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

TL;DR

This work tackles the problem of computing the tightest outer ellipsoid that upper-bounds the intersection of time-varying ellipsoids measured across a network. It introduces a novel distributed, discrete-time reformulation of the centralized outer Löwner-John SDP, enabling each node to solve a local SDP and mutually adjust its estimate using neighbor information. Theoretical results guarantee finite-time convergence in the static case and finite-time bounded tracking in the dynamic case, along with robustness and boundedness of estimates; a continuity-based argument underpins the dynamic guarantees. The approach is demonstrated on illustrative simulations and integrated into a distributed Kalman filter, where it yields improved mean-square performance over state-of-the-art consensus-based methods, highlighting practical impact for robust distributed estimation and control.

Abstract

This paper presents the first discrete-time distributed algorithm to track the tightest ellipsoids that outer approximates the global dynamic intersection of ellipsoids. Given an undirected network, we consider a setup where each node measures an ellipsoid, defined as a time-varying positive semidefinite matrix. The goal is to devise a distributed algorithm to track the tightest outer approximation of the intersection of all the 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. 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.

Figures (6)

• Figure 1: Problem setting: (top) each node in graph $\mathcal{G}$ represents a time-varying ellipsoid described by the matrix $\mathbf{P}_i[k] \quad \forall i \in \{1, 2, 3, 4\}$. The goal is to cooperate to find the smallest ellipsoid that outer approximates their intersection (black ellipsoid for $\mathbf{Q}^*[k]$); (bottom) each node (e.g., node $2$) only has access to its neighboring information, leading to conservative approximations of $\mathbf{Q}^*[k]$ (black ellipsoid for $\mathbf{Q}_2[k]$). In this work we propose a distributed method to make $\mathbf{Q}_i[k]$ converge to $\mathbf{Q}^*[k]$ using local information.
• Figure 2: An example of the importance of robustness (Theorem \ref{['th:main']}) in our problem. Throughout the paper, we combine the representation of the ellipsoids in $\mathbb{R}^n$ (left column) and in $\mathbb{S}^n_{+}$ projected to $\mathbb{R}^2$ (right column). The latter helps understanding the relationships between the input ellipsoids $\mathbf{P}_i[k]^{-1}$, their local outer Löwner-John solutions $\mathbf{Q}_i[k+1]$ and the global optimum $\mathbf{Q}^*[k]$. When $\mathcal{C}_i[k] \subseteq \mathcal{C}^*[k]$ (top), the local estimate $\mathbf{Q}_i[k+1]$ is always contained in both the local and global feasible set, which implies that the node is able to reconstruct an approximation of $\mathbf{Q}^*[k]$ and the global intersection of ellipsoids, even if $f(\mathbf{Q}_i[k-1]) < f(\mathbf{Q}^*[k])$. On the other hand, when $\mathcal{C}_i[k] \nsubseteq \mathcal{C}^*[k]$ (bottom), the local optimum estimate $\mathbf{Q}_i[k+1]$ might be out of the global feasible set. As shown on the right, if $f(\mathbf{Q}_i[k]) < f(\mathbf{Q}^*[k+1])$, then the node is not able to track the outer approximation of the intersection because the current estimate $\mathbf{Q}_i[k]$ is smaller than $\mathbf{Q}^*[k+1]$ in the sense of $f$. Moreover, $\mathbf{Q}_i[k]$ will propagate across the network, blocking all nodes from correctly estimating $\mathbf{Q}^*[k+1]$ at subsequent time steps.
• Figure 3: Illustrative visualization of the cases 3) and 4) of the proof of Lemma \ref{['lem:global:optim']}. The figure leverages the representation of ellipsoids in $\mathbb{S}^n_{+}$ projected to $\mathbb{R}^3$ and $\mathbb{R}^2$ introduced in Fig. \ref{['fig:robustness']}.
• Figure 4: Illustrative example of Algorithm \ref{['al:algorithm']} applied to a static problem. At each time step, the error between the eigenvalues of the global optimum matrix and the local estimates $|q^i_j[k] - q^*_j|$ decreases until the error becomes zero in finite time (in this case, $K=3$). The top left pannel shows the initial ellipsoids at each node and the optimal solution from the original centralized Löwner-John method. The top right pannel shows, for node $1$, its estimate after using Algorithm \ref{['al:algorithm']} at $k=1$ and the ellipsoids exchanged with its neighbors. The bottom right pannel depicts the estimate at node $1$ and $k=2$ after using again Algorithm \ref{['al:algorithm']}, depicting that node $1$ already recovers the desired $\mathbf{Q}^*$. The bottom left pannel depicts the evolution of the error with time for all the nodes.
• Figure 5: Illustrative example of Algorithm \ref{['al:algorithm']} applied to a dynamic problem. The plot depicts the time evolution of the error between the eigenvalues of the global optimum matrix and the average of the local estimates $|\tilde{q}_j[k] - q^*_j|$, for $j\in\{1,2\}$.

Theorems & Definitions (15)

• Theorem 1
• Remark 1
• Remark 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4