CSU EAST BAY
DEPARTMENT OF MATHEMATICS AND
COMPUTER SCIENCE
COLLOQUIUM
Wednesday, May 3, 2006; Noon-1pm Sc N108
Speaker:
Dr. Boumediene Hamzi, Visiting Research Asst. Professor
Dept of Mathematics, UC Davis
The Controlled Center Dynamics
In this talk we present the "Controlled Center Dynamics" which is the control theory analog of the center manifold theory of dynamical systems.
The center manifold theorem can be viewed as a model reduction technique for a nonlinear dynamics around a n equilibrium where one or more eigenvalues of its linear part are on the imaginary axis. If the rest of the eigenvalues are in the open left half plane then the local asymptotic stability of the equilibrium is decided by the local asymptotic stability of the dynamics on the center manifold. This leads to a reducti on of the dimension of the dynamics that needs to be analyzed to determine local asymptotic stability of the equilibrium. ^M
For a nonlinear control system around an equilibrium, the local asymptotic stability of the linear controllable directions can be easily achieved by linear feedback. Therefore the stabilizability of the whole system should depend on a reduced order model that corresponds to the stabilizability of the linearly uncontrollable directions. The controlled center dynamics technique formalizes this intuition.
We show, using normal forms under the feedback group, how the stabilizability of the overall system can be reduced to the stabilizability of the dynamics on a controlled center manifold. Part of the feedback is used to stabilize the linearly stabilizable directions and the other part is used to shape the center manifold. The shape of the center manifold determines the dynamics on it and the goal is to shape the center manifold so that its dynamics is locally asymptotically stable. We illustrate this approach by stabilizing systems with a transcontrollable, a fold, and a Hopf control bifurcations.
Pizza and soda will be served for those attending!