15
impulse type terms we have
J dj
I a. T- u(t) =0 t > 0 (2-18)
j=0 J dtJ K
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
x(t) = 3x(t) + x2(t) + (2x2(t) + 4)w(t) + u(t)
y(t) = x (t)
(2-19)
where we desire y*(t) = r*(t)
constant. Thus
djSincot and the disturbance is
x (t) = o^sintot
k, .
w (t) = ou
(2-20)
where and are constants. Substituting (2-20) into (2-19) yields
the following
12 2 *
ctjWCOSwt = 3a^sin)t +(l-COS2ait) + [a^ (l-cos2a>t) + 4]a^ + U (t)
(2-21)