sol3.09 - ECE521 Linear Systems Fall 2009 Homework 3...

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Unformatted text preview: ECE521 Linear Systems Fall 2009 Homework 3 Solutions Problem 1: Consider a square matrix A for which det( A ) = 0 . Is it possible that det( e tA ) = 0 for some finite t > ? Justify your answer. We know that matrix exponential is a nonsingular matrix hence its determinant cannot be invertible. Problem 2: Solve ˙ x ( t ) =   1 0 − 1 0 1 1 0 − 1   x ( t ) +   e- 2 t e- 2 t   , x (0) =   1   Is the solution bounded as t → ∞ ? Let A be the system matrix. We have that A =   1 0 − 1 0 1 1 0 − 1   =   1 0 − 1 0 0 1 0 − 1   +   0 0 0 0 1 0 0 0 0   = B + C. Clearly B and C comute so we have e tA = e tB e tC . We note that C is diagonal and B 2 = 0 hence e tB = I 3 + tB =   1 + t − t 1 t 0 1 − t   e tC =   1 e t 1   e tA =   1 + t − t e t t 1 − t   Thus the solution to the IVP is x ( t ) =   1 + t + e- 2 t e- 2 t t   Clearly when t → ∞ we have that || x ( t ) || is unbounded. 2 Problem 3: Consider the system ˙ x = parenleftbigg 1 2 − 1 parenrightbigg x + parenleftbigg 1 2 − 1 1 1 2 parenrightbigg u Find u ( t ) which drives the system from state x (0) = [1 1] T to x (1) = [0 0] T . In class we derived a formula for control u which drives the system from the initial state x ( t ) to the final state x ( t f ) interms of the Gramian M r (if it is nonsingular) and state transition matrix Φ( t,t ). The formula is u ( t ) = B T Φ( t,t ) T M- 1 R ( x ( t f ) − Φ( t,t ) x ( t )) First we calculate the reachability Gramian, M = M R (0 ,t,A,B ) = integraldisplay 1 e ( t- τ ) A BB T e ( t- τ ) A T dτ = integraldisplay 1 parenleftbigg e ( t- τ ) / 2 e- ( t- τ ) parenrightbiggparenleftbigg...
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sol3.09 - ECE521 Linear Systems Fall 2009 Homework 3...

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