Optimal control of dynamical systems and structures under stochastic uncertainty: Stochastic optimal feedback control

Consider a dynamic mechanical control systems or structure under stochastic uncertainty, as e.g. the active control of a mechanical structure under stochastic applied dynamic loadings. Optimal controls, being most insensitive with respect to random parameter variations, are determined by finding stochastic optimal controls, i.e., controls minimizing the expected total costs composed of the costs arising along the trajectory, the costs for the control (correction), and possible terminal costs. The problem is modeled in the framework of optimal control under stochastic uncertainty, where the process differential equation depends on certain random parameters having a given probability distribution. Since by computing stochastic optimal controls, random parameter variations are incorporated into the optimal control design, most insensitive or robust controls are obtained. Based on the stochastic Hamiltonian of the optimal control problem under stochastic uncertainty, the class of ''H-minimal controls'' is determined first by solving a finite-dimensional stochastic program for the minimization of the expected Hamiltonian with respect to the input u(t) at time t. Having a H-minimal control, a two-point boundary value problem with random parameters is formulated for the computation of optimal state-and costate trajectories. Inserting then these trajectories into the H-minimal control, stochastic optimal controls are found, or at least stationary controls satisfying the necessary optimality conditions for a stochastic optimal control. Numerical solutions of the two-point boundary value problem are obtained by (i) Discretization of the underlying probability distribution of the random parameters, and (ii) Taylor expansion of the expected total costs and the expected Hamiltonian with respect to the random parameter vector at its expectation. The method is illustrated by the stochastic optimal regulation of a robot.