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

Abstract

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 satisfyd^{m+1}\omega(t)/dt^{m+1} = 0, i.e., represented asmth-degree polynomials in timetwith unknown coefficients. In this way, the optimal controlleru^{0}(t)was obtained as the sum of: 1) a linear combination of the state variablesx_{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\rhoth-degree linear differential equationD: d^{\rho}\omega(t)/dt^{\rho} + \beta_{\rho}d^{\rho-1}\omega(t)/dt^{\rho-1}+...+\beta_{2}d\omega/dt + \beta_{1}\omega=0where the coefficients\beta_{i}, i = 1,...,\rho, are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulationx(t) \approx 0in the face of any and every external disturbance function\omega(t)which satisfies the given differential equationD-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.