lead lagbode handout
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EE 3CL4, §91 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
EE3CL4:Introduction to Linear Control Systems
Section 9: Design of Lead and Lag Compensators usingFrequency Domain Techniques
Tim Davidson
McMaster University
Winter 2014
EE 3CL4, §92 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Outline
1 Frequency Domain Approach to Compensator Design
2 LeadCompensators
3 LagCompensators
EE 3CL4, §94 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Frequency domain analysis
• Analyze closed loop using open loop transfer functionL(s) = Gc(s)G(s)H(s).
• Nyquist’s stability criterion• Gain margin: 1
|L(jωx )| , where ωx isthe frequency at which ∠L(jω) reaches −180◦
• Phase margin, φpm: 180◦ + ∠L(jωc), where ωc isthe frequency at which |L(jω)| equals 1
• Damping ratio: φpm = f (ζ)• Roughly speaking, settling time decreases with
increasing bandwidth of the closed loop
EE 3CL4, §95 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Bode diagram
L(jω) =1
jω(1 + jω)(1 + jω/5)
• Gain margin ≈ 15 dB
• Phase margin ≈ 43◦
EE 3CL4, §96 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Compensators and Bodediagram
• We have seen the importance of phase margin• If G(s) does not have the desired margin,
how should we choose Gc(s) so thatL(s) = Gc(s)G(s) does?
• To begin, how does Gc(s) affect the Bode diagram• Magnitude:
20 log10(|Gc(jω)G(jω)|
)= 20 log10
((|Gc(jω)|
)+ 20 log10
(|G(jω)|
)• Phase:
∠Gc(jω)G(jω) = ∠Gc(jω) + ∠G(jω)
EE 3CL4, §98 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Lead Compensators
• Gc(s) = Kc(s+z)s+p , with |z| < |p|, alternatively,
• Gc(s) = Kcα
1+sατ1+sτ , where p = 1/τ and α = p/z > 1
• Bode diagram (in the figure, K1 = Kc/α):
EE 3CL4, §99 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Lead Compensation
• What will lead compensation, do?• Phase is positive: might be able to increase phase
margin φpm
• Slope is positive: might be able to increase thecross-over frequency, ωc , (and the bandwidth)
EE 3CL4, §910 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Lead Compensation
• Gc(s) = Kcα
1+sατ1+sτ
• By making the denom. real, can show that∠Gc(jω) = atan
(ωτ(α−1)1+α(ωτ)2
)• Max. occurs when ω = ωm = 1
τ√α
=√
zp
• Max. phase angle satisfies tan(φm) = α−12√α
• Equivalently, sin(φm) = α−1α+1
• At ω = ωm, we have |Gc(jωm)| = Kc/√α
EE 3CL4, §911 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Bode Design Principles
• Set the loop gain so that desired steady-state errorconstants are obtained
• Insert the compensator to modify the transientproperties:
• Damping: through phase margin• Response time: through bandwidth
• Compensate for the attenuation of the lead network, ifappropriate
To maximize impact of phase lead, want peak of phase nearωc of the compensated open loop
EE 3CL4, §912 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Design Guidelines1 For uncompensated (i.e., proportionally controlled)
closed loop, set gain Kp so that steady-state errorconstants of the closed loop meet specifications
2 Evaluate the phase margin, and the amount of phaselead required.
3 Add a little “safety margin” to the amount of phase lead4 From this, determine α using sin(φm) = α−1
α+15 To maintain steady-state error const’s, set Kc = Kpα6 Determine (or approximate) the frequency at which
KpG(jω) has magnitude −10 log10(α).7 If we set ωm of the compensator to be this frequency,
then Gc(jωm)G(jωm) = 1 (or ≈ 1). Hence, thecompensator will provide its maximum phasecontribution at the appropriate frequency
8 Choose τ = 1/(ωm√α). Hence, p = ωm
√α.
9 Set z = p/α.10 Compensator: Gc(s) = Kc(s+z)
s+p .
EE 3CL4, §913 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Example
• Type 1 plant of order 2: G(s) = 5s(s+2)
• Design goals:• Steady-state error due to a ramp input less than 5% of
velocity of ramp• Phase margin at least 45◦ (implies a damping ratio)
• Steady state error requirement implies Kv = 20.• For prop. controlled Type 1 plant: Kv = lims→0 sKpG(s).
Hence Kp = 8.• To find phase margin of prop. controlled loop we need
to find ωc , where |KpG(jωc)| =∣∣ 40
jωc(jωc+2)
∣∣ = 1
• ωc ≈ 6.2rad/s• Evaluate ∠KpG(jω) = −90◦ − atan(ω/2) at ω = ωc
• Hence φpm, prop = 18◦
EE 3CL4, §914 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Example
• φpm, prop = 18◦. Hence, need 27◦ of phase lead• Let’s go for a little more, say 30◦
• So, want peak phase of lead comp. to be 30◦
• Solving α−1α+1 = sin(30◦) yields α = 3. Set Kc = 3× 8
• Since 10 log10(3) = 4.8 dB we should choose ωm to bewhere 20 log10
(∣∣ 40jωm(jωm+2)
∣∣) = −4.8 dB
• Solving this equations yields ωm = 8.4rad/s• Therefore z = ωm/
√α = 4.8, p = αz = 14.4
• Gc(s) = 24(s+4.8)s+14.4
• Gc(s)G(s) = 120(s+4.8)s(s+2)(s+14.4) , actual φpm = 43.6◦
• Goal can be achieved by using a larger target foradditional phase, e.g., α = 3.5
EE 3CL4, §915 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Bode Diagram
EE 3CL4, §916 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Step Response
EE 3CL4, §917 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Ramp Response
EE 3CL4, §918 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Ramp Response, detail
EE 3CL4, §920 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Lag Compensators• Gc(s) = Kc(s+z)
s+p , with |p| < |z|, alternatively,
• Gc(s) = Kcα(1+sτ)1+sατ , where z = 1/τ and α = z/p > 1
• Bode diagrams of lag compensators for two differentαs, in the case where Kc = 1/α
EE 3CL4, §921 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
What will lag compensation do?
• Since zero and pole are typically close to the origin,phase lag aspect is not really used.
• What is useful is the attenuation above ω = 1/τ :gain is −20 log10(α), with little phase lag
• Can reduce cross-over frequency, ωc , without addingmuch phase lag
• Tends to reduce bandwidth
EE 3CL4, §922 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Qualitative example
• Uncompensated system has small phase margin• Phase lag of compensator does not play a large role• Attenuation of compensator does:ωc reduced by about a factor of a bit more than 3
• Increased phase margin is due to the natural phasecharacteristic of the plant
EE 3CL4, §923 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Bode Design Principles
For lag compensators:• Set the loop gain so that desired steady-state error
constants are obtained• Insert the compensator to modify the phase margin:
• Do this by reducing the cross-over frequency• Observe the impact on response time
Basic principle: Set attenuation to reduce ωc far enough sothat uncompensated open loop has desired phase margin
EE 3CL4, §924 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Design Guidelines1 For uncompensated (i.e., proportionally controlled)
closed loop, set gain Kp so that steady-state errorconstants of the closed loop meet specifications
2 Evaluate the phase margin, analytically, or using aBode diagram. If that is insufficient. . .
3 Determine ω′c , the frequency at which theuncompensated open loop, KpG(jω), has a phasemargin equal to the desired phase margin plus 5◦.
4 Design a lag comp. so that the gain of the compensatedopen loop, Gc(jω)G(jω), at ω = ω′c is 0 dB
• Choose Kc = Kp/α so that steady-state error const’sare maintained
• Place zero of the comp. around ω′c/10 so that at ω′c weget almost all the attenuation available from the comp.
• Choose α so that 20 log10(α) = 20 log10(|KpG(jω′c)|).With that choice and Kc = Kp/α, |Gc(jω′c)G(jω′c)| ≈ 1
• Place the pole at p = z/α• Compensator: Gc(s) = Kc(s+z)
s+p
EE 3CL4, §925 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Example, same set up as leaddesign
• Type 1 plant of order 2: G(s) = 5s(s+2)
• Design goals:• Steady-state error due to a ramp input less than 5% of
velocity of ramp• Phase margin at least 45◦ (implies a damping ratio)
• Steady state error requirement implies Kv = 20.• For prop. controlled Type 1 plant: Kv = lims→0 sKpG(s).
Hence Kp = 8.• To find phase margin of prop. controlled loop we need
to find ωc , where |KpG(jωc)| =∣∣ 40
jωc(jωc+2)
∣∣ = 1
• ωc ≈ 6.2rad/s• Evaluate ∠KpG(jω) = −90◦ − atan(ω/2) at ω = ωc
• Hence φpm, prop = 18◦
EE 3CL4, §926 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Example
• Since want phase margin to be 45◦, we set ω′c such that∠G(jω′c) = −180◦ + 45◦ + 5◦ = −130◦. =⇒ ω′c ≈ 1.5
• To make the open loop gain at this frequency equal to 0 dB,the required attenuation is 20 dB. Actual curves are around2 dB lower than the straight line approximation shown
• Hence α = 10. Set Kc = Kp/α = 0.8
• Zero set to be one decade below ω′c ; z = 0.15
• Pole is z/α = 0.015.
• Hence Gc(s) = 0.8(s+0.15)s+0.015
EE 3CL4, §927 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Example: Comp’d open loop
• Compensated open loop: Gc(s)G(s) = 4(s+0.15)s(s+2)(s+0.015)
• Numerical evaluation:• new ωc = 1.58• new phase margin = 46.8◦• By design, Kv remains 20
EE 3CL4, §928 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Step Response
EE 3CL4, §929 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Ramp Response
EE 3CL4, §930 / 30
Tim Davidson
FrequencyDomainApproach toCompensatorDesign
LeadCompensators
LagCompensators
Ramp Response, detail
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