where w is the fundamental frequency.
Given a positive integer K, let uK(t) denote the truncation
K
uK(t) ° + E akcos(kwt + k) (2-10)
k= 1
The truncation UK(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 xK(t) denote the solution to
XK(t) = f(xK(t), uK(t), wp(t)) (2-11)
(assuming the solution xK(t) exists)
If lix p(t) - xK W)1 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 lix p(t) - XK W11is for a given value of K. Also note that in practice there is always some error, so that demanding lix p(t) - xK(t)l = 0 is not reasonable.
We now mention an important practical point which was overlooked in the preceeding discussion. For ix p(t) - xK M11 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