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Research Papers

Periodic Signal Tracking for Lightly Damped Systems

[+] Author and Article Information
Rick van der Maas

Control Systems Technology Group,
Department of Mechanical Engineering,
Eindhoven University of Technology,
Eindhoven 5600 MB, The Netherlands
e-mail: rick@vdmaas.eu

Tarunraj Singh

Department of Mechanical and
Aerospace Engineering,
University at Buffalo,
Buffalo, NY 14260

Maarten Steinbuch

Control Systems Technology Group,
Department of Mechanical Engineering,
Eindhoven University of Technology,
Eindhoven 5600 MB, The Netherlands

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received August 31, 2015; final manuscript received December 4, 2016; published online March 24, 2017. Assoc. Editor: Douglas Bristow.

J. Dyn. Sys., Meas., Control 139(6), 061007 (Mar 24, 2017) (8 pages) Paper No: DS-15-1407; doi: 10.1115/1.4035454 History: Received August 31, 2015; Revised December 04, 2016

The focus of this paper is on the development of time-delay filters to accomplish tracking of periodic signals with zero phase errors. The class of problems addressed include systems whose dynamics are characterized by lightly damped modes. A general approach for the zero-phase tracking of periodic inputs is presented followed by an illustration of single harmonic tracking of underdamped second-order systems with relative degree two. A general formulation of the approach is then posed for higher-order systems and systems including zeros. The paper concludes with the illustration of enforcing constraints to desensitize the time-delay filter to uncertainties in the location of the poles of the system and forcing frequencies. A numerical practical design case based on a medical X-ray system is used to illustrate the potential of the proposed technique.

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Figures

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Fig. 1

Time-delay controlled open-loop structure of a second-order system

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Fig. 2

Upper: resulting responses; reference signal (black solid), unshaped (plant only) (blue solid), point-to-point shaper (black dashed–dotted), and harmonic shaper (red dashed). Lower: resulting errors.

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Fig. 3

Input signals: reference signal (solid), point-to-point shaper (dashed-dotted), and harmonic shaper (dashed)

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Fig. 4

Frequency domain representations. Left: plant P(s) (blue solid), Gptp(s) (red dashed) and Hptp(s)=Gptp(s)P(s) (black dashed-dotted). Right: plant P(s) (blue), Gh(s) (red dashed) and Hh(s)=Gh(s)P(s) (black dashed-dotted). The vertical black lines indicated the evaluated harmonic in the reference signal.

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Fig. 5

Illustration of the controller parameters as a function of a variation of the forcing frequency according to the prefilter structure indicated in Fig. 1: A0 (blue), A1 (red), and A2 (black)

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Fig. 6

Upper: resulting responses; reference signal (black solid), unshaped (plant only) (blue solid), and harmonic shaper (red dashed). Lower: resulting errors.

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Fig. 7

Frequency domain representations; Left: plant P(s) (blue solid), Gnp(s) (red dashed) where the subscript np indicates “no pole compensation,” i.e., only compensation for the excited harmonic, and Hnp(s)=Gnp(s)P(s) (black dashed-dotted); Right: plant P(s) (blue), Gh(s) (red dashed), and Hh(s)=Gh(s)P(s) (black dashed–dotted) with the subscript h indicating the harmonic time-delay filter. The plant has a pole at 1 Hz, while the pure-sinusoidal reference has an harmonic at 0.8 Hz (encircled).

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Fig. 8

Steady-state errors: Gh(s) (solid) and Gnp(s) (dashed)

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Fig. 9

Absolute errors due to uncertainties: Gh(s) (solid), Gω(s) (robust implementation for plant variations; red dashed–dotted), Gω̂(s) (robust implementation for forcing frequency variations; green dash-circles), and GR(s) (blue dashed); left: variations in ω̂; right: variations in model parameter

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Fig. 10

Upper: resulting responses; reference signal (black solid), unshaped (plant only) (blue solid), and harmonic shaper (red dashed); lower left, error for time 0–10 s; lower right, error for time 10–20 s (zoomed)

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Fig. 11

Upper:resulting responses; reference signal (black solid), unshaped (plant only) (blue solid), and harmonic shaper (red dashed); lower: resulting errors

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Fig. 12

Control setup of Philips Xper Allura FD20 system and implemented control scheme

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Fig. 13

Frequency domain representations; left: plant P(s) (blue solid), H(s)=P(s)Gh(s) (red dashed-dotted), H(s)=P(s)GhR(s) (green dotted); right: zoomed

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Fig. 14

Upper: resulting time-domain responses; reference signal (black dashed), unshaped (plant only) (red solid), shaped results (blue solid); lower: corresponding error signals. Left: Normal hTDF (L = 12); right: robust hTDF (L = 16). Gray area: worst-case uncertainty bound for ±10% variations.

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Fig. 15

Resulting gains over a wide range of frequencies Ai ∀i=1,…,L given T; upper: normal hTDF; lower robust hTDF

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