Funnel model predictive control for nonlinear systems with relative degree one. (English) Zbl 1514.93033
Summary: We show that funnel MPC, a novel model predictive control (MPC) scheme, allows tracking of smooth reference signals with prescribed performance for nonlinear multi-input multioutput systems of relative degree one with stable internal dynamics. The optimal control problem solved in each iteration of funnel MPC resembles the basic idea of penalty methods used in optimization. To this end, we present a new stage cost design to mimic the high-gain idea of (adaptive) funnel control. We rigorously show initial and recursive feasibility of funnel MPC without imposing terminal conditions or other requirements like a sufficiently long prediction horizon.
MSC:
| 93B45 | Model predictive control |
| 93C10 | Nonlinear systems in control theory |
| 49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
Keywords:
model predictive control; funnel control; reference tracking; nonlinear systems; initial feasibility; recursive feasibilityReferences:
| [1] | Aydiner, E., Müller, M. A., and Allgöwer, F., Periodic reference tracking for nonlinear systems via model predictive control, in 2016 European Control Conference (ECC), , IEEE, Piscataway, NJ, 2016, pp. 2602-2607, doi:10.1109/ECC.2016.7810682. |
| [2] | Berger, T., Dennstädt, D., Ilchmann, A., and Worthmann, K., Funnel MPC for nonlinear systems with relative degree one, Math. Optim. Control., to appear. |
| [3] | Berger, T., Ilchmann, A., and Ryan, E. P., Funnel control of nonlinear systems, Math. Control Signals Systems, 33 (2021), pp. 151-194. · Zbl 1461.93246 |
| [4] | Berger, T., Kästner, C., and Worthmann, K., Learning-based Funnel-MPC for output-constrained nonlinear systems, IFAC-PapersOnLine, 53 (2020), pp. 5177-5182. |
| [5] | Berger, T., Otto, S., Reis, T., and Seifried, R., Combined open-loop and funnel control for underactuated multibody systems, Nonlinear Dynam., 95 (2019), pp. 1977-1998, doi:10.1007/s11071-018-4672-5. · Zbl 1432.93102 |
| [6] | Boccia, A., Grüne, L., and Worthmann, K., Stability and feasibility of state constrained MPC without stabilizing terminal constraints, Systems Control Lett., 72 (2014), pp. 14-21. · Zbl 1297.93068 |
| [7] | Byrnes, C. I. and Isidori, A., Asymptotic stabilization of minimum phase nonlinear systems, IEEE Trans. Automat. Control, 36 (1991), pp. 1122-1137. · Zbl 0758.93060 |
| [8] | Coron, J.-M., Grüne, L., and Worthmann, K., Model predictive control, cost controllability, and homogeneity, SIAM J. Control Optim., 58 (2020), pp. 2979-2996. · Zbl 1453.93083 |
| [9] | Di Cairano, S. and Borrelli, F., Reference tracking with guaranteed error bound for constrained linear systems, IEEE Trans. Automat. Control, 61 (2016), pp. 2245-2250, doi:10.1109/TAC.2015.2491738. · Zbl 1359.93272 |
| [10] | Esterhuizen, W., Worthmann, K., and Streif, S., Recursive feasibility of continuous-time model predictive control without stabilising constraints, IEEE Control Syst. Lett., 5 (2020), pp. 265-270. |
| [11] | Falugi, P. and Mayne, D. Q., Getting robustness against unstructured uncertainty: A tube-based MPC approach, IEEE Trans. Automat. Control, 59 (2013), pp. 1290-1295. · Zbl 1360.93292 |
| [12] | Grüne, L. and Pannek, J., Nonlinear Model Predictive Control: Theory and Algorithms, Springer, London, 2017, doi:10.1007/978-0-85729-501-9. · Zbl 1429.93003 |
| [13] | Hackl, C. M., Non-Identifier Based Adaptive Control in Mechatronics-Theory and Application, Springer, Cham, Switzerland, 2017. · Zbl 1371.93002 |
| [14] | Ilchmann, A., Non-Identifier-Based High-Gain Adaptive Control, Springer, London, 1993, doi:10.1007/BFb0032266. · Zbl 0786.93059 |
| [15] | Ilchmann, A., Ryan, E. P., and Sangwin, C. J., Tracking with prescribed transient behaviour, ESAIM: Control Optim. Calc. Var., 7 (2002), pp. 471-493. · Zbl 1044.93022 |
| [16] | Ilchmann, A. and Trenn, S., Input constrained funnel control with applications to chemical reactor models, Systems Control Lett., 53 (2004), pp. 361-375. · Zbl 1157.93426 |
| [17] | Köhler, J., Müller, M. A., and Allgöwer, F., Nonlinear reference tracking: An economic model predictive control perspective, IEEE Trans. Automat. Control, 64 (2019), pp. 254-269, doi:10.1109/TAC.2018.2800789. · Zbl 1423.93146 |
| [18] | Köhler, J., Müller, M. A., and Allgöwer, F., A nonlinear model predictive control framework using reference generic terminal ingredients, IEEE Trans. Automat. Control, 65 (2020), pp. 3576-3583, doi:10.1109/TAC.2019.2949350. · Zbl 1533.93179 |
| [19] | Köhler, J., Soloperto, R., Müller, M. A., and Allgöwer, F., A computationally efficient robust model predictive control framework for uncertain nonlinear systems, IEEE Trans. Automat. Control, 66 (2020), pp. 794-801. · Zbl 1536.93213 |
| [20] | Liberzon, D. and Trenn, S., The bang-bang funnel controller, in Proceedings of the 49th IEEE Conference on Decision and Control, , Atlanta, GA, IEEE, Piscataway, NJ, 2010, pp. 690-695. |
| [21] | Limon, D., Bravo, J., Alamo, T., and Camacho, E., Robust MPC of constrained nonlinear systems based on interval arithmetic, IEE Proc. Control Theory Appl., 152 (2005), pp. 325-332. |
| [22] | Mayne, D. Q., Seron, M. M., and Raković, S., Robust model predictive control of constrained linear systems with bounded disturbances, Automatica J. IFAC, 41 (2005), pp. 219-224. · Zbl 1066.93015 |
| [23] | Qin, S. J. and Badgwell, T. A., A survey of industrial model predictive control technology, Control Eng. Pract., 11 (2003), pp. 733-764. |
| [24] | Rawlings, J. B., Mayne, D. Q., and Diehl, M., Model Predictive Control: Theory, Computation, and Design, Vol. 2, Nob Hill Publishing, Madison, WI, 2017. |
| [25] | Sakamoto, N., When does stabilizability imply the existence of infinite horizon optimal control in nonlinear systems?, Automatica, 147 (2023), p. 110706, doi:10.1016/j.automatica.2022.110706. · Zbl 1505.93212 |
| [26] | Seifried, R. and Blajer, W., Analysis of servo-constraint problems for underactuated multibody systems, Mech. Sci., 4 (2013), pp. 113-129. |
| [27] | Senfelds, A. and Paugurs, A., Electrical drive DC link power flow control with adaptive approach, in Proceedings of the 55th International Scientific Conference on Power and Electrical Engineering of Riga Technical University, , Riga, Latvia, IEEE, Piscataway, NJ, 2014, pp. 30-33. |
| [28] | Singh, S., Majumdar, A., Slotine, J.-J., and Pavone, M., Robust online motion planning via contraction theory and convex optimization, in 2017 IEEE International Conference on Robotics and Automation (ICRA), , IEEE, Piscataway, NJ, 2017, pp. 5883-5890. |
| [29] | Sussmann, H., Limitations on the stabilizability of globally-minimum-phase systems, IEEE Trans. Automat. Control, 35 (1990), pp. 117-119. · Zbl 0708.93070 |
| [30] | Sussmann, H. and Kokotovic, P., The peaking phenomenon and the global stabilization of nonlinear systems, IEEE Trans. Automat. Control, 36 (1991), pp. 424-440. · Zbl 0749.93070 |
| [31] | Walter, W., Ordinary Differential Equations, Springer, New York, 1998. · Zbl 0991.34001 |
| [32] | Wills, A. G. and Heath, W. P., Barrier function based model predictive control, Automatica J. IFAC, 40 (2004), pp. 1415-1422, doi:10.1016/j.automatica.2004.03.002. · Zbl 1077.93024 |
| [33] | Yu, S., Maier, C., Chen, H., and Allgöwer, F., Tube MPC scheme based on robust control invariant set with application to Lipschitz nonlinear systems, Systems Control Lett., 62 (2013), pp. 194-200. · Zbl 1259.93048 |
| [34] | Yuan, M., Manzie, C., Good, M., Shames, I., Keynejad, F., and Robinette, T., Bounded error tracking control for contouring systems with end effector measurements, in 2019 IEEE International Conference on Industrial Technology (ICIT), , IEEE, Piscataway, NJ, 2019, pp. 66-71, doi:10.1109/ICIT.2019.8755064. |
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