Optimization and Systems Theory Seminar
Monday, December 7, 1998, 11.00-12.00, Room 3733, Lindstedtsvägen 25

Dr. Bertina Ho-Mock-Qai
Centre d'Etudes de la Navigation Aerienne
Toulouse, France

A nonlinear approach to robust stabilization

The robust stabilization of a family of control systems consists in finding a control law which stabilizes all the systems in the family. We present a new approach to robust stabilization which enables us to establish the following results:

Countable families of systems

• Given any countable family of general nonlinear systems, that can be individually asymptotically stabilized, there exists a $C^0$ state feedback law which stabilizes (not asymptotically) all the systems in the family.
• Given any finite family of stabilizable LTI (linear time invariant) systems, there exists a $C^0$ nonlinear time-varying state feedback law which exponentially asymptotically stabilizes all the systems in the family.
• We have obtained sufficient conditions for the existence of a time-varying feedback law which asymptotically stabilizes all the systems of a finite family of nonlinear systems.

• Given any family of stabilizable LTI systems which depends continuously on real parameters that lie in closed intervals, there exists a continuous nonlinear time-invariant state feedback law which stabilizes (not asymptotically) all the systems in the family.
• Given any family of stabilizable LTI systems which depends continuously on real parameters that lie in closed intervals, there exists a continuous nonlinear time-varying state feedback law which robustly exponentially stabilizes the family.
• We have constructed continuous time-invariant feedback laws which robustly stabilize certain uncountable families of 2-dimensional nonlinear systems, which cannot be robustly stabilized by means of $C^1$ feedback.