A learning- and scenario-based MPC design for nonlinear systems in LPV framework with safety and stability guarantees. (English) Zbl 07909705
Summary: This paper presents a learning- and scenario-based model predictive control (MPC) design approach for systems modelled in the linear parameter-varying (LPV) framework. Using input-output data collected from the system, a state-space LPV model with uncertainty quantification is first learned through the variational Bayesian inference neural network (BNN) approach. The learned probabilistic model is assumed to contain the true dynamics of the system with a high probability and is used to generate scenarios that ensure safety for a scenario-based MPC. Moreover, to guarantee stability and enhance the performance of the closed-loop system, a parameter-dependent terminal cost and controller, as well as a terminal robust positive invariant set are designed. Numerical examples will be used to demonstrate that the proposed control design approach can ensure safety and achieve desired control performance.
MSC:
| 93B45 | Model predictive control |
| 93C10 | Nonlinear systems in control theory |
| 68T07 | Artificial neural networks and deep learning |
Keywords:
safe scenario-based model predictive control; learning-based control design; linear parameter-varying framework; Bayesian neural networksReferences:
| [1] | Abbas, H. S., Tóth, R., Petreczky, M., Meskin, N., Mohammadpour Velni, J., & Koelewijn, P. J. (2021). LPV modeling of nonlinear systems: A multi-path feedback linearization approach. International Journal of Robust and Nonlinear Control, 31(18), 9436-9465. · Zbl 1527.93174 |
| [2] | Aswani, A., Gonzalez, H., Sastry, S. S., & Tomlin, C. (2013). Provably safe and robust learning-based model predictive control. Automatica, 49(5), 1216-1226. · Zbl 1319.93047 |
| [3] | Bao, Y., Chan, K. J., Mesbah, A., & Velni, J. M. (2022). Learning-based adaptive-scenario-tree model predictive control with probabilistic safety guarantees using Bayesian neural networks. In 2022 American control conference (ACC) (pp. 3260-3265). IEEE. |
| [4] | Bao, Y., Chan, K. J., Mesbah, A., & Velni, J. M. (2023). Learning-based adaptive-scenario-tree model predictive control with improved probabilistic safety using robust Bayesian neural networks. International Journal of Robust and Nonlinear Control, 33(5), 3312-3333. · Zbl 1532.93152 |
| [5] | Bao, Y., & Mohammadpour Velni, J. (2022). Safe control of nonlinear systems in LPV framework using model-based reinforcement learning. International Journal of Control, 96(4), 1079-1090. . |
| [6] | Bao, Y., Mohammadpour Velni, J., & Shahbakhti, M. (2020). An online transfer learning approach for identification and predictive control design with application to RCCI engines. In Dynamic systems and control conference (Vol. 84270, pp. V001T21A003). ASME. |
| [7] | Bao, Y., Mohammadpour Velni, J., & Shahbakhti, M. (2021). Epistemic uncertainty quantification in state-space LPV model identification using Bayesian neural networks. IEEE Control Systems Letters, 5(2), 719-724. |
| [8] | Bao, Y., Velni, J. M., Basina, A., & Shahbakhti, M. (2020). Identification of state-space linear parameter-varying models using artificial neural networks. IFAC-PapersOnLine, 53(2), 5286-5291. |
| [9] | Blei, D. M., Kucukelbir, A., & McAuliffe, J. D. (2017Feb). Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518), 859-877. |
| [10] | Blundell, C., Cornebise, J., Kavukcuoglu, K., & Wierstra, D. (2015, June). Weight uncertainty in neural network. In International conference on machine learning (pp. 1613-1622). PMLR. |
| [11] | Bonzanini, A. D., Paulson, J. A., & Mesbah, A. (2020). Safe learning-based model predictive control under state- and input-dependent uncertainty using scenario trees. In 2020 59th IEEE conference on decision and control (CDC) (pp. 2448-2454). IEEE. |
| [12] | Calafiore, G. C., & Fagiano, L. (2013). Stochastic model predictive control of LPV systems via scenario optimization. Automatica, 49(6), 1861-1866. · Zbl 1360.93644 |
| [13] | Casavola, A., Famularo, D., & Franze, G. (2008). A predictive control strategy for norm-bounded LPV discrete-time systems with bounded rates of parameter change. International Journal of Robust and Nonlinear Control: IFAC-Affiliated Journal, 18(7), 714-740. · Zbl 1284.93146 |
| [14] | Clevert, D. A., Unterthiner, T., & Hochreiter, S. (2015). Fast and accurate deep network learning by exponential linear units (elus). arXiv preprint arXiv:1511.07289. |
| [15] | Cox, P. B. (2018). Towards efficient identification of linear parameter-varying state-space models [Unpublished doctoral dissertation]. Eindhoven University of Technology. |
| [16] | Defourny, B. (2010). Machine learning solution methods for multistage stochastic programming [PhD diss.]. University of Liege. https://www.lehigh.edu/defourny/PhDthesis_B_Defourny.pdf. |
| [17] | Ellis, M., Liu, J., & Christofides, P. D. (2017). Economic model predictive control. Springer, 5(7), 65. · Zbl 1405.93004 |
| [18] | Gilbert, E. G., & Tan, K. T. (1991). Linear systems with state and control constraints: The theory and application of maximal output admissible sets. IEEE Transactions on Automatic Control, 36(9), 1008-1020. · Zbl 0754.93030 |
| [19] | Gowal, S., Dvijotham, K., Stanforth, R., Bunel, R., Qin, C., Uesato, J., Arandjelovic, R., Mann, T., & Kohli, P. (2018). On the effectiveness of interval bound propagation for training verifiably robust models. arXiv preprint arXiv:1810.12715. |
| [20] | Hanema, J. (2018). Anticipative model predictive control for linear parameter-varying systems [Unpublished doctoral dissertation]. Technische Universiteit Eindhoven, Eindhoven, The Netherlands. |
| [21] | Hanema, J., Lazar, M., & Tóth, R. (2020). Heterogeneously parameterized tube model predictive control for LPV systems. Automatica, 111, 108622. · Zbl 1430.93063 |
| [22] | Hanema, J., Tóth, R., & Lazar, M. (2021). Stabilizing non-linear model predictive control using linear parameter-varying embeddings and tubes. IET Control Theory & Applications, 15(10), 1404-1421. |
| [23] | Hastie, T., Tibshirani, R., & Friedman, J. (2009). Model assessment and selection. In The elements of statistical learning (pp. 219-259). Springer. · Zbl 1273.62005 |
| [24] | Hewing, L., Wabersich, K. P., Menner, M., & Zeilinger, M. N. (2020). Learning-based model predictive control: Toward safe learning in control. Annual Review of Control, Robotics, and Autonomous Systems, 3(1), 269-296. |
| [25] | Høyland, K., Kaut, M., & Wallace, S. W. (2003). A heuristic for moment-matching scenario generation. Computational Optimization and Applications, 24(2/3), 169-185. · Zbl 1094.90579 |
| [26] | Høyland, K., & Wallace, S. W. (2001). Generating scenario trees for multistage decision problems. Management Science, 47(2), 295-307. · Zbl 1232.91132 |
| [27] | Ji, X., Zhu, S., Wang, S., & Zhang, S. (2005). A stochastic linear goal programming approach to multistage portfolio management based on scenario generation via linear programming. Iie Transactions, 37(10), 957-969. |
| [28] | Koller, T., Berkenkamp, F., Turchetta, M., Boedecker, J., & Krause, A. (2019). Learning-based model predictive control for safe exploration and reinforcement learning. arXiv preprint arXiv: 1906.12189. |
| [29] | Koller, T., Berkenkamp, F., Turchetta, M., & Krause, A. (2018, December). Learning-based model predictive control for safe exploration. In 2018 IEEE conference on decision and control (CDC) (pp. 6059-6066). IEEE. |
| [30] | Lee, J., & Yu, Z. (1997). Worst-case formulations of model predictive control for systems with bounded parameters. Automatica, 33(5), 763-781. · Zbl 0878.93025 |
| [31] | Liu, H., Ong, Y. S., Shen, X., & Cai, J. (2020). When Gaussian process meets big data: A review of scalable GPs. IEEE Transactions on Neural Networks and Learning Systems, 31(11), 4405-4423. |
| [32] | Lloyd, S. (1982). Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2), 129-137. · Zbl 0504.94015 |
| [33] | Lucia, S., Finkler, T., & Engell, S. (2013). Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty. Journal of Process Control, 23(9), 1306-1319. |
| [34] | Luo, Q., Nguyen, A. T., Fleming, J., & Zhang, H. (2021). Unknown input observer based approach for distributed tube-based model predictive control of heterogeneous vehicle platoons. IEEE Transactions on Vehicular Technology, 70(4), 2930-2944. |
| [35] | Maiworm, M., Bäthge, T., & Findeisen, R. (2015). Scenario-based model predictive control: Recursive feasibility and stability. IFAC-PapersOnLine, 48(8), 50-56. |
| [36] | Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789-814. · Zbl 0949.93003 |
| [37] | Mesbah, A. (2018). Stochastic model predictive control with active uncertainty learning: A survey on dual control. Annual Reviews in Control, 45, 107-117. |
| [38] | Morato, M. M., Normey-Rico, J. E., & Sename, O. (2020). Model predictive control design for linear parameter varying systems: A survey. Annual Reviews in Control, 49, 64-80. |
| [39] | Nguyen, H.-N. (2014). Set theoretic methods in control. In Constrained control of uncertain, time-varying, discrete-time systems (pp. 7-42). Springer International Publishing. · Zbl 1301.93003 |
| [40] | Pandey, A., & de Oliveira, M. C. (2017). Quadratic and poly-quadratic discrete-time stabilizability of linear parameter-varying systems. IFAC-PapersOnLine, 50(1), 8624-8629. |
| [41] | Rizvi, S. Z., Mohammadpour Velni, J., Abbasi, F., Tóth, R., & Meskin, N. (2018). State-space LPV model identification using kernelized machine learning. Automatica, 88, 38-47. · Zbl 1379.93094 |
| [42] | Shapiro, A. (2003). Monte Carlo sampling methods. Handbooks in Operations Research and Management Science, 10, 353-425. · Zbl 1115.90001 |
| [43] | Wicker, M., Laurenti, L., Patane, A., & Kwiatkowska, M. (2020). Probabilistic safety for Bayesian neural networks. arXiv preprint arXiv:2004.10281. |
| [44] | Wills, A., & Ninness, B. (2012). System identification of linear parameter varying state-space models. In P. Lopes dos Santos, T. P. Azevedo Perdicoulis, & C. Novara (Eds.), Linear parameter-varying system identification: New developments and trends (pp. 295-315). World Scientific. · Zbl 1297.93178 |
| [45] | Xu, D., Chen, Z., & Yang, L. (2012). Scenario tree generation approaches using K-means and LP moment matching methods. Journal of Computational and Applied Mathematics, 236(17), 4561-4579. · Zbl 1245.05106 |
| [46] | Zhang, H., Weng, T. W., Chen, P. Y., Hsieh, C. J., & Daniel, L. (2018). Efficient neural network robustness certification with general activation functions. Advances in Neural Information Processing Systems, 31. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
