Averaging, integral of special function

[psl86]

Homework Statement

Integrate the right hand side of the state equation to find the average system

$$\dot{x_1} = x_2$$
$$\dot{x_2} = -x_1 + 1 -2u(0.5sat(x_2) + 0.5 - p(t))$$

Homework Equations

u(s) = 0 for s<0, u(s) = 1 for s>=0
p(t) is periodic with T = 1
$$f_{avg} =\frac{1}{T}\int_0^Tf(x,t)$$

The Attempt at a Solution

$$\dot{x_1} = x_2$$
$$\dot{x_2} = -x1 - sat(x_2)$$

 

[psl86]

Im almost 100% certain that this is suppose to end up as

x1_dot = x2
x2_dot = -x1 - sat(x2)

but I can't make the argument..

 

[psl86]

No one in here with a steady averaging-theory brain?