Abstract
This paper describes a distributed optimal control problem of a multi-agent nonlinear system, where each agent interacts with others through their control input signals. The individual optimization problem is formulated and a solution for each agent is derived considering an inverse optimal control approach. To address the interaction between the agents, a coordination method based on a game played by each agent with their neighbours at each sampling interval is proposed. Using small-gain arguments, we derive conditions under which the proposed game converges to an equilibrium point. Simulation results for linear and nonlinear agents are presented illustrating situations in which the convergence condition holds, and others where it does not hold, suggesting (for the examples described) that the convergence results are tight.
This work was performed within the framework of the project HARMONY, Distributed Optimal Control for Cyber-Physical Systems Applications, financed by FCT under contract AAC n2/SAICT/2017-031411, project IMPROVE-POCI-01-0145-FEDER-031823, and pluriannual INESC-ID funding UID/CEC/50021/2019.
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References
Borrelli, F., Keviczky, T.: Distributed LQR design for identical dynamically decoupled systems. IEEE Trans. Autom. Control 53(8), 1901–1912 (2008). https://doi.org/10.1109/TAC.2008.925826
Sun, H., Liu, Y., Li, F., Niu, X.: A survey on optimal consensus of multi-agent systems. Proceedings - 2017 Chinese Automation Congress, CAC 2017, 2017 Jan, pp. 4978–4983(2017). https://doi.org/10.1109/CAC.2017.8243662
Venkat, A.N., Hiskens, I.A., Rawlings, J.B., Wright, S.J.: Distributed MPC strategies with application to power system automatic generation control. IEEE Trans. Control Sys. Technol. 16(6), 1192–1206 (2008)
Lemos, J.M., Pinto, L.R., Rato, L.M., Rijo, M.: Multivariable and distributes LQR control of a water delivery canal. J. Irrig. Drain. Eng. 139(10), 855–863 (2013)
Boucekkine, R., Veliov, V. M., Augeraud-v, E.: Distributed Optimal Control Models in Environmental Economics: A Review (No. 1902) (2019). https://ideas.repec.org/p/aim/wpaimx/1902.html
Wrzaczek, S., Kuhn, M., Prskawetz, A., Feichtinger, G.: The Reproductive Value in Distributed Optimal Control Models (No. 0906) (2009). https://ideas.repec.org/p/vid/wpaper/0904.html
Foderaro, G., Ferrari, S., Wettergren, T.A.: Distributed optimal control for multi-agent trajectory optimization. Automatica 50(1), 149–154 (2014). https://doi.org/10.1016/j.automatica.2013.09.014
Kantaros, Y., Zavlanos, M.M.: Distributed optimal control synthesis for multi-robot systems under global temporal tasks. In: Proceedings - 9th ACM/IEEE International Conference on Cyber-Physical Systems, ICCPS 2018, pp. 162–173 (2018). https://doi.org/10.1109/ICCPS.2018.00024
Lemos, J.M., Pinto, L.: Distributed linear-quadratic control of serially chained systems. IEEE Control Syst. Mag. 26–38 (2012). https://doi.org/10.1109/MCS.2012.2214126
Mota, J.F.C., Xavier, J.M.F., Aguiar, P.M.Q., Puschel, M.: Distributed optimization with local domains: applications in MPC and network flows. IEEE Trans. Autom. Control 60(7), 2004–2009 (2015). https://doi.org/10.1109/TAC.2014.2365686
Maestre, J.M., de La Pena, D.M., Camacho, E.F.: Distributed model predictive control based on a cooperative game. Optimal Control Appl. Methods 32, 153–176 (2010). https://doi.org/10.1002/oca
Al-tamimi, A., Lewis, F.L., Abu-khalaf, M.: Discrete-time nonlinear HJB solutions using approximate dynamic programming: convergence proof. IEEE Trans. Syst. Man Cybern.-Part B 38(4), 943–949 (2008)
Ornelas-Tellez, F., Sanchez, E.N., Loukianov, A.G., Rico, J.J.: Robust inverse optimal control for discrete-time nonlinear system stabilization. Eur. J. Control 20(1), 38–44 (2014). https://doi.org/10.1016/j.ejcon.2013.08.001
Khalil, H.: Nonlinear Systems, 3rd edn. Prentice Hall (2002)
Jiang, Z., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869 (2001)
Jiang, Z.-P., Wang, Y.: Small gain theorems on input-to-output stability. In: Proceedings of the 3rd International DCDIS Conference, pp. 220–224 (2003)
Dashkovskiy, S., Rüffer, B., Wirth, F.R.: An ISS small gain theorem for general networks. Math. Control Signals Syst. 19, 93–122 (2007)
Zhongping, J., Yuandan, L., Yuan, W.: Nonlinear small-gain theorems for discrete-time large-scale systems. In: Proceedings of the 27th Chinese Control Conference, pp. 704–708 (2008)
Jiang, Z.-P., Wang, Y.: A generalization of the nonlinear small-gain theorem for large-scale complex systems. In: Proceedings of the 7th World Congress on Intelligent Control and Automation, vol. 1, pp. 1188–1193 (2008)
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Belfo, J.P., Aguiar, A.P., Lemos, J.M. (2022). Convergence of a Distributed Optimal Control Coordination Method via the Small-Gain Theorem. In: Zattoni, E., Simani, S., Conte, G. (eds) 15th European Workshop on Advanced Control and Diagnosis (ACD 2019). ACD 2019 2018. Lecture Notes in Control and Information Sciences - Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-030-85318-1_23
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