A Distributionally Robust LQR for Systems with Multiple Uncertain Players

In this paper, we study the robust linear quadratic regulator (LQR) problem for a class of discrete-time dynamical systems composed of several uncertain players with unknown or ambiguous distribution information. A distinctive feature of the assumed model is that each player is prescribed by a nominal probability distribution and categorized according to an uncertainty level of confidence. Our approach is based on minimax optimization. By following a dynamic programming approach a closed-form expression of the robust control policy is derived. The effect of ambiguity on the performance of the LQR is studied via a sequential hierarchical game with one leader and several followers. The equilibrium solution is obtained through a maximizing, time-varying probability distribution characterizing each player’s optimal policy. The behavior of the proposed method is demonstrated through an application to a drop-shipping retail fulfillment model.