# PID Control

PID control is a feedback method used to control systems. Its benefits, mathematical form, and effects of each term are discussed below.

## Benefits

• Past, Current, and Future

PID control considers past, current, and the future behavior of the system.

• Proportional control (P) removes the current error.
• Integral control (I) removes the offset (past error).
• Derivative control (D) tries to predict how the system behaves in the future.

• Logical Reasoning

Since PID is a linear controller, its behavior can be explained and analyzed logically. For example, it is possible to explain the effects of changing PID parameters. It is also possible to analyze whether the system stays stable when its characteristics change. These are difficult with counterparts such as fuzzy control and neural network.

• Intuitive

The effects of each PID parameter can be explained qualitatively.

• On-site Tuning

PID controller can be fine-tuned on-site. It is usually difficult to make a model that perfectly matches the actual process. Having the ability to adjust the controller after deployment is an important factor to consider.

## Mathematical Form

Standard form:

 m(t) Control output e(t) Error e(t) = SV(t) - PV(t) SV(t): target value PV(t): measured value PB Proportional band (%) TI Integral time TD Derivative time

Laplace form:

Block diagram:

## Proportional Term

Proportional-only controller:

If there is a constant error c, the proportional controller keeps outputting a value proportional to that error.

## Integral Term

Intagral-only controller:.

If there is a constant error c, the integrator controller outputs a value that increases over time. The rate at which the output increases is proportional to the error:

Output as a function of time:

m(TI) = c so TI can be interpreted as the time it takes for the integrator output to reach c.

Integrator strength increases as TI becomes smaller.

## Derivative Term

Derivative-only controller:

If there is a constant error that increases at a constant rate c, then the controller outputs a value that is proportional to this rate:

Output as a function of time:

e(TD) = cTD can be interpreted as the time it takes for the error to match the derivative controller output (assuming an error that increases at a constant rate).

Derivative strength increases as TD becomes larger.