ECE-320: Linear Control Systems Homework 6 Due: Thursday January 21, 2009 at the beginning of class 1) Consider the plant 013()0.5pGsssαα==++ where 3 is the nominal value of 0α and 0.5 is the nominal value of1α. In this problem we will investigate the sensitivity of closed loop systems with various types of controllers to these two parameters. We will assume we want the settling time of our system to be 0.5 seconds and the steady state error for a unit step input to be less than 0.1. a) (ITAE Model Matching) Since this is a first order system, we will use the first order ITAE model, ()oooGssωω=+ i) For what value of oωwill we meet the settling time requirements and the steady state error requirements? ii) Determine the corresponding controller . ( )cGs iii) Show that the closed loop transfer function (using the parameterized form of and the controller from part ii) is ()pGs0108( 0.5)3()8( ) ( 0.5)3osGsss sααα+=++ + iv) Show that the sensitivity of to variations in ( )oGs0α is given by 008GsSsα=+ v) Show that the sensitivity of to variations in ( )oGs1α is given by 120.58.5 4oGsSssα−=++b) (Proportional Control) Consider a proportional controller, with 2.5pk=. i) Show that the closed loop transfer function is 0102.5()2.5oGssααα=++ ii) Show that the sensitivity of to variations in ( )oGs0α is given by 000.58GsSsα+=+ iii) Show that the sensitivity of to variations in ( )oGs1α is given by 10.58oGSsα−=+ c) (Proportional+Integral Control) Consider a PI controller with 4pk= and . 40ik = i) Show that the closed loop transfer function is 0104( 10)()()4(1osGsss s 0)ααα+=++ + ii) Show that the sensitivity of to variations in ( )oGs0α is given by 002(0.5)12.5 120GssSssα+=++ iii) Show that the sensitivity of to variations in ( )oGs1α is given by 120.512.5 120oGsSssα−=++ d) Using Matlab, simulate the unit step response of each type of controller. Plot all responses on one graph. Use different line types and a legend. Turn in your plot and code. g) Using Matlab and subplot, plot the sensitivity to 0α for each type of controller on one graph at the top of the page, and the sensitivity to 1α on one graph on the bottom of the page. Be sure to use different line types and a legend. Turn in your plot and code. Only plot up to about 8 Hz (50 rad/sec) using a semilog scale with the sensitivity in dB (see below). Do not make separate graphs for each system!In particular, these results should show you that the model matching method, which essentially tries and cancel the plant, are generally more sensitive to getting the plant parameters correct than the PI controller for low frequencies. However, for higher frequencies the methods are all about the same. Hint: If 22()210sTsss=++, plot the magnitude of the frequency response using: T = tf([2 0],[1 2 10]); w = logspace(-1,1.7,1000); [M,P]= bode(T,w); Mdb = 20*log10(M(:)); semilogx(w,Mdb); grid; xlabel('Frequency (rad/sec)'); ylabel('dB'); 2) Consider the following characteristic equations 232432() 1() 1() 1ssbsssbscss s bs cs dsΔ=++Δ=+ ++Δ=+ + ++ a) Show that for the 2nd order system we need for no RHP poles 0b > b) Show that for the 3rd order system we need , , and for no RHP poles 0b > 0c > 10bc −> c) Show that for the 4th order system we need , , and for no RHP poles 0b > 0c > 0d >220bcd d b−−> 3) For , 32() 2 2sss sΔ=+++ a) determine if there are any poles in the RHP b) if possible factor the characteristic equation and determine all of the poles 4) For , 432() 2 4 6 3ss s s sΔ=+ + ++ a) determine if there are any poles in the RHP b) if possible factor the characteristic equation and determine all of the poles5) Consider the following control system with plant 21()1pGsss=++ ()pGs()cGs+ - a) For the integral controller ()ckGss=, use the Routh array to show that there are no poles in the RHP for 0k 1<<. Verify your results using Matlab (either sisotool or the rlocus command). b) For the PI controller (()cks zGss)+=, with , show that for no RHP poles we must have and , or for we must have 0z >1z <0k >1z >101kz<<−. Determine the factors of . Verify your results using Matlab (either sisotool or the rlocus command). ()sΔ 6) Consider the following control system with plant 21()21pGsss=++ ()pGs()cGs+ -a) For the integral controller ()ckGss=, use the Routh array to show that there are no poles in the RHP for 0k 2<<. Verify your results using Matlab (either sisotool or the rlocus command). b) For the PI controller (()cks zGss)+=, with , show that for no RHP poles we must have02 and , or for we must have 0z >z<< 0k >2z >202kz<<− Determine the factors of ()sΔ. Verify your results using Matlab (either sisotool or the rlocus
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