D = QB (2-54)
By the assumption p < q there exists a row vector z' E Rq which is orthogonal to the range D. Hence
z'D = 0 (2-55)
Consequently, by letting v' = z'Q it is evident that (2-50) holds.
In the previous sections, it was shown that stability of the closed-loop transient system NCT is a key requirement in the solution of the nonlinear servomechansim problem. To achieve such stability, the conditions that the pair (A, B) be controllable and the pair (A, K2) be observable were shown to be crucial. Hence, from Proposition 2.4 we can conclude that the number of blocks q in the internal model system must be such that q < min(m,p).
Now recall a major earlier assumption; namely, the open-loop input u*(t) which forces the nonlinear system N to track r*(t) satisfies the differential equation (2-4). Ultimately, such an input is generated by
the internal model system as can be seen from Theorem 2.1 or equation (2-35). If u*(t) E Rm is arbitrary (aside from satisfying (2-4)) then it is not difficult to see that the internal model system must have at least m blocks. In otherwords, we must have the condition q > m.
Because of earlier condition that q < min(m,p) and because of the
assumption that m > p we are forced to consider systems with m = p.
*The modes of the various elements of u*(t) have a one to one corresondence with the eigenvalues of the C matrices which mak up the block diagonal A matrix. Since we assume all m elements of u (t) are independent of one another then m separate blocks will be needed in A to insure this independence. For further insight, see the proof of
Proposition 2.2.