1. Time-Domain ODE
y''(t) - y'(t) - 2y(t) = f(t)
Assume: y(0-) = 1, y'(0-) = 0
2. Laplace Transformation
[s2Y(s) - s(1) - 0] - [sY(s) - 1] - 2Y(s) = F(s)
Grouping terms...
(s2 - s - 2)Y(s) = F(s) + s - 1
3. Transfer Function & Characteristic Modes
H(s) = 1s2 - s - 2 = 1(s-2)(s+1)
Char. Poly D(s): s2 - s - 2 = (s-2)(s+1)
Poles: p1 = 2, p2 = -1
Modes: e2t, e-t
Stability:
Unstable
4. Total Response Decomposition Y(s)
Solving for Y(s)...
Y(s) =
F(s)s2 - s - 2
YZS(s)
+
s - 1s2 - s - 2
YZI(s)
Important Conceptual Connections
- The Transfer Function is a Zero-State Object: H(s) strictly maps the input F(s) to the output Y(s). It is defined by assuming all initial conditions are zero.
- Shared Denominator: Notice how the characteristic polynomial D(s) forms the denominator for both the Transfer Function H(s) and the Zero-Input Response YZI(s).
- Characteristic Modes: Because the denominator is shared, the system's fundamental building blocks (the characteristic modes, like e-t or sin(ωt)) dictate how the system responds to both external inputs and its own stored energy.