final - ME254/ECE217C 2009 Final Exam(corrected again and...

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ME254/ECE217C, Final Exam (corrected again and again) Winter 2009 This is a take home exam, it is due back at 4pm, Thursday 3/19 in Prof. Bamieh’s mailbox (sharp deadline). You can consult the notes that have been handed out in class or on the course web page, but no other references are allowed. You may not discuss these problems with others in class until you have turned your work in. Please write legibly and make full use of logical reasoning. Have a good spring break. 1. Consider the time-invariant system: ˙ x = Ax + Bu y = Cx + Du . Find the minimum energy control over the interval ( -∞ , 0], i.e. { u ( t ); - ∞ < t 0 } , required to set up the initial state x (0) = x o . In other words, solve min Z 0 -∞ u T ( t ) u ( t ) dt, such that x (0) = x o , and x ( -∞ ) = 0. Find the solution of this problem in terms of the (infinite horizon) Grammians of the system, your answer should include the optimal function u , and the optimal value of the performance index. 2. Consider the following linear system with a scalar state ˙ x = x + u, and the non-quadratic performance objective J = 1 2 Z 0 ( x 4 + u 2 ) dt.
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