[Paper Review] Optimization of the $H_\infty$-norm of Dynamic Flow Networks

Alexander Johansson, Jieqiang Wei+3 more|arXiv (Cornell University)|Oct 9, 2017

Interconnection Networks and Systems18 references3 citations

TL;DR

This paper formulates an optimization framework to minimize the $\mathcal{H}_\infty$-norm of dynamic flow networks—modeling distribution systems like pipe networks—by allocating edge weights (e.g., pipe capacities). Using linear matrix inequalities (LMIs), it shows that for single-input single-output (SISO) systems, the $\mathcal{H}_\infty$-norm equals the effective resistance between source and sink nodes, and derives an upper bound in terms of the graph's algebraic connectivity, enabling robust design under uncertainty.

ABSTRACT

In this paper, we study the $H_\infty$-norm of linear systems over graphs, which is used to model distribution networks. In particular, we aim to minimize the $H_\infty$-norm subject to allocation of the weights on the edges. The optimization problem is formulated with LMI (Linear-Matrix-Inequality) constraints. For distribution networks with one port, i.e., SISO systems, we show that the $H_\infty$-norm coincides with the effective resistance between the nodes in the port. Moreover, we derive an upper bound of the $H_\infty$-norm, which is in terms of the algebraic connectivity of the graph on which the distribution network is defined.

Motivation & Objective

• To minimize the $\mathcal{H}_\infty$-norm of linear dynamic flow networks, which quantifies robustness to external disturbances.
• To develop a convex optimization framework using linear matrix inequalities (LMIs) for allocating edge weights (e.g., pipe capacities) to achieve minimal $\mathcal{H}_\infty$-norm.
• To establish a connection between the $\mathcal{H}_\infty$-norm and graph-theoretic quantities such as effective resistance and algebraic connectivity.
• To provide a suboptimal but robust design strategy when full knowledge of the input matrix $E$ is unavailable.

Proposed method

• Formulates the $\mathcal{H}_\infty$-norm minimization problem as a convex optimization problem with LMI constraints on edge weights.
• Uses the property that for state-space symmetric systems, the $\mathcal{H}_\infty$-norm is attained at zero frequency, simplifying analysis.
• Interprets the Riccati inequality as a definiteness condition for a Laplacian matrix on a signed graph (with positive and negative edge weights).
• For SISO systems with $E = e_i - e_j$, proves that the $\mathcal{H}_\infty$-norm equals the effective resistance between nodes $i$ and $j$.
• Derives an upper bound on the $\mathcal{H}_\infty$-norm in terms of the algebraic connectivity $\lambda_2(L_w)$ of the underlying graph.
• Solves the optimization numerically using YALMIP with a total capacity constraint $\sum \omega_i = c$.

Experimental results

Research questions

• RQ1 How can the $\mathcal{H}_\infty$-norm of a dynamic flow network be minimized through optimal allocation of edge weights?
• RQ2 What is the relationship between the $\mathcal{H}_\infty$-norm and effective resistance in SISO dynamic flow networks?
• RQ3 Can an upper bound on the $\mathcal{H}_\infty$-norm be derived using only the algebraic connectivity of the graph?
• RQ4 How can robustness be ensured when the input matrix $E$ is partially unknown?

Key findings

• For SISO dynamic flow networks, the $\mathcal{H}_\infty$-norm is exactly equal to the effective resistance between the source and sink nodes.
• The optimal allocation of pipe capacities in the numerical example yields $w_{12}^* = 0.6$, $w_{24}^* = 0.4$, and a minimal $\mathcal{H}_\infty$-norm of $\gamma^* = 5$.
• The $\mathcal{H}_\infty$-norm is verified numerically by showing $\|y(t)\|_2 \leq \gamma^* \|d(t)\|_2$ under a test input signal.
• An upper bound on the $\mathcal{H}_\infty$-norm is derived that depends only on the algebraic connectivity $\lambda_2(L_w)$, enabling robust design under uncertainty.
• Maximizing algebraic connectivity under a total capacity constraint provides a suboptimal but bounded solution when $E$ is unknown.