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Unformatted text preview: EE263 S. Lall 2011.06.03.03 Final exam 201011 Solutions This is a 24hour takehome exam. You may use any books, lecturenotes or computer programs that you wish, but you may not discuss this exam with anyone until Wednesday June 8, after everyone has taken it. The only exception is that you can ask the TA’s or Sanjay Lall for clarification. Web page for typos and datafiles. The web page http://junction.stanford.edu/~lall/ee263/final contains any needed files, and it should always contain all known typos; check this before contacting one of us for clarification. Please note that we have tried pretty hard to make the exam unambiguous and clear, so we are unlikely to say much. If you choose to send out an email for clarification, please use the staff email address [email protected] so that you can get reply as soon as possible. Since you have 24 hours, we expect your solutions to be legible, neat, and clear. Do not hand in your rough notes, and please try to simplify your solutions as much as possible. Good Luck! 1. Some short answer problems. 60211 Please provide short answers for each of these problems. Each of the answers to these questions can be stated with a few words or with one mathematical expression. Do not provide work or justification; we will not read long answers. Each part is equally weighted. (a) You would like to determine two unknown quantities x = bracketleftbig x 1 x 2 bracketrightbig T using one of two systems. The first system takes two measurements of the form y = A 1 x + v A 1 = I bardbl v bardbl = 1 The second system takes two measurements of the form y = A 2 x + v A 2 = bracketleftbigg 1 1 − 1 1 bracketrightbigg bardbl v bardbl = 1 In both cases, v is a noise parameter that must lie on a unit ball. The estimation error is the worstcase value of bardbl x − x est bardbl . Which system gives the smaller error? For your answer we just want to see first system or second system . (b) You are solving a controls problem. You have a desired target location y des , and you would like to choose your input x so that y des = Ax , where A ∈ R 2 × 3 . You evaluate the leastnorm solution x opt and find that bardbl x opt bardbl = 1. You give the solution to your boss, who tells you that they are actually looking for a special solution for x that has a norm of exactly two, so that bardbl x special bardbl = 2. Give an expression for x special in terms of x opt and the SVD of A . Your expression should be simple and compact. (c) A and B are both matrices with condition number of 100. We would like to see the maximum and minimum possible condition numbers of AB , and the maximum and minimum possible condition numbers of A + B . For your answer, give us these four numbers; no justification is necessary....
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 Summer '08
 BOYD,S
 Numerical Analysis, Optimization, condition number, Lall, S. Lall, boomerang constraint

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