Inverse Dynamic Games With Process Noise and Unknown Target States: A Linear Estimation Approach

The inverse dynamic games problem is to model expert demonstrations by identifying the underlying cost functions of multiple agents from observed trajectories of their dynamic game interactions. This article investigates discrete-time, finite-horizon linear-quadratic (LQ) problems where both the state weight matrix and input weight matrix are unknown, with the presence of both process noise and observation noise. In addition, each player's cost function incorporates a player-specific, unknown linear term with respect to the state. Under this framework, first, sufficient conditions are established for the solvability of the weight matrices. Subsequently, it is proved that the inverse dynamic games problem involving heterogeneous unknown target states is structurally identifiable, unaffected by process noise. Building on the necessary conditions for Nash equilibrium solutions in forward problems, the estimation of the cost function parameters is formulated as a nontrivial solution to a homogeneous linear estimation problem, which can be implemented in a distributed manner. Furthermore, the proposed estimator achieves statistical consistency under the influence of observation noise. The effectiveness is illustrated through a multivehicle spring-coupled dynamic game and an interactive steering control scenario.