**Homepage**

## 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.