Control of Rabinovich chaotic system using sliding mode ... · Rabinovich system was determined...

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2014; 28:1413–1421Published online 25 November 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.2450

Control of Rabinovich chaotic system using sliding mode control

Uğur Erkin Kocamaz1,*,†, Yılmaz Uyaroğlu2 and Hakan Kizmaz2

1Department of Computer Technologies, Vocational School of Karacabey, Uludağ University, Bursa, Turkey2Department of Electrical & Electronics Engineering, Faculty of Engineering, Sakarya University, Sakarya, Turkey

SUMMARY

This paper investigates the control of the continuous time Rabinovich chaotic system with the sliding modecontrol method. Based on the properties of the sliding mode theory, the controllers are designed and addedto the nonlinear Rabinovich system. Numerical simulations show that the Rabinovich chaotic system can beregulated to its equilibrium points in the state space by using the sliding mode controllers, which verifiesall the theoretical analyses. Simulation results of the proposed sliding mode control strategy have been alsocompared with the passive control method, and their performances are discussed. Copyright © 2013 JohnWiley & Sons, Ltd.

Received 16 April 2013; Revised 13 July 2013; Accepted 21 October 2013

KEY WORDS: Rabinovich chaotic system; sliding mode control; passive control; chaos control

1. INTRODUCTION

Hubler introduced the adaptive control of chaotic systems in 1989 [1]. Then, Ott, Grebogi, andYorke presented a method called OGY for controlling chaotic systems in 1990 [2]. Since thesepioneering studies, the control of chaotic systems has become one of the major research areas fornonlinear systems. Several significant chaotic attractors are also discovered [3, 4]. Then, effectivetypes of control methods have been applied to the chaotic systems such as active [5, 6], passive[7], linear feedback [8], nonlinear feedback [9], time-delayed feedback [10], adaptive [11], back-stepping design [12], and impulsive [13] controls. Sliding mode control is one of the importantchaos control methods; it is a nonlinear control method that effectively controls uncertain nonlinearsystems [14–16]. The dynamic performance of this method is determined by the prescribed mani-fold or sliding surface where a switching structure maintains the control. It provides noncontinuouscontrol by enforcing the system states to stay on the sliding surface [17]. Recently, the slidingmode control method has been successfully implemented for the control of Lorenz [18], Chua [19],Rössler [20], Duffing-Holmes [21], and many other chaotic systems [22, 23]. It is also applied forthe synchronization of chaotic systems [24–26]. The number of controllers generally equals to thenumber of state variables, but some methods such as passive control and linear feedback control canmaintain the control of a chaotic system with only one controller [27, 28]. Recently, the single con-troller approaches have been gaining importance because of its easiness in applications and low-costproduction [29–31].

In 1978, Pikovski, Rabinovich, and Trakhtengerts introduced a Rabinovich differential system[32]. This continuous time chaotic system resembles the well-known Lorenz chaotic system in someproperties. The bifurcation diagrams and dynamical behaviors of the Rabinovich chaotic system

*Correspondence to: Uğur Erkin Kocamaz, Department of Computer Technologies, Vocational School of Karacabey,Uludağ University, 16700 Karacabey, Bursa, Turkey.

†E-mail: [email protected]

Copyright © 2013 John Wiley & Sons, Ltd.

1414 U. E. KOCAMAZ, Y. UYAROĞLUAND H. KIZMAZ

were investigated in some papers [33, 34]. A new 4D hyperchaotic system was constructed fromthe Rabinovich system in 2010 [35]. To the knowledge of the authors, the control of chaos in theRabinovich system was determined only by the passive control method [36].

Motivated by the previous chaos control studies, in this study, further investigation on the controlof the continuous time Rabinovich chaotic system is explored by using the sliding mode controlmethod. First, a brief description of a nonlinear Rabinovich system is given in Section 2. Then,sliding mode controllers are employed for achieving the control of the Rabinovich chaotic systemto its equilibrium points in Section 3. In this section, the passive controlled Rabinovich systemis also described. Afterward, numerical simulations are demonstrated to confirm and compare theeffectiveness of the sliding mode and passive control methods for the control of Rabinovich chaoticsystem in Section 4. Finally, conclusions are given in Section 5.

2. SYSTEM DESCRIPTION

The differential equations of the Rabinovich chaotic system is

Px D�axC hy C y´,Py D hx � by � x´,Ṕ D �d´C xy,

(1)

where x, y, and ´ are state variables, and a, b, d , and h are positive constant parameters [36].The Rabinovich system exhibits a chaotic behavior when the parameter values are taken as a D 4,b D 1, d D 1, and h D 6.75 with the initial conditions x.0/ D 5.5, y.0/ D �1.25, and ´.0/ D 8.4.According to these parameters, the Rabinovich chaotic system has three real equilibrium points:E0.0, 0, 0/, E1.4.6119, 1.3979, 6.4469/, and E2.�4.6119,�1.3979, 6.4469/. The time series of theRabinovich chaotic system are shown in Figure 1, the 2D phase plots are shown in Figure 2, and the3D phase plane is shown in Figure 3.

(a) (b)

(c)

Figure 1. Time series of the Rabinovich chaotic system for the (a) x, (b) y, and (c) ´ signals.

Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2014; 28:1413–1421DOI: 10.1002/acs

CONTROL OF RABINOVICH CHAOTIC SYSTEM USING SLIDING MODE 1415

(a) (b)

(c)

Figure 2. Phase plots of the Rabinovich chaotic system in the (a) x–y, (b) x–´, and (c) y–´ phase planes.

Figure 3. x–y–´ phase plane of the Rabinovich chaotic system.

3. CONTROL FOR THE RABINOVICH CHAOTIC SYSTEM

3.1. Sliding mode control for the Rabinovich chaotic system

Sliding mode control can be applied to system (1) for controlling the chaos in the Rabinovichsystem. The controlled Rabinovich system is described as follows:

Px D�axC hy C yĆ u1,Py D hx � by � xĆ u2,Ṕ D �dĆ xy C u3,

(2)



where u1, u2, and u3 are the control inputs. The fixed point is denoted as ( Nx, Ny, Ń/, then the trajec-tory error states could be determined as e1 D x � Nx, e2 D y � Ny, and e3 D ´ � Ń . Therefore, thestate variables can be defined as x D e1C Nx, y D e2C Ny, and ´D e3C Ń . The error state dynamicequations of system (2) can be obtained as

Pe1 D�a.e1C Nx/C h.e2C Ny/C .e2C Ny/.e3C Ń/C u1,Pe2 D h.e1C Nx/� b.e2C Ny/� .e1C Nx/.e3C Ń/C u2,Pe3 D�d.e3C Ń/C .e1C Nx/.e2C Ny/C u3.

(3)

Then, the error dynamics in Equation (3) become

Pe1 D�ae1C .hC Ń/e2C Nye3C e2e3 � a NxC h Ny C Ny Ń C u1,Pe2 D .h� Ń/e1 � be2 � Nxe3 � e1e3C h Nx � b Ny � Nx Ń C u2,Pe3 D Nye1C Nxe2 � de3C e1e2 � d Ń C Nx Ny C u3.

(4)

When the error dynamics in Equation (4) are asymptotically stable on the zero equilibrium point,the �a NxCh NyC Ny Ń , h Nx� b Ny � Nx Ń , and �d Ń C Nx Ny equations equal to 0. This implies that the errorstate dynamic equations can be simplified as

Pe1 D�ae1C .hC Ń/e2C Nye3C e2e3C u1,Pe2 D .h� Ń/e1 � be2 � Nxe3 � e1e3C u2,Pe3 D Nye1C Nxe2 � de3C e1e2C u3.

(5)

The error dynamics (5) can be regularized in the matrix notation as

Pe D AeC �.e/C u, (6)

where

AD

24�a hC Ń Nyh� Ń �b � NxNy Nx �d

35 , �.e/D

24

e2e3�e1e3e1e2

35 , uD

24u1u2u3

35 . (7)

The control signal u is defined as

uD��.e/CBv, (8)

where B is chosen so that (A, B/ could be controllable. Thus, B is taken as

B D�0 1 0

�T. (9)

The sliding surface should be selected so that the system dynamics can remain stable. In orderto obtain this sliding surface, the system is transformed into a regular form, and the sliding sur-face coefficients are computed by using a regular form [37]. If Equation (8) is substituted intoEquation (6), then the system altered to the following linear form:

Pe D AeCBv, (10)

where A 2 Rnxn, B 2 Rnxm, e 2 Rm, and v 2 Rm. The error dynamics of system (10) should beseparated into two subsystems, and one of them should include a control signal. In order to transformthe system into its regular form, a nonsingular transformation is used as follows:

´D Te, (11)

where T is a nonsingular transformation matrix. If Equation (11) is substituted into the linear form,the following alternative system, which consists of two subsystems, is revealed as

Ṕ1 D A11´1CA12´2,Ṕ2 D A21´1CA22´2CLv, (12)



where L is a gain matrix. After that, according to system (12), the sliding surface design isconsidered as

s.t/D S´D S1´1C S2´2 D 0, (13)

where S1 2 R1x.n�m/, and S2 2 R1. In Equation (13), if ´2 is taken alone and substituted intosystem (12), the following equation is acquired

Ṕ1 D ŒA11 �A12S�12 S1� ´1, (14)

which gives the ideal sliding motion. The dynamics of ´2 depend on ´1 in terms of mathematicalrelationship. Hence, the stabilization of ´1 allows the system to be stable. According to the dynam-ics of ´1, the eigenvalues of the expression A11 – A12S�12 S1 should be on the left-half s-planeso that the dynamics of ´1 could be asymptotically stable. In order to calculate S�12 S1, the polereplacement and the optimal control techniques can be used. S2 may be selected arbitrary on thecondition that it is not singular. Then, S1 is calculated according to S2.

s.t/D S´D STe D Ce.C D ST . (15)

The eigenvalues of A11 – A12S�12 S1 have been chosen on the left-half s-plane. Afterward, S�12 has

been selected as the identity matrix, and then, S1 is calculated [38]. From Equation (15), the slidingsurface vector C has been determined as [8/9 1 0], [0 1 1], and [0 1 �1] for the E0, E1, and E2equilibrium points, respectively. Then, the sliding surface for the zero equilibrium point becomes

s D Ce D�8=9 1 0

�e D 8=9 e1C e2, (16)

which renders the sliding mode state equation asymptotically stable.From the property of the sliding mode control theory,

v.t/D�.CB/�1 ŒC.kI CA/eC qsign.s/� . (17)

Now, the v.t/ control signal for the zero equilibrium point becomes

v.t/D�..8=9/ � .k � a/C h� Ń/ � .x � Nx/ � ..8=9/ � .hC Ń/C k � b/ � .y � Ny/� ..8=9/ � Ny � Nx/ � .´� Ń/� q � sign..8=9/ � .x � Nx/C y � Ny/. (18)

A large value of k can cause chattering, and an appropriate value of q speeds up the reaching timeto the sliding surface and also reduces the chattering.

Hence, the required sliding mode control signal is obtained as in Equation (8), where �.e/ and Bare defined as in Equations (7) and (9), respectively:

u1 D�.y � Ny/ � .´� Ń/,u2 D .x � Nx/ � .´� Ń/C v.t/,u3 D�.x � Nx/ � .y � Ny/.

(19)

Thus, the sliding mode controlled Rabinovich system (2) is completed with Equations (18) and (19).

3.2. Passive control for the Rabinovich chaotic system

Emiroglu and Uyaroglu have investigated the passive control of the Rabinovich chaotic system in2010 [36]. The controlled Rabinovich system is designed by

Px D�axC hy C y´,Py D hx � by � xĆ u,Ṕ D �dĆ xy,

(20)

where the passive controller uD�2hx�xĆy.b�˛/Cv. Simulation results show that consideringthe passive control parameter as ˛ D 10 tends toward better performance [36].



4. NUMERICAL SIMULATIONS

Numerical simulations are performed to demonstrate the sliding mode control of the Rabinovichchaotic system. The fourth-order Runge–Kutta method has been used in all numerical simulationswith the time step being equal to 0.001. The parameters of the Rabinovich system are taken as aD 4,b D 1, d D 1, and hD 6.75 with the initial conditions x.0/D 5.5, y.0/D�1.25, and ´.0/D 8.4 sothat it exhibits a chaotic behavior. The sliding mode gains have been selected as k D 10 and q D 0.1,and the passive control parameter is taken as ˛ D 10. The controllers are activated at t D 20 in allsimulations.

According to the selected parameters of the Rabinovich system, there exist three equilibriumpoints. The first one is the zero equilibrium point, so the sliding mode control . Nx, Ny, Ń/ fixed pointsare taken as Nx D 0, Ny D 0, and Ń D 0, and the passive control parameter is v D 0. The slidingsurface vector C is considered as [8/9 1 0]. The simulation results for the control of the Rabinovichchaotic system to the zero equilibrium point by using the sliding mode control and passive controlmethods are shown in Figure 4.

As expected, the signals of the controlled Rabinovich system that are shown in Figure 4converge to the zero equilibrium point asymptotically. The control is provided at t > 26 by usingboth the sliding mode control and passive control methods. As seen in Figure 4, the sliding modecontrol method performs slightly better than the passive control method for the control of theRabinovich chaotic system.

The other equilibrium points of the Rabinovich system are E1.4.6119, 1.3979, 6.4469/ andE2.�4.6119,�1.3979, 6.4469/. The sliding surface vector C is considered as [0 1 1] and [0 1 �1]for E1 and E2 equilibrium points, respectively. The Rabinovich chaotic system stabilizes toward anequilibrium point with the passive controller by adjusting the v parameter. For the former equilib-rium point, v must be taken as v D 104.5743, and for the latter, v D �104.5743 when the otherpassive control parameter ˛ equals to 10. The simulation results for the control of the Rabinovichchaotic system to the E1 and E2 equilibrium points by using the sliding mode control and passivecontrol methods are shown in Figures 5 and 6, respectively.

As expected, the signals of the controlled Rabinovich system that are shown in Figures 5 and 6converge to the equilibrium points asymptotically. While the control is provided at t > 22.5 by using

(a) (b)

(c)

Figure 4. The time response of the states for the control of the Rabinovich chaotic system to E0.0, 0, 0/equilibrium point with controllers activated at t D 20 for the (a) x, (b) y, and (c) ´ signals.



(a) (b)

(c)

Figure 5. The time response of states for the control of the Rabinovich chaotic system toE1.4.6119, 1.3979, 6.4469/ equilibrium point with controllers activated at t D 20 for the (a) x, (b) y, and

(c) ´ signals.

(a) (b)

(c)

Figure 6. The time response of the states for the control of the Rabinovich chaotic system toE2.�4.6119,�1.3979, 6.4469/ equilibrium point with controllers activated at t D 20 for the (a) x, (b)

y, and (c) ´ signals.

the passive control method, it is reached when t > 21.5with the sliding mode control method. There-fore, this comparison validates that the sliding mode control method performs better than the passivecontrol method for the control of the Rabinovich chaotic system to E1.4.6119, 1.3979, 6.4469/ andE2.�4.6119,�1.3979, 6.4469/ equilibrium points. On the other hand, the passive control methoduses only one controller that provides easiness in implementation and low-cost production. Hence,



it can be concluded that the sliding mode control method is more successful than the passive controlmethod for the control of the Rabinovich chaotic system because of using more controllers.

5. CONCLUSION

The aim of this paper is to investigate the control of the continuous time Rabinovich chaotic systemby means of the sliding mode control method. According to the sliding mode control methodology,the controllers have been constructed to stabilize the nonlinear Rabinovich system toward its equilib-rium points. Theoretical analyses are also confirmed by numerical simulations. Related Figures 4–6show that both the sliding mode controllers and the passive controller have achieved the control ofthe Rabinovich chaotic system with an appropriate time period. The proposed sliding mode con-trol method regulates the system to its equilibrium points more effectively than the passive controlmethod, so it is more appropriate for the control of the Rabinovich chaotic system.

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