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

Feedback Shaping of the Waterbed Effect and Transient Improvement in Feedforward Control Allocation Xu Chen Masayoshi Tomizuka CML Sponsors’ Meeting 2014 Mechanical Engineering Mechanical Engineering

Practical servo challenges and opportunities in HDDs (exaggerated demo) Internal disturbance External disturbance

Feedback local loop shaping 5 0 -5 baseline -10 Magnitude (dB) -15 w/ LLS -20 -25 -30 -35 -40 -45 -50 2 3 4 10 10 10 Frequency (Hz) -3 w/o compensation A scaled PES spectrum 1.5 x 10 under audio vibration Magnitude 1 0.5 0 500 1000 1500 2000 2500 3000 Frequency (Hz)

Theory: all-stabilizing local loop shaping Theorem : • 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 Much simplified design on Q • Great for adaptive control and loop shaping Mechanical Engineering

Achieved loop shapes Wide-band audio-vibration rejection 0 Small amplification at other frequencies -20 -40 Magnitude (dB) -60 solid: baseline dashed: w/ LLS -80 -100 -120 4kHz -140 500Hz -160 1 2 3 4 10 10 10 10 Frequency (Hz)

Achieved loop shapes Enhanced repetitive control for harmonic cancellation 10 0 -10 -20 Magnitude (dB) -30 w/ LLS -40 baseline -50 -60 -70 -80 1 2 3 4 10 10 10 10 Frequency (Hz) 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.

Benchmark I: rejection of disk flutter & fan noise Benchmark simulation w/o compensation 0.04 20 w/ compensation 3  = 8.40 %TP PES (%TP) 10 w/o compensation 3  = 14.52 %TP 0.035 0 0.03 -10 0.025 Magnitude -20 0 5 10 15 20 0.02 w/ compensation 20 0.015 PES (%TP) 10 0.01 0 0.005 -10 -20 0 0 5 10 15 20 500 1000 1500 2000 2500 Revolution Frequency (Hz) 80 Gain (dB) 60 40 20 0 1 2 3 4 10 10 10 10 180 Phase (degree) 90 0 -90 -180 1 2 3 4 10 10 10 10 Frequency (Hz)

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

Simulation and experimental results 0.04 0.04 Residual force [V] Residual force [V] Open loop Open loop 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 5 10 15 20 25 30 35 0 5 10 15 20 25 30 0.04 0.04 Residual force [V] Residual force [V] Closed loop Closed loop 0.02 0.02 0 0 -0.02 -0.02 -0.04 -0.04 5 10 15 20 25 30 35 0 5 10 15 20 25 30 Time [sec] Time [sec] simulation experiment Spectral density of the plant output Spectral density of the plant output -40 -40 open loop open loop -45 -45 closed loop closed loop -50 -50 -55 -55 -60 -60 dB [Vrms] dB [Vrms] -65 -65 -70 -70 -75 -75 -80 -80 -85 -85 -90 -90 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 Frequency [Hz] Frequency [Hz]

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Reaching and controlling the feedback limitations 10 10 5 0 0 -5 Magnitude (dB) Magnitude (dB) -10 -10 -15 -20 baseline -20  = 0.945 Detail at 3000 Hz -25  = 0.993 -30 -30 baseline  = 0.945 -35  = 0.993 -40 1 2 3 -40 10 10 10 1 2 3 10 10 10 Frequency (Hz) Frequency (Hz) Direct and flexible control of the waterbed effect Mechanical Engineering

Mathematical benefits of inverse parameterization • Proposed approximate coprime factorization • Sensitivity function Affine Q parameterization “Plant-independent” Q design

Flexible control of the waterbed effect: zero modulation 1- z - m Q ( z -1 ) 5 Pole-Zero Map 1 0 0.5  /T Magnitude (dB) 0.6  /T 0.4  /T 0.8 0.1 0.7  /T 0.3  /T -5 0.2 0.3 0.6 -10 0.8  /T 0.2  /T 0.4 0.5 0.4 0.6 -15 enhanced Imaginary Axis 0.7 0.9  /T 0.1  /T 0.8 baseline 0.2 0.9 -20 0 1 2 3 4  /T 10 10 10 10 10 0  /T Q ( z -1 ) -0.2 0.9  /T 0.1  /T 0 -0.4 -5 0.8  /T 0.2  /T Magnitude (dB) -0.6 -10 0.7  /T 0.3  /T -15 -0.8 0.6  /T 0.4  /T 0.5  /T -20 -1 -1 -0.5 0 0.5 1 -25 Real Axis -30 Flexible magnitude constraints 0 1 2 3 4 10 10 10 10 10 Frequency (Hz) Direct control via the Q design Mechanical Engineering

Practical feedback challenges in HDDs All-stabilizing local loop shaping Feed- Control of the “waterbed” effect back Adaptive (audio) vibration rejection Feedforward Transient improvement

Central idea of online frequency adaptation NF log plot: Amplitude Spectrum 5 1 log plot: Amplitude Spectrum 0.9 0 1 Magnitude (dB) 0.8 0.9 -5 0.7 0.8 Signal Power 0.7 -10 0.6 Signal Power 0.6 0.5 -15 0.5 45 0.4 0.4 0.3 0.3 Phase (deg) 0.2 0 0.2 0.1 0.1 0 0 2 3 4 10 10 10 2 3 4 -45 10 10 10 2 3 4 Frequency [Hz] 10 10 10 Frequency [Hz] Frequency (Hz) • 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 Mechanical Engineering

Output-error adaptation Output-error adaptation

Equation-error adaptation Equation-error adaptation

Adaptive audio-vibration rejection result -3 -3 w/o compensation w/o compensation 3 x 10 3 x 10 Magnitude Magnitude 2 2 1 1 0 0 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000 -3 -3 w/ compensation w/ compensation 3 x 10 3 x 10 Magnitude Magnitude 2 2 1 1 0 0 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000 Frequency (Hz) Frequency (Hz) -3 -3 w/o compensation w/o compensation 1.5 x 10 1.5 x 10 Magnitude Magnitude 1 1 0.5 0.5 0 0 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000 -3 -3 w/ compensation w/ compensation 1.5 x 10 1.5 x 10 Magnitude Magnitude 1 1 0.5 0.5 0 0 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000 Frequency (Hz) Frequency (Hz)

Feedforward control allocation General feedforward allocation Topic II:

Transfer function algebra Disturbance source Disturbance source HDD Mechanical path 1: HDD Mechanical path 1: P P spindle, actuator, etc spindle, actuator, etc Plant and Feedback d Plant and d Actuator Compensator Actuator u ff y = - PES u ff PES P C P -1 Feedback Compensator C HDD closed-loop system

Transfer function algebra • Controller are commonly designed to have stable zeros (general loop-shaping principle) Zeros are stable from Routh test e.g. • Controller poles are often marginally stable More stable zeros give better transient performance (distribution theory) Mechanical Engineering

Evaluation: half-sine shock resistance Feedforward injection Feedforward injection 30 30 w/ compensation w/ compensation w/o compensation w/o compensation 20 20 10 10 PES (%TP) PES (%TP) 0 0 -10 -10 -20 -20 -30 -30 0 1 2 3 4 5 0 1 2 3 4 5 Revolution Revolution 60 60 w/ compensation w/ compensation 40 40 w/o compensation w/o compensation 20 20 PES (%TP) PES (%TP) 0 0 -20 -20 -40 -40 -60 -60 0 1 2 3 4 5 0 1 2 3 4 5 Revolution Revolution