H∞ output tracking fuzzy control for nonlinear systems with time-varying delay

Applied Soft Computing 12 (2012) 2963–2972

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H∞ output tracking fuzzy control for nonlinear systems with time-varying delay Peter Liu a,∗ , Tung-Sheng Chiang b a b

Department of Electrical Engineering, Tamkang University, 25137 Danshui Dist., New Taipei City, Taiwan Department of Electrical Engineering, Ching Yun University, 320 Zhongli City, Taoyuan, Taiwan

a r t i c l e

i n f o

Article history: Received 16 September 2011 Received in revised form 2 March 2012 Accepted 19 April 2012 Available online 16 May 2012 Keywords: Time-delay systems T–S fuzzy Virtual desired variables Linear matrix inequalities

a b s t r a c t This paper proposes an observer-based output tracking control via virtual desired reference model for a class of nonlinear systems with time-varying delay and disturbance. First, the Takagi–Sugeno fuzzy model represents the nonlinear system with time-varying delay and disturbance. Then we design an observer to estimate immeasurable states and controller to drive the error between estimated state and virtual desired variables (VDVs) to zero such that the overall control output tracking system has H∞ control performance. Using Lyapunov–Krasovskii functional, we derive sufficient conditions for stability. The advantages of the proposed output control system are (i) systematic approach to derive VDVs for controller design; (ii) relaxes need for real reference model; (iii) drops need for information of equilibrium; (iv) relaxed condition is provided via three-step procedure to find observer and controller gain. We carry out simulation using a continuous stirred tank reactor system where the effectiveness of the proposed controller is demonstrated by satisfactory numerical results. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Tracking control is important in practical applications, e.g., robotic control, servo-motor control, missile control, etc. For nonlinear systems, especially in recent years, control using Takagi–Sugeno (T–S) fuzzy model [1] has attracted wide attention. This is due to the systematic approach to analysis and design of controllers where multiple objectives can be considered in a unified manner [2–15]. The controller design using parallel distributed compensation (PDC) and stability analysis are carried out using Lyapunov direct method which reformulates the control objectives into linear matrix inequality problems (LMIPs) [16–18]. Meanwhile, when immeasurable states exist, studies [19–21] provide a T–S fuzzy observer-based stabilization schemes with conditions in LMIPs. However, except for some works [22–24] which address tracking, most results only focus on the stabilization problem. In details, for regulation, the works [22,25] uses feed-forward compensation of linear control theory. Meanwhile for tracking, the works [22,24] consider the command signal as disturbance to the closed-loop system where a robust control scheme attenuates the tracking error residual. In addition, combining the linear regulation theory and PDC, the works [25,26] achieve the control objective of static or varying output. However, the study [23] has noted that high interaction among plant and controller rules will lead to failure. For output tracking design, the works [27,28] propose a novel

∗ Corresponding author. Tel.: +886 2 2621 5656x2582; fax: +886 2 2620 9814. E-mail addresses: [email protected] (P. Liu), [email protected] (T.-S. Chiang). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.04.025

concept of virtual desired variables (VDVs). The advantages of VDV approach are (i) unifies the stability and tracking design problem, (ii) relaxes the need for a real reference model, and (iii) achieves tracking of time-varying signals. Another challenging problem in control is when time-delay exists in systems. Time delays frequently occur in many practical systems, e.g., chemical processes, nuclear reactors, long transmission lines, and telecommunication. Since time-delay is a main cause of instability and poor performance, control of such systems has received considerable attention. Generally speaking, previous stability analysis can be classified as delay-dependent [29,30] and delay-independent [31,32]. The delay-dependent methods need information on the exact delay duration. On the other hand, the delay-independent methods do not need any information of the delay making these approaches more suitable for practical applications. However, most time-delay system applications deal with stabilizing to a equilibrium. Recently, the work [33] discusses robust output tracking with time-varying delays assuming ideal conditions where premise variables between observer and controller fuzzy rules are matched and actual reference signal exists. In this paper, we are motivated to propose a novel output tracking control of a time-varying reference signal via a VDV approach based on the T–S fuzzy model for time-varying delay systems with disturbances without needing information of equilibrium. First, we represent the nonlinear system with time-varying delay and disturbances into a T–S fuzzy model. Next, we design a set of VDVs. Third, we combine the concept of VDVs controller synthesis and observer-based estimation to develop an output tracking controller. Using Lyapunov’s direct method, we derive sufficient

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P. Liu, T.-S. Chiang / Applied Soft Computing 12 (2012) 2963–2972

conditions and formulate LMIs which guarantees H∞ control performance. We then illustrate the approach on a continuous stirred tank reactor to validate the aforementioned claims. Note that comparing to the work [33], we relax the need for assuming perfect matching of premise variables between the observer and controller. This further improves the applicability of the proposed method on real world applications where observer premise variables only rely on estimated states. In addition, we use a virtual reference model from the solution of the plan itself as the stated reference needs an actual model. The remaining paper is organized as follows. In Section 2, we formulate the control problem. In Section 3, we design the observer-based output tracking control. In Section 4, we explain the VDV design. Section 5 shows the results of the numerical simulation. Finally, some concluding remarks are made in Section 6.

and a control input



u(t) = b−1 0

x˙ dn (t) −

r 

 i (z(t))[Ai,n xd (t)+Adi,n xd (t − )]

+ uk (t),

i=1

(4) where xd (t) is the th (for =1, 2, . . ., n − 1) element of xd (t); Ai, and Adi, is the th row of Ai and Adi matrix, respectively. We can then implement the control as (4) whereas the virtual desired vector xd (t) satisfying (3) will be determined later. Define tracking error xh (t) = x(t) − xd (t). The fuzzy controller based on PDC is as follows: Controller Rule i : IF z1 (t) is F1i and · · · and zf (t) is Ffi THEN uk (t) = −Ki (x(t) − xd (t)),

2. Nonlinear system with time-varying delay and disturbance

where Ki are constant control gains to be determined. The inferred output of the controller

Based on fuzzy modeling methods [34,35], we can describe a class of nonlinear systems with time-varying delay as the T–S fuzzy model:

IF z1 (t) is F1i and · · · and zf (t) is Ffi THEN ˙ x(t) =

Ai x(t) + Adi x(t − (t)) + Bu(t) + w(t)

y(t) =

Cx(t) + v(t)

x(t) =

ϕ(t), t ∈ [− 0]

x˙ h (t) =

(1)

y(t) = Cx(t) + v(t),

r

wi (z(t))/

r

i=1

z2 (t)

wi (z(t)) with

···

zn (t)]T ,

wi (z(t)) =

f

and

F (z(t)). j=1 ji

i (z(t)) = Note

that

 (z(t)) = 1 for all t, where i (z(t)) ≥ 0, for i = 1, 2, . . ., r, are i=1 i the normalized weights. For simplicity we assume B = [0 0 · · · 0 b0 ]T with scalar b0 = / 0 in system (1) in a region of interest x ∈ x . We first consider the simplest case of (i) all state variables are measurable; and (ii) time delay is constant and known, i.e. (t) = . We now introduce a set of VDVs xd (t) = [xd1 (t) xd2 (t)· · · xdn (t)]T which lead to a virtual reference model

x˙ d (t) =

r 







i (z(t)) Ai xd (t) + Adi xd (t − ) + B(u(t) − uk (t)) ,

tf

i=1

xhT (t)Rxh (t) dt ≤

1 2



tf

w(t)2 dt,

0

Theorem 1. A controller (4) and a set of VDVs such that the system (6) is stabilizable to a prescribed  > 0 with H∞ performance, if there exist a symmetric positive definite matrix P, matrices Ki (i = 1, 2, . . ., r) and  > 0 satisfying the following LMIs:

⎢ ⎣

Ai X + XATi + BM i + MiT BT + Q + 2 I

(∗)

(∗)

XATdi

−Q

(∗)

X

0

−R−1

J = xhT (t)Rxh (t) − r 

1 T w (t)w(t) + V˙ 2

i (z(t))xh (t)

i=1 (HiT P

(2)

(3)

⎥ ⎦ < 0,

Proof. Consider a Lyapunov–Krasovskii functional t xhT (t)Pxh (t) + t− xhT ( )xh ( ) d . Define a function

(7)

V=

(8)

(8)

+ PH i + R +  + 2 PP + PAdi −1 ATdi P)xh (t),

where we use the fact 2xhT (t)Pw(t) ≤ 2 xhT (t)PPxTh (t) + (1/2 )wT (t)w(t). If the following matrix inequality HiT P + PH i + R +  + 2 PP + PAdi −1 ATdi P < 0

i (z(t))[Ai, xd (t) + Adi, xd (t − )]

⎤

where X = P−1 , Mi = Ki P−1 , Q = XX, and (∗) is denotes the transposed elements in the symmetric positions.

≤

where uk (t) is the fuzzy controller to be determined. Decomposing (2), we have a constraint of kinematics 0 = x˙ d (t) −

(6)

where tf is terminal time of control;  is a prescribed value denoting the effect of w(t) on xh (t); and R is a symmetric positive definite matrix.

i=1

r 

i (z(t))[Hi xh (t) + Adi xh (t − ) + w(t)],

0

⎡

i (z(t))[Ai x(t) + Adi x(t − (t)) + Bu(t) + w(t)]

z(t) = [z1 (t)

r 

where Hi = Ai − BKi . The control objective is to drive all system states track the virtual desired states with H∞ performance

i=1

where

(5)

i=1

where x(t) ∈ Rn , u(t) ∈ R and y(t) are state, control input and measured output, respectively; Ai , B, Adi , C are matrices with appropriate dimensions; z1 (t) ∼ zf (t) are the premise variables which are states (or combination of states); Fji (j = 1, 2, . . ., f) are the fuzzy sets; r is the number of fuzzy rules; y(t) is the output; ϕ(t) is the initial condition; (t) is the time-varying delay with upper bound  0 , i.e. (t) ≤  0 ; and w(t), v(t) are external disturbance and sensor measurement disturbance, respectively. Using the singleton fuzzifier, product fuzzy inference and weighted average defuzzifier, the inferred output r 

i (z(t))Ki (x(t) − xd (t)),

i=1

leading to the error dynamics

Plant Rule i :

˙ x(t) =

r 

uk (t) = −

(9)

then J < 0. By integrating (8) from 0 to tf , the inequality holds, tf T T 2 0

[xh (t)Rxh (t) − (1/ )w(t) w(t) + V˙ ]dt ≤ 0. For initial condition

P. Liu, T.-S. Chiang / Applied Soft Computing 12 (2012) 2963–2972

V(0) = 0, we have V (tf ) + 0.

V(tf ) ≥ 0,

Since

t

 tf 0

we

[xhT (t)Rxh (t) − (1/2 )w(t)T w(t)]dt ≤ arrive

with

 tf

xhT (t)xh (t)dt 0

Now, the fuzzy controller’s rules are

≤

Controller Rule i :

(1/2 ) 0 f w(t)2 dt. Therefore, the tracking control has H∞ performance. Applying Schur’s complement, we have (7).  Remark 1. Note that if w(t) = 0, the system (6) is asymptotically stable if there exists a symmetric and positive definite matrix P, some matrices Ki (i = 1, 2, . . ., r) and Q = XX > 0 satisfying the following LMIs:



Ai X + XATi + BM i + MiT BT + Q

(∗)

XATdi

−Q



IF z1 (t) is F1i and · · · and zf (t) is Ffi THEN uk (t) = −Ki (ˆx(t) − xd (t)), where Ki are the control gains to be determined. The inferred output of the controller r 

uk (t) = − <0

2965

i (z(t))Ki (ˆx(t) − xd (t)).

(15)

i=1

(10)

Therefore, the error dynamic

where X = P−1 , Fi = Ki X and Q = XX.

x˙ h (t) =

r 

i (z(t))[(Ai − BK i )xh (t) + Adi xh (t − m ) + Li Ce(t) + Li v(t)].

i=1

3. Robust output tracking controller design

Hence, the closed-loop error system Consider the more practical case assuming some states are immeasurable and time-varying delay is unknown. We consider two relationships of premise variables between plant and observer: matching or mismatched.

x˙ e =

r 

i (z(t))[Gi xe (t) + Mi xe (t − ) + Ei w(t)],

where xe (t) = [eT (t) xhT (t)]T , w(t) = [v(t)T w(t)T ]T ,



3.1. Matching premise variables Consider here the observer’s premise variable z(t) is same as the plant. In this case, the proposed fuzzy observer rules are Observer Rule i :

=

Gi

 =

Mi

Ai − Li C

Ai xˆ (t) + Adi xˆ (t − m ) + Bu(t) + Li (y(t) − yˆ (t))

yˆ (t) =

C xˆ (t)

where xˆ (t) ∈ Rn and yˆ (t) ∈ R are state and output estimation, respectively;  m is a suitable value of time-delay (where in practical situations, this suitable value is tunable via trial and error); and Li are observer gains to be determined. Therefore, the inferred output r 

xˆ˙ (t) =



i (z(t)) Ai xˆ (t) + Adi xˆ (t − m ) + Bu(t)

+Li (y(t) − yˆ (t)) yˆ (t) = C xˆ (t).

(11)

Define estimation error e(t) = x(t) − xˆ (t), where the time derivative



0

Adi

r

r 

i (z(t))[Ai xd (t) + Adi xd (t − ) + B(u(t) − uk (t)).

(12)

Decomposing (12), the constraint of kinematics 0 = x˙ d (t) −



tf

xeT (t)Rxe (t) dt

(13)

i=1

and control input



x˙ dn (t) −

r 

0

.

(∗) (∗) −1 I 2

0

⎤

⎥ ⎥<0 ⎦

m

1 ≤ 2 



tf

  w(t)2 dt

(17)

0

J

=

xeT (t)Rxe (t) − r 

1 w(t)T w(t) + V˙ 2



i (z(t))xe (t) GiT P + PGi + R +  + PM i −1 MiT P T

(18)



+2 PE i EiT P xe (t) where we use the facts xeT (t)PM i xe (t − m ) ≤ xeT (t)PM i −1 MiT Pxe (t) + 2 xeT (t − m )xe (t − m ) and

 i (z(t))[Ai,n xd (t) + Adi,n xd (t − )

Li



with tf is terminal time of control,  is a prescribed value denotes the effect of w(t) on xe (t), and R is a positive definite matrix. Define a function

i=1

i (ˆz (t))[Ai, xd (t) + Adi, xd (t − m )],

I

Proof. Consider a Lyapunov–Krasovskii functional as t V (t) = xeT (t)Pxe (t) + t− xeT ( )xe2 ( ) d . The control objective is

≤

r

−Li

where (∗) is denotes the transposed elements in the symmetric positions.

i=1

u(t) =

, Ei =

EiT P

0

w(t) =  (z(t))Adi [x(t − (t)) − xˆ (t − m )] + w(t). where i=1 i Design a virtual reference model

b−1 0



⎡ GT P + PG + R +  (∗) i i ⎢ TP − M ⎢ i ⎣



i (z(t))[(Ai − Li C)e(t) + Adi e(t − m ) − Li v(t) + w(t)],

i=1

x˙ d (t) =



0

required to satisfy

r

˙ = e(t)

Ai − BK i

Li C

Theorem 2. A controller (14) and a set of VDVs such that the closeloop fuzzy system (16) is stabilizable to a prescribed  > 0 with H∞ performance, if there exist a symmetric and positive definite matrix P, some matrices Ki (i = 1, 2, . . ., r) and  > 0 satisfying the following LMIs:

(11)



i=1



0

Adi

IF z1 (t) is F1i and · · · and zf (t) is Ffi THEN xˆ˙ (t) =

(16)

i=1

T

+ uk (t).

i=1

(14)

2xeT (t)PE i w(t) ≤ 2 xeT (t)PE i EiT Pxe (t) + (1/2 )w (t)w(t). condition GiT P + PGi + R +  + PM i −1 MiT P + 2 PE i EiT P < 0

If

the

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P. Liu, T.-S. Chiang / Applied Soft Computing 12 (2012) 2963–2972

and a new control input

holds, then J < 0. By integrating (18) from 0 to tf , we obtain



tf



xeT (t)Rxe (t) −

0

1 w(t)T w(t) + V˙ 2

initial

Assuming

 tf

condition



V(0) = 0,

(19)

we

T [xeT (t)Rxe (t) − (1/2 )w (t)w(t)] dt ≤ 0. Since 0 2 t t arrive with 0 f xeT (t)Rxe (t) dt ≤ (1/2 ) 0 f w(t) that the overall system (16) has H∞ performance.

  





dt ≤ 0.

u(t) =

V(tf ) ≥ 0,

i (ˆz (t))[Ai,n xd (t) + Adi,n xd (t − m )] (24)

The error dynamics x˙ h (t) =

3.2. Mismatched premise variables



+ uk (t).

we

  dt. This means

x˙ dn (t) −

r  i=1

V (tf ) +

have

b−1 0

r 

i (ˆz (t))[(Ai − BK i )xh (t) + Adi xh (t − ) + Li Ce(t) + Li v(t)].

i=1

Consider here the premise variables of observer depends on estimated states zˆ (t) which leads to membership functions i (ˆz (t)) = / i (z(t)). The fuzzy observer rules are

Hence, the closed-loop error system x˙ e =

Observer Rule i :

=

where h(t) = [h1 (t)T 0]T ,

Ai xˆ (t) + Adi xˆ (t − m ) + Bu(t) + Li (y(t) − yˆ (t))

yˆ (t) =



C xˆ (t)

where zˆ1 (t)∼ˆzf (t) are the observer’s premise variables composed of estimated states. The inferred output r 

xˆ˙ (t) =

i (ˆz (t))[Ai xˆ (t) + Adi xˆ (t − m )

yˆ (t) = C xˆ (t).

(20)

The error dynamics i (ˆz (t))[(Ai − Li C)e(t) + Adi e(t − m )−Li v(t)]+w(t)+h1 (t),

i=1

where h1 (t) =

r

r

(i (z(t) − i (ˆz (t))[Ai x(t) + Adi x(t − (t))] and i=1

w(t) =  (ˆz (t))Adi [x(t − (t)) − xˆ (t − m )] + w(t). i=1 i the time derivative of tracking error



=

 =

Ai − Li C

Ai − BK i

Li C Adi

0

0

Adi





, Ei =

I

Li

0



tor Tji and premise variables z(t), zˆ (t) in the region of interest. Since premise variables are states or combination of states, the relationship Fji (x(t)) − Fji (ˆx(t)) = Tji (x(t) − xˆ (t)). Therefore the function error is proportional to the estimation error where

i (x(t)) − i (ˆx(t))

=

f 

f 

k=2 f −2

k=3

Fki (x(t)) + F1i (ˆx(t))T2i e(t))

T1i e(t)

Therefore +· · · +



f −1 

+

i=1

− x˙ d (t).

(21)

The fuzzy controller rules are

=

Fki (x(t))

Fki (ˆx(t))T(f −1)i e(t)Ffi (x(t))

k=1

Fki (ˆx(t))Tfi e(t)

k=1 Ti e(t)

for some bounded function vector i . In light of the above relationship, we have

Controller Rule i : IF zˆ1 (t) is F1i and · · · and zˆf (t) is Ffi THEN

h2 (t) =

uk (t) = −Ki (ˆx(t) − xd (t)), where Ki are control gains to be determined. Then the inferred output of the controller



−Li

Remark 2. Note that the membership functions Fji (z(t)) satisfy Fji (z(t)) − Fji (ˆz (t)) = Tji (z(t) − zˆ (t)) for some bounded function vec-

i (ˆz (t))[Ai xˆ (t) + Adi xˆ (t − m ) + Bu(t) + Li Ce(t) + Li v(t)]

r

uk (t) = −



0

r

x˙ h (t) =

(25)

(20)

i=1

r 

Gi

Mi

+Bu(t) + Li (y(t) − yˆ (t))]

˙ = e(t)

i (ˆz (t))[Gi xe (t) + Mi xe (t − ) + Ei w(t) + h(t)]

i=1

IF zˆ1 (t) is F1i and · · · and zˆf (t) is Ffi THEN xˆ˙ (t)

r 

r 

(i (x(t)) − i (ˆx(t)))[Ai x(t) + Adi x(t − (t))]

i=1 r

=



[(Ai x(t) + Adi x(t − (t))) Ti ]e(t).

i=1

i (ˆz (t))Ki (ˆx(t) − xd (t)).

(22)

i=1

Suppose an upper bound for x(t), x(t − (t)) and (t) exists in the region of interest. The term h1 (t) therefore satisfies [36,37]

Design a virtual reference model x˙ d (t) =

r 

hT1 (t)h1 (t) ≤ eT (t)U T Ue(t) i (ˆz (t))[Ai xd (t) + Adi xd (t − ) + B(u(t) − uk (t)).

i=1

Therefore the constraint of kinematics 0 = x˙ d (t) −

r  i=1

i (ˆz (t))[Ai, xd (t) + Adi, xd (t − m )]

(23)

with a symmetric positive-definite matrix U depend on Ti , x(t) and x(t − (t)). We can attenuate h1 (t) from affecting control performance by choosing suitable observer gains Li and controller gains Ki shown as follows. Theorem 3. A controller (24) and a set of VDVs such that system (25) is stabilizable to a prescribed  > 0 with H∞ performance, if there exist

P. Liu, T.-S. Chiang / Applied Soft Computing 12 (2012) 2963–2972

a symmetric and positive definite matrix P, some matrices Ki (i = 1, 2, . . ., r) and  > 0 satisfying the following LMIs:

⎡ ⎢ ⎢ ⎢ ⎣

T

GiT P + PGi + R +  + PP + U U

(∗)

⎤

(∗)

⎥ − (∗) ⎥ < 0 ⎥ ⎦ −1

MiT P EiT P

0

2

(26)

I

where (∗) is denotes the transposed elements in the symmetric positions and U =block-diag{U0}. Proof. Consider a Lyapunov–Krasovskii functional V = t xeT (t)Pxe (t) + t− xeT ( )xe ( ) d . The control objective must satisfy (17). Taking the derivative of Lyapunov–Krasovskii functional and applying (18) along with (25), the function J = xeT (t)Rxe (t) −

≤

r 

1 w(t)T w(t) + V˙ 2

(27)

i (ˆz (t))xe (t)[Gi + P MiT −1 Mi P + 2 xeT (t)PE i EiT P]xe (t), (27) T

i=1 T

where Gi = GiT P + PGi + R +  + PP + U U. Since xeT (t)PM i xe (t − m ) ≤ xeT (t)PM i −1 MiT Pxe (t) + xeT (t − m )xe (t − m ) 2xeT (t)PE i w(t)

≤

2 xeT (t)PE i EiT Pxe (t)

2xeT (t)Ph(t)

≤

xeT (t)PPxe (t)

+

1 T + 2 w (t)w(t) 

xeT (t)U

T

Uxe (t).

If the condition (26) holds then J < 0. By integrating (27) from 0 to tf , we obtain (19). By assuming initial condition V(0) = 0, we have V (tf ) +

 tf

(xeT (t)Rxe (t) − (1/2 )w(t)T w(t)) dt ≤ 0. Since 0 2 tf T t xe (t)Rxe (t) dt ≤ (1/2 ) 0 f w(t) dt. This 0

  



V(tf ) ≥ 0, we have means that the overall system has H∞ performance.

 

3.3. Procedure of determining observer and controller gain We will now show the procedure for determining the observer and controller gains for the most complex case where time-varying delay exist and premise variables between observer and plant are mismatched. Note that for more ideal cases, we can follow the same design procedures. From Theorem 3, we can determine the observer gains Li and controller gains Ki and P > 0,  > 0 satisfying (26) for a given  > 0, R = RT > 0 and U. We provide two procedures, two step or three step to solve the LMIs (26). Two-step procedure: The concept of two-steps procedure to determine Ki , Li , P and  are introduced by [38]. The main idea begins from defining P,  and R as block-diagonal form, i.e. P = block − diag{P1 P2 },  = block − diag{1 2 }, and R = block − diag{R1 R2 }. According to (26), we therefore have

⎡

i

(∗)

(∗)

(∗)

(∗)

(∗)

⎢ (∗) (∗) (∗) (∗) ⎢ P2 Li C i ⎢ ⎢ AT P 0 −1 (∗) (∗) (∗) ⎢ di 1 ⎢ T ⎢ 0 Adi P2 0 −2 (∗) (∗) ⎢ ⎢ −1 ⎢ −LT P LT P 0 0 I (∗) ⎢ i 1 i 2 2 ⎢ ⎣ −1 P1

0

0

0

0

⎤

2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ I

where i = (Ai − Li C)T P1 + P1 (Ai − Li C) + R1 + 1 + P1 P1 + UT U i = (Ai − BKi )T P2 + P2 (Ai − BKi ) + P2 P2 + R2 + 2 .

Step 1 The matrix inequalities (28) implies

⎡ i ⎢ AT P ⎢ di 1 ⎣

−LiT P1

(∗)

(∗)

−1

(∗)

0

−1 I 2

and

⎤

⎥ ⎥ < 0. ⎦

(29)

From Schur’s complement, we have (Ai − Li C)T P1 + P1 (Ai − Li C) + ATdi P2 + P1 P1 + 2 P1 Li LiT P1 < 0 and R1 + 1 + U T U + P2 Adi −1 2

⎡ AT P1 + P1 Ai − Ni C − C T N T + R1 + 1 + U T U

(∗)

(∗)

ATdi P1

−2

(∗)

(∗) ⎥

−LiT P1

0

−1 I 2

(∗) ⎦

P1

0

0

i

⎢ ⎢ ⎢ ⎣

⎤

(∗)

i

⎥ ⎥ < 0,

(30)

−I

where Ni = P1 Li . Step 2 According to (28) and by the Schur’s complement, we have

⎡

i

(∗)

(∗)

(∗)

(∗)

(∗)

(∗)

⎢ ⎢ Li C i (∗) (∗) (∗) (∗) ⎢ ⎢ T 0 −1 (∗) (∗) (∗) ⎢ Adi P1 ⎢ ⎢ 0 ATdi X2 0 −Q2 (∗) (∗) ⎢ ⎢ ⎢ T −1 ⎢ −Li P1 LiT 0 0 I (∗) ⎢ 2 ⎢ ⎢ −1 ⎢ P1 0 0 0 0 I ⎣ 2 0

X2

0

0

0

(∗) (∗) (∗) (∗) (∗)

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(31)

−R2−1

0

where X2 = P2−1 , i = Ai X2 + X2 ATi − BF i − FiT BT + Q2 + I, Fi = Ki X2 , and Q2 = X2 2 X2 . Once (30) is feasible, the controller gains Ki = Fi P2 , 2 = P2 Q2 P2 . Then, if (31) is feasible, the observer gains Li = P1−1 Ni . Three-steps procedure: In this procedure, the matrices P and  do not need to be block-diagonal, making LMIs (26) more feasible with computational load tradeoff. ATdi P2 < 0, we have by Schur’s Since (28) imply i + P2 Adi −1 2 complement

⎡

i

(∗)

(∗)

⎢ T ⎣ Adi X2 −Q2

(∗)

X2

⎤

⎥ ⎦ < 0.

(32)

−R2−1

0

Step 1 Solve (30) for observer gain Li ; Step 2 Solve (32) for controller gains Ki ; Step 3 From Step 1 and Step 2, if the gain Ki and Li are available, then solve matrices P > 0 and  > 0 satisfying the following LMIs:

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

T

GiT P + PGi + R +  + U U

PM i

PE i

P

MiT P

−

0

0

EiT P

0

−1 I 2

0

0

P (28)

2967

P ∈ R2n×2n

⎤

⎥ ⎥ ⎥ ⎥<0 0 ⎥ ⎦

−I

 ∈ R2n×2n .

where and Note that in the three-steps procedure, the matrices P1 , P2 , 1 , and 2 ∈ Rn×n in Step 1 and Step 2. Meanwhile the matrices P ∈ R2n×2n and  ∈ R2n×2n in Step 3. Remark 3. The two-step procedure is more suitable and when (26) is satisfied. However, since R, P and  are in block-diagonal form, the procedure is more conservative in comparison to the three-step procedure.

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P. Liu, T.-S. Chiang / Applied Soft Computing 12 (2012) 2963–2972

4. Virtual desired variables design Analogous to the above discussion, the most complex case is when we consider time-varying delay with mismatched premise variables between observer and plant. We will show in the following the detailed procedure for designing VDVs xd (t). Note that for more ideal cases, we can follow the same design procedures. We design the VDVs according to the following equation yd (t) = xdi (t)

x˙ d (t) =

r 

VDVs

xˆ (t )

y (t )

Nonliear time-delay system

(33)

y (t ) − yˆ (t )

yˆ (t )

i (ˆz (t))[Ai, xd (t) + Adi, xd (t − )],

(34)

Observer

i=1

xˆ (t )

where =1, 2, . . ., n − 1. By solving the constraint (33) and (34), we obtain VDVs xd (t) and the tracking error xh (t). Therefore we can design the controller

 u(t) =

y d (t )

xd (t )

b−1 0 r 

−

x˙ dn (t) −

r 

xd (t )

u (t )

Controller

 i (ˆz (t))[Ai,n xd (t) + Adi,n xd (t − )

i=1

Fig. 1. The overall structure of the observer-based output tracking control via VDVs approach.

(35)

i (ˆz (t))Ki (ˆx(t) − xd (t)).

5. Simulation results

i=1

To implement the control input (35), we require the following information, (i) derivative signal x˙ dn (t); (ii) state estimation xˆ (t); suitable time delay  m . Three cases of output tracking are discussed as follows. Case 1 (reference output equals desired state): Since desired output yd (t) = xdn (t) and y˙ d (t) ∈ L2 , we solve (34) for VDVs and use observer (20) to estimate immeasurable states. Therefore, the controller can be implemented as (35). Case 2 (reference output is not equal to the desired state): Since yd (t) = / xdn (t), controller design might not be as straightforward. To cope with this, we introduce an approximate signal x˙ dn (t) ≈ (xdn (t) − xdn (t − ts ))/ts , where ts is the sampling period. We may then attenuate the arising approximation errors by suitably choosing observer gains Li and controller gains Ki . Case 3 (output regulation): This is an ideal case where the desired state vector xd is constant and xˆ (t) = x(t) = xd such that i (z(t)) = i (ˆz (t)), xd (t − ) = xd and x(t − (t)) = xd . Thus, r 

i (zd ))[Ai, xd (t) + Adi, xd (t − )] = 0,  = 1, 2, . . . , n − 1 (36)

i=1

where zd = xd . There exists n − 1 equations in (36). If we assign the desired output yd (t) = xdn , then x˙ dn (t) = 0. Note that the desired state xd should be properly chosen such that xd ∈ x . According the analysis above, the overall structure of the controlled output tracking control system in Fig. 1. Therefore, the design procedure for controlled output tracking control is summarized as follows: Design procedure: Step 1 Construct the T–S fuzzy model for the nonlinear timedelay system in (1). Step 2 Given an attenuation level , follow the three-steps procedure to obtained controller and observer gain. Step 3 Design the VDVs from (33) and (34), such that the error state xh (t) is obtained. Step 4 Implement the controller as (35). According the analysis above, the design constraint of VDVs and controller are for various cases are summarized in Table 1.

Consider a continuous stirred tank reactor (CSTR) from [39] represented in dimensionless variables



x˙ 1 (t) = f1 (x(t)) + 1 −



x˙ 2 (t) = f2 (x(t)) + 1 −

1

1

 

x1 (t − (t)) x1 (t − (t)) + ˇu(t)

where f1 (x(t)) =

−1 x1 (t) + Da (1 − x1 (t)) exp

f2 (x(t)) = (



x2 (t) 1 + x2 (t)/0

−1 + ˇ)x2 (t) + HDa (1 − x1 (t)) exp





x2 (t) 1 + x2 (t)/0

 .

The state x1 (t) corresponds to the conversion rate of the reaction 0 ≤ x1 (t) ≤ 1; and x2 (t) > 0 is the dimensionless temperature. We set x2 (t) ∈2 with 2 = {x2 | 0.1 ≤ x2 ≤ 6}. The parameters  0 = 20, H = 8, ˇ = 0.3, Da = 0.072, = 0.8, and  = 2. Applying the design methodology mentioned in previous sections, we arrive with the following: 1 Define membership functions w11 = (g1 − Step w21 = (g2 − d22 )/(d21 − d12 )/(d11 − d12 ), w12 = 1 − w11 , d22 ), w22 = 1 − w21 , where g1 = exp (x2 (t)/(1 + x2 (t)/ 0 )), g2 = exp (x2 (t)/(1 + x2 (t)/ 0 ))/x2 , d11 = max g1 = 101.0267, d12 = min g1 = 1.1046, x2 ∈2

x2 ∈2

d21 = max g2 = 16.8378, x2 ∈2

min g1 = 2.5789. Therefore the subsystem matrices

x2 ∈2

A1



=

 A2

=

 A3

=

 A4

=

(−1/ ) − Da d11

Da d21

−HDa d11

−(1/ + ˇ) + HDa d21

(−1/ ) − Da d11

Da d22

−HDa d11

−(1/ + ˇ) + HDa d22

(−1/ ) − Da d12

Da d21

−HDa d12

−(1/ + ˇ) + HDa d21

(−1/ ) − Da d12

Da d22

−HDa d12

−(1/ + ˇ) + HDa d22

   

and

d22 =

P. Liu, T.-S. Chiang / Applied Soft Computing 12 (2012) 2963–2972

2969

Table 1 VDVs and controller designs for various conditions. Conditions

VDVs

Controller





r

x˙ d (t) = States are all available with constant time delay 

b−1 0

u(t) =

i (z(t)) Ai, xd (t)

x˙ dn (t) −

r 

i (z(t))



i=1



i=1



×[Ai,n xd (t) + Adi,n xd ](t − )

+Adi, xd (t − )

+uk (t)

r 

uk (t) = −

i=1

x˙ d (t) = Partial states are immeasurable with matched premise variables i (ˆz (t)) = i (z(t)) and time-varying delay

r 



u(t)

i (z(t)) Ai, xd (t)

b−1 0

=

x˙ dn (t) −

r 

i (z(t))



i=1



i=1

i (z(t)) × Ki (x(t) − xd (t)).



× [Ai,n xd (t) + Adi,n xd (t − m )]

+Adi, xd (t − m )

+uk (t)

r 

uk (t) = −

i=1

x˙ d (t) = Partial states are immeasurable with mismatched premise / i (z(t)) variables i (ˆz (t)) = and time-varying delay

r 



u(t)

i (ˆz (t)) Ai, xd (t)

= b−1 0

x˙ dn (t) −

r 

i (ˆz (t))



i=1



i=1

i (z(t))Ki (ˆx(t) − xd (t))



×[Ai,n xd (t) + Adi,n xd (t − )]

+Adi, xd (t − )

+uk (t)

r 

uk (t) = −

i (ˆz (t))Ki (ˆx(t) − xd (t))

i=1

 Ad =

 



1 − 1/

0

0

1 − 1/

,B=

where ref is the multi-step signal (each step size is fixed). Since x˙ d1 (t) and xd1 (t − ) are available, from (37), we have

0

ˇ

Step 2: Solve controller and observer gains following the three-step procedure. Given an attenuation level  = 1.1, matrices R = 0.01diag(1, 1, 1, 1), U = diag(0.01, 0.01), the solved controller gains K1 = [ −175.2199 44.9349 ], K2 = [ −180.2105 16.2964 ], K3 = [ 8.2168 42.8092 ], K4 = [ 3.3857 14.2047 ], observer gains L1 = [ 16.3823 L3 = [ 16.4024 matrices

⎡

0.8749

⎢ −0.0805 ⎣ 0.0202

P=⎢

0.0040

⎡

0.5758

T

165.0860 ] , L2 = [ 16.2812

T 165.2433 ] ,

−0.0805

L4 = [ 16.3008 0.0202

0.0191

0.0045

0.0045

0.0757

0.0071

−0.0011

−0.1192

−0.0121

0.0040

T

164.2247 ] ,

T 164.3701 ] ,

⎤

−0.0194

−0.0071 ⎥

⎥

−0.0011 ⎦





u(t)=ˇ

−1

x˙ d2 (t) −

r 



i (z(t))[Ai,2 xd (t) + Adi,2 xd (t − )]

+ uk (t).

i=1

0.0077 0.0369

⎤ 2.6

0.0217

2.4

Step 3: Ensure desired state satisfies constraint

2.2

r

0 = x˙ d1 (t) −

Therefore we have xh (t) = xˆ (t) − xd (t). Step 4: Implement the controller

For this simulation, the external disturbance and measured noise are chosen as uniformly random noise with amplitude 0.01. The time-varying delay is shown in Fig. 2. The simulation results of the H∞ control are shown in Figs. 3–7 with desired controlled output

⎢ −0.1192 0.2807 −0.0124 −0.0268 ⎥ ⎥ ⎣ −0.0121 −0.0124 0.1057 −0.0194 ⎦ . −0.0268

x˙ d1 (t) − (−1/ − Da g) . Da z/x2 (t)

and

=⎢

0.0369

xd2 (t) =

i (z(t))[Ai,1 xd (t) + Adi,1 xd (t − )].

(37)

2 1.8

i=1

We consider the desired output to be sine wave yd (t) = xd1 (t) = 0.5 + 0.2 sin (t) or multi-step signal. To ensure a smooth multi-step signal, we assign xd1 (t) = 4xr1 and x˙ d1 (t) = 4xr2 (t) with reference state xr1 and xr2 . The reference dynamic system x˙ r1 (t) = xr2 (t)

(38)

x˙ r2 (t) = −4xr1 (t) − 2xr2 (t) + ref

(38)

1.6 1.4 1.2 1

0

5

10

15

20

25

Fig. 2. Time-varying delay.

30

35

40

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P. Liu, T.-S. Chiang / Applied Soft Computing 12 (2012) 2963–2972

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

5

10

15

20

25

30

35

0

40

0

5

10

15

20

25

30

35

40

time

time 0.4

0.6 0.4

0.2

0.2 0

0 −0.2

0

5

10

15

20

25

30

35

40

−0.2

0

5

10

15

20

25

30

35

40

time

time Fig. 5. Response of (a) x1 (t) and xd1 (t) (b) xh1 (t).

Fig. 3. Response of (a) x1 (t) and xˆ 1 (t) (b) e1 (t).

5

sine wave xd1 (t) = 0.5 + 0.2 sin (t). Note that we consider the more complex case of mismatched premise variables for all of the simulations. Fig. 3(a) shows x1 (t), xˆ 1 (t) and Fig. 3(b) shows estimation error e1 (t). Fig. 4(a) shows x2 (t), xˆ 2 (t) and Fig. 4(b) shows estimation error e2 (t). From Figs. 3 and 4, the result validate the proposed observer design (20). Fig. 5(a) shows x1 (t), xd1 (t) and Fig. 5(b) shows tracking error xh1 (t). Fig. 6(a) shows x2 (t), xd2 (t) and Fig. 6(b) shows tracking error xh2 (t). Fig. 7 is the control input u(t). From Figs. 5–7, the results validate the proposed controller design (24). The results of the multi-step reference signal are shown in Figs. 8–12. Fig. 8(a) shows x1 (t), xˆ 1 (t) and Fig. 8(b) shows the estimation error e1 (t). Fig. 9(a) shows x2 (t), xˆ 2 (t) and Fig. 9(b) shows estimation error e2 (t). From Figs. 3 and 4, the result validate the proposed observer design (20). Fig. 10(a) shows x1 (t), xd1 (t) and Fig. 10(b) shows tracking error xh1 (t). Fig. 11(a) shows x2 (t), xd2 (t) and Fig. 11(b) shows tracking error xh2 (t). Fig. 12 is the control signal. From Figs. 10–12, the results validate the proposed controller design (24). From the above simulation results, we can verify the proposed control method achieve H∞ performance for nonlinear time-delay

4 3 2 1 0

0

10

15

20

25

30

35

40

time 1

0

−1

−2

0

5

10

15

20

25

30

35

40

time Fig. 6. Response of (a) x2 (t) and xd2 (t) (b) xh2 (t).

5

40

4

35

3

30

2

5

25

1 20

0

0

5

10

15

20

25

30

35

40

time

15

1

10

0.5

5 0

0

−5

−0.5 −1

−10

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

Fig. 4. Response of (a) x2 (t) and xˆ 2 (t) (b) e2 (t).

30

35

40

time

time Fig. 7. Control signal u(t).

P. Liu, T.-S. Chiang / Applied Soft Computing 12 (2012) 2963–2972

0.8

6

0.6

4

0.4

2

0.2

0

0

0

5

10

15

20

25

30

35

40

−2

time 0.8

0

5

10

2971

15

20

25

30

35

40

time 2

0.6

1 0.4 0.2

0

0

−1

−0.2

0

5

10

15

20

25

30

35

40

time

−2

0

5

10

15

20

25

30

35

40

time

Fig. 8. Response of (a) x1 (t) and xˆ 1 (t) (b) e1 (t).

Fig. 11. Response of (a) x2 (t) and xd2 (t) (b) xh2 (t).

5 4 10

3 2

5

1 0

0

0

5

10

15

20

25

30

35

40

time

−5

2 −10

1 −15

0 −20

−1 −2

−25

0

5

10

15

20

25

30

35

40

0

5

10

15

20

25

time

30

35

40

time

Fig. 12. Control signal u(t).

Fig. 9. Response of (a) x2 (t) and xˆ 2 (t) (b) e2 (t).

0.8

systems with disturbance. This is aligned with the claims of Theorem 3.

0.6 0.4

6. Conclusions 0.2 0

0

5

10

15

20

25

30

35

40

time 0.8 0.6 0.4 0.2

This paper has presented an observer-based via VDV output control approach for a class of nonlinear systems with time-varying delay and disturbances where the advantages are: (i) systematic approach to derive VDVs for controller can be design; (ii) relaxes need for real reference model; (iii) drops need for information of equilibrium; (iv) relaxed condition is provided via three-step procedure is provided to find observer and controller gain. Finally, simulation results on a CSTR with sine-wave and multi-step reference illustrate the expected performance.

0 −0.2

0

5

10

15

20

25

30

35

40

time Fig. 10. Response of (a) x1 (t) and xd1 (t) (b) xh1 (t).

Acknowledgements This work was supported by National Science Council, R.O.C., under grant NSC 100-2221-E032-024 and NSC 98-2221-E-231-004.

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P. Liu, T.-S. Chiang / Applied Soft Computing 12 (2012) 2963–2972

References [1] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems, Man, and Cybernetics 15 (1985) 116–132. [2] M. Cococcioni, B. Lazzerini, F. Marcelloni, On reducing computational overhead in multi-objective genetic Takagi–Sugeno fuzzy systems, Applied Soft Computing 11 (2011) 675–688. [3] R. Rajesh, M. Kaimal, T–S fuzzy model with nonlinear consequence and PDC controller for a class of nonlinear control systems, Applied Soft Computing 7 (2007) 772–782. [4] R. Ketata, Y. Rezgui, N. Derbel, Stability and robustness of fuzzy adaptive control of nonlinear systems, Applied Soft Computing 11 (2011) 166–178. [5] C.-W. Chen, M.H.L. Wang, J.-W. Lin, Managing target the cash balance in construction firms using a fuzzy regression approach, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems (2009) 667–684. [6] C.-W. Chen, K. Yeh, K.F.-R. Liu, Adaptive fuzzy sliding mode control for seismically excited bridges with lead rubber bearing isolation, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems (2009) 705–727. [7] M.-L. Lin, C.-W. Chen, Q.-B. Wang, Y. Cao, J.-Y. Shih, Y.-T. Lee, C.-Y. Chen, S. Wang, Fuzzy model-based assessment and monitoring of desertification using MODIS satellite imagery, Engineering Computations: International Journal for Computer-Aided Engineering 26 (2009) 745–760. [8] C.-W. Chen, The stability of an oceanic structure with T–S fuzzy models, Mathematics and Computers in Simulation 80 (2009) 402–426. [9] C.-W. Chen, Modeling and control for nonlinear structural systems via a NNbased approach, Expert Systems with Applications 36 (2009) 4765–4772. [10] M.-L. Lin, C.-W. Chen, Application of fuzzy models for the monitoring of ecologically sensitive ecosystems in a dynamic semi-arid landscape from satellite imagery, Engineering Computations 27 (2010). [11] C.-Y. Chen, J.-W. Lin, W.-I. Lee, C.-W. Chen, Fuzzy control for an oceanic structure: a case study in time-delay TLP system, Journal of Vibration and Control 16 (2010) 147–160. [12] C.-W. Chen, Modeling and fuzzy PDC control and its application to an oscillatory TLP structure, Mathematical Problems in Engineering 2010 (2010) 13. [13] K. Yeh, C.-W. Chen, Stability analysis of interconnected fuzzy systems using the fuzzy Lyapunov method, Mathematical Problems in Engineering 2010 (2010) 10. [14] C.-W. Chen, Application of fuzzy-model-based control to nonlinear structural systems with time delay: an LMI method, Journal of Vibration and Control 16 (2010) 1651–1672. [15] K. Yeh, C.-Y. Chen, C.-W. Chen, Robustness design of time-delay fuzzy systems using fuzzy Lyapunov method, Applied Mathematics and Computation 205 (2008) 568–577. [16] B.-S. Chen, C.-S. Tseng, H.-J. Uang, Robustness design of nonlinear dynamic systems via fuzzy linear control, IEEE Transactions on Fuzzy Systems 7 (1999) 571–585. [17] F. Zheng, Q.-G. Wang, T.H. Lee, X. Huang, Robust PI controller design for nonlinear systems via fuzzy modeling approach, IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans 31 (2001) 666–675. [18] I. Zamani, M.H. Zarif, On the continuous-time Takagi–Sugeno fuzzy systems stability analysis, Applied Soft Computing 11 (2011) 2102–2116, The Impact of Soft Computing for the Progress of Artificial Intelligence. [19] K. Tanaka, T. Ikeda, H. Wang, Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs, IEEE Transactions on Fuzzy Systems 6 (1998) 250–265.

[20] S.K. Nguang, P. Shi, H∞ fuzzy output feedback control design for nonlinear systems: an LMI approach, IEEE Transactions on Fuzzy Systems 11 (2003) 331–340. [21] J. Yoneyama, H∞ output feedback control for fuzzy systems with immeasurable premise variables: discrete-time case, Applied Soft Computing 8 (2008) 949–958. [22] C.-S. Tseng, B.-S. Chen, H.-J. Uang, Fuzzy tracking control design for nonlinear dynamic systems via T–S fuzzy model, IEEE Transactions on Fuzzy Systems 9 (2001) 381–392. [23] H.J. Lee, J.B. Park, Y.H. Joo, Comments on “output tracking and regulation of nonlinear system based on Takagi–Sugeno fuzzy model”, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 33 (2003) 521–523. [24] C.-S. Tseng, Model reference output feedback fuzzy tracking control design for nonlinear discrete-time systems with time-delay, IEEE Transactions on Fuzzy Systems 14 (2006) 58–70. [25] P. Korba, R. Babuska, H. Verbruggen, P. Frank, Fuzzy gain scheduling: controller and observer design based on Lyapunov method and convex optimization, IEEE Transactions on Fuzzy Systems 11 (2003) 285–298. [26] X.-J. Ma, Z.-Q. Sun, Output tracking and regulation of nonlinear system based on Takagi–Sugeno fuzzy model, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 30 (2000) 47–59. [27] K.-Y. Lian, C.-H. Chiang, H.-W. Tu, LMI-based sensorless control of permanentmagnet synchronous motors, IEEE Transactions on Industrial Electronics 54 (2007) 2769–2778. [28] K.-Y. Lian, J.-J. Liou, Output tracking control for fuzzy systems via output feedback design, IEEE Transactions on Fuzzy Systems 14 (2006) 628–639. [29] Y.-Y. Cao, Y.-X. Sun, C. Cheng, Delay-dependent robust stabilization of uncertain systems with multiple state delays, IEEE Transactions on Automatic Control 43 (1998) 1608–1612. [30] L. Guo, H∞ output feedback control for delay systems with nonlinear and parametric uncertainties, IEE Proceedings – Control Theory and Applications 149 (2002) 226–236. [31] C.-Y. Lu, J.S.-H. Tsai, G.-J. Jong, T.-J. Su, An LMI-based approach for robust stabilization of uncertain stochastic systems with time-varying delays, IEEE Transactions on Automatic Control 48 (2003) 286–289. [32] R. Luo, L.-Y. Chung, Stabilization for linear uncertain system with time latency, IEEE Transactions on Industrial Electronics 49 (2002) 905–910. [33] C. Lin, Q.-G. Wang, T.H. Lee, H∞ output tracking control for nonlinear systems via T–S fuzzy model approach, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 36 (2006) 450–457. [34] K.-Y. Lian, C.-S. Chiu, T.-S. Chiang, P. Liu, LMI-based fuzzy chaotic synchronization and communications, IEEE Transactions on Fuzzy Systems 9 (2001) 539–553. [35] K.-Y. Lian, T.-S. Chiang, C.-S. Chiu, P. Liu, Synthesis of fuzzy model-based designs to synchronization and secure communications for chaotic systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 31 (2001) 66–83. [36] K.-Y. Lian, H.-W. Tu, J.-J. Liou, Stability conditions for LMI-based fuzzy control from viewpoint of membership functions, IEEE Transactions on Fuzzy Systems 14 (2006) 874–884. [37] K.-Y. Lian, C.-H. Chiang, H.-W. Tu, LMI-based sensorless control of permanentmagnet synchronous motors, IEEE Transactions on Industrial Electronics 54 (2007) 2769–2778. [38] J.-C. Lo, M.-L. Lin, Observer-based robust H∞ control for fuzzy systems using two-step procedure, IEEE Transactions on Fuzzy Systems 12 (2004) 350–359. [39] Y.-Y. Cao, P. Frank, Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach, IEEE Transactions on Fuzzy Systems 8 (2000) 200–211.