Bounded parametric control of random vibrations

The dynamic programming approach is used to study a feedback control problem for a randomly excited single-degree-of-freedom system. The available actuator for control can provide temporal stiffness variations for the system, which are of a bounded magnitude. An analytical solution is obtained for the corresponding Hamilton-Jacobi-Bellman equation for the expected response energy, which should be minimized according to the integral cost criterion. While this solution is valid within some parts of the phase plane only, it extends to the whole phase plane with increasing time-interval of control. Thus the exact explicit expression for the optimal control law is obtained for the case where steady-state response is to be controlled. This law requires feedback-controlled switching stiffness between the given bounds. Stationary random vibration of the system with this control law is studied then, by a direct energy balance approach and by a stochastic averaging method. The latter of these provides a more extensive description of the response, which may provide reliability estimates for the controlled system, but this asymptotic approach is valid for the limiting case of a weak control only. The direct energy balance predicts only the expected response energy, or the mean square displacement. However, its range of applicability is expected to be less restrictive with respect to the maximal magnitude of the system's stiffness variations: the non-dimensional variations may not be much smaller than unity. The analytical studies are extended to the case of a controlled system with a nonlinear restoring force.