On second-order cone positive systems

TL;DR

This paper introduces a second-order cone-based certificate for external positivity of linear systems that overcomes realization dependency issues inherent in internal positivity certificates, with applications in system approximation, control design, and model reduction.

Contribution

It proposes a realization-independent, tractable sufficient certificate for external positivity based on second-order cones, expanding applicability and reducing conservatism compared to existing methods.

Findings

The certificate is applicable to a wide class of systems without special realizations.

It enables the construction of externally positive system approximations.

It facilitates the design of controllers ensuring external positivity and preserves this property during model reduction.

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 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 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 present one appears to be favourable 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 demonstrated 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. Lastly, we present modifications to generalized balanced truncation such that external positivity is preserved where our certificate applies.