Nonlinear Dynamics (2005) 40: 107–117
c Springer 2005
Stabilizing Unstable Periodic Orbits via Universal Input–Output Delayed-Feedback Control CHYUN-CHAU FUH1,∗ , HSUN-HENG TSAI2 , and PI-CHENG TUNG3 1 Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan, R.O.C.; 2 Department
of Biosystems Engineering, National Pingtung University of Science and Technology, Pingtung 91201, Taiwan, R.O.C.; 3 Department of Mechanical Engineering, National Central University, Chung-Li, 32054, Taiwan, R.O.C.; ∗Author for correspondence (e-mail:
[email protected]; fax: 886-2-24636061) (Received: 22 July 2004; accepted: 17 September 2004)
Abstract. Estimating the approximated linear state variable equation around an unstable periodic orbit (UPO) is difficult in most chaotic systems since the measurable signal is generally only in the form of a scalar output signal. Therefore, conventional state delayed feedback control (DFC) methods may be unfeasible in practice, particularly for high-dimensional systems. Consequently, this study proposes an input–output delayed-feedback control (IODFC) method. Initially, the approximated input–output description around the desired UPO is estimated using the delay coordinates. Subsequently, a (2n − 1)-dimensional controllable canonical state equation is reconstructed, in which the state vector consists of the delayed input and the delayed output. All the eigenvalues of the (2n − 1)-dimensional system can therefore be assigned to be inside the unit circle using the pole placement technique. The proposed method requires only that the system be both controllable and observable around the UPO. Key words: chaos, control, Poincar´e map, pole placement
1. Introduction Chaos is an interesting phenomenon which occurs only in a certain class of nonlinear dynamic systems. In some engineering applications, chaos is regarded as an undesirable behavior since it can cause performance degradation and can restrict the operating range of dynamic systems. Therefore, developing strategies for controlling chaos, based on the features of chaotic motion, is highly important. Chaotic systems share a common characteristic, namely the presence of strange attractors, in which dense unstable periodic orbits (UPOs) are embedded. Therefore, it is possible to select an appropriate unstable periodic orbit and to then stabilize the desired periodic orbit using a near zero steady control force (or a small parameter perturbation). Ott, Grebogi, and York (OGY) proposed just such a strategy for controlling chaos in 1990 [1]. Over the last decade, many theoretical and experimental approaches for controlling chaos have been developed [2, 3]. In 1992, Pyragas [4] proposed an alternate approach for controlling chaos, referred to as delayed feedback control (DFC). In this approach, the control signal was established according to the difference between the T-time delayed state and the current state, where T denotes the period of the desired orbit. Chen and Yu [5] derived the sufficient conditions for the stabilization and tracking problems of DFC systems. Although the DFC method is very simple, it has been demonstrated that it suffers an important limitation [6–9]. Specifically, if the Jacobian matrix of the linearized system around the UPO has an odd number of real eigenvalues greater than unity, then the DFC cannot stabilize the UPO. Contributed
by Prof. S.W. Shaw.
108 C.-C. Fuh et al. Many modified or extended DFC methods have been developed to overcome this limitation [10–15]. Ushio and Yamamoto [10] proposed a DFC method involving the nonlinear estimation of stabilized orbits, and provided a synthesis method for the feedback gain using linear matrix inequalities and robust control theory. Konishi and Kokame [11] presented an observer-based DFC approach which also successfully overcame the inherent failing of the DFC method. Yamamoto et al. [12] proposed a dynamic DFC method whose dimension was no greater than that of the chaotic system. Yamamoto et al. [13] proposed the use of a recursive DFC method to stabilize the UPOs of discrete-time chaotic systems. The recursive DFC can be designed without the requirement to solve any non-convex optimization problem the dynamic DFC may have. Although the modified DFC methods presented above can overcome the odd number limitation of the conventional DFC approach, their feedback gains are dependent on the Jacobian matrices around the desired UPO. This implies that state space parameter identification is necessary. However, the states of practical chaotic systems are generally inaccessible and the measurable signal may only be a scalar output signal. Therefore, state DFC methods may be unfeasible in practice, particularly for the case of high-dimensional systems. The extended delayed-feedback control (EDFC) method has been studied extensively in recent years [14–19]. This method utilizes many previous states of the system in a form closely related to the amplitude of light reflected from a Fabry-P´erot interferometer. However, the EDFC method also has an inherent weakness, namely controlling a single input high-dimensional chaotic system is problematic since only the scalar gain, K, or the parameter, r, can be tuned. This paper proposes an input–output delayed-feedback control (IODFC) method. The method assumes that the exact dynamic equations of the chaotic system are unknown and that only one scalar output signal is measurable. The delay coordinates [20] are utilized to reconstruct the strange attractor. Initially, the approximate input–output difference equation around the desired UPO is estimated using the parameter identification technique. Subsequently, the approximation is represented by a (2n − 1)dimensional controllable canonical state equation, in which the state vector consists of the delayed input and delayed output. Hence, the pole placement technique [21] can be employed to assign all the eigenvalues of the (2n − 1)-dimensional system to be inside the unit circle. The proposed method does not limit the location of the eigenvalues of the linearized system around the UPO, but simply requires that the system be both controllable and observable. Controllability is a prerequisite for controlling chaos no matter which controller is used, while observability implies that the strange attractor can be reconstructed from the delay coordinates. The IODFC method developed in this study is not only applicable to single-input/single-output (SISO) systems, but can also be extended to the case of multi-input/multi-output (MIMO) high-dimensional systems.
2. Input–Output Delayed-Feedback Control Without any loss of generality, consider the case of a SISO nonlinear map described by the following difference equations: x(k + 1) = f(x(k), p(k)),
(1a)
y(k) = h(x(k)),
(1b)
where the state vector x(k) ∈ Rn is inaccessible; the scalar output signal y(k) ∈ R is measurable; p is an accessible parameter which can be externally perturbed; and functions f and h are continuously
Stabilizing Unstable Periodic Orbits
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¯ The small parameter differentiable, at least locally in the neighborhood of the fixed point (x, p) = (x, ¯ p). ¯ perturbation (input) near p¯ is defined as u(k) = p(k) − p. The tracking error is defined as: ε(k) = x(k) − x, ¯
(2)
e(k) = y(k) − y¯ = h(x(k)) − h(x). ¯
(3)
The linear approximation of the above system around the fixed point (or UPO) is given by: ε(k + 1) = Aε(k) + bu(k), e(k) = cε(k),
(4a) (4b)
where:
∂f A= , ∂x x=x¯ p= p¯ ∂f b= , ∂ p x = x¯ p= p¯ ∂h T c = . ∂x x = x¯
(5) (6) (7)
p = p¯
It is noted that the parameter matrices A, b, and cT cannot be estimated since the functions f and h are unknown, and the state vector ε (or x) is assumed to be inaccessible. It is assumed that the system is controllable and observable, i.e., the controllability matrix, U = [b
Ab
···
An−1 b ],
and the observability matrix, V = [ cT
AT cT
···
(An−1 )T cT ],
both have the rank n. If the dynamic system of Equation (4) is controllable, the feedback u(k) = kε(k), where k is a 1 × n real constant vector, permits the eigenvalues of A + bk to be arbitrarily assigned, provided that the complex conjugate eigenvalues are assigned in pairs. However, in most experimental situations, the exact nonlinear dynamic equations of a chaotic system are generally unknown. Furthermore, the only measurable signal may be a scalar output signal, e.g., y(·). In this case, delay coordinates [20] can be employed to represent the evolution of the system and to extract the quantities necessary for its control. The delay coordinates used to reconstruct the chaotic attractors can be classified into two types, namely those which are based on the continuous output signal (flow) denoted by y(t), and those which are based on the discrete output signal (Poincar´e map) denoted by y(k). Economizing memory is a fundamental advantage of using Poincar´e maps. For example, if the Poincar´e map is used, a single state (or delayed-output) vector is sufficient to save an unstable period-1 orbit embedded in the strange attractor. Conversely, if the continuous output signal (flow) is employed, it is necessary to store many state (or delayed-output) vectors to ensure that the reconstructed trajectory is sufficiently smooth. Consequently, the current analysis and design of controllers is based on discrete-time dynamical systems (Poincar´e map).
110 C.-C. Fuh et al. Since the tracking error, e(k), defined by Equation (3), and the small parameter perturbation, u(k), are available, the parameter identification technique can be used to estimate the following nominal transfer function: E(z) = cT (zIn − A)−1 b U (z) β1 z −1 + β2 z −2 + · · · + βn z −n = . 1 + α1 z −1 + α2 z −2 + · · · + αn z −n
G(z) =
(8)
It is noted that the denominator and the numerator share no common factor since the assumption has been made that the system is both controllable and observable. Equation (8) yields the following input–output difference equation: e(k) =
n
(−αi e(k − i) + βi u(k − i)) + v(k − 1),
(9)
i=1
where v(·) denotes the effect of measurement noise and process noise. The purpose of this study is to design a feedback controller, consisting of the unique measurable signal series y(k) (or e(k)), to stabilize the fixed point x¯ (i.e. such that the tracking error, e(k), converges to zero). Initially, the delayed-output vector is defined in n-dimensional delay coordinates as follows: e(k) = [ e(k) e(k − 1) · · · e(k − n + 1) ]T ,
(10)
where k denotes the sampling instant and is an integer in the range −∞ to +∞. Assume that enough input–output data around the fixed point have been collected. We then define the following matrices: −e(k1 − 1) · · · −e(k1 − n) u(k1 − 1) · · · u(k1 − n) .. .. .. .. .. .. X= (11) . . . . . . −e(km − 1) · · · −e(km − n) u(km − 1) · · · u(km − n) z = [ e(k1 − 1) · · · e(km − 1) ]T θ = [ α1
· · · αn
β1
· · · βn ]
(12) T
(13)
where m n and ki are arbitrary integers. Using the least-squares estimation, the optimal estimate of θ can be solved as: θˆ = (XT X)−1 XT z.
(14)
It is worth noting that the noise does not only affect the accuracy of parameter identification but reduces the control performance. In general, the variance of θˆ increases as noise level increases; most control algorithm can not stabilize a dynamic system with too large noise. However, even an enough accuracy model has been estimated and the noise level is small, it may be impossible to use the feedback u(k) = ke(k) directly to stabilize Equations (4) or (9). For example, consider a 2-dimensional system with the following approximation: e(k) + 5e(k − 1) + e(k − 2) = u(k − 2).
(15)
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If an output delayed-feedback controller (ODFC) is designed with the form: u(k) = ke(k) = k1 e(k) + k2 e(k − 1),
(16)
the control signal, u(k), is accessible since the entries of vector e(k) are dependent only on the measurable output signal, y(·). Substituting the control law given in Equation (16) into Equation (15) yields the following characteristic equation: z 3 + 5z 2 + (1 − k1 )z − k2 = 0.
(17)
It is noted that the open-loop system of Equation (15) is a 2-dimensional difference equation. However, the closed-loop system, with a delayed output controller, becomes a 3-dimensional system. In this example, it is obvious that irrespective of the manner in which the feedback gains, k1 and k2 , are selected, it is impossible to assign all the eigenvalues within the unit circle. Clearly, the origin cannot be stabilized. In fact, using delayed-output feedback in a discrete system is analogous to using an integrator in a continuous system. Using integrators increases the dimensions of the closed-loop system. For example, the dimensions of a discrete system will increase at least 3 if a delayed signal of e(k − 3) is used in the feedback controller. This paper considers a more general case and defines a (2n − 1)-dimensional augmented vector (or input–output vector) as follows: w(k) = [ w1 (k) · · · wn (k) | wn+1 (k) · · · w2n−1 (k) ]T ≡ [ e(k − n + 1)
···
e(k) | u(k − n + 1) · · · u(k − 1) ]T .
(18)
It is noted that the entries of the input–output vector w(k) are accessible. Using the augmented vector, Equations (4) or (9) can be reconstructed in the following form: ˜ ˜ ˜ w(k + 1) = Aw(k) + bu(k) + dv(k) ˜
˜
˜
A11 A˜ 12 b1 d1 = w(k) + u(k) + v(k), ˜ ˜ ˜ A21 A22 b2 d˜ 2 e(k) = cw(k) ˜ = 1 01×(2n−2) w(k),
(19a) (19b)
where:
0
0 ˜ A11 = ... 0 −αn
0
0 ˜ A12 = ... 0 βn
1
0
··· 0
0
0 .. . 0
1 .. . 0
··· .. . ···
0 .. . 1
−αn−1
−αn−2
···
0 .. . 0
0
0
··· 0
0 .. .
0 .. .
··· .. .
0
0
··· 0
−α2 0 0 .. . 0
βn−1
βn−2
· · · β3
β2
0 .. .
−α1
n×(n−1)
,
,
(20)
n×n
(21)
112 C.-C. Fuh et al. A˜ 21 = 0(n−1)×n , 0 1 0 0 A˜ 22 = ... ... 0 0
b˜ 1 = b˜ 2 = d˜ 1 =
0
0 ··· 0
0
(22)
0 0 .. .. . . 0 ··· 0 1
1 ··· .. . . . .
0 ··· 0
0
0(n−1)×1 β1 0(n−2)×1 1 0(n−1)×1
0
,
(23)
(n−1)×(n−1)
,
(24)
,
(25)
,
(26)
1
d˜ 2 = 0(n−1)×1 .
(27)
˜ Since Equation (19) is represented by a controllable canonical form, the feedback, u(k) = kw(k), can be applied to assign all of its eigenvalues to arbitrary locations. This study proposes placing all the eigenvalues of Equation (19) at zero. According to the pole placement technique [21], the feedback gain for assigning all eigenvalues to zero is given by: k˜ = [ a2n−1
a2n−2
· · · a1 ](UM)−1 ,
(28)
where the matrices, U and M, are defined as: U = [ b˜
˜ b˜ A
2 A˜ b˜
n−1 · · · A˜ b˜ ],
(29)
and
a2n−2
a 2n−3 M = ... a1 1
a2n−3
· · · a1
a2n−4 .. .
··· .. .
1 .. .
1
···
0
0 .. . . 0
0
···
0
0
1
(30)
˜ Here, a1 , a2 , . . ., a2n−1 are the coefficients of the characteristic polynomial of A, ˜ = z 2n−1 + a1 z 2n−2 + · · · + a2n−1 . det(zI − A)
(31)
Using the feedback gain defined by Equation (28), Equation (19) becomes: ˜ ˜ w(k + 1) = (A˜ + b˜ k)w(k) + dv(k),
(32)
Stabilizing Unstable Periodic Orbits
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˜ has the eigenvalue 0, with multiplicity 2n − 1. Since the where the resulting matrix, A¯ = A˜ + b˜ k, characteristic polynomial of A¯ is given by: (λ) = λ2n−1
(33)
and r
w(k + r ) = A¯ w(k),
(34)
the Cayley–Hamilton theorem implies that: ¯ r = 0, A
∀r ≥ 2n − 1.
(35)
Hence, the trajectory will converge to the desired fixed point following a maximum of 2n − 1 iterations if the real trajectory of the chaotic system falls in the neighborhood of the desired fixed point and v(k) = 0. This result is very different from that of the OGY method, in which the convergence of the tracking error may be very slow since the orbit approaches the fixed point at a geometrical rate, λs .
3. Numerical Simulation Example 1: The following H´enon map is considered: x1 (k + 1) = − p(k)x12 (k) + x2 (k) + 1, x2 (k + 1) = q x1 (k), y(k) = x1 (k), where p(k) = p¯ + u(k) and u(k) is the control signal. The corresponding parameters are p¯ = 1.4 and q = 0.3. With such parameters, an uncontrolled system will exhibit a chaotic attractor. The principal characteristic of a chaotic system is the presence of a dense set of unstable periodic orbits embedded in a strange attractor. It is assumed that the exact nonlinear dynamic equation of the chaotic system is unknown and that only the output signal y(k) is measurable. In the current example, the period-8, period1, and period-2 orbits are successively stabilized. Specifically, in the range k = 1–3000, the proposed method is applied to stabilize period-8, from 3001–6000, to stabilize period-1, and from 6001–9000, to ¯ The transfer functions of the linear stabilize period-2. The tracking error is defined as e(k) = y(k) − y. approximations of the chaotic system near period-8, period-1, and period-2 are estimated, respectively, as: E(z) = U (z) E(z) Period−1: = U (z) E(z) Period−2: = U (z)
Period−8:
15.5824z −1 + 0.8234z −2 , 1 + 33.6301z −1 − 0.0010z −2 −0.3986z −1 , 1 + 1.7678z −1 − 0.3000z −2 −1.4949z −1 + 0.4086z −2 . 1 + 3.0400z −1 − 0.0900z −2
114 C.-C. Fuh et al.
Figure 1. Simulation results for the H´enon map, where the control signal varies in the range |u(k)| < 0.2. From k = 1–3000, the proposed method is applied to stabilize period-8, from 3001–6000, to stabilize period-1, from 6001–9000, to stabilize period-2. (a) Time response of y(k); (b) Control signal u(k); (c)–(e) Stabilized unstable periodic orbits embedded in the strange attractor.
According to the proposed method, the delayed input–output vector is taken to be: w(k) = [ e(k − 1)
e(k)
u(k − 1) ]T .
The gains of the IODFC are designed as: Period =8: k˜ = [ −0.0001 2.1616 −0.0529 ], Period =1: k˜ = [ 0.7526 −4.4349 0 ], Period =2: k˜ = [ −0.0557 −1.8644 0.2528 ]. The corresponding result is shown in Figure 1, in which the control signal is varied in the range of |u(k)| < 0.2 and the controller is turned on only when e(k)2 < 0.25. Example 2: Since physical systems are generally continuous in nature, they can be described by differential equations. The feasibility of the method proposed in this study can be verified by considering the following Duffing equation: x˙ 1 = x2 , x˙ 2 = x1 − x13 − 0.25x2 + 0.3 sin(ωt) + u, y = x1 ,
Stabilizing Unstable Periodic Orbits
115
where ω = 1 and u is the control signal. It is assumed that the exact nonlinear dynamic equation of the chaotic system is unknown, but that the period of the external force is known and the output signal, y(t), is measurable. The series y(k) denotes a sampling of the output signal every 2π /ω seconds (i.e., the period of the external force). Since the proposed controller is designed based on the Poincar´e map, the control input is varied for each map (i.e., 2π /ω seconds). In the current example, the period-3, period-1, and period-5 orbits are successively stabilized. In the range k = 1–3000, the proposed method is applied to stabilize period-3, from 3001–6000, to stabilize period-1, and from 6001–9000, to stabilize period-5. The transfer functions of the linear approximation of the chaotic system near period-3, period-1, and period-5 are estimated, respectively, as: E(z) = U (z) E(z) Period =1: = U (z) E(z) Period =5: = U (z) Period =3:
−85.8679z −1 − 56.6970z −2 , 1 + 117.8560z −1 + 0.0054z −2 −13.7114z −1 + 13.3490z −2 , 1 + 7.9994z −1 + 0.2016z −2 180.8079z −1 − 4.9266z −2 . 1 − 43.6002z −1 + 0.4980z −2
The augmented vector is of the form: w(k) = [ e(k − 1) e(k) u(k − 1) ]T .
Figure 2. Simulation result for the Duffing equation, where the control signal varies in the range |u(k)| < 0.2. From k = 1– 3000, the proposed method is applied to stabilize period-3, from 3001–6000, to stabilize period-1, from 6001–9000, to stabilize period-5. (a) Time response of y(k); (b) Control signal u(k); (c)–(e) Stabilized unstable periodic orbits embedded in the strange attractor.
116 C.-C. Fuh et al. The gains of the IODFC are as follows: Period =3: k˜ = [ −0.0001 −1.3803 −0.6640 ], Period =1: k˜ = [ −0.0131 −0.5199 0.8703 ], Period =5: k˜ = [ 0.0028 −0.2413 0.0273 ]. Figure 2 presents the simulation result, in which the control signal is restricted to vary only in the range of |u(k)| < 0.2, and the controller is turned on only when e(k)2 < 0.25.
4. Conclusion This paper has proposed an input–output delayed-feedback controller to stabilize UPOs embedded within a chaotic attractor. The proposed approach commences by estimating the approximated transfer function around the desired UPO using delay coordinates. A (2n−1)-dimensional controllable canonical state equation is then reconstructed, in which the state vector consists of the delayed input and the delayed output. Subsequently, the pole placement technique is applied to assign all the eigenvalues of the (2n − 1)-dimensional system within the unit circle. Since the proposed method requires only the input and output time series, it is applicable for practical physical systems, in which the full state is generally inaccessible and only a scalar output signal is measurable. The proposed method does not limit the location of the eigenvalues of the linearized model around the UPO, but simply requires that the system be controllable and observable. The former is a basic condition for controlling chaos, irrespective of the type of controller used. Meanwhile, the latter implies that the linearized model can be reconstructed by delay coordinates. Clearly, according to linear control theory, these two conditions can on occasions be relaxed to stabilizable and detectable, respectively. The proposed IODFC method is not only applicable to single-input/single-output (SISO) systems, but can also be extended to multiinput/multi-output (MIMO) high-dimensional systems.
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