Science topics: Mathematical SciencesNonlinear Systems

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# Nonlinear Systems - Science topic

Nonlinear Systems are theoretic and applied results in nonlinear system theory, non-linear models and nonlinear models.

Questions related to Nonlinear Systems

Dear collegues,

I am doing a system modeling by software and I am not sure how to introduce the system and validate the results of the system.

I would like that somebody could conctact me to solve this doubt.

Thank you so much!!

A universal controller with fixed parameters is designed by the derivatives balance method. Controller simultaneously can handle the problems of stabilization, adaptation, reference tracking, and disturbances rejection for both linear and nonlinear systems without controller parameters retuning.

more info about derivatives balance technique https://t.me/universalcontrol

Kindly provide examples with MATLAB command

thanks

I mean chaotic flows. That is possible for chaotic maps.

A set of differential equations are provided which represents the non linear system.How can we convert it into a plant model to develop a controller? There are five state variables.I've shown only two equations in the sample image

My nonlinear model in state space is

xdot= Ax + Bu+ Phi(x), where phi(x) is nonlinearity in the system

I am looking for papers/articles concentrated on indirect MRAC design. Any directions are appreciated.

Since the poisson ratio is negative for auxetic material.. what is the response of auxetic material subject to impact load in the nonlinear system? either impact force will be isolate or amplify or just remain constant with but in different direction.

My goal is to solve the nonlinear system of differential or integral equations in Maple.

I have attached some explanations.

In other words, I mean whether nonlinear systems in Maple can only be solved with 'solve' command.

Thank you

What kind of nonlinear strategies can be utilized as the best controller in the condition that there is no information about the model of the system?

Consider that the model of the nonlinear system is unavailable and there is no information about it, regardless of estimating the model, which means that the model is very complicated and we can not estimate it properly. In the mentioned situation, what kind of controller could have better performance in the same conditions?

For instance, robust controllers such as sliding mode controllers have good performance.

I want to train a nonlinear system with MLP neural network in PyTorch , how can found the optimum value of hyper parameters such as the number of hidden layers and nodes?

Hi

How can i correct this error?? I think it's about matrix dimensions for port e.

Error in default port dimensions function of S-function 'FeedbackLinearization/Controller'. This function does not fully set the dimensions of output port 2

I'm running a simulation based on feedback linearization control method that comes from a paper attached below.

the model is also attached.

Anyone help me, helps a poor student. (if it makes sense lol)

For example, Broyden’s method, Newton iteration method, ... .

I did not write much program with it. Also, I do not have much time.

I have attached some explanations.

Thank you

He is the author of one of the most important works of of the early twenty century in the qualitative theory.

Howdy!

I am in the process of modeling a very complex manufacturing process that involves a mixture (solid & liquid) with mechanical, thermal, optical and chemical reactions happening simultaneously. In top of that, I also need to take into account stick-slip boundary conditions and a free boundary that evolves and eventually encounters an obstacle (a mold).

I already derived a highly nonlinear system of PDEs that models its behavior and I am about to finish the numerical simulations of a linearized version of the aforementioned problem using Matlab only. Trying to solve the general problem with Matlab would be extremely difficult thus I am now in the search of the best software package that can take into account all the previous phenomena. Any suggestions would be greatly appreciated.

Thank you!

The System is found in the article attached. Steps or codes to transform to an LPV system

Suppose we have a lipschitz nonlinear system with a disturbance input. The disturbance is of decaying exponential like nature with an upper bound known.

System: x_dot = f(x) + g(x)u(t) + d(t)

where, x:state; u(t): control input; d(t): Disturbance input

How can we design nonlinear observer for such system?

I am currently working on a project on local nonlinear effects. Comfortably solved the problem using harmonic balance method. Then I move on to write my own code based on Newton Raphson method. Tried various suggestion and nothing sort out. If any one have the algorithm for modal analysis of nonlinear structures using Newton Raphson please share. Most code i found online are for nonlinear modal analysis, having nonlinear eigenvalue and not for nonlinear systems.

Thanks

Of course, a simple nonlinear system can be solved with "solve" in Maple.

But if the system is complex, "solve" cannot be solved.

Thank you

Suppose that I have a system of nonlinear equations F(X) = [f1(X), f2(X), ..., fn(X)]^T where X = [x1, x2, ..., xn]^T and F : R^n to R^n. How to find X as the solutions (maybe multiple) of F(X)=0 using deep learning?

Anyone have the recommendation of references about this problem?

Thank you.

I faced such a problem. I have a nonlinear system for control synthesis and I should compare not only my controllers but also a linear version of my system to describe the legitimacy of this linearization. But it never occurred to me how to compare it in numerical. We often do it in a frequency domain for linear systems comparing bandwidth, gain, or phase margin. And we have a numerical result. But I can't do the Laplace transform (for example) because of no superposition principle being. I heard about nonlinear Fourier transform but I doubt that it could help me

Dear Professor,

I found a very small analytical function. It calculates the same result as Feigenbaum Scenario.

I.e. formula 15 in the article (M. J. Feigenbaum,

*Universal behavior in nonlinear systems*, Physica 7D (1983) 16-39).x(i+1)= 4*lambda*x(i)*{1-x(i)}

So, we will use y = f(x).

It was amazing.

Do you know about this function? If not, and if you need it, I can make a gift as a volunteer.

I am trying to model a system which consists of a second order linear part (for example a set of mass spring damper with variable parameters for spring and damper) and a non-linear part which is a cascade of a derivative, a time delay, a static non-linearity (e.g. half-wave rectifier) and a low-pass system. I want to use neural networks for this purpose. I have tried time delay neural networks which were not successful. I am now trying to use RNNs (Recurrent Neural Networks). Since I am relatively new to this subject, I was wondering which network architecture are suitable for this purpose? (I have to use a limited number of parameters)

There has been rich literature describing how to reformalize stochastic MPC in linear systems into deterministic optimization. However for nonlinear system, propagation of uncertainty through time seems very complicated, so most literature I find solve nonlinear stochastic MPC with probabilistic constraints using sampling based approaches. I am new to nonlinear stochastic MPC, so I am wondering whether there's any existing library/toolbox for nonlinear stochastic MPC with probabilistic constraints? Or if there's no such convenient tools, based on your practical experience, which paper do you recommend me so that I can implement the algorithms myself?

I wonder if we’re using the « contraposition logic method » on symmetries, Noether theorems:

Is it correct to conclude from

**If a system is symmetric then it cannot generate more energy per Noether theorem**then*If a system generates more energy then it is not symmetric (i.e. broken symmetry)*I have the nonlinear systems of Khalil, however some definitions are no so clear, is there a newer book with Matlab examples

i am designing a gain scheduled MPC controller for a non linear system, the design process is described below.

1. linearize the non-linear system at various operating points and obtain the linear models ( say i represent the non linear system with 7 linear models over the entire operating range of the system)

2. Design separate MPC controllers for each linear models

question is

1. can global stability be assured by establishing local stability of each of 7 mpc controllers separately?

2. how an extensive stability analysis of the system can be done?

By dynamical systems, I mean systems that can be modeled by ODEs.

For linear ODEs, we can investigate the stability by eigenvalues, and for nonlinear systems as well as linear systems we can use the Lyapunov stability theory.

I want to know is there any other method to investigate the stability of dynamical systems?

I have a complicated 4th order nonlinear differential equation for a system. I am going to design a nonlinear controller for this system. is it possible to reduce it into a second-order system? how? do I need to reduce the '' model order '' in order to change it into a simpler form?

I didn't find a chapter about the nonlinear system in MRAS authentic books. It would be appreciated if you could help me with this.

Bifurcation diagrams are very useful to evaluate the dynamical behavior of nonlinear dynamical systems. In chaos literature, I notice that some authors draw bifurcation diagrams by removing the first 1000 seconds of data. I like to understand the reason behind this. Any help on this is highly appreciated. Thank you!

I have a nonlinear system, I linearized it using the Jacobian method and got the A, B, C, D matrices for state-space representation. How could I take into account the change of the equilibrium point - i.e. the point where the system is linearized - during a simulation? I tried to solve it by subtracting u0 from the input and adding y0 to the output of the system - please see the attached picture -, but the results are incorrect because the operating point where the system is linearized is changing.

A closed-loop system (with feedback), and "actuator saturation". The poles and zeros of the system change when the system goes into "actuator saturation" behavior?

Thanks a lot for your feedback

Dear friends and colleagues! I have designed a Simulink model of an inverted pendulum control system. In order to study the LQR I have also linearised the model and obtained the K gain matrix for the controller, which works well for the linear system, now I am hoping to apply it to the non-linear system as well. However, I do not quite understand the reference for the control system.

The system has 4 states: position of the cart, pendulum angle and their derivatives. I am modeling a data acquisition system, so I assume I can only have encoders measuring the position and angle, and then I model software derivatives I could perform on a computer. Then I have a block to get u=K1*x1+ K2*x2+K3*x3+K4*x4. That seems to be my input in the system. I, however, assume there must be some comparison block, but I can not seem to wrap my head around it. In the diagram I have attached to this question you can see the reference is set to zero, but I did it merely to have some refernece in general. But what is this zero? What if my desired states are not zero? What if i want to move my reference to some other point during execution pf the program? Which one do I move? If I still want to stabilise the pendulum but not at x=0, but somewhere at x=ref?

a) Can subharmonics ever appear in linear systems?

b) Is it possible for subharmonics to appear in p.e. 2-DOF nonlinear systems? What are the conditions for this to happen? For example, this does not happen in the periodically forced Duffing oscillator (typical nonlinear system).

Thank you in advance!

Hi everyone

I'm involved in designing the optimal controller for a class of nonlinear system. I have designed the controller in the state feedback form (u=-Kx), but there are some constraints in both the input control signals and the state variables. How can I add these constraints on the controllers?! For example, I have a six order nonlinear system with three inputs and these constraints: (x1>0, x2>0, x4>=0, x5>0 and 0<=u1,u2,u3<=1). The controller is based on LQR. Is there anyone here to share a sample m-code with me?

ANN, which has the ability to model nonlinear systems, could be used to predict BW from age.

I am currently working on a model from a circuit that includes diodes and time varying terms. I have the experimental data and I want to find the best parameters to fit the data with my model obtained from physical laws.

Thank you!

could I apply adaptive fuzzy (wang 1993.)for switched nonlinear systems or it needs new challenges?

Conference Paper Stable adaptive fuzzy control of nonlinear systems

What are the possible methods to solve non-linear system of ordinary differential equations of n equations containing n variables?

List possible analytical method through which we could solve the nonlinear system. Also list possible numerical methods as well.

Share any material or article or book or even videos you know worth related about solving nonlinear equations.

Thanks.

Hi guys !

I have a fundamental question regarding the fitness criterion when performing a model estimation for system identification. This far, I have obtained a good model for a driving simulator identification task. In this post, I want to discuss the SISO vertical model. The training data has been obtained by an acceleration sensor which delivers the vertical acceleration signal. Measurements have been performed in a frequency range of 0.2-20Hz. It is obvious that for small frequencies the sensor noise is considerable.

After estimating the (nonlinear) system models, I have calculated the goodnessOfFit using the NRMSE and NMSE focus. I have found that the NMSE yields significantly better fit results compared to the NRMSE. I have understood that the NRMSE is the better choice for describing a 1:1 fit, either the model fits perfectly or it doesn't. However, the NMSE seems to punish outliners less harsh than the NRMSE does.

Thus, I want to argument that my model is good based upon a NMSE measure. The reason being is that for the lower frequency parts the identified model is robust against noise and does not model it. Therefore, it is obvious that the model will not represent the noise in the training signal and is better described by a NMSE criterion than a NRMSE one.

The bigger plot (lolimot_noise_training.jpg) shows what i mean. Training was done using a local linear model approach with a concatenation of single sines as training data. The model was cross-validated using sweep and noise test signals.

The original training signal (unzoomed) can be found in the second plot (copy.jpg).

To put the plot in numbers:

Fit using NRMSE for the training signal : 87%

Fit using NMSE for the training signal : 98%

Similar values for the test/cross-validation signals.

So my final question is: What do you think of this argumentation. From the plot it is clear that the obtained model is sufficient and good enough to fully represent the system dynamics. I would like to pursue my argumentation using the NMSE as a measure of fit instead using the NRMSE. Though, I have found that in most literature the NRMSE was used most often, however being performed on more 'clinical' training data without too much of a noise.

Thank you very much !

Hi

I m looking for good book that explain how to find the control law for nonlinear system with n order

Thank you

For extended Kalman filter, we need to do matrix differentiation for converting the non linear system to a linear system using first order Taylor approximation

For 2D Henon map,

LE1 = 0.42312

LE2 = -1.6271

Lyapunov Dim = 1 + (0.42312/1.6271) = 1.26

For modified 2D Henon map,

LE1 = 1.1795

LE2 = 0.6926

Lyapunov Dim = ???

I think, it would be 2, But I am not confirm. Please help to make it clear. Will the supporting reason or logic be same for an n-D hyperchaotic continous system?

How could we calculate the zero dynamics of a MIMO nonlinear system with non-zero initial condition?

I am dealing with a nonlinear system in which having a positive value on the inputs (cable tension) is one of the necessary condition for the system to be active. I would like to know how can the zero dynamics of the system be calculated in a non-zero initial condition.

greetings and regards all, do you think that Butterfly effect is a proved scientific fact, (i.e does small change in one state of a deterministic nonlinear system results in large differences in a later state)?

I have come across few of the nonlinear equations like Mathieu,duffing...etc,where i found a stable and unstable limit cycles in a parametric spaces. what is the physics behind these stable and unstable limit cycles, particularly unstable limit cycles? what makes them to appear/exist in a system?

Suppose we have a nonlinear system:

x_dot=A x + B u(t-T) + phi(x) + d(t)

y=Cx

where x:state, u:control input, d:exogenous disturbance (bounded), phi(x): nonlinear vector (satisfy local Lipschitz), A: system matrix (with uncertain parameters), T:time delay, y;

output.

Assumptions:

i. Exogenous disturbance bounded.

ii. Parameter uncertainty bounded

iii. Nonlinear function satisfies locally Lipschitz condition

Constraints:

i. control , u>=0, for all time,t >0 and u<=U_max

ii. states, x_i>=0, for all time t>0

Q/ How to design a robust MPC that will account for the input time-delay uncertain nonlinear system?

Given a nonlinear dynamical system

\dot{x} = f(x)

y = h(x)

whose output y has dimension larger than the state vector x. Is the system necessarily observable? What are the other necessary or sufficient conditions?

When the system is not full state feedback, it contains inner state and outer state. The outer state is stable by a feedback gain, like u = -kx. Question: 1. How can I prove the inner state is stable or unstable?2. If the simulation results of the outer state is stable, can it prove the stability of zero dynamics? 3. I'm interested in the equilibrium point(not zero), and is it right that the zero dynamics of the system is $\dot_\eta = f(\eta, \xi=\xi_d)$, ?

What is the difference between nonlinear PID and linear PID.

If you share your experiences, I'll be glad .

Good day.

I am trying to determine the equilibrium points in the astrodynamics system, but the equilibrium condition is a highly nonlinear system of equations. I have tried the 'fsolve' in Matlab, but it is very sensitive to the initial guess of the solution and is lack of robustness. So I am wondering whether there is any better solver in Matlab or any other software package.

Give examples , papers and books .

I have seen people are using step response model, integrator model etc.

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<u<10

-0.2<du<0.2

0<Y<8 here Y is a nonlinear constraint and a vector

In the well-known nonlinear system reference

**H.Khalil, Nonlinear Systems Third Edition**

in chapter

**13.Feedback linearization,**Khalil developed a theory for feedback linearization(Theorem 13.2)Does this theorem also implies state controllability of nonlinear systems in the form of x_dot=f(x)+g(x)*u?

How to design sliding mode controller for non linear system

Hi everyone,

I would like to simulate an NMPC controller for a nonlinear, 3dof robotic arm, constrained system in Matlab/Simulink. So far, I have not been successful in finding a good example. I would like to build a SIMULINK model using the NMPC block... Would it be a good idea to linearize the nonlinear system first? Are there any good examples on how to use the SIMULINK NMPC block, specifically for a robotic arm?

Thank you in advanced!

for deterministic systems, with defining proper terminal constraint , terminal cost and local controller we can prove the recursive feasibility and stability of nonlinear system under model predictive control. For stochastic nonlinear system it is impossible to do that since we do not have bounded sets for states.

what is the framework for establishing the recursive feasibility and stability of MPC for stochastic nonlinear systems?

In nonlinear systems, we know several bifurcations (i.e. Saddle-node, Pitchfork, Transcritical, and Hopf). The question is: does there exist a specific bifurcation that merges two "stable" limit cycles to one "stable" (and probably with larger amplitude) limit cycle?

I would like to implement IMC (tune the PID parameters) considering a third order integrator system. Most of the existing papers discuss first or second order systems or simplify the system model.

E=[1 0 0;0 1 0;0 0 0]

ExDot=Ax(t)+B(u)+f(x,u)

according to f(x,u),I can't calculate x(3) simply.

and Idea?

is there any article about control of nonlinear system input to dead-zone or saturation with sliding mode or backstepping theory?

How to build discrete time Linear Parameter Varying Sytems (LPV System from a Nonlinear System ?What is Scheduling in LPV and how to choose it?

what about the Nouval Nonlinear technics to control nonlinear system??

The concept of flatness and flat outputs was introduced in :

Fliess, Michel, et al. "Flatness and defect of non-linear systems: introductory theory and examples." International journal of control 61.6 (1995): 1327-1361.

How should one interpret the term 'flat system'?

Is it possible to do Frequency Response Analysis of a nonlinear system in ANSYS Workbench?

Is it correct that linear PID control can not be used for trajectory tracking of nonlinear systems?

Can anyone give me a comprehensive explanation?

I am wondering to enforce state dissipation to stabilize nonlinear or linear systems. Assume a nonlinear control system as xdot=-x

^{3}+u. Then to forcefully dissipate x as exponentially by setting: x = x0*exp(-a*t), where a is dissipation rate, so xdot = -a*x0*exp(-a*t), hence from the evolution dynamics; xdot=-x^{3}+u, we have the control variable as u= xdot+x^{3}= -a*x0*exp(-a*t) + (x0*exp(-a*t))^{3},or in state feedback format; u(x)=-a*x+x

^{3}.This is a time-varying open-loop control and in other format a state feedback strategy. So what is your idea? How do you think about that? Does it worth as a new control methodology?!

Consider a general nonlinear system with output y(t). Suppose that you know an observer design that approximates its state by using the measurements y(t). In the presence of sensor nonlinearities, you do not have access to y(t) anymore but only to a transformed psi(y(t)), where psi is a local diffeomorphism with inverse difficult to compute.

To verify if the system satisfies the observability property. Several techniques and tools have been developed to study whether a nonlinear system is observable or not. Generally, the observability property study of a nonlinear system are depended values when the expression of determinant (D) are canceled for limited points but not for all operation modes such as in the case of

*complex*expression of determinant.In discrete-time affine nonlinear systems, the optimized solution sometimes can not guarantee the stability of the closed-loop system. How to solve this problem?

I am interested in the application of derivative-free methods, in particular pattern search algorithms, to the identification of nonlinear systems. I have found many application papers that use this kind of methods to identify the parameters of the plant, but I have not found too much information about the advantages of using those algorithms with respect to gradient-based methods. Obviously the results will depend on your model structure, but is there any paper comparing both alternatives in general?