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Elements of Feedback Control

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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 ... – PowerPoint PPT presentation

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Title: Elements of Feedback Control


1
Open 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.
2
Prototype 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
3
Closed-loop BW to wn ratio
BW1.4wn
z
4
Prototype 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
5
Prototype 2nd order system closed-loop frequency
response Mr vs z
6
Percentage Overshoot in closed-loop step response
z
7
Percentage Overshoot in closed-loop step response
Mr
8
Percentage Overshoot in closed-loop step response
Mr in dB
9
Phase Margin
PM 100z
z
10
PMMp 70 line
Percentage Overshoot in closed-loop step response
Phase Margin in degrees
11
0.2
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
12
Open-loop wgc to wn ratio
wgc0.7wn
z
13
In the range of good zeta, PM is about 100z
z0.1
0.2
0.3
0.4
w/wn
14
Important 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

15
Desired Bode plot shape
16
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 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)

18
PD Controller
19
KP/KD
20log(KP)
Place wgcd here
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  • 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
22
Variation
  • 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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Lead Controller Design
30
plead
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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32
Lead 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

33
Lead 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

34
n50000 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)
35
After design
Before design
36
Closed-loop Bode plot by
margin(ncl1.414,dcl)
Magnitude plot shifted up 3dB So, gc is BW
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n50 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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n50 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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