14
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
K
UK(t) = aQ + £ a^cos( knit + <(>k) (2-12)
Taking Laplace transforms of both sides we get
Ms) = K^ (2-13)
l\ l\ rt 0 0
s[ n (s + K ai )]
k=l
where N(s) is a polynomial in s. Equation (2-13) can be expressed as
s[ n (s2 + k2a)2)]UK(s) = N(s) (2-14)
Next, by writing
we have
s[ n (s2+ k2o>2)]
k=l
J
£ a.s3 J = 2K + 1
j=0 3
[ £ a.sJ] U,,(s) = N(s)
j=0 3 K
(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