Chaos, Solitons and Fractals 31 (2007) 712–717

www.elsevier.com/locate/chaos

Model-reference control of chaotic systems

Ahmet Ucar *

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 systemto follow a desired reference system. The control strategy is developed within the general framework of the nonlinearmodel-reference control systems. Lyapunov stability is used to ensure the global stability of the error dynamics repre-sents 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 thechaotic systems is their sensitivity to the sates initial conditions. The trajectories of the chaotic systems starting fromany pair of arbitrary close positions will diverge exponentially and will become more and more uncorrelated withthe time. This property leads to loss of information about the system states in the future. Therefore chaos is requiredto be removed to prevent catastrophic situations. Furthermore, controlling chaos, that is to convert chaotic oscillationsinto 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. Severalcontrol methods have been proposed and implemented. For the methodologies and reviews these methods (see [2–5] andthe references therein). These methods can be considered into two categories: feedback and nonfeedback methods. Inthe feedback methods, the actual trajectory in the phase space of the system is monitored and some feedback processesare employed to force and to maintain the trajectory in the desired mode [2–11]. Hence the feedback methods do notchange the chaotic systems. They stabilize unstable periodic orbits on chaotic attractors by control signal. However, inthe nonfeedback methods, some system property or knowledge of the system is used to modify or exploit chaotic behav-ior [2–5,12–14]. This results to change the chaotic systems slightly, mainly by a small permanent to shift system param-eters, 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 frame-work of model-reference control systems. In the following section, the basic principles of model reference control sys-tems are presented. In Section 3, model-reference control is used to stabilize chaotic behaviors of a chaotic system.

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

* Tel.: +90 424 2370000x5214; fax: +90 424 2415526.E-mail address: aucar1@firat.edu.tr.

A. Ucar / Chaos, Solitons and Fractals 31 (2007) 712–717 713

Numerical simulations are provided in Section 4 to illustrate and verify of the method. Finally, concluding remarks aregiven 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 modelthat will produce the desired output [15,16]. The model-reference control (MRC) objective is to develop a control strat-egy which forces the plant dynamics to follow the dynamic of an ideal model. The ideal model need not be actual hard-ware and can be only a mathematical model simulated on a computer. In the MRC system, the output of the model andthat of the controlled system are compared and the error vector, e is used to generate the control signals. The MRC hasbeen used to obtain acceptable performance in some very difficult control problems involving the systems contain non-linearity and/or time varying parameters [15,16]. Here MRC is used for controlling chaotic behaviors and its advanta-ges 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 vectorand f 2 Rn · Rr! Rn is a nonlinear vector-valued function. Here and throughout the paper, in a general discussion wesimply assume all the necessary conditions on the vector-valued function f such that the system is well-posed and has aunique 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 systemsate vector, x, and a desired model states vector, xd goes to zero as the time tends to infinity. Here the MRC is consid-ered for satisfying this performance. The block diagram of the closed loop configuration of MRC for chaotic system isshown in Fig. 1. The desired model can be a linear or a nonlinear system. Consider the desired model is linear and in thefollowing form:

_xd ¼ Axd þ Bv ð2Þ

where xd 2 Rn is the desired state vector and v 2 Rr is the input vector. Assume that the eigenvalues of A have negativereal 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 sys-tem 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 followingerror dynamics

_e ¼ Aeþ 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

limt!1keðtÞk ¼ 0 ð6Þ

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 fordesigning the control vector. Let us assume that the form of the Lyapunov function is

Desired Reference

SystemController

xd Chaotic Systems

v u x

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

714 A. Ucar / Chaos, Solitons and Fractals 31 (2007) 712–717

V ðeÞ ¼ eTPe 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 _eTPeþ eTP _e ¼ eTAT þ xTAT � f Tðx; u; tÞ þ vTBT� �

Peþ eTP eAþ xA� f ðx; u; tÞ þ vB½ �¼ eTðATP þ PAÞeþ N ð8Þ

where N is a scalar quantity that will be determined by control vector and is in the following form;

N ¼ 2eTP 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 thecondition 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 differentialequations that depend on three positive real parameters. The Genesio–Tesi system is one of paradigms of chaos since itcaptures many features of chaotic systems. The dynamic equations of the system with the control signal u is

_x1

_x2

_x3

264

375 ¼

0 1 0

0 0 1

x1 � c �b �a

264

375

x1

x2

x3

264

375þ

0

0

1

264375u ð10Þ

where a, b and c are positive constants. The system has two equilibrium points, one at origin xeq1 = (0,0,0) and theother at xeq2 = (c, 0,0). The stability for the origin of (10) requires ab < c. However, the system given in (10) exhibitscomplex dynamics includes chaos. Without control signal, u = 0 keeping the system parameters at c = 2b = 6, initialcondition 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, lim-it cycle, multiple periodic solutions and chaotic behaviors. The phase portraits of x1 and x2 of the system (10) that showdifferent behavior are depicted in Fig. 2 for a range of the system parameter a. In these numerical simulations, the sixth-order 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. Furtherdecreasing 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 thestates of a chaotic behavior of the Genesio–Tesi system towards the desired model states. It is assumed that the threeparameters a, b and c of the Genesio–Tesi system (10) are c = 2b = 6 and a = 1.12. Consider the following referencemodel

_xd1

_xd2

_xd3

264

375 ¼

0 1 0

0 0 1

�cm �bm �am

264

375

xd1

xd2

xd3

264

375þ

0

0

cm

264

375v ð11Þ

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, thechaotic system states will follow the desired reference model states and stabilize the states of error system (4) so that theerror states converge to zero as time t goes to infinity. For this end we propose the following control law for the system(10):

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.

A. Ucar / Chaos, Solitons and Fractals 31 (2007) 712–717 715

u ¼ ðc� cmÞx1 þ ðb� bmÞx2 þ ða� amÞx3 þ cmv� x21 sgnðp13e1 þ p23e2 þ p33e3Þ ð12Þ

where sgn is sign function:

sgnðp13e1 þ p23e2 þ p33e3Þ ¼1 if ðp13e1 þ p23e2 þ p33e3Þ > 0

�1 if ðp13e1 þ p23e2 þ p33e3Þ < 0

�ð13Þ

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

�Q ¼ ATP 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 suchthat the algebraic Eq. (14) hold.

For the error vector

e1 ¼ xd1 � x1

e2 ¼ xd2 � x2

e3 ¼ xd3 � x3

ð15Þ

consider the following Lyapunov function.

Fig.

716 A. Ucar / Chaos, Solitons and Fractals 31 (2007) 712–717

V ðeÞ ¼ e1 e2 e3½ �

p11 p12 p13

p12 p22 p23

p13 p23 p33

2664

3775

e1

e2

e3

2664

3775 ð16Þ

The derivative with respect to time leads Eq. (8). Substituting control law (12) into (8) results the scalar quantityN

N ¼ 2ðp13e1 þ p23e2 þ p33e3Þð�x21 þ x2

1 sgnðp13e1 þ p23e2 þ p33e3ÞÞ

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 Gene-sio–Tesi system (10) behaves chaotically for c = 2b = 6, a = 1.12 and u = 0. In order to have eigenvalues of the refer-ence model matrix A at k1 = k2 = k3 = �5 in the complex plane, the reference model parameters bm = 5am = 75 andcm = 125 are chosen. For Q = diag{1,1,1} in the Lyapunov function (14) results the following P matrix;

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. Ucar / Chaos, Solitons and Fractals 31 (2007) 712–717 717

P ¼�0:4266 0:5 1:088

0:5 �1:088 0:5

1:088 0:5 �11:6

264

375

Substituting the chosen parameters in the reference model, the Genesio–Tesi system and Lyapunov equation leads thefollowing control law

u ¼ �119x1 � 72x2 � 13:88x3 þ 175v� x21 sgnð1:088e1 þ 0:5e2 � 11:6e3Þ

For this particular choice, the conditions in Theorem 1; the eigenvalues of the error system must be negative real orcomplex 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 ref-erence model. The simulation results are depicted in Fig. 3(a)–(c) for error vector (15), respectively, when the controlsignal is activated at s = 200 s. Fig. 3(a)–(c) shows the error signals e1, e2, and e1 converge to zero, respectively, whenthe controller activated at s = 200 s. This illustrates that convergence of chaotic behavior to the desired behavior of thereference 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 stabilityLyapunov 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 errordynamics can be modified by the linear part of the controller. Numerical simulations are provided to show the effec-tiveness of the proposed method.

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