Bounded parametric control of random vibrations

Abstract

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 sti¬ness 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 sti¬ness 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 sti¬ness 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.

Energy Balance for Random Vibrations of Piecewise-Conservative Systems

“…system (14) as well (which has nothing to do with swings), just to describe the timing for stepwise variations of parameters. This law has been shown in reference [3] to be the optimal one for the so-called &&long-term'' response control for system (14) in case q"k, namely, it provides minimal expected steady state response energy among all laws with the given bound on magnitude R. Whilst the proof of optimality was obtained by a solution to the relevant Hamilton}Jacobi}Bellman PDE [3], it can also be obtained for other cases by using asymptotic theory for small R, as has been done for the case of swings in reference [6].…”

“…The di!erence for the non-linear case is in the RHS of this equation, where ¹ should now depend on H, that is, on the instantaneous starting energy value of the response cycle, even if it is predicted approximately as a natural cycle duration for a system without random excitation. For a slightly non-linear system with smooth non-linearity, with ¹(H) being linear in H, the linear part may be included into the RHS of equation (1), together, with the constant one [3]. As long as the energy loss in the LHS depends on the same H, the mean response energy can be predicted indeed.…”

Energy Partition Oscillator and Necessary and Sufficient Conditions of Energy Localization

“…Such type of explicit solution has been known for quite a long time with no relation to any physical system [11], however it was used recently in some physical and mechanical applications in a phenomenological way [12], [13]. Although solution (15) holds only for the linearized model (5), it nevertheless helps to clarify specifics of the behavior of phase variables in nonlinear cases.…”

“…where θ 0 is the amplitude of θ, whereas another constant can be introduced into the phase φ(t) = ε sec θ 0 |G|Ωt as an arbitrary temporal shift admitted by equation (13). Such type of explicit solution has been known for quite a long time with no relation to any physical system [11], however it was used recently in some physical and mechanical applications in a phenomenological way [12], [13].…”

Optimal Vibration Control for Structural Quasi-Hamiltonian Systems with Noised Observations

“…29 The stochastic optimal controls for linear and nonlinear systems have been studied and many control strategies have been presented. [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] However, the stochastic optimal control for a nonlinear system with a noised observation was only considered in several studies. 43 Under a specified condition, the separation theorem was applied to convert the nonlinear stochastic system with a noised observation into a completely observable linear system for determining optimal control, but the application is strongly limited.…”