16 - FrequencyLimitations16 filled

# 16 - FrequencyLimitations16 filled - ME575 Handouts A Brief...

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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 – Frequency-Domain Limitations School of Mechanical Engineering Purdue University ωn 10ωn Frequency (rad/sec) Slide 1 FREQUENCY-DOMAIN 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 – Frequency-Domain 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 one-DOF 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 open-loop 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 – Frequency-Domain 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 – Frequency-Domain 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 one-DOF 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 open-loop 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 – Frequency-Domain 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 open-loop poles, the integral of log opensensitivity is required to be greater than zero, which makes sensitivity minimization more difficult ME575 Session 16 – Frequency-Domain 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 one-DOF 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 open-loop 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 – Frequency-Domain 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 – Frequency-Domain 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 one-DOF 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 open-loop 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 – Frequency-Domain 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 open-loop 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 – Frequency-Domain Limitations School of Mechanical Engineering Purdue University Slide 10 5 ...
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