Desired Bode plot shape Ess requirement Noise requirement 0 -90 -180 0dB gcd High low-freq-gain for steady state tracking Low high-freq-gain for noise.

Desired Bode plot shape

Ess requirement

Noise requirement

0

-90

-180

0dB

wgcd

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

w

w

PMd

Mid frequency

Desired Bode plot shape

0

-90

-180

0dB

wgc

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

w

w

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

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)

Ph

ase

(deg

)M

agn

itu

de

(dB

)

-10

0

10

20

30

40

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

10-1

100

101

-180

-135

-90

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

Mp=10%

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (sec)

Am

plit

ud

e

Unit Step Response

ts=3.65 tp=0.508

Mp=60.4%

ess tolerance band: +-2%

td=0.159

tr=0.19

yss=1

ess=0

Bode Diagram

Frequency (rad/sec)

Ph

ase

(deg

)M

agn

itu

de

(dB

)

-40

-20

0

20


10-1

100

101

-180

-135

-90

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Am

plit

ud

e

Unit Step Response

ts=3.98 tp=2.82

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

td=0.883

tr=1.33

yss=1

ess=0

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

20

30

40

50

60

70

Mag

nit

ud

e (d

B)

10-2

10-1

100

101

102

0

45

90

Ph

ase

(deg

)

Bode Diagram

Frequency (rad/sec)

20*log(KP)

KP/KD

Place wgcd here

Bad for noise

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

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);

2

200Example: ( )

4 4

When ( ) 1, 16 , 64%

Want: 16%

o

G ss s

C s PM Mp

Mp

C(s) G(s)

ssssG

23 3.002.0

1)(

Example

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

Can use Lead or PD

Bode Diagram

Frequency (rad/sec)

Ph

ase

(deg

)M

agn

itu

de

(dB

)

-100

-80

-60

-40

-20

0

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

10-1

100

101

102

-270

-225

-180

-135

-90

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Am

plit

ud

e

Unit Step Response

ts=2.68 tp=3

Mp=-1.06%


td=0.841

tr=1.52

yss=1

ess=0

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)

Ph

ase

(deg

)M

agn

itu

de

(dB

)

-60

-40

-20

0

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

100

101

102

-180

-135

-90

0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Am

plit

ud

e

Unit Step Response

ts=0.655 tp=0.461


td=0.154

tr=0.225

yss=1

ess=0

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)

Ph

ase

(deg

)M

agn

itu

de

(dB

)

-80

-60

-40

-20

0

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

100

101

102

-270

-225

-180

-135

-90

0.5 1 1.5 2 2.5 3 3.5 4 4.5

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Am

plit

ud

e

Unit Step Response

ts=3.17 tp=0.814

Mp=38.9%


td=0.317

tr=0.317

yss=1

ess=0

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)

Ph

ase

(deg

)M

agn

itu

de

(dB

)

-60

-40

-20

0


100

101

102

-180

-150

-120

-90

0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Am

plit

ud

e

Unit Step Response

ts=1.56 tp=0.728


td=0.236

tr=0.321

yss=1

ess=0

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)

Ph

ase

(deg

)M

agn

itu

de

(dB

)

-60

-40

-20

0


100

101

102

-180

-135

-90

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Am

plit

ud

e

Unit Step Response

ts=1.07 tp=0.695


td=0.22

tr=0.305

yss=1

ess=0

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 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

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; 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);

-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 = 20/100; 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;

-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

)


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

)


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

)


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


td=1.56

tr=1.95

yss=1

ess=0

Lead design

C(s) G(s)

40( )

( 2)G s

s s

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

Lag design

C(s) G(s)

50( )

(0.02 1)G s

s 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 steady state tracking Low high-freq-gain for noise.

Documents

pm d pm

noise slide

p t d slide

ngc slide

pmd pm

gj gcd slide

dcl slide

high frequency slide