Recursive filtering algorithms for certain systems with nonlinearities of piecewise-linear type

A partly observable multidimensional sequence $(\theta_t,\xi_t)$, $t=0,1,\dots$, is described bу recurrent equations where the unobservable component $\xi_t$ varies in a piecewise linear way and the additive disturbances are of the white noise type. When the equation of the unobservable component does not contain random disturbances, the filtering problem is accurately solved by closed recurrent equations for the a posteriori probability density and for a mean square-optimal filter.