Alexander L. Fradkov
Institute for Problems of Mechanical Engineering
Russian Academy of Sciences
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).
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.