Application of PI controller with example

A PI controller produces a control signal proportional to both the error and the integral of the error:

$$u(t) = K_p e(t) + K_i \int_0^t e(t)\,dt, \qquad G_c(s) = K_p + \frac{K_i}{s} = \frac{K_p\left(s + K_i/K_p\right)}{s}$$

It adds a pole at the origin and a zero at $s = -K_i/K_p$.

Why it is applied

• The pole at the origin raises the system type by one. A type-0 plant gets zero steady-state error to a step; a type-1 plant gets zero error to a ramp.
• It removes the steady-state offset that a pure P controller always leaves.
• The zero, placed close to the origin, limits the extra phase lag, so stability is not badly affected.
• It acts like a lag compensator (low-pass), so it also filters high-frequency noise.
• Drawbacks: slightly slower response, possibly more overshoot, and integral wind-up when the actuator saturates.

Example: speed control of a DC motor

Let the motor (speed output) be $G(s) = \dfrac{10}{s+2}$ with unity feedback (type 0).

• With P control $K_p = 4$: $K_P = \lim_{s\to0} \dfrac{40}{s+2} = 20$, so the step error $e_{ss} = \dfrac{1}{1+20} = 0.048$, i.e. a 4.8% speed offset that grows when load torque is applied.
• With PI control $G_c = 4 + \dfrac{8}{s}$:

$$G_c G = \frac{4(s+2)}{s}\cdot\frac{10}{s+2} = \frac{40}{s}$$

The system becomes type 1, $K_P = \infty$ and $e_{ss} = 0$. The closed loop is $\dfrac{40}{s+40}$, a fast first-order response with time constant 0.025 s and no offset, even under constant load disturbances.

Other common uses: liquid-level and flow control, temperature control of furnaces, pressure control in process plants and voltage regulators, wherever zero steady-state offset is essential.