Non-linear stochastic control via stationary response design

This paper presents a strategy for controlling externally excited stochastic systems with uncertain parameters. The control objective is to drive the system response from an arbitrary initial distribution to a prescribed stationary probability density function (PDF). This problem can be interpreted as the stochastic stabilization about a PDF. The control consists of a non-linear feedback part and a switching term inspired by robust sliding control concepts. The control of a one-dimensional stochastic process with a Gaussian target-PDF is used to illustrate the approach. The control performance is evaluated by studying the time evolution of the first and second order moments of the response. The dependence of the response on the number of feedback terms and the rate of convergence to the stationary PDF are studied numerically. Motivated by the fast convergence observed, a feedback controller with time-varying gains is applied to the problem of tracking a moving PDF. Monte Carlo simulations validate very well the control for both stabilization and tracking problems.