Title: Elements of Feedback Control
1Open 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.
2Prototype 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 Mr0.5dB for
z0.6 Mr3dB for z0.5 Mr7dB for z0.4
w/wn
3Closed-loop BW to wn ratio
BW1.4wn
z
4Prototype 2nd order system closed-loop frequency
response
z0.1
When z lt0.5 ? visible resonance peak near wwn
When z gt0.6 ?no visible resonance peak
0.2
0.3
No resonance for z lt 0.7 Mrlt0.5 dB for
z0.6 Mr1.2 dB for z0.5 Mr2.5 dB for z0.4
wwn
w/wn
5Prototype 2nd order system closed-loop frequency
response Mr vs z
6Percentage Overshoot in closed-loop step response
z
7Percentage Overshoot in closed-loop step response
Mr
8Percentage Overshoot in closed-loop step response
Mr in dB
9Phase Margin
PM 100z
z
10PMMp 70 line
Percentage Overshoot in closed-loop step response
Phase Margin in degrees
110.2
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
12Open-loop wgc to wn ratio
wgc0.7wn
z
13In the range of good zeta, PM is about 100z
z0.1
0.2
0.3
0.4
w/wn
14Important relationships
- Closed-loop BW are very close to wn
- Open-loop gain cross over wgc (0.650.8) wn,
- When z lt 0.6, wr and wn are close
- 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
15Desired Bode plot shape
16Proportional 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
17- 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)
18PD Controller
19KP/KD
20log(KP)
Place wgcd here
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21- n0 0 1 d0.02 0.3 1 0
- figure(1) clf margin(n,d)
- Mp 10/100
- zeta sqrt((log(Mp))2/(pi2(log(Mp))2))
- PMd zeta 100 3
- tr 0.3 w_n1.8/tr w_gcd w_n
- PM angle(polyval(n,jw_gcd)/polyval(d,jw_gcd))
- phi PMdpi/180-PM Td tan(phi)/w_gcd
- KP 1/abs(polyval(n,jw_gcd)/polyval(d,jw_gcd))
- KP KP/sqrt(1Td2w_gcd2) KDKPTd
- ngc conv(n, KD KP)
- figure(2) margin(ngc,d)
- figure(3) stepchar(ngc, dngc)
Could be a little less
22Variation
- 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 KDKPTD
23- n0 0 5 d0.02 0.3 1 0
- figure(1) clf margin(n,d)
- Mp 10/100
- zeta sqrt((log(Mp))2/(pi2(log(Mp))2))
- PMd zeta 100 18
- GM,PM,wgc,wpcmargin(n,d)
- phi (PMd-PM)pi/180 Td tan(phi)/wgc
- Kp1 KdKpTd
- ngc conv(n, Kd Kp)
- figure(2) margin(ngc,d)
- figure(3) stepchar(ngc, dngc)
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29Lead Controller Design
30plead
zlead
20log(Kzlead/plead)
Goal select z and p so that max phase lead is at
desired wgc and max phase lead PM
defficiency!
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32Lead Design
- From specs gt PMd and wgcd
- From plant, draw Bode plot
- Find PMhave 180 angle(G(jwgcd)
- DPM PMd - PMhave a few degrees
- Choose aplead/zlead so that fmax DPM and it
happens at wgcd
33Lead design example
- Plant transfer function is given by
- n50000 d1 60 500 0
- Desired design specifications are
- Step response overshoot lt 16
- Closed-loop system BWgt20
34n50000 d1 60 500 0 Gtf(n,d) figure(1)
margin(G) Mp_d 16/100 zeta_d 0.5 or
calculate from Mp_d PMd 100zeta_d
3 BW_d20 w_gcd BW_d0.7 Gwgcevalfr(G,
jw_gcd) PM piangle(Gwgc) phimax
PMdpi/180-PM alpha(1sin(phimax))/(1-sin(phimax
)) zlead w_gcd/sqrt(alpha) pleadw_gcdsqrt(alp
ha) Ksqrt(alpha)/abs(Gwgc) ngc conv(n, K1
zlead) dgc conv(d, 1 plead) figure(1)
hold on margin(ngc,dgc) hold off ncl,dclfeed
back(ngc,dgc,1,1) figure(2) step(ncl,dcl)
35After design
Before design
36Closed-loop Bode plot by
margin(ncl1.414,dcl)
Magnitude plot shifted up 3dB So, gc is BW
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38n50 d1/5 1 0 figure(1) clf margin(n,d)
grid hold on Mp 20/100 zeta
sqrt((log(Mp))2/(pi2(log(Mp))2)) PMd zeta
100 10 ess2ramp 1/200 Kvd1/ess2ramp Kva
n(end)/d(end-1) Kzp Kvd/Kva figure(2)
margin(Kzpn,d) grid GM,PM,wpc,wgcmargin(Kzp
n,d) w_gcdwgc phimax (PMd-PM)pi/180 alpha
(1sin(phimax))/(1-sin(phimax)) zw_gcd/sqrt(alph
a) pw_gcdsqrt(alpha) ngc conv(n,
alphaKzp1 z) dgc conv(d, 1
p) figure(3) margin(tf(ngc,dgc))
grid ncl,dclfeedback(ngc,dgc,1,1) figure(4)
step(ncl,dcl) grid figure(5)
margin(ncl1.414,dcl) grid
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45n50 d1/5 1 0 figure(1) clf margin(n,d)
grid hold on Mp 20/100 zeta
sqrt((log(Mp))2/(pi2(log(Mp))2)) PMd zeta
100 10 ess2ramp 1/200 Kvd1/ess2ramp Kva
n(end)/d(end-1) Kzp Kvd/Kva figure(2)
margin(Kzpn,d) grid GM,PM,wpc,wgcmargin(Kzp
n,d) w_gcdwgc phimax (PMd-PM)pi/180 alpha
(1sin(phimax))/(1-sin(phimax)) zw_gcd/alpha.25
sqrt(alpha) pw_gcdalpha.75
sqrt(alpha) ngc conv(n, alphaKzp1 z) dgc
conv(d, 1 p) figure(3) margin(tf(ngc,dgc))
grid ncl,dclfeedback(ngc,dgc,1,1) figure(4)
step(ncl,dcl) grid figure(5)
margin(ncl1.414,dcl) grid
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