Date of final exam: 26/02/2001

E-mail: giovanardi@dsi.unifi.it

Tutor: Prof.  A. Tesi, Università di Firenze

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Stability analysis and control of periodic solutions in nonlinear systems   ___________________________________________________________________________________________________________

Advisor:

Prof.  A. Tesi, Università di Firenze

Summary of the thesis

The study of the properties of periodic solutions, despite being a "classical" topic in itself, is nowadays the subject of a renewed interest in nonlinear systems engineering. One of the reasons is the importance they have for the understanding of mechanisms and routes to complex and chaotic behavior, as well as for the control of such phenomena.

The key idea of so-called "low-energy" methods for feedback control of chaos is to stabilize one among the infinite unstable periodic orbits coexisting in the chaotic attractor. To this purpose, the approach known as "delayed feedback control" has rapidly become popular in recent years, especially thanks to its simplicity. However, systematic and comprehensive design tools for controllers of this type are still lacking.

The goal of the present work is to tackle the stability problem for families of periodic solutions in forced nonlinear systems by developing methodologies for stability analysis which, when given a control-oriented interpretation, can also serve as guidelines for the design of delayed feedback controllers capable of improving their stability properties. The main tools used are typical of nonlinear control systems engineering, namely linearization of the system around the periodic solution of interest and subsequent application of sufficient conditions for stability of the resulting linear periodic system.

The stability conditions to be verified turn out to be scalar whenever the considered nonlinear system can be recast in Lur'e form, while have a multivariable form for nonlinear systems belonging to a more general class. For their derivation both classical absolute stability results (circle criterion and extensions) and more recent tools such as those based on Integral Quadratic Constraints (IQCs) and Linear Matrix Inequalities (LMI) have been used. As a further original contribution, by specializing to the periodic case some existing results valid for generically time-varying linear systems, we derive specific criteria which provide an explicit estimate of the degree of exponential stability of the periodic solutions.
 

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