Extended Luenberger Observer Design Using Algorithmic Differentiation

We consider a nonlinear single-output system \[ \dot{x}=f(x),\quad y=h(x) \] with the state vector x, the scalar output y, the vector field $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and the scalar field $h:\mathbb{R}^{n}\to\mathbb{R}$. The observer suggested by Bestle and Zeitz [1-3] has the form \[ \dot{\hat{x}}=f(\hat{x})+l(\hat{x})\cdot(y-h(\hat{x})) \] with the state-dependend gain vector $l:\mathbb{R}^{n}\to\mathbb{R}^{n}$. The gain is computed in three steps:

  1. Assuming that that maps f and h are sufficiently smooth, we can compute the nonlinear observability matrix as \[ Q(x)=\begin{pmatrix}dh(x)\\ dL_{f}h(x)\\ \vdots\\ dL_{f}^{n-1}h(x) \end{pmatrix}, \] where $L_{f}h(x)=h^{\prime}(x)\cdot f(x)$ denotes the Lie derivative of $h$ along $f$.
  2. Then, we calculate the so-called starting vector field $v:\mathbb{R}^{n}\to\mathbb{R}^{n}$ by \[ v(x)=Q^{-1}(x)e_{n}, \] where $e_{n}=(0,\ldots,0,1)^{T}$ is the $n$th unit vector. The starting vector field is the last row of the inverse observability matrix.
  3. The observer gain vector is obtained by a generalization of Ackermann's formula for nonlinear systems \[ l(x)=p_{0}v(x)+p_{1}ad_{-f}g(x)+\cdots p_{n-1}ad_{-f}^{n-1}g(x)+ad_{-f}^{n}g(x), \] where $p_{0},\ldots p_{n-1}$ are the coefficients of the characteristic polynomial \[ s^{n}+p_{n-1}s^{n-1}+\cdots p_{1}s+p_{0} \] of the linear part. The roots have to be chosen such that the nonlinearities are dominated by the linear part. Recall that \[ ad_{f}g(x)=[f,g](x)=g^{\prime}(x)f(x)-f^{\prime}(x)g(x) \] denotes the Lie bracket of the vector fields $f$ and $g$.

The computation of the observer gain is usually carried out symbolically, which can result in large expressions. We suggested an alternative approach to compute the gain using automatic or algorithmic differentiation [4-7].

Examples:

References/Publications:

  1. Bestle, D. & Zeitz, M.: Canonical form observer design for non-linear time-variable systems.
    Int. J. Control, 1983, 38, 419-431
  2. Birk, J. & Zeitz, M.: Extended Luenberger observer for non-linear multivariable systems.
    Int. J. Control, 1988, 47, 1823-1836
  3. Zeitz, M.: The extended Luenberger observer for nonlinear systems.
    Systems & Control Letters, 1987, 9, 149-156
  4. Röbenack, K.; Reinschke, K. J.: Nonlinear Observer Design using Automatic Differentiation.
    In : Corliss, G.; Faure, C.; Griewank, A.; Hascoët, L.; Naumann, U. (Hrsg.),
    Automatic Differentiation: From Simulation to Optimization, Springer, 2002, Kapitel 15, 137-142
  5. Röbenack, K.: Computation of the Observer Gain for Extended Luenberger Observers Using Automatic Differentiation.
    IMA Journal of Mathematical Control and Information, 2004, 21, 33-47
  6. Röbenack, K.: Beobachterentwurf für nichtlineare Zustandssysteme mit Hilfe des Automatischen Differenzierens.
    Shaker Verlag, 2003 (ISBN: 978-3-8322-1069-4)
  7. Röbenack, K.: Regler- und Beobachterentwurf für nichtlineare Systeme mit Hilfe des Automatischen Differenzierens
    Shaker Verlag, 2005 (ISBN: 978-3-8322-4414-9)