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].

Publications

  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.
    SYSTEM THEORY, CONTROL AND COMPUTING JOURNAL, 1(1), 88-94.
  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.