Elements of Feedback Control

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

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Desired Bode plot shape
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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

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• 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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PD Controller
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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
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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

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• 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
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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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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

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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

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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)
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After design
Before design
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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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