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# Introduction to Feedback Control Systems CMPE 241

UCSC

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This 10 page Class Notes was uploaded by Buck Ankunding on Monday September 7, 2015. The Class Notes belongs to CMPE 241 at University of California - Santa Cruz taught by Staff in Fall. Since its upload, it has received 43 views. For similar materials see /class/182226/cmpe-241-university-of-california-santa-cruz in Computer Engineering at University of California - Santa Cruz.

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Date Created: 09/07/15

EEl 54CMPE24l Winter 2008 Due 6PM 21Feb2008 Homework 6 Root Locus Design Problems are from Franklin Powell Emami Feedback Control of Dmamic Systems 5Lh Edition FPE Midterm is in Kresge 327 14Feb08 Noon open book open notes 1 FPE 522 2 FPE 536 3 FPE 541 l sz 1 work but I want you to do most of this by hand in order to develop your intuition 4 Given the system Gs Note you may use MATLAB to check your a Design a lead controller With its zero at the origin Ks K 3 p unity feedback such that the dominant second order roots are at 72 r 2j Use root locus techniques to do this design What is the resulting K s 57 Sketch the locus of roots against the loop gain for the design generated in a above Show asymptotes center of asymptotes departure angles etc Where appropriate EEl 54CMPE24l Winter 2008 Due 6PM 28Feb2008 Homework 7 Bode Plots 1 Sketch the Bode diagrams frequency response plots for the following systems First sketch the asymptotes and then add the values at the key break points on the plots 1 ss 5 a Gs 1 1quot Fm Hi 3LI 2 A lead compensator is de ned by Ds K K 5 Where the s b s Jaw center frequency is a and the ratio of pole location to zero location is a ie g 1 a Plot the magnitude and phase plots for the lead compensator Ds S 0395 Note that the peak in the phase occurs at the center s frequency on a log scale a 05 2 l In this case 2 4 and 1 a 1 Find the value ofthe peak in phase lead for Evalues of2 4 10 and 50 1 1 57 suggest just evaluating the complex number rather than trying to do it graphically and its OK to use MATLAB EEl 54CMPE24l Winter 2007 Due 6PM 21Feb2007 Homework 7 Bode Plots 1 Sketch the Bode diagrams frequency response plots for the following systems First sketch the asymptotes and then add the values at the key break points on the plots 1 ss 5 a Gs 1 1quot Fm Hi 3LI 2 A lead compensator is de ned by Ds K K 5 Where the s b s Jaw center frequency is a and the ratio of pole location to zero location is a ie g 1 a Plot the magnitude and phase plots for the lead compensator Ds S 0395 Note that the peak in the phase occurs at the center s frequency on a log scale a 05 2 l In this case 2 4 and 1 a 1 Find the value ofthe peak in phase lead for Evalues of2 4 10 and 50 1 1 57 suggest just evaluating the complex number rather than trying to do it graphically and its OK to use MATLAB EE154CMPE24l Winter 2008 Due 6PM 13Mar2008 Homework 9 Final Review 1 ReRead Chapters 1 through 6 in FPE again Review your notes and lectures as required 2 Graduate Students ONLY Write up a LaTEX or equivalent document detailing the equations of motion for the inverted pendulum transformation of these equations into a transfer function or state space representation your design requirements and your controller designs Leave the section for experimental results blank This report is due at the beginning of the final exam The following questions are from a practice nal You should be able to do the entire thing in a single three hour set Try to set some time aside at least three uninterrupted hours to attempt it A er doing it take the time to redo every problem until you have it right and understand it 3 Consider proportional control gain stabilization for the following plant Gs 3 i 3 sile2s3 sz4szsi i a Determine the range of stabilizing K assuming unity feedback using RouthHurwitz techniques 57 Plot the root locus of the system both 180 and 0 compute breakaway points asymptotes centroid j wcrossings and stable ranges for K 0 Plot the Bode Plot of the open loop system denoting any gain and phase margins if they exist P Plot the Nyquist plot of the open loop system again denoting gain and phase margin for both negative and positive feedback configurations Hint GQ I 180 and remember that unstable poles have different phase plots than stable ones 4 Consider the system KGs set up in unity feedback KGs That S S a a Sketch the root locus With respect to K assuming a 1 b Sketch the root locus with respect to a assuming K10 Him put the characteristic equation into Evan s Form Do not bother with the calculation of breakin or breakaway points just provide the basic plot asymptotes departure angles arrival angles centroid etc 5 Find the transfer inction YxRs for the following two systems and note that there are several Ways to compute these transfer functions l 6 Cons1der a Simple inverted pendulum modeled as Gs 2 2 1 Assume s s unity feedback and design a controller that stabilizes this plant H int Root Locus is probably most helpful although a Bode design can work as well Do not spend your time ne tuning the control parameters just nd a compensator strategy that works 7 Consider the plant Gs 3 in a unity feedback system 4 33 3 a Sketch the Root Locus showing asymptotes departure angles etc Using graphical techniques find the approximate gainK at the stability boundary crossing the jmaXis Fquot Find a compensator that stabilizes the system with a bandwidth of approximately 10 rads or greater Find approximate values for any parameters EE154CMPE24l Winter 2007 Due 6PM 13Mar2007 Homework 9 Final Review 1 ReRead Chapters 1 through 6 in FPE again Review your notes and lectures as required 2 Graduate Students ONLY Write up a LaTEX or equivalent document detailing the equations of motion for the inverted pendulum transformation of these equations into a transfer function or state space representation your design requirements and your controller designs Leave the section for experimental results blank The following questions are from a practice nal You should be able to do the entire thing in a single three hour set Try to set some time aside at least three uninterrupted hours to attempt it A er doing it take the time to redo every problem until you have it right and understand it 3 Consider proportional control gain stabilization for the following plant 3 3 GO sils2Xs3 sz4szs76I 33 Determine the range of stabilizing K assuming unity feedback using RouthHurwitz techniques 57 Plot the root locus of the system both 180 and 0 compute breakaway points asymptotes centroid jwcrossings and stable ranges for K 0 Plot the Bode Plot of the open loop system denoting any gain and phase margins if they exist P Plot the Nyquist plot of the open loop system again denoting gain and phase margin for both negative and positive feedback con gurations Hint GQ I 180 and remember that unstable poles have different phase plots than stable ones 4 Consider the system KGs set up in unity feedback KGV That I S a a Sketch the root locus with respect to K assuming a 1 b Sketch the root locus with respect to a assuming K10 Him ut the characteristic equation into Evan s Form Do not bother With the calculation of breakin or breakaway points just provide the basic plot asymptotes departure angles arrival angles centroid et 5 Find the transfer function YsRs for the following two systems and note that there are several Ways to compute these transfer inctions 6 Consider a simple inverted pendulum modeled as GS 2 1 1 Assume s s unity feedback and design a controller that stabilizes this plant H int Root Locus is probably most helpful although a Bode design can work as well Do not spend your time ne tuning the control parameters just nd a compensator strategy that works 7 Consider the plant Gs 3 in a unity feedback system s s i 3 a Sketch the Root Locus showing asymptotes departure angles etc Using graphical techniques find the approximate gainK at the stability boundary crossing the jmaXis Fquot Find a compensator that stabilizes the system with a bandwidth of approximately 10 rads or greater Find approximate values for any parameters EEl 54CMPE24l Winter 2007 Due 6PM 19Feb2007 Homework 6 Root Locus Design Problems are from Franklin Powell Emami Feedback Control of Dmamic Systems 5Lh Edition FPE l FPE 522 N FPE 536 U FPE 541 l sz 1 work but I want you to do most of this by hand in order to develop your intuition 5 Given the system Gs Note you may use MATLAB to check your a Design a lead controller With its zero at the origin Ks K 3 p unity feedback such that the dominant second order roots are at s 72 r 2j Use root locus techniques to do this design What is the resulting K s 57 Sketch the locus of roots against the loop gain for the design generated in a above Show asymptotes center of asymptotes departure angles etc Where appropriate

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