Solving for u*(t) gives
u*(t) = lwCOSwt - 3aisinwt + a,22 )cos2wt - 12(+ -42
(2-22)
The characteristic polynomial for U*(s) is
(s2 + 4w2)(s2+w2)(s) = s' + 5w2s3 + 4w4s (2-23)
Hence, both u*(t) and x*(t) satisfy
d5 2 d3 4 d
- (.) + 5w - (.) + 4w a- (.) : 0 (2-24)
dt dt
Now that assumptions (A.1) and (A.2) have been justified, we proceed by introducing an internal model system. In the literature, an internal model system is usually taken as a system which replicates the dynamics of the reference and disturbance signals. Here it will take on a slightly different meaning which is made more precise by the following definitions. Let C . Rrxr and T e R r be defined as follows
0 I 0 . . . 0 0
0 0 1 . . . 0 0
C := . . . • T := . (2-25)
0 0 0 . 1 0
-a0 -a, -a2 . . . -ar 1
with the coefficients aj, i = 0,1, ., r-1 defined by (2-4).