⚡ electrical engineering

Model-Based and Data-Driven Hierarchical Control and Topology Co-Design for Robust Networked Systems

This paper proposes model-based and data-driven hierarchical control and topology co-design strategies for robust networked systems that ensure global dissipativity and optimize interconnection costs by solving linear matrix inequalities, with applications demonstrated in DC microgrids.

Original authors: Shirantha Welikala, Zihao Song, Hai Lin, Panos J. Antsaklis

Published 2026-06-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shirantha Welikala, Zihao Song, Hai Lin, Panos J. Antsaklis

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 large orchestra where every musician (a "subsystem") is playing their own instrument. In a traditional setup, a single conductor stands on a podium, sees everyone, and shouts instructions to the whole group at once. This works well for small groups, but if you have 1,000 musicians, the conductor gets overwhelmed, the sound takes too long to travel, and if the conductor trips, the whole orchestra stops.

This paper proposes a smarter way to run the orchestra, and it does so in two different ways: one where the conductor knows the sheet music perfectly (Model-Based), and one where the conductor has never seen the sheet music but has recorded the musicians playing for hours (Data-Driven).

Here is the breakdown of their solution using everyday analogies:

The Big Problem: Chaos and Cost

In complex systems like power grids (microgrids), robots, or traffic, you have many parts connected together.

The Challenge: You need to keep the whole system stable (like keeping the music in tune) even when things go wrong (disturbances like a sudden noise or a power surge).
The Old Way: Designing the connections (topology) and the controllers (the rules musicians follow) separately often leads to messy, expensive, or inefficient results. It's like trying to fix a broken car engine by guessing which part is wrong without a manual, or by hiring a mechanic who tries to fix the whole car at once instead of fixing one cylinder at a time.

The Solution: A Two-Step "Hierarchical" Dance

The authors suggest a Hierarchical Co-Design approach. Think of this as a two-step dance:

Step 1 (Local): Each musician practices alone to ensure they can play their part perfectly, no matter what. They become "dissipative."
- What is Dissipativity? Imagine a shock absorber on a car. If you hit a bump, the shock absorber soaks up the energy so the car doesn't bounce wildly. "Dissipativity" is the mathematical way of saying, "This part of the system is good at soaking up bad energy (disturbances) so it doesn't ruin the whole show."
Step 2 (Global): Once every musician is a pro at soaking up their own bumps, the conductor designs the connections between them. Who talks to whom? The goal is to use the fewest possible phone lines (communication links) to keep the whole orchestra in sync, while ensuring the whole group remains stable.

Approach 1: The "Model-Based" Method (Knowing the Rules)

This is for when you have the "blueprints" or the exact mathematical equations of how every subsystem works.

How it works: The engineers use a set of mathematical rules (called Linear Matrix Inequalities or LMIs) to calculate the perfect local practice routine for each musician and the perfect communication map for the group.
The Benefit: It's like having a GPS that knows the exact road conditions. It guarantees the system will be stable and robust. It avoids the messy, trial-and-error loops that usually waste time and computing power.

Approach 2: The "Data-Driven" Method (Learning from Experience)

This is the paper's big innovation. What if you don't have the blueprints? What if the system is too complex to write down in equations?

The Scenario: Imagine you don't know how the musicians' instruments work, but you have a recording of them playing for a long time.
The Trick: The authors use a clever mathematical tool called the Matrix S-Lemma. Think of this as a "safety net." Even though we don't know the exact rules of the instruments, we know that the "noise" or "disturbances" hitting them aren't infinite—they are bounded (like a volume knob that can't go past 10).
How it works: The system looks at the recorded data (input and output trajectories). It asks, "If the noise is within these limits, what controller guarantees the system stays safe?" It designs the controller directly from the data, skipping the need to figure out the underlying physics first.
The Benefit: It works even when the system is a "black box." It's robust against the unknown, provided the unknown isn't infinitely wild.

The Real-World Test: The DC Microgrid

To prove this works, the authors tested it on a DC Microgrid (a small, local power grid, like a neighborhood with solar panels and batteries).

The Goal: Keep the voltage steady (like keeping the water pressure in pipes constant) and share the electrical load fairly among all the generators.
The Result:
- They designed a system where local controllers handled the immediate bumps (like a sudden cloud covering a solar panel).
- They then designed a communication network that told the generators how to talk to each other.
- The Win: Their method created a sparser network. Instead of every generator talking to every other generator (which is expensive and slow), they found a way to have fewer connections that still kept the whole grid stable and efficient.

Summary

This paper gives us a new toolkit to manage complex networks:

If you know the math: Use the Model-Based approach to design local safety nets and a global communication map simultaneously.
If you only have data: Use the Data-Driven approach to learn the safety nets and communication map directly from recordings, assuming the chaos isn't too extreme.

The result is a system that is robust (can handle shocks), efficient (uses fewer connections), and scalable (works for small groups and huge networks alike), all without needing a single, overwhelmed central boss.

Technical Summary: Model-Based and Data-Driven Hierarchical Control and Topology Co-Design for Robust Networked Systems

1. Problem Statement

The paper addresses the challenge of designing large-scale networked systems composed of interconnected linear subsystems subject to external disturbances. The core objective is to jointly synthesize three components:

Local Controllers: Feedback laws for individual subsystems.
Distributed Global Controllers: Coordination mechanisms utilizing information exchanged between neighboring subsystems.
Interconnection Topology: The communication or physical structure linking the subsystems.

The design must ensure global dissipativity (a robustness metric encompassing stability, $L_2$ -gain, and passivity) from disturbance inputs to performance outputs while optimizing the cost of the interconnection topology (e.g., communication sparsity).

A significant limitation in existing literature is the reliance on precise knowledge of subsystem dynamics, which is often unavailable in real-world applications. Furthermore, many existing methods treat the topology as fixed or require centralized, non-convex, and iterative design procedures that lack scalability. This paper seeks to develop a unified framework that:

Performs hierarchical co-design (local controllers $\to$ global controllers + topology).
Operates under model-based assumptions (known dynamics) and data-driven assumptions (unknown dynamics, known trajectory data).
Maintains compositionality and decentralizability by relying on Linear Matrix Inequalities (LMIs).

2. Methodology

The proposed approach is structured hierarchically and is presented in two parallel tracks: model-based and data-driven.

A. Theoretical Foundations

The framework relies on dissipativity theory, specifically $X$ -dissipativity, which generalizes properties like passivity and finite-gain $L_2$ -stability.

Local Level: Each subsystem is designed to satisfy a local dissipativity certificate ( $X_i$ -dissipativity).
Global Level: These local certificates are propagated to the network level to guarantee global dissipativity ( $Y$ -dissipativity) via the interconnection matrix $M$ .
Tools: The design utilizes the Schur complement, congruence principles, and the Matrix S-lemma (for handling quadratic constraints in data-driven settings).

B. Model-Based Hierarchical Design

This approach assumes full knowledge of subsystem dynamics ( $A_i, B_i, C_i$ ).

Local Controller Synthesis: A local state-feedback controller ( $K_i$ ) is designed for each subsystem to enforce a specific local dissipativity property ( $X_i$ -dissipativity) with constraints $X_{i}^{11} > 0$ and $X_{i}^{22} < 0$ . This is formulated as an LMI problem.
Refined Local Design: To improve the feasibility of the subsequent global stage, the local design is augmented with necessary conditions derived from the global LMI. This involves a change of variables to bias the local dissipativity certificates toward global compatibility.
Global Co-Design: Using the fixed local certificates, a distributed global controller ( $K$ ) and the communication topology are co-designed. The global controller is defined as $\bar{u} = Ky$ , where the sparsity of $K$ (and thus the communication links) is optimized. This stage solves a single LMI problem to enforce global dissipativity while minimizing an objective function that promotes sparsity ( $\| \bar{K} \|_1$ ) and optimizes performance bounds.

C. Data-Driven Hierarchical Design

This approach assumes subsystem dynamics are unknown, but rich input-state-output trajectory data are available.

Data Consistency: The set of all possible system models consistent with the observed data and bounded disturbances is characterized using a Quadratic Matrix Inequality (QMI).
Disturbance Modeling: Unknown disturbances are assumed to be bounded by a quadratic matrix inequality (relaxing conventional bounds).
Local Controller Synthesis: The local controller is synthesized directly from data. The Matrix S-lemma is employed to convert the requirement that the controller must work for all models consistent with the data into a tractable LMI. This ensures robustness against unknown dynamics and disturbances.
Global Co-Design: Similar to the model-based case, the global controller and topology are co-designed. The unknown system matrices ( $B$ ) in the global LMI are handled by reformulating the problem using the data-consistency QMIs and the Matrix S-lemma, resulting in a robust LMI condition that does not require explicit system identification.

3. Key Contributions

The paper outlines four primary contributions:

Hierarchical Co-Design Formulation: A novel framework for networked linear systems that jointly synthesizes local controllers, distributed global controllers, and interconnection topologies under global dissipativity constraints.
Model-Based LMI Framework: A two-stage design process where local feedback enforces subsystem dissipativity, which is then leveraged to co-design distributed controllers and topologies with network-level guarantees.
Direct Data-Driven Counterpart: A method that derives robust LMI conditions using only subsystem trajectory data and quadratic disturbance bounds, bypassing the need for explicit system identification.
Case Study Application: Demonstration of the proposed methods on a DC Microgrid (DCMG) control problem, aiming for robust voltage regulation and current sharing.

4. Results and Application

The paper validates the proposed methods using a DC Microgrid case study involving multiple Distributed Generators (DGs) and resistive lines.

Objective: Achieve robust voltage regulation and proportional current sharing while designing the communication topology.
Comparison: The proposed dissipativity-based hierarchical design was compared against a standard stabilizing global controller.
Findings:
- The proposed method successfully enforced the desired dissipativity certificates (voltage regulation and current sharing).
- Topology Optimization: Unlike the benchmark stabilizing controller, which required a fully connected communication topology, the proposed hierarchical design yielded a sparser communication topology.
- Performance: Both controllers achieved the desired steady-state regulation, but the proposed method did so with fewer communication links, demonstrating the efficacy of the joint topology and control optimization.

5. Significance and Claims

The paper claims that its primary significance lies in addressing the gap between scalable control design and topology synthesis for networked systems.

Scalability and Compositionality: By using a hierarchical approach based on dissipativity, the method avoids centralized, non-convex, and iterative design processes. It retains compositionality, allowing local guarantees to be systematically propagated to global guarantees.
Robustness to Uncertainty: The data-driven extension is significant because it provides theoretical guarantees for stability and performance without requiring precise system models, which is crucial for complex real-world systems where modeling is difficult.
Unified Framework: The paper provides a unified tractable framework (via LMIs) that handles the joint design of controllers and topology under both model-based and data-driven settings, a combination that the authors note has been largely unexplored in existing literature.

The authors maintain a modest tone, noting that while the preliminary version of this work addressed analysis and topology synthesis, this paper extends the scope to include the synthesis of local controllers and the handling of unknown dynamics via data-driven methods. The results are illustrated on a specific DC microgrid example, with the implication that the methodology is applicable to other networked systems like vehicular platoons and supply chains, though specific results for those domains are not presented in this text.