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 ( ) u ( t- ) d Z t | g ( ) || u ( t- ) | d M Z t | g ( ) | d 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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lecture6 - MAE 171A: Dynamic Systems Control Lecture 6:...

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