Robustness of uncertain systems : globally optimal Lyapunov function

The Lyapunov direct method is utilized to determine the robustness bounds for nonlinear, time-variant uncertainies p[subscript i]. Determination of the robustness bounds consists of two principal steps: (i) generation of a Lyapunov function and (ii) determination of the bounds based on the generated Lyapunov function. Presently in robustness investigations, a Lyapunov function is generated by inserting the nominal matrix to the Lyapunov equation and setting Q as identity matrix. The objective of this study is to utilize structural features of the uncertainties to develop a recursive algorithm for the generation of the globally optimal quadratic Lyapunov function. The proposed method is seemingly an improvement with respect to those reported in recent literature in three senses: i) ease of application, given an interactive program which requires only system matrices as inputs; ii) provision of improved estimates of the robustness bounds; and iii) extendability of the procedure to the design of robust controllers. The algorithm and the program prepared (in MATLAB) are presented. Several examples are considered for purposes of the comparison of robustness bounds estimates. Examples are demonstrated to show the superiority of the robustness bounds estimated by the proposed method over those obtained by small gain theorem. In a number of cases, the estimated robustness bounds are proven to be the exact robustness bounds