Process Controllers

Overview

Engineering reference data for Process Controllers in process control.

Key Formulas

PID Controller

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

Proportional-Integral-Derivative control.

Transfer Function

$G(s) = \frac{K}{\tau s + 1}$

First-order system.

Variables

Symbol	Description	Unit
$K_p$	Proportional gain	—
$K_i$	Integral gain	1/s
$K_d$	Derivative gain	s
$\tau$	Time constant	s

Basic Controller Components

A basic process controller consists of the following components:

• Measuring Unit: Includes an appropriate instrument (e.g., temperature transmitter, pressure transmitter) to measure the state of the process.
• Set Point Device: Used to input the desired value for the process variable.
• Comparator: Compares the measured value with the set point to calculate the error or deviation.
• Control Unit: Calculates the required output magnitude and direction to compensate for the error.
• Output Unit: Converts the controller's output signal into a physical action (e.g., adjusting a control valve, operating a motor).

Controller Principles

Control units are generally built upon three fundamental control principles:

• Proportional control
• Integral control
• Derivative control

These can be used individually or combined in various ways (e.g., PI, PD, PID) to achieve desired system performance.

Proportional Controller (P-Controller)

The P-Controller produces an output action proportional to the error between the set point and the measured process value.

The output is given by: $OP = -k_P E_r$

Where:

• $OP$ = output of the proportional controller
• $k_P$ = proportional gain (or action factor)
• $E_r$ = error (set point - measured value)

The proportional gain $k_P$ can be related to the proportional band $P$: $k_P = \frac{100}{P}$

Key Properties:

• A high gain $k_P$ makes the system respond quickly but risks oscillation and instability.
• A low gain $k_P$ results in a large steady-state error.
• Proportional control reduces rise time and decreases, but does not eliminate, steady-state error.

Integral Controller (I-Controller)

The I-Controller output is proportional to the accumulated error over time, which eliminates steady-state offset.

The output is given by: $O_I = -k_I \int E_r \, dt$

Where:

• $O_I$ = output of the integral controller
• $k_I$ = integral gain
• $\int E_r \, dt$ = integral of the error over time

Key Properties:

• Eliminates steady-state error over time.
• Tends to respond slowly initially.
• Can degrade transient response and may lead to instability.
• Susceptible to integral windup during sustained errors (e.g., plant shutdowns), causing the output to saturate at its limits (e.g., ±100%).

Derivative Controller (D-Controller)

The D-Controller output is proportional to the rate of change of the error, providing predictive action.

The output is given by: $O_D = -k_D \frac{dE_r}{dt}$

Where:

• $O_D$ = output of the derivative controller
• $k_D$ = derivative gain
• $\frac{dE_r}{dt}$ = rate of change of the error over time

Key Properties:

• Improves system stability, reduces overshoot, and improves transient response.
• Is never used alone because it allows large steady-state errors over the long term.
• Can amplify noise due to its sensitivity to rapid changes.

PID Controller Response Effects

The following table summarizes the general effect of each control mode on key performance metrics.

References

• Original Source