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Stability Notes

# These variables have qualitative interpretation for

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Unformatted text preview: , x '(0) = 0 dt dt c m F = bu(t ) X ( s) = b k m τ2 = k K= where c c1 = 2ζτ ⇒ ζ = k 2 mk The time constant is τ , the process gain is K, and the damping coefficient is ζ . (These variables have qualitative interpretation for response.) Stability? Factor the denominator to see if the roots have positive (unstable) or negative (stable) real parts: τ 2 s 2 + 2ζτ s + 1 = Quadratic formula: s= ζ 2 −1 − b ± b2 − 4 ac −2ζτ ± ( 2ζτ )2 − 4τ 2 −ζ = = ± 2 2a τ τ 2τ If ζ 2 > 1 , then the roots are purely real (not complex): If ζ 2 = 1 , then there are two repeated roots: If ζ 2 < 1 , these roots are complex, and they will be complex conjugates of each other. If τ > 0,ζ > 0 , then the second ­order linear system is stable. Example: Impulse response for 0 <| ζ |≤ 1 : K U ( s) τ s + 2ζτ s + 1 X ( s) = 22 U ( s) = 1 X ( s) =...
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