Obtain the closed loop transfer function for multipleloop sy

Obtain the closed loop transfer function for multiple-loop system by using MATLAB.

1 to Ris) 481 + 48 S-10 8s+5 +2s +1 sA2+2 SA3+28

Solution

%% Obtaining Closed loop transfer function for the given block
clear
clc
K = 10;
%% Inner loop formed by G1 , G2 , H1 which is -ve feedback loop Ga(s) = G1*G2/(1+H1*G1*G2)
G1 = tf([1],[1 -10])
G2 = tf([1 0],[1 0 48])
H1 = tf([8 5],[1 2 1])
disp(\'LOOP 1 Transfer function\')
Ga = feedback(G1*G2,H1)
%% Cascade inner loop with first loop which is formed using G3 , H2 +Ve
%% feedback loop Gb(s) = G3/1-G3*H2
G3 = tf([1],[1 0 0])
H2 = 150
disp(\'LOOP 2 Transfer function\')
Gb = feedback(G3,-H2)
%% Outerloop with H3 as -ve feedback G(s) = Ga*Gb/1+H3*Ga*Gb
H3 = tf([1 0 2],[1 0 0 28])
disp(\'LOOP 3 Transfer function\')
G = feedback(Ga*Gb,H3)
%% Transfer function of overall system T(s) Y(s)/R(s) = K*G(s)
disp(\'Over all Transfer function\')
T = K*G

OUTPUT

Transfer function:
1
------
s - 10


Transfer function:
s
--------
s^2 + 48


Transfer function:
8 s + 5
-------------
s^2 + 2 s + 1

LOOP 1 Transfer function

Transfer function:
s^3 + 2 s^2 + s
--------------------------------------------
s^5 - 8 s^4 + 29 s^3 - 386 s^2 - 907 s - 480


Transfer function:
1
---
s^2

H2 =

150

LOOP 2 Transfer function

Transfer function:
1
---------
s^2 - 150


Transfer function:
s^2 + 2
--------
s^3 + 28

LOOP 3 Transfer function

Transfer function:

s^6 + 2 s^5 + s^4 + 28 s^3 + 56 s^2 + 28 s
--------------------------------------------------------------------------------------------------------------------------------------------------------------------
s^10 - 8 s^9 - 121 s^8 + 842 s^7 - 5481 s^6 + 54033 s^5 + 158844 s^4 - 75193 s^3   + 1.608e006 s^2 + 3.809e006 s + 2.016e006
  

Over all Transfer function

Transfer function:

10 s^6 + 20 s^5 + 10 s^4 + 280 s^3 + 560 s^2 + 280 s
--------------------------------------------------------------------------------------------------------------------------------------------------------------------
s^10 - 8 s^9 - 121 s^8 + 842 s^7 - 5481 s^6 + 54033 s^5 + 158844 s^4 - 75193 s^3 + 1.608e006 s^2 + 3.809e006 s + 2.016e006