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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 R* 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
() + vi(')(r'1) + + n(-)<1) + = 0 (3-2)
where the characteristic roots of (3-2) are assumed to be in the closed
right half-plane. We shall let i = 1, 2 r denote the
distinct characteristic roots of (3-2) where r < r due to multi
plicities. 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 time-
invariant 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.