[PPT] - Feedback Shaping of the Waterbed Effect and Transient Improvement PowerPoint Presentation

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Mechanical Engineering Mechanical Engineering

Xu Chen Masayoshi Tomizuka

CML Sponsors’ Meeting 2014

Feedback Shaping of the Waterbed Effect and Transient Improvement in Feedforward Control Allocation

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Practical servo challenges and opportunities in HDDs

Internal disturbance

(exaggerated demo)

External disturbance

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Feedback local loop shaping

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5 Frequency (Hz) Magnitude (dB)

baseline w/ LLS

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Magnitude w/o compensation

Frequency (Hz) A scaled PES spectrum under audio vibration

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Mechanical Engineering

Coprime factorizations: P = N/D, C = X/Y, NX+DY = 1
Any stabilizing controller can be formed as:
S:={stable, proper, and rational transfer functions}
stability
Great for adaptive control and loop shaping

Theory: all-stabilizing local loop shaping

Theorem:

Much simplified design on Q

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Frequency (Hz) Magnitude (dB)

solid: baseline dashed: w/ LLS

Achieved loop shapes

Small amplification at other frequencies 4kHz 500Hz Wide-band audio-vibration rejection

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Achieved loop shapes

Enhanced repetitive control for harmonic cancellation

X. Chen and M. Tomizuka, “New Repetitive Control with Improved Steady-state Performance and Accelerated Transient,” IEEE Transactions on Control

Systems Technology, vol. 21, no. 3, doi:10.1109/TCST.2013.2253102.

X. Chen and M. Tomizuka, “An Enhanced Repetitive Control Algorithm using the Structure of a Disturbance Observer,” in Proceedings of 2012

IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Taiwan, Jul. 11-14, 2012, pp. 490-495.

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Benchmark I: rejection of disk flutter & fan noise

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10 20 w/o compensation PES (%TP) 5 10 15 20

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10 20 w/ compensation Revolution PES (%TP) 500 1000 1500 2000 2500 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Frequency (Hz) Magnitude w/ compensation 3 = 8.40 %TP w/o compensation 3 = 14.52 %TP

Benchmark simulation

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20 40 60 80 Gain (dB) 10

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Benchmark II: active suspension

I.D. Landau, Benchmark on adaptive regulation European Journal of Control 2013, July European Control Conference 2013, July

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Frequency [Hz] dB [Vrms] Spectral density of the plant output

pen loop

closed loop 50 100 150 200 250 300 350 400

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Frequency [Hz] dB [Vrms] Spectral density of the plant output

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closed loop

Simulation and experimental results

simulation experiment

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0.02 0.04 Residual force [V] Open loop 5 10 15 20 25 30 35

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0.02 0.04 Time [sec] Residual force [V] Closed loop 5 10 15 20 25 30

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0.02 0.04 Residual force [V] Open loop 5 10 15 20 25 30

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0.02 0.04 Time [sec] Residual force [V] Closed loop

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(Copy of summary page)

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Mechanical Engineering

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5 10 Frequency (Hz) Magnitude (dB) baseline  = 0.945  = 0.993 10

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10 Frequency (Hz) Magnitude (dB) baseline  = 0.945  = 0.993 Detail at 3000 Hz

Reaching and controlling the feedback limitations

Direct and flexible control of the waterbed effect

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Proposed approximate coprime factorization
Sensitivity function

Affine Q parameterization

“Plant-independent” Q design

Mathematical benefits of inverse parameterization

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Mechanical Engineering

Flexible control of the waterbed effect: zero modulation

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Magnitude (dB) Frequency (Hz) Q(z-1) enhanced baseline

Pole-Zero Map Real Axis Imaginary Axis

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0.2 0.4 0.6 0.8 1 0.1/T 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T

/T

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1/T 0.2/T 0.3/T 0.4/T 0.5/T 0.6/T 0.7/T 0.8/T 0.9/T

/T

Flexible magnitude constraints Direct control via the Q design

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Feed- back

Control of the “waterbed” effect Practical feedback challenges in HDDs Adaptive (audio) vibration rejection

Feedforward

All-stabilizing local loop shaping Transient improvement

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Mechanical Engineering

Central idea of online frequency adaptation

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 log plot: Amplitude Spectrum Frequency [Hz] Signal Power

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 log plot: Amplitude Spectrum Frequency [Hz] Signal Power

Choices for parameter adaptation algorithms (PAA):
Equation-error methods: simple, guaranteed convergence in the

noise-free case

Output-error methods: good performance in noisy environments

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Output-error adaptation

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Equation-error adaptation

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Adaptive audio-vibration rejection result

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Magnitude w/o compensation 500 1000 1500 2000 2500 3000 0.5 1 1.5 x 10

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Frequency (Hz) w/ compensation Magnitude 500 1000 1500 2000 2500 3000 0.5 1 1.5 x 10

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Magnitude w/o compensation 500 1000 1500 2000 2500 3000 0.5 1 1.5 x 10

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Frequency (Hz) w/ compensation Magnitude 500 1000 1500 2000 2500 3000 1 2 3 x 10

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Magnitude w/o compensation 500 1000 1500 2000 2500 3000 1 2 3 x 10

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Frequency (Hz) w/ compensation Magnitude 500 1000 1500 2000 2500 3000 1 2 3 x 10

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Magnitude w/o compensation 500 1000 1500 2000 2500 3000 1 2 3 x 10

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Frequency (Hz) w/ compensation Magnitude

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Feedforward control allocation

General feedforward allocation

Topic II:

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P C P

1

Plant and Actuator Feedback Compensator

PES

HDD Mechanical path 1: spindle, actuator, etc d Disturbance source uff

P P

Plant and Actuator HDD Mechanical path 1: spindle, actuator, etc

C

Feedback Compensator Disturbance source d

y = -PES

uff

HDD closed-loop system

Transfer function algebra

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Mechanical Engineering

Transfer function algebra

Controller are commonly designed to have

stable zeros (general loop-shaping principle)

Controller poles are often marginally stable

Zeros are stable from Routh test More stable zeros give better transient performance (distribution theory) e.g.

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20 40 60 Revolution PES (%TP) w/ compensation w/o compensation

Evaluation: half-sine shock resistance

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10 20 30 Revolution PES (%TP) w/ compensation w/o compensation

Feedforward injection Feedforward injection