# 3. Controllers and controller types

## 3.2 Continuous controllers

## 3.2.1 Proportional controller (P)

The P controller is the most basic controller type. The manipulating variable is directly proportional to the **control difference e = w - x**.

The disadvantage is the remaining** control deviation e**_{p}. This is an inevitable result of the transfer function: Since the manipulating variable y remains, the remaining control deviation ** e**_{p} cannot disappear either.

The remaining control deviation** e**_{p} can be reduced by increasing the **control gain** **K**_{p} or reducing the **proportional band x**_{p} . However, this simultaneously increases the control system’s tendency to oscillation and it can become unstable.

Step response with different controller gain ** K**_{p}** **

## 3.2.2 Proportional-Derivative controller (PD)

On a PD controller, in addition to the P component a **component proportional to the speed **is fed back. This component has a **damping influence**.

Therefore, for PD controllers a higher control gain **K**_{p} can be selected without the system beginning to oscillate.

Thus, a PD controller generally has a lower remaining control deviation ** e**_{p} than a pure P controller.

## 3.2.3 Proportional-Integral controller (PI)

With a PI controller, an integral component is fed back in addition to the P component.

This eliminates a remaining control deviation ** e**_{p}.

As soon as the control difference e is not equal to zero, the integral sums up the deviations over time and the controller output variable y rises continuously.

Finally, the controller output variable y is of the exact value at which the remaining control deviation ** e**_{p} becomes zero.

The integral component reduces the stability and the P component must be reduced accordingly.

This video explains the Proportional Integral (PI) controllers in a clear and detailed way. It includes an explanation of class excises.

## 3.2.4 Proportional-Integral-Derivative controller (PID)

The PID controller is the most common controller type. The manipulating variable is calculated from the control difference e as follows:

The integral component means that this controller type, like the PI controller, has no remaining control deviation** e**_{p} .

The derivative component is proportional to the speed and thus has a damping effect.

Control system with **PID controller**

## 3.2.5 Direction of controller action

The direction of control action defines the direction in which the controller output variable y changes with an input signal. We differentiate between **direct and inverse (reverse) direction of control action**. The familiar level control provides an example. The level in the tank can be regulated by changing the inflow or the drainage.

**Control using the inflow**Level (x) rises: inflow (y) must be restricted.

An**increase**in the controlled variable x results in a**reduction**in the manipulating variable y: The direction of control action is**inverse (reverse)**.

**Control using the drainage**

Level (x) rises: drainage (y) musst be increased.

An **increase** in the controlled variable x results in an **increase** in the manipulating variable y: The direction of control action is **direct**.

Direction of control action and operating point correction using the example of a P controller:

Direction of control action on a pneumatic control valve

The direction of control action can normally be adjusted on the controller. However, it is often also possible to change the direction of control action on the actuator, e.g. on a pneumatic control valve. Depending on the design, the air pressure opens or closes the valve.

Sometimes, this behaviour is defined by safety requirements. In this case, the valve should take on a certain safety-related prescribed position in case of failure of the auxiliary energy (pressure drop).

Example of direct control action on a pneumatic control valve

https://instrumentationtools.com/control-valve-working-animation/