Optimal bounded control of steady-state random vibrations

A SDOF system is considered, which is excited by a white-noise random force. The systemʹs response is controlled by a force of bounded magnitude, with the aim of minimizing integral of the expected response energy over a given period of time. The integral to be minimized satisfies the Hamilton–Jacobi–Bellman (HJB) equation. An analytical solution of this PDE is obtained within a certain outer part of the phase plane. This solution is analyzed for large time intervals, which correspond to the limiting steady-state random vibration. The analysis shows the outer domain expanding onto the whole phase plane in the limit, implying that the simple dry-friction control law is the optimal one for steady-state response. The resulting value of the (unconditional) expected response energy, for the case of a stationary excitation, is also obtained. It matches with the corresponding result of energy balance analysis, as obtained by direct application of the SDE Calculus, as well as that of stochastic averaging for the case where the magnitude of dry friction force and intensity of excitation are both small. A general expression for mean absolute value of the response velocity is also obtained using the SDE calculus. Certain reliability predictions both for first-passage and fatigue-type failures are also derived for the optimally controlled system using the stochastic averaging method. These predictions are compared with their counterparts for the system with a linear velocity feedback and same r.m.s. response, thereby illustrating the price to be paid for the bounds on control force in terms of the reduced reliability of the system.