Studies on Stability and Stabilization of Randomly Switched Systems

This thesis presents a study on stability analysis and stabilizing controller synthesis of randomly switched systems. These systems have two ingredients: a family of nonlinear subsystems and a random switching signal that specifies which subsystem is active at each time instant. In broad strokes, the approach pursued here consists of identifying key properties of the switching signal and the family of subsystems, and finding conditions to connect these two sets of properties such that the switched system has some desirable stability characteristics. The method of multiple Lyapunov functions is employed in conjunction with some statistical properties of the switching signal for the analysis. The results apply to situations where traditional methods involving infinitesimal generators are difficult to apply, either due to insufficient information about the properties of the switching signal, or due to nontrivial dependence on its past history. Some of the results have conceptual parallels in deterministic switched systems theory. Stability in the presence of exogenous deterministic inputs is also considered, properties analogous to input-to-state stability are proposed, and sufficient conditions are established under which a randomly switched system exhibits these properties. Stabilizing controllers are synthesized for randomly switched systems with control inputs; the analysis results are utilized in conjunction with multiple control-Lyapunov functions and universal formulas for feedback stabilization of nonlinear systems. This approach lends a modular structure to the synthesis stage and facilitates the usage of standard off-the-shelf controllers.