Extended-State Backward Iteration for Stackelberg Dynamic Games: Application to a 2-DOF Flexible Robot

This paper proposes a general framework for hierarchical dynamic games based on an Iterative Derivation of Optimal Policies (IDOP). The main theoretical result, stated in Theorem 1, reformulates the game using an extended state that includes the adjoint variables of all players. This enables a backward procedure in which the instantaneous optimal gain of each active player is computed while accounting for higher-priority strategies. A dedicated operator is introduced to compactly represent and solve the coupled Riccati equations arising from the Hamilton-Jacobi-Bellman framework. The method is generic and applicable to a broad class of hierarchical decision problems. Its effectiveness is demonstrated through two numerical examples and an experimental validation on a real two-degree-of-freedom (2-DOF) flexible serial robot.