16
Solving for u (t) gives
fr OI p 1
U (t) = a^toCOSiot 3a^Sl'no)t + (-g + agJcosZwt (tj- + 2) 4ct2
(2-22)
The characteristic polynomial for U (s) is
(s2 + 4(o2) (s2+oj2) (s) = s^ + 5u2s^ + 4<j/s (2-23)
Hence, both u (t) and x (t) satisfy
^ () + 52-4 () + () = 0
dt3 dt3 az
(2-24)
Now that assumptions (A.l) 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 e Rrxr and t e Rr be defined as follows
II
0
"0 1 0 ... 0
0 0 1 ... 0
   
   
, t : =
1
0 0  
1
   
0 0 0 ... 1
_-a0 ~al -a2    "ar-l
r
1* 0 
1
with the coefficients aj, j = 0,1, ...,
r-1 defined by (2-4).