Observability
Brief introduction to observability for linear dynamical systems
Using full state feedback i.e. $\mathbf{u} = - \mathbf{K}\mathbf{x}$, we can modify the behavior of a controllable system. However, it is not always possible to have full-state measurements of the state vector $\mathbf{x}$. In this case, we have to estimat it. This is only possible when the system observable [1]. In this post we will have a brief view of observability.
Recall that we deal with linear systems of the form
$$\frac{d\mathbf{x}}{dt} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}, ~~ \mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u}$$
In this case, observability is similar to controlability [1]. Briefly, a system is observable if it is possible to estimate any state $\boldsymbol{\xi} \in \mathbb{R}^n$ from a history of measurements $\mathbf{y}(t)$ [1]. The observability matrix $\mathbf{\cal{O}}$ allows us to determin entirely whether a system is observable or not [1]. It is defined as
$$\mathcal{\cal{O}} = \begin{bmatrix}\mathbf{C} \\ \mathbf{CA} \\ \mathbf{C}\mathbf{A}^2 \\ \vdots \\ \mathbf{C}\mathbf{A}^{n-1} \end{bmatrix}$$
Where $n$ is the number of state variables. Specifically, if the rows of the matrix span $\mathbb{R}^n$ then it is possible to estimate any full-dimensional state vector $\mathbf{x} \in \mathbb{R}^n$ from the time-history of $\mathbf{y}(t)$ [1].
If a system is observable, then it is possible to design the eignevalues of the estimated dynamics to have properties such as noise attenuation and fast estimation [1]. Finally, note that observability matrix is the transpose of the controllability matrix $\mathbf{\cal{C}}$.
- Steven L. Brunton, J. Nathan Kutz,
Data-Driven Science and Engineering. Machine Learning, Dynamical System and Control, Cambridge University Press.