Hybrid Synchronization of Identical Hyperchaotic Systems via Novel Sliding Control with Application to Hyperchaotic Vaidyanathan- Volos System
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1 I J C T A, 9(38), 2016, pp Iteratioal Sciece Press Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol with Applicatio to Hyperchaotic Vaidyaatha- Volos System S. Sivaperumal* ABSTRACT First, this paper proposes a geeral procedure for the hybrid sychroizatio of idetical hyperchaotic systems usig ovel slidig mode cotrol. The geeral result derived usig ovel slidig mode cotrol method is established via Lyapuov stability theory. As a applicatio of the geeral result, the problem of hybrid sychroizatio of idetical hyperchaotic Vaidyaatha-Volos systems (2016) is studied ad a ew slidig mode cotroller is derived. Numerical simulatios with MATLAB have bee show to illustrate all the mai results derived i this work. Keywords: Hyperchaos, hyperchaos sychroizatio, hybrid sychroizatio, slidig mode cotrol. 1. INTRODUCTION A chaotic system is commoly defied as a oliear dissipative dyamical system that is highly sesitive to eve small perturbatios i its iitial coditios [1]. I other words, a chaotic system is a oliear dyamical system with at least oe positive Lyapuov expoet. Some paradigms of chaotic systems ca be listed as Areodo system [4], Sprott systems [5], Che system [6], Lü-Che system [7], Liu system [8], Cai system [9], Tiga system [10], etc. I the last two decades, may ew chaotic systems have bee also discovered like Li system [11], Sudarapadia systems [12-13], Vaidyaatha systems [14-33], Pehliva systems [34-35], Pham systems [36-37], Jafari system [38], etc. Hyperchaotic systems are the chaotic systems with more tha oe positive Lyapuov expoet. They have importat applicatios i cotrol ad commuicatio egieerig. Some recetly discovered 4-D hyperchaotic systems are hyperchaotic Vaidyaatha systems [39-40], hyperchaotic Vaidyaatha-Azar system [41], hyperchaotic Sampath system [42], etc, Chaos theory has several applicatios i a variety of fields such as oscillators [43-44], chemical reactors [45-58], biology [59-80], ecology [81-82], eural etworks [83-84], robotics [85-86], memristors [87-89], fuzzy systems [90-91], etc. The problem of cotrol of a chaotic system is to fid a state feedback cotrol law to stabilize a chaotic system aroud its ustable equilibrium [92-93]. Some popular methods for chaos cotrol are active cotrol [94-98], adaptive cotrol [99-100], slidig mode cotrol [ ], etc. If a particular chaotic system is called the master or drive system ad aother chaotic system is called the slave or respose system, the the idea of hybrid sychroizatio is to completely sychroize oe part * School of Electrical ad Computig, Vel Tech Uiversity, Avadi, Cheai, Tamil Nadu, Idia, sivaperumals@gmail.com.
2 262 S. Sivaperumal of the systems (amely, the odd states) ad ati-sychroize the other part of the systems (amely, the eve states) so that both complete ad ati-sychroizatio persist i the process of sychroizatio of the master ad slave systems. The sychroizatio of chaotic systems has applicatios i secure commuicatios [ ], cryptosystems [ ], ecryptio [ ], etc. The chaos sychroizatio problem has bee paid great attetio i the literature ad a variety of impressive approaches have bee proposed. Sice the pioeerig work by Pecora ad Carroll [ ] for the chaos sychroizatio problem, may differet methods have bee proposed i the cotrol literature such as active cotrol method [ ], adaptive cotrol method [ ], sampled-data feedback cotrol method [ ], time-delay feedback approach [152], backsteppig method [ ], slidig mode cotrol method [ ], etc. I this paper, ew results have bee derived for the hybrid sychroizatio of idetical hyperchaotic systems usig ovel slidig cotrol. The slidig mode cotrol has advatages of low sesitivity to parameter variatios i the plat ad disturbaces affectig the plat. I Sectio 2, we describe the hybrid sychroizatio of idetical hyperchaotic systems. I Sectio 3, we derive a geeral result for the hybrid sychroizatio of idetical hyperchaotic systems usig ovel slidig mode cotrol. I Sectio 4, we describe the hyperchaotic Vaidyaatha-Volos system ([42], 2016) ad its qualitative properties. The phase portraits of the hyperchaotic Vaidyaatha-Volos system are described usig MATLAB. I Sectio 5, we describe the slidig mode cotroller desig for the hybrid sychroizatio of the hyperchaotic Vaidyaatha-Volos systems usig ovel slidig mode cotrol ad its umerical simulatios usig MATLAB. Sectio 6 cotais a summary of the mai results derived i this paper. 2. HYBRID SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC SYSTEMS I this sectio, we provide a problem statemet for the hybrid sychroizatio of idetical hyperchaotic systems. As the master or drive system, we cosider the hyperchaotic system give by x Ax ( x) (1) I Eq. (1), x R deotes the state of the system, A deotes the matrix of system parameters ad cotais the oliear parts of the system. As the slave or respose system, we take the cotrolled hyperchaotic system give by I Eq. (2), y R deotes the state of the system ad y Ay ( y) u (2) u R is the cotrol. The hybrid sychroizatio error betwee the hyperchaotic systems (1) ad (2) is defied as yi xi if i is odd ei yi xi if i is eve A simple calculatio yields the error dyamics as (3) e Ae ( x, y) u (4) Thus, the hybrid sychroizatio problem for the hyperchaotic systems (1) ad (2) ca be defied as follows: Fid a cotroller u so as to reder the hybrid sychroizatio error e(t) to be globally asymptotically stable for all values of e(0) R, i.e.
3 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol lim e( t) 0 for all e(0) R. (5) t 3. A NOVEL SLIDING CONTROLLER DESIGN First, we set the desig by settig the cotrol as I Eq. (6), u( t) ( x, y) Bv( t) (6) B R is chose such that (A, B) is completely cotrollable. By substitutig (6) ito (4), we get the closed-loop error dyamics e Ae Bv (7) The system (7) is a liear time-variat cotrol system with sigle iput v. We start the slidig cotroller desig by defiig the slidig variable as s( e) Ce c e c e c e, (8) where 1 C R is a costat vector to be determied. The slidig maifold is defied as the hyperplae We shall assume that a slidig motio occurs o the hyperplae S. I slidig mode, the followig equatios must be satisfied: We assume that S e R : s( e) Ce 0. (9) s 0 ad s CAe CBv 0 (10) CB 0 (11) The slidig motio is iflueced by the equivalet cotrol derived from (10) as v t CB CAe t (12) ( ) ( ) 1 ( ) eq By substitutig (12) ito (7), we obtai the equivalet error dyamics i the slidig phase as where 1 e Ae ( CB) CAe Ee, (13) 1 E [ I B( CB) C] A (14) We ote that E is idepedet of the cotrol ad has at most ( 1) ozero eigevalues, depedig o the chose switchig surface, while the associated eigevectors belog to ker(c) Sice (A, B) is cotrollable, we ca use slidig cotrol theory to choose B ad C so that E has ay desired ( 1) stable eigevalues. This shows that the dyamics i the slidig mode is globally asymptotically stable. Fially, for the slidig cotroller desig, we apply a ovel slidig cotrol law, viz. s ks qs 2 sg( s) (15) I Eq. (15), sg(.) deotes the sig fuctio ad the slidig mode cotrol costats k > 0, q > 0 are foud i such a way that the slidig coditio is satisfied ad that the slidig motio will occur.
4 264 S. Sivaperumal as By combiig equatios (10), (12) ad (15), we fially obtai the slidig mode cotrol (SMC) v(t) Next, we establish the mai result of this sectio. 1 2 v( t) ( CB) [ C( ki A) e qs sg( s)] (16) Theorem 1. The slidig mode cotroller law defied by (6) achieves global ad asymptotic hybrid sychroizatio of the idetical hyperchaotic systems (1) ad (2) for all iitial coditios x(0), y(0) R, where v is defied by the ovel slidig cotrol law (16), 1 B R is such that (A, B) is cotrollable, 1 C R is such that CB 0 ad that the matrix E defied by (14) has ( 1) stable eigevalues. Proof. Upo substitutio of the cotrol laws (6) ad (16) ito the error dyamics (4), we get the closedloop error dyamics as 1 2 e Ae B( CB) C( ki A) e qs sg( s) We shall show that the error system (17) is globally asymptotically stable by cosiderig the quadratic Lyapuov fuctio The slidig mode motio is characterized by the equatios (17) 1 2 V ( e) s ( e) (18) 2 s( e) 0 ad s ( e) 0 (19) By the choice of E, the dyamics i the slidig mode is globally asymptotically stable. Whe s( e) 0, V ( e) 0. Also, whe s( e) 0, differetiatig V alog the error dyamics (17) or the equivalet dyamics (15), we get 2 3 V ( e) ss ks qs sg( s) 0. (20) Hece, by Lyapuov stability theory [174], the error dyamics (17) is globally asymptotically stable for all e(0) R. This completes the proof. 4. HYPERCHAOTIC VAIDYANATHAN-VOLOS SYSTEM Hyperchaotic Vaidyaatha-Volos system ([42], 2016) is described by the 4-D dyamics x 1 a( x2 x1 ) x4 x bx px x x x 3 x1x2 cx3 x 4 x1 x (21) where x1, x2, x3, x 4 are state variables ad a, b, c, p are positive parameters of the system. The Vaidyaatha-Volos system (21) exhibits a hyperchaotic attractor for the values
5 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol a 12, b 36, c 5, p 12 (22) The Lyapuov expoets of the system (21) are umerically obtaied with MATLAB as L , L , L 0, L (23) As there are two positive Lyapuov expoets i (23), the system (21) is hyperchaotic. Sice the sum of the Lyapuov expoets i (23) is egative, the hyperchaotic Vaidyaatha-Volos system is dissipative. The Lyapuov dimesio of the hyperchaotic Vaidyaatha-Volos system (21) is foud as D L L L L L4 (24) For simulatios, the iitial values of the Vaidyaatha-Volos system (21) are take as x (0) 1.8, x (0) 1.6, x (0) 1.2, x (0) 1.4 (25) Figures 1-4 show the 3-D view of the hyperchaotic attractor of the Vaidyaatha-Volos system (21) i ( x1, x2, x 3), ( x1, x2, x4), ( x1, x3, x4) ad ( x2, x3, x 4) spaces, respectively. Figure 1: 3-D view of hyperchaotic Vaidyaatha-Volos system i (x 1, x 2, x 3 ) space
6 266 S. Sivaperumal Figure 2: 3-D view of hyperchaotic Vaidyaatha-Volos system i (x1, x2, x4) space Figure 3: 3-D view of hyperchaotic Vaidyaatha-Volos system i (x1, x3, x4) space
7 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol Figure 4: 3-D view of hyperchaotic Vaidyaatha-Volos system i (x 2, x 3, x 4 ) space 5. HYBRID SYNCHRONIZATION OF HYPERCHAOTIC VAIDYANATHAN-VOLOS SYSTEMS VIA NOVEL SLIDING CONTROL I this sectio, we describe ovel slidig mode cotrol desig for the hybrid sychroizatio of idetical hyperchaotic Vaidyaatha-Volos systems. As the master system, we cosider the hyperchaotic Vaidyaatha-Volos system give by x 1 a( x2 x1 ) x4 x bx px x x x 3 x1x2 cx3 x 4 x1 x (26) where x 1, x 2, x 3, x 4 are state variables ad a, b, c, d, p, q are positive parameters. As the slave system, we cosider the hyperchaotic Vaidyaatha-Volos system give by y 1 a( y2 y1) y4 u1 y by py y y u y 3 y1 y2 cy3 u3 y 4 y1 y2 u (27) where y 1, y 2, y 3, y 4 are state variables ad u 1, u 2, u 3, u 4 are the cotrols.
8 268 S. Sivaperumal The hybrid sychroizatio error betwee the systems (26) ad (27) is defied by e1 y1 x1 e y x e y x e y x (28) The the error dyamics is obtaied as e 1 a( e2 e1 ) e4 2ax2 2x4 u1 e be e 2 bx p( y y x x ) u e 3 ce3 y1 y2 x1 x2 u3 e 4 e1 e2 2x1 u (29) I matrix form, we ca write the error dyamics (29) as where e Ae ( x, y) u, (30) a a 0 1 2ax2 2x4 u1 b bx1 p( y1 y3 x1 x3 ) u, (, ), 2 A x y u 0 0 c 0 y1 y2 x1 x 2 u x u 1 4 (31) First, we set u as where B is selected such that (A, B) is completely cotrollable. We choose B as u( t) ( x, y) Bv( t), (32) 1 1 B 1 1 We choose the parameters of the Vaidyaatha-Volos systems as i the hyperchaotic case, viz. The slidig mode variable is selected as (33) a 12, b 36, c 5, p 12 (34) s Ce e e e e e (35) which reders the slidig motio globally asymptotically stable. Next, we take the slidig mode gais as
9 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol From Eq. (16) of Sectio 3, we obtai the ovel slidig cotrol v as k 6 ad q 0.2. (36) v e e e e s s (37) sg( ) As a applicatio of Theorem 1 to the idetical hyperchaotic Vaidyaatha-Volos systems, we obtai the followig mai result of this sectio. Theorem 2. The idetical hyperchaotic Vaidyaatha-Volos systems (26) ad (27) are globally ad 4 asymptotically hybrid sychroized for all iitial coditios x(0), y(0) R with the slidig cotroller u defied by (32), where ( x, y) ad B are defied by (31) adis defied by (37). For umerical simulatio, we take the parameter values as i the hyperchaotic case, i.e. a 12, b 36, c 5, p 12 As a iitial coditio for the master system (25), we take x (0) 5.2, x (0) 2.3, x (0) 4.5, x (0) As a iitial coditio for the slave system (26), we take y (0) 1.4, y (0) 5.8, y (0) 2.6, y (0) Figures 5-8 depict the hybrid sychroizatio of the hyperchaotic Vaidyaatha-Volos systems. Figure 9 depicts the time-history of the hybrid sychroizatio errors. Figure 5: Hybrid sychroizatio of the states x 1 ad y 1
10 270 S. Sivaperumal Figure 6: Hybrid sychroizatio of the states x 2 ad y 2 Figure 7: Hybrid sychroizatio of the states x 3 ad y 3
11 Hybrid Sychroizatio of Idetical Hyperchaotic Systems via Novel Slidig Cotrol Figure 8: Hybrid sychroizatio of the states x 4 ad y 4 Figure 9: Time history of the hybrid sychroizatio errors
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SLIDING CONTROLLER DESIGN FOR THE HYBRID CHAOS SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC XU SYSTEMS
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