PID Controller: What Each Action Does

Short answer

The PID controller calculates its output with three terms: one proportional to the error, one proportional to its accumulation over time and one proportional to its rate of change. This guide explains what each term corrects, the parallel and standard forms of the equation, the effect of raising each gain, the derivative filter and the most common usage mistakes.

What a PID is for

The PID controller (proportional, integral, derivative) is the most widely used algorithm in industry for regulating temperature, flow, pressure, level and speed. It is found in panel controllers, in PLC function blocks, in distributed control systems and inside variable frequency drives, which use it, for example, to hold the pressure of a pumping network. When a single loop is not enough, cascade, ratio and feedforward control are explained in cascade, ratio and feedforward control.

It starts from a simple idea: measure the error between the setpoint and the process variable, e = SP − PV, and calculate the output to the valve or the drive with three combined rules. The three rules look at the error from three angles: how large it is now, how much has accumulated and where it is heading.

The equation

There are two ways to write it, and it is worth recognizing both because manufacturers use one or the other.

Parallel (ideal) form:

u(t) = Kp · e(t) + Ki · ∫ e(τ) dτ + Kd · de/dt

Standard (or ISA) form:

u(t) = Kp · [ e(t) + (1/Ti) · ∫ e(τ) dτ + Td · de/dt ]

The relationship between them is Ti = Kp / Ki and Td = Kd / Kp. In the standard form, Ti and Td are measured in units of time (seconds or minutes). That makes it easier to picture their physical meaning. Ti is the time the integral action takes to match the effect of the proportional action for a constant error. Td is the prediction horizon of the derivative.

Three practical warnings:

Some controllers use repeats per minute instead of integral time. It is the inverse of Ti and is easy to mix up.
In the standard form the gain Kp multiplies all three terms, so changing it also changes the integral and the derivative. In the parallel form each gain acts separately.
Some manufacturers use proportional band (PB) instead of gain: a band of 50 % is approximately equivalent to a gain of 2.

What each action does

Proportional (P)

The output is proportional to the error at that instant. A large error produces a large correction. It is fast and intuitive, but it has two limits:

With low gain, the loop is slow and leaves a steady-state error (offset), because to hold a non-zero output an error must exist.
With high gain, the loop responds fast but oscillates and can become unstable.

Integral (I)

It accumulates the error as long as it exists. While the PV does not equal the SP, the integral keeps growing and pushes the output until the error vanishes. It is the action that eliminates offset. In exchange:

It makes the loop slower and more oscillatory.
If the output reaches a physical limit, for example a fully open valve, the integral keeps accumulating and causes integral windup, the subject of another guide in this series.

Derivative (D)

It measures the rate of change of the error and anticipates where it is going. If the PV rises quickly toward the SP, the derivative brakes before it overshoots. It reduces overshoot and stabilizes loops with delay. Its problems:

It amplifies high-frequency noise in the measurement. Small noise with fast changes produces an erratic output that moves the valve needlessly.
It does not eliminate steady-state error; it only acts when there is change.
On a step change in the SP, the derivative of the error gives a very high spike (derivative kick).

Effect of raising each gain

The table summarizes the qualitative effect on the step response in a typical process:

When increasing	Rise time	Overshoot	Settling time	Steady-state error	Stability
Kp	Smaller	Larger	Little change	Smaller	Worsens
Ki	Smaller	Larger	Larger	Eliminated	Worsens
Kd	Little change	Smaller	Smaller	No theoretical effect	Improves if small

These trends are a guide and not a law: they depend on the process, and the actions interact with each other.

Usual combinations

Type	When it is used	Remark
P	Tank level where some variation is tolerated	Simple, with offset
PI	Flow, pressure, level, speed	The most common in plants
PID	Temperature and slow processes with little noise	Needs a derivative filter
PD	Positioning and some servos	Rarely in processes

Plant experience shows that the great majority of loops work with PI. Derivative is added when there is a clear reason and the noise can be filtered.

Improvements real controllers add

A textbook PID is rarely the one used in the field. Industrial devices add:

Derivative filter. A low-pass filter limits the amplification of noise. There is a trade-off: heavy filtering takes away the effect of the derivative.
Derivative on measurement. Instead of differentiating the error, the PV is differentiated. That way a step change in the SP does not produce the derivative spike.
Setpoint weighting. The weight of the SP in the proportional term is reduced to smooth the response to setpoint changes without affecting disturbance rejection.
Output dead band. Avoids moving the valve for tiny changes, which reduces wear.
Output limits and anti-windup. Essential for the integral term (see the guide on integrator windup).
Bumpless transfer between manual and automatic.
Gain scheduling. Different parameter sets according to the operating point.

Numerical example (illustrative)

A tank temperature loop is used, with made-up values. Standard form, Kp = 2 (%/°C), Ti = 100 s, Td = 0, an instant after an SP change from 60 to 65 °C, with the PV still at 60 °C.

Initial error: e = +5 °C.
Proportional term: 2 × 5 = 10 % immediate change in the output.
Integral term: grows with time. With the error constant at 5 °C, after 100 s the integral contributes Kp · (1/Ti) · e · t = 2 × (1/100) × 5 × 100 = an additional 10 %. That is, in a time Ti the integral matches what the proportional contributed, which gives the physical meaning of Ti.
As the PV rises, the error falls and both terms shrink. When the PV reaches 65 °C, the error is zero and the output stays at the value the integrator accumulated, the one needed to hold the new SP.

With these numbers it is clear why a small Ti means strong integral action: the integral recovers the error quickly, but with a risk of oscillation.

How to recognize each behavior on the trend

When looking at the PV, SP and output chart, each action leaves a signature that allows diagnosis without touching anything:

A PV that settles below or above the SP and does not move indicates a lack of integral action, or integral turned off.
A slow, regular oscillation with a long period usually comes from an integral time that is too short, or a high gain with a slow process.
A fast oscillation, with the output jumping from one extreme to the other, points to excessive gain or to the derivative amplifying noise.
An output that jitters while the PV is still signals measurement noise feeding the derivative.
A PV that reaches the SP very slowly, without overshooting, suggests low gain or a long integral time.

Reading the trend before changing parameters saves tests and avoids making the loop worse.

Common mistakes

Raising the gain to remove offset. What removes it is the integral. More gain only brings you closer to the stability limit.
Confusing Ti with repeats per minute. A correct value on one device becomes an unstable loop on another.
Using derivative on a flow loop. The natural noise of flow produces a nervous output and wears the actuator.
Copying parameters from another manufacturer. The units and the form of the equation may differ.
Ignoring the output limits. Without them, the integral grows beyond what the actuator can deliver.
Tuning a loop with a defective valve. If the valve has high friction or backlash, no tuning fixes it.
Not recording the parameters. After a failure, nobody knows what values the loop had.

Checklist

Controller equation form identified (parallel or standard) and units of Ti and Td known.
Correct direction of action (direct or reverse).
Output limits and anti-windup protection active.
Derivative filter configured, or derivative turned off if not needed.
Derivative on measurement and SP weighting reviewed.
Manual-to-automatic bumpless transfer tested.
Gain, integral time and derivative time values documented.
Response to a small step verified with the real process.

Engineering support

A PID loop is tuned well when the process is known, and ruined when numbers are copied from another plant. If you need to define the control strategy, the instrumentation and the documentation of a project, see our engineering design and studies service, delivered remotely and worldwide. If what you need is to check the loops with the process running before handover, see testing and commissioning, in Colombia or elsewhere on request. For inquiries, use contact with the process diagram.

References

Wikipedia, PID controller (equation forms, effect of the gains, derivative filter).
Woolf, P., Chemical Process Dynamics and Controls, University of Michigan, chapter on PID control (CC BY 3.0).
Kuphaldt, T. R., Lessons in Industrial Instrumentation, chapters on closed-loop control and process dynamics (CC BY 4.0).
IEC 61131-3, PLC programming languages (control function blocks).

Frequently asked questions

Is derivative action always needed?

No. In flow loops, liquid pressure loops and noisy level loops, PI is the usual choice. Derivative helps in processes with large delay and little noise, such as some temperature loops, but it amplifies measurement noise and wears the valve.

What is the difference between gain and proportional band?

They are the same idea expressed inversely. Proportional band is the percentage change in the process variable that drives the output from 0 to 100 %, and it is approximately 100 divided by the gain when both are expressed in percent. Manufacturers use one or the other.

Are the PID in a PLC and in a panel controller the same?

They calculate essentially the same thing, but they differ in the form of the equation, in the units of Ti and Td, in the derivative filter and in the handling of the setpoint. A tuning value from one device does not transfer as is to another.

What does integral action do when the error is zero?

It holds the output where it is. The integral stores the accumulated value the controller needs to keep the process at the setpoint. That is why the output of a stable controller does not return to zero when the error vanishes.