In this post, I’ll explore the basics of observability in nonlinear systems—a topic central to my research.
What Is Observability?
Observability refers to whether a system’s internal state can be inferred from its outputs over time.
For a nonlinear system:
$$ \dot{x}(t) = f(x(t), u(t)),$$ $$ y(t) = h(x(t)), $$
where:
- $x(t) \in \mathbb{R}^n$ is the state,
- $u(t) \in \mathbb{R}^m$ is the input,
- $y(t) \in \mathbb{R}^p$ is the output.
We say the system is locally observable at $x_0$ if different initial states around $x_0$ produce different outputs.
Lie Derivatives
A powerful tool in nonlinear observability is the Lie derivative. For a scalar output $y = h(x)$, we define:
$$ L_f h(x) = \frac{\partial h}{\partial x} f(x). $$
We compute higher-order Lie derivatives as needed.
Example
Let:
$$ \dot{x}_1 = x_2, $$ $$\dot{x}_2 = -x_1, $$ $$y = x_1$$
This is a harmonic oscillator. You can verify that it is observable, since $y = x_1$, and differentiating gives access to $x_2$.
Summary
Observability is critical in designing sensors and state estimators. Future posts will explore:
- The observability Gramian
- Empirical observability
- Sensor placement algorithms
Thanks for reading! Let me know if there’s a topic you’d like me to dive deeper into.