⚡ electrical engineering

Robust Data-Driven Nash Equilibrium Seeking under Partial-Decision Information

This paper proposes a robust, data-driven framework that enables decentralized Nash equilibrium seeking in multi-agent systems with unknown linear or nonlinear dynamics, exogenous disturbances, and partial-decision information by synthesizing stabilizing controllers directly from noisy input-state data via semi-definite programs.

Original authors: Linqi Wang, Yifei Li, Wenjie Liu, Yuzhou Wei, Gang Wang, Lihua Xie

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Linqi Wang, Yifei Li, Wenjie Liu, Yuzhou Wei, Gang Wang, Lihua Xie

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 group of drones trying to fly in a perfect formation while avoiding wind gusts, but with a twist: none of them can see what the others are doing directly. They can only talk to their immediate neighbors, and they don't know the exact physics of how their own engines or the wind affect them. They also have strict rules to follow, like "everyone must stay at exactly 30 meters altitude."

This paper presents a clever new way for these drones (or any group of agents) to figure out the perfect, stable arrangement—called a Nash Equilibrium—without needing a manual, a blueprint, or a crystal ball.

Here is the breakdown of their solution using simple analogies:

1. The Problem: The "Blind Orchestra"

Usually, to get a group of agents to agree on a strategy, you need to know two things:

The Rules: How the system moves (the math of the drone).
The Score: What everyone else is doing right now.

In this paper, the "orchestra" is playing in the dark.

Partial Information: Each drone only hears its neighbors, not the whole band.
Unknown Dynamics: They don't know their own weight, engine power, or how the wind pushes them.
Disturbances: There is external noise (wind) messing things up.
Strict Rules: They must satisfy specific constraints (like staying at a fixed height).

If they tried to guess the math or ask for the full list of everyone's moves, the system would fail or be too slow.

2. The Solution: The "Data-Driven Detective"

Instead of trying to figure out the physics first, the authors say: "Let's just watch what happens when we push the buttons."

They propose a framework that acts like a detective who solves a crime by looking at the evidence (data) rather than knowing the criminal's history (the model).

The Four Pillars of the Framework

The authors built a "control system" with four distinct tools working together:

The "Goal Calculator" (NE Model):
Imagine each drone has a calculator that constantly asks, "Am I happy with my current position compared to my neighbors?" If not, it calculates a "regret" score. This calculator uses a special math trick (KKT conditions) to ensure they don't just move randomly but move toward a specific, stable agreement where no one wants to change their mind.
The "Whisper Network" (Communication Protocol):
Since a drone can't see everyone, it needs to guess what the distant drones are doing. The authors created a "whisper network." Each drone asks its neighbors, "What do you think the others are doing?" and passes that guess along. Over time, these guesses become accurate enough that every drone effectively "knows" the whole group's position, even though they only talked to their neighbors.
The "Wind Shield" (Internal Model):
The wind (disturbance) is annoying. To cancel it out, the system uses an "Internal Model." Think of this as a noise-canceling headphone for the drone. If the wind blows in a predictable pattern (like a steady gust or a wave), the drone builds a mental copy of that pattern and pushes back against it perfectly, keeping the flight smooth.
The "Data-Driven Pilot" (The Controller):
This is the magic part. Usually, you need the exact equations of the drone to write the code that controls it. Here, the authors use Semi-Definite Programs (SDPs).
- The Analogy: Imagine you want to teach a robot to walk. Instead of writing a physics textbook for the robot, you just make it walk 50 times while recording the data. You then feed that messy, noisy video into a computer program that says, "Okay, based on these 50 tries, here is the exact button-pressing pattern that worked best."
- The paper proves that even if the data is noisy (like a shaky camera), this method guarantees the drones will eventually stop wobbling and settle into the perfect formation.

3. The "Nonlinear" Twist

The paper also tackles a harder version: what if the drones have weird, non-linear quirks (like a motor that behaves differently when it gets hot)?

They use a technique called "Lifting." Imagine taking a 2D drawing of a curve and stretching it into 3D so it looks like a straight line. This allows them to use the same data-driven math on complex, wiggly systems as long as those wiggles follow certain predictable rules (like being "bounded" or "smooth").

4. The Results: Proof in the Simulation

The authors tested this on two scenarios:

UAV Network: Six drones trying to reach specific waypoints while staying at a fixed altitude, despite wind.
Rotary-Wing Formation: Five helicopters with complex, non-linear physics trying to coordinate.

The Outcome:

The drones successfully found the perfect formation (the Nash Equilibrium).
They ignored the wind and stayed at the correct altitude.
Crucially: Their performance was almost identical to a "perfect" system that did know all the physics equations. The data-driven method worked just as well without needing the manual.

Summary

This paper is about teaching a group of strangers to dance in perfect sync without a choreographer, without knowing the music's tempo, and without seeing the other dancers. They do it by:

Listening to their immediate neighbors to guess the whole room's moves.
Recording their own steps to learn the rhythm on the fly.
Using a "noise-canceling" trick to ignore the wind.
Trusting the data they collected to guide them to the perfect, stable dance formation.

The paper claims this works mathematically and proves it with computer simulations, showing that you don't need to know the "rules of physics" to control complex, connected systems—you just need good data and the right algorithm.

Technical Summary: Robust Data-Driven Nash Equilibrium Seeking under Partial-Decision Information

Problem Formulation
This paper addresses the challenge of decentralized Nash equilibrium (NE) seeking in multi-agent systems characterized by unknown linear (and subsequently nonlinear) dynamics, exogenous disturbances, equality constraints, and partial-decision information. In this setting, agents aim to minimize local quadratic cost functions subject to linear equality constraints. However, unlike standard game-theoretic setups, agents do not have direct access to the decisions of all other agents; they only possess local estimates derived from a connected communication graph. Furthermore, the system dynamics are unknown, and agents are subject to disturbances generated by known exosystems (for the linear case) or constant disturbances (for the nonlinear case). The objective is to design decentralized controllers that steer the system outputs to the unique constrained NE without relying on explicit system identification.

Methodology
The proposed framework reformulates the constrained NE seeking problem as a cooperative output regulation problem. This is achieved through a unified architecture integrating four key components:

NE Model: A dynamical system is introduced to generate a surrogate for the tracking error. This model utilizes local estimates of the decision profile and incorporates Karush-Kuhn-Tucker (KKT) conditions to handle equality constraints. The regulation error is defined based on the pseudo-gradient of the cost function and the Lagrange multipliers.
Communication Protocol: To address partial-decision information, a consensus-based communication protocol is employed. Agents exchange local estimates of the decision profile to reconstruct the unavailable components of the global decision vector required for gradient evaluation.
Internal Model: To reject exogenous disturbances, an internal model principle is applied. For linear systems, this involves a dynamic internal model driven by the exosystem matrix $E_i$ . For constant disturbances, this reduces to an integral control structure.
Data-Driven Synthesis: Instead of identifying system matrices ( $A_i, B_i, C_i, D_i$ $A_{i}, B_{i}, C_{i}, D_{i}$ ), the controllers are synthesized directly from noisy input-state data collected offline.
- Linear Case: The problem is cast into a semi-definite program (SDP). By leveraging the Fundamental Lemma and factorizing the unknown disturbance data, the authors derive conditions (involving matrices $P_i, S_i$ and scalars $\tau_i, \chi_i, \alpha_i$ ) that guarantee closed-loop stability and asymptotic convergence to the NE. The feasibility of these SDPs ensures the existence of a stabilizing gain $K_i$ computed solely from data.
- Nonlinear Case: The framework is extended to a class of nonlinear systems with constant disturbances. The nonlinearities are handled via a "lifted" representation and constrained by quadratic inequalities (S-procedure). The synthesis involves solving Linear Matrix Inequalities (LMIs) that account for the nonlinear terms and measurement noise.

Key Contributions
The paper makes three primary contributions:

Unified Framework: It develops a control framework that integrates an NE model, a communication protocol, an internal model, and a data-driven controller. This reformulates the NE seeking problem into a cooperative output regulation problem, explicitly handling equality constraints and partial-decision information.
Robust Linear Synthesis: For unknown linear systems subject to disturbances from known exosystems, it proposes a data-driven design method based on SDPs. This method guarantees closed-loop stability and asymptotic convergence to the constrained NE using only noisy input-state data, without requiring exact system matrices.
Nonlinear Extension: It extends the direct data-driven approach to a class of nonlinear systems with constant disturbances. By combining integral control with quadratic constraints on nonlinearities, the paper provides decentralized data-driven NE seeking with asymptotic convergence guarantees.

Results
The efficacy and robustness of the proposed method are validated through numerical simulations:

UAV Network (Linear): A network of six unmanned aerial vehicles with unknown linear dynamics and wind disturbances (generated by an exosystem) was simulated. The agents successfully converged to the constrained NE (maintaining a specific altitude while tracking individual waypoints) using controllers synthesized from noisy offline data. Comparisons showed performance comparable to a model-based controller using exact system matrices.
Rotary-Wing Formation (Nonlinear): A formation of five rotary-wing aerial vehicles with heterogeneous nonlinear dynamics (involving sinusoidal terms) and constant disturbances was tested. The data-driven controller successfully drove the outputs to the constrained NE, with regulation errors converging asymptotically to zero.

Significance and Claims
The paper positions its work as a solution to fundamental challenges in dynamic games where system models are unavailable, disturbances are present, and information is incomplete.

Distinction from Existing Work: The authors clarify that their results are not a straightforward combination of existing methods. Unlike previous data-driven game approaches (e.g., Wang et al., 2026b) that focused on unconstrained games with aligned structures, this work handles equality constraints and partial-decision information, requiring the explicit incorporation of KKT dynamics and distributed protocols.
Model Independence: Unlike model-based approaches (e.g., Guo and De Persis, 2021; Romano and Pavel, 2025), this framework operates without exact system matrices. It accommodates dynamic exosystems for linear cases and extends to nonlinear systems with constant disturbances, broadening the applicability of data-driven NE seeking.
Robustness: The framework explicitly accounts for measurement noise and external disturbances, providing formal guarantees for stability and convergence in the presence of these uncertainties.

The paper concludes by noting that future work will address inequality constraints, time-varying dynamics, and completely unknown exosystem matrices.