正实引理与广义正有理函数所有实现的构造
The positive real lemma and construction of all realizations of generalized positive rational functions
AI总结：
将正实引理推广到非最小实现的复矩阵值广义正有理函数，并利用该结果给出所有状态空间实现的构造方法，进而通过Lyapunov包含划分实现子集并刻画可通过静态输出反馈变为广义正的系统。
摘要：
本文将著名的正实引理（也称为Kalman-Yakubovich-Popov引理）推广到考虑非最小实现时的复矩阵值广义正有理函数。然后利用这一结果，为广义正函数的所有（不一定最小）状态空间实现提供了一种简单的构造过程。作为副产品，我们将所有状态空间实现划分为子集：每个子集由满足相同Lyapunov包含的一组矩阵标识，从而形成一个凸可逆锥，简称cic。此外，这种方法使我们能够刻画可通过静态输出反馈变为广义正的系统。通过Lyapunov包含的表述建议引入一个与相同系统矩阵相关联的不同维数有理函数的等价类。
英文摘要：
We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a by-product, we partition all state space realizations into subsets: Each is identified with a set of matrices satisfying the same Lyapunov inclusion and thus form a convex invertible cone, cic in short. Moreover, this approach enables us to characterize systems which may be brought to be generalized positive through static output feedback. The formulation through Lyapunov inclusions suggests the introduction of an equivalence class of rational functions of various dimensions associated with the same system matrix.
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