Control Systems: New Approaches to Analysis and Design

This dissertation deals with two open problems in control theory. The first problem concerns the synthesis of fixed structure controllers for Linear Time Invariant (LTI) systems. The problem of synthesizing fixed structure/order controllers has practical importance when simplicity, hardware limitations, or reliability in the implementation of a controller dictates a low order of stabilization. A new method is proposed to simplify the calculation of the set of fixed structure stabilizing controllers for any given plant. The method makes use of computational algebraic geometry techniques and sign-definite decomposition method. Although designing a stabilizing controller of a fixed structure is important, in many practical applications it is also desirable to control the transient response of the closed loop system. This dissertation proposes a novel approach to approximate the set of stabilizing Proportional-Integral-Derivative (PID) controllers guaranteeing transient response specifications. Such desirable set of PID controllers can be constructed upon an application of Widder's theorem and Markov-Lukacs representation of non-negative polynomials. The second problem explored in this dissertation handles the design and control of linear systems without requiring the knowledge of the mathematical model of the system and directly from a small set of measurements, processed appropriately. The traditional approach to deal with the analysis and control of complex systems has been to describe them mathematically with sets of algebraic or differential equations. The objective of the proposed approach is to determine the design variables directly from a small set of measurements. In particular, it will be shown that the functional dependency of any system variable on any set of system design parameters can be determined by a small number of measurements. Once the functional dependency is obtained, it can be used to extract the values of the design parameters