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 Mr1dB for
z0.6 Mr3dB for z0.5 Mr7dB for z0.4
w/wn
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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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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

• 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

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Important 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

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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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n1 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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KP/KD
20log(KP)
Place wgcd here
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PD Control

• C(s)KP KDs KP(1TDs)
• For fixing wgcd and PMd
• Compute wgcd from tr, td, etc
• Compute PMd from Mp
• Compute f PMd PM_at_wgcd
• Compute TD tan(f)/wgcd
• KP 1/sqrt(1Td2wgcd2)/abs(G(jwgcd))
• KDKPTD

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Example
C(s)
G(s)
Want maximum overshoot lt 10 rise
time lt 0.3 sec
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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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Less than spec
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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 10
• 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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• 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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