Extended Luenberger Observer for nonlinear system control


Herman (who commented for this posting) told me that there are two different versions of ELO. One is just to linearize a nonlinear function (this posting will handle it), and the other is using Lie-algebraic approach (refer to M. Zeitz 1987 “The extended Luenberger Observer”).

Thank you Herman.


This article is to explain the use of Luenberger observer for nonlinear system control. In other words, it is Extended Luenberger Observer, (ELO, just like Kalman Filter (KF), and Extended KF).

The basic idea is to linearize nonlinear system around the interesting point. The below is the description of ELO and how to select gain values for Extended Luenberger Observer. The below description assumes that you already know about Luenberger Observer for linear system. If you don’t know visit here.



I wish this can help your understanding about Luenberger observer, if you have any question, please leave me a comment below.


I am Youngmok Yun, and writing about robotics theories and my research.

My main site is http://youngmok.com, and Korean ver. is  http://yunyoungmok.tistory.com.


7 thoughts on “Extended Luenberger Observer for nonlinear system control

  1. Pingback: Luenberger observer for linear system control | Youngmok Yun: Roboticist in The Univ. of Texas at Austin

  2. Herman

    Hello Youngmok Yun,

    that what you explained above i actally know as a “Observer Design via Linearization”. Do you know the work of M. Zeitz 1987 “The extended Luenberger Observer”? That is about a Lie-algebraic approach, however without the necessity of integration of differential equations and inversion of non-linear equations, i.e. no state transformation is needed. Are you familiar with the differences of that two approaches in regard of convergence and performance?

    Regards, Herman

  3. admin Post author

    Oh.. You are right, I just checked the paper which you just let me know, Zeitz’s work, and it turns out that my post is wrong. I just thought that the relation between Luenberger Observer and ELO are same with the relation between KF and EKF. I just applied the way of EKF to the LO.

    After studying more, I will modify the post to be corrected.

    Thank you very much, Herman.


  4. Herman

    Hello Mok, i think your post is not wrong, just another definition. There are some papers about ELO design related to your post (See C. Elmas 1996 or Jooho Song 2000). But in the contrary there do not exist articles (or maybe hardly to find) that investigate the Zeitz’s approach in some applications… For me it would be interesting finding out differences in performance etc…

    Kind Regards and success in your research, Herman

    1. youngmok yun

      Hello Herman,
      Thank you very much for your kind explanation. I am happy to hear that my explanation is also not wrong.

      I will modify my posting to explain this story.

      Observer is always a difficult issue, especially for nonlinear and uncertain system. So my favorite is EKF or UKF instead of LO or ELO although their computation is little more expensive.

      Anyhow, I also wish your success in your reserch.


  5. Simon K


    is the current above representation now finally wrong or true? At the moment I try to estimate Rotor-temperature of induction machines with the help of ELO but still I need some good figure of whats really going on at the ELO.


    1. admin Post author

      Just there are two different versions of ELO. You can use this method. It is also a very common idea of ELO.


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