Kybernetika 61 no. 3, 429-446, 2025

LMI-based nonlinear observer design for a class of nonlinear systems modeled with differential algebraic equations

Miguel Bernal, Antonio Sala and Antonio GonzálezDOI: 10.14736/kyb-2025-3-0429

Abstract:

This work presents a novel methodology to design nonlinear observers for a class of systems modeled as differential algebraic equations. The proposal is based on writing both the system and the observer as nonlinear descriptor redundancy representations subject to algebraic restrictions; then the nonlinear observation error system is written in an explicit incremental form via suitable factorization techniques. A redundant Lyapunov function is then employed to guarantee asymptotic stability of the estimation error; linearity of the Lyapunov function and its time derivative with respect to the observer gains and Lyapunov function terms, allows gridding or convex treatment of expressions via linear matrix inequalities. Physical examples are presented to illustrate the proposal effectiveness against former methodologies.

Keywords:

linear matrix inequality, nonlinear observer, descriptor redundancy, differential algebraic equations

Classification:

93B53, 93B50, 93C10, 93C15, 93D05

paper.pdf

References:

  1. J. Álvarez, J. Servín, J. A. Díaz and M. Bernal: Differential algebraic observer-based trajectory tracking for parallel robots via linear matrix inequalities. Int. J. Systems Sci. 53 (2022), 10, 2149-2164.   DOI:10.1080/00207721.2022.2043482
  2. J. C. Arceo, J. Alvarez, C. Armenta, J. Lauber, S. Cremoux, E. Simoneau-Buessinger and M. Bernal: Novel solutions on model-based and model-free robotic-assisted ankle rehabilitation. Arch. Control Sci. 31 (2021), 1, 5-27.   DOI:10.24425/acs.2021.136878
  3. J. C. Arceo, M. Sánchez, V. Estrada-Manzo and M. Bernal: Convex stability analysis of nonlinear singular systems via linear matrix inequalities. IEEE Trans. Automat. Control 64 (2018), 4, 1740-1745.   DOI:10.1109/TAC.2018.2854651
  4. T. Berger: On observers for nonlinear differential-algebraic systems. IEEE Trans. Automat. Control 64 (2019), 5, 2150-2157.   DOI:10.1109/TAC.2018.2866438
  5. M. Bernal, A. Sala, Z. Lendek and T. M. Guerra: Analysis and Synthesis of Nonlinear Control Systems: A Convex Optimsation Approach. Springer, Cham 2022.   CrossRef
  6. H. G. Bock and V. Schulz: Mathematical aspects of CFD-based optimization. In: Optimization and Computational Fluid Dynamics, Springer 2008, pp. 61-78.   DOI:10.1007/978-3-540-72153-6\_3
  7. S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan: Linear matrix inequalities in system and control theory. Studies in Applied Mathematics 15, Philadelphia 1994.   DOI:10.1137/1.9781611970777
  8. P. T. Cardin, P. R. da Silva and M. A. Teixeira: Implicit differential equations with impasse singularities and singular perturbation problems. Israel J. Math. 189 (2012), 307-322.   DOI:10.1007/s11856-011-0169-3
  9. G. R. Duan: Analysis and Design of Descriptor Linear Systems. Springer-Verlag, New York 2010.   CrossRef
  10. P. Gahinet, A. Nemirovskii, A. J. Laub and M. Chilali: The LMI control toolbox. In: Proc. 1994 33rd IEEE Conference on Decision and Control 3 IEEE 1994, pp. 2038-2041.   DOI:10.1109/cdc.1994.411440
  11. T. Gonzalez, M. Bernal, A. Sala and B. Aguiar: Cancellation-based nonquadratic controller design for nonlinear systems via Takagi-Sugeno models. IEEE Trans. Cybernet. 47 (2016), 9, 2628-2638.   DOI:10.1109/TCYB.2016.2589543
  12. T. M. Guerra, V. C. de Oliveira, D. Berdjag, Ch. Lv and A. T. Nguyen: Fault tolerant observer design for a class of nonlinear systems with corrupted outputs. Int. J. Robust Nonlinear Control 34 (2024), 13, 8825-8843.   DOI:10.1002/rnc.7446
  13. T. M. Guerra, V. Estrada-Manzo and Z. Lendek: Observer design for Takagi-Sugeno descriptor models: An LMI approach. Automatica 52 (2015), 154-159.   DOI:10.1016/j.automatica.2014.11.008
  14. MathWorks Inc.: Symbolic Math Toolbox. Natick, Massachusetts 2019.   CrossRef
  15. H. Khalil: Nonlinear Control. Prentice Hall, New Jersey 2014.   DOI:10.1145/2743015
  16. Z. Lendek, T. M. Guerra, R.Babuska and B. {De-Schutter}: Stability analysis and nonlinear observer design using Takagi-Sugeno fuzzy models. Studies Fuzziness Soft Computing. Springer-Verlag, 2011.   CrossRef
  17. J.-P. Merlet: Parallel robots. Springer Science Business Media 128, 2006.   CrossRef
  18. N. S. Nedialkov, J. D. Pryce and G. Tan: Algorithm 948: Daesaa Matlab tool for structural analysis of differential-algebraic equations: Software. ACM Trans. Math. Software (TOMS) 41 (2015), 2, 1-14.   CrossRef
  19. H. Ohtake, K. Tanaka and H. O. Wang: Fuzzy modeling via sector nonlinearity concept. Integrated Computer-Aided Engrg. 10 (2003), 4, 333-341.   DOI:10.3233/ICA-2003-10404
  20. C. C. Pantelides: The consistent initialization of differential-algebraic systems. SIAM J. Scientific Statist. Comput. 9 (1988), 2, 213-231.   DOI:10.1016/S0003-2670(00)81358-6
  21. C. C. Pantelides, D. Gritsis, K. R. Morison and R. W. H. Sargent: The mathematical modelling of transient systems using differential-algebraic equations. Computers Chemical Engrg. 12 (1988), 5, 449-454.   DOI:10.1016/0098-1354(88)85062-2
  22. D. Quintana, V. Estrada-Manzo and M. Bernal: An exact handling of the gradient for overcoming persistent problems in nonlinear observer design via convex optimization techniques. Fuzzy Sets Systems 416 (2021), 125-140.   DOI:10.1016/j.fss.2020.04.012
  23. P. J. Rabier, W. C and Rheinboldt: Nonholonomic motion of rigid mechanical systems from a DAE viewpoint. SIAM, 2000.   CrossRef
  24. P. J. Rabier and W. C. Rheinboldt: Theoretical and numerical analysis of differential-algebraic equations. 2002.   CrossRef
  25. R. Riaza: Differential-algebraic Systems: Analytical Aspects and Circuit Applications. World Scientific, 2008.   CrossRef
  26. R. Robles, A. Sala, M. Bernal and T. González: Subspace-based Takagi-Sugeno modeling for improved LMI performance. IEEE Trans. Fuzzy Systems 25 (2016), 4, 754-767.   DOI:10.1109/TFUZZ.2016.2574927
  27. A. Sadeghzadeh and R. Tóth: Linear parameter-varying embedding of nonlinear models with reduced conservativeness. IFAC-PapersOnLine 53 (2020), 2, 4737-4743.   DOI:10.1016/j.ifacol.2020.12.598
  28. A. Sala: Computer control under time-varying sampling period: An LMI gridding approach. Automatica 41 (2005), 12, 2077-2082.   DOI:10.1016/j.automatica.2005.05.017
  29. A. Sala and C. Arino: Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem. Fuzzy Sets Systems 158 (2007), 24, 2671-2686.   DOI:10.1016/j.fss.2007.06.016
  30. C. W. Scherer: LMI relaxations in robust control. European J. Control 12 (2006), 1, 3-29.   DOI:10.3166/ejc.12.3-29
  31. M. Schoukens and R. Tóth: Linear parameter varying representation of a class of MIMO nonlinear systems. IFAC-PapersOnLine 51 (2018), 26, 94-99.   DOI:10.1016/j.ifacol.2018.11.162
  32. K. Takaba, N. Morihira and T. Katayama: A generalized Lyapunov theorem for descriptor system. Systems Control Lett. 24 (1995), 1, 49-51.   DOI:10.1016/0167-6911(94)00041-S
  33. K. Tanaka and H. O. Wang: Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. John Wiley and Sons, New York 2001.   CrossRef
  34. T. Taniguchi, K. Tanaka and H. O. Wang: Fuzzy descriptor systems and nonlinear model following control. IEEE Trans. Fuzzy Systems 8 (2000), 4, 442-452.   DOI:10.1109/91.868950
  35. G. Zheng, D. Efimov, F. J. Bejarano, W. Perruquetti and H. Wang: Interval observer for a class of uncertain nonlinear singular systems. Automatica 71 (2016), 159-168.   DOI:10.1016/j.automatica.2016.04.002