代数Riccati不等式的另一种方法
Yet another approach to the Algebraic Riccati Inequality
摘要：
给出了可控和不可控系统代数Riccati不等式（ARI）解集的秩刻画。假设相应的代数Riccati方程（ARE）解的存在性，在不假设符号能控性的情况下，刻画了能控/不能控系统ARI解的有界/无界性质.作为我们观察的结果，我们得到了Willems的结果$K_{min}\leq K\leq K_{max}$的情况下，一个ARI的可控系统，并探讨了一些结构上的极值解。我们还考虑了好奇的情况下，无法控制的纯虚特征值和行为的解决方案集的ARI。特别是，我们证明了一个系统是可控的，当且仅当一个ARI的解决方案是有界的。此外，我们还研究了系统矩阵的特征值在复平面上的位置对ARI解集行为的影响。此外，我们还得到了能控系统ARI解的秩参数化。
英文摘要：
We give a rank characterization of the solution set of algebraic Riccati inequality (ARI) for both controllable and uncontrollable systems. Assuming an existence of a solution of the corresponding algebraic Riccati equation (ARE), we characterize the boundedness/unboundedness properties of solutions of ARI for controllable/uncontrollable systems without any assumption on sign controllability. As a consequence of our observations, we obtain Willems' result $K_{min}\leq K\leq K_{max}$ for an ARI in the case of controllable systems and explore some structure on the extremal solutions. We also consider the curious case of uncontrollable purely imaginary eigenvalues and the behavior of the solution set of ARI. In particular, we show that a system is controllable if and only if the set of solutions of an ARI is bounded. In addition, we study the effect of the position of eigenvalues of the system matrix in the complex plane on the behavior of the solution set of ARIs. Furthermore, we obtain a rank parametrization for solutions of ARI for controllable systems.
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