Abstract

This dissertation includes two main parts. In the first part, the main contribution is
to use an inverse optimality approach to analytically solve the Hamilton-Jacobi-Bellman
equation of a third order nonlinear optimal control problem for which the dynamics are
affine and the cost is quadratic in the input. One special advantage of this work is that
the solution is directly obtained for the control input without finding a value function
first. However, the value function can be obtained after one solves for the control input
and it is shown to be at least a local Lyapunov function. Furthermore, the developed
controller is combined with a Continuous-Discrete Extended Kalman Filter (CDEKF) as
an approach to deal with noisy measurements and provide an estimate of the states for
feedback. The proposed technique is illustrated by its application to a path following
problem of a Wheeled Mobile Robot (WMR).
The main contribution of the second part of this thesis is the development of two
recursive state estimation algorithms for discrete-time piecewise affine (PWA) singular
systems with simulation evidence that the idea works for both uncorrelated and correlated
process and measurement noise. The proposed algorithms are derived based on successive
QR decompositions and Maximum Likelihood (ML) estimation theory. Numerical examples
are presented for the case of a PWA system with an unknown input, transformed to a
PWA singular system.