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.