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On extended state based Kalman filter design for a class of nonlinear time-varying uncertain systems

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This paper considers the filtering problem for a class of multi-input multi-output systems with nonlinear time-varying uncertain dynamics, random process and measurement noise. An extended state based Kalman filter, with the idea of timely estimating the unknown dynamics, is proposed for better robustness and higher estimation precision. The stability of the proposed filter is rigorously proved for nonlinear timevarying uncertain system with weaker stability condition than the extended Kalman filter, i.e., the initial estimation error, the uncertain dynamics and the noises are only required to be bounded rather than small enough. Moreover, quantitative precision of the proposed filter is theoretically evaluated. The proposed algorithm is proved to be the asymptotic unbiased minimum variance filter for constant uncertainty. The simulation results of some benchmark examples demonstrate the feasibility and effectiveness of the method.

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  1. 1

    Anderson B D O, Moore J B. Optimal Filtering. Upper Saddel River: Prentice Hall, 1995

  2. 2

    Grewal M S, Andrews A P. Kalman Filtering: Theorey and Practice. Upper Saddel River: Prentice Hall, 1993

  3. 3

    Guo L. Estimating time-varying parameters by the Kalman filter based algorithm: stability and convergence. IEEE Trans Autom Control, 1990, 35: 141–147

  4. 4

    Chen C, Liu Z X, Guo L. Performance bounds of distributed adaptive filters with cooperative correlated signals. Sci China Inf Sci, 2016, 59: 112202

  5. 5

    Ljung L. Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems. IEEE Trans Autom Control, 1979, 24: 36–50

  6. 6

    Deyst J, Price C. Conditions for asymptotic stability of the discrete minimum-variance linear estimator. IEEE Trans Autom Control, 1968, 13: 702–705

  7. 7

    Reif K, Gunther S, Yaz E, et al. Stochastic stability of the discrete-time extended Kalman filter. IEEE Trans Autom Control, 1999, 44: 714–728

  8. 8

    Reif K, Gunther S, Yaz E, et al. Stochastic stability of the continuous-time extended Kalman filter. IEEE Proc Control Theory Appl, 2000, 147: 45–52

  9. 9

    Julier S J, Uhlmann J K. Unscented filtering and nonlinear estimation. Proc IEEE, 2004, 92: 401–422

  10. 10

    Zhang W, Chen B S, Tseng C S. Robust H1 filtering for nonlinear stochastic systems. IEEE Trans Signal Process, 2005, 53: 589–598

  11. 11

    Zhu Y M. From Kalman filtering to set-valued filtering for dynamic systems with uncertainty. Commun Inf Syst, 2012, 12: 97–130

  12. 12

    Zorzi M. Robust Kalman filtering under model perturbations. IEEE Trans Autom Control, 2017, 62: 2902–2907

  13. 13

    Xie L, Soh Y C, de Souza C E. Robust Kalman filtering for uncertain discrete-time systems. IEEE Trans Autom Control, 1994, 39: 1310–1314

  14. 14

    Liu Q Y, Wang Z D, He X, et al. A resilient approach to distributed filter design for time-varying systems under stochastic nonlinearities and sensor degradation. IEEE Trans Signal Process, 2017, 65: 1300–1309

  15. 15

    Simon D. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. Hoboken: John Wiley & Sons, 2006

  16. 16

    Shaked U, Berman N. H1 nonlinear filtering of discrete-time processes. IEEE Trans Signal Process, 1995, 43: 2205–2209

  17. 17

    Su Q Y, Huang Y, Jiang Y G, et al. Quasi-consistent fusion navigation algorithm for DSS. Sci China Inf Sci, 2018, 61: 012201

  18. 18

    Sayed A H. A framework for state-space estimation with uncertain models. IEEE Trans Autom Control, 2001, 46: 998–1013

  19. 19

    Seo J, Yu M J, Park C G, et al. An extended robust H1 filter for nonlinear constrained uncertain systems. IEEE Trans Signal Process, 2006, 54: 4471–4475

  20. 20

    Scholte E, Campbell M E. A nonlinear set membership filter for online applications. Int J Robust Nonlin Control, 2003, 13: 1337–1358

  21. 21

    Huang Y, Xu K K, Han J Q, et al. Flight control design using extended state observer and non-smooth feedback. In: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, 2001. 223–228

  22. 22

    Liu L, Wang D, Peng Z H, et al. Saturated coordinated control of multiple underactuated unmanned surface vehicles over a closed curve. Sci China Inf Sci, 2017, 60: 070203

  23. 23

    Tang S, Zhang L, Qian S K, et al. Second-order sliding mode attitude controller design of a small-scale helicopter. Sci China Inf Sci, 2016, 59: 112209

  24. 24

    Su J B, Ma H Y, Qiu W B, et al. Task-independent robotic uncalibrated hand-eye coordination based on the extended state observer. IEEE Trans Syst Man Cybern, 2004, 34: 1917–1922

  25. 25

    Liu H X, Li S H. Speed control for PMSM servo system using predictive functional control and extended state observer. IEEE Trans Ind Electron, 2012, 59: 1171–1183

  26. 26

    Dong L, Avanesian D. Drive-mode control for vibrational MEMS gyroscopes. IEEE Trans Ind Electron, 2009, 56: 956–963

  27. 27

    Sun L, Dong J Y, Li D H, et al. A practical multivariable control approach based on inverted decoupling and decentralized active disturbance rejection control. Ind Eng Chem Res, 2016, 55: 2008–2019

  28. 28

    Gao Z Q. Scaling and bandwidth-parameterization based controller tuning. In: Proceedings of the American Control Conference, Denver, 2003

  29. 29

    Bai W Y, Xue W C, Huang Y, et al. The extended state filter for a class of multi-input multi-output nonlinear uncertain hybrid systems. In: Proceedings of the 33rd Chinese Control Conference (CCC), Nanjing, 2014. 2608–2613

  30. 30

    Bai W Y, Xue W C, Huang Y, et al. Extended state filter design for general nonlinear uncertain systems. In: Proceedings of the 54th Annual Conference of the Society of Instrument and Control Engineers of Japan (SICE), Hangzhou, 2015. 712–717

  31. 31

    Guo B Z, Zhao Z L. On the convergence of an extended state observer for nonlinear systems with uncertainty. Syst Control Lett, 2011, 60: 420–430

  32. 32

    Han J Q. From PID to active disturbance rejection control. IEEE Trans Ind Electron, 2009, 56: 900–906

  33. 33

    Bougerol P. Kalman filtering with random coefficients and contractions. SIAM J Control Optim, 1993, 31: 942–959

  34. 34

    Zhao C, Guo L. PID controller design for second order nonlinear uncertain systems. Sci China Inf Sci, 2017, 60: 022201

  35. 35

    Bar-Shalom Y, Li X R, Kirubarajan T. Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software. Hoboken: John Wiley & Sons, 2004

  36. 36

    Shamma J S, Tu K Y. Approximate set-valued observers for nonlinear systems. IEEE Trans Autom Control, 1997, 42: 648–658

  37. 37

    Zhou B, Qian K, Ma X D, et al. A new nonlinear set membership filter based on guaranteed bounding ellipsoid algorithm. Acta Autom Sin, 2013, 39: 146–154

  38. 38

    Li X R, Jilkov V P. A survey of maneuvering target tracking — part II: ballistic target models. In: Proceedings of SPIE Conference on Signal and Data Processing of Small Targets, San Diego, 2001

  39. 39

    Austin J W, Leondes C T. Statistically linearized estimation of reentry trajectories. IEEE Trans Aerosp Electron Syst, 1981, 54–61

  40. 40

    Costa P J. Adaptive model architecture and extended Kalman-Bucy filters. IEEE Trans Aerosp Electron Syst, 1994, 30: 525–533

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Correspondence to Wenchao Xue.

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Bai, W., Xue, W., Huang, Y. et al. On extended state based Kalman filter design for a class of nonlinear time-varying uncertain systems. Sci. China Inf. Sci. 61, 042201 (2018). https://doi.org/10.1007/s11432-017-9242-8

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  • Kalman filter
  • extended state observer
  • nonlinear time-varying uncertain system
  • unbiased minimum variance filter
  • active disturbance rejection control