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 usefulness 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.

[1]  Ruzena Bajcsy,et al.  Safe semi-autonomous control with enhanced driver modeling , 2012, 2012 American Control Conference (ACC).

[2]  Anca D. Dragan,et al.  Planning for Autonomous Cars that Leverage Effects on Human Actions , 2016, Robotics: Science and Systems.

[3]  Ilya Kolmanovsky,et al.  Stable Hierarchical Model Predictive Control Using an Inner Loop Reference Model , 2011 .

[4]  Patrizio Colaneri,et al.  Hierarchical model predictive control , 2007, 2007 46th IEEE Conference on Decision and Control.

[5]  Antonio Ferramosca,et al.  Cooperative distributed MPC for tracking , 2013, Autom..

[6]  Patrizio Colaneri,et al.  An MPC approach to the design of two-layer hierarchical control systems , 2010, Autom..

[7]  Sean P. Meyn,et al.  Building thermal model reduction via aggregation of states , 2010, Proceedings of the 2010 American Control Conference.

[8]  Bart De Schutter,et al.  Toward System-Optimal Routing in Traffic Networks: A Reverse Stackelberg Game Approach , 2015, IEEE Transactions on Intelligent Transportation Systems.

[9]  Shankar Mohan,et al.  Convex computation of the reachable set for hybrid systems with parametric uncertainty , 2016, 2016 American Control Conference (ACC).

[10]  David E. Culler,et al.  Energy-Efficient Building HVAC Control Using Hybrid System LBMPC , 2012, ArXiv.

[11]  Claire Tomlin,et al.  Game-theoretic routing of GPS-assisted vehicles for energy efficiency , 2011, Proceedings of the 2011 American Control Conference.

[12]  Alexandre M. Bayen,et al.  Stackelberg Routing on Parallel Networks With Horizontal Queues , 2012, IEEE Transactions on Automatic Control.

[13]  Anil Aswani,et al.  Behavioral modeling in weight loss interventions , 2019, Eur. J. Oper. Res..

[14]  Zuo-Jun Max Shen,et al.  Inverse Optimization with Noisy Data , 2015, Oper. Res..

[15]  Humberto González,et al.  Zoned HVAC control via PDE-constrained optimization , 2015, 2016 American Control Conference (ACC).

[16]  B. Jose,et al.  Closed-Loop Stackelberg Strategies with Applications in the Optimal Control of Multilevel Systems , 2022 .

[17]  David Q. Mayne,et al.  Invariant approximations of the minimal robust positively Invariant set , 2005, IEEE Transactions on Automatic Control.

[18]  David E. Culler,et al.  Identifying models of HVAC systems using semiparametric regression , 2012, 2012 American Control Conference (ACC).

[19]  David E. Culler,et al.  Reducing Transient and Steady State Electricity Consumption in HVAC Using Learning-Based Model-Predictive Control , 2012, Proceedings of the IEEE.

[20]  Panagiotis D. Christofides,et al.  Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control , 2005, Proceedings of the 2005, American Control Conference, 2005..

[21]  Heinrich von Stackelberg,et al.  Stackelberg (Heinrich von) - The Theory of the Market Economy, translated from the German and with an introduction by Alan T. PEACOCK. , 1953 .

[22]  Yaman Arkun,et al.  Quasi-Min-Max MPC algorithms for LPV systems , 2000, Autom..

[23]  Riccardo Scattolini,et al.  Architectures for distributed and hierarchical Model Predictive Control - A review , 2009 .

[24]  Marcello Farina,et al.  Distributed non-cooperative MPC with neighbor-to-neighbor communication , 2011 .

[25]  Manfred Morari,et al.  An iterative decentralized MPC algorithm for large-scale nonlinear systems , 2009 .

[26]  Christopher L.E. Swartz,et al.  A Bilevel Programming Formulation for Dynamic Real-time Optimization∗ , 2015 .

[27]  Bruce H. Krogh,et al.  Distributed model predictive control , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[28]  Francesco Borrelli,et al.  A distributed predictive control approach to building temperature regulation , 2011, Proceedings of the 2011 American Control Conference.

[29]  Anil Aswani,et al.  Duality Approach to Bilevel Programs with a Convex Lower Level , 2016, 2018 Annual American Control Conference (ACC).

[30]  David Q. Mayne,et al.  Robust time-optimal control of constrained linear Systems , 1997, Autom..

[31]  Anil Aswani,et al.  Behavioral analytics for myopic agents , 2017, Eur. J. Oper. Res..

[32]  E. Gilbert,et al.  Theory and computation of disturbance invariant sets for discrete-time linear systems , 1998 .

[33]  Henrik Madsen,et al.  A bilevel model for electricity retailers' participation in a demand response market environment , 2013 .

[34]  Patrice Marcotte,et al.  An overview of bilevel optimization , 2007, Ann. Oper. Res..

[35]  Stephen J. Wright,et al.  Stability and optimality of distributed model predictive control , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[36]  Jose B. Cruz,et al.  An approach to discrete-time incentive feedback Stackelberg games , 2002, IEEE Trans. Syst. Man Cybern. Part A.

[37]  Ilya V. Kolmanovsky,et al.  Stable hierarchical model predictive control using an inner loop reference model and lambda-contractive terminal constraint sets , 2013, Autom..

[38]  Marcello Farina,et al.  Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems , 2012, Autom..

[39]  Minghui Zhu,et al.  Stackelberg-game analysis of correlated attacks in cyber-physical systems , 2011, Proceedings of the 2011 American Control Conference.