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Distributed-Order Dynamic Systems pp 39–58Cite as

Distributed-Order Filtering and Distributed-Order Optimal Damping

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Abstract

The idea of using the distributed-order differential equation first proposed by Caputo in (1969) is at least mathematically interesting as demonstrated in the previous chapters. However, people may question its usefulness in engineering practice. In this chapter, we included two generic applications. One is on distributed order signal processing and the other is on optimal distributed damping. We hope these two initial applications can serve as motivating examples to further the investigation in distributed order dynamics systems, signal processing, modeling and controls.

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Notes

  1. 1.

    It follows from the residue of \(\frac{\mathrm{ e}^{st}(s^{-a}-s^{-b})}{\ln (s)}\) which equals zero at \(s=\infty \), that the path integral of it along \(s\rightarrow \infty \) is vanished for \(b\le 1.\)

  2. 2.

    When \(s=0\), DC gain of \(\int _a^b\frac{1}{(s+\lambda )^\alpha }\mathrm{ d}\alpha =\int _a^b\frac{1}{\lambda ^\alpha }\mathrm{ d}\alpha =\frac{1}{\ln \lambda }\left(\frac{1}{\lambda ^a}-\frac{1}{\lambda ^b}\right)\). So, unity gain requires gain scaling factor \(\frac{\lambda ^{a+b}\ln \lambda }{\lambda ^b-\lambda ^a}.\)

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Authors and Affiliations

  1. Department of Automation, Tsinghua University, Beijing, People’s Republic of China

    Zhuang Jiao

  2. Department of Electrical and Computer Engineering, Utah State University, Old Main Hill, CSOIS 4120, Logan, UT, 84322-4120, USA

    YangQuan Chen

  3. BERG Faculty, Technical University of Kosice, B. Nemcovej 3, 04200, Kosice, Slovakia

    Igor Podlubny

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  1. Zhuang Jiao
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  2. YangQuan Chen
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  3. Igor Podlubny
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Correspondence to YangQuan Chen .

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Jiao, Z., Chen, Y., Podlubny, I. (2012). Distributed-Order Filtering and Distributed-Order Optimal Damping. In: Distributed-Order Dynamic Systems. SpringerBriefs in Electrical and Computer Engineering(). Springer, London. https://doi.org/10.1007/978-1-4471-2852-6_4

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  • DOI: https://doi.org/10.1007/978-1-4471-2852-6_4

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-2851-9

  • Online ISBN: 978-1-4471-2852-6

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