ECE-320 Linear Control SystemsHomework 3Due Date: Tuesday September 21Problems:1 For the following transfer functions:a) H(s) =s2+ 3s4+ 3s3+ 6s2+ 7s + 3b) H(s) =3s4+ 3s3+ 6s2+ 7s + 3c) H(s) =43s + 5s4+ 14s3+ 44s2+ 43s + 5d) H(s) =s2+ s + 0.0125s5+ 1.75s4+ 1.475s3+ 0.625s2+ 0.1375s + 0.01251) Using the pole command in Matlab, find the poles.2) Estimate the settling time (4 time constants) Complex poles generally have the form −1/τ ±jωd= −σ ± jωdwhere τ is the time constant. Compute the settling time associated with eachpole. The system settling time is largest settling time.3) The pole(s) associated with the largest settling time is (are) the dominant poles. Determinethe dominant poles for each system.4) Plot the step response of the system using Matlab for 30 seconds. Use the subplot commandto plot all four graphs on one page. Using the formulas derived in class estimate the positionerror for each system and identify the position error on the graphs (draw on the graph to showyou know what the position error is.)5) Plot the ramp response of the system using Matlab for 30 seconds. Use the subplot commandto plot all four graphs one one page. Using the formulas derived in class estimate the velocityerror for each system and identify the velocity error on the graphs (draw on the graph to showyou know what the velocity error is).Turn in your Matlab plots for each system!Ans. Ts= 8, 8, 29.9 and 16. One of these is not a very good approximation. The formula for Ts(Ts= 4τ = 4 /σ) assumes an ideal second order system. It is just a guideline., ep= 0 for all.ev= 2.333, 2.333, 0, 69.02 For the following transfer functions:I) determine the characteristic polynomialII) determine the characteristic modesIII) determine if the system is stable, unstable, or marginally stable.a) H(s) =s + 1(s)(s + 2)(s + 10)b) H(s) =(s − 1)(s + 2)(s + 1)2(s2+ s + 1)c) H(s) =1s2(s + 1)d) H(s) =s2− 2(s − 1)(s + 2)(s2+ 1)e) H(s) =1(s2+ 2)(s + 1)3 For each of the following groups of poles for a given transfer function, determine the form ofthe corresponding impulse response (time domain function) :a) −1, −2, −5 ± 3jb) 1, −2 ± 2j, −1 (repeated)c) 0 (repeated), −1 ± j (repeated)(If a pole is repeated, it is a double pole)4 For the following systems, determine the value of Gpfso that the position error of the closedloop system is zero. (Hint: Think of the easiest way to do this, you never need to write out theclosed loop transfer function as the ratio of two polynomials. You most definitely do not needMaple, and will not have Maple on an exam.)Then simulate each system (using Matlab) for astep resp onse. Run each simulation until the system comes to steady state. Use subplot to putall four plots on one sheet of paper (you may want to type orient tall so more of the page isused.R(s)-Gpf-±°²¯-s+2s+1-1s+3-Y (s)6+-R(s)-Gpf-±°²¯-s + 2-5s2+2s+3-Y (s)6+-R(s)-Gpf-±°²¯-1s-5s2+2s+3-Y (s)6+-R(s)-Gpf-±°²¯-10s-s+2s2+2s+10-Y (s)46+-Ans. Gpf=52, 1.3, 1, 45 Assume we have the unity feedback system shown below, which models the one degree of free-dom systems you have been using in the lab. Gc(s) is a controller that you will be implementing,ωnand ζ are estimates obtained using either time-domain or frequency domain methods, andKclgis the system closed loop gain. Note that we are modelling the motor as contributing onlya gain to the system, and we are lumping both the plant’s gain and the motor gain togetherinto one parameter.-½¼¾»-Gc(s)-Kclg1ω2ns2+2ζωns+1-6+-Show that for a proportional controller,Gc(s) = kp,the steady state output yssdue to a step input of amplitude Amp is given byyss=Amp kpKclg1 + kpKclgwhich can be rewritten asKclg=ysskp1Amp − yssThis is the expression we will use in lab to estimate the closed loop system gain Kclg.6 Consider the following system, with plant Gp(s) =1s+1and controller Gc(s).-½¼¾»-Gc(s)-1s+1-6+-a) Determine the controller so that the closed loop system matches a second order zero p ositionerror ITAE optimal system, i.e., so that the closed lo op transfer functions isG0(s) =ω20s2+ 1.4ω0s + ω20Ans. Gc(s) =ω20(s+1)s(s+1.4ω0). Note that there is a pole/zero cancellation between the controller andthe plant.b) Determine the controller so that the closed loop system matches a second order zero velocityerror ITAE optimal system, i.e., so that the closed lo op transfer functions isG0(s) =3.2ω0s + ω20s2+ 3.2ω0s + ω20Ans. Gc(s) =(3.2ω0s+ω20)(s+1)s2. Note that there is a pole/zero cancellation between the controllerand the plant.c) Simulate the step response system in part (a) for ω0= 10, 20, 40 and 80 for 1 second. Usesubplot to put all four graphs on one sheet of
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