Identification Methods for Ordinal Potential Differential Games

TL;DR

This work addresses the identification of ordinal potential differential games (OPDGs) for linear-quadratic (LQ) differential games, enabling a single potential cost to reproduce the Nash equilibrium of multi-agent dynamics. It introduces two LMIs-based methods: Trajectory-Free Optimization (TFO), which identifies an OPDG using only cost data and minimizes the condition number of the potential cost, and Weakly Trajectory-Dependent Optimization (WTDO), which relaxes trajectory-free requirements to leverage partial trajectory information when TFO is infeasible. The methods are validated through academic and engineering examples, showing faster convergence, accurate trajectory reproduction, and robustness to measurement noise compared with a prior approach (Input-Dependent Optimization, IDO). The results broaden the applicability of OPDGs to practical cooperative control problems and motivate future work on cooperative learning controllers and real-human interaction validation.

Abstract

This paper introduces two new identification methods for linear quadratic (LQ) ordinal potential differential games (OPDGs). Potential games are notable for their benefits, such as the computability and guaranteed existence of Nash Equilibria. While previous research has analyzed ordinal potential static games, their applicability to various engineering applications remains limited. Despite the earlier introduction of OPDGs, a systematic method for identifying a potential game for a given LQ differential game has not yet been developed. To address this gap, we propose two identification methods to provide the quadratic potential cost function for a given LQ differential game. Both methods are based on linear matrix inequalities (LMIs). The first method aims to minimize the condition number of the potential cost function's parameters, offering a faster and more precise technique compared to earlier solutions. In addition, we present an evaluation of the feasibility of the structural requirements of the system. The second method, with a less rigid formulation, can identify LQ OPDGs in cases where the first method fails. These novel identification methods are verified through simulations, demonstrating their advantages and potential in designing and analyzing cooperative control systems.

Figures (5)

• Figure 1: The illustration of the general idea of the potential games, where the original game is replaced by a (fictitious) potential function. The optimum of this potential function provides the NE of the original game.
• Figure 2: A schematic representation of the trajectory independency of the optimization in a three-dimensional space, where $\left(\boldsymbol{e}^1,\boldsymbol{e}^2,\boldsymbol{e}^3\right)$ are the basis vectors of the coordinate system. The vectors $\boldsymbol{v}^{(p)}$ and $\boldsymbol{v}^{(i)}$ show in the same direction and thus are linearly dependent. This means that the condition (\ref{['necessary_opdg_simplified']}) is automatically fulfilled at $t_1$ as well as at any other time $t_2$.
• Figure 3: Comparison of the system state trajectories of the original game (solid lines) and the trajectories of the potential game (dashed lines)
• Figure 4: The derivatives of the Hamiltonian functions of the two players in accordance to \ref{['eq:opdg_def']}
• Figure 5: Comparison of the system state trajectories of the original game (solid lines) and the trajectories of the potential game (dashed lines)

Theorems & Definitions (9)

• Definition 1: Nash Equilibrium
• Definition 2: Exact Potential Differential Game 1996_PotentialGames_monderer
• Definition 3: Ordinal Potential Differential Game 2021_PotentialDifferentialGames_varga
• Lemma 1: Sufficient Condition of an OPDG 2021_PotentialDifferentialGames_varga
• proof
• Remark 1
• Lemma 2: Necessary Condition for the feasibility of the TFO for OPDGs
• proof
• Remark 2