   # Fig. 2. Block Diagram Of Continuous Pid Controller

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Fig.2. Block diagram of Continuous PID Controller
What the PID controller does is basically is to act on the variable to be manipulated through a proper combination of the three control actions that is the P control action, I control action and D control action. The P action is the control action that is proportional to the actuating error signal, which is the difference between the input and the feedback signal. The I action is the control action which is proportional to the integral of the actuating error signal. Finally the D action is the control action which is proportional to the derivative of the actuating error signal. With the combination of all the three actions, the continuous PID can be realized. This type of controller is widely
On that particular situation Zeigler-Nichols first rule have to be applied; first method is based on the step response of the plant. But Zeigler-Nichols second method is applied when the output shows sustained oscillation , . From the response below, the system under study is indeed oscillatory and hence the Z-N tuning rule based on critical gain (Kcr) and critical period (Pcr). The system under study above has a following block diagram.

Fig.3.Block Diagram of Controller and Plant

Since the Ti = ∞ and Td = 0, this can be reduced to the transfer function of:
The value of Kp that makes the system marginally stable, so that sustained oscillation occurs, can be obtained using Routh’s Stability Criterion. The characteristic equation for the closed-loop system is:
From Routh’s Stability Criterion, the value of Kp that makes the system marginally stable can be determined as follows:
1 5
6 Kp
Hence the new equation becomes:
This can be simplified to
From the above simplification, the sustained oscillation can be reduced into: ω2=5 or ω=√5. The period of the sustained oscillation, Pcr, can now calculated as:

The Ziegler-Nichols Second Frequency Method [6-8] table, shown below, gives suggested tuning parameters for various types of PID controllers based on the critical period.
Recommended PID Value Setting

Hence from the above table, the values of the PID tuning parameters Kp, Ti and Td is as follow:
Kp = 0.6 X 30=18
Ti = 0.5 X 2.8099=1.405
Td = 0.125 X 2.8099=0.351
The transfer function of the PID controller with all the parameters is given as:

From the above transfer function, we can see that the PID controller has pole at the origin and double zeroes at s = -1.4235. The block diagram of the control system with PID controller is as follows.

Fig.4. Close Loop Transfer function

The above system can be reduced to single block using MATLAB function. Below the simplified system is given.

The block diagram of reduced system is shown in Fig.5.

Fig.5. Simplified reduce System
The overall function with its feedback can be calculated

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