22
In order to verify (2-35) we first note that by assumption (A.l)
there is an initial state x*(0) = x* and an input u*(t) such that e(t)
= 0 for all t > 0. Hence, it must be shown that for some initial
^ it ^
state n (0) = nQ of the internal model system, the input u (t) can be
produced by feedback of the form
u*(t) = -KlX*(t) K2n*(t) (2-36)
From assumption (A.2) we know that the elements of u*(t) and x*(t) will
satisfy the differential equation (2-4) (or equivalently, equation
(2-30)). Also observe that because e(t) = 0 in (2-35), the internal
model system is completely decoupled from the original system. This
decoupling allows us to apply Proposition 2.2. Specifically, we may
verify (2-36) by letting z(t) = -u*(t) K^x*(t) in Proposition 2.2.
This completes the proof.
We have shown that if certain conditions have been met, then when
ic it
the exogenous signals r (t) and w (t) are acting on the closed-loop
it it
system NC, there exists an initial state [xQ, n ] such that perfect
tracking occurs. However, if the initial state [x(0), n(0)] differs
it it
from [x f nQ], the resulting state trajectory [x(t), n(t)] may not
it ^
converge to [x (t), n (t)] as t > . To achieve (asymptotic)
it ft
tracking, we want [x(t), n(t)] to converge to [x (t), n (t)] for some
range of initial states [x(0), n(0)]. This leads to the following
notation.