Use of PID controllers

A PID controller combines proportional, integral and derivative actions on the error. It is the most widely used controller in industry (process control of temperature, pressure, flow, level; motor speed and position control) because it can fix both transient and steady-state behaviour with three tunable gains.

$$u(t)=K_pe(t)+K_i\int_0^te\,dt+K_d\frac{de}{dt},\qquad G_c(s)=K_p+\frac{K_i}{s}+K_ds=\frac{K_ds^2+K_ps+K_i}{s}$$

        +-->[  Kp  ]---+
 e ---->+-->[ Ki/s ]--(+)--> u ---> Plant ---> c
        +-->[ Kd s ]---+

PID adds one pole at the origin and two zeros to the open loop.

Role of each term

Uses / advantages

1. Zero steady-state error to step inputs (and ramp for a type-1 plant) due to the integral term.
2. Good transient response: derivative action limits overshoot introduced by P and I.
3. Better relative stability (PD part acts like a lead compensator, PI part like a lag; PID behaves like a lag-lead compensator).
4. Simple tuning without an exact model, e.g. Ziegler–Nichols rules: $K_p=0.6K_{cr}$, $T_i=0.5P_{cr}$, $T_d=0.125P_{cr}$.
5. Available as standard industrial hardware and in PLC/DCS software.

Limitations

• D action amplifies noise, so a filtered derivative is used.
• Integral wind-up when the actuator saturates (anti-wind-up needed).

Example: in a boiler temperature loop, P gives quick heating, I removes the final offset from the set point, and D stops the temperature overshooting when the setpoint is changed.