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?

  • $\begingroup$ I've never heard that term (probably because I am not a controls engineer) but what is zero-state observability? $\endgroup$ Apr 14, 2016 at 16:02
  • 1
    $\begingroup$ @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. $\endgroup$ Apr 14, 2016 at 16:19

1 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.

  • $\begingroup$ What is the difference to zero-state detectably? $\endgroup$
    – Carlos
    May 17, 2023 at 17:54

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