For nonlinear systems, we usually check local observability by constructing the observability matrix and checking its rank. You need to use Lie derivatives to do that. Check out Slotine's "Applied Nonlinear Control" book.

Cheers

Thanks Mohammed, this helps a lot!

Dear Jan

Observability for a nonlinear system can be made locally effectively. But the observability criterion is similar to that of linear case.

You must calculate the derivative of the output (by using the Lie derivative),

you must then build the observability matrix and requires that the rank of this matrix is full (= n, the dimension the state of the system). In Linear case, you must derive not much than (n-1) times, because of the Cayley Hamilton Theorem.

For Non linear case, unfortunately this theorem doesn't exists. If it works up the order (n-1), this is perfect, otherwise you must compute the derivatives up to having a full order .

Question: until what order?

Answer: we do not know, it depends on the system.

I suggest you book Hassan Khalil "Nonlinear Systems."

In your case the system is of order 1, you do no need to calculate the observability matrix. It is a left invertibility problem.

The output is always > = 0 (positive or zero) so, starting from the output, we obtain

x = sqrt (y) 2.

Dear Jan,

The first step is to linearize the nonlinear system into its linear equivalent using Lie derivative. Secondly, the available observability approaches of linear systems are used. In this regard in addition to the references above, I recommend to read the chapter :

http://www.me.berkeley.edu/ME237/6_cont_obs.pdf

that comprises a very simple explanations of discovering the observability of a NN system.

Your system is globally observable, i.e. any two distinct states can be distinguished by applying some (e.g. piece-constant control). A system is (globally) observable if and only if the functions from the observation algebra distinguish distinct states. To construct observation algebra you take Lie derivatives of the output with respect to the dynamics (for constant u) and substitute into polynomials. The first Lie derivative gives 2(x-2)(x+u)=2(x-2)^2+2(x-2)(2+u). Subtracting 2(x-2)^2, which already is in the observation algebra and taking u=0, we get 4(x-2). This function distinguishes any two distinct states. Of course globally observable system is also locally observable. It is good to recall that the rank condition is only sufficient for local observability.

That's interesting Zbigniew, because by using a linearization of the system and the Lie derivatives, the rank condition is not fulfilled for the state x=2.

Dear Jan,

Sorry, but my response was truncated, …

I said :

The output is always > = 0 (positive or zero) so, starting from the output, we obtain

x = sqrt (y) + 2.

So you can obtain the state directly from you measurement output.

The following References may meet some of your your Expectations on Observability and Contolability of Non Linear Systems.

1. Local observability of nonlinear systems

Z. Bartosiewicz

Politechnika Bia lostocka

Wiejska 45, Bia lystok

Poland

E-mail: [email protected]

Abstract

A necessary and sufficient condition for local observability of analytic

systems is presented. For smooth systems this condition is only sufficient,

but it is still weaker than the Hermann-Krener’s rank condition. It is

expressed with the language of ideals of germs of analytic or smooth functions nd real radicals of such ideals. The condition may be practically

checked in many cases.

Key words. Nonlinear control system. Local observability. Real radical.

Germ. Rank condition.

2. citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.48.1810...‎

by ED Sontag - ‎Cited by 140 - ‎Related articles

in the context of local nonlinear controllability. 1. ... gives a proof in the (continuous) case with m>1 but with extra assumptions on ...

3. https://www.math.ucdavis.edu/~krener/1-25/10.IEEE

4. citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.48.1810...‎

by ED Sontag - ‎Cited by 140 - ‎Related articles

in the context of local nonlinear controllability. 1. ... gives a proof in the (continuous) case with m>1 but with extra assumptions on A. These ..... nonlinear systems.

Copy of item 1 above is attached.

P.S.

A comment to Mohamed Djemai proposition: Your formula x = sqrt (y) + 2 works only for x>2. If x

Dear Bartosiewicz,

Thank you for your comments, of course you are right. My response was not complete.

It can be done by making an observabitlity matrix using lee algebra and simply look for the full rank of the matrix as is the case in linear systems.

Another book, which has not been suggested so far in the answers is the following:

Nonlinear Control Systems by Alberto Isidori.

The difference between the book by Khalil and the one from Isidori is that the first gives a more engineering approach while the second one uses concept of differential geometry as (distribution, codistribution, annhilators and so on).

Specifically for the observability analysis of a nonlinear system go to section 1.9.

I hope it helps.