(t) Z _t

t−^τ_l x^T(s)ds Z _t

t−_m^hx^T(s)ds

P

˙

x^T(t) x^T(t)−x^T t−τ

l

x^T(t)−x^T

t−h m

T

+γ₁^T(t)Q₁γ₁(t)−γ₁^T

t−τ l

Q₁γ₁

t−τ l

+γ₂^T(t)Q₂γ2(t)−γ₂^T

t−h m

Q₂γ2

t−h

m

, (9)

V˙2(t,x(t),x_t(θ)) =x^T(t) τ

lR11+h mR21

x(t)

− Z t

t−^τ

l

x^T(s)R₁₁x(s)ds− Z t

t−_m^h x^T(s)R₂₁x(s)ds +x˙^T(t)

τ lR₁₂+h

mR₂₂

x(t)˙ − Z t

t−^τ

l

˙

x^T(s)R₁₂x(s)˙ ds

− Z _t

t−_m^hx˙^T(s)R₂₂x(s)˙ ds, (10)

V˙₃(t,x(t),x_t(θ)) =x˙^T(t) 1

2 τ

l

2Z₁+1 2

h m

2Z₂

˙ x(t)

− Z 0

−^τ

l

Z t t+θ

˙

x^T(s)Z₁x(s)˙ dsdθ

− Z 0

−_m^h Z t

t+θ

˙

x^T(s)Z₂x(s)˙ dsdθ. (11) By applying Lemma2in each integral term of (10) and (11), we obtain

− Z t

t−^τ

l

x^T(s)R₁₁x(s)ds≤ −l τ

Z t t−^τ

l

x^T(s)dsR₁₁

· Z _t

t−^τ_l

x(s)ds,

(12)

− Z _t

t−_m^h

x^T(s)R₂₁x(s)ds≤ −m h

Z _t t−_m^h

x^T(s)dsR₂₁

· Z t

t−^h

m

x(s)ds,

(13)

− Z 0

−^τ

l

Z t t+θ

˙

x^T(s)Z₁x(s)dsdθ˙ ≤

−2 l

τ 2Z 0

−^τ

l

Z t t+θ

˙

x^T(s)dsdθZ₁ Z 0

−^τ

l

Z t t+θ

˙ x(s)dsdθ

≤ −2 l

τ 2

τ lx^T(t)−

Z t t−^τ_l

x^T(s)dsdθ

·Z₁ τ

lx(t)− Z _t

t−^τ_l

x(s)dsdθ

,

(14)

− Z ₀

−_m^h Z _t

t+θ

x˙^T(s)Z₂x(s)dsdθ˙ ≤

−2m h

2Z ₀

−_m^h Z _t

t+θ

˙

x^T(s)dsdθZ₂ Z ₀

−_m^h Z _t

t+θ

˙ x(s)dsdθ

≤ −2m h

2h mx^T(t)−

Z t t−_m^h

x^T(s)ds

·Z₂ h

mx(t)− Z t

t−_m^h

x(s)ds

.

(15)

Substituting (9) – (15) in (8), we have V˙(t,x(t),x_t(θ)) =2

x^T(t)

Z _t t−^τ

l

x^T(s)ds Z _t

t−_m^h x^T(s)ds

·P

˙

x^T(t)x^T(t)−x^T t−τ

l

x^T(t)−x^T

t−h m

T

+γ₁^T(t)Q₁γ1(t)−γ₁^T t−τ

l

Q₁γ1

t−τ

l

+γ₂^T(t)Q₂γ2(t)−γ₂^T t−h

m

Q₂γ2

t− h

m

+x^T(t) τ

lR₁₁+h mR₂₁

x(t) +x˙^T(t) τ lR₁₂+h

mR₂₂ +1

2 τ

l 2

Z1+1 2

h m

2

Z2

! x(t˙ )

−l τ

Z _t t−^τ_l

x^T(s)dsR₁₁ Z _t

t−^τ_l

x(s)ds

−m h

Z t t−_m^h

x^T(s)dsR₂₁ Z t

t−_m^h

x(s)ds−

Z t t−^τ_l

˙

x^T(s)R₁₂x(s)ds˙

− Z _t

t−_m^h

˙

x^T(s)R₂₂x(s)˙ ds−2 l

τ 2

τ lx^T(t)

− Z t

t−^τ_l

x^T(s)dsdθ

Z₁ τ

lx(t)− Z t

t−^τ_l

x(s)dsdθ

−2m h

2h mx^T(t)−

Z t t−_m^h

x^T(s)ds

Z₂ h

mx(t)

− Z t

t−^h

m

x(s)ds

. (16)

574 R. Sakthivel et al.·Robust Control for Fuzzy Nonlinear Uncertain Systems On the other hand, for any appropriately dimensioned

matricesL₁, L₂and for matricesM_j satisfyingM_j<

R_j2, j=1,2, we have 2ζ^T(t)L₁

x(t)−x

t−τ

l −

Z _t t−^τ_l

˙ x(s)ds

=0, (17) 2ζ^T(t)L₂

x(t)−x

t−h

m

− Z _t

t−_m^h

˙ x(s)ds

=0, (18) τ

lζ^T(t)L₁M₁⁻¹L^T₁ζ(t)− Z t

t−^τ_l ζ^T(t)L₁R⁻¹₁₂L^T₁ζ(t)ds

≥0,

(19) h

mζ^T(t)L₂M₂⁻¹L^T₂ζ(t)− Z t

t−^h

m

ζ^T(t)L₂R⁻¹₂₂L^T₂ζ(t)ds

≥0,

(20)

whereζ^T(t) =

γ₁^T(t)x^T(t−τ)γ₂^T(t)^Rt−

h m

t−^(m+1)h_m x^T(s)ds R_t

t−^τ_l x^T(s)ds x^T t−_m^h

˙ x^T(t)

. Also forζ₂^T(t) =

γ₁^T(t) x^T(t−τ) γ₂^T(t) x˙^T(t) and any matrixN= [N₁^TN₂^TN₃^TN₄^T]^T, in whichN1∈R^ln×ln, N₂∈R^n×n,N₃∈R^mn×mn, andN₄∈R^n×n, the following equality holds:

2

r

∑

i=1

η_i(θ(t))ζ₂^T(t)N

A_i(t)x(t) +A_di(t)x(t−τ) +A_hi(t)

Z t t−h

x(s)ds−x(t)˙

=0,

(21)

⇒

r

∑

i=1

η_i(θ(t))ζ^T(t) sym(W_N^T

1NW_N2) +ε_1i⁻¹W_N^T

1NH_iH_i^TN^TW_N1+ε_1iE^TE

ζ(t) =0.

(22)

It follows from (16) and (17) – (22) that

V˙(t,x(t),x_t(θ))≤

r i=1

∑

ηi(θ(t))ζ^T(t)

"

Ω+τ

lL₁M₁⁻¹L^T₁ +h

mL₂M₂⁻¹L^T₂+ε_1i⁻¹W_N^T

1NH_iH_i^TN^TW_N1 +ε_1iE^TE

# ζ(t)−

Z t t−^τ

l

ζ^T(t)L₁+x˙^T(s)R₁₂ R⁻¹₁₂

·

L^T₁ζ(t) +R12x(s)˙

ds (23)

− Z _t

t−_m^h

ζ^T(t)L₂+x˙^T(s)R₂₂ R⁻¹₂₂

L^T₂ζ(t) +R22x(s)˙ ds

≤

r

∑

i=1

ηi(θ(t))ζ^T(t)

Ω+τ

lL₁M₁⁻¹L^T₁+h

mL₂M₂⁻¹L^T₂ +ε_1i⁻¹W_N^T

1NH_iH_i^TN^TW_N1+ε1iE^TE

ζ(t).

Then, by using Lemma1(Schur complement) in (23), it is easy to obtainΨ in (7). Hence, ifΨ <0 then V˙(t)<0. Therefore, by the Lyapunov–Krasovski the- orem [27], the closed loop fuzzy system (5) is robustly asymptotically stable. This completes the proof.

Next we present the controller design method for the nonlinear fuzzy system (4) through the following The- orem:

Theorem 2. Let Assumptions1and2hold. For given integers m,l ≥1, closed loop system (4) is robustly asymptotically stable, if there exist symmetric posi- tive definite matricesR˜_1j >0, R˜_2j>0, Z˜_j>0, j= 1,2, 3n×3n matrix P˜ >0, ln×ln matrices Q˜₁>

0, Q˜2>0, any appropriately dimensioned matrices X,M˜₁,M˜₂,L˜₁,L˜₂and scalars n_j, j=1,2, . . . ,(l+m+

5),ε_ji>0, j=1,2, such that the following LMIs hold for i,k=1,2, . . . ,r, (k≤i):

Ψ˜ =





















 Ω˜

qτ lL˜₁

qh

mL˜₂ ε1iW_N^T

1nHˆ _i XE¯^T √

2XG¯^T ε2iW_N^T

1nAˆ _{f i}

∗ −M˜₁ 0 0 0 0 0

∗ ∗ −M˜₂ 0 0 0 0

∗ ∗ ∗ −ε_1iI 0 0 0

∗ ∗ ∗ ∗ −ε_1iI 0 0

∗ ∗ ∗ ∗ ∗ −ε_2iI 0

∗ ∗ ∗ ∗ ∗ ∗ −ε_2iI























<0, (24)

wherenˆ=col(n_j), Ω˜ =W_Q^TQW˜ _Q+W_R^T

1R˜₁W_R1+W_R^T

2R˜₂W_R2+W_Z^TZW˜ _Z+sym W_P^T

1PW˜ _P2+LW˜ _L+W_N^T

1nXˆ W¯_N2 ,

P˜=





P˜₁₁ P˜₁₂ P˜₁₃

∗ P˜₂₂ P˜₂₃

∗ ∗ P˜₃₃



,

Q˜=diag{Q˜₁,−Q˜₁,Q˜₂,−Q˜₂}, L˜=L˜₁L˜₂ , R˜₁=diag{R˜₁₁, −R˜₁₁, R˜₁₂},

R˜₂=diag{R˜₂₁, −R˜₂₁, R˜₁₂}, Z˜=diag{Z˜1,Z˜2,−Z˜1, −Z˜2}, W¯_N2 =h

A_i+B_iK_k 0_n,(l−1)n A_di

m

z }| {

A_hi. . .A_hi 0_3n −I_ni , G¯=h

G1i 0_n,(l−1)n G2i 0_n,(m+4)ni , E¯=

E1i+E4iK 0_n,(l−1)n E2i

m

z }| { E3i, . . . ,E3i 0n,4n

,

and other parameters are same as given in Theorem1.

Proof.By following the similar steps of Theorem1for the nonlinear fuzzy system (4), from the equality (21), we get

2

r

∑

i=1

ηi(θ(t))

r

∑

k=1

ηk(θ(t))ζ₂^T(t)Nh A_i(t) +B_i(t)K_k

x(t) +A_di(t)x(t−τ) +A_hi(t) Z t

t−h

x(s)ds +Af_ifi(x(t),x(t−τ),t)−x(t˙ )i

=0. (25)

For any positive scalarε2iand from Assumption2, we have

2

r

∑

i=1

η_i(θ(t))ζ₂^T(t)NA_{f i}f_i(x(t),x(t−τ))

≤

r i=1

∑

ηi(θ(t))ε_2iζ₂^T(t)NA_{f i}A^T_{f i}N^Tζ2(t)

+ε_2i⁻¹

|G_1ix(t)|+|G_2ix(t−τ)|2

≤

r

∑

i=1

η_i(θ(t))ε_2iζ₂^T(t)NA_{f i}A^T_{f i}N^Tζ₂(t) +2ε_2i⁻¹

x^T(t)G^T_1iG1ix(t) +x^T(t−τ)G^T_2iG2ix(t−τ) . Then (25) becomes

r

∑

i=1

η_i(θ(t))

r

∑

k=1

η_k(θ(t))ζ^T(t)h

sym(W_N^T

1NW_N2) +ε1iW_N^T

1NH_iH_i^TN^TW_N1+ε_1i⁻¹E¯^TE¯ +ε_2iW_N^T

1NA_{f i}A^T_{f i}N^TW_N^T

1+2ε_2i⁻¹G¯^TG¯i

ζ(t) =0. (26) It follows from (8) – (20) and (26) that

V˙(t,x(t),x_t(θ))≤

r

∑

i=1

η_i(θ(t))

r

∑

k=1

η_k(θ(t))

Ω (27) +τ

lL₁M₁⁻¹L^T₁+ h

mL₂M₂⁻¹L^T₂+ε_1i⁻¹W_N^T

1NH_iH_i^TN^TW_N1 +ε1iE¯^TE¯+ε_2iW_N^T₁NAf iA^T_{f i}N^TW_N^T₁+2ε_2i⁻¹G¯^TG¯

ζ(t), where

Ω=W_P^T

1PW¯ _P2+W_Q^TQW_Q+W_R^T

1R₁W_R1+W_R^T

2R₂W_R2 +W_Z^TZW_Z+sym LW_L+W_N^T

1NW¯_N2 .

By using Lemma1 (Schur’s complement), (27) be- comes

V˙(t)≤

r i=1

∑

ηi(θ(t))

r k=1

∑

ηk(θ(t))ζ^T(t)Ψ ζ¯ (t), (28) where

Ψ¯ =





















 Ω

qτ lL1

qh

mL2 ε1iW_N^T

1NHi E¯^T √

2 ¯G^T ε2iW_N^T

1NAf i

∗ −M₁ 0 0 0 0 0

∗ ∗ −M₂ 0 0 0 0

∗ ∗ ∗ −ε_1iI 0 0 0

∗ ∗ ∗ ∗ −ε_1iI 0 0

∗ ∗ ∗ ∗ ∗ −ε_2iI 0

∗ ∗ ∗ ∗ ∗ ∗ −ε_2iI























. (29)

For any scalars n_j >0, let N_j =n_jN,¯ j=1,2, . . . , (l+m+5) and applying the congruent transfor- mation to (29) with ˆN = diag{N,˜ I,I,I,I}, where N˜=diag{N¯⁻¹, . . . ,N¯⁻¹} ∈R^(l+m+7).Furthermore, for

simplicity we chooseX =N¯⁻¹,X PX^T=P,˜ X QX^T= Q,˜ X R1X^T=R˜1,X R2X^T=R˜2,X ZX^T=Z˜etc., we ob- tain ˜Ψin (24). If ˜Ψ<0 then we get ˙V(t)<0 and hence closed loop system (4) is robustly asymptotically sta-

576 R. Sakthivel et al.·Robust Control for Fuzzy Nonlinear Uncertain Systems ble. Thus, the fuzzy nonlinear system (2) is robustly

asymptotically stabilizable by the state feedback con- troller given by (3) with gain matricesK_k=Y_kX⁻¹,k= 1,2, . . . ,r. The proof is completed.

4. Numerical Simulation

In this section, we present two numerical exam- ples to demonstrate the effectiveness of the proposed method.

Example 1. Consider the following uncertain TS fuzzy system together with the ith rule given as fol- lows:

Plant Rule 1:IFθ1(t) =x2(t)/0.5 is about 0THEN

˙

x(t) =A₁(t)x(t) +A_d1(t)x(t−τ) +A_h1(t) Z t

t−h

x(s)ds. Plant Rule 2 : IF θ1(t) =x₂(t)/0.5 is less than or about−πorgreater than or aboutπTHEN

x(t) =˙ A₂(t)x(t) +A_d2(t)x(t−τ) +A_h2(t) Z t

t−h

x(s)ds. The membership functions are shown in Figure1. Then the final output of the uncertain TS fuzzy system is inferred as given below:

x(t) =˙

r i=1

∑

η_i(θ(t))

"

A_i(t)x(t) +A_di(t)x(t−τ)

+A_hi(t) Z t

t−hx(s)ds

# ,

(30)

−3 −2 −1 0 1 2 3

−1

−0.5 0 0.5 1 1.5 2

Fig. 1 (colour online). Membership functions for Example1.

where

η₁(x₂(t)) =







0 for|x₂|>π/2,

2

πx₂+1 for −π/2≤x₂<0,

−²

πx2+1 for 0≤x2≤π/2, η2(x₂(t)) =







1 for|x₂|>π/2,

−²

πx₂ for −π/2≤x₂<0,

2

πx₂ for 0≤x₂≤π/2, and the parameters are

A₁=

−2 0 0 −0.9

, A₂=

−2 0 0 −1.4

, A_d1=

−1 0

−1 −1

,A_d2=

−1 0

−1 −1.5

, A_h1=A_h2=

δ 0

0 δ

,H_i=I, F_i(t) =

sin(t) −cos(t)

−cos(t) −sin(t)

, E_1i=E_2i=E_3i=βI, i=1,2.

If the partitioning numberlandmare given, by solving LMI conditions in Theorem1via MATLAB LMI tool- box, one can easily obtain the feasible solutions. If we take the values for distributed delay same as discrete Table 1. Calculated upper bounds ofτ=hforβ=0.

δ 0.1 0.3 0.5 0.7

whenm=l=1 1.6659 1.2455 1.0565 0.9386 whenm=l=2 2.4067 1.6987 1.4033 1.2254 whenm=l=3 2.8376 1.9603 1.6054 1.3939 whenm=l=4 3.1859 2.1859 1.7832 1.5198

Table 2. Calculated upper bounds ofτ=hforδ=0.3

β 0.1 0.3 0.5

whenm=l=1 1.0817 0.8029 0.5968 whenm=l=2 1.4606 1.0609 0.7652 whenm=l=3 1.6465 1.1919 0.8346 whenm=l=4 1.8451 1.3353 0.9269

Table 3. Calculated upper bounds ofτforδ=0.

β 0 0.3

In [20] 2.2615 0.8168

In [26] 2.3675 1.0658

Theorem 1whenm=l=1 2.4483 1.2859 Theorem 1whenm=l=2 4.1320 2.0374 Theorem 1whenm=l=3 5.2511 2.8298 Theorem 1whenm=l=4 6.1416 2.9812