40
Proof: We shall show the existence of u*(t) and x*(t).
Let
u*(t) = -Kx*(t) + u(t) (3-3)
where K e Rmxn and Tj(t) e Rm are still to be defined. Also let the
Laplace transforms of r*(t), w*(t), x*(t), and U(t) be denoted as
R (s), W (s), X (s), and U(s) respectively. Then, if tracking is to
occur, it can be verified using (3-1) that the following relationships
must hold.
X*(s) = [sI-F]_1x* + [sI-F]-1Glj(s) + [sI-F]_1EW* (s) (3-4)
and
R*(s) = H[sI-F]-1x* + H[sI-F]-1G(s) + H[sI-F]_1EW*(s) (3-5)
where
F:= F GK (3-6)
Now consider the following conditions
(1) m > p
(2) rank [ x.I-F] = n for all x^ i = 1,2, ..., r
(3) rank H[X^I-F]G = p for all X^ i = 1,2 r
where x.., i = 1,2 r are the characteristic roots of the linear
differential equation (3-2).
When (1), (2), and (3) are satisfied then both (3-4) and (3-5) will
hold true (i.e., tracking will occur). Furthermore, when these three