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MIT6_241JS11_midterm

# MIT6_241JS11_midterm - modeled in continuous time by the...

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6.241 Spring 2011 Midterm Exam 3/16/2011 You have a total of three hours to complete this exam. These three hours can be chosen at your convenience. Problem 1 Let A C n × n , and B C m × m . Show that X ( t ) = e At X (0) e Bt is the solution to X ˙ = AX + XB . Problem 2 Given two non-zero vectors v, w R n . Does there exist a matrix A such that v = Aw and 1. σ max ( A ) = v T v/w T w ? 2. A 1 = v / w ? Prove or disprove each case separately. Problem 3 1 Let A < 1. Show that ( I A ) 1 � ≥ 1 + A . Problem 4 Use the projection theorem to solve the problem: min x R n { x T Qx : Ax = b } , 1

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where Q is a positive-definite n × n matrix, A is a m × n real matrix, with rank m < n , and b is a real m -dimensional vector. Is the solution unique? Problem 5 Consider a single-input discrete-time LTI system, described by 1 1 0 x [ k + 1] = x [ k ] + u [ k ] 0 1 1 y [ k ] = x [ k ] , and the initial condition x [0] = 0. Given M > 1, what is the maximum value of y [ M ] 2 that can be attained with an input of “unit energy,”, i.e., such that u [0] 2 + u [1] 2 + . . . + u [ M 1] 2 = 1? What is the input that attains such value? How would your answer change if you were to double M , i.e., M 2 M ? Problem 6 Consider a physical system whose
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Unformatted text preview: modeled, in continuous time, by the diﬀer-ential equation x ˙ = Ax + Bu. Assume that you have two sensors. The frst sensor yields measurements y 1 = C 1 x For t = 0 , 1 , 2 , 3 ,... , and the second sensor yields measurements y 2 = C 2 x For t = 0 , 2 , 4 ,... . Assuming that u ( t ) = u ( ± t ² ), For all t ≥ 0, derive a discrete-time state-space model For the system, relating the inputs at times ( u (0) ,u (1) ,u (2) ,... ) to the outputs at times ( y (0) ,y (1) ,y (2) ,... ). 2 MIT OpenCourseWare http://ocw.mit.edu 6.241J / 16.338J Dynamic Systems and Control Spring 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT6_241JS11_midterm - modeled in continuous time by the...

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