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

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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 > 0 ? 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 0 1 0 1 x ( t ) + e - 2 t e - 2 t 0 , x (0) = 1 0 0 Is the solution bounded as t → ∞ ? Let A be the system matrix. We have that A = 1 0 1 0 1 0 1 0 1 = 1 0 1 0 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 0 t 0 1 0 t 0 1 t e tC = 1 0 0 0 e t 0 0 0 1 e tA = 1 + t 0 t 0 e t 0 t 0 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.
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2 Problem 3: Consider the system ˙ x = parenleftbigg 1 2 0 0 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 0 ) to the final state x ( t f ) interms of the Gramian M r (if it is nonsingular) and state transition matrix Φ( t, t 0 ). The formula is u ( t ) = B T Φ( t, t 0 ) T M - 1 R ( x ( t f ) Φ( t, t 0 ) x ( t 0 )) First we calculate the reachability Gramian, M = M R (0 , t, A, B ) = integraldisplay 1 0 e ( t - τ ) A BB T e ( t - τ ) A T = integraldisplay 1 0 parenleftbigg e ( t - τ ) / 2 0 0 e - ( t - τ ) parenrightbiggparenleftbigg 5 4 0 0 5 4 parenrightbiggparenleftbigg e ( t - τ ) / 2 0 0 e - ( t - τ ) parenrightbigg = 5 4 integraldisplay 1 0 parenleftbigg e ( t - τ ) 0 0 e - 2( t - τ ) parenrightbigg = 5 4 parenleftbigg e t 1 0 0 1 2 (1 e - 2 t ) parenrightbigg
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