- Heaviside step function
| \(H(s)\) | \(h(t)\) | \(\text{init cond}\)1 |
|---|---|---|
| \(1/s\) | \(u(t)\) | - |
| \(1/s^2\) | \(r(t) = t \cdot u(t)\) | - |
| \(\dfrac{1}{s+a}\) | \(e^{-at} \cdot u(t)\) | \(A\cdot e^{-at}\) |
| \(\dfrac{1}{s}\cdot\dfrac{1}{s+a}\) | \(\dfrac{1}{a} [1-e^{-at}] u(t)\) | - |
| \(\dfrac{s^2}{s^2+b^2}\) | - | - |
| \(\dfrac{s}{s^2+b^2}\) | \(\cos bt \cdot u(t)\) | - |
| \(\dfrac{bs}{(s+a)^2+b^2}\) | - | - |
| \(\dfrac{b}{(s+a)^2+b^2}\) | \(e^{-at} \sin bt \cdot u(t)\) | - |
- Responses With Initial Condition
For transfer function \(H(s)\), input \(X(s)\) (usually step input \(X(s) = 1/s\)). The full response with initial condition \(A\) is
\[y(t) = F^{-1}(H(s)X(s)) + A \cdot ic(t)\]Initial condition function only depends on the denominator of the transfer function. Such function can be added to the response with appropriate coefficient, to obtain the correct initial condition. ↩