Stochastic Invariance and Aperiodic Control for Uncertain Constrained Systems

Abstract: Uncertainties and constraints are present in most control systems. For example, robot motion planning and building climate regulation can be modeled as uncertain constrained systems. In this thesis, we develop mathematical and computational tools to analyze and synthesize controllers for such systems.As our first contribution, we characterize when a set is a probabilistic controlled invariant set and we develop tools to compute such sets. A probabilistic controlled invariantset is a set within which the controller is able to keep the system state with a certainprobability. It is a natural complement to the existing notion of robust controlled invariantsets. We provide iterative algorithms to compute a probabilistic controlled invariantset within a given set based on stochastic backward reachability. We prove that thesealgorithms are computationally tractable and converge in a finite number of iterations. The computational tools are demonstrated on examples of motion planning, climate regulation, and model predictive control.As our second contribution, we address the control design problem for uncertain constrained systems with aperiodic sensing and actuation. Firstly, we propose a stochastic self-triggered model predictive control algorithm for linear systems subject to exogenous disturbances and probabilistic constraints. We prove that probabilistic constraint satisfaction, recursive feasibility, and closed-loop stability can be guaranteed. The control algorithm is computationally tractable as we are able to reformulate the problem into a quadratic program. Secondly, we develop a robust self-triggered control algorithm for time-varying and uncertain systems with constraints based on reachability analysis. In the particular case when there is no uncertainty, the design leads to a control system requiring minimum number of samples over finite time horizon. Furthermore, when the plant is linear and the constraints are polyhedral, we prove that the previous algorithms can be reformulated as mixed integer linear programs. The method is applied to a motion planning problem with temporal constraints.