Where is w=0+, where is w=0-, where is w=+inf, where is w=-inf, what is the system type, what is the relative order of the TF, how should you complete.

• Where is w=0+, where is w=0-, • where is w=+inf, where is w=-inf, • what is the system type, • what is the relative order of the TF, • how should you complete the nyquist plot, • what are P/N/Z values as in the nyquist criterion, • is the closed-loop system stable, • what the is the phase margin, • by how much can the gain be varied without affecting

stability? • how many gain cross-over points and how many phase

cross-over points are there?

-20 -15 -10 -5 0 5 10

-10

-5

0

5

10

-2 -1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

-0.2 -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

G(s)

Open vs Closed Loop Frequency Response And Frequency Domain Specifications

C(s)

Goal: 1) Define typical “good” freq resp shape for closed-loop 2) Relate closed-loop freq response shape to step response shape 3) Relate closed-loop freq shape to open-loop freq resp shape 4) Design C(s) to make C(s)G(s) into “good” shape.

10-1

100

101

-50

-40

-30

-20

-10

0

10

20

30

40

50

=0.1

0.2

0.3

No resonancefor <= 0.7

Mr=1dB for =0.6

Mr=3dB for =0.5

Mr=7dB for =0.4

For small zeta,resonance freqis about n

BW ranges from0.5wn to 1.5n

For good z range,BW is 0.8 to 1.1 n

So take BW = n

Prototype 2nd order system closed-loop frequency response

n

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750

0.5

1

1.5

2

2.5

3

Mr in value

Mr in dB

Prototype 2nd order system closed-loop frequency response

Mr vs

10-2

10-1

100

101

102

-200

-150

-100

-50

0

50

100

150

n

z=0.1 0.2 0.3

0.4

gc

In the range of good zeta,gc is about 0.65 times to 0.8 times n

10-2

10-1

100

101

102

-180

-170

-160

-150

-140

-130

-120

-110

-100

-90

n

=0.10.2

0.3

0.4

In the range of good zeta,PM is about 100*

Important relationships• Prototype n, open-loop gc, closed-loop BW

are all very close to each other• When there is visible resonance peak, it is

located near or just below n,

• This happens when <= 0.6• When >= 0.7, no resonance• determines phase margin and Mp:

0.4 0.5 0.6 0.7

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

• gc determines n and bandwidth

– As gc ↑, ts, td, tr, tp, etc ↓

• Low frequency gain determines steady state tracking:– L.F. magnitude plot slope/(-20dB/dec) = type– L.F. asymptotic line evaluated at = 1: the

value gives Kp, Kv, or Ka, depending on type

• High frequency gain determines noise immunity

Important relationships

Desired Bode plot shape

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 = n and choose KP as above

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

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 = 10/100;zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));PMd = zeta * 100 + 3;semilogx(V(1:2), [PMd-180 PMd-180],':r');%get desired w_gcx=ginput(1); w_gcd = x(1);KP = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));figure(2); margin(KP*n,d);figure(3); stepchar(KP*n, d+KP*n);

Bode Diagram

Frequency (rad/sec)

Ph

ase

(deg

)M

agn

itu

de

(dB

)

-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

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

n=[1]; d=[1/5/50 1/5+1/50 1 0];figure(1); clf; margin(n,d);%proportional control design:figure(1); hold on; grid; V=axis;Mp = 10/100;zeta = sqrt((log(Mp))^2/(pi^2+(log(Mp))^2));PMd = zeta * 100 + 3;semilogx(V(1:2), [PMd-180 PMd-180],':r');%get desired w_gcx=ginput(1); w_gcd = x(1);Kp = 1/abs(polyval(n,j*w_gcd)/polyval(d,j*w_gcd));Kv = Kp*n(1)/d(3); ess=0.01; Kvd=1/ess;z = w_gcd/5; p = z/(Kvd/Kv);ngc = conv(n, Kp*[1 z]); dgc = conv(d, [1 p]);figure(1); hold on; margin(ngc,dgc);[ncl,dcl]=feedback(ngc,dgc,1,1);figure(2); step(ncl,dcl); grid;figure(3); margin(ncl*1.414,dcl); grid;

-100

-50

0

Mag

nit

ud

e (d

B)

10-1

100

101

102

103

-270

-225

-180

-135

-90

Ph

ase

(deg

)

Bode DiagramGm = 34.8 dB (at 15.8 rad/sec) , Pm = 77.8 deg (at 0.981 rad/sec)

Frequency (rad/sec)

-100

-50

0

50M

agn

itu

de

(dB

)

100

101

102

103

-270

-225

-180

-135

-90

Ph

ase

(deg

)

Bode DiagramGm = 13.9 dB (at 36.8 rad/sec) , Pm = 53 deg (at 13.3 rad/sec)

Frequency (rad/sec)

-200

-100

0

100

200M

agn

itu

de

(dB

)

10-4

10-2

100

102

-270

-180

-90

Ph

ase

(deg

)

Bode DiagramGm = 26 dB (at 15 rad/sec) , Pm = 51.7 deg (at 2.31 rad/sec)

Frequency (rad/sec)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4Step Response

Time (sec)

Am

plit

ud

e

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 = n and choose KP as above

C(s) Gp(s)

21

21)(psps

zszsKsC

42

2

1

31

35

46

RR

R

R

RR

RR

RRK