Skip to main content
SpringerLink
Log in

Partial Observer Canonical Form Design Method for Single-output Affine Nonlinear System with Simple Validation Conditions

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

The problem concerning observer error linearization is to find a new state coordinate transformation (diffeomorphism) to transform the underlying nonlinear system into a Partial Observer Canonical Form (POCF). Since the complexity of the existing verification conditions associated with the existence of POCF, this paper develops a two-steps method for calculating POCF to simplify the verification conditions. This method also shows that POCF is equivalent to a compound diffeomorphism that consists of two coordinate transformations. One of them is a diffeomorphism which transforms the system into an observable canonical form based on maximum invariant distribution, and the other one is a diffeomorphism of transforming the observable subsystem into an observer linearization form. With the help of this method, a part of the original conditions are replaced by a group of new conditions that are easier to be verified, and another part of the original conditions are proved to be redundant and could not be verified anymore. At last, three examples are used to demonstrate the validity of this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Liu, M. Zhong, Y. Liu, and Z. Yang, “An observer and post filter based scheme for fault estimation of nonlinear systems,” International Journal of Control, Automation, and Systems, vol. 18, no. 8, pp. 1956–1964, 2020.

    Article  Google Scholar 

  2. A. S. S. Abadi, P. A. Hosseinabadi, and S. Mekhilef, “Fuzzy adaptive fixed-time sliding mode control with state observer for a class of high-order mismatched uncertain systems,” International Journal of Control, Automation, and Systems, vol. 18, no. 10, pp. 1956–1964, 2020.

    Article  Google Scholar 

  3. M. S. de Oliveira and R. L. Pereira, “Improved LMI conditions for unknown input observer design of discrete-time LPV systems,” International Journal of Control, Automation, and Systems, vol. 18, no. 10, pp. 2543–2551, 2020.

    Article  Google Scholar 

  4. M. Elbuluk and C. Li, “Sliding mode observer for wide-speed sensorless control of PMSM drives,” Proc. of IEEE Industry Applications Conference, pp. 480–485, 2003.

  5. M. Gan and C. Wang, “An adaptive nonlinear extended state observer for the sensorless speed control of a PMSM,” Mathematical Problems in Engineering, vol. 1, pp. 1–14, 2015.

    Google Scholar 

  6. H. Cai and J. Huang, “The leader-following attitude control of multiple rigid spacecraft systems,” Automatica, vol. 50, no. 4, pp. 1109–1115, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  7. X. Peng and Z. Geng, “Distributed observer-based leader-follower attitude consensus control for multiple rigid bodies using rotation matrices,” International Journal of Robust and Nonlinear Control, vol. 29, no. 14, pp. 4755–4774, 2019.

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Niu, C. K. Ahn, H. Li, and M. Liu, “Adaptive control for stochastic switched nonlower triangular nonlinear systems and its application to a one-link manipulator,” IEEE Transactions on Systems Man & Cybernetics: Systems, vol. 48, no. 10, pp. 1701–1714, 2017.

    Article  Google Scholar 

  9. H. Xu, J. Wang, H. Wang, and B. Wang, “Distributed observers design for a class of nonlinear systems to achieve omniscience asymptotically via differential geometry,” International Journal of Robust and Nonlinear Control, vol. 31, pp. 6288–6313, 2021.

    Article  MathSciNet  Google Scholar 

  10. Y. Liu, X. Zong, Q. Jian, S. Li, and X. Cheng, “A nonlinear observer for activated sludge wastewater treatment process: Invariant observer,” Asian Journal of Control, vol. 22, pp. 1670–1678, 2020.

    Article  MathSciNet  Google Scholar 

  11. L. Zhang and G. H. Yang, “Observer-based fuzzy adaptive sensor fault compensation for uncertain nonlinear strict-feedback systems,” IEEE Transactions on Fuzzy Systems, vol. 26, no. 4, pp. 2301–2310, 2018.

    Article  Google Scholar 

  12. L. An and G. H. Yang, “Decentralized adaptive fuzzy secure control for nonlinear uncertain interconnected systems against intermittent dos attacks,” IEEE Transactions on Cybernetics, vol. 49, no. 3, pp. 827–838, 2019.

    Article  Google Scholar 

  13. F. Zhu, Observer Research for Nonlinear Control System, Ph.D. Dissertation, Shanghai Jiao Tong University, 2003.

  14. Z. Duan and C. Kravaris, “Nonlinear observer design for two-time-scale systems,” Asian Journal of Control, vol. 66, no. 6, pp. 1–15, 2020.

    Google Scholar 

  15. K. Reif, F. Sonnemann, and R. Unbehauen, “An EKF-based nonlinear observer with a prescribed degree of stability,” Automatica, vol. 34, no. 9, pp. 1119–1123, 1998.

    Article  MATH  Google Scholar 

  16. S. Afshar, K. Morris, and A. Khajepour, “State-of-charge estimation using an EKF-based adaptive observer,” IEEE Transactions on Control Systems Technology, vol. 27, no. 5, pp. 1907–1923, 2019.

    Article  Google Scholar 

  17. C. Unsal and P. Kachroo, “Sliding mode measurement feedback control for antilock braking systems,” IEEE Transactions on Control Systems Technology, vol. 7, no. 2, pp. 271–281, 1999.

    Article  Google Scholar 

  18. H. Du, S. S. Ge, and J. K. Liu, “Adaptive neural network output feedback control for a class of non-affine non-linear systems with unmodelled dynamics,” IET Control Theory & Applications, vol. 5, no. 3, pp. 465–477, 2011.

    Article  MathSciNet  Google Scholar 

  19. H. N. Wu and H. X. Li, “Robust adaptive neural observer design for a class of nonlinear parabolic PDE systems,” Journal of Process Control, vol. 21, no. 8, pp. 1172–1182, 2011.

    Article  Google Scholar 

  20. W. Cong and D. J. Hill, “Deterministic learning and nonlinear observer design,” Asian Journal of Control, vol. 12, no. 6, pp. 714–724, 2010.

    Article  MathSciNet  Google Scholar 

  21. H. K. Khalil, “High-gain observers in nonlinear feedback control,” International Journal of Robust and Nonlinear Control, vol. 24, no. 6, pp. 993–1015, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Lei and H. K. Khalil, “High-gain-predictor-based output feedback control for time-delay nonlinear systems,” Automatica, vol. 71, pp. 324–333, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Lei and H. K. Khalil, “Feedback linearization for nonlinear systems with time-varying input and output delays by using high-gain predictors,” IEEE Transactions on Automatic Control, vol. 61, no. 8, pp. 2262–2268, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Wang, D. Astolfi, L. Marconi, and H. Su, “High-gain observers with limited gain power for systems with observability canonical form,” Automatica, vol. 75, no. C, pp. 16–23, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. J. Krener and A. Isidori, “Linearization by output injection and nonlinear observers,” Systems & Control Letters, vol. 3, no. 1, pp. 47–52, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. J. Krener and W. Respondek, “Nonlinear observers with linearizable error dynamics,” SIAM Journal on Control & Optimization, vol. 23, no. 2, pp. 197–216, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  27. X. H. Xia and W. B. Gao, “Nonlinear observer design by observer error linearization,” SIAM Journal on Control & Optimization, vol. 27, no. 1, pp. 199–216, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  28. K. Nam, “An approximate nonlinear observer with polynomial coordinate transformation maps,” IEEE Transactions on Automatic Control, vol. 42, no. 4, pp. 522–527, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. G. Lee, “Verifiable conditions for multioutput observer error linearizability,” IEEE Transactions on Automatic Control, vol. 62, no. 9, pp. 4876–4883, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  30. D. Boutat, A. Benali, H. Hammouri, and K. Busawon, “New algorithm for observer error linearization with a diffeomorphism on the outputs,” Automatica, vol. 45, no. 10, pp. 2187–2193, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  31. H. G. Lee, K. D. Kim, and H. T. Jeon, “Restricted dynamic observer error linearizability,” Automatica, vol. 53, pp. 171–178, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  32. H. G. Lee and H. Hong, “New conditions for nonlinear observer error linearizability with computer programming,” International Journal of Control, Automation, and Systems, vol. 13, no. 6, pp. 1544–1549, 2015.

    Article  Google Scholar 

  33. H. G. Lee, “Verifiable conditions for discrete-time multioutput observer error linearizability,” IEEE Transactions on Automatic Control, vol. 64, no. 4, pp. 1632–1639, 2019.

    Article  MathSciNet  MATH  Google Scholar 

  34. H. G. Lee and H. Hong, “Remarks on discrete-time multioutput nonlinear observer canonical form,” International Journal of Control, Automation, and Systems, vol. 16, no. 5, pp. 2569–2574, 2018.

    Article  Google Scholar 

  35. A. Isidori, Nonlinear Control Systems, Wiley Interscience, 1995.

  36. D. Boutat and K. Busawon, “On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form,” International Journal of Control, vol. 84, no. 1, pp. 94–106, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Ltaief, M. Farza, T. Menard, T. Maatoug, M. M’Saad, and Y. Koubaa, “High gain observer design for a class of MIMO non uniformly observable uncertain systems,” Proc. of International Conference on Sciences and Techniques of Automatic Control and Computer Engineering, pp. 817–821, 2016.

  38. M. Farza, T. Ménard, A. Ltaief, I. Bouraoui, M. MiSaad, and T. Maatoug, “Extended high gain observer design for a class of MIMO non-uniformly observable systems,” Automatica, vol. 86, pp. 138–146, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  39. D. Noh, N. H. Jo, and J. J. Seo, “Nonlinear observer design by dynamic observer error linearization,” IEEE Transactions on Automatic Control, vol. 49, no. 10, pp. 1746–1753, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  40. J. Back, K. T. Yu, and H. S. Jin, “Dynamic observer error linearization,” Automatica, vol. 42, no. 12, pp. 2195–2200, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  41. H. Trinh, T. Fernando, and S. Nahavandi, “Partial-state observers for nonlinear systems,” IEEE Transactions on Automatic Control, vol. 51, no. 11, pp. 1808–1812, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  42. N. Jo and J. Seo, “Observer design for non-linear systems that are not uniformly observable,” International Journal of Control, vol. 75, no. 5, pp. 369–380, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  43. K. Roebenack and A. F. Lynch, “Observer design using a partial nonlinear observer canonical form,” International Journal of Applied Mathematics & Computer Science, vol. 16, no. 3, pp. 333–343, 2006.

    MathSciNet  MATH  Google Scholar 

  44. P. Dufour, S. Flila, and H. Hammouri, “Observer design for mimo non-uniformly observable systems,” IEEE Transactions on Automatic Control, vol. 57, no. 2, pp. 511–516, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  45. R. Tami, G. Zheng, D. Boutat, D. Aubry, and H. Wang, “Partial observer normal form for nonlinear system,” Automatica, vol. 64, no. C, pp. 54–62, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  46. L. Dianpu, Theoretical Basis of Nonlinear Control Systems, Tsinghua University Press, Beijing, 2014.

    Google Scholar 

  47. W. Chen, An Introduction to Differential Manifold, High Education Press, Beijing, 1998.

    Google Scholar 

  48. J. Cheng, Y. Shan, J. Cao, and J. H. Park, “Nonstationary control for T-S fuzzy Markovian switching systems with variable quantization density,” IEEE Transactions on Fuzzy Systems, vol. 29, no. 6, pp. 1375–1385, 2021.

    Article  Google Scholar 

  49. J. Cheng, W. Huang, H.-K. Lam, J. Cao, and Y. Zhang, “Fuzzy-model-based control for singularly perturbed systems with nonhomogeneous Markov switching: A dropout compensation strategy,” IEEE Transactions on Fuzzy Systems, vol. 30, no. 2, pp. 530–541, 2022.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Control Science and Engineering, Shandong University, Jinan, 250061, China

    Haotian Xu

  2. Department of Automation, Shanghai Jiao Tong University, Shanghai, China

    Jingcheng Wang

  3. Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai, China

    Jingcheng Wang

  4. Shanghai Engineering Research Center of Intelligent Control and Management, Shanghai, China

    Jingcheng Wang

  5. Autonomous Systems and Intelligent Control International Joint Research Center, Xi’an Technological University, Xi’an, Shaanxi, 710021, China

    Jingcheng Wang

Authors
  1. Haotian Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jingcheng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jingcheng Wang.

Additional information

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by National Natural Science Foundation of China (No. 61633019, 61533013).

Haotian Xu received his B.S. degree in the School of Mathematics, Shandong University, China, in 2016; and a Ph.D. degree with the Department of Automation, Shanghai Jiao Tong University, Shanghai, China, 2022. He is working as postdoctoral in control theory and engineering from Shandong University, Jinan, China (2022-). He is the independent reviewer of international journals: IEEE Transactions on Neural Networks and Learning Systems, IEEE Robotics and Automation Letters (RA-L), International Journal of Electrical Power and Energy Systems, et. al; and international conferences: IFAC World Congress, and International Conference on Industrial Informatics. He is also the assistant reviewer of IEEE Transactions on Automatic Control. His current research interests include distributed observers, distributed-observer-based distributed control law, and nonlinear observers.

Jingcheng Wang received his B.S. and M.S. degrees from Northwestern Polytechnic University, Xi’an, China, in 1992 and 1995, respectively, and a Ph.D. degree form Zhejiang University, Hangzhou, China, in 1998. He is a Former Research Fellow with Alexander von Humboldt Foundation, Rostock University, Rostock, Germany, and he is currently a Professor with Shanghai Jiao Tong University, Shanghai, China. His current research interests include robust control, intelligent control, and real-time control and simulation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, H., Wang, J. Partial Observer Canonical Form Design Method for Single-output Affine Nonlinear System with Simple Validation Conditions. Int. J. Control Autom. Syst. 20, 2211–2221 (2022). https://doi.org/10.1007/s12555-020-0887-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-020-0887-6

Keywords

Search

Navigation