Branislav Rehák

# Abstract:

This paper presents a computational procedure for the design of an observer of a nonlinear system. Outputs can be delayed, however, this delay must be known and constant. The characteristic feature of the design procedure is computation of a solution of a partial differential equation. This equation is solved using the finite element method. Conditions under which existence of a solution is guaranteed are derived. These are formulated by means of theory of partial differential equations in $L^2$-space. Three examples demonstrate viability of this approach and provide a comparison with the solution method based on expansions into Taylor polynomials.

# Keywords:

finite element method, nonlinear observer, delayed-output system

93C10, 65P99

# References:

1. G. Birkhoff and S. M. Lane: A Survey of Modern Algebra. CRC Press, 2017.   DOI:10.1201/9781315275499
2. A. Borri, F. Cacace, A. De Gaetano, A. Germani, C. Manes, P. Palumbo, S. Panunzi and P. Pepe: Luenberger-like observers for nonlinear time-delay systems with application to the artificial pancreas: The attainment of good performance. IEEE Control Syst. 37 (2017), 4, 33-49.   DOI:10.1109/mcs.2017.2696759
3. F. Cacace, A. Germani and C. Manes: An observer for a class of nonlinear systems with time varying observation delay. Systems Control Lett. 59 (2010), 305-312.   DOI:10.1016/j.sysconle.2010.03.005
4. F. Cacace, A. Germani and C. Manes: A chain observer for nonlinear systems with multiple time-varying measurement delays. SIAM J. Control Optim. 52 (2014), 1862-1885.   DOI:10.1137/120876472
5. S. Čelikovský and B. Rehák: Output regulation problem with nonhyperbolic zero dynamics: Femlab-based approach. IFAC Proc. Vol. 37 (2004), 21, 651-656.   CrossRef
6. S. Čelikovský, J. A. Torres-Munoz and A. Dominguez-Bocanegra: Adaptive high gain observer extension and its application to bioprocess monitoring. Kybernetika 54 (2018), 1, 155-174.   DOI:10.14736/kyb-2018-1-0155
7. M. Farza, O. Hernández-González, T. Ménard, B. Targui, M. M'Saad and C.-M. Astorga-Zaragoza: Cascade observer design for a class of uncertain nonlinear systems with delayed outputs. Automatica 89 (2018), 125-134.   DOI:10.1016/j.automatica.2017.12.012
8. A. Germani, C. Manes and P. Pepe: A new approach to state observation of nonlinear systems with delayed output. IEEE Trans. Automat Control 47 /2002), 1, 96-101.   CrossRef
9. N. Kazantzis and C. Kravaris: Nonlinear observer design using Lyapunov's auxiliary theorem. Systems Control Lett. 34 (1998), 241-247.   DOI:10.1016/s0167-6911(98)00017-6
10. N. Kazantzis and R. Wright: Nonlinear observer design in the presence of delayed output measurements. Systems Control Lett. 54 (2005), 877-886.   DOI:10.1016/j.sysconle.2004.12.005
11. H. Khalil: Nonlinear Systems. Prentice Hall, New Jersey 2001.   CrossRef
12. V. Lynnyk and B. Rehák: Design of a nonlinear observer using the finite element method with application to a biological system. Cybernet. Physics 8 (2019), 292-297.   DOI:10.35470/2226-4116-2019-8-4-292-297
13. B. Rehák: Alternative method of solution of the regulator equation: {L2} -space approach. Asian J. Control 14 (2011), 1150-1154.   DOI:10.1002/asjc.416
14. B. Rehák: Sum-of-squares based observer design for polynomial systems with a known fixed time delay. Kybernetika 51 (2015), 856-873.   DOI:10.14736/kyb-2015-5-0856
15. B. Rehák: Observer design for a time delay system via the {Razumikhin} approach. Asian J. Control 19 (2017), 6, 2226-2231.   DOI:10.1002/asjc.1507
16. B. Rehák: Finite-element based observer design for nonlinear systems with delayed measurements. IFAC-PapersOnLine 51 (2018), 14, 201-206.   CrossRef
17. B. Rehák, J. Orozco-Mora, S. Čelikovský and J. Ruiz-León: Real-time error-feedback output regulation of nonhyperbolically nonminimum phase system. In: 2007 American Control Conference 2007, pp. 3789-3794.   DOI:10.1109/acc.2007.4282643
18. B. Rehák and S. Čelikovský: Numerical method for the solution of the regulator equation with application to nonlinear tracking. Automatica 44 (2008), 5, 1358-1365.   DOI:10.1016/j.automatica.2007.10.015
19. B. Rehák, S. Čelikovský, J. Ruiz-León and J. Orozco-Mora: A comparison of two {Fem}-based methods for the solution of the nonlinear output regulation problem. Kybernetika 45 (2009), 427-444.   CrossRef
20. H.-G. Roos, M. Stynes and L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin 1996.   DOI:10.1007/978-3-662-03206-0
21. N. Sakamoto and B. Rehák: Iterative methods to compute center and center-stable manifolds with application to the optimal output regulation problem. In: Proc. 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando 2011.   DOI:10.1109/cdc.2011.6161089
22. N. Sakamoto, B. Rehák and K. Ueno: Nonlinear Luenberger observer design via invariant manifold computation. In: Proc. 19th IFAC World Congress, 2014, Cape Town 2014.   DOI:10.3182/20140824-6-za-1003.01103
23. A. T. Tran, S. Suzuki and N. Sakamoto: Nonlinear optimal control design considering a class of system constraints with validation on a magnetic levitation system. IEEE Control Systems Lett. 1 (2017), 2, 418-423.   DOI:10.1109/lcsys.2017.2717932
24. Y. Yu and Y. Shen: Robust sampled-data observer design for Lipschitz nonlinear systems. Kybernetika 54 (2018), 4, 699-717.   DOI:10.14736/kyb-2018-4-0699