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### Control System: Block Diagrams Reduction using MATLAB

Most of the circuits in Control System today are represented by simple blocks that help us understand the function of each block in a better way. Is also helps the designers to easily make amendments in the circuit for better functionality and testing purpose. But the problem with Block Diagrams is that having blocks and their feedbacks makes the transfer function on the system to tedious to calculate.

Here we are going to study block reduction using MATLAB. The blocks connected in series, parallel and as feedbacks are at times very tedious to compute. MATLAB allows solving of such blocks directly using some functions that is being discussed below with the help of the example. Here we have to calculate C(s)/R(s), that is taken as T(s).

The MATLAB code for the above problem is:

num1 = [1 2];
den1 = [3 1 0];
G1 = tf(num1, den1)                      %Making G1 as the tranfer function
G2 = tf( [2], [1 7] )
G3 = tf( [1 5], [1 6 3 ] )
G4 = tf( [1], [1 0] )

T1 = parallel(G1, G2)                   %as G1 and G2 are in parallel
T2 = series(T1, G3)                       %as T1 and G3 are in series
T = feedback(T2, G4, -1)            %as G4 is the negative feedback

Here we use the tf() function to get the transfer function
parallel() and series() functions according to the requirement
and the feedback() function for feedback.

The output for the above code is as follows:

s + 2
———
3 s^2 + s

Transfer function:
2
—–
s + 7

Transfer function:
s + 5
————-
s^2 + 6 s + 3

Transfer function:
1

s

Transfer function:
7 s^2 + 11 s + 14
——————–
3 s^3 + 22 s^2 + 7 s

Transfer function:
7 s^3 + 46 s^2 + 69 s + 70
—————————————–
3 s^5 + 40 s^4 + 148 s^3 + 108 s^2 + 21 s

Transfer function:
7 s^4 + 46 s^3 + 69 s^2 + 70 s
——————————————————-
3 s^6 + 40 s^5 + 148 s^4 + 115 s^3 + 67 s^2 + 69 s + 70

Here we can see that the transfer function for the block diagram is very complex and tedious to deduce. Which can be obtained by using MATLAB very easily.