# Nonlinear Systems

5
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.

• Yin Chao asked a question:
New
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).

27
How can I calculate the Lyapunov exponent?

In Mathematics the Lyapunov exponent of a dynamical systems is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Is there some new algorithms  for calculate the Lyapunov exponent?

Yes, this what I find in some publications. They declare that the system is chaotic from the scrambled graph.

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.

10
Is complex network truly nonlinear?
Personally, I take complex network (nodes can be dynamical systems) as a nonlinear system. But seems the work in this field are using linear methods extensively. From community structure detection algorithm to master stability method, they can all be boiled down to linear algebra (spectrum theory, more specifically).
So, is complex network a nonlinear system, or just a complex linear system?

Essentially non-linearity means interaction. Then dynamics of some system will depend on this interaction, e.g., dx/dt=a x*y for systems x and y. Now suppose that one of these variables changes very little. Then we can consider it as constant. In this case we arrive at the linear system. In short non-linearity is the generalization of the linear  dynamics for the real world, for the world of interaction. Complex systems are nonlinear, but  this does not mean such systems can not be analysed with the techniques from the well understood linear systems theory under some approximation.Any modelling is always implies some approximation, that is  it is really simpler than the real world.

3
What could be the best possible control strategy for a 2 I/P 3O/P nonlinear system?

I have a nonlinear level control system with 2 I/Ps (flow) and 3 O/Ps (tank levels). All the Inputs and outputs are coupled in such a way that they almost equally affect all the 3 outputs i.e the relative gain array matrix has all the elements of nearly same values. How the control is achieved when the each of three outputs has different time varying reference signals (set points).

You could achieved, the required control by using readymade microprocessor kit(with two 8255, ADC and DAC)by needed programing. Refer my papers. After achieving reliability, could go for, needed readymade microcontroller, would be cheaper, reliable and accurate..

2
For a mixed nonlinear distribution, what is the efficient sampling method?

We have a distribution function which is neither discrete nor continuous, but both.  The continuous portion is nonlinear as well.

What would be an efficient method of sampling from such function, table lookup, rejection methods or ....?

What kind of research is it?

13
What is the easiest way to determine if a parameter is identifiable in a nonlinear model?

Imagine we have a nonlinear model of the following form:

Xdot = F(x,p,u)

Y= h(x,p,u)

where x is the system states, p are system parameters, u is the system input, Y is the system output and f(.) and h(.) are nonlinear mapping process and measurement functions.

Can anyone help me with what would be the easiest way to figure out if a parameter in p set is identifiable from the outputs of the system being measured. I am looking for the fastest and easiest way possible with possible proof available.

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.

7
Is there any complete non-metrizable semi-linear uniform space?

See the attached file.

Thank you Dr. Suad. Yes you are write. we should take unionwith delta.

7
How can I determine whether a given point (in the phase space) lies in the basin of attraction of an attractor?
1. For a nonlinear ODE systems, dx/dt=f(x), if one trajectory gets as close as possible to an attractor, i.e., |x(T)-x*|<eps (where x(T) is the value of the solution x(t) at the terminal time T for a given initial value x0, x* is the state vector of an attractor, and eps is a very small number), does it imply that x0  lies in the basin of attraction of an attractor?
2. Although the definition of basin of attraction is as follows. For each such attractor, its basin of attraction is the set of initial conditions leading to long-time behavior that approaches that attractor. However, to my knowledge, nonlinear systems often display different types of attractors, such as fixed points, periodic attractors, and strange attractors. Except for the basin boundary with a smooth curve, there exist some fractal basin boundaries and riddled basins of attraction. As I understand them, for these basins with complex boundaries, if |x(T)-x*|<eps, the point x0 may not lie in the basin of attraction of that attractor. Is it true?
3. Are there some theories or conditions that can guarantee one point definitely lies in the basin of attraction for any different type of attractors?

I have only considered what might be called elementary systems and strange attractors have not arisen. I use the separatrices to determine the basin of attraction. If the system has several critical values, often there are separatrices which are determined essentially by the saddle points. These separatrices divide the phase plane into distinct regions of behavior of the system. In this sense, they become the boundary of the basin of attraction for the appropriate critical point. If this comment is to elementary or off the mark, please excuse it.

8
What are good books for control of linear and nonlinear systems?

Dear all,

I would like to study regarding control of linear and nonlinear systems in detail. So, please suggest me some books which can provide in-depth knowledge regarding it. Thank you.

Here is a valuable edition to the collection already presented by our colleagues:

S. Boyd, L. Ghaoui, E. Feron, and V. Balakishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, ISBN 0-89871-334-X (1994).

There may exist an updated edition, book displays a very rich bibliography. The book shows strong connections of the subject matter with optimization, dynamical systems as well as the classical themes (Lyapunov theory).

4
What are the current research tends in the control of nonlinear systems?
Theses days, I am doing literature review about the new trends in control of nonlinear systems, but it looks like this is a very broad subject. What are the new research areas in the control of such systems?
4
Can anyone help with a convergence and initial condition problem (non linear solver)?

Actually I have a problem is running the numerical simulation.The solution did not converges and at the end give an error that the initial conditions are inconsistent. It will be highly appreciated if someone help me in this regard. I have spend weeks but didn't solved this error. Attached is the COMSOL file.

Best Regards

Sikandar Khan

Thanks all for your kind replies. It help me a lot in solving my problems.

regards

Sikandar

37
If you have a time-dependent parabolic equation what is the best method to solve it other than finite difference method?
The equation is u_t=a(t)u_xx+b(t)u_x+c(t)u+f(x,t) with initial condition and Direichet boundary conditions as follows: u(x,0)=u_0(x) and u(0,t)=u_1(t), u(1,t)=u_2(t).

Dear all thank you so much for all the answers that really great feedback, and I will take a huge benefit.

6
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

3
Does the conjugacy condition of linear conjugate gradient methods still holds for nonlinear conjugate gradient methods?

For linear CG method the search directions must satisfy the conjugacy condition

d_i'Hd_j=0, i not equal to j.

It is a fact that for a nonlinear CG method the above equation does not hold, since the Hessian changes at different iterations.

With this in mind, I am asking whether there is any modified conjugacy condition for the directions of nonlinear CG methods. If there is no such condition, then why are we using  the word  "conjugate" for nonlinear CG  methods.

No, up to now there is no method with modified conjugacy condition.

The name comes from the fact that the iterations of these methods (linear vs non-linear) are formally identical. In each iteration, the new search direction of the non-linear method is conjugate to the old search direction.

7
Is there any 2D system with a limit cycle and without any equilibrium?
Consider a general 2D system:
x' = f(x,y)
y’ = g(x,y)
Do you know any such system which has a limit cycle while there is no equilibrium in it? For example the famous Van Der Pol is not an answer; Because, although it has a limit cycle, it has an equilibrium in the origin (0,0)
x' = y
y’ = ay(1-x^2)-x

Dear Dr. Taghavi,

Regards,

8
Is there any 2D system (flow) with more than one limit cycle?
Consider a general 2D system:
x' = f(x,y)
y’ = g(x,y)
Do you know any such system (preferably a simple one, ideally quadratic) which has more than one limit cycle? I would prefer it if there was at most one unstable equilibrium.

The  method of construction of a quadratic system with 4 limit cycles is   based on the following:

They pick a quadratic sytem with 2 hyperbolic limit cycles with disjoint interior. Existence of these 2 limit cycles  is  a consequence of Poincare-Bendixson theorem. One of thses 2 limit cycles suround a weak focus of order 2. So after perturbation we obtain two  additional  (smal) limit cycles. So we have  4 limit cycles with (3,1) distribution

It is unknown whether there is  a quadratic  system with more than 4 limit  cycle. It is also unknown that whether there is a (2,2) distribution of limit cycles for quadratic systems.

27
How can I evaluate a bifurcation diagram?

In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied.  A bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a bifurcation parameter in the system.   How to evaluate a bifurcation diagram ?

Thanks for all giving valuable answers and suggestions

15
How might I determine a Lyapunov Function and the corresponding region of attraction for a nonlinear system??

When designing a Lyapunov-type controller for a nonlinear system, how should the region of attraction be determined?

Is there any general approach to determine the Lyapunov function with respect to system dynamics?

I was wondering if someone could help me.

There is no  generally accepted approach. However some methods  exist, you can find them in D.R. Merkin, Introduction to the Theory of Stability, Springer-Verlag, 1997. N.G. Chetayev, The Stability of Motion, Pergamon Press, 1961.

5
Does a chimera exist only in a network of coupled nonlinear systems and not in single nonlinear system?

When I see the literature about the chimera state, it exists only in the network of coupled nonlinear oscillator. They define the chimera such as the existence of coherent state and in-coherent state simultaneously. My question is that, if an attractor of a system has both the coherent and incoherent state, should we call it as chimera state? Like some portion of the attractor is chaotic and rest becomes regular.

Dear All, This is (see figure) what I  was talking about. You can see some portion of the attractor have regular structure and remaining of the become chaotic.

15
What are the state of art of the estimation methods?

I know that the title of question covers a wide area. Currently I am trying to apply Kalman Filter to estimate some parameters of a highly nonlinear system. I took a look at the papers and articles, but I could not see any obvious advances.

If anyone can recommend me a book or a recent review article which includes and explains the state of art estimation methods, I will be glad.