SIAM Journal on Control and Optimization, Volume 59, Issue 4, Page 2717-2739, January 2021.
Internal positivity offers a computationally cheap certificate for external (input-output) positivity of a linear time-invariant system. However, the drawback with this certificate lies in its realization dependency. First, computing such a realization requires finding a polyhedral cone with a potentially high number of extremal generators that lifts the dimension of the state-space representation, significantly. Second, not all externally positive systems possess an internally positive realization. Third, in many typical applications such as controller design, system identification, and model order reduction, internal positivity is not preserved. To overcome these drawbacks, we present a tractable sufficient certificate of external positivity based on second-order cones. This certificate does not require any special state-space realization: if it succeeds with a possibly non-minimal realization, then it will do so with any minimal realization. While there exist systems where this certificate is also necessary, we also demonstrate how to construct systems, where both second-order and polyhedral cones as well as other certificates fail. Nonetheless, in contrast to other realization independent certificates, the second-order-cone one appears to be favorable in terms of applicability and conservatism. Three applications are representatively discussed to underline its potential. We show how the certificate can be used to find externally positive approximations of nearly externally positive systems and demonstrate that this may help to reduce system identification errors. The same algorithm is used then to design state-feedback controllers that provide closed-loop external positivity, a common approach to avoid over- and undershooting of the step response. Last, we present modifications to generalized balanced truncation such that external positivity is preserved for those systems, where our certificate applies.

中文翻译：

---

二阶锥正系统

SIAM Journal on Control and Optimization，第 59 卷，第 4 期，第 2717-2739 页，2021 年 1 月。
内部正性为线性时不变系统的外部（输入-输出）正性提供了计算成本低廉的证明。但是，此证书的缺点在于其实现依赖性。首先，计算这样的实现需要找到一个多面体锥体，该锥体具有潜在的大量极值生成器，可以显着提升状态空间表示的维度。其次，并非所有外部积极的系统都具有内部积极的实现。第三，在控制器设计、系统辨识、模型降阶等许多典型应用中，内部积极性没有得到保留。为了克服这些缺点，我们提出了一个基于二阶锥体的易于处理的外部正性证明。此证书不需要任何特殊的状态空间实现：如果它以可能的非最小实现成功，那么它将以任何最小实现实现。虽然存在也需要此证书的系统，但我们还演示了如何构建二阶和多面体锥以及其他证书都失败的系统。尽管如此，与其他实现独立证书相比，二阶锥证书在适用性和保守性方面似乎更有利。有代表性地讨论了三个应用程序以强调其潜力。我们展示了如何使用证书来查找几乎外部正系统的外部正近似，并证明这可能有助于减少系统识别错误。然后使用相同的算法来设计提供闭环外部积极性的状态反馈控制器，避免阶跃响应过冲和下冲的常用方法。最后，我们提出了对广义平衡截断的修改，以便为我们的证书适用的那些系统保留外部积极性。