Stability and stabilization of nonlinear discontinuous systems

This thesis presents some new results on stability and stabilization of discontinuous nonlinear systems. Stability conditions are formulated as constrained convex optimization problems, and can be tested using efficient tools of linear programming and convex optimization involving linear matrix inequalities.;Complex dynamic systems are considered, which are composed of subsystems with uncertain interconnections. A distinct feature of our investigation is the assumption that uncertain interconnection matrices reside in matrix polytopes. To establish polytopic connective stability we introduce the concept of parameter-dependent vector Liapunov functions. Both continuously differentiable and Lipschitz-type functions are introduced and their differences discussed in the new setting. The connective stability conditions are stated as convex optimization problems in terms of convex M-matrices and the vertices of the uncertainty polytopes. The new results are applied to formulations of decentralized robust control laws for polytopic interconnected systems when generalized matching conditions are satisfied.;We also consider the robust quadratic stability and feedback stabilization of a class of linear constant systems under additive nonlinear time-varying perturbations in both continuous and discrete-time domains. The perturbations are uncertain functions constrained within quadratic bounds. We formulate a convex optimization problem of maximizing the estimate of the bound on the perturbation which the stable nominal system can tolerate without going unstable. An important benefit of the new formulation is that it can accommodate unstable linear plants. A design of feedback control laws can be carried out to stabilize the plant and, at the same time, maximize the bounds on perturbations. We generalize our method to include interconnected systems where the interconnections between subsystems are considered as perturbations. The crucial assumption is the presence of decentralized information structure constraints which limits the feedback control laws to locally available states at the subsystem level. The optimization procedure generates control laws that simultaneously stabilize the overall system and maximize the bounds on the uncertain interconnection terms.