Control of chaos in nonlinear circuits and systems

Nonlinear circuits and systems, such as electronic circuits (Chapter 5), power converters (Chapter 6), human brains (Chapter 7), phase lock loops (Chapter 8), sigma delta modulators (Chapter 9), etc, are found almost everywhere. Understanding nonlinear behaviours as well as control of these circuits and systems are important for real practical engineering applications. Control theories for linear circuits and systems are well developed and almost complete. However, different nonlinear circuits and systems could exhibit very different behaviours. Hence, it is difficult to unify a general control theory for general nonlinear circuits and systems. Up to now, control theories for nonlinear circuits and systems are still very limited. The objective of this book is to review the state of the art chaos control methods for some common nonlinear circuits and systems, such as those listed in the above, and stimulate further research and development in chaos control for nonlinear circuits and systems. This book consists of three parts. The first part of the book consists of reviews on general chaos control methods. In particular, a time-delayed approach written by H. Huang and G. Feng is reviewed in Chapter 1. A master slave synchronization problem for chaotic Lur’e systems is considered. A delay independent and delay dependent synchronization criteria are derived based on the H performance. The design of the time delayed feedback controller can be accomplished by means of the feasibility of linear matrix inequalities. In Chapter 2, a fuzzy model based approach written by H.K. Lam and F.H.F. Leung is reviewed. The synchronization of chaotic systems subject to parameter uncertainties is considered. A chaotic system is first represented by the fuzzy model. A switching controller is then employed to synchronize the systems. The stability conditions in terms of linear matrix inequalities are derived based on the Lyapunov stability theory. The tracking performance and parameter design of the controller are formulated as a generalized eigenvalue minimization problem which is solved numerically via some convex programming techniques. In Chapter 3, a sliding mode control approach written by Y. Feng and X. Yu is reviewed. Three kinds of sliding mode control methods, traditional sliding mode control, terminal sliding mode control and non-singular terminal sliding mode control, are employed for the control of a chaotic system to realize two different control objectives, namely to force the system states to converge to zero or to track desired trajectories. Observer based chaos synchronizations for chaotic systems with single nonlinearity and multi-nonlinearities are also presented. In Chapter 4, an optimal control approach written by C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao is reviewed. Systems with nonparametric regression with jump points are considered. The rough locations of all the possible jump points are identified using existing kernel methods. A smooth spline function is used to approximate each segment of the regression function. A time scaling transformation is derived so as to map the undecided jump points to fixed points. The approximation problem is formulated as an optimization problem and solved via existing optimization tools. The second part of the book consists of reviews on general chaos controls for continuous-time systems. In particular, chaos controls for Chua’s circuits written by L.A.B. Tôrres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes are discussed in Chapter 5. An inductorless Chua’s circuit realization is presented, as well as some practical issues, such as data analysis, mathematical modelling and dynamical characterization, are discussed. The tradeoff among the control objective, the control energy and the model complexity is derived. In Chapter 6, chaos controls for pulse width modulation current mode single phase H-bridge inverters written by B. Robert, M. Feki and H.H.C. Iu are discussed. A time delayed feedback controller is used in conjunction with the proportional controller in its simple form as well as in its extended form to stabilize the desired periodic orbit for larger values of the proportional controller gain. This method is very robust and easy to implement. In Chapter 7, chaos controls for epileptiform bursting in the brain written by M.W. Slutzky, P. Cvitanovic and D.J. Mogul are discussed. Chaos analysis and chaos control algorithms for manipulating the seizure like behaviour in a brain slice model are discussed. The techniques provide a nonlinear control pathway for terminating or potentially preventing epileptic seizures in the whole brain. The third part of the book consists of reviews on general chaos controls for discrete-time systems. In particular, chaos controls for phase lock loops written by A.M. Harb and B.A. Harb are discussed in Chapter 8. A nonlinear controller based on the theory of backstepping is designed so that the phase lock loops will not be out of lock. Also, the phase lock loops will not exhibit Hopf bifurcation and chaotic behaviours. In Chapter 9, chaos controls for sigma delta modulators written by B.W.K. Ling, C.Y.F. Ho and J.D. Reiss are discussed. A fuzzy impulsive control approach is employed for the control of the sigma delta modulators. The local stability criterion and the condition for the occurrence of limit cycle behaviours are derived. Based on the derived conditions, a fuzzy impulsive control law is formulated so that the occurrence of the limit cycle behaviours, the effect of the audio clicks and the distance between the state vectors and an invariant set are minimized supposing that the invariant set is nonempty. The state vectors can be bounded within any arbitrary nonempty region no matter what the input step size, the initial condition and the filter parameters are. The editors are much indebted to the editor of the World Scientific Series on Nonlinear Science, Prof. Leon Chua, and to Senior Editor Miss Lakshmi Narayan for their help and congenial processing of the edition

Design and Implementation of Secure Chaotic Communication Systems

Chaotic systems have properties such as ergodicity, sensitivity to initial conditions/parameter mismatches, mixing property, deterministic dynamics, structure complexity, to mention a few, that map nicely with cryptographic requirements such as confusion, diffusion, deterministic pseudorandomness, algorithm complexity. Furthermore, the possibility of chaotic synchronization, where the master system (transmitter) is driving the slave system (receiver) by its output signal, made it probable for the possible utilization of chaotic systems to implement security in the communication systems. Many methods like chaotic masking, chaotic modulation, inclusion, chaotic shift keying (CSK) had been proposed however, many attack methods later showed them to be insecure. Different modifications of these methods also exist in the literature to improve the security, but almost all suffer from the same drawback. Therefore, the implementation of chaotic systems in security still remains a challenge. In this work, different possibilities on how it might be possible to improve the security of the existing methods are explored. The main problem with the existing methods is that the message imprint could be found in the dynamics of the transmitted signal, therefore by some signal processing or pattern classification techniques, etc, allow the exposition of the hidden message. Therefore, the challenge is to remove any pattern or change in dynamics that the message might bring in the transmitted signal

Design and implementation of secure chaotic communication systems

Chaotic systems have properties such as ergodicity, sensitivity to initial conditions/parameter mismatches, mixing property, deterministic dynamics, structure complexity, to mention a few, that map nicely with cryptographic requirements such as confusion, diffusion, deterministic pseudorandomness, algorithm complexity. Furthermore, the possibility of chaotic synchronization, where the master system (transmitter) is driving the slave system (receiver) by its output signal, made it probable for the possible utilization of chaotic systems to implement security in the communication systems. Many methods like chaotic masking, chaotic modulation, inclusion, chaotic shift keying (CSK) had been proposed however, many attack methods later showed them to be insecure. Different modifications of these methods also exist in the literature to improve the security, but almost all suffer from the same drawback. Therefore, the implementation of chaotic systems in security still remains a challenge. In this work, different possibilities on how it might be possible to improve the security of the existing methods are explored. The main problem with the existing methods is that the message imprint could be found in the dynamics of the transmitted signal, therefore by some signal processing or pattern classification techniques, etc, allow the exposition of the hidden message. Therefore, the challenge is to remove any pattern or change in dynamics that the message might bring in the transmitted signal.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area

Nonlinear Time-Frequency Control Theory with Applications

Nonlinear control is an important subject drawing much attention. When a nonlinear system undergoes route-to-chaos, its response is naturally bounded in the time-domain while in the meantime becoming unstably broadband in the frequency-domain. Control scheme facilitated either in the time- or frequency-domain alone is insufficient in controlling route-to-chaos, where the corresponding response deteriorates in the time and frequency domains simultaneously. It is necessary to facilitate nonlinear control in both the time and frequency domains without obscuring or misinterpreting the true dynamics. The objective of the dissertation is to formulate a novel nonlinear control theory that addresses the fundamental characteristics inherent of all nonlinear systems undergoing route-to-chaos, one that requires no linearization or closed-form solution so that the genuine underlying features of the system being considered are preserved. The theory developed herein is able to identify the dynamic state of the system in real-time and restrain time-varying spectrum from becoming broadband. Applications of the theory are demonstrated using several engineering examples including the control of a non-stationary Duffing oscillator, a 1-DOF time-delayed milling model, a 2-DOF micro-milling system, unsynchronized chaotic circuits, and a friction-excited vibrating disk. Not subject to all the mathematical constraint conditions and assumptions upon which common nonlinear control theories are based and derived, the novel theory has its philosophical basis established in the simultaneous time-frequency control, on-line system identification, and feedforward adaptive control. It adopts multi-rate control, hence enabling control over nonstationary, nonlinear response with increasing bandwidth ? a physical condition oftentimes fails the contemporary control theories. The applicability of the theory to complex multi-input-multi-output (MIMO) systems without resorting to mathematical manipulation and extensive computation is demonstrated through the multi-variable control of a micro-milling system. The research is of a broad impact on the control of a wide range of nonlinear and chaotic systems. The implications of the nonlinear time-frequency control theory in cutting, micro-machining, communication security, and the mitigation of friction-induced vibrations are both significant and immediate

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Nonlinear Time-Frequency Control Theory with Applications

Nonlinear control is an important subject drawing much attention. When a nonlinear system undergoes route-to-chaos, its response is naturally bounded in the time-domain while in the meantime becoming unstably broadband in the frequency-domain. Control scheme facilitated either in the time- or frequency-domain alone is insufficient in controlling route-to-chaos, where the corresponding response deteriorates in the time and frequency domains simultaneously. It is necessary to facilitate nonlinear control in both the time and frequency domains without obscuring or misinterpreting the true dynamics. The objective of the dissertation is to formulate a novel nonlinear control theory that addresses the fundamental characteristics inherent of all nonlinear systems undergoing route-to-chaos, one that requires no linearization or closed-form solution so that the genuine underlying features of the system being considered are preserved. The theory developed herein is able to identify the dynamic state of the system in real-time and restrain time-varying spectrum from becoming broadband. Applications of the theory are demonstrated using several engineering examples including the control of a non-stationary Duffing oscillator, a 1-DOF time-delayed milling model, a 2-DOF micro-milling system, unsynchronized chaotic circuits, and a friction-excited vibrating disk. Not subject to all the mathematical constraint conditions and assumptions upon which common nonlinear control theories are based and derived, the novel theory has its philosophical basis established in the simultaneous time-frequency control, on-line system identification, and feedforward adaptive control. It adopts multi-rate control, hence enabling control over nonstationary, nonlinear response with increasing bandwidth ? a physical condition oftentimes fails the contemporary control theories. The applicability of the theory to complex multi-input-multi-output (MIMO) systems without resorting to mathematical manipulation and extensive computation is demonstrated through the multi-variable control of a micro-milling system. The research is of a broad impact on the control of a wide range of nonlinear and chaotic systems. The implications of the nonlinear time-frequency control theory in cutting, micro-machining, communication security, and the mitigation of friction-induced vibrations are both significant and immediate