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# ps5-soln - Problem Set 5 Solutions January 3 2010 Problem 1...

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January 3, 2010 Problem 1 Combine G 6 and G 7 yielding G 6 G 7 . Add G 4 and obtain the following diagram: R ( s ) C ( s ) + + + - - G 1 G 8 G 5 G 2 G 4 + G 6 G 7 G 3 Figure 1: Initial block diagram. Next combine G 3 and G 4 + G 6 G 7 , as shown in Figure 2. Push G 5 to the left past the picko± point. See Figure 3 for this arrangement. Notice that the feedback is in parallel form. Thus the equivalent feedback, H eq ( s ) = G 2 G 5 + G 3 ( G 4 + G 6 G 7 )+ G 8 . Since the forward path transfer function is G ( s ) = G eq ( s ) = G 1 G 5 , the closed-loop transfer function is T ( s ) = G eq ( s ) 1 + G eq ( s ) H eq ( s ) . Hence, T ( s ) = G 5 G 1 1 + G 1 ( G 8 G 5 + G 7 G 6 G 5 G 3 + G 5 G 4 G 3 + G 2 ) Problem 2 Combine the feedback with G 6 and combine the parallel G 2 and G 3 . Displayed below in Figure 4. Move G 2 + G 3 to the left past the picko± point. Shown is Figure 5. 1

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ps5-soln - Problem Set 5 Solutions January 3 2010 Problem 1...

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