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Sparse Representation for Sampled-Data \(H^\infty \) Filters

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Realization and Model Reduction of Dynamical Systems

Abstract

We consider the problem of discretization of analog filters and propose a novel method based on sampled-data \(H^\infty \) control theory with sparse representation. For optimal discretization, we adopt minimization of the \(H^\infty \) norm of the error system between a (delayed) target analog filter and a digital system consisting of an ideal sampler, the zero-order hold, and an FIR (finite impulse response) filter. Also, for digital implementation, we propose a sparse representation of the FIR filter to reduce the number of nonzero coefficients with the \(\ell ^1\) norm regularization. We show that this multi-objective optimization is reducible to a convex optimization problem, which can be solved efficiently by numerical computation. We then extend the design method to multi-rate filters, and show a design example. We also give an application to the feedback filter design of delta-sigma modulators.

Supported in part by the JSPS grant JP20H02172 and 20K21008; Supported in part by the JSPS grant JP19H02161.

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Notes

  1. 1.

    A real-rational transfer function is a rational function of s with real coefficients.

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Author information

Authors and Affiliations

  1. The University of Kitakyushu, Hibikino 1-1, Wakamatsu, Kitakyushu, Fukuoka, 808-0135, Japan

    Masaaki Nagahara

  2. Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Yutaka Yamamoto

Authors
  1. Masaaki Nagahara
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  2. Yutaka Yamamoto
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Corresponding author

Correspondence to Masaaki Nagahara .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Christopher Beattie

  2. Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  3. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Mark Embree

  4. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Serkan Gugercin

  5. IMT Lille Douai, Douai Cedex, France

    Sanda Lefteriu

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Nagahara, M., Yamamoto, Y. (2022). Sparse Representation for Sampled-Data \(H^\infty \) Filters. In: Beattie, C., Benner, P., Embree, M., Gugercin, S., Lefteriu, S. (eds) Realization and Model Reduction of Dynamical Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-95157-3_23

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