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• 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