Kybernetika 62 no. 3, 349-372, 2026

On the advantages of using virtual outputs to design nonlinear unknown-input MIMO observers: a novel LMI-based solution

Daniel Quintana and Miguel BernalDOI: 10.14736/kyb-2026-3-0349

Abstract:

Sufficient conditions in the form of linear matrix inequalities for design of a novel nonlinear unknown-input MIMO observer are given in this paper. The proposed scheme uses knowledge of real and virtual outputs in order to express the unknown input dynamics, based on which an extended observer-error system is obtained. Asymptotic stability of this system is ensured by means of exact factorization of the error signal, convex rewriting of expressions in a region of interest, finite-time estimation of a set of time derivatives of the system outputs, and the direct Lyapunov method. As examples show, the proposal is able to successfully reconstruct states, inputs, faults, and disturbances, where former methodologies fail.

Keywords:

linear matrix inequality, nonlinear observer, virtual outputs, direct Lyapunov method

Classification:

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

paper.pdf

References:

  1. H. Alwi, Ch. Edwards and A. Marcos: Fault reconstruction using a LPV sliding mode observer for a class of LPV systems. J. Franklin Inst. 349 (2012), 2, 510-530.   DOI:10.1016/j.jfranklin.2011.06.026
  2. N. Bedioui, R- Houimli and M. Besbes: Simultaneous sensor and actuator fault estimation for continuous-time polytopic LPV system. Int. J. Systems Sci. 50 (2019), 6, 1290-1302.   DOI:10.1080/00207721.2019.1599078
  3. M. Bernal, A. Sala, Z. Lendek and T. M. Guerra: Analysis and Synthesis of Nonlinear Control Systems. Springer, 2022.   CrossRef
  4. S. Bhattacharyya: Observer design for linear systems with unknown inputs. IEEE Trans. Automat. Control 23 (1978), 3, 483-484.   DOI:10.1109/TAC.1978.1101758
  5. S. {Boyd}, L. E. Ghaoui, E. Feron and V. Belakrishnan: Linear Matrix Inequalities in System and Control Theory, volume 15. SIAM: Studies In Applied Mathematics, Philadelphia 1994.   CrossRef
  6. W.-S. Chua, J. Ch. L. Chan, Ch. P. Tan, E. K. P. Chong and S. Saha: Robust fault reconstruction for a class of nonlinear systems. Automatica 113 (2020), 108718.   DOI:10.1016/j.automatica.2019.108718
  7. P. H. S. Coutinho, I. Bessa, W. B. Xie, A. T. Nguyen and R. Martínez Palhares: A sufficient condition to design unknown input observers for nonlinear systems with arbitrary relative degree. Int. J. Robust Nonlinear Control 32 (2022), 15, 8331-8348.   DOI:10.1002/rnc.6273
  8. X. Fan and M. Arcak: Observer design for systems with multivariable monotone nonlinearities. Systems Control Lett. 50 (2003), 4, 319-330.   DOI:10.1016/S0167-6911(03)00170-1
  9. P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali: LMI Control Toolbox. Math. Works, Natick 1995.   CrossRef
  10. P. Gasga, D. Quintana, S. Gomez-Penate and M. Bernal: Robust fault reconstruction using sliding mode observers with nonlinear nominal systems. ISA Transactions, 2023.   CrossRef
  11. J. Guzman, F.-R. López-Estrada, V. Estrada-Manzo and G. Valencia-Palomo: Actuator fault estimation based on a proportional-integral observer with nonquadratic Lyapunov functions. Int. J. Systems Sci. 52 (2021), 9, 1938-1951.   DOI:10.1080/00207721.2021.1873451
  12. H. Hammouri and Z. Tmar: Unknown input observer for state affine systems: A necessary and sufficient condition. Automatica 46 (2010), 2, 271-278.   DOI:10.1016/j.automatica.2009.11.004
  13. I- Hosseini, M. Fiacchini, P. Karimaghaee and A. Khayatian: Optimal reset unknown input observer design for fault and state estimation in a class of nonlinear uncertain systems. J. Franklin Inst. 357 (2020), 5, 2978-2996.   DOI:10.1016/j.jfranklin.2019.12.008
  14. D. Ichalal and T.-M. Guerra: Decoupling unknown input observer for nonlinear quasi-LPV systems. In: 2019 IEEE 58th Conference on Decision and Control (CDC), IEEE 2019, pp. 3799-3804.   DOI:10.1109/cdc40024.2019.9029339
  15. D. Ichalal and S. Mammar: On unknown input observers of linear systems: Asymptotic unknown input decoupling approach. IEEE Trans. Automat. Control 65 (2019), 3, 1197-1202.   DOI:10.1109/TAC.2019.2924375
  16. D. Ichalal, B. Marx, J- Ragot and D. Maquin: Fault detection, isolation and estimation for Takagi-Sugeno nonlinear systems. J. Franklin Inst. 351 (2014), 7, 3651-3676.   DOI:10.1016/j.jfranklin.2013.04.012
  17. A. {Isidori}: Nonlinear Control Systems. Third Edition. Springer, London 1995.   CrossRef
  18. Z. Lendek, T. M Guerra, R. Babuška and B. De-Schutter: Stability Analysis and Nonlinear Observer Design Using Takagi-Sugeno Fuzzy Models. Springer-Verlag, 2010.   CrossRef
  19. A. {Levant}: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76 (2003), 9-10, 924-941.   DOI:10.1080/0020717031000099029
  20. A. Levant and M. Livne: Robust exact filtering differentiators. European J. Control 55 (2020), 33-44.   DOI:10.1016/j.ejcon.2019.08.006
  21. H. Li, S. Wang, H. Shi, L. Wang, Ch. Su and P. Li: Robust asynchronous fuzzy predictive fault-tolerant tracking control for nonlinear multi-phase batch processes with time-varying reference trajectories. Engrg. Appl. Artific. Intel. 133 (2024), 108415.   DOI:10.1016/j.engappai.2024.108415
  22. B. Marx, D. Ichalal, J. Ragot, D. Maquin and S. Mammar: Unknown input observer for LPV systems. Automatica 100 (2019), 67-74.   DOI:10.1016/j.automatica.2018.10.054
  23. J. A. Moreno, E. Rocha-Cózatl and A. V. Wouwer: A dynamical interpretation of strong observability and detectability concepts for nonlinear systems with unknown inputs: application to biochemical processes. Bioprocess Biosystems Engrg. 37 (2014), 37-49.   DOI:10.1007/s00449-013-0915-5
  24. A. M. Pertew, H. J. Marquez and Q. Zhao: Design of unknown input observers for Lipschitz nonlinear systems. In: Proc. American Control Conference, IEEE 2005, pp. 4198-4203.   DOI:10.1109/acc.2005.1470637
  25. A. M. Pertew, H. J. Marquez and Q. Zhao: LMI-based sensor fault diagnosis for nonlinear Lipschitz systems. Automatica 43 (2007), 8, 1464-1469.   DOI:10.1016/j.automatica.2007.01.015
  26. Inc Quanser: Mechatronics Control Kit User's Manual (Instructor). Mathworks, inc, Natick 2006.   CrossRef
  27. 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
  28. D. Quintana, V. Estrada-Manzo and M. Bernal: An LMI-Based Unknown Input Observer for Nonlinear Systems. In: Congreso Nacional de Control Automático, Mexico 2023, pp. 3799-3804.   CrossRef
  29. D. Quintana, V. Estrada-Manzo and M. Bernal: Fault detection and isolation via a novel convex optimization scheme. IEEE Latin America Trans. 17 (2019), 07, 1096-1101.   DOI:10.1109/TLA.2019.8931196
  30. D. Quintana, V. Estrada-Manzo and M. Bernal: Overcoming non-convexity of the robust nonlinear output feedback problem: An adaptive linear matrix inequality solution. Int. J. Robust Nonlinear Control 32 (2022), 18, 9829-9848.   DOI:10.1002/rnc.6348
  31. D. Quintana, V. Estrada-Manzo and M. Bernal: No excuses to avoid observing unstable nonlinear systems: an LMI-based discontinuous solution. IEEE Trans. Automat. Control (2024).   DOI:10.1109/tac.2024.3393838
  32. C. {Scherer}. \newblock{Linear Matrix Inequalities in Control Theory.} \newblock{Delf University and Delf 2004.}:     CrossRef
  33. R. Seeber and H. Haimovich: Optimal robust exact differentiation via linear adaptive techniques. Automatica 148 (2023), 110725.   DOI:10.1016/j.automatica.2022.110725
  34. H. Shi, L. Zuo, Sh. Wang, Y. Yuan, Ch. Su and P. Li: Robust predictive fault-tolerant switching control for discrete linear systems with actuator random failures. Computers Chemical Engrg. 181 (2024), 108554.   DOI:10.1016/j.compchemeng.2023.108554
  35. K. Tanaka and H. O. Wang: Fuzzy Control Systems Design and Analysis: A linear matrix inequality approach. John Wiley, New York 2001.   CrossRef
  36. E. Tavasolipour, J. Poshtan and S. Shamaghdari: A new approach for robust fault estimation in nonlinear systems with state-coupled disturbances using dissipativity theory. ISA Trans. 114 (2021), 31-43.   DOI:10.1016/j.isatra.2020.12.040
  37. A. Vargas, J- A. Moreno and A. Vande Wouwer: Super-twisting estimation of a virtual output for extremum-seeking output feedback control of bioreactors. J. Process Control 35 (2015), 41-49.   DOI:10.1016/j.jprocont.2015.08.003
  38. M. Vidyasagar: Nonlinear Systems Analysis. SIAM, 2002.   CrossRef
  39. X. Wang, Ch. P. Tan, L. Liu and Q. Qi: A novel unknown input interval observer for systems not satisfying relative degree condition. Int. J. Robust Nonlinear Control 31 (2021), 7, 2762-2782.   DOI:10.1002/rnc.5407
  40. F. Xu, J. Tan, Y. Wang, X. Wang, B. Liang and B. Yuan: Robust fault detection and set-theoretic uio for discrete-time LPV systems with state and output equations scheduled by inexact scheduling variables. IEEE Trans. Automat. Control 64 (2019), 12, 4982-4997.   DOI:10.1109/TAC.2019.2902611
  41. X. G. Yan and Ch. Edwards: Nonlinear robust fault reconstruction and estimation using a sliding mode observer. Automatica 43 (2007), 9, 1605-1614.   DOI:10.1016/j.automatica.2007.02.008
  42. X. Zhang, M. M. Polycarpou and T. Parisini: A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems. IEEE Trans. Automat. Control 47 (2002), 4, 576-593.   DOI:10.1109/9.995036