lecture6

lecture6 - MAE 171A Dynamic Systems Control Lecture 6...

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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.

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lecture6 - MAE 171A Dynamic Systems Control Lecture 6...

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