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

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?

• I've never heard that term (probably because I am not a controls engineer) but what is zero-state observability? – wwarriner Apr 14 '16 at 16:02
• @starrise> The system x' = f(x, u) y = h(x, u) with f(0, 0) = h(0, 0) = 0 is said to be zero-state observable if no solution of x' = f(x, 0) can stay identically in S = {x ∈ Rn | h(x, 0) = 0}, other than x(t) = 0. – Sardar_Usama Apr 14 '16 at 16:19

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.