### Optimization and Systems Theory Seminar

April 28, 1997, 14.00-15.00, Room 3721, Lindstedtsvägen 25

** Alexander L. Fradkov **

Institute for Problems of Mechanical Engineering

Russian Academy of Sciences

St.Petersburg

Russia

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Discrete and continuous adaptive control of nonlinear oscillations and chaos

In the first part of the talk a unified approach to adaptive control
of periodic and chaotic oscillations in nonlinear continuous-time systems
is presented. The approach is based on the so called speed-gradient method
(changing the control parameters proportionally to the gradient of the
derivative of the given objective functional along the controlled system
trajectories).

The speed-gradient method is extended to achieve both conventional
control objectives (regulation and tracking) and specific goals like
excitation (swinging) and syncronization of oscillations (see [1-3]).
Stability and robustness issues both for general case and for Hamiltonian
systems are addressed. Then multigoal and constrained problems are
considered. Special attention is paid to the problem of controlling and
synchronizing chaos.

The second part of the talk is devoted to examination and extension
of the so called OGY method of controlling chaos. The rigorous analysis is
given based on the new concept "controlled Poincare map" and on the method
of recursive goal inequalities suggested by V.Yakubovich in 1966. The
obtained control algorithms solve the tracking problem for the recurrent
trajectories under bounded disturbances.

The theoretical results are illustrated by examples: mechanical
(pendulum-like) oscillatory systems as well as some classical chaotic
systems (Duffing equation, Chua's circuit, forced brusselator).

**References**

1. A.L.Fradkov, Swinging control of nonlinear oscillations.
International J. Control, v.64, No 6, 1996, pp. 1189-1202.

2. A.L. Fradkov, A.Yu. Pogromsky,
Speed-gradient control of chaotic continuous-time systems.
IEEE Trans.Circuits and Systems, part I, v.43, No11 pp. 907-913.

3. A.L. Fradkov, A.Yu. Pogromsky,
Methods of nonlinear and adaptive control of chaotic systems.
13th IFAC World Congress, San-Francisco, July 1996, v.E, pp. 185-190.

Calendar of seminars

*Last update: April 28, 1997 by
Anders Forsgren,
andersf@math.kth.se.
*