A randomized relaxation method to ensure feasibility in stochastic control of linear systems subject to state and input constraints

Abstract

We consider a linear system affected by an additive stochastic disturbance and address the design of a finite horizon control policy that is optimal according to some cost criterion and accounts also for probabilistic constraints on both the input and state variables. The resulting policy can be implemented over a receding horizon according to the model predictive control strategy. Such a possibility, however, is hampered by the fact that a feasibility issue may arise when recomputing the policy. Infeasibility indeed can occur if the disturbance has unbounded support and the state is required to remain in a bounded set. In this paper, we propose a solution to this issue that is based on the introduction of a constraint relaxation that becomes effective only when the original problem turns out to be unfeasible. This is obtained via a cascade of two probabilistically-constrained optimization problems where, in the first one, performance is neglected and the policy is designed to fully recover feasibility or -if this is not possible-to determine the minimum level of relaxation which is needed to recover feasibility; in the second step, such a minimum relaxation level is imposed while optimally (re-)tuning the control policy parameters. Both problems are solved through a computationally tractable scenario-based scheme using a finite number of disturbance realizations and providing an approximate solution that satisfies with high confidence the original probabilistic constraints of the cascade.

New results on resource sharing problems with random agent arrivals and an application to economic dispatch in power systems

“…To address this problem, we view it under a data driven lens and provide a confidence interval for a probabilistic sensitivity index given by the probability that the optimal resource share changes upon the arrival of a new agent. Such a result is based on a combination of tools from linear programming, duality theory, and the so called scenario approach, [17], [18], [19], especially the scenario approach with constraint relaxation, [20], [21]. However, the standard scenario approach theory is based on the existence of a certain subset of the samples with an a priori fixed cardinality that produces the same solution that is obtained when all the samples are employed.…”