Distributed Discrete-time Dynamic Outer Approximation of the Intersection of Ellipsoids

This paper presents the first discrete-time distributed algorithm to track the tightest ellipsoids that outer approximates the global dynamic intersection of ellipsoids. The ellipsoids are defined as time-varying positive definite matrices. On the other hand, given an undirected network, each node is equipped with one of these ellipsoids. The solution is based on a novel distributed reformulation of the original centralized semi-definite outer Löwner-John program, characterized by a non-separable objective function and global constraints. We prove finite-time convergence to the global minima of the centralized problem in the static case and finite-time bounded tracking error in the dynamic case. Moreover, we prove boundedness of estimation in the tracking of the global optimum and robustness in the estimation against time-varying inputs. As a by-product, the proposed algorithm extends min/max dynamic consensus algorithms to positive definite matrices. We illustrate the properties of the algorithm with different simulated examples, including a distributed estimation showcase where our proposal is integrated into a distributed Kalman filter to surpass the state-of-the-art in mean square error performance.