Stability Certificates for Receding Horizon Games

TL;DR

This work develops the first general stability analysis for Receding Horizon Games (RHG), uniting dissipativity theory and monotone operator theory to certify closed-loop stability under coupling constraints and dynamics. The authors derive an LMI-based sufficient condition, requiring a positive definite Lyapunov-like matrix $P$ and nonnegative scalars $\lambda_1,\lambda_2$, under which the interconnection of an LTI plant with a non-cooperative game yields a globally asymptotically stable equilibrium and recursive feasibility, without terminal ingredients. A distributed stability certificate is also provided for decoupled agent dynamics, enabling local verification via per-agent LMIs. The theoretical results are corroborated by numerical experiments on a battery charging game, demonstrating the effectiveness of the certificates and practical tuning guidelines, including scenarios where RHG can destabilize otherwise stable systems if not carefully configured. Overall, the paper offers a rigorous, computationally tractable framework to guarantee stability in multi-agent RHG deployments with coupling and provides actionable guidance for design and tuning in realistic applications.

Abstract

Game-theoretic MPC (or Receding Horizon Games) is an emerging control methodology for multi-agent systems that generates control actions by solving a dynamic game with coupling constraints in a receding-horizon fashion. This control paradigm has recently received an increasing attention in various application fields, including robotics, autonomous driving, traffic networks, and energy grids, due to its ability to model the competitive nature of self-interested agents with shared resources while incorporating future predictions, dynamic models, and constraints into the decision-making process. In this work, we present the first formal stability analysis based on dissipativity and monotone operator theory that is valid also for non-potential games. Specifically, we derive LMI-based certificates that ensure asymptotic stability and are numerically verifiable. Moreover, we show that, if the agents have decoupled dynamics, the numerical verification can be performed in a scalable manner. Finally, we present tuning guidelines for the agents' cost function weights to fulfill the certificates and, thus, ensure stability.

Figures (12)

• Figure 1: In RHG, control actions are generated by solving a finite-horizon dynamic game, in a receding-horizon fashion.
• Figure 2: Feedback interconnection between LTI system $\Sigma_1$ and the static nonlinearity $\phi(\cdot)$ in $\Sigma_2$.
• Figure 3: Feasibility regions of Corollary \ref{['cor:StabCondLocal']} (i) as a function of $A^v$, $\lambda_1^v$, $\mu$, and $W^v$ for fixed $K=10$, $B^v = 1$ and $\lambda_2^v = 0$. The plots were generated using the RegionPlot function of Mathematica.
• Figure 4: Evolution of the closed-loop system in Section \ref{['ss:IE']}, for different choices of the state weighs $W^v$ in the RHG feedback law.
• Figure 5: Schematic of households owning a private battery and being connected to the power grid through a point of common coupling.

Theorems & Definitions (24)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• proof
• Proposition 3
• proof
• Corollary 1
• proof