Optimal polynomial filter of degrees 3 and 4 and its application to an automotive system

Optimal regulator for third degree polynomial systems

“…where the matrix function is the solution of the following equation dual to the variance equation (5) with the terminal condition Q(T)= Y. The binary operation * has been introduced in Section 3, and q(t) =(ql(t),q2(t) , ...,qn( t)) denotes the vector consisting of the diagonal elements of Q(t).…”

“…In this section, the optimal filtering equations for a polynomial state equation of degree 3 over linear observations (obtained in [5]) are briefly reminded for reference purposes. Let an unobservable random process x(t) satisfy a polynomial equation of third degree and linear observations are given by:…”

“…Thus, the last optimal regulator should exist and be dual to the optimal filter in all cases where the optimal filter (or, vice versa, the optimal control) exists in a closed finitedimensional form. This proposition is now applied to a third order polynomial system, for which the optimal filter has already been obtained (see [5]). …”

“…However, taking into account that the optimal control problem can be solved in the linear case applying the duality principle to the solution of the optimal filtering problem, this paper exploits the same idea for designing the optimal control in a polynomial system with linear control input, using the optimal filter for polynomial system states over linear observations. Based on the obtained polynomial filter of the third degree [5], the optimal regulator for a polynomial system of degree 3 with linear control input and quadratic criterion is obtained in a closed form, finding the optimal regulator gain matrix as dual transpose to the optimal filter gain one and constructing the optimal regulator gain equation as dual to the variance equation in the optimal filter. The results obtained by virtue of the duality principle could be rigorously verified through the general equations of [3] or [4] applied to a specific polynomial case, although the physical duality seems obvious: if the optimal filter exists in a closed from, the optimal closed-form regulator should also exist, and vice versa.…”