20112ee141_1_EE141_hw2_sol

20112ee141_1_EE141_hw2_sol - EE141 Principles of Feedback...

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Unformatted text preview: EE141 Principles of Feedback Control (Winter 2010) Solutions to Homework 2 Problem 3.21 (a) The block diagram is simplified to: Figure 1: Block diagram for 3.21(a) where G * 2 = G 2 1- G 2 H 2 G * 3 = G 3 1- G 3 H 3 Thus Y ( s ) R ( s ) = G 1 (1 + G * 2 ) 1 + G 1 (1 + G * 2 ) G * 3 = G 1 (1- G 2 H 2 + G 2 )(1- G 3 H 3 ) (1- G 2 H 2 )(1- G 3 H 3 ) + G 1 G 3 (1- G 2 H 2 + G 2 ) (b) We move the summer on the right past the integrator to get b 1 s and repeat to get ( b 2 + b 1 s ) s . Meanwhile we apply the feedback rule to the first inner loop to get 1 s + a 1 as shown in the figure and repeat for the second and third loops to get: Figure 2: Block diagram for 3.21(b) Thus Y ( s ) R ( s ) = b 1 s 2 + b 2 s + b 3 s 3 + a 1 s 2 + a 2 s + a 3 1 (c) Applying block diagram reduction: reduce innermost loop, shift b 2 to the b 3 node, reduce next innermost loop and continue systematically to obtain: Figure 3: Block diagram for 3.21(c) Figure 4: Block diagram for 3.21(c) Figure 5: Block diagram for 3.21(c) Thus Y ( s ) R ( s ) = b 1 s 2 + ( b 2 + b 1 a 1 ) s + b 3 + b 2 a 1 + b 1 a 2 s 3 + a 1 s 2 + a 2 s + a 3 (d) The block diagram is simplified to: 2 Figure 6: Block diagram for 3.21(d)Figure 6: Block diagram for 3....
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This note was uploaded on 07/02/2011 for the course EE 141 taught by Professor Balakrishnan during the Spring '07 term at UCLA.

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20112ee141_1_EE141_hw2_sol - EE141 Principles of Feedback...

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