In order to verify (2-35) we first note that by assumption (A.1) there is an initial state x*(O) = xo and an input u *(t) such that e(t) = 0 for all t > 0. Hence, it must be shown that for some initial state n*(O) = no of the internal model system, the input u *(t) can be produced by feedback of the form
u*(t) = -K1x*(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) - Klx*(t) in Proposition 2.2. This completes the proof.
We have shown that if certain conditions have been met, then when the exogenous signals r*(t) and w*(t) are acting on the closed-loop system NC, there exists an initial state [x , no] such that perfect tracking occurs. However, if the initial state [x(O), n(O)] differs from [xo, no , the resulting state trajectory [x(t), n(t)] may not converge to [x (t), n (t)] as t + . To achieve (asymptotic)
tracking, we want [x(t), n(t)] to converge to [x*(t), n*(t)] for some range of initial states [x(O), n(O)]. This leads to the following
notation.