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Stability

# 1 1 2 y t km 1 e t cos 1 2 t sin 1

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Unformatted text preview: + BU ( s) Y ( s) = CX ( s) + DU ( s) This is all the same as before, but need to use matrix operations for inverse. ( sI − A) X ( s) = BU ( s) X ( s) = ( sI − A)−1 B U ( s) Y ( s) = C ( sI − A)−1 BU ( s) + DU ( s) G ( s) = Y ( s) = C ( sI − A)−1 B + D U ( s) So ( sI − A) must be invertible if G(s) is to be bounded. This is the same criterion as computing the eigenvalues above. Recap on second ­order systems: A. Oscillations • ζ > 0 , overdamped, no natural oscillation • ζ = 1 , critically damped, no natural oscillation, repeated roots • 0 < ζ ≤ 1 , underdamped, sines and cosines in the solution B. Stability: Determined by Re(s). If Re(s) > 0 for any root s, then unstable. Else stable. Finished Lecture 16 here. Class act...
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