since later, a stabilizing feedback law will be developed for the actual implementation.
The following example shows how a differential equation of the form given by (2-4) is derived from a truncated Fourier series
Example 2.1
Suppose
uK(t) M o + k cos(kwt + k)
k=1k
(2-12)
Taking Laplace transforms of both sides we get
U K(s) =
N(s)
S[ I (S2 + K2w2
(2-13)
where N(s) is a polynomial in s. Equation (2-13) can be expressed as
K 2 22
s[ II (s + k W )]UK(s) = N(s)
k=1
(2-14)
Next, by writing
K 2 22
sE H (s2+ kw)] w as , J = 2K + 1
k=1 j=O
J ajsj] UK(S) = N(s)
j=O
(2-15) (2-17)
Now taking inverse Laplace transforms and noting that since N(s) is a polynomial in s and hence has an inverse Laplace transform consisting of
we have