l06_modal_decomp

l06_modal_decomp - Matrix Diagonalization Suppose A is...

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Matrix Diagonalization Suppose A is diagonizable with independent eigenvectors V = [ v 1 , . . . , v n ] use similarity transformations to diagonalize dynamics matrix x ˙ = Ax x ˙ d = A d x d λ 1 Δ = Λ = A d V 1 AV = · λ n Corresponds to change of state from x to x d = V 1 x System response given by e At , look at power series expansion At = V Λ tV 1 2 ( At ) 2 = ( V Λ tV 1 ) V Λ tV 1 = V Λ t 2 V 1 ( At ) n = V Λ n t n V 1 1 At e = I + At + ( At ) 2 + . . . 2 1 2 2 V 1 V I + Λ + Λ t + . . . = 2 λ 1 t e · λ n t e V e Λ t V 1 V 1 = V = 1

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± ² Taking Laplace transform, 1 s λ 1 ( sI A ) 1 = V V 1 · 1 s λ n R i n i =1 s λ i where the residue R i = v i w i T , and we deFne = T w 1 , V 1 = . . V = v 1 . . . v n . T w n Note that the w i are the left eigenvectors of A associated with the right eigenvectors v i λ 1 λ 1 · V 1 A = V
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l06_modal_decomp - Matrix Diagonalization Suppose A is...

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