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 \omega(t) which are not directly measurable but which can be assumed to satisfy d^{m+1}\omega(t)/dt^{m+1} = 0 , i.e., represented as m th-degree polynomials in time t with unknown coefficients. In this way, the optimal controller u^{0}(t) was obtained as the sum of: 1) a linear combination of the state variables x_{i}, i = 1,2,...,n , plus 2) a linear combination of the first (m + 1) 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 \omega(t) which are known only to satisfy a given \rho th-degree linear differential equation D: d^{\rho}\omega(t)/dt^{\rho} + \beta_{\rho}d^{\rho-1}\omega(t)/dt^{\rho-1}+...+\beta_{2}d\omega/dt + \beta_{1}\omega=0 where the coefficients \beta_{i}, i = 1,...,\rho , are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulation x(t) \approx 0 in the face of any and every external disturbance function \omega(t) which satisfies the given differential equation D -even steady-state periodic or unstable functions \omega(t) . 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.