Skip to main content

Advertisement

Springer Nature Link
Log in

A New Hamilton–Jacobi Differential Game Framework for Nonlinear Estimation and Output Feedback Control

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

In this paper, we develop a new framework for designing state estimators/filters and output measurement feedback controllers for affine nonlinear systems in state space. The problems are formulated as zero-sum differential games, and sufficient conditions for their solvability are given in terms of Hamilton–Jacobi–Isaacs equations (HJIEs). These HJIEs are new, in the sense that they are both state-dependent and measurement output dependent. This allows for the filter and observer gains to be optimized over all possible nonlinear gains. Examples and simulation results are also presented to support the theory.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. \(T^{*}{\mathcal X}\), \(T{\mathcal Y}\) represent the cotangent and tangent bundles of \({\mathcal X}\) and \({\mathcal Y}\) with coordinates \((\hat{x},\hat{p}_{1})\), \((y,\dot{y})\), respectively.

  2. Where \({\mathcal L}_{f}(.)\) is the Lie-derivative operator along f.

References

  1. M.D.S. Aliyu, Nonlinear \(\cal{H}_{\infty }\)Control Hamiltonian Systems and Hamilton–Jacobi Equations (CRC-Taylor and Francis, Boca-Raton, FL, 2011)

  2. B.D.O. Anderson, J.B. Moore, Optimal Filtering (Prentice Hall, Upper Saddle River, 1979)

    MATH  Google Scholar 

  3. I. Arasaratnan, S. Haykin, Cubature Kalman filters. IEEE Trans. Autom. Control 54(6), 1254–1269 (2009)

    Article  MathSciNet  Google Scholar 

  4. B. Balaji, Universal nonlinear filtering using Feynman path integrals-I: the continuous–discrete model with additive noise. IEEE Trans. Aerosp. Electron. Syst. 48(3), 1935–1960 (2012)

    Article  Google Scholar 

  5. T. Basar, P. Bernhard, \({\cal{H}}_{\infty }\)Optimal Control and Related Minimax Designx. (Birkhauser, New York, 1991)

  6. T. Basar, G.J. Olsder, Dynamic Noncooperative Game Theory (Academic Press, Cambridge, NY, 1982)

    MATH  Google Scholar 

  7. N. Berman, U. Shaked, \(\cal{H}_{\infty }\) nonlinear filtering. Int. J. Robust Nonlinear Control 6, 281–296 (1996)

    Article  Google Scholar 

  8. R.S. Bucy, Linear and nonlinear filtering. IEEE Proc. 58(6), 854–864 (1970)

    Article  MathSciNet  Google Scholar 

  9. X. Chen, X. Luo, S.S.-T. Yau, Direct method for time-varying nonlinear filtering problems. IEEE Trans. Aerosp. Electron. Syst. 53(2), 630–639 (2017)

    Article  Google Scholar 

  10. L.C. Evans, Partial Differential Equations (Graduate Studies in Maths, AMS, 1998)

  11. A. Isidori, A. Astolfi, \({\cal{H}}_{\infty }\) control via measurement-feedback for affine nonlinear systems. IEEE Trans. Autom. Control 4, 553–574 (1994)

    MathSciNet  Google Scholar 

  12. A. Isidori, \({\cal{H}}_{\infty }\) control via measurement-feedback for affine nonlinear systems. Int. J. Robust Nonlinear Control 4, 553–574 (1994)

    Article  MathSciNet  Google Scholar 

  13. S.J. Julier, J.K. Ulhmann, H.F. Durrant-Whyte, A new method for nonlinear transformation of means and covariances in filters and estimators. IEEE Trans. Autom. Control 45(3), 472–482 (2000)

    Article  MathSciNet  Google Scholar 

  14. S.J. Julier, J.K. Uhlmann, Unscented filtering and nonlinear estimation. Proc. IEEE 92(3), 401–422 (2004)

    Article  Google Scholar 

  15. C. Kalender, A. Schottl, Sparse grid-based nonlinear filtering. IEEE Trans. Aerosp. Electron. Syst. 49(4), 2386–2396 (2013)

    Article  Google Scholar 

  16. D.S. Kalogerias, A. Petropulu, Asymptotically optimal discrete-time nonlinear filters from stochastically convergent state estimation process approximations. IEEE Trans. Signal Process. 63(13), 3522–3536 (2015)

    Article  MathSciNet  Google Scholar 

  17. P. Khargonekhar, M. Rotea, E. Baeyens, Mixed \({\cal{H}}_{2}/{\cal{H}}_{\infty }\) filtering. Int. J. Robust Nonlinear Control 6(4), 313–330 (1996)

    Article  Google Scholar 

  18. A.J. Krener, Necessary and sufficient conditions for worst-case \(({\cal{H}}_{\infty }\)) control and estimation. J. Math. Syst. Estim. Control 4(4), 1–25 (1994)

    MathSciNet  Google Scholar 

  19. H.J. Kushner, Nonlinear filtering: the exact dynamical equation satisfied by the conditional mode. IEEE Trans. Autom. Control 12(3), 262–267 (1967)

    Article  Google Scholar 

  20. Y. Liu, H. Wang, C. Hou, UKF based nonlinear filtering using minimum entropy criterion. IEEE Trans Signal Proccess. 61(20), 4988–4999 (2013)

    Article  MathSciNet  Google Scholar 

  21. X. Luo, S.S.-T. Yau, Complete real-time solution of the general nonlinear filtering problem without memory. IEEE Trans. Autom. Control 58(10), 2562–2578 (2013)

    MathSciNet  Google Scholar 

  22. D. Marelli, M. Fu, B. Ninness, Asymptotic optimality of the maximum-likelihood Kalman filter for Bayesian tracking with multiple nonlinear sensors. IEEE Trans. Signal Proccess. 63(17), 4502–4515 (2015)

    Article  MathSciNet  Google Scholar 

  23. A. Mohammadi, K.N. Plataniotis, Complex-valued Gaussian filter for nonlinear filtering of nongaussian/non-circular noise. IEEE Trans. Signal Process. Lett. 22(4), 440–445 (2015)

    Article  Google Scholar 

  24. R.E. Mortensen, Maximum-likelihood recursive nonlinear filtering. J. Optim. Theory Appl. 2(6), 386–394 (1968)

    Article  MathSciNet  Google Scholar 

  25. K.M. Nagpal, P. Khargonekhar, Filtering and smoothing in an \(\cal{H}_{\infty }\) setting. IEEE Trans. Autom. Control 36(2), 152–166 (1991)

    Article  MathSciNet  Google Scholar 

  26. S.K. Nguang, M. Fu, Robust nonlinear \({\cal{H}}_{\infty }\) filtering. Automatica 32(8), 1195–1199 (1996)

    Article  MathSciNet  Google Scholar 

  27. H.H. Niemann, J. Stoustrup, B. Shafai, S. Beale, LTR design of proportional-integral observers. Int. J. Robust Nonlinear Contr. 5, 671–693 (1995)

    Article  MathSciNet  Google Scholar 

  28. R. Postoyan, P. Tabuada, D. Nesic, A. Anta, A framework for the event triggered stabilization of nonlinear systems. IEEE Trans. Autom. Control 60(4), 982–996 (2015)

    Article  MathSciNet  Google Scholar 

  29. M.L. Psiaki, Gaussian mixture nonlinear filtering with resampling for mixand narrowing. IEEE Trans. Signal Proccess. 64(21), 5499–5512 (2016)

    Article  MathSciNet  Google Scholar 

  30. J, Qiu, K. Sun, T. Wang, H. Gao, Observer-based fuzzy adaptive event-triggered control for pure-feedback nonlinear systems. IEEE Trans Fuzzy Systems, To appear (2019). https://doi.org/10.1109/TFUZZ.2019.2895560

  31. R. Radhakrishnan, A. Yadav, P. Date, S. Bhaumik, A new method of generating sigma points and weights for nonlinear filter. IEEE Control Syst. Mag. 2(3), 519–524 (2018)

    Article  Google Scholar 

  32. S. Sastry, Nonlinear Systems: Analysis (Springer-Verlag, Berlin, 1999). Stability and Control

    Book  Google Scholar 

  33. N. Sen, Generation of Conditional densities in nonlinear filtering for infinite-dimensional systems. IEEE Trans. Autom. Control 63(7), 1868–1882 (2018)

    Article  MathSciNet  Google Scholar 

  34. F. Wang, J. Zhang, B. Lin, X. Li, Two-stage particle filter for nonlinear Baysian estimation. IEEE Access: Spec. Sect. Multimed. Anal. Int. Things 6, 13803–13809 (2018)

    Article  Google Scholar 

  35. W.H. Zhang, B.S. Chen, C.C. Tseng, Robust \(\cal{H}_{\infty }\) filtering for nonlinear stochastic systems. IEEE Trans. Signal Process. 53(2), 589–598 (2005)

    Article  MathSciNet  Google Scholar 

  36. M. Zhong, D. Guo, D. Zhou, A Krein space approach to \(H_{\infty }\) filtering for discrete-time nonlinear systems. IEEE Trans. Circuits Syst. I 61(9), 2644–2652 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, King Faisal University, P. O. Box 400, Al-Ahsa, 31982, Saudi Arabia

    M. D. S. Aliyu

Authors
  1. M. D. S. Aliyu
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to M. D. S. Aliyu.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aliyu, M.D.S. A New Hamilton–Jacobi Differential Game Framework for Nonlinear Estimation and Output Feedback Control. Circuits Syst Signal Process 39, 1831–1852 (2020). https://doi.org/10.1007/s00034-019-01229-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-019-01229-4

Keywords