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.

中文翻译：

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二阶和高阶描述符系统的正真实性

本文关注的是二阶和高阶描述符系统的正真实性问题。首先，在没有任何线性化的情况下，建立了二阶描述符系统分别是严格正实数和扩展严格正实数的充分必要条件。应用正实性与最优控制理论之间的关系，所提出的正实引理方程的解可以用二阶广义Riccati方程的对称半正定解来表示。然后，利用多项式矩阵分解技术，在高阶描述子系统的原始系数矩阵的基础上，还给出了扩展严格正实数引理。此外，给出了线性矩阵不等式条件，可以有效地检验系统的正实在性和扩展的严格正实在性。最后，提供了三个数值例子来验证所开发的理论结果的有效性。