Alexander L. Fradkov

Introduction to control of oscillations and chaos

Mathematics of nonlinear control mathematics of oscillations and chaos methods of nonlinear and adaptive control of oscillations control of oscillatory and chaotic systems applications.

Control of chaos: Methods and applications in engineering

Abstract A survey of the emerging field termed “control of chaos” is given. Several major branches of research are discussed in detail: feedforward or “nonfeedback control” (based on periodic excitation of the system); “OGY method” (based on linearization of the Poincare map), “Pyragas method” (based on a time-delay feedback), traditional control engineering methods including linear, nonlinear and adaptive control, neural networks and fuzzy control. Some unsolved problems concerning the justification of chaos control methods are presented. Other directions of active research such as chaotic mixing, chaotization, etc. are outlined. Applications in various fields of engineering are discussed.

Control of Chaos: Methods and Applications. I. Methods

The problems and methods of control of chaos, which in the last decade was the subject of intensive studies, were reviewed. The three historically earliest and most actively developing directions of research such as the open-loop control based on periodic system excitation, the method of Poincare map linearization (OGY method), and the method of time-delayed feedback (Pyragas method) were discussed in detail. The basic results obtained within the framework of the traditional linear, nonlinear, and adaptive control, as well as the neural network systems and fuzzy systems were presented. The open problems concerned mostly with support of the methods were formulated. The second part of the review will be devoted to the most interesting applications.

On self-synchronization and controlled synchronization

An attempt is made to give a general formalism for synchronization in dynamical systems encompassing most of the known definitions and applications. The proposed set-up describes synchronization of interconnected systems with respect to a set of functionals and captures peculiarities of both self-synchronization and controlled synchronization. Various illustrative examples are given.

Exponential feedback passivity and stabilizability of nonlinear systems

Abstract Motivated by N.Krasovskii’s characterisation of exponential stability, the concept of exponential passivity is introduced. It is shown that to make a nonlinear system with factorisable high-frequency gain matrix exponentially passive via either state or output feedback, exponential minimum phaseness and invertibility conditions are necessary and sufficient. These conditions also guarantee exponential output feedback stabilisability. This result extends previous results concerning linear systems.

Swinging control of nonlinear oscillations

The speed-gradient method of control design used previously for problems of regulation and tracking is extended to oscillating systems with energy-based objective functions. The concept of swinging control is introduced, meaning achievement of arbitrary large level of the objective function by arbitrary small control level. The existence of swinging control for hamiltonian systems is established. Simulation results for pendulum swinging problem are demonstrated.

Adaptive observer-based synchronization for communication

The problem of synchronizing two nonlinear systems (transmitter and receiver) is considered. A simple design of an adaptive observer for estimating the unknown parameters of the transmitter is proposed based on the design of quadratic Lyapunov function for the error system. The results are illustrated by an example of signal transmission based on a pair of synchronizing Chua circuits.

Adaptive synchronization of chaotic systems based on speed gradient method and passification

A problem of synchronizing two nonlinear multidimensional systems with unknown parameters is considered. Two general procedures for adaptive synchronization law design based on speed-gradient method are proposed. Conditions ensuring the synchronization are given. The second procedure is based on passification of error system (making it passive by feedback). The results are illustrated by examples: synchronizing a pair of Chuas circuits and a pair of circuits with tunnel diodes. Computer simulation results confirming theoretical analysis are given.

Control of Chaos: Methods and Applications. II. Applications

Reviewed were the problems and methods for control of chaos, which in the last decade was the subject of intensive studies. Consideration was given to their application in various scientific fields such as mechanics (control of pendulums, beams, plates, friction), physics (control of turbulence, lasers, chaos in plasma, and propagation of the dipole domains), chemistry, biology, ecology, economics, and medicine, as well as in various branches of engineering such as mechanical systems (control of vibroformers, microcantilevers, cranes, and vessels), spacecraft, electrical and electronic systems, communication systems, information systems, and chemical and processing industries (stirring of fluid flows and processing of free-flowing materials)).

Time domain interpretations of frequency domain inequalities on (semi)finite ranges

Many of the significant results in systems and control literature rely on characterizations of system properties in terms of frequency domain inequalities (FDIs) and/or time domain inequalities (TDIs). Classical FDIs, required to hold on the entire frequency range, have been interpreted by equivalent TDIs so that satisfaction of one implies that of the other. Recent developments have addressed FDIs within (semi)finite frequency ranges to increase flexibility in the system analysis and synthesis. This paper provides necessary and sufficient conditions for a general FDI to hold within a restricted frequency range in terms of TDIs to be satisfied by every inputs within a certain class specified by a matrix-valued integral quadratic constraint.