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2007final2

# 2007final2 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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�� �� 16.323, #32 April 11, 2007 16.323 Final Exam This is a closed-book exam, but you are allowed 3 pages of notes (both sides). You have 3 hours. There are six 6 questions with the relative values clearly marked. Some handy formulas: u ( t ) = arg min H ( x , u , p , t ) u ( t ) ∈U B u R 1 B T = 0 A T P ss + P ss A + R xx P ss uu u P ss = R 1 B T K ss uu u P ss a b a + d ± a 2 2 ad + d 2 + 4 bc λ i = c d 2 ( i ) P N = H ( ii ) F k = R k + B k T P k +1 B k 1 B k T P k +1 A k ( iii ) P k = Q k + F k T R k F k + { A k + B k F k } T P k +1 { A k + B k F k }
1. (15%) Consider a model of a first order unstable plant with the state equation x ˙( t ) = ax ( t ) au ( t ) a > 0 and a new cost functional J = e 2 bt x 2 ( t ) + ρu 2 ( t ) dt ρ > 0 , b 0 0 It can be shown that, with a change of variables in the system to e bt x ( t ) x ˜( t ) and bt e u ( t ) u ˜( t ), then the only real modification required to solve for the static LQR con- troller with this

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2007final2 - MIT OpenCourseWare http/ocw.mit.edu 16.323...

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