Consider the linear time-invariant system
L: x(t) = Fx(t) + Gu(t) + Ew(t)
y(t) = Hx(t) (3-1)
e(t) = r(t) - y(t)
where x(t) e Rn is the state, u(t) E Rm is the input, w(t) E Rd is a disturbance, y(t) e RP is the output, and e(t) e RP is the error which arises in tracking the reference signal r(t) e RP. Conditions shall be given as to when it is possible to design a controller such that e(t) + 0 as t + -. It is assumed that the elements of the reference r(t) as well as the disturbance w(t) satisfy the linear differential equation
(.)(r) + yrl(.)(r-1) + . + y1(.)(1) + yO(.) = 0 (3-2)
where the characteristic roots of (3-2) are assumed to be in the closed right half-plane. We shall let Xi, i = 1, 2, ., F denote the
distinct characteristic roots of (3-2) where F 4 r due to multiplicities. The following well known result gives conditions under which the linear servomechanism problem can be solved.
Theorem 3.1: Assume the state x(t) is available for feedback. A
necessary and sufficient condition that there exists a linear timeinvariant controller for (3-1) such that e(t) + 0 as t +- for all
r(t) and w(t) with elements satisfying (3-2) is that the following two conditions both hold.