Observability of Nonlinear Systems

Problem Statement

Consider a nonlinear state space system

\[\begin{array}{ccrl} \dot{x}&=&f(x) & \text{(system dynamics)}\\ y&=&h(x) & \text{(measured output)} \end{array}\]

with the state vector $x$ and the measured output $y$. The system is described by a vector field $f$ and a scalar field $h$. The concept of observability describes the possibility of reconstructing the state $x$ from output measurement.

Lie Derivatives, Observability Map, Observability Matrix

The Lie derivatives of a function $h$ along a vector field $f$ are defined by

\[L^{k+1}_fh(x)=L_f L_f^k h(x) \quad\text{with}\quad L_f h(x)=\frac{\partial h}{\partial x}(x)f(x) \quad\text{and}\quad L_f^0h(x)=h(x). \label{eq:lie-der}\]

We combine the first $N$ Lie derivatives in the observability map defined by

\[q_N(x)= \begin{pmatrix} h(x) \\ L_fh(x) \\ \cdots\\ L_f^{N-1}h(x) \end{pmatrix}.\]

The system is (locally) observable if the observability map is (locally) injective for some $N\in\mathbb{N}$.

The observability matrix is the Jacobian of the observability map. Therefore, the rows of the observability matrix consists of the gradients of the Lie derivatives:

\[Q_N(x)=\frac{\partial}{\partial x} q_N(x) = \begin{pmatrix} \frac{\partial}{\partial x}h(x) \\ \frac{\partial}{\partial x}L_fh(x) \\ \cdots\\ \frac{\partial}{\partial x}L_f^{N-1}h(x) \end{pmatrix}.\]

If the observability matrix has full column rank for some $N\in\mathbb{N}$, then the system is locally observable due to the inverse function theorem.

Conducted Research on Observability

An efficient method to compute the observability matrix with automatic differentiation was developed in [1]. In [2] we utilized quantifier elimination to verify local and global observability. A decidable criterion for local and global observability of polynomial systems was derived in [3]. This approach was illustrated on some example systems in [4,5,6]. An extension to the algebraic identifiability is discussed in [7].


  1. Röbenack, K., Reinschke, K. J. (2000, September). An efficient method to compute Lie derivatives and the observability matrix for nonlinear systems.
    In Proc. Int. Symposium on Nonlinear Theory and its Applications (NOLTA) (Vol. 2, pp. 625-628).
  2. Röbenack, K., Voßwinkel, R. (2019, October). Formal verification of local and global observability of polynomial systems using quantifier elimination.
    In 2019 23rd International Conference on System Theory, Control and Computing (ICSTCC) (pp. 314-319). IEEE.
  3. Gerbet, D., Röbenack, K. (2020). On global and local observability of nonlinear polynomial systems: A decidable criterion.
    at-Automatisierungstechnik, 68(6), 395-409.
  4. Gerbet, D., Röbenack, K. (2020, October). Nonlinear observability for polynomial systems: Computation and examples.
    In 2020 24th International Conference on System Theory, Control and Computing (ICSTCC) (pp. 425-432). IEEE.
  5. Gerbet, D., & Röbenack, K. (2021). On the nonlinear observability of polynomial dynamical systems.
  6. Gerbet, D., & Röbenack, K. (2023). On the observability of embedded polynomial dynamical systems. TU Ilmenau.
  7. Gerbet, D., & Röbenack, K. (2021). An Algebraic Approach to Identifiability.
    Algorithms, 14(9), 255.