⚡ electrical engineering

Dissipativity-Based Data-Driven Decentralized Control of Interconnected Systems

This paper proposes a unified, data-driven decentralized control framework for stabilizing interconnected discrete-time linear systems by deriving local dissipativity-based conditions and global stability criteria formulated as linear matrix inequalities, with demonstrated effectiveness and scalability in microgrid applications.

Original authors: Taiki Nakano, Ahmed Aboudonia, Jaap Eising, Andrea Martinelli, Florian Dörfler, John Lygeros

Published 2026-07-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Taiki Nakano, Ahmed Aboudonia, Jaap Eising, Andrea Martinelli, Florian Dörfler, John Lygeros

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a massive orchestra where every musician (a "subsystem") is playing their own instrument, but they are all connected by invisible strings. If one musician plays too loudly or out of sync, it can throw off the entire group, causing the music to descend into chaos. The challenge is that the conductor doesn't know the sheet music for any of the instruments, doesn't know how the strings are tied, and the musicians are playing in a noisy room where they can't hear each other perfectly.

This paper presents a new way to teach these musicians how to play in harmony without a central conductor, using only the sound they hear from their immediate neighbors and a bit of trial-and-error data.

Here is the breakdown of their approach using simple analogies:

1. The Problem: The Unknown Orchestra

In the real world, we often have large networks of things working together—like a smart power grid (microgrid) with many solar panels and batteries, or a fleet of drones.

The Catch: We don't have the blueprints (mathematical models) for how these individual parts work.
The Noise: The data we collect is messy, like trying to record a conversation in a hurricane.
The Goal: We need to design a controller for each part so that the entire network stays stable, even if we don't know exactly how they are connected to each other.

2. The Secret Weapon: "Energy Rules" (Dissipativity)

The authors use a concept called Dissipativity. Think of this as a rule about "energy exchange."

Imagine each musician is a battery. They can store energy (sound) and release it.
Dissipativity is a guarantee that a musician never creates more energy than they are given. They might store some, but they never generate a "super-sound" that explodes out of nowhere.
If every musician follows this rule, and they are connected in a specific way, the whole orchestra is guaranteed to stay calm and not spiral into chaos.

3. The Solution: A Two-Step "Data-Driven" Recipe

Instead of building a model first (which is like trying to draw a map of a city before you've ever visited it), the authors propose a data-driven approach. They look at the raw data (the recordings of the noise and signals) to figure out the rules.

Step A: The Local Rule (The Individual Musician)
Each subsystem looks at its own data (what it did and what it heard from neighbors) and solves a math puzzle (called a Linear Matrix Inequality, or LMI).

The Goal: Find a controller (a set of instructions) that forces that specific subsystem to obey the "Energy Rule" (Dissipativity).
The Magic: They treat the "Energy Rule" itself as a flexible variable. Instead of guessing the rule, they let the math find the best possible rule that fits the noisy data.

Step B: The Global Harmony (The Whole Orchestra)
Once every subsystem has its own "Energy Rule," the authors check if these rules work well together.

They use a second math puzzle to ensure that if everyone follows their local rule, the entire network will be stable.
Crucially, this check only requires local information. You don't need to know the whole map of the orchestra; you just need to know how your neighbors are behaving.

4. Special Case: The "Diffusive" Connection

The paper also looks at a specific type of connection called Diffusive Coupling.

Analogy: Imagine a group of people standing in a circle, each holding a bucket of water. If one person's bucket is full and their neighbor's is empty, water flows from the full one to the empty one until they are equal.
In this scenario, the authors developed a special, faster algorithm. They calculate the "maximum possible flow" between neighbors based on the data and use that to guarantee stability. It's like saying, "Even if the water flows as fast as physically possible, the system won't flood."

5. The Results: Faster and Smarter

The authors tested this on a simulated Microgrid (a network of 50 power units).

Efficiency: Their method worked with very little data (just 50 data points per unit) and took milliseconds to compute on a laptop.
Comparison: They compared it to a "Centralized" method (where one giant computer tries to control everything). The centralized method needed much more data (up to 800 points) and took much longer to compute.
Robustness: Even when the data was very noisy or the connections were complex, their decentralized method kept working, while the centralized method struggled or failed.

Summary

In short, this paper teaches a group of unknown, noisy, interconnected machines how to stabilize themselves. Instead of needing a central brain with a perfect map of the world, each machine learns its own "energy rules" from local data. As long as everyone follows their own rules, the whole group stays safe and stable. It's like teaching a chaotic crowd to dance in perfect unison by giving each person a simple, local rhythm to follow, without ever needing to tell them the steps for the whole dance.

Technical Summary: Dissipativity-Based Data-Driven Decentralized Control of Interconnected Systems

Problem Formulation
The paper addresses the stabilization of interconnected discrete-time linear time-invariant (LTI) systems where the local subsystem models and the global interconnection structure are both unknown. The systems are subject to process and measurement noise, as well as interconnection noise. The control objective is to design local state-feedback controllers ( $u_i = K_i x_i$ ) for each subsystem such that the nominal global closed-loop system is asymptotically stable. Crucially, the design must be decentralized, meaning each controller is synthesized using only local data and knowledge of its immediate neighbors, without requiring global system information or centralized coordination. The authors specifically consider two scenarios: general interconnected systems with unknown interconnection weights and systems interconnected via diffusive coupling (where interconnection inputs depend on the difference between neighboring outputs).

Methodology
The proposed approach leverages dissipativity theory to create a unified pipeline for data-driven control. The methodology proceeds in two main stages, both formulated as Linear Matrix Inequalities (LMIs):

Local Dissipativity Synthesis:
For each subsystem, the authors derive a data-driven LMI condition (Theorem 1) to compute a state-feedback gain $K_i$ . This gain ensures that the local closed-loop subsystem is dissipative with respect to a quadratic supply rate defined by matrices $(F_i, G_i, H_i)$ .
- The method utilizes the Fundamental Lemma from behavioral systems theory to relate collected data trajectories (state, input, interconnection input, and output) to the set of possible system matrices consistent with that data.
- Noise is modeled using Quadratic Matrix Inequalities (QMIs), allowing the controller to be robust to noise-corrupted data without explicit system identification.
- The supply rate matrices $(F_i, G_i, H_i)$ are treated as joint decision variables rather than fixed parameters, allowing the local design to adapt to the global stability requirements.
Decentralized Stability Verification:
The paper establishes conditions under which the dissipativity of local subsystems guarantees the asymptotic stability of the global interconnected system.
- General Interconnections: Using a matrix S-lemma and dual QMI formulations, the authors derive a data-driven decentralized LMI (Theorem 2) that ensures global stability. This condition relies on the local interconnection data (neighbor outputs and local interconnection inputs) and the supply rate matrices derived in the first step.
- Diffusive Coupling: For systems with diffusive coupling, the authors propose a specific stability condition (Proposition 5) dependent on the weighted degree of the interconnection graph. Since the interconnection weights are unknown, they formulate a quadratic program (Theorem 3) to compute the tightest upper bound on the weighted degree consistent with the observed interconnection data. This bound is then used in the stability LMI.

Key Contributions
The paper makes the following specific contributions:

Data-Driven Local Dissipativity: Derivation of a necessary and sufficient LMI condition to synthesize a state-feedback controller that renders an unknown, noisy subsystem dissipative, treating supply rate matrices as decision variables.
Data-Driven Global Stability: Proposal of decentralized LMI conditions that guarantee global stability based solely on the dissipativity of local subsystems and local interconnection data, without requiring knowledge of the global interconnection matrix $M$ .
Unified Pipeline: A combined algorithm (Algorithm 1) that solves the local and global LMIs simultaneously to produce decentralized controllers.
Diffusive Coupling Extension: A specialized algorithm (Algorithm 2) for diffusive coupling that computes data-driven bounds on interconnection weights to ensure stability.
Scalability Demonstration: Numerical validation showing that the proposed decentralized methods require significantly less data and computational time compared to centralized structured data-driven control approaches.

Results
The effectiveness of the algorithms is validated through numerical examples involving a DC microgrid with 50 distributed generation units (DGUs).

Stability: Both Algorithm 1 (general case) and Algorithm 2 (diffusive coupling) successfully stabilize the microgrid, driving currents and voltages to the origin.
Robustness: The controllers maintain stability in the presence of noise satisfying the assumed QMI bounds, demonstrating input-to-state stability.
Feasibility and Scalability:
- In tests with 100 random system instances, Algorithm 1 maintained 100% feasibility across varying data lengths, noise levels, and interconnection densities.
- Algorithm 2 showed reduced feasibility under high noise or sparse data, attributed to the conservatism introduced by computing the upper bound of the weighted degree.
- Comparison: When compared to a centralized structured data-driven controller, the proposed decentralized algorithms achieved feasibility with much shorter data lengths (50 samples vs. 600+ samples) and significantly lower computation times (milliseconds vs. ~24 seconds).

Significance and Claims
The paper claims to provide a unified, data-driven, and decentralized framework for stabilizing interconnected systems without requiring system identification or global information. The significance lies in the ability to handle unknown interconnection structures and noise-corrupted data simultaneously, a challenge often difficult to address with existing methods. By treating supply rate matrices as decision variables, the authors bridge the gap between local dissipativity design and global stability analysis. The authors modestly note that while the approach is effective for linear systems with quadratic supply rates, future work is needed to address nonlinear dynamics, dynamic interconnections, and online adaptation. The work is positioned as a step toward scalable control of large-scale cyber-physical systems like power grids and robotic networks.