## Controller

The controller determines the value of a controlled variable, compares the actual value to the reference value, helps in determining the deviation, and produces a control signal that will reduce the deviation to the smallest possible value i.e either to Zero. Controllers can be electrical, pneumatic, hydraulic, electromechanical, or electronic types. Hydraulic controllers are used for controlling heavy loads, pneumatic controllers are suitable for shop floor applications. The selection of a particular type of controller depends on the nature of the plant, and operating conditions such as safety, cost, availability, accuracy, weight, and size.

The controller consists of an error detector and amplifier. The transducer converts the output variable to another suitable variable such as displacement, pressure, or electrical signals, and can be used for comparing the output to the reference input signal.

**e=r-b [difference between controlled variable and set point(references point)]**

**Proportional controller(P)**

There is a continuous linear relation between the output of the controller m(manipulated variable) and actuating error signal e(deviation). Basically proportional controller is an amplifier with adjustable gain.

*Kp is called proportional gain or proportional sensitivity.*

Consider the liquid level system. The float lever is directly connected to the control valve. When the level of the liquid rise, the valve close in proportionate amount this reduces the inflow of the vessel and vice versa. The inverse of proportional gain or proportional sensitivity is the proportional band and is defined as the change in level necessary to operate the value through the full stroke.

**Integral controller (I)**

The output of the controller is changed at a rate that is proportional to the actuating error signal e(t). The integral control action is “Reset control”.

** The inverse of Ki is called integral time T1**.

*Integral time is defined as the time of change of output caused by a unit change of actuating error signal.*From the figure, it is clear that,

For positive error, the output of the controller is ram(positive).

For Zero error, there is no change in the output of the controller.

For negative error, the output of the controller is a negative ramp.

**Derivative controller (D)**

The output of the controller depends on the rate of change of actuating error signal e(t). The derivative controller is also called rate control.

when the error is zero or constant the output of the controller will be Zero. Therefore, this type of controller cannot be used alone. In this type of controller, the gain should be small.

**Proportional plus Integral controller (PI)**

This controller is a combination of proportional and integral control actions.

The parameters Kp and Ti are adjustable. Ti is called integral time and the inverse of integral time is defined as the number of times per minute that the proportional part of the response is duplicated called the reset rate of **“repeats per minute”.**

The error varies at t=t1. The output of the controller suddenly changes to Mp due to proportional control action, after that controller output changes linearly with respect to time at a rate of Kp/Ti.

For unit step (t1=0) the response is shown below. The proportionality sensitivity Kp affects both the proportional and integral parts of the action.

**[Note: In the figure above T1=Ti]**

**Proportional plus derivative controller (PD)**

The combination of proportional and integral controllers is called a PD controller. In this type of controller derivative control action is added in series to proportional controller.

Td is called derivative time and is defined as the time interval by which the rate action advances the effect of the proportional control action or is defined as the amount of lead, expressed in units of time, that the control action is given.

If the actuating error signal e(t) is ramp function at t=t1.The derivative mode causes a step md at t1 and the proportional mode causes a rise of mp equal to md at t2.

PD control action reduces the rise time, faster response, improves the bandwidth, and improves the damping.

For unit ramp input, the figure is shown below:

**Proportional Integral Derivative controller (PID Controller)**

The combination of proportional, integral, and derivative control action is called a PID controller. This type of controller is also called three action controller.

This is the transfer function of the PID controller.

*Kp is the proportional gain*

*Ti is the integral time*

*Td is the derivative time.*

*Actuating error signal e=At where A is a constant and t is the time.*

The proportional part of the control action repeats the change of error(lower straight line). The derivative of the control action adds an increment of output so that the proportional plus derivative action is shifted ahead in time(middle straight line). The integral parts add a further increment of output proportional to the area under the deviation line.

**References**

- https://www.theengineeringconcepts.com/process-controllers-and-its-types/
- https://www.elprocus.com/the-working-of-a-pid-controller/
- https://www.electrical4u.com/types-of-controllers-proportional-integral-derivative-controllers/