How to prove the observability condition for the following state sytem?

Question

A linear state system is given with the following matrices $\boldsymbol{A}$ and $\boldsymbol{C}$. $$ \boldsymbol{A} = \begin{bmatrix} μ & 1 & 0 & \dots & 0 \\ 0 & μ & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 &\ldots & 1 \\ 0 & 0 & 0 &\ldots & μ \end{bmatrix}$$$$ \boldsymbol{C} = \begin{bmatrix} c_1 & c_2 & \dots & c_n \\ \end{bmatrix} $$ And the observability matrix $\boldsymbol{\mathcal{O}}$ is equal to

\begin{bmatrix} C\\ CA\\ CA^2\\ \vdots\\ CA^{n-1} \end{bmatrix} How can we show that the rank of the observability matrix is $n$ (the state system is observable) if and only if $c_1$ is nonzero? Remark: empirically, I found that the determinant of $\boldsymbol{\mathcal{O}}$ is equal to ${c_1}^n$, however, I need analytical proof.

Answer 1

Kalman's observability criterion states that the observability matrix $\boldsymbol{\mathcal{O}}$ has to have full rank in order to guarantee observability for the linear time-invariant system. Your observability matrix $\boldsymbol{\mathcal{O}}$ is a square matrix. And a square matrix has a full rank if it is invertible, which is equivalent to the statement that the determinant is not equal to zero.

You have found out that the determinant is equal to (in order to prove this you can use mathematical induction)

$$\det \boldsymbol{\mathcal{O}} = c_1^n.$$

And $c_1\neq 0$ guarantees that it is not equal to $0$. Hence, $c_1\neq 0$ guarantees observability.

Note, that for practical purposes a very small value of $c_1$ is not desirable as the determinant comes closer to the value of zero. Which might mean that practically your system is unobservable.

Answer 2

Checking the rank of the observability matrix $\mathcal{O}$ is equivalent to the Hautus test, which for observability can be formulated as

$$ \text{rank} \begin{bmatrix} C \\ A - \lambda\,I \end{bmatrix}=n \quad \forall\,\lambda \in \mathbb{C} $$

with $A\in\mathbb{R}^{n \times n}$.

In this case it is easier to proof observability using the Hautus test. Since the given $A$ is just one Jordan block with eigenvalues $\mu$, then the matrix, whose rank it tested, might only lose rank when $\lambda = \mu$. Performing this substitution gives

$$ \begin{bmatrix} C \\ A - \mu\,I \end{bmatrix} = \begin{bmatrix} c_1 & c_2 & c_3 &\cdots & c_n \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & 0 \\ 0 & 0 & \cdots & 0 & 1 \\ 0 & 0 & \cdots & 0 & 0 \end{bmatrix} $$

From here it is straightforward to show that the rank of that matrix is only equal to $n$ if $c_1 \neq 0$.