In general form of F and h, obviously there is not any general way to determine identifiability. There may be infinite number of different combinations for u and p vectors; e.g. in y = (a.b) x. However, if u ranges in a small interval, linearization may help to realize it. For given model structure, I can also suggest simulation of the system with various parameter sets ranging in a predetermined interval known based on physical definitions of the parameters and our pre knowledge about the system. If the simulations result in similar outputs/trajectories, probably there are parameters that are not identifiable. 

However, correctness of the modelling procedure is very important. For instance, if friction is a parameter in a model, it cannot be negative, and if the equations are developed based on physics and correct rules, logically it should be identifiable by measuring velocity.

In summary, your question is so wide and vague, and generally speaking, the answer I think is just vast number of simulations.   

You have not defined the system or the type of parameters. A parameter may be physical or a result of the model. If the parameter is something like viscosity or Reynolds number, the definition and calculation depend upon other parameters. You may be able to solve for the parameters at fixed points, but not generally. There is not enough information to provide advice.

Thank you for your answers.

To make the case more clear, here is the model I am working on. It is a High Fidelity Reaction Wheel model from the link attached here (pp. 168-173)

I would like to know if parameters Kt (in omega_dot equation) and Vbus (in f3 equation) are identifiable and how to prove if they are or not.

Based on the results presented in this thesis they should be but there is no proof provided on this matter.

I want to be able to back my results later when I am done with estimating these parameters to be able to analytically say these parameters were identifiable from the system model and system outputs (measurements = I and omega)

http://spectrum.library.concordia.ca/976146/1/MR45478.pdf

Have a look at :

Keesman, Karel J.: System Identification: An Introduction., 1st edition, springer, 2011, (ISBN 978‐0‐85729‐ 521‐7).

Once I wanted to solve this problem for the Lorenz system. The idea is to replace this problem by the problem of optimal stabilization. You first need to build a functional characterizing the deviation of the resulting solution of differential equations from a given function on the interval time. This function will describe the experimental data. It can get a trigonometric approximation. You can use the Zubov-method for solving the problem of optimal stabilization.

Article Lectures on control theory

Also you can use the Afanas’ev-Dzyuba-method for solving the problem of optimal stabilization.

Article On a suboptimal control of nonlinear systems via quadratic criteria

Dear Afshin Rahimi,

in general there is no easy way to determine parameter identifiability. However, there are well established methods for checking both structural and practical identifiability. Please have a look at http://www.fcc.chalmers.se/software/other-software/identifiabilityanalysis/, which apart from our Mathematica package for structural identifiability analysis includes a list of papers documenting the package and comparing with other methods repeated here for your convenience:

Conference Paper An Efficient Method for Structural Identifiability Analysis ...

Article Minimal output sets for identifiability

Article Comparison of approaches for parameter identifiability analy...

This is similar to the standard observability query for nonlinear systems. Evaluation of Lie Derivatives is a sure shot way to conclude local observability and also identifiability of parameters. In general the approach is to augment the state with the parameters, setting their derivatives to 0 (since parameters are assumed constant) and then evaluating subsequent Lie derivatives. An online search with keywords "observability, nonlinear systems" will yield several results.

Dear Afshin,

The attached and recently accepted paper may interest you because it exactly addresses the issue you raise. It  will appear in Automatica in July/August. In general, identifiability is a a hard question to solve, but in the paper the symbolic computations are guided by a singular value decomposition of the parametric output sensitivity matrix and this helps A LOT.

Kind regards, Hans Stigter

Article A Fast Algorithm to Assess Local Structural Identifiability

There is a procedure based on orthogonalization of jacobian matrix which can rank the parameters in order their estimability from given set of experimental data. I have used this technique and it's pretty straightforward.

Yao, K. Z., Shaw, B. M., Kou, B., McAuley, K. B., & Bacon, D. W. (2003). Modeling ethylene/butene copolymerization with multi‐site catalysts: parameter estimability and experimental design. Polymer Reaction Engineering, 11(3), 563-588.

Hi Afshin,

There are also a lot of numerical ways, such as Bootstrap and Profile likelyhood approach, for dealing with parameter identifiablity problem in non-linear deterministic models. For detail, i would recommend the papers below.

Article Comparison of approaches for parameter identifiability analy...

Conference Paper Uncertainty Analysis for Non-identifiable Dynamical Systems:...

The papers mentioned in other replies cover a good part of the most relevant literature for solving this problem. From a structural identifiability point of view, it is worth mentioning also the COMBOS method by DiStefano's lab, which included a web implementation: 

http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0110261  

And maybe also the GenSSI toolbox: 

http://bioinformatics.oxfordjournals.org/content/27/18/2610

Note that, although these methods were developed for biological applications, they may be applied to nonlinear dynamic systems in general.

An algorithm of identification of nonlinear system parameters

Article On Algorithm of Identification of Nonlinear System Parameters