Two-component controller design to safeguard data-driven predictive control

References

[1] T. Martin and F. Allgöwer, “Data-driven system analysis of nonlinear systems using polynomial approximation,” IEEE Trans. Automat. Control, vol. 69, no. 7, pp. 4261–4274, 2024, https://doi.org/10.1109/tac.2023.3321212.Suche in Google Scholar

[2] J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. M. De Moor, “A note on persistency of excitation,” Syst. Control Lett., vol. 54, no. 4, pp. 325–329, 2005, https://doi.org/10.1016/j.sysconle.2004.09.003.Suche in Google Scholar

[3] J. Coulson, J. Lygeros, and F. Dörfler, “Data-enabled predictive control: In the shallows of the DeePC,” in 18th European Control Conference (ECC), 2019.10.23919/ECC.2019.8795639Suche in Google Scholar

[4] T. Faulwasser, R. Ou, G. Pan, P. Schmitz, and K. Worthmann, “Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again,” Annu. Rev. Control, vol. 55, pp. 92–117, 2023. https://doi.org/10.1016/j.arcontrol.2023.03.005.Suche in Google Scholar

[5] C. Verhoek, R. Tóth, S. Haesaert, and A. Koch, “Fundamental lemma for data-driven analysis of linear parameter-varying systems,” in 60th IEEE Conference on Decision and Control (CDC), 2021.10.1109/CDC45484.2021.9683151Suche in Google Scholar

[6] M. Alsalti, J. Berberich, V. G. Lopez, F. Allgöwer, and M. A. Müller, “Data-based system analysis and control of flat nonlinear systems,” in 60th IEEE Conference on Decision and Control (CDC), 2021.10.1109/CDC45484.2021.9683327Suche in Google Scholar

[7] P. Schmitz, T. Faulwasser, P. Rapisarda, and K. Worthmann, “A continuous-time fundamental lemma and its application in data-driven optimal control,” Syst. Control Lett., vol. 194, p. 105950, 2024, https://doi.org/10.1016/j.sysconle.2024.105950.Suche in Google Scholar

[8] D. Bertsekas, Reinforcement Learning and Optimal Control, vol. 1, Belmont, Massachusetts, Athena Scientific, 2019.Suche in Google Scholar

[9] B. Kiumarsi, K. G. Vamvoudakis, H. Modares, and F. L. Lewis, “Optimal and autonomous control using reinforcement learning: A survey,” IEEE Transact. Neural Networks Learn. Syst., vol. 29, no. 6, pp. 2042–2062, 2017, https://doi.org/10.1109/tnnls.2017.2773458.Suche in Google Scholar PubMed

[10] F. Dörfler, J. Coulson, and I. Markovsky, “Bridging direct and indirect data-driven control formulations via regularizations and relaxations,” IEEE Trans. Automat. Control, vol. 68, no. 2, pp. 883–897, 2022, https://doi.org/10.1109/tac.2022.3148374.Suche in Google Scholar

[11] B. O. Koopman, “Hamiltonian systems and transformation in Hilbert space,” Proc. Natl. Acad. Sci., vol. 17, no. 5, pp. 315–318, 1931, https://doi.org/10.1073/pnas.17.5.315.Suche in Google Scholar PubMed PubMed Central

[12] B. O. Koopman and J. von Neumann, “Dynamical systems of continuous spectra,” Proc. Natl. Acad. Sci. U. S. A., vol. 18, no. 3, pp. 255–263, 1932, https://doi.org/10.1073/pnas.18.3.255.Suche in Google Scholar PubMed PubMed Central

[13] S. L. Brunton, M. Budišić, E. Kaiser, and J. N. Kutz, “Modern Koopman theory for dynamical systems,” SIAM Review, vol. 64, no. 2, pp. 229–340, 2022, https://doi.org/10.1137/21m1401243.Suche in Google Scholar

[14] S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, and C. Schütte, “Data-driven approximation of the Koopman generator: Model reduction, system identification, and control,” Phys. D: Nonlinear Phenom., vol. 406, p. 132416, 2020, https://doi.org/10.1016/j.physd.2020.132416.Suche in Google Scholar

[15] M. Williams, I. Kevrekidis, and C. Rowley, “A data-driven approximation of the koopman operator: Extending dynamic mode decomposition,” J. Nonlinear Sci., vol. 25, no. 6, pp. 1307–1346, 2015, https://doi.org/10.1007/s00332-015-9258-5.Suche in Google Scholar

[16] A. Mauroy, Y. Susuki, and I. Mezic, Koopman Operator in Systems and Control, Cham, Switzerland, Springer, 2020.10.1007/978-3-030-35713-9Suche in Google Scholar

[17] S. L. Brunton, B. W. Brunton, J. L. Proctor, and J. N. Kutz, “Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control,” PloS One, vol. 11, no. 2, p. e0150171, 2016, https://doi.org/10.1371/journal.pone.0150171.Suche in Google Scholar PubMed PubMed Central

[18] A. Surana, “Koopman operator based observer synthesis for control-affine nonlinear systems,” in 55th IEEE Conference on Decision and Control (CDC), 2016.10.1109/CDC.2016.7799268Suche in Google Scholar

[19] M. Korda and I. Mezić, “Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control,” Automatica, vol. 93, pp. 149–160, 2018, https://doi.org/10.1016/j.automatica.2018.03.046.Suche in Google Scholar

[20] D. Bruder, X. Fu, and R. Vasudevan, “Advantages of bilinear Koopman realizations for the modeling and control of systems with unknown dynamics,” IEEE Rob. Autom. Lett., vol. 6, no. 3, pp. 4369–4376, 2021, https://doi.org/10.1109/lra.2021.3068117.Suche in Google Scholar

[21] L. C. Iacob, R. Tóth, and M. Schoukens, “Koopman form of nonlinear systems with inputs,” Automatica, vol. 162, 2024, p. 111525, https://doi.org/10.1016/j.automatica.2024.111525.Suche in Google Scholar

[22] M. Korda and I. Mezić, “On convergence of extended dynamic mode decomposition to the Koopman operator,” J. Nonlinear Sci., vol. 28, no. 2, pp. 687–710, 2018, https://doi.org/10.1007/s00332-017-9423-0.Suche in Google Scholar

[23] I. Mezić, “On numerical approximations of the Koopman operator,” Mathematics, vol. 10, no. 7, p. 1180, 2022, https://doi.org/10.3390/math10071180.Suche in Google Scholar

[24] C. Zhang and E. Zuazua, “A quantitative analysis of Koopman operator methods for system identification and predictions,” C. R. Mec., vol. 351, no. S1, pp. 721–751, 2023, https://doi.org/10.5802/crmeca.138.Suche in Google Scholar

[25] F. Nüske, S. Peitz, F. Philipp, M. Schaller, and K. Worthmann, “Finite-data error bounds for Koopman-based prediction and control,” J. Nonlinear Sci., vol. 33, no. 14, 2023, https://doi.org/10.1007/s00332-022-09862-1.Suche in Google Scholar

[26] F. M. Philipp, M. Schaller, S. Boshoff, S. Peitz, F. Nüske, and K. Worthmann, “Variance representations and convergence rates for data-driven approximations of Koopman operators,” ArXiv preprint arXiv:2402.02494, 2024.Suche in Google Scholar

[27] L. Bold, L. Grüne, M. Schaller, and K. Worthmann, “Data-driven MPC with stability guarantees using extended dynamic mode decomposition,” IEEE Trans. Automat. Control, vol. 70, no. 1, pp. 534–541, 2024. https://doi.org/10.1109/tac.2024.3431169.Suche in Google Scholar

[28] R. Strässer, M. Schaller, K. Worthmann, J. Berberich, and F. Allgöwer, “Koopman-based feedback design with stability guarantees,” IEEE Trans. Automat. Control, vol. 70, no. 1, pp. 355–370, 2024. https://doi.org/10.1109/tac.2024.3425770.Suche in Google Scholar

[29] I. Mezić, “Analysis of fluid flows via spectral properties of the Koopman operator,” Annu. Rev. Fluid. Mech., vol. 45, no. 1, pp. 357–378, 2013. https://doi.org/10.1146/annurev-fluid-011212-140652.Suche in Google Scholar

[30] I. Mezić, “Spectral properties of dynamical systems, model reduction and decompositions,” Nonlinear Dyn., vol. 41, nos. 1–3, pp. 309–325, 2005, https://doi.org/10.1007/s11071-005-2824-x.Suche in Google Scholar

[31] D. Goswami and D. A. Paley, “Bilinearization, reachability, and optimal control of control-affine nonlinear systems: A Koopman spectral approach,” IEEE Trans. Automat. Control, vol. 67, no. 6, pp. 2715–2728, 2021, https://doi.org/10.1109/tac.2021.3088802.Suche in Google Scholar

[32] S. Klus, F. Nüske, and B. Hamzi, “Kernel-based approximation of the Koopman generator and Schrödinger operator,” Entropy, vol. 22, no. 7, p. 722, 2020, https://doi.org/10.3390/e22070722.Suche in Google Scholar PubMed PubMed Central

[33] F. M. Philipp, M. Schaller, K. Worthmann, S. Peitz, and F. Nüske, “Error bounds for kernel-based approximations of the Koopman operator,” Appl. Comput. Harmon. Anal., vol. 71, p. 101657, 2024, https://doi.org/10.1016/j.acha.2024.101657.Suche in Google Scholar

[34] F. Köhne, F. M. Philipp, M. Schaller, A. Schiela, and K. Worthmann, “L∞-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition,” SIAM J. Appl. Dyn. Syst., 2024, To appear (arXiv preprint arXiv:2403.18809).10.1137/24M1650120Suche in Google Scholar

[35] L. Bold, F. M. Philipp, M. Schaller, and K. Worthmann, “Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability,” Preprint arXiv:2412.02811, 2024.Suche in Google Scholar

[36] J. Garcıa and F. Fernández, “A comprehensive survey on safe reinforcement learning,” J. Mach. Learn. Res., vol. 16, no. 1, 2015.Suche in Google Scholar

[37] L. Hewing, K. P. Wabersich, M. Menner, and M. N. Zeilinger, “Learning-based model predictive control: Toward safe learning in control,” Annu. Rev. Control, Robot., Autonom. Syst., vol. 3, no. 1, pp. 269–296, 2020, https://doi.org/10.1146/annurev-control-090419-075625.Suche in Google Scholar

[38] M. Osborne, H.-S. Shin, and A. Tsourdos, “A review of safe online learning for nonlinear control systems,” in International Conference on Unmanned Aircraft Systems (ICUAS), 2021.10.1109/ICUAS51884.2021.9476765Suche in Google Scholar

[39] M. Chen and C. J. Tomlin, “Hamilton–Jacobi reachability: Some recent theoretical advances and applications in unmanned airspace management,” Annu. Rev. Control, Robot., Autonom. Syst., vol. 1, no. 1, 2018.10.1146/annurev-control-060117-104941Suche in Google Scholar

[40] A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: Theory and applications,” in 18th European Control Conference (ECC), 2019.10.23919/ECC.2019.8796030Suche in Google Scholar

[41] T. J. Perkins and A. G. Barto, “Lyapunov design for safe reinforcement learning,” J. Mach. Learn. Res., vol. 3, pp. 803–832, 2002.Suche in Google Scholar

[42] A. Aswani, H. Gonzalez, S. S. Sastry, and C. Tomlin, “Provably safe and robust learning-based model predictive control,” Automatica, vol. 49, no. 5, pp. 1216–1226, 2013, https://doi.org/10.1016/j.automatica.2013.02.003.Suche in Google Scholar

[43] K. P. Wabersich and M. N. Zeilinger, “A predictive safety filter for learning-based control of constrained nonlinear dynamical systems,” Automatica, vol. 129, p. 109597, 2021, https://doi.org/10.1016/j.automatica.2021.109597.Suche in Google Scholar

[44] K. P. Wabersich and M. N. Zeilinger, “Predictive control barrier functions: Enhanced safety mechanisms for learning-based control,” IEEE Trans. Automat. Control, vol. 68, no. 5, pp. 2638–2651, 2022, https://doi.org/10.1109/tac.2022.3175628.Suche in Google Scholar

[45] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, West Sussex, England, John Wiley & Sons, 2005.Suche in Google Scholar

[46] M. Araki and H. Taguchi, “Two-degree-of-freedom PID controllers,” Int. J. Control Autom. Syst., vol. 1, no. 4, 2003.Suche in Google Scholar

[47] J. Sánchez, A. Visioli, and S. Dormido, “A two-degree-of-freedom PI controller based on events,” J. Process Control, vol. 21, no. 4, pp. 639–651, 2011, https://doi.org/10.1016/j.jprocont.2010.12.001.Suche in Google Scholar

[48] S. Drücker, L. Lanza, T. Berger, T. Reis, and R. Seifried, “Experimental validation for the combination of funnel control with a feedforward control strategy,” Multibody Syst. Dyn., vol. 63, pp. 105–123, 2024. https://doi.org/10.1007/s11044-024-09976-2.Suche in Google Scholar

[49] P. Schmitz, L. Lanza, and K. Worthmann, “Safe data-driven reference tracking with prescribed performance,” in 27th International Conference on System Theory, Control and Computing (ICSTCC), IEEE, 2023, pp. 454–460.10.1109/ICSTCC59206.2023.10308521Suche in Google Scholar

[50] S. Gottschalk, L. Lanza, K. Worthmann, and K. Lux-Gottschalk, “Reinforcement learning for docking maneuvers with prescribed performance,” IFAC-PapersOnLine, vol. 58, no. 17, pp. 196–201, 2024, https://doi.org/10.1016/j.ifacol.2024.10.163.Suche in Google Scholar

[51] T. Johannink, et al.., “Residual reinforcement learning for robot control,” in IEEE International Conference on Robotics and Automation (ICRA), IEEE, 2019.10.1109/ICRA.2019.8794127Suche in Google Scholar

[52] L. Lanza, et al.., “Sampled-data funnel control and its use for safe continual learning,” Syst. Control Lett., vol. 192, p. 105892, 2024. https://doi.org/10.1016/j.sysconle.2024.105892.Suche in Google Scholar

[53] T. Berger, A. Ilchmann, and E. P. Ryan, “Funnel control of nonlinear systems,” Math. Control, Signals, Syst., vol. 33, pp. 151–194, 2021, https://doi.org/10.1007/s00498-021-00277-z.Suche in Google Scholar

[54] A. Ilchmann, E. P. Ryan, and C. J. Sangwin, “Tracking with prescribed transient behaviour,” ESAIM Control, Optim. Calc. Var., vol. 7, pp. 471–493, 2002. https://doi.org/10.1051/cocv:2002064.10.1051/cocv:2002064Suche in Google Scholar

[55] G. BastosJr, R. Seifried, and O. Brüls, “Analysis of stable model inversion methods for constrained underactuated mechanical systems,” Mech. Mach. Theor., vol. 111, pp. 99–117, 2017, https://doi.org/10.1016/j.mechmachtheory.2017.01.011.Suche in Google Scholar

[56] C. De Persis and P. Tesi, “Formulas for data-driven control: Stabilization, optimality, and robustness,” IEEE Trans. Automat. Control, vol. 65, no. 3, pp. 909–924, 2019, https://doi.org/10.1109/tac.2019.2959924.Suche in Google Scholar

[57] C. De Persis and P. Tesi, “Learning controllers for nonlinear systems from data,” Annu. Rev. Control, vol. 56, p. 100915, 2023, https://doi.org/10.1016/j.arcontrol.2023.100915.Suche in Google Scholar

[58] T. Liu and Z.-P. Jiang, “Event-based control of nonlinear systems with partial state and output feedback,” Automatica, vol. 53, pp. 10–22, 2015, https://doi.org/10.1016/j.automatica.2014.12.027.Suche in Google Scholar

[59] J. Lunze and D. Lehmann, “A state-feedback approach to event-based control,” Automatica, vol. 46, no. 1, pp. 211–215, 2010, https://doi.org/10.1016/j.automatica.2009.10.035.Suche in Google Scholar

[60] D. E. Quevedo, V. Gupta, W.-J. Ma, and S. Yüksel, “Stochastic stability of event-triggered anytime control,” IEEE Trans. Automat. Control, vol. 59, no. 12, pp. 3373–3379, 2014, https://doi.org/10.1109/tac.2014.2351952.Suche in Google Scholar

[61] C. P. Bechlioulis and G. A. Rovithakis, “A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems,” Automatica, vol. 50, no. 4, pp. 1217–1226, 2014, https://doi.org/10.1016/j.automatica.2014.02.020.Suche in Google Scholar

[62] C. P. Bechlioulis and G. A. Rovithakis, “Prescribed performance adaptive control for multi-input multi-output affine in the control nonlinear systems,” IEEE Trans. Automat. Control, vol. 55, no. 5, pp. 1220–1226, 2010, https://doi.org/10.1109/tac.2010.2042508.Suche in Google Scholar

[63] T. Berger, D. Dennstädt, L. Lanza, and K. Worthmann, “Robust funnel model predictive control for output tracking with prescribed performance,” SIAM J. Control Optim., vol. 62, no. 4, pp. 2071–2097, 2024, https://doi.org/10.1137/23m1551195.Suche in Google Scholar

[64] A. Mauroy and J. Goncalves, “Koopman-based lifting techniques for nonlinear systems identification,” IEEE Trans. Automat. Control, vol. 65, no. 6, pp. 2550–2565, 2019, https://doi.org/10.1109/tac.2019.2941433.Suche in Google Scholar

[65] A. Mauroy and J. Goncalves, “Linear identification of nonlinear systems: A lifting technique based on the Koopman operator,” in 55th IEEE Conference on Decision and Control (CDC), 2016.10.1109/CDC.2016.7799269Suche in Google Scholar

[66] P. van Goor, R. Mahony, M. Schaller, and K. Worthmann, “Reprojection methods for Koopman-based modelling and prediction,” in 62nd IEEE Conference on Decision and Control (CDC), 2023.10.1109/CDC49753.2023.10383796Suche in Google Scholar

[67] J. L. Proctor, S. L. Brunton, and J. N. Kutz, “Dynamic mode decomposition with control,” SIAM J. Appl. Dyn. Syst., vol. 15, no. 1, pp. 142–161, 2016, https://doi.org/10.1137/15m1013857.Suche in Google Scholar

[68] H. H. Asada and J. A. Solano-Castellanos, “Control-coherent Koopman modeling: A physical modeling approach,” arXiv preprint arXiv:2403.16306, 2024.10.1109/CDC56724.2024.10886771Suche in Google Scholar

[69] S. Peitz, S. E. Otto, and C. W. Rowley, “Data-driven model predictive control using interpolated Koopman generators,” SIAM J. Appl. Dyn. Syst., vol. 19, no. 3, pp. 2162–2193, 2020, https://doi.org/10.1137/20m1325678.Suche in Google Scholar

[70] C. Folkestad and J. W. Burdick, “Koopman NMPC: Koopman-based learning and nonlinear model predictive control of control-affine systems,” in IEEE International Conference on Robotics and Automation (ICRA), 2021.10.1109/ICRA48506.2021.9562002Suche in Google Scholar

[71] A. Narasingam, S. H. Son, and J.S.-I. Kwon, “Data-driven feedback stabilisation of nonlinear systems: Koopman-based model predictive control,” Int. J. Control, vol. 96, no. 3, pp. 770–781, 2023, https://doi.org/10.1080/00207179.2021.2013541.Suche in Google Scholar

[72] M. Rosenfelder, L. Bold, H. Eschmann, P. Eberhard, K. Worthmann, and H. Ebel, “Data-driven predictive control of nonholonomic robots based on a bilinear Koopman realization: Data does not replace geometry,” arXiv preprint arXiv:2411.07192, 2024.Suche in Google Scholar

[73] L. Grüne, J. Pannek, M. Seehafer, and K. Worthmann, “Analysis of unconstrained nonlinear MPC schemes with time varying control horizon,” SIAM J. Control Optim., vol. 48, no. 8, pp. 4938–4962, 2010, https://doi.org/10.1137/090758696.Suche in Google Scholar

[74] A. Boccia, L. Grüne, and K. Worthmann, “Stability and feasibility of state constrained MPC without stabilizing terminal constraints,” Syst. Control Lett., vol. 72, pp. 14–21, 2014, https://doi.org/10.1016/j.sysconle.2014.08.002.Suche in Google Scholar

[75] L. Bold, M. Rosenfelder, H. Eschmann, H. Ebel, and K. Worthmann, “On Koopman-based surrogate models for non-holonomic robots,” IFAC-PapersOnLine, vol. 58, no. 21, pp. 55–60, 2024, https://doi.org/10.1016/j.ifacol.2024.10.142.Suche in Google Scholar

[76] M. Korda and I. Mezić, “Optimal construction of Koopman eigenfunctions for prediction and control,” IEEE Trans. Automat. Control, vol. 65, no. 12, pp. 5114–5129, 2020, https://doi.org/10.1109/tac.2020.2978039.Suche in Google Scholar

[77] M. O. Williams, C. W. Rowley, and I. G. Kevrekidis, “A kernel-based method for data-driven koopman spectral analysis,” J. Comput. Dynam., vol. 2, no. 2, 2016.Suche in Google Scholar

[78] H. Wendland, Scattered Data Approximation, vol. 17, Cambridge, England, Cambridge University Press, 2004.10.1017/CBO9780511617539Suche in Google Scholar

[79] L. Bold, M. Schaller, I. Schimperna, and K. Worthmann, “Kernel EDMD for data-driven nonlinear Koopman MPC with stability guarantees,” Preprint arxiv:2501.08709, 2025.Suche in Google Scholar

[80] R. Soloperto, J. Köhler, and F. Allgöwer, “A nonlinear MPC scheme for output tracking without terminal ingredients,” IEEE Trans. Automat. Control, vol. 68, no. 4, pp. 2368–2375, 2022, https://doi.org/10.1109/tac.2022.3173494.Suche in Google Scholar