International Journal of Instrumentation and Control Systems (IJICS) is a Quarterly open access peer-reviewed journal that publishes articles which contribute new results in all areas of Instrumentation Engineering and Control Systems. The journal focuses on all technical and practical aspects of Instrumentation Engineering and Control Systems. The goal of this journal is to bring together researchers and practitioners from academia and industry to focus on advanced instrumentation engineering, control systems and automation concepts and establishing new collaborations in these areas.
Authors are solicited to contribute to this journal by submitting articles that illustrate research results, projects, surveying works and industrial
experiences that describe significant advances in the Instrumentation Engineering
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG AND HYPERCHAOTIC LI SYSTEMS WITH UN...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
ADAPTIVESYNCHRONIZER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC ZH...ijitcs
This paper derives new adaptive synchronizers for the hybrid synchronization of hyperchaotic Zheng
systems (2010) and hyperchaotic Yu systems (2012). In the hybrid synchronization design of master and
slave systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the
other part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the
process of synchronization. The research problem gets even more complicated, when the parameters of the
hyperchaotic systems are not known and we handle this complicate problem using adaptive control. The
main results of this research work are established via adaptive control theory andLyapunov stability
theory. MATLAB plotsusing classical fourth-order Runge-Kutta method have been depictedfor the new
adaptive hybrid synchronization results for the hyperchaotic Zheng and hyperchaotic Yu systems.
ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU ...ijait
This paper derives new adaptive results for the hybrid synchronization of hyperchaotic Xi systems (2009)
and hyperchaotic Li systems (2005). In the hybrid synchronization design of master and slave systems, one
part of the systems, viz. their odd states, are completely synchronized (CS), while the other part, viz. their
even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process of
synchronization. The research problem gets even more complicated, when the parameters of the
hyperchaotic systems are unknown and we tackle this problem using adaptive control. The main results of
this research work are proved using adaptive control theory and Lyapunov stability theory. MATLAB
simulations using classical fourth-order Runge-Kutta method are shown for the new adaptive hybrid
synchronization results for the hyperchaotic Xu and hyperchaotic Li systems.
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC ZHEN...ijscai
This paper deals with a new research problem in the chaos literature, viz. hybrid synchronization of a
pair of chaotic systems called the master and slave systems. In the hybrid synchronization design of
master and slave systems, one part of the systems, viz. their odd states, are completely synchronized (CS),
while the other part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS coexist in the process of synchronization. This research work deals with the hybrid synchronization of
hyperchaotic Zheng systems (2010) and hyperchaotic Yu systems (2012). The main results of this hybrid
synchronization research work have been proved using Lyapunov stability theory. Numerical examples of
the hybrid synchronization results are shown along with MATLAB simulations for the hyperchaotic
Zheng and hyperchaotic Yu systems.
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU AN...Zac Darcy
The synchronization of chaotic systems treats a pair of chaotic systems, which are usually called as master
and slave systems. In the chaos synchronization problem, the goal of the design is to synchronize the states
of master and slave systems asymptotically. In the hybrid synchronization design of master and slave
systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the other
part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process
of synchronization. This research work deals with the hybrid synchronization of hyperchaotic Xi systems
(2009) and hyperchaotic Li systems (2005). The main results of this hybrid research work are established
with Lyapunov stability theory. MATLAB simulations of the hybrid synchronization results are shown for
the hyperchaotic Xu and Li systems.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems.
Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM ijcseit
The hyperchaotic Liu system (Wang and Liu, 2006) is one of the important models of four-dimensional
hyperchaotic systems. This paper investigates the adaptive chaos control and synchronization of
hyperchaotic Liu system with unknown parameters. First, adaptive control laws are designed to stabilize
the hyperchaotic Liu system to its unstable equilibrium at the origin based on the adaptive control theory
and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Liu systems with unknown parameters. Numerical simulations
are presented to demonstrate the effectiveness of the proposed adaptive chaos control and
synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF SPROTT-I SYSTEM WITH UNKNOWN PARAMETERSijscai
This paper derives new results for the adaptive control and synchronization design of the Sprott-I chaotic system (1994), when the system parameters are unknown. First, we build an adaptive controller to stabilize the Sprott-I chaotic system to its unstable equilibrium at the origin. Then we build an adaptive
synchronizer to achieve global chaos synchronization of the identical Sprott-I chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Sprott-I chaotic system have been established using adaptive control theory and Lyapunov stability
theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Sprott-I chaotic system.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG AND HYPERCHAOTIC LI SYSTEMS WITH UN...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
ADAPTIVESYNCHRONIZER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC ZH...ijitcs
This paper derives new adaptive synchronizers for the hybrid synchronization of hyperchaotic Zheng
systems (2010) and hyperchaotic Yu systems (2012). In the hybrid synchronization design of master and
slave systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the
other part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the
process of synchronization. The research problem gets even more complicated, when the parameters of the
hyperchaotic systems are not known and we handle this complicate problem using adaptive control. The
main results of this research work are established via adaptive control theory andLyapunov stability
theory. MATLAB plotsusing classical fourth-order Runge-Kutta method have been depictedfor the new
adaptive hybrid synchronization results for the hyperchaotic Zheng and hyperchaotic Yu systems.
ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU ...ijait
This paper derives new adaptive results for the hybrid synchronization of hyperchaotic Xi systems (2009)
and hyperchaotic Li systems (2005). In the hybrid synchronization design of master and slave systems, one
part of the systems, viz. their odd states, are completely synchronized (CS), while the other part, viz. their
even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process of
synchronization. The research problem gets even more complicated, when the parameters of the
hyperchaotic systems are unknown and we tackle this problem using adaptive control. The main results of
this research work are proved using adaptive control theory and Lyapunov stability theory. MATLAB
simulations using classical fourth-order Runge-Kutta method are shown for the new adaptive hybrid
synchronization results for the hyperchaotic Xu and hyperchaotic Li systems.
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC ZHEN...ijscai
This paper deals with a new research problem in the chaos literature, viz. hybrid synchronization of a
pair of chaotic systems called the master and slave systems. In the hybrid synchronization design of
master and slave systems, one part of the systems, viz. their odd states, are completely synchronized (CS),
while the other part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS coexist in the process of synchronization. This research work deals with the hybrid synchronization of
hyperchaotic Zheng systems (2010) and hyperchaotic Yu systems (2012). The main results of this hybrid
synchronization research work have been proved using Lyapunov stability theory. Numerical examples of
the hybrid synchronization results are shown along with MATLAB simulations for the hyperchaotic
Zheng and hyperchaotic Yu systems.
ACTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU AN...Zac Darcy
The synchronization of chaotic systems treats a pair of chaotic systems, which are usually called as master
and slave systems. In the chaos synchronization problem, the goal of the design is to synchronize the states
of master and slave systems asymptotically. In the hybrid synchronization design of master and slave
systems, one part of the systems, viz. their odd states, are completely synchronized (CS), while the other
part, viz. their even states, are completely anti-synchronized (AS) so that CS and AS co-exist in the process
of synchronization. This research work deals with the hybrid synchronization of hyperchaotic Xi systems
(2009) and hyperchaotic Li systems (2005). The main results of this hybrid research work are established
with Lyapunov stability theory. MATLAB simulations of the hybrid synchronization results are shown for
the hyperchaotic Xu and Li systems.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems.
Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM ijcseit
The hyperchaotic Liu system (Wang and Liu, 2006) is one of the important models of four-dimensional
hyperchaotic systems. This paper investigates the adaptive chaos control and synchronization of
hyperchaotic Liu system with unknown parameters. First, adaptive control laws are designed to stabilize
the hyperchaotic Liu system to its unstable equilibrium at the origin based on the adaptive control theory
and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos
synchronization of identical hyperchaotic Liu systems with unknown parameters. Numerical simulations
are presented to demonstrate the effectiveness of the proposed adaptive chaos control and
synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF SPROTT-I SYSTEM WITH UNKNOWN PARAMETERSijscai
This paper derives new results for the adaptive control and synchronization design of the Sprott-I chaotic system (1994), when the system parameters are unknown. First, we build an adaptive controller to stabilize the Sprott-I chaotic system to its unstable equilibrium at the origin. Then we build an adaptive
synchronizer to achieve global chaos synchronization of the identical Sprott-I chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Sprott-I chaotic system have been established using adaptive control theory and Lyapunov stability
theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Sprott-I chaotic system.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
Adaptive Controller Design For The Synchronization Of Moore-Spiegel And Act S...ijcsa
In this paper, we design adaptive controllers for the global chaos synchronization of identical MooreSpiegel systems (1966), identical ACT systems (1981) and non-identical Moore-Spiegel and ACT chaotic systems with unknown parameters. Our adaptive synchronization results derived in this paper for
uncertain Moore-Spiegel and ACT systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical uncertain Moore-Spiegel and ACT chaotic
systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the global chaos synchronization of the uncertain chaotic systems derived in this paper.
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF LÜ-LIKE ATTRACTORijcseit
This paper derives new results for the adaptive chaos stabilization and synchronization of Lü-like attractor
with unknown parameters. The Lü-like attractor is one of the recently discovered 3-scroll chaotic systems,
which was proposed by D. Li (2007). First, adaptive control laws are determined to stabilize the Lü-like
attractor to its unstable equilibrium at the origin. These adaptive laws are established using Lyapunov
stability theory. Then adaptive synchronization laws are determined so as to achieve global chaos
synchronization of identical Lü-like attractors with unknown parameters. Numerical simulations are
presented to validate and demonstrate the effectiveness of the proposed adaptive control and
synchronization schemes for the Lü-like attractor.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU’S FOUR-WING CHAOTIC SYSTEM WITH C...IJCSEA Journal
This paper investigates the adaptive chaos control and synchronization of Liu’s four-wing chaotic system with cubic nonlinearity (Liu, 2009) and unknown parameters. First, we design adaptive control laws to stabilize the Liu’s four-wing chaotic system with cubic nonlinearity to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Next, we derive adaptive control laws to achieve global chaos synchronization of identical Liu’s four-wing chaotic systems with cubic nonlinearity and unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive chaos control and synchronization schemes.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTI...IJCSEIT Journal
This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
hyperchaotic Xu systems.
SLIDING CONTROLLER DESIGN FOR THE GLOBAL CHAOS SYNCHRONIZATION OF IDENTICAL H...ijait
This paper establishes new results for the sliding controller design for the global chaos synchronization of identical hyperchaotic Yujun systems (2010). Hyperchaotic systems are chaotic nonlinear systems having more than one positive Lyapunov exponent. Because of the complex dynamics properties of hyperchaotic system such as high capacity, high security and high efficiency, they are very useful in secure
communication devices and data encryption. Using sliding mode control theory and Lyapunov stability theory, a general result has been obtained for the global chaos synchronization of identical chaotic nonlinear systems. As an application of this general result, this paper designs a sliding controller for the
global chaos synchronization of hyperchaotic Yujun systems. Numerical results and simulations are shown to validate the proposed sliding controller design and demonstrate its effectiveness in achieving global chaos synchronization of hyperchaotic Yujun systems.
STATE FEEDBACK CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF SPROTT-H SYSTEMijistjournal
This paper investigates the problem of state feedback controller design for the output regulation of SprottK chaotic system, which is one of the simple, classical, three-dimensional chaotic systems discovered by J.C. Sprott (1994). Explicitly, we have derived new state feedback control laws for the constant tracking problem for the Sprott-H system. The state feedback control laws have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott H chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-H chaotic system.
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
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ACTIVE CONTROLLER DESIGN FOR THE GENERALIZED PROJECTIVE SYNCHRONIZATION OF TH...ijait
This paper discusses the design of active controllers for generalized projective synchronization (GPS) of identical Wang 3-scroll chaotic systems (Wang, 2009), identical Dadras 3-scroll chaotic systems (Dadras and Momeni, 2009) and non-identical Wang 3-scroll system and Dadras 3-scroll system. The synchronization results (GPS) derived in this paper for the 3 scroll chaotic systems have been derived using active control method and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active control method is very effective and convenient for
achieving the generalized projective synchronization (GPS) of the 3-scroll chaotic systems addressed in this paper. Numerical simulations are provided to illustrate the effectiveness of the GPS synchronization results derived in this paper.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Ch...ijtsrd
In this paper, a class of uncertain chaotic and non-chaotic systems is firstly introduced and the state observation problem of such systems is explored. Based on the time-domain approach with integral and differential equalities, an exponential state observer for a class of uncertain nonlinear systems is established to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be calculated correctly. Finally, numerical simulations are presented to exhibit the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Chaotic Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20219.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20219/exponential-state-observer-design-for-a-class-of-uncertain-chaotic-and-non-chaotic-systems/yeong-jeu-sun
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
We give an elementary exposition of a method to obtain the infinitesimal point symmetries of Lagrangians.Besides, we exhibit the Lanczos approach to Noether’s theorem to construct the first integral associated with each symmetry.
MSC 2010:49S05, 58E30, 70H25, 70H33
Research on 4-dimensional Systems without Equilibria with ApplicationTELKOMNIKA JOURNAL
Recently chaos-based encryption has been obtained more and more attention. Chaotic systems
without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs)
because there does not exist corresponding chaos criterion theorem on such systems. This paper
proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions
introduces a chaotic system without equilibria. Using this system and the generalized chaos
synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization
(8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results
show that there are no significant correlations between the key stream and the perturbed key streams
generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As
an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image.
The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar
to those generated via ideal chaotic PRNGs.
Simultaneous State and Actuator Fault Estimation With Fuzzy Descriptor PMID a...Waqas Tariq
In this paper, Takagi-Sugeno (T-S) fuzzy descriptor proportional multiple-integral derivative (PMID) and Proportional-Derivative (PD) observer methods that can estimate the system states and actuator faults simultaneously are proposed. T-S fuzzy model is obtained by linearsing satellite/spacecraft attitude dynamics at suitable operating points. For fault estimation, actuator fault is introduced as state vector to develop augmented descriptor system and robust fuzzy PMID and PD observers are developed. Stability analysis is performed using Lyapunov direct method. The convergence conditions of state estimation error are formulated in the form of LMI (linear matrix inequality). Derivative gain, obtained using singular value decomposition of descriptor state matrix (E), gives more design degrees of freedom together with proportional and integral gains obtained from LMI. Simulation study is performed for our proposed methods.
Solving output control problems using Lyapunov gradient-velocity vector functionIJECEIAES
This paper describes a controller and observer parameter definition approach in one input-one output (closed-loop) control systems using Lyapunov gradient-velocity vector function. Construction of the vector function is based on the gradient nature of the control systems and the parity of the vector functions with the potential function from the theory of catastrophe. Investigation of the closed-loop control system’s stability and solution of the problem of controller (determining the coefficient of magnitude matrix) and observer (calculation of the matrix elements of the observing equipment) synthesis is based on the direct methods of Lyapunov. The approach allows to select parameters based on the requested characteristics of the system.
International Journal of Computer Science, Engineering and Information Techno...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
Adaptive Controller and Synchronizer Design for Hyperchaotic Zhou System with...Zac Darcy
In this paper, we establish new results for the adaptive controller and synchronizer design for the
hyperchaotic Zhou system (2009), when the parameters of the system are unknown. Using adaptive control theory and Lyapunov stability theory, we first design an adaptive controller to stabilize the hyperchaotic Zhou system to its unstable equilibrium at the origin. Next, using adaptive control theory and Lyapunov stability theory, we design an adaptive controller to achieve global chaos synchronization
of the identical hyperchaotic Zhou systems with unknown parameters. Simulations have been provided for adaptive controller and synchronizer designs to validate and illustrate the effectiveness of the schemes.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC PANG AND HYPERCHAOTIC WANG-CHEN SYSTEMS ...ijctcm
Hyperchaotic systems are chaotic systems having more than one positive Lyapunov exponent and they have
important applications in secure data transmission and communication. This paper applies active control
method for the synchronization of identical and different hyperchaotic Pang systems (2011) and
hyperchaotic Wang-Chen systems (2008). Main results are proved with the stability theorems of Lypuanov
stability theory and numerical simulations are plotted using MATLAB to show the synchronization of
hyperchaotic systems addressed in this paper.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
Adaptive Controller Design For The Synchronization Of Moore-Spiegel And Act S...ijcsa
In this paper, we design adaptive controllers for the global chaos synchronization of identical MooreSpiegel systems (1966), identical ACT systems (1981) and non-identical Moore-Spiegel and ACT chaotic systems with unknown parameters. Our adaptive synchronization results derived in this paper for
uncertain Moore-Spiegel and ACT systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical uncertain Moore-Spiegel and ACT chaotic
systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the global chaos synchronization of the uncertain chaotic systems derived in this paper.
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF LÜ-LIKE ATTRACTORijcseit
This paper derives new results for the adaptive chaos stabilization and synchronization of Lü-like attractor
with unknown parameters. The Lü-like attractor is one of the recently discovered 3-scroll chaotic systems,
which was proposed by D. Li (2007). First, adaptive control laws are determined to stabilize the Lü-like
attractor to its unstable equilibrium at the origin. These adaptive laws are established using Lyapunov
stability theory. Then adaptive synchronization laws are determined so as to achieve global chaos
synchronization of identical Lü-like attractors with unknown parameters. Numerical simulations are
presented to validate and demonstrate the effectiveness of the proposed adaptive control and
synchronization schemes for the Lü-like attractor.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU’S FOUR-WING CHAOTIC SYSTEM WITH C...IJCSEA Journal
This paper investigates the adaptive chaos control and synchronization of Liu’s four-wing chaotic system with cubic nonlinearity (Liu, 2009) and unknown parameters. First, we design adaptive control laws to stabilize the Liu’s four-wing chaotic system with cubic nonlinearity to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Next, we derive adaptive control laws to achieve global chaos synchronization of identical Liu’s four-wing chaotic systems with cubic nonlinearity and unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive chaos control and synchronization schemes.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS VIA ACTI...IJCSEIT Journal
This paper investigates the anti-synchronization of identical hyperchaotic Bao systems (Bao and Liu,
2008), identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009) and non-identical hyperchaotic Bao
and hyperchaotic Xu systems. Active nonlinear control has been deployed for the anti- synchronization of
the hyperchaotic systems addressed in this paper and the main results have been established using
Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active
nonlinear control method is very effective and convenient to achieve anti-synchronization of identical and
non-identical hyperchaotic Bao and hyperchaotic Xu systems. Numerical simulations have been provided to
validate and demonstrate the effectiveness of the anti-synchronization results for the hyperchaotic Cao and
hyperchaotic Xu systems.
SLIDING CONTROLLER DESIGN FOR THE GLOBAL CHAOS SYNCHRONIZATION OF IDENTICAL H...ijait
This paper establishes new results for the sliding controller design for the global chaos synchronization of identical hyperchaotic Yujun systems (2010). Hyperchaotic systems are chaotic nonlinear systems having more than one positive Lyapunov exponent. Because of the complex dynamics properties of hyperchaotic system such as high capacity, high security and high efficiency, they are very useful in secure
communication devices and data encryption. Using sliding mode control theory and Lyapunov stability theory, a general result has been obtained for the global chaos synchronization of identical chaotic nonlinear systems. As an application of this general result, this paper designs a sliding controller for the
global chaos synchronization of hyperchaotic Yujun systems. Numerical results and simulations are shown to validate the proposed sliding controller design and demonstrate its effectiveness in achieving global chaos synchronization of hyperchaotic Yujun systems.
STATE FEEDBACK CONTROLLER DESIGN FOR THE OUTPUT REGULATION OF SPROTT-H SYSTEMijistjournal
This paper investigates the problem of state feedback controller design for the output regulation of SprottK chaotic system, which is one of the simple, classical, three-dimensional chaotic systems discovered by J.C. Sprott (1994). Explicitly, we have derived new state feedback control laws for the constant tracking problem for the Sprott-H system. The state feedback control laws have been derived using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott H chaotic system has important applications in Electronics and Communication Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-H chaotic system.
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
ACTIVE CONTROLLER DESIGN FOR THE GENERALIZED PROJECTIVE SYNCHRONIZATION OF TH...ijait
This paper discusses the design of active controllers for generalized projective synchronization (GPS) of identical Wang 3-scroll chaotic systems (Wang, 2009), identical Dadras 3-scroll chaotic systems (Dadras and Momeni, 2009) and non-identical Wang 3-scroll system and Dadras 3-scroll system. The synchronization results (GPS) derived in this paper for the 3 scroll chaotic systems have been derived using active control method and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active control method is very effective and convenient for
achieving the generalized projective synchronization (GPS) of the 3-scroll chaotic systems addressed in this paper. Numerical simulations are provided to illustrate the effectiveness of the GPS synchronization results derived in this paper.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Ch...ijtsrd
In this paper, a class of uncertain chaotic and non-chaotic systems is firstly introduced and the state observation problem of such systems is explored. Based on the time-domain approach with integral and differential equalities, an exponential state observer for a class of uncertain nonlinear systems is established to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be calculated correctly. Finally, numerical simulations are presented to exhibit the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "Exponential State Observer Design for a Class of Uncertain Chaotic and Non-Chaotic Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20219.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20219/exponential-state-observer-design-for-a-class-of-uncertain-chaotic-and-non-chaotic-systems/yeong-jeu-sun
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
We give an elementary exposition of a method to obtain the infinitesimal point symmetries of Lagrangians.Besides, we exhibit the Lanczos approach to Noether’s theorem to construct the first integral associated with each symmetry.
MSC 2010:49S05, 58E30, 70H25, 70H33
Research on 4-dimensional Systems without Equilibria with ApplicationTELKOMNIKA JOURNAL
Recently chaos-based encryption has been obtained more and more attention. Chaotic systems
without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs)
because there does not exist corresponding chaos criterion theorem on such systems. This paper
proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions
introduces a chaotic system without equilibria. Using this system and the generalized chaos
synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization
(8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results
show that there are no significant correlations between the key stream and the perturbed key streams
generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As
an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image.
The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar
to those generated via ideal chaotic PRNGs.
Simultaneous State and Actuator Fault Estimation With Fuzzy Descriptor PMID a...Waqas Tariq
In this paper, Takagi-Sugeno (T-S) fuzzy descriptor proportional multiple-integral derivative (PMID) and Proportional-Derivative (PD) observer methods that can estimate the system states and actuator faults simultaneously are proposed. T-S fuzzy model is obtained by linearsing satellite/spacecraft attitude dynamics at suitable operating points. For fault estimation, actuator fault is introduced as state vector to develop augmented descriptor system and robust fuzzy PMID and PD observers are developed. Stability analysis is performed using Lyapunov direct method. The convergence conditions of state estimation error are formulated in the form of LMI (linear matrix inequality). Derivative gain, obtained using singular value decomposition of descriptor state matrix (E), gives more design degrees of freedom together with proportional and integral gains obtained from LMI. Simulation study is performed for our proposed methods.
Solving output control problems using Lyapunov gradient-velocity vector functionIJECEIAES
This paper describes a controller and observer parameter definition approach in one input-one output (closed-loop) control systems using Lyapunov gradient-velocity vector function. Construction of the vector function is based on the gradient nature of the control systems and the parity of the vector functions with the potential function from the theory of catastrophe. Investigation of the closed-loop control system’s stability and solution of the problem of controller (determining the coefficient of magnitude matrix) and observer (calculation of the matrix elements of the observing equipment) synthesis is based on the direct methods of Lyapunov. The approach allows to select parameters based on the requested characteristics of the system.
International Journal of Computer Science, Engineering and Information Techno...ijcseit
In chaos theory, the problem anti-synchronization of chaotic systems deals with a pair of chaotic systems
called drive and response systems. In this problem, the design goal is to drive the sum of their respective
states to zero asymptotically. This problem gets even more complicated and requires special attention when
the parameters of the drive and response systems are unknown. This paper uses adaptive control theory
and Lyapunov stability theory to derive new results for the anti-synchronization of hyperchaotic Wang
system (2008) and hyperchaotic Li system (2005) with uncertain parameters. Hyperchaotic systems are
nonlinear dynamical systems exhibiting chaotic behaviour with two or more positive Lyapunov exponents.
The hyperchaotic systems have applications in areas like oscillators, lasers, neural networks, encryption,
secure transmission and secure communication. The main results derived in this paper are validated and
demonstrated with MATLAB simulations.
Adaptive Controller and Synchronizer Design for Hyperchaotic Zhou System with...Zac Darcy
In this paper, we establish new results for the adaptive controller and synchronizer design for the
hyperchaotic Zhou system (2009), when the parameters of the system are unknown. Using adaptive control theory and Lyapunov stability theory, we first design an adaptive controller to stabilize the hyperchaotic Zhou system to its unstable equilibrium at the origin. Next, using adaptive control theory and Lyapunov stability theory, we design an adaptive controller to achieve global chaos synchronization
of the identical hyperchaotic Zhou systems with unknown parameters. Simulations have been provided for adaptive controller and synchronizer designs to validate and illustrate the effectiveness of the schemes.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC PANG AND HYPERCHAOTIC WANG-CHEN SYSTEMS ...ijctcm
Hyperchaotic systems are chaotic systems having more than one positive Lyapunov exponent and they have
important applications in secure data transmission and communication. This paper applies active control
method for the synchronization of identical and different hyperchaotic Pang systems (2011) and
hyperchaotic Wang-Chen systems (2008). Main results are proved with the stability theorems of Lypuanov
stability theory and numerical simulations are plotted using MATLAB to show the synchronization of
hyperchaotic systems addressed in this paper.
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS ...ijistjournal
This paper derives new results for the global chaos synchronization of identical hyperchaotic Qi systems (2008), identical hyperchaotic Jha systems (2007) and non-identical hyperchaotic Qi and Jha systems. Active nonlinear control is the method adopted to achieve the complete synchronization of the identical and different hyperchaotic Qi and Jha systems. Our stability results derived in this paper are established using Lyapunov stability theory. Numerical simulations are shown to validate and illustrate the effectiveness of the synchronization results derived in this paper.
ANALYSIS AND SLIDING CONTROLLER DESIGN FOR HYBRID SYNCHRONIZATION OF HYPERCHA...IJCSEA Journal
Hybrid synchronization of chaotic systems is a research problem with a goal to synchronize the states of master and slave chaotic systems in a hybrid manner, namely, their even states are completely synchronized (CS) and odd states are anti-synchronized. This paper deals with the research problem of hybrid synchronization of chaotic systems. First, a detailed analysis is made on the qualitative properties of hyperchaotic Yujun system (2010). Then sliding controller has been derived for the hybrid synchronization of identical hyperchaotic Yujun systems, which is based on a general hybrid result derived in this paper.MATLAB simulations have been shown in detail to illustrate the new results derived for the hybrid synchronization of hyperchaotic Yujun systems. The results are proved using Lyapunov stability theory.
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...ijistjournal
In this paper, we apply adaptive control method to derive new results for the global chaos synchronization of 4-D chaotic systems, viz. identical Lorenz-Stenflo(LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and non-identical LS and Qi systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the identical and non-identical, uncertain LS and Qi 4-D chaotic systems.
HYBRID SLIDING SYNCHRONIZER DESIGN OF IDENTICAL HYPERCHAOTIC XU SYSTEMS ijitjournal
In this paper, new results have been obtained via sliding mode control for the hybrid chaos synchronization
of identical hyperchaotic Xu systems (Xu, Cai and Zheng, 2009). In hybrid synchronization of master and
slave systems, the odd states are completely synchronized, while the even states are anti-synchronized. The
stability results derived in this paper for the hybrid synchronization of identical hyperchaotic Xu systems
are established using Lyapunov stability theory. MATLAB simulations have been shown for the numerical
results to illustrate the hybrid synchronization schemes derived for the identical hyperchaotic Xu systems.
THE ACTIVE CONTROLLER DESIGN FOR ACHIEVING GENERALIZED PROJECTIVE SYNCHRONIZA...ijait
This paper discusses the design of active controllers for achieving generalized projective synchronization (GPS) of identical hyperchaotic Lü systems (Chen, Lu, Lü and Yu, 2006), identical hyperchaotic Cai systems (Wang and Cai, 2009) and non-identical hyperchaotic Lü and hyperchaotic Cai systems. The synchronization results (GPS) for the hyperchaotic systems have been derived using active control method and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active control method is very effective and convenient for achieving the GPS of the
hyperchaotic systems addressed in this paper. Numerical simulations are provided to illustrate the effectiveness of the GPS synchronization results derived in this paper.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
Anti-Synchronization Of Four-Scroll Chaotic Systems Via Sliding Mode Control IJITCA Journal
In this paper, new results are derived for the anti-synchronization of identical Liu-Chen four-scroll chaotic systems (Liu and Chen, 2004) and identical Lü-Chen-Cheng four-scroll chaotic systems (Lü,Chen and Cheng, 2004) by sliding mode control. The stability results derived in this paper for the antisynchronization of identical four-scroll chaotic systems are established using sliding mode control and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve anti-synchronization of the identical four-scroll chaotic systems. Numerical simulations are shown to illustrate and validate the antisynchronization schemes derived in this paper for the identical four-scroll systems
Adaptive Control and Synchronization of Hyperchaotic Cai Systemijctcm
The hyperchaotic Cai system (Wang, Cai, Miao and Tian, 2010) is one of the important paradigms of fourdimensional hyperchaotic systems. This paper investigates the adaptive control and synchronization of hyperchaotic Cai system with unknown parameters. First, adaptive control laws are designed to stabilize the hyperchaotic Cai system to its unstable equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Cai systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes
Global Chaos Synchronization of Hyperchaotic Pang and Hyperchaotic Wang Syste...CSEIJJournal
This paper investigates the global chaos synchronization of identical hyperchaotic Wang systems, identical
hyperchaotic Pang systems, and non-identical hyperchaotic Wang and hyperchaotic Pang systems via
adaptive control method. Hyperchaotic Pang system (Pang and Liu, 2011) and hyperchaotic Wang system
(Wang and Liu, 2006) are recently discovered hyperchaotic systems. Adaptive control method is deployed
in this paper for the general case when the system parameters are unknown. Sufficient conditions for global
chaos synchronization of identical hyperchaotic Pang systems, identical hyperchaotic Wang systems and
non-identical hyperchaotic Pang and Wang systems are derived via adaptive control theory and Lyapunov
stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control
method is very convenient for the global chaos synchronization of the hyperchaotic systems discussed in
this paper. Numerical simulations are presented to validate and demonstrate the effectiveness of the
proposed synchronization schemes.
Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems IJECEIAES
In this paper, the synchronization problem of T chaotic system and Lu chaotic system is studied. The parameter of the drive T chaotic system is considered unknown. An adaptive projective lag control method and also parameter estimation law are designed to achieve chaos synchronization problem between two chaotic systems. Then Lyapunov stability theorem is utilized to prove the validity of the proposed control method. After that, some numerical simulations are performed to assess the performance of the proposed method. The results show high accuracy of the proposed method in control and synchronization of chaotic systems.
THE DESIGN OF ADAPTIVE CONTROLLER AND SYNCHRONIZER FOR QI-CHEN SYSTEM WITH UN...IJCSEA Journal
This paper investigates the design problem of adaptive controller and synchronizer for the Qi-Chen system (2005), when the system parameters are unknown. First, we build an adaptive controller to stabilize the QiChen chaotic system to its unstable equilibrium at the origin. Then we build an adaptive synchronizer to achieve global chaos synchronization of the identical Qi-Chen chaotic systems with unknown parameters. The results derived for adaptive stabilization and adaptive synchronization for the Qi-Chen chaotic system are established using adaptive control theory and Lyapunov stability theory. Numerical simulations have been shown to demonstrate the effectiveness of the adaptive control and synchronization schemes derived in this paper for the Qi-Chen chaotic system.
The International Journal of Information Technology, Control and Automation (...IJITCA Journal
The International Journal of Information Technology, Control and Automation (IJITCA) is a Quarterly open access peer-reviewed journal that publishes articles which contribute new results in all areas of Information Technology (IT), Control Systems and Automation Engineering. The journal focuses on all technical and practical aspects of IT, Control Systems and Automation with applications in real-world engineering and scientific problems. The goal of this journal is to bring together researchers and practitioners from academia and industry to focus on information technology, control engineering, automation, modeling concepts and establishing new collaborations in these areas.
Authors are invited to contribute to this journal by submitting articles that illustrate research results, projects, surveying works and industrial experiences that describe significant advances in Information Technology, Control Systems and Automation.
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM cseij
The hyperchaotic Qi system (Chen, Yang, Qi and Yuan, 2007) is one of the important models of fourdimensional hyperchaotic systems. This paper investigates the adaptive stabilization and synchronization of hyperchaotic Qi system with unknown parameters. First, adaptive control laws are designed to stabilize the hyperchaotic Qi system to its equilibrium point at the origin based on the adaptive control theory and Lyapunov stability theory. Then adaptive control laws are derived to achieve global chaos synchronization of identical hyperchaotic Qi systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization schemes.
HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC S...IJITCA Journal
This paper investigates the hybrid chaos synchronization of uncertain 4-D chaotic systems, viz. identical Lorenz-Stenflo (LS) systems (Stenflo, 2001), identical Qi systems (Qi, Chen and Du, 2005) and nonidentical
LS and Qi systems. In hybrid chaos synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while the even states of the two systems are antisynchronized so that complete synchronization (CS) and anti-synchronization (AS) co-exist in the
synchronization of the two systems. In this paper, we shall assume that the parameters of both master and slave systems are unknown and we devise adaptive control schemes for the hybrid chaos synchronization using the estimates of parameters for both master and slave systems. Our adaptive synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to achieve hybrid synchronization of identical and non-identical LS and Qi systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptivesynchronization schemes for the identical and non-identical uncertain LS and Qi 4-D chaotic systems.
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About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
International Journal of Instrumentation and Control Systems (IJICS)
1. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013
DOI : 10.5121/ijics.2013.3201 1
ADAPTIVE CONTROLLER DESIGN FOR THE
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC
YANG AND HYPERCHAOTIC PANG SYSTEMS
Sundarapandian Vaidyanathan1
1
Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
Avadi, Chennai-600 062, Tamil Nadu, INDIA
sundarvtu@gmail.com
ABSTRACT
In the anti-synchronization of chaotic systems, a pair of chaotic systems called drive and responsesystems
are considered, and the design goal is to drive the sum of their respective states to zero asymptotically. This
paper derives new results for the anti-synchronization of hyperchaotic Yang system (2009) and
hyperchaotic Pang system (2011) with uncertain parameters via adaptive control. Hyperchaotic systems
are nonlinear chaotic systems withtwo or more positive Lyapunov exponents and they have applications in
areas like neural networks, encryption, secure data transmission and communication. The main results
derived in this paper are illustrated with MATLAB simulations.
KEYWORDS
Hyperchaos, Adaptive Control, Anti-Synchronization, Hyperchaotic Systems.
1. INTRODUCTION
Since the discovery of a hyperchaotic system by O.E.Rössler ([1], 1979), hyperchaotic systems
are known to have characteristics like high security, high capacity and high efficiency.
Hyperchaotic systems are chaotic systems having two or morepositive Lyapunov exponents. They
are applicable in several areas like oscillators [2], neural networks [3], secure communication [4-
5], data encryption [6], chaos synchronization [7], etc.
The synchronization problem deals with a pair of chaotic systems called the drive and response
chaotic systems, where the design goal is to drive the difference of their respective states to zero
asymptotically [8-9].
The anti-synchronization problem deals with a pair of chaotic systems called the drive and
response systems, where the design goal is to drive the sum of their respective states to zero
asymptotically.
The problems of synchronization and anti-synchronization of chaotic and hyperchaotic systems
have been studied via several methods like active control method [10-12], adaptive control
method [13-15],backstepping method [16-19], sliding control method [20-22] etc.
This paper derives new results for the adaptive controller design for the anti-synchronization of
hyperchaotic Yang systems ([23], 2009) and hyperchaotic Pang systems ([24], 2008) with
2. International Journal of Instrumentation and Control Systems (IJICS) Vol.3, No.2 , April 2013
2
unknown parameters. The main results derived in this paper were proved using adaptive control
theory [25] and Lyapunov stability theory [26].
2. PROBLEM STATEMENT
The drive system is described by the chaotic dynamics
( )x Ax f x= + (1)
where A is the n n× matrix of the system parameters and : n n
f →R R is the nonlinear part.
The response system is described by the chaotic dynamics
( )y By g y u= + + (2)
where B is the n n× matrix of the system parameters, : n n
g →R R is the nonlinear part and
n
u ∈R is the active controller to be designed.
For the pair of chaotic systems (1) and (2), the design goal of the anti-synchronization problemis
to construct a feedback controller ,u which anti-synchronizes their states for all (0), (0) .n
x y ∈R
Theanti-synchronization erroris defined as
,e y x= + (3)
Theerror dynamics is obtained as
( ) ( )e By Ax g y f x u= + + + + (4)
The design goal is to find a feedback controller uso that
lim ( ) 0
t
e t
→∞
= for all (0)e ∈Rn
(5)
Using the matrix method, we consider a candidate Lyapunov function
( ) ,T
V e e Pe= (6)
where P is a positive definite matrix.
It is noted that : n
V →R R is a positive definite function.
If we find a feedback controller uso that
( ) ,T
V e e Qe= − (7)
whereQ is a positive definite matrix, then : n
V → R R is a negative definite function.
Thus, by Lyapunov stability theory [26], the error dynamics (4) is globally exponentially stable.
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When the system parameters in (1) and (2) are unknown, we need to construct a parameter update
law for determining the estimates of the unknown parameters.
3. HYPERCHAOTIC SYSTEMS
The hyperchaotic Yang system ([23], 2009) is given by
1 2 1
2 1 1 3 4
3 3 1 2
4 1 2
( )x a x x
x cx x x x
x bx x x
x dx x
= −
= − +
= − +
= − −
(8)
where , , ,a b c d are constant, positive parameters of the system.
The Yang system (8) exhibits a hyperchaotic attractor for the parametric values
35, 3, 35, 2, 7.5a b c d = = = = = (9)
The Lyapunov exponents of the system (8) for the parametric values in (9) are
1 2 3 40.2747, 0.1374, 0, 38.4117 = = = = − (10)
Since there are two positive Lyapunov exponents in (10), the Yang system (8) is hyperchaotic for
the parametric values (9).
The phase portrait of the hyperchaotic Yang system is described in Figure 1.
The hyperchaotic Pang system ([24], 2011) is given by
1 2 1
2 2 1 3 4
3 3 1 2
4 1 2
( )
( )
x x x
x x x x x
x x x x
x x x
= −
= − +
= − +
= − +
(11)
where , , , are constant, positive parameters of the system.
The Pang system (11) exhibits a hyperchaotic attractor for the parametric values
36, 3, 20, 2 = = = = (12)
The Lyapunov exponents of the system (9) for the parametric values in (12) are
1 2 3 41.4106, 0.1232, 0, 20.5339 = = = = − (13)
Since there are two positive Lyapunov exponents in (13), the Pang system (11) is hyperchaotic
for the parametric values (12).
The phase portrait of the hyperchaotic Pang system is described in Figure 2.
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Figure 1. The Phase Portrait of the Hyperchaotic Yang System
Figure 2. The Phase Portrait of the Hyperchaotic Pang System
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4. ADAPTIVE CONTROL DESIGN FOR THE ANTI-SYNCHRONIZATION OF
HYPERCHAOTIC YANG SYSTEMS
In this section, we design an adaptive controller for the anti-synchronization of two identical
hyperchaotic Yang systems (2009) with unknown parameters.
Thedrive system is the hyperchaotic Yangdynamicsgiven by
1 2 1
2 1 1 3 4
3 3 1 2
4 1 2
( )x a x x
x cx x x x
x bx x x
x dx x
= −
= − +
= − +
= − −
(14)
where , , , ,a b c d are unknown parameters of the system and 4
x∈ R is the state.
The response system is the controlled hyperchaotic Yangdynamics given by
1 2 1 1
2 1 1 3 4 2
3 3 1 2 3
4 1 2 4
( )y a y y u
y cy y y y u
y by y y u
y dy y u
= − +
= − + +
= − + +
= − − +
(15)
where
4
y∈R is the state and 1 2 3 4, , ,u u u u are the adaptivecontrollers to be designed.
For the anti-synchronization, the error e is defined as
1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (16)
Then we derive the error dynamics as
1 2 1 1
2 1 4 1 3 1 3 2
3 3 1 2 1 2 3
4 1 2 4
( )e a e e u
e ce e y y x x u
e be y y x x u
e de e u
= − +
= + − − +
= − + + +
= − − +
(17)
The adaptive controller to achieve anti-synchronization is chosen as
1 2 1 1 1
2 1 4 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 1 2 4 4
ˆ( ) ( )( )
ˆ( ) ( )
ˆ( ) ( )
ˆ ˆ( ) ( ) ( )
u t a t e e k e
u t c t e e y y x x k e
u t b t e y y x x k e
u t d t e t e k e
= − − −
= − − + + −
= − − −
= + −
(18)
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In Eq. (18), , ( 1,2,3,4)ik i = are positive gains and ˆ ˆ ˆˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t t are estimates for the
unknown parameters , , , ,a b c d respectively.
By the substitution of (18) into (17), the error dynamics is simplified as
1 2 1 1 1
2 1 2 2
3 3 3 3
4 1 2 4 4
ˆ( ( ))( )
ˆ( ( ))
ˆ( ( ))
ˆ ˆ( ( )) ( ( ))
e a a t e e k e
e c c t e k e
e b b t e k e
e d d t e t e k e
= − − −
= − −
= − − −
= − − − − −
(19)
As a next step, we define the parameter estimation errors as
ˆ ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de t a a t e t b b t e t c c t e t d d t e t t = − = − = − = − = − (20)
Upon differentiation, we get
ˆ ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c de t a t e t b t e t c t e t d t e t t = − = − = − = − = −
(21)
Substituting (20) into the error dynamics (19), we obtain
1 2 1 1 1
2 1 2 2
3 3 3 3
4 1 2 4 4
( )a
c
b
d
e e e e k e
e e e k e
e e e k e
e e e e e k e
= − −
= −
= − −
= − − −
(22)
We consider the candidate Lyapunov function
( )2 2 2 2 2 2 2 2 2
1 2 3 4
1
2
a b c dV e e e e e e e e e= + + + + + + + + (23)
Differentiating (23) along the dynamics (21) and (22), we obtain
( )
( ) ( ) ( )
2 2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 3
1 2 1 4 2 4
ˆˆ( )
ˆ ˆˆ
a b
c d
V k e k e k e k e e e e e a e e b
e e e c e e e d e e e
= − − − − + − − + − −
+ − + − − + − −
(24)
In view of (24), we choose the following parameter update law:
1 2 1 5 1 4 8
2
3 6 2 4 9
1 2 7
ˆˆ ( ) ,
ˆ ˆ,
ˆ
a d
b
c
a e e e k e d e e k e
b e k e e e k e
c e e k e
= − + = − +
= − + = − +
= +
(25)
Next, we prove the following main result of this section.
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Theorem 4.1 The adaptive control law defined by Eq. (18) along with the parameter update law
defined by Eq. (25) achieveglobal and exponential anti-synchronization of the identical
hyperchaotic Yang systems (14) and (15)with unknown parameters for all initial conditions
4
(0), (0) .x y ∈R Moreover, the parameter estimation errors ( ), ( ), ( ), ( ), ( )a b c de t e t e t e t e t
globally and exponentially converge to zero for all initial conditions.
Proof.The proof is via Lyapunov stability theory [26] by taking V defined by Eq. (23) as the
candidate Lyapunov function. Substituting the parameter update law (25) into (24), we get
2 2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8 9( ) a b c dV e k e k ek e k e k e k e k e k e k e= − −− − − − − − − (26)
which is a negative definite function on 9
.R This completes the proof.
Next, we illustrate our adaptive anti-synchronization results with MATLAB simulations.The
classical fourth order Runge-Kutta method with time-step 8
10h −
= has been used to solve the
hyperchaotic Yang systems (14) and (15) with the nonlinear controller defined by (18).
The feedback gains in the adaptive controller (18) are taken as 4, ( 1, ,9).ik i= =
The parameters of the hyperchaotic Yang systems are taken as in the hyperchaotic case, i.e.
35, 3, 35, 2, 7.5a b c d = = = = =
For simulations, the initial conditions of the drive system (14) are taken as
1 2 3 4(0) 7, (0) 16, (0) 23, (0) 5x x x x= = = − = −
Also, the initial conditions of the response system (15) are taken as
1 2 3 4(0) 34, (0) 8, (0) 28, (0) 20y y y y= = − = = −
Also, the initial conditions of the parameter estimates are taken as
ˆ ˆ ˆˆ ˆ(0) 12, (0) 8, (0) 7, (0) 5, (0) 4a b c d = = = − = =
Figure 3 depicts the anti-synchronization of the identical hyperchaotic Yang systems.
Figure 4 depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e e
Figure 5 depicts the time-history of the parameter estimation errors , , , , .a b c de e e e e
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Figure 3. Anti-Synchronization of Identical Hyperchaotic Yang Systems
Figure 4. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e
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Figure 5. Time-History of the Parameter Estimation Errors , , , ,a b c de e e e e
5. ADAPTIVE CONTROL DESIGN FOR THE ANTI-SYNCHRONIZATION OF
HYPERCHAOTIC PANG SYSTEMS
In this section, we design an adaptive controller for the anti-synchronization of two identical
hyperchaotic Pang systems (2011) with unknown parameters.
Thedrive system is the hyperchaotic Pangdynamics given by
1 2 1
2 2 1 3 4
3 3 1 2
4 1 2
( )
( )
x x x
x x x x x
x x x x
x x x
= −
= − +
= − +
= − +
(27)
where , , , are unknown parameters of the system and 4
x∈ R is the state.
The response system is the controlled hyperchaotic Pangdynamics given by
1 2 1 1
2 2 1 3 4 2
3 3 1 2 3
4 1 2 4
( )
( )
y y y u
y y y y y u
y y y y u
y y y u
= − +
= − + +
= − + +
= − + +
(28)
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where 4
y∈R is the state and 1 2 3 4, , ,u u u u are the adaptivecontrollers to be designed.
For the anti-synchronization, the error e is defined as
1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (29)
Then we derive the error dynamics as
1 2 1 1
2 2 4 1 3 1 3 2
3 3 1 2 1 2 3
4 1 2 4
( )
( )
e e e u
e e e y y x x u
e e y y x x u
e e e u
= − +
= + − − +
= − + + +
= − + +
(30)
The adaptive controller to achieve anti-synchronization is chosen as
1 2 1 1 1
2 2 4 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 1 2 4 4
ˆ( )( )
ˆ( )
ˆ( )
ˆ( )( )
u t e e k e
u t e e y y x x k e
u t e y y x x k e
u t e e k e
= − − −
= − − + + −
= − − −
= + −
(31)
In Eq. (31), , ( 1,2,3,4)ik i = are positive gains and ˆ ˆˆ ˆ( ), ( ), ( ), ( )t t t t are estimates for the
unknown parameters , , , respectively.
By the substitution of (31) into (30), the error dynamics is simplified as
1 2 1 1 1
2 2 2 2
3 3 3 3
4 1 2 4 4
ˆ( ( ))( )
ˆ( ( ))
ˆ( ( ))
ˆ( ( ))( )
e t e e k e
e t e k e
e t e k e
e t e e k e
= − − −
= − −
= − − −
= − − + −
(32)
As a next step, we define the parameter estimation errors as
ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )e t t e t t e t t e t t = − = − = − = − (33)
Upon differentiation, we get
ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )e t t e t t e t t e t t = − = − = − = −
(34)
Substituting (33) into the error dynamics (32), we obtain
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1 2 1 1 1
2 2 2 2
3 3 3 3
4 1 2 4 4
( )
( )
e e e e k e
e e e k e
e e e k e
e e e e k e
= − −
= −
= − −
= − + −
(35)
We consider the candidate Lyapunov function
( )2 2 2 2 2 2 2 2
1 2 3 4
1
2
V e e e e e e e e = + + + + + + + (36)
Differentiating (36) along the dynamics (34) and (35), we obtain
( )
( ) ( )
2 2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 3
2
2 1 4 1 2
ˆˆ( )
ˆˆ ( )
V k e k e k e k e e e e e e e
e e e e e e e
= − − − − + − − + − −
+ − + − + −
(37)
In view of (37), we choose the following parameter update law:
1 2 1 5
2
3 6
2
2 7
1 4 1 2 8
ˆ ( )
ˆ
ˆ
ˆ ( )
e e e k e
e k e
e k e
e e e e k e
= − +
= − +
= +
= − + +
(38)
Next, we prove the following main result of this section.
Theorem 5.1 The adaptive control law defined by Eq. (31) along with the parameter update law
defined by Eq. (38) achieve global and exponential anti-synchronization of the identical
hyperchaotic Pang systems (27) and (28) with unknown parameters for all initial conditions
4
(0), (0) .x y ∈R Moreover, the parameter estimation errors ( ), ( ), ( ), ( )e t e t e t e t globally and
exponentially converge to zero for all initial conditions.
Proof.The proof is via Lyapunov stability theory [26] by taking V defined by Eq. (36) as the
candidate Lyapunov function. Substituting the parameter update law (38) into (37), we get
2 2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8 9( ) a b c dV e k e k ek e k e k e k e k e k e k e= − −− − − − − − − (39)
which is a negative definite function on 9
.R This completes the proof.
Next, we illustrate our adaptive anti-synchronization results with MATLAB simulations. The
classical fourth order Runge-Kutta method with time-step 8
10h −
= has been used to solve the
hyperchaotic Pang systems (27) and (28) with the nonlinear controller defined by (31).
The feedback gains in the adaptive controller (31) are taken as 4, ( 1, ,8).ik i= =
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The parameters of the hyperchaotic Pang systems are taken as in the hyperchaotic case, i.e.
36, 3, 20, 2 = = = =
For simulations, the initial conditions of the drive system (27) are taken as
1 2 3 4(0) 17, (0) 22, (0) 11, (0) 25x x x x= = − = − =
Also, the initial conditions of the response system (28) are taken as
1 2 3 4(0) 24, (0) 18, (0) 24, (0) 17y y y y= = − = = −
Also, the initial conditions of the parameter estimates are taken as
ˆ ˆˆ ˆ(0) 3, (0) 4, (0) 27, (0) 15 = = − = =
Figure 6depicts the anti-synchronization of the identical hyperchaotic Pang systems.
Figure 7depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e e
Figure 8 depicts the time-history of the parameter estimation errors , , , .e e e e
Figure 6. Anti-Synchronization of Identical Hyperchaotic Pang Systems
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Figure 7. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e
Figure 8. Time-History of the Parameter Estimation Errors , , ,e e e e
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6. ADAPTIVE CONTROL DESIGN FOR THE ANTI-SYNCHRONIZATION OF
HYPERCHAOTIC YANG AND HYPERCHAOTIC PANG SYSTEMS
In this section, we design an adaptive controller for the anti-synchronization of non-identical
hyperchaotic Yang (2009) and hyperchaotic Pang systems (2011) with unknown parameters.
Thedrive system is the hyperchaotic Yangdynamics given by
1 2 1
2 1 1 3 4
3 3 1 2
4 1 2
( )x a x x
x cx x x x
x bx x x
x dx x
= −
= − +
= − +
= − −
(40)
where , , , ,a b c d are unknown parameters of the system and 4
x∈ R is the state.
The response system is the controlled hyperchaotic Pangdynamics given by
1 2 1 1
2 2 1 3 4 2
3 3 1 2 3
4 1 2 4
( )
( )
y y y u
y y y y y u
y y y y u
y y y u
= − +
= − + +
= − + +
= − + +
(41)
where , , , are unknown parameters,
4
y∈R is the state and 1 2 3 4, , ,u u u u are the
adaptivecontrollers to be designed.
For the anti-synchronization, the error e is defined as
1 1 1 2 2 2 3 3 3 4 4 4, , ,e e e ey x y x y x y x= + = + = + = + (42)
Then we derive the error dynamics as
1 2 1 2 1 1
2 2 4 1 1 3 1 3 2
3 3 3 1 2 1 2 3
4 1 2 1 2 4
( ) ( )
( )
e y y a x x u
e y e cx y y x x u
e y bx y y x x u
e y y dx x u
= − + − +
= + + − − +
= − − + + +
= − + − − +
(43)
The adaptive controller to achieve anti-synchronization is chosen as
1 2 1 2 1 1 1
2 2 4 1 1 3 1 3 2 2
3 3 3 1 2 1 2 3 3
4 1 2 1 2 4 4
ˆ ˆ( )( ) ( )( )
ˆ ˆ( ) ( )
ˆˆ( ) ( )
ˆˆ ˆ( )( ) ( ) ( )
u t y y a t x x k e
u t y e c t x y y x x k e
u t y b t x y y x x k e
u t y y d t x t x k e
= − − − − −
= − − − + + −
= + − − −
= + + + −
(44)
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In Eq. (44), , ( 1,2,3,4)ik i = are positive gains, ˆ ˆ ˆˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t t are estimates for the
unknown parameters , , , ,a b c d respectively, and ˆ ˆˆ ˆ( ), ( ), ( ), ( )t t t t are estimates for the
unknown parameters , , , respectively.
By the substitution of (44) into (43), the error dynamics is simplified as
1 2 1 2 1 1 1
2 2 1 2 2
3 3 3 3 3
4 1 2 1 2 4 4
ˆ ˆ( ( ))( ) ( ( ))( )
ˆ ˆ( ( )) ( ( ))
ˆˆ( ( )) ( ( ))
ˆˆ ˆ( ( ))( ) ( ( )) ( ( ))
e t y y a a t x x k e
e t y c c t x k e
e t y b b t x k e
e t y y d d t x t x k e
= − − + − − −
= − + − −
= − − − − −
= − − + − − − − −
(45)
As a next step, we define the parameter estimation errors as
ˆ ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
a b c de t a a t e t b b t e t c c t e t d d t e t t
e t t e t t e t t e t t
= − = − = − = − = −
= − = − = − = −
(46)
Upon differentiation, we get
ˆ ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
a b c de t a t e t b t e t c t e t d t e t t
e t t e t t e t t e t t
= − = − = − = − = −
= − = − = − = −
(47)
Substituting (46) into the error dynamics (45), we obtain
1 2 1 2 1 1 1
2 2 1 2 2
3 3 3 3 3
4 1 2 1 2 4 4
( ) ( )
( )
a
c
b
d
e e y y e x x k e
e e y e x k e
e e y e x k e
e e y y e x e x k e
= − + − −
= + −
= − − −
= − + − − −
(48)
We consider the candidate Lyapunov function
( )2 2 2 2 2 2 2 2 2 2 2 2 2
1 2 3 4
1
2
a b c dV e e e e e e e e e e e e e = + + + + + + + + + + + + (49)
Differentiating (49) along the dynamics (47) and (48), we obtain
2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 3 3 2 1
4 1 4 2 1 2 1 3 3
2 2 4 1 2
ˆˆ ˆ( )
ˆ ˆˆ ˆ( )
ˆˆ+ ( )
a b c
d
V k e k e k e k e e e x x a e e x b e e x c
e e x d e e x e e y y e e y
e e y e e y y
= − − − − + − − + − − + −
+ − − + − − + − − + − −
− + − + −
(50)
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In view of (50), we choose the following parameter update law:
1 2 1 5 1 2 1 10
3 3 6 3 3 11
2 1 7 2 2 12
4 1 8 4 1 2 13
ˆˆ ( ) , ( )
ˆ ˆ,
ˆˆ ,
ˆ ˆ, ( )
a
b
c
d
a e x x k e e y y k e
b e x k e e y k e
c e x k e e y k e
d e x k e e y y k
= − + = − +
= − + = − +
= + = +
= − + = − + +
4 2 9
ˆ
e
e x k e
= − +
(51)
Theorem 6.1 The adaptive control law defined by Eq. (44) along with the parameter update law
defined by Eq. (51) achieve global and exponential anti-synchronization of the non-identical
hyperchaotic Yang system (40) and hyperchaotic Pang system (41) with unknown parameters for
all initial conditions 4
(0), (0) .x y ∈R Moreover, all the parameter estimation errors globally and
exponentially converge to zero for all initial conditions.
Proof.The proof is via Lyapunov stability theory [26] by taking V defined by Eq. (49) as the
candidate Lyapunov function. Substituting the parameter update law (51) into (50), we get
2 2 2 2 2 2 2 2 2
1 1 2 2 3 3 4 4 5 6 7 8 9
2 2 2 2
10 11 12 13
( ) a b c dV e k e k ek e k e k e k e k e k e k e
k e k e k e k e
= − −− − − − − − −
− − − −
(52)
which is a negative definite function on 13
.R This completes the proof.
Next, we illustrate our adaptive anti-synchronization results with MATLAB simulations. The
classical fourth order Runge-Kutta method with time-step 8
10h −
= has been used to solve the
hyperchaotic systems (40) and (41) with the nonlinear controller defined by (44). The feedback
gains in the adaptive controller (31) are taken as 4, ( 1, ,8).ik i= =
The parameters of the two hyperchaoticsystems are taken as in the hyperchaotic case, i.e.
35, 3, 35, 2, 7.5, 36, 3, 20, 2a b c d = = = = = = = = =
For simulations, the initial conditions of the drive system (40) are taken as
1 2 3 4(0) 29, (0) 14, (0) 23, (0) 9x x x x= = = − = −
Also, the initial conditions of the response system (41) are taken as
1 2 3 4(0) 14, (0) 18, (0) 29, (0) 14y y y y= = − = = −
Also, the initial conditions of the parameter estimates are taken as
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ˆ ˆ ˆˆ ˆ(0) 9, (0) 4, (0) 3, (0) 8, (0) 4,
ˆ ˆˆ ˆ(0) 6, (0) 2, (0) 11, (0) 9
a b c d
= = = − = = −
= = = = −
Figure 9depicts the anti-synchronization of the non-identical hyperchaotic Yang and hyperchaotic
Pang systems. Figure 10depicts the time-history of the anti-synchronization errors 1 2 3 4, , , .e e e e
Figure 11 depicts the time-history of the parameter estimation errors , , , ,e .a b c de e e e Figure 12
depicts the time-history of the parameter estimation errors , , , .e e e e
Figure 9. Anti-Synchronization of Hyperchaotic Yang and Hyperchaotic Pang Systems
Figure 10. Time-History of the Anti-Synchronization Errors 1 2 3 4, , ,e e e e
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Figure 11. Time-History of the Parameter Estimation Errors , , , ,a b c de e e e e
Figure 12. Time-History of the Parameter Estimation Errors , , ,e e e e
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7. CONCLUSIONS
In this paper, we have used adaptive control to derive new results for the anti-synchronization of
hyperchaotic Yang system (2009) and hyperchaotic Pang system (2011) with unknown
parameters. Main results of anti-synchronization design results for hyperchaotic systems
addressed in this paper were proved using adaptive control theory and Lyapunov stability theory.
Hyperchaotic systems have important applications in areas like secure communication, data
encryption, neural networks, etc.MATLAB simulations have been shown to validate and
demonstrate the adaptive anti-synchronization results for hyperchaotic Yang and hyperchaotic
Pang systems.
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Author
Dr. V. Sundarapandian earned his D.Sc. in Electrical and Systems Engineering
from Washington University, St. Louis, USA in May 1996. He is Professor and
Dean of the R & D Centre at Vel Tech Dr. RR & Dr. SR Technical University,
Chennai, Tamil Nadu, India. So far, he has published over 300 research works in
refereed international journals. He has also published over 200 research papers in
National and International Conferences. He has delivered Key Note Addresses at
many International Conferences with IEEE and Springer Proceedings. He is an India
Chair of AIRCC. He is the Editor-in-Chief of the AIRCC Control Journals –
International Journal of Instrumentation and Control Systems, International Journal
of Control Theory and Computer Modeling,International Journal of Information Technology, Control and
Automation, International Journal of Chaos, Control, Modelling and Simulation, and International Journal
of Information Technology, Modeling and Computing. His research interests are Control Systems, Chaos
Theory, Soft Computing, Operations Research, Mathematical Modelling and Scientific Computing. He has
published four text-books and conducted many workshops on Scientific Computing, MATLAB and
SCILAB.