desired bode plot shape - iowa state university

Desired Bode plot shape

Ess requirement

Noise requirement

0

-90

-180

0dB

ωgcd

High low-freq-gain for steady state tracking Low high-freq-gain for noise attenuation Sufficient PM near ωgc for stability

ω

ω

PMd

Mid frequency

Desired Bode plot shape

0

-90

-180

0dB

ωgc

High low freq gain for steady state tracking Low high freq gain for noise attenuation Sufficient PM near ωgc for stability

ω

ω

Low freq

High freq

Want high gain

Want low gain

Mid freq

Want sufficient Phase margin

Use low pass filters

Use PI or lag control

Use lead or PD control

PM+Mp=70

C(s) Gp(s)

( )( )( )( )!

!

21

21)(pspszszsKsC

−−

−−=

Controller design with Bode

From specs: => desired Bode shape of Gol(s) Make Bode plot of Gp(s) Add C(s) to change Bode shape Get closed loop system Run 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 ωn •  Open-loop gain cross over ωgc ≈

(0.65~0.8)* ωn, •  When ζ <= 0.6, ωr and ωn are close •  When ζ >= 0.7, no resonance •  ζ determines phase margin and Mp:

ζ 0.4 0.5 0.6 0.7 PM 44 53 61 67 deg ≈100ζ Mp 25 16 10 5 %

PM+Mp ≈70

Mid frequency requirements •  ωgc is critically important

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

•  Hence it determines tr, td directly

•  PM at ωgc controls ζ– Mp ≈ 70 – PM

•  PM and ωgc together controls σ and ωd – Determines ts, tp

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

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 è ωgc – Overshoot/resonance è PMd

•  Use PD or lead to achieve PMd@ ωgc •  Use overall gain K to enforce ωgc •  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 ωgcd to be the freq at

which PM is sufficient, and KP = 1/|G(jωgcd)| – For fixing speed: from td, tr, tp, or ts

requirement, find out ωn, let ωgcd = (0.65~0.8)*ωn and KP = 1/|G(jωgcd)|

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 percentage PMd = 70-Mp + 3; semilogx(V(1:2), [PMd-180 PMd-180],':r'); %get desired w_gc x=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)

Phas

e (d

eg)

Mag

nitu

de (d

B)

-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

plitu

de

Unit Step Response

ts=3.65 tp=0.508

Mp=60.4%

ess tolerance band: +-2%

td=0.159

tr=0.19

yss=1ess=0

Bode Diagram

Frequency (rad/sec)

Phas

e (d

eg)

Mag

nitu

de (d

B)

-40

-20

0

20

40Gm = Inf, Pm = 63.31 deg (at 1.0055 rad/sec)

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

plitu

de

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 sK

KC j K K j K jK

C j K

KK

K KC j jK K

ω ω ω

ω

ω ω ω−

= + = +

= + = +

=

⎛ ⎞+ + ⎜ ⎟

⎝ ⎠

∠ =∠ + =

PD Controller

20

30

40

50

60

70

Mag

nitu

de (d

B)

10-2 10-1 100 101 1020

45

90

Phas

e (d

eg)

Bode Diagram

Frequency (rad/sec)

20*log(KP)

KP/KD

Place ωgcd 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 sG s C s G s C s G s

ω

ω

ω

ω

π ω

ω

=

= +

Δ = − +

= Δ

= +

= ⇒ = +

= +

i

(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

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)

Phas

e (d

eg)

Mag

nitu

de (d

B)

-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

plitu

de

Unit Step Response

ts=2.68 tp=3

Mp=-1.06%

ess tolerance band: +-2%

td=0.841

tr=1.52

yss=1

ess=0

n=[0 0 1]; d=[0.02 0.3 1 0]; figure(1); clf; margin(n,d); 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; ngc = conv(n, [KD KP]); figure(2); margin(ngc,d); figure(3); mystep(ngc, d+ngc);

Could be a little less

Bode Diagram

Frequency (rad/sec)

Phas

e (d

eg)

Mag

nitu

de (d

B)

-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

plitu

de

Unit Step Response

ts=0.655 tp=0.461


td=0.154

tr=0.225

yss=1

ess=0

Less than spec

PD control design Variation •  Restricted to using KP = 1 •  Meet Mp requirement

•  Find ωgc and PM •  Find PMd

•  Let φ = PMd – PM + (a few degrees) •  Compute TD = tan(φ)/wgcd

•  KP = 1; KD=KPTD

n=[0 0 5]; 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 + 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(2); margin(ngc,d); figure(3); stepchar(ngc, d+ngc);

2

200Example: ( )4 4

When ( ) 1, 16 , 64%Want: 16%

o

G ss s

C s PM MpMp

=+ +

= = =

≤

lead

lead

pszsKsC

+

+=)(

Lead Controller Design

00 >>> Kzp leadlead

lead

lead

pjzjKjC

+

+=

ωω

ω )(

0)(tan)(tan

)()()(

11 >−=

+∠−+∠=∠

−−

leadlead

leadlead

pz

pjzjjCωω

ωωω

101520253035404550

Mag

nitu

de (d

B)

10-2 10-1 100 101 102 1030

30

60

90

Phas

e (d

eg)

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 dpz

zp −− −=φ

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

ϕα

ϕ+

=−

gcd gcd/ , *lead leadz pω α ω α= =

Lead Design •  From specs => PMd and ωgcd

•  From plant, draw Bode plot •  Find PMhave = 180 + angle(G(jωgcd) •  ΔPM = PMd - PMhave + a few degrees •  Choose α=plead/zlead so that φmax = ΔPM

and it happens at ωgcd

( ) 1gcdgcdgcd

gcdgcd

max

max

)/()()(

*,/

sin1sin1

−++=

==

−

+=

leadlead

leadlead

pjjGzjK

pz

ωωω

αωαω

φφ

α

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_d PMd = 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

agni

tude

(dB

)

10-1 100 101 102 103-270

-225

-180

-135

-90

Phas

e (d

eg)

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

Frequency (rad/sec)

Before design After design

-150

-100

-50

0

50M

agni

tude

(dB

)

10-1 100 101 102 103 104-270

-180

-90

0

Phas

e (d

eg)

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 3dB So, 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

1.4 Step Response

Time (sec)

Am

plitu

de

Lead design tuning example C(s) G(s)

2

1( )( 5)

G ss s

=+

Design specifications: rise time <=2 sec overshoot <16%

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

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

n=[1]; d=[1 5 0 0]; G=tf(n,d); Mp_d = 16; %in percentage PMd = 70 - Mp_d + 4; %use Mp + PM =70 tr_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); grid figure(2); margin(G); hold on; margin(ngc,dgc); hold off; grid

-200

-100

0

100

200M

agni

tude

(dB)

10-3 10-2 10-1 100 101 102 103-270

-225

-180

-135

-90

Phas

e (d

eg)

Bode DiagramGm = 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)

Ampl

itude

Unit Step Response

ts=20.9 tp=5.85


td=1.56

tr=1.95

yss=1

ess=0

desired bode plot shape - iowa state university

Documents