# Types of Controllers | Proportional Integral and Derivative Controllers

Before I introduce you about various controllers in detail, it is very essential to know the uses of controllers in the theory of control systems. The important uses of the controllers are written below:

1. Controllers improve steady state accuracy by decreasing the steady state errors.
2. As the steady state accuracy improves, the stability also improves.
3. They also help in reducing the offsets produced in the system.
4. Maximum overshoot of the system can be controlled using these controllers.
5. They also help in reducing the noise signals produced in the system.
6. Slow response of the over damped system can be made faster with the help of these controllers.

Now what are controllers? A controller is one which compares controlled values with the desired values and has a function to correct the deviation produced.

## Types of Controllers

Let us classify the controllers. There are mainly two types of controllers and they are written below:
Continuous Controllers: The main feature of continuous controllers is that the controlled variable (also known as the manipulated variable) can have any value within the range of controller’s output. Now in the continuous controller’s theory, there are three basic modes on which the whole control action takes place and these modes are written below. We will use the combination of these modes in order to have a desired and accurate output.

1. Proportional controllers.
2. Integral controllers.
3. Derivative controllers.
4. Combinations of these three controllers are written below:

5. Proportional and integral controllers.
6. Proportional and derivative controllers.

Now we will discuss each of these modes in detail.

### Proportional Controllers

We cannot use this types of controllers at anywhere and with each type controller, there are certain conditions that must be fulfilled. With proportional controllers there are two conditions and these are written below:

1. Deviation should not be large; it means there should be less deviation between the input and output.
2. Deviation should not be sudden.

Now we are in a condition to discuss proportional controllers, as the name suggests in a proportional controller the output (also called the actuating signal) is directly proportional to the error signal. Now let us analyze proportional controller mathematically. As we know in proportional controller output is directly proportional to error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where, Kp is proportional constant also known as controller gain.
It is recommended that Kp should be kept greater than unity. If the value of Kp is greater than unity (>1), then it will amplify the error signal and thus the amplified error signal can be detected easily.

Now let us discuss some advantages of proportional controller.

1. Proportional controller helps in reducing the steady state error, thus makes the system more stable.
2. Slow response of the over damped system can be made faster with the help of these controllers.

Now there are some serious disadvantages of these controllers and these are written as follows:

1. Due to presence of these controllers we get some offsets in the system.
2. Proportional controllers also increases the maximum overshoot of the system.

### Integral Controllers

As the name suggests in integral controllers the output (also called the actuating signal) is directly proportional to the integral of the error signal. Now let us analyze integral controller mathematically. As we know in an integral controller output is directly proportional to the integration of the error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where, Ki is integral constant also known as controller gain. Integral controller is also known as reset controller.

Due to their unique ability they can return the controlled variable back to the exact set point following a disturbance that’s why these are known as reset controllers.

It tends to make the system unstable because it responds slowly towards the produced error.

### Derivative Controllers

We never use derivative controllers alone. It should be used in combinations with other modes of controllers because of its few disadvantages which are written below:

1. It never improves the steady state error.
2. It produces saturation effects and also amplifies the noise signals produced in the system.

Now, as the name suggests in a derivative controller the output (also called the actuating signal) is directly proportional to the derivative of the error signal. Now let us analyze derivative controller mathematically. As we know in a derivative controller output is directly proportional to the derivative of the error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where, Kd is proportional constant also known as controller gain. Derivative controller is also known as rate controller.

The major advantage of derivative controller is that it improves the transient response of the system.

### Proportional and Integral Controller

As the name suggests it is a combination of proportional and an integral controller the output (also called the actuating signal) is equal to the summation of proportional and integral of the error signal. Now let us analyze proportional and integral controller mathematically. As we know in a proportional and integral controller output is directly proportional to the summation of proportional of error and integration of the error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where, Ki and kp proportional constant and integral constant respectively.

### Proportional and Derivative Controller

As the name suggests it is a combination of proportional and a derivative controller the output (also called the actuating signal) is equals to the summation of proportional and derivative of the error signal. Now let us analyze proportional and derivative controller mathematically. As we know in a proportional and derivative controller output is directly proportional to summation of proportional of error and differentiation of the error signal, writing this mathematically we have,

Removing the sign of proportionality we have,

Where, Kd and kp proportional constant and derivative constant respectively.