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) and the input u*(k) will satisfy the difference equation
s(k+r) + drIs(k+r-1) + . + dls(k+l) + d0s(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) = adn(k) + Bde(k)
(5-18)
e(k) = r(k) - y(k)
The matrices Ad and Bd are defined as
Ad = T-1 block diag. [Cd, Cd, ., Cd] T (5-19)
Bd = T-1 block diag. [T, T, .,T (5-20)
where