Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for...

transcript

Desired Bode plot shape

Ess requirement

Noise requirement

High low-freq-gain for steady state trackingLow high-freq-gain for noise attenuationSufficient PM near wgc for stability

Mid frequency

Desired Bode plot shape

High low freq gain for steady state trackingLow high freq gain for noise attenuationSufficient PM near wgc for stability

Low freq

High freq

Want high gain

Want low gain

Mid freq

Want sufficientPhase margin

Use low pass filters

Use PI or lag control

Use lead or PD control

PM+Mp=70

C(s) Gp(s)

21)(psps

zszsKsC

Controller design with Bode

From specs: => desired Bode shape of Gol(s)Make Bode plot of Gp(s) Add C(s) to change Bode shapeGet closed loop systemRun step response, or sinusoidal response

Mr and BW are widely used

Closed-loop phase resp. rarely used

Important relationships• Closed-loop BW are very close to wn

• Open-loop gain cross over wgc ≈ (0.65~0.8)* wn,

• When z <= 0.6, wr and wn are close

• When z >= 0.7, no resonance• z determines phase margin and Mp:

z 0.4 0.5 0.6 0.7

PM 44 53 61 67 deg ≈100z Mp 25 16 10 5 %

PM+Mp ≈70

Mid frequency requirements• wgc is critically important

– It is approximately equal to closed-loop BW– It is approximately equal to wn

• Hence it determines tr, td directly

• PM at wgc controls z– Mp 70 – PM

• PM and wgc together controls s and wd – Determines ts, tp

• Need wgc at the right frequency, and need sufficient PM at wgc

Low frequency requirements• Low freq gain slope and/or phase

determines system type• Height of at low frequency determine error

constants Kp, Kv, Ka• Which in turn determine ess

• Need low frequency gain plot to have sufficient slope and sufficient height

High frequency requirements• Noise is always present in any system• Noise is rich in high frequency contents• To have better noise immunity, high

frequency gain of system must be low

• Need loop gain plot to have sufficient slope and sufficiently small value at high frequency

Overall Loop shaping strategy• Determine mid freq requirements

– Speed/bandwidth wgc

– Overshoot/resonance PMd

• Use PD or lead to achieve PMd@ wgc

• Use overall gain K to enforce wgc

• PI or lag to improve steady state tracking– Use PI if type increase neede – Use lag if ess needs to be reduced

• Use low pass filter to reduce high freq gain

Proportional controller design• Obtain open loop Bode plot• Convert design specs into Bode plot req.• Select KP based on requirements:

– For improving ess: KP = Kp,v,a,des / Kp,v,a,act

– For fixing Mp: select wgcd to be the freq at which PM is sufficient, and KP = 1/|G(jwgcd)|

– For fixing speed: from td, tr, tp, or ts requirement, find out wn, let wgcd = (0.65~0.8)*wn and KP = 1/|G(jwgcd)|

clear all;n=[0 0 40]; d=[1 2 0];figure(1); clf; margin(n,d);%proportional control design:figure(1); hold on; grid; V=axis;Mp = 10; %overshoot in percentagePMd = 70-Mp + 3;semilogx(V(1:2), [PMd-180 PMd-180],':r');%get desired w_gcx=ginput(1); w_gcd = x(1);KP = 1/abs(evalfr(tf(n,d),j*w_gcd));figure(2); margin(KP*n,d);figure(3); mystep(KP*n, d+KP*n);

Bode Diagram

Frequency (rad/sec)

50Gm = Inf, Pm = 17.964 deg (at 6.1685 rad/sec)

G(s)=40/s(s+2)

Mp=10%

0 1 2 3 4 5 60

Time (sec)

Unit Step Response

ts=3.65 tp=0.508

Mp=60.4%

ess tolerance band: +-2%

td=0.159

tr=0.19

Bode Diagram

Frequency (rad/sec)

1 2 3 4 5 60

Time (sec)

Unit Step Response

ts=3.98 tp=2.82

Mp=6.03%ess tolerance band: +-2%

td=0.883

tr=1.33

( ) (1 )

Gain: 20 log(| ( ) |) 20 log( )

20 log( 1

Phase: ( ) (1 ) tan ( )

DP D P

KC s K K s K s

KC j K K j K j

K KC j j

PD Controller

Bode Diagram

Frequency (rad/sec)

20*log(KP)

Place wgcd here

Bad for noise

From specs, find and

a few degrees

tan( ) /

1/ (1 ) ( )

( ) ( ) ( ) / 1 ( ) ( )

Perform c.l. step response, tune C

P D s j

D D P P D

PM angle G j

PM PMd PM

K T s G s

K T K C s K K s

G s C s G s C s G s

(s) as needed

PD control design

n=[0 0 1]; d=[0.02 0.3 1 0];

figure(1); clf; margin(n,d);

Mp = 10/100;

zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));

PMd = zeta * 100 + 3;

tr = 0.3; w_n=1.8/tr; w_gcd = w_n;

PM = angle(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));

phi = PMd*pi/180-PM; Td = tan(phi)/w_gcd;

KP = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));

KP = KP/sqrt(1+Td^2*w_gcd^2); KD=KP*Td;

ngc = conv(n, [KD KP]);

figure(2); margin(ngc,d);

figure(3); mystep(ngc, d+ngc);

Could be a little less

PD control design Variation

• Restricted to using KP = 1

• Meet Mp requirement

• Find wgc and PM

• Find PMd

• Let f = PMd – PM + (a few degrees)

• Compute TD = tan(f)/wgcd

• KP = 1; KD=KPTD

n=[0 0 5]; d=[0.02 0.3 1 0];

Mp = 10/100;

zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));

PMd = zeta * 100 + 18;

[GM,PM,wgc,wpc]=margin(n,d);

phi = (PMd-PM)*pi/180; Td = tan(phi)/wgc;

Kp=1; Kd=Kp*Td;

ngc = conv(n, [Kd Kp]);

figure(3); stepchar(ngc, d+ngc);

200Example: ( )

When ( ) 1, 16 , 64%

Want: 16%

G ss s

C s PM Mp

C(s) G(s)

23 3.002.0

Example

Want: maximum overshoot <= 10% rise time <= 0.3 sec

Can use Lead or PD

Bode Diagram

Frequency (rad/sec)

20Gm = 23.522 dB (at 7.0711 rad/sec), Pm = 73.367 deg (at 0.9768 rad/sec)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

Time (sec)

Unit Step Response

ts=2.68 tp=3

Mp=-1.06%

td=0.841

tr=1.52

n=[0 0 1]; d=[0.02 0.3 1 0];

Mp = 10; %overshoot in percentage

PMd = 70 – Mp + 3;

tr = 0.3; w_n=1.8/tr; w_gcd = w_n;

PM = angle(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));

phi = PMd*pi/180-PM; Td = tan(phi)/w_gcd;

KP = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));

KP = KP/sqrt(1+Td^2*w_gcd^2); KD=KP*Td;

figure(3); mystep(ngc, d+ngc);

Could be a little less

Bode Diagram

Frequency (rad/sec)

20Gm = Inf, Pm = 62.116 deg (at 6 rad/sec)

0.2 0.4 0.6 0.8 1 1.2 1.40

Time (sec)

Unit Step Response

ts=0.655 tp=0.461

td=0.154

tr=0.225

Less than spec

Variation

• Restricted to using KP = 1

• Meet Mp requirement

• Find wgc and PM

• Find PMd

• Let f = PMd – PM + (a few degrees)

• Compute TD = tan(f)/wgcd

• KP = 1; KD=KPTD

KP=5; n=KP*[0 0 1]; d=[0.02 0.3 1 0];

figure(3); stepchar(n, d+n);

Bode Diagram

Frequency (rad/sec)

20Gm = 9.5424 dB (at 7.0711 rad/sec), Pm = 32.613 deg (at 3.7468 rad/sec)

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (sec)

Unit Step Response

ts=3.17 tp=0.814

Mp=38.9%

td=0.317

tr=0.317

KP=5; n=KP*[0 0 1]; d=[0.02 0.3 1 0];

Mp = 10;

PMd = 70 – Mp + 10;

KP=1; KD=KP*Td;

Bode Diagram

Frequency (rad/sec)

0.5 1 1.5 2 2.50

Time (sec)

Unit Step Response

ts=1.56 tp=0.728

td=0.236

tr=0.321

KP=5; n=KP*[0 0 1]; d=[0.02 0.3 1 0];

Mp = 10;

PMd = 70 – Mp + 18;

KP=1; KD=KP*Td;

Bode Diagram

Frequency (rad/sec)

0 0.5 1 1.5 20

Time (sec)

Unit Step Response

ts=1.07 tp=0.695

td=0.22

tr=0.305

Lead Controller Design

00 Kzp leadlead

0)(tan)(tan

)()()(

leadlead

pjzjjC

101520253035404550

)Bode Diagram

Frequency (rad/sec)

leadlead zp

20log(Kzlead/plead)lead

)(tan)(tan 11max lea d

Goal: select z and p so that max phase lead is at desired wgc and max phase lead = PM defficiency!

1 sinLet

gcd gcd/ , *lead leadz p

Lead Design • From specs => PMd and wgcd

• From plant, draw Bode plot• Find PMhave = 180 + angle(G(jwgcd)

• DPM = PMd - PMhave + a few degrees

• Choose a=plead/zlead so that fmax = DPM and it happens at wgcd

gcdgcdgcd

gcdgcd

)/()()(

leadlead

pjjGzjK

Lead design example• Plant transfer function is given by:

• n=[50000]; d=[1 60 500 0];

• Desired design specifications are:– Step response overshoot <= 16%– Closed-loop system BW>=20;

n=[50000]; d=[1 60 500 0]; G=tf(n,d);figure(1); margin(G); Mp_d = 16/100; zeta_d =0.5; % or calculate from Mp_dPMd = 100*zeta_d + 3;BW_d=20;w_gcd = BW_d*0.7; Gwgc=evalfr(G, j*w_gcd);PM = pi+angle(Gwgc);phimax= PMd*pi/180-PM;alpha=(1+sin(phimax))/(1-sin(phimax));zlead= w_gcd/sqrt(alpha);plead=w_gcd*sqrt(alpha);K=sqrt(alpha)/abs(Gwgc);ngc = conv(n, K*[1 zlead]);dgc = conv(d, [1 plead]);figure(1); hold on; margin(ngc,dgc); hold off;[ncl,dcl]=feedback(ngc,dgc,1,1);figure(2); step(ncl,dcl);

Bode DiagramGm = 13.8 dB (at 38.3 rad/sec) , Pm = 53 deg (at 14 rad/sec)

Frequency (rad/sec)

Before designAfter design

Bode DiagramGm = 8.8 dB (at 38.3 rad/sec) , Pm = 40.6 deg (at 25.2 rad/sec)

Frequency (rad/sec)

Closed-loop Bode plot by:

Magnitude plot shifted up 3dBSo, gc is BW

margin(ncl*1.414,dcl);

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1.4Step Response

Time (sec)

Alternative use of lead• Use Lead and Proportional together

– To fix overshoot and ess– No speed requirement

1.Select K so that KG(s) meet ess req.

2.Find wgc and PM, also find PMd

3.Determine phi_max, and alpha

4.Place phi_max a little higher than wgc

maxgc gc

1 sin/ , *

lead lead

lead lead lead

p s z s zC s K K

z s p s p

Alternative lead design example• Plant transfer function is given by:

• n=[50]; d=[1/50 1 0];

• Desired design specifications are:– Step response overshoot <= 20%– Steady state tracking error for ramp input <=

1/200;

– No speed requirements– No settling concern– No bandwidth requirement

n=[50]; d=[1/5 1 0];figure(1); clf; margin(n,d); grid; hold on;Mp = 20/100; zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));PMd = zeta * 100 + 10;ess2ramp= 1/200; Kvd=1/ess2ramp;Kva = n(end)/d(end-1); Kzp = Kvd/Kva;figure(2); margin(Kzp*n,d); grid;[GM,PM,wpc,wgc]=margin(Kzp*n,d);w_gcd=wgc; phimax = (PMd-PM)*pi/180;alpha=(1+sin(phimax))/(1-sin(phimax));z=w_gcd/alpha^.25; %sqrt(alpha); %phimax located higherp=w_gcd*alpha^.75; %sqrt(alpha); %than wgcngc = conv(n, alpha*Kzp*[1 z]); dgc = conv(d, [1 p]);figure(3); margin(tf(ngc,dgc)); grid;[ncl,dcl]=feedback(ngc,dgc,1,1);figure(4); step(ncl,dcl); grid;figure(5); margin(ncl*1.414,dcl); grid;

Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 18 deg (at 15.4 rad/sec)

Frequency (rad/sec)

Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 9.04 deg (at 31.4 rad/sec)

Frequency (rad/sec)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

1.4Step Response

Time (sec)

Frequency (rad/sec)

Lead design tuning example

C(s) G(s)

( 5)G s

Design specifications: rise time <=2 secovershoot <16%

4. Go back and take wgcd = 0.6*wn so that tr is not too small

Desired tr < 2 secWe had tr = 1.14 in the previous 4 designs

n=[1]; d=[1 5 0 0]; G=tf(n,d);Mp_d = 16; %in percentagePMd = 70 - Mp_d + 4; %use Mp + PM =70tr_d = 2; wnd = 1.8/tr_d; w_gcd = 0.6*wnd;Gwgc=evalfr(G, j*w_gcd);PM = pi+angle(Gwgc);phimax= PMd*pi/180-PM;alpha=(1+sin(phimax))/(1-sin(phimax));zlead= w_gcd/sqrt(alpha);plead=w_gcd*sqrt(alpha);K=sqrt(alpha)/abs(Gwgc);ngc = conv(n, K*[1 zlead]);dgc = conv(d, [1 plead]);[ncl,dcl]=feedback(ngc,dgc,1,1);stepchar(ncl,dcl); gridfigure(2); margin(G); hold on; margin(ngc,dgc); hold off; grid

)Bode Diagram

Gm = 22.3 dB (at 3.45 rad/sec) , Pm = 60 deg (at 0.54 rad/sec)

Frequency (rad/sec)

0 5 10 15 20 25 30 350

Time (sec)

eUnit Step Response

ts=20.9 tp=5.85

td=1.56

tr=1.95

Lead design

C(s) G(s)

( 2)G s

Design specifications: Keep tr, td, butreduce overshoot to <16%

Lag design

C(s) G(s)

(0.02 1)G s

• Desired design specifications are:– Step response overshoot <= 20%– Steady state tracking error for ramp

input <= 1/200;

Alternative lead design example• Plant transfer function is given by:

• n=[50]; d=[1/50 1 0];

• Desired design specifications are:– Step response overshoot <= 20%– Steady state tracking error for ramp input <=

1/200;

– No speed requirements– No settling concern– No bandwidth requirement

Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for...

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