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midsol09 - MAE 143A Signals and Systems Prof M Krstic...

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MAE 143A Signals and Systems Prof. M. Krstic MIDTERM July 20, 2009 NAME: SOLUTIONS Closed book. One sheet (both sides) of handwritten notes allowed. Present your reasoning and calculations clearly. Random or inconsistent etchings will not be graded. Write answers only in the blue book. The problems are not ordered by difficulty. Total points: 50. Time: 2.5 hours.
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Problem 1. (8 points) Find the state space representation for the following circuit: C 1 R 1 R 2 C 2 R 3 u y + + + + Solution: Denote the voltages on the capacitors by v 1 and v 2 . We have C 1 ˙ v 1 u v 1 R 1 = 0 (1) C 2 ˙ v 2 y v 2 R 2 = 0 (2) C 2 ˙ v 2 u y R 3 = 0 (3) From (2) and (3) we get y v 2 R 2 = u y R 3 y = R 3 R 2 + R 3 v 2 + R 2 R 2 + R 3 u Using this expression for y , we get C 1 ˙ v 1 = 1 R 1 v 1 + 1 R 1 u C 2 ˙ v 2 = 1 R 2 + R 3 v 2 + 1 R 2 + R 3 u State space representation is bracketleftbigg ˙ v 1 ˙ v 2 bracketrightbigg = bracketleftbigg 1 C 1 R 1 0 0 1 C 2 ( R 2 + R 3 ) bracketrightbigg bracketleftbigg v 1 v 2 bracketrightbigg + bracketleftbigg 1 C 1 R 1 1 C 2 ( R 2 + R 3 ) bracketrightbigg u y = bracketleftbig 0 R 3 R 2 + R 3 bracketrightbig bracketleftbigg v 1 v 2 bracketrightbigg + R 2 R 2 + R 3 u 2
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Problem 2. (8 points) Consider the following system ˙ x 1 = 2 x 1 + x 2 2 (1 + u ) ˙ x 2 = x 2 + 2 x 1 (9 2 x 3 ) x 2 2 + 9 + x 1 u ˙ x 3 = 4 x 3 + 3 + x 2 (This system does not come from any physical application but its nonlinearities are typical for chemical and biological systems.)
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