Partitioning and Observability in Linear Systems via Submodular Optimization

TL;DR

The paper addresses partitioning and observability in large-scale linear time-invariant (LTI) networks by casting partitioning as a submodular maximization problem under a partition matroid and then solving the sensor-placement (SP) problem on the partitioned network. It leverages observability Gramian-based metrics, proves modularity and submodularity properties under partitioning, and uses the continuous multilinear extension with the continuous greedy algorithm to obtain high-quality solutions with a (1-1/e) guarantee. The authors derive theoretical bounds relating the observability of the global system to that of the sum of partitioned subsystems and validate these results on combustion-reaction networks, demonstrating substantial computational savings without sacrificing observability performance. The work offers a scalable framework for decentralized state estimation and sensor allocation in large dynamical networks, with potential extensions to actuator placement and distributed implementations.

Abstract

Network partitioning has gained recent attention as a pathway to enable decentralized operation and control in large-scale systems. This paper addresses the interplay between partitioning, observability, and sensor placement (SP) in dynamic networks. The problem, being computationally intractable at scale, is a largely unexplored, open problem in the literature. To that end, the paper's objective is designing scalable partitioning of linear systems while maximizing observability metrics of the subsystems. We show that the partitioning problem can be posed as a submodular maximization problem -- and the SP problem can subsequently be solved over the partitioned network. Consequently, theoretical bounds are derived to compare observability metrics of the original network with those of the resulting partitions, highlighting the impact of partitioning on system observability. Case studies on networks of varying sizes corroborate the derived theoretical bounds.

Figures (11)

• Figure 1: (a) A system of measured internal state, $v\in \mathcal{V}$ (measurable space), depicting interactions of a dynamical system and (b) the subsequent subsystems $\mathcal{S}_i \subseteq \mathcal{V}$. The nodes represent the system states and edges represent the internal state connections. The colored boxes represent the nodes that belong to a particular subsystem. The interactions between the subsystems remain after system partitioning.
• Figure 2: Matroid Constraints: For any (a) submodular maximization problem, a matroid constraint (b) satisfies: (i) the null property, (ii) the heredity property and (iii) augmentation property. Two common matroid constraints are the (c) uniform matroid $\mathcal{I}_c$ and (d) partition matroid $\mathcal{I}_p$.
• Figure 3: Overview of the system partitioning problem under $\mathbf{P2}$ (Sections \ref{['sec:problemformulation']}–\ref{['sec:main']}), followed by SP under $\mathbf{P3}$ (Sections \ref{['sec:SNS']}). The continuous greedy algorithm can be utilized to solve $\mathbf{P2}$ and $\mathbf{P3}$ (Section \ref{['sec:multilinear']}) for an LTI system such as a linearized combustion reaction network (Section \ref{['sec:simulation']}).
• Figure 4: (a) Disjoint selection of state $v$ from ground set $\mathcal{V}$. (b) Disjoint selection from ground set $\mathcal{X}=\mathcal{C}\times \mathcal{V}$ by duplicating each element $v\in\mathcal{V}$ for each subsystem $i\in\mathcal{C}$; selection under a partition matroid.
• Figure 5: (a) Mapping of state subsets $\mathcal{S}_i$ to outputs $\boldsymbol{y}_{\mathcal{S}_i}[k]$; each $v_i$ corresponds to a row $\mathbf{c}_v$ of $\mathbf{C}$. (b) Composition of the full observability Gramian $\boldsymbol{W}_o$ from subsystem Gramians $\boldsymbol{W}_{\mathcal{S}_i}$ and the corresponding matrix structure in each summation term of the subsystem Gramians.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Lemma 1
• Proposition 1
• Corollary 1
• Lemma 2
• Theorem 1
• Corollary 2
• Corollary 3
• Corollary 4