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Unformatted text preview: ME575 Handouts A Brief Look at Bandwidth (Frequency at which the magnitude curve is 3 dB below its value at low frequencies)
For pole at s = −ζωn±jωd:
For pole at s = −a: ωBW = a
40 0
3 ωBW = ωn 1 − 2ζ 2 + 4ζ 4 − 4ζ 2 + 2 20
Phase (deg); Magnitude (dB) Phase (deg); Magnitude (dB) 0
20 40 0 45 20
40
60
0
45
90 135
90 180 0.01a 0.1a a 10a 100a 0.1ωn Frequency (rad/sec)
ME575 Session 16 – FrequencyDomain Limitations School of Mechanical Engineering
Purdue University ωn 10ωn Frequency (rad/sec)
Slide 1 FREQUENCYDOMAIN DESIGN LIMITATIONS
• Bode’s Integral Constraints on Sensitivity
– Water Bed Effect for systems with or without RHP
poles • Integral Constraints on Complementary
Sensitivity
– Systems with or without RHP zeros • Poisson Integral Constraint on Sensitivity
– Systems with RHP zeros • Poisson Integral Constraint on Complementary
Sensitivity
– Systems with RHP poles
ME575 Session 16 – FrequencyDomain Limitations School of Mechanical Engineering
Purdue University Slide 2 1 ME575 Handouts Bode’s Integral Constraints on Sensitivity
• Lemma 9.1 (Water Bed Effect)
Consider stable CL system with oneDOF controller configuration and
open loop TF given by L(s) = G(s)C(s) = e − sτL(s), τ ≥ 0
where L(s) is a rational TF of relative degree r = nL − mL > 0.
Assume that L(s) has no openloop poles in open RHP. Then, the
sensitivity function satisfies ∫ ∞ 0 where ⎧0
⎪
ln S ( j ω ) d ω = ⎨
π
⎪− κ 2
⎩ for τ > 0 for τ = 0 κ ≡ lim s → ∞ sL(s) ME575 Session 16 – FrequencyDomain Limitations School of Mechanical Engineering
Purdue University Slide 3 PID Control of DC Motor Position
Sensitivity Functions with and without actuation delay
Plant
ln S ( jω)
623
Log sensitivity function Go (s) = s(s + 3.5) PlD 1 Controller C1(s) = 8.3s2 + 205s + 2054
s(s + 117) Without delay
L1(s) = Go (s)C1(s)
With actuation delay L1T (s) = e−τsGo (s)C1(s)
τ = 0.01sec
PlD 3 Controller 34s2 + 1550s + 26729
s(s + 237)
All log sensitivity functions have equal negative and positive integral areas ! Frequency ω ME575 Session 16 – FrequencyDomain Limitations School of Mechanical Engineering
Purdue University C3 (s) = Slide 4 2 ME575 Handouts Bode’s Integral Constraints on Sensitivity
• Lemma 9.2 (Unstable Open Loop Poles)
Consider stable CL system with oneDOF controller configuration and
open loop TF given by L(s) = G(s)C(s) = e − sτL(s), τ ≥ 0
where L(s) is a rational TF of relative degree r = nL − mL > 0. Assume that L(s) has unstable openloop poles at p1,… p N . Then,
the sensitivity function satisfies ∫ ∞ 0 ∫ ∞ 0 N ln S ( j ω ) d ω = π ∑ Re {p i }, for τ > 0 i =1 ln S ( j ω ) d ω = −κ N
π
+ π ∑ Re {p i },
2
i =1 for τ = 0 where κ ≡ lim s → ∞ sL(s)
ME575 Session 16 – FrequencyDomain Limitations School of Mechanical Engineering
Purdue University Slide 5 Bode’s Integral Constraints on Sensitivity
Bode
• Conclusions
– Independent of controller design, low sensitivity in
certain prescribed frequency bands will result in a
sensitivity larger than one in other frequency bands
– With unstable openloop poles, the integral of log
opensensitivity is required to be greater than zero, which
makes sensitivity minimization more difficult ME575 Session 16 – FrequencyDomain Limitations School of Mechanical Engineering
Purdue University Slide 6 3 ME575 Handouts Integral Constraints on Complementary Sensitivity
• Lemma 9.3 (Minimum Phase Systems or No Unstable Zeros)
(Minimum
Consider stable CL system with oneDOF controller configuration and
open loop TF given by L(s) = G(s)C(s) = e − sτL(s), τ ≥ 0
where L(s) is a rational TF of relative degree r = nL − mL > 1 and has
at least one free integrator (i.e., L(0)−1 = 0 ) . Assume that L(s) has
no openloop zeros in open RHP. Then, the complementary sensitivity
function satisfies ∫ ∞ 0− where 1
π
ln T ( jω ) d ω = πτ −
2
2K V
ω K V ≡ lim s 0 sL(s) Note: ∫ ∞ 0 − 1
ln T ( jω ) d ω =
ω2 ME575 Session 16 – FrequencyDomain Limitations ∫ ∞ 0 ⎛ 1⎞
ln T ⎜ j ⎟ dv,
⎝ v⎠ v= 1
ω School of Mechanical Engineering
Purdue University Slide 7 PID Control of DC Motor Position
Complementary Sensitivity Functions
Log complementary sensitivity function ln T ( jω) Plant Go (s) = 623
s(s + 3.5) PlD1 Controller C1(s) = 8.3s2 + 205s + 2054
s(s + 117) Without delay
L1(s) = Go (s)C1(s)
With actuation delay L1T (s) = e−τsGo (s)C1(s)
τ = 0.01sec
PlD3 Controller C3 (s) = 34s2 + 1550s + 26729
s(s + 237) Inverse Frequency v = 1/ ω
All log complementary sensitivity functions have equal negative and positive integral
areas w.r.t. v = 1/ω when τ = 0, and larger positive integral area when τ > 0 !
ME575 Session 16 – FrequencyDomain Limitations School of Mechanical Engineering
Purdue University Slide 8 4 ME575 Handouts Integral Constraints on Complementary Sensitivity
• Lemma 9.4 (Nonminimum Phase Systems or Unstable Zeros)
Consider stable CL system with oneDOF controller configuration and
open loop TF given by L(s) = G(s)C(s) = e − sτL(s), τ ≥ 0
where L(s) is a rational TF of relative degree r = nL − mL > 1 and has
at least one free integrator (i.e., L(0)−1 = 0 ) . Assume that L(s) has
unstable openloop zeros at z1,… , z M . Then, the complementary
sensitivity function satisfies ∫ ∞ 0− where M
1
π
1
ln T ( jω ) d ω = πτ −
+ π∑
2
ω
2K V
i =1 z i K V ≡ lim s → 0 sL(s) Note: ∫ ∞ 0− 1
ln T ( jω ) d ω =
ω2 ME575 Session 16 – FrequencyDomain Limitations ∫ ∞ 0 ⎛ 1⎞
ln T ⎜ j ⎟ dv,
⎝ v⎠ v= 1
ω School of Mechanical Engineering
Purdue University Slide 9 Bode’s Integral Constraints on Sensitivity
Bode
• Conclusions
– Independent of controller design, low complementary
sensitivity in certain prescribed frequency bands will
result in a complementary sensitivity larger than one in
other frequency bands
– In general, negative integral is unavoidable at high
frequency range for noise attenuation while zero value is
desirable at low frequency range for command following
and disturbance rejection
– The appearance of unstable openloop zeros adds more
openpositive value to the integral of log complementary
sensitivity, and thus makes the allocation of
complementary sensitivity in the frequency domain more
difficult
ME575 Session 16 – FrequencyDomain Limitations School of Mechanical Engineering
Purdue University Slide 10 5 ...
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 Fall '10
 Meckl

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