Transformation and Symbolic Calculations in Filtering and Control

Abstract: This thesis deals with several aspects of optimal control and filtering problems. A fundamental problem in the optimal control theory is the design of a regulatorfora linear system which minimizes a quadratic cost function characterizing the control effort and the deviations of the system from the ideal operation. On the other hand, a basic problem in signal analysis is the optimal estimation of a useful signal from observations corrupted by additive noise. The solutions of both problems depend in a crucial way on solutions of the matrix Riccati differential equation.This work is an effort to unify the theoretical analysis and the numerical or symbolic calculation of solutions of the Riccati differential equation (RDE) as well as the algebraic Riccati equation (ARE) via a matrix transformation. It is shown that the most important issues evolving around the Riccati equation solution can be completely chara:cterized by submatrices of a transformation matrix. Not only the necessary and sufficient conditions for the existence of a solution but also an explicit expression of the solution are obtained from this computable matrix transformation. The transformation matrix can be calculated whether the solution exists or not. It can clearly be seen from this transformation matrix that the solution of the ARE can be explicitly expressed via the submatrices even if the system is not stabilizable (in the optimal filtering context). Furthermore, the Riccati differential equation can also be solved analytically in terms of these submatrices. The criterion, which ensures a solution of the Riccati differential equation to converge to the stabilizing solution, or the strong solution, are established via the same transformation matrix and with a more relaxed requirement on the initial condition than existing results. The criterion is proven to be sufficient and necessary, thus extending existing convergence results.The technique of the matrix transformation is shown to be very useful for exploring the Riccati equation associated with linear time varying systems. We show that an important class of linear time varying systems can be transformed using an appropriate time varying matrix transformation to a linear time invariant form. Hence, instead of attempting to solve the RDE with time varying coefficients, its time invariant correspondence may be solved symbolically.In applying the theoretical analysis for practical systems, a central question is how well-established theories for linear systems can be applied to practical nonlinear systems. As an effort to bridge the gap, we extend linear optimal filtering to some nonlinear problems. The technique proposed in the thesis is applied to solve the problem of transformer saturation in protective relaying for power systems. Simulation results are provided to illustrate the accuracy of the method.