Any ics0 ut ut yt yt bibo stable system asymptotic

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Unformatted text preview: bounded output. Any ICs=0 u(t) u(t) y(t) y(t) BIBO stable system Asymptotic stability : Asymptotic Wind-induced vibration Wind- Collapsed! Any ICs generates y(t) converging to zero. y(t) 2008… ICs u(t)=0 u(t)=0 y(t) y(t) Asymp. stable Asymp. system 5 6 Stability condition in s-domain Some terminologies (Proof omitted, and not required) (Proof For a system represented by a transfer function G(s), Ex. Zero : roots of n(s) Zero system is BIBO stable Pole : roots of d(s) Pole All the poles of G(s) are in the open left half of the complex plane. Characteristic polynomial : d(s) Characteristic system is asymptotically stable Characteristic equation : d(s)=0 Characteristic 7 8 Second order impulse responseUnderdamped and Undamped “Idea” of stability condition Example … Overdamped … Critically damped Asym. Stability: Asym. (U(s)=0) U(s)=0) … Underdamped … Undamped BIBO Stability: (y(0)=0) Bounded if Re(α)>0 (α)> 9 Second order impulse response – Underdamped and Undamped Changing / Fixed 10 Second order impulse response – Underdamped and Undamped Changing / Fixed Impulse Response Impulse Response 4 3 3 2 2 Amplitude 5 4 Amplitude 5 1 0 1 0 -1 -1 -2 -2 -3 -4 -3 0 2 4 6 Time (sec) 8 10 -4 12 6 0 2 4 6 Time (sec) 8 10 12 6 4 4 2 2 0 0 -2 -2 -4 -6 -4 11 -5 0 5 -6 12 -5 0 5 Second order impulse response – Underdamped and Undamped Changing / Fixed Remarks on stability For a general system (nonlinear etc.), BIBO For stability condition a...
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