Minimum Energy Estimation Applied to the Lorenz Attractor

  • Arthur J. KrenerEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 29)


Minimum Energy Estimation is a way of filtering the state of a nonlinear system from partial and inexact measurements. It is a generalization of Gauss’ method of least squares. Its application to filtering of control systems goes back at least to Mortensen who called it Maximum Likelyhood Estimation. For linear, Gaussian systems it reduces to maximum likelihood estimation (aka Kalman Filtering) but this is not true for nonlinear systems. We prefer the name Minimum Energy Estimation (MEE) that was introduced by Hijab. Both Mortensen and Hijab dealt with systems in continuous time, we extend their methods to discrete time systems and show how Taylor polynomial techniques can lessen the computational burden. The degree one version is equivalent to the Extended Kalman Filter in Information form. We apply this and the degree three version to problem of estimating the state of the three dimensional Lorenz Attractor from a one dimensional measurement.


Infinite horizon stochastic optimal control Finite horizon stochastic optimal control Stochastic Hamilton–Jacobi–Bellman equations Stochastic algebraic Riccati equations Stochastic differential Riccati equations Stochastic linear quadratic regulator 


  1. 1.
    Anderson, B.D.O., Moore, J.B.: Optimal Filtering. Prentice Hall, Englewood Cliffs (1979)Google Scholar
  2. 2.
    Chen, C.-T.: Introduction to Linear System Theory. Holt, Rinehart and Winston, New York (1970)Google Scholar
  3. 3.
    Gelb, A.: Applied Optimal Estimation, MIT Press, Cambridge (1974)Google Scholar
  4. 4.
    Hijab, O.B.: Minimum energy estimation. PhD Thesis, University of California, Berkeley (1980)Google Scholar
  5. 5.
    Julier, S.J., Uhlmann, J.K., Durrant-Whyte, H.F.: A new approach to filtering nonlinear systems. In: Proceedings of American Control Conference (1995)Google Scholar
  6. 6.
    Kazantis, N., Kravaris, C.: Nonlinear observer design using Lyapunov’s auxiliary theorem. Syst. Control Lett. 34, 241–247 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Khalil, H.K.: High-Gain Observers in Nonlinear Feedback Control. SIAM, Philadelphia (2017)CrossRefGoogle Scholar
  8. 8.
    Krener, A.J.: The convergence of the minimum energy estimator. In: Kang, W., Xiao, M., Borges, C. (eds.) New Trends in Nonlinear Dynamics and Control, and Their Applications, pp. 187–208. Springer, Heidelberg (2003)Google Scholar
  9. 9.
    Krener, A.J.: Minimum energy estimation and moving horizon estimation. In: Proceedings of the 2015 CDC (2015)Google Scholar
  10. 10.
    Krener, A.J., Respondek, W.: Nonlinear observers with linearizable error dynamics. SIAM J. Control Optim. 23, 197–216 (1985)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mortensen, R.E.: Maximum likelihood recursive nonlinear filtering. J. Optim. Theory Appl. 2, 386–394 (1968)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Naval Postgraduate SchoolMontereyUSA

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