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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.

The original bode plot The new factor log j1 + G({!) We are most concerned with the e ect of the new term 20logj1 + G({!)j Speci cally, as 1 + G({!) !0 lim 1+G({!)!0 20logj1 + G({!)j= 1 An unstable mode!-80-70-60-50-40-30-20-10 0 10 20 Magnitude (dB) 10-1 10 0 10 1 10 2-180-135-90-45 0 Phase (deg) Bode Diagram Frequency (rad/sec) Figure :Open ...

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