Abstract
This paper presents a framework for the competitive analysis of Model Predictive Controllers (MPC). Competitive analysis means evaluating the relative performance of the MPC as compared to other controllers. Concretely, we associate the MPC with a regret value which quantifies the maximal difference between its cost and the cost of any alternative controller from a given class. Then, the problem we tackle is that of determining whether the regret value is at most some given bound. Our contributions are both theoretical as well as practical: (1) We reduce the regret problem for controllers modeled as hybrid automata to the reachability problem for such automata. We propose a reachability-based framework to solve the regret problem. Concretely, (2) we propose a novel CEGAR-like algorithm to train a deep neural network (DNN) to clone the behavior of the MPC. Then, (3) we leverage existing reachability analysis tools capable of handling hybrid automata with DNNs to check bounds on the regret value of the controller.
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Notes
- 1.
Note that if X contains more variables than just x, this function is not unique.
- 2.
Our toolchain splits each interval into n equally large segments and adds all points in the resulting lattice. In our experiments, we use \(n = 4\).
- 3.
All graphs and numbers can be reproduced using scripts from: https://doi.org/10.5281/zenodo.8255730.
References
Abadi, M., et al.: TensorFlow: a system for large-scale machine learning. In: keeton, K., Roscoe, T. (eds.) 12th USENIX Symposium on Operating Systems Design and Implementation, OSDI 2016, Savannah, GA, USA, 2–4 November 2016, pp. 265–283. USENIX Association (2016). https://www.usenix.org/conference/osdi16/technical-sessions/presentation/abadi
Bak, S., Bogomolov, S., Johnson, T.T.: HYST: a source transformation and translation tool for hybrid automaton models. In: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control, pp. 128–133 (2015)
Bratko, I., Urbančič, T., Sammut, C.: Behavioural cloning: phenomena, results and problems. IFAC Proc. Vol. 28(21), 143–149 (1995)
Chen, T., Chen, H.: Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Trans. Neural Netw. 6(4), 911–917 (1995)
Chen, X., Sankaranarayanan, S.: Reachability analysis for cyber-physical systems: are we there yet? In: Deshmukh, J.V., Havelund, K., Perez, I. (eds.) NASA Formal Methods, NFM 2022. LNCS, vol. 13260. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-06773-0_6
Clavière, A., Dutta, S., Sankaranarayanan, S.: Trajectory tracking control for robotic vehicles using counterexample guided training of neural networks. In: Benton, J., Lipovetzky, N., Onaindia, E., Smith, D.E., Srivastava, S. (eds.) Proceedings of the Twenty-Ninth International Conference on Automated Planning and Scheduling, ICAPS 2018, Berkeley, CA, USA, 11–15 July 2019, pp. 680–688. AAAI Press (2019). https://ojs.aaai.org/index.php/ICAPS/article/view/3555
Fantoni, I., Lozano, R., Lozano, R.: Non-linear Control for Underactuated Mechanical Systems. Springer, London (2002). https://doi.org/10.1007/978-1-4471-0177-2
Frehse, G., et al.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_30
Gillis, J., Vandewal, B., Pipeleers, G., Swevers, J.: Effortless modeling of optimal control problems with rockit. In: 39th Benelux Meeting on Systems and Control, Elspeet, The Netherlands, 10 March 2020–12 March 2020 (2020)
Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? J. Comput. Syst. Sci. 57(1), 94–124 (1998). https://doi.org/10.1006/jcss.1998.1581
Hertneck, M., Köhler, J., Trimpe, S., Allgöwer, F.: Learning an approximate model predictive controller with guarantees. IEEE Control. Syst. Lett. 2(3), 543–548 (2018). https://doi.org/10.1109/LCSYS.2018.2843682
Hunter, P., Pérez, G.A., Raskin, J.: Reactive synthesis without regret. Acta Informatica 54(1), 3–39 (2017). https://doi.org/10.1007/s00236-016-0268-z
Ivanov, R., Carpenter, T., Weimer, J., Alur, R., Pappas, G., Lee, I.: Verisig 2.0: verification of neural network controllers using Taylor model preconditioning. In: Silva, A., Leino, K.R.M. (eds.) CAV 2021. LNCS, vol. 12759, pp. 249–262. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-81685-8_11
Julian, K.D., Lopez, J., Brush, J.S., Owen, M.P., Kochenderfer, M.J.: Policy compression for aircraft collision avoidance systems. In: 2016 IEEE/AIAA 35th Digital Avionics Systems Conference (DASC), pp. 1–10. IEEE (2016)
LaValle, S.M.: Planning Algorithms. Cambridge University Press (2006)
Muvvala, K., Amorese, P., Lahijanian, M.: Let’s collaborate: regret-based reactive synthesis for robotic manipulation. In: 2022 International Conference on Robotics and Automation, ICRA 2022, Philadelphia, PA, USA, 23–27 May 2022, pp. 4340–4346. IEEE (2022). https://doi.org/10.1109/ICRA46639.2022.9812298
Ross, S., Bagnell, D.: Efficient reductions for imitation learning. In: Teh, Y.W., Titterington, D.M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, AISTATS 2010. JMLR Proceedings, Chia Laguna Resort, Sardinia, Italy, 13–15 May 2010, vol. 9, pp. 661–668. JMLR.org (2010). http://proceedings.mlr.press/v9/ross10a.html
Ross, S., Gordon, G.J., Bagnell, D.: A reduction of imitation learning and structured prediction to no-regret online learning. In: Gordon, G.J., Dunson, D.B., Dudík, M. (eds.) Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, AISTATS 2011. JMLR Proceedings, Fort Lauderdale, USA, 11–13 April 2011, vol. 15, pp. 627–635. JMLR.org (2011). http://proceedings.mlr.press/v15/ross11a/ross11a.pdf
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Bellis, S., Denil, J., Krishnamurthy, R., Leys, T., Pérez, G.A., Raha, R. (2023). A Framework for the Competitive Analysis of Model Predictive Controllers. In: Bournez, O., Formenti, E., Potapov, I. (eds) Reachability Problems. RP 2023. Lecture Notes in Computer Science, vol 14235. Springer, Cham. https://doi.org/10.1007/978-3-031-45286-4_11
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