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:

- Controllers improve the steady state accuracy by decreasing the steady state error.
- As the steady state accuracy improves, the stability also improves.
- Controllers also help in reducing the unwanted offsets produced by the system.
- Controllers can control the maximum overshoot of the system.
- Controllers can help in reducing the noise signals produced by the system.
- Controllers can help to speed up the slow response of an over damped system.

Now that’s all well and all but… what are controllers? Well, a controller is the mechanism which seeks to minimise the difference between the actual value of a system (i.e. the process variable) and the desired value of the system (i.e. the set point). There are various types of controllers, which will be discussed in detail below.

## Types of Controllers

There are two main types of controllers: continuous controllers, and discontinuous controllers.

In discontinuous controllers, the manipulated variable changes between discrete values. Depending on how many different states the manipulated variable can assume, a distinction is made between two-position, threeposition and multiposition controllers. Compared to continuous controllers, discontinuous controllers operate on very simple, switching final controlling elements.

The main feature of continuous controllers is that the controlled variable (also known as the manipulated variable) can have any value within controller’s output range. Now in the continuous controller theory, there are three basic modes on which the whole control action takes place, which are:

**Proportional controllers**.**Integral controllers**.**Derivative controllers**.

We use the combination of these modes to control our system such that the process variable is equal to the setpoint (or as close as we can get it). These three types of controllers can be combined into new controllers:

- Proportional and integral controllers (PI Controller)
- Proportional and derivative controllers (PD Controller)
- Proportional integral derivative control (PID Controller)

Now we will discuss each of these control modes in detail below.

### Proportional Controllers

All controllers have a specific use case to which they are best suited. We cannot jus through any type of controller at any system and expect a good result – there are certain conditions that must be fulfilled. For a **proportional controller** there are two conditions and these are written below:

- The deviation should not be large; i.e. there should be not be a large deviation between the input and output.
- The 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, K_{p} is proportional constant also known as controller gain.

It is recommended that K_{p} should be kept greater than unity. If the value of K_{p} is greater than unity (>1), then it will amplify the error signal and thus the amplified error signal can be detected easily.

#### Advantages of Proportional Controller

Now let us discuss some advantages of proportional controller.

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

#### Disadvantages of Proportional Controller

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

- Due to presence of these controllers we get some offsets in the system.
- 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, K_{i} is integral constant also known as controller gain. Integral controller is also known as reset controller.

#### Advantages of Integral 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.

#### Disadvantages of Integral Controller

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:

- It never improves the steady state error.
- 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, K_{d} is proportional constant also known as controller gain. Derivative controller is also known as rate controller.

#### Advantages of Derivative 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, K_{i} and k_{p} proportional constant and integral constant respectively.

Advantages and disadvantages are the combinations of the advantages and disadvantages of proportional and integral controllers.

### 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, K_{d} and k_{p} proportional constant and derivative constant respectively.

Advantages and disadvantages are the combinations of advantages and disadvantages of proportional and derivative controllers