Hi Francisco.

First, the qualitative behaviour of the discretized system (e.g., being asymptotically stable when the continuous system is stable) depends on the stability bounds of the scheme you use for discretizing. For example, in the case of the Euler scheme, in presence of real negative eigenvalues of the linearized system of magnitude $|\lambda| \le M$, the time step should satisfy $h < 2/M$. In case of complex eigenvalues, you should refer to the "absolute stability region" (or simply "stability region") of the scheme in the complex plane. As a first guess, however, you should try reducing the time step h.

Second, discretization affects the geometric properties of the system, in particular the reachability set. To the best of my knowledge, an extensive review on the topic has been written some years ago by F. Lempio and V. Veliov (it should be the only paper with these two authors, you can easily spot it on Scholar).

I take it that you have some data and you are trying to fit the differential equations to this data. The problem might be in the accuracy or stability of the solution or in the method you are using to do the fit. Try reducing the step size h.  Try using different methods of fitting. Try using excel's solver which has a genetic algorithm based optimiser in the newer versions of excel.

https://www.bricsys.com/static/protected/common/forumAttach/28635/Curve_Fitting.pdf

If it does not work, use a different optimising algorithm eg levenberg-marquart algorithm

which you can do in r.

http://www.r-bloggers.com/learning-r-parameter-fitting-for-models-involving-differential-equations/

Dear Loopinder and Roberto

I am not just interested in the numerical solution, but also interested in the difference equations in order to design a discrete observer.  If you see the attached file in the question, I have a set of ODEs and I need a set of Difference Equations. I already try by reducing h, and with Rubnge-kutta instead of Euler, but I cannot fit the continuos model. 

Can you please put the data you are trying to fit in an excel file and paste it here.

Dear Francisco,

I think Runge-Kutta method (4rth order) for a system of five differential equations is so complicated, you have to compute many k-constants for each step.

What about using Runge-Kutta method of lower order, or what about using finite difference method to discretize your system?

Loopiner I am not trying to fit a set of data, I am trying to fit a model, for Inputs U1=18, U2=5. So the idea is to transform the continuos time ODE given in the png file to a difference equations. Then, design a discrete-time controller with the discrete model. For this I need to validate the discrete model, which I am trying in Matlab, nevertheless, I can match the continuos time model

Dear Francisco,

it is impossible to give any precise hint if you don't provide the values for all the constants appearing in the continuous model, as well as the reason why the discrete system does not match the expected behaviour.

Dear Francisco. We can approximated the derivative with the next expression:

\dot{x} = \dfracc{x((k+1)T) - x(kT)}{T} where T it's the sample period. So for your first equation:

x1((k+1)T)=T*[a1* \dfracc{u1(kT) - x2(kT)}{z1} + \mu x1(kT)^2] + x1(kT).

It looks as the Euler algorithm but it works. I hope to help. Bye and good luck :)

Dear Carlos, In fact, is the  forward Euler algorithm, the problem with the Euler is that the error increases every sample time,  I tried previously with this, but the problem it doesn't work,  that is the reason I am trying to find a different method. Thanks. 

The basic method is the Lagrange approximation.

Another approximation is to use implicit or implicit methods.

It is possible to use equal or unequal spaced increments.

The equal space increments produce approximation like Euler, trapezoidal or higher order approximations.

see Prof. Gear notes in Celaya.

djr

Hello Francisco Ronay Lopez Estrada:   Why don't you use the Runge Kutta 4th order method that is explained in most numerical text books. The Euler method is very crude except when it is accompanied by a corrector technique.