midterm_f07

midterm_f07 - EE221A Linear System Theory Midterm Test...

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Unformatted text preview: EE221A Linear System Theory Midterm Test Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2007 10/16/07, 9.30-11.00am Your answers must be supported by analysis, proof, or counterexample. There are 6 questions: approximate points for each question are indicated. The total points for the test are 30. USEFUL FORMULAE: 1. For the Linear Quadratic Optimization problem min u J , where: J = 1 2 ( integraldisplay t 1 t ( || u || 2 2 + || Cx || 2 2 ) dt + x ( t 1 ) T Sx ( t 1 )) ˙ x = Ax + Bu the optimal control is given by u ( t ) =- B T ˜ X ( t ) X- 1 ( t ) x ( t ), ∀ t ∈ [ t ,t 1 ] where X ( t ) , ˜ X ( t ) solve the backwards Hamiltonian linear matrix differential equation: d dt bracketleftbigg X ( t ) ˜ X ( t ) bracketrightbigg = bracketleftbigg A ( t )- B ( t ) B T ( t )- C T ( t ) C ( t )- A T ( t ) bracketrightbiggbracketleftbigg X ( t ) ˜ X ( t ) bracketrightbigg (1) with X ( t 1 ) = I and ˜ X ( t 1 ) = S ....
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midterm_f07 - EE221A Linear System Theory Midterm Test...

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