#### Controller is used to decrease the errors of control signal to zero or somewhere close to zero.

It can be classified into *six* different types namely:

- ON – OFF control
- Proportional control
- Integral control
- Proportional – plus – Integral control (P – I)
- Proportional – plus – Derivative control (P – D)
- Proportional – plus – Integral – plus – Derivative control (P – I – D)

According to the application, anyone of these controllers can be used. The functions of each controller are described briefly below.

### ON – OFF control:

The element in the ON – OFF controller offers *two control methods* such as:

- Complete OFF
- Complete ON

m(t) = M_{1}, if e(t) > 0

m(t) = M_{2}, if e(t) < 0

Where,

- m(t) denotes the
*control signal*created by the controller. - e(t) denotes the
*error*on the controller. - In most of the cases either M
_{1}or M_{2}will be 0.

The purpose of an ON – OFF control is to protect the controller from swinging with *very high frequency*. This is made possible by moving the error through several ranges before the operation starts. Here, the range is considered as the *differential gap*.

### Proportional control:

A control signal produced by this controller is *proportional* to the error. It is basically used as an amplifier by means of a gain (K_{p}). This is represented as:

m(t) = K_{p} e(t)

The transfer function will be:

M(s) / E(s) = K_{p}

The proportional controller will be best suited for providing *smooth control action*.

### Integral control:

A control signal produced by the integral controller is *altered* at a rate proportional to the error (i.e.) the control signal maximizes quickly if the error is big, and the control signal maximizes slowly if the error is small. This can be represented as:

m(t) = K_{i} ∫ e(t) dt

Here, the K_{i} denotes the *integrator gain*.

The transfer function will be:

M(s) / E(s) = K_{i} / s

Here, the 1/s is used for integration.

### Proportional – plus – Integral control (PI):

The PI controller is used to *overcome* two major issues. They are:

- The integral control is capable of offering
*zero errors*, but it is set back with its*slow response*. - The proportional control provides
*error*while counteracting a load on the system.

This can be represented as K_{p}.

m(t) = K_{p }e(t) + K_{p }/ T_{i} ∫ e(t) dt

Here,

K_{p }is used to adjust the proportional and integrator gain.

T_{i} is used to adjust only the integrator gain.

The transfer function will be:

M(s) / E(s) = K_{p} (1 + 1 / T_{i} s)

### Proportional – plus – Derivative control (PD):

The control signal produced by the PD controller is proportional to the rate of change of the error. This method is used rarely because of its incapability to provide output without the change of error. An advantage is that it can give changes with *faster responses*. This can be represented as

m(t) = K_{p} e(t) + K_{p }T_{d} de(t) / dt

The transfer function will be:

M(s) / E(s) = K_{p} (1 + T_{d}s)

### Proportional – plus – Integral – plus – Derivate control (PID):

The PID controller integrates *three control actions*, and it is the *most frequently* used controller. It is because of its fast response, and low steady – state error. This controller can be represented as

m(t) = K_{p} e(t) + K_{p} / T_{i } ∫ e(t) dt + K_{p} T_{d }de(t) / dt

The transfer function will be:

M(s) / E(s) = K_{p} (1 + 1 / T_{i}s + T_{d}s)