Title: Where is w=0 , where is w=0-,
1- Where is w0, where is w0-,
- where is winf, 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?
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4Open vs Closed Loop Frequency Response And
Frequency Domain Specifications
G(s)
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.
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6Prototype 2nd order system closed-loop frequency
response
For small zeta, resonance freq is about wn BW
ranges from 0.5wn to 1.5 wn For good z range, BW
is 0.8 to 1.1 wn So take BW wn
z0.1
0.2
0.3
No resonance for z lt 0.7 Mr1dB for
z0.6 Mr3dB for z0.5 Mr7dB for z0.4
w/wn
7Prototype 2nd order system closed-loop frequency
response Mr vs z
80.2
z0.1
0.3
0.4
wgc
In the range of good zeta, wgc is about 0.65
times to 0.8 times wn
w/wn
9In the range of good zeta, PM is about 100z
z0.1
0.2
0.3
0.4
w/wn
10Important relationships
- Prototype wn, open-loop wgc, closed-loop BW are
all very close to each other - When there is visible resonance peak, it is
located near or just below wn, - This happens when z lt 0.6
- When z gt 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
11Important relationships
- wgc determines wn and bandwidth
- As wgc ?, 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 w 1 the
value gives Kp, Kv, or Ka, depending on type - High frequency gain determines noise immunity
12Desired Bode plot shape
13Proportional 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 wn and
choose KP as above
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16- clear all
- n0 0 40 d1 2 0
- figure(1) clf margin(n,d)
- proportional control design
- figure(1) hold on grid Vaxis
- Mp 10/100
- zeta sqrt((log(Mp))2/(pi2(log(Mp))2))
- PMd zeta 100 3
- semilogx(V(12), PMd-180 PMd-180,'r')
- get desired w_gc
- xginput(1) w_gcd x(1)
- KP 1/abs(polyval(n,jw_gcd)/polyval(d,jw_gcd))
- figure(2) margin(KPn,d)
- figure(3) stepchar(KPn, dKPn)
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19n1 d1/5/50 1/51/50 1 0 figure(1) clf
margin(n,d) proportional control
design figure(1) hold on grid Vaxis Mp
10/100 zeta sqrt((log(Mp))2/(pi2(log(Mp))2)
) PMd zeta 100 3 semilogx(V(12),
PMd-180 PMd-180,'r') get desired
w_gc xginput(1) w_gcd x(1) Kp
1/abs(polyval(n,jw_gcd)/polyval(d,jw_gcd)) Kv
Kpn(1)/d(3) ess0.01 Kvd1/ess z w_gcd/5
p z/(Kvd/Kv) ngc conv(n, Kp1 z) dgc
conv(d, 1 p) figure(1) hold on
margin(ngc,dgc) ncl,dclfeedback(ngc,dgc,1,1)
figure(2) step(ncl,dcl) grid figure(3)
margin(ncl1.414,dcl) grid
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32Proportional 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 wn and
choose KP as above
33C(s)
Gp(s)
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