Question

H4 R(s) C(s) G1 G2 G3 G4 H2 H3 H1 ...


Find the total transfer function of the system based on the block diagram of a multi-loop control system given in the figure.

Transcribed: H4 C(s) R(s) G1 G2 G3 G4 H2 H3 H1

Answer

The given block diagram is:

Electrical Engineering homework question answer, step 1, image 1

Moving the take off point between G3 and G4 to the right of G4. So the modified diagram is

Electrical Engineering homework question answer, step 2, image 1

Solving the feedback loop (1)

TF1=G3G41+G3G4H2

The modified block diagram is:

Electrical Engineering homework question answer, step 3, image 1

Solving the feedback loop (2)

TF2=G3G41+G3G4H21-G3G4H31+G3G4H2=G3G41+G3G4H2-G3G4H3

The modified block diagram is:

Electrical Engineering homework question answer, step 4, image 1

Solving the feedback loop (3)

TF3=G2G3G41+G3G4H2-G3G4H31-G2G3G4H4G41+G3G4H2-G3G4H3=G2G3G41+G3G4H2-G3G4H3-G2G3H4

The modified block diagram is:

Electrical Engineering homework question answer, step 5, image 1

Solving the feedback loop (3)

TF4=G1G2G3G41+G3G4H2-G3G4H3-G2G3H41+G1G2G3G4H11+G3G4H2-G3G4H3-G2G3H4=G1G2G3G41+G3G4H2-G3G4H3-G2G3H4+G1G2G3G4H1

Therefore, the total transfer function of the block diagram is

TF=CsRs=G1G2G3G41+G3G4H2-G3G4H3-G2G3H4+G1G2G3G4H1

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