13
where m is the fundamental frequency.
Given a positive integer K, let u^(t) denote the truncation
K
UK(t) = aQ + E a^cos(kcot + <ยก>k) (2-10)
k-1
The truncation u^(t) satisfies a differential equation of the form given
by (2-4). Example 2.1 will show how to obtain the specific differential
equation using Laplace transform theory.
Now let X|<(t) denote the solution to
xK(t) = f(xK(t), uK(t), wp(t)) (2-11)
(assuming the solution x^(t) exists)
If HXp(t) x^(t)ii is suitably small for t > 0, the assumption
that the input satisfies (2-4) is reasonable. Often, either by using
simulations or actual tests, it is possible to determine apriori how
small iiXp(t) x|^(t)ii is for a given value of K. Also note that in
practice there is always some error, so that demanding
nx (t) x(t)n = 0 is not reasonable.
P k
We now mention an important practical point which was overlooked in
the preceeding discussion. For iixp(t) x^(t) ii to be suitably small,
the nonlinear system N must be stable in the sense that bounded inputs
give bounded outputs. If this is not the case, it would be necessary to
use a pre-stabilizing feedback so that the unstable portion of xK(t)
could be eliminated. This allows one to make the most meaningful
assessment of how "good" the input uK(t) acutally is. The use of such a
stabilizing feedback would be needed only in simulations and testing