Abstract
The stability of the dynamical system is associated with the concept of Region of Attraction (ROA), whose accurate estimation opens the door to multidisciplinary approaches involving control theory and machine learning. The Lyapunov theory provides sufficient conditions for stability and it can be applied to derive the ROA. However, finding the appropriate Lyapunov functions for accurate ROA estimation often is a major issue. The inherent region may be overly tight or, in the case of a multi-dimensional dynamical system, be difficult to understand in virtue of the inherent mathematical complexity (e.g., polynomials with high degree). The use of explainable machine learning overcomes this issue, by exploiting the model intelligibility to describe the ROA in terms of states. In this perspective, explainable machine learning and Lyapunov stability theory are jointly studied to let the ROA be intelligible and to simplify the optimization procedure for constructing positively invariant estimates of the ROA. Results on the Van der Pol oscillator show how this may lead to larger ROAs than via traditional methods.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Rulex data analytics platform; www.rulex.ai.
- 2.
Code and datasets are available at https://github.com/mopamopa/Liapunov-Logic-Learning-Machine, [28].
References
Khalil H (2002) Nonlinear systems, 3rd edn. Prentice Hall, Upper Saddle River
Valmorbida G, Anderson J (2017) Region of attraction estimation using invariant sets and rational lyapunov functions. Automatica 75:37–45. http://www.sciencedirect.com/science/article/pii/S0005109816303387
Korda M, Henrion D, Jones CN (2013) Controller design and region of attraction estimation for nonlinear dynamical systems
Zhai C, Nguyen HD (2019) Region of attraction for power systems using gaussian process and converse lyapunov function – part i: theoretical framework and off-line study
Lun YZ, D’Innocenzo A, Smarra F, Malavolta I, Benedetto MDD (2019) State of the art of cyber-physical systems security: an automatic control perspective. J Syst Softw 149:174–216. http://www.sciencedirect.com/science/article/pii/S0164121218302681
Mongelli M, Muselli M, Ferrari E, Fermi A (2018) Performance validation of vehicle platooning via intelligible analytics. IET Cyber-Phys Syst: Theory & Appl 10:2018
Road vehicles safety of the intended functionality pd iso pas 21448:2019, International Organization for Standardization, Geneva, CH, Standard (2019)
Glassman E, Desbiens AL, Tobenkin M, Cutkosky M, Tedrake R (2012) Region of attraction estimation for a perching aircraft: a lyapunov method exploiting barrier certificates. In: IEEE international conference on robotics and automation, pp 2235–2242
Mongelli M, Ferrari E, Muselli M (2019) Achieving zero collision probability in vehicle platooning under cyber attacks via machine learning. In: 2019 4th international conference on system reliability and safety (ICSRS)
Jones M, Mohammadi H, Peet MM (2017) Estimating the region of attraction using polynomial optimization: a converse lyapunov result. In: 2017 IEEE 56th annual conference on decision and control (CDC), pp 1796–1802
Zečević AI, Šiljak DD (2010) Regions of attraction. Springer, Boston, pp 111–141
Burchardt H, Ratschan S (2007) Estimating the region of attraction of ordinary differential equations by quantified constraint solving. In: Proceedings of the 3rd WSEAS international conference on DYNAMICAL SYSTEMS and CONTROL (CONTROL’07). WSEAS Press, pp 241–246
Mongelli M, Ferrari E, Muselli M, Scorzoni A (2019) Accellerating prism validation of vehicle platooning through machine learning. In: 2019 4th international conference on system reliability and safety (ICSRS)
Guidotti R, Monreale A, Ruggieri S, Turini F, Giannotti F, Pedreschi D (2018) A survey of methods for explaining black box models. ACM Comput Surv 51(5):93:1–93:42. http://doi.acm.org/10.1145/3236009
Noroozi N, Paknoosh K, Fatemeh S, Hamed J (2008) Generation of lyapunov functions by neural networks. Lect Notes Eng Comput Sci 2170:07
Petridis V, Petridis S (2006) Construction of neural network based lyapunov functions 01(2006):5059–5065
Chang Y-C, Roohi N, Gao S (2019) Neural lyapunov control 01:2019
Richards SM, Berkenkamp F, Krause A (2018) The lyapunov neural network: adaptive stability certification for safe learning of dynamic systems. arXiv:1808.00924
Diao R, Vittal V, Logic N (2010) Design of a real-time security assessment tool for situational awareness enhancement in modern power systems. IEEE Trans Power Syst 25:957–965
He M, Zhang J, Vittal V (2013) Robust online dynamic security assessment using adaptive ensemble decision-tree learning. IEEE Trans Power Syst 28(4):4089–4098
Bastani O, Pu Y, Solar-Lezama A (2018) Verifiable reinforcement learning via policy extraction. arXiv:1805.08328
Kozarev A, Quindlen J, How J, Topcu U (2016) Case studies in data-driven verification of dynamical systems. In: Proceedings of the 19th international conference on hybrid systems: computation and control HSCC 2016, 04 2016, pp 81–86
Khalil HK (2002) Nonlinear systems, 3rd ed. Prentice-Hall, Upper Saddle River; the book can be consulted by contacting: PH-AID: Wallet, Lionel. https://cds.cern.ch/record/1173048
Papachristodoulou A, Anderson J, Valmorbida G, Prajna S, Seiler P, Parrilo P (2013) Sostools version 3.00 sum of squares optimization toolbox for matlab, 10
Cangelosi D, Muselli M et al (2014) Use of attribute driven incremental discretization and logic learning machine to build a prognostic classifier for neuroblastoma patients. BMC Bioinf 15(suppl):5
Muselli M, Ferrari E (2011) Coupling logical analysis of data and shadow clustering for partially defined positive boolean function reconstruction. IEEE Trans Knowl Data Eng 23(1):37–50
Sturm JF (1999) Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones. Opt Methods Softw 11(1–4):625–653. https://doi.org/10.1080/10556789908805766
Mongelli M, Orani V, Git repository of liapunov logic learning machine. https://github.com/mopamopa/Liapunov-Logic-Learning-Machine
Acknowledgements
The authors gratefully acknowledge colleagues Elisabetta Punta and Fabrizio Dabbene for suggestions on Lyapunov stability theory and Marco Muselli for the usage of LLM with zero error and inherent value ranking.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Maurizio, M., Vanessa, O. (2021). Stability Certification of Dynamical Systems: Lyapunov Logic Learning Machine. In: Thampi, S.M., Lloret Mauri, J., Fernando, X., Boppana, R., Geetha, S., Sikora, A. (eds) Applied Soft Computing and Communication Networks. Lecture Notes in Networks and Systems, vol 187. Springer, Singapore. https://doi.org/10.1007/978-981-33-6173-7_15
Download citation
DOI: https://doi.org/10.1007/978-981-33-6173-7_15
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-33-6172-0
Online ISBN: 978-981-33-6173-7
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)