- Email: [email protected]
Delay-dependent robust H∞ control for uncertain fuzzy hyperbolic systems with multiple delays
Available online at www.sciencedirect.com
Progress in Natural Science 18 (2008) 97–104
Delay-dependent robust H1 control for uncertain fuzzy hyperbolic systems with multiple delays Huaguang Zhang
a,*
, Qingxian Gong
a,b
, Yingchun Wang
a
a b
School of Information Science and Engineering, Northeastern University, Shenyang 110004, China Department of Electrical Engineering, Changchun Institute of Technology, Changchun 130012, China Received 7 July 2007; received in revised form 10 August 2007; accepted 10 September 2007
Abstract The robust H1 control problem was considered for a class of fuzzy hyperbolic model (FHM) systems with parametric uncertainties and multiple delays. First, FHM modeling method was presented for time-delay nonlinear systems. Then, by using Lyapunov–Krasovskii approaches, delay-dependent sufficient condition for the existence of a kind of state feedback controller was proposed, which was expressed as linear matrix inequalities (LMIs). The controller can guarantee that the resulting closed-loop system is robustly asymptotically stable with a prescribed H1 performance level for all admissible uncertainties and time-delay. Finally, a simulation example was provided to illustrate the effectiveness of the proposed approach. 2007 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. Keywords: Fuzzy hyperbolic model; Robust H1 control; Asymptotical stability; Linear matrix inequality; Uncertain time-delay system
1. Introduction Recently, fuzzy control has attracted much attention owing to its superior approximation and generalization abilities to handle complex and ill-defined systems. During the last few years, many significant results of fuzzy control have been achieved. In particular, many significant results have been obtained from the Takagi–Sugeno (T-S) fuzzy model (see Refs. [1–4] and literatures therein). Fuzzy logic system is usually used as approximator to approximate the nonlinear system or the controller, and the Lyapunov second method is used to analyze the stability of fuzzy system, where most results are formulated as convex optimization techniques with linear matrix inequality (LMI) constraints. Recently, a new continuous-time fuzzy model, called the fuzzy hyperbolic model (FHM), has been proposed [5–7]. Being the same as the T-S fuzzy model, FHM can be used *
Corresponding author. Tel.: +86 24 83867762; fax: +86 24 836877623. E-mail address: [email protected] (H.G. Zhang).
to establish the model for a certain unknown complex system. Compared with T-S model, no structure identification or completeness design of premise variables space is required when an FHM is used. Thus, FHM can be obtained as a better representation of the nonlinear system. Moreover, FHM can be seen as a neural network model, and so we can learn the model parameter by the back-propagation (BP) algorithm [5]. On the other hand, time-delays exist in various engineering systems such as chemical process, which are usually a source of instability and frequently lead to poor control performances. And, uncertainties are also unavoidable in control systems due to modeling errors, measurement errors, and so on. They usually influence the practical control performances. Therefore, the problems of stability analysis and controllers’ design for dynamic time-delay uncertain systems are practically important and have attracted considerable attention over the past years, see for example Refs. [2–4,8–15] and the references therein. These results can be classified into two types: delay-independent
1002-0071/$ - see front matter 2007 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. doi:10.1016/j.pnsc.2007.09.002
98
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
results as reported in Refs. [8,9], and delay-dependent ones as described in Refs. [2–4,10–15]. The delay-independent approach provides a controller which can stabilize a system irrespective of the size of the delay, while the delay-dependent stabilization is concerned with the size of the time delay and usually provides an upper bound of the timedelay. It is well known that delay-independent criteria often cause more conservativeness because of ignoring the information of the size of the delay, especially when the delay is very small. In addition, H1 control problem was considered and different approaches were derived in Refs. [13,14]. Ref. [16] focused on eliminating disturbance about H1 control problem. In Refs. [17,18], H1 control problem was studied for FHM systems without time-delay and uncertainties. In Ref. [19], the guaranteed cost control problem was concerned for a class of time-delay FHM systems, and the delay-independent sufficient condition for the existence of controller was obtained. However, how to model FHM for a time-delay system was not discussed. Moreover, the sufficient conditions of controllers were independent of the time-delay, and so the results might be conservative. To our knowledge, the problem of H1 control for uncertain time-delay FHM systems is still open and remains unsolved. Motivated by this, in this paper, we will investigate H1 control problem for a class of FHM systems with multiple delays and parametric uncertainties. Firstly, the FHM modeling method for a nonlinear timedelay system is presented briefly. Secondly, by choosing a novel Lyapunov–Krasovskii functional, the sufficient condition for the solvability of the delay-dependent robust H1 control problem is obtained in terms of LMIs. A robust memoryless feedback fuzzy controller is constructed based on FHM, which can render the corresponding closed-loop system asymptotically stable with a specified level of disturbance attenuation for all admissible uncertainties and disturbance. The LMI feasible problem can be efficiently solved by the convex optimization techniques with global convergence. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approach. Besides the H1 control problem, it is worth pointing out that there are more advantages in the controller design. Compared with our earlier work [18,19], more information of the FHM system and the size of time-delay are taken into account. Therefore, the presented results are less conservative. 2. Preliminaries and problem formulation In this section, we give the method for constructing FHM of a nonlinear time-delay system, and formulate the main problem and definitions.
time-delay nonlinear system, similarly, we address the following definition. Definition 1. Given a plant with n input variables x = [x1(t) . . . xn(t)]T (the superscript ‘‘T’’ stands for transpose of a vector or a matrix) and n output variables x_ ¼ ½_x1 ðtÞ . . . x_ n ðtÞT , we call the following group of fuzzy rule base hyperbolic type fuzzy rule base (HFRB), if it satisfies the following three conditions: (i) For each output variable x_ l ; l ¼ 1; 2; . . . ; n, the corresponding group of fuzzy rules has the following form: Rl: IF x1(t) is F x1 , x2(t) is F x2 , . . ., and xn(t) is F xn ; x1(t s1) is F x1;1 , x2(t s1) is F x2;1 , . . ., and xn(t s1) is F xn;1 ; . . .; x1(t sr) is F x1;r , x2(t sr) is F x2;r , . . ., and xn(t sr) is F xn;r ; THEN x_ l ¼ clF x þ clF x þ þ clF xn þ 1 2 clF x þ clF x þ þ clF x þ þ clF x þ clF x þ þ 1;1
2;1
1;r
n;1
2;r
clF xn;r . Here r is the number of time delays, sh > 0 is a constant denoting a time delay, h = 1, . . ., r; F xi , F xi;1 ; . . ., and F xi;r are fuzzy sets of xi, which include P xi ; P xi;h (positive) and N xi ; N xi;h (negative), respectively; and xi(t) = /i(t), "t 2 [s, 0], with s = max(s1, . . ., sr), where /i(t) is a known continuous initial function, i = 1, 2, . . ., n. (ii) The constant terms clF x and clF x in the THEN-part i i;h correspond to F xi and F xi;h in the IF-part, respectively. That is, if the linguistic values of the F xi , F xi;h term in the IF-part are P xi , P xi;h , respectively, the corresponding clF x , clF x must appear as clP x , clP x in the THENi i i;h i;h part, respectively; if the linguistic values of the F xi , F xi;h terms in the IF-part are N xi , N xi;h , respectively, the corresponding clF x , clF x must appear as clN x , i i i;h clN x in the THEN-part, respectively; if there are no i;h F xi , F xi;h in the IF-part, clFxi and clF x do not appear i;h in the THEN-part (h = 1, . . ., r). (iii) There are two fuzzy rules in each rule base. Thus, there are a total of 2(r+1)n input variable combinations of all the possible P xi , P xi;h , N xi and N xi;h in the IF-part (h = 1, . . ., r). Remark 1. It is clear that we need n HFRBs to describe a plant with n output variables. Proposition 1. Given n HFRBs, if we define the membership functions of P xi , N xi , P xi;h and N xi;h as: 1
2
1
2
lP xi ðxi ðtÞÞ ¼ e2ðxi ðtÞki Þ
lN xi ðxi ðtÞÞ ¼ e2ðxi ðtÞþki Þ 1
2
1
2
lP x ðxi ðt sh ÞÞ ¼ e2ðxi ðtsh Þki Þ
ð1Þ
i;h
lN xi;h ðxi ðt sh ÞÞ ¼ e2ðxi ðtsh Þþki Þ 2.1. FHM for nonlinear time-delay system We follow the FHM modeling method for general nonlinear systems given in Refs. [5,6]. To model FHM for a
where i = 1, 2, . . ., n; h = 1, 2, . . ., r; ki are positive constants, then we can derive the whole system as the following form
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
_ ¼ S þ A tanhðKxðtÞÞ þ xðtÞ
r X
Ah tanhðKxðt sh ÞÞ
ð2Þ
h¼1
where P S = [s1, s2, . . ., sn]T is a constant vector with si ¼ nj¼1 ciP x þciN x Pn Pr ciP xj;h þciN xj;h j j þ j¼1 h¼1 2 2 2
q11
6 6 q21 6 A¼6 6 .. 6 . 4 qn1 2 qh11 6 6 qh21 6 Ah ¼ 6 6 .. 6 . 4 qhn1
q12
. . . q1n
3
7 . . . q2n 7 7 7 .. 7; .. . 7 . 5 . . . qnn
q22 .. . qn2 qh12
...
qh22
...
.. .
.. .
qhn2
...
qh1n
with qij ¼
ciP x ciN x j
j
½ DAðtÞ DA1 ðtÞ DAr ðtÞ ¼ MFðtÞ½ N 0 N 1 N r
2
3 with qkij ¼
ciP x ciN x j;h
j;h
2
qhnn
(h = 1, . . ., r; i,j = 1, . . ., n) K = diag(k1, . . ., kn) (diag(. . .) denotes a diagonal or a block diagonal matrix), and tanh(Kx) is defined by T
tanhðk 2 x2 Þ
tanhðk n xn Þ
Proof. The proof follows from Refs. [5,6].
h
Remark 2. In Proposition 1, if ciP x ; ciN x ðj ¼ 1; 2; . . . ; n; j;h
r X
where M, N0 and Nh(h = 1, . . ., r) are known real constant matrices of appropriate dimension, and F(t) is an unknown matrix function satisfying FT(t)F(t) 6 I (I is an identity matrix with appropriate dimension). Such parametric uncertainties are said to be admissible. For convenience, let DA: = DA(t), DAh: = DAh(t) (h = 1, 2, . . ., r). The aim of this study is to construct a memoryless feedback control law for system (4) and (5) in FHM form as uðtÞ ¼ G tanhðKxðtÞÞ
Ah tanhðKxðt sh ÞÞ
ð3Þ
h¼1
There is no essential difference between the control of (2) and (3), so we will discuss the problem based on time-delay FHM described in (3).
ð7Þ
where G 2 Rmn is a gain matrix to be determined, such that the resulting closed-loop system _ ¼ ðA þ DA þ BGÞ tanhðKxðtÞÞ xðtÞ r X þ ðAh þ DAh Þ tanhðKxðt sh ÞÞ þ B w wðtÞ
ð8Þ
h¼1
j;h
h ¼ 1; 2; . . . ; rÞ do not appear in the rules, they should be 0. We call (2) a time-delay fuzzy hyperbolic model. From Definition 1, if we set clP x and clN x , clP x and i i i;h l cN x ðl; i ¼ 1; 2; . . . ; n; h ¼ 1; 2; . . . ; rÞ negative to each i;h other, respectively, we can obtain a homogeneous FHM _ ¼ A tanhðKxðtÞÞ þ xðtÞ
where xðtÞ 2 Rn ; uðtÞ 2 Rm ; zðtÞ 2 Rp denote the state vector, input vector and controlled output vector, respectively; wðtÞ 2 Rq denotes the exogenous disturbance, which can be unknown but belongs to L2[0, 1); A 2 Rnn ; Ah 2 Rnn ; B 2 Rnm ; B w 2 Rnq ; C 2 Rpn , Dw 2 Rpq are known real constant matrices; sh > 0 (h = 1, . . ., r) are the time-delay constants, s = max(s1, . . ., sr). The initial condition /(t) is given by initial vector function, which is continuous for s 6 t 6 0; DAðtÞ 2 Rnn and DAh ðtÞ 2 Rnn are timevarying parametric uncertainty matrices and satisfy ð6Þ
7 qh2n 7 7 7 .. 7; . 7 5
tanhðKxÞ ¼ ½ tanhðk 1 x1 Þ
99
zðtÞ ¼ C tanhðKxðtÞÞ þ Dw wðtÞ
ð9Þ
satisfies the following requirements: (i) the closed-loop system is asymptotically stable in the absence of exogenous disturbance (that is w(t) ” 0); (ii) under the zero-initial condition, for any nonzero w(t) 2 L2[0, 1) (i.e. w(t) is square integrable over [0, 1)), the controlled output satisfies Z
1 T
z ðtÞzðtÞ dt < c 0
2
Z
1
wT ðtÞwðtÞ dt
0
for any given disturbance attenuation level c > 0.
1.2. Robust H1 control problem We consider synthetically a class of nonlinear systems described as the following form with multiple delays and uncertainties: _ ¼ ðA þ DAðtÞÞ tanhðKxðtÞÞ xðtÞ r X þ ðAh þ DAh ðtÞÞ tanhðKxðt sh ÞÞ
2. Main results Lemma 1 [20]. Given appropriate dimension matrices M, E and F satisfying FTF 6 I, for any real scalar e > 0, the following result holds MFE þ E T F T M T 6 eMM T þ e1 E T E
h¼1
þ BuðtÞ þ Bw wðtÞ zðtÞ ¼ C tanhðKxðtÞÞ þ Dw wðtÞ xðtÞ ¼ /ðtÞ;
8t 2 ½s; 0
ð4Þ ð5Þ
Lemma 2. For any constant positive-definite symmetric matrix J 2 Rmm , scalars b > 0 and j > 0, and the vector function t : [b j, b] fi Rm·1 such that the integrations in the following are well defined, then
100
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
Z
b
tT ðsÞJtðsÞ ds P
j
Z
b
tðsÞ ds
bj
T Z J
bj
b
T
H11 ¼ KAP þ PAT K þ KBL þ ðBLÞ K þ rQ1 þ
tðsÞ ds
T
sh Q 2
h¼1
bj
Proof. Since J > 0 is a constant matrix, " nonzero c = [c1, c2, . . ., cm+1]T: = [c1, qT]T, we have 1
r X
þ
r X
eh KMM T K
h¼0
H12 ¼ KA1 P
1
ðc1 tðsÞ þ J qÞ Jðc1 tðsÞ þ J qÞ P 0
KA2 P
KAr P
H13 ¼ ½ Q3 Q3 Q3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
that is,
r
T
T
T 1
T
c1 t ðsÞJðc1 tðsÞÞ þ c1 q tðsÞ þ c1 t ðsÞq þ q J q P 0
ð10Þ
H22 ¼ diagðQ1 ; Q1 ; . . . ; Q1 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r
which is equivalent to T t ðsÞJtðsÞ tT ðsÞ cT cP0 tðsÞ J 1
ð11Þ
Since c is an arbitrary nonzero vector, from inequality (11), it follows that T t ðsÞJtðsÞ tT ðsÞ P0 ð12Þ tðsÞ J 1
ð13Þ
Using Schur complement [21], we have Z b T Z b tT ðsÞJtðsÞ ds j1 tðsÞ ds J tðsÞ ds P 0
b
bj
or Z j
bj
b
tT ðsÞJtðsÞ ds P bj
Z
bj
b
tðsÞ ds
T Z J
bj
b
tðsÞ ds
ð15Þ
Theorem 1. The system (15) is robustly asymptotically stabilizable, if there exist a diagonal matrix P > 0, matrices Qk > 0ðk ¼ 1; 2; 3Þ and L, and scalars ei > 0 (i = 0, 1, . . ., r) such that the following LMI holds 3 2 H11 H12 H13 0 PN T0 7 6 H22 H23 H24 0 7 6 7 6 7<0 6 ð16Þ H 0 0 33 7 6 7 6 5 4 H44 0 where
e0 I
H44 ¼ diagðe1 I; e2 I; . . . ; er IÞ
Moreover, the controller can be chosen as (7) with G ¼ LP1 . Proof. Consider feedback controller (7) for system (15), and the resulting closed-loop system is
ð17Þ
h¼1
Choose a Lyapunov–Krasovskii functional candidate for the time-delay system (17) as follows
h
h¼1
1 1 H33 ¼ diagðs1 1 Q 2 ; s2 Q 2 ; . . . ; sr Q 2 Þ
ð14Þ
Now, we analyze the robust stabilization problem of (4) when w(t) = 0, that is, r X _ ¼ ðA þ DAÞ tanhðKxðtÞÞ þ xðtÞ ðAh þ DAh Þ tanhðKxðt sh ÞÞ þ BuðtÞ
H24 ¼ diagðPN T1 ; PN T2 ; . . . ; PN Tr Þ
_ ¼ ðA þ DA þ BGÞ tanhðKxðtÞÞ xðtÞ r X þ ðAh þ DAh Þ tanhðKxðt sh ÞÞ
bj
The proof of Lemma 2 is completed.
r
P > 0 means that P is a symmetric matrix and positive definite, and the notation * stands for the transposed block of the symmetric matrix.
Integrating (12) from b j to b yields 2R R T 3 b b tT ðsÞJtðsÞ ds tðsÞ ds bj bj 4 5P0 Rb 1 tðsÞ ds jJ bj Z
H23 ¼ diagðQ3 ; Q3 ; . . . ; Q3 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
V ðtÞ ¼ 2
n X
pi lnðcoshðk i xi ðtÞÞÞ
i¼1
þ
r Z X h¼1
þ
r X h¼1
þ
t
tanhT ðKxðsÞÞQ1 tanhðKxðsÞÞ ds
tsh
Z
0
Z
sh
tanhT ðKxðaÞÞQ2 tanhðKxðaÞÞ da db tþb
r Z t X
T tanhðKxðsÞÞ ds
tsh
h¼1
Q3
t
Z
t
tanhðKxðsÞÞ ds
ð18Þ
tsh
where scalars pi > 0 (i = 1, . . . ,n); matrices Qk > 0 (k = 1, 2, 3); and ki (i = 1, . . ., n) are defined in (1). Then the time-derivative of V(t) along the solution of (17) gives
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
V_ ðtÞjð17Þ ¼ 2 tanhT ðKxðtÞÞKP ðA þ DA þ BGÞ tanhðKxðtÞÞ r X
þ
V_ ðtÞjð17Þ 6 tanhT ðKxðtÞÞ KPðA þ BGÞ þ ðA þ BGÞT PK
!
þrQ1 þ
r X
sh Q 2 þ
h¼1
ðAh þ DAh Þ tanhðKxðt sh ÞÞ
h¼1
r X
! T
eh KPMM PK
r X
2tanhT ðKxðtÞÞKPAh
h¼1
tanhT ðKxðtÞÞQ1 tanhðKxðtÞÞ
tanhðKxðt sh ÞÞ þ
h¼1
T
tanh ðKxðt sh ÞÞQ1 tanhðKxðt sh ÞÞ
r X
þ
t
tanhðKxðsÞÞds þ
r X
sh tanhT ðKxðtÞÞQ2 tanhðKxðtÞÞ
t T
tsh
þ
h¼1
h¼1
tanh ðKxðsÞÞQ2 tanhðKxðsÞÞ ds
r X
Z
t
tanhðKxðsÞÞds þ
tsh
2ðtanhðKxðtÞÞ tanhðKxðt sh ÞÞÞ
s1 h Q2
T
Z
ð19Þ
tanhðKxðsÞÞ ds
where P = diag(p1, p2, . . ., pn). From Lemma 1 and (6), we have
P11 P12 P13
T 6 tanhT ðKxðtÞÞðe0 KPMM T PK þ e1 0 N 0 N 0 Þ tanhðKxðtÞÞ
ð20Þ 2 tanhT ðKxðtÞÞKP
r X
P33
P11 ¼ KPðA þ BGÞ þ ðA þ BGÞT PK þ rQ1 þ Xr T þ eh KPMM T PK þ e1 0 N0 N0 h¼0
Xr h¼1
sh Q2
r
1 1 P33 ¼ diagðs1 1 Q 2 ; s2 Q 2 ;. .. ; sr Q 2 Þ
h¼1 T e1 h tanh ðKxðt
sh ÞÞN Th N h
tanhðKxðt sh ÞÞÞ
ð21Þ
where eh (h = 0, 1, . . ., r) are positive scalars. Moreover, from Lemma 2, for h = 1, 2, . . ., r, we have Z t tanhT ðKxðsÞÞQ2 tanhðKxðsÞÞ ds tsh
T
t
tanhðKxðsÞÞ ds
Z
t
tanhðKxðsÞÞ ds
Q2 tsh
ð22Þ Substituting (20)–(22) into (19), we obtain
T 1 T 1 T P22 ¼ diagðQ1 þ e1 1 N 1 N 1 ;Q 1 þ e2 N 2 N 2 ; . .. ;Q 1 þ er N r N r Þ
P23 ¼ diagðQ3 ;Q3 ; . .. ; Q3 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
r X 6 ðeh tanhT ðKxðtÞÞKPMM T PK tanhðKxðtÞÞ
tsh
7 P22 P23 5
P13 ¼ ½ Q3 Q3 Q3 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
h¼1
6
3
r
2 tanhT ðKxðtÞÞKPMFðtÞN h tanhðKxðt sh ÞÞ
Z
tanhðKxðsÞÞds 6 fT ðtÞPfðtÞ
P12 ¼ ½ KPA1 KPA2 KPAr
DAh tanhðKxðt sh ÞÞ
h¼1
s1 h
tsh
t
fT ðtÞ ¼ tanhT ðKxðtÞÞ wT ðtÞ gT ðtÞ wT ðtÞ ¼ tanhT ðKxðt s1 ÞÞ tanhT ðKxðt sr ÞÞ T R T Rt t T g ðtÞ ¼ tanhðKxðsÞÞds tanhðKxðsÞÞds ts1 tsr 6 P ¼ 4
¼ 2 tanhT ðKxðtÞÞKPMFðtÞN 0 tanhðKxðtÞÞ
þ
tanhðKxðsÞÞds
where
2
2 tanhT ðKxðtÞÞKPDA tanhðKxðtÞÞ
¼
h¼1
T
t
ð23Þ
t tsh
r X
Z
r Z X
tsh
h¼1
Q3
tanhT ðKxðt sh ÞÞ
T ðQ1 þ e1 h N h N h Þ tanhðKxðt sh ÞÞ r X 2tanhT ðKxðt sh ÞÞðQ3 Þ þ
h¼1
Z
2tanhT ðKxðtÞÞQ3
h¼1
Z
tsh
r X
T þ e1 0 N0 N0
h¼0
tanhðKxðtÞÞ þ
r X
þ
101
The sufficient condition of V_ ðtÞjð17Þ < 0 for all nonzero fT(t) is that ð24Þ
P<0
Pre- and post-multiplying diagðP1 ; P1 ; . . . ; P1 Þ to (24), |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2rþ1
and letting P ¼ P1 , Qk ¼ P1 Qk P1 ðk ¼ 1; 2; 3Þ and L = GP1, we can obtain (16) by Schur complement. It implies that the system (17) is asymptotically stable for any admissible uncertainties. The proof of Theorem 1 is completed. h Remark 3. Theorem 1 presents a sufficient condition of robust stabilization for a class of uncertain time-delay
102
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
FHM systems. The first item of Lyapunov–Krasovskii functional (18) in the proof is different from that in the existing results P (Refs. [18,19] for example), which was often chosen as 2 ni¼1 pkii lnðcoshðk i xi ðtÞÞÞ. Thus, the corresponding result is dependent on K, which is the coefficient matrix of system state in the hyperbolic function. Obviously, it may reduce conservativeness because more information of the system is taken into account, and it can guarantee the designed controller reaches better performance. Now we give the main theorem of the robust H1 control problem.
Consider system (8), (9) and the initial condition, (26) can be shown as J ðT Þ
¼ 6
H11
6 6 6 6 6 6 6 6 6 6 6 4
H12
H13
H14
0
PN T0
PC
H22
H23 H33
0 0
H25 0
0 0
0 0
0
H44
0
0
0
H55
0 e0 I
0 0
I
ðzT ðtÞzðtÞ c2 wT ðtÞwðtÞ þ V_ ðtÞjð8Þ Þ dt V ðT Þ
0
ðzT ðtÞzðtÞ c2 wT ðtÞwðtÞ þ V_ ðtÞjð8Þ Þ dt
where V(t) is defined in (18). V_ ðtÞjð8Þ ¼ 2 tanhT ðKxðtÞÞKPððA þ DA þ BGÞ tanhðKxðtÞÞ þ
r X ðAh þ DAh Þ tanhðKxðt sh ÞÞ þ B w wðtÞÞ h¼1
þ
r X ðtanhT ðKxðtÞÞQ1 tanhðKxðtÞÞ h¼1
3 T 7 7 7 7 7 7 7<0 7 7 7 7 5
0
RT
ð27Þ
Theorem 2. Given a scalar c > 0, for system (4) and (5), an H1 controller with disturbance attenuation level c can be chosen in form (7), if there exist a diagonal matrix P > 0, matrices Qk > 0ðk ¼ 1; 2; 3Þ and L, and scalars ei > 0 (i = 0, 1, . . ., r) such that the following LMI holds 2
RT
tanhT ðKxðt sh ÞÞQ1 tanhðKxðt sh ÞÞÞ r X þ sh tanhT ðKxðtÞÞQ2 tanhðKxðtÞÞ h¼1
ð25Þ
Z
T
tanh ðKxðsÞÞQ2 tanhðKxðsÞÞ ds
tsh
r X þ 2ðtanhðKxðtÞÞ tanhðKxðt sh ÞÞÞT h¼1
where H11 ¼ KAP þ PA K þ KBL þ ðBLÞ K þ rQ1 r r X X þ sh Q 2 þ eh KMM T K h¼1
Q3
T
T
h¼0
H12 ¼ KA1 P KA2 P KAr P H13 ¼ ½ Q3 Q3 Q3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
r
Z
tanhðKxðsÞÞ ds
t
ð28Þ
tsh
Similar to that in proof of Theorem 1, substituting (9) and (28) into (27), we obtain
J ðT Þ 6
Z
T
nT ðtÞPnðtÞ dt
ð29Þ
0
T
H14 ¼ KB w þ PC Dw H22 ¼ diagðQ1 ; Q1 ; . . . ; Q1 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
t
where
r
H23 ¼ diagðQ3 ; Q3 ; . . . ; Q3 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r
H25 ¼ diagðPN T1 ; PN T2 ; . . . ; PN Tr Þ 1 1 H33 ¼ diagðs1 1 Q 2 ; s2 Q 2 ; . . . ; sr Q 2 Þ
H44 ¼ c2 I þ DTw Dw H55 ¼ diagðe1 I; e2 I; . . . ; er IÞ And, the gain of FHM controller is G ¼ LP1 . Proof. To establish the H1 performance for the corresponding closed-loop system of (4), (5) and (7) under zero initial condition, we introduce Z T J ðT Þ ¼ ðzT ðtÞzðtÞ c2 wT ðtÞwðtÞÞ dt ð26Þ 0
for any nonzero w(t) 2 L2 [0, 1), where T > 0.
nT ðtÞ ¼ ½ fT ðtÞ wT ðtÞ ; where fðtÞis defined inð23Þ; 2 3 P11 P12 P13 P14 6 P22 P23 0 7 6 7 P¼6 7 4 P33 0 5 P44 r X T P11 ¼ KP ðA þ BGÞ þ ðA þ BGÞ PK þ rQ1 þ sh Q 2 h¼1
þ
r X
T eh KPMM T PK þ e1 0 N 0N
h¼0
P12 ¼ ½ KPA1 KPA2 KPAr P13 ¼ ½ Q3 Q3 Q3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r
0
þ CTC
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
P14 ¼ KPBw þ C T Dw P22 ¼ diagðQ1 þ þ
T e1 1 N 1 N 1 ; Q1
þ
T e1 2 N 2 N 2 ; . . . ; Q1
T e1 r N r N r ÞP23
¼ diagðQ3 ; Q3 ; . . . ; Q3 ÞP33 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r
¼
1 1 diagðs1 1 Q2 ; s2 Q2 ; . . . ; sr Q2 ÞP44
The sufficient condition of J(T) < 0 for all nonzero n(t) is ð30Þ
P<0 1
1
1
Pre- and post-multiplying diagðP ; P ; . . . ; P ; IÞ to the |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 2rþ1
P ¼ P 1 , left-hand side of (30) and letting 1 1 1 Qk ¼ P Qk P ðk ¼ 1; 2; 3Þ and L = GP , we can obtain the result (25) by Schur complement. It implies that J(T) < 0 is satisfied, that is, for any T > 0 and any nonzero w(t) 2 L2 [0, 1) Z T Z T zT ðtÞzðtÞ dt < c2 wT ðtÞwðtÞ dt ð31Þ 0
0
which yields Z Z 1 T 2 z ðtÞzðtÞ dt < c 0
1
wT ðtÞwðtÞ dt 0
In addition, it is easy to see that (16) holds from (25). That is, the closed-loop system is robustly asymptotically stable when w(t) = 0. The proof of Theorem 2 is completed. 3. Illustrative example In this section, we provide an example to illustrate the effectiveness of the presented result. Consider the following time-delay nonlinear system x_ 1 ðtÞ ¼ 0:1x31 ðtÞ 0:0125x1 ðt sÞ 0:02x2 ðtÞ 0:67x2 ðtÞ 0:1x32 ðt sÞ 0:005x2 ðt sÞ 20u1 ðtÞ þ w1 ðtÞ x_ 2 ðtÞ ¼ x1 ðtÞ þ u2 ðtÞ
IF x1(t) is P x1 and x2(t) is N x2 , THEN zðtÞ ¼ Dx1 Dx2 ; IF x1(t) is N x1 and x2(t) is N x2 , THEN zðtÞ ¼ Dx1 Dx2 ; Here, r = 1. We choose membership functions of P xi ; N xi (i = 1,2), P dx1 and N dx1 as (1), and s1 = s = 0.5 s, the initial condition /(t) = [0.75 0.5]T, "t 2 [0.5, 0]. Then, using neural network BP algorithm [6], we obtain the FHM of (32) as (4) and (5), with T
¼ c2 I þ DTw Dw
ð32Þ
zðtÞ ¼ 0:0241 sinðx1 ðtÞÞ þ 0:0523x2 ðtÞ xðtÞ ¼ /ðtÞ; 8t 2 ½s; 0 Setting the number of fuzzy sets for every variable to be two (positive, negative), we have the HFRBs for x_ i ðtÞði ¼ 1; 2Þ and z(t) of the nominal system as follows: IF x1(t) is P x1 , and x2(t) is P x2 , and x1(t s) is P dx1 and x2(t s) is P dx2 , THEN x_ i ¼ C ix1 þ C ix2 þ C idx1 þ C idx2 ; IF x1(t) is N x1 , and x2(t) is P x2 , and x1(t s) is P dx1 and x2(t s) is P dx2 , THEN x_ i ¼ C ix1 þ C ix2 þ C idx1 þ C idx2 ; IF x1(t) is N x1 , and x2(t) is N x2 , and x1(t s) is N dx1 and x2(t s) is N dx2 , THEN x_ i ¼ C ix1 C ix2 C idx1 C idx2 ; IF x1(t) is P x1 and x2(t) is P x2 , THEN zðtÞ ¼ Dx1 þ Dx2 ; IF x1(t) is N x1 and x2(t) is P x2 , THEN zðtÞ ¼ Dx1 þ Dx2 ;
103
T
xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ ; uðtÞ ¼ ½ u1 ðtÞ u2 ðtÞ ; 0:0472 7:5773 ; wðtÞ ¼ w1 ðtÞ; A ¼ 2:8487 0:0617 0:3520 0:2173 20 0 ; B¼ ; A1 ¼ 0:0329 0:0440 0 1 1 ; C ¼ ½ 0:0484 0:1 ; Dw ¼ 0; and Bw ¼ 0 K ¼ diagð0:3586; 0:0970Þ: T
Let M ¼ ½ 0:2 0:2 ; N 0 ¼ ½ 0:1 0:2 ; N 1 ¼ ½ 0:2 0:1 and c = 0.6. Utilizing Theorem 2, we can obtain a set of feasible solutions as follows: 1:4935 0:0001 P ¼ diagð1:3141; 1:3192Þ; Q1 ¼ ; 0:0001 1:4935 0:8961 0:0000 0:2987 0 Q2 ¼ ; Q3 ¼ ; 0:0000 0:8961 0 0:2987 0:2437 0:2394 L¼ ; and e0 ¼ e1 ¼ 1:4935: 15:4632 17:6312 Then, the feedback controller can bechosen in form (7) 0:1854 0:1815 1 with G ¼ LP ¼ . 11:7675 13:3655 In the same way, to realize the robust H1 controller, we can also choose the following group of HFRBs: IF x1(t) is P x1 and x2(t) is P x2 , THEN u1(t) = 0.0039; IF x1(t) is P x1 and x2(t) is N x2 , THEN u1(t) = 0.3669; IF x1(t) is N x1 and x2(t) is P x2 , THEN u1(t) = 0.3669; IF x1(t) is N x1 and x2(t) is N x2 , THEN u1(t) = 0.0039; IF x1(t) is P x1 and x2(t) is P x2 , THEN u2(t) = 1.5980; IF x1(t) is P x1 and x2(t) is N x2 , THEN u2(t) = 25.1330; IF x1(t) is N x1 and x2(t) is P x2 , THEN u2(t) = 25.1330; IF x1(t) is N x1 and x2(t) is N x2 , THEN u2(t) = 1.5980. Set F(t) = sint, and an exogenous disturbance w(t) = 0.1 et sin(5t). The simulation results of the corresponding closed-loop system are shown in Figs. 1 and 2, which demonstrate the effectiveness of the proposed design method. 4. Conclusions In this paper, the robust H1 control problem was considered for a class of FHM systems with uncertainties and multiple delays. The modeling method of FHM was given for time-delay nonlinear system. Via choosing a new Lyapunov–Krasovskii functional, the condition for the existence of robust H1 controller was obtained in terms of LMI, which depends on the coefficients of the state in
104
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT0421).
1 x1(t )
States
x2(t )
0.5
References 0
–0.5
0
1
2
3
4
5
6
7
Time (s) 4 u1(t )
Control inputs
3
u2(t )
2 1 0 –1 –2
0
1
2
3
4
5
6
7
Time (s)
Fig. 1. The trajectories of states (a) and control inputs (b).
0.08 z(t ) w(t )
Output and disturbance input
0.06
0.04
0.02
0
–0.02
–0.04
0
1
2
3
4
5
6
7
Time (s)
Fig. 2. The trajectories of output z(t) and disturbance input w(t).
the hyperbolic functions and the information of time-delay. Therefore, it may lead to less conservativeness. The feedback controller was constructed based on FHM, which was also realized easily by hyperbolic fuzzy rules. The controller can guarantee that the resulting closed-loop system is robustly asymptotically stable with a prescribed H1 disturbance attenuation performance for all admissible uncertainties. The simulation example has illustrated the effectiveness of the proposed approach. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 60534010, 60572070, 60774048, and 60728307), the Funds for Creative Research Groups of China (Grant No. 60521003)
[1] Jia XC, Zhang NN. Reliable fuzzy control with domain guaranteed cost for fuzzy systems with actuator failures. Prog Nat Sci 2004;14(12):1109–15. [2] Chen B, Liu XP, Tong SC. Delay-dependent stability analysis and control synthesis of fuzzy dynamic systems with time delay. Fuzzy Sets Syst 2006;157(16):2224–40. [3] Zhang HG, Wang YC. Guaranteed cost control for uncertain stochastic fuzzy systems with time delay. Prog Nat Sci 2007;17(1): 95–101. [4] Lin C, Wang QG, Lee TH. Delay-dependent LMI conditions for stability and stabilization of T-S fuzzy systems with bounded timedelay. Fuzzy Sets Syst 2006;157(9):1229–47. [5] Zhang HG, Quan YB. Modeling, identification and control of a class of nonlinear systems. IEEE Trans Fuzzy Syst 2001;49(2): 349–54. [6] Zhang HG, He XQ. Fuzzy adaptive control theory and its application. Beijing: Beijing Aeronautics and Astronomics University Press; 2002, p. 127–32 [in Chinese]. [7] Zhang HG, Quan YB. Modeling and control based on fuzzy hyperbolic model. Acta Autom Sin [in Chinese] 2000;26(6): 729–35. [8] Kapila V, Haddad WM. Memoryless H1 controller for discrete time systems with time delay. Automatica 1998;34(9):1141–4. [9] Esfahani SH, Petersen IR. An LMI approach to output-feedback guaranteed cost control for uncertain time-delay systems. Int J Robust Nonlinear Control 2000;10(2):157–74. [10] Cao YY, Sun YX, Cheng CW. Delay-dependent robust stabilization of uncertain systems with multiple state delays. IEEE Trans Automatic Control 1998;43(11):1608–12. [11] De Souza CE, Li X. Delay-dependent robust H1 control of uncertain linear state-delayed systems. Automatica 1999;35(7):1313–21. [12] Fridman E, Shaked U. An improved stabilization method for linear time-delay systems. IEEE Trans Automatic Control 2002;47(11): 1931–7. [13] Fridman E, Shaked U. A descriptor ststem approach to H1 control of linear time-delay systems. IEEE Trans Automatic Control 2002;47(2):253–70. [14] Suplin V, Fridman E, Shaked U. H1 control of linear uncertain timedelay systems – a projection approach. IEEE Trans Automatic Control 2006;51(4):680–5. [15] Xu SY, Lam J, Zou Y. New results on delay-dependent robust H1 control for systems with time-varying delays. Automatica 2006;42(2):343–8. [16] He JB, Wang QG, Lee TH. H1 disturbance attenuation for state delayed systems. Syst Control Lett 1998;33(2):105–14. [17] Margaliot M, Langholz G. A new approach to fuzzy modeling and control of discrete-time systems. IEEE Trans Fuzzy Syst 2003;11(4):486–94. [18] Yang J, Liu DR, Feng J, et al. Controller design for a class of nonlinear systems based on fuzzy hyperbolic model. In: Proceedings of the sixth world congress on intelligent control and automation. Dalian, China: IEEE; 2006. p. 873–7. [19] Lun SX, Zhang HG. Delayed-independent fuzzy hyperbolic guaranteed cost control design for a class of nonlinear continuous-time systems with uncertainties. Acta Autom Sin 2005;31(5):720–6. [20] Wang Y, Xie LH, De Souza CE. Robust control of a class of uncertain nonlinear systems. Syst Control Lett 1992;19(2): 139–49. [21] Yu L. Robust control – the method of linear matrix inequalities. Beijing: Tsinghua University Press; 2002, p. 8–9 [in Chinese].
Progress in Natural Science 18 (2008) 97–104
Delay-dependent robust H1 control for uncertain fuzzy hyperbolic systems with multiple delays Huaguang Zhang
a,*
, Qingxian Gong
a,b
, Yingchun Wang
a
a b
School of Information Science and Engineering, Northeastern University, Shenyang 110004, China Department of Electrical Engineering, Changchun Institute of Technology, Changchun 130012, China Received 7 July 2007; received in revised form 10 August 2007; accepted 10 September 2007
Abstract The robust H1 control problem was considered for a class of fuzzy hyperbolic model (FHM) systems with parametric uncertainties and multiple delays. First, FHM modeling method was presented for time-delay nonlinear systems. Then, by using Lyapunov–Krasovskii approaches, delay-dependent sufficient condition for the existence of a kind of state feedback controller was proposed, which was expressed as linear matrix inequalities (LMIs). The controller can guarantee that the resulting closed-loop system is robustly asymptotically stable with a prescribed H1 performance level for all admissible uncertainties and time-delay. Finally, a simulation example was provided to illustrate the effectiveness of the proposed approach. 2007 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. Keywords: Fuzzy hyperbolic model; Robust H1 control; Asymptotical stability; Linear matrix inequality; Uncertain time-delay system
1. Introduction Recently, fuzzy control has attracted much attention owing to its superior approximation and generalization abilities to handle complex and ill-defined systems. During the last few years, many significant results of fuzzy control have been achieved. In particular, many significant results have been obtained from the Takagi–Sugeno (T-S) fuzzy model (see Refs. [1–4] and literatures therein). Fuzzy logic system is usually used as approximator to approximate the nonlinear system or the controller, and the Lyapunov second method is used to analyze the stability of fuzzy system, where most results are formulated as convex optimization techniques with linear matrix inequality (LMI) constraints. Recently, a new continuous-time fuzzy model, called the fuzzy hyperbolic model (FHM), has been proposed [5–7]. Being the same as the T-S fuzzy model, FHM can be used *
Corresponding author. Tel.: +86 24 83867762; fax: +86 24 836877623. E-mail address: [email protected] (H.G. Zhang).
to establish the model for a certain unknown complex system. Compared with T-S model, no structure identification or completeness design of premise variables space is required when an FHM is used. Thus, FHM can be obtained as a better representation of the nonlinear system. Moreover, FHM can be seen as a neural network model, and so we can learn the model parameter by the back-propagation (BP) algorithm [5]. On the other hand, time-delays exist in various engineering systems such as chemical process, which are usually a source of instability and frequently lead to poor control performances. And, uncertainties are also unavoidable in control systems due to modeling errors, measurement errors, and so on. They usually influence the practical control performances. Therefore, the problems of stability analysis and controllers’ design for dynamic time-delay uncertain systems are practically important and have attracted considerable attention over the past years, see for example Refs. [2–4,8–15] and the references therein. These results can be classified into two types: delay-independent
1002-0071/$ - see front matter 2007 National Natural Science Foundation of China and Chinese Academy of Sciences. Published by Elsevier Limited and Science in China Press. All rights reserved. doi:10.1016/j.pnsc.2007.09.002
98
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
results as reported in Refs. [8,9], and delay-dependent ones as described in Refs. [2–4,10–15]. The delay-independent approach provides a controller which can stabilize a system irrespective of the size of the delay, while the delay-dependent stabilization is concerned with the size of the time delay and usually provides an upper bound of the timedelay. It is well known that delay-independent criteria often cause more conservativeness because of ignoring the information of the size of the delay, especially when the delay is very small. In addition, H1 control problem was considered and different approaches were derived in Refs. [13,14]. Ref. [16] focused on eliminating disturbance about H1 control problem. In Refs. [17,18], H1 control problem was studied for FHM systems without time-delay and uncertainties. In Ref. [19], the guaranteed cost control problem was concerned for a class of time-delay FHM systems, and the delay-independent sufficient condition for the existence of controller was obtained. However, how to model FHM for a time-delay system was not discussed. Moreover, the sufficient conditions of controllers were independent of the time-delay, and so the results might be conservative. To our knowledge, the problem of H1 control for uncertain time-delay FHM systems is still open and remains unsolved. Motivated by this, in this paper, we will investigate H1 control problem for a class of FHM systems with multiple delays and parametric uncertainties. Firstly, the FHM modeling method for a nonlinear timedelay system is presented briefly. Secondly, by choosing a novel Lyapunov–Krasovskii functional, the sufficient condition for the solvability of the delay-dependent robust H1 control problem is obtained in terms of LMIs. A robust memoryless feedback fuzzy controller is constructed based on FHM, which can render the corresponding closed-loop system asymptotically stable with a specified level of disturbance attenuation for all admissible uncertainties and disturbance. The LMI feasible problem can be efficiently solved by the convex optimization techniques with global convergence. Finally, a simulation example is provided to illustrate the effectiveness of the proposed approach. Besides the H1 control problem, it is worth pointing out that there are more advantages in the controller design. Compared with our earlier work [18,19], more information of the FHM system and the size of time-delay are taken into account. Therefore, the presented results are less conservative. 2. Preliminaries and problem formulation In this section, we give the method for constructing FHM of a nonlinear time-delay system, and formulate the main problem and definitions.
time-delay nonlinear system, similarly, we address the following definition. Definition 1. Given a plant with n input variables x = [x1(t) . . . xn(t)]T (the superscript ‘‘T’’ stands for transpose of a vector or a matrix) and n output variables x_ ¼ ½_x1 ðtÞ . . . x_ n ðtÞT , we call the following group of fuzzy rule base hyperbolic type fuzzy rule base (HFRB), if it satisfies the following three conditions: (i) For each output variable x_ l ; l ¼ 1; 2; . . . ; n, the corresponding group of fuzzy rules has the following form: Rl: IF x1(t) is F x1 , x2(t) is F x2 , . . ., and xn(t) is F xn ; x1(t s1) is F x1;1 , x2(t s1) is F x2;1 , . . ., and xn(t s1) is F xn;1 ; . . .; x1(t sr) is F x1;r , x2(t sr) is F x2;r , . . ., and xn(t sr) is F xn;r ; THEN x_ l ¼ clF x þ clF x þ þ clF xn þ 1 2 clF x þ clF x þ þ clF x þ þ clF x þ clF x þ þ 1;1
2;1
1;r
n;1
2;r
clF xn;r . Here r is the number of time delays, sh > 0 is a constant denoting a time delay, h = 1, . . ., r; F xi , F xi;1 ; . . ., and F xi;r are fuzzy sets of xi, which include P xi ; P xi;h (positive) and N xi ; N xi;h (negative), respectively; and xi(t) = /i(t), "t 2 [s, 0], with s = max(s1, . . ., sr), where /i(t) is a known continuous initial function, i = 1, 2, . . ., n. (ii) The constant terms clF x and clF x in the THEN-part i i;h correspond to F xi and F xi;h in the IF-part, respectively. That is, if the linguistic values of the F xi , F xi;h term in the IF-part are P xi , P xi;h , respectively, the corresponding clF x , clF x must appear as clP x , clP x in the THENi i i;h i;h part, respectively; if the linguistic values of the F xi , F xi;h terms in the IF-part are N xi , N xi;h , respectively, the corresponding clF x , clF x must appear as clN x , i i i;h clN x in the THEN-part, respectively; if there are no i;h F xi , F xi;h in the IF-part, clFxi and clF x do not appear i;h in the THEN-part (h = 1, . . ., r). (iii) There are two fuzzy rules in each rule base. Thus, there are a total of 2(r+1)n input variable combinations of all the possible P xi , P xi;h , N xi and N xi;h in the IF-part (h = 1, . . ., r). Remark 1. It is clear that we need n HFRBs to describe a plant with n output variables. Proposition 1. Given n HFRBs, if we define the membership functions of P xi , N xi , P xi;h and N xi;h as: 1
2
1
2
lP xi ðxi ðtÞÞ ¼ e2ðxi ðtÞki Þ
lN xi ðxi ðtÞÞ ¼ e2ðxi ðtÞþki Þ 1
2
1
2
lP x ðxi ðt sh ÞÞ ¼ e2ðxi ðtsh Þki Þ
ð1Þ
i;h
lN xi;h ðxi ðt sh ÞÞ ¼ e2ðxi ðtsh Þþki Þ 2.1. FHM for nonlinear time-delay system We follow the FHM modeling method for general nonlinear systems given in Refs. [5,6]. To model FHM for a
where i = 1, 2, . . ., n; h = 1, 2, . . ., r; ki are positive constants, then we can derive the whole system as the following form
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
_ ¼ S þ A tanhðKxðtÞÞ þ xðtÞ
r X
Ah tanhðKxðt sh ÞÞ
ð2Þ
h¼1
where P S = [s1, s2, . . ., sn]T is a constant vector with si ¼ nj¼1 ciP x þciN x Pn Pr ciP xj;h þciN xj;h j j þ j¼1 h¼1 2 2 2
q11
6 6 q21 6 A¼6 6 .. 6 . 4 qn1 2 qh11 6 6 qh21 6 Ah ¼ 6 6 .. 6 . 4 qhn1
q12
. . . q1n
3
7 . . . q2n 7 7 7 .. 7; .. . 7 . 5 . . . qnn
q22 .. . qn2 qh12
...
qh22
...
.. .
.. .
qhn2
...
qh1n
with qij ¼
ciP x ciN x j
j
½ DAðtÞ DA1 ðtÞ DAr ðtÞ ¼ MFðtÞ½ N 0 N 1 N r
2
3 with qkij ¼
ciP x ciN x j;h
j;h
2
qhnn
(h = 1, . . ., r; i,j = 1, . . ., n) K = diag(k1, . . ., kn) (diag(. . .) denotes a diagonal or a block diagonal matrix), and tanh(Kx) is defined by T
tanhðk 2 x2 Þ
tanhðk n xn Þ
Proof. The proof follows from Refs. [5,6].
h
Remark 2. In Proposition 1, if ciP x ; ciN x ðj ¼ 1; 2; . . . ; n; j;h
r X
where M, N0 and Nh(h = 1, . . ., r) are known real constant matrices of appropriate dimension, and F(t) is an unknown matrix function satisfying FT(t)F(t) 6 I (I is an identity matrix with appropriate dimension). Such parametric uncertainties are said to be admissible. For convenience, let DA: = DA(t), DAh: = DAh(t) (h = 1, 2, . . ., r). The aim of this study is to construct a memoryless feedback control law for system (4) and (5) in FHM form as uðtÞ ¼ G tanhðKxðtÞÞ
Ah tanhðKxðt sh ÞÞ
ð3Þ
h¼1
There is no essential difference between the control of (2) and (3), so we will discuss the problem based on time-delay FHM described in (3).
ð7Þ
where G 2 Rmn is a gain matrix to be determined, such that the resulting closed-loop system _ ¼ ðA þ DA þ BGÞ tanhðKxðtÞÞ xðtÞ r X þ ðAh þ DAh Þ tanhðKxðt sh ÞÞ þ B w wðtÞ
ð8Þ
h¼1
j;h
h ¼ 1; 2; . . . ; rÞ do not appear in the rules, they should be 0. We call (2) a time-delay fuzzy hyperbolic model. From Definition 1, if we set clP x and clN x , clP x and i i i;h l cN x ðl; i ¼ 1; 2; . . . ; n; h ¼ 1; 2; . . . ; rÞ negative to each i;h other, respectively, we can obtain a homogeneous FHM _ ¼ A tanhðKxðtÞÞ þ xðtÞ
where xðtÞ 2 Rn ; uðtÞ 2 Rm ; zðtÞ 2 Rp denote the state vector, input vector and controlled output vector, respectively; wðtÞ 2 Rq denotes the exogenous disturbance, which can be unknown but belongs to L2[0, 1); A 2 Rnn ; Ah 2 Rnn ; B 2 Rnm ; B w 2 Rnq ; C 2 Rpn , Dw 2 Rpq are known real constant matrices; sh > 0 (h = 1, . . ., r) are the time-delay constants, s = max(s1, . . ., sr). The initial condition /(t) is given by initial vector function, which is continuous for s 6 t 6 0; DAðtÞ 2 Rnn and DAh ðtÞ 2 Rnn are timevarying parametric uncertainty matrices and satisfy ð6Þ
7 qh2n 7 7 7 .. 7; . 7 5
tanhðKxÞ ¼ ½ tanhðk 1 x1 Þ
99
zðtÞ ¼ C tanhðKxðtÞÞ þ Dw wðtÞ
ð9Þ
satisfies the following requirements: (i) the closed-loop system is asymptotically stable in the absence of exogenous disturbance (that is w(t) ” 0); (ii) under the zero-initial condition, for any nonzero w(t) 2 L2[0, 1) (i.e. w(t) is square integrable over [0, 1)), the controlled output satisfies Z
1 T
z ðtÞzðtÞ dt < c 0
2
Z
1
wT ðtÞwðtÞ dt
0
for any given disturbance attenuation level c > 0.
1.2. Robust H1 control problem We consider synthetically a class of nonlinear systems described as the following form with multiple delays and uncertainties: _ ¼ ðA þ DAðtÞÞ tanhðKxðtÞÞ xðtÞ r X þ ðAh þ DAh ðtÞÞ tanhðKxðt sh ÞÞ
2. Main results Lemma 1 [20]. Given appropriate dimension matrices M, E and F satisfying FTF 6 I, for any real scalar e > 0, the following result holds MFE þ E T F T M T 6 eMM T þ e1 E T E
h¼1
þ BuðtÞ þ Bw wðtÞ zðtÞ ¼ C tanhðKxðtÞÞ þ Dw wðtÞ xðtÞ ¼ /ðtÞ;
8t 2 ½s; 0
ð4Þ ð5Þ
Lemma 2. For any constant positive-definite symmetric matrix J 2 Rmm , scalars b > 0 and j > 0, and the vector function t : [b j, b] fi Rm·1 such that the integrations in the following are well defined, then
100
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
Z
b
tT ðsÞJtðsÞ ds P
j
Z
b
tðsÞ ds
bj
T Z J
bj
b
T
H11 ¼ KAP þ PAT K þ KBL þ ðBLÞ K þ rQ1 þ
tðsÞ ds
T
sh Q 2
h¼1
bj
Proof. Since J > 0 is a constant matrix, " nonzero c = [c1, c2, . . ., cm+1]T: = [c1, qT]T, we have 1
r X
þ
r X
eh KMM T K
h¼0
H12 ¼ KA1 P
1
ðc1 tðsÞ þ J qÞ Jðc1 tðsÞ þ J qÞ P 0
KA2 P
KAr P
H13 ¼ ½ Q3 Q3 Q3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
that is,
r
T
T
T 1
T
c1 t ðsÞJðc1 tðsÞÞ þ c1 q tðsÞ þ c1 t ðsÞq þ q J q P 0
ð10Þ
H22 ¼ diagðQ1 ; Q1 ; . . . ; Q1 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r
which is equivalent to T t ðsÞJtðsÞ tT ðsÞ cT cP0 tðsÞ J 1
ð11Þ
Since c is an arbitrary nonzero vector, from inequality (11), it follows that T t ðsÞJtðsÞ tT ðsÞ P0 ð12Þ tðsÞ J 1
ð13Þ
Using Schur complement [21], we have Z b T Z b tT ðsÞJtðsÞ ds j1 tðsÞ ds J tðsÞ ds P 0
b
bj
or Z j
bj
b
tT ðsÞJtðsÞ ds P bj
Z
bj
b
tðsÞ ds
T Z J
bj
b
tðsÞ ds
ð15Þ
Theorem 1. The system (15) is robustly asymptotically stabilizable, if there exist a diagonal matrix P > 0, matrices Qk > 0ðk ¼ 1; 2; 3Þ and L, and scalars ei > 0 (i = 0, 1, . . ., r) such that the following LMI holds 3 2 H11 H12 H13 0 PN T0 7 6 H22 H23 H24 0 7 6 7 6 7<0 6 ð16Þ H 0 0 33 7 6 7 6 5 4 H44 0 where
e0 I
H44 ¼ diagðe1 I; e2 I; . . . ; er IÞ
Moreover, the controller can be chosen as (7) with G ¼ LP1 . Proof. Consider feedback controller (7) for system (15), and the resulting closed-loop system is
ð17Þ
h¼1
Choose a Lyapunov–Krasovskii functional candidate for the time-delay system (17) as follows
h
h¼1
1 1 H33 ¼ diagðs1 1 Q 2 ; s2 Q 2 ; . . . ; sr Q 2 Þ
ð14Þ
Now, we analyze the robust stabilization problem of (4) when w(t) = 0, that is, r X _ ¼ ðA þ DAÞ tanhðKxðtÞÞ þ xðtÞ ðAh þ DAh Þ tanhðKxðt sh ÞÞ þ BuðtÞ
H24 ¼ diagðPN T1 ; PN T2 ; . . . ; PN Tr Þ
_ ¼ ðA þ DA þ BGÞ tanhðKxðtÞÞ xðtÞ r X þ ðAh þ DAh Þ tanhðKxðt sh ÞÞ
bj
The proof of Lemma 2 is completed.
r
P > 0 means that P is a symmetric matrix and positive definite, and the notation * stands for the transposed block of the symmetric matrix.
Integrating (12) from b j to b yields 2R R T 3 b b tT ðsÞJtðsÞ ds tðsÞ ds bj bj 4 5P0 Rb 1 tðsÞ ds jJ bj Z
H23 ¼ diagðQ3 ; Q3 ; . . . ; Q3 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
V ðtÞ ¼ 2
n X
pi lnðcoshðk i xi ðtÞÞÞ
i¼1
þ
r Z X h¼1
þ
r X h¼1
þ
t
tanhT ðKxðsÞÞQ1 tanhðKxðsÞÞ ds
tsh
Z
0
Z
sh
tanhT ðKxðaÞÞQ2 tanhðKxðaÞÞ da db tþb
r Z t X
T tanhðKxðsÞÞ ds
tsh
h¼1
Q3
t
Z
t
tanhðKxðsÞÞ ds
ð18Þ
tsh
where scalars pi > 0 (i = 1, . . . ,n); matrices Qk > 0 (k = 1, 2, 3); and ki (i = 1, . . ., n) are defined in (1). Then the time-derivative of V(t) along the solution of (17) gives
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
V_ ðtÞjð17Þ ¼ 2 tanhT ðKxðtÞÞKP ðA þ DA þ BGÞ tanhðKxðtÞÞ r X
þ
V_ ðtÞjð17Þ 6 tanhT ðKxðtÞÞ KPðA þ BGÞ þ ðA þ BGÞT PK
!
þrQ1 þ
r X
sh Q 2 þ
h¼1
ðAh þ DAh Þ tanhðKxðt sh ÞÞ
h¼1
r X
! T
eh KPMM PK
r X
2tanhT ðKxðtÞÞKPAh
h¼1
tanhT ðKxðtÞÞQ1 tanhðKxðtÞÞ
tanhðKxðt sh ÞÞ þ
h¼1
T
tanh ðKxðt sh ÞÞQ1 tanhðKxðt sh ÞÞ
r X
þ
t
tanhðKxðsÞÞds þ
r X
sh tanhT ðKxðtÞÞQ2 tanhðKxðtÞÞ
t T
tsh
þ
h¼1
h¼1
tanh ðKxðsÞÞQ2 tanhðKxðsÞÞ ds
r X
Z
t
tanhðKxðsÞÞds þ
tsh
2ðtanhðKxðtÞÞ tanhðKxðt sh ÞÞÞ
s1 h Q2
T
Z
ð19Þ
tanhðKxðsÞÞ ds
where P = diag(p1, p2, . . ., pn). From Lemma 1 and (6), we have
P11 P12 P13
T 6 tanhT ðKxðtÞÞðe0 KPMM T PK þ e1 0 N 0 N 0 Þ tanhðKxðtÞÞ
ð20Þ 2 tanhT ðKxðtÞÞKP
r X
P33
P11 ¼ KPðA þ BGÞ þ ðA þ BGÞT PK þ rQ1 þ Xr T þ eh KPMM T PK þ e1 0 N0 N0 h¼0
Xr h¼1
sh Q2
r
1 1 P33 ¼ diagðs1 1 Q 2 ; s2 Q 2 ;. .. ; sr Q 2 Þ
h¼1 T e1 h tanh ðKxðt
sh ÞÞN Th N h
tanhðKxðt sh ÞÞÞ
ð21Þ
where eh (h = 0, 1, . . ., r) are positive scalars. Moreover, from Lemma 2, for h = 1, 2, . . ., r, we have Z t tanhT ðKxðsÞÞQ2 tanhðKxðsÞÞ ds tsh
T
t
tanhðKxðsÞÞ ds
Z
t
tanhðKxðsÞÞ ds
Q2 tsh
ð22Þ Substituting (20)–(22) into (19), we obtain
T 1 T 1 T P22 ¼ diagðQ1 þ e1 1 N 1 N 1 ;Q 1 þ e2 N 2 N 2 ; . .. ;Q 1 þ er N r N r Þ
P23 ¼ diagðQ3 ;Q3 ; . .. ; Q3 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
r X 6 ðeh tanhT ðKxðtÞÞKPMM T PK tanhðKxðtÞÞ
tsh
7 P22 P23 5
P13 ¼ ½ Q3 Q3 Q3 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
h¼1
6
3
r
2 tanhT ðKxðtÞÞKPMFðtÞN h tanhðKxðt sh ÞÞ
Z
tanhðKxðsÞÞds 6 fT ðtÞPfðtÞ
P12 ¼ ½ KPA1 KPA2 KPAr
DAh tanhðKxðt sh ÞÞ
h¼1
s1 h
tsh
t
fT ðtÞ ¼ tanhT ðKxðtÞÞ wT ðtÞ gT ðtÞ wT ðtÞ ¼ tanhT ðKxðt s1 ÞÞ tanhT ðKxðt sr ÞÞ T R T Rt t T g ðtÞ ¼ tanhðKxðsÞÞds tanhðKxðsÞÞds ts1 tsr 6 P ¼ 4
¼ 2 tanhT ðKxðtÞÞKPMFðtÞN 0 tanhðKxðtÞÞ
þ
tanhðKxðsÞÞds
where
2
2 tanhT ðKxðtÞÞKPDA tanhðKxðtÞÞ
¼
h¼1
T
t
ð23Þ
t tsh
r X
Z
r Z X
tsh
h¼1
Q3
tanhT ðKxðt sh ÞÞ
T ðQ1 þ e1 h N h N h Þ tanhðKxðt sh ÞÞ r X 2tanhT ðKxðt sh ÞÞðQ3 Þ þ
h¼1
Z
2tanhT ðKxðtÞÞQ3
h¼1
Z
tsh
r X
T þ e1 0 N0 N0
h¼0
tanhðKxðtÞÞ þ
r X
þ
101
The sufficient condition of V_ ðtÞjð17Þ < 0 for all nonzero fT(t) is that ð24Þ
P<0
Pre- and post-multiplying diagðP1 ; P1 ; . . . ; P1 Þ to (24), |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2rþ1
and letting P ¼ P1 , Qk ¼ P1 Qk P1 ðk ¼ 1; 2; 3Þ and L = GP1, we can obtain (16) by Schur complement. It implies that the system (17) is asymptotically stable for any admissible uncertainties. The proof of Theorem 1 is completed. h Remark 3. Theorem 1 presents a sufficient condition of robust stabilization for a class of uncertain time-delay
102
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
FHM systems. The first item of Lyapunov–Krasovskii functional (18) in the proof is different from that in the existing results P (Refs. [18,19] for example), which was often chosen as 2 ni¼1 pkii lnðcoshðk i xi ðtÞÞÞ. Thus, the corresponding result is dependent on K, which is the coefficient matrix of system state in the hyperbolic function. Obviously, it may reduce conservativeness because more information of the system is taken into account, and it can guarantee the designed controller reaches better performance. Now we give the main theorem of the robust H1 control problem.
Consider system (8), (9) and the initial condition, (26) can be shown as J ðT Þ
¼ 6
H11
6 6 6 6 6 6 6 6 6 6 6 4
H12
H13
H14
0
PN T0
PC
H22
H23 H33
0 0
H25 0
0 0
0 0
0
H44
0
0
0
H55
0 e0 I
0 0
I
ðzT ðtÞzðtÞ c2 wT ðtÞwðtÞ þ V_ ðtÞjð8Þ Þ dt V ðT Þ
0
ðzT ðtÞzðtÞ c2 wT ðtÞwðtÞ þ V_ ðtÞjð8Þ Þ dt
where V(t) is defined in (18). V_ ðtÞjð8Þ ¼ 2 tanhT ðKxðtÞÞKPððA þ DA þ BGÞ tanhðKxðtÞÞ þ
r X ðAh þ DAh Þ tanhðKxðt sh ÞÞ þ B w wðtÞÞ h¼1
þ
r X ðtanhT ðKxðtÞÞQ1 tanhðKxðtÞÞ h¼1
3 T 7 7 7 7 7 7 7<0 7 7 7 7 5
0
RT
ð27Þ
Theorem 2. Given a scalar c > 0, for system (4) and (5), an H1 controller with disturbance attenuation level c can be chosen in form (7), if there exist a diagonal matrix P > 0, matrices Qk > 0ðk ¼ 1; 2; 3Þ and L, and scalars ei > 0 (i = 0, 1, . . ., r) such that the following LMI holds 2
RT
tanhT ðKxðt sh ÞÞQ1 tanhðKxðt sh ÞÞÞ r X þ sh tanhT ðKxðtÞÞQ2 tanhðKxðtÞÞ h¼1
ð25Þ
Z
T
tanh ðKxðsÞÞQ2 tanhðKxðsÞÞ ds
tsh
r X þ 2ðtanhðKxðtÞÞ tanhðKxðt sh ÞÞÞT h¼1
where H11 ¼ KAP þ PA K þ KBL þ ðBLÞ K þ rQ1 r r X X þ sh Q 2 þ eh KMM T K h¼1
Q3
T
T
h¼0
H12 ¼ KA1 P KA2 P KAr P H13 ¼ ½ Q3 Q3 Q3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
r
Z
tanhðKxðsÞÞ ds
t
ð28Þ
tsh
Similar to that in proof of Theorem 1, substituting (9) and (28) into (27), we obtain
J ðT Þ 6
Z
T
nT ðtÞPnðtÞ dt
ð29Þ
0
T
H14 ¼ KB w þ PC Dw H22 ¼ diagðQ1 ; Q1 ; . . . ; Q1 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
t
where
r
H23 ¼ diagðQ3 ; Q3 ; . . . ; Q3 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r
H25 ¼ diagðPN T1 ; PN T2 ; . . . ; PN Tr Þ 1 1 H33 ¼ diagðs1 1 Q 2 ; s2 Q 2 ; . . . ; sr Q 2 Þ
H44 ¼ c2 I þ DTw Dw H55 ¼ diagðe1 I; e2 I; . . . ; er IÞ And, the gain of FHM controller is G ¼ LP1 . Proof. To establish the H1 performance for the corresponding closed-loop system of (4), (5) and (7) under zero initial condition, we introduce Z T J ðT Þ ¼ ðzT ðtÞzðtÞ c2 wT ðtÞwðtÞÞ dt ð26Þ 0
for any nonzero w(t) 2 L2 [0, 1), where T > 0.
nT ðtÞ ¼ ½ fT ðtÞ wT ðtÞ ; where fðtÞis defined inð23Þ; 2 3 P11 P12 P13 P14 6 P22 P23 0 7 6 7 P¼6 7 4 P33 0 5 P44 r X T P11 ¼ KP ðA þ BGÞ þ ðA þ BGÞ PK þ rQ1 þ sh Q 2 h¼1
þ
r X
T eh KPMM T PK þ e1 0 N 0N
h¼0
P12 ¼ ½ KPA1 KPA2 KPAr P13 ¼ ½ Q3 Q3 Q3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r
0
þ CTC
H.G. Zhang et al. / Progress in Natural Science 18 (2008) 97–104
P14 ¼ KPBw þ C T Dw P22 ¼ diagðQ1 þ þ
T e1 1 N 1 N 1 ; Q1
þ
T e1 2 N 2 N 2 ; . . . ; Q1
T e1 r N r N r ÞP23
¼ diagðQ3 ; Q3 ; . . . ; Q3 ÞP33 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r
¼
1 1 diagðs1 1 Q2 ; s2 Q2 ; . . . ; sr Q2 ÞP44
The sufficient condition of J(T) < 0 for all nonzero n(t) is ð30Þ
P<0 1
1
1
Pre- and post-multiplying diagðP ; P ; . . . ; P ; IÞ to the |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 2rþ1
P ¼ P 1 , left-hand side of (30) and letting 1 1 1 Qk ¼ P Qk P ðk ¼ 1; 2; 3Þ and L = GP , we can obtain the result (25) by Schur complement. It implies that J(T) < 0 is satisfied, that is, for any T > 0 and any nonzero w(t) 2 L2 [0, 1) Z T Z T zT ðtÞzðtÞ dt < c2 wT ðtÞwðtÞ dt ð31Þ 0
0
which yields Z Z 1 T 2 z ðtÞzðtÞ dt < c 0
1
wT ðtÞwðtÞ dt 0
In addition, it is easy to see that (16) holds from (25). That is, the closed-loop system is robustly asymptotically stable when w(t) = 0. The proof of Theorem 2 is completed. 3. Illustrative example In this section, we provide an example to illustrate the effectiveness of the presented result. Consider the following time-delay nonlinear system x_ 1 ðtÞ ¼ 0:1x31 ðtÞ 0:0125x1 ðt sÞ 0:02x2 ðtÞ 0:67x2 ðtÞ 0:1x32 ðt sÞ 0:005x2 ðt sÞ 20u1 ðtÞ þ w1 ðtÞ x_ 2 ðtÞ ¼ x1 ðtÞ þ u2 ðtÞ
IF x1(t) is P x1 and x2(t) is N x2 , THEN zðtÞ ¼ Dx1 Dx2 ; IF x1(t) is N x1 and x2(t) is N x2 , THEN zðtÞ ¼ Dx1 Dx2 ; Here, r = 1. We choose membership functions of P xi ; N xi (i = 1,2), P dx1 and N dx1 as (1), and s1 = s = 0.5 s, the initial condition /(t) = [0.75 0.5]T, "t 2 [0.5, 0]. Then, using neural network BP algorithm [6], we obtain the FHM of (32) as (4) and (5), with T
¼ c2 I þ DTw Dw
ð32Þ
zðtÞ ¼ 0:0241 sinðx1 ðtÞÞ þ 0:0523x2 ðtÞ xðtÞ ¼ /ðtÞ; 8t 2 ½s; 0 Setting the number of fuzzy sets for every variable to be two (positive, negative), we have the HFRBs for x_ i ðtÞði ¼ 1; 2Þ and z(t) of the nominal system as follows: IF x1(t) is P x1 , and x2(t) is P x2 , and x1(t s) is P dx1 and x2(t s) is P dx2 , THEN x_ i ¼ C ix1 þ C ix2 þ C idx1 þ C idx2 ; IF x1(t) is N x1 , and x2(t) is P x2 , and x1(t s) is P dx1 and x2(t s) is P dx2 , THEN x_ i ¼ C ix1 þ C ix2 þ C idx1 þ C idx2 ; IF x1(t) is N x1 , and x2(t) is N x2 , and x1(t s) is N dx1 and x2(t s) is N dx2 , THEN x_ i ¼ C ix1 C ix2 C idx1 C idx2 ; IF x1(t) is P x1 and x2(t) is P x2 , THEN zðtÞ ¼ Dx1 þ Dx2 ; IF x1(t) is N x1 and x2(t) is P x2 , THEN zðtÞ ¼ Dx1 þ Dx2 ;
103
T
xðtÞ ¼ ½ x1 ðtÞ x2 ðtÞ ; uðtÞ ¼ ½ u1 ðtÞ u2 ðtÞ ; 0:0472 7:5773 ; wðtÞ ¼ w1 ðtÞ; A ¼ 2:8487 0:0617 0:3520 0:2173 20 0 ; B¼ ; A1 ¼ 0:0329 0:0440 0 1 1 ; C ¼ ½ 0:0484 0:1 ; Dw ¼ 0; and Bw ¼ 0 K ¼ diagð0:3586; 0:0970Þ: T
Let M ¼ ½ 0:2 0:2 ; N 0 ¼ ½ 0:1 0:2 ; N 1 ¼ ½ 0:2 0:1 and c = 0.6. Utilizing Theorem 2, we can obtain a set of feasible solutions as follows: 1:4935 0:0001 P ¼ diagð1:3141; 1:3192Þ; Q1 ¼ ; 0:0001 1:4935 0:8961 0:0000 0:2987 0 Q2 ¼ ; Q3 ¼ ; 0:0000 0:8961 0 0:2987 0:2437 0:2394 L¼ ; and e0 ¼ e1 ¼ 1:4935: 15:4632 17:6312 Then, the feedback controller can bechosen in form (7) 0:1854 0:1815 1 with G ¼ LP ¼ . 11:7675 13:3655 In the same way, to realize the robust H1 controller, we can also choose the following group of HFRBs: IF x1(t) is P x1 and x2(t) is P x2 , THEN u1(t) = 0.0039; IF x1(t) is P x1 and x2(t) is N x2 , THEN u1(t) = 0.3669; IF x1(t) is N x1 and x2(t) is P x2 , THEN u1(t) = 0.3669; IF x1(t) is N x1 and x2(t) is N x2 , THEN u1(t) = 0.0039; IF x1(t) is P x1 and x2(t) is P x2 , THEN u2(t) = 1.5980; IF x1(t) is P x1 and x2(t) is N x2 , THEN u2(t) = 25.1330; IF x1(t) is N x1 and x2(t) is P x2 , THEN u2(t) = 25.1330; IF x1(t) is N x1 and x2(t) is N x2 , THEN u2(t) = 1.5980. Set F(t) = sint, and an exogenous disturbance w(t) = 0.1 et sin(5t). The simulation results of the corresponding closed-loop system are shown in Figs. 1 and 2, which demonstrate the effectiveness of the proposed design method. 4. Conclusions In this paper, the robust H1 control problem was considered for a class of FHM systems with uncertainties and multiple delays. The modeling method of FHM was given for time-delay nonlinear system. Via choosing a new Lyapunov–Krasovskii functional, the condition for the existence of robust H1 controller was obtained in terms of LMI, which depends on the coefficients of the state in
104
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and the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT0421).
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the hyperbolic functions and the information of time-delay. Therefore, it may lead to less conservativeness. The feedback controller was constructed based on FHM, which was also realized easily by hyperbolic fuzzy rules. The controller can guarantee that the resulting closed-loop system is robustly asymptotically stable with a prescribed H1 disturbance attenuation performance for all admissible uncertainties. The simulation example has illustrated the effectiveness of the proposed approach. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 60534010, 60572070, 60774048, and 60728307), the Funds for Creative Research Groups of China (Grant No. 60521003)
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