Bounded nonlinear stochastic control based on the probability distribution for the sdof oscillator

In the response control of building structures subjected to strong excitations, the controller saturation problem needs be considered in the design stage. The nonlinearity of saturated controllers, however, makes the exact probabilistic evaluation of control effectiveness difficult. In this study, a stochastic nonlinear control algorithm with bounded control force is proposed and the closed-loop system's joint probability density function (PDF) governed by the reduced Fokker-Planck equation is derived. Based on the derived joint PDF, the bound of the control force that restricts the first-passage failure probability of the displacement response to a prescribed value is determined. Numerical analyses results show that the proposed controller reduces the maximum required control force considerably compared to the linear controller for the same displacement reduction, and the proposed approximation of the failure probability is considerably accurate.