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Applied Mathematics and Computation 196 (2008) 200–206 www.elsevier.com/locate/amc
LMI optimization approach to stabilization of Genesio–Tesi chaotic system via dynamic controller Ju H. Park
a,*
, O.M. Kwon b, S.M. Lee
c
a
Robust Control and Nonlinear Dynamics Laboratory, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea b School of Electrical and Computer Engineering, 12 Gaeshin-Dong, Heungduk-Gu, Chungbuk National University, Cheongju, Republic of Korea c Platform Verification Division, BcN Business Unit, KT Co. Ltd., Daejeon, Republic of Korea
Abstract In this paper, a design method for a new controller to control the Genesio–Tesi chaotic systems is proposed. In this work, a dynamic outputs feedback controller for the system is developed for the first time. For stability analysis, a well-known Lyapunov stability theorem combining with LMI (linear matrix inequality) optimization approach is utilized. A numerical simulation is presented to show the usefulness of the proposed control scheme. 2007 Elsevier Inc. All rights reserved. Keywords: LMI optimization; Genesio–Tesi system; Chaos; Stabilization; Dynamic controller
1. Introduction Chaos is very interesting nonlinear phenomenon and has applications in many areas such as biology, economics, signal generator design, secure communication, many other engineering systems, and so on. Because a nonlinear system in the chaotic state is very sensitive to its initial condition and chaos causes often irregular behavior in practical systems, chaos is sometimes undesirable. Thus, one may wish to avoid and eliminate such behavior. Therefore, many chaotic systems including Chen system, Rossler system, Lorenz system, unified chaotic systems, and so on, have been extensively analyzed over the past decades [1–13]. However, most of control problems for various chaotic systems is only focused on the design of static controller without dynamics. In some situations, there is a strong need to construct dynamic controller instead of static controller in order to obtain better performance and dynamical behavior of state response. To the best knowledge of authors, the topic of dynamic output feedback control for chaotic systems has not been investigated yet. This paper considers the stabilization of a class of nonlinear chaotic systems, Genesio–Tesi system, studied in [14]. We propose a dynamic controller to control the system using the Lyapunov stability theory. The *
Corresponding author. E-mail address: [email protected] (J.H. Park).
0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.05.045
J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206
201
existence condition of such controller is derived in terms of linear matrix inequalities (LMIs). The LMI can be easily solved by various convex optimization algorithms [17]. Especially, in order to solve the LMIs, we utilize the LMI control toolbox which implements the interior-point algorithms with fast convergence [16]. Through the paper, Rn denotes n-dimensional Euclidean space, and Rnm is the set of all n · m real matrices. X < 0 means that X is a real symmetric negative definitive matrix. I denotes the identity matrix with appropriate dimensions. kÆk refers to Euclidean vector norm or the induced matrix 2-norm. The organization of this paper is as follows. In Section 2, the problem statement and controller design method are presented. In Section 3, we provide a numerical example to demonstrate the usefulness of the proposed method. Finally concluding remark is given. 2. Controller design The Genesio–Tesi system, proposed by Genesio and Tesi [14], is one of paradigms of chaos since it captures many features of chaotic systems. It includes a simple square part and three simple ordinary differential equations that depend on three positive real parameters. The dynamic equation of the system is as follows: 8 > < x_ 1 ¼ x2 ; x_ 2 ¼ x3 ; ð1Þ > : 2 x_ 3 ¼ cx1 bx2 ax3 þ x1 ; where xi, i = 1, 2, 3 are state variables, and a, b and c are the positive real constants satisfying ab < c. For instance, the system is chaotic for the parameters a = 1.2, b = 2.92, c = 6. For the initial condition (x1, x2, x3) = (0.2, 0.3, 0.1), the chaotic motion of the system is illustrated in Figs. 1 and 2. In order to control the chaotic behavior in Genesio–Tesi system, we have the following control system: x_ ðtÞ ¼ AxðtÞ þ DAxðtÞ þ BuðtÞ;
ð2Þ
yðtÞ ¼ CxðtÞ;
10 x1(t) 5 0
0
5
10
15
20
25
30
35
40
10 x (t) 2
5 0
0
5
10
15
20
25
30
35
40
20 x (t) 3
10 0
0
5
10
15
20
25
t(sec)
Fig. 1. Chaotic motions.
30
35
40
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J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206 10
15
8 10 6
5
2
x3
x2
4
0 0
0
2
x
4
6
8
0
5
10
x
1
2
Fig. 2. Chaotic attractors on (x1, x2) and (x2, x3) planes.
where x(t) is the state vector, u(t) is the feedback control law, y(t) is the output vector, B 2 Rnm is the input matrix, C 2 Rpn is the output matrix, and 2 3 2 3 2 3 2 3 0 1 0 0 0 0 0 x1 6 7 6 7 6 7 6 7 0 1 5; B ¼ 4 0 5; DA ¼ 4 0 0 0 5: xðtÞ ¼ 4 x2 5; A ¼ 4 0 x1 0 0 c b a a x3 Note that the real parameter a in input matrix B is a weighting parameter. Now, in order to stabilize system Eq. (1), let’s consider the following dynamic output feedback controllers: _ ¼ Ac nðtÞ þ Bc yðtÞ; nðtÞ
ð3Þ
uðtÞ ¼ C c nðtÞ;
where nðtÞ 2 Rn is the controller state, and Ac, Bc and Cc are gain matrices with appropriate dimensions to be determined later. Applying this controller Eq. (3) to system Eq. (1) results in the closed-loop system z_ ðtÞ ¼ AzðtÞ þ DAzðtÞ;
ð4Þ
where
xðtÞ zðtÞ ¼ ; nðtÞ
A A¼ Bc C
BC c ; Ac
DA DA ¼ 0
0 : 0
Remark 1. Since, a chaotic system has bounded trajectories, there exists a constant m such that kDAk ¼ kDAk ¼ kx1 k 6 m: Then, we have the following main result.
ð5Þ
J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206
203
Theorem 1. For given a constant m, there exists a dynamic output feedback controller Eq. (3) for stabilization b B; b satisfying b C of the origin of the system Eq. (1) if there exist positive-definite matrices S, Y and matrices A; the following LMIs: " # bþC b T BT þ 2mY b T þ 2mI AY þ YAT þ B C AþA 0:
Proof. Let us consider the following legitimate Lyapunov functional candidate: V ¼ zT ðtÞPzðtÞ;
ð8Þ
where P > 0. Taking the time derivative of V along the solution of Eq. (1), we have dV ¼ 2zT ðtÞP z_ ðtÞ ¼ zT ðtÞ P A þ AT P zðtÞ þ 2zT ðtÞP DAzðtÞ dt 6 zT ðtÞ P A þ AT P þ 2mP zðtÞ zT ðtÞRzðtÞ:
ð9Þ
Thus, if the inequality R < 0 holds, there exists a positive scalar c such that dV 6 ckxðtÞk2 : dt
ð10Þ
In the matrix R, the matrix P > 0 and the controller parameters Ac, Bc and Cc, which included in the matrix A, are unknown and occur in nonlinear fashion. Hence, the inequality R < 0 cannot be considered as an linear matrix inequality problem. In the following, we will use a method of changing variables such that the inequality can be solved as convex optimization algorithm [15]. First, partition the matrix P and its inverse as S N Y M 1 P¼ ; P ¼ ; ð11Þ NT U MT W where S; Y 2 Rnn are positive-definite matrices, and M and N are invertible matrices. Note that the equality P1P = I gives that MN T ¼ I YS:
ð12Þ
Define F1 ¼
Y
I
MT
0
;
F2 ¼
I
S
ð13Þ
:
0 NT
Then, it follows that PF 1 ¼ F 2 ;
F T1 PF 1 ¼ F T1 F 2 ¼
Y
I
I
S
> 0:
ð14Þ
Now, postmultiplying and premultiplying the matrix inequality, R < 0, by the matrix F T1 and by its transpose, respectively, gives F T2 AF 1 þ F T1 AT F 2 þ 2mF T2 F 1 < 0:
ð15Þ
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J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206
By utilizing the relation Eqs. (11)–(14), it can be easily obtained that the inequality Eq. (15) is equivalent to C1 C2 < 0; ð16Þ H C3 where C1 ¼ AY þ YAT þ BC c M T þ MC Tc BT þ 2mY ; C2 ¼ A þ YAT S þ MC Tc BT S þ YC T BTc N T þ MATc N T þ 2mI; C3 ¼ SA þ NBc C þ AT S þ C T BTc N T þ 2mS: By defining a new set of variables as follows: b ¼ SAY þ SB C b þ BCY b A þ NAc M T ; b ¼ NBc ; B b ¼ Cc M T ; C
ð17Þ
then, Eq. (16) is simplified to the LMI Eq. (6). Finally, the LMI Eq. (7) is equivalent to the positiveness of P. This completes the proof. h Remark 2. The problem of Theorem 1 is to determine whether the problem is feasible or not. It is called the b B b and feasibility problem. The solutions of the problem can be found by solving eigenvalue problem in S; Y ; A; b which is a convex optimization problem. Note that a locally optimal point of a convex optimization probC, lem with strictly convex objective is globally optimal [17]. Various efficient convex optimization algorithms can be used to check whether the matrix inequalities Eq. (6) and Eq. (7) is feasible. In this paper, in order to solve LMIs, we utilize Matlab’s LMI Control Toolbox [16], which implements state-of-the-art interior-point algorithms, which is significantly faster than classical convex optimization algorithms [17]. Remark 3. Given any solution of the LMIs Eqs. (6) and (7) in Theorem 1, a corresponding controller of the form Eq. (3) will be constructed as follows: • Compute the invertible matrices M and N satisfying Eq. (12) using matrix algebra. • Utilizing the matrices M and N obtained above, solve the system of Eq. (17) for Bc, Cc and Ac (in this order). 3. Numerical example Consider the Genesio–Tesi chaotic system with: 2 3 2 3 0 1 0 0 6 7 6 7 A¼4 0 0 1 5; B ¼ 4 0 5; 6 2:9 1:2 10 2 3 0 0 0 6 7 C ¼ ½ 3 0:5 0 ; DA ¼ 4 0 0 0 5; xð0Þ ¼ ½0:2 0:3 0:1T : x1
ð18Þ
0 0
As shown in Fig. 1, the system has chaotic behavior when the control input does not apply. Before designing the dynamic controller Eq. (3) to control the Genesio–Tesi system to the origin, we assume that the magnitude of state vector x1 is bounded as jx1j 6 6. Now, we design a suitable dynamic output feedback controller of the form Eq. (3) for system Eq. (18), which guarantees the asymptotic stability of the closed-loop system. By applying Theorem 1 to the system
J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206 6
205
x1(t)
4 2 0
0
5
10
15
20
25
30
35
40
x2(t)
0
0
5
10
15
20
25
30
35
40
x3(t)
100 0
0
5
10
15
20
25
30
35
40
t(sec)
Fig. 3. System trajectories under the control.
Eq. (18) and checking the feasibility of the LMIs Eqs. (6) and (7), we can find that the LMIs are feasible via LMI Control Toolbox [16] and obtain a possible set of solutions of the LMIs: 2 3 2 3 1:5697 0:1829 0:0170 0:0003 0:0022 0:0107 6 7 6 7 S ¼ 104 4 0:1829 0:0290 0:0022 5; Y ¼ 104 4 0:0022 0:0186 0:1298 5; 0:0170
0:0022 0:0002
2
12:0000 0:0000 6 b ¼ 4 1:0000 12:0000 A 0:0000 b ¼ 10 ½ 0:0095 C 4
1:0000 0:1208
6:0000
3
7 2:9000 5; 10:8000
0:0107 0:1298 2 3 1:9212 7 b ¼ 104 6 B 4 0:3780 5;
1:5637
0:0368
8:1185 :
Then, by further calculation in light of Remark 3, we have a possible stabilizing dynamic output feedback controller for the system Eq. (18): 2 3 2 3 11:9784 12:9033 3:9342 137:8396 6 7 6 7 Ac ¼ 4 12:0370 21:3254 7:0490 5; Bc ¼ 4 356:5511 5; C c ¼ ½ 5:7045 16:7121 1:1843 : 1:9307
5:0234
20:8518
81:3638
The simulation result is illustrated in Fig. 3 in which the control input is applied at t = 20 s. In the figures, one can see that the system is indeed well stabilized to the origin. 4. Concluding remark In this paper, we consider the controlled Genesio–Tesi chaotic systems. We have proposed a novel dynamic feedback control scheme for asymptotic stability with respect to origin using the Lyapunov stability theory. Finally, a numerical simulation is provided to show the effectiveness of our method.
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J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206
Acknowledgement This research was supported by the Yeungnam University Research Grants in 2007. References [1] E. Ott, C. Grebogi, JA. Yorke, Controlling chaos, Phys. Rev. Lett. 64 (1990) 1196–1199. [2] H. Lu, Z. He, Chaotic behavior in first-order autonomous continuous-time systems with delay, IEEE Trans. Circuit Syst. 43 (1996) 700–702. [3] Y. Tian, F. Gao, Adaptive control of chaotic continuous-time systems with delay, Physica D 117 (1998) 1–12. [4] X.P. Guan, C.L. Chen, H.P. Peng, Z.P. Fan, Time-delayed feedback control of time-delay chaotic systems, Int. J. Bifurcat. Chaos 13 (2003) 193–205. [5] J. Sun, Delay-dependent stability criteria for time-delay chaotic systems via time-delay feedback control, Chaos Soliton. Fract. 32 (2007) 725–1734. [6] M. Sun, L. Tian, S. Jiang, J. Xu, Feedback control and adaptive control of the energy resource chaotic system, Chaos Soliton. Fract. 21 (2004) 143–150. [7] G. Chen, Chaos on some controllability conditions for chaotic dynamics control, Chaos Soliton. Fract. 8 (1997) 1461–1470. [8] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170 (1992) 421–428. [9] J.H. Park, O.M. Kwon, LMI optimization approach to stabilization of time-delay chaotic systems, Chaos Soliton. Fract. 23 (2005) 445–450. [10] J.H. Park, Stability criterion for synchronization of linearly coupled unified chaotic systems, Chaos Soliton. Fract. 23 (2005) 1319– 1325. [11] Y. Wang, Z.H. Guan, H.O. Wang, Feedback an adaptive control for the synchronization of Chen system via a single variable, Phys. Lett. A 312 (2003) 34–40. [12] J.H. Park, Controlling chaotic systems via nonlinear feedback control, Chaos Soliton. Fract. 23 (3) (2005) 1049–1054. [13] M. Chen, Z. Han, Controlling and synchronizing chaotic Genesio system via nonlinear feedback control, Chaos Soliton. Fract. 17 (2003) 709–716. [14] R. Genesio, A. Tesi, A harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, Automatica 28 (1992) 531–548. [15] C. Scherer, P. Gahinet, M. Chilali, Multiobjective output feedback control via LMI optimization, IEEE Trans. Autom. Control 42 (1997) 896–911. [16] P. Gahinet et al., LMI control toolbox user’s guide, The Mathworks, Masachusetts, 1995. [17] S. Boyd et al., Linear matrix inequalities in systems and control theory, SIAM, Philadelphia, 1994.
Applied Mathematics and Computation 196 (2008) 200–206 www.elsevier.com/locate/amc
LMI optimization approach to stabilization of Genesio–Tesi chaotic system via dynamic controller Ju H. Park
a,*
, O.M. Kwon b, S.M. Lee
c
a
Robust Control and Nonlinear Dynamics Laboratory, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea b School of Electrical and Computer Engineering, 12 Gaeshin-Dong, Heungduk-Gu, Chungbuk National University, Cheongju, Republic of Korea c Platform Verification Division, BcN Business Unit, KT Co. Ltd., Daejeon, Republic of Korea
Abstract In this paper, a design method for a new controller to control the Genesio–Tesi chaotic systems is proposed. In this work, a dynamic outputs feedback controller for the system is developed for the first time. For stability analysis, a well-known Lyapunov stability theorem combining with LMI (linear matrix inequality) optimization approach is utilized. A numerical simulation is presented to show the usefulness of the proposed control scheme. 2007 Elsevier Inc. All rights reserved. Keywords: LMI optimization; Genesio–Tesi system; Chaos; Stabilization; Dynamic controller
1. Introduction Chaos is very interesting nonlinear phenomenon and has applications in many areas such as biology, economics, signal generator design, secure communication, many other engineering systems, and so on. Because a nonlinear system in the chaotic state is very sensitive to its initial condition and chaos causes often irregular behavior in practical systems, chaos is sometimes undesirable. Thus, one may wish to avoid and eliminate such behavior. Therefore, many chaotic systems including Chen system, Rossler system, Lorenz system, unified chaotic systems, and so on, have been extensively analyzed over the past decades [1–13]. However, most of control problems for various chaotic systems is only focused on the design of static controller without dynamics. In some situations, there is a strong need to construct dynamic controller instead of static controller in order to obtain better performance and dynamical behavior of state response. To the best knowledge of authors, the topic of dynamic output feedback control for chaotic systems has not been investigated yet. This paper considers the stabilization of a class of nonlinear chaotic systems, Genesio–Tesi system, studied in [14]. We propose a dynamic controller to control the system using the Lyapunov stability theory. The *
Corresponding author. E-mail address: [email protected] (J.H. Park).
0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.05.045
J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206
201
existence condition of such controller is derived in terms of linear matrix inequalities (LMIs). The LMI can be easily solved by various convex optimization algorithms [17]. Especially, in order to solve the LMIs, we utilize the LMI control toolbox which implements the interior-point algorithms with fast convergence [16]. Through the paper, Rn denotes n-dimensional Euclidean space, and Rnm is the set of all n · m real matrices. X < 0 means that X is a real symmetric negative definitive matrix. I denotes the identity matrix with appropriate dimensions. kÆk refers to Euclidean vector norm or the induced matrix 2-norm. The organization of this paper is as follows. In Section 2, the problem statement and controller design method are presented. In Section 3, we provide a numerical example to demonstrate the usefulness of the proposed method. Finally concluding remark is given. 2. Controller design The Genesio–Tesi system, proposed by Genesio and Tesi [14], is one of paradigms of chaos since it captures many features of chaotic systems. It includes a simple square part and three simple ordinary differential equations that depend on three positive real parameters. The dynamic equation of the system is as follows: 8 > < x_ 1 ¼ x2 ; x_ 2 ¼ x3 ; ð1Þ > : 2 x_ 3 ¼ cx1 bx2 ax3 þ x1 ; where xi, i = 1, 2, 3 are state variables, and a, b and c are the positive real constants satisfying ab < c. For instance, the system is chaotic for the parameters a = 1.2, b = 2.92, c = 6. For the initial condition (x1, x2, x3) = (0.2, 0.3, 0.1), the chaotic motion of the system is illustrated in Figs. 1 and 2. In order to control the chaotic behavior in Genesio–Tesi system, we have the following control system: x_ ðtÞ ¼ AxðtÞ þ DAxðtÞ þ BuðtÞ;
ð2Þ
yðtÞ ¼ CxðtÞ;
10 x1(t) 5 0
0
5
10
15
20
25
30
35
40
10 x (t) 2
5 0
0
5
10
15
20
25
30
35
40
20 x (t) 3
10 0
0
5
10
15
20
25
t(sec)
Fig. 1. Chaotic motions.
30
35
40
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J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206 10
15
8 10 6
5
2
x3
x2
4
0 0
0
2
x
4
6
8
0
5
10
x
1
2
Fig. 2. Chaotic attractors on (x1, x2) and (x2, x3) planes.
where x(t) is the state vector, u(t) is the feedback control law, y(t) is the output vector, B 2 Rnm is the input matrix, C 2 Rpn is the output matrix, and 2 3 2 3 2 3 2 3 0 1 0 0 0 0 0 x1 6 7 6 7 6 7 6 7 0 1 5; B ¼ 4 0 5; DA ¼ 4 0 0 0 5: xðtÞ ¼ 4 x2 5; A ¼ 4 0 x1 0 0 c b a a x3 Note that the real parameter a in input matrix B is a weighting parameter. Now, in order to stabilize system Eq. (1), let’s consider the following dynamic output feedback controllers: _ ¼ Ac nðtÞ þ Bc yðtÞ; nðtÞ
ð3Þ
uðtÞ ¼ C c nðtÞ;
where nðtÞ 2 Rn is the controller state, and Ac, Bc and Cc are gain matrices with appropriate dimensions to be determined later. Applying this controller Eq. (3) to system Eq. (1) results in the closed-loop system z_ ðtÞ ¼ AzðtÞ þ DAzðtÞ;
ð4Þ
where
xðtÞ zðtÞ ¼ ; nðtÞ
A A¼ Bc C
BC c ; Ac
DA DA ¼ 0
0 : 0
Remark 1. Since, a chaotic system has bounded trajectories, there exists a constant m such that kDAk ¼ kDAk ¼ kx1 k 6 m: Then, we have the following main result.
ð5Þ
J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206
203
Theorem 1. For given a constant m, there exists a dynamic output feedback controller Eq. (3) for stabilization b B; b satisfying b C of the origin of the system Eq. (1) if there exist positive-definite matrices S, Y and matrices A; the following LMIs: " # bþC b T BT þ 2mY b T þ 2mI AY þ YAT þ B C AþA 0:
Proof. Let us consider the following legitimate Lyapunov functional candidate: V ¼ zT ðtÞPzðtÞ;
ð8Þ
where P > 0. Taking the time derivative of V along the solution of Eq. (1), we have dV ¼ 2zT ðtÞP z_ ðtÞ ¼ zT ðtÞ P A þ AT P zðtÞ þ 2zT ðtÞP DAzðtÞ dt 6 zT ðtÞ P A þ AT P þ 2mP zðtÞ zT ðtÞRzðtÞ:
ð9Þ
Thus, if the inequality R < 0 holds, there exists a positive scalar c such that dV 6 ckxðtÞk2 : dt
ð10Þ
In the matrix R, the matrix P > 0 and the controller parameters Ac, Bc and Cc, which included in the matrix A, are unknown and occur in nonlinear fashion. Hence, the inequality R < 0 cannot be considered as an linear matrix inequality problem. In the following, we will use a method of changing variables such that the inequality can be solved as convex optimization algorithm [15]. First, partition the matrix P and its inverse as S N Y M 1 P¼ ; P ¼ ; ð11Þ NT U MT W where S; Y 2 Rnn are positive-definite matrices, and M and N are invertible matrices. Note that the equality P1P = I gives that MN T ¼ I YS:
ð12Þ
Define F1 ¼
Y
I
MT
0
;
F2 ¼
I
S
ð13Þ
:
0 NT
Then, it follows that PF 1 ¼ F 2 ;
F T1 PF 1 ¼ F T1 F 2 ¼
Y
I
I
S
> 0:
ð14Þ
Now, postmultiplying and premultiplying the matrix inequality, R < 0, by the matrix F T1 and by its transpose, respectively, gives F T2 AF 1 þ F T1 AT F 2 þ 2mF T2 F 1 < 0:
ð15Þ
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J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206
By utilizing the relation Eqs. (11)–(14), it can be easily obtained that the inequality Eq. (15) is equivalent to C1 C2 < 0; ð16Þ H C3 where C1 ¼ AY þ YAT þ BC c M T þ MC Tc BT þ 2mY ; C2 ¼ A þ YAT S þ MC Tc BT S þ YC T BTc N T þ MATc N T þ 2mI; C3 ¼ SA þ NBc C þ AT S þ C T BTc N T þ 2mS: By defining a new set of variables as follows: b ¼ SAY þ SB C b þ BCY b A þ NAc M T ; b ¼ NBc ; B b ¼ Cc M T ; C
ð17Þ
then, Eq. (16) is simplified to the LMI Eq. (6). Finally, the LMI Eq. (7) is equivalent to the positiveness of P. This completes the proof. h Remark 2. The problem of Theorem 1 is to determine whether the problem is feasible or not. It is called the b B b and feasibility problem. The solutions of the problem can be found by solving eigenvalue problem in S; Y ; A; b which is a convex optimization problem. Note that a locally optimal point of a convex optimization probC, lem with strictly convex objective is globally optimal [17]. Various efficient convex optimization algorithms can be used to check whether the matrix inequalities Eq. (6) and Eq. (7) is feasible. In this paper, in order to solve LMIs, we utilize Matlab’s LMI Control Toolbox [16], which implements state-of-the-art interior-point algorithms, which is significantly faster than classical convex optimization algorithms [17]. Remark 3. Given any solution of the LMIs Eqs. (6) and (7) in Theorem 1, a corresponding controller of the form Eq. (3) will be constructed as follows: • Compute the invertible matrices M and N satisfying Eq. (12) using matrix algebra. • Utilizing the matrices M and N obtained above, solve the system of Eq. (17) for Bc, Cc and Ac (in this order). 3. Numerical example Consider the Genesio–Tesi chaotic system with: 2 3 2 3 0 1 0 0 6 7 6 7 A¼4 0 0 1 5; B ¼ 4 0 5; 6 2:9 1:2 10 2 3 0 0 0 6 7 C ¼ ½ 3 0:5 0 ; DA ¼ 4 0 0 0 5; xð0Þ ¼ ½0:2 0:3 0:1T : x1
ð18Þ
0 0
As shown in Fig. 1, the system has chaotic behavior when the control input does not apply. Before designing the dynamic controller Eq. (3) to control the Genesio–Tesi system to the origin, we assume that the magnitude of state vector x1 is bounded as jx1j 6 6. Now, we design a suitable dynamic output feedback controller of the form Eq. (3) for system Eq. (18), which guarantees the asymptotic stability of the closed-loop system. By applying Theorem 1 to the system
J.H. Park et al. / Applied Mathematics and Computation 196 (2008) 200–206 6
205
x1(t)
4 2 0
0
5
10
15
20
25
30
35
40
x2(t)
0
0
5
10
15
20
25
30
35
40
x3(t)
100 0
0
5
10
15
20
25
30
35
40
t(sec)
Fig. 3. System trajectories under the control.
Eq. (18) and checking the feasibility of the LMIs Eqs. (6) and (7), we can find that the LMIs are feasible via LMI Control Toolbox [16] and obtain a possible set of solutions of the LMIs: 2 3 2 3 1:5697 0:1829 0:0170 0:0003 0:0022 0:0107 6 7 6 7 S ¼ 104 4 0:1829 0:0290 0:0022 5; Y ¼ 104 4 0:0022 0:0186 0:1298 5; 0:0170
0:0022 0:0002
2
12:0000 0:0000 6 b ¼ 4 1:0000 12:0000 A 0:0000 b ¼ 10 ½ 0:0095 C 4
1:0000 0:1208
6:0000
3
7 2:9000 5; 10:8000
0:0107 0:1298 2 3 1:9212 7 b ¼ 104 6 B 4 0:3780 5;
1:5637
0:0368
8:1185 :
Then, by further calculation in light of Remark 3, we have a possible stabilizing dynamic output feedback controller for the system Eq. (18): 2 3 2 3 11:9784 12:9033 3:9342 137:8396 6 7 6 7 Ac ¼ 4 12:0370 21:3254 7:0490 5; Bc ¼ 4 356:5511 5; C c ¼ ½ 5:7045 16:7121 1:1843 : 1:9307
5:0234
20:8518
81:3638
The simulation result is illustrated in Fig. 3 in which the control input is applied at t = 20 s. In the figures, one can see that the system is indeed well stabilized to the origin. 4. Concluding remark In this paper, we consider the controlled Genesio–Tesi chaotic systems. We have proposed a novel dynamic feedback control scheme for asymptotic stability with respect to origin using the Lyapunov stability theory. Finally, a numerical simulation is provided to show the effectiveness of our method.
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