30
as the number of outputs. In this section, further insight into this
assumption is presented by showing its relation to the chosen controller
structure. In addition, sufficient conditions will be given to allow
one to consider a system with more inputs than outputs. The case where
the input dimension is less than that of the output shall not be
considered since, in this circumstance, a solution to the servo-
mechansism problem does not generally exist. The intuitive reason for
this is that it requires at least p independent inputs to control p
degrees of freedom independently.
Let us now consider a nonlinear system with input u(t) e Rm and
output y(t) e RP, where m > p. Assume that the controller is im
plemented in essentially the same way as the previously discussed
controller except now consider changing the dimension of the internal
model system. It is assumed that the matrix A of the internal model
system has q blocks on the diagonal rather than p blocks as before.
Consequently, we now have A e Rclrxclr, B e R^P and l<2 e RmX(lr. The
exact change in the A matrix is shown by the following equation
A = T-1 block diag. [C, C, ..., C] T (2-49)
y -j
q blocks
where C is again defined by (2-25). The corresponding change in the B
matrix does not need to be considered in this analysis.
Proposition 2.4: Given the triple (A, B, 1^) with A e R