On feedback design and risk allocation in chance constrained control

This paper considers the problem of planning for linear, Gaussian systems, and extends existing chance constrained optimal control solutions. Due to the imperfect knowledge of the system state caused by process uncertainty and sensor noise, the system constraints cannot be guaranteed to be satisfied and consequently must be considered probabilistically. Therefore they are formulated as convex constraints on a Gaussian random variable, with the violation probability of all the constraints guaranteed to be below a threshold. Previous work considered optimizing the feedback controller to shape the uncertainty of the system to facilitate the satisfaction of the stochastic constraints. The joint constraints were bounded using an ellipsoidal relaxation technique which assigns uniform risk to each constraint. However this results in a large amount of conservatism degrading the performance of the overall system. Instead of using the ellipsoidal relaxation technique, this work bounds the joint constraints using Boole´s inequality which results in a tighter approximation. The conservatism is further reduced by optimizing the risk assigned to each constraint along with the feedback controller. A two-stage optimization algorithm is proposed that alternates between optimizing the feedback controller and the risk allocation until convergence. This solution methodology is shown to reduce the conservatism in previous approaches and improve the performance of the overall system.