This paper derives new results for the hybrid synchronization of identical hyperchaotic Liu systems (Liu, Liu and Zhang, 2008) via sliding mode control. In hybrid synchronization of master and slave systems, the odd states of the two systems are completely synchronized, while their even states are antisynchronized. The stability results derived in this paper for the hybrid synchronization of identical hyperchaotic Liu systems have been proved 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 anti- synchronization of the identical hyperchaotic Liu systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Liu 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 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.
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
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...ijcsa
This paper derives new results for the design of sliding mode controller for the hybrid synchronization of identical hyperchaotic Chen systems (Jia, Dai and Hui, 2010). The synchronizer results derived in this paper for the hybrid synchronization of identical hyperchaotic Chen 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 hybrid synchronization of the
identical hyperchaotic Chen systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Chen systems.
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.
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.
Hybrid Chaos Synchronization of Hyperchaotic Newton-Leipnik Systems by Slidin...ijctcm
This paper investigates the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya, 2010) by sliding mode control. The stability results derived in this paper for the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems are established using Lyapunov stability theory. Hybrid synchronization of hyperchaotic Newton-Leipnik systems is achieved through the complete synchronization of first and third states of the systems and the anti-synchronization of second and fourth states of the master and slave systems. Since the Lyapunov exponents are not required for these calculations, the sliding mode control is very effective and convenient to achieve hybrid chaos synchronization of the identical hyperchaotic Newton-Leipnik systems. Numerical simulations are shown to validate and demonstrate the effectiveness of the synchronization schemes 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.
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
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...ijcsa
This paper derives new results for the design of sliding mode controller for the hybrid synchronization of identical hyperchaotic Chen systems (Jia, Dai and Hui, 2010). The synchronizer results derived in this paper for the hybrid synchronization of identical hyperchaotic Chen 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 hybrid synchronization of the
identical hyperchaotic Chen systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Chen systems.
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.
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.
Hybrid Chaos Synchronization of Hyperchaotic Newton-Leipnik Systems by Slidin...ijctcm
This paper investigates the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya, 2010) by sliding mode control. The stability results derived in this paper for the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems are established using Lyapunov stability theory. Hybrid synchronization of hyperchaotic Newton-Leipnik systems is achieved through the complete synchronization of first and third states of the systems and the anti-synchronization of second and fourth states of the master and slave systems. Since the Lyapunov exponents are not required for these calculations, the sliding mode control is very effective and convenient to achieve hybrid chaos synchronization of the identical hyperchaotic Newton-Leipnik systems. Numerical simulations are shown to validate and demonstrate the effectiveness of the synchronization schemes derived in this paper.
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.
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.
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.
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.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This paper presents synchronization of a four-dimensional autonomous hyperchaotic system based on the
generalized augmented Lü system. Based on the Lyapunov stability theory an active control law is derived
such that the two four-dimensional autonomous hyper-chaotic systems are to be synchronized. Numerical
simulations are presented to demonstrate the effectiveness of the synchronization schemes.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This paper presents synchronization of a four-dimensional autonomous hyperchaotic system based on the
generalized augmented Lü system. Based on the Lyapunov stability theory an active control law is derived
such that the two four-dimensional autonomous hyper-chaotic systems are to be synchronized. Numerical
simulations are presented to demonstrate the effectiveness of the synchronization schemes.
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.
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 A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic 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 highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic 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 highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
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.
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.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This paper presents synchronization of a four-dimensional autonomous hyperchaotic system based on the generalized augmented Lü system. Based on the Lyapunov stability theory an active control law is derived such that the two four-dimensional autonomous hyper-chaotic systems are to be synchronized. Numerical simulations are presented to demonstrate the effectiveness of the synchronization schemes.
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
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic
systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations
are presented to demonstrate the effectiveness of the synchronization schemes.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic systems. Generalized backstepping method is similarity to backstepping. Backstepping method is used only to strictly feedback systems but generalized backstepping method expands this class. Numerical simulations are presented to demonstrate the effectiveness of the synchronization schemes.
This paper presents synchronization of a new autonomous hyperchaotic system. The generalized
backstepping technique is applied to achieve hyperchaos synchronization for the two new hyperchaotic systems. Generalized back-stepping method is similarity to backstepping. Backstepping method is used only
to strictly feedback systems but generalized back-stepping method expands this class. Numerical simulations are presented to demonstrate the effectiveness of the 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.
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.
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.
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.
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.
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.
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.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This paper presents synchronization of a four-dimensional autonomous hyperchaotic system based on the
generalized augmented Lü system. Based on the Lyapunov stability theory an active control law is derived
such that the two four-dimensional autonomous hyper-chaotic systems are to be synchronized. Numerical
simulations are presented to demonstrate the effectiveness of the synchronization schemes.
SYNCHRONIZATION OF A FOUR-WING HYPERCHAOTIC SYSTEMijccmsjournal
This paper presents synchronization of a four-dimensional autonomous hyperchaotic system based on the
generalized augmented Lü system. Based on the Lyapunov stability theory an active control law is derived
such that the two four-dimensional autonomous hyper-chaotic systems are to be synchronized. Numerical
simulations are presented to demonstrate the effectiveness of the synchronization schemes.
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.
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 A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic 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 highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTORijistjournal
Recently, a novel three-dimensional highly chaotic attractor has been discovered by Srisuchinwong and Munmuangsaen (2010). This paper investigates the adaptive control and synchronization of this highly chaotic attractor with unknown parameters. First, adaptive control laws are designed to stabilize the highly chaotic 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 highly chaotic systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes.
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.
ANTI-SYNCHRONIZATION OF HYPERCHAOTIC WANG AND HYPERCHAOTIC LI SYSTEMS WITH UN...ijcseit
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HYBRID SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEMS VIA SLIDING MODE CONTROL
1. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
1
HYBRID SYNCHRONIZATION OF HYPERCHAOTIC
LIU SYSTEMS VIA SLIDING MODE CONTROL
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 results for the hybrid synchronization of identical hyperchaotic Liu systems (Liu,
Liu and Zhang, 2008) via sliding mode control. In hybrid synchronization of master and slave systems,
the odd states of the two systems are completely synchronized, while their even states are anti-
synchronized. The stability results derived in this paper for the hybrid synchronization of identical
hyperchaotic Liu systems have been proved 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 anti- synchronization of the identical hyperchaotic Liu systems. Numerical
simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper
for the identical hyperchaotic Liu systems.
KEYWORDS
Sliding Mode Control, Hybrid Synchronization, Hyperchaos, Hyperchaotic Liu System.
1. INTRODUCTION
Chaotic systems are nonlinear dynamical systems that are highly sensitive to initial conditions.
The sensitive nature of chaotic systems is commonly called as the butterfly effect [1].
Chaos synchronization is a phenomenon that occurs when two or more chaotic oscillators are
coupled or when a chaotic oscillator drives another chaotic oscillator. Because of the butterfly
effect which causes the exponential divergence of the trajectories of two identical chaotic
systems started with nearly the same initial conditions, synchronizing two chaotic systems is
seemingly a very challenging problem.
Hyperchaotic system is usually defined as a chaotic system with more than one positive
Lyapunov exponent. The first hyperchaotic system was discovered by O.E. Rössler (1979).
Since hyperchaotic system has the characteristics of high capacity, high security and high
efficiency, it has the potential of broad applications in nonlinear circuits, secure
communications, lasers, neural networks, biological systems and so on.
Thus, the studies on hyperchaotic systems, viz. control, synchronization and circuit
implementation are very challenging problems in the chaos literature.
In most of the chaos synchronization approaches, the master-slave formalism is used. If a
particular chaotic system is called the master system and another chaotic system is called the
slave system, then the idea of the anti-synchronization is to use the output of the master system
to control the slave system so that the states of the slave system have the same amplitude but
opposite signs as the states of the master system asymptotically.
2. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
2
Since the pioneering work by Pecora and Carroll ([2], 1990), chaos synchronization problem
has been studied extensively and intensively in the literature [2-17]. Chaos theory has been
applied to a variety of fields such as physical systems [3], chemical systems [4], ecological
systems [5], secure communications [6-8], etc.
In the last two decades, various schemes have been successfully applied for chaos
synchronization such as PC method [2], OGY method [9], active control method [10-14],
adaptive control method [15-20], time-delay feedback method [21], backstepping design
method [22], sampled-data feedback method [23], etc.
In hybrid synchronization of master and slave systems, the odd states are completely
synchronized (CS) and the even states are anti-synchronized (AS). The co-existence of CS and
AS in the synchronization enhances the security of secure communication systems.
In this paper, we derive new results based on the sliding mode control [24-28] for the hybrid
synchronization of identical hyperchaotic Liu systems ([29], Liu, Liu and Zhang, 2008). In
robust control systems, the sliding mode control method is often adopted due to its inherent
advantages of easy realization, fast response and good transient performance as well as its
insensitivity to parameter uncertainties and external disturbances.
This paper has been organized as follows. In Section 2, we describe the problem statement and
our methodology using sliding mode control (SMC). In Section 3, we discuss the hybrid
synchronization of identical hyperchaotic Liu systems. In Section 4, we summarize the main
results obtained in this paper.
2. PROBLEM STATEMENT AND OUR METHODOLOGY USING SMC
Consider the chaotic system described by
( )
x Ax f x
= +
& (1)
where n
x∈R is the state of the system, A is the n n
× matrix of the system parameters and
: n n
f →
R R is the nonlinear part of the system.
We consider the system (1) as the master or drive system.
As the slave or response system, we consider the following chaotic system described by the
dynamics
( )
y Ay f y u
= + +
& (2)
where n
y ∈R is the state of the system and m
u ∈R is the controller to be designed.
We define the hybrid synchronization error as
if is odd
if is even
i i
i
i i
y x i
e
y x i
−
=
+
(3)
3. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
3
Then the error dynamics is obtained as
( , )
e Ae x y u
η
= + +
& (4)
The objective of the anti-synchronization problem is to find a controller u such that
lim ( ) 0
t
e t
→∞
= for all (0) .
n
e ∈R (5)
To solve this problem, we first define the control u as
( , )
u x y Bv
η
= − + (6)
where B is a constant gain vector selected such that ( , )
A B is controllable.
Substituting (6) into (4), the error dynamics simplifies to
e Ae Bv
= +
& (7)
which is a linear time-invariant control system with single input .
v
Thus, the original anti-synchronization problem can be replaced by an equivalent problem of
stabilizing the zero solution 0
e = of the system (7) by a suitable choice of the sliding mode
control. In the sliding mode control, we define the variable
1 1 2 2
( ) n n
s e Ce c e c e c e
= = + + +
L (8)
where [ ]
1 2 n
C c c c
= L is a constant vector to be determined.
In the sliding mode control, we constrain the motion of the system (7) to the sliding manifold
defined by
{ }
| ( ) 0
n
S x s e
= ∈ =
R
which is required to be invariant under the flow of the error dynamics (7).
When in sliding manifold ,
S the system (7) satisfies the following conditions:
( ) 0
s e = (9)
which is the defining equation for the manifold S and
( ) 0
s e =
& (10)
which is the necessary condition for the state trajectory ( )
e t of (7) to stay on the sliding
manifold .
S
4. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
4
Using (7) and (8), the equation (10) can be rewritten as
[ ]
( ) 0
s e C Ae Bv
= + =
& (11)
Solving (11) for ,
v we obtain the equivalent control law
1
eq ( ) ( ) ( )
v t CB CA e t
−
= − (12)
where C is chosen such that
0.
CB ≠
Substituting (12) into the error dynamics (7), we obtain the closed-loop dynamics as
1
( )
e I B CB C Ae
−
= −
& (13)
The row vector C is selected such that the system matrix of the controlled dynamics
1
( )
I B CB C A
−
−
is Hurwitz, i.e. it has all eigenvalues with negative real parts.
Then the controlled system (13) is globally asymptotically stable.
To design the sliding mode controller for (7), we apply the constant plus proportional rate
reaching law
sgn( )
s q s k s
= − −
& (14)
where sgn( )
⋅ denotes the sign function and the gains 0,
q > 0
k > are determined such that the
sliding condition is satisfied and sliding motion will occur.
From equations (11) and (14), we can obtain the control ( )
v t as
[ ]
1
( ) ( ) ( ) sgn( )
v t CB C kI A e q s
−
= − + + (15)
which yields
[ ]
[ ]
1
1
( ) ( ) , if ( ) 0
( )
( ) ( ) , if ( ) 0
CB C kI A e q s e
v t
CB C kI A e q s e
−
−
− + + >
=
− + − <
(16)
Theorem 2.1. The master system (1) and the slave system (2) are globally and asymptotically
hybrid synchronized for all initial conditions (0), (0) n
x y R
∈ by the feedback control law
( ) ( , ) ( )
u t x y Bv t
η
= − + (17)
where ( )
v t is defined by (15) and B is a column vector such that ( , )
A B is controllable. Also, the
sliding mode gains ,
k q are positive.
5. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
5
Proof. First, we note that substituting (17) and (15) into the error dynamics (4), we obtain the
closed-loop error dynamics as
[ ]
1
( ) ( ) sgn( )
e Ae B CB C kI A e q s
−
= − + +
& (18)
To prove that the error dynamics (18) is globally asymptotically stable, we consider the
candidate Lyapunov function defined by the equation
2
1
( ) ( )
2
V e s e
= (19)
which is a positive definite function on .
n
R
Differentiating V along the trajectories of (18) or the equivalent dynamics (14), we get
2
( ) ( ) ( ) sgn( )
V e s e s e ks q s s
= = − −
& & (20)
which is a negative definite function on .
n
R
This calculation shows that V is a globally defined, positive definite, Lyapunov function for the
error dynamics (18), which has a globally defined, negative definite time derivative .
V
&
Thus, by Lyapunov stability theory [30], it is immediate that the error dynamics (18) is globally
asymptotically stable for all initial conditions (0) .
n
e ∈R
This means that for all initial conditions (0) ,
n
e R
∈ we have
lim ( ) 0
t
e t
→∞
=
Hence, it follows that the master system (1) and the slave system (2) are globally and
asymptotically hybrid synchronized for all initial conditions (0), (0) .
n
x y ∈R
This completes the proof.
3. HYBRID SYNCHRONIZATION OF IDENTICAL HYPERCHAOTIC LIU
SYSTEMS VIA SLIDING MODE CONTROL
3.1 Theoretical Results
In this section, we apply the sliding mode control results derived in Section 2 for the hybrid
synchronization of identical hyperchaotic Liu systems ([29], Liu et al. 2008).
Thus, the master system is described by the 4D dynamics
1 2 1
2 1 1 3 4
3 1 2 3 4
4 1 2
( )
x a x x
x bx x x x
x x x cx x
x dx x
= −
= + −
= − − +
= +
(21)
6. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
6
where 1 2 3 4
, , ,
x x x x are state variables and , , ,
a b c d are positive, constant parameters of the
system.
The Liu system (21) is hyperchaotic when the parameters are chosen as
10, 35, 1.4
a b c
= = = and 5.
d =
Figure 1 depicts the phase portrait of the hyperchaotic Liu system.
Figure 1. Phase Portrait of the Hyperchaotic Liu System
7. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
7
The slave system is described by the controlled hyperchaotic Liu dynamics
1 2 1 1
2 1 1 3 4 2
3 1 2 3 4 3
4 1 2 4
( )
y a y y u
y by y y y u
y y y cy y u
y dy y u
= − +
= + − +
= − − + +
= + +
(22)
where 1 2 3 4
, , ,
y y y y are state variables and 1 2 3 4
, , ,
u u u u are the controllers to be designed.
The hybrid synchronization error is defined by
1 1 1
2 2 2
3 3 3
4 4 4
e
e
e
e
y x
y x
y x
y x
= −
= +
= −
= +
(23)
The error dynamics is easily obtained as
1 2 1 2 1
2 1 4 1 1 3 1 3 2
3 3 4 4 1 2 1 2 3
4 1 2 1 4
( ) 2
2
2
2
e a e e ax u
e be e bx y y x x u
e ce e x y y x x u
e de e dx u
= − − +
= − + + + +
= − + − − + +
= + + +
(24)
We write the error dynamics (24) in the matrix notation as
( , )
e Ae x y u
η
= + +
(25)
where
0 0
0 0 1
,
0 0 1
1 0 0
a a
b
A
c
d
−
−
=
−
2
1 1 3 1 3
4 1 2 1 2
1
2
2
( , )
2
2
ax
bx y y x x
x y
x y y x x
dx
η
−
+ +
=
− − +
and
1
2
3
4
u
u
u
u
u
=
(26)
The sliding mode controller design is carried out as detailed in Section 2.
First, we set u as
( , )
u x y Bv
η
= − + (27)
where B is chosen such that ( , )
A B is controllable.
8. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
8
We take B as
1
1
1
1
B
=
(28)
In the hyperchaotic case, the parameter values are taken as
10, 35, 1.4
a b c
= = = and 5.
d =
The sliding mode variable is selected as
[ ] 1 2 3
2 1 1 0 2
s Ce e e e e
= = − = + − (29)
which makes the sliding mode state equation asymptotically stable.
We choose the sliding mode gains as 6
k = and 0.2.
q =
We note that a large value of k can cause chattering and an appropriate value of q is chosen to
speed up the time taken to reach the sliding manifold as well as to reduce the system chattering.
From Eq. (15), we can obtain ( )
v t as
1 2 3 4
( ) 13.5 13 2.3 0.1sgn( )
v t e e e e s
= − − + + − (30)
Thus, the required sliding mode controller is obtained as
( , )
u x y Bv
η
= − + (31)
where ( , ),
x y B
η and ( )
v t are defined as in the equations (26), (28) and (30).
By Theorem 2.1, we obtain the following result.
Theorem 3.1. The identical hyperchaotic Liu systems (21) and (22) are globally and
asymptotically hybrid synchronized for all initial conditions with the sliding mode controller
u defined by (31).
3.2 Numerical Results
For the numerical simulations, the fourth-order Runge-Kutta method with time-step 8
10
h −
= is
used to solve the hyperchaotic Liu systems (21) and (22) with the sliding mode controller
u given by (31) using MATLAB.
In the hyperchaotic case, the parameter values are given by
10, 35, 1.4
a b c
= = = and 5.
d =
9. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
9
The sliding mode gains are chosen as
6
k = and 0.2.
q =
The initial values of the master system (21) are taken as
1 2 3 4
(0) 4, (0) 9, (0) 16, (0) 10
x x x x
= = = = −
The initial values of the slave system (22) are taken as
1 2 3 4
(0) 21, (0) 12, (0) 6, (0) 2
y y
y y
= = = = −
Figure 2 illustrates the hybrid synchronization of the hyperchaotic Liu systems (21) and (22).
Figure 3 illustrates the time-history of the synchronization errors 1 2 3 4
, , , .
e e e e
Figure 2. Hybrid Synchronization of Identical Hyperchaotic Liu Systems
10. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.1, No.1, September 2012
10
Figure3. Time-History of the Hybrid Synchronization Error
4. CONCLUSIONS
In this paper, we have deployed sliding mode control (SMC) to achieve hybrid synchronization
for the identical hyperchaotic Liu systems (2008). Our hybrid synchronization results for the
identical hyperchaotic Liu systems have been proved 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 hybrid synchronization for the identical hyperchaotic
Liu systems. Numerical simulations are also shown to illustrate the effectiveness of the hybrid
synchronization results derived in this paper using the sliding mode control.
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Author
Dr. V. Sundarapandian obtained his Doctor of Science degree in Electrical and
Systems Engineering from Washington University, St. Louis, USA in May 1996.
He is a Professor at the R D Centre at Vel Tech Dr. RR Dr. SR Technical
University, Chennai, Tamil Nadu, India. He has published over 270 refereed
international journal publications. He has published over 170 papers in National
and International Conferences. He is the Editor-in-Chief of the AIRCC Journals -
International Journal of Instrumentation Control Systems, International
Journal of Control Systems Computer Modelling, International Journal of
Information Technology, Control Automation, and International Journal of
Chaos, Control, Modelling Simulation. His research interests are Linear and
Nonlinear Control Systems, Chaos Theory and Control, Soft Computing, Optimal
Control, Operations Research, Mathematical Modelling and Scientific
Computing. He has delivered several Key Note Lectures on Control Systems,
Chaos Theory, Scientific Computing, Mathematical Modelling, MATLAB and
SCILAB.