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Root Locus of Zeros of Discrete Time Systems as a Function of Sample Rate

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Abstract

Root locus plots are one of the basic design tools in classical control. They help the designer tune control gains which appear linearly in the coefficients of the closed loop characteristic polynomial. And they give considerable intuition to the designer, based on the simple rules that root loci must follow. When designing a control system, one wants to know where the zeros are, but when designing a digital control system new issues appear. The original zero locations when mapped to discrete time are functions of the new parameter, the sample time T (as well as the pole locations). In addition, new zeros are usually introduced by the discretization process. The purpose of this paper is to give a general understanding of the nature of root loci of discrete time transfer function zeros as a function of this parameter T. We consider the complete range of values from T equal zero to infinity to understand the full plot. Reasonable sample rates will only use part of the plots. The characteristic polynomial coefficients are nonlinear functions of T so the usual root locus rules do not apply. One can be amazed at how the usual root locus rules are repeatedly violated, and what new kinds of unexpected behavior can be observed.

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Acknowledgments

This work has been partially supported by the China Fundamental Research Funds for the Central Universities (No. 3082015NS2015027).

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Authors and Affiliations

  1. College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, China

    Wenxiang Zhou

  2. Columbia University, MC4703, 500 West 120th Street, New York, NY, 10027, USA

    Richard W. Longman

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  1. Wenxiang Zhou
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  2. Richard W. Longman
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Correspondence to Wenxiang Zhou.

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Zhou, W., Longman, R.W. Root Locus of Zeros of Discrete Time Systems as a Function of Sample Rate. J Astronaut Sci 67, 164–187 (2020). https://doi.org/10.1007/s40295-019-00164-5

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  • DOI: https://doi.org/10.1007/s40295-019-00164-5

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