Definition: Given the system N described by (2-2), suppose that for a particular r*(t) and a particular w*(t) assumptions (A.1) and (A.2) both hold. Then an internal model system of r*(t) and w*(t) with respect to N is a system of the following form:
*(t) = An(t) + Be(t)
e(t) = r (t) - y(t) = H[x (t) - x(t)] (2-26)
where
A = T-1 block diag. CC, C, ., C] T (2-27)
p blocks
B = T-1 block diag. [T, T, ., T] (2-28)
p blocks
where the state n(t) E Rpr, T is an arbitrary nonsingular matrix, B 6 Rprxp, and the pair (A,B) is completely controllable. In practice, T is usually taken as the identity matrix.
Roughly speaking, the above internal model system is seen to contain p copies of the dynamics of the state and input signals which must occur during tracking. This is different from the internal model system used in the linear servomechanism problem where only the dynamics of the reference and disturbance signals are included. The dynamics of the reference and disturbance will inevitably be included in (2-26); however, the nonlinear structure of N may necessitate the introduction of additional dynamics.