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.
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.
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.
ADAPTIVE CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF HYPERCHAOTIC YANG ...ijics
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.
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.
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.
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.
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 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.
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.
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.
ADAPTIVE CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF HYPERCHAOTIC YANG ...ijics
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.
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.
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.
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.
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 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.
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
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.
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.
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.
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
Output Regulation of SPROTT-F Chaotic System by State Feedback ControlAlessioAmedeo
This paper solves the output regulation problem of Sprott-F chaotic system, which is one of the classical chaotic systems discovered by J.C. Sprott (1994). Explicitly, for the constant tracking problem, new state feedback control laws have been derived for regulating the output of the Sprott-F chaotic system. Our controller design has been carried out using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott-F chaotic system has important applications in many areas of Science and Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-F chaotic system.
The issues about maneuvering target track prediction were discussed in this paper. Firstly, using Kalman filter which based on current statistical model describes the state of maneuvering target motion, thereby analyzing time range of the target maneuvering occurred. Then, predict the target trajectory in real time by the improved gray prediction model. Finally, residual test and posterior variance test model accuracy, model accuracy is accurate.
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.
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
optimal solution method of integro-differential equaitions under laplace tran...INFOGAIN PUBLICATION
In this paper, Laplace Transform method is developed to solve partial Integro-differential equations. Partial Integro-differential equations (PIDE) occur naturally in various fields of science. Engineering and Social Science. We propose a max general form of linear PIDE with a convolution Kernal. We convert the proposed PIDE to an ordinary differential equation (ODE) using the LT method. We applying inverse LT as exact solution of the problems obtained. It is observed that the LT is a simple and reliable technique for solving such equations. The proposed model illustrated by numerical examples.
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.
Lesson 14: Derivatives of Exponential and Logarithmic Functions (Section 021 ...Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
International Journal of Instrumentation and Control Systems (IJICS)ijcisjournal
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
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.
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
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.
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.
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.
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
Output Regulation of SPROTT-F Chaotic System by State Feedback ControlAlessioAmedeo
This paper solves the output regulation problem of Sprott-F chaotic system, which is one of the classical chaotic systems discovered by J.C. Sprott (1994). Explicitly, for the constant tracking problem, new state feedback control laws have been derived for regulating the output of the Sprott-F chaotic system. Our controller design has been carried out using the regulator equations of C.I. Byrnes and A. Isidori (1990). The output regulation of the Sprott-F chaotic system has important applications in many areas of Science and Engineering. Numerical simulations are shown to illustrate the effectiveness of the control schemes proposed in this paper for the output regulation of the Sprott-F chaotic system.
The issues about maneuvering target track prediction were discussed in this paper. Firstly, using Kalman filter which based on current statistical model describes the state of maneuvering target motion, thereby analyzing time range of the target maneuvering occurred. Then, predict the target trajectory in real time by the improved gray prediction model. Finally, residual test and posterior variance test model accuracy, model accuracy is accurate.
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.
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
optimal solution method of integro-differential equaitions under laplace tran...INFOGAIN PUBLICATION
In this paper, Laplace Transform method is developed to solve partial Integro-differential equations. Partial Integro-differential equations (PIDE) occur naturally in various fields of science. Engineering and Social Science. We propose a max general form of linear PIDE with a convolution Kernal. We convert the proposed PIDE to an ordinary differential equation (ODE) using the LT method. We applying inverse LT as exact solution of the problems obtained. It is observed that the LT is a simple and reliable technique for solving such equations. The proposed model illustrated by numerical examples.
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ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION OF HYPERCHAOTIC XU AND HYPERCHAOTIC LI SYSTEMS
1. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April2013
DOI : 10.5121/ijait.2013.3202 17
ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID
SYNCHRONIZATION OF HYPERCHAOTIC XU AND
HYPERCHAOTIC LI 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
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.
KEYWORDS
Hybrid Synchronization, Adaptive Control, Chaos, Hyperchaos, Hyperchaotic Systems.
1. INTRODUCTION
The hyperchaotic system was first discovered by the German scientist, O.E. Rössler ([1], 1979). It
is a nonlinear chaotic system having two or more positive Lyapunov exponents. Hyperchaotic
systems have many attractive features and hence they are applicable in areas like neural networks
[2], oscillators [3], communication [4-5], encryption [6], synchronization [7], etc.
For the synchronization of chaotic systems, there are many methods available in the chaos
literature like OGY method [8], PC method [9], backstepping method [10-12], sliding control
method [13-15], active control method [16-17], adaptive control method [18-19], sampled-data
feedback control [20], time-delay feedback method [21], etc.
In the hybrid synchronization of a pair of chaotic systems called the master and slave systems,
one part of the systems, viz. the odd states, are completely synchronized (CS), while the other
part of the systems, viz. the even states, are anti-synchronized so that CS and AS co-exist in the
process of synchronization of the two systems.
This paper focuses upon adaptive controller design for the hybrid synchronization of hyperchaotic
Xu systems ([22], 2009) and hyperchaotic Li systems ([23], 2005) with unknown parameters. The
2. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April2013
18
main results derived in this paper have been proved using adaptive control theory [24] and
Lyapunov stability theory [25].
2. ADAPTIVE CONTROL METHODOLOGY FOR HYBRID SYNCHRONIZATION
The master 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 slave system is described by the chaotic dynamics
( )y By g y u= + + (2)
where B is the n n× matrix of the system parameters and : n n
g →R R is the nonlinear part
For the pair of chaotic systems (1) and (2), the hybrid synchronization error is defined as
, if is odd
, if is even
i i
i
i i
y x i
e
y x i
−
=
+
(3)
The error dynamics is obtained as
1
1
( ) ( ) ( ) if is odd
( ) ( ) ( ) if is even
n
ij j ij j i i i
j
i n
ij j ij j i i i
j
b y a x g y f x u i
e
b y a x g y f x u i
=
=
− + − +
=
+ + + +
∑
∑
(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)
where Q is a positive definite matrix, then : n
V → R R is a negative definite function.
Thus, by Lyapunov stability theory [25], the error dynamics (4) is globally exponentially stable.
Hence, the states of the chaotic systems (1) and (2) will be globally and exponentially
hybrid synchronized for all initial conditions (0), (0) .n
x y ∈R When the system parameters are
unknown, we use estimates for them and find a parameter update law using Lyapunov approach.
3. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April2013
19
3. HYPERCHAOTIC SYSTEMS
The hyperchaotic Xu system ([22], 2009) has the 4-D dynamics
1 2 1 4
2 1 1 3
3 3 1 2
4 1 3 2
( )x a x x x
x bx rx x
x cx x x
x x x dx
= − +
= +
= − −
= −
(8)
where , , , ,a b c r d are constant, positive parameters of the system.
The Xu system (8) exhibits a hyperchaotic attractor for the parametric values
10, 40, 2.5, 16, 2a b c r d= = = = = (9)
The Lyapunov exponents of the system (8) for the parametric values in (9) are
1 2 3 41.0088, 0.1063, 0, 13.6191 = = = = − (10)
Since there are two positive Lyapunov exponents in (10), the Xu system (8) is hyperchaotic for
the parametric values (9).
The strange attractor of the hyperchaotic Xu system is displayed in Figure 1.
The hyperchaotic Li system ([23], 2005) has the 4-D dynamics
1 2 1 4
2 1 1 3 2
3 3 1 2
4 2 3 4
( )x x 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 Li system (11) exhibits a hyperchaotic attractor for the parametric values
35, 3, 12, 7, 0.58 = = = = = (12)
The Lyapunov exponents of the system (11) for the parametric values in (12) are
1 2 3 40.5011, 0.1858, 0, 26.1010 = = = = − (13)
Since there are two positive Lyapunov exponents in (13), the Li system (11) is hyperchaotic for
the parametric values (12).
The strange attractor of the hyperchaotic Li system is displayed in Figure 2.
4. International Journal of Advanced Information Technology (IJAIT) Vol. 3, No.2, April2013
20
Figure 1. The State Portrait of the Hyperchaotic Xu System
Figure 2. The State Portrait of the Hyperchaotic Li System
4. ADAPTIVE CONTROL DESIGN FOR THE HYBRID SYNCHRONIZATION OF
HYPERCHAOTIC XU SYSTEMS
In this section, we design an adaptive controller for the hybrid synchronization of two identical
hyperchaotic Xu systems (2009) with unknown parameters.
The hyperchaotic Xu system is taken as the master system, whose dynamics is given by
1 2 1 4
2 1 1 3
3 3 1 2
4 1 3 2
( )x a x x x
x bx rx x
x cx x x
x x x dx
= − +
= +
= − −
= −
(14)
where , , , ,a b c d r are unknown parameters of the system and 4
x∈ R is the state of the system.
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The hyperchaotic Xu system is also taken as the slave system, whose dynamics is given by
1 2 1 4 1
2 1 1 3 2
3 3 1 2 3
4 1 3 2 4
( )y a y y y u
y by ry y u
y cy y y u
y y y dy u
= − + +
= + +
= − − +
= − +
(15)
where 4
y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptive controllers to be designed using
estimates ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t r t of the unknown parameters , , , ,a b c d r , respectively.
For the hybrid synchronization, the error e is defined as
1 1 1 2 2 2 3 3 3 4 4 4, , ,e y x e y x e y x e y x= − = + = − = + (16)
A simple calculation gives the error dynamics
1 2 2 1 4 4 1
2 1 1 1 3 1 3 2
3 3 1 2 1 2 3
4 2 1 3 1 3 4
( ) 2
( ) ( )
e a y x e e x u
e b y x r y y x x u
e ce y y x x u
e de y y x x u
= − − + − +
= + + + +
= − − + +
= − + + +
(17)
Next, we choose a nonlinear controller for achieving hybrid synchronization as
1 2 2 1 4 4 1 1
2 1 1 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 2 1 3 1 3 4 4
ˆ( )( ) 2
ˆ ˆ( )( ) ( )( )
ˆ( )
ˆ( )
u a t y x e e x k e
u b t y x r t y y x x k e
u c t e y y x x k e
u d t e y y x x k e
= − − − − + −
= − + − + −
= + − −
= − − −
(18)
In Eq. (18), , ( 1,2,3,4)ik i = are positive gains and ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t r t are estimates of the
unknown parameters , , , ,a b c d r , respectively.
By the substitution of (18) into (17), the error dynamics is simplified as
1 2 2 1 1 1
2 1 1 1 3 1 3 2 2
3 3 3 3
4 2 4 4
ˆ( ( ))( )
ˆ ˆ( ( ))( ) ( ( ))( )
ˆ( ( ))
ˆ( ( ))
e a a t y x e k e
e b b t y x r r t y y x x k e
e c c t e k e
e d d t e k e
= − − − −
= − + + − + −
= − − −
= − − −
(19)
Next, we define the parameter estimation errors as
ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c d re t a a t e t b b t e t c c t e t d d t e t r r t= − = − = − = − = − (20)
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Differentiating (20) with respect to ,t we get
ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )a b c d re t a t e t b t e t c t e t d t e t r t= − = − = − = − = −
(21)
In view of (20), we can simplify the error dynamics (19) as
1 2 2 1 1 1
2 1 1 1 3 1 3 2 2
3 3 3 3
4 2 4 4
( )
( ) ( )
a
b r
c
d
e e y x e k e
e e y x e y y x x k e
e e e k e
e e e k e
= − − −
= + + + −
= − −
= − −
(22)
We take the quadratic Lyapunov function
( )2 2 2 2 2 2 2 2 2
1 2 3 4
1
,
2
a b c d rV e e e e e e e e e= + + + + + + + + (23)
which is a positive definite function on
9
.R
When we differentiate (22) along the trajectories of (19) and (21), we get
2 2 2 2
1 1 2 2 3 3 4 4 1 2 2 1 2 1 1
2
3 2 4 2 1 3 1 3
ˆˆ( ) ( )
ˆˆ ˆ( )
a b
c d r
V k e k e k e k e e e y x e a e e y x b
e e c e e e d e e y y x x r
= − − − − + − − − + + −
+ − − + − − + + −
(24)
In view of Eq. (24), we take the parameter update law as
2
1 2 2 1 5 2 1 1 6 3 7
2 4 8 2 1 3 1 3 9
ˆˆ ˆ( ) , ( ) ,
ˆ ˆ, ( )
a b c
d r
a e y x e k e b e y x k e c e k e
d e e k e r e y y x x k e
= − − + = + + = − +
= − + = + +
(25)
Theorem 4.1 The adaptive control law (18) along with the parameter update law (25), where
,( 1,2, ,9)ik i = are positive gains, achieves global and exponential hybrid synchronization of
the identical hyperchaotic Xu systems (14) and (15), where ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )a t b t c t d t r t are estimates
of the unknown parameters , , , , ,a b c d r respectively. Moreover, all the parameter estimation
errors converge to zero exponentially for all initial conditions.
Proof. We prove the above result using Lyapunov stability theory [25].
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 9a b c d rV k e k e k e k e k e k e k e k e k e= − − − − − − − − − (26)
which is a negative definite function on
9
.R
This shows that the hybrid synchronization errors 1 2 3 4( ), ( ), ( ), ( )e t e t e t e t and the parameter
estimation errors ( ), ( ), ( ), ( ), ( )a b c d re t e t e t e t e t are globally exponentially stable for all initial
conditions.
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This completes the proof.
Next, we demonstrate our hybrid synchronization results with MATLAB simulations.
The classical fourth order Runge-Kutta method with time-step 8
10h −
= has been applied to solve
the hyperchaotic Xu systems (14) and (15) with the adaptive nonlinear controller (18) and the
parameter update law (25). The feedback gains are taken as 5, ( 1,2, ,9).ik i= =
The parameters of the hyperchaotic Xu systems are taken as in the hyperchaotic case, i.e.
10, 40, 2.5, 16, 2a b c r d= = = = =
For simulations, the initial conditions of the hyperchaotic Xu system (14) are chosen as
1 2 3 4(0) 11, (0) 7, (0) 9, (0) 5x x x x= = = = −
Also, the initial conditions of the hyperchaotic Xu system (15) are chosen as
1 2 3 4(0) 3, (0) 4, (0) 6, (0) 12y y y y= = = − =
Also, the initial conditions of the parameter estimates are chosen as
ˆ ˆˆ ˆ ˆ(0) 4, (0) 11, (0) 2, (0) 5, (0) 16a b c d r= = = − = − =
Figure 3 depicts the hybrid synchronization of the identical hyperchaotic Xu systems.
Figure 4 depicts the time-history of the hybrid synchronization errors 1 2 3 4, , , .e e e e
Figure 5 depicts the time-history of the parameter estimation errors , , , , .a b c d re e e e e
Figure 3. Hybrid Synchronization of Identical Hyperchaotic Xu Systems
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Figure 4. Time-History of the Hybrid Synchronization Errors 1 2 3 4, , ,e e e e
Figure 5. Time-History of the Parameter Estimation Errors , , , ,a b c d re e e e e
5. ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION
DESIGN OF HYPERCHAOTIC LI SYSTEMS
In this section, we design an adaptive controller for the hybrid synchronization of two identical
hyperchaotic Li systems (2005) with unknown parameters.
The hyperchaotic Li system is taken as the master system, whose dynamics is given by
1 2 1 4
2 1 1 3 2
3 3 1 2
4 2 3 4
( )x x 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 of the system.
The hyperchaotic Li system is also taken as the slave system, whose dynamics is given by
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1 2 1 4 1
2 1 1 3 2 2
3 3 1 2 3
4 2 3 4 4
( )y y y y u
y y y y y u
y y y y u
y y y y u
= − + +
= − + +
= − + +
= + +
(28)
where 4
y ∈ R is the state and 1 2 3 4, , ,u u u u are the adaptive controllers to be designed using
estimates ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )t t t t t of the unknown parameters , , , , , respectively.
For the hybrid synchronization, the error e is defined as
1 1 1 2 2 2 3 3 3 4 4 4, , ,e y x e y x e y x e y x= − = + = − = + (29)
A simple calculation gives the error dynamics
1 2 2 1 4 4 1
2 1 1 2 1 3 1 3 2
3 3 1 2 1 2 3
4 4 2 3 2 3 4
( ) 2
( )
e y x e e x u
e y x e y y x x u
e e y y x x u
e e y y x x u
= − − + − +
= + + − − +
= − + − +
= + + +
(30)
Next, we choose a nonlinear controller for achieving hybrid synchronization as
1 2 2 1 4 4 1 1
2 1 1 2 1 3 1 3 2 2
3 3 1 2 1 2 3 3
4 4 2 3 2 3 4 4
ˆ( )( ) 2
ˆ ˆ( )( ) ( )
ˆ( )
ˆ( )
u t y x e e x k e
u t y x t e y y x x k e
u t e y y x x k e
u t e y y x x k e
= − − − − + −
= − + − + + −
= − + −
= − − − −
(31)
In Eq. (31), , ( 1,2,3,4)ik i = are positive gains.
By the substitution of (31) into (30), the error dynamics is simplified as
1 2 2 1 1 1
2 1 1 2 2 2
3 3 3 3
4 4 4 4
ˆ( ( ))( )
ˆ ˆ( ( ))( ) ( ( ))
ˆ( ( ))
ˆ( ( ))
e t y x e k e
e t y x t e k e
e t e k e
e t e k e
= − − − −
= − + + − −
= − − −
= − −
(32)
Next, we define the parameter estimation errors as
ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( )
ˆ ˆ( ) ( ), ( ) ( )
e t t e t t e t t
e t t e t t
= − = − = −
= − = −
(33)
Differentiating (33) with respect to ,t we get
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ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )e t t e t t e t t e t t e t t = − = − = − = − = −
(34)
In view of (33), we can simplify the error dynamics (32) as
1 2 2 1 1 1
2 1 1 2 2 2
3 3 3 3
4 4 4 4
( )
( )
e e y x e k e
e e y x e e k e
e e e k e
e e e k e
= − − −
= + + −
= − −
= −
(35)
We take the quadratic Lyapunov function
( )2 2 2 2 2 2 2 2 2
1 2 3 4
1
,
2
V e e e e e e e e e = + + + + + + + + (36)
which is a positive definite function on
9
.R
When we differentiate (35) along the trajectories of (32) and (33), we get
2 2 2 2 2
1 1 2 2 3 3 4 4 1 2 2 1 3
2 2
2 2 1 1 4
ˆˆ( )
ˆˆ ˆ( )
V k e k e k e k e e e y x e e e
e e e e y x e e
= − − − − + − − − + − −
+ − + + − + −
(37)
In view of Eq. (37), we take the parameter update law as
2 2
1 2 2 1 5 3 6 2 7
2
2 1 1 8 4 9
ˆˆ ˆ( ) , ,
ˆ ˆ( ) ,
e y x e k e e k e e k e
e y x k e e k e
= − − + = − + = +
= + + = +
(38)
Theorem 5.1 The adaptive control law (31) along with the parameter update law (38), where
,( 1,2, ,9)ik i = are positive gains, achieves global and exponential hybrid synchronization of
the identical hyperchaotic Li systems (27) and (28), where ˆ ˆˆ ˆ ˆ( ), ( ), ( ), ( ), ( )t t t t t are
estimates of the unknown parameters , , , , , respectively. Moreover, all the parameter
estimation errors converge to zero exponentially for all initial conditions.
Proof. We prove the above result using Lyapunov stability theory [25].
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 9V k e k e k e k e k e k e k e k e k e = − − − − − − − − − (39)
which is a negative definite function on
9
.R
This shows that the hybrid synchronization errors 1 2 3 4( ), ( ), ( ), ( )e t e t e t e t and the parameter
estimation errors ( ), ( ), ( ), ( ), ( )e t e t e t e t e t are globally exponentially stable for all initial
conditions. This completes the proof.
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Next, we demonstrate our hybrid synchronization results with MATLAB simulations.
The classical fourth order Runge-Kutta method with time-step 8
10h −
= has been applied to solve
the hyperchaotic Li systems (27) and (28) with the adaptive nonlinear controller (31) and the
parameter update law (38). The feedback gains are taken as 5, ( 1,2, ,9).ik i= =
The parameters of the hyperchaotic Li systems are taken as in the hyperchaotic case, i.e.
35, 3, 12, 7, 0.58 = = = = =
For simulations, the initial conditions of the hyperchaotic Li system (27) are chosen as
1 2 3 4(0) 6, (0) 7, (0) 15, (0) 22x x x x= = − = = −
Also, the initial conditions of the hyperchaotic Li system (28) are chosen as
1 2 3 4(0) 12, (0) 4, (0) 9, (0) 6y y y y= = = = −
Also, the initial conditions of the parameter estimates are chosen as
ˆ ˆˆ ˆ ˆ(0) 7, (0) 8, (0) 2, (0) 9, (0) 12 = = = − = − =
Figure 6 depicts the hybrid synchronization of the identical hyperchaotic Li systems.
Figure 7 depicts the time-history of the hybrid 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 e
Figure 6. Hybrid Synchronization of Identical Hyperchaotic Li Systems
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Figure 7. Time-History of the Hybrid Synchronization Errors 1 2 3 4, , ,e e e e
Figure 8. Time-History of the Parameter Estimation Errors , , , ,e e e e e
6. ADAPTIVE CONTROLLER DESIGN FOR THE HYBRID SYNCHRONIZATION
DESIGN OF HYPERCHAOTIC XU AND HYPERCHAOTIC LI SYSTEMS
In this section, we design an adaptive controller for the hybrid synchronization of hyperchaotic
Xu system (2009) and hyperchaotic Li system (2005) with unknown parameters.
The hyperchaotic Xu system is taken as the master system, whose dynamics is given by
1 2 1 4
2 1 1 3
3 3 1 2
4 1 3 2
( )x a x x x
x bx rx x
x cx x x
x x x dx
= − +
= +
= − −
= −
(40)
where , , , ,a b c d r are unknown parameters of the system.
The hyperchaotic Li system is also taken as the slave system, whose dynamics is given by
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1 2 1 4 1
2 1 1 3 2 2
3 3 1 2 3
4 2 3 4 4
( )y y y y u
y y y y y u
y y y y u
y y y y u
= − + +
= − + +
= − + +
= + +
(41)
where , , , , are unknown parameters and 1 2 3 4, , ,u u u u are the adaptive controllers.
For the hybrid synchronization, the error e is defined as
1 1 1 2 2 2 3 3 3 4 4 4, , ,e y x e y x e y x e y x= − = + = − = + (42)
A simple calculation gives the error dynamics
1 2 1 2 1 4 4 1
2 1 2 1 1 3 1 3 2
3 3 3 1 2 1 2 3
4 4 2 2 3 2 3 4
( ) ( )e y y a x x y x u
e y y bx rx x y y u
e y cx y y x x u
e y dx y y x x u
= − − − + − +
= + + + − +
= − + + + +
= − + + +
(43)
Next, we choose a nonlinear controller for achieving hybrid synchronization as
1 2 1 2 1 4 4 1 1
2 1 2 1 1 3 1 3 2 2
3 3 3 1 2 1 2 3 3
4 4 2 2 3 2 3 4 4
ˆ ˆ( )( ) ( )( )
ˆˆ ˆ ˆ( ) ( ) ( ) ( )
ˆ ˆ( ) ( )
ˆˆ( ) ( )
u t y y a t x x y x k e
u t y t y b t x r t x x y y k e
u t y c t x y y x x k e
u t y d t x y y x x k e
= − − + − − + −
= − − − − + −
= − − − −
= − + − − −
(44)
where , ( 1,2,3,4)ik i = are positive gains.
By the substitution of (44) into (43), the error dynamics is simplified as
1 2 1 2 1 1 1
2 1 2 1 1 3 2 2
3 3 3 3 3
4 4 2 4 4
ˆ ˆ( ( ))( ) ( ( ))( )
ˆˆ ˆ ˆ( ( )) ( ( )) ( ( )) ( ( ))
ˆ ˆ( ( )) ( ( ))
ˆˆ( ( )) ( ( ))
e t y y a a t x x k e
e t y t y b b t x r r t x x k e
e t y c c t x k e
e t y d d t x k e
= − − − − − −
= − + − + − + − −
= − − + − −
= − − − −
(45)
Next, we define the parameter estimation errors as
ˆ ˆˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
ˆˆ ˆˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
ˆ ˆ( ) ( ), ( ) ( )
a b c d
r
e t a a t e t b b t e t c c t e t d d t
e t r r t e t t e t t e t t
e t t e t t
= − = − = − = −
= − = − = − = −
= − = −
(46)
Differentiating (46) with respect to ,t we get
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ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
ˆ ˆˆ ˆ ˆ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )
a b c d re t a t e t b t e t c t e t d t e t r t
e t t e t t e t t e t t e t t
= − = − = − = − = −
= − = − = − = − = −
(47)
In view of (46), we can simplify the error dynamics (45) as
1 2 1 2 1 1 1
2 1 2 1 1 3 2 2
3 3 3 3 3
4 4 2 4 4
( ) ( )a
b r
c
d
e e y y e x x k e
e e y e y e x e x x k e
e e y e x k e
e e y e x k e
= − − − −
= + + + −
= − + −
= − −
(48)
We take the quadratic Lyapunov function
( )2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 2 3 4
1
,
2
a b c d rV e e e e e e e e e e e e e e = + + + + + + + + + + + + + (49)
which is a positive definite function on 14
.R
When we differentiate (48) along the trajectories of (45) and (46), we get
2 2 2 2
1 1 2 2 3 3 4 4 1 2 1 2 1 3 3
4 2 2 1 3 1 2 1 3 3
2 2 2 1
ˆˆ ˆ( )
ˆ ˆˆˆ ( )
ˆˆ
a b c
d r
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 x r e e y y e e y
e e y e e y e
= − − − − + − − − + − + −
+ − − + − + − − + − −
+ − + − +
4 4
ˆe y −
(50)
In view of Eq. (50), we take the parameter update law as
1 2 1 5 2 1 6 3 3 7
4 2 8 2 1 3 9 1 2 1 10
3 3 11 2 2 12
ˆˆ ˆ( ) , , ,
ˆ ˆˆ, , ( ) ,
ˆ ˆ, ,
a b c
d r
a e x x k e b e x k e c e x k e
d e x k e r e x x k e e y y k e
e y k e e y k e
= − − + = + = +
= − + = + = − +
= − + = +
2 1 13
4 4 14
ˆ ,
ˆ
e y k e
e y k e
= +
= +
(51)
Theorem 6.1 The adaptive control law (44) along with the parameter update law (51), where
,( 1,2, ,14)ik i = are positive gains, achieves global and exponential hybrid synchronization of
the hyperchaotic Xu system (40) hyperchaotic Li system (41), where ˆ( ),a t ˆ( ),b t ˆ( ),c t ˆ( ),d t ˆ( ),r t
ˆ( ),t ˆ( ),t ˆ( ),t ˆ( ),t ˆ( )t are estimates of the unknown parameters , , , , ,a b c d r , , , , ,
respectively. Moreover, all the parameter estimation errors converge to zero exponentially for all
initial conditions.
Proof. We prove the above result using Lyapunov stability theory [25].
Substituting the parameter update law (51) into (50), we get
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31
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 2
10 11 12 13 14
a b c d rV k e k e k e k 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 14
.R
This shows that the hybrid synchronization errors 1 2 3 4( ), ( ), ( ), ( )e t e t e t e t and the parameter
estimation errors ( ), ( ), ( ), ( ), ( ),a b c d re t e t e t e t e t ( ), ( ), ( ), ( ), ( )e t e t e t e t e t are globally
exponentially stable for all initial conditions. This completes the proof.
Next, we demonstrate our hybrid synchronization results with MATLAB simulations.
The classical fourth order Runge-Kutta method with time-step 8
10h −
= has been applied to solve
the hyperchaotic Li systems (27) and (28) with the adaptive nonlinear controller (31) and the
parameter update law (38). The feedback gains are taken as 5, ( 1,2, ,9).ik i= =
The parameters of the hyperchaotic Xu and Li systems are taken as in the hyperchaotic case, i.e.
10, 40, 2.5, 16, 2, 35, 3, 12, 7, 0.58a b c r d = = = = = = = = = =
For simulations, the initial conditions of the hyperchaotic Xu system (27) are chosen as
1 2 3 4(0) 12, (0) 4, (0) 8, (0) 20x x x x= = − = = −
Also, the initial conditions of the hyperchaotic Li system (28) are chosen as
1 2 3 4(0) 21, (0) 7, (0) 19, (0) 12y y y y= = = = −
Also, the initial conditions of the parameter estimates are chosen as
ˆ ˆˆ ˆ ˆ(0) 7, (0) 14, (0) 12, (0) 8, (0) 15
ˆ ˆˆ ˆ ˆ(0) 12, (0) 5, (0) 4, (0) 11, (0) 6
a b c d r
= − = = = − =
= = = = = −
Figure 9 depicts the hybrid synchronization of hyperchaotic Xu and hyperchaotic Li systems.
Figure 10 depicts the time-history of the hybrid synchronization errors 1 2 3 4, , , .e e e e
Figure 11 depicts the time-history of the parameter estimation errors , , , .a b c de e e e
Figure 12 depicts the time-history of the parameter estimation errors , , , , .e e e e e
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Figure 9. Hybrid Synchronization of Hyperchaotic Xu and Lu Systems
Figure 10. Time-History of the Hybrid Synchronization Errors 1 2 3 4, , ,e e e e
Figure 11. Time-History of the Parameter Estimation Errors , , ,a b c de e e e
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Figure 12. Time-History of the Parameter Estimation Errors , , , ,e e e e e
7. CONCLUSIONS
In this paper, using adaptive control method, we derived new results for the adaptive controller
design for the hybrid synchronization of hyperchaotic Xu systems (2009) and hyperchaotic Li
systems (2005) with unknown parameters. Using Lyapunov control theory, adaptive control laws
were derived for globally hybrid synchronizing the states of identical hyperchaotic Xu systems,
identical hyperchaotic Li systems and non-identical hyperchaotic Xu and Li systems. MATLAB
simulations were displayed in detail to demonstrate the adaptive hybrid synchronization results
derived in this paper for hyperchaotic Xu and Li systems.
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