Optimal bounded control of random vibration and hybrid solutions to dynamic programming equations

Classical linear optimization theory for stochastic control problems may lead to very large required control forces in the actuators, thereby making active control of vibration unfeasible. Bounds on the available control forces make the problem nonlinear. A possible approach to optimal bounded control for randomly vibrating systems is based on dynamic programming and the Hamilton-Jacobi-Bellman (or HJB) partial differential equation. The major difficulty with this approach, related to necessity for solving the basic PDE within infinite domain, is resolved by using a hybrid solution method. Exact analytical solutions are obtained for certain outer domains and used to obtain matching boundary conditions for the (bounded in velocity) inner domains, where the basic PDE is solved numerically. The solutions are presented for SDOF systems with white-noise excitation and minimized functional being the expected response energy either by a given time instant or integrated within a given operation time. These solutions are extended to the case of a (strongly nonlinear) system with impacts against a rigid barrier. Special attention is given to an important special case where the steady-state response to a random excitation is to be controlled. A universal property of optimality according to the integral cost functional is proved for a simple "dry-friction" control law for this case of a "long-term" control. Certain analytical predictions for stationary response of the optimally controlled system are presented together with some reliability estimates. Extensions to MDOF systems are developed via transformation to modal coordinates. Whenever control forces can be implemented in terms of the modal coordinates, the complete reduction to a set of solutions for SDOF systems is possible. If, however, the control forces can be applied to the original generalized coordinates only, the resulting optimal control law may become unfeasible because some constraints may become violated. A reasonable procedure for developing a certain "semioptimal" control laws is described for this case. This problem does not arise, however, for the case of a long-term control.