# Nonlinear Systems

1
How to solve linear couple equations by Differential Quadrature method?

HI.how can i solve this linear couple equations by Differential Quadrature method and gain value omega which is unknown?

thanks

file is attached below

Hello,  first rewrite the system using a matrix, and the unknowns :

f1, f'1,  f2, f'2,   f3, f'3, f"3, f^(3)3  : 8 functions.

Then integrate with a numerical or symbolic system (to get the eigenspaces / values) and have the general solution.

Then identify your parameter using the BCS.

I am not sure there is a unique solution for w.

Best Regards,

Gabriel

1
Strange nonchaotic attractors in autonomous system are possible?

I read the comment on" Strange nonchaotic attractors in autonomous and periodically driven systems by Pikovsy et al.  The appearance of SNA is associated with destruction of torus. If the autonomous system or self-oscillator undergoes torus and multi-frequency quasiperiodic regime in dissipative systems having dimension of greater than or equal to three. In such systems, is their any possibility of existence of  SNA(Strange nonchaotic attractors). If it possible how we can prove it?

Dear Chithra.

I provided the experimental investigation of the strange nonchaotic attractors in the non-autonomous systems, as example - the our article "EXPERIMENTAL OBSERVATION OF DYNAMICS NEAR THE TORUS-DOUBLING TERMINAL CRITICAL POINT", Bezruchko B.P., Kuznetsov S.P., Seleznev Y.P. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 2000. Т. 62. № 6 B. С. 7828-7830.

I think, that in autonomous systen there are no the strange nonchaotic attractor, because the transition from 2D torus to chaos is associated wiht some tipical bifurcations as result the SNA can not exist.  More over, there are no the methods of the indefication of the SNA in the autonomous systems. But  for  the non-autonomous systems the method of rational aproximations show very good result.

Dear Chithra, I will be very glad to answer on any your questions.

Sorry my english.

6
Is there any generalization to the p-laplacian of the work of Diaz, J. I. et al. in the link below?

Is there any generalization to the p-laplacian of the work: Diaz, J. I., Morel, J. M., & Oswald, L. (1987). An elliptic equation with singular nonlinearity. Communications in Partial Differential Equations, 12(12), 1333-1344.?

I will be very grateful if you can let me know about any result related to the problem -Δ_{p}u+(1/(u^{α}))=f.

• Source
##### Article: An elliptic equation with singular nonlinearity

Full-text · Article · Jan 1987 · Communications in Partial Differential Equations

I am deeply grateful to Professor Abdellaoui, B; who showed me this reference.

Hai, D. D. (2010). Singular boundary value problems for the p-Laplacian. Nonlinear Analysis: Theory, Methods & Applications, 73(9), 2876-2881.

4
Can anyone suggest good reference for the time optimal nonlinear control?

I am looking for a reference for a time optimal control for nonlinear system, preferably with some applied examples instead of just theory..

Prajakta,

You may want to take a look at H. Hermes and J. P. La Salle, Functional Analysis and Time Optimal Control. (Mathematics in Science and Engineering, Volume 56). VIII + 136 S. m. Abb. New York/London 1969. Academic Press.

http://onlinelibrary.wiley.com/doi/10.1002/zamm.19720520711/abstract

8
What is the best program i can use it to calculate the Fox H function approximation series ?

The series have form as in file attached

Do not exist numerical methods to calculate general Fox functions, only some particular cases can be calculated. Therefore you can not find routines to calculate such general function.

21
I want to study Nonlinear control, can any body recommend a book or Videotutorials?

I have the nonlinear systems of Khalil, however some definitions are no so clear, is there a newer book with Matlab examples

Nonlinear Control Systems and Power System Dynamics
Authors: Qiang Lu, Yuanzhang Sun, Shengwei Mei

you follow this book. It is a very good book.

5
What are some examples of experimental investigations of fractional order dynamical systems?

How much experimental work has been done on fractional dynamical systems? I recall hearing of some electronic oscillators that are modeled by fractional equations. In that case, the circuits were expressly designed to fit the equations. Are there any examples that are less contrived?

See the papers of Duarte Valério, Richard Magin, or Tenreiro Machado. See my paper in attach

• ##### Article: Fractional model of an electrochemical capacitor
[Hide abstract]
ABSTRACT: The fractional model of the electrochemical capacitor (EC) and its potential relaxation are presented. The potential relaxation occurs after charge or discharge current interruption. The EC fractional model is based on the fractional order transfer function that was obtained by means of least squares fitting of the EC impedance data. The inverse Laplace transform is used to obtain the EC impulse response. By using of the EC impulse response the EC charge and discharge simulation were performed.
No preview · Article · Dec 2014 · Signal Processing
9
Is there any relation between fractional calculus and physical concepts?
Nowadays, fractional calculus has been introduced in many fields. However, I have not seen a relation between physics and fractional calculus. In other words, is there any natural or physical phenomenon whose dynamics may be given with fractional order differential equations?

Hi Javid,

If you want partial answer to your question(related to Diffusion Phenomena),you may refer the following papers

1.S. Havlin And  D. Ben-Avraham, Diffusion in Disordered Media, Advances in physics,51,1,(2002),187-292

2.R. R. Nigmatullin, To the Theoretical Explaination of the ”Universal Response”,
Phys. Stat. Sol.(b) 123, 739-745, 1984

3 Nigmatullin, On the Theory of Relaxation for System with ”Remnant Memory”, Phys. Stat. Sol.(b) 124, 389-393, 1984.106
4R. R. Nigmatullin, The Realization of the Generalized Transfer Equation in a Medium with Fractal Geometry, Phys. Stat. Sol.(b)133, 425-430, 1986.

with regards

DNYAN

6
Are the phase plane analysis applicaple for systems as X'=AX+BU ?

Hello, everyone,

I try to find a phase portrait of first order systems of the forme X'=AX+BU, and so, for trying to explain the effect of the choice of the initial conditions to the convergence, any suggestions ??

With thanks !!

The phase plane analysis holds for second order systems which are also autonomous hence u must be constant. If A is nonsingular you have a single equilibrium point defined by $\bar{X} = - A^{-1}Bu$. This equilibrium can be saddle, focus, node, stable or unstable or other  - which are not structurally stable. This follows from the analysis of the eigenvalues of A. To draw the phase trajectories - use the computer programs!

11
Which tool you prefer for non linear analysis and why?

I just want to know which tool is used more for the fea non linear simulations. And the reason behind why we use that tool.

One of the main tools in approximating the solutions of nonlinear (scalar or matrix, or operator) equations is that of applying the successive approximation method (contraction principle). Another powerful iterative method is Newton's method (especially for equations P(x)=0, where P is convex and monotone). Both methods mentioned above allow  estimating the error.

2
Is there any solution for random analysis of a partial nonlinear vibration system?

is there any solution for random analysis of a partial nonlinear vibration system?

Hi

I advice you to see these documents. You will find what you need.

Best regards

8
How can I estimate the uncertainty of parameters?

I design robust controller and robust observer for nonlinear system using
norm bounded parameters uncertainty. I want to verify how controller and
observer works. I have nominal system in typical state-space form:
x(k+1) = Ax(k)+Bu(k) + g(x)
y(k) = Cx(k);
Is It possible to estimate the uncertainty of parameters (find matrix H, E1 and E2) for system in form
x(k+1) = (A+HFE1) (kB+HFE2)u(k) + g(x)
y(k) = Cx(k);
base of nominal model and real measurements data form real system?

Regards

http://www.scopus.com/record/display.uri?eid=2-s2.0-84907861447&origin=resultslist&sort=plf-f&src=s&st1=porcel+galvéz&sid=0D55198F2AA0EB108D13EC08136B739E.WeLimyRvBMk2ky9SFKc8Q%3a100&sot=b&sdt=b&sl=26&s=AUTHOR-NAME%28porcel+galvéz%29&relpos=0&relpos=0&citeCnt=0&searchTerm=AUTHOR-NAME%28porcel+galvéz%29
6
Is there any theorem / lemma/ theory regarding closed form expressions which says that we can find out some nth derivative of a function?

Consider a function

x_dot= f (x),  its 1st derivative can be written as   x(1)=f(x),

And its 2nd derivative can be x(2)=f '(x). x_dot,

And recursively, we can find out x(n) nth derivative of the x_dot= f(x) in the case if f(x) is linear, which is a reason for the formation of matrix exponential (eAT) If A is a linear matrix in f (x).

Or

One can also say that if f (x) results in a closed form expression for its Taylor expansion. Then nth derivative can be written. My question is that expression can be written for nonlinear systems if they come to have a closed form expression in their Taylor expansion.

Of course if your function is analytic, there is also Cauchy's integral formula, though I don't know if it qualifies for a "closed-form" expression (it is often pretty useful).

6
How can I find the relation between observer gain and time delay?

because at starting the convergence is bigger so i want to solve this problem by find relation between time delay and observer gain?

THanks @Philippe for replayed .

5
Is there any approach towards finding all infinite solutions of a set of nonlinear equations when the number of unknowns is more than equations?

We know that when the number of nonlinear equations is the same as unknowns, we can simply find the solutions by inserting different starting points. I have 3 nonlinear equations with 4 unknowns, with some bound constraints. How can I see if there is a solution to the problem?

A standard method would be to investigate the system of nonlinear equations for the existence of symmetry.

3
It is true that a homogeneous function of degree k has topological degree less than or equal to k?
Let $F:R^2\to R^2$ be a homogeneous function of degree $k\in Z_+$ (i.e., $F(tx)=t^kF(x)$, t>0) such that $F^{-1}(0) = 0$. What conditions should we impose that F has topological degree less than or equal to k?
This is true if F is homogeneous polynomial!
An example that fails is $F(r\,\cos(\theta),r\,\sin(\theta))=(r\cos(n\theta),r\,\sin(n\theta))$!

To expand on @Jost Eschenburg answer, to see that F is a homogeneous polynomial if F is smooth and homogeneous of degree k, write

F(x) =  F(tx)/t^k

= t^{-k} F_0 +  t^{-k+1}F_1(x)  +  ... tF_{k-1}(x) + F_k(x) +  t^{-k}R(tx)

where the F_i are the homogenous polynomials of the Taylor expansion. The lefthand side is independent of t and lim_{t \to 0} t^{-k}R(tx) = 0. It follows that all the terms in the Taylor expansion $F_i$ for $i < k$  are zero and

F(x) = lim_{t \to 0} (F_k(x) + t^{-k}R(tx)) = F_k(x).

1
The attach is an output signal of second-order nonlinear system and corresponding input is envelope of vibration singal of bearing.
• The output signal is enhanced by stochastic resonance in nonlinear systems or it is unrelated to input singal ? Why is it a modualted or sine-like signal?

Yours sincerely

J-F Antoine

3
What is the best tool/method for optimization that involves multiple nonlinear variables?

I have been reading about optimization for steel trusses. However, anyone experienced in the actual process and would not mind to discuss more with me? Is it true to advancement of FEM has made design of steel trusses a lot easier (where structural engineers can easily play around with different sections) that causes the older optimization method being forgotten?

Hi Patrick, there are many ways for analysis of truss structures. The most common method is to use the Finite Element method, simply because it is accurate enough and formulation and analysis of truss problems with the finite element method are very straight forward. About truss optimization, when the problem is indeterminate any change in elements stiffness results in redistribution of loads in elements. As you mentioned, the basic design method for this case is to change elements, perform the analysis (FEM), obtain the elements loads/stresses and nodal displacements and check the design criteria. If you are thinking of using more sophisticated mathematical optimization methods, you can find a lot of them in the literature. The following paper and MATLAB codes might help:

https://www.researchgate.net/publication/260155936_Design_of_space_trusses_using_modified_teachinglearning_based_optimization

http://www.mathworks.com/matlabcentral/fileexchange/51202-teaching-learning-based-algorithm-for-truss-optimization

http://www.mathworks.com/matlabcentral/fileexchange/51250-truss-optimization-with-matlab-genetic-algorithm--ga--function

• Source
##### Article: Design of space trusses using modified teaching–learning based optimization
[Hide abstract]
ABSTRACT: A modified teaching–learning-based optimization (TLBO) algorithm is applied to fixed geometry space trusses with discrete and continuous design variables. Designs generated by the modified TLBO algorithm are compared with other popular evolutionary optimization methods. In all cases, the objective function is the total weight of the structure subjected to strength and displacement limitations. Designs are evaluated for fitness based on their penalized structural weight, which represents the actual truss weight and the degree to which the design constraints are violated. TLBO is conceptually modeled on the two types of pedagogy within a classroom: class-level learning from a teacher and individual learning between students. TLBO uses a relatively simple algorithm with no intrinsic parameters controlling its performance and can easily handle a mixture of both continuous and discrete design variables. Without introducing any additional algorithmic parameters, the modified TLBO algorithm uses a fitness-based weighted mean in the teaching phase and a refined student updating process. The computational performance of TLBO designs for several benchmark space truss structures is presented and compared with classical and evolutionary optimization methods. Optimization results indicate that the modified TLBO algorithm can generate improved designs when compared to other population-based techniques and in some cases improve the overall computational efficiency.
Full-text · Article · Mar 2014 · Engineering Structures
5
How to prove the stability of nonlinear systems, in general ?

From that I want to prove the stability of the reduced system that I obtained by project the original system using state transformation

We can plot the waveform of the output signal for a known input signal in time domain and see whether it is of finite duration or infinite duration.

• Yin Chao asked a question:
Open
Are there any models with linear dynamic equation and nonlinear measurement equation which are not the BOT model or the CT model?

Could Any one help me? thanks!

x(k) = F* x(k-1) + v(k-1)

y(k) = h(x(k)) + w(k)

NOTE:

BOT stands for the Bearings Only Tracking, in which the dynamic eq. is a Constant velocity(CV) and measurement eq. is atan2(y-y0, x-x0).

CT stands for the Constant Turn rate.

5
How do I identify a reduced order system from the response of a higher one?

Dear all,

I work on a nonlinear system after linearization we get a higer order one we want to find a reduced order system that approximate rather accurately the original one. I need your help

Hi

I advice you to see this document. You will find what you need.

I hope that I helped you, let us know if you have another questions or you need more details.

With best regards

4
Is there any formal way to represent a nonlinear infinite dimensional systems ?

A formal representation of systems governed by ODEs is:

dot{x} = f(x,u)

y        = g(x)

Is there any formal way to define a nonlinear system with PDEs?

A good starting point is the excellent textbook "Partial Differential Equations" by L.C. Evans. Part III of the book is devoted to nonlinear pde's.

@book{evans2010partial,
title={Partial Differential Equations},
author={Evans, L.C.},
isbn={9780821849743},
lccn={2009044716},
year={2010},
publisher={American Mathematical Society}
}

1
How can identifying multiple model nonlinear systems?

Multiple Model Adaptive Control With Mixing

Matthew Kuipers and Petros Ioannou, Fellow, IEEE

We do so using symbolic regression, in particular genetic programming. See our paper list. I can provide a preprint at the end of this week.

3
How do I identify a reduced order system from the response of a higher one?

Dear all,

I work on a nonlinear system after linearization we get a higer order one we want to find a reduced order system that approximate rather accurately the original one. I need your help

Thanks-the paper, however, doesn't seem related to the question. Indeed, given the equations (1)-(6), why not simply solve them, numerically, and see what the results look like? The analysis in the paper doesn't seem very clear on what its objective is.

6
How can I find equilibrium point of n particle autonomous system using MATLAB where each particle's dynamics is influenced by other?

I have a n particle system in the consideration which follows certain dynamics such that dynamics of particle Influenced bu particle i+1. Further they are non linear system. In such scenario, how can I find equilibrium condition? In other words I am looking for a method to find equilibrium points for non linear autonomous system.

Edit: As asked some of the members who replied to the thread, equations of the system looks like this: x(j) = d*(  x'(j+1) - x'(j)  ) / (  x(j+1)  - x(j) ) + e*( f (  x(j+1) - x(j) )  - x'(j) ) where f(x) can be assumed to sigmoid function for any function that has similar characteristics. d and e are some constant parameters.

Edit2: x'(j) is the derivative of x(j).

2
What is the most versatile approach for predicting uncertainties in guidance and tracking of generic nonlinear systems, i.e. x_dot = f(x,u)?

What is the most versatile approach for predicting uncertainties in guidance and tracking of generic nonlinear systems, i.e. x_dot = f(x,u)? For example, estimation of uncertainty ensembles, modeling randomness and (un)stationarity.

Thank you

Turgay

Hi,

A potential approach is the use of extended Kalman filter. In practice you will linearize the model equations and propagate the uncertainty. If the EKF is not working an approach can be to use the unscented Kalman filter which will increase the computational complexity but has been shown to give satisfying results. Some of our past work was in this direction.

Kind regards

11
Why do ordinary differential equation (ODE) models of cancer suggest different behaviors for cancer cells?

For validation part of my study, I need a comparison between my model of ductal carcinoma in situ (DCIS) and ODE models of this area. But, I’m really confused because ordinary differential equation (ODE) models of cancer have suggested different behaviors for tumor and immune cell populations. For example, the below behaviors are reported by the survey of Eftimie et al. (2011) [1]:

• Tumor size decreases exponentially through interactions with the immune cells.
• Tumor size decreases at first. Then, the decay of immune cells leads to an exponentially increase in it again.
• Tumor size decays in an oscillatory manner.
• Tumor size grows in an oscillatory manner.

I don’t understand the reason of the difference! And, I don’t know which behavior is right. Could anyone possibly help me, please?

------------------------------

[1] Eftimie, Raluca, Jonathan L. Bramson, and David JD Earn. "Interactions between the immune system and cancer: a brief review of non-spatial mathematical models." Bulletin of Mathematical Biology 73.1 (2011): 2-32.

I agree with Joseph Malinzi. It might depend on the chosen parameters to obtain a different behavior of course, and certainly you can use a logistic or Gompertz growth term in your model equation, but as Dr. Malinzi says, this has nothing to do with the interaction with the T cells.

7
What should I do if my Fuzzy sliding mode controller does not provide stability to the system?

Fuzzy sliding mode controller does not provide stability to the nonlinear system. how we can solve it, if SMC be stable.

I definitely disagree with the statement of Monica Patrascu "fuzzy never guarantees stability". A countless number of  papers has been published especially on "Sliding mode fuzzy control" or "Fuzzy sliding mode control" (see Palm 1992,1994) and on TS-fuzzy control systems (see K.Tanaka, 1992). The reason for instability of a fuzzy SMC can be manifold as it also could be for a classical one. Sources of instability could be a wrong choice of the output scaling factor or the neglection of  the finite sample time of the system.

Another thing is when the fuzzy controller is a little expert system  describing indirectly  the system to be controlled (e.g. a chemical process) and the operator's expert knowledge at the same time. In this case a stability analysis is quite difficult to accomplish and an according guarantee for stability is hard to be given.