A new set of tools for the stability analysis and robust control design for nonlinear systems is introduced in this dissertation. The tools have a wide range of applicability, covering systems with common types of hysteresis, as well as systems with memoryless forms such as saturation, or slope-restricted nonlinearities (e.g., parameter uncertainty or gain variation), and can be used to guarantee stability for systems with both nonlinearities and additional norm-bounded uncertainty. These robust stability tests are developed using a combination of passivity and dissipation theories, and are presented in both graphical (Nyquist) and numerical form using linear matrix inequalities (LMIs). The LMI formulation yields tests that are efficiently solved with existing software packages, and allows the extension to the case of multiple nonlinearities. In particular, an asymptotic stability test is developed using LMIs for systems with multiple hysteresis nonlinearities. The invariant set for such systems is shown in general to be a polytopic region of the state space. This analysis

This paper presents an iterative algorithm for solving the parametric robust H2 controller synthesis problem and analyzes the convergence properties of the algorithm on several examples. Iterative procedures are normally applied to a large class of robust control design problems in which the formulation naturally leads to bilinear matrix inequalities (BMIs). It is di cult to make concrete statements about the behavior of these iterative algorithms, except that it is often conjectured that the cost in each step of the solution procedure is reduced, which implies that the algorithms should converge to a local minimum. Similar di culties exist for the new LMIbased iterative algorithm that we haverecently proposed to solve the BMIs that occur in robust H2 control design. The e ectiveness of the new algorithm has already been demonstrated on several numerical examples. This paper adds an important component tothediscussion on the convergence of the new algorithm by verifying that it e ciently converges to the optimal solution. In the process, we provide some new key insights on the proposed design technique which indicate that it exhibits properties similar to the D{K iteration of the complex =Km-synthesis. 1

Recent papers have demonstrated the e ectiveness of our iterative algorithm using linear matrix inequalities (LMI's) on several parametric robust H1 control designs. This paper presents two additional important components to the discussion on the behavior of the new LMI-based iterative algorithm: a comparison study between the LMI synthesis technique and the existing iterative approaches of the complex and mixed =Km synthesis, and a convergence analysis of this new algorithm. The results indicate that the Popov H1 controller synthesis provides a viable alternative for designing real parametric robust controllers and exhibits properties similar to the D{K and D,G{K iteration of the complex and mixed =Km synthesis. The key potential advantage of using the LMI approach is the elimination of the curve- tting for the D and G scaling functions. 1