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Research Papers

Phase Compensation Strategies for Modulated-Demodulated Control With Application to Pulsed Jet Injection

[+] Author and Article Information
Cory Hendrickson

 Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, CA 90095

Robert T. M’Closkey

 Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, CA 90095rtm@seas.ucla.edu

J. Dyn. Sys., Meas., Control 134(1), 011024 (Dec 06, 2011) (9 pages) doi:10.1115/1.4004768 History: Received November 06, 2010; Revised May 12, 2011; Published December 06, 2011; Online December 06, 2011

Modulated-demodulated control is an effective method for asymptotic disturbance rejection and reference tracking of periodic signals, however, conventional static phase compensation often limits the loop gain in order to avoid sensitivity function peaking in a neighborhood of the frequencies targeted for rejection or tracking. This paper introduces dynamic phase compensation for modulated-demodulated control which improves disturbance rejection characteristics by inverting the plant phase in a neighborhood of the control frequency. Dynamic phase compensation is implemented at baseband which enables the use of low-bandwidth compensators to invert high frequency dynamics. Both static and dynamic phase compensation methods are used to demonstrate a novel application of repetitive control for pulsed jet injection. In this application pulsing an injectant has been shown to produce advantageous effects such as increased mixing in many energy generation and aerospace systems. The sharpness of the pulse can have a large impact on the effectiveness of control. Modulated-demodulated control is used to maximize the sharpness of a pulsed jet of air using active forcing by tracking a square wave in the jet’s temporal velocity profile.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

The “Dynamic phase” modulated-demodulated controller for asymptotic reference tracking of a single sinusoid at frequency ωo

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Figure 2

Pulsed jet injection experimental setup. A piston, actuated by a modal shaker, is used to actively control the temporal velocity profile of a jet at the nozzle exit.

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Figure 3

Actuation system frequency response. The magnitude roll-off after the plenum mode at 1.7 kHz produces an actuator with an approximate bandwidth of 2 kHz. The dashed line represents the bandwidth for control around a single frequency with ωc  = 50 Hz. The linear phase delay represents a significant transport lag which makes high-gain, wide bandwidth control impossible to achieve.

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Figure 4

An example of dynamic phase compensators for plant phase inversion in a neighborhood around the control frequency (ωo  = 100 Hz). Solid line—empirical, dashed line—model fit. (a) Hd and (b) Hx . The empirical data are fit up to 42 Hz using an eighth order and sixth order model for Hd and Hx , respectively.

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Figure 7

Envelope of the response of a ωo  = 100 Hz dynamic phase controller to a 1.4 ms−1 step input. The controller’s gain is determined by a specified time constant of τ = 0.050 s to be K=20/|P(jωo)|.

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Figure 8

Loop transfer function comparison of static phase compensation (solid) to dynamic phase compensation (dashed) for control at ωo  = 100 Hz with measured time constant τ = 0.045 s

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Figure 9

Nyquist plot comparison of static phase compensation (thick solid line) to dynamic phase compensation (thin solid line) for control at ωo  = 100 Hz with measured time constant τ = 0.045 s

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Figure 10

Sensitivity function comparison of static phase compensation at 100 Hz (solid line), 200 Hz (dashed line), 300 Hz (dot-dashed line), and 400 Hz (dotted line) with measured time constant τ = 0.045 s

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Figure 11

Sensitivity function comparison of dynamic phase compensation at 100 Hz (solid line), 200 Hz (dashed line), 300 Hz (dot-dashed line), and 400 Hz (dotted line) with measured time constant τ = 0.045 s

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Figure 12

Hotwire noise spectrum for the open-loop (case a) and closed-loop (case b) system when the reference signal coefficients are zero. Each controller creates a deep notch in the noise spectrum in the closed-loop case. The mean jet velocity is 8 ms−1 .

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Figure 13

Suppression of harmonic distortion in response to a 100 Hz single tone input ((a) open-loop and (b) closed-loop)

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Figure 14

Elimination of intermodulation distortion in response to a 300 Hz and 400 Hz dual tone input ((a) open-loop and (b) closed-loop)

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Figure 5

Details of pulses produced with a repetition rate of 100 Hz and α = 20% (case a), and α = 40% (case b). One period of an ideal square wave (dashed line) is compared to the ideal square wave truncated at 20 sinusoids (thick solid line) and the empirical square wave (thin solid line).

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Figure 6

Square wave spectrum (a) α = 20% and (b) α = 40%. The empirical square wave spectrum at each forcing frequency (circles) is identical to the ideal square wave Fourier series (Xs).

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