9. Frequency-Domain Analysis
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1 9.1 Sinusidal resnses 9. Frequency-Dmain Analysis We give meaning t the steady-state resnse f systems t sinusidal inuts, which is called as the frequency resnse. Fr the resnse, the transfer functin is evaluted as siω. r(t Acsω t R(s r (t c (t st e H(s st H(se C(s r(t e c(t iωt He iωt r(t c(t Re Re { i ω e t } { } He iωt r(t Acsω t (t A H cs( ω t + ϕ H (i ω c DEU-MEE 5017 Advanced Autmatic Cntrl Assc.Prf.Dr.Levent Malgaca, Fall 2015
2 Examle 9.1: MATLAB: Find the resnse fr the cnsidered system t an inut f 7cs3t. G(s R(s C(s 5 s + 5 s3*i; ng5;dg[1 5]; giwlyval(ng,s / lyval(dg,s; gaabs(giw,gfiangle(giw MATLAB gives the results as ga , gfi r(t7cs3t c(t7*0.8575cs(3t cs(3t
3 Examle 9.1: (Cntinue We can lt the sinusidal inut and resnse by MATLAB. clc;clear; a7;w3; sw*i; ng5;dg[1 5]; giwlyval(ng,s / lyval(dg,s; gaabs(giw,gfiangle(giw t2*i/w;dtt/20; t0:dt:5*t; r7*cs(w*t; lt(t,r;hld n; ca*ga*cs(w*t+gfi; lt(t,c,'r';
4 Examle 9.2: R(s G(s 2 C(s s MATLAB: Find the resnse fr the cnsidered system t an inut f 7cs3t * 0.5* 5s + 5 clc;clear; a7;w3; ksi0.5;wnsqrt(5; sw*i; ng5;dg[1 2*ksi*wn wn^2]; giwlyval(ng,s / lyval(dg,s; gaabs(giw,gfiangle(giw t2*i/w;dtt/20; t0:dt:5*t; r7*cs(w*t; lt(t,r;hld n; ca*ga*cs(w*t+gfi; lt(t,c,'r'; r(t7cs3t c(t7*0.5249cs(3t
5 Examle 9.2: (Cntinue ω 3 ω 5 ω 2
6 9.2 Frequency resnses The frequency resnse has meaning far beynd the calculatin f the time resnse t sinusids. We define the frequency resnse functin t be a the functin G fr 0 ω. Fr a given value f ω, G is a cmlex number. Thus the functin G is a cmlex functin. Cnsider the frequency resnse f systems (Examle Cnsider Examle 9.1. Plt the frequency resnse f the system. 5 G(s MATLAB: s + 5 Examle 9.3: clc;clear; t1/5; wb1/t; w0:0.05:10*wb; sw*i; ng5;dg[1 5]; giwlyval(ng,s./ lyval(dg,s; magabs(giw;fiangle(giw; sublt(2,1,1;lt(w,mag; sublt(2,1,2;lt(w,fi*180/i;
7 Examle 9.4: Recnsider Examle 9.2. Plt the frequency resnse f the system. G 2 (s s 5 + 2* 0.5* 5s + 5 MATLAB: clc;clear; ksi0.5;wnsqrt(5; w0:0.05:3*wn; sw*i; ng5;dg[1 2*ksi*wn wn^2]; giwlyval(ng,s./ lyval(dg,s; magabs(giw;fiangle(giw; sublt(2,1,1;hld n;lt(w,mag,'k'; sublt(2,1,2;hld n;lt(w,fi*180/i,'k';
8 9.3 Bde diagram Bde diagram is a lt f magnitude f the l transfer functin KG in decibels and the hase f KG in degrees, all versus frequency ω. The stability f the clsed l system can be determined by bserving the behaviur f these lts. R(s + E (s C(s K G (s H(s KG 1 + KG (s (s 1 + KG (s 0 KG (s 1 s iω KG 1 G KG 1 Magnitude: Phase : G ϕ G KG Bde lts M 20lg G 20lg KG 20lg(K + 20lg(G [db]
9 Examle 9.5: R(s + E (s C(s K G (s 8 G(s 2 s(s + 5(s + 2s + 18 MATLAB: clc;clear;clse all w0:0.05:1000; k1;ng8;dgcnv([1 5 0],[1 2 18]; dhlyadd(k*ng,dg;rts(dh sw*i; hiwlyval(k*ng,s./ lyval(dg,s; magabs(hiw;fiangle(hiw; bde(k*ng,dg %hld n;lt(w,20*lg10(mag,'k'; %return figure(2 sublt(2,1,1;hld n;lt(lg10(w,20*lg10(mag,'k'; sublt(2,1,2;hld n;lt(lg10(w,fi*180/i,'k';
10 9.4 Stability analysis: (Gain margin, hase margin M(dB M 20lg G ϕ G Magnitude lt Phase lt 0 a ϕ G M lg ω Gain margin (G M : The difference (in db between 0 db and system gain, cmuted at the frequency where the hase is b 180 P M lg ω Phase margin (P M : The difference (in between the system hase and 180 ο, cmuted at the frequency where the gain is 1 (lg A system is stable if the gain and hase margins are sitive. G M P M 0 ( a b ( 180
11 Margin calculatins: KG lane Im Gain margin (G M : Im G 0 ω ω a GM : Phase crssver freq. [rad / s] ReG 1 a ReG 1 (i ω 1 a ϕ M Re G M 1 20lg ReG G M 20lg1 20lg ReG G M 20lg ReG [db] Nyquist lt Phase margin (P M : G 1 ω g ω g : Gain crssver freq. [rad / s] G tan 1 ImG ReG g g [deg ree]
12 Examle 9.6: Obtain Bde lts, find the gain and hase margins by MATLAB. R(s + MATLAB: E (s C(s K G (s clc;clear;clse all k1;ng2500;dgcnv([1 0],cnv([1 5],[1 50] ; gtf(k*ng,dg bde(g margin(g [gm,m,w,wg]margin(g K: Cntrl gain 2500 G (s s(s + 5(s + 50 MATLAB gives these results, gm5.5, m , wg rad/s, w rad/s K c K 5.5 K c K 5 c 5. G M 20lg( db
13 Examle 9.7: Recnsider Examle 9.6. Calculate the gain and hase margins.
14 9.5 Cmansatrs (Design in frequency dmain G M and ϕ M can be chsen as G M 6 db and P M 65 t design in frequency-dmain. Phase-lead cmanstr: Relatinshi between the hase margin and daming rati: P M (deg ξ 100 s + z G cm (s s + Imrve daming rati Imrve bandwith (faster resnse a 1 Increase hase margin sinϕ m a ats G cm (s a > Ts ωm at Phase-lag cmanstr: Imrve steady-state errr Decrease hase margin 1 (1 + ats G cm (s a (1 + Ts a < 1 s + z G cm (s s + R(s + P z > z < M 2 tan 1 ϕ ϕz ϕ s lane 1 T 2ξ 2 1 at 2ξ ξ K G (s G cm (s iω E (s C(s 2
15 Examle 9.8: Recnsider Examle 9.6. Design a cmensatr system and simulate results by MATLAB in time and frequency dmain with/withut the cmensatr.
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