Transcription of Systems Analysis and Control
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Systems Analysis and Control
Systems Analysis and ControlMatthew M. PeetArizona State UniversityLecture 21: Stability Margins and Closing the LoopOverviewIn this Lecture, you will learn:Closing the Loop Effect on Bode Plot Effect on StabilityStability Effects Gain Margin Phase Margin BandwidthEstimating closed -Loop Performance using Open-Loop Data Damping Ratio Settling Time Rise TimeM. PeetLecture 21: Control Systems2 / 31 ReviewRecall: Frequency ResponseInput:u(t) =Msin( t+ )Output:Magnitude and Phase Shifty(t) =|G( )|Msin( t+ + G( ))02468101214161820 2 1 Simulation ResultsTime (sec)AmplitudeFrequency Response tosin tis given byG( )M. PeetLecture 21: Control Systems3 / 31 ReviewRecall: Bode PlotDefinition Bode Plotis a pair oflog-logand semi-logplots:1. Magnitude Plot:20 log10|G( )| 2. Phase Plot: G( ) Bite-Size Chucks: G( )= i Gi( )M. PeetLecture 21: Control Systems4 / 31 Complex Poles and ZerosWe left off withComplex Poles:G(s) =1((s n)2+ 2 s n+ 1)M.
The Quadratic Approximation gives the Closed-Loop Damping Ratio as M = tan 1 q 2 2 2 + p 4 4 + 1 M is from the Open-Loop Data! A Handy approximation is = M 100 Only valid out to ˘=:7. Given M, we nd closed-loop . M. Peet Lecture 21: Control Systems 17 / 31
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