Design & Compensation via Root Locus [email protected] 1 Contents • Compensation – Lead Compensation – Lag Compensation – Lag-Lead Compensation Compensation • Used to Improve the performance of stable or unstable systems. Compensator Gc(S) Plant G(S) H(S) Lead Compensation • Used to Improve the Transient behavior of the system. 1 (S ) T Gc ( S ) K c 1 (S ) T where 1 Compensator Plant G(S) Gc(S) H(S) Lead Compensation • Consider the Following Unity feedback system R (S ) k=4 p=[0 -2 ] z=[]; usys=zpk(z,p,k) rlocus(usys) csys=feedback(usys,1) [wn b]=damp(csys) sgrid(b,wn) axis([-3 0 -3 3 ]) 4 S ( S 2) C (S ) Lead Compensation contd… Design Requirements • It is desired to increase ωn to 4 rad/sec without changing damping ratio of closed loop poles. R (S ) k=4 p=[0 -2 ] z=[]; usys=zpk(z,p,k) rlocus(usys) wn=[2 4]; b=[0.5 0.5]; sgrid(b,wn) axis([-5 0 -5 5 ]) 4 S ( S 2) C (S ) Lead Compensation contd… Lead Compensator (Follow the Class Work) ( S 2.9) Gc ( S ) 4.68 ( S 5.4) kc=4.68 pc=-5.4 zc=-2.9 lc=zpk(zc,pc,kc) ltiview(lc) Lead Compensation contd… Pzmap of Lead Compensator Pole-Zero Map 1 0.98 0.96 0.92 0.85 0.7 0.45 0.8 0.991 0.6 0.4 0.998 Imag Axis 0.2 0 5 4 3 2 1 -0.2 -0.4 0.998 -0.6 0.991 -0.8 0.98 -1 -5 0.96 -4 0.92 -3 0.85 -2 0.7 -1 0.45 0 Real Axis Lead Compensation contd… Bode plot of Lead Compensator Bode Diagram 14 Magnitude (dB) 13 12 11 10 9 8 20 Phase (deg) 15 10 5 0 -1 10 0 1 10 10 2 10 Frequency (rad/sec) Lead Compensation contd… Compensated Vs Uncompensated System Root locus of Compensated & uncompensated Systems ( S 2.9) G ( S )Gc ( S ) 18.7 S ( S 2)(S 5.4) 4 G( S ) S ( S 2) b=0.5; wn=[2 4]; ku=4 pu=[0 -2 ] zu=[ ]; usys=zpk(zu,pu,ku) k=18.7 p=[0 -2 -5.4] z=-2.9 csys=zpk(z,p,k) rlocus(usys, ‘g’,csys,’b’) Sgrid(b,wn) Lead Compensation contd… Compensated Vs Uncompensated System Root locus of Compensated & uncompensated Systems Root Locus 10 8 6 Uncompensated 4 compensated 4 0.5 2 Imag Axis 2 0 -2 2 -4 0.5 4 -6 -8 -10 -5 -4 -3 -2 -1 0 Real Axis Lead Compensation contd… Compensated Vs Uncompensated System Response of compensated & Uncompensated system G( S )Gc ( S ) 18.7 S 54.23 3 1 G( S )Gc ( S ) S 7.4S 2 29.5S 54.23 G(S ) 4 2 1 G ( S ) S 2S 4 numu=4; denu=[1 2 4]; numc=[18.7 54.23]; denc=[1 7.4 29.5 54.23]; csys=tf(numc,denc); usys=tf(numu,denu); ltiview(usys, ‘g:’,csys,’b’) Lead Compensation contd… Compensated Vs Uncompensated System Pzmap of Compensated & Uncompensated System Pole-Zero Map 4 3 Uncompensated compensated 2 Imag Axis 1 0 -1 -2 -3 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 Real Axis Lead Compensation contd… Compensated Vs Uncompensated System Step response of Compensated & Uncompensated System Step Response 1.4 1.2 Amplitude Uncompensated compensated 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 Time (sec) Lead Compensation contd… Compensated Vs Uncompensated System Bode plot of Compensated & Uncompensated System Bode Diagram 20 -20 -40 -60 -80 0 -45 Phase (deg) Uncompensated compensated Magnitude (dB) 0 -90 -135 -180 -1 10 0 1 10 10 2 10 Frequency (rad/sec) Lead Compensation contd… Compensated Vs Uncompensated System Steady State Error of Compensated & Uncompensated System According to final value theorem K v lim SG ( S ) S 0 Steady state error is given by In MATLAB 1 ess Kv limit(f(x),x,0) Returns the limit of a function f(x) as x 0. Lead Compensation contd… Compensated Vs Uncompensated System Steady State Error of Compensated & Uncompensated System clear 4 syms s G( S ) S ( S 2) numu=4 denu=s^2+2*s ( S 2.9) Gu=numu/denu G( S )Gc ( S ) 18.7 S ( S 2)(S 5.4) numc=18.7*s+54.23 denc=s^3+7.4*s^2+10.8*s Gc=numc/denc kvu=limit(s*Gu,s,0) kvc=limit(s*Gc,s,0) essu=1/kvu essc=1/kvc Lead Compensation contd… Exercise#1 Repeat the same tutorial using SISO Design tool. Exercise#2 Design another lead compensator for the same system and compare the results. End of tutorial You can download this tutorial from: http://imtiazhussainkalwar.weebly.com/control-system-design-and-analysis.html