1 / 26 · First-order notes · pp. 1, 3
First-Order System
Consider the first-order system
G(s) = as + aa is a real constant.
First order: one derivative, ẏ + ay = au.
From the differential equation to G(s)
With zero initial state, (s + a)Y(s) = aU(s), so G(s) = Y(s)/U(s) = a/(s + a).
Pole
s + a = 0 → p = −a
Stability
p < 0 ⇔ a > 0
A negative pole gives exponential decay.
For the response curves that follow, a > 0.
We use zero initial state. For the impulse response, this means the state before the impulse is zero.