Robust control of stochastic nonlinear systems.

摘要

This thesis investigates several topics involving robust control of stochastic nonlinear systems in strict-feedback form, namely stochastic stabilization, risk-sensitive stochastic control, locally optimal controller design, and constrained minimax optimal control.; The first topic is the stabilizability problem for general nonlinear stochastic dynamic systems. The concept of stochastic input-to-state stability is introduced and applied to singularly perturbed systems. Based on time scale decomposition, a result of the “total stability” type is obtained; i.e., if the fast subsystem and the slow subsystem are both input-to-state stable with respect to disturbances, then this property continues to hold for the full order system as long as the singular perturbation parameter is sufficiently small and stochastic small gain conditions are satisfied. The result holds for a broad class of disturbances, and resembles similar results for deterministic systems.; The second topic studied involves the optimization and control of stochastic nonlinear systems. For a class of risk-sensitive stochastic control problems with system dynamics in strict-feedback form, we obtain through a constructive derivation state-feedback controllers that are both locally optimal and globally inverse optimal, which further lead to closed-loop system trajectories that are bounded in probability. Local optimality implies that a linearized version of these controllers solve a linear exponential quadratic Gaussian problem, and global inverse optimality says that there exists an appropriate cost function according to which these controllers are optimal.; The third topic studied involves the constrained minimax optimization problem for a class of stochastic nonlinear systems in strict-feedback form, where in addition to the standard Wiener process there is a norm-bounded unknown disturbance driving the system. The bound on the disturbance is a stochastic integral quadratic constraint, and it is also related to the constraint on the relative entropy between the uncertainty probability measure and the reference probability measure on the original probability space. Within this structure, by first converting the original constrained optimization problem into an unconstrained one (a stochastic differential game) and then making use of the duality relationship between stochastic games and risk-sensitive stochastic control, we obtain a minimax state-feedback control law that is both locally optimal and globally inverse optimal. Furthermore, the closed-loop system is absolutely stable in the presence of stochastic uncertainty disturbances.