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Unformatted text preview: MAE 171A: Dynamic Systems Control Lecture 6: Stability Goele Pipeleers Overview of the Lectures so Far • L1: review of the Laplace transform and transfer function • L2: deriving a transfer function model from a physical model • L3: system poles, zeros and impulse response • L4: system step response and relation with pole locations • L5: effect of zeros of system response 1 In This Lecture System Stability • definitions • role of the poles • analyzing stability without computing the poles • control design for stability 2 Stability: Various Definitions Bounded Input – Bounded Output (BIBO) Stability • definition: a system is called BIBO stable if every bounded input results in a bounded output • equivalent condition 1: a system with impulse response g ( t ) is BIBO stable if and only if Z ∞  g ( t )  dt < ∞ 3 – proof of sufficiency: suppose  u ( t )  ≤ M < ∞ , then  y ( t )  = Z t g ( τ ) u ( t τ ) dτ ≤ Z t  g ( τ )  u ( t τ )  dτ ≤ M Z t  g ( τ )  dτ • equivalent condition 2: a system is BIBO stable if and only if all its poles have a negative real part 4 “Autonomous” Stability (Lyapunov Stability) • definition: a system is called autonomously stable if the autonomous response from every initial condition is bounded • equivalent condition 1: a system with impulse response g ( t ) is au tonomously stable if and only if  g ( t )  ≤ M < ∞ , ∀ t • equivalent condition 2: a system is autonomously stable if and only if all its poles have a nonpositive real part and the poles on the imaginary axis are singular • systems that are autonomously stable but not BIBO stable are sometimes called neutrally stable 5 System Poles and Impulse Response From Lecture 3 • system G ( s ) with poles p i has an impulse response g ( t ) =...
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This note was uploaded on 08/08/2011 for the course MAE 171 taught by Professor Pipeleers during the Summer '11 term at UCLA.
 Summer '11
 Pipeleers
 Laplace

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