Low-rank approximation with convex constraints

Researchers: Christian Grussler, Anders Rantzer, Pontus Giselsson, Andrey Ghulchak

Funding: LCCC Linnaeus center

Model Order Reduction of Postive Systems:

Transportation networks, biological systems as well as heat transfer model are only a few examples for systems with the fundemental property of operating with positively measured inputs and outputs only. Typically these systems are large-scale and one way of overcoming this issue in control and simulations is to approximate the systems with the help of so-called Model Order Redcution (MOR). Unfortuantely, stanard MOR-methods do not preserve positivity and by that may lead to false conclusions in simulations as well as controler design.

Research in Posivity Preserving Model Order Reduction has been conducted earlier, however with strong conservatism regarding dimensionality and errors. Our main goal is to supply new approximaton strategies with the incentive of weakening the current conservatism, e.g. by considering ellispoidal cone invariant systems.


Low-rank approximation with convex constraints

Model order reduction that is preserving external positivity is essentially equivalent to a low-rank approximation of an infinite-dimensional Hankel-matrix under the preservation of the Hankel-structure and the non-negativity. However, even for finite matrices it is unkown how to find an optimal low-rank approximation that preserves convex constraints. Instead heuristics, like the nuclear-norm regularization method are the state of the art. 

Our main goal is to fill this gap and give deterministic solutions that do not depend on a regularization parameter. Our main tool are the so-called "Low-rank Inducing Norm". Those norms are SDP-representable and have cheap computational proximal mappings. 

Software for Low-Rank Inducing norms can be found at: