On the synthesis of fixed order stabilizing controllers

In this dissertation, we consider two problems concerning the synthesis of fixed order controllers for Single Input, Single Output systems. The first problem deals with the synthesis of absolutely stabilizing fixed order controllers for Lure-Postnikov systems. The second problem deals with the synthesis of fixed order stabilizing controllers directly from the empirical frequency response data and from some coarse information of the plant. Lure-Postnikov systems are frequently encountered in mechanical engineering applications. Analytical tools for synthesizing stabilizing fixed structure controllers, such as the PID controllers examining the absolute stability of Lure-Postnikov systems, have recently been studied in the literature. However, tools for synthesizing controllers of arbitrary order have not been studied yet. We propose a systematic method for synthesizing absolutely stabilizing controllers of arbitrary order for the Lure-Postnikov systems. Our approach is based on recent results in the literature on approximation of the set of stabilizing controller parameters that render a family of real and complex polynomials Hurwitz. We provide an example of a robotic system to illustrate the procedure developed. Exact analytical models of plants may not be readily available for controller design. The current approach is to synthesize controllers through the identification of the analytical model of the plant from empirical frequency response data. In this dissertation, we depart from this conventional approach. We seek to synthesize controllers directly (i.e. without resort to identification) from the empirical frequency response data of the plant and coarse information about it. The coarse information required is the number of nonminimum phase zeros of the plant(or the number of poles of the plant with positive real parts) and the frequency range beyond which the phase response of the LTI plant does not change appreciably and the amplitude response goes to zero. We also assume that the LTI plant does not have purely imaginary zeros or poles. The method of synthesizing stabilizing controllers involves the use of generalized Hermite-Biehler theorem for counting the roots of rational functions and the use of recently developed Sum-of-Squares techniques for checking the nonnegativity of a polynomial in an interval through the Markov-Lucaks theorem. The method does not require an explicit analytical model of the plant that must be stabilized or the order of the plant, rather, it only requires the empirical frequency response data of the plant. The method also allows for measurement errors in the frequency response of the plant. We illustrate the developed procedure with an example. Finally, we extended the technique to the synthesis of controllers of arbitrary order that also guarantee performance specifications such as the phase margin and gain margin.