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).