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 servomechansism 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 implemented 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 : Rqrxqr, B E Rqrxp and K2 . Rmxqr. The
exact change in the A matrix is shown by the following equation
A = T-1 block diag. [C, C, ., C] T (2-49)
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, K2) with A . Rqrxqr defined by
(2-49), B . Rqrxp arbitrary, and K2 . Rmxqr arbitrary. The following properties are true.