Preface, Notations 1.Introduction to Time-Delay Systems I.Frequency-Domain Approach 2.Systems with Commensurate Delays 3.Systems withIncommensurate Delays 4.Robust Stability Analysis II.Time Domain Approach 5.Systems with Single Delay 6.Robust Stability Analysis 7.Systems with Multiple and Distributed Delays III.Input-Output Approach 8.Input-output stability A.Matrix Facts B.LMI and Quadratic Integral Inequalities Bibliography Index

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Stability of Time-Delay Systems

1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

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The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

From the Publisher:
A comprehensive treatment of model-based fuzzy control systems
This volume offers full coverage of the systematic framework for the stability and design of nonlinear fuzzy control systems. Building on the Takagi-Sugeno fuzzy model, authors Tanaka and Wang address a number of important issues in fuzzy control systems, including stability analysis, systematic design procedures, incorporation of performance specifications, numerical implementations, and practical applications.
Issues that have not been fully treated in existing texts, such as stability analysis, systematic design, and performance analysis, are crucial to the validity and applicability of fuzzy control methodology. Fuzzy Control Systems Design and Analysis addresses these issues in the framework of parallel distributed compensation, a controller structure devised in accordance with the fuzzy model.
This balanced treatment features an overview of fuzzy control, modeling, and stability analysis, as well as a section on the use of linear matrix inequalities (LMI) as an approach to fuzzy design and control. It also covers advanced topics in model-based fuzzy control systems, including modeling and control of chaotic systems. Later sections offer practical examples in the form of detailed theoretical and experimental studies of fuzzy control in robotic systems and a discussion of future directions in the field.
Fuzzy Control Systems Design and Analysis offers an advanced treatment of fuzzy control that makes a useful reference for researchers and a reliable text for advanced graduate students in the field.

Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach

Controlling chaos

Recurrence plots for the analysis of complex systems

An adaptive dynamic state feedback controller for stabilizing and tracking unknown steady states of dynamical systems is considered. We prove that the steady state can never be stabilized if the system and controller as a whole has an odd number of real positive eigenvalues. For two-dimensional systems, this topological limitation states that only an unstable focus or node can be stabilized with a stable controller, and a stabilization of a saddle requires the presence of an unstable degree of freedom in a feedback loop. The use of the controller to stabilize saddles and unstable spirals is demonstrated numerically with several models: a pendulum driven with a constant torque, the Lorenz system, and an electrochemical Ni dissolution system.

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Adaptive stabilization of unstable steady states of nonlinear dynamical systems

We propose an adaptive modification of the delayed feedback control (DFC) method that automatically finds the optimal feedback gain of the DFC systems. The system under control is perturbed by a Gaussian white noise with a low amplitude, and the variance of the delayed difference is estimated by a simple integrator. The problem of minimization of the variance is substituted by a minimization of some quantity that represents a generalization of the Lyapunov exponent in the presence of noise. The generalization is derived by considering a simple Langevin equation. The numerical simulations for the controlled Rössler system show that the adaptively obtained optimal feedback gains are in good quantitative agreement with the corresponding exact values.

Adaptive Delayed Feedback Control Algorithm With an Iterated Feedback Gain in the Presence of Noise

The phase reduction method for a limit cycle oscillator subjected to a strong amplitude-modulated high-frequency force is developed. An equation for the phase dynamics is derived by introducing a new, effective phase response curve. We show that if the effective phase response curve is everywhere positive (negative), then an entrainment of the oscillator to an envelope frequency is possible only when this frequency is higher (lower) than the natural frequency of the oscillator. Also, by using the Pontryagin maximum principle, we have derived an optimal waveform of the perturbation that ensures an entrainment of the oscillator with minimal power. The theoretical results are demonstrated with the Stuart-Landau oscillator and model neurons.

Phase reduction of a limit cycle oscillator perturbed by a strong amplitude-modulated high-frequency force

Intracavity frequency-doubled solid-state lasers exhibit intensity fluctuations of their light output, which are cause by nonlinear dynamical processes. Up to now, there are different solutions to this problem, but they reduce the output power, increase the size of the laser and/or make them more complicated to assemble. One focus of current research in nonlinear dynamics is derivation of control strategies from mathematical models and their experimental realization. We suggest a method to stabilize the output power by means of an electronic feedback of the emitted infrared light intensity to the pump power. First we show the theoretical predictions of a recently published stability analysis of a rate equation model with feedback. The presented experimental observation show systematic deviations from theory. This makes it necessary to refine the model to explain the deviations. The refinement has direct impact on the improvement of the feedback loop and, therefore, on the application of such a control scheme.

Kestutis Pyragas

Papers

Adaptive stabilization of unstable steady states of nonlinear dynamical systems

Adaptive Delayed Feedback Control Algorithm With an Iterated Feedback Gain in the Presence of Noise

Phase reduction of a limit cycle oscillator perturbed by a strong amplitude-modulated high-frequency force

Stabilization of the output power of intracavity frequency-doubled lasers

Using Time-Delay Feedback for Control and Synchronization of Dynamical Systems