Desired Bode plot shape

0

-90

-180

0dB

gc

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

Low freqess, type

High freqNoise immu

Want high gain

Want low gain

Mid freqSpeed, BW

Want sufficientPhase margin

Use low pass filters

Use PI or lag control

Use lead or PD control

PM+Mp=70

Mr, Mp

Overall Loop shaping strategy• Determine mid freq requirements

– Speed/bandwidth gc

– Overshoot/resonance PMd

• Use PD or lead to achieve PMd@ gc

• Use overall gain K to enforce gc

• 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 gcd to be the freq at which PM is sufficient, and KP = 1/|G(jgcd)|

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

2

1

( ) (1 )

( ) (1 )

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

20 log( 1

Phase: ( ) (1 ) tan ( )

DP D P

P

DP D P

P

P

D

P

D D

P P

KC s K K s K s

K

KC j K K j K j

K

C j K

K

K

K KC j j

K K

PD Controller

gcd

gcd

gcd

gcd

gcd

gcd

From specs, find and

( )

a few degrees

tan( ) /

1/ (1 ) ( )

; ( )

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

Perform c.l. step response, tune C

d

D

P D s j

D D P P D

cl

PM

PM angle G j

PM PMd PM

T 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

PD control design Variation

• Restricted to using KP = 1

• Meet Mp requirement

• Find gc and PM

• Find PMd

• Let = PMd – PM + (a few degrees)

• Compute TD = tan()/wgcd

• KP = 1; KD=KPTD

lead

lead

ps

zsKsC

)(

Lead Controller Design

00 Kzp leadlead

lead

lead

pj

zjKjC

)(

0)(tan)(tan

)()()(

11

leadlead

leadlead

pz

pjzjjC

101520253035404550

Mag

nit

ud

e (d

B)

10-2

10-1

100

101

102

103

0

30

60

90

Ph

ase

(deg

)Bode Diagram

Frequency (rad/sec)

zlead

plead

leadlead zp

20log(Kzlead/plead)lead

leadz

p

)(tan)(tan 11max lea d

lea d

lea d

lea d

pz

zp

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

max

max

1 sinLet

1 sin

gcd gcd/ , *lead leadz p

Lead Design • From specs => PMd and gcd

• From plant, draw Bode plot

• Find PMhave = 180 + angle(G(jgcd)

• PM = PMd - PMhave + a few degrees

• Choose =plead/zlead so that max =PM and it happens at gcd

1

gcdgcdgcd

gcdgcd

max

max

)/()()(

*,/

sin1

sin1

leadlead

leadlead

pjjGzjK

pz

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; PMd = 70 – Mp_d ;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);

-100

-50

0

50M

agn

itu

de

(dB

)

10-1

100

101

102

103

-270

-225

-180

-135

-90

Ph

ase

(deg

)

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

Frequency (rad/sec)

Before designAfter design

-150

-100

-50

0

50M

agn

itu

de

(dB

)

10-1

100

101

102

103

104

-270

-180

-90

0

Ph

ase

(deg

)

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

0.2

0.4

0.6

0.8

1

1.2

1.4Step Response

Time (sec)

Am

plit

ud

e

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

max

1 sin/ , *

1 sin

( )

lead lead

lead lead lead

lead lead lead

z p

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;PMd = 70 – Mp + 7;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;

-40

-20

0

20

40

60M

agn

itu

de

(dB

)

10-1

100

101

102

-180

-135

-90

Ph

ase

(deg

)

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

Frequency (rad/sec)

-20

0

20

40

60M

agn

itu

de

(dB

)

10-1

100

101

102

-180

-150

-120

-90

Ph

ase

(deg

)

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

Frequency (rad/sec)

-50

0

50

Mag

nit

ud

e (d

B)

10-1

100

101

102

103

104

-180

-150

-120

-90

Ph

ase

(deg

)

Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 52.3 deg (at 50.1 rad/sec)

Frequency (rad/sec)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.2

0.4

0.6

0.8

1

1.2

1.4Step Response

Time (sec)

Am

plit

ud

e

-80

-60

-40

-20

0

20M

agn

itu

de

(dB

)

100

101

102

103

104

-180

-135

-90

-45

0

Ph

ase

(deg

)

Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 77.5 deg (at 82.2 rad/sec)

Frequency (rad/sec)

-100

-50

0

50

Mag

nit

ud

e (d

B)

10-1

100

101

102

103

104

-180

-135

-90

Ph

ase

(deg

)

Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 52.3 deg (at 50.1 rad/sec)

Frequency (rad/sec)

Lead design tuning example

C(s) G(s)

2

1( )

( 5)G s

s 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

-200

-100

0

100

200M

ag

nit

ud

e (

dB

)

10-3

10-2

10-1

100

101

102

103

-270

-225

-180

-135

-90

Ph

ase (

deg

)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

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Am

pli

tud

eUnit Step Response

ts=20.9 tp=5.85

Mp=16.5%ess tolerance band: +-2%

td=1.56

tr=1.95

yss=1

ess=0