89
First consider the internal model system. Since it is to be
implemented digitally a discrete-time model is required. Rather than
discretizing the continuous-time internal model system it is convenient
to reformulate the problem in terms of discrete-time signals. If
assumption (A.2) holds then the elements of both the sampled state x (k)
fa
and the input u (k) will satisfy the difference equation
s(k+r) + d^sik+r-l) + ... + d^s (k+1) + dQs(k) = 0 (5-17)
where s(j), j = k, k+1, ..., k+r denotes either an element of x (j) or
an element of u*(j). This result is readily obtained using z-transform
theory and later an example will be given demonstrating how to obtain
the scalars dj, j = 0,1,..., r-1 using z-transforms.
The internal model system then takes the form
n(k+l) = A^nCk) + B^k)
(5-18)
e(k) = r(k) y(k)
The matrices Ad and Bd are defined as
Ad = T1 block diag. [Cd, Cd, ..., Cd] T (5-19)
Bd = T"1 block diag. [t, t, ..., x] (5-20)
where