Positive Realness of Second-order and High-order Descriptor Systems
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Positive Realness of Second-order and High-order Descriptor Systems Liping Zhang1 · Guoshan Zhang1 Received: 11 June 2019 / Revised: 2 May 2020 / Accepted: 5 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper is concerned with the positive realness problem for second-order and highorder 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. Keywords Generalized positive real lemma · Descriptor system · Linear quadratic optimal control · Matrix decomposition
1 Introduction Positive realness is an important property in linear circuit network and system control theory, which has found many applications in the realization theory of electric networks, absolute stability analysis and inverse optimal control problem [1,6,25,32,33]. It is well known that the descriptor system provides more natural mathematical descriptions than the normal linear system in the modeling of electrical circuit systems,
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Guoshan Zhang [email protected] Liping Zhang [email protected]
1
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
Circuits, Systems, and Signal Processing
mechanical systems and other areas [13,23]. Over the last decade, the positive realness analysis and positive real control problems for descriptor systems have aroused the scholars great attention. For instance, [38] first proposed extended strictly positive real conditions for continuous-time and discrete-time descriptor systems based on the generalized Riccati equation and inequality. [34] established a new extended strictly positive real condition for discrete descriptor systems by means of strict LMIs technique, which is different from the non-strict case in [38]. To improve the previous results in [35,38] presented another version of positive real lemma for descriptor systems. But the above articles mainly focus on the impulse-free descriptor system, [7] removed the extra assumption on the system and put forward the extension version of Kalman-Yakubovich-Popov(KYP) lemma in the behavioral framework. Applying the similar idea, [26,27] investigated a mor
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