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 5(t) . 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.
1*I1
X*(s) = [sI-F] x° + [slI-F]-G(s) + [sl-F]- EW*(s) (3-4)
and
R*(s) = H[sI-F]1 x + H[sI-FI 1GU(s) + H[sI-]-EW*(s) (3-5)
0
where
F:= F - GK (3-6)
Now consider the following conditions
(1) m > p
(2) rank [ x.I-F] = n for all xi , i : 1,2, .,
(3) rank HEXi I-FIG = p for all Xi ' i = 1,2, .,
where xi, 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