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157afinal - ω g c Determine the resonant-peak M pω the...

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TAKE-HOME EXAM - R ( s ) + - h - E ( s ) C ( s ) - P ( s ) - Y ( s ) 6 input: r ( t ) , output: y ( t ) error: e ( t ) = r ( t ) - y ( t ) open-loop transfer-function: G ( s ) = P ( s ) C ( s ) Let P ( s ) = 40 ( s + 12 ) ( s + 4 )( s 2 + 6 s + 34 ) i) First consider a proportional controller, i.e., let C ( s ) = K p . a) Find the range of K p for closed-loop stability. b) Determine the steady-state error due to a unit-step input r ( t ) = 1( t ). c) Sketch the root-locus. d Sketch the Nyquist plot with K p = 1. e) Sketch the Bode-plot with K p = 1. f) Determine the gain-margin GM, phase-margin PM, phase-crossover frequency ω p and gain-crossover frequency ω g . ii) Now design a controller C ( s ) such that the closed-loop system is stable, the steady-state error due to a unit-step input is zero , and the phase-margin PM is at least 45 o . a) For the controller you designed, sketch the root-locus, the Nyquist plot, the Bode-plot. b) Determine the gain-margin GM, phase-margin PM, phase-crossover frequency ω p and gain-crossover frequency
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Unformatted text preview: ω g . c) Determine the resonant-peak M pω , the resonant frequency ω r and the bandwidth ω B . d) Determine the percent-overshoot PO of the unit-step response and the approximate settling time t s . e) Now suppose that there is a delay in the system, i.e.,that G ( s ) is multiplied by the delay transfer-function e-sT d . What is the maximum time-delay T d that your system can tolerate such that the closed-loop system is still stable? The design for part (ii) can be the simplest design that satisfies the criteria. You can use MATLAB as a design aid, to compute, to obtain details of plots and to verify your design. All plots and sketches should be hand-sketched, but you can verify them using MATLAB. Printed and/or typed reports are not accepted....
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