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HWCE4_Solution - H W/CE#4 Due Wednesday 1 Suppose we have...

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HW/CE#4 Due Wednesday, March 24, 2010 1. Suppose we have the following specifications: M p :: 40% settling time t s :::: 5 sec rise time t r :::: 1.0 sec Choose phase margin PM and crossover frequency We so that we meet the above. Use the "more exact" tabulations on the second page of the reference sheets. Hint: start with ~ ~ 0.30. , ) 3./ PH - '0
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2. Assume G(s) =KL(s), shown below, is the open-loop gain in a unity feedback system. (Note that this is the same G(s) you did the root locus and Bode plot for in the last homework exercise.) Make hand sketches showing the Nyquist plots for: K G( s) = KL( s) = -------:::------ S(S2 + IDs ,+ 125) For K = 1000, 1250, and 1500. Show critical points (crossover values on the real axis) for all three cases. Describe closed loop system stability for the 3 cases, i.e., marginally stable, unstable, etc. Verify your sketches by submitting MATLAB generated Nyquist plots. Gf;w) --7 0 2:;0° G'~'~ ).-:, 0 L0./(") ~ "-#" I:. (- to,;r" -j (/~:w-(jl») - = .Fe ~ '-( ..... ) I +J J- i Gt~)J IN) cAt <f J.. (I ~ r iJ - /j'~) ~ ~Ih;; ~ i 6-L/w)] = ~ Itt f~~J : _ (), 8) . - I. ~; - /,::l. -= oSP M i --~-r-~~-4------'-----~ !.)() s; It ~e t:T : ~~ r~ ( /~ "~I'-.,f ! 61'
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1 » numL=[l]; » denL=[110 1250]; » L=tf(numL,denL) Transfer function: sl\3 + 10 sl\2 + 125 s » nyquist(1000* L) » nyquist(1250*L) » nyquist(1500* L) Nyquist Diagram ------r------~--~-- 4 -, .. -.-- - -.- .. -.-. . _-. 3 2 '.,. ~ I) - - - - - - - - - - - - - - - - - - - - - -+ --- -;:- -~ ..... ::::-::::- -:::::;-:- ::::- -:::-:::- -:-::- ~- -~- ~- -~-~- ~- -~-~- -~-::-:::- -:::-;:- -::;-~-;J- '01 on E -1 _L_._ __ . __ ._ __ .. -4 .-.- ......... - ....... ------1.------.--- ....... -1:; -1 -os o Real Axis K = 1000 '-,-
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(I) ~ i 'Ol ;D E 2 0 - - - - - - - - - - - - - - - - - - ;:- -~*""o::c::-::=:- z- -~-:-::- ~- -~-~-~-~- :- -~-~-~- ~- -~-~-~- ~- -~-~-~- ~- :- -~-::-~- ~- -~-~-:;:- ~- :;- -1 -2 -3 -4 L- --L __ l!e.. -L. __ _ -1 -0.5 u -1 .5 Nyquist Diagram Real Axis K = 1250 '.,
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Nyquist Diagram 4,----------,--r----------.----------~ 3 2 '., ------1--------------------------------------------- .-, -~ " .:i .. 4 L-.
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