impulse type terms we have
S di
. a. JuK(t) = 0 t > 0 (2-18)
j=O j dt
This is exactly of the form given by (2-4).
We now give further motivation for the assumption that u*(t) satisfies equation (2-4) by showing an example of a nonlinear system where this is indeed the case.
Example 2.2
Consider the system
k(t) = 3x(t) + x 2(t) + (2x 2(t) + 4)w(t) + u(t)
y(t) = x(t) (2-19)
where we desire y*(t) = r*(t) = 1 sinwt and the disturbance is
constant. Thus
x (t) = c1sinwt
w*(t) = a2 (2-20)
where a 1 and a2 are constants. Substituting (2-20) into (2-19) yields the following
1 2 2
tlwCOSot = 3oisinwt + l1-cos2 t) + [ I (l-cos2ot) + 4]a2 + u*(t)
(2-21)