Robustness analysis of uncertain linear systems and robust stabilization of uncertain delayed systems.

摘要

This dissertation focuses on two main aspects. One, developing new tools for the robustness analysis of uncertain linear systems. Two, the synthesis of robust controllers for uncertain delay systems.; Traditional methods to analyze the robust stability of linear systems have depended on structured singular value bounds. However, this approach neglects phase dependence of the uncertainty and results in undue conservatism. In this dissertation, the concept of the Nyquist robust-stability margin, is introduced for characterizing the closed-loop stability of uncertain systems. The approach makes direct use of Nyquist domain arguments and is based on the analysis of the perturbed eigenvalue loci, hence avoiding undue conservatism through the use of singular-value upper bounds. A key element in the new approach is Critical Direction Theory applied to uncertainties in the Nyquist plane. The critical direction method is based on recognizing that, at any given frequency on the Nyquist plane, there is only one direction of perturbation of relevance to the stability analysis. This allows the characterization of robust stability margins for uncertain systems characterized by irregular perturbation templates, a problem that poses significant challenges to other analysis methods. Examples of practical relevance are given to illustrate the application of the new theory. Using the new approach, the problem of assessing robust stability and computing stability margins for SISO systems with affine complex parametric uncertainties is tackled successfully. Exact analytical results are derived for geometrically simple uncertainty sets such as ellipses and rectangles.; In the later part of the dissertation, the synthesis of robust controllers for state-delayed and input-delayed systems is considered. In particular, Sliding Mode Control is chosen as the technique of choice, as it possesses the combination of robustness and performance guarantees that one seeks in a control system. Robust stability to the chosen perturbation characterization is rigorously proven. Practical difficulties in implementation are pointed out and ways to overcome these hurdles are presented. Some open questions in the literature are brought out and theoretical analysis and analytical answers are presented. Finally, ideas and interesting formulations are presented for future work.