hw6_f09 - UNIVERSITY OF CALIFORNIA AT BERKELEY Department...

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UNIVERSITY OF CALIFORNIA AT BERKELEY Department of Mechanical Engineering ME132 Dynamic Systems and Feedback Fall 2009 Homework #6 Due: Friday, October 09 Midterm 1 information: Date: Monday, October 12 Time: 6:15pm - 7:15pm Location: 2060 VLSB Details: * Closed book, closed notes * One sheet (front and back) of handwritten notes allowed * One hour exam 1. Find state-space realizations for the following input-output differential equations, i.e. write the differential equations in the form ˙ x ( t ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) + Du ( t ) where A, B, C , and D are matrices of appropriate dimensions. (a) y (4) + 4 y (3) + 10 y (2) + 7 y (1) - 2 y = u (3) + 5 u (2) - 2 u (1) - u (b) y (3) + 5 y (2) - 4 y (1) = 3 u (3) - 2 u (1) + u 2. Find the free response of the differential equation: y (2) + 4 y (1) + 4 y = 3 u (2) - 2 u (1) + u with initial conditions y (0) = 1 , y (1) (0) = 0. 3. Consider the transfer function: H ( s ) = 3 s 3 + 2 s + 1 ( s 2 + 4 s + 5) 2 (a) Find the poles and zeros of H ( s ). (b) Find the form of the free response. (c) Is H ( s ) stable?
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