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Design of Imaginary Spectrum of LTI Systems with Delays to Manipulate Stability Regions

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Time Delay Systems

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 7))

Abstract

This chapter is on the design problem of linear time-invariant (LTI) systems with delays. Our recent studies on the use of algebraic techniques, namely resultant and iterated discriminants operations, in connection with the well-known Rekasius transformation implemented on the system characteristic equation already revealed that it is indeed possible to compute the exact range of the imaginary spectrum of such systems. This know-how, which is the key toward understanding the stabilty/instability decomposition of the system, is utilized here to craft the imaginary spectrum of LTI systems with multiple delays, specifically with the aim to manipulate stability regions in a systematic manner in the delay parameter space.

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Notes

  1. 1.

    Here we assume that the system indeed exhibits crossings. Otherwise \(\varOmega \) is an empty set, and the system is delay-independent stable, or unstable.

  2. 2.

    These observations lead to a general proof that, except for possibly some special points in \(T_v\), for a feasible \(s=j\omega \) solution to exist the hypersurfaces formed by \(R_{21}\) and the numerator of \(R_1\) do not intersect in the L-dimensional \(T_v\) parameter space [27].

  3. 3.

    Notice that sweeping the frequency \(\omega \) is not equivalent to sweeping \(T_v\), \(\omega \tau _v\), or \(\omega T_v\) due to the intricate nonlinear relationships between (10) and (11), see also [5, 26].

  4. 4.

    The number of positive real roots of polynomials can be assessed following algebraic tools. If this number is zero and \(\omega \ne 0\), then \(\varOmega \) is an empty set hence the system maintains its stable/unstable characteristic irrespective of delays [13]. See also [5, 7, 17] for multiple delay treatments.

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Northeastern University, 360 Huntington Avenue, Boston, MA, 02115, USA

    Rifat Sipahi

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  1. Rifat Sipahi
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Correspondence to Rifat Sipahi .

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Editors and Affiliations

  1. Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest, Hungary

    Tamás Insperger

  2. Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA

    Tulga Ersal

  3. Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan, USA

    Gábor Orosz

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Sipahi, R. (2017). Design of Imaginary Spectrum of LTI Systems with Delays to Manipulate Stability Regions. In: Insperger, T., Ersal, T., Orosz, G. (eds) Time Delay Systems. Advances in Delays and Dynamics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-53426-8_9

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  • DOI: https://doi.org/10.1007/978-3-319-53426-8_9

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  • Online ISBN: 978-3-319-53426-8

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