  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 L\"owner-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.
