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Design of neural high-gain observers for autonomous nonlinear systems using universal differential equations

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Abstract

The main goal of this paper is to introduce universal high-gain observers for nonlinear autonomous systems in observability canonical form. After a brief review of observability concepts for nonlinear autonomous systems and of results taken from the literature about universal differential equations, a universal high-gain observer for autonomous nonlinear systems is proposed. Its design is carried out by using universal differential equations both to estimate the dynamics in observability canonical form of the plant and to design the (time-varying) gain of the observer. Different training methods are proposed to efficiently tune the universal differential equations involved in the design. The practical effectiveness of this observer is demonstrated through several numerical examples.

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Data Availability

All data and materials are available upon request.

Code Availability

All the code used to carry out the numerical simulations reported in this work is available upon request.

References

  1. Abdollahi F, Talebi HA, Patel RV (2006) A stable neural network-based observer with application to flexible-joint manipulators. IEEE Trans Neural Netw 17(1):118–129

    Article  Google Scholar 

  2. Adhyaru DM (2012) State observer design for nonlinear systems using neural network. Appl Soft Comput 12(8):2530–2537

    Article  Google Scholar 

  3. Alfieri V, Pedicini C, Possieri C (2020) Design of a neural virtual sensor for the air and charging system in a diesel engine. IFAC-PapersOnLine 53(2):14,061-14,066

    Article  Google Scholar 

  4. Allaire G (2015) A review of adjoint methods for sensitivity analysis, uncertainty quantification and optimization in numerical codes. Ingénieurs de l’Automobile 836:33–36

  5. Alvarez M, Luengo D, Lawrence ND (2009) Latent force models. In: Artificial Intelligence and Statistics, PMLR, pp 9–16

  6. Arbi A (2021) Novel traveling waves solutions for nonlinear delayed dynamical neural networks with leakage term. Chaos, Solitons Fractals 152(111):436

    MathSciNet  MATH  Google Scholar 

  7. Arbi A, Cao J, Alsaedi A (2018) Improved synchronization analysis of competitive neural networks with time-varying delays. Nonlin Anal: Model Control 23(1):82–107

    Article  MathSciNet  MATH  Google Scholar 

  8. Arbi A, Guo Y, Cao J (2020) Convergence analysis on time scales for HOBAM neural networks in the Stepanov-like weighted pseudo almost automorphic space. Neural Computing and Applications pp 1–15

  9. Bezanson J, Edelman A, Karpinski S, et al (2017) Julia: a fresh approach to numerical computing

  10. Brüggemann S, Possieri C (2020) On the use of difference of log-sum-exp neural networks to solve data-driven model predictive control tracking problems. IEEE Control Syst Lett 5(4):1267–1272

    Article  MathSciNet  Google Scholar 

  11. Chen RTQ, Rubanova Y, Bettencourt J, et al (2018) Neural ordinary differential equations. In: 32nd International Conference on Neural Information Processing Systems, p 6572–6583

  12. Ciccarella G, Dalla Mora M, Germani A (1993) A Luenberger-like observer for nonlinear systems. Int J Control 57(3):537–556

    Article  MathSciNet  MATH  Google Scholar 

  13. Cybenko G (1989) Approximation by superpositions of a sigmoidal function. Math Control, Signals Syst 2(4):303–314

    Article  MathSciNet  MATH  Google Scholar 

  14. Esfandiari F, Khalil H (1989) Observer-based control of uncertain linear systems: recovering state feedback robustness under matching condition. In: American Control Conference, pp 931–936

  15. Esfandiari F, Khalil HK (1987) Observer-based design of uncertain systems: recovering state feedback robustness under matching conditions. In: Proceedings of Allerton Conference, pp 97–106

  16. Esfandiari F, Khalil HK (1992) Output feedback stabilization of fully linearizable systems. Int J Control 56(5):1007–1037

    Article  MathSciNet  MATH  Google Scholar 

  17. Gauthier JP, Kupka I (2001) Deterministic observation theory and applications. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  18. Hindmarsh AC, Brown PN, Grant KE et al (2005) SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans Math Softw 31(3):363–396

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu J, Wu W, Ji B, et al (2021) Observer design for sampled-data systems via deterministic learning. IEEE Transactions on Neural Networks and Learning Systems

  20. Hu Y, Boker S, Neale M et al (2014) Coupled latent differential equation with moderators: simulation and application. Psychol Methods 19(1):56

    Article  Google Scholar 

  21. Inouye Y (1977) On the observability of autonomous nonlinear systems. J Math Anal Appl 60(1):236–247

    Article  MathSciNet  MATH  Google Scholar 

  22. Khalil HK (2002) Nonlin Syst, 3rd edn. Prentice Hall, USA

    Google Scholar 

  23. Khalil HK (2017) High-gain observers in nonlinear feedback control. SIAM, USA

    Book  MATH  Google Scholar 

  24. Kingma DP, Ba J (2014) Adam: a method for stochastic optimization, arXiv:1412.6980

  25. Lakhal A, Tlili A, Braiek NB (2010) Neural network observer for nonlinear systems application to induction motors. Int J Control Autom 3(1):1–16

    Google Scholar 

  26. Leu YG, Lee TT, Wang WY (1999) Observer-based adaptive fuzzy-neural control for unknown nonlinear dynamical systems. IEEE Trans Syst, Man, Cybern, Part B (Cybern) 29(5):583–591

    Article  Google Scholar 

  27. Levenberg K (1944) A method for the solution of certain non-linear problems in least squares. Quart. Appl Math 2(2):164–168

    Article  MathSciNet  MATH  Google Scholar 

  28. Li DP, Liu YJ, Tong S et al (2018) Neural networks-based adaptive control for nonlinear state constrained systems with input delay. IEEE Trans Cybern 49(4):1249–1258

    Article  Google Scholar 

  29. Menini L, Possieri C, Tornambe A (2015) Sinusoidal disturbance rejection in chaotic planar oscillators. Int J Adapt Control Signal Process 29(12):1578–1590

    Article  MathSciNet  MATH  Google Scholar 

  30. Rackauckas C, Ma Y, Martensen J et al (2020) Universal differential equations for scientific machine learning. arXiv preprint arXiv:2001.04385

  31. Saberi A, Sannuti P (1988) Observer design for loop transfer recovery and for uncertain dynamical systems. In: American Control Conference, IEEE, pp 803–808

  32. Sengupta B, Friston KJ, Penny WD (2014) Efficient gradient computation for dynamical models. NeuroImage 98:521–527

    Article  Google Scholar 

  33. Shen Q, Shi P, Zhu J et al (2019) Neural networks-based distributed adaptive control of nonlinear multiagent systems. IEEE Trans Neural Netw Learn Syst 31(3):1010–1021

    Article  MathSciNet  Google Scholar 

  34. Sprott JC (2010) Elegant chaos: algebraically simple chaotic flows. World Scientific, Singapore

    Book  MATH  Google Scholar 

  35. Stumfoll J, Yao J, Balakrishnan S (2020) Neural network based discrete time modified state observer: Stability analysis and case study. In: 2020 American Control Conference (ACC), IEEE, pp 2484–2489

  36. Tornambe A (1992) High-gain observers for non-linear systems. Int J Syst Sci 23(9):1475–1489

    Article  MATH  Google Scholar 

  37. Traore BB, Kamsu-Foguem B, Tangara F (2018) Deep convolution neural network for image recognition. Ecol Inform 48:257–268

    Article  Google Scholar 

  38. Vargas JR, Hemerly EM (2000) Neural adaptive observer for general nonlinear systems. In: American Control Conference, IEEE, pp 708–712

  39. Young T, Hazarika D, Poria S et al (2018) Recent trends in deep learning based natural language processing [review article]. IEEE Comput Intell Mag 13(3):55–75

    Article  Google Scholar 

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Funding

C.P. acknowledges partial financial support by Regione Lazio through Research Program POR FESR LAZIO 2014-2020 - GRUPPI DI RICERCA 2020 under Grant GeCoWEB A0375-2020-36616, project OPENNESS.

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Author notes

  1. Corrado Possieri and Antonio Tornambe have contributed equally to this study.

Authors and Affiliations

  1. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma Tor Vergata, Via del Politecnico, 1, 00133, Roma, Italy

    Francesco Gismondi, Corrado Possieri & Antonio Tornambe

  2. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, Consiglio Nazionale delle Ricerche (IASI-CNR), Via dei Taurini, 19, 00185, Roma, Italy

    Corrado Possieri

Authors
  1. Francesco Gismondi
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  2. Corrado Possieri
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  3. Antonio Tornambe
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All authors contributed equally to this work.

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Correspondence to Corrado Possieri.

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Gismondi, F., Possieri, C. & Tornambe, A. Design of neural high-gain observers for autonomous nonlinear systems using universal differential equations. Int. J. Dynam. Control 10, 1794–1806 (2022). https://doi.org/10.1007/s40435-022-00941-5

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  • DOI: https://doi.org/10.1007/s40435-022-00941-5

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