Estimation and Control with Cubic Nonlinearities

The nonlinearities in a dynamic system and its measurement equations are assumed to be cubic and "small," i.e., all proportional to a single scalar small paramter ϵ. The optimal digital nonlinear feedback control law, in the case of perfect state knowledge, as well as in the case of noisy measurements, are carried through the first power of ϵ. In the first case the control law involves cubic as well as linear terms in the state. In the second case it is necessary to take into account the non-Gaussian character of the conditional distribution of the state, given the measurements at any stage. The optimal control law now involves higher moments of the conditional state distribution as well as terms linear and cubic in the maximum likelihood state estimate.