# 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 Phase-lead and Phase-lag compensators using Bode Plot Method. Phase-lead 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 phase-lead compensator for the system G ( s) Steady-state 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 open-loop 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 Phase-lead design procedure: i.) ii.) Choose the DC gain constant K such that the steady-state 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 phase-lead 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 gain-crossover frequency ω m. The phase-lead 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 phase-lead compensator based on the values obtained in steps (i.) thru (iii.). The transfer function of a phase-lead 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 phase-lead ω m α compensator design of the following:.4s C( s) =.23s Phase-lead compensator simulation results: Matlab Simulation clear all; wm = 4.5; % gain-crossover frequency alpha =.3; % phase-lead compensator parameter T = /wm/sqrt(alpha); % phase-lead compensator time constant K = ; % DC compensator gain % Phase-lead compensator C(s) cnum = K*[T ]; cden = [T*alpha ]; % Open-loop sys G(s) gnum = []; gden = [ ]; % Unity-Gain Feedback Loop H(s) hnum = []; hden = []; % Open-loop sys C(s)*G(s) numo = conv(cnum,gnum); deno = conv(cden,gden); % Closed-loop 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 phase-lead 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 -8-2 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. Phase-lag compensator design using Bode Plot Method =, such that the steady-state s ( s ) error is less than. for a unit ramp input and a percent overshoot less than 25%. Goal: Design a phase-lag compensator for the system G ( s) Steady-state error specification As computed in (.), K. Percent overshoot specification o As obtained in (.), PMcomp 45. Phase-lag design procedure: i.) ii.) Choose the DC gain constant K such that the steady-state 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 gain-crossover 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 phase-lag compensator must provide 2 log α = or α = =. iv.) Calculate the parameters of the phase-lag compensator based on the values obtained in steps (i.) thru (iii.). The transfer function of a phase-lag 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 gain-crossover frequency ω m. This leads to a phase-lag compensator design of the following:.9s C ( s) =. 9s Phase-lead compensator simulation results: Matlab Simulation clear all; wm =.84; % gain-crossover frequency alpha = ; % phase-lag compensator parameter T = /wm; % phase-lead compensator time constant K = ; % DC compensator gain % Phase-lead compensator C(s) cnum = K*[T ]; cden = [T*alpha ]; % Open-loop sys G(s) gnum = []; gden = [ ]; % Unity-Gain Feedback Loop H(s) hnum = []; hden = []; % Open-loop sys C(s)*G(s) numo = conv(cnum,gnum); deno = conv(cden,gden); % Closed-loop 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 phase-lag 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 -8-2 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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