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Int. J. Modelling, Identification and Control, Vol. 23, No. 1, 2015
Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system Sundarapandian Vaidyanathan* Research and Development Centre, Vel Tech University, Avadi, Chennai-600 062, Tamil Nadu, India Email: [email protected] *Corresponding author
Sivaperumal Sampath Department of Electronics and Communication Engineering, Vel Tech University, Avadi, Chennai-600 062, Tamil Nadu, India Email: [email protected]
Ahmad Taher Azar Faculty of Computers and Information, Benha University, Egypt Email: [email protected] Abstract: Synchronisation of chaotic systems is an important research problem in chaos theory. In this research work, a novel sliding mode control method is proposed for the global chaos synchronisation of identical chaotic systems. The general result derived using novel sliding mode control method is established using Lyapunov stability theory. As an application of the general result, the problem of global chaos synchronisation of identical Zhu chaotic systems (2010) is studied and a new sliding mode controller is derived. Numerical simulations have been shown to illustrate the phase portraits of Zhu chaotic system and the sliding mode controller design for the global chaos synchronisation of identical Zhu chaotic systems. Keywords: chaos; chaotic systems; sliding mode control; SMC; Lyapunov stability theory; chaos synchronisation; Zhu system. Reference to this paper should be made as follows: Vaidyanathan, S., Sampath, S. and Azar, A.T. (2015) ‘Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system’, Int. J. Modelling, Identification and Control, Vol. 23, No. 1, pp.92–100. Biographical notes: Sundarapandian Vaidyanathan is a Professor and Dean at the R&D Centre, Vel Tech University, Chennai, India. He received his DSc in Electrical and Systems Engineering from Washington University, St. Louis, USA in 1996. His current research focuses on linear and nonlinear control systems, chaotic and hyperchaotic systems, chaos control and synchronisation, computational science and robotics. He has published over 110 Scopus-indexed research publications. He has delivered plenary lectures on control systems and chaos theory in many international conferences. He has conducted many workshops on mathematical modelling using MATLAB and SCILAB. Sivaperumal Sampath received his ME (2007) in VLSI Design and BE (2005) in Electronics and Communication Engineering from Anna University, Chennai, India. He is working as an Assistant Professor in the Department of Electronics and Communications Engineering at Vel Tech University, Chennai, India. He has published many Scopus-indexed papers in international journals and conferences. His current research interests are communications, control systems, robotics and VLSI design. Ahmad Taher Azar received his MSc (2006) in System Dynamics and PhD (2009) in Adaptive Neuro-Fuzzy Systems from Faculty of Engineering, Cairo University (Egypt). He is currently working as an Assistant Professor, Faculty of Computers and Information, Benha University, Egypt. He is the Editor-in-Chief of two journals published by IGI Global, USA. He is also an Associate Editor of IEEE Transactions on Neural Networks and Learning Systems. His research
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Global chaos synchronisation of identical chaotic systems via novel sliding mode control method
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interests include: soft computing, machine learning, rough set, mathematical modelling and computer simulation, automatic control systems, systems engineering, systems dynamics, process control, robotics and systems, intelligent control systems and computational intelligence.
1
Introduction
Chaotic systems are dissipative nonlinear dynamical systems having at least one positive Lyapunov exponent. The Lyapunov exponent of a chaotic system is a measure of the divergence of points which are initially very close and this can be used to quantify chaotic systems (Leonov, 2008). Each nonlinear dynamical system has a spectrum of Lyapunov exponents, which are equal in number to the dimension of the state space. The largest Lyapunov exponent of a nonlinear dynamical system is called the maximal Lyapunov exponent (MLE). An important paradigm of a 3-D chaotic system was discovered by Lorenz (1963) while he was studying a 3-D weather model. Subsequently, many chaotic systems were found in the literature such as Rössler (1976) system, ACT system (Arneodo et al., 1980), Sprott (1994) systems, Chen (1999) system, Lü system (Lü and Chen, 2002), Liu system (Liu et al., 2004), Cai system (Cai and Tan, 2007), Tigan system (Tigan and Opris, 2008), Zhu system (Zhu et al., 2010), Pan system (Pan et al., 2010), Sundarapandian system (Sundarapandian, 2013a, 2013b, 2013c), Li-Wu system (Li et al., 2013), etc. Chaos theory has applications like chemical reactions (Li and Zhu, 2004; Villegas et al., 2012), lasers (Antonelli and Mecozzi, 2009; Hu et al., 2011), vibrations (Liu et al., 2008; Luo and Lv, 2009), turbines (Asgari et al., 2013), combustion engines (Wendeker et al., 2003), power systems (Yu et al., 2003), robotics (Volos et al., 2012), etc. In communications, chaos has applications like cryptosystems (Usama et al., 2010; Rhouma and Belghith, 2010), secure communications (Suzuki and Imai, 2006; Chen and Min, 2008; Fallahi and Leung, 2010), etc. Next, we describe the complete synchronisation of chaotic systems (A) and (B) as follows. If the chaotic system (A) is called the master or drive system and the controlled chaotic system (B) is called the slave or response system, then the complete synchronisation aims to control the slave system (B) so that the states of the slave system (B) track the states of the master system (A) asymptotically. Various control schemes have been developed to investigate the global chaos synchronisation problem such as PC method (Pecora and Carroll, 1990), OGY method (Ott et al., 1990), active control method (Ho and Hung, 2002; Chen, 2005; Sundarapandian, 2011a; Sarasu and Sundarapandian, 2011; Zhang et al., 2012), adaptive control method (Liao and Tsai, 2000; Sundarapandian and Karthikeyan, 2011; Sundarapandian, 2011b, 2013d; Sundarapandian and Pehlivan, 2012), backstepping method (Yu and Zhang, 2004; Park, 2006; Sundarapandian, 2013e), time-delay feedback method (Guo and Zhong, 2009), sliding mode control (SMC) (Zarrabi et al., 2012; Shang and
Wang, 2013; Zen et al., 2013; Wu et al., 2014; Kaur and Janardhanan, 2014), etc. In this paper, we use a novel sliding control method for deriving a general result for the global chaos synchronisation of identical chaotic systems using SMC theory. The sliding control method is an effective control tool having advantages of low sensitivity to parameter variations in the plant and disturbances affecting the plant. In the SMC design, the control dynamics has two sequential modes, viz. the reaching mode and the sliding mode. In accordance with this general design, the sliding controller design consists of two parts: a hyperplane design and controller design. First, a hyperplane is designed using the pole-placement approach in the modern control theory and then a controller is designed using the sliding condition. The stability of the overall system is guaranteed by the sliding condition and by a stable hyperplane. The rest of the paper is organised as follows. In Section 2, we discuss the problem statement for the complete synchronisation problem of identical chaotic systems. In Section 3, we derive a general result for the global chaos synchronisation of identical chaotic systems using novel sliding control method. In Section 4, we describe the Zhu chaotic system (Zhu et al., 2010) and its properties. Phase portraits of the Zhu system are described using MATLAB. In Section 5, we describe the sliding mode controller design for the identical Zhu chaotic systems using novel sliding control method and its numerical simulations using MATLAB. Finally, conclusions are given in Section 6.
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Problem statement
This section gives a problem statement for the global chaos synchronisation of identical chaotic systems. As the master or drive system, we take the chaotic system given by x = Ax + f ( x)
(1)
In equation (1), x ∈ Rn denotes the state of the system, A ∈ Rn×n denotes the matrix of system parameters and f(x) ∈ Rn contains the nonlinear parts of the system. As the slave or response system, we take the controlled chaotic system given by y = Ay + f ( y ) + u
(2)
In equation (2), y ∈ Rn denotes the state of the system and u is the control. For the complete synchronisation of systems (1) and (2), the synchronisation error is defined as e= y−x
(3)
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A simple calculation yields the error dynamics as e = Ae + ψ ( x, y ) + u,
(4)
veq (t ) = −(CB ) −1 CAe(t )
where
ψ ( x, y ) = f ( y ) − f ( x).
(5)
Thus, the complete synchronisation problem can be defined as follows: find a controller u(x, y) so as to render the synchronisation error e(t) to be globally asymptotically stable for all values of e(0) ∈ Rn, i.e., lim e(t ) = 0 for all e(0) ∈ R n .
t →∞
3
The sliding motion is influenced by the equivalent control derived from (11) as
(6)
A novel SMC method for the global chaos synchronisation of chaotic systems
First, we set the design by setting the control as u (t ) = −ψ ( x, y ) + Bv(t )
(7)
In equation (7), B ∈ Rn is chosen such that (A, B) is completely controllable. By substituting (7) into (4), we get the closed-loop error dynamics e = Ae + Bv
(8)
The system (8) is a linear time-variant control system with single input v. Hence, we have converted the origin problem of complete chaos synchronisation of the identical chaotic systems (1) and (2) into an equivalent control problem of globally stabilising the error system (8) by a suitable choice of the feedback control (7). We start the sliding controller design by defining the sliding variable as s (e) = Ce = c1e1 + c2 e2 + " + cn en ,
(9)
where C ∈ R1×n is a constant vector to be determined. The sliding manifold S is defined as the hyperplane
{
}
S = e ∈ R n : s (e) = Ce = 0 .
(10)
We shall assume that a sliding motion occurs on the hyperplane S. In sliding mode, the following equations must be satisfied:
s ≡ 0 and s ≡ CAe + CBv = 0
(11)
We assume that CB ≠ 0
(12)
(13)
By substituting (13) into (8), we obtain the equivalent error dynamics in the sliding phase as e = Ae − (CB) −1 CAe = Ee,
(14)
where E = ⎡⎣ I − B (CB ) −1 C ⎤⎦ A
(15)
We note that E is independent of the control and has at most (n – 1) non-zero eigenvalues, depending on the chosen switching surface, while the associated eigenvectors belong to ker(C). Since (A, B) is controllable, we can use sliding control theory (Utkin, 1977) to choose B and C so that E has any desired (n – 1) stable eigenvalues. This shows that the dynamics in the sliding mode is globally asymptotically stable. Finally, for the sliding controller design, we apply a novel sliding control law, viz. s = −ks − qs 2 sgn( s )
(16)
In equation (16), sgn(⋅) denotes the sign function and the SMC constants k > 0, q > 0 are found in such a way that the sliding condition is satisfied and that the sliding motion will occur. By combining equations (11), (13) and (16), we finally obtain the SMC v(t) as
v(t ) = −(CB ) −1[C (kI + A)e + qs 2 sgn( s)]
(17)
Next, we establish the main result of this section. Theorem 1: The sliding mode controller law defined by (7) achieves global and asymptotic synchronisation of the identical chaotic systems (1) and (2) for all initial conditions x(0), y(0) ∈ Rn, where v is defined by the novel sliding control law (16), C ∈ Rn×1 is such that (A, B) is controllable, C ∈ R1×n is such that CB ≠ 0 and that the matrix E defined by (15) has (n – 1) stable eigenvalues. Proof: Upon substitution of the control laws (7) and (17) into the error dynamics (4), we get the closed-loop error dynamics as e = Ae − B (CB ) −1 ⎡⎣C (kI + A)e + qs 2 sgn( s) ⎤⎦
(18)
We shall show that the error system (18) is globally asymptotically stable by considering the quadratic Lyapunov function V ( e) =
1 2 s ( e) 2
(19)
Global chaos synchronisation of identical chaotic systems via novel sliding mode control method The sliding mode motion is characterised by the equations s (e) = 0 and s(e) = 0
(20)
By the choice of E, the dynamics in the sliding mode is globally asymptotically stable. When s(e) ≠ 0, V(e) > 0. Also, when s(e) ≠ 0, differentiating V along the error dynamics (18) or the equivalent dynamics (16), we get V (e) = ss = −ks 2 − qs 3 sgn( s ) < 0.
(21)
Hence, by Lyapunov stability theory (Leonov, 2008), the error dynamics (18) is globally asymptotically stable for all e(0) ∈ Rn. This completes the proof.
4
Properties of the Zhu chaotic system
The 2-D projections of the phase-portrait of the Zhu system (22) are depicted in Figures 2 to 4. The system (22) has Lyapunov exponents in the chaotic case as L1 = 0.6747, L2 = 0, L3 = −4.0738
The Lyapunov dimension of the Zhu system is found as DL = 2 +
L1 + L2 = 2.1656 | L3 |
The dynamics of the Lyapunov exponents is shown in Figure 5. Figure 2
Phase portrait of the Zhu system in x1 – x2 plane (see online version for colours)
Figure 3
Phase portrait of the Zhu system in x2 – x3 plane (see online version for colours)
(22)
x3 = −cx3 + x1 x2
where x1, x2, x3 are the state variables and a, b, c are positive parameters. The Zhu system (22) is chaotic when the parameter values are taken as a = 1.5, b = 2.5, c = 4.9
(23)
The 3-D phase portrait of the Zhu system (22) is depicted in Figure 1, when the parameter values are chosen as in (23) and the initial conditions are taken as follows: x1 (0) = 1.2, x2 (0) = 0.8, x3 (0) = 1.4 Figure 1
Phase portrait of the Zhu chaotic system in R3 (see online version for colours)
(24)
(25)
The MLE of the Zhu system is L1 = 0.6747.
This section describes the qualitative properties of the Zhu chaotic system (Zhu et al., 2010), which is a new 3-D chaotic system. The Zhu system is described by the dynamics x1 = − x1 − ax2 + x2 x3 x2 = bx2 − x1 x3
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Figure 4
Phase portrait of the Zhu system in x1 – x3 plane (see online version for colours)
where x1, x2, x3 are the state variables and a, b, c are positive parameters. As the slave system, we consider the controlled Zhu chaotic system given by y1 = − y1 − ay2 + y2 y3 + u1 y 2 = by2 − y1 y3 + u2
(27)
y3 = −cy3 + y1 y2 + u3
where y1, y2, y3 are the state variables and u1, u2, u3 are the controls. The complete synchronisation error is defined by e1 = y1 − x1 e2 = y2 − x2 e3 = y3 − x3
(28)
P (ω , θ , γ ) = 1 +
Then the error dynamics is obtained as Figure 5
e1 = −e1 − ae2 + y2 y3 − x2 x3 + u1 e2 = be2 − y1 y3 + x1 x3 + u2
The dynamics of the Lyapunov exponents (see online version for colours)
(29)
e3 = −ce3 + y1 y2 − x1 x2 + u3
In matrix form, we can write the error dynamics (28) as e = Ae + ψ ( x, y ) + u,
(30)
where ⎡ −1 −a 0 ⎤ A = ⎢⎢ 0 b 0 ⎥⎥ , ⎢⎣ 0 0 −c ⎥⎦
(31)
⎡ y2 y3 − x2 x3 ⎤ ψ ( x, y ) = ⎢⎢ − y1 y3 + x1 x3 ⎥⎥ , ⎢⎣ y1 y2 − x1 x2 ⎥⎦
(32)
and
5
Sliding mode controller design for the global chaos synchronisation of Zhu systems
This section describes the sliding mode controller design for the global chaos synchronisation of Zhu (2010) chaotic systems based on the novel method described by Theorem 1 in Section 3. As the master system, we consider the Zhu chaotic system given by x1 = − x1 − ax2 + x2 x3 x2 = bx2 − x1 x3 x3 = −cx3 + x1 x2
(26)
⎡ u1 ⎤ u = ⎢⎢u2 ⎥⎥ ⎢⎣ u3 ⎥⎦
(33)
We follow the procedure given in Section 3 for the construction of the novel sliding controller to achieve complete synchronisation of the identical Zhu systems (26) and (27). First, we set u as u (t ) = −ψ ( x, y ) + Bv(t ),
(34)
where B is selected such that (A, B) is completely controllable.
Global chaos synchronisation of identical chaotic systems via novel sliding mode control method A simple choice of B is ⎡1⎤ B = ⎢⎢1⎥⎥ . ⎢⎣1⎥⎦
The controllability calculated as follows.
(35)
matrix
Z = [ B AB A2 B]
⎡1 −2.5 −1.25 ⎤ Z = ⎢⎢1 2.5 6.25⎥⎥ ⎢⎣1 −4.9 24.01⎥⎦
is
Figures 6 to 8 show the complete synchronisation of the Zhu chaotic systems (25) and (26). Figure 9 shows the time-history of the synchronisation errors e1, e2, e3. Figures 6 to 8 show that the states (x1, y1), (x2, y2) and (x3, y3) are synchronised in two seconds, while Figure 9 shows that the chaos synchronisation errors e1, e2, e3 decay to zero in two seconds. Figure 6
Synchronisation of Zhu systems: states x1 and y1 (see online version for colours)
Figure 7
Synchronisation of Zhu systems: states x2 and y2 (see online version for colours)
(36)
The controllability matrix Z has full rank. Thus, it follows that (A, B) is completely controllable. The Zhu system displays a strange attractor when the parameter values are taken as a = 1.5, b = 2.5, c = 4.9
(37)
The sliding mode variable is selected as s = Ce = [1 −2.1 1] e
(38)
which renders the sliding motion globally asymptotically stable. Next, we take the sliding mode gains as k = 5 and q = 0.2.
(39)
From equation (17) of Section 3, we obtain the novel sliding control v as v = 40e1 − 172.5e2 + e3 + 2 s 2 sgn( s )
(40)
As an application of Theorem 1 to the identical Zhu chaotic systems, we obtain the following main result of this section. Theorem 2: The identical Zhu chaotic systems (26) and (27) are globally and asymptotically synchronised for all initial conditions x(0), y(0) ∈ R3 with the sliding controller u defined by (34), where ψ(x, y) is defined by (32), B is defined by (35), and v is defined by (40). For numerical simulations, we use MATLAB software for solving systems of differential equations using the classical fourth order Runge-Kutta method with step size h = 10–8. The parameters of the Zhu chaotic systems (25) and (26) are taken as in the chaotic case, viz., a = 1.5, b = 2.5, c = 4.9
As an initial condition for the Zhu system (25), we take x1 (0) = 2.7, x2 (0) = 1.5, x3 (0) = 2.4
As an initial condition for the Zhu system (26), we take y1 (0) = −5.6, y2 (0) = 4.3, y3 (0) = 8.7
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Figure 8
Synchronisation of Zhu systems: states x3 and y3 z (see online version for colours)
can be applied for other chaotic systems as well and the efficiency of this procedure in chaos applications may be studied.
References
Figure 9
6
Time-history of the synchronisation errors e1, e2, e3 (see online version for colours)
Conclusions
In this research work, a novel SMC result has been proposed for the complete synchronisation of a pair of identical chaotic systems. Lyapunov stability theory has been used to prove this main result of the work. Next, as an application of the main result, a sliding controller has been designed for achieving complete chaos synchronisation of identical Zhu chaotic systems (Zhu et al., 2010). Numerical simulations using MATLAB have been provided to illustrate all the Zhu chaotic systems and the novel sliding mode controller design for the global chaos synchronisation of Zhu chaotic systems. As we have presented a general procedure using novel SMC for the complete synchronisation of identical chaotic systems, this procedure
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Zarrabi, M.R., Farahi, M.H., Effati, S. and Koshkouei, A.J. (2012) ‘Sliding mode controllers for second order and extended Heisenberg systems’, Int. J. Modelling, Identification and Control, Vol. 16, No. 4, pp.345–352. Zen, R., Liu, J., Wu, X., Wu, X., Zhu, Q., Wang, Y. and Nouri, H. (2013) ‘Synchronise a new class of noise-perturbed hyper-chaotic systems by adaptive sliding mode control’, Int. J. Modelling, Identification and Control, Vol. 20, No. 2, pp.164–173.
Zhang, Z., Wang, Z. and Shen, H. (2012) ‘Synchronisation control of two identical Liu systems with known and unknown parameters’, Int. J. Modelling, Identification and Control, Vol. 17, No. 2, pp.166–172. Zhao, J. and Lu, J. (2008) ‘Using sampled-data feedback control and linear feedback synchronization in a new hyperchaotic system’, Chaos, Solitons and Fractals, Vol. 35, No. 2, pp.376–382. Zhu, C., Liu, Y. and Guo, Y. (2010) ‘Theoretic and numerical study of a new chaotic system’, Intelligent Information Management, Vol. 2, pp.104–109.
Int. J. Modelling, Identification and Control, Vol. 23, No. 1, 2015
Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system Sundarapandian Vaidyanathan* Research and Development Centre, Vel Tech University, Avadi, Chennai-600 062, Tamil Nadu, India Email: [email protected] *Corresponding author
Sivaperumal Sampath Department of Electronics and Communication Engineering, Vel Tech University, Avadi, Chennai-600 062, Tamil Nadu, India Email: [email protected]
Ahmad Taher Azar Faculty of Computers and Information, Benha University, Egypt Email: [email protected] Abstract: Synchronisation of chaotic systems is an important research problem in chaos theory. In this research work, a novel sliding mode control method is proposed for the global chaos synchronisation of identical chaotic systems. The general result derived using novel sliding mode control method is established using Lyapunov stability theory. As an application of the general result, the problem of global chaos synchronisation of identical Zhu chaotic systems (2010) is studied and a new sliding mode controller is derived. Numerical simulations have been shown to illustrate the phase portraits of Zhu chaotic system and the sliding mode controller design for the global chaos synchronisation of identical Zhu chaotic systems. Keywords: chaos; chaotic systems; sliding mode control; SMC; Lyapunov stability theory; chaos synchronisation; Zhu system. Reference to this paper should be made as follows: Vaidyanathan, S., Sampath, S. and Azar, A.T. (2015) ‘Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system’, Int. J. Modelling, Identification and Control, Vol. 23, No. 1, pp.92–100. Biographical notes: Sundarapandian Vaidyanathan is a Professor and Dean at the R&D Centre, Vel Tech University, Chennai, India. He received his DSc in Electrical and Systems Engineering from Washington University, St. Louis, USA in 1996. His current research focuses on linear and nonlinear control systems, chaotic and hyperchaotic systems, chaos control and synchronisation, computational science and robotics. He has published over 110 Scopus-indexed research publications. He has delivered plenary lectures on control systems and chaos theory in many international conferences. He has conducted many workshops on mathematical modelling using MATLAB and SCILAB. Sivaperumal Sampath received his ME (2007) in VLSI Design and BE (2005) in Electronics and Communication Engineering from Anna University, Chennai, India. He is working as an Assistant Professor in the Department of Electronics and Communications Engineering at Vel Tech University, Chennai, India. He has published many Scopus-indexed papers in international journals and conferences. His current research interests are communications, control systems, robotics and VLSI design. Ahmad Taher Azar received his MSc (2006) in System Dynamics and PhD (2009) in Adaptive Neuro-Fuzzy Systems from Faculty of Engineering, Cairo University (Egypt). He is currently working as an Assistant Professor, Faculty of Computers and Information, Benha University, Egypt. He is the Editor-in-Chief of two journals published by IGI Global, USA. He is also an Associate Editor of IEEE Transactions on Neural Networks and Learning Systems. His research
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Global chaos synchronisation of identical chaotic systems via novel sliding mode control method
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interests include: soft computing, machine learning, rough set, mathematical modelling and computer simulation, automatic control systems, systems engineering, systems dynamics, process control, robotics and systems, intelligent control systems and computational intelligence.
1
Introduction
Chaotic systems are dissipative nonlinear dynamical systems having at least one positive Lyapunov exponent. The Lyapunov exponent of a chaotic system is a measure of the divergence of points which are initially very close and this can be used to quantify chaotic systems (Leonov, 2008). Each nonlinear dynamical system has a spectrum of Lyapunov exponents, which are equal in number to the dimension of the state space. The largest Lyapunov exponent of a nonlinear dynamical system is called the maximal Lyapunov exponent (MLE). An important paradigm of a 3-D chaotic system was discovered by Lorenz (1963) while he was studying a 3-D weather model. Subsequently, many chaotic systems were found in the literature such as Rössler (1976) system, ACT system (Arneodo et al., 1980), Sprott (1994) systems, Chen (1999) system, Lü system (Lü and Chen, 2002), Liu system (Liu et al., 2004), Cai system (Cai and Tan, 2007), Tigan system (Tigan and Opris, 2008), Zhu system (Zhu et al., 2010), Pan system (Pan et al., 2010), Sundarapandian system (Sundarapandian, 2013a, 2013b, 2013c), Li-Wu system (Li et al., 2013), etc. Chaos theory has applications like chemical reactions (Li and Zhu, 2004; Villegas et al., 2012), lasers (Antonelli and Mecozzi, 2009; Hu et al., 2011), vibrations (Liu et al., 2008; Luo and Lv, 2009), turbines (Asgari et al., 2013), combustion engines (Wendeker et al., 2003), power systems (Yu et al., 2003), robotics (Volos et al., 2012), etc. In communications, chaos has applications like cryptosystems (Usama et al., 2010; Rhouma and Belghith, 2010), secure communications (Suzuki and Imai, 2006; Chen and Min, 2008; Fallahi and Leung, 2010), etc. Next, we describe the complete synchronisation of chaotic systems (A) and (B) as follows. If the chaotic system (A) is called the master or drive system and the controlled chaotic system (B) is called the slave or response system, then the complete synchronisation aims to control the slave system (B) so that the states of the slave system (B) track the states of the master system (A) asymptotically. Various control schemes have been developed to investigate the global chaos synchronisation problem such as PC method (Pecora and Carroll, 1990), OGY method (Ott et al., 1990), active control method (Ho and Hung, 2002; Chen, 2005; Sundarapandian, 2011a; Sarasu and Sundarapandian, 2011; Zhang et al., 2012), adaptive control method (Liao and Tsai, 2000; Sundarapandian and Karthikeyan, 2011; Sundarapandian, 2011b, 2013d; Sundarapandian and Pehlivan, 2012), backstepping method (Yu and Zhang, 2004; Park, 2006; Sundarapandian, 2013e), time-delay feedback method (Guo and Zhong, 2009), sliding mode control (SMC) (Zarrabi et al., 2012; Shang and
Wang, 2013; Zen et al., 2013; Wu et al., 2014; Kaur and Janardhanan, 2014), etc. In this paper, we use a novel sliding control method for deriving a general result for the global chaos synchronisation of identical chaotic systems using SMC theory. The sliding control method is an effective control tool having advantages of low sensitivity to parameter variations in the plant and disturbances affecting the plant. In the SMC design, the control dynamics has two sequential modes, viz. the reaching mode and the sliding mode. In accordance with this general design, the sliding controller design consists of two parts: a hyperplane design and controller design. First, a hyperplane is designed using the pole-placement approach in the modern control theory and then a controller is designed using the sliding condition. The stability of the overall system is guaranteed by the sliding condition and by a stable hyperplane. The rest of the paper is organised as follows. In Section 2, we discuss the problem statement for the complete synchronisation problem of identical chaotic systems. In Section 3, we derive a general result for the global chaos synchronisation of identical chaotic systems using novel sliding control method. In Section 4, we describe the Zhu chaotic system (Zhu et al., 2010) and its properties. Phase portraits of the Zhu system are described using MATLAB. In Section 5, we describe the sliding mode controller design for the identical Zhu chaotic systems using novel sliding control method and its numerical simulations using MATLAB. Finally, conclusions are given in Section 6.
2
Problem statement
This section gives a problem statement for the global chaos synchronisation of identical chaotic systems. As the master or drive system, we take the chaotic system given by x = Ax + f ( x)
(1)
In equation (1), x ∈ Rn denotes the state of the system, A ∈ Rn×n denotes the matrix of system parameters and f(x) ∈ Rn contains the nonlinear parts of the system. As the slave or response system, we take the controlled chaotic system given by y = Ay + f ( y ) + u
(2)
In equation (2), y ∈ Rn denotes the state of the system and u is the control. For the complete synchronisation of systems (1) and (2), the synchronisation error is defined as e= y−x
(3)
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A simple calculation yields the error dynamics as e = Ae + ψ ( x, y ) + u,
(4)
veq (t ) = −(CB ) −1 CAe(t )
where
ψ ( x, y ) = f ( y ) − f ( x).
(5)
Thus, the complete synchronisation problem can be defined as follows: find a controller u(x, y) so as to render the synchronisation error e(t) to be globally asymptotically stable for all values of e(0) ∈ Rn, i.e., lim e(t ) = 0 for all e(0) ∈ R n .
t →∞
3
The sliding motion is influenced by the equivalent control derived from (11) as
(6)
A novel SMC method for the global chaos synchronisation of chaotic systems
First, we set the design by setting the control as u (t ) = −ψ ( x, y ) + Bv(t )
(7)
In equation (7), B ∈ Rn is chosen such that (A, B) is completely controllable. By substituting (7) into (4), we get the closed-loop error dynamics e = Ae + Bv
(8)
The system (8) is a linear time-variant control system with single input v. Hence, we have converted the origin problem of complete chaos synchronisation of the identical chaotic systems (1) and (2) into an equivalent control problem of globally stabilising the error system (8) by a suitable choice of the feedback control (7). We start the sliding controller design by defining the sliding variable as s (e) = Ce = c1e1 + c2 e2 + " + cn en ,
(9)
where C ∈ R1×n is a constant vector to be determined. The sliding manifold S is defined as the hyperplane
{
}
S = e ∈ R n : s (e) = Ce = 0 .
(10)
We shall assume that a sliding motion occurs on the hyperplane S. In sliding mode, the following equations must be satisfied:
s ≡ 0 and s ≡ CAe + CBv = 0
(11)
We assume that CB ≠ 0
(12)
(13)
By substituting (13) into (8), we obtain the equivalent error dynamics in the sliding phase as e = Ae − (CB) −1 CAe = Ee,
(14)
where E = ⎡⎣ I − B (CB ) −1 C ⎤⎦ A
(15)
We note that E is independent of the control and has at most (n – 1) non-zero eigenvalues, depending on the chosen switching surface, while the associated eigenvectors belong to ker(C). Since (A, B) is controllable, we can use sliding control theory (Utkin, 1977) to choose B and C so that E has any desired (n – 1) stable eigenvalues. This shows that the dynamics in the sliding mode is globally asymptotically stable. Finally, for the sliding controller design, we apply a novel sliding control law, viz. s = −ks − qs 2 sgn( s )
(16)
In equation (16), sgn(⋅) denotes the sign function and the SMC constants k > 0, q > 0 are found in such a way that the sliding condition is satisfied and that the sliding motion will occur. By combining equations (11), (13) and (16), we finally obtain the SMC v(t) as
v(t ) = −(CB ) −1[C (kI + A)e + qs 2 sgn( s)]
(17)
Next, we establish the main result of this section. Theorem 1: The sliding mode controller law defined by (7) achieves global and asymptotic synchronisation of the identical chaotic systems (1) and (2) for all initial conditions x(0), y(0) ∈ Rn, where v is defined by the novel sliding control law (16), C ∈ Rn×1 is such that (A, B) is controllable, C ∈ R1×n is such that CB ≠ 0 and that the matrix E defined by (15) has (n – 1) stable eigenvalues. Proof: Upon substitution of the control laws (7) and (17) into the error dynamics (4), we get the closed-loop error dynamics as e = Ae − B (CB ) −1 ⎡⎣C (kI + A)e + qs 2 sgn( s) ⎤⎦
(18)
We shall show that the error system (18) is globally asymptotically stable by considering the quadratic Lyapunov function V ( e) =
1 2 s ( e) 2
(19)
Global chaos synchronisation of identical chaotic systems via novel sliding mode control method The sliding mode motion is characterised by the equations s (e) = 0 and s(e) = 0
(20)
By the choice of E, the dynamics in the sliding mode is globally asymptotically stable. When s(e) ≠ 0, V(e) > 0. Also, when s(e) ≠ 0, differentiating V along the error dynamics (18) or the equivalent dynamics (16), we get V (e) = ss = −ks 2 − qs 3 sgn( s ) < 0.
(21)
Hence, by Lyapunov stability theory (Leonov, 2008), the error dynamics (18) is globally asymptotically stable for all e(0) ∈ Rn. This completes the proof.
4
Properties of the Zhu chaotic system
The 2-D projections of the phase-portrait of the Zhu system (22) are depicted in Figures 2 to 4. The system (22) has Lyapunov exponents in the chaotic case as L1 = 0.6747, L2 = 0, L3 = −4.0738
The Lyapunov dimension of the Zhu system is found as DL = 2 +
L1 + L2 = 2.1656 | L3 |
The dynamics of the Lyapunov exponents is shown in Figure 5. Figure 2
Phase portrait of the Zhu system in x1 – x2 plane (see online version for colours)
Figure 3
Phase portrait of the Zhu system in x2 – x3 plane (see online version for colours)
(22)
x3 = −cx3 + x1 x2
where x1, x2, x3 are the state variables and a, b, c are positive parameters. The Zhu system (22) is chaotic when the parameter values are taken as a = 1.5, b = 2.5, c = 4.9
(23)
The 3-D phase portrait of the Zhu system (22) is depicted in Figure 1, when the parameter values are chosen as in (23) and the initial conditions are taken as follows: x1 (0) = 1.2, x2 (0) = 0.8, x3 (0) = 1.4 Figure 1
Phase portrait of the Zhu chaotic system in R3 (see online version for colours)
(24)
(25)
The MLE of the Zhu system is L1 = 0.6747.
This section describes the qualitative properties of the Zhu chaotic system (Zhu et al., 2010), which is a new 3-D chaotic system. The Zhu system is described by the dynamics x1 = − x1 − ax2 + x2 x3 x2 = bx2 − x1 x3
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Figure 4
Phase portrait of the Zhu system in x1 – x3 plane (see online version for colours)
where x1, x2, x3 are the state variables and a, b, c are positive parameters. As the slave system, we consider the controlled Zhu chaotic system given by y1 = − y1 − ay2 + y2 y3 + u1 y 2 = by2 − y1 y3 + u2
(27)
y3 = −cy3 + y1 y2 + u3
where y1, y2, y3 are the state variables and u1, u2, u3 are the controls. The complete synchronisation error is defined by e1 = y1 − x1 e2 = y2 − x2 e3 = y3 − x3
(28)
P (ω , θ , γ ) = 1 +
Then the error dynamics is obtained as Figure 5
e1 = −e1 − ae2 + y2 y3 − x2 x3 + u1 e2 = be2 − y1 y3 + x1 x3 + u2
The dynamics of the Lyapunov exponents (see online version for colours)
(29)
e3 = −ce3 + y1 y2 − x1 x2 + u3
In matrix form, we can write the error dynamics (28) as e = Ae + ψ ( x, y ) + u,
(30)
where ⎡ −1 −a 0 ⎤ A = ⎢⎢ 0 b 0 ⎥⎥ , ⎢⎣ 0 0 −c ⎥⎦
(31)
⎡ y2 y3 − x2 x3 ⎤ ψ ( x, y ) = ⎢⎢ − y1 y3 + x1 x3 ⎥⎥ , ⎢⎣ y1 y2 − x1 x2 ⎥⎦
(32)
and
5
Sliding mode controller design for the global chaos synchronisation of Zhu systems
This section describes the sliding mode controller design for the global chaos synchronisation of Zhu (2010) chaotic systems based on the novel method described by Theorem 1 in Section 3. As the master system, we consider the Zhu chaotic system given by x1 = − x1 − ax2 + x2 x3 x2 = bx2 − x1 x3 x3 = −cx3 + x1 x2
(26)
⎡ u1 ⎤ u = ⎢⎢u2 ⎥⎥ ⎢⎣ u3 ⎥⎦
(33)
We follow the procedure given in Section 3 for the construction of the novel sliding controller to achieve complete synchronisation of the identical Zhu systems (26) and (27). First, we set u as u (t ) = −ψ ( x, y ) + Bv(t ),
(34)
where B is selected such that (A, B) is completely controllable.
Global chaos synchronisation of identical chaotic systems via novel sliding mode control method A simple choice of B is ⎡1⎤ B = ⎢⎢1⎥⎥ . ⎢⎣1⎥⎦
The controllability calculated as follows.
(35)
matrix
Z = [ B AB A2 B]
⎡1 −2.5 −1.25 ⎤ Z = ⎢⎢1 2.5 6.25⎥⎥ ⎢⎣1 −4.9 24.01⎥⎦
is
Figures 6 to 8 show the complete synchronisation of the Zhu chaotic systems (25) and (26). Figure 9 shows the time-history of the synchronisation errors e1, e2, e3. Figures 6 to 8 show that the states (x1, y1), (x2, y2) and (x3, y3) are synchronised in two seconds, while Figure 9 shows that the chaos synchronisation errors e1, e2, e3 decay to zero in two seconds. Figure 6
Synchronisation of Zhu systems: states x1 and y1 (see online version for colours)
Figure 7
Synchronisation of Zhu systems: states x2 and y2 (see online version for colours)
(36)
The controllability matrix Z has full rank. Thus, it follows that (A, B) is completely controllable. The Zhu system displays a strange attractor when the parameter values are taken as a = 1.5, b = 2.5, c = 4.9
(37)
The sliding mode variable is selected as s = Ce = [1 −2.1 1] e
(38)
which renders the sliding motion globally asymptotically stable. Next, we take the sliding mode gains as k = 5 and q = 0.2.
(39)
From equation (17) of Section 3, we obtain the novel sliding control v as v = 40e1 − 172.5e2 + e3 + 2 s 2 sgn( s )
(40)
As an application of Theorem 1 to the identical Zhu chaotic systems, we obtain the following main result of this section. Theorem 2: The identical Zhu chaotic systems (26) and (27) are globally and asymptotically synchronised for all initial conditions x(0), y(0) ∈ R3 with the sliding controller u defined by (34), where ψ(x, y) is defined by (32), B is defined by (35), and v is defined by (40). For numerical simulations, we use MATLAB software for solving systems of differential equations using the classical fourth order Runge-Kutta method with step size h = 10–8. The parameters of the Zhu chaotic systems (25) and (26) are taken as in the chaotic case, viz., a = 1.5, b = 2.5, c = 4.9
As an initial condition for the Zhu system (25), we take x1 (0) = 2.7, x2 (0) = 1.5, x3 (0) = 2.4
As an initial condition for the Zhu system (26), we take y1 (0) = −5.6, y2 (0) = 4.3, y3 (0) = 8.7
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Figure 8
Synchronisation of Zhu systems: states x3 and y3 z (see online version for colours)
can be applied for other chaotic systems as well and the efficiency of this procedure in chaos applications may be studied.
References
Figure 9
6
Time-history of the synchronisation errors e1, e2, e3 (see online version for colours)
Conclusions
In this research work, a novel SMC result has been proposed for the complete synchronisation of a pair of identical chaotic systems. Lyapunov stability theory has been used to prove this main result of the work. Next, as an application of the main result, a sliding controller has been designed for achieving complete chaos synchronisation of identical Zhu chaotic systems (Zhu et al., 2010). Numerical simulations using MATLAB have been provided to illustrate all the Zhu chaotic systems and the novel sliding mode controller design for the global chaos synchronisation of Zhu chaotic systems. As we have presented a general procedure using novel SMC for the complete synchronisation of identical chaotic systems, this procedure
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