ROBUST CONTROL FOR UNCERTAIN FUZZY SYSTEMS WITH

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 52 No. 5 2009, 673-686

ROBUST CONTROL FOR UNCERTAIN FUZZY SYSTEMS WITH BOTH DISTRIBUTED DELAYS AND INPUT DELAYS Yongmin Li1 , Lingli Wu2 , Yuming Chu3 § , Shengyuan Xu4 1,4 School

of Automation Nanjing University of Science and Technology Jiangsu, Nanjing, 210094, P.R. CHINA 2 School of Educational Science and Technology Huzhou Teachers College Zhejiang, Huzhou, 313000, P.R. CHINA 3 Department of Mathematics Huzhou Teachers College Zhejiang, Huzhou, 313000, P.R. CHINA 3 e-mail: [email protected] Abstract: This paper deals with the problems of robust control for uncertain Takagi-Sugeno (T-S) fuzzy systems with both distributed delays and input delays. The uncertainties are time-varying but norm-bounded. The purpose is to design state feedback fuzzy controllers such that the resulting closed-loop system is robustly stable. Sufficient conditions in the form of linear matrix inequality (LMI) for the solvability of the problem are obtained by utilizing a proper Lyapunov functional together with some matrix transformation techniques. These LMIs can easily be solved in Matlab Toolbox, and thus give an convenient way to construct the desired controller. AMS Subject Classification: 93D09 Key Words: distributed delays, linear matrix inequality, robust control, Takagi-Sugeno (T-S) fuzzy systems

1. Introduction In the last three decades, fuzzy systems in the Takagi-Sugeno (T-S) model Received:

July 24, 2008

§ Correspondence

author

c 2009 Academic Publications

674

Y. Li, L. Wu, Y. Chu, S. Xu

[11] is a powerful tool for modeling complex nonlinear systems such as the inverted pendulum systems [7] and chaotic systems [6]. And thus there has been dramatic progress in the study of stability analysis and controller disign of this model. It is known that, by using the T-S fuzzy model, a nonlinear system can be described as a weighted sum of some simple linear subsystems and then can be stabilized by a model-based fuzzy control. Many stability and control issues related to the T-S fuzzy systems have been studied in the past two decades. In [12, 16], some results on the stability analysis and stabilization synthesis for T-S fuzzy systems were obtained; while the H∞ fuzzy control problem was investigated in [4, 8, 9]. When parametric uncertainties appear in a T-S fuzzy system, the robust stability problem was addressed in [10], where the stability conditions were expressed in terms of LMIs. Sufficient conditions for the solvability of the robust H∞ fuzzy control problem for uncertain T-S fuzzy systems were proposed in [5] by using the algebraic Riccati inequalitybased approach and the LMI-based approach, respectively. On the other hand, it has been shown that the existence of time delays is often one of the main causes of instability and poor performance of a control system [3]. Therefore, the study of stability analysis and control synthesis for time-delay systems has attracted a great deal of attention over the past years. For example, via different approaches, the authors in [2] have investigated the H∞ controller design problem for linear time-delay systems of retarded type. The design of state feedback H∞ controllers for linear neutral time-delay systems was realized in [14] by solving some algebraic Riccati inequalities; this was further extended to neutral delay systems with norm-bounded parametric uncertainties in [15] based on the linear matrix inequality (LMI) approach. In fact, distributed delays always arise when the number of summands in a system is increased and the differences between neighboring argument values are decreased. Also, the existence of distributed delays in a fuzzy delay system may make the problem of fuzzy controller design more complicated and difficult to be solved by traditional method. To the best of our knowledge, so far, no results on the robust control of uncertain T-S fuzzy systems with both distributed delays and input delays are available in the literature, which is still open and remains unsolved. This motivates the present study. In this paper, we are concerned with the problems of robust control for uncertain T-S fuzzy neutral systems with both distributed delays and input delays. The uncertainties are assumed to be time-varying norm-bounded. The purpose is to design the a state feedback fuzzy controller guaranteeing the robust stability of the resulting closed-loop system. Sufficient conditions for

ROBUST CONTROL FOR UNCERTAIN FUZZY SYSTEMS...

675

the solvability of the problem are obtained in terms of certain LMIs, which can be easily implemented by using standard numerical algorithms [1]. Notation. Throughout this paper, for real symmetric matrices X and Y , X ≥ Y (respectively, X > Y ) means that the matrix X − Y is positive semi-definite (respectively, positive definite). I is an identity matrix with appropriate dimension. The superscript “T ” represents the transpose of a matrix. The notation “∗” is used as an ellipsis for terms that are induced by symmetry. Matrices, if the dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Problem Formulation Consider the following uncertain T-S fuzzy system with both distributed delays and input time delays: Plant Rule i: IF s1 (t) is µi1 and s2 (t) is µi2 . . . and sg (t) is µig , i = 1, 2, . . . , r, THEN Z t x(s)ds x(t) ˙ = Ai (t)x(t) + Ahi (t)x(t − τ ) + Adi (t) t−τ

+ B1i (t)u(t) + B2i (t)u(t − τ ), (2.1) x(t) = φ(t),

t ∈ [−τ, 0],

(2.2)

where Ai (t) = Ai +∆Ai (t), Ahi (t) = Ahi +∆Ahi (t), Adi (t) = Adi +∆Adi (t), B1i (t) = B1i +∆B1i (t), B2i (t) = B2i +∆B2i (t); µij is the fuzzy set and r is the number of IF-THEN rules; x(t) ∈ Rn is the state; u(t) ∈ Rm is the control input; τ > 0 are constant time-delay of the fuzzy system; φ(t) is the continuously differentiable initial function on [−τ, 0]; s1 (t), s2 (t), . . . , sg (t) are the premise variables. Throughout this paper, it is assumed that the premise variables do not depend on the input variables u(t). Ai , Ahi , Ad1 , B1i , B2i are known real constant matrices; ∆Ai (t), ∆Ahi (t), ∆Adi (t), ∆B1i (t) and ∆B2i (t) are real-valued unknown matrices representing time-varying parameter uncertainties, and are assumed to be of the form [∆Ai (t), ∆Ahi (t), ∆Adi (t), ∆B1i (t), ∆B2i (t)] = Mi Fi (t)[N1i , N2i , N3i , N4i , N5i ], (2.3) where Mi , N1i , N2i , N3i , N4i and N5i are known real constant matrices of appropriate dimensions and Fi (·) : N 7→ Rl1 ×l2 is an unknown time-varying matrix

676


function satisfying Fi (t)T Fi (t) ≤ I, i = 1, 2, · · · , r.

(2.4)

The uncertain matrices ∆Ai (t), ∆Ahi (t), ∆Adi (t), ∆B1i (t) and ∆B2i (t) are said to be admissible if both (2.3) and (2.4) hold. By using a center-average defuzzier, product inference, and singleton fuzzifier, the dynamic fuzzy model in (2.1) and (2.2) can be represented by r X hi (s(t)){Ai (t)x(t) + Ahi (t)x(t − τ ) x(t) ˙ = (2.5) Σ: i=1 +Adi (t)α(t) + B1i (t)u(t) + B2i (t)u(t − τ )}, x(t) = φ(t),

t ∈ [−τ, 0],

(2.6)

where ω i (s(t)) hi (s(t)) = Pr , j=1 ω j (s(t))

ω i (s(t)) =

g Y

µij (sj (t)),

j=1 t

s(t) = [s1 (t) s2 (t) . . . sg (t)],

α(t) =

Z

x(s)ds,

t−τ

and µij (sj (t)) is the grade of membership of sj (t) in µij . Then, it can be seen that r X ω i (s(t)) ≥ 0, i = 1, 2, . . . , r, ω j (s(t)) > 0, ∀t > 0, i=1

and hi (s(t)) > 0, i = 1, 2, . . . , r,

r X

hi (s(t)) = 1,

∀t > 0.

i=1

Throughout this paper, we shall use the following definitions: Definition 2.1. The uncertain fuzzy delay system (Σ) is said to be robustly stable if the equilibrium solution of system Σ is globally uniformly asymptotically stable for all admissible uncertainties ∆Ai (t), ∆Ahi (t), ∆Adi (t), ∆B1i (t) and ∆B2i (t). In the present study, by the parallel distributed compensation (PDC) technique [12], we are interested in designing a fuzzy state feedback controller in the following form: Control Rule i: IF s1 (t) is µi1 and s2 (t) is µi2 . . . and sg (t) is µig THEN u (t) = Ki x (t) ,

(2.7)

where Ki ∈ Rm×n is the controller gain to be determined. Then, the overall

ROBUST CONTROL FOR UNCERTAIN FUZZY SYSTEMS... fuzzy state feedback controller is given by r X hi (s(t))Ki x(t). u(t) =

677

(2.8)

i=1

The objective of this paper is to design a state feedback fuzzy controller in the form of (2.8) such that the resulting closed-loop system is robustly asymptotically stable. Before concluding this section, we introduce the following lemmas, which will be used in the derivation of our main results in the next sections. Lemma 2.1. (see [13]) Let A, D, S, W and F be real matrices with appropriate dimensions such that Y > 0 and F T F ≤ I. Then we have the following: (1) For any scalar ǫ > 0 and vectors x and y of appropriate dimensions, 2xT DFSy ≤ ǫ−1 xT DD T x + ǫy T S T Sy. (2) For any scalar ǫ > 0 such that W − ǫDDT > 0, (A + DFS)T W −1 (A + DFS) ≤ AT (W − ǫDDT )−1 A + ǫ−1 S T S.

3. Robust Stabilization In this section, we will give some sufficient conditions for the solvability of the robust control stabilization problem formulated in the previous section. First we give the stability analysis for system Σ. Theorem 3.1. The uncertain fuzzy delay system Σ is robustly stable if there exist matrices P > 0, P1 > 0, P2 > 0, and scalars ǫi > 0, such that the following LMIs hold for all i = 1, 2, · · · , r,   TN T Ωi P Ahi + ǫi N1i 2i P Adi + ǫi N1i N3i P Mi TN − P TN  ∗  ǫi N2i ǫi N2i 2i 1 3i   < 0, (3.1) T  ∗ ∗ δi N3i N3i − P2 0  ∗ ∗ ∗ −ǫi I where T Ωi = P Ai + ATi P + P1 + τ32 P2 + ǫi N1i N1i .

(3.2)

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Proof. To establish the robust stability of the system Σ, we consider (2.5) with u(t) ≡ 0 and u(t − τ ) ≡ 0; that is, r X hi (s(t)){Ai (t)x(t) + Ahi (t)x(t − τ ) + Adi (t)α(t)} . (3.3) x(t) ˙ = i=1

For this system, we define the following Lyapunov functional candidate: V (t) = V0 (t) + V1 (t) + V2 (t) + V3 (t),

(3.4)

with Z

t

x(s)T P1 x(s)ds, V0 (t) = x(t) P x(t), V1 (t) = t−τ Z t Z t Z t T V2 (t) = x(θ) dθ P2 x(θ)dθ ds, T

t−τ τ

Z

V3 (t) =

s

ds

0

Z

s

t

(θ + s − t)x(θ)T P2 x(θ)dθ.

t−s

Then, the time derivative of V (t) along the trajectory of the system (3.3) is given by V˙ (t) = V˙ 0 (t) + V˙ 1 (t) + V˙ 2 (t) + V˙ 3 (t), (3.5) where V˙ 0 (t) = 2x(t)T P x(t), ˙ T V˙ 1 (t) = x(t) P1 x(t) − x(t − τ1 )T P1 x(t − τ1 ), and V˙ 2 (t) =

Z

t

t−τ

o nR Rt t ∂ [ s x(θ)T dθ]P2 [ s x(θ)dθ] ∂t

=2 =2

Z

Z

t

T

x(t) P2 t−τ t Z θ

t−τ

=2

t−τ Z t

Z

t

ds −

Z

t

T

x(θ) dθP2

t−τ

Z

t

x(θ)dθ t−τ

x(θ)dθds − α(t)T P2 α(t)

s

x(t)T P2 x(θ)dsdθ − α(t)T P2 α(t) x(t)T P2 x(θ)(θ − t + τ )dθ − α(t)T P2 α(t). (3.6)

t−τ

Similar to (3.6), it can be verified that Z t τ2 V˙ 3 (t) = x(t)T P2 x(t) − x(θ)T P2 x(θ)(θ − t + τ )dθ. 2 t−τ By Lemma 2.1, we have


V˙ 2 (t) ≤

Z

t

679

(θ − t + τ )[x(t)T P2 x(t)

t−τ

τ2 + x(θ)T P2 x(θ)]dθ − α(t)T P2 α(t) = x(t)T P2 x(t) 2 Z t x(θ)T P2 x(θ)(θ − t + τ )dθ − α(t)T P2 α(t). + t−τ

Then, there holds V˙ (t) ≤ 2x(t)T P x(t) ˙ + x(t)T (P1 + τ 2 P2 )x(t) −x(t − τ )T P1 x(t − τ ) − α(t)T P2 α(t).

(3.7)

It follows from (3.7) and Lemma 2.1 that r P 2x(t)T P x(t) ˙ = 2x(t)T P hi (s(t)){Ai (t)x(t) i=1

+Ahi (t)x(t − τ ) + Adi (t)α(t)} r P = 2 hi (s(t))x(t)T P [Ai x(t) + Ahi x(t − τ ) i=1

+Adi α(t)] + 2 ≤ 2

r P

r P

¯i η(t) hi (s(t))x(t)T P Mi Fi (t)N

(3.8)

i=1

hi (s(t))x(t)T P A¯i η(t)

i=1 r P

+

i=1

T T hi (s(t))[ǫ−1 i x(t) P Mi Mi P x(t)

¯T N ¯i η(t)], +ǫi η(t)T N i where ¯i = [N1i , N2i , N3i ], A¯i = [Ai , Ahi , Adi ], N η(t) = [x(t)T , x(t − τ )T , α(t)T ]T . Hence, with the support of the above conditions, we have V˙ (t) ≤ 2x(t)T P x(t) ˙ + x(t)T (P1 + τ 2 P2 )x(t) −x(t − τ )T P1 x(t − τ ) − α(t)T P2 α(t) r X hi (s(t))x(t)T P A¯i η(t) = 2 i=1 r X

+

i=1

T T T ¯T ¯ hi (s(t)) ǫ−1 i x(t) P Mi Mi P x(t) + ǫi η(t) Ni Ni η(t)

+x(t)T (P1 + τ 2 P2 )x(t) − x(t − τ )T P1 x(t − τ ) − α(t)T P2 α(t) r h i X ˜ + ǫi N ¯iT N ¯i η(t), hi (s(t))η(t)T Ω = i=1

680


where T P Ai + ATi P + P1 + τ 2 P2 + ǫ−1 i P Mi Mi P ˜ = Ω ∗ ∗



 P Ahi P Adi −P1 0 . ∗ −P2

So, by the Schur complement formula, it follows from (3.1) that ˜ + ǫi N ¯TN ¯i < 0. Ω i

This means V˙ (t) ≤ −akx(t)k2 , where h i ˜ + ǫi N ¯T N ¯i > 0. a = min λmin − Ω i 1≤i≤r

Therefore, by [3] and Definition 2.1, it can be seen that system Σ is robustly stable. This completes the proof. Now, using the controller in the form (2.8), we give the robust stabilization results in the following theorem. Theorem 3.2. The uncertain fuzzy neutral delay system in Σ is robustly stabilizable if there exist matrices X > 0, X1 > 0, X2 > 0, Yi , and scalars ǫijl > 0, such that following LMIs hold for 1 ≤ i ≤ j ≤ r, l = 1, 2, · · · , r,   Υijl Hijl Adij X2 Gij Gji 2X 2τ X  ∗ −2X1 − 4X 0 Gijl Gjil 0 0    T T  ∗ ∗ −2X2 X2 N3i X2 N3j 0 0     ∗ ∗ ∗ −ǫijl I 0 0 0    < 0, (3.9)  ∗ ∗ ∗ ∗ −ǫijl I 0 0     ∗ ∗ ∗ ∗ ∗ −2X1 0  ∗ ∗ ∗ ∗ ∗ ∗ −2X2 where Adij

= Adi + Adj ,

Υijl = Hij +

T Hij

Hij = (Ai + Aj )X + B1i Yj + B1j Yi , + ǫijl Mij MijT ,

Hijl = Ahi X + Ahj X + B2i Yl + B2j Yl , Gij

T T = XN1i + Yj N4i ,

T T Gijl = XN2i + Yl N5i ,

Mij = [Mi , Mj ].

In this case, the desired stabilizing fuzzy controller can be chosen in the form of (2.8) with the state feedback gains Ki = Yi X −1 ,

i = 1, 2, · · · , r.

(3.10)

Proof. Let P = X −1 , P1 = X1−1 , P2 = X2−1 , then pre- and post-multiplying


681

the LMI in (3.9) by diag(P, P, P2 , I, I, I, I), and denote ˜ ij = Ai + Aj + B1i Kj + B1j Ki , H ˜ ijl = P H ˜ ij + H ˜ T P + ǫijl P Mij M T P, Υ ij ij ˜ Hijl = Ahi + Ahj + B2i Kl + B2j Kl , T T T T ˜ ij = N1i ˜ ijl = N2i G + Kj N4i , G + Kl N5i , Q = −2P P1−1 P − 4P, thenwe have ˜ ijl P H ˜ ijl P Adij Υ  ∗ Q 0   ∗ −2P2  ∗   ∗ ∗ ∗   ∗ ∗ ∗   ∗ ∗ ∗ ∗ ∗ ∗

˜ ij ˜ ji G G 2I 2τ I ˜ ˜ Gijl Gjil 0 0 T T N3i N3j 0 0 −ǫijl I 0 0 0 ∗ −ǫijl I 0 0 ∗ ∗ −2P1−1 0 ∗ ∗ ∗ −2P2−1



      < 0,    

(3.11)

Meanwhile, it is easy to verify that (P1 − P )P1−1 (P1 − P ) ≥ 0, that is −2P P1−1 P − 4P ≥ −2P1 ,

(3.12)

so, from (3.11) and (3.12), using the Schur complement formula, we have  ˆ ˜ ijl P Adij ˜ ij ˜ ji  Υijl P H G G  ∗ ˜ ijl ˜ jil  −2P1 0 G G   T T  ∗  < 0, (3.13) ∗ −2P2 N3i N3j    ∗  ∗ ∗ −ǫijl I 0 ∗ ∗ ∗ ∗ −ǫijl I ˜ ˜ ˜ ˜ where Hij , Hijl , Gij , Gijl are defined in (3.11) and T ˜ ijl = P H ˜ ij + H ˜ ij Υ P + ǫijl P Mij MijT P + 2(P1 + τ 2 P2 ).

Now applying the state feedback controller in (2.8) with the controller gains

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given in (3.10) to system Σ, we obtain the following closed-loop system: r P x(t) ˙ = hi (s(t)){Ai (t)x(t) + Ahi (t)x(t − τ ) i=1

+Adi (t)α(t) + B1i (t) +B2i =

r P

r P

r P

hj (s(t))Kj x(t)

j=1

hl (s(t − τ ))(t)Kl x(t − τ )}

l=1 r r P P

(3.14)

hi (s(t))hj (s(t))hl (s(t − τ )){Ai (t)x(t)

i=1 j=1 l=1

+Ahi (t)x(t − τ )Adi (t)α(t) +B1i (t)Kj x(t) + B2i (t)Kl x(t − τ )}. By the same Lyapunov functional candidate as in (3.4), the time derivative of V (t) along the trajectory of the system (3.14) is similar to (3.7). Note that 2x(t)T P x(t) ˙ r r P r P P hi (s(t))hj (s(t))hl (s(t − τ ))2x(t)T P [(Ai (t) = i=1 j=1 l=1

+B1i (t)Kj )x(t) + (Ahi + B2i (t)Kl )x(t − τ ) + Adi (t)α(t)] r r r P P P = hl (s(t − τ )) hi (s(t))hj (s(t)){x(t)T P [(Ai (t) l=1

i=1 j=1

+B1i (t)Kj )x(t) + (Ahi + B2i (t)Kl )(t)x(t − τ ) + Adi (t)α(t)] +[(Ai (t) + B1i (t)Kj )x(t) + (Ahi +B2i (t)Kl )x(t − τ ) + Adi (t)α(t)]T P x(t)} r r P r P P hl (s(t − τ )) hi (s(t))hj (s(t)){x(t)T P (A¯ijl = 21 i=1 j=1 l=1 +A¯jil )η(t) + η(t)T P (A¯ijl + A¯jil )T x(t) + x(t)T P (∆A¯ijl (t) +∆A¯jil (t))η(t) + η(t)T P (∆A¯ijl (t) + ∆A¯jil (t))T x(t)} r r P r P P hl (s(t − τ )) hi (s(t))hj (s(t)){η(t)T P T (A¯ijl = 21 i=1 j=1 l=1 +A¯jil )η(t) + η(t)T (A¯ijl + A¯jil )T Pη(t) + η(t)T P T (∆A¯ijl (t) +∆A¯jil (t))η(t) + η(t)T (∆A¯ijl (t) + ∆A¯jil (t))T Pη(t)} r r P r P P = 21 hl (s(t − τ )) hi (s(t))hj (s(t))η(t)T Wijl (t)η(t) , i=1 j=1 l=1

(3.15)

where η(t) is given in (3.8) and A¯jil = [Ai + B1i Kj , Ahi + B2i Kl , Adi ], A¯jil = [∆Ai (t) + ∆B1i (t)Kj , ∆Ahi (t) + ∆B2i (t)Kl , ∆Adi (t)], P = [P, 0, 0], Wijl = P(A¯ijl + A¯jil ) + (A¯ijl + A¯jil )T P,


683

∆Wijl (t) = P T (∆A¯ijl (t) + ∆A¯jil (t)) + (∆A¯ijl (t) + ∆A¯jil (t))T P, Wijl (t) = Wijl + ∆Wijl (t). By Lemma 2.2 and (3.15), we obtain ∆Wijl (t)

T F (t)T M T P = P T Mij Fij (t)Nijl + Nijl ij ij T N , ≤ ǫijl P T Mij Mij P + ǫ−1 N ijl ijl ijl

(3.16)

where Fij (t) =

Fi (t) 0 0 Fi (t)

Nijl (t) =

N1i + N4i Kj N2i + N5i Kl N3i N1j + N4j Ki N2j + N5j Kl N3j

,

.

So, from (3.15) and (3.16), we get r r P r P P hl (s(t − τ )) hi (s(t))hj (s(t)) 2x(t)T P x(t) ˙ ≤ 21 i=1 j=1 l=1

(3.17)

T η(t)T [Wijl + ǫijl P T Mij Mij P + ǫ−1 ijl Nijl Nijl ]η(t) .

Then similar to (3.7) we obtain V˙ (t) ≤ 2x(t)T P x(t) ˙ + x(t)T (P1 + τ 2 P2 )x(t) −x(t − τ )T P1 x(t − τ ) − α(t)T P2 α(t) r r r P P P hl (s(t − τ )) hi (s(t))hj (s(t))η(t)T Γijl η(t), ≤ 21 i=1 j=1 l=1

(3.18)

where

 P1 + τ 2 P2 0 0 Φ =  0 −P1 0 , 0 0 −P2 

T Γijl = Wijl + ǫijl P T Mij Mij P + ǫ−1 ijl Nijl Nijl + 2Φ.

It is easy to verify that Γijl =  Υijl Hijl Adij X2 Gij Gji  ∗ −2X − 4X 0 G G 1 ijl jil  T X NT  ∗ ∗ −2X X N 2 2 2 3i 3j   ∗ ∗ ∗ −ǫ I 0 ijl   ∗ ∗ ∗ ∗ −ǫijl I   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

2X 2τ X 0 0 0 0 0 0 0 0 −2X1 0 ∗ −2X2



     . (3.19)    

684


Then by that (3.13) means r r P r P P hl (s(t − τ )) hi (s(t))hj (s(t))η(t)T Γijl η(t) V˙ (t) ≤ 21 i=1 j=1 l=1 r r P 1 P = 2 hl (s(t − τ )) hi (s(t))2 η(t)T Γiil η(t) +

l=1 r P

l=1

≤ 0 which implies

i=1

hl (s(t − τ ))

P

(3.20)

hi (s(t))hj (s(t))η(t)T Γijl η(t)

1≤i