Nonlinear Systems - Science topic

The primary difference between FDI (Fault Detection and Isolation) based FTC (Fault Tolerant Control) schemes and fault estimation based FTC schemes is the way they address faults in a system.

FDI-based FTC schemes involve detecting and isolating faults in a system and then taking appropriate actions to compensate for the faults. These actions may include reconfiguring the system or switching to a backup system. The FDI approach is based on monitoring the system's inputs and outputs to detect changes in the system's behavior that may indicate the presence of a fault. Once a fault is detected, the system's control algorithm is modified to compensate for the fault.

On the other hand, fault estimation based FTC schemes focus on estimating the fault magnitude and then using this estimate to compensate for the fault. The estimation of the fault magnitude is usually done using a model of the system, and the fault estimation algorithm is designed to minimize the difference between the predicted system behavior and the actual system behavior. Once the fault magnitude is estimated, the system's control algorithm is modified to compensate for the fault.

In summary, FDI-based FTC schemes focus on detecting and isolating faults, while fault estimation based FTC schemes focus on estimating the fault magnitude. Both approaches can be effective in providing fault tolerance for a system, and the choice between them depends on the specific requirements of the system and the nature of the faults that need to be addressed

Nonlinear systems are widely used in different industries, including chemical engineering, aerospace, electrical engineering, robotics, and biomedical engineering. These systems are used to model, analyze, and control complex processes and systems, such as chemical reactions, aircraft flight, power grids, robot motion, and biological systems. Nonlinear systems allow engineers and scientists to ensure stable and efficient performance and develop control strategies for various applications.

Hi Ricky, Yes, it is possible to approximate a discrete-time, time-invariant input-affine nonlinear system with a piecewise affine (PWA) system and guarantee that all trajectories of the nonlinear system are also contained in this PWA system. This technique is known as PWA approximation or PWA overapproximation.

The idea behind PWA approximation is to partition the state space of the nonlinear system into a finite number of polyhedral regions, and then construct a linear system for each region. The resulting PWA system is an overapproximation of the nonlinear system because it contains more regions than necessary, and each linear system is only valid within its corresponding region. However, if the regions are chosen carefully, the PWA system can provide a good approximation of the nonlinear system for many practical purposes.

To construct a PWA approximation of a nonlinear system, we first need to partition the state space into a finite number of polyhedral regions. This can be done using a variety of techniques, such as clustering algorithms or Voronoi tessellation. Once the regions have been defined, we can construct a linear system for each region by linearizing the nonlinear system around a set of operating points within each region. The resulting PWA system can be expressed as:

x(k+1) = Ai x(k) + Bi u(k) + ci, i = 1,2,...,N

where x(k) is the state vector, u(k) is the input vector, and N is the number of regions. The matrices Ai, Bi, and ci are the state transition, input, and offset matrices for each region, respectively.

To guarantee that all trajectories of the nonlinear system are contained in the PWA system, we need to ensure that the PWA system is a valid overapproximation of the nonlinear system for all possible initial conditions and inputs. This can be done using Lyapunov-based techniques or set-based methods, which involve computing the control invariant set for each state space representation and using the intersection of all sets as an approximation of the control invariant set of the nonlinear system.

The control invariant set is the set of initial states for which there exists a control input that can keep the system within a desired region of the state space. This set can be computed using a variety of techniques, such as linear programming or semidefinite programming. Once the control invariant sets have been computed for each region, we can compute the intersection of all sets to obtain an approximation of the control invariant set of the nonlinear system.

PWA approximation is a useful technique for approximating nonlinear systems with a finite number of linear systems. By partitioning the state space into a finite number of polyhedral regions and constructing a linear system for each region, we can obtain a good overapproximation of the nonlinear system. To ensure that the PWA system is a valid overapproximation for all initial conditions and inputs, we need to compute the control invariant set for each state space representation and use the intersection of all sets as an approximation of the control invariant set of the nonlinear system ;)

The solutions of non-linear differential equations, with reasonable boundary conditions can simply be obtained using MAPLE SOFTWARE

First you understand and start from Eulers methods explicitly and implicitly then Rk4 will be easy for understanding. You can also solve system of ode by built-in code of ode in Matlab are Ode45, ode23, for linear and for non-linear differential equations shortly.

These codes basically used the Rk4 method and Rk nystrom method explicitly and implicitly.

If you want to solve nonlinear differential equations (coupled or uncoupled) you can use 'odeint' function in python language. It uses the RK4 method of numerical integration. To use this function you have to define the nonlinear functions f(x,y), g(x,y) etc from the equations dx/dt=f(x,y) and dy/dt=g(x,y) and then supply these to odeint function as arguments. Along with this, you need to define a time range of integration and initial conditions for x and y.

You can start by solving the differential equation of the simple harmonic oscillator by odeint function in python. The system is defined by two coupled 1st order differential equations: dx/dt=y and dy/dt=-w*w*x. Where w is the frequency of the oscillator. Though it is not nonlinear but you can use this method for nonlinear differential equations also.

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

In general, you could employ a finite difference approach and solve the problem numerically. There are a variety of good examples on the Matlab Central QA boards. I would start there.

السلام عليكم

Hi Anamika Gautam

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 = G_b = 1, with the initial condition {x₁(0) = 1, x₂(0) = 0.5, x₃(0) = –0.5, x₄(0) = –1}, and u = 0.

Dear Prof. Ankur Gajjar,

Please refer to my work other than the suggestion of Dear Scientists on the solution of linear equations in the form of AX=B.

Kind regards

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.

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.

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.

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 ( )

Hi,

file name "sfun_ abcaaaaa.c" is not available in the folder mentioned.

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.

Also check https://www.genealogy.math.ndsu.nodak.edu/id.php?id=38108

+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

Hy Daniel

Why don't you try Log-Linearization around the steady stat ?

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):

Very specialised problem -

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.

Thank you for your help.

Have a look at these two preprints.

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.

Dear

Actually , I do not know about this kind of function but its really

amazing so could you please send to me the referred paper ...I will be grateful.

I am interesting

Many thanks

The multi-layer perceptron, the radial basis function network and the functional-link network

For more information:

S. CHEN &S. A. BILLINGS

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

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

For Video tutorials , we have:

Introduction | Nonlinear Control Systems: https://www.youtube.com/watch?v=Xgnwn0G9qoo

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.

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.).

Dear Amirreza ,

You can read it in the following link:

Article

Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective

Archives of Computational Methods in Engineering volume 21, pages331–358(2014)

Hello, you can also use simple dynamic model (ex. second order) to represent the behaviour of your system.

Dear Sundarapandian Vaidyanathan

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.

Dear Ádám Domina,

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

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

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.

see

Farey sequence in the appearance of subharmonic Shapiro steps

Odavić, Jovan, Mali, Petar, Tekić, JasminaJournal:Physical Review EYear:2015

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)

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:

1.https://www.researchgate.net/publication/264422491_Using_Artificial_Neural_Network_to_Predict_Body_Weights_of_Rabbits

2. https://dl4.globalstf.org/products-page/proceedings/vetsci/application-of-artificial-neural-network-to-predict-body-weight-in-madras-red-sheep-artificial-neural-network-in-prediction-of-body-weight/

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

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

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.

Hi I am Professor Soheil Sayed Hosseini See my post on satellite control article nmce is availebel

You can follow NPTL lectures for your topic .

Thank you so much.

Yes, the principle generalizes. If all the LEs are positive, then the Lyapunov dimension is equal to the number of variables (and LEs).

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.

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

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).

@ Richard Epenoy

Thank you very much Sir for your response.

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!

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

Hi ,P.K.Karmakar.

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.

Thankyou Maam

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.

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

Use command 'ctrb'. Please follow the below mentioned link for more details. https://in.mathworks.com/help/control/ref/ctrb.html;jsessionid=82108156791a9c9a9b26ea22601d

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.

I agree Prof. Yew-Chung Chak

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 .

Dear Dr. Guillermo Valencia-Palomo

bounded uncertainty is a robust approach not stochastic approach.

Try x'' = ax - x^3 + (1 - x'^2)x' in the vicinity of a = 1.87525 (the Rayleigh-Duffing two-well oscillator).

Dear Fateme Bakhshande

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.

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

Hi Fatiha,

Depending on the classes of nonlinear systems, there are several well-known control schemes for controlling strongly nonlinear processes, such as follows:

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.

You should read the book

LEVINE - Analysis and control of nonlinear systems. A flatness-based approach (2009, Springer-Verlag Berlin Heidelberg)

where the concept is clearly explained.

Attached file should get you started on this subject.

If this answer worked for you please recommend it.

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:

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 .