12
Furthermore, assume that xp(t) and wp(t) are periodic with a common
period of T. Under these conditions, for the same inital state xp(0)
and disturbance wp(t), there exists a periodic input up(t) having a
period of T which results in the state trajectory xp(t).
Proof: Make the definition
uT( t)
u(t)
0
0 < t < T
otherwise
Then let
co
u (t) = £ uT(t-nT)
p n=0 1
(2-6)
(2-7)
Since xp(0) = xp(T) = xp(2T) = ... and wp(t) repeats itself over every
interval T, the result is obvious.
Let xp(t), up(t), and wp(t) be periodic with a common period T and
assume that the following differential equation is satisfied.
xp(t) = f(xp(t), up(t), wp(t)) (2-8)
In Proposition 2.1 we have already asserted that a periodic up(t) will
exist whenever xp(t) and wp(t) are periodic with a common period.
Assuming up(t) is integrable over any period, let the Fourier series
expansion of up(t) be
00
Un(t) = a + E a. cos(kwt + <(>. )
P 0 K K
(2-9)