Tracking control of non-linear stochastic systems by using path cross-entropy and Fokker-Planck equation

Abstract

The stochastic control problem is defined in terms of state probability. Clearly, the system is designed in such a manner that its state probability tracks the desired state probability of the reference system. The tracking criterion to be minimized is the path cross-entropy (or relative entropy or Kullback entropy) of the two probability density functions, and the problem then turns out to be a distributed parameter one in which the state dynamical equation is the Fokker-Planck equation. The explicit solution of the conjugate equation is obtained by using expansions in Hermite's polynomials. In the special case of neighbouring-optimal control, a slight extension of the LQG approach is proposed, by using a cost function which is a weighted combination of the path cross-entropy and a control quadratic term. Future improvements are discussed.