Dear Amirreza ,

You can read it in the following link:

Article

Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective

• Ulrike Baur, Peter Benner & Lihong Feng

Archives of Computational Methods in Engineering volume 21, pages331–358(2014)

Hi Amirreza,

Model reduction techniques that might suit your problem are highly linked to your own problem structure. For instance, questions you might need to address before choosing a specific framework are:

* which class of inputs are you considering in your problem?

* which is your specific control objective?

* which are the stability properties of your system?

* In which 'sense' you are trying to approximate your system?

* etc.

A word of caution with respect to the concept of 'simplicity' that you infer in your question. For linear systems, this is often understood in terms of the dimension of the system, i.e. a reduced system is simpler if its state-space realization has fewer states. For nonlinear systems, this dimensional argument is not necessarily appropriate, as one has to take into consideration also the complexity of the functions involved in its representation. Furthermore, the notion of dimension has, for nonlinear systems, a local nature. In other words, the dimension of a 'good' reduced order model may depend upon the initial condition.

I can suggest the following monograph, which not only describes a specific framework for nonlinear model reduction, but also provides a number of references to other techniques.

Book Nonlinear Model Reduction by Moment Matching

Hope this helps.

This is like asking “can I fit a straight line to a ... ?” Yes. You can fit a straight line to anything!

for your problem ... just eye-ball the impulse or step response of your high-order non linear system. What is the frequency of oscillation (roughly)? What is the rate of decay (roughly)? Those Two parameters are enough to define your 2nd order linear system.

in general, for this type of operation, Furier is used,

but you can also try to subdivide the resulting graph into fundamental harmonics, without reducing the degree, an attractive trend of the function that you can work on in reliable approximation comes out.

It is only idea!

see you

C.

Hi Amirreza Tootchi ,

is it possible to know what you are trying to model? This because as was pointed out by Nicolás Faedo , the suitability of the model order reduction technique might depend on the type of nonlinearity you have. For instance, it not the same a nonlinearity coming from a chemical reactor where some exponential factors could appear in the reaction terms than the nonlinearity that arises when modeling power electronics converters.

Regards,

Jorge

Dear Amirreza Tootchi,

I suggest you to see links and attached files on topic.

https://www.degruyter.com/view/journals/nleng/6/4/article-p301.xml?language=en

Article Model order reduction for bilinear control systems with inho...

Conference Paper Truly Nonlinear Model-Order Reduction Techniques

Chapter Model Order Reduction of Nonlinear Systems in Circuit Simula...

Best regards

You may do that by repetitive phase space transformation on the derivative scales

The simplest rigorous and systematic answer to most questions on model order reduction is to read my magnum opus "Model emergent dynamics in complex systems" http://bookstore.siam.org/mm20/ This book explores how to derive relatively simple dynamical equations that model complex physical interactions. The objectives are to use sound theory to explore algebraic techniques, develop interesting applications, and discover general modeling principles.

Alternatively, one way to immediately proceed is write your fourth-order DEs as a system of first-order DEs and submit them to my web service for analysis:

http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php

However, the form you submit builds in assumptions about your domain of interest and the expected quasi-balance in the system. Further, you mention you want to design a controller which suggests that you need methodologies suitable for non-autonomous systems. If so, instead you should submit to

http://www.maths.adelaide.edu.au/anthony.roberts/sdesm.php

as although originally intended for stochastic systems, the analysis holds for all sorts of non-autonomous effects such as a controller.

Hi, you can also try Article Kernel Methods for the Approximation of Nonlinear Systems

It's an essentially linear method applied to nonlinear systems.

Dear Jorge Humberto Urrea-Quintero Nicolás Faedo A. J. Roberts Boumediene Hamzi P. K. Karmakar Mohamed-Mourad Lafifi Hugh Lachlan Kennedy Mohammad Mahdi Share Pasand

Omar H. Abdalla

Thank you all for your responses. I appreciate it truly. but about the question you did ask, I have to say the system I have modeled already is a pneumatic servo positioning system. I have attached the extracted fourth-order equation to this massage. I am going to design a nonlinear Lyapunov based model reference adaptive control for the aim of trajectory tracking. as you might aware, working with a 4th order system is complicated. whether analyzing it or designing a control system for them. it's not easy to obtain a control law within writing the stability theorem by candidating a Lyapunov function with 4 states. that is why I am somehow thinking of reducing the order of my system.

in addition, I need some good examples or papers which represent nonlinear control systems for high order systems (n > 3).

I want to thank you beforehand for your responses...

Hi, the method I mentioned should work for these systems. Just check the paper. I also explain it at https://www.youtube.com/watch?v=WxwH7Lk1gWc

Hi, I hope the following methods are useful to you for the model order reduction of interval systems.

1) A Quadratic method for Nonlinear Model Order Reduction developed by the MIT researchers. Authors developed a quadratic reduction method based on Krylon subspace obtained from the linearized anzlysis. The method verified on the nonlinear resistor network.

2) Empirical Gramians computed by generalizing linear Gramians onto nonlinear systems. (Ref: Empirical Gramian-based spatial basis functions for model reduction of nonlinear distributed parameter systems.)

If you need any papers, kindly send me a private message.

All the best

@Nahid Habib, Recently I have published a paper on " Model order reduction of interval systems using an arithmetic operation ", International Journal of systems science, Vol. 51, Issue 5, 2020.

The paper is an extension of the differentiation method for large scale interval systems and also an alternative approach to the existing differentiation method. I have a little knowledge regarding the reduction of nonlinear systems.

Could you please explain how can we apply the differentiation approach to the model order reduction of nonlinear systems.

There are too many ways for model order reductions, I suggest you linearization and via a general way and via Lie Algebra techniques. For a system of eqs or matrices you can use PCA/ SVD.