Scalable Outer Approximation of Minkowski Sums of Matrix Ellipsoids for Data-Driven Control

TL;DR

This paper introduces a scalable, LMI-free method for outer approximation of Minkowski sums of matrix ellipsoids, enhancing data-driven control under uncertainty.

Contribution

It develops an analytical solution for one approximation criterion and an efficient MM algorithm for another, improving scalability and computational efficiency.

Findings

The method outperforms standard solvers in computational efficiency.

It provides exact solutions for the sum of squared semi-axes criterion.

The MM algorithm converges monotonically to stationary points.

Abstract

Matrix ellipsoids provide a standard framework for representing bounded uncertainties in data-driven control. Since noise models for sequential observations are naturally represented as the Minkowski sum of multiple matrix ellipsoids, applying existing robust control methods, which typically assume a single ellipsoidal set, requires a tight outer approximation. While techniques based on linear matrix inequalities (LMI) are applicable, their computational cost grows quadratically with the data length, limiting their scalability. This paper investigates the optimal outer approximation problem under two criteria: the sum of squared semi-axes and the volume. We propose an LMI-free approach by introducing a parameterized family of bounding matrix ellipsoids. Specifically, we derive an exact analytical solution for the first criterion and develop an efficient majorization-minimization (MM) algorithm for the second. The proposed MM algorithm employs a first-order approximation of the log-determinant function to provide closed-form update rules, ensuring monotonic convergence to the set of stationary points. Numerical experiments demonstrate that our method offers significantly higher computational efficiency and scalability than standard interior-point solvers.