ps1-soln - Problem Set 1 Solutions January 24, 2010 Problem...

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January 24, 2010 Problem 1 The mathematical model is u - v C + L di L dt = 0 i L + C dV C dt + h ( v C ) = 0 . The state space model is dx 1 dt = 1 L x 2 - 1 L u dx 2 dt = - 1 C x 1 - 1 C h ( x 2 ) . Problem 2 Free-body diagram : there are two masses, m 1 and m 2 , hence we will draw two diagrams: M 1 M 2 b ( ˙ X 1 - ˙ X 2 ) u k 2 ( X 1 - X 2 ) k 3 X 2 b ( ˙ X 1 - ˙ X 2 ) k 2 ( X 1 - X 2 ) k 1 X 1 Figure 1: Free-body diagrams Note that, when x 1 > x 2 and hence x 1 - x 2 > 0, the spring k 2 pushes m 1 to the left, and m 2 to the right. Hence the orientation of the forces in the free-body diagram. A similar reasoning holds for the damper b . Applying Newton’s law to the free-body diagram we get: m 1 ¨ x 1 = - k 1 x 1 - k 2 ( x 1 - x 2 ) - b ( ˙ x 1 - ˙ x 2 ) + u m 2 ¨ x 2 = - k 3 x 2 + k 2 ( x 1 - x 2 ) + b ( ˙ x 1 - ˙ x 2 ) NOTE : Suppose that one wants to position m 2 at a desired location, i.e., to control x 2 . In this case, the control input is the force u , the output is
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ps1-soln - Problem Set 1 Solutions January 24, 2010 Problem...

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