A general partitioning strategy for non-centralized control

TL;DR

The paper tackles partitioning large-scale networks for non-centralized control by introducing a generalized framework that first selects fundamental system units (FSUs) and then aggregates them into composite system units (CSUs) to form a control partition. Central to the approach is an equivalent graph representation of the dynamical system and a global partition index $p^{\text{idx}}(\mathcal{P})$ that balances intra- and inter-CSU interactions, with a granularity parameter $\alpha$ controlling aggregation level. The framework supports both algorithmic (greedy) and optimization-based (IQP) partitioning and is validated on distributed model predictive control (DMPC) for linear and hybrid systems, showing significant reductions in computation time and cost while preserving or improving performance. These results demonstrate a scalable path to efficiently deploy distributed control in complex large-scale systems, with practical implications for privacy, communication, and computational resource management.

Abstract

Partitioning is a fundamental challenge for non-centralized control of large-scale systems, such as hierarchical, decentralized, distributed, and coalitional strategies. The problem consists of finding a decomposition of a network of dynamical systems into system units for which local controllers can be designed. Unfortunately, despite its critical role, a generalized approach to partitioning applicable to every system is still missing from the literature. This paper introduces a novel partitioning framework that integrates an algorithmic selection of fundamental system units (FSUs), considered indivisible entities, with an aggregative procedure, either algorithmic or optimization-based, to select composite system units (CSUs) made of several FSUs. A key contribution is the introduction of a global network metric, the partition index, which quantitatively balances intra- and inter-CSU interactions, with a granularity parameter accounting for the size of CSUs, allowing for their selection at different levels of aggregation. The proposed method is validated through case studies in distributed model predictive control (DMPC) for linear and hybrid systems, showing significant reductions in computation time and cost while maintaining or improving control performance w.r.t. conventional strategies.

Figures (8)

• Figure 1: Representations of the equivalent graph of a system with 20 input (in red) and 100 states (in cyan) variables. In (Fig. \ref{['fig:Network_States_100_Inputs_20network_map']}) the concentric degree-based representation, where input nodes are positioned in the most outer part of a circle, while state nodes on concentric rings representing their degree, with the most outer being degree one, and growing of one each ring toward the center. In (Fig. \ref{['fig:Network_States_100_Inputs_20atomic_control_agents']}) the representation of the FSUs after the application of the selection procedure. Each of the 17 FSU is represented by the green area, where dynamic relationships among the variables in the same FSU are represented by with black arrows, and among the FSUs by green arrows.
• Figure 2: Graphs related to the partitioning of a modular network with 64 FSUs. The strength of the connection is represented by the thickness of the links. Partitions $\mathcal{P}_1$-$\mathcal{P}_4$ are obtained for values $\alpha = 10^6, 10^4, 10^2, 1$ respectively.
• Figure 3: Comparison of the computation times (Fig. \ref{['fig:computationtimes-linear']}), and computation costs (Fig. \ref{['fig:computationcost-linear']}) for the different partitions. For the former, the gray trend line highlights a diminishing return in computation speed as the number of cores increases, while for the latter the almost linear relation between the number of cores and their associated computation cost.
• Figure 4: Evolution of the inputs (Fig. \ref{['fig:inputPWA']}) and of the states (Fig. \ref{['fig:statePWA']}) for the network of PWA FSUs. For the former, all signals are approximately the same and they respect the constraints.For the latter, when the value of a state crosses the threshold at zero, we have a change of dynamics, which is reflected in a different behavior. Accordingly, the input signal changes too, but reaches saturation before the state is able to reproduce the reference.
• Figure 5: Evolution of the maximum error (Fig. \ref{['fig:Error']}) and absolute value (Fig. \ref{['fig:MaxState']}) across all CSUs for different partitions.

Theorems & Definitions (14)

• Lemma 1
• proof
• Definition 1: Subsystem
• Definition 2: Composite System Unit
• Definition 3: Fundamental System Unit
• Definition 4: Aggregation Operation
• Proposition 1
• proof
• Definition 5: Control Partition of a Dynamical System
• Remark 1