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Model-reference control of chaotic systems Ahmet Uc¸ar

*

Department of Electrical and Electronics Engineering, Firat University, Elazıg 23119, Turkey Accepted 10 October 2005

Communicated by Prof. A. Helal

Abstract In this paper, the problem of controlling chaotic systems is studied. A control law is introduced for a chaotic system to follow a desired reference system. The control strategy is developed within the general framework of the nonlinear model-reference control systems. Lyapunov stability is used to ensure the global stability of the error dynamics represents the diﬀerence between the desired and chaotic systems. 2005 Elsevier Ltd. All rights reserved.

1. Introduction Chaos has been observed in many practical engineering and natural systems. A fundamental characteristic of the chaotic systems is their sensitivity to the sates initial conditions. The trajectories of the chaotic systems starting from any pair of arbitrary close positions will diverge exponentially and will become more and more uncorrelated with the time. This property leads to loss of information about the system states in the future. Therefore chaos is required to be removed to prevent catastrophic situations. Furthermore, controlling chaos, that is to convert chaotic oscillations into desired regular motion, is important for many engineering applications. Controlling chaos was ﬁrst introduced by Ott et al. [1] and since then it has been studied from various angles. Several control methods have been proposed and implemented. For the methodologies and reviews these methods (see [2–5] and the references therein). These methods can be considered into two categories: feedback and nonfeedback methods. In the feedback methods, the actual trajectory in the phase space of the system is monitored and some feedback processes are employed to force and to maintain the trajectory in the desired mode [2–11]. Hence the feedback methods do not change the chaotic systems. They stabilize unstable periodic orbits on chaotic attractors by control signal. However, in the nonfeedback methods, some system property or knowledge of the system is used to modify or exploit chaotic behavior [2–5,12–14]. This results to change the chaotic systems slightly, mainly by a small permanent to shift system parameters, in order to obtain desired behaviors. The aim of this paper is to revisit the concept of control of chaotic system via nonlinear state feedback in the framework of model-reference control systems. In the following section, the basic principles of model reference control systems are presented. In Section 3, model-reference control is used to stabilize chaotic behaviors of a chaotic system. *

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0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.040

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Numerical simulations are provided in Section 4 to illustrate and verify of the method. Finally, concluding remarks are given in Section 5.

2. Model-reference control of chaotic systems One useful method for specifying system performance for nonlinear and uncertain systems is by means of a model that will produce the desired output [15,16]. The model-reference control (MRC) objective is to develop a control strategy which forces the plant dynamics to follow the dynamic of an ideal model. The ideal model need not be actual hardware and can be only a mathematical model simulated on a computer. In the MRC system, the output of the model and that of the controlled system are compared and the error vector, e is used to generate the control signals. The MRC has been used to obtain acceptable performance in some very diﬃcult control problems involving the systems contain nonlinearity and/or time varying parameters [15,16]. Here MRC is used for controlling chaotic behaviors and its advantages are explored for chaotic systems. Consider the following nonlinear system that exhibits chaotic behaviors x_ ¼ f ðx; u; tÞ

ð1Þ

where x 2 Rn is the state vector and assume is available at the system output. In (1), u 2 Rr is the external input vector and f 2 Rn · Rr ! Rn is a nonlinear vector-valued function. Here and throughout the paper, in a general discussion we simply assume all the necessary conditions on the vector-valued function f such that the system is well-posed and has a unique solution within a certain region of interest in the state-space for given initial condition x0 = x(t0) and t P t0 P 0. It is desired to design the control vector, u for the system deﬁned in (1) such that the error vector between the system sate vector, x, and a desired model states vector, xd goes to zero as the time tends to inﬁnity. Here the MRC is considered for satisfying this performance. The block diagram of the closed loop conﬁguration of MRC for chaotic system is shown in Fig. 1. The desired model can be a linear or a nonlinear system. Consider the desired model is linear and in the following form: x_ d ¼ Axd þ Bv

ð2Þ

n

r

where xd 2 R is the desired state vector and v 2 R is the input vector. Assume that the eigenvalues of A have negative real part so that the reference system is asymptotically stable. In order to obtain control vector u let us deﬁne the error vector between the model state vector and the chaotic system sate vector as e ¼ xd x

ð3Þ

Subtracting the chaotic system (1) from desired model (2), which includes the control vector, we obtain the following error dynamics e_ ¼ Ae þ Ax f ðx; u; tÞ þ Bv

ð4Þ

The control vector u is the state feedback type, uðtÞ ¼ gðx; e; tÞ

ð5Þ

and designed such that the tracking control goal lim keðtÞk ¼ 0

ð6Þ

t!1

is achieved. It is clear from Eqs. (4) and (6) that the control problem is converted to the asymptotic stability of origin of error dynamics. Lyapunov function can be constructed and can be applied to obtain rigorous mathematical techniques for designing the control vector. Let us assume that the form of the Lyapunov function is

v

Desired Reference System

xd

Controller

u

Chaotic Systems

x

Fig. 1. Block diagram for model-reference control of chaotic systems.

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A. Uc¸ar / Chaos, Solitons and Fractals 31 (2007) 712–717

V ðeÞ ¼ eT Pe

ð7Þ

where P is a symmetric positive deﬁnite matrix. The derivative of V(e) along the solution of error dynamics (4) gives V_ ðeÞ ¼ e_ T Pe þ eT P e_ ¼ eT AT þ xT AT f T ðx; u; tÞ þ vT BT Pe þ eT P ½eA þ xA f ðx; u; tÞ þ vB ¼ eT ðAT P þ PAÞe þ N

ð8Þ

where N is a scalar quantity that will be determined by control vector and is in the following form; N ¼ 2eT P ðAx f ðx; u; tÞ þ BvÞ

ð9Þ

The assumed V(e) function is a Lyapunov function if 1. (ATP + PA) = Q is a negative-deﬁned matrix. 2. The control vector u can be chosen to make the scalar quantity N nonpositive. Then, the equilibrium state e = 0 is asymptotically stable in the large. Since the eigenvalues of the model system matrix A are assumed to have negative real parts, the condition 1 can be always met by a proper chose of P. Thus the problem is reduced to choose an appropriate control vector u to satisfy the condition 2 so that N is either zero or negative. We shall illustrate the application of the present approach to control of chaotic behavior of the Genesio–Tesi system [17,18] which is belong to a large class of continuous time chaotic systems.

3. MRC of Genesio–Tesi system Genesio–Tesi proposed a nonlinear system in [17] includes a simple square part and three simple ordinary diﬀerential equations that depend on three positive real parameters. The Genesio–Tesi system is one of paradigms of chaos since it captures many features of chaotic systems. The dynamic equations of the system with the control signal u is 32 3 2 3 2 3 2 x1 0 1 0 0 x_ 1 76 7 6 7 6 7 6 ð10Þ 0 1 54 x2 5 þ 4 0 5u 4 x_ 2 5 ¼ 4 0 x1 c b a 1 x_ 3 x3 where a, b and c are positive constants. The system has two equilibrium points, one at origin xeq1 = (0, 0, 0) and the other at xeq2 = (c, 0, 0). The stability for the origin of (10) requires ab < c. However, the system given in (10) exhibits complex dynamics includes chaos. Without control signal, u = 0 keeping the system parameters at c = 2b = 6, initial condition at (x1(0), x2(0), x3(0)) = (0.5, 0, 0) and changing the parameter a within the range of 3 > a > 1 leads stable, limit cycle, multiple periodic solutions and chaotic behaviors. The phase portraits of x1 and x2 of the system (10) that show diﬀerent behavior are depicted in Fig. 2 for a range of the system parameter a. In these numerical simulations, the sixthorder Runge–Kutta method is used to solve uncontrolled Genesio–Tesi system with adaptive step-size algorithm. Fig. 2(a) and (b) show the system has stable solution and a limit cycle for a = 2.5 and a = 1.9, respectively. Further decreasing a leads to two periods, multiple periods solutions, chaotic behaviors and eventually unstable. Fig. 1(c) and (d), respectively, shows two periods and chaotic solution for a = 1.28 and a = 1.12. Here the design problem is to synthesize a model following controller that always generates a signal that forces the states of a chaotic behavior of the Genesio–Tesi system towards the desired model states. It is assumed that the three parameters a, b and c of the Genesio–Tesi system (10) are c = 2b = 6 and a = 1.12. Consider the following reference model 2 3 2 32 3 2 3 x_ d1 xd1 0 1 0 0 6 7 6 76 7 6 7 0 1 54 xd2 5 þ 4 0 5v ð11Þ 4 x_ d2 5 ¼ 4 0 x_ d3 xd3 cm bm am cm where the system parameters cm, bm and am are positive and are chosen such that the system is stable. Without control u = 0 in (10), the two systems (10) and (11) have diﬀerent behaviors. That is the systems in (10) and (11) exhibit chaotic and stable behaviors, respectively. However, with a suitable control vector u(t) = g(x, e, t) 5 0, the chaotic system states will follow the desired reference model states and stabilize the states of error system (4) so that the error states converge to zero as time t goes to inﬁnity. For this end we propose the following control law for the system (10):

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Fig. 2. Phase portrait of x1 and x2 of the system (10) without control, u = 0 with the parameters c = 2b = 6 and initial conditions (x1(0), x2(0), x3(0)) = (0.5, 0.0): (a) stable behavior for a = 2.5; (b) a limit cycle for a = 1.9; (c) two periods solution for a = 1.28; (d) chaotic behavior for a = 1.12.

u ¼ ðc cm Þx1 þ ðb bm Þx2 þ ða am Þx3 þ cm v x21 sgnðp13 e1 þ p23 e2 þ p33 e3 Þ

ð12Þ

where sgn is sign function: sgnðp13 e1 þ p23 e2 þ p33 e3 Þ ¼

if

ðp13 e1 þ p23 e2 þ p33 e3 Þ > 0

1 if

ðp13 e1 þ p23 e2 þ p33 e3 Þ < 0

1

ð13Þ

and pij is i, j elements of the positive deﬁnite matrix P = PT deﬁned in Lyaponov function (7) satisﬁes the following Lyaponov equation for the reference model (11) Q ¼ AT P þ PA

ð14Þ

Then, we have the following theorem. Theorem 1. The chaotic system (10) will globally and asymptotically follow the desired stable system (11) if the control law u = g(x, e, t) defined in (12) is chosen. Proof. From the stability assumption on the model-reference system in (11), there exists a positive deﬁnite P = PT such that the algebraic Eq. (14) hold. For the error vector e1 ¼ xd1 x1 e2 ¼ xd2 x2 e3 ¼ xd3 x3 consider the following Lyapunov function.

ð15Þ

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A. Uc¸ar / Chaos, Solitons and Fractals 31 (2007) 712–717

2

V ðeÞ ¼ ½ e1

e2

p11 6 e3 6 4 p12 p13

p12 p22 p23

32

3 e1 76 7 6 7 p23 7 54 e2 5 p33 e3 p13

ð16Þ

The derivative with respect to time leads Eq. (8). Substituting control law (12) into (8) results the scalar quantity N N ¼ 2ðp13 e1 þ p23 e2 þ p33 e3 Þðx21 þ x21 sgnðp13 e1 þ p23 e2 þ p33 e3 ÞÞ and that is nonpositive and ensures asymptotic stability of the error dynamics (4) by Lyapunov stability theory.

h

4. Numerical simulation In this subsection, numerical simulations are given to verify the proposed method. As seen from Fig. 2(d) that Genesio–Tesi system (10) behaves chaotically for c = 2b = 6, a = 1.12 and u = 0. In order to have eigenvalues of the reference model matrix A at k1 = k2 = k3 = 5 in the complex plane, the reference model parameters bm = 5am = 75 and cm = 125 are chosen. For Q = diag{1, 1, 1} in the Lyapunov function (14) results the following P matrix;

Fig. 3. The time response of the error signals of error vector (15) with model following control (12) applied at a time s = 200 s.

A. Uc¸ar / Chaos, Solitons and Fractals 31 (2007) 712–717

2 6 P ¼4

0:4266 0:5

0:5 1:088

1:088

0:5

717

3 1:088 7 0:5 5 11:6

Substituting the chosen parameters in the reference model, the Genesio–Tesi system and Lyapunov equation leads the following control law u ¼ 119x1 72x2 13:88x3 þ 175v x21 sgnð1:088e1 þ 0:5e2 11:6e3 Þ For this particular choice, the conditions in Theorem 1; the eigenvalues of the error system must be negative real or complex with negative real parts are satisﬁed. Thus leading to chaotic system (10) follows the desired reference model (11). For the simulation, the initial conditions of the chaotic system (10) and reference model are taken as (x1(0), x2(0), x3(0)) = (0.5, 0, 0) and (xd1(0), xd2(0), xd3(0)) = (0, 1, 0.5), respectively. Zero input, v = 0 is taken for the reference model. The simulation results are depicted in Fig. 3(a)–(c) for error vector (15), respectively, when the control signal is activated at s = 200 s. Fig. 3(a)–(c) shows the error signals e1, e2, and e1 converge to zero, respectively, when the controller activated at s = 200 s. This illustrates that convergence of chaotic behavior to the desired behavior of the reference model (11) is achieved within a ﬁnite time.

5. Conclusion Here, a model-reference control is developed successfully for stabilizing chaotic system. For the closed loop stability Lyapunov method is used. The controller is designed such that the chaotic system (10) follows the desired model (11) within a desired ﬁnite time. The controller has linear and nonlinear parts. The speed of the convergence time of error dynamics can be modiﬁed by the linear part of the controller. Numerical simulations are provided to show the eﬀectiveness of the proposed method.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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