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Positive Realness of Second-order and High-order Descriptor Systems

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Abstract

This paper is concerned with the positive realness problem for second-order and high-order descriptor systems. First, without any linearization, necessary and sufficient conditions are established under which the second-order descriptor systems are strictly positive real and extended strictly positive real, respectively. Applying the relations between the positive realness and the optimal control theory, the solutions of the proposed positive real lemma equations can be represented by the symmetric positive semi-definite solutions for the second-order generalized Riccati equations. Then, employing polynomial matrix decomposition techniques, the extended strictly positive real lemma of high-order descriptor systems is also presented based on the original coefficient matrices of the system. Furthermore, linear matrix inequality conditions are given that can effectively test the positive realness and the extended strictly positive realness of the system. Finally, three numerical examples are provided to verify the effectiveness of the developed theoretical results.

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Acknowledgements

This work was supported by National Natural Science Foundation of China under grant 61473202.

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  1. School of Electrical and Information Engineering, Tianjin University, Tianjin, 300072, China

    Liping Zhang & Guoshan Zhang

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  1. Liping Zhang
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  2. Guoshan Zhang
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Correspondence to Guoshan Zhang.

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Zhang, L., Zhang, G. Positive Realness of Second-order and High-order Descriptor Systems. Circuits Syst Signal Process 39, 5882–5905 (2020). https://doi.org/10.1007/s00034-020-01449-z

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  • DOI: https://doi.org/10.1007/s00034-020-01449-z

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