Stability

# b k x s 2 u s g su s 2 2 u s ms cs

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Unformatted text preview: r growth Second ­order equations: k Consider the Laplace approach first: d 2x dx m 2 + c + kx = bu(t ), x (0) = 0, x '(0) = 0 dt dt 2 s mX ( s) + scX ( s) + kX ( s) = bU ( s) c m F = bu(t ) b K X ( s) = 2 U ( s) = G ( s)U ( s) = 2 2 U ( s) ms + cs + k τ s + 2ζτ s + 1 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 = as 2 + bs + c = ( s + b1 )( s + b2 ) 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): • • The second term always has a smaller magnitude than the first The stability of the system is determined by the signs on ζ ,τ • “Overdamped” If ζ 2 = 1 , then there are two repeated roots: • The stabi...
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## This note was uploaded on 04/09/2014 for the course CHBE 4400 taught by Professor Staff during the Spring '08 term at Georgia Tech.

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