Short note: PD and PI controller

PD controller

The control signal is proportional to the error and its rate of change:

$$u(t) = K_p e(t) + K_d \frac{de(t)}{dt}, \qquad G_c(s) = K_p + K_d s = K_d\left(s + \frac{K_p}{K_d}\right)$$

• Adds a zero at $s = -K_p/K_d$; acts like a lead compensator (high-pass).
• Anticipates the error, so it adds damping: reduces overshoot, rise time and settling time.
• Improves stability and increases bandwidth.
• System type is unchanged, so steady-state error is not removed.
• Amplifies high-frequency noise because of the derivative action.

PI controller

The control signal is proportional to the error and its integral:

$$u(t) = K_p e(t) + K_i \int e(t)\,dt, \qquad G_c(s) = K_p + \frac{K_i}{s} = \frac{K_p(s + K_i/K_p)}{s}$$

• Adds a pole at the origin and a zero; acts like a lag compensator (low-pass).
• Raises the system type by one, so the steady-state error to a step (type-0 plant) becomes zero.
• Reduces bandwidth; the response becomes slower and may overshoot more.
• Can reduce relative stability if $K_i$ is too large; suffers integral wind-up.
• Filters high-frequency noise.