Controller Principle

Proportional term

We cannot use types of controllers at anywhere, 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,

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, then it will amplify the error signal and thus the amplified error signal can be detected easily.

In place of the proportional-action coefficient Kp, the old term ‘proportional band’ is frequently used
in literature which is represented by the parameter RB[%]. The parameter is converted as follows:

Characteristic of the proportional mode :
1. If the error is zero, the output is a constant equal to (P(o))
2. Normal value of p(0) 50%
3. There is a band of error about zero of magnitude PB within which the output is not saturated at 0% or 100%
4. The tighten of RB will act as a control to open and close.

Integral term

An integral controller provides an output rate of change that is determined by the magnitude of the
error and the integral constant. The controller has the unique ability to return
the process back to the exact set point. The integral control mode is not normally used
by itself because of its slow response to an error signal. There are 2 equation in integral:

 Equation 1

Equation 2

The main problem faced by a proportional controller is a permanent error occurs when there are changes in load. To remove the controller integral limit can be used as a controller is called reset action controllers. This type has the controller output is proportional to the error.
Figure 4.4 (a) shows the relationship between the rate of change of the controller output error for two different values of K1. Figure 4.4 (b) shows how the error was fixed value of K1 produce different controller output value (P).

 Figure 4.4 (a)

Figure 4.4 (b)

                                                                                                                                

Derivative term

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,

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

P-I 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,

Where Ki and kp proportional constant and integral constant respectively.
Advantages and disadvantages are the combinations of the advantages and disadvantages of proportional and integral controllers.

P-D 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,

Where Kd and kp proportional constant and derivative constant respectively.
Advantages and disadvantages are the combinations of advantages and disadvantages of proportional and derivative controllers

P-I-D Controller
P-I-D controller has the optimum control dynamics including zero steady state error, fast response (short rise time), no oscillations and higher stability. The necessity of using a derivative gain component in addition to the PI controller is to eliminate the overshoot and the oscillations occurring in the output response of the system. One of the main advantages of the P-I-D controller is that it can be used with higher order processes including more than single energy storage. The formula for PID control as shown below:

The controller can handle a fixed offset error and also prevent cyclic process. The effects of his actions is shown in Figure 4.7 (e)

Figure 4.7(e)

Solve Practical Problem using Equation

CONTROLLER P

CONTROLLER I

CONTROLLER D

CONTROLLER P+I

CONTROLLER P+D

ADVANTAGE & DISADVANTAGE PID CONTROLLER

Advantages of Proportional(P) 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(P) controller.

• To achieve stability there will we some offsets in the system. 
• Offset can be reduce by increasing multiplication value of  Kp that is very high.

Advantages of Integral(I) controller. 

• can overcome offset problem in the system.
• the system still can be control whether it have change in load

Disadvantages of Integral(I) controller.

• Only suitable to use in the system that have big changes in load.

Advantages of Derivative(D) controller.

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

Disadvantages of Derivative(D) controller.

• Using the derivative control mode is a bad idea when the process variable (PV) has a lot of noise.