Further study of the linear regulator with disturbances--The case of vector disturbances satisfying a linear differential equation

In a previous paper [1], the conventional optimal linear regulator theory was extended to accommodate the case of external input disturbances which are not directly measurable but which can be assumed to satisfy , i.e., represented as th-degree polynomials in time with unknown coefficients. In this way, the optimal controller was obtained as the sum of: 1) a linear combination of the state variables , plus 2) a linear combination of the first time integrals of certain other linear combinations of the state variables. In the present paper, the results obtained in [1] are generalized to accommodate the case of unmeasurable disturbances which are known only to satisfy a given th-degree linear differential equation where the coefficients , are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulation in the face of any and every external disturbance function which satisfies the given differential equation -even steady-state periodic or unstable functions . An essentially different method of deriving this result, based on stabilization theory, is also described, In each cases the results are extended to the case of vector control and vector disturbance.