Optimal Structural Control under Stochastic Uncertainty: Robust Optimal Open-Loop Feedback Control

Abstract

AbstractOptimal feedback controls of PD- or PID-type can be approximated very efficiently by optimal open-loop feedback controls based on optimal open-loop controls. Extending the standard construction, stochastic optimal open-loop feedback controls are constructed by taking into account still the random parameter variations in the control system. Hence, corresponding to the standard deterministic case, stochastic optimal open-loop feedback controls are obtained by computing first stochastic optimal open-loop controls on the remaining time intervals tb ≤ t ≤ tf with an arbitrary intermediate starting time point tb, t0 ≤ tb ≤ tf. Evaluating these controls at the corresponding intermediate starting time points tb only, a stochastic optimal open-loop feedback control law is obtained. Stochastic optimal open-loop controls are determined by using a stochastic Hamiltonian approach. Hence, after introducing the stochastic Hamiltonian of the control problem under stochastic uncertainty, the H-minimal controls are determined and the related two-point boundary value problem with random parameters is solved. In order to apply the solution technique in real-time for two-point boundary value problems developed in former papers, also to problems with a non constant system matrices, the occurring fundamental matrix is computed approximatively by appropriate approximations of the system matrix, as e.g. by i) constant system matrices, ii) piecewise constant system matrices, and iii) by discretization of the probability distribution in case of a random system matrix. Error estimates are given.