How can I check whether a nonlinear system is zero-state observable?

Question

Given a nonlinear system, such as:

$$\begin{align} x_1' &= x_2 \\ x_2' &= −x_1^3 + u \\ y &= x_2 \end{align}$$

How can I check the zero-state observability of the system?

Answer 1

I've found the answer.

To check if a system is zero state observable, put $u=0$ and check whether $x=0$ when $y=0$. If yes, it is zero-state observable. Otherwise not!

For the given system, by putting $u=0$ and $y=0$, we see that $x_2=0$, therefore $x'_2=0$ and thus $-x_1^3=0$ or $x_1=0 \implies x=0$. Thus it is zero-state observable.