A moment-based approach for nonlinear stochastic tracking control

摘要

This paper describes a new stochastic control methodology for nonlinear affine systems subject to bounded parametric and functional uncertainties. The primary objective of this method is to control the statistical nature of the state of a nonlinear system to designed (attainable) statistical properties (e.g., moments). This methodology involves a constrained optimization problem for obtaining the undetermined control parameters, where the norm of the error between the desired and actual stationary moments of state responses is minimized subject to constraints on moments corresponding to a stationary distribution. To overcome the difficulties in solving the associated Fokker–Planck equation, generally experienced in nonlinear stochastic control and filtering problems, an approximation using the direct quadrature method of moments is proposed. In this approach, the state probability density function is expressed in terms of a finite collection of Dirac delta functions, and the partial differential equation can be converted to a set of ordinary differential equations. In addition to the above mentioned advantages, the state process can be non-Gaussian. The effectiveness of the method is demonstrated in an example including robustness with respect to predefined uncertainties and able to achieve specified stationary moments of the state probability density function.