Control Synthesis for Bilevel Linear Model Predictive Control

Distributed model predictive control (MPC) is either cooperative or competitive, and control-theoretic properties have been less studied in the competitive (e.g., game theory) setting. This paper studies MPC with linear dynamics and a Stackelberg game structure: Given a fixed lower-level linear MPC (LoMPC) controller, the bilevel linear MPC (BiMPC) controller chooses inputs to steer LoMPC knowing that LoMPC is optimizing with respect to a different cost function. After defining LoMPC and BiMPC, we give examples to demonstrate how interconnections in a dynamic Stackelberg game can lead to loss/gain (as compared to the same system being centrally controlled) of controllability or stability. Then, we give sufficient conditions under an arbitrary finite MPC horizon for stabilizability of BiMPC, and develop an approach to synthesize a stabilizing BiMPC controller. Next, we define two (a duality-based technique and an integer-programming-based technique) reformulations to numerically solve the optimization problem associated with BiMPC, prove equivalence of these reformulations to BiMPC, and demonstrate their useful by simulations of a synthetic system and a case study of an electric utility changing electricity prices to perform demand response of a home's air conditioner controlled by a linear MPC.