Time-varying Feedback Systems Design Via Diophantine Equation Order Reduction

Diophantine equation plays an important role in the design and synthesis of feedback compensators. Many methods have been developed to solve the Diophantine equation. This dissertation develops a new systematic approach of solving a linear time-varying Diophantine equation. This approach is based upon successively reducing the order of the Diophantine equation by Euclidean algorithm. Euclidean algorithm for solving for both time-invariant and time-varying Diophantine equations for directly determining both the quotient and the remainder associated with the division of one polynomial by another is presented. The coprimeness (right or left) of two Polynomial Differential Operators is needed to guarantee, in general, the existence of solutions of the respective Diophantine equation. The illustrative examples are given. This dissertation also develops a procedure of setting up canonical forms for linear time-varying, single-input single-output systems. The starting point is a differential equation description of the system. Two canonical forms are considered: observability and observer forms. Initial condition conversions between the canonical forms and the differential equation description are also derived.