Design of Control Systems for Compensation

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Detailed discussion on the design of control systems for both incremental and retroactive compensation, with examples of related problems. Exploration of phase-lead and phase-lag compensation, analyzing the impact on polar plots and Bode diagrams to differentiate between phase-lead and phase-lag systems.

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KONTROL SSTEMLER TASARIMI lerlemeli Kompenzasyon Gerilemeli Kompenzasyon lerlemeli-Gerilemeli Kompenzasyon rnek Problemler 23.02.2025 Dr. Nurdan Bilgin 20210-2021 1

Faz ilerlemeli kompenzasyon Faz ilerlemeli kompenzat r n genel formu ? +1 ? +1 ?? + 1 ??? + 1= ?? ? ??? = ??? , (0 < ? < 1) ?? Yandaki ekil Kc= 1 olmak zere ??? + 1 ???? + 1 ???? = ??? Sin zoidal transfer fonksiyonunu kutupsal grafi ini g stermektedir. G r ld gibi ??maksimum faz a s n g stermektedir. 1 ? =1 ? 1 + ? 2 sin??= 1 + ? 2 23.02.2025 Dr. Nurdan Bilgin 2020-2021 2

Gerilemeli Kompenzasyon Gerilemeli kompanzat r n genel formu a a daki gibiydi. ? +1 ? +1 ?? + 1 ??? + 1= ?? ? ??? = ??? , (? > 1) ?? 23.02.2025 Dr. Nurdan Bilgin 20210-2021 3

Faz Gerilemeli vsFaz lerlemeli Faz lerlemeli Faz Gerilemeli Faz gerilemeli ve faz ilerlemeli sistemlerin kutupsal yer e rileri birbirinin simetri idir. 23.02.2025 Dr. Nurdan Bilgin 20210-2021 4

Faz Gerilemeli vsFaz lerlemeli Faz lerlemeli Faz Gerilemeli Faz gerilemeli ve faz ilerlemeli kompanzat rlerin bode diyagramlar na bak ld nda faz gerilemeli kompanzat r n al ak ge iren, faz ilerlemeli kompanzat r n ise y ksek ge iren filtre oldu u g r l r. 23.02.2025 Dr. Nurdan Bilgin 20210-2021 5

FAZ GERLEMEL-LERLEMEL KOMPENZASYON ekilde g sterilen ekilde ba lanm a a daki faz gerilemeli-ilerlemeli kompenzat r kulland m z varsayal m: ? +1 ? +1 ?1 ?2 1 ??2 ??= ?? ? +? ? + ?1 Bu denklemde ? > 1 ve ? > 1. (??sabitinin faz gerilemeli-ilerlemeli kompenzat r n ilerletici k sm na ait oldu unu d n n z) Faz gerilemeli-ilerlemeli kompenzat r n tasar m n bu b l mde ? = ? olma durumu i in inceleyece iz. 23.02.2025 Dr. Nurdan Bilgin 20210-2021 6

FAZ GERLEMEL-LERLEMEL KOMPENZASYON ? > 1 ve ? > 1 art ? +1 ? +1 ?1 ?2 1 ??2 ??= ?? ? +? ? + ?1 A a daki terim. ? +1 =1 ?1? + 1 ?1 ?? + 1 ?1 (? > 1) ? +? ? ?1 Faz ilerlemeli devrenin etkilerini retir. Faz gerilemeli devrenin etkilerini ise a a daki terim retir. ? +1 ?2 1 ??2 ?2? + 1 ??2? + 1 = ? ? + 23.02.2025 Dr. Nurdan Bilgin 20210-2021 7

FAZ GERLEMEL-LERLEMEL KOMPENZASYON Faz gerilemeli-ilerlemeli kompanzat r; 0 < ? < ?1i in faz gerilemeli kompanzat r gibi ?1< ? < i in faz ilerlemeli kompanzat r gibi davranmaktad r. 1 Grafikler ??= 1,? = ? = 10 ?? ?2= 10?1i in izilmi tir. ?1= ?1?2 23.02.2025 Dr. Nurdan Bilgin 20210-2021 8

rnek 23.02.2025 Dr. Nurdan Bilgin 20210-2021 9

[val,idx]=min(abs(magdb-(-GME))); minValGME=magdb(idx); minw=w(idx); IT=minw*sqrt(alfa);ITa=IT/alfa;Kc=K/alfa; numc=Kc*[1 IT];denc=[1 ITa]; sysc=tf(numc,denc);sysp=tf(num,den); sys_son=sysp*sysc; figure; bode(sys_son); [Gms,Pms,Wgms,Wpms] = margin(sys_son); GmdB = 20*log10(Gms); [GmdB,Pms,Wgms,Wpms] sysun_CL=feedback(sysp,1); sysc_CL=feedback(sys_son,1); t = 0:0.01:6; figure; [c1,x1] = step(sysun_CL,t); [c2,x2] = step(sysc_CL,t); plot(t,c1,'.',t,c2,'-');legend('Uncompensated System','Compensated System') grid; title('Unit-Step Responses of Uncompensated System and Compensated System') xlabel('t Sec'); ylabel('Outputs') sysint=tf(1,[1 0]); figure; [c1,x1] = step(sysint*sysun_CL,t); [c2,x2] = step(sysint*sysc_CL,t); plot(t,c1,'.',t,c2,'-',t,t,'--'); grid; title('Unit-Ramp Responses of Uncompensated System and Compensated System'); xlabel('t Sec'); ylabel('Outputs') legend('Uncompensated System','Compensated System') clear all;clc;close all; %%Spesifikasyonlar kv=10; % PMC=25; GMC=8; a=5; %Transfer fonksiyonu nums=a*[1 0];num=a; den=conv(conv([1 1],[0.1 1]),[1 0]); %den=[1 1 0] %bode(num,den); sys=tf(nums,den); [Num,Den] = tfdata(sys,'v'); syms s sys_syms=poly2sym(Num,s)/poly2sym(Den,s); lim = limit(sys_syms,s,0);lim=double(lim); K=kv/lim;num1=K*a; sys1=tf(num1,den); w = logspace(-2,3,1000); figure; bode(sys1,w); [mag,ph]=bode(sys1,w); [Gm,Pm,Wgm,Wpm] = margin(sys1); ang=PMC-Pm+10; syms alf eq=sin(ang*pi/180)*(1+alf)-(1-alf); alfa=vpa(solve(eq),4);alfa=double(alfa); GME=1/sqrt(alfa);GME=20*log10(GME); for i=1:length(w) magdb(i)=20*log10(mag(i)); phaseA(i)=ph(i); end 23.02.2025 Dr. Nurdan Bilgin 20210-2021 10

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[val,idx]=min(abs(magdb-(abs(20*log10(Gm))+21)));%magdb-(20*log10(Gm)+20)[val,idx]=min(abs(magdb-(abs(20*log10(Gm))+21)));%magdb-(20*log10(Gm)+20) minValGME=magdb(idx); minw=w(idx);minw=round(minw,1); IT=minw;ITa=IT*B numc=conv([1 IT2],[1 IT]); denc=conv([1 ITB2],[1 ITa]); figure; bode(numc,denc); title('Faz Gerilemeli- lerlemeli Kompanzat r n Bode Diyagram ') figure; bode(sys1,w);hold on; sys2=tf(conv(numc,num),conv(denc,den)) bode(sys2,w);legend('sys1_uncompan.','sys2_comp.'); [Gm,Pm,Wgm,Wpm] = margin(sys2) sysc_CL=feedback(sys2,1); t = 0:0.01:20; figure; [c2,x2] = step(sysc_CL,t); plot(t,c2,'-'); grid; title('Unit-Step Responses of Compensated System') xlabel('t Sec'); ylabel('Outputs') sysint=tf(1,[1 0]); figure; [c2,x2] = step(sysint*sysc_CL,t); plot(t,c2,'-',t,t,'--'); grid; title('Unit-Ramp Responses of Compensated System'); xlabel('t Sec'); ylabel('Outputs') clear all;clc;close all; %%Spesifikasyonlar kv=20; % PMC=40; GMC=5; %Transfer fonksiyonu nums=[1 0]; den=conv(conv([1 1],[1 6]),[1 0]); %den=[1 1 0] %bode(num,den); sys=tf(nums,den); [Num,Den] = tfdata(sys,'v'); syms s sys_syms=poly2sym(Num,s)/poly2sym(Den,s); lim = limit(sys_syms,s,0);lim=double(lim); K=kv/lim;num=K; sys1=tf(num,den); w = logspace(-1,1,1000); [mag,ph]=bode(sys1,w); [Gm,Pm,Wgm,Wpm] = margin(sys1);i=1;B=10;Qm=1; while Qm<PMC Qm=asin((B-1)/(B+1))*180/pi; end IT2=Wgm/10;ITB2=IT2/B; for i=1:length(w) magdb(i)=20*log10(mag(i)); phaseA(i)=ph(i); end 23.02.2025 Dr. Nurdan Bilgin 20210-2021 16