Laplace transform and transfer functions
A tool that turns differential equations into multiplication and division by s. How to read a system's behaviour from a single transfer function.
Why use it
Differential equations get harder to solve every time you connect another block. After a Laplace transform, differentiation becomes multiplying by s and integration becomes dividing by s, so it turns into algebra.
| Time domain | s domain |
|---|---|
| dy/dt | s·Y(s) |
| ∫y dt | Y(s)/s |
| Step input | 1/s |
Transfer function
The ratio of output to input, G(s) = Y(s) / U(s) (zero initial conditions)
- First-order system: G(s) = K / (τs + 1)
- Integrator: G(s) = 1/s
- PID controller: C(s) = Kp + Ki/s + Kd·s
What you read from a transfer function
- Poles (values of s where the denominator = 0): the shape of the response and stability. Stable if all are in the left half-plane (real part < 0)
- DC gain (substitute s = 0): the output/input ratio in steady state
- Order: what order the system is
Practical sense
In practice, rather than doing inverse transforms by hand, you put the transfer function into MATLAB/Simulink or Python (the control library) and plot the step response and Bode plot. What matters is the feel for how a system will move, just from looking at its transfer function.