Science topic

# 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
Question
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!!
if you have the algebraic equations available you might try GAMS www.gams.com
As a young engineer you should train yourself in Python programming. Forget about fortran and all the old school stuff.
Question
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.
Adaptive controllers are not needed, because there is already a universal controller. It does not need to be adjusted when changing the parameters of any linear or non-linear control plant. https://t.me/universalcontrol/95 https://t.me/universalcontrol/40
Question
Kindly provide examples with MATLAB command
thanks
x_1dot = x2.g_1+f_1
x_2dot = u.g_2+f_2
y = x_1
May be this Nonlinear system is not controllable if x_2.g_1 is counteracted by f_1
Question
I mean chaotic flows. That is possible for chaotic maps.
What makes it possible to deduce or suppose the existence of chaos are the positivé exponents of lyaponuv and not the eigenvalues ​​(multipliers) associated with a fixed point or a cycle.
Question
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
Dear Suhaib Jaleel:
Topics Solved:
--Solve Nonlinear System Without and Including Jacobian.
-- Large Sparse System of Nonlinear Equations with Jacobian.
--Large System of Nonlinear Equations with Jacobian Sparsity Pattern.
--Nonlinear Systems with Constraints.
--Solver-Based Optimization Problem Setup.
--Equation Solving Algorithms
Best wishes....
Question
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.
x4 is not controllable, but stabilizable if p5 > 0.
x3 and x2 are controllable.
If x4 is stable, then x1 is controllable.
You can run a basic stability test for parameters p1 = p2 = p3 = p4 = p5 = Gb = 1, with the initial condition {x1(0) = 1, x2(0) = 0.5, x3(0) = –0.5, x4(0) = –1}, and u = 0.
Question
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.
Mostafa Ranjbar The contribution of auxetic materials and structures comes from their unique deformation pattern (lateral contraction when they are compressed). This behaviour lead to densify the material under the impact are in which it leads to more dissipation of the kinetic energy. One factor should be consider accurately in which its the geometrical parameters of these materials/structures.
Question
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
Dear Ma Ka
I suggest you to start your excursion into Matlab, by learning it from a class fellow or friend who is conversant with Maple or Matlab languages.
Learning Matlab by self study is a bit difficult, however, interactive videos available online, are as well a good option.
Question
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.
Hi Mohammad Javad Mirzaei for estimation of state of a nonlinear system you can use High Gain Observer. If you know some of the states or the out of the system you can apply High Gain Observer. For linear System we can use linear observer or Kalman filter but in nonlinear case these two are mostly failed. So, its better to use high gain observer.
Question
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?
Well, no specific answer in this regard; you can select a range of layers (number of layers), and got your results. Then you can use the ROC curve against the number of layers and the effectiveness achieved. An example of the ROC curve used is here ( )
Question
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)
Hi,
file name "sfun_ abcaaaaa.c" is not available in the folder mentioned.
Question
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
The type of nonlinear system need be clarified. Is it system of scalar nonlinear equations? system of nonlinear BVPs of ODEs or PDEs? System of nonlinear IBVPs of ODEs or PDEs...?
In case of the usual system of nonlinear scalar equations, the matlab symbolic tool (command) ''solve'' may be used. Numerical methods are Newton method, Halley's method, Chebyshev's method....There are references with matlab codes using the iteration methods.
Question
He is the author of one of the most important works of of the early twenty century in the qualitative theory.
Question
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!
+1 to Konstantinos answer. I was able to solve rather sophisticated nonlinear dynamic problem with GetFEM++
As for your question, mechanical, thermal, optical and chemical reactions sounds for me as elastic operator(div \sigma=rhs; ) and many Laplace operators for temperature, concentration, etc in one equation system.
You may solve this in GetFem with your full control at each stage
Question
The System is found in the article attached. Steps or codes to transform to an LPV system
Hy Daniel
Why don't you try Log-Linearization around the steady stat ?
Question
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?
Disturbance is sometimes helpful while dealing with multicollinearity. Also helpful may be keeping the number of degrees of freedom minimum.
Maybe you can alernatively consider the recursive least squares algorithm (RLS). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), adquired data is weighted according to its age, with increased weight given to the most recent data.
Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter (pilot plant). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):
Question
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
Very specialised problem -
Question
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
Hello to all the nobles.
Thank you everyone for your inputs.
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.
Question
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?
Thank you.
Have a look at these two preprints.
Question
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
First, we cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking at the qualitative dynamics of a system. Equilibrium points– steady states of the system– are an important feature that we look for. Many systems settle into a equilibrium state after some time, so they might tell us about the long-term behavior of the system.
Equilibrium points can be stable or unstable: put loosely, if you start near an equilibrium you might, over time, move closer (stable equilibrium) or away (unstable equilibrium) from the equilibrium. Physicists often draw pictures that look like hills and valleys: if you were to put a ball on a hill top and give it a push, it would roll down either side of the hill. If you were to put a ball at the bottom of a valley and push it, it would fall back to the bottom of the valley.
Question
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.
Dear
amazing so could you please send to me the referred paper ...I will be grateful.
I am interesting
Many thanks
Question
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)
The multi-layer perceptron, the radial basis function network and the functional-link network
Neural networks for nonlinear dynamic system modelling and identification
S. CHEN &S. A. BILLINGS
Question
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?
1- Stochastic Model Predictive Control: An Overview and Perspectives for Future Research
DOI: 10.1109/MCS.2016.2602087
2- Stochastic model predictive control — how does it work?
By: panelTor Aksel N.HeirungJoel A.Paulson
3- Stochastic Nonlinear Model Predictive Control with Probabilistic Constraints Ali Mesbah1 , Stefan Streif , Rolf Findeisen , and Richard D. Braatz
Question
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)
Hello Albert:
This aspect has become quite important in Grand Unified Theories.
My original analysis will indicate that spatially local asymmetries generate energies to make global symmetries, while in the process time will become asymmetric. What will keep space and time transferring energies to drive symmetry? My answers based on my findings of the dimension 5 will be that "sense". A lot more knowledge of quantum relativity physics with advancing Standard Model and Theory of Everything as Superstring Theory, Supersymmetry, among others are necessary to get a handle on this universal principle.
Thank you for sharing this research. We will have ongoing communications.
My articles here on RESEARCHGATE about macro time asymmetry relativistically and micro time reversibility quantumwise will tell how nature seems to oscillate between symmetry and energy!!
Sincerely,
Rajan Iyer ENGINEERINGINC INTERNATIONAL OPERATIONAL TEKNET EARTH GLOBAL
Question
I have the nonlinear systems of Khalil, however some definitions are no so clear, is there a newer book with Matlab examples
For Video tutorials , we have:
Introduction | Nonlinear Control Systems: https://www.youtube.com/watch?v=Xgnwn0G9qoo
Question
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?
Generally, the global stability has to be inferred from extensive simulations and mainly depends on how the controller is scheduled i.e., use interpolation, linearization scheduling or other methods, how many and which operating points to use as a basis for the scheduling, etc.,
While considering the classical gain scheduling methods, the key issues are i) the fact that transitions among operating conditions are not addressed in the design, and ii) shortcomings of the linearization’s. Hence, extensive offline testing is required to establish global stability and performance guarantees. An often-used rule-of-thumb states that the scheduling variable should only vary slowly to prohibit the introduction of additional dynamics.
Generally, the scheduling is based intuitively on physical variables in the plant. In case of the quasi-LPV, a coordinate transformation has to be applied. As this approach is conservative, this might lead to infeasible controller design problem, i.e. no feasible controller may be available. Quasi-LPV techniques on the other hand, enable global stability as well as performance and robustness analysis.
Question
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?
An alternative method of demonstrating stability is given by Vasile Mihai POPOV, a great scientist of Romanian origin, who settled in the USA.
The theory of hyperstability (it has been renamed the theory of stability for positive systems) belongs exclusively to him ... (1965).
See Yakubovic-Kalman-Popov theorem, Popov-Belevitch-Hautus criterion, etc.
If the Liapunov (1892) method involves "guessing the optimal construction" of the Liapunov function to obtain a domain close to the maximum stability domain, Popov's stability criterion provides the maximum stability domain for nonlinearity parameters in the system (see Hurwitz , Aizerman hypothesis, etc.).
Question
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?
Dear Amirreza ,
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)
Question
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.
Hello, you can also use simple dynamic model (ex. second order) to represent the behaviour of your system.
Question
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!
As you mentioned, the bifurcation diagram allows us to analyze the dynamics of the systems, in this case, chaotic systems. The idea in eliminating a certain amount of data in the time series (no matter if are 1000 secs, 1500 points in the time series, or whatever) before constructing the bifurcation diagrams, lies in eliminating the transitory state of the system and focus on what it is known as steady-state dynamics. I think that is what is referred to in the articles you describe, where the amount of time referred to is not so relevant, and it would suffice to say that the dynamics of the steady-state is analyzed, or that they have not been considered the transitory states for the construction of the bifurcation diagrams.
Question
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.
First of all, you need to make sure that the Jacobian matrix evaluated at the equilibrium state is hyperbolic. ( The eigenvalues has non zero real parts) Then you can apply the Hartman-Grobman theorem. That is, the linear system and nonlinear system behave the same in the neighborhood of the equilibrium point. So, the main point is to determine the eigenvalues of the Jacobian matrix which essential to study the stability of your system.
Best regards
Question
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
While the definition of poles and zeros are well-established for linear differential systems, its 'extension' to the nonlinear case is far from being trivial, so I'm not entirely sure what you mean by 'pole' or 'zero'.
That said, it might be the case that you are designing a model-based controller considering a process represented by a linear system (obtained, for instance, via Jacobian linearisation), while you also want to take into account actuator limitations, in terms of a saturation function. If this is the case, then I suggest you take a look at AntiWind-up techniques (see, for instance, http://cse.lab.imtlucca.it/~bemporad/teaching/ac/pdf/AC2-09-AntiWindup.pdf)
Hope this helps
Question
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?
Hi, so far, it seems you are on the right path. However, I have some remarks:
1. what's your control goal? I assume is the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. Then that desired position and angle (with respect to your model) giving you the upright position are your controller reference (could be 180° or 0°, it depends which angle you took as the reference to develop your model).
2. Because you are thinking in implementation and as you said the only measurements available will be position and angle, then you would need an observer to estimate the other two states (derivatives). I recommend a Kalman Filter.
3. In order to better understand how to use your linear approximation to control your non-linear model, the following paper could help (which explain how to translate your linear model to the operating point of the nonlinear system):
Regards.
Question
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).
see
Farey sequence in the appearance of subharmonic Shapiro steps
Odavić, Jovan, Mali, Petar, Tekić, JasminaJournal:Physical Review EYear:2015
Question
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?
I highly recommend you read "Predictive control with constraints" by J.M.Maciejowski. It has the best introduction and code samples for MPC for a control designer who is new to the subject. It's not something I am able to code up right at the moment, but there are excellent examples in the book.
The book has a website here (old Cambridge archive), but I don't know how much of the m code is online (www-control.eng.cam.ac.uk/jmm/mpcbook/mpcbook.html)
Question
ANN, which has the ability to model nonlinear systems, could be used to predict BW from age.
Dear Ananta Kumar Das,
Yeah, the MLPNN (multilayer perceptron neural networks) can easily be used to predict the body weight from age.
Especially, the LM (Levenberg Marquardt) or BR (Bayesian regularisation) based MLPNN can be more productive to estimate the body weight from age. However, it's interesting to see how training model accuracy varies for different age inputs having constant body weight output. To make a system more robust, you have to take huge dataset (may be 1000 or more).
You can visit these links for more clarification:
Furthermore, the training of network models are implicitly dependent on many parameters like number of epochs, number of layers, number of hidden neurons in the hidden layer and most importantly the division ratio of training dataset.
The procedure to apply MLPNN models are given in following link:
Regards
Question
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!
Dear A.-J. Guel-Cortez,
One of the effective ways to parameters' identification is the Tikhonov regularization method. This method can be successfully applied for identification of nonlinear system parameters. If by some reasons you have not possibility to apply this method yourself, I hope that I can help you if you send me your parametrized nonlinear system.
Best regards,
Sharif Guseynov
Question
could I apply adaptive fuzzy (wang 1993.)for switched nonlinear systems or it needs new challenges?
Question
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.
Finding a closed form solution is difficult to say the least. However, there is a rich theory on the behavior of such solutions. There is plenty of work on numerical analysis of this subject. However, there is a wealth of work on the behavior of such systems - particularly their asymptotic properties. For autonomous systems, i.e., x_dot=f(x), where f is independent of t, at a point where f(x0)=|= 0, locally a solution curve through x0 behaves like the vector field f. The questions arises about the solution near a equilibrium point. f(x0)=0. This gives rise to the stable, unstable and central manifolds. That is locally the equation is approximated by the linear equation x_dot= Df*x. The eigenvalues of Df(x0) determine the behavior of the solutions and define the stable manifold ( associated with eigenvalues with negative real part), unstable manifold (associated with eigenvalues with positive real part) and central manifold (associated with eigenvalues with zero real part). A point x0 is a hyperbolic equilibrium if f(x0)=0 and none of the eigenvalues of Df(x0) have zero real part.
The famous theorem of Hartman and Grobman ( developed at the same time, Hartman in the US and Grobman in the former Soviet Union unbeknown to one another) give a complete characterization of hyperbolic fixed points. There has been much work on analyzing the central manifold since the work of G. D. Birkhoff and still on going today and has led the concept of a dynamical system and is extremely active.
For the case where the system is non-automous, i.e., f = f(t, x), there is a wealth of literature examining the asymptotic properties of the solutions of the equation. This field of endeavor goes by the name "asymptotic integration." In this case often the methods used establish existence of a solution and to derive proprieties come from functional analysis, fixed point theorems, topological arguments, etc.
A good place to start is Phil Hartman's, classic reference, known by many as the "tomb", "Ordinary Differential Equations." Also a good place to start with dynamic systems is the work of Steve Smale, Charlie Pugh and the references there in their work.
Question
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 am Professor Soheil Sayed Hosseini See my post on satellite control article nmce is availebel
Question
Hi
I m looking for good book that explain how to find the control law for nonlinear system with n order
Thank you
Question
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
Thank you so much.
Question
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?
Yes, the principle generalizes. If all the LEs are positive, then the Lyapunov dimension is equal to the number of variables (and LEs).
Question
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.
We discuss zero dynamics of a multivariable system (zero dynamics of multiple input, multiple output "MIMO" system):
x˙ = Ax + Bu y = Cx,
where (A, B, C) is minimal and A is n × n. From now we always assume that both B and C have full rank. For the time being we assume that the number of inputs and the number of the outputs are the same (a square system), namely B is n × m and C is m × n. Correspondingly we also have the frequency domain representation as: G(s) = C(sI − A) −1B.
Unfortunately we do not have a straightforward way to extend the concept of transmission zero for a single-input, single-output (SISO) system to the MIMO case.
When a system has non-minimum phase zeros, high-gain feedback (e.g. F matrix with large norm) cannot be used to stabilize the system, since it can be shown that some of the poles of the feedback system tend to its zeros as the matrix F increases. This is a similar phenomenon as that what occurs with root-locus diagrams. Consequently, the selection of feedback matrices in the non-minimum phase case is very delicate, since feedback should be high enough to have some effect but simultaneously low enough to avoid instabilities. Finally, it can be shown that the sensitivity of a non-minimum phase system to disturbances acting at the input of the plant is severely limited both when the open-loop plant has unstable poles and non-minimum phase zeros.
I hope it is helpful to you.
All the best.
Question
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)?
The chaotic behaviour that is the basis of the classic "butterfly effect" can be described as the apparent randomness. It results from the extreme sensitivity to the initial conditions. Typically, a deterministic dynamic system of the form: dX / dt = F (X, t), where X is a vector can exhibit chaotic behaviour after a slight perturbation of the initial conditions. This fact was discovered by H. Poincaré in the early 1900s during investigation of a three-body problem. The concept was rediscovered more than 60 years later by Edward Lorenz after the analysis of a spectral form of the equations of dynamic meteorology. Lorenz presented a very beautiful definition capturing the essential property of chaos:
“Chaos: When the present determines the future, but the approximate present does not approximately determine the future”
One of the most famous manifestations of chaos is the chaotic advection discovered by Aref in the 1960s. I have attached figures illustrating the tangled web of tracer filaments created by the atmospheric flow. Such butterfly effects are observed in atmospheric transport processes. This fact has been confirmed many times during the observation of the radioactivity transported during major nuclear accidents such as Chernobyl and Fukushima.
In general, the effects of chaos are not always obvious in all physical systems because of the presence of truly stochastic forcing. A good example of such a disturbance is thermal noise. To represent this fact, we must add the random disturbances (noise) on the right side of the dynamic system:
dX / dt = F (X, t) + f '
where X and f 'are vectors.
Adding noise changes the phase portrait of the system. In the case of a classical "Lorenz butterfly" of atmospheric dynamics, the addition of diffusion generally decreases the chaotic effects.
In conclusion, the "butterfly effect" is a scientifically proven fact, although its intensity depends on the role of stochastic noise.
Reference
Question
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?
Dear colleagues, this question must be answered by part. What is a center? What is a periodic solution? In the case of a physical system, we are talking about a regime of "work" that is maintained over time, which does not affect exogenous conditions, that is, it is stable. Therefore a limit cycle, is an "ideal" regime (not achievable in finite time), to which the other work regimes are close enough (or move away in the case of an unstable limit cycle, or approach and move away if it is semi-stable), without ever reaching it. The electrical devices that you have in our houses in general comply with this rule (a refrigerator, for example).
Question
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?
@ Richard Epenoy
Thank you very much Sir for your response.
Question
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?
While for integrable systems the initial conditions determine everything, since they define the values of the conserved charges, that define the dynamics completely, for deterministic chaotic systems, the initial conditions are irrelevant, since the attractor doesn't depend on them. The exponential sensitivity to the initial conditions, that has been promoted as a defining feature of deterministic chaos, in fact, is a transient effect, until the attractor has been reached.
The fact that the attractor has measure zero doesn't imply that it can't be observed, however!
Question
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 $\dot_\eta = f(\eta, \xi=\xi_d)$, ?
Dear Tang Alisa:
Even in the previous case with an inner dynamics, I believe the paper "Necessary and Sufficient Condition for Asymptotic Stability of Nonlinear ODE’s" it is useful.
Please do not hesitate to contact me if you need a copy of the paper.
Best regards
Andrés García
Question
What is the difference between nonlinear PID and linear PID.
Good day.
Hi ,P.K.Karmakar.
Question
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.
Yue Wang - thanks for the complement, but the title should be just "Dr."
For the homotopy method, I suggest you check out these packages: PHCpack, PHClab, and Hom4PS-3.
Question
I have seen people are using step response model, integrator model etc.
Hi,
Here is a complete workshop on how to implement MPC and MHE in MATALB.
The workshop shows a complete explanation of the implementation with coding examples. the codes are also provided.
I thought of sharing it here as it might be helpful.
Question
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
Hi,
Here is a complete workshop on how to implement MPC and MHE in MATALB.
The workshop shows a complete explanation of the implementation with coding examples. the codes are also provided
Question
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?
In this theorem first asumptions implies controllability of system.
Proof of this condition is shown for example in: Hunt, L.R., “Sufficient Conditions for Controllability”, IEEE Trans. Circuit and Systems, May 1982.
Question
How to design sliding mode controller for non linear system
I agree Prof. Yew-Chung Chak
Question
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?
Hi, unfortunately in order to operate a nonlinear system with simulink, it is much better to exert simplification first and then use Mpc block . But simplification can not always be the best way. Consequently, in my opinion, it is much beter to do it with a code script and for example fmincon function to optimize the controlling inputs. For this method you need the nonlinear steady states equations .
Question
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?
bounded uncertainty is a robust approach not stochastic approach.
Question
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?
Try x'' = ax - x^3 + (1 - x'^2)x' in the vicinity of a = 1.87525 (the Rayleigh-Duffing two-well oscillator).
Question
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.
You have an output regulation problem and you need to use Internal Model Principle (IMP). Assuming that your output is the first state and the reference to be track is a sinus, you have two options:
1- Use a dynamic controller: the dynamics of your controller should contain a 1-copy of the internal model of the reference signal. If you have p outputs in general, you need p-copy of the internal model. Then, the controller signal is a feedback from its internal state (the controller is dynamics so it has an internal state) and the state (or the output) of your system. This approach is robust.
2- Use a feedforward approach: Design a stabilizing controller for your system assuming that the reference is zero. Then, solve the output regulation equation to obtain the feedforward part. The controller is a sum of these two by some small modifications. Note that this method is not robust if you have uncertainty in the system.
A handy book on this topic is  Nonlinear Output Regulation: Theory and Applications", by Jie Huang.
Question
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?
tahnk you verry mucha to answer to my quetion.
i have a problem to simulate the state vectors of a switched singular systems with time-varying delay in the form of Eix(k+1) = Aix(k) +Adix(k-d(k)) , and i hope you to help me to solve this problem that caused also to simulate the state vectors of stabilization and filtering of switched singular sysytems with time delay
Question
is there any article about control of nonlinear system input to dead-zone or saturation with sliding mode or backstepping theory?
Question
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?
Question
what about the Nouval Nonlinear technics to control nonlinear system??
Hi Fatiha,
Depending on the classes of nonlinear systems, there are several well-known control schemes for controlling strongly nonlinear processes, such as follows:
1. Feedback linearization
2. Integrator backstepping
3. Passivity-based conntrol
4. Control Lyapunov function
5. Sliding mode control
If you identify an unsolved control problem, then you may come up with a novel nonlinear control design. Of course, you can improve an existing nonlinear control scheme. Or, you can create a totally new way of analyzing the stability of nonlinear control systems.
Question
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'?
LEVINE - Analysis and control of nonlinear systems. A flatness-based approach (2009, Springer-Verlag Berlin Heidelberg)
where the concept is clearly explained.
Question
Is it possible to do Frequency Response Analysis of a nonlinear system in ANSYS Workbench?
Attached file should get you started on this subject.
Question
Is it correct that linear PID control can not be used for trajectory tracking of nonlinear systems?
Can anyone give me a comprehensive explanation?
PID controller is used to control linear systems which are obtained about a single operating point. If the system operating region is small, we can use the robust control approaches to design linear robust controllers where the nonlinearities are treated as model uncertainties. However, most of real (industrial) systems exhibit nonlinear behavior. Therefore, the operating region is large, the above mentioned controller synthesis may be inapplicable. For this reason, the controller design for nonlinear systems has attracted considerable in the last years. Several solutions have been proposed in the literature. Among them, we cite Gain scheduling approach which has a wide range of use in industrial applications. Gain scheduling is used when a single set of controller gains does not provide desired performance and stability throughout the entire range of operating conditions for the plant. The design of controller based on gain-scheduled approach consists in the following steps:
• Linearize nonlinear system at different operating points to obtain linear models that describe system behavior in all operating domain.
• Tune controller gains for all the linear system models using for example PID controllers.
• Implement a gain-scheduled controller architecture
The main advantage of gain scheduling is that it conserves the advantages of linear controller design methods such as PID control which is the most popular control strategy in industrial applications .
See for example, the following :
Question
I am wondering to enforce state dissipation to stabilize nonlinear or linear systems. Assume a nonlinear control system as xdot=-x3+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=-x3+u, we have the control variable as u= xdot+x3= -a*x0*exp(-a*t) + (x0*exp(-a*t))3,
or in state feedback format; u(x)=-a*x+x3.
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?!
Question
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.
Hi Francisco,
If you speak about a diffeomorphism, by definition, it has an inverse (and its derivative exists), so you can state this fact as true for general results even if it is hard to compute.
For this particular example, psi(y)=sin(y)+2y, you can use psi(y) +psi''(y) (second derivative) to compute y(t) (or, at least, a numeric approximation).
Question
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.
Dear Zoheir,
you are correct. One alternative is to look at a condition number (of the observability matrix) averaged along a trajectory (hence over a large number of operating conditions), instead of just computing the rank of the observability matrix. Christophe Letellier and I have done work on this for the last 20 years. The following papers are good starting points:
An approximate approach based on data can be found here:
Regards.
Luis Aguirre
Question
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?
Dear Yu Lu,
I suggest you to see links and attached files in topic.
- Model predictive control of constrained piecewise affine discrete-time ...
- H∞ model predictive control for constrained discrete-time piecewise ...
- On the stability and robustness of non-smooth nonlinear MPC
- Model Predictive Control of Nonlinear Input-Affine Systems with ...
Best regards
Question