The closed-loop control scheme proposed to solve the nonlinear servomechanism problem has, incorporated into the feedback, the internal model system of the disturbance and reference signals with respect to the nonlinear system N. The implementation of this closed-loop controller is shown in Figure 2-1. The equations modeling the closedloop system are the following:
NC: (t) = f(x(t), u(t), w (t))
A(t) = An(t) + BH[x*(t) -x(t)]
(2-29)
u(t) = -Kx(t) - K2n(t)
e(t) = H[x*(t) -x(t)]
where K1 e Rpxn and K2 Rpxpr are constant feedback matrices. It is assumed that the state x(t) is available for feedback.
We will show in Theorem 2.1 that there exists an initial state for the closed-loop system NC such that tracking occurs with e(t) = 0 for t > 0. The following proposition shall be required in the proof of this theorem.
Proposition 2.2 Let z(t) e RP be any vector with elements satisfying the linear differential equation
(.)(r) + a rl(.)(r-1) + . + a,(.)(1) + ao(.) = 0 (2-30)
and let C e Rrxr be a matrix whose eigenvalues, including multiplicities, exactly match the characteristic roots of (2-30). If in
addition, the pair (A, K2) is observable with the constant matrix K2 . Rpxpr and A . Rprxpr defined by