Robust fuzzy observer-based fuzzy control design for nonlinear systems with persistent bounded disturbances: A novel decoupled approach

Fuzzy Sets and Systems 160 (2009) 2824 – 2843 www.elsevier.com/locate/fss

Robust fuzzy observer-based fuzzy control design for nonlinear systems with persistent bounded disturbances: A novel decoupled approach Chung-Shi Tsenga,∗ , Bor-Sen Chenb , Yen-Fang Lia a Department of Electrical Engineering, Ming Hsin University of Science and Technology, Hsin-Feng 30401, Taiwan b Department of Electrical Engineering, National Tsing Hua University, Hsin-Chu 30043, Taiwan

Received 7 October 2007; received in revised form 14 October 2008; accepted 11 February 2009 Available online 21 February 2009

Abstract Unlike the L2 -gain (H∞ ) control case, the L∞ -gain control problem is dealing with persistent bounded disturbances. This study introduces L∞ -gain fuzzy observer-based fuzzy control design, where the premise variables depend on the state variables estimated by a fuzzy observer, for nonlinear systems via T–S fuzzy models. The fuzzy control design for this case is more flexible but much more complex than that for the case where the premise variables do not depend on the state variables estimated by a fuzzy observer. A novel decoupled method is proposed in this study to transform the non-LMI conditions into some LMI forms. By the proposed decoupled method and the genetic algorithm, the L∞ -gain fuzzy observer-based fuzzy control problem can be easily solved by an LMI-based method. © 2009 Elsevier B.V. All rights reserved. Keywords: L∞ -gain control; T–S fuzzy model; Fuzzy observer-based fuzzy controller; Decoupled method, persistent bounded disturbances; LMI

1. Introduction In the last decade, L∞ -gain (L1 ) optimal control has been introduced to reject the bounded disturbance by minimizing the maximum (peak) amplitude of the tracking error in the time domain [1]. For the linear systems, the duality theory and linear programming techniques have been proposed to deal with the L∞ -gain optimal control design problem by solving a set of linear equations [1–6]. This approach is very complicated, especially for the high-order MIMO systems [5]. On the other hand, for the nonlinear systems, there are no efficient algorithms to solve the L∞ -gain control problem for the nonlinear dynamic systems [7–10]. However, in practical control cases, the systems are inherently nonlinear and the external disturbances are persistent. In this situation, it is more appealing to develop a feasible method to solve the L∞ -gain optimal control for the nonlinear systems with persistent bounded disturbances. Recently there are many studies concerned with the systematic synthesis of output feedback control design for nonlinear systems using their Takagi–Sugeno (T–S) fuzzy models. The output feedback control design for nonlinear ∗ Corresponding author. Tel.: +886 3 5593142.

E-mail addresses: [email protected] (C.-S. Tseng), [email protected] (B.-S. Chen), [email protected] (Y.-F. Li). 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.02.006

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systems using their T–S fuzzy models is more general and complex than the state feedback one. When dealing with output feedback control design, either a static output feedback case or a dynamic output feedback (fuzzy observerbased) case is considered [11,12]. For the static output feedback case, the premise variables of both T–S fuzzy model and controller depend on the system outputs only [13,14]. This assumption is very restricted since, sometimes, the information of the system outputs is not enough to construct the T–S fuzzy model for a nonlinear system. For the dynamic output feedback case, a fuzzy observer is also involved as well as the fuzzy controller. In other words, for the fuzzy observer-based output feedback control design, the premise variables depend on the state variables. The dependence of the premise variables on the state variables makes it necessary to consider two cases for the fuzzy observer design. One is that the premise variables do not depend on the state variables estimated by a fuzzy observer, which is denoted as case A. In other words, some state variables should be measurable to serve as premise variables for both fuzzy observer and fuzzy controller in this case [15–18]. However, sometimes the measurable state variables may not be used as the premise variables. Moreover, it happens that none of the state variables are measurable for some systems. The other is that the premise variables depend on the state variables estimated by a fuzzy observer, which is denoted as case B [19–21]. The control design for the latter case (case B) is more flexible but much more complex than that for the former case (case A) [22]. For the dynamic output stabilization problem, the case A leads to a separation principle between fuzzy controller and fuzzy observer. However, the separation principle is not available for T–S fuzzy model with uncertainties [23–25], stabilization with robust performance such as H∞ performance and L∞ -gain performance [26–29], or case B [21]. In [29], the stabilization problem with L∞ -gain (robust) performance for case A is considered. In [21], the stabilization problem for case B, which gives sufficient conditions in terms of LMI problems, is first addressed. Note that the sufficient conditions provided by [21] are not strictly LMI since four scalars should be specified in advance. Unlike the L2 -gain (H∞ ) control case, which reduces the influence of the energy of external disturbance on the energy of output signal as small as possible, in this study, an L∞ -gain fuzzy observer-based fuzzy controller for case B is proposed to reduce the influence of the peak (i.e., L∞ -norm) of external disturbance on the peak (i.e., L∞ -norm) of the output signal as small as possible. Therefore, the L∞ -gain fuzzy observer-based fuzzy controller may be more suitable for some practical applications because control engineers may concern more about the peak (amplitude) than the total energy of the perturbed signal. A novel decoupled method is proposed in this study to transform the non-LMI conditions into some LMI forms. This robust stabilization problem can be characterized in terms of linear matrix inequality problem (LMIP) if six scalars can be specified in advance. By the proposed decoupled method, the L∞ -gain fuzzy observer-based fuzzy control problem for case B can be easily solved by an LMI-based method. In general, it is a difficult task to solve the LMIP with six pre-specified scalars. Fortunately, this problem can be easily solved by using the genetic algorithm (GA). The proposed decoupled method is very simple and easy to follow. The main contributions of this paper are stated as follows: (1) The L∞ -gain fuzzy observer-based fuzzy control problem, where the premise variables depend on the state variables estimated by a fuzzy observer, for nonlinear dynamic systems is studied for the first time. (2) A novel decoupled method is proposed to transform the non-LMI conditions into some LMI forms. By the proposed decoupled method and the GA, the fuzzy control gains and the fuzzy observer gains can be obtained simultaneously. (3) Both the stability and L∞ -gain disturbance rejection performance are guaranteed for the fuzzy control systems. The paper is organized as follows: some notations and definitions are given in Section 2. In Section 3, L∞ -gain fuzzy observer-based fuzzy control design for nonlinear dynamic systems with persistent bounded disturbances is introduced, while some simulation results are presented in Section 4. Finally, concluding remarks are made in Section 5. In what follows,     T M11 ∗ M11 M21  . M21 M22 M21 M22 2. Notation and definition Before the description of the fuzzy control problem, some notations and definitions are stated below:  |x(t)|  x T (t)x(t) for x(t) ∈ R n ,

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x(t)∞  sup |x(t)| for x(t) ∈ R n , t

x(t) ∈ L∞ if x(t)∞ < ∞. Remark 1. For the linear systems [3,10], x(t)∞ where H : L∞ → L∞ . ∈L∞ (t)∞

H 1  sup

The physical meaning of H 1 (L1 -norm or L∞ -gain) is the worst ratio of the peak of the output signal x(t) to the peak of all possible bounded input signal (t). If we can make H 1  , then the peak of the output signal x(t) can be attenuated by a level  under the effect of the peak of the persistently bounded input signal (t). 3. L∞ -gain fuzzy observer-based fuzzy control design for nonlinear dynamic systems A fuzzy linear dynamic model has been proposed by Takagi and Sugeno [30] to represent local linear input/output relations of nonlinear systems. The fuzzy linear model is described by fuzzy If–Then rules and will be employed here to deal with the robust L∞ -gain control problem for the nonlinear system. The i-th rule of the fuzzy linear model for a nonlinear system is of the following form [26,27]: Plant Rule i : If

z 1 (t) is Fi1 and z 2 (t) is Fi2 . . . and z g (t) is Fig x(t) ˙ = Ai x(t) + B1i u(t) + B2i w(t)

Then

(1)

y(t) = C1i x(t) + C2i v(t) for i = 1, 2, . . . , L ,

where x(t) = [x1 (t), . . . , xn (t)]T ∈ R n×1 denotes the vector of the states, u(t) = [u 1 (t), . . . , u m (t)]T ∈ R m×1 denotes the vector of the control inputs; w(t) = [w1 (t), . . . , w p (t)]T ∈ R p×1 denotes the vector of the bounded external disturbances; y(t) = [y1 (t), . . . , yr (t)]T ∈ R r ×1 denotes the vector of the outputs; v(t) = [v1 (t), . . . , vq (t)]T ∈ R q×1 denotes the vector of the bounded measurement disturbances; Fi j is the fuzzy set, Ai ∈ R n×n , B1i ∈ R n×m , B2i ∈ R n× p , C1i ∈ R r ×n , C2i ∈ R r ×q ; L is the number of If–Then rules; and z 1 (t), z 2 (t), . . . , z g (t) are the premise variables. The defuzzification outputs of the fuzzy system are inferred as follows: L L  (z(t))(Ai x(t) + B1i u(t) + B2i w(t))  = x(t) ˙ = i=1 i h i (z(t))[Ai x(t) + B1i u(t) + B2i w(t)], L i=1 i (z(t)) i=1 L y(t) =

i=1 i (z(t))(C 1i x(t) + C 2i v(t)) L i=1 i (z(t))

=

L 

h i (z(t))[C1i x(t) + C2i v(t)],

(2)

i=1

where i (z(t)) =

g 

 (z(t)) , z(t) = [z 1 (t), z 2 (t), . . . , z g (t)] Fi j (z j (t)) h i (z(t)) =  L i i=1 i (z(t)) j=1

and Fi j (z j (t)) is the grade of membership of z j (t) in Fi j . It is assuming that i (z(t)) 0 and L  i=1

for all t.

i (z(t)) > 0 for i = 1, 2, . . . , L ,

(3)

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Therefore, we get h i (z(t))0 for i = 1, 2, . . . , L

(4)

and L 

h i (z(t)) = 1.

(5)

i=1

Suppose the following fuzzy observer is proposed to deal with the state estimation of nonlinear system (1): Observer Rule j : zˆ 1 (t) is Fˆ j1 and zˆ 2 (t) is Fˆ j2 . . . and zˆ g (t) is Fˆ jg ˙ˆ = A j x(t) x(t) ˆ + B1 j u(t) + L j (y(t) − yˆ (t)),

If Then

(6)

L where x(t) ˆ is an estimate of x(t), yˆ (t) = s=1 h s (ˆz (t))C1s x(t), ˆ and L j is the observer gain for the j-th observer rule. The overall fuzzy observer is represented as follows: ˙ˆ = x(t)

L 

h j (ˆz (t))(A j x(t) ˆ + B1 j u(t) + L j (y(t) − yˆ (t)))

j=1

=

L 

L 

h i (z(t))

i=1

h j (ˆz (t))

L 

h s (ˆz (t))[A j x(t) ˆ + B1 j u(t) + L j (C1i x(t) + C2i v(t) − C1s x(t))], ˆ

(7)

s=1

j=1

where zˆ (t) is an estimate of z(t). Remark 2. In general, there are two cases for fuzzy observer-based controller design. For case A, the premise variables do not depend on the state variables estimated by a fuzzy observer. For case B, the premise variables depend on the state variables estimated by a fuzzy observer. In general, case B is more general and more complex than case A. In this paper, the case B is studied. The details can be found in [21,22]. Hence, the fuzzy observer-based fuzzy controller is proposed as u(t) =

L 

h s (ˆz (t))K s x(t), ˆ

(8)

s=1

where K s is the control gain for the s-th controller rule. Let us denote the estimation errors as e(t) = x(t) − x(t). ˆ

(9)

By differentiating (9), we get ˙ˆ = e(t) ˙ = x(t) ˙ − x(t)

L 

h i (z(t))

L 

h i (z(t))

i=1

=

L  i=1

h i (z(t))

L 

h j (ˆz (t))

j=1 L  j=1

h s (ˆz (t))(Ai x(t) + B1i K s x(t) ˆ + B2i w(t))

s=1

i=1

−

L 

h j (ˆz (t))

L 

h s (ˆz (t))[A j x(t) ˆ + B1 j K s x(t) ˆ + L j (C1i x(t) + C2i v(t) − C1s x(t))] ˆ

s=1 L 

h s (ˆz (t)){[(Ai − A j ) + (B1i − B1 j )K s

s=1

+L j (C1s − C1i )]x(t) + [(A j − L j C1s ) − (B1i − B1 j )K s ]e(t) + B2i w(t) − L j C2i v(t)}.

(10)

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After some manipulation, the augmented system can be expressed as the following compact form:      L L L    Ai + B1i K s −B1i K s x(t) x(t) ˙ h i (z(t)) h j (ˆz (t)) h s (ˆz (t)) = Si1js Si2js e(t) e(t) ˙ s=1 i=1 j=1    0 w(t) B2i , + B2i −L j C2i v(t)

(11)

where Si1js = [(Ai − A j ) + (B1i − B1 j )K s + L j (C1s − C1i )],

Si2js = ( A j − L j C1s ) − (B1i − B1 j )K s .

Let us denote     Ai + B1i K s −B1i K s x(t) (t) = , Ai js = , Si1js Si2js e(t)     0 w(t) B2i Bi j = , d(t) = , B2i −L j C2i v(t)

(12)

for all i, j, s = 1, 2, . . . , L , then the augmented system in (11) can be expressed as follows: ˙ (t) =

L 

h i (z(t))

i=1

L 

h j (ˆz (t))

j=1

L 

h s (ˆz (t))[Ai js (t) + Bi j d(t)].

(13)

s=1

The optimal L ∞ -gain fuzzy control problem is to specify the control input u in (8) such that the following minimax problem is achieved for the nonlinear systems in (13) [7,8,10]: (t)∞ . d(t)∈L∞ d(t)∞

min sup u

(14)

However, it is very difficult to solve the minimax problem in (14) for the nonlinear system in (13) directly, the suboptimal approach via minimizing the upper bound of the L∞ -gain is proposed in this study. Let us consider the L∞ -gain performance as follows: The L∞ -gain control problem is said to be solved if there exists a control law such that the following L∞ performance with (0) = 0 (t)∞  , ∀d(t) ∈ L∞ , d(t)∞

(15)

where  is a disturbance attenuation level or (t)∞  d(t)∞ , ∀d(t) ∈ L∞

(16)

can be achieved. Remark 3. If we can make (t)∞ /d(t)∞ , then the peak of the state (t) can be attenuated by a level  under the effect of the peak of the persistently bounded input signal d(t). If the initial condition is considered, i.e., (0)  0, the L∞ -gain in (16) is modified as follows: (t)∞   |(0)| + d(t)∞ ,

(17)

where  and  are some positive scalars. Remark 4. The L∞ -gain performance in (16) is equivalent to the bounded input and bounded state stability. (Moreover, the L∞ -gain performance in (17) is called L∞ -stable with finite gain.)

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The purpose of this study is to determine a fuzzy controller in (8) for the closed-loop system in (13) with the guaranteed L∞ -gain performance in (17) for all d(t) ∈ L∞ . Thereafter, the attenuation level  can be minimized so that the ratio of (t)∞ /d(t)∞ is reduced as small as possible. First, one useful lemma is stated below. Lemma 1. If a real scalar function (t) satisfies the following differential inequality: (t) ˙ − (t) + v(t),

(18)

where  > 0 and > 0 then (t) e−t (0) +



t

e− v(t − ) d .

(19)

0

Proof. The proof is trivial.  Then, we obtain the following main result: Theorem 1. In the augmented nonlinear system (13), if there exist a symmetric positive-definite matrix P and two positive scalars  and c such that the following matrix inequalities: 

Ai js + Ais j 2

T

 P+P

Ai js + Ais j 2

+

1 P Bi j BiTj P + P < 0 c

(20)

hold for all i, j s (i, j, s = 1, 2, . . . , L), then the L∞ -gain control performance in (17) is guaranteed for an attenuated   level  = c/ min (P), and  = max (P)/ min (P), where max (P) and min (P) denote the maximal and minimal eigenvalues of P, respectively. Proof. From (13), we obtain d 1/2 |P (t)|2 = ˙ T (t)P(t) + T (t)P ˙ (t) dt ⎤T ⎡ L L L    h i (z(t)) h j (ˆz (t)) h s (ˆz (t))(Ai js (t) + Bi j d(t))⎦ P(t) = ⎣ i=1

⎡

+T (t)P ⎣

L 

h i (z(t))

i=1

=

L 

s=1

j=1

h i (z(t))

i=1

L 

L 

h j (ˆz (t))

L 

⎤ h s (ˆz (t))(Ai js (t) + Bi j d(t))⎦

s=1

j=1

h j (ˆz (t))

L 

h s (ˆz (t))[(Ai js (t))T P(t) + T (t)P Ai js (t)

s=1

j=1

+d T (t)BiTj P(t) + T (t)P Bi j d(t)] =

L 

h i (z(t))

i=1

L  j=1

h j (ˆz (t))

L 

 h s (ˆz (t)) ( Ai js (t))T P(t) + T (t)P Ai js (t)

s=1

T  √ √ 1 T 1 T − √  (t)P Bi j − cd(t) √  (t)P Bi j − cd(t) c c  1 + T (t)P Bi j BiTj P(t) + cd(t)T d(t) c 

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C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843



L 

h i (z(t))

i=1

L 

h j (ˆz (t))

L 

 h s (ˆz (t)) ( Ai js (t))T P(t) + T (t)P Ai js (t)

s=1

j=1



1 + T (t)P Bi j BiTj P(t) + cd(t)T d(t) c     L L L    1 T T T h i (z(t)) h j (ˆz (t)) h s (ˆz (t))  (t) Ai js P + P Ai js + P Bi j Bi j P + P (t) = c i=1

s=1

j=1

− (t)P(t) + cd(t) d(t)    L L   Ai j j + Ai j j T Ai j j + Ai j j 2 T = h i (z(t)) h j (ˆz (t))  (t) P+P 2 2 T

i=1

T

j=1

  1 T + P Bi j Bi j P + P (t) c

+2

L 

h i (z(t))

i=1

L 

h j (ˆz (t))

L 





h s (ˆz (t))  (t) T

s=1

j=1

Ai js + Ais j 2

T

 P+P

Ai js + Ais j 2



  1 + P Bi j BiTj P + P (t) − T (t)P(t) + cd(t)T d(t), c where  and c are positive scalars. By the matrix inequalities in (20), we obtain d 1/2 |P (t)|2  − T (t)P(t) + cd T (t)d(t) = −|P 1/2 (t)|2 + c|d(t)|2 . dt By Lemma 1, we obtain |P 1/2 (t)|2  e−t |P 1/2 (0)|2 + c



e−t |P 1/2 (0)|2 + c|d(t − )|2

 sup = sup

∈[0,t]

e− |d(t − )|2 d

0



∈[0,t]

t

(21)



t

 e− d

0

 c e−t |P 1/2 (0)|2 + |d(t − )|2 (1 − e−t ) . 



From above, we have min (P)|(t)|2  |P 1/2 (t)|2  sup

∈[0,t]



 c e−t |P 1/2 (0)|2 + |d(t − )|2 (1 − e−t ) . 

Performing “supt∈[0,∞) ” to both sides of the above inequality, we obtain   c sup min (P)|(t)|2  sup |P 1/2 (t)|2  sup sup e−t |P 1/2 (0)|2 + |d(t − )|.2 (1 − e−t )  t∈[0,∞) t∈[0,∞) t∈[0,∞) ∈[0,t]  |P 1/2 (0)|2 +

c sup |d(t)|2  t∈[0,∞)

(22)

The above inequality implies that c c min (P)(t)2∞ P 1/2 (t)2∞ |P 1/2 (0)|2 + d(t)2∞  max (P)|(0)|2 + d(t)2∞ .  

(23)

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2831

Hence, max (P) (t)2∞  min (P) or

 (t)∞ 

|(0)| +

c

2

max (P) |(0)| + min (P)

 min (P) 

 d(t)2∞ 

max (P) |(0)| + min (P)



c  min (P)

2 d(t)∞

c d(t)∞ ,  min (P)

(24)

for all d(t) ∈ L∞ . This completes the proof.  To obtain better performance, the control problem can be formulated as the following minimization problem so that the L∞ -gain performance in (17) is reduced as small as possible: c min  min (P) subject to

 > 0, c > 0, P = P T > 0 and (20).

For the convenience of design, let   P11 0 . P= 0 P22

(25)

(26)

By (26), we obtain   Ai js + Ais j Ai js + Ais j T P + P = M1 + M1T + [0 P22 ]T M2 + M2T [0 P22 ] 2 2   0 0 T T  M1 + M1 + 4 + −1 4 M2 M2 , 2 0 P22 where 4 > 0, ⎡

⎤ P11 ( Ai + B1i K s ) −P11 B1i K s M1 = ⎣ Y j (C1s − C1i ) + Ys (C1 j − C1i ) (P22 A j − Y j C1s ) + (P22 As − Ys C1 j ) ⎦ , P22 (Ai − A j ) + 2 2   (B1i − B1 j )K s + (B1i − B1s )K j (B1i − B1 j )K s + (B1i − B1s )K j , Y j = P22 L j . M2 = − 2 2

Then (20) holds, if  M1 +

M1T

+ 4

0

0

0

2 P22

 T + −1 4 M2 M2 +

1 P Bi j BiTj P + P < 0. c

By the Schur complements, (27) is equivalent to   ⎡ 0 0 T + P ∗ ∗ ⎢ M 1 + M 1 + 4 2 0 P22 ⎢ ⎢ ⎢ M2 −4 I ∗ ⎣ T 0 −cI Bi j P

(27)

⎤ ⎥ ⎥ ⎥ < 0. ⎥ ⎦

(28)

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Note that, by (26), the matrix inequalities in (28) are equivalent to the following matrix inequalities: ⎤ ⎡ ∗ ∗ ∗ ∗

11 + P11 ⎥ ⎢ T 2 ⎢ − 12 + 21 22 + P22 + 4 P22 ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎥ ⎢

31 − 31 −4 I ∗ ∗ ⎥ < 0, ⎢ ⎥ ⎢ T T ⎢ 24 0 −cI ∗ ⎥

14 ⎦ ⎣ T 0 −25 0 0 −cI

(29)

where T , 11 = P11 ( Ai + B1i K s ), 12 = P11 B1i K s ,

11 = 11 + 11

Y j (C1s − C1i ) + Ys (C1 j − C1i ) , 2 (B1i − B1 j )K s + (B1i − B1s )K j , = 2

21 = P22 (Ai − A j ) +

31

T

14 = P11 B2i , 22 = 22 + 22 ,

(P22 A j − Y j C1s ) + (P22 As − Ys C1 j ) , 2 = P22 B2i , 25 = Y j C2i and

22 = 24

Y j = P22 L j for all i, j s (i, j, s = 1, . . . , L). Since four parameters P11 , P22 , K s , and L j and scalars , c, and 4 should be determined from (29), there are no effective algorithms for solving them simultaneously. In the following, a novel decoupled method is provided to solve P11 , P22 , K s , and L j simultaneously. Note that (29) can be decoupled as follows: ⎡ ⎤ ∗ ∗ ∗ ∗

11 + P11 ⎢ ⎥ T 2 ⎢ − 12 + 21 22 + P22 + 4 P22 ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ ⎥

31 − 31 −4 I ∗ ∗ ⎥ ⎢ ⎢ ⎥ T T ⎢ 24 0 −cI ∗ ⎥

14 ⎣ ⎦ T 0 −25 0 0 −cI ⎤ ⎡ ∗ ∗ ∗ ∗

11 + P11 + 3 I ⎥ ⎢ T ⎢ −1 P11 ∗ ∗ ∗⎥ − 12 ⎥ ⎢ ⎥ ⎢

31 − 31 −4 I ∗ ∗⎥ =⎢ ⎥ ⎢ ⎢ T 0 0 − 2c I ∗ ⎥

14 ⎦ ⎣ 0 0 0 0 0 ⎤ ⎤ ⎡ ⎡ 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −3 I ⎥ ⎢ ⎢ 0  + ( +  )P +  P 2 ∗ ∗ ⎥ 22 2 22 1 11 ∗ ⎥ ⎢ 21 −2 P22 + 4 P22 ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎢ 0 0 ∗ ∗ ⎥ (30) +⎢ 0 0 ∗ ∗⎥ ⎥, ⎥+⎢ 0 ⎢0 ⎥ ⎥ ⎢ ⎢ c T 24 0 −2 I ∗ ⎦ ⎣ 0 ⎣0 0 0 0 ∗⎦ 0

T −25

0

where 1 , 2 , and 3 are some positive scalars. The following useful lemma is introduced first.

0

−cI

0

0

0 0 0

C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843

Lemma 2. If ⎡ ∗ a11 ∗ ⎢ ⎢ a21 a22 ∗ ⎢ ⎢a ⎣ 31 a32 a33

∗

⎤

⎤ ⎡ b11 ∗ ∗ ⎥ ∗ ⎥ ⎥ ⎥ < 0, ⎢ ⎣ b21 b22 ∗ ⎦ < 0 and ⎥ ∗ ⎦ b31 b32 b33 a44

a41 a42 a43

2833



c11

∗

c21 c22

 <0

(31)

then ⎡

a11

∗

∗

∗

∗

⎤

⎡

∗

0

∗

∗

⎢a ⎢ ∗ ∗⎥ ⎢ 21 a22 ∗ ⎥ ⎢ 0 b11 ∗ ∗ ⎢ ⎥ ⎢ ⎢ a31 a32 a33 ∗ ∗ ⎥ + ⎢ 0 0 0 ∗ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎣ a41 a41 a43 a44 ∗ ⎦ ⎣ 0 b21 0 b22 0

0

0

0

0

∗

⎤

⎡

c11

∗

∗ ∗ ∗

⎤

⎥ ⎢ ∗ ⎥ ⎥ ⎢ c21 c22 ∗ ∗ ∗ ⎥ ⎥ ⎢ ⎥ ⎢ ∗ ⎥ 0 0 ∗ ∗⎥ ⎥+⎢ 0 ⎥ < 0. ⎥ ⎢ ⎥ 0 0 0 ∗⎦ ∗ ⎦ ⎣ 0

0 b31 0 b32 b33

0

0

(32)

0 0 0

Proof. The proof is trivial.  By Lemma 2, it is obvious that if ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎣

11 + P11 + 3 I

∗

∗

T − 12

−1 P11

31

− 31

T

14

0

∗

⎤

⎥ ∗ ⎥ ⎥ < 0, −4 I ∗ ⎥ ⎦ c 0 −2 I ∗

22 + ( + 2 )P22 + 1 P11

∗

T 24

− 2c I

T −25

0

∗

(33)

⎤

⎥ ∗ ⎦<0

(34)

−cI

and 

−3 I

∗

21

2 −2 P22 + 4 P22

 <0

(35)

then (29) (or (20)) holds. Remark 5. Note that (33) is related to the controller part (the parameters are P11 and K s ) and (34) and (35) are related to the observer part (the parameters are P22 , L j , and P11 ), respectively. Although the parameter P11 is still included 2 is still included in (35), four parameters P , P , K , and L can be determined in (34) and nonlinear parameter P22 11 22 s j simultaneously by the following arrangement. Note that, by the Schur complements [31], (33) is equivalent to ⎡ ⎤ ∗ ∗ ∗ ∗ 11 + X 11 ⎢ ⎥ T ⎢ −12 −1 X 11 ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ 31 −31 −4 I ∗ ∗ ⎥ ⎢ ⎥ < 0, ⎢ ⎥ c T ⎢ 0 0 −2 I ∗ ⎥ 14 ⎣ ⎦ −1 X 11 0 0 0 −3 I

(36)

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C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843

where −1 T , 11 = 11 + 11 , X 11 = P11

11 = Ai X 11 + B1i Ws , (B1i − B1 j )Ws + (B1i − B1s )W j , 2 and Ws = K s X 11 ,

12 = B1i Ws , 31 = 14 = B2i

(34) is equivalent to ⎡ ∗ ∗ 22 + ( + 2 )P22 ⎢ c T −2 I ∗ 24 ⎢ ⎢ ⎢ T −25 0 −cI ⎣ I

0

and (35) is equivalent to ⎡ ∗ −3 I ⎢  ⎣ 21 −2 P22 0

P22

∗ ∗

0

∗ ∗ ∗ −−1 1 X 11

⎤ ⎥ ⎥ ⎥<0 ⎥ ⎦

(37)

⎤ ⎥ ⎦ < 0.

(38)

−−1 4 I

Therefore, if (36)–(38) hold, then (29) (or (20)) holds. Based on the analysis above, the minimization problem in (25) is rewritten as follows: c Find {X 11 , P22 , Ws , Y j , , c, 1 , 2 , 3 , 4 } such that is as small as possible  min (P) ⎧ T T ⎪ ⎨ X 11 = X 11 > 0, P22 = P22 > 0, Ws , Y j ,  > 0, subject to (39) c > 0, 1 > 0, 2 > 0, 3 > 0, 4 > 0, ⎪ ⎩ (36), (37), and (38). Remark 6. Note that the L∞ -gain fuzzy observer-based fuzzy control problem in (39) is not a strict LMI problem since it is still a bilinear form of six scalars , c, 1 , 2 , 3 and 4 and becomes a standard LMIP, if , c, 1 , 2 , 3 −1 −1 −1 and 4 are given in advance. Therefore, four parameters P11 = X 11 , P22 , K s = Ws X 11 , and L j = P22 Y j can be determined simultaneously from (36), (37), and (38), if , c, 1 , 2 , 3 and 4 are given in advance. Software packages such as LMI optimization toolbox in Matlab [32] can be employed to easily solve the LMIP in (39). Also, note that c/ min (P) is not linear in c, , and P. Hence, we cannot solve (39) by minimizing c/ min (P) directly, even that , c, 1 , 2 , 3 and 4 can be given in advance. In general, it is not an easy task to determine , c, 1 , 2 , 3 and 4 beforehand to solve the LMIP in (39). In this study, two design procedures based on the GA are proposed to deal with the L∞ -gain fuzzy observer-based fuzzy control problem in (39) since GA, which can simultaneously evaluate many points in the parameters space, is a very powerful searching algorithm based on the mechanics of natural selection and natural genetics. A GA in its simplest form uses three operations: reproduction, crossover, and mutation. More details about GA can be found in [33]. According to the analysis above, the L∞ -gain fuzzy observer-based fuzzy control problem via fuzzy control scheme is summarized as follows: Design Procedure 1: Step 1: Construct the fuzzy plant rules in (1). Step 2: Collect parent sets (the first generation) of feasible solutions for the parameters , c, 1 , 2 , 3 and 4 (using the Design Procedure 2 in the following section). With the binary coding method, these feasible parameters , c, 1 , 2 , 3 and 4 are coded as binary strings with the same lengths as that in the Design Procedure 2.

C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843

2835

Step 3: Solve the LMIP in (39) with the parameters c, , 1 , 2 , 3 , and 4 corresponding to a binary string using the LMI toolbox in Matlab by searching the minimal value of c/ min (P). If the LMIP is infeasible for the corresponding string, this string is escaped from the current generation. Step 4: Calculate the fitness value for each passed string: In this step, the fitness value is calculated based on the attenuation level c/ min (P). The fitness value is inverse proportional to the value of c/ min (P). Step 5: Create offspring strings to form a new generation by some simple GA operators like reproduction, crossover, and mutation: In this step, (a) strings are selected in a mating pool from the qualified (passed) strings with probabilities proportional to their fitness values, (b) and then crossover process is applied with a probability equal to a prescribed crossover rate, (c) and finally mutation process is applied with a probability equal to a prescribed mutation rate. Repeat (a)–(c) until enough strings are generated to form the next generation. Step 6: Repeat Steps 3–5 for several generations until a stop criterion (goal) is met (for example, the attenuation level  does not change for several generations). Step 7: Based on the parameters (c, , 1 , 2 , 3 , and 4 ) obtained from above steps, one can obtain the attenuation −1 −1 −1 level c/ min (P) and the corresponding X 11 , P22 , Ws , Y j (thus P11 = X 11 , K s = Ws X 11 and L j = P22 Y j can also be obtained), simultaneously. Step 8: Construct the fuzzy observer in (7). Step 9: Construct the fuzzy controller in (8). Remark 7. In this study, two design procedures (Design Procedure 1 and Design Procedure 2) based on the GA are proposed to deal with the L∞ -gain fuzzy observer-based fuzzy control problem in (39). In the Design Procedure 1, a parent generation is needed to search the minimal value of c/ min (P). The parent generation can be obtained by the Design Procedure 2 (in the following section). By this way, the searching of the minimal value of c/ min (P) can be more efficient. 4. Simulation examples To illustrate the proposed fuzzy control approach, a control problem of stabilizing a chaotic system is considered in this study. For this example, the Lorenz’s equation with input and output terms is given by x˙1 x˙2 x˙3 y1 y2

= −10x1 + 10x2 + u(t) + 0.1w1 , = 28x1 − x2 − x1 x3 + 0.1w2 , = x1 x2 − 8/3x3 + 0.1w3 , = 10x1 + 2x2 + 0.01v1 , = 5x2 + x3 + 0.01v2 ,

(40)

where external disturbance w = [w1 , w2 , w3 = [50 sin(10t), 50 cos(10t), 50[sin(10t) + ∈ L∞ and the measurement disturbance v = [v1 , v2 ]T = [N1 , N2 ]T ∈ L∞ , where N1 and N2 are assumed to be periodic square wave with period (2/5) and magnitude ±200. In this example, the Lorenz’s equation can be exactly represented by the following two-rule fuzzy model under x 1 ∈ [− , ] [22,34]: ]T

Rule 1: IF x1 is F1 THEN x˙ = A1 x + B11 u + B21 w, and y = C11 x + C21 v. Rule 2: IF x1 is F2 THEN x˙ = A2 x + B12 u + B22 w, and y = C12 x + C22 v, where

⎡

⎤ −10 10 0 ⎢ ⎥ A1 = ⎣ 28 −1 − ⎦ , 0 −8/3

B1i = [1 0 0]T ,

⎡

⎤ −10 10 0   10 2 0 ⎢ ⎥ ⎦ , C1i = A2 = ⎣ 28 −1 , 0 5 1 0 − −8/3

B2i = 0.1I3×3 and C2i = 0.01I2×2 (i = 1, 2),

where F1 = 21 (1 + x1 / ) = h 1 (x1 ), F2 = 21 (1 − x1 / ) = h 2 (x1 ), and = 30.

cos(10t)]]T

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C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843

In the second step of the Design Procedure 1, one has to collect parent sets (the first generation) of feasible solutions for the parameters , c, 1 , 2 , 3 and 4 first. The following procedure introduces how to find parent sets of feasible solutions for the parameters. By observing the following matrix inequalities: ⎤ ⎡ ∗ ∗ ∗

11 + P11 + 3 I ⎥ ⎢ T ⎢ −1 P11 ∗ ∗ ⎥ − 12 ⎥ < I, ⎢ (41) ⎥ ⎢

− − I ∗ 31 31 4 ⎦ ⎣ T 0 0 − 2c I

14 ⎤ ⎡ ∗ ∗ 22 + ( + 2 )P22 + 1 P11 ⎥ ⎢ T 24 − 2c I ∗ ⎦ < I (42) ⎣ T −25

and



−3 I

∗

21

2 −2 P22 + 4 P22

0

−cI

 < I,

(43)

where  is a scalar, if  < 0 then (36)–(38) hold. Therefore,  can be a performance index (as smaller as better) for the feasibility of solving (36)–(38). To avoid random search for  < 0, the GA can also be applied to efficiently find a set of feasible solutions for the parameters. The design procedure for searching parent sets of feasible solutions for the parameters , c, 1 , 2 , 3 and 4 is proposed as follows. Design Procedure 2: Step 1: Generate randomly a population of binary strings: With the binary coding method, the parameters , c, 1 , 2 , 3 and 4 are coded as binary strings with length L 1 , L 2 , L 3 , L 4 , L 5 , and L 6 (may be different for specified resolution in the parameter space), respectively. Step 2: Solve the following eigenvalue problem (EVP): min subject to

 ⎧ T T ⎪ ⎨ X 11 = X 11 > 0, P22 = P22 > 0, Ws , Y j ,  > 0, c > 0, 1 > 0, 2 > 0, 3 > 0, 4 > 0, ⎪ ⎩ (41), (42), and (43),

(44)

with the parameters c, , 1 , 2 , 3 , and 4 corresponding to a binary string using the LMI toolbox in Matlab by searching the minimal value of . Step 3: Calculate the fitness value for each string: In this step, the fitness value is calculated based on the value of . The fitness value is inverse proportional to the value of . Step 4: Create offspring strings to form a new generation by some simple GA operators like reproduction, crossover, and mutation: In this step, (a) strings are selected in a mating pool from the qualified (passed) strings with probabilities proportional to their fitness values, (b) and then crossover process is applied with a probability equal to a prescribed crossover rate, (c) and finally mutation process is applied with a probability equal to a prescribed mutation rate. Repeat (a)–(c) until enough strings are generated to form the next generation. Step 5: Repeat Steps 2–4 for generations until  < 0 is met. The solutions of c, , 1 , 2 , 3 , and 4 corresponding to  < 0 are the feasible parameters for (36)–(38). Save this set of solutions. Repeat Steps 2–5 until enough sets of feasible solutions for the parameters of c, , 1 , 2 , 3 , and 4 are collected. These feasible solutions are collected as the parent generation (the first generation) for the Design Procedure 1. Remark 8. Note that the Design Procedure 2 has the tendency of finding the solutions for c, , 1 , 2 , 3 , and 4 toward the direction of  < 0 using the GA. To employ GA in the Design Procedure 2, the parameter space for , c, 1 , 2 , 3 , and 4 should be given in advance. To find c/ min (P) as small as possible, smaller c is expected. By observing from (36) to (38), one notices

C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843

2837

that (36)–(38) may not be feasible for large values of , 1 , 2 , 3 , and 4 . Consequently, the parameter space for , c, 1 , 2 , 3 , and 4 is chosen within a certain wide range as follows: 0 <  < 100, 0 < c < 100, 0 < 1 < 100, 0 < 2 < 100, 0 < 3 < 100, 0 < 4 < 100. Then, based on Design Procedure 2, 30 sets of parent generation (the first generation) of the feasible parameters c, , 1 , 2 , 3 , and 4 are obtained first. With the first generation, the parameters of c, , 1 , 2 , 3 , and 4 such that c/ min (P) is as small as possible can be further determined using the Design Procedure 1. In the following examples, binary strings with length L = 15 bits for the parameters , c, 1 , 2 , 3 and 4 are coded and crossover rate and mutation rate are set to 1.0 and 0.01, respectively. Example 1. According to the Design Procedure 1, the LMIP in (39) is solved using the GA with the LMI optimization toolbox in Matlab as follows. The following table shows the tendency of the attenuation level  with respect to the solutions of c, , 1 , 2 , 3 , and 4 . 1

2

3

4



c



33.6589

56.151

69.6799

1.0001 × 10−6

3.5666

31.431

3.476

23.921

1.2043 × 10−4

1.3227

1.9377

3.0728

0.3809

6.7276 × 10−5

2.6366

0.0425

2.9511

25.928

0.2063

1.0526 × 10−6

2.1784

0.0191

2.934

25.0194

1.9529 × 10−4

7.6216 × 10−5

2.7757

2.0441 × 10−5

2.9294

29.3292

2.5279 × 10−4

6.012 × 10−5

2.188

2.4551 × 10−5

2.8131

36.2024

7.9274 × 10−5

1.5781 × 10−5

2.0197

8.1217 × 10−6

2.7376

34.1564

5.5871 × 10−5

1.5225 × 10−5

2.4389

6.252 × 10−6

2.6957

32.5344

35.0313

3.6136 × 10−5

3.7089 × 10−5

2.3401

4.1684 × 10−6

2.692

32.0784

36.0006

2.5176 × 10−5

3.898 × 10−5

2.4917

3.1328 × 10−6

2.6692

32.4541

34.9669

4.3209 × 10−5

1.9115 × 10−5

2.1939

4.5416 × 10−6

2.6578

35.7268

6.7502 × 10−5

1.0002 × 10−5

2.2971

7.5343 × 10−6

2.6498

32.2502

34.7685

30.6552

25.5094

31.5319 31.2651 32.4758 31.6262 33.2107

31.9462

In the last case of 1 = 31.9462, 2 = 35.7268, 3 = 6.7502 × 10−5 , 4 = 1.0002 × 10−5 ,  = 2.2971, and c = 7.5343 × 10−6 on the above table, ⎡ ⎤ 2.3243 × 104 −2.2608 × 103 5.6680 × 100 ⎢ ⎥ X 11 = ⎣ −2.2608 × 103 3.5026 × 104 −5.0571 × 100 ⎦ , ⎡

5.6680 × 100 1.5093 × 10−2

⎢ P22 = ⎣ 2.1727 × 10−10

−5.0571 × 100 2.1727 × 10−10 4.6869 × 10−7

−1.3601 × 10−10 2.7224 × 10−10

3.4372 × 104 −1.3601 × 10−10

⎤

⎥ 2.7224 × 10−10 ⎦ , 4.6719 × 10−7

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C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843



c

= 2.6498  min (P)   max (P) 0.0151 = and  = = 179.7456 min (P) 4.6714 × 10−7 =

are obtained by the GA. The observer gains are found to be ⎡ ⎡ ⎤ 4.0186 × 101 1.5360 × 10−1 6.4441 × 101 ⎢ ⎢ ⎥ L 1 = ⎣ 2.6255 × 105 6.4575 × 105 ⎦ , L 2 = ⎣ 4.1173 × 105 4.3912 × 102

1.2900 × 105

−1.0863 × 103

−3.2489 × 10−1

⎤

⎥ 1.0381 × 106 ⎦ , 2.0815 × 105

and the control gains are found to be K 1 = [−31.9357 − 28.9284 − 0.0114], K 2 = [−31.9357 − 28.9284 − 0.0114]. Remark 9. The parameter  obtained in this example may not be minimum since the proposed GA searches the minimal value of c/ min (P) from the solution space provided by the LMI solver (“feasp”) in Matlab. The LMI solver computes arbitrary solutions (if exist) for the parameters in a free manner. The LMI solver may not find the optimal combination (solutions) of the parameters for GA to obtain the minimal value of c/ min (P). Example 2. Obviously, high observer gains are obtained in Example 1. To avoid high observer gains, one can put additional LMI constraints to constrain the location of eigenvalues of ( A j − L j C1s ) within left half circle (on complex plane) with radius r > 0 as follows: ⎡ ⎤ −r P22 ∗ ⎢ ⎥  ⎣ (P22 A j − Y j C1s ) + (P22 As − Ys C1 j ) T ⎦<0 −r P22 2 for j  s ( j, s = 1, 2) [35]. If r = 500 is chosen, then  = 2.0578, c = 7.4847 × 10−6 , 1 = 32.6507, 2 = 46.5549, 3 = 5.8854 × 10−5 , 4 = 7.1310 × 10−5 , ⎡ 2.8904 × 104 ⎢ X 11 = ⎣ −2.7092 × 103 ⎡

4.8061 × 104

9.1536 × 10−3 −1.0210 × 10−2 1.2909 × 10−2

⎢ P22 = ⎣ 5.5048 × 10−9 

−2.7092 × 103

5.5048 × 10−9 7.6040 × 10−7

−2.7670 × 10−8 5.1042 × 10−11 c

9.1536 × 10−3

⎥ −1.0210 × 10−2 ⎦ , 4.7593 × 104 −2.7670 × 10−8

⎤

⎥ 5.1042 × 10−11 ⎦ , 7.6017 × 10−7

= 2.1874  min (P)   max (P) 0.0129 and  = = 130.3150 = min (P) 7.6016 × 10−7 =

⎤

C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843

2839

15 x1 estimated x1hat

10 states

5 0 −5 −10 −15 −20 0

5

10

15

time (sec) 15 estimated error

10

estimated error for x1

5 0 −5 −10 −15 −20 0

5

10

15

time (sec) Fig. 1. The trajectories of (a) x1 (solid line) and xˆ1 (dash-dot (red) line) and (b) the estimated error between x1 and xˆ1 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

are obtained by the proposed GA. The observer gains are found to be ⎡ ⎡ ⎤ 3.2527 × 100 6.6575 × 10−1 3.2531 × 100 ⎢ ⎢ ⎥ L 1 = ⎣ 3.0764 × 102 −2.5089 × 101 ⎦ , L 2 = ⎣ 1.7766 × 102 −1.1913 × 103

4.8228 × 102

−1.2172 × 103

6.8119 × 10−1

⎤

⎥ 2.7120 × 101 ⎦ , 4.8192 × 102

hence, the fuzzy observer is constructed as follows: ˙ˆ = x(t)

2 

h j (xˆ1 (t))(A j x(t) ˆ + B1 j u(t) + L j (y(t) − yˆ (t))).

j=1

The control gains are found to be K 1 = [−32.6506 − 27.7371 − 2.3234 × 10−5 ], K 2 = [−32.6506 − 27.7371 − 2.3234 × 10−5 ], hence, the fuzzy observer-based fuzzy controller is constructed as follows: u(t) =

2 

h s (xˆ1 (t))K s x(t). ˆ

s=1

Remark 10. In this example, the estimated state variable xˆ1 (t) is used as premise variable to construct the proposed fuzzy observer and fuzzy controller. The trajectories of x1 , and xˆ1 , the trajectories of x2 and xˆ2 , and the trajectories of x3 and xˆ3 for Example 2 are shown in Figs. 1, 2, and 3, respectively. The control signal (and the fuzzy observer) is applied at t0 = 3 and shown in Fig. 4.

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C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843

20 x2 estimated x2 hat

states

10 0 −10 −20 −30 0

5

10

15

time (sec)

estimated error

20 estimated error for x2

10 0 −10 −20 −30 0

5

10

15

time (sec) Fig. 2. The trajectories of (a) x2 (solid line) and xˆ2 (dash-dot (red) line) and (b) the estimated error between x2 and xˆ2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

50 x3 estimated x3hat

40 states

30 20 10 0 −10 0

5

10

15

time (sec)

estimated error

50 estimated error for x3

40 30 20 10 0 −10 0

5

10

15

time (sec)

Fig. 3. The trajectories of (a) x3 (solid line) and xˆ3 (dash-dot (red) line) and (b) the estimated error between x3 and xˆ3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843

2841

50 control input

40 30

control signal

20 10 0 −10 −20 −30 −40 −50 0

5

10

15

time (sec)

Fig. 4. The control signal.

From these simulation results, the proposed fuzzy observer-based fuzzy control clearly results in a desired L∞ -gain performance. Remark 11. For the above simulation results, the ratio of the peak of (t) to the peak of d(t) for t  10 in this example is (t)∞ 5.5319 = = 0.0186 <  = 2.1874. d(t)∞ 295.8039 The above L∞ -gain performance is conservative. This is because the min (P) = 7.6016 × 10−7 is very small in this example. Example 3. It is noted that large value of = 30 brings some conservatism on the above two examples. If = 5 is chosen, then  = 2.6675, c = 5.1987 × 10−6 , 1 = 29.5004, 2 = 28.2417, 3 = 3.6044 × 10−5 , 4 = 4.3652 × 10−5 , ⎡ ⎤ 5.7909 × 104 −3.3615 × 104 4.4950 × 101 ⎢ ⎥ X 11 = ⎣ −3.3615 × 104 9.1371 × 104 −4.1805 × 101 ⎦ , ⎡

4.4950 × 101 1.2488 × 10−2

⎢ P22 = ⎣ 3.1998 × 10−8

−4.1805 × 101

5.5678 × 104

3.1998 × 10−8 −1.8776 × 10−7 8.5049 × 10−6

−1.8776 × 10−7 1.7136 × 10−8

⎤

⎥ 1.7136 × 10−8 ⎦ ,

8.4090 × 10−6

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C.-S. Tseng et al. / Fuzzy Sets and Systems 160 (2009) 2824 – 2843



c

= 0.4787  min (P)   max (P) 0.0125 and  = = 38.5430 = min (P) 8.4060 × 10−6

=

are obtained by the proposed GA. The observer gains and control gains are found to be ⎡

11.6864

−2.5722

⎤

⎢ ⎥ L 1 = ⎣ 132.0864 18.7898 ⎦ , −612.2859 249.9502

⎡

12.8243

−2.9888

⎤

⎢ ⎥ L 2 = ⎣ 128.5392 20.3523 ⎦ −672.5422 272.3002

and K 1 = [−29.4497 − 24.9817 − 0.0021], K 2 = [−29.4497 − 24.9817 − 0.0021]. The ratio of the peak of (t) to the peak of d(t) for t  10 in this example is 4.9041 (t)∞ = 0.0166 <  = 0.4787. = d(t)∞ 295.8039 Remark 12. Comparing with Example 2, obviously the above L∞ -gain performance is less conservative. 5. Conclusions In this study, L∞ -gain fuzzy observer-based fuzzy control design, where the premise variables depend on the state variables estimated by a fuzzy observer, for nonlinear systems is considered for the first time via T–S fuzzy models. Both the stability and L∞ -gain disturbance rejection performance are guaranteed for the fuzzy control system. A novel decoupled method is proposed to solve the robust stabilization problem, which can be characterized in terms of linear matrix inequality problem (LMIP) if six scalars can be specified in advance. The GA is proposed to solve this special LMIP. By the proposed decoupled method and the GA, the fuzzy control gains and the fuzzy observer gains can be obtained simultaneously. Since the peak of the system is a greater concern than the energy in some design cases, the proposed L∞ -gain fuzzy observer-based fuzzy control for nonlinear dynamic systems is more appealing in some practical applications. Several simulation results have confirmed the elimination of the peak of the output signal against the persistent bounded disturbances for the nonlinear system by the proposed fuzzy L∞ -gain control scheme. References [1] M. Vidyasagar, Optimal rejection of persistent bounded disturbances, IEEE Trans. Autom. Control 31 (6) (1986) 527–534. [2] M. Vidyasagar, Further results on the optimal rejection of persistent bounded disturbances, IEEE Trans. Autom. Control 36 (6) (1991) 642–652. [3] M.A. Dahleh, J.B. Pearson Jr., L1 -optimal compensators for continuous-time systems, IEEE Trans. Autom. Control 32 (10) (1987) 889–895. [4] A.A. Stoorvogel, Nonlinear L1 -optimal controllers for linear systems, IEEE Trans. Autom. Control 40 (4) (1995) 694–696. [5] M.A. Dahleh, J.B. Pearson Jr., l1 -Optimal feedback controllers for MIMO discrete-time systems, IEEE Trans. Autom. Control 32 (4) (1987) 314–322. [6] M.A. Dahleh, I.J. Diaz-Bobillo, Control of Uncertain Systems: A Linear Programming Approach, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1995. [7] W.-M. Lu, Rejection of persistent L∞ -bounded disturbances for nonlinear systems, IEEE Trans. Autom. Control 43 (12) (1998) 1692–1702. [8] R.J.P. de Figueiredo, G. Chen, Optimal disturbance rejection for nonlinear control systems, IEEE Trans. Autom. Control 34 (12) (1989) 1242–1248. [9] J.S. Shamma, Nonlinear state-feedback for l1 optimal control, Syst. Control Lett. 21 (1993) 265–270. [10] J.S. Shamma, M.A. Dahleh, Time-varying versus time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization, IEEE Trans. Autom. Control 36 (7) (1991) 838–847.

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