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Finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems
Expert Systems with Applications 37 (2010) 6963–6967
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems Dedong Yang *, Kai-Yuan Cai National Key Laboratory of Science and Technology on Holistic Control, School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
a r t i c l e
i n f o
Keywords: Finite-time stability (FTS) Quantized control Fuzzy control Linear matrix inequality (LMI)
a b s t r a c t This paper considers the problem of finite-time quantized guaranteed cost fuzzy control for continuoustime nonlinear systems. Firstly, the definition on finite-time stability (FTS) for continuous-time nonlinear systems is provided and we give a novel and explicit interpretation for finite-time quantized guaranteed cost control. Secondly, sufficient conditions for the existence of state feedback controller are derived in terms of linear matrix inequities (LMIs), which guarantee the requirements of the provided performance criterion. The related optimization problem is also offered to minimize the guaranteed cost performance bound. Finally, an illustrative example is presented to show the validity of the proposed scheme. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Since there exist many cases where the state values are restrained, we generally need to ensure that these state values are allowable. In order to deal with these cases, the concept of finitetime stability (FTS) (or short-time stability) can be introduced into our eyeshot. As illuminated in Amato, Ariola, and Dorato (2001), a system is said to be finite-time stable if, once we fix a time-interval, its state does not exceed certain bound during this time-interval. Some early results on FTS can be found in Dorato (1961), Weiss and Infante (1967). More recently the problem of finite-time control for both continuous-time and discrete-time linear systems has been investigated in Amato and Ariola (2005), Amato, Ariola, and Cosentino (2006), Amato et al. (2001) via linear matrix inequities (LMIs) technique. In the existing results, FTS is mainly associated with linear systems. How to solve the FTS problem for nonlinear systems is still a difficult problem. Recently, Takagi–Sugeno (T–S) fuzzy model proposed in Takagi and Sugeno (1985) is widely applied in various industrial control fields because of its simple structure with local dynamics. The typical approach named as parallel distributed compensation (PDC) (Wang, Tanaka, & Griffin, 1996) is also developed to design the fuzzy controllers. Moreover, besides the stabilization problem, the guaranteed cost fuzzy control was extendedly investigated to stabilize the controlled systems while providing an upper bound on a given performance index by many scholars in the last decade, such as Chen and Liu (2005), Chen, Liu, Tong, and Lin (2007), Guan and Chen (2004), Jiang and Han (2007). * Corresponding author. Tel.: +86 010 82315087; fax: +86 010 82317328. E-mail address: [email protected] (D. Yang). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.03.024
The output signals of the controller are generally assumed to be returned directly to the plant without data loss in the classical feedback control theories. In practice, however, this is not true in many real applications since all data transmissions cannot be performed with infinite precision in computer control systems and quantization always exists. In recent years, considering the limited communication capacity, the problem of quantized feedback control has been investigated by many researchers, such as Brockett and Liberzon (2000), Elia and Mitter (2001), Fu and Xie (2005), Liberzon (2003). Moreover, how to design a quantized feedback controller for nonlinear systems is still an interesting issue. In this paper, we propose a finite-time quantized guaranteed cost fuzzy control scheme for continuous-time nonlinear systems. Its objective is to find a suitable quantized controller such that the provided performance criterion is satisfied. As far as we know, up to date, the finite-time quantized guaranteed cost control for continuous-time nonlinear systems has not been considered in the existing results. The remainder of this paper is organized as follows: Basic problem formulation is introduced in Section 2. The finite-time quantized guaranteed cost controller via state feedback is designed in Section 3. Section 4 provides an illustrative example to demonstrate the effectiveness of the proposed scheme. Finally, concluding remarks are made in Section 5. In the following sections, the identity matrices and zero matrices are denoted by I and 0, respectively. X T denotes the transpose of matrix X. Rn denotes the n-dimensional Euclidean space. Rþ denotes the positive real number. The notation * always denotes the symmetric block in one symmetric matrix. The standard notation > (< ) is used to denote the positive (negative)-definite ordering of matrices. Inequality X > Y shows that the matrix X Y is positive
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definite. kmax ðÞ and kmin ðÞ indicate the maximum and minimum eigenvalues of the matrix, respectively.
Considering the above quantization behavior, we can obtain the following quantized fuzzy controller:
2. Problem formulation
ðtÞ ¼ f u
r X
! hi ðhðtÞÞK i xðtÞ :
ð3Þ
i¼1
Consider a general class of continuous-time nonlinear systems which can be represented by the following T–S fuzzy dynamic model.
hi ðhðtÞÞ½Ai xðtÞ þ Bi uðtÞ;
ð1Þ
i¼1
P where hi ðhðtÞÞ ¼ li ðhðtÞÞ= ri¼1 li ðhðtÞÞ;
li ðhðtÞÞ ¼ Ppj¼1 Nij ðhj ðtÞÞ and
N ij ðhj ðtÞÞ is the grade of membership of hj ðtÞ in Nij . It is assumed that P li ðhðtÞÞ P 0, and ri¼1 li ðhðtÞÞ > 0 for all t. Then we can see that Pr hi ðhðtÞÞ P 0 and i¼1 hi ðhðtÞÞ ¼ 1. According to the conventional PDC concept, we design the following fuzzy controller via state feedback: Controller Rule i IF h1 ðtÞ is N i1 ; . . ., and hp ðtÞ is N ip THEN uðtÞ ¼ K i xðtÞ
r X
_ xðtÞ ¼
hi ðhðtÞÞK i xðtÞ;
ð4Þ
hi ðhðtÞÞK i xðtÞ:
ð2Þ
i¼1
It is assumed that the output signals of fuzzy controller (2) are passed via a quantizer and the quantizer is denoted as f ðÞ ¼ ½f1 ðÞ; f2 ðÞ; . . . ; fm ðÞT , which is assumed to be symmetric, that is, f ðv Þ ¼ f ðv Þ. The set of quantized levels is described by
r X
ðtÞ ¼ AxðtÞ; hi ðhðtÞÞ½Ai xðtÞ þ Bi u
ð5Þ
i¼1
P P where A ¼ ri¼1 rj¼1 hi ðhðtÞÞhj ðhðtÞÞ½Ai þ Bi ðI þ KÞK j . Next we will provide a novel concept on FTS for continuoustime nonlinear systems. It is formalized through the following definition. Similar description can be found in Amato et al. (2001). Definition 1. The closed-loop continuous-time nonlinear system (5) is said to be finite-time stable (FTS) with respect to ðc1 ; c2 ; T; RC Þ where 0 < c1 < c2 ; T 2 Rþ and RC > 0 if
xT ð0ÞRC xð0Þ 6 c1 ) xT ðtÞRC xðtÞ < c2 ;
8 t 2 ð0; T:
Remark 1. Asymptotic stability and FTS are independent concepts each other as it was described in Amato et al. (2001). In certain cases, asymptotic stability can be also considered as an additional requirement while restricting our attention on a finite-time interval. Furthermore, for (5), we provide the following cost function associated with the above definition:
J¼
where i ¼ 1; 2; . . . ; r is the number of controller rules, K i 2 Rmn is the controller gain matrix. Thus, the fuzzy controller can be expressed as follows:
uðtÞ ¼
r X
where K ¼ diagfK1 ; K2 ; . . . ; Km g and Ks 2 ½r; r; s ¼ 1; . . . ; m. The closed-loop system (1) can be reconstructed under the quantized fuzzy controller (4) as follows:
where i ¼ 1; 2; . . . ; r is the number of fuzzy rules, xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input, Ai 2 Rnn and Bi 2 Rnm are known constant matrices, of the ith subsystem, h1 ðtÞ; h2 ðtÞ; . . . ; hp ðtÞ are the premise variables, N ij is the fuzzy set ðj ¼ 1; 2; . . . ; pÞ. By using singleton fuzzifier, product inference and center-average defuzzifier, the fuzzy dynamic model is expressed by r X
ðtÞ ¼ ðI þ KÞ u
i¼1
Plant Rule i IF h1 ðtÞ is N i1 ; . . ., and hp ðtÞ is N ip THEN _ xðtÞ ¼ Ai xðtÞ þ Bi uðtÞ
_ xðtÞ ¼
Moreover, (3) can be expressed as
Z
T
T ðtÞQ 2 u ðtÞ dt; xT ðtÞQ 1 xðtÞ þ u
ð6Þ
0
where Q 1 > 0 and Q 2 > 0 are given weighting matrices or given positive scalars. The following novel and explicit interpretation for finite-time quantized guaranteed cost control is given by: Definition 2. For (5), if there exists a quantized fuzzy control law (4) and a scalar w0 such that the closed-loop system is finite-time stable and the value of the cost function (6) satisfies J < w0 , then w0 is said to be a guaranteed cost bound and the designed control law (4) is said to be a finite-time quantized guaranteed cost fuzzy control law.
X ¼ fui ; i ¼ 0; 1; 2; . . .g [ f0g: According to Elia and Mitter (2001) and Fu and Xie (2005), a quantizer is called logarithmic if the set of quantized levels is characterized by
X ¼ fui ; ui ¼ qi u0 ; i ¼ 1; 2; . . .g [ fu0 g [ f0g; 0 < qi < 1;
u0 > 0:
For the logarithmic quantizer, the associated quantizer f ðÞ is defined as follows:
8 1 1 > < ui ; if 1þr ui < v 6 1r ui ; f ðv Þ ¼ 0; if v ¼ 0; > : f ðv Þ; if v < 0: where
r¼
1q : 1þq
v > 0;
3. Main results In this section, the problem of finite-time quantized guaranteed cost fuzzy control via state feedback is studied according to (5). Some results are provided as follows. Theorem 1. The closed-loop continuous-time nonlinear system (5) is finite-time stable with respect to ðc1 ; c2 ; T; RC Þ where 0 < c1 < c2 ; T 2 Rþ ; RC > 0, and has the guaranteed cost bound W if there exist a scalar a P 0, matrix P > 0 2 Rnn and the controller gain matrix K j 2 Rmn such that the following conditions are satisfied
Wii < 0; 1 6 i 6 r; Xij < 0; 1 6 i < j 6 r;
ð7Þ
c1 c2 ; 6 kmin ðPÞ kmax ðPÞeaT
ð8Þ
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D. Yang, K.-Y. Cai / Expert Systems with Applications 37 (2010) 6963–6967
Z
T
e 1 ðAi þ Bi ðI þ KÞK i Þ a P e 1 þ P e 1 þ Wii ¼ ðAi þ Bi ðI þ KÞK i ÞT P T e 1 T 1 e Q 1 þ K i ðI þ KÞQ 2 ðI þ KÞK i , Xij ¼ ðAi þ Bi ðI þ KÞK j Þ P þ P ðAi þ Bi
VðxðTÞÞ Vðxð0ÞÞ a
e 1 ðAj þ Bj ðI þ KÞK i Þ 2a P e 1 þ P e 1 þ ðI þ KÞK j Þ þ ðAj þ Bj ðI þ KÞK i ÞT P ð1=2Þ T T e 2Q 1 þ K ðI þ KÞQ 2 ðI þ KÞK i þ K ðI þ KÞQ 2 ðI þ KÞK j , and P ¼ R
Moreover, J < kmax ðP1 Þc1 eaT can be obtained by considering Definition 1. It is easy to see that the guaranteed cost function satisfies
where
i
ð1=2Þ
PRC
j
C
VðxðtÞÞdt < J:
0
J < W ¼ kmax ðP1 Þc1 eaT :
. The guaranteed cost bound is described by:
W ¼ kmax ðP1 Þc1 eaT :
The proof is completed.
e 1 xðtÞ where P e ¼ Rð1=2Þ PRð1=2Þ with Proof. Let VðxðtÞÞ ¼ xT ðtÞ P C C P > 0 and assume the time derivative of VðxðtÞÞ along the solution of system (5) to satisfy the following condition for all t 2 ð0; T
Remark 2. If the conditions (7) and (8) in Theorem 1 are satisfied with a = 0, the continuous-time nonlinear system (5) is finite-time stable with respect to ðc1 ; c2 ; T; RC Þ and it is also asymptotically stable.
_ VðxðtÞÞ < aVðxðtÞÞ;
ð9Þ
where a P 0 is a scalar. Dividing both sides of (9) by VðxðtÞÞ and integrating from 0 to t, with t 2 ð0; T, we obtain
ln
h
VðxðtÞÞ < at: Vðxð0ÞÞ
ð10Þ
From (10), we can obtain the following inequality: 1=2 1 1=2 1 1=2 T at xT ðtÞR1=2 C P RC xðtÞ < x ð0ÞRC P RC xð0Þe :
ð11Þ
Remark 3. In existing results, the initial condition is generally needed to be known (Chen & Liu, 2005; Chen et al., 2007; Guan & Chen, 2004; Jiang & Han, 2007). Whereas the guaranteed cost bound obtained in Theorem 1 only depends on some parameters associated with FTS. In other words, we need not to know the exact initial condition. This is true that these conditions (7) and (8) in Theorem 1 have not been represented the form of LMIs. The following result about the feasible problem of LMIs is provided.
Now we have 1=2
1=2
ð12Þ
1=2
1=2
ð13Þ
xT ðtÞRC P1 RC xðtÞ P kmin ðP1 ÞxT ðtÞRC xðtÞ; xT ð0ÞRC P1 RC xð0Þ 6 kmax ðP1 ÞxT ð0ÞRC xð0Þ 6 kmax ðP1 Þc1 : Combining (11)–(13), we have
xT ðtÞRC xðtÞ <
kmax ðP 1 Þc1 eat 1
kmin ðP Þ
¼ kmax ðPÞeat
c1 kmin ðPÞ
c1 6 kmax ðPÞe : kmin ðPÞ aT
Consider xT ðtÞRC xðtÞ < c2 for all t 2 ð0; T in Definition 1. The condi: _ aVðxðtÞÞ tion (8) in Theorem 1 is obtained. Denoting DV¼VðxðtÞÞ T and utilizing the inequality K i ðI þ KÞQ 2 ðI þ KÞK j þ K Tj ðI þ KÞ Q 2 ðI þ KÞK i 6 K Ti ðI þ KÞQ 2 ðI þ KÞK i þ K Tj ðI þ KÞQ 2 ðI þ KÞK j , we can obtain the following result:
DV ¼
r X
r X
i¼1
j¼1
h
e 1 hi ðhðtÞÞhj ðhðtÞÞxT ðtÞ ðAi þ Bi ðI þ KÞK j ÞT P
i e 1 ðAi þ Bi ðI þ KÞK j Þ a P e 1 xðtÞ þP 6
r X
hi ðhðtÞÞhi ðhðtÞÞxT ðtÞWii xðtÞ
i¼1
þ
r1 X r X i¼1
j>i
r X r X i¼1
hi ðhðtÞÞhj ðhðtÞÞxT ðtÞXij xðtÞ hi ðhðtÞÞhj ðhðtÞÞxT ðtÞðQ 1 þ K Ti ðI þ KÞQ 2 ðI þ KÞK j ÞxðtÞ;
j¼1
Theorem 2. The closed-loop continuous-time nonlinear system (5) is finite-time stable with respect to ðc1 ; c2 ; T; RC Þ where 0 < c1 < c2 ; T 2 Rþ ; RC > 0, and has the guaranteed cost bound W if there exist scalars a P 0; c > 0, matrices P > 0 2 Rnn ; R > 0 2 Rnn and matrix Sj 2 Rmn such that the following LMIs are satisfied
Nii < 0; 1 6 i 6 r; Hij < 0; 1 6 i < j 6 r;
ð14Þ
c1 aT e cI 6 P 6 R; c2 R 6 cI;
ð15Þ ð16Þ
where
2
6 6 Nii ¼ 6 6 6 4 2 6 6 6 6 6 Hij ¼ 6 6 6 6 6 4
e P
STi
Q 1 1
0
C
N1;1
STi
3
7 0 7 7 7; 0 7 5 12 I
H1;1
e P
STi
STj
STi
12 Q 1 1
0
0
0
C
0
0
C
0
12 I
STj
3
7 0 7 7 7 0 7 7; 7 0 7 7 7 0 5
12 I
e T þ Ai P e T þ Ai P e þ ST BT þ Bi Si a P e þ Bi RBT ; H1;1 ¼ PA e þ PA e Tþ N1;1 ¼ PA i i i i i j
where T e 1
Wii ¼ ðAi þ Bi ðI þ KÞK i Þ P
e 1 ðAi þ Bi ðI þ KÞK i Þ þP
e 1 þ Q 1 þ K T ðI þ KÞQ 2 ðI þ KÞK i ; aP i Xij ¼ ðAi þ Bi ðI þ KÞK j ÞT Pe 1 þ Pe 1 ðAi þ Bi ðI þ KÞK j Þ e 1 þ P e 1 e 1 ðAj þ Bj ðI þ KÞK i Þ 2a P þ ðAj þ Bj ðI þ KÞK i ÞT P þ 2Q 1 þ K Ti ðI þ KÞQ 2 ðI þ KÞK i þ K Tj ðI þ KÞQ 2 ðI þ KÞK j : From the above derivation, we can obtain that if Wii < 0 and Xij < 0 hold, then DV < 0 for any nonzero xðtÞ, i.e., (9) is true. After Integrating actions from 0 to T, the following result is obtained:
e þ ST BT þ Bi Sj þ ST BT þ Bj Si 2a P e þ Bi RBT þ Bj RBT ; C ¼ Q 1 þ R; Aj P i j j i i j 2 ð1=2Þ ð1=2Þ 2 e e 1 . The guaranteed cost bound R ¼ r I; P ¼ R PR and K j ¼ Sj P C
C
is described by:
W ¼ kmax ðP1 Þc1 eaT : Proof. Use Schur formula (Boyd, Ghauoi, Feron, & Balakrishnan, e I and set 1994) for Wii < 0, left and right multiply diag½ P; e Si ¼ K i P. Utilize Schur formula again and matrix inequality X T Y þ Y T X 6 X T X þ Y T Y, the first inequality condition of (14) in Theorem 2 is obtained. Similarly, we also obtain the second
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inequality condition of (14). It is easy to check that condition (8) can be guaranteed by assuming the following conditions
1.2 1
ecI 6 P 6 R; c1
c2 ; eaT R 6 cI;
e
6
where 0 < e 6 1; c > 0 and matrix R > 0. The conditions (15) and (16) in Theorem 2 can be obtained. The proof is completed. h
0.8 0.6 0.4 0.2
Remark 4. It is slight different from the proofs in Amato et al. (2001) that a free positive scalar c is introduced to relax the restricted conditions. We can obtain more broad scope of solutions via adjusting the free parameter. It is explicit that LMI Feasibility Problem (from Corollary 9) (Amato et al., 2001, p. 1462) is a special case of (15) when c ¼ 1. Generally, we need to minimize the guaranteed cost performance bound under some given conditions. The optimization problem about the continuous-time nonlinear system (5) is described as follows:
Minimize k; Subject to LMIsð14Þ—ð16Þ; P > 0; R > 0 and P k1 I P 0 where 0 < c1 < c2 ; T 2 Rþ ; RC > 0; a P 0 and c > 0: ð17Þ
0 −0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 1. Response of state xðc1 ¼ 1; c2 ¼ 5; RC ¼ I; T ¼ 30Þ.
When the initial condition is chosen as xð0Þ ¼ ½100T , the response of state x is showed in Fig. 1. From the above example, it can be seen that the requirements of the provided performance criterion are guaranteed. The continuous-time nonlinear system is finite-time stable with respect to ðc1 ; c2 ; T; RC Þ and it is also asymptotically stable because of a = 0.
4. An illuminated example
5. Conclusions
In this section, we will design a finite-time quantized guaranteed cost fuzzy controller for the following continuous-time nonlinear system as in Tanaka, Ikeda, and Wang (1998). Lorenz Equation with Input Term:
In this study, a finite-time quantized guaranteed cost fuzzy control scheme for continuous-time nonlinear systems has been investigated. The whole closed-loop system satisfies the requirements of the provided performance criterion. An illustrative example is given to illustrate the effectiveness of the proposed scheme.
x_ 1 ðtÞ ¼ ax1 ðtÞ þ ax2 ðtÞ þ uðtÞ; x_ 2 ðtÞ ¼ cx1 ðtÞ x2 ðtÞ x1 ðtÞx3 ðtÞ; x_ 3 ðtÞ ¼ x1 ðtÞx2 ðtÞ bx3 ðtÞ; where a, b, and c are constants, u (t) is the control input. Assume that x1 ðtÞ 2 ½dd and d > 0. Then, the following fuzzy model will exactly represents the nonlinear equation under x1 ðtÞ 2 ½dd. _ Rule 1: IF x1 ðtÞ is M1 , THEN xðtÞ ¼ A1 xðtÞ þ BuðtÞ, _ ¼ A2 xðtÞ þ BuðtÞ, Rule 2: IF x1 ðtÞ is M2 , THEN xðtÞ where xðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; x3 ðtÞT ,
2
3 2 3 2 3 a a 0 a a 0 1 6 7 6 7 6 7 6 7 6 7 6 A1 ¼ 4 c 1 d 5 A2 ¼ 4 c 1 d 5 B ¼ 4 0 7 5; 0 d b 0 d b 0 1 x1 ðtÞ 1 x1 ðtÞ M 2 ðx1 ðtÞÞ ¼ : M 1 ðx1 ðtÞÞ ¼ 1þ 1 2 d 2 d In this paper, a ¼ 10; b ¼ 8=3; c ¼ 28, and d = 30. Moreover, Q 1 and Q 2 in the cost function (6) are given as Q 1 ¼ diag½11 and Q 2 ¼ 1. Setting c1 ¼ 1; c2 ¼ 5; RC ¼ I; a ¼ 0; c ¼ 0:01; T ¼ 30 and r ¼ 0:4, we solve the optimization problem (17). The minimum of the guaranteed cost bound and the controller gain matrices are obtained as follows:
W min ¼ 480:6030; K 1 ¼ ½ 128:9082 43:4983 0:0026 ; K 2 ¼ ½ 128:9082 43:4983 0:0026 :
Acknowledgement This work was supported in part by the National Science Foundation of China under Grant 60904066. References Amato, F., & Ariola, M. (2005). Finite-time control of discrete-time linear systems. IEEE Transactions on Automatatic Control, 50(5), 724–729. Amato, F., Ariola, M., & Cosentino, C. (2006). Finite-time stabilization via dynamic output feedback. Automatica, 42(2), 337–342. Amato, F., Ariola, M., & Dorato, P. (2001). Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica, 37(9), 1459–1463. Boyd, S., Ghauoi, L. E., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Philadelphia, PA: SIAM. Brockett, R. W., & Liberzon, D. (2000). Quantized feedback stabilization of linear systems. IEEE Transactions on Automatic Control, 45(7), 1279–1289. Chen, B., & Liu, X. P. (2005). Fuzzy guaranteed cost control for nonlinear systems with time-varying delay. IEEE Transactions on Fuzzy Systems, 13(2), 238–249. Chen, B., Liu, X. P., Tong, S. C., & Lin, C. (2007). Guaranteed cost control of T–S fuzzy systems with state and input delays. Fuzzy Sets and Systems, 158(20), 2251–2267. Dorato, P. (1961). Short time stability in linear time-varying systems. In Proceedings of the IRE international convention record (Part 4, pp. 83–87). Elia, N., & Mitter, S. K. (2001). Stabilization of linear systems with limited information. IEEE Transactions on Automatic Control, 46(9), 1384–1400. Fu, M. Y., & Xie, L. H. (2005). The sector bound approach to quantized feedback control. IEEE Transactions on Automatic Control, 50(11), 1698–1711. Guan, X. P., & Chen, C. L. (2004). Delay-dependent guaranteed cost control for T–S fuzzy systems with time delays. IEEE Transactions on Fuzzy Systems, 12(2), 236–249. Jiang, X., & Han, Q. L. (2007). On guaranteed cost fuzzy control for nonlinear systems with interval time-varying delay. IET Control Theory and Application, 1(6), 1700–1710. Liberzon, D. (2003). On stabilization of linear systems with limited information. IEEE Transactions on Automatic Control, 48(2), 304–307.
D. Yang, K.-Y. Cai / Expert Systems with Applications 37 (2010) 6963–6967 Takagi, T., & Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, 15(1), 116–132. Tanaka, K., Ikeda, T., & Wang, H. O. (1998). Unified approach to controlling chaos via an LMI-based fuzzy control system design. IEEE Transactions on Circuits and Systems I, 45(10), 1021–1040.
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems Dedong Yang *, Kai-Yuan Cai National Key Laboratory of Science and Technology on Holistic Control, School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
a r t i c l e
i n f o
Keywords: Finite-time stability (FTS) Quantized control Fuzzy control Linear matrix inequality (LMI)
a b s t r a c t This paper considers the problem of finite-time quantized guaranteed cost fuzzy control for continuoustime nonlinear systems. Firstly, the definition on finite-time stability (FTS) for continuous-time nonlinear systems is provided and we give a novel and explicit interpretation for finite-time quantized guaranteed cost control. Secondly, sufficient conditions for the existence of state feedback controller are derived in terms of linear matrix inequities (LMIs), which guarantee the requirements of the provided performance criterion. The related optimization problem is also offered to minimize the guaranteed cost performance bound. Finally, an illustrative example is presented to show the validity of the proposed scheme. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Since there exist many cases where the state values are restrained, we generally need to ensure that these state values are allowable. In order to deal with these cases, the concept of finitetime stability (FTS) (or short-time stability) can be introduced into our eyeshot. As illuminated in Amato, Ariola, and Dorato (2001), a system is said to be finite-time stable if, once we fix a time-interval, its state does not exceed certain bound during this time-interval. Some early results on FTS can be found in Dorato (1961), Weiss and Infante (1967). More recently the problem of finite-time control for both continuous-time and discrete-time linear systems has been investigated in Amato and Ariola (2005), Amato, Ariola, and Cosentino (2006), Amato et al. (2001) via linear matrix inequities (LMIs) technique. In the existing results, FTS is mainly associated with linear systems. How to solve the FTS problem for nonlinear systems is still a difficult problem. Recently, Takagi–Sugeno (T–S) fuzzy model proposed in Takagi and Sugeno (1985) is widely applied in various industrial control fields because of its simple structure with local dynamics. The typical approach named as parallel distributed compensation (PDC) (Wang, Tanaka, & Griffin, 1996) is also developed to design the fuzzy controllers. Moreover, besides the stabilization problem, the guaranteed cost fuzzy control was extendedly investigated to stabilize the controlled systems while providing an upper bound on a given performance index by many scholars in the last decade, such as Chen and Liu (2005), Chen, Liu, Tong, and Lin (2007), Guan and Chen (2004), Jiang and Han (2007). * Corresponding author. Tel.: +86 010 82315087; fax: +86 010 82317328. E-mail address: [email protected] (D. Yang). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.03.024
The output signals of the controller are generally assumed to be returned directly to the plant without data loss in the classical feedback control theories. In practice, however, this is not true in many real applications since all data transmissions cannot be performed with infinite precision in computer control systems and quantization always exists. In recent years, considering the limited communication capacity, the problem of quantized feedback control has been investigated by many researchers, such as Brockett and Liberzon (2000), Elia and Mitter (2001), Fu and Xie (2005), Liberzon (2003). Moreover, how to design a quantized feedback controller for nonlinear systems is still an interesting issue. In this paper, we propose a finite-time quantized guaranteed cost fuzzy control scheme for continuous-time nonlinear systems. Its objective is to find a suitable quantized controller such that the provided performance criterion is satisfied. As far as we know, up to date, the finite-time quantized guaranteed cost control for continuous-time nonlinear systems has not been considered in the existing results. The remainder of this paper is organized as follows: Basic problem formulation is introduced in Section 2. The finite-time quantized guaranteed cost controller via state feedback is designed in Section 3. Section 4 provides an illustrative example to demonstrate the effectiveness of the proposed scheme. Finally, concluding remarks are made in Section 5. In the following sections, the identity matrices and zero matrices are denoted by I and 0, respectively. X T denotes the transpose of matrix X. Rn denotes the n-dimensional Euclidean space. Rþ denotes the positive real number. The notation * always denotes the symmetric block in one symmetric matrix. The standard notation > (< ) is used to denote the positive (negative)-definite ordering of matrices. Inequality X > Y shows that the matrix X Y is positive
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definite. kmax ðÞ and kmin ðÞ indicate the maximum and minimum eigenvalues of the matrix, respectively.
Considering the above quantization behavior, we can obtain the following quantized fuzzy controller:
2. Problem formulation
ðtÞ ¼ f u
r X
! hi ðhðtÞÞK i xðtÞ :
ð3Þ
i¼1
Consider a general class of continuous-time nonlinear systems which can be represented by the following T–S fuzzy dynamic model.
hi ðhðtÞÞ½Ai xðtÞ þ Bi uðtÞ;
ð1Þ
i¼1
P where hi ðhðtÞÞ ¼ li ðhðtÞÞ= ri¼1 li ðhðtÞÞ;
li ðhðtÞÞ ¼ Ppj¼1 Nij ðhj ðtÞÞ and
N ij ðhj ðtÞÞ is the grade of membership of hj ðtÞ in Nij . It is assumed that P li ðhðtÞÞ P 0, and ri¼1 li ðhðtÞÞ > 0 for all t. Then we can see that Pr hi ðhðtÞÞ P 0 and i¼1 hi ðhðtÞÞ ¼ 1. According to the conventional PDC concept, we design the following fuzzy controller via state feedback: Controller Rule i IF h1 ðtÞ is N i1 ; . . ., and hp ðtÞ is N ip THEN uðtÞ ¼ K i xðtÞ
r X
_ xðtÞ ¼
hi ðhðtÞÞK i xðtÞ;
ð4Þ
hi ðhðtÞÞK i xðtÞ:
ð2Þ
i¼1
It is assumed that the output signals of fuzzy controller (2) are passed via a quantizer and the quantizer is denoted as f ðÞ ¼ ½f1 ðÞ; f2 ðÞ; . . . ; fm ðÞT , which is assumed to be symmetric, that is, f ðv Þ ¼ f ðv Þ. The set of quantized levels is described by
r X
ðtÞ ¼ AxðtÞ; hi ðhðtÞÞ½Ai xðtÞ þ Bi u
ð5Þ
i¼1
P P where A ¼ ri¼1 rj¼1 hi ðhðtÞÞhj ðhðtÞÞ½Ai þ Bi ðI þ KÞK j . Next we will provide a novel concept on FTS for continuoustime nonlinear systems. It is formalized through the following definition. Similar description can be found in Amato et al. (2001). Definition 1. The closed-loop continuous-time nonlinear system (5) is said to be finite-time stable (FTS) with respect to ðc1 ; c2 ; T; RC Þ where 0 < c1 < c2 ; T 2 Rþ and RC > 0 if
xT ð0ÞRC xð0Þ 6 c1 ) xT ðtÞRC xðtÞ < c2 ;
8 t 2 ð0; T:
Remark 1. Asymptotic stability and FTS are independent concepts each other as it was described in Amato et al. (2001). In certain cases, asymptotic stability can be also considered as an additional requirement while restricting our attention on a finite-time interval. Furthermore, for (5), we provide the following cost function associated with the above definition:
J¼
where i ¼ 1; 2; . . . ; r is the number of controller rules, K i 2 Rmn is the controller gain matrix. Thus, the fuzzy controller can be expressed as follows:
uðtÞ ¼
r X
where K ¼ diagfK1 ; K2 ; . . . ; Km g and Ks 2 ½r; r; s ¼ 1; . . . ; m. The closed-loop system (1) can be reconstructed under the quantized fuzzy controller (4) as follows:
where i ¼ 1; 2; . . . ; r is the number of fuzzy rules, xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input, Ai 2 Rnn and Bi 2 Rnm are known constant matrices, of the ith subsystem, h1 ðtÞ; h2 ðtÞ; . . . ; hp ðtÞ are the premise variables, N ij is the fuzzy set ðj ¼ 1; 2; . . . ; pÞ. By using singleton fuzzifier, product inference and center-average defuzzifier, the fuzzy dynamic model is expressed by r X
ðtÞ ¼ ðI þ KÞ u
i¼1
Plant Rule i IF h1 ðtÞ is N i1 ; . . ., and hp ðtÞ is N ip THEN _ xðtÞ ¼ Ai xðtÞ þ Bi uðtÞ
_ xðtÞ ¼
Moreover, (3) can be expressed as
Z
T
T ðtÞQ 2 u ðtÞ dt; xT ðtÞQ 1 xðtÞ þ u
ð6Þ
0
where Q 1 > 0 and Q 2 > 0 are given weighting matrices or given positive scalars. The following novel and explicit interpretation for finite-time quantized guaranteed cost control is given by: Definition 2. For (5), if there exists a quantized fuzzy control law (4) and a scalar w0 such that the closed-loop system is finite-time stable and the value of the cost function (6) satisfies J < w0 , then w0 is said to be a guaranteed cost bound and the designed control law (4) is said to be a finite-time quantized guaranteed cost fuzzy control law.
X ¼ fui ; i ¼ 0; 1; 2; . . .g [ f0g: According to Elia and Mitter (2001) and Fu and Xie (2005), a quantizer is called logarithmic if the set of quantized levels is characterized by
X ¼ fui ; ui ¼ qi u0 ; i ¼ 1; 2; . . .g [ fu0 g [ f0g; 0 < qi < 1;
u0 > 0:
For the logarithmic quantizer, the associated quantizer f ðÞ is defined as follows:
8 1 1 > < ui ; if 1þr ui < v 6 1r ui ; f ðv Þ ¼ 0; if v ¼ 0; > : f ðv Þ; if v < 0: where
r¼
1q : 1þq
v > 0;
3. Main results In this section, the problem of finite-time quantized guaranteed cost fuzzy control via state feedback is studied according to (5). Some results are provided as follows. Theorem 1. The closed-loop continuous-time nonlinear system (5) is finite-time stable with respect to ðc1 ; c2 ; T; RC Þ where 0 < c1 < c2 ; T 2 Rþ ; RC > 0, and has the guaranteed cost bound W if there exist a scalar a P 0, matrix P > 0 2 Rnn and the controller gain matrix K j 2 Rmn such that the following conditions are satisfied
Wii < 0; 1 6 i 6 r; Xij < 0; 1 6 i < j 6 r;
ð7Þ
c1 c2 ; 6 kmin ðPÞ kmax ðPÞeaT
ð8Þ
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Z
T
e 1 ðAi þ Bi ðI þ KÞK i Þ a P e 1 þ P e 1 þ Wii ¼ ðAi þ Bi ðI þ KÞK i ÞT P T e 1 T 1 e Q 1 þ K i ðI þ KÞQ 2 ðI þ KÞK i , Xij ¼ ðAi þ Bi ðI þ KÞK j Þ P þ P ðAi þ Bi
VðxðTÞÞ Vðxð0ÞÞ a
e 1 ðAj þ Bj ðI þ KÞK i Þ 2a P e 1 þ P e 1 þ ðI þ KÞK j Þ þ ðAj þ Bj ðI þ KÞK i ÞT P ð1=2Þ T T e 2Q 1 þ K ðI þ KÞQ 2 ðI þ KÞK i þ K ðI þ KÞQ 2 ðI þ KÞK j , and P ¼ R
Moreover, J < kmax ðP1 Þc1 eaT can be obtained by considering Definition 1. It is easy to see that the guaranteed cost function satisfies
where
i
ð1=2Þ
PRC
j
C
VðxðtÞÞdt < J:
0
J < W ¼ kmax ðP1 Þc1 eaT :
. The guaranteed cost bound is described by:
W ¼ kmax ðP1 Þc1 eaT :
The proof is completed.
e 1 xðtÞ where P e ¼ Rð1=2Þ PRð1=2Þ with Proof. Let VðxðtÞÞ ¼ xT ðtÞ P C C P > 0 and assume the time derivative of VðxðtÞÞ along the solution of system (5) to satisfy the following condition for all t 2 ð0; T
Remark 2. If the conditions (7) and (8) in Theorem 1 are satisfied with a = 0, the continuous-time nonlinear system (5) is finite-time stable with respect to ðc1 ; c2 ; T; RC Þ and it is also asymptotically stable.
_ VðxðtÞÞ < aVðxðtÞÞ;
ð9Þ
where a P 0 is a scalar. Dividing both sides of (9) by VðxðtÞÞ and integrating from 0 to t, with t 2 ð0; T, we obtain
ln
h
VðxðtÞÞ < at: Vðxð0ÞÞ
ð10Þ
From (10), we can obtain the following inequality: 1=2 1 1=2 1 1=2 T at xT ðtÞR1=2 C P RC xðtÞ < x ð0ÞRC P RC xð0Þe :
ð11Þ
Remark 3. In existing results, the initial condition is generally needed to be known (Chen & Liu, 2005; Chen et al., 2007; Guan & Chen, 2004; Jiang & Han, 2007). Whereas the guaranteed cost bound obtained in Theorem 1 only depends on some parameters associated with FTS. In other words, we need not to know the exact initial condition. This is true that these conditions (7) and (8) in Theorem 1 have not been represented the form of LMIs. The following result about the feasible problem of LMIs is provided.
Now we have 1=2
1=2
ð12Þ
1=2
1=2
ð13Þ
xT ðtÞRC P1 RC xðtÞ P kmin ðP1 ÞxT ðtÞRC xðtÞ; xT ð0ÞRC P1 RC xð0Þ 6 kmax ðP1 ÞxT ð0ÞRC xð0Þ 6 kmax ðP1 Þc1 : Combining (11)–(13), we have
xT ðtÞRC xðtÞ <
kmax ðP 1 Þc1 eat 1
kmin ðP Þ
¼ kmax ðPÞeat
c1 kmin ðPÞ
c1 6 kmax ðPÞe : kmin ðPÞ aT
Consider xT ðtÞRC xðtÞ < c2 for all t 2 ð0; T in Definition 1. The condi: _ aVðxðtÞÞ tion (8) in Theorem 1 is obtained. Denoting DV¼VðxðtÞÞ T and utilizing the inequality K i ðI þ KÞQ 2 ðI þ KÞK j þ K Tj ðI þ KÞ Q 2 ðI þ KÞK i 6 K Ti ðI þ KÞQ 2 ðI þ KÞK i þ K Tj ðI þ KÞQ 2 ðI þ KÞK j , we can obtain the following result:
DV ¼
r X
r X
i¼1
j¼1
h
e 1 hi ðhðtÞÞhj ðhðtÞÞxT ðtÞ ðAi þ Bi ðI þ KÞK j ÞT P
i e 1 ðAi þ Bi ðI þ KÞK j Þ a P e 1 xðtÞ þP 6
r X
hi ðhðtÞÞhi ðhðtÞÞxT ðtÞWii xðtÞ
i¼1
þ
r1 X r X i¼1
j>i
r X r X i¼1
hi ðhðtÞÞhj ðhðtÞÞxT ðtÞXij xðtÞ hi ðhðtÞÞhj ðhðtÞÞxT ðtÞðQ 1 þ K Ti ðI þ KÞQ 2 ðI þ KÞK j ÞxðtÞ;
j¼1
Theorem 2. The closed-loop continuous-time nonlinear system (5) is finite-time stable with respect to ðc1 ; c2 ; T; RC Þ where 0 < c1 < c2 ; T 2 Rþ ; RC > 0, and has the guaranteed cost bound W if there exist scalars a P 0; c > 0, matrices P > 0 2 Rnn ; R > 0 2 Rnn and matrix Sj 2 Rmn such that the following LMIs are satisfied
Nii < 0; 1 6 i 6 r; Hij < 0; 1 6 i < j 6 r;
ð14Þ
c1 aT e cI 6 P 6 R; c2 R 6 cI;
ð15Þ ð16Þ
where
2
6 6 Nii ¼ 6 6 6 4 2 6 6 6 6 6 Hij ¼ 6 6 6 6 6 4
e P
STi
Q 1 1
0
C
N1;1
STi
3
7 0 7 7 7; 0 7 5 12 I
H1;1
e P
STi
STj
STi
12 Q 1 1
0
0
0
C
0
0
C
0
12 I
STj
3
7 0 7 7 7 0 7 7; 7 0 7 7 7 0 5
12 I
e T þ Ai P e T þ Ai P e þ ST BT þ Bi Si a P e þ Bi RBT ; H1;1 ¼ PA e þ PA e Tþ N1;1 ¼ PA i i i i i j
where T e 1
Wii ¼ ðAi þ Bi ðI þ KÞK i Þ P
e 1 ðAi þ Bi ðI þ KÞK i Þ þP
e 1 þ Q 1 þ K T ðI þ KÞQ 2 ðI þ KÞK i ; aP i Xij ¼ ðAi þ Bi ðI þ KÞK j ÞT Pe 1 þ Pe 1 ðAi þ Bi ðI þ KÞK j Þ e 1 þ P e 1 e 1 ðAj þ Bj ðI þ KÞK i Þ 2a P þ ðAj þ Bj ðI þ KÞK i ÞT P þ 2Q 1 þ K Ti ðI þ KÞQ 2 ðI þ KÞK i þ K Tj ðI þ KÞQ 2 ðI þ KÞK j : From the above derivation, we can obtain that if Wii < 0 and Xij < 0 hold, then DV < 0 for any nonzero xðtÞ, i.e., (9) is true. After Integrating actions from 0 to T, the following result is obtained:
e þ ST BT þ Bi Sj þ ST BT þ Bj Si 2a P e þ Bi RBT þ Bj RBT ; C ¼ Q 1 þ R; Aj P i j j i i j 2 ð1=2Þ ð1=2Þ 2 e e 1 . The guaranteed cost bound R ¼ r I; P ¼ R PR and K j ¼ Sj P C
C
is described by:
W ¼ kmax ðP1 Þc1 eaT : Proof. Use Schur formula (Boyd, Ghauoi, Feron, & Balakrishnan, e I and set 1994) for Wii < 0, left and right multiply diag½ P; e Si ¼ K i P. Utilize Schur formula again and matrix inequality X T Y þ Y T X 6 X T X þ Y T Y, the first inequality condition of (14) in Theorem 2 is obtained. Similarly, we also obtain the second
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inequality condition of (14). It is easy to check that condition (8) can be guaranteed by assuming the following conditions
1.2 1
ecI 6 P 6 R; c1
c2 ; eaT R 6 cI;
e
6
where 0 < e 6 1; c > 0 and matrix R > 0. The conditions (15) and (16) in Theorem 2 can be obtained. The proof is completed. h
0.8 0.6 0.4 0.2
Remark 4. It is slight different from the proofs in Amato et al. (2001) that a free positive scalar c is introduced to relax the restricted conditions. We can obtain more broad scope of solutions via adjusting the free parameter. It is explicit that LMI Feasibility Problem (from Corollary 9) (Amato et al., 2001, p. 1462) is a special case of (15) when c ¼ 1. Generally, we need to minimize the guaranteed cost performance bound under some given conditions. The optimization problem about the continuous-time nonlinear system (5) is described as follows:
Minimize k; Subject to LMIsð14Þ—ð16Þ; P > 0; R > 0 and P k1 I P 0 where 0 < c1 < c2 ; T 2 Rþ ; RC > 0; a P 0 and c > 0: ð17Þ
0 −0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 1. Response of state xðc1 ¼ 1; c2 ¼ 5; RC ¼ I; T ¼ 30Þ.
When the initial condition is chosen as xð0Þ ¼ ½100T , the response of state x is showed in Fig. 1. From the above example, it can be seen that the requirements of the provided performance criterion are guaranteed. The continuous-time nonlinear system is finite-time stable with respect to ðc1 ; c2 ; T; RC Þ and it is also asymptotically stable because of a = 0.
4. An illuminated example
5. Conclusions
In this section, we will design a finite-time quantized guaranteed cost fuzzy controller for the following continuous-time nonlinear system as in Tanaka, Ikeda, and Wang (1998). Lorenz Equation with Input Term:
In this study, a finite-time quantized guaranteed cost fuzzy control scheme for continuous-time nonlinear systems has been investigated. The whole closed-loop system satisfies the requirements of the provided performance criterion. An illustrative example is given to illustrate the effectiveness of the proposed scheme.
x_ 1 ðtÞ ¼ ax1 ðtÞ þ ax2 ðtÞ þ uðtÞ; x_ 2 ðtÞ ¼ cx1 ðtÞ x2 ðtÞ x1 ðtÞx3 ðtÞ; x_ 3 ðtÞ ¼ x1 ðtÞx2 ðtÞ bx3 ðtÞ; where a, b, and c are constants, u (t) is the control input. Assume that x1 ðtÞ 2 ½dd and d > 0. Then, the following fuzzy model will exactly represents the nonlinear equation under x1 ðtÞ 2 ½dd. _ Rule 1: IF x1 ðtÞ is M1 , THEN xðtÞ ¼ A1 xðtÞ þ BuðtÞ, _ ¼ A2 xðtÞ þ BuðtÞ, Rule 2: IF x1 ðtÞ is M2 , THEN xðtÞ where xðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ; x3 ðtÞT ,
2
3 2 3 2 3 a a 0 a a 0 1 6 7 6 7 6 7 6 7 6 7 6 A1 ¼ 4 c 1 d 5 A2 ¼ 4 c 1 d 5 B ¼ 4 0 7 5; 0 d b 0 d b 0 1 x1 ðtÞ 1 x1 ðtÞ M 2 ðx1 ðtÞÞ ¼ : M 1 ðx1 ðtÞÞ ¼ 1þ 1 2 d 2 d In this paper, a ¼ 10; b ¼ 8=3; c ¼ 28, and d = 30. Moreover, Q 1 and Q 2 in the cost function (6) are given as Q 1 ¼ diag½11 and Q 2 ¼ 1. Setting c1 ¼ 1; c2 ¼ 5; RC ¼ I; a ¼ 0; c ¼ 0:01; T ¼ 30 and r ¼ 0:4, we solve the optimization problem (17). The minimum of the guaranteed cost bound and the controller gain matrices are obtained as follows:
W min ¼ 480:6030; K 1 ¼ ½ 128:9082 43:4983 0:0026 ; K 2 ¼ ½ 128:9082 43:4983 0:0026 :
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