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Figure 1 Problem 3 For the system shown in the above figure, where G(s) a) Draw a Bode diagram of the open-loop transfer function G(s) when K 10. b) On your plot, indicate the crossover frequencies, PM, and GM. Is the closed-loop system stable with K-10? c) Determine the value of K such that the phase margin is 30°. What are the gain margin and the crossover frequencies with this K? Note: You can finish problems 2-3 with the help of MATLAB commands rlocus, bode, and margin

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Solution in a Giren open doop transfer fination for Kalo GIS) = K s(sht st4) = 10 s(s?tst4) - Assajo 8 Gejows Juo (G. 10° + jAs Gain Margin = -1.958 dB (Negative) * 1 To obiain Gain cross-over freprency - Equating Gepool to f. magnitude of - w Jt way12-609 881 5-90 - toring and ope) = -90 - [188 - Toni (.507] Lulja) = -98-189 + 42.90 P.M = 180+ Laywon lw=wige rispe = 10x +Sketching the Bode prot- IMde 27.9408 10dB < Magnitude peat 1 .000 20-621 : 39-4a13 01 2 26095 To yw 1-okupa Zurpos. ♡ Phase| eqvwtx » ਲੰਮਣ -- 1st ਜੇ -- Jime ( = 150° > $( ) : 60 : 3 9 ਚ 4 - JS = 9 ੦ ੧ + 0 ੧ • + - 4 0 ( solicy - 0 = 179- 6 -2,04 (00 K = 288 o ke 3146411 To achiene P.M = 300 value of k is equal to 3.4641 Now, G.M = doleg coin law As, Wee is independentMATLAB CODE clc clear all close all syms s K=10 G=tf([O O O K],[1 1 4 0]) % Defining the open loop transfer function bode (G)Rode Diagram System: G Gain Margin (dB): -7.96 At frequency (rad/s): 2 Closed loop stable? No Magnitude (dB) Phase (deg) - -Bode Diagram -.-.-.-.-.-.-.-.- - - - - - - - - System: G2 Gain Margin (dB): 1.25 At frequency (rad/s): 2 Closed loop stable?

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