7:34 This lab aims to design proportional controller for certain specifications using root locus and use to study the st

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7:34 This lab aims to design proportional controller for certain specifications using root locus and use to study the stability of systems. The root locus of an open-loop transfer function His is a plot of the locations locus) of all possible closed loop poles with proportional gain K and unity feedback The closed-loop transfer function is Yix) KINA) 71-KUC and thus the poles of the closed-loop poles of the closed-loop system are values of such that 1 KH-0. If we write so-bisi/ais), then this equation has the form et n-order of acal and moeder of bol the order of a polynomial is the highest power of that appears in it). We will consider all positive values of K. In the listit an-9, the poles of the cloand-loop system are ais-0 or the poles of Hist. In the limit as the poles of the closed-loop system are bisi-0 or the aeros of His No matter what we pick K to be, the closed-loop system must always have spoles, where n is the number of poles of His). The root locus must have n branches, each branch starts at a pole of His) and goes to a zero of Hist. If He has more poles than zeros mand we say that His has zeros at infinity. In this case, the limit of His as a 00 is zero. The number of acros at infinity is nm, the number of poles minus the number of seros, and is the number of tranches of the root locus that go to infinity (asymptotes). Since the mot locus is actually the locations of all possible closed-loop poles, from the root locus we can select a in such that our closed-loop system will perform the way we want. If any of the selected pales are on the right half plate, the closed-loop system will be unstable. The poles that are

7:32 & closest to the imaginary sods have the greatest influence on the closed-loop responser, so even though the system has three or four poies, it may will act like a second or even first order system depending on the locations of the dominant pol in Figure shows the transient repustwieria of second order system for a unit step. It is possible to design controller st certain percent overshoot IPO and peak time using rostlocus. PO and the are functions of the ming and Dequency 10-10-11-2 (11) and (13) 1.1 Required Matish Commands help functionname gives detailed information rootsinum-find polynomial roots rlocus-calculates and plot the locus of the roots of 1-0 K.polca-ricindy-find the gain and its corresponding poles for a given roets locus sgrifa, Wajdraw lines boundaries of art and W feedback.t-computes a closed-loop model 0 for the feedback loop 2.1 Finding Roots and Root Locus For the given following transfer functions, calculate poles and serse and show..codes. results spoles and acros and comment on the stability of system in the banks. For each plot the root forus. Huse roots, t) and rious functional 2 of 5

7:32 & Pules and [5] 2²+4² +6+1 G() 4+2 s²+2+1 GO)=- 2.2. A feedback control system with certals design criteria Consider an open-loop system which has a transfer function of ru Design a feedback control system with proportional (gain KS 150 less than 55 and s than arc Follow the steps given below Open a new serie in Matlab and use sts) to write the transfer function Hariable named sys. Mat root locus. Lastly, limit your ass using ania 22.3-15 15). Write your code in the following Man After you run your codes, you will have the plot given below To satisfy the design criteria Find using equation 1.1 in order to have a percent overshoot PO less than P 3 of 5 Code Subdy

7:32 & Find using gantion 1.2 in order to have a peak time than According to your slutions write appropriata ) and wnm) in towing Malah code and add it to the script. You will see the following plat zeta- was grid(Zeta) on the plot you get, the two dimed lines indicate pale locations with your in between these lines, the poles will have higher than & The semicircle indicates pole locations with a natural frequency a Going back to our problem, to make the overshoot less than 5%, the poles have to be in between the two white dotted lines. And to make the peak time shorter than 1 second the poles have to be outside of the white dotted semicinde. So now we know only the part of the locus outside of the semicircle and in between the two lines are acceptable. At the poles in this location are in the left-half plane, so the closed will be stable We need a proportional controller to save the pales to the desired regen. Use the rociad command in MATLAB to choose the desired poles on the locus. Click on the plot the point where you want the closed-loop pole to be. Select the points indicated in the pittoy the design criteris K.pl find In order to find the wep response, you need to know the closed-loop transfer function. Write Kinte code below and apply to your sm Love any c NO 6

7:33 & X- feedback stepays l Prove your design with indink using transfer function Mock to construct and apply step ingut. Show the block diagram and output