PID controller block diagram, transfer function, characteristics

A PID controller produces a control signal that is the sum of three terms: proportional to the error, to the integral of the error, and to the derivative of the error.

$$u(t) = K_p e(t) + K_i \int_0^t e(t)\,dt + K_d \frac{de(t)}{dt}$$

Block diagram

              +----------+
          +-->|   Kp     |----+
          |   +----------+    |
 r  +  e  |   +----------+  + v  u   +-------+   c
 --->(O)--+-->|  Ki / s  |-->(O)---->| Plant |--+-->
     -^   |   +----------+    ^      +-------+  |
      |   |   +----------+    |                 |
      |   +-->|  Kd s    |----+                 |
      |       +----------+                      |
      +-----------------------------------------+

Transfer function

$$G_c(s) = \frac{U(s)}{E(s)} = K_p + \frac{K_i}{s} + K_d s = \frac{K_d s^2 + K_p s + K_i}{s}$$

or, in standard form, $G_c(s) = K_p\left(1 + \dfrac{1}{T_i s} + T_d s\right)$ with integral time $T_i = K_p/K_i$ and derivative time $T_d = K_d/K_p$.

It adds one pole at the origin and two zeros to the open-loop transfer function.

Characteristics

• Raises the system type by one, so steady-state error to a step is zero.
• Derivative action adds damping and improves stability and transient response.
• Gives good overall performance; used in most industrial loops (temperature, pressure, flow, motor speed).
• Tuning is done by trial, by Ziegler–Nichols rules or by root locus/frequency methods.
• In practice the derivative term is filtered, $K_d s/(1+\tau s)$, to limit noise, and anti-windup is added to the integral term.