- Email: [email protected]
Repetitive learning control of continuous chaotic systems
Chaos, Solitons and Fractals 22 (2004) 161–169 www.elsevier.com/locate/chaos
Repetitive learning control of continuous chaotic systems Maoyin Chen b
a,*
, Yun Shang b, Donghua Zhou
a
a Department of Automation, Tsinghua University, Beijing 100084, PR China College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710068, PR China
Accepted 29 December 2003
Abstract Combining a shift method and the repetitive learning strategy, a repetitive learning controller is proposed to stabilize unstable periodic orbits (UPOs) within chaotic attractors in the sense of least mean square. If nonlinear parts in chaotic systems satisfy Lipschitz condition, the proposed controller can be simplified into a simple proportional repetitive learning controller. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Recently considerable effort has been made to the control of chaotic systems, including the stabilization of the UPOs within chaotic attractors [3–18,21,22]. Among many methods, the time-delayed feedback control (TDFC) method is an effective approach to stabilize the UPOs. This method utilizes the difference between the states and the delayed states as an input control provided that the delayed time is determined as the period of the UPO to be stabilized. Unfortunately, the TDFC method has its inherent limitation. When analyzing the stability of the TDFC controlled system, Ushio found out that the TDFC method has a property, called by odd number characteristic exponent property [11]. If a UPO has odd number real eigenvalues greater than one, it cannot be stabilized by the TDFC method. This property holds for all kinds of TDFC controllers, including exponential and extended TDFC controllers [11–13]. Since this property is a main limitation for the application of the TDFC method, overcoming this limitation has been a topic in chaos control field [3,6,10,14–18]. Except that static and dynamic feedback methods can effectively resolve this limitation [14,18], the optimal principle is also a useful tool to resolve this limitation [15]. In addition, the integration of the repetitive learning principle and the time-delayed chaos control technique can effectively stabilize the UPOs of chaotic systems [6]. This paper considers the stabilization of the UPOs of continuous chaotic systems. Using a shift method and the repetitive learning strategy, a repetitive learning controller is proposed. First, we extend a shift method (Ref. [2]) to arbitrary time-delayed chaotic system. Second, a repetitive learning controller is designed based on the repetitive learning strategy and the least mean square principle. This controller can effectively stabilize UPOs in the sense of least mean square. At last, the proposed controller can be simplified into a simple proportional repetitive learning controller if nonlinear parts in chaotic systems satisfy Lipschitz condition.
*
Corresponding author. E-mail address: [email protected] (M. Chen).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.01.008
162
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
2. Problem description Consider the following controlled chaotic system x_ ðtÞ ¼ AxðtÞ þ f ðyðtÞÞ þ BuðtÞ;
ð1Þ
yðtÞ ¼ CxðtÞ;
where x 2 Rn1 is the state vector, y 2 Rp1 is the output, uðtÞ 2 Rm1 is the control vector, and f ðÞ is a nonlinear function vector on the output. Moreover, A 2 Rnn , C 2 Rpn and the gain B 2 Rnm is column-full rank. When there exists no control input, Eq. (1) behaves chaotically. In this paper our aim is to stabilize the UPOs embedded in the chaotic attractor of Eq. (1). In order to keep the ergodicity of chaos motion, control input ui ðtÞ ði ¼ 1; 2; . . . ; mÞ are limited by the same saturated law, i.e., 8 ui ðtÞ > e; < e; sate ðui ðtÞÞ ¼ ui ðtÞ; jui ðtÞj 6 e; ð2Þ : e; ui ðtÞ < e in which e > 0 stands for the maximum amplitude of control action ui ðtÞ. Hence Eq. (1) becomes x_ ðtÞ ¼ AxðtÞ þ f ðyðtÞÞ þ B sate ðuðtÞÞ; yðtÞ ¼ CxðtÞ:
ð3Þ
In general many chaotic systems have the same form as Eq. (1), including Chua system 2 3 2 32 3 2 3 x_ 1 a a 0 x1 af ðyÞ 4 x_ 2 5 ¼ 4 1 1 1 54 x2 5 þ 4 0 5; x_ 3 0 b 0 0 x3 where yðtÞ ¼ ½ 1 0 2 3 2 x_ 1 0 6 7 6 4 x_ 2 5 ¼ 4 1 x_ 3 0
0 xðtÞ and f ðyÞ ¼ by þ 1=2ða bÞðjy þ 1j jy 1jÞ. In addition, R€ ossler system 32 3 2 3 x1 1 1 0 76 7 6 7 a 0 54 x2 5 þ 4 0 5; x3 0 b c þ x1 x3
ð4Þ
ð5Þ
y ¼ ½ 0 0 1 x can be transformed 2 3 2 0 n_ 1 6_ 7 6 4 n2 5 ¼ 4 1 1 n_ 3
into 1 a 0
32 3 2 3 n1 0 en3 76 7 6 7 0 54 n2 5 þ 4 0 5; n3 0 b þ ce n3
ð6Þ
g ¼ ½ 0 0 1 n via the transformation ½ n1
n2
n3 ¼ ½ x1
x2
lnðx3 Þ and g ¼ lnðyÞ.
3. Controller design 3.1. A shift method of chaotic systems with delay Consider a controlled chaotic system with a s-time-delayed saturated control input, given by x_ ðtÞ ¼ AxðtÞ þ f ðxðt sÞÞ þ B sate ðuðt sÞÞ:
ð7Þ
Here it is worth noting that matrix A is not necessarily stable. When there exists no control input in Eq. (2), a shift method is proposed to predict the states of Eq. (7) [2]. A nonlinear filter can shift arbitrarily complex wave produced by a chaotic system with a delayed feedback backwards in time. However, this method can be only applied to the system with a stable matrix. Here it is generalized to arbitrary time-delayed system with a delayed control input, described by Eq. (7). The predictor is constructed as
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
x_ ðtÞ ¼ ðA þ LÞxðtÞ þ f ðxðt s0 ÞÞ Lxðt s0 Þ þ Buðt s0 Þ
163
ð8Þ
with a suitable matrix L and a delayed time s0 , not necessarily equivalent to s. The time error and the state error are defined as s00 ¼ s s0
and
es00 ðtÞ ¼ xðtÞ xðt s00 Þ;
respectively. In the linear saturated area, the error system on the error es00 ðtÞ is e_ s00 ðtÞ ¼ AxðtÞ þ f ðxðt sÞÞ þ Buðt sÞ ðA þ LÞxðt s00 Þ f ðxðt s00 s0 ÞÞ þ Lxðt s00 s0 Þ Buðt s00 s0 Þ ¼ ðA þ LÞed 00 ðtÞ:
ð9Þ
Whether matrix A is stable or not, we can choose a matrix L to make Eq. (9) be stable. If matrix L is chosen as L ¼ A þ kIn ðk < 0Þ, Eq. (9) can be exponentially stable. So lim ðxðtÞ xðt s00 ÞÞ ¼ lim ðxðtÞ xðt s þ s0 ÞÞ ¼ 0
t!1
t!1
ð10Þ
for arbitrary constant s0 > 0. Therefore, in the linear saturated area, we conclude that Case 1: When s0 > s, the limit limt!1 ðxðtÞ xðt þ ðs0 sÞÞ ¼ 0 holds, which means that Eq. (8) can predict the past states of Eq. (7). Case 2: When s0 < s, the limit limt!1 ðxðtÞ xðt ðs s0 ÞÞ ¼ 0 shows that Eq. (8) is a future state predictor of Eq. (7). Case 3: When s0 ¼ s, the limit limt!1 ðxðtÞ xðtÞÞ ¼ 0 implies Eq. (8) is a current state predictor of Eq. (7). Remark 1. Matrix L can be chosen according to the well-known pole-placement algorithm [20]. Choosing a matrix D 2 Rnl ðl < nÞ such that the pair ðA; DÞ is controllable, and calculating the matrix E 2 Rln to make all the eigenvalues of A þ DE be in left open complex plane, hence L ¼ DE. 3.2. Design of the repetitive learning controller In Ref. [6] a time-delayed controller is proposed to stabilize the UPOs based on the invariant manifold of the chaotic systems. The integration of the repetitive learning strategy and the time-delayed chaos control technique can learn appropriate control action from each learning cycles. In this section, we also apply the repetitive learning strategy to stabilize the UPOs. For Eq. (3), a state predictor is given by x_ ðtÞ ¼ ðA þ LCÞxðtÞ þ f ðyðt sÞÞ Lyðt sÞ þ Buðt sÞ; ð11Þ where matrix L can be chosen such that A þ LC is stable [20]. From Section 3.1, we have lim ðxðtÞ xðt þ sÞÞ ¼ 0;
t!1
equivalently, lim ðxðt sÞ xðtÞÞ ¼ 0
t!1
which means that the state xðtÞ of Eq. (11) is the estimation value of the delayed state xðt sÞ of Eq. (3). Therefore once the state xðtÞ tends to the state xðtÞ asymptotically, the s-periodic UPO can be stabilized in the linear saturated area. Now we design a repetitive learning controller to make xðtÞ approach xðtÞ. Let eðtÞ ¼ xðtÞ xðtÞ: In the linear saturated area, the error system is expressed as e_ ðtÞ ¼ AxðtÞ þ f ðyðtÞÞ þ BuðtÞ ðA þ LCÞxðt sÞ f ðyðt sÞÞ þ Lyðt sÞ Buðt sÞ ¼ ðA þ LCÞðxðtÞ xðtÞÞ þ f ðyðtÞÞ f ðyðt sÞÞ þ Lðyðt sÞ yðtÞÞ þ BðuðtÞ uðt sÞÞ: In the right hand side of Eq. (12), if the term f ðyðtÞÞ f ðyðt sÞÞ þ Lðyðt sÞ yðtÞÞ þ BðuðtÞ uðt sÞÞ
ð12Þ
164
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
is zero, the limit lim ðxðtÞ xðtÞÞ ¼ 0
t!1
holds. Therefore, controller uðtÞ should satisfy f ðyðtÞÞ f ðyðt sÞÞ þ Lðyðt sÞ yðtÞÞ þ BðuðtÞ uðt sÞÞ ¼ 0:
ð13Þ
Denoting u ðtÞ ¼ uðtÞ uðt sÞ; Eq. (13) becomes f ðyðtÞÞ f ðyðt sÞÞ þ Lðyðt sÞ yðtÞÞ þ Bu ðtÞ ¼ 0:
ð14Þ
Since B is column-full rank, the least mean square solution of Eq. (14) is u ðtÞ ¼ ðBT BÞ 1 BT ½LðyðtÞ yðt sÞÞ þ f ðyðt sÞÞ f ðyðtÞÞ : Therefore
uðtÞ ¼ uðt sÞ þ ðBT BÞ 1 BT ½LðyðtÞ yðt sÞÞ þ f ðyðt sÞÞ f ðyðtÞÞ ; uðtÞ ¼ 0; t < T ;
ð15Þ
t PT;
ð16Þ
where T is sufficiently large such that the s-periodic orbit can be approximated more accurately. Remark 2. Controller (16) is a repetitive learning controller with a learning period s [1]. Compared with iterative learning method [19], this controller need not reset the initial condition during each learning cycle. During the interval ½ks; ðk þ 1Þs , we have d ðxðtÞ xðtÞÞ ¼ ðA þ LCÞðxðtÞ xðtÞÞ: dt So the initial error satisfies lim kxðksÞ xðksÞk ¼ 0:
k!1
Remark 3. Controller (16) is an optimal control law in the sense of least mean square, which makes the distance between the state xðtÞ and the delayed state xðt sÞ be little as soon as possible. If follows that uðkÞ ¼ uðk sÞ when the state xðtÞ approaches the delayed state xðt sÞ as time tends to infinity. Remark 4. Controller (16) does not embody information on Eq. (11), which is only useful to construct controller (16) and doesn’t join in the application of this controller. That is to say, Eq. (11) is an auxiliary system to design controller (16). Remark 5. It is well-known that TDFC method has its inherent limitation, called by odd number characteristic exponent property. In this paper, controller (16) can stabilize arbitrary UPO in the sense of least mean square. It means that controller (16) can overcome the inherent limitation of TDFC. 3.3. Simplification of the repetitive learning controller Suppose the nonlinear part f ðÞ satisfies Lipschitz condition, i.e., kf ðx1 Þ f ðx2 Þk < rkx1 x2 k;
8x1;2 2 Rn ;
where r > 0 is a Lipschitz constant. In this case, the predictor on Eq. (3) is constructed as x_ ðtÞ ¼ ðA þ LCÞxðtÞ þ f ðy Þ Lyðt sÞ þ Buðt sÞ; y ðtÞ ¼ CxðtÞ: Eq. (17) can still estimate the state of Eq. (3) at t s instant in the linear area if matrix L is chosen suitably.
ð17Þ
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
165
Using Eqs. (3) and (17), it follows that d ðxðtÞ xðt þ sÞÞ ¼ ðA þ LCÞðxðtÞ xðt þ sÞÞ þ f ðxðtÞÞ f ðxðt þ sÞÞ: dt
ð18Þ
Let a Lyapunov function be V ðxðtÞ; xðt þ sÞÞ ¼ ðxðtÞ xðt þ sÞÞT P ðxðtÞ xðt þ sÞÞ; where matrix P ¼ P T > 0 satisfies the following Riccati inequality ðA þ LCÞT P þ P ðA þ LCÞ þ r2 PP þ In < 2dP
ð19Þ
with a positive constant d. Therefore V_ ðxðtÞ; xðt þ sÞÞ ¼ ðxðtÞ xðt þ sÞÞT ½ðA þ LCÞT P þ P ðA þ LCÞ þ 2½f ðxðtÞÞ f ðxðt þ sÞ T P ðxðtÞ xðt þ sÞÞ: Since 2½f ðxðtÞÞ f ðxðt þ sÞ T P ðxðtÞ xðt þ sÞÞ 6 ðxðtÞ xðt þ sÞÞT ðr2 PP þ In ÞðxðtÞ xðt þ sÞÞ; we have V_ ðxðtÞ; xðt þ sÞÞ ¼ ðxðtÞ xðt þ sÞÞT ½ðA þ LCÞT P þ P ðA þ LCÞ þ r2 PP þ In ðxðtÞ xðt þ sÞÞ < 2dðxðtÞ xðt þ sÞÞT P ðxðtÞ xðt þ sÞÞ 6 0: This means that Eq. (18) is asymptotically stable, that is, the state xðtÞ of Eq. (17) can estimate the delayed state xðt sÞ of Eq. (3). The dynamics on the error xðtÞ xðtÞ is described by d ðxðtÞ xðtÞÞ ¼ ðA þ LCÞðxðtÞ xðtÞÞ þ f ðxðtÞÞ f ðxðtÞÞ; dt whose stability can be ensured by the following Lyapunov function V1 ðxðtÞ; xðtÞÞ ¼ ðxðtÞ xðtÞÞT P ðxðtÞ xðtÞÞ with matrix P satisfying Ineq. (19). Denoting X ¼ PL, Ineq. (19) becomes AT P þ CX T þ PA þ XC T þ r2 PP þ In < 2dP ; which is equivalent to T A P þ CX T þ PA þ XC T þ In þ 2dP P
P r12 In
ð20Þ
< 0:
ð21Þ
According to the method proposed in Ref. [6], we can derive the matrix solution X , P ¼ P T > 0. Then L ¼ P 1 X . So controller (16) can be simplified into
uðtÞ ¼ uðt sÞ þ ðBT BÞ 1 BT ½LðyðtÞ yðt sÞÞ ; t P T ; ð22Þ uðtÞ ¼ 0; t < T : Remark 6. Controller (22) is a proportional repetitive leaning controller, which is the linear iteration on the difference between the output and the delayed output. Compared with the method proposed in Ref. [8], this controller can also effectively resolve the inherent limitation of the TDFC method in the sense of least mean square.
4. Numerical simulations In this section Chua and R€ ossler systems are illustrated to verify the effectiveness of the repetitive learning controller. In the following suppose that the controllers in the first and second simulations are activated since T ¼ 30 s and T ¼ 40 s, respectively.
166
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
Example 1. Chua system The dynamics of Chua system is described by Eq. (4), in which parameters are chosen as a ¼ 10, b ¼ 14:87, a ¼ 1:27 and b ¼ 0:68. In this simulation an initial condition is chosen as, ½0:2; 0:1; 0:2 T and the gain matrix is B ¼ ½ 1 1 1 T . Assume that the maximum control value is e ¼ 0:6 and the UPO to be stabilized is s ¼ 1 periodic. Because ðA; CÞ is observable, matrix L is chosen as ½ 2; 1:313; 9:496 . Therefore the repetitive learning controller is
uðtÞ ¼ uðt 1Þ 13 ½6:183ðyðtÞ yðt 1ÞÞ þ aðf ðyðtÞÞ f ðyðt 1ÞÞÞ ; t P 30; ð23Þ uðtÞ ¼ 0; t < 30: The orbit of the controlled Chua system in three-dimensional space is plotted in Fig. 1. In order to show the effectiveness of controller (23), the relation between the iteration number N and the total error during each cycle Z 30þðN þ1Þ jyðtÞ yðt 1Þj dt ðN ¼ 1; 2; . . .Þ eN ¼ 30þN
is shown by Fig. 2. From this figure, we know that the error decreases to zero after several iterations. Therefore the oneperiodic UPO is stabilized. Example 2. R€ ossler system The dynamics of R€ ossler system is given by Eq. (5), which can be transformed into Eq. (6) via a suitable transformation. In this simulation a ¼ 0:2, b ¼ 5:7, c ¼ 0:2, and e ¼ 6. An initial condition is chosen as ð2; 2; 2ÞT and B ¼ ½ 1 1 1 T . If matrix L is ½ 12:24 4:248 6:2 T , the repetitive learning controller for Eq. (6) is constructed as ( h i yðtÞ 1 1 þ 0:2 yðt 5Þ þ ðyðtÞ yðt 5ÞÞ ; t P 40; yðtÞ uðtÞ ¼ uðt 1Þ þ 13 14:192 ln yðt 5Þ ð24Þ uðtÞ ¼ 0:5; t < 40: The histories of three states of the controlled R€ ossler system are shown by Fig. 3. The relation between the iteration number N and the error during each cycle Z 40þðN þ5Þ jyðtÞ yðt 5Þj dt ðN ¼ 1; 2; . . .Þ eN ¼ 40þN
is plotted in Fig. 4. From these figures the controlled R€ ossler system is stabilized at the T ¼ 5 periodic orbit.
x1
5
0
-5 0
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
1
x2
0.5 0 -0.5 -1 0
x3
5
0
-5 0
t
Fig. 1. The orbits of three states of the controlled Chua system.
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
167
8
7
6
error
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
N
Fig. 2. The relation between iteration number N and the error eN .
20
x1
10 0 -10
0
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
10
x2
0 -10 -20
0
30
x3
20 10 0 -10
0
t
Fig. 3. The orbits of three states of the controlled R€ ossler system.
168
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169 60
50
40
error
30
20
10
0
-10
1
2
3
4
5
6
7
N
Fig. 4. The relation between iteration number N and the error eN .
5. Conclusion Using the repetitive learning strategy and a shift method, this paper proposes a repetitive learning controller to stabilize UPOs within chaotic attractors in the sense of least mean square. If nonlinear parts in chaotic systems satisfy Lipschitz condition, the proposed controller can be simplified into a simple proportional repetitive learning controller.
Acknowledgements Prof. Zhou thanks the main support by the NSFC (Grant No. 60025307, 60234010), partial support by the national 863 program, RFDP (Grant No. 20020003063) and the national 973 program (Grant No. 2002CB312200) of China.
References [1] Tsai M, Anwar G, Tomizka M. Discrete time repetitive control for robot manipulators. In: Proc 1988 IEEE Int Conf Robot Automation, Philadelphia, PA, vol. 3. 1988. p. 341–7. [2] Voss HU. A backward time shift filter for nonlinear delayed-feedback systems. Phys Lett A 2001;279:207–14. [3] Konishi K, Kokame H. Observer-based delayed-feedback control for discrete-time chaotic systems. Phys Lett A 1998;248:359–68. [4] Liao TL, Huang NS. An observer-based approach for chaotic synchronization with applications to secure communications. IEEE Trans Circuits Syst I 1999;46:1144–50. [5] Nijmeijer H, Mareels IMY. An observer looks at synchronization. IEEE Trans Circuits Syst I 1997;44:882–9. [6] Song Y, Yu X, Chen G, Tian Y. Time delayed repetitive learning control for chaotic systems. Int J Bifur Chaos 2002;12:1057–65. [7] Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. [8] Paragas K. Continuous control of chaos by self-controlling feedback. Phys Lett A 1992;170:421–8. [9] Yagasaki K, Yozumi T. A new approach for controlling chaotic dynamical systems. Phys Lett A 1998;238:349–57. [10] Yu X, Chen G, Yang X, et al. An invariant-manifold-based method for chaos control. IEEE Trans Circuits Syst I 2001;48:930–7. [11] Ushio T. Limitation of delayed feedback control in nonlinear discrete-time systems. IEEE Trans Circuits Syst I 1996;43:815–6. [12] Nakajima H. On analytical properties of delayed feedback control. Phys Lett A 1997;232:207–10.
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
169
[13] Just W, Reibold E, Benner H, et al. Limits of time-delayed feedback control. Phys Lett A 1999;254:158–64. [14] Yamamoto S, Hino T, Ushio T. Dynamic delayed feedback controllers for chaotic discrete-time systems. IEEE Trans Circuits Syst I 2001;48:785–9. [15] Tian Y. An optimization approach to locating and stabilizing unstable periodic orbits of chaotic systems. Int J Bifur Chaos 2002;12:1163–72. [16] Nakajima H. Delayed feedback control with state predictor for continuous-time chaotic systems. Int J Bifur Chaos 2002;12:1067– 77. [17] Hino T, Yamamoto Y, Ushio T. Stabilization of unstable periodic orbits of chaotic discrete-time systems using prediction-based feedback control. Int J Bifur Chaos 2002;12:436–9. [18] Kittel A, Parisi J, Pyragas K. Delayed feedback control of chaos by self-adapted delay time. Phys Lett A 1995;198:433–6. [19] Jang TJ, Ahn HS, Choi CH. Iterative learning control for discrete-time nonlinear systems. Int J Syst Sci 1994;25:1179–89. [20] Chen CT. Linear system theory and design. New York: Holt, Rinehart and Winston; 1984. [21] Feki M. An adaptive feedback control of linearizable chaotic systems. Chaos, Solitons & Fractals 2003;15:883–90. [22] Yu YG, Zhang SC. Controlling uncertain L€ u system using backstepping design. Chaos, Solitons & Fractals 2003;15:897–902.
Repetitive learning control of continuous chaotic systems Maoyin Chen b
a,*
, Yun Shang b, Donghua Zhou
a
a Department of Automation, Tsinghua University, Beijing 100084, PR China College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710068, PR China
Accepted 29 December 2003
Abstract Combining a shift method and the repetitive learning strategy, a repetitive learning controller is proposed to stabilize unstable periodic orbits (UPOs) within chaotic attractors in the sense of least mean square. If nonlinear parts in chaotic systems satisfy Lipschitz condition, the proposed controller can be simplified into a simple proportional repetitive learning controller. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Recently considerable effort has been made to the control of chaotic systems, including the stabilization of the UPOs within chaotic attractors [3–18,21,22]. Among many methods, the time-delayed feedback control (TDFC) method is an effective approach to stabilize the UPOs. This method utilizes the difference between the states and the delayed states as an input control provided that the delayed time is determined as the period of the UPO to be stabilized. Unfortunately, the TDFC method has its inherent limitation. When analyzing the stability of the TDFC controlled system, Ushio found out that the TDFC method has a property, called by odd number characteristic exponent property [11]. If a UPO has odd number real eigenvalues greater than one, it cannot be stabilized by the TDFC method. This property holds for all kinds of TDFC controllers, including exponential and extended TDFC controllers [11–13]. Since this property is a main limitation for the application of the TDFC method, overcoming this limitation has been a topic in chaos control field [3,6,10,14–18]. Except that static and dynamic feedback methods can effectively resolve this limitation [14,18], the optimal principle is also a useful tool to resolve this limitation [15]. In addition, the integration of the repetitive learning principle and the time-delayed chaos control technique can effectively stabilize the UPOs of chaotic systems [6]. This paper considers the stabilization of the UPOs of continuous chaotic systems. Using a shift method and the repetitive learning strategy, a repetitive learning controller is proposed. First, we extend a shift method (Ref. [2]) to arbitrary time-delayed chaotic system. Second, a repetitive learning controller is designed based on the repetitive learning strategy and the least mean square principle. This controller can effectively stabilize UPOs in the sense of least mean square. At last, the proposed controller can be simplified into a simple proportional repetitive learning controller if nonlinear parts in chaotic systems satisfy Lipschitz condition.
*
Corresponding author. E-mail address: [email protected] (M. Chen).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.01.008
162
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
2. Problem description Consider the following controlled chaotic system x_ ðtÞ ¼ AxðtÞ þ f ðyðtÞÞ þ BuðtÞ;
ð1Þ
yðtÞ ¼ CxðtÞ;
where x 2 Rn1 is the state vector, y 2 Rp1 is the output, uðtÞ 2 Rm1 is the control vector, and f ðÞ is a nonlinear function vector on the output. Moreover, A 2 Rnn , C 2 Rpn and the gain B 2 Rnm is column-full rank. When there exists no control input, Eq. (1) behaves chaotically. In this paper our aim is to stabilize the UPOs embedded in the chaotic attractor of Eq. (1). In order to keep the ergodicity of chaos motion, control input ui ðtÞ ði ¼ 1; 2; . . . ; mÞ are limited by the same saturated law, i.e., 8 ui ðtÞ > e; < e; sate ðui ðtÞÞ ¼ ui ðtÞ; jui ðtÞj 6 e; ð2Þ : e; ui ðtÞ < e in which e > 0 stands for the maximum amplitude of control action ui ðtÞ. Hence Eq. (1) becomes x_ ðtÞ ¼ AxðtÞ þ f ðyðtÞÞ þ B sate ðuðtÞÞ; yðtÞ ¼ CxðtÞ:
ð3Þ
In general many chaotic systems have the same form as Eq. (1), including Chua system 2 3 2 32 3 2 3 x_ 1 a a 0 x1 af ðyÞ 4 x_ 2 5 ¼ 4 1 1 1 54 x2 5 þ 4 0 5; x_ 3 0 b 0 0 x3 where yðtÞ ¼ ½ 1 0 2 3 2 x_ 1 0 6 7 6 4 x_ 2 5 ¼ 4 1 x_ 3 0
0 xðtÞ and f ðyÞ ¼ by þ 1=2ða bÞðjy þ 1j jy 1jÞ. In addition, R€ ossler system 32 3 2 3 x1 1 1 0 76 7 6 7 a 0 54 x2 5 þ 4 0 5; x3 0 b c þ x1 x3
ð4Þ
ð5Þ
y ¼ ½ 0 0 1 x can be transformed 2 3 2 0 n_ 1 6_ 7 6 4 n2 5 ¼ 4 1 1 n_ 3
into 1 a 0
32 3 2 3 n1 0 en3 76 7 6 7 0 54 n2 5 þ 4 0 5; n3 0 b þ ce n3
ð6Þ
g ¼ ½ 0 0 1 n via the transformation ½ n1
n2
n3 ¼ ½ x1
x2
lnðx3 Þ and g ¼ lnðyÞ.
3. Controller design 3.1. A shift method of chaotic systems with delay Consider a controlled chaotic system with a s-time-delayed saturated control input, given by x_ ðtÞ ¼ AxðtÞ þ f ðxðt sÞÞ þ B sate ðuðt sÞÞ:
ð7Þ
Here it is worth noting that matrix A is not necessarily stable. When there exists no control input in Eq. (2), a shift method is proposed to predict the states of Eq. (7) [2]. A nonlinear filter can shift arbitrarily complex wave produced by a chaotic system with a delayed feedback backwards in time. However, this method can be only applied to the system with a stable matrix. Here it is generalized to arbitrary time-delayed system with a delayed control input, described by Eq. (7). The predictor is constructed as
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
x_ ðtÞ ¼ ðA þ LÞxðtÞ þ f ðxðt s0 ÞÞ Lxðt s0 Þ þ Buðt s0 Þ
163
ð8Þ
with a suitable matrix L and a delayed time s0 , not necessarily equivalent to s. The time error and the state error are defined as s00 ¼ s s0
and
es00 ðtÞ ¼ xðtÞ xðt s00 Þ;
respectively. In the linear saturated area, the error system on the error es00 ðtÞ is e_ s00 ðtÞ ¼ AxðtÞ þ f ðxðt sÞÞ þ Buðt sÞ ðA þ LÞxðt s00 Þ f ðxðt s00 s0 ÞÞ þ Lxðt s00 s0 Þ Buðt s00 s0 Þ ¼ ðA þ LÞed 00 ðtÞ:
ð9Þ
Whether matrix A is stable or not, we can choose a matrix L to make Eq. (9) be stable. If matrix L is chosen as L ¼ A þ kIn ðk < 0Þ, Eq. (9) can be exponentially stable. So lim ðxðtÞ xðt s00 ÞÞ ¼ lim ðxðtÞ xðt s þ s0 ÞÞ ¼ 0
t!1
t!1
ð10Þ
for arbitrary constant s0 > 0. Therefore, in the linear saturated area, we conclude that Case 1: When s0 > s, the limit limt!1 ðxðtÞ xðt þ ðs0 sÞÞ ¼ 0 holds, which means that Eq. (8) can predict the past states of Eq. (7). Case 2: When s0 < s, the limit limt!1 ðxðtÞ xðt ðs s0 ÞÞ ¼ 0 shows that Eq. (8) is a future state predictor of Eq. (7). Case 3: When s0 ¼ s, the limit limt!1 ðxðtÞ xðtÞÞ ¼ 0 implies Eq. (8) is a current state predictor of Eq. (7). Remark 1. Matrix L can be chosen according to the well-known pole-placement algorithm [20]. Choosing a matrix D 2 Rnl ðl < nÞ such that the pair ðA; DÞ is controllable, and calculating the matrix E 2 Rln to make all the eigenvalues of A þ DE be in left open complex plane, hence L ¼ DE. 3.2. Design of the repetitive learning controller In Ref. [6] a time-delayed controller is proposed to stabilize the UPOs based on the invariant manifold of the chaotic systems. The integration of the repetitive learning strategy and the time-delayed chaos control technique can learn appropriate control action from each learning cycles. In this section, we also apply the repetitive learning strategy to stabilize the UPOs. For Eq. (3), a state predictor is given by x_ ðtÞ ¼ ðA þ LCÞxðtÞ þ f ðyðt sÞÞ Lyðt sÞ þ Buðt sÞ; ð11Þ where matrix L can be chosen such that A þ LC is stable [20]. From Section 3.1, we have lim ðxðtÞ xðt þ sÞÞ ¼ 0;
t!1
equivalently, lim ðxðt sÞ xðtÞÞ ¼ 0
t!1
which means that the state xðtÞ of Eq. (11) is the estimation value of the delayed state xðt sÞ of Eq. (3). Therefore once the state xðtÞ tends to the state xðtÞ asymptotically, the s-periodic UPO can be stabilized in the linear saturated area. Now we design a repetitive learning controller to make xðtÞ approach xðtÞ. Let eðtÞ ¼ xðtÞ xðtÞ: In the linear saturated area, the error system is expressed as e_ ðtÞ ¼ AxðtÞ þ f ðyðtÞÞ þ BuðtÞ ðA þ LCÞxðt sÞ f ðyðt sÞÞ þ Lyðt sÞ Buðt sÞ ¼ ðA þ LCÞðxðtÞ xðtÞÞ þ f ðyðtÞÞ f ðyðt sÞÞ þ Lðyðt sÞ yðtÞÞ þ BðuðtÞ uðt sÞÞ: In the right hand side of Eq. (12), if the term f ðyðtÞÞ f ðyðt sÞÞ þ Lðyðt sÞ yðtÞÞ þ BðuðtÞ uðt sÞÞ
ð12Þ
164
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
is zero, the limit lim ðxðtÞ xðtÞÞ ¼ 0
t!1
holds. Therefore, controller uðtÞ should satisfy f ðyðtÞÞ f ðyðt sÞÞ þ Lðyðt sÞ yðtÞÞ þ BðuðtÞ uðt sÞÞ ¼ 0:
ð13Þ
Denoting u ðtÞ ¼ uðtÞ uðt sÞ; Eq. (13) becomes f ðyðtÞÞ f ðyðt sÞÞ þ Lðyðt sÞ yðtÞÞ þ Bu ðtÞ ¼ 0:
ð14Þ
Since B is column-full rank, the least mean square solution of Eq. (14) is u ðtÞ ¼ ðBT BÞ 1 BT ½LðyðtÞ yðt sÞÞ þ f ðyðt sÞÞ f ðyðtÞÞ : Therefore
uðtÞ ¼ uðt sÞ þ ðBT BÞ 1 BT ½LðyðtÞ yðt sÞÞ þ f ðyðt sÞÞ f ðyðtÞÞ ; uðtÞ ¼ 0; t < T ;
ð15Þ
t PT;
ð16Þ
where T is sufficiently large such that the s-periodic orbit can be approximated more accurately. Remark 2. Controller (16) is a repetitive learning controller with a learning period s [1]. Compared with iterative learning method [19], this controller need not reset the initial condition during each learning cycle. During the interval ½ks; ðk þ 1Þs , we have d ðxðtÞ xðtÞÞ ¼ ðA þ LCÞðxðtÞ xðtÞÞ: dt So the initial error satisfies lim kxðksÞ xðksÞk ¼ 0:
k!1
Remark 3. Controller (16) is an optimal control law in the sense of least mean square, which makes the distance between the state xðtÞ and the delayed state xðt sÞ be little as soon as possible. If follows that uðkÞ ¼ uðk sÞ when the state xðtÞ approaches the delayed state xðt sÞ as time tends to infinity. Remark 4. Controller (16) does not embody information on Eq. (11), which is only useful to construct controller (16) and doesn’t join in the application of this controller. That is to say, Eq. (11) is an auxiliary system to design controller (16). Remark 5. It is well-known that TDFC method has its inherent limitation, called by odd number characteristic exponent property. In this paper, controller (16) can stabilize arbitrary UPO in the sense of least mean square. It means that controller (16) can overcome the inherent limitation of TDFC. 3.3. Simplification of the repetitive learning controller Suppose the nonlinear part f ðÞ satisfies Lipschitz condition, i.e., kf ðx1 Þ f ðx2 Þk < rkx1 x2 k;
8x1;2 2 Rn ;
where r > 0 is a Lipschitz constant. In this case, the predictor on Eq. (3) is constructed as x_ ðtÞ ¼ ðA þ LCÞxðtÞ þ f ðy Þ Lyðt sÞ þ Buðt sÞ; y ðtÞ ¼ CxðtÞ: Eq. (17) can still estimate the state of Eq. (3) at t s instant in the linear area if matrix L is chosen suitably.
ð17Þ
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
165
Using Eqs. (3) and (17), it follows that d ðxðtÞ xðt þ sÞÞ ¼ ðA þ LCÞðxðtÞ xðt þ sÞÞ þ f ðxðtÞÞ f ðxðt þ sÞÞ: dt
ð18Þ
Let a Lyapunov function be V ðxðtÞ; xðt þ sÞÞ ¼ ðxðtÞ xðt þ sÞÞT P ðxðtÞ xðt þ sÞÞ; where matrix P ¼ P T > 0 satisfies the following Riccati inequality ðA þ LCÞT P þ P ðA þ LCÞ þ r2 PP þ In < 2dP
ð19Þ
with a positive constant d. Therefore V_ ðxðtÞ; xðt þ sÞÞ ¼ ðxðtÞ xðt þ sÞÞT ½ðA þ LCÞT P þ P ðA þ LCÞ þ 2½f ðxðtÞÞ f ðxðt þ sÞ T P ðxðtÞ xðt þ sÞÞ: Since 2½f ðxðtÞÞ f ðxðt þ sÞ T P ðxðtÞ xðt þ sÞÞ 6 ðxðtÞ xðt þ sÞÞT ðr2 PP þ In ÞðxðtÞ xðt þ sÞÞ; we have V_ ðxðtÞ; xðt þ sÞÞ ¼ ðxðtÞ xðt þ sÞÞT ½ðA þ LCÞT P þ P ðA þ LCÞ þ r2 PP þ In ðxðtÞ xðt þ sÞÞ < 2dðxðtÞ xðt þ sÞÞT P ðxðtÞ xðt þ sÞÞ 6 0: This means that Eq. (18) is asymptotically stable, that is, the state xðtÞ of Eq. (17) can estimate the delayed state xðt sÞ of Eq. (3). The dynamics on the error xðtÞ xðtÞ is described by d ðxðtÞ xðtÞÞ ¼ ðA þ LCÞðxðtÞ xðtÞÞ þ f ðxðtÞÞ f ðxðtÞÞ; dt whose stability can be ensured by the following Lyapunov function V1 ðxðtÞ; xðtÞÞ ¼ ðxðtÞ xðtÞÞT P ðxðtÞ xðtÞÞ with matrix P satisfying Ineq. (19). Denoting X ¼ PL, Ineq. (19) becomes AT P þ CX T þ PA þ XC T þ r2 PP þ In < 2dP ; which is equivalent to T A P þ CX T þ PA þ XC T þ In þ 2dP P
P r12 In
ð20Þ
< 0:
ð21Þ
According to the method proposed in Ref. [6], we can derive the matrix solution X , P ¼ P T > 0. Then L ¼ P 1 X . So controller (16) can be simplified into
uðtÞ ¼ uðt sÞ þ ðBT BÞ 1 BT ½LðyðtÞ yðt sÞÞ ; t P T ; ð22Þ uðtÞ ¼ 0; t < T : Remark 6. Controller (22) is a proportional repetitive leaning controller, which is the linear iteration on the difference between the output and the delayed output. Compared with the method proposed in Ref. [8], this controller can also effectively resolve the inherent limitation of the TDFC method in the sense of least mean square.
4. Numerical simulations In this section Chua and R€ ossler systems are illustrated to verify the effectiveness of the repetitive learning controller. In the following suppose that the controllers in the first and second simulations are activated since T ¼ 30 s and T ¼ 40 s, respectively.
166
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
Example 1. Chua system The dynamics of Chua system is described by Eq. (4), in which parameters are chosen as a ¼ 10, b ¼ 14:87, a ¼ 1:27 and b ¼ 0:68. In this simulation an initial condition is chosen as, ½0:2; 0:1; 0:2 T and the gain matrix is B ¼ ½ 1 1 1 T . Assume that the maximum control value is e ¼ 0:6 and the UPO to be stabilized is s ¼ 1 periodic. Because ðA; CÞ is observable, matrix L is chosen as ½ 2; 1:313; 9:496 . Therefore the repetitive learning controller is
uðtÞ ¼ uðt 1Þ 13 ½6:183ðyðtÞ yðt 1ÞÞ þ aðf ðyðtÞÞ f ðyðt 1ÞÞÞ ; t P 30; ð23Þ uðtÞ ¼ 0; t < 30: The orbit of the controlled Chua system in three-dimensional space is plotted in Fig. 1. In order to show the effectiveness of controller (23), the relation between the iteration number N and the total error during each cycle Z 30þðN þ1Þ jyðtÞ yðt 1Þj dt ðN ¼ 1; 2; . . .Þ eN ¼ 30þN
is shown by Fig. 2. From this figure, we know that the error decreases to zero after several iterations. Therefore the oneperiodic UPO is stabilized. Example 2. R€ ossler system The dynamics of R€ ossler system is given by Eq. (5), which can be transformed into Eq. (6) via a suitable transformation. In this simulation a ¼ 0:2, b ¼ 5:7, c ¼ 0:2, and e ¼ 6. An initial condition is chosen as ð2; 2; 2ÞT and B ¼ ½ 1 1 1 T . If matrix L is ½ 12:24 4:248 6:2 T , the repetitive learning controller for Eq. (6) is constructed as ( h i yðtÞ 1 1 þ 0:2 yðt 5Þ þ ðyðtÞ yðt 5ÞÞ ; t P 40; yðtÞ uðtÞ ¼ uðt 1Þ þ 13 14:192 ln yðt 5Þ ð24Þ uðtÞ ¼ 0:5; t < 40: The histories of three states of the controlled R€ ossler system are shown by Fig. 3. The relation between the iteration number N and the error during each cycle Z 40þðN þ5Þ jyðtÞ yðt 5Þj dt ðN ¼ 1; 2; . . .Þ eN ¼ 40þN
is plotted in Fig. 4. From these figures the controlled R€ ossler system is stabilized at the T ¼ 5 periodic orbit.
x1
5
0
-5 0
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
1
x2
0.5 0 -0.5 -1 0
x3
5
0
-5 0
t
Fig. 1. The orbits of three states of the controlled Chua system.
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
167
8
7
6
error
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
N
Fig. 2. The relation between iteration number N and the error eN .
20
x1
10 0 -10
0
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
10
20
30
40
50
60
70
80
10
x2
0 -10 -20
0
30
x3
20 10 0 -10
0
t
Fig. 3. The orbits of three states of the controlled R€ ossler system.
168
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169 60
50
40
error
30
20
10
0
-10
1
2
3
4
5
6
7
N
Fig. 4. The relation between iteration number N and the error eN .
5. Conclusion Using the repetitive learning strategy and a shift method, this paper proposes a repetitive learning controller to stabilize UPOs within chaotic attractors in the sense of least mean square. If nonlinear parts in chaotic systems satisfy Lipschitz condition, the proposed controller can be simplified into a simple proportional repetitive learning controller.
Acknowledgements Prof. Zhou thanks the main support by the NSFC (Grant No. 60025307, 60234010), partial support by the national 863 program, RFDP (Grant No. 20020003063) and the national 973 program (Grant No. 2002CB312200) of China.
References [1] Tsai M, Anwar G, Tomizka M. Discrete time repetitive control for robot manipulators. In: Proc 1988 IEEE Int Conf Robot Automation, Philadelphia, PA, vol. 3. 1988. p. 341–7. [2] Voss HU. A backward time shift filter for nonlinear delayed-feedback systems. Phys Lett A 2001;279:207–14. [3] Konishi K, Kokame H. Observer-based delayed-feedback control for discrete-time chaotic systems. Phys Lett A 1998;248:359–68. [4] Liao TL, Huang NS. An observer-based approach for chaotic synchronization with applications to secure communications. IEEE Trans Circuits Syst I 1999;46:1144–50. [5] Nijmeijer H, Mareels IMY. An observer looks at synchronization. IEEE Trans Circuits Syst I 1997;44:882–9. [6] Song Y, Yu X, Chen G, Tian Y. Time delayed repetitive learning control for chaotic systems. Int J Bifur Chaos 2002;12:1057–65. [7] Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. [8] Paragas K. Continuous control of chaos by self-controlling feedback. Phys Lett A 1992;170:421–8. [9] Yagasaki K, Yozumi T. A new approach for controlling chaotic dynamical systems. Phys Lett A 1998;238:349–57. [10] Yu X, Chen G, Yang X, et al. An invariant-manifold-based method for chaos control. IEEE Trans Circuits Syst I 2001;48:930–7. [11] Ushio T. Limitation of delayed feedback control in nonlinear discrete-time systems. IEEE Trans Circuits Syst I 1996;43:815–6. [12] Nakajima H. On analytical properties of delayed feedback control. Phys Lett A 1997;232:207–10.
M. Chen et al. / Chaos, Solitons and Fractals 22 (2004) 161–169
169
[13] Just W, Reibold E, Benner H, et al. Limits of time-delayed feedback control. Phys Lett A 1999;254:158–64. [14] Yamamoto S, Hino T, Ushio T. Dynamic delayed feedback controllers for chaotic discrete-time systems. IEEE Trans Circuits Syst I 2001;48:785–9. [15] Tian Y. An optimization approach to locating and stabilizing unstable periodic orbits of chaotic systems. Int J Bifur Chaos 2002;12:1163–72. [16] Nakajima H. Delayed feedback control with state predictor for continuous-time chaotic systems. Int J Bifur Chaos 2002;12:1067– 77. [17] Hino T, Yamamoto Y, Ushio T. Stabilization of unstable periodic orbits of chaotic discrete-time systems using prediction-based feedback control. Int J Bifur Chaos 2002;12:436–9. [18] Kittel A, Parisi J, Pyragas K. Delayed feedback control of chaos by self-adapted delay time. Phys Lett A 1995;198:433–6. [19] Jang TJ, Ahn HS, Choi CH. Iterative learning control for discrete-time nonlinear systems. Int J Syst Sci 1994;25:1179–89. [20] Chen CT. Linear system theory and design. New York: Holt, Rinehart and Winston; 1984. [21] Feki M. An adaptive feedback control of linearizable chaotic systems. Chaos, Solitons & Fractals 2003;15:883–90. [22] Yu YG, Zhang SC. Controlling uncertain L€ u system using backstepping design. Chaos, Solitons & Fractals 2003;15:897–902.