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Second-order cone certification of external positivity

Grussler, C and Rantzer, A Second-order cone certification of external positivity. (Unpublished)

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Abstract

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 the its realization dependency. Firstly, computing such a realization requires to find a polyhedral cone with a potentially high number of extremal generators that lifts the dimension of the state-space representation, significantly. Secondly, not all externally positive systems posses an internally positive realization. Thirdly, 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 certificate of external positivity based on finding second-order cones. In particular, the certificate is guaranteed to succeed, whenever it is feasible for any minimal realization. Nonetheless, it is also revealed that neither second-order cones nor polyhedral cones can provide a necessary certificate. Finally, we exemplify the potential of our certificate based on three applications. We discuss how our certificate an be used to find externally positive approximations of nearly externally positive systems, which as demonstrated can help to reduce system identification errors. The same algorithm is used then to design state-feedback controllers that avoid over- and undershooting. Lastly, we present modifications to generalized balanced truncation such that external positivity is preserved.

Item Type: Article
Uncontrolled Keywords: math.OC math.OC 49N35, 93B30, 93B15, 93B55, 93B52, 49N05
Subjects: UNSPECIFIED
Divisions: Div F > Control
Depositing User: Cron Job
Date Deposited: 27 Jun 2019 01:18
Last Modified: 02 Jun 2020 08:11
DOI: