On Data-based Nash Equilibria in LQ Nonzero-sum Differential Games

TL;DR

This work addresses Nash equilibria in infinite-horizon linear-quadratic nonzero-sum differential games, proposing data-based solutions that require no full model knowledge. By leveraging Willems' lemma for continuous-time systems and persistently excited (PCPE) data, the authors derive data-based representations of closed-loop dynamics and solve the coupled algebraic Riccati equations (AREs) in both deterministic (full-state) and stochastic (observer-based) settings. The main contributions include (i) data-based expressions that connect offline Hankel-like data to stabilizing controllers, (ii) an iterative data-driven procedure that is equivalent to the traditional model-based ARE solver, and (iii) a stochastic NZS framework with finite-dimensional observers validated via numerical simulation. The practical impact lies in enabling model-free, data-driven Nash design for multiagent systems with guaranteed equilibrium structure, while identifying persistence-of-excitation requirements and outlining avenues for relaxing offline data assumptions and extending to graphical games.

Abstract

This paper considers data-based solutions of linear-quadratic nonzero-sum differential games. Two cases are considered. First, the deterministic game is solved and Nash equilibrium strategies are obtained by using persistently excited data from the multiagent system. Then, a stochastic formulation of the game is considered, where each agent measures a different noisy output signal and state observers must be designed for each player. It is shown that the proposed data-based solutions of these games are equivalent to known model-based procedures. The resulting data-based solutions are validated in a numerical experiment.

Figures (2)

• Figure 1: Performance of the proposed data-based controller.
• Figure 2: State estimation of $x_1$ for both players