Christian Grußler

Nadav Barak, Christian Grussler

Data-Enabled Predictive Control (DeePC) has recently emerged as a framework for controlling unknown systems from data. However, its performance relies on the relevance of the collected data, and as such, disturbances lead to inevitable errors. This paper addresses this problem by proposing an augmentation of DeePC using Model-Free Disturbance Observer with Online Modification (MFDOOM). The method corrects output predictions based on previous prediction errors using a dedicated continuously updated Hankel matrix. We compare our method, both theoretically and through simulation, to other recent algorithms designed for time-varying systems in the DeePC framework. It is shown that for disturbances that can be modeled as the output of an autonomous linear time-invariant system, this approach can reduce tracking error and online-update burden compared with existing online DeePC variants.

Christian Grussler, Pontus Giselsson

Low-rank inducing unitarily invariant norms have been introduced to convexify problems with low-rank/sparsity constraint. They are the convex envelope of a unitary invariant norm and the indicator function of an upper bounding rank constraint. The most well-known member of this family is the so-called nuclear norm. To solve optimization problems involving such norms with proximal splitting methods, efficient ways of evaluating the proximal mapping of the low-rank inducing norms are needed. This is known for the nuclear norm, but not for most other members of the low-rank inducing family. This work supplies a framework that reduces the proximal mapping evaluation into a nested binary search, in which each iteration requires the solution of a much simpler problem. This simpler problem can often be solved analytically as it is demonstrated for the so-called low-rank inducing Frobenius and spectral norms. Moreover, the framework allows to compute the proximal mapping of compositions of these norms with increasing convex functions and the projections onto their epigraphs. This has the additional advantage that we can also deal with compositions of increasing convex functions and low-rank inducing norms in proximal splitting methods.

Christian Grussler, Pontus Giselsson

We analyze the local convergence of proximal splitting algorithms to solve optimization problems that are convex besides a rank constraint. For this, we show conditions under which the proximal operator of a function involving the rank constraint is locally identical to the proximal operator of its convex envelope, hence implying local convergence. The conditions imply that the non-convex algorithms locally converge to a solution whenever a convex relaxation involving the convex envelope can be expected to solve the non-convex problem.