# Nonlinear Systems

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.

Eric Pigeon · Université de Caen Basse-Normandie

I think the same things that Nadege Kabache

How could I make a MATLAB code to design a PID controller?

I'm trying to design a PID controller to control a nonlinear system, I finished all the dynamics and error (e) equations, P, I, D; but I couldn't find some useful tips, to start writing my code.

One could found some ready code from internet, but I want to understand how one could be able to make such things; since I'm a bit new to this area!!

Any one can suggest some tutorials, or informative resources?

Hazim Hashim Tahir · Ministry of Science and Technology, Iraq

Well done عفية عليك يا امين; practice makes perfect.

How we can to reduce the output oscillation magnitude in Fuzzy sliding mode controller?

One of the important challenge in intelligent nonlinear control is overshoot. How we can to reduce/eliminate it?

Farzin Piltan · IRANIAN CENTER OF ADVANCED SCIENCE AND TECHNOLOGY

I have about 50 papers related to sliding mode topics, you can follow my research and please don't any hesitate if you have any question.

Regarding to above question if fuzzy type control be PID I think we also can to reduce the output gain updating factor's of PD parts.

Does anyone have experience with an extended kalman filter without process noise?

I am currently working on an extended kalman filter for a nonlinear system.

And I am very confused.

Assume there is a nonlinear system without process noise but with measurement noise. Then when using EKF to estimate the state of this system, how can I handle Q in EKF.

For example:

x(k+1)= x(k) ; or x(k+1)= x(k) + u(k)

and observer equation
y(k)=x²(k) + v(k)

u(k) is the input and v(k) is the measurement noise.

When EKF is used for the estimate of the state, should I set Q=0?

Jihong Yu · Université Paris-Sud 11

Hi all, thanks very much for your help.

I got the mean of Q.  There is no relationship between process noize and Q in the nonlinear systems. It is in the linear systems where Q is the covariance of process noise.

Is linear analysis in controller design tool (in MATLAB simulink) applicable to unstable systems?

Hi every body,

I have a nonlinear system model in simulink. This system is unstable and I want to design a controller to stabilize it. Due to this, I want to linearize this model and then design a stabilizer controller. To do this, I use the linearization ability of simulnk, but Is linear analysis in controller design (in MATLAB simulnk) applicable to unstable systems ?

Afef Fekih · University of Louisiana at Lafayette

Yes, it's definitely applicable.

Solve y^3 + y''' = 0?
Can anyone suggest an analytical method to solve y^3 + y''' = 0? The solution should satisfy the boundary conditions y tends to plus or minus one as x tends to plus or minus infinity. The solution should also be bounded everywhere. The tanh method doesn't work, but perhaps some variant of it does.

Hello everybody---

Thanks for all of your help with this ODE. Your input showed that the correct ODE to look at was somewhat different than the one I asked about here. The paper that came out of this line of research is attached. Hopefully you'll find it interesting.

Best regards,

How can I solve a^2 y = y^3 - y''' - y'?
Can anyone suggest an analytical method to solve the ODE a^2 y = y^3 - y''' - y', where a is a positive constant? The solution should have y(0) = 0. Also y -> a and y' -> 0 as x -> infinity.

Hello everybody---

You can find the published version of our paper

How can you solve NMPC objective functions with nonlinear constraint in MATLAB?
NMPC- Nonlinear model predictive control X_dot=f(x,u) Y=C*x objective function: min J = (Y-Ys)^2+du^2+u^2 w.r.t u constraints are : 0
Himour Yassine · Université de Khemis Miliana

Check this funciton "nmpc.m" and examples of use...

http://numerik.mathematik.uni-bayreuth.de/~lgruene/nmpc-book/matlab_nmpc.html

What orthogonal polynomials best fit the Volterra-Wiener nonlinear series?

To reduce the total number of parameters estimated in nonlinear system identification using Volterra-Wiener functions, laguerre polynomials are often used. Is this orthogonal set of functions the optimal choice for parameter compression or are there any better orthogonal functions for this task?

Simone Orcioni · Università Politecnica delle Marche

Polynomial expansions lead to block-structured models, as shown in Fig. 1 of [1]. In [1] you can also find a comparison between LET (Laguerre expansion technique) and FOA (fast orthogonal algorithm). You can see also [2] about Laguerre expansion.

These models (so as Volterrra series) can lead to a matrix formulation that is linear in the coefficients and that can be solved with LMS or RLS methods. In the limit of the approximation given by the expansion, these estimation methods (LMS,RLS) give good results, mostly if you have a reduced number of parameter to be estimated.

But an issue that can affect these estimation methods is the locality.

In simple terms, if you choose an input with high variance you stimulate well high order nonlinearity, resulting in well estimated high order coefficients and pourly estimated lower order (linear part included). The opposite if you use a low energy input.

This problem also affect orthogonal methods like that of Lee-Schetzen, that does not approximate the kernels but have a greater number of coefficients with respect of block-structured methods.

In [3] you can find a proposal for the mitigation of the problem of locality.

The main obstacle to the use of an orthogonal method like that of Lee-Schetzen or that proposed in [3] is given by the system in study. If you can give in input to your system a white noise, the preferred choice is an orthogonal methods, like [3]. If you have a limited and non gaussian inputs for your system, I would choose FOA.

[1] David T. Westwick, Bela Suki, and Kenneth R. Lutchen, “Sensitivity analysis of kernel estimates: Implications in nonlinear physiological system identification,” Annals of Biomedical Engineering, vol. 26, pp. 488–501, 1998.

[2] Vasilis Z. Marmarelis, “Modeling methodology for nonlinear physiological systems,” Annals of Biomedical Engineering, vol. 25, pp. 239–251, 1997.

[3] Orcioni, Simone, "Improving the approximation ability of Volterra series identified with a cross-correlation method", Nonlinear Dynamics, Volume 78, Issue 4, pp 2861-2869, December 2014.

To control any nonlinear system; what is the difference between fuzzy sliding mode controller and sliding mode fuzzy controller?

To design a controller for nonlinear and uncertain systems we have two choice: the first one is fuzzy sliding mode controller and the second one is sliding mode fuzzy controller. I'd like to know that which one is better and why?

Rainer Palm · Örebro universitet

Before I wrote my paper "Slinding mode fuzzy control" I was wondering why a diagonal Mamdani  FC with rules like  IF error=PB and change of error=NS THEN u=NB  work so well and robustly with regard to uncertainties even for a nonlinear system whose parameters are not completely known in advance. The reason is that this kind of diagonal FC works as an SMC with a nonlinear boundary layer.  However the main point is to choose the input scaling factors for error and change of error and the output gain accordingly so that stability, performance and robustness are guaranteed. In the following this SMFC was enhanced by the equivalent control which serves as a kind of compensation term for  well known parts of the system.

In my opinion, now a  controller is called  "FSMC"  for all fuzzy controllers with a sliding mode term included which is also the case for the SMFC. So I can't see a significant difference between the two.

What is the relationship between Moving Horizon Estimation (MHE) and EKF?

Now I want to solve a navigation problem using both MHE and EKF. I found that MHE has a better performance than EKF for highly nonlinear system. Can anyone tell me the reason? What is the relationship between these for estimation method? Are they identical to each other under certain assumptions?

Flávio Eler De Melo · University of Liverpool

The MHE arises as a maximum a posteriori (MAP) probability estimator (maximisation of the log posterior probability density), whereas the EKF is a suboptimal solution in the minimum mean square error (MMSE) sense. If the state and measurement processes are linear Gaussian without constraints, the "one-step" solutions coincide.

The MHE can tackle highly nonlinear problems where the state and uncertainty may have constraints, but at a higher computational cost and some practical difficulties such as approximating the arrival cost. The EKF is computationally simple but it requires both the system nonlinearities to be mild and the variances of the noises to be small (in order to keep the linearisation valid).

Depending on the nature of the system, I would rather consider either the UKF or sequential Monte-Carlo samplers (e.g., particle filters).

What is wave-number in Rayleigh Benard instability?

Hi all.

I have just begun with the field of instability and going through texts of Francois Charru.

While going through Rayleigh Benard thermal instability (where, to best of what I have understood, the condition when due to temperature difference the advection will be prominent i.e. movement to fluid will start) I came across the following attached plot. I was unable to get what the wave number represents here?

Like in Taylor stability, it represents the wavelength of the perturbation, but what does k stand for here?

If it stands for perturbation wavelength, then what is perturbation here?

Is it different diameter particles/fluid elements exposed to heat surface?

Kheng Ka Tan · HELP University College

Dear Deewakar,

Pl try to read Lord Rayleigh's 1916 original paper, a classic available free on internet, and also Chandrashekar's book for  more modern treatment, then go back to the definition for the dimensionless wave number and Lord Rayleigh's original derivation, you will have a better understanding of its mathematical and true physical sense.

kk

L. Rayleigh, On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Phil. Mag., 32:529–46, 1916.

Subrahmanyan Chandrasekhar (1982). Hydrodynamic and Hydromagnetic Stability (Dover). ISBN 0-486-64071-X

P.G. Drazin and W.H. Reid (2004). Hydrodynamic Stability, second edition (Cambridge University Press).

How can we define Capabilities of system identification for nonlinear systems?
What does it mean when a nonlinear system can be identified?
What terms do we need to identify a nonlinear system?
Dmitry Kovriguine · Russian Academy of Sciences. Industrial Eng. Institute. Nizhniy Novgorod

Geysers behave periodically, and not almost such, and not so such, - a simple model relates to a linear piecewise point maps with unknown parameters. Let these are identified from time series, then one can set a billboard with a sketch of given dynamical system as Poincare maps, which can be understood almost for everyone, say, in Yellowstone, to show advances in science, in prediction of picturesque beauties in the Nature.

What are the methods for checking stability of zero dynamics of a nonlinear system?

Input-Output feedback linearization method for a nonlinear system requires the check of stability of zero dynamics or internal dynamics.

Zigang Pan · N/A

Dr. Castanos's response is on the mark.  I just have one more suggestion.  After you obtain the zero dynamics, you can apply Lyapunov function analysis, linear approximation to study the local stability of the zero dynamics.  A more powerful local stability analysis tool for nonlinear systems is "center manifold analysis", which you can find in Prof. Khalil's book "Nonlinear Systems".  It is like the linearization except it can deal with cases where the linearization method is indeterminant.

How can we design FPGA-Based sliding mode controller?

One of the important problem to design and implement Sliding Mode controller is dynamical based FPGA-part, so what's your idea about it?

Any ideas about the stability of two solutions in a nonlinear system?
Suppose a non linear system has at least 2 positive periodic solutions in a bounded domain. If one can show that the system is globally asymptotically stable, then is it a contradiction to the previous statement? If not, what can one say about the other solution? Can anybody suggest me any relevant references?
Mohammed Lamine Moussaoui · University of Science and Technology Houari Boumediene

Dear Santanu Biswas,

To be Convergent the Solution has to be Stable and Consistent (see lax theorem). The choice of the Discretization Step(s) defines a Domain of Stability this gives you several solutions for each choice. But the solution must be unique as it is stated by Fletcher in his book.

What are the recent advances, issues to be addressed, and scope of research in the area of stability analysis of multi-agent control systems?
What are the recent advances, issues to be addressed, and scope of research in the area of stability analysis of multi-agent control systems during intermittent and/or permanent sensor faults or loss of observation?
Please shed some light on this area.
In need of help.
Jose Gonzalez de Durana · University of the Basque Country (Spain)

First of all I would like to know what "stability analysis of multi-agent control systems" means (I do not know any study on it).

What is the recent iteration method to solve a nonlinear system of equation?
Using some iteration method, we can solve a nonlinear system of equation. I know lots of old methods like, Newton Raphson, Gauss newton, marquardt, levenberg marquardt etc. I want to know about some upgraded methods.
James Lundeen · Illinois State University

Marquardt Levenberg paper published in 1961. At that time, IBM 360 memory was very limited. CPU speed slow too compared with today. That method should be put in a museum where it belongs. To use it, you must guess your final answers. This leads to non-linear least squares regression settling on local rather than global minima. Finite element analysis assures global minima least squares best fit and does not use initial guesses of anything.

Is there a Hamiltonian model for the perception of music?
The musical experience appears to contain a gap in our understanding: We have strong physical theory for the mechanics of vibration, musical theory for a range of compositional or improvisational styles, and some understanding of the processes in the brain that are stimulated by music. However, I am not aware of the mechanism by which these processes converge to the interpretation by which "music has charms to soothe a savage breast." I am wondering if the physical processes that we understand have some sort of mental mirror that "resonates" (pardon the pun) within us. Have any of you studied this?
Federico Zertuche · Universidad Nacional Autónoma de México

Sorry that's a stupid question

How to test the Observability of "Discrete time" Nonlinear system?

Is it related to the rank of the Jacobian matrix? I know in continuous time this is true, but I am not sure about discrete time. Any references and links are welcome.

Qudrat Khan · COMSATS Institute of Information Technology

The reachability and observability is a very famous in discrete time system. You can find your answer in the book "Linear System Theory" by Wilson J. Rugh, Second edition p. 462-472.   Having studied this chapter you will find your answer.

Is it possible to have a chaotic system with one equilibrium in which the real part of all the eigenvalues are positive?
I mean chaotic flows. That is possible for chaotic maps.
Julien Clinton Sprott · University of Wisconsin–Madison

Such systems exist, but they are rare. I have found such an example in a 3-D system of ODEs with quadratic nonlinearities, and a publication describing it is in preparation.

How is robustness of a nonlinear controller based on contraction theory and incremental stability approach? Are there any physical experiments?
I am working on the first draft of my PhD proposal about a class of nonlinear multi-physics systems. I would like to know if there are real experiments with control schemes based on contraction theory and incremental stability approach.

Hi,

2) http://web.mit.edu/nsl/www/preprints/stochastic_contraction09.pdf

or a simple pendulum will do.

• What are lyapunov based disturbance rejection approaches for nonlinear systems?

Just tell me their names to search on the web.

I have searched but results were so confusing because some of the results were for linear systems.

Jorge Luis Barahona-Avalos · Universidad Tecnológica de la Mixteca

Hi Salman:

Lyapunov's approach to deal with the disturbance rejection in nonlinear systems is not the only way to counteract the adverse effects which could cause such disruption in the stability and / or system performance.

I recommend that you will revise the work done several years ago by Dr. Gao and his colleagues, whose area of ​​research is related to the active disturbance rejection control (ADRC).

Either way, I append to this comment a related article with what you are investigating, hoping you find it useful. I'll also add some publications on active disturbance rejection which can be found here on ResearchGate.

Best regards from Oaxaca, Mexico

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.
Nikos Lazarides · University of Crete

In case you are still interested in a simple example of a 2D flow with more than one limit cycles, you could have a look in the paper in:

http://arxiv.org/pdf/1001.3240v1.pdf

the modified van der Pol oscillator eq. (1) has from one to three limit cycles; in the latter case, two of them are stable and one unstable!

Regards,

Nikos

Is there an existing work concerning the (global) controllabity of nonlinear systems at most affine in state (dx/dt=A(t,u)x+B(t,u))?

But non linear with respect to the control?

Fotsa Mbogne David Jaures · University of Ngaoundéré

I encounter some difficulties in the exploitation of  the attached documents. Is it possible to have more precise (english) references? Thank you all.

How can I find the integrals directly from Lie point symmetries in ODEs?
From the vector fields, we can find the invariants by solving the characteristic equation associated with the vector field. Whether this invariant is exactly same as the first integral admitted by the given ODE?
Artur Sergyeyev · Silesian University in Opava

If your ODE is Lagrangian, then under certain conditions you can construct integrals from the symmetries using the Noether theorem.

On the other hand, you can construct first integrals for your ODE by solving the characteristic equation associated with the characteristic vector field of this ODE rather than the characteristic equation associated with the vector fields of the Lie point symmetries of your ODE.

What is the difference between iSMC and Fuzzy SMC?

I'd like to design iSMC for a nonlinear system such as robot manipulator but I'd like to know about i-SMC.

A.-J. Muñoz-Vazquez · Center for Research and Advanced Studies of the National Polytechnic Institute

Prof Tiwari,

I know the integral SMC, and I kwnow that it is robust to bounded disturbances, BUT this robustness is not provided by the reaching phase elimination, instead the reaching-phase avoidance is provided by the robustness, and not the converse. The robustness is associated with the permissible gain in the nonlinear control part, and it is a direct consequence that this nonlinear term appears in the derivative of an auxiliary sliding manifold, then the equivalent controller is equal to the inverse additive of the disturbance effects.

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?
Wondimu W Teka · Indiana University-Purdue University Indianapolis

Hi Javid,

Fractional calculus has been observed on neuronal spiking activities. For example, check the following papers.

I need to solve a numerically stochastic differential equation with additive colored noise.
Mahmood Dadkhah · Payame Noor University

Hi

Visit the:

1. "http://www.photonics.umbc.edu/Menyuk/Phase-Noise/kasdin_ProcIEEE_950501n.pdf"

2. search the avaxhome.ws site for books.

regards

• Does anyone know some conservative chaotic systems (3-D flows)?
I know Sprott-A (Nose-Hoover system), and a system Heidel and Zhang investigated in “Nonchaotic and chaotic Behavior in Three-Dimensional Quadratic Systems: Five-One Conservative Cases” (which I think originally was reported by Sprott in 2000). Does anybody know another 3-D chaotic conservative flow?
Sundarapandian Vaidyanathan · Vel Tech - Technical University

There is a chapter titled "Chaos in Conservative Systems" by M. Lakshmanan et al. in their Springer book "Nonlinear Dynamics" (2003). In the introduction of this Chapter, they refer to three systems as examples of conservative chaotic flows:

(1) Henon-Heiles system (1964)

(2)  Conservative Duffing Oscillator under periodic forcing

(3) Standard Map (Discrete Chaos)

As system (4), some friends here already mentioned "Sprott-A system" (1994), which is also known as "Nose-Hoover system".

As the Lyapunov exponents are ordered in non-increasing order, for a 3-D conservative chaotic system, we must have L1 + L2 + L3 = 0. As L2 = 0, it follows that L3 = - L1.

My interest is about 4-D conservative chaotic systems. The Lyapunov exponents of such systems have the nice property that L2 = L3 = 0 and L4 = - L1. I am currently working on finding 4-D conservative chaotic systems. (It is not possible for hyperchaos in 4-D conservative chaotic systems).

Regards,

Sundar