EE 4343/ Control System Design Project LECTURE 10


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1 Copyright S. Ikenaga 998 All rights reserved EE 4343/ Control System Design Project LECTURE EE 4343/5329 Homepage EE 4343/5329 Course Outline Design of Phaselead and Phaselag compensators using Bode Plot Method. Phaselead compensator design using Bode Plot Method =, such that the steadystate error is less than. for a unit ramp input and a % overshoot less than s ( s ) 25%. Goal: Design a phaselead compensator for the system G ( s) Steadystate error specification K KV = lim sg( s) = lim s = K s s s s e ss = K V = <. K ( ) K % overshoot specification Recall the relationship between % overshoot and damping ratio (ζ ) which is given by and is shown in Figure. % Overshoot = e ζπ ζ 2  
2 Figure. % Overshoot vs. Damping Ratio % Overshoot damping ratio ζ Then, the relationship between phase margin (PM) and damping ratio (ζ ) for the special case of openloop transfer function G( s) PM = tan 2 n ω = which is given by s ( s + 2ζω ) 2ζ = 4ζ 2 n 2ζ maintains that the phase margin of the system should be greater than obtain a percent overshoot less than 25% and is shown in Figure 2. 2 o 45 to Phase Margin (PM) damping ratio ζ Figure2. Phase Margin vs. Damping Ratio
3 Phaselead design procedure: i.) ii.) Choose the DC gain constant K such that the steadystate error specification is met. From above, we know K must be greater than or equal to, so let K =. Obtain the gain margin and phase margin plots of the system along with the DC gain constant K found in (i.) to determine the amount of phase lead φ m needed to realize the required phase margin so that the percent overshoot specification is met. Bode Diagrams 6 Gm = Inf, Pm=7.964 deg. (at rad/sec) 4 2 Phase (deg); Magnitude (db) Frequency (rad/sec) Figure 3. Bode plot of system G( s) K. From Figure 3., the PM of the system PM of the system as o PMuncomp 2. Thus, choosing the o PM comp = 45, then the additional amount of phase lead o φ m = PM comp PMuncomp = 25. Now that φ m has been determined, the parameter α of the phaselead compensator can be chosen from Figure 2.4 in Appendix A, which has been chosen to be α =. 3 which corresponds to a maximum phase lead of 33. iii.) The maximum phase lead φ m must be added around the new gaincrossover frequency ω m. The phaselead compensator contributes a gain around log(.3) = 5.2dB at the new ω m ; therefore, one must determine the frequency at which the system has a magnitude log(.3) = 5.2dB. Thus, ωm should equal this frequency so that it becomes the new db crossover frequency in the system. From inspection of Figure 3, the magnitude of the system equals 5.2dB at the frequency ω = 4.5 rad sec. Let ω = 4.5 rad sec. m iv.) Calculate the parameters of the phaselead compensator based on the values obtained in steps (i.) thru (iii.). The transfer function of a phaselead compensator is given as s T jωt C( s) = or C( jω ) = with α < α s αt jωαt o  3 
4 where T =. Thus, for α =. 3, T =.4sec. This leads to a phaselead ω m α compensator design of the following:.4s C( s) =.23s Phaselead compensator simulation results: Matlab Simulation clear all; wm = 4.5; % gaincrossover frequency alpha =.3; % phaselead compensator parameter T = /wm/sqrt(alpha); % phaselead compensator time constant K = ; % DC compensator gain % Phaselead compensator C(s) cnum = K*[T ]; cden = [T*alpha ]; % Openloop sys G(s) gnum = []; gden = [ ]; % UnityGain Feedback Loop H(s) hnum = []; hden = []; % Openloop sys C(s)*G(s) numo = conv(cnum,gnum); deno = conv(cden,gden); % Closedloop sys [gnumc,gdenc] = feedback(k*gnum,gden,hnum,hden,); [numc,denc] = feedback(numo,deno,hnum,hden,); bode(cnum,cden);  4 
5 Bode Diagrams 3 25 Phase (deg); Magnitude (db) Frequency (rad/sec) Figure 4. Bode plot of phaselead compensator C ( s)
6 sys = tf(k*gnum,gden); sys2 = tf(numo,deno); [mag,ph,w]=bode(k*gnum,gden,logspace(,2,5)); [mag2,ph2,w]=bode(numo,deno,logspace(,2,5)); subplot(2); semilogx(w,2*log(mag),'r', w,2*log(mag2),'b'); title('bode Diagrams'); ylabel('magnitude (db)'); legend('','', ); subplot(22); semilogx(w,ph,'r', w,ph2,'b'); ylabel('phase (deg)'); xlabel('frequency (rad/sec)'); legend('','', ); 5 Bode Diagrams Magnitude (db) Phase (deg) system system 82 Frequency (rad/sec) Figure 5. Bode plots of and systems. figure; sysc = tf(gnumc,gdenc); sys2c = tf(numc,denc); step(sysc,sys2c);grid; legend('','',);  6 
7 Step Response Amplitude Time (sec.) Figure 6. Step response of and system
8 t=:.:5; y = t; [y,x]=step(gnumc,conv(gdenc,[ ]),t); [y2,x2]=step(numc,conv(denc,[ ]),t); [y3,x3]=step(numc,denc,t); [y4,x4]=step(gnumc,gdenc,t); plot(t,y,'r',t,y2,'b',t,y,'g');grid; xlabel('time (sec)'); title('unit Ramp Input response'); legend('', '', 'desired',); 5 Unit Ramp Input response desired Time (sec) Figure 7. Response of and systems due to unit ramp input
9 2. Phaselag compensator design using Bode Plot Method =, such that the steadystate s ( s ) error is less than. for a unit ramp input and a percent overshoot less than 25%. Goal: Design a phaselag compensator for the system G ( s) Steadystate error specification As computed in (.), K. Percent overshoot specification o As obtained in (.), PMcomp 45. Phaselag design procedure: i.) ii.) Choose the DC gain constant K such that the steadystate error specification is met. From above, we know K must be greater than or equal to, so let K =. Obtain the gain margin and phase margin plots of the system along with the DC gain constant K found in (i.) to estimate the frequency at which the PM of 5 occurs. Denote this frequency as the new gaincrossover frequency ω. From Figure 8., let ω =.84 rad sec. m m o Bode Diagrams 6 Gm = Inf, Pm=7.964 deg. (at rad/sec) 4 2 Phase (deg); Magnitude (db) ω m 8  Frequency (rad/sec) Figure 8. Bode plot of system G( s) K
10 iii.) Determine the magnitude of system at ω =.84 rad sec. From Figure 8., the magnitude of the system at ω =.84 rad sec is 2 db. To bring the magnitude curve down to db at ( ) 2 db 2 2 ω m m m, the phaselag compensator must provide 2 log α = or α = =. iv.) Calculate the parameters of the phaselag compensator based on the values obtained in steps (i.) thru (iii.). The transfer function of a phaselag compensator is given as s T jωt C( s) = or C( jω ) = with α > α s αt jωαt where T =.9 sec. This is to ensure that the frequency at ω = is one decade ω T = m below the new gaincrossover frequency ω m. This leads to a phaselag compensator design of the following:.9s C ( s) =. 9s Phaselead compensator simulation results: Matlab Simulation clear all; wm =.84; % gaincrossover frequency alpha = ; % phaselag compensator parameter T = /wm; % phaselead compensator time constant K = ; % DC compensator gain % Phaselead compensator C(s) cnum = K*[T ]; cden = [T*alpha ]; % Openloop sys G(s) gnum = []; gden = [ ]; % UnityGain Feedback Loop H(s) hnum = []; hden = []; % Openloop sys C(s)*G(s) numo = conv(cnum,gnum); deno = conv(cden,gden); % Closedloop sys [gnumc,gdenc] = feedback(k*gnum,gden,hnum,hden,); [numc,denc] = feedback(numo,deno,hnum,hden,);  
11 bode(cnum,cden); Bode Diagrams 2 5 Phase (deg); Magnitude (db) Frequency (rad/sec) Figure 9. Bode plot of phaselag compensator C ( s).  
12 sys = tf(k*gnum,gden); sys2 = tf(numo,deno); [mag,ph,w]=bode(k*gnum,gden,logspace(,2,5)); [mag2,ph2,w]=bode(numo,deno,logspace(,2,5)); subplot(2); semilogx(w,2*log(mag),'r', w,2*log(mag2),'b'); title('bode Diagrams'); ylabel('magnitude (db)'); legend('','', ); subplot(22); semilogx(w,ph,'r', w,ph2,'b'); ylabel('phase (deg)'); xlabel('frequency (rad/sec)'); legend('','', ); 5 Bode Diagrams Magnitude (db) Phase (deg) system system 82 Frequency (rad/sec) Figure. Bode plots of and systems
13 sysc = tf(gnumc,gdenc); sys2c = tf(numc,denc); step(sysc,sys2c);grid; legend('', '',); Step Response Amplitude Time (sec.) Figure. Step response of and system
14 t=:.:5; y = t; [y,x]=step(gnumc,conv(gdenc,[ ]),t); [y2,x2]=step(numc,conv(denc,[ ]),t); [y3,x3]=step(numc,denc,t); [y4,x4]=step(gnumc,gdenc,t); plot(t,y,'r',t,y2,'b',t,y,'g');grid; xlabel('time (sec)'); title('unit Ramp Input response'); legend('', '', 'desired',); 5 Unit Ramp Input response desired Time (sec) Figure 2. Response of and systems due to unit ramp input
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