A.G. Krener, University of California, Davis, U.S.A.

1. Topic: The Global Convergence of the Minimum Energy Estimator and the Local Convergence of the Extended Kalman Filter

Time: June 13, 2003, 2 pm
Place: Building 19, Room 316

Abstract: We shall discuss the problem of constructing an observer for a nonlinear system and introduce the minimum energy estimator (MEE). From this we shall derive the Extended Kalman Filter (EKF). Then we discuss the global convergence of the MEE and the local convergence of the EKF.

2. Topic: Control Bifurcations

Abstract: A parametrized nonlinear differential equation can have multiple equilibria as the parameter is varied. A local bifurcation of a parametrized differential equation occurs at an equilibrium where there is a change in the topological character of the nearby solution curves. This typically happens because some eigenvalues of the parametrized linear approximating differential equation cross the imaginary axis and there is a change in stability of the equilibrium. The topological nature of the solutions is unchanged by smooth changes of state coordinates so these may be used to bring the differential equation into Poincar\'{e} normal form. From this normal form, the type of the bifurcation can be determined. For differential equations depending on a single parameter, the typical ways that the system can bifurcate are fully understood, e.g., the fold (or saddle node), the transcritical and the Hopf bifurcation. A nonlinear control system has multiple equilibria typically parametrized by the set value of the control. A control bifurcation of a nonlinear system typically occurs when its linear approximation loses stabilizability. The ways in which this can happen are understood through the appropriate normal forms. These are the normal forms under quadratic and cubic change of state coordinates and invertible state feedback. The system need not be linearly controllable. We introduce some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations.