Advertisement

Exponential nonlinear observer based on the differential state-dependent Riccati equation

  • Hossein BeikzadehEmail author
  • Hamid D. Taghirad
Regular Papers

Abstract

This paper presents a novel nonlinear continuous-time observer based on the differential state-dependent Riccati equation (SDRE) filter with guaranteed exponential stability. Although impressive results have rapidly emerged from the use of SDRE designs for observers and filters, the underlying theory is yet scant and there remain many unanswered questions such as stability and convergence. In this paper, Lyapunov stability analysis is utilized in order to obtain the required conditions for exponential stability of the estimation error dynamics. We prove that under specific conditions, the proposed observer is at least locally exponentially stable. Moreover, a new definition of a detectable state-dependent factorization is introduced, and a close relation between the uniform detectability of the nonlinear system and the boundedness property of the state-dependent differential Riccati equation is established. Furthermore, through a simulation study of a second order nonlinear model, which satisfies the stability conditions, the promising performance of the proposed observer is demonstrated. Finally, in order to examine the effectiveness of the proposed method, it is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines. The simulation results verify how effectively this modification can increase the region of attraction and the observer error decay rate.

Keywords

Detectability direct method of Lyapunov exponential stability nonlinear observer region of attraction state-dependent Riccati equation (SDRE) technique 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. G. Luenberger. Observers for multivariable systems. IEEE Transactions on Automatic Control, vol. 11, no. 2, pp. 190–197, 1966.CrossRefGoogle Scholar
  2. [2]
    R. E. Kalman, R. S. Bucy. New results in linear filtering and prediction theory. Basic Engineering, vol. 83, no. 1, pp. 95–108, 1961.MathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Azemi, E. Yaz. Comparative study of several nonlinear stochastic estimators. In Proceedings of the 38th IEEE Conference on Decision and Control, IEEE, Phoenix, USA, vol. 5, pp. 4549–4554, 1999.Google Scholar
  4. [4]
    B. L. Walcott, M. J. Corless, S. H Zak. Comparative study of nonlinear state-observation technique. International Journal of Control, vol. 45, no. 6, pp. 2109–2132, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    L. N. Zhou, C. Y. Yang, Q. L. Zhang. Observers for descriptor systems with slop-restricted nonlinearities. International Journal of Automation and Computing, vol. 7, no. 4, pp. 472–478, 2010.CrossRefGoogle Scholar
  6. [6]
    J. Chen, E. Prempain, Q. H. Wu. Observer-based nonlinear control of a torque motor with perturbation estimation. International Journal of Automation and Computing, vol.3, no. 1, pp. 84–90, 2006.CrossRefGoogle Scholar
  7. [7]
    A. Gelb. Applied Optimal Estimation, Cambridge, UK: The MIT Press, 2001.Google Scholar
  8. [8]
    A. H. Jazwinski. Stochastic Processes and Filtering Theory, New York, USA: Academic Press, 1972.Google Scholar
  9. [9]
    V. J. Aidala. Kalman filter behavior in bearing-only tracking applications. IEEE Transactions on Aerospace and Electronic Systems, vol.AES-15, no. 1, pp. 29–39, 1979.CrossRefGoogle Scholar
  10. [10]
    C. P. Mracek, J. R. Cloutier, C. A. D’souza. A new technique for nonlinear estimation. In Proceedings of the 1996 IEEE International Conference on Control Applications, IEEE, Dearborn, USA, pp. 338–343, 1996.Google Scholar
  11. [11]
    J. R. Cloutier, D. T. Stansbery. The capabilities and art of state-dependent Riccati equation-based design. In Proceedings of American Control Conference, IEEE, Anchorage, USA, vol. 1, pp. 86–91, 2002.Google Scholar
  12. [12]
    J. R. Cloutier, C. N. D’souza, C. P. Mracek. Nonlinear regulation and nonlinear H-infinity control via the state-dependent Riccati equation technique. Part 1, theory; Part 2, Examples. In Proceedings of the 1st International Conference on Nonlinear Problems in Aviation and Aerospace, Daytona Beach, USA, pp. 117–141, 1996.Google Scholar
  13. [13]
    H. Beikzadeh, H. D. Taghirad. Nonlinear sensorless speed control of PM synchronous motor via an SDRE observercontroller combination. In Proceedings of the 4th IEEE Conference on Industrial Electronics and Applications, IEEE, Xi’an, China, pp. 3570–3575, 2009.CrossRefGoogle Scholar
  14. [14]
    Z. Qu, J. R. Cloutier, C. P. Mracek. A new sub-optimal nonlinear control design technique-SDARE. In Proceedings of the 13th IFAC World Congress, San Francisco, USA, pp. 365–370, 1996.Google Scholar
  15. [15]
    D. Haessig, B. Friedland. State dependent differential Riccati equation for nonlinear estimation and control. In Proceedings of the 15th IFAC Triennial World Congress, Barcelona, Spain, 2002. [Online], Available: http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2002/data/content/02294/2294.pdf, January 11, 2011.
  16. [16]
    T. Cimen, A. O. Merttopcuoglu. Asymptotically optimal nonlinear filtering: Theory and examples with application to target state estimation. In Proceedings of the 17th IFAC World Congress, Seoul, Korea, 2008. [Online], Available: http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac2008/data/papers/3636.pdf, January 11, 2011.
  17. [17]
    T. Cimen. Systematic and effective design of nonlinear feedback controllers via the state-dependent Riccati equation (SDRE) method. Annual Reviews in Control, vol. 34, no. 1, pp. 32–51, 2010.CrossRefGoogle Scholar
  18. [18]
    V. Pappano, B. Friedland. Analytical solution for a separate state and parameter SDRE observer for a CSI-fed induction motor. In Proceedings of the 1998 IEEE International Conference on Control Applications, IEEE, Trieste, Italy, vol. 2, pp. 1286–1291, 1998.Google Scholar
  19. [19]
    I. Y. Bar-Itzhack, R. R. Harman, D. Choukroun. State-dependent pseudo-linear filters for spacecraft attitude and rate estimation. In Proceedings of AIAA Guidance, Navigation and Control Conference and Exhibit, Monterey, USA, pp. 259–269, 2002.Google Scholar
  20. [20]
    H. T. Banks, H. D. Kwon, J. A. Toivanen, H. T. Tran. A state-dependent Riccati equation-based estimator approach for HIV feedback control. Optimal Control Applications and Methods, vol. 27, no. 2, pp. 93–121, 2006.MathSciNetCrossRefGoogle Scholar
  21. [21]
    H. T. Banks, B. M. Lewis, H. T. Tran. Nonlinear feedback controllers and compensators: A state-dependent Riccati equation approach. Computational Optimization and Applications, vol. 37, no. 2, pp. 177–218, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    B. M. Lewis. Optimal Control and Shape Design: Theory and Applications, Ph. D. dissertation, Department of Mathematics, North Carolina State University, Raleigh, 2003.Google Scholar
  23. [23]
    C. M. Ewing. An analysis of the state-dependent Riccati equation nonlinear estimation technique. In Proceedings of the AIAA Guidance, Navigation and Control Conference, Denver, USA, 2000.Google Scholar
  24. [24]
    N. F. Palumbo, T. D. Jackson. Integrated missile guidance and control system: A state-dependent Riccati differential equation approach. In Proceedings of the 1999 IEEE International Conference on Control Applications, IEEE, Hawaii, USA, vol. 1, pp. 243–248, 1999.Google Scholar
  25. [25]
    C. Jaganath, A. Ridley, D. S. Bernstein. A SDRE-based asymptotic observer for nonlinear discrete-time systems. In Proceedings of the 2005 American Control Conference, IEEE, Portland, USA, vol. 5, pp. 3630–3635, 2005.CrossRefGoogle Scholar
  26. [26]
    H. Beikzadeh, H. D. Taghirad. Robust H filtering for nonlinear uncertain systems using state-dependent Riccati equation technique. In Proceedings of the 48th IEEE Conference on Decision and Control, IEEE, Shanghai, PRC, pp. 4438–4445, 2009.Google Scholar
  27. [27]
    H. Beikzadeh, H. D. Taghirad. Stability analysis of the discrete-time difference SDRE state estimator in a noisy environment. In Proceedings of the 2009 IEEE International Conference on Control and Automation, IEEE, Christchurch, New Zealand, pp. 1751–1756, 2009.CrossRefGoogle Scholar
  28. [28]
    K. Reif, R. Unbehauen. The extended Kalman filter as an exponential observer for nonlinear systems. IEEE Transactions on Signal Processing, vol. 47, no. 8, pp. 2324–2328, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    K. Reif, F. Sonnemann, R. Unbehauen. An EKF-based nonlinear observer with a prescribed degree of stability. Automatica, vol. 34, no. 9, pp. 1119–1123, 1998.zbMATHCrossRefGoogle Scholar
  30. [30]
    S. R. Kou, D. L. Elliot, T. J. Tarn. Exponential observers for nonlinear dynamic systems. Information and Control, vol. 29, no. 3, pp. 204–216, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    B. D. Anderson, J. B. Moore. Linear Optimal Control, Englewood Cliffs, USA: Prentice-Hall, 1971.zbMATHGoogle Scholar
  32. [32]
    B. D. Anderson. Exponential data weighting in the Kalman-Bucy filter. Information Sciences, vol. 5, pp. 217–230, 1973.zbMATHCrossRefGoogle Scholar
  33. [33]
    J. S. Baras, A. Bensoussan, M. R. James. Dynamic observers as asymptotic limits of recursive filters: Special cases. SIAM Journal on Applied Mathematics, vol. 48, no. 5, pp. 1147–1158, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    J. R. Cloutier. State dependent Riccati equation techniques: An overview. In Proceedings of the American Control Conference, IEEE, Albuquerque, New Mexico, vol. 2, pp. 932–936, 1997.Google Scholar
  35. [35]
    S. P. Banks, K. J. Manha. Optimal control and stabilization for nonlinear systems. IMA Journal of Mathematical Control and Information, vol. 9, no. 2, pp. 179–196, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    M. Vidyasagar. Nonlinear Systems Analysis, 2nd ed., Englewood Cliffs, USA: Prentice Hall, 1993.zbMATHGoogle Scholar
  37. [37]
    J. W. Grizzle, Y. Song. The extended Kalman filter as a local asymptotic observer for nonlinear discrete-time systems. Journal of Mathematical Systems Estimation and Control, vol. 5, no. 1, pp. 59–78, 1995.MathSciNetzbMATHGoogle Scholar
  38. [38]
    J. J. Deyst, C. F. Price. Conditions for asymptotic stability of the discrete minimum-variance linear estimator. IEEE Transactions on Automatic Control, vol. 13, no. 6, pp. 702–705, 1968.MathSciNetCrossRefGoogle Scholar
  39. [39]
    K. D. Hammett, C. D. Hall, D. B. Ridgely. Controllability issues in nonlinear the state-dependent Riccati equation control. Journal of Guidance Control & Dynamics, vol. 21, no. 5, pp. 767–773, 1998.CrossRefGoogle Scholar
  40. [40]
    L. Salvatore, S. Stasi, L. Tarchioni. A new EKF-based algorithm for flux estimation in induction machines. IEEE Transactions on Industrial Electronics, vol. 40, no. 5, pp. 496–504, 1993.CrossRefGoogle Scholar

Copyright information

© Institute of Automation, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of AlbertaEdmontonCanada
  2. 2.Faculty of Electrical and Computer EngineeringK. N. Toosi University of TechnologyTehranIran

Personalised recommendations