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I am working on dynamic state estimation of power system for IEEE 14 bus system using Kalman filter.
1. I was taking F as unity matrix, but estimation was very poor when compared with RSCAD measurements data. However, it works when i took simulated measurements, estimation was better.
2. Is there any method to get F matrix of the dynamic system.
3. Is there any other error problem might be there.
Please Suggest.
Thanks
Sumanta Nanda
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Thank you so much Hassan Nasser sir.
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I need to use mathematical software program for mathematical modelling such as; modelling of dynamic systems etc.
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Matlab works well for "engineering" types of problems. However, Matlab often requires a specialized add on "toolbox." The base price of Matlab is pretty expensive and the add on toolboxes are expensive.
For general purpose work, Wolfram Research Mathematica will address more general problems than Matlab in that it has the computational elements and symbolic manipulation support.
The upside of both Matlab and Mathematica is they both have very good debugger support built into the package. Both Matlab and Mathematica are "free format" programming languages. There is a Mathematica called MATLink that allows Matlab functions and functionality to be used within Mathematica. It provides the best of both worlds.
As others pointed out Python can be used. The downsides for me for Python is the lack of a high quality debugger as one has in both Matlab and Mathematica and it is a fixed format language. That is the column in which one starts a line is important. The formatting is quite rigid. For those of us that are old enough to remember the fixed format languages such as Fortran II and IV, the benefits of a free format language are obvious. For those that need to trace the values of a variable though a complex algorithm, a robust debugger is important. Unless a lot as changed the debugger in Python levels a lot to be desired. The biggest upside of Python is it is open source and "free."
A GNU open source Matlab equivalent is Cctave. It is designed to be a drop in replacement for Matlab. However, the editor is not as sophisticated nor as robust as Matlab and neither is the debugger. On the other hand you get 85% of the capability and it is free. Code developed in Octave will run in Matlab and visa versa.
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I have a dynamic system including some nonlinearitoes, and want to control its output (Y). The concept of the control is to use MPC for reference output signal (y*) planning then another controlller will be utilized to provide an input (u) to track the reference output (y*). The process should be completed within 300 ms.
There are a wide variety of control method avaliable in the literature, and for a non expert in system control and control theory, it is not an easy task to choose from.
I am looking after a control method that can be combined with MPC to produce a stable controller and perfume the task within the alloted time.
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Thank you for your valuable suggestions and the materials provided.
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I created Electric vehicle dynamic model in Matlab Simulink.IT is showing the perfect result. But I am facing the difficulties when I am integrating that EV in any system or MG. Its either not running or its taking too much time to run.How to solve this issue,can anyone find the solution please let me know.
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It is known that in order to quantify chaos in a dynamic system, the maximum Lyapunov exponent must be detected, but before applying Lyapunov detection, is there any way to know at a glance a dynamic system is chaotic ?
Is it enough to see that the phase space is fully and disorderly filled up? It happens that in a dynamic system I am computer simulating I get a fully and completely disordered state space. I want to know if this is an indication of chaotic behavior.
Thanks in advance to those who answer !!!
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simple way that a system is said to be chaotic is if it shows a sensitive dependence on the choice of the initial conditions.
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Regarding the analysis of a dynamic system without wavelets and with wavelets, I obtained values. My question is, if I want to find the optimal answer from the obtained values(D parameter), what optimal methods or references are used for the dynamic system?
Also, if I want to analyze the error according to the values obtained (D parameter), what are the methods or references for error detection in dynamic system?
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thanks for the articles.
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Hi,
I am doing research on differential equations but came around the term dynamic system and dynamical system. Some papers says dynamic systems and some other says dynamical systems. I can't figure what the exact technical difference between these two terms or are they just two different words used with the same meaning. Thanks in Advance
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Dear Dr. Muhammad Usman
These two expressions are the same. Maybe your mean is difference equations and differential equations.
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When the solution of a dynamic system is obtained in time series, to analyze that solution in the frequency domain, we can use the power spectrum (one of the system displays in frequency), but MATLAB does not have a Fourier transform signal plot like power spectrum plot for the time domain,
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According to recent results, how could it be analyzed the log log figure of Lorenz system by wavelets ?
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Chaos is a long-term non-periodic behavior in a deterministic system that is dependent
Shows sensitivity to initial conditions.
The operating environment of chaos is dynamic systems. A dynamic system consists of a single phase space or a fuzzy state whose coordinates determine the dynamic state of the system using dynamic laws.
A dynamic system can be Intermittent or chaotic. Dynamic systems (Lorenz-Rossler) are called strange attractors because they are a set of all paths that converge toward a fixed point, a finite loop, or so on.
Attractors are highly sensitive to initial conditions and are called strange because they consist of a fractal set.
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It could be a code which consists lorenz attractor and interpolation points in order to produce these figures,
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Can any body suggest me a straightforward way with low computational cost, for obtaining LCO amplitude and frequency for multi-dimensional phase space?
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Dear Mohsen,
an old but useful way is to use the Describing Function method. If I am not mistaken Dr. Ronilson Rocha has used this method with chaotic circuits and systems with good success.
Regards.
Luis
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With Fourier relations, the Lorenz and Rossler dynamic system was formulated (alpha) based on fixed parameters, and analyzed with fixed parameters to describe the chaotic behavior,
Is it possible to create an image of dynamical systems to analyze the the concept of fractals and chaotic behavior?
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According to the previous discussion (power law transformation method),
After determining the solution area, using nth degree of polynomials , the fitting of the curve is done according to the points on the histogram, and at the end, based on the obtained curve, we can analyze the fractal of the curve.
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I am looking for Python packages which represent a good alternative to Matlab's System Identification Toolbox (or at least for parts of it). It would be great if you could recommend Python packages for linear and nonlinear system identification where you have already gained extensive experience. What are your experiences with e.g. SIPPY or SysIdentPy? Are there better and more comprehensive Python packages?
I especially plan to use the identified models for MPC and NMPC. Thanks in advance!
Best regards,
Günther
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I have two matrices U and W, each containing the eigenvectors of two respective and independent dynamic physical systems. In other words, U and W each span respective subspaces (eigenspaces). My scope is to show that U and W belong to some mutual vector space V.
More specifically:
a) what does the angle between subspace U and subspace W tell me about their relationship in terms of a hypothetical mutual vector space V?
b) if U and W are intersecting, how do I compute the subspace in which the intersection lives?
c) if V exists, how do I compute its basis?
How do I approach a), b) and c) in Python?
Apologies if these questions may overlap eachother. Many thanks in advance!
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From what I understand about your question, the matrices U and W hold subsets of the entire set of eigenvectors (of some systems). If both sets of eigenvectors have the same dimension, they belong to the same space, i.e. the one spanned by the identity matrix of that dimension.
a) The angle between the subspaces tells their relative orientation in the vector space V.
b) To find the intersection, you can assume a linear combination of the columns of U and another linear combination of the columns of W and equate them:
c1 U1 + c2 U2 + ... = d1 W1 + d2 W2 + ...
The solution to this (i.e. the linear combination formed using the coefficients ci or di) gives a subspace that is common to both the given subspaces.
c) The basis of V can simply be the identity matrix of dimension n, where n is the dimension of the vectors in U and W.
The figure shows an example where the eigenvectors (the columns of U and W) form a 2-D subspace of the 3-D vector space V. They intersect along a line, i.e. a 1-D subspace.
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As I know, the ordinary differential equation (ODE), xdot= -x^3+u, where x is the state variable, and u the control variable, is the control system associated to a falling object in atmosphere with viscous drag. I am not sure to be correct on that! Please comment on that!.
Update 1: xdot= -x^3+u, is called the hyper-sensitive system.
c.f.: A Collection of Optimal Control Test Problems: John T Betts.
Another example is velocity control for aircrafts in horizontal flight, which has an ODE evolution:
xdot=-x^2+u. Notice the attachment picked from:
Optimal Control with Engineering Applications; By: Hans Peter Geering.
I want to also know the real model associated to the control system described by the ODE: xdot= x^3+u. I guess more probably, this is associated to electrical systems.
Update 2: My own intuition says, positively damped systems as:
x_dot+x^3=u
are mechanical. Meanwhile, negatively damped systems as:
x_dot-x^3=u
are electrical.
You can yourself find some other examples in this regard.
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Thanks for showing the disturbance suppression capability of your proposed controller. However, I'm unsure what you meant by "injecting a step-function as a disturbance input". The disturbance is a sinusoidal signal.
I have tuned the PD gains so that you can compare both of them meaningfully. The setpoint or the reference target value for the process variable (x) of the double integrator is 1. The double integrator is also loaded with a sinusoidal disturbance of 0.1*sin(t).
x'' = - Kp*(x - 1) - Kd*x' + 0.1*sin(t); x(0) = 0; x'(0) = 0
where Kp = 0.002304, and Kd = 0.096.
The final value of the process variable (x) oscillates within the magnitudes of 1 ± 0.1.
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What are temporal limits in determining brain metastability? In other words, how long does the minimum length of the scan segment using fMRI have to be for the result to be of any value?
I'm particularly interested in measuring metastability during decision making. If I have two different decision conditions, which are shown alternately in the fMRI paradigm (so let's say condition A and B, and paradigm look like it: ABABABAB....), will "slicing" the data and combining them later produce reliable results in the context of metastability estimation (so I could calculate metastability for A and B condition separately)?
I came across a publication where a similar operation was performed (Alderson, TH, Bokde, AL, Kelso, JS, Maguire, L. and Coyle, D., 2020. Metastable neural dynamics underlies cognitive performance across multiple behavioral paradigms. Human brain mapping, 41 ( 12), pp. 3212-3234.). However, it concerned cutting out fixation elements between trials and joining whole blocks. I, on the other hand, want to cut and join the trials, then compare the level of metastability between the two conditions. Is it possible, or not really good idea?
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I hope this article will help you regarding metastability.
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I want to know that we have a real system with fractional order state space model
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Dear Professor Sabatier
First, I'd like to thank you so for asking these kind of challenging questions. These kinds of questions are very important for advancing this field. As you said, our life is full of doubts. Now this question arise that: How can we trust to different integer differential models? The only sin of fractional derivative is that it has been introduced after integer derivative. Why should we accept that the first derivative (u') denotes the velocity? Why another fractional derivative (such u^{0999}) is not?
As we can check in many considerable works, fractional derivatives modify many old results in integer derivatives. As you said, you often use fractional models to build and implement very physical and useful systems (eg: battery state observers, hydrogen generator, ...). We need a comparison in study of these different natural phenomena and so a standard basic theory for the comparison which is usually integer derivative models while we can doubt it. I fee that we could improve our initial mental attitude. Again, I appreciate you.
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Dear colleagues, friends, and professors,
As we know, we have very strong analytical approaches to control theory. Any dynamic decision-making process that its variables change in time could be characterized by state-space and/or state-action representations. However, we see very few control viewpoints for solving electricity market problems. I would like to invite you to share your thoughts about the opportunities, and limitations of such a viewpoint.
Thank you and kind regards,
Reza.
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Hello everyone, I hope you have a good day,
As we all know, the lateral dynamic system of vehicles has two output, lateral error and heading error, and we have one input, which is steering angle, I always have one big problem:
How to Design a Controller to have zero steady-state error, when I have XY reference path?
I designed a controller to track the heading, but when the vehicle gets departed from the path, as it does not have any sense of lateral error, it will not come back to the path, it will just follow the heading with some offset.
I read a lot of papers in this area, but none of them talked about XY reference paths.
I add a photo to clear up some points, please check the attached file.
Thanks in advanced,
Arash
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I am working on system identification for general dynamic systems.
The idea is that I think it can be well-presented using regression using the input and output data.
Do you know anyone who worked on this before, or maybe, you can direct me to published work of something similar?
Thank you very much
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In addition to the mentioned references, I suggest reading the first chapter of a famous book "System Identification; Theory for the user" (Written by Ljung 1999) on pages 1-12, where you find a brief mathematical introduction to the regression method.
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solved
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The fastest way is by fast cosine (or sine) transforms (depending on the boundary conditions). It's just a matter of arranging the iterations.
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Measuring dependency of chaotic systems on initial values by Lyapunov exponent, an original trajectory along with a perturbed one is needed, but I cannot understand the connection with the following article, (fig.3 of the article).
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thanks for your answer.
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Nature or the earth is not static but dynamic in nature. I am very interested to the possible reason behind this.
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Any system that has memory in which the current outputs (aka outcomes) are not mere results of the current inputs but also depend on past inputs and outputs is called a dynamic system. Nature is very well a dynamic system!
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Whath does a pair of equal floquet multipliers signify in stability analysis
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Pl. go through attached article.
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In the paper which is attached in simulation results section, four sigmoid activation functions are introduced but all of them are for scalar input (and using such sigmoids the method just works for systems with one state variable ) so if one decides to apply this controller (Neuro-adaptive controller) to a dynamic system with higher orders(2,3,....),what should be the form of sigmoid activation functions?
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What's the similarity of these systems? How can I identify them?
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Dear Liming,
I suggest you tackle this using root locus (RL). First, for practical reasons, do not think of a purely D control action (which is not implementable). I suggest, for the sake of investigation, to think of a phase-lead controller. If you put the zero of this phase-lead at frequency \omega=0 rad/s, then you have a pure derivative term plus one pole at higher frequencies to end up with a proper rational function (or even two poles, to become strictly proper). Now think of a system with one unstable open-loop pole and draw the RL. Then you immediately see that this phase-lead controller is able to stabilize the system for a sufficiently *large* gain. You can test other classes of systems in this insightful way.
Regards.
Luis
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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.
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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.
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Hi, Is it possible to implement these algorithms for dynamic state problem where the values are not static and continuously changing with respect to time?
And which algorithm is most suitable for dynamic state problem?
Please provide me your valuable answers. Thanks
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I answer your question with a video: https://www.youtube.com/watch?v=JsbFkgJZLyc
Regards!
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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 $\dot_\eta = f(\eta, \xi=\xi_d)$, ?
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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
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I have a nonlinear model that I have applied feedback linearization to. However, the llinearized model no longer had valid physics. Can this issue be avoided? Could this issue be resolved if the dynamics of the system are represented in a different coordinate system?
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normally FL method has 2 parts; first to remove nonlinearity and second for linear part that you will add some terms according to relative degree. I didn't understand what do you mean by "agreement with newton's law", but if your nonlinear system is controllable and you do not have any unstable residual dynamics, this method is working for you.
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Change is nature, and climate is changing. Human capacity in boosting and controlling climate change is still a debatable issue and has the scope of investigation. Whatever the level of human intervention in boosting climate change, human capacity of controlling climate change is very limited. We may slowdown the rate of changing climate but for sure, we cannot prevent it. Climate change is inevitable. The only realistic way to fight against climate change is adaptation.
For a sustainable adaptation, we must understand both positive and negative effects of climate change. Earth is a complete dynamic system and always trying to keep the system in a balance. Hence, when climate change is bringing future challenges for us, it is also bringing equal amount opportunities. For a better future, we must identify both challenges and opportunities.
Can you see any opportunities in climate change?
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I am not sure (I think no one) that climate change is 100% anthropocentric. if we want to think that climate change is 100% , we need to think the entire universe a isolated static system, where only human has the power of influence others. We also should not forget that Human are part of nature. If what we are doing is bad than we must not survive
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In "Simultaneous Stabilization and Tracking of Nonholonomic Mobile Robots: A Lyapunov-Based Approach", the authors stated that the nonholonomy of mobile robots allows control of the system with less control inputs. A mobile robot is underactuated in nature, and it's often controlled with less number of control inputs than the system's degree of freedom (if dynamic is ignored). How does the nonholonomy of the system helps with that?
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Dear Samira Eshghi,
I suggest you to see links and attached files on topic.
Feedback Control of a Nonholonomic Car-like Robot - LAAS
Exponential stabilization of mobile robots with nonholonomic constraints
Control of a Nonholonomic Mobile Robot: Backstepping ... - CiteSeerX
Path planning and control of non-holonomic mobile robots
Best regards
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I'm writing a project of energy reduction in a base station. Please I need guide on how to model a base station with respect to its energy consumption.
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Dear Sunday
you d better analyze the station in energy consumption viewpoint and define the energy consumption of every part. then you can decide whether on part can be decreased or not.
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I have thermal dynamic system for shell and tube heat exchanger but i dont sure wheather from thermal dynamic system can design a form of shell and tube heat exchanger transfer function or not. Can someone help me with these confusion?
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Saad Najeeb Shehab and Abbas J Jubear pls show me the example how to convert thermal dynamic to transfer function
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A) As evolutionary-NeoSchumpeterians (or complexity-oriented) economists, we conceive the economy as a dynamic system in which scattered heterogeneous and boundedly-rational agents interact. Local and global interactions involving feedbacks and domain-specific connections involve producing, investing, consuming, distributing incomes, trading in general, learning, innovating, entry/exit, etc. And the ongoing development of the specifc dynamics we propose to explore a problem generate "EMERGENT PROPERTIES".
B) The methodologies we use range from verbal logical arguments (which of course can be genuinely complex) to complex ABMs, passing through non-linear highly stylized models, replicator dynamics and evolving complex networks with the afore-mentioned components.
C)The specific methodology used is not innocous. Thus, whereas verbal arguments involving real complexity are often almost inestricable, ABMs are a bit more enlightening (the less so the higher the scale), and, in my opinion, the subset of low-scale ABMs, enriched-replicator dynamics, networks and non-linear styled complex models are the best. They often even allow for closed-form quasi-exhaustive analysis.
D) The problem is how should we pass from the results we obtain in our theory, to the posing of policy recommendations to be implemented within a reality which we perceive as emerging from a complex system?
Notice that there are two sources of complexity (2 complex realms involved):
1)The inherent complexity of the real system under scrutiny.
2)The often black-boxed complexity of the theory we propose.
We know that even small differences between two evolving complex systems can make a huge difference in their outcomes. If we assume (as we should) that we can never access the "real complex mechanism underlying reality" (just we should aspire to approach it, at least in social sciences), we should be very prudent in our policy prescriptions.
E) The solution prescribed by those using simple models (mainstream economic models or simple statistical models) is not valid, since they begin by assuming that reality is SIMPLE (instead of complex), and they falsely avoid the problem. Why should social reality be simple in its functioning? The historical record of crisis and social distorsions, and the analogies with natural systems point out to a clear failure of the standard approach. Thus, if we accept complexity:
How do you address the issue of double complextity 1) and 2)?
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Surely this is an extremely interesting and important theme, but I have no enough time today to discuss the points in detail. The following are just my impressions I felt when reading Francisco Fatas-Villafranca and Isabel Almudi 's commnets:
(1) It may not very wise to consider at the same time policy advisers' misconducts and epistemological questions of complex systems and complexity. I understand that you have actual problems on this points, but I feels that epistemology must precede good policy recommendation question.
(2) Fransisco thinks of two realms. I wonder if this is a good framework when we examine complex systems and behaviors in a complex system. I have been thinking that we should better distinguish three "realms" (?) of complexity:
(a) complexity of an objective system (e.g. national or world economy)
(b) behavior of humans in a complex situation (.e.g. how we behave in a complex world.What does the complexity means for us, human beings? )
(c) theory or models of complex systems (e.g. economics and economic of complexity in particular)
(3) Simple rules or behaviors may engender complex processes.
See for example Ronald A. Heiner (1983) The Origin of Predictable Behavior. The American Economic Review 73(4): 560-595.
(4) My general impression vis-à-vis Santa Fe Institute is that they lack a good theory of human or animal behaviors. Fransisco and Isabel may be right when they claim that they are considering too simply the correspondence between real economy and their models. ABM or ABS must be used mainly to understand the working of complex systems, but we should refrain at least at the actual stage of our discipline from using them as means of predictions. Direct application of their simulations is extremely misleading when we do not understand the real process and characteristics of the real system. ABS is a necessary and hopeful mean of economics research but we should also keep in mind that it is still in a embryonic prematured stage.
See my paper: A Guided Tour of the Backside of Agent-Based Simulation.
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Dear colleagues,
I am trying to design a recurrent neural network to predict patients' length of stay as output. I have different types of data (numeric, categorical, free text, etc..) and I want to present the data as a time series( to capture the temporal dynamic of the system). It means I want the model to know when each piece of information is collected.
Here is my question:
1- What are the methods to feed the data into the model?
2- What are the advantages and disadvantages of each method?
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Dear Ali Kamali Mohammadzadeh
I have a research paper presently under review relating to your question. In my opinion, I would suggest that you have 3 inputs and one output to your Artificial neural network (ANN); the inputs being monthly temperature, monthly relative humidity and the corresponding months. Your output would be the monthly electricity consumption. One way of coding the months would be say Jan as ‘1’, Feb as ‘2’, March as ‘3’......Nov as ‘11’ and Dec as ‘12’. Note that including corresponding months as input attributes make much sense considering that the different months significantly influence electricity consumption. Going further, you will have to normalize the 3 inputs to the range 0 to 1 before feeding them as inputs to the ANN.
Input data coding
Note that you will have to normalize the 3 inputs using the range (or maximum) value for each input correspondingly. Never use a single value to normalize the 3 different inputs, as you create bias in the structure of the data. For example, to normalize the month inputs, you could use the maximum value '12' to divide all months input attributes. You find a suitable value for the temperature and relative humidity too, and then proceed with normalization just as for the months input data.
Output data coding
Method 1: You may normalize your desired output data (monthly electricity consumption) to the range 0 to 1, in which case you'd been using Log-Sigmoid activation function in your output layer. If you choose this method, note that you would have to rescale the outputs back to actual values when you test the trained network. I was absolutely fine with this option in my paper.
Method 2: You may use the monthly electricity consumption as they are (actual values) as desired outputs and then use a linear activation function in your output layer. Note that this option may make learning extremely difficult.
Training and Testing ANN
After you have successfully preparing the data. i.e. normalization of inputs and may be output; you'd want to divide your data into training and testing data; optionally, you could have a validation data. Also, select a suitable ration for training data: testing data. One more important point, in order to reduce the chance of the ANN only memorizing (over-fitting) your training data, you'd probably want to randomize the training data first before training.
Hidden layer
You will have to heuristically determine the suitable number of hidden neurons for properly learning the task. For example, increasing the number of hidden neurons gradually. Also, note that the activation function used in the hidden layer should allow the learning of non-linear functions; hence, you could use the Tangent-Sigmoid or Log-Sigmoid function, though I would strongly recommend the later in your situation.
Other network training heuristics
You still have to determine parameters such as learning rate, momentum rate, maximum number of epochs, required MSE (error).
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Imagine we have an ODE system
x_dot=[f1(x,u), f2(x,u), f3(x,u),....fn(x,u)]
where f1,..,fn are nonlinear functions of control input u and states x,
x is member of R^n and u is member of R^m
under which conditions we can change the dynamic of the system to arbitrary dynamic x_dot=[f1_des(x) f2_des(x),...,fn_des(x)]
where f1_des(x) f2_des(x),...,fn_des(x) are arbitrary pre-defined nonlinear functions.
is input to state controllability enough to do that?
Example
consider the 2-D system:
x1_dot=f1(x1,x2)+g1(x1,x2)*u1
x2_dot=f2(x1,x2)+g2(x1,x2)*u2
by choosing
u1=(f1_des(x1,x2)-f1(x1,x2))/g1(x1,x2)
u2=(f2_des(x1,x2)-f2(x1,x2))/g2(x1,x2)
we can change the dynamic of the system to
x1_dot=f1_des(x1,x2)
x2_dot=f2_des(x1,x2)
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Yes, it is. Controllability, or more precise input-to-state controllability, means that you can force all states behave as you desire, even for a single input.
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I am trying to use Model Reference Adaptive Control (MRAC) based on Lyapunov's rule on a system. My question is: should the reference model be assumed or is there a systematic procedure for determining its parameters in relation to the dynamism of the system in question.
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Dear Demilade . . .
Adapting/adjusting the controller parameters for such cases (where plant parameters were not accurately known) is called adaptive control. so as it can be infer, you should have an acceptable approximation of your dynamic system to tune the adaptive control parameters.
recently i have read a paper about this subject which i want to suggest it to you to read it. its about how to design the law for your MRAC.
it might be helpful.
best regards
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It is clear that a dynamic system might exhibit chaotic behavior (complexity), if internal feedback loops (nonlinear dynamics) exist among its interacting components for a certain range of control parameter values and the system is far from thermodynamic equilibrium (e.g. oscillating reactions). My question is: Is there a systematic methodology (algorithm) to find the ranges of values for which the system exhibits chaos?
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For these purposes bifurcation diagrams are often used.
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If I have a dynamic system, what are the differences between linear and rotational disturbances (single and multiple)? Also, what are the differences between disturbances on the dynamic system and disturbances on the motor torque system?
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Ziad Sobih Thank you so much
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In a simplest case imagine we have a continuous finite-dimensional dynamic system described by and ODE
x_dot=f(x) (1) ,
Is it possible to prove the asymptotic stability of (1) by investigating the discrete time version of (1)
x(k+1)=F(x(k)) (2)
((2) might be written by RK4 discretization, Euler first order descretization or any other descretization scheme)
if it is possible to do that, what about infinite dimensional system resulting from space descritization of PDEs?
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Dear Ali,
your question is interesting. In general, the discrete version will depend on two fundamental choices: the discretization scheme used (there are several) and the discretization time. Notice that it is typicaly not sufficient just to replace t for k and \dot{x(t)} with x(k+1). The stability of discretized models will usually depend on the discretization time. This feature is obviously beyond the continuous-time counterpart.
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Task allocation problem is of critical importance in manufacturing industry, and determines the effectiveness and efficiency of advanced manufacturing systems. In order to be adaptable to the needs and future challenges that industry are facing, like rapid changes of demands and requirements, the cost of the entire production system have to be minimized. This performance start by optimizing the sequences of tasks and resource allocations through the system, and this can be handled by using computers optimization algorithms.
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This is essentially the same problem addressed in mu paper "Adaptive submodel selection in hybrid models" doi: 10.3389/fenvs.201500058
The paper describes a strategy for changing the mix of submodels used to represent a complex system in response to the efficiencies and needs of the current components.
This extends the work described in the paper "Increasing model efficiency by changing model representations", which is availabe on researchgate. A mathemarical structure is developed in the supplementary data which might help (also on RG).
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Does this problem have a clear and straightforward algorithmic solution or
the resolution of this problem involve the use of heuristics?
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The bifurcation diagrams and dynamical maps can do the trick. Just consider whole parameters' space of the system. Actually, excluding extra specific cases (e.g. hidden attractors), it is computationally adequate task. Depends on resolution, of course, but some basic knowledge about system dynamics can be obtained easily enough.
I advice to use first more rapid but less precise algorithms instead of direct LLE or Lyapunov Spectrum calculation, which are really computationally heavy methods.
You can find some simple examples here:
Please remember, that in discrete simulations, timestep is also a system parameter:
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My work is mainly experiment-based research. Moving a step further in the advanced analysis, can you please help me with the following questions?
1- Do you think this topic is linked with dynamic systems analysis? if yes: how this analysis should be done?
2- What kind of theoretical analysis (based on differential equations formulation) could be added to my research (especially to the vortex's stability and/or stochastic factors)?
3- What's your best suggestion for making sure that the results obtained (from experiments) are dependable? (Validation by CFD?)
Every single answer is important to me.
Thank you very much.
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Vortex flows are ubiquitous at all scales of matter organization, from quantum systems to large structures of the universe. In the most general mathematical sense, it is useful to look at these structures in a unified way. When trying to organize my ideas in this field, I have encountered a book on the general theory of vortices that I recommend as a valuable source of information placing the subject in a multidisciplinary context; for the synopsis please see:
Regarding the research suggestions, I agree with the previous comments, but I can add some specific answers:
>>Do you think this topic is linked with dynamic systems analysis? if yes: how this analysis should be done?<<
The answer is definitely yes. You can consider the following paper as an illustration of the methods derived from Dynamical Systems Theory.
>>What kind of theoretical analysis (based on differential equations formulation) could be added to my research (especially to the vortex's stability and/or stochastic factors)?<<
The theory of stable and unstable manifolds discussed in the reference above. It is also useful to consult a book by Ottino: The kinematics of mixing: stretching, chaos and transport
>>What's your best suggestion for making sure that the results obtained (from experiments) are dependable? (Validation by CFD?)<<
The best way to obtain reliable results is to set an experiment as carefully as possible. CFD calculations are generally validated by experiment. However, the use of CFD to validate the experimental results is very useful (I always look at numerical simulations as a parallel experiment).
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How can we characterize a chaotic system having two equilibrium points with a negative and two zero eigenvalue?
\lambda_1=-A and \lambda_{2,3}=0.
Is it a non-hyperbolic system? or there is another class of such systems?
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Thanks for your great help Andrey Shilnikov I will investigate this right now
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As we know, coherent groups of synchronous generators can be identified and converted to an equivalent dynamic model to simplify a given power system and still maintain the dynamics of the system. However, the Type-3 and Type-4 wind farms do not possess much inertia and hence cannot replicate the behaviour of synchronous generators. What approaches can still be made to aggregate the wind farms present in a region and connect it to a single transmission bus, such that the dynamics of the system is still maintained? Any help on this would be much appreciated.
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Modern generators - using any source, wind, solar, fossil - are all themselves increasingly complex and use algorithms and computers to 'optimise' their performance. So I don't think you can aggregate them and 'simplify' the result - if you try to do this your results will be deceptive, and will not closely (or at all) match the behaviour of such systems in the real world. Seek instead to define the transfer functions of each farm and come up with specifications of their control algorithms that ensure that this transfer function is maintained across all likely operating conditions. Best of luck!
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Hi,
I have a 3D figure of around 1000 points and the shape of the figure is very similar to a bifurcation. The question is how can I generate the related dynamic equation of these points?
Please let me know if you have any idea about using MATLAB codes as well.
Thanks
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I suggest to fit a multiple (3-pl) regression equation to the 1000 points in 3D by least squares methods or by orthogonal polynomials.
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Dear Dr ,MARIUS-F. DANCA
I wonder I if I can have a thertical backround or Matlab code of Lyapunov exponents of (combined / hybridization /coupled) dynamic systems (Chaotic)
Best regards
Karima. Amara Korba
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There is our last paper with theoretical background of LE computation fro ODE
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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.
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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
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differences of dynamic systems with system thinking
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One is a fraud, the other isn't. Systems thinking, unless you went to MIT, is a cognitive paradigm and useful in virtually any endeavor. If you went to MIT or read Forrester, Sengé, Richmond, et al. then systems thinking is system dynamics. Meadows wrote some useful concepts, but she was the exception in that crowd. The thing they don't tell you in their books is that system dynamics is a very limited, almost useless modelling tool that looks really cool, but can't be modified easily to reflect change or non-linearity. In other words, both the bias and the variance of the model are very high. The verification may be reasonable, but the validity tends to be bad.
Some examples of this failure: Jay Forrester's World Dynamics. I encourage you to read the reviews such as "Measurement without data." Peak oil. This was a system dynamics model crafted at great expense by some really smart economists and petrol-heads and predicted world oil production would peak about 15 years ago. Didn't happen. In fact production is up far beyond the simulation. So, what good was the system dynamics simulation? not much.
For a good insight on what systems thinking really is and how it is useful, check the work of Russell Ackoff. He has some good lectures online. C. West Churchman was his graduate advisor I believe, also worth reading.
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I used the lagrangian energy methods and reach the ODE equations of motions and want to submit them to numerical integration in MATLAB to model the dynamics of the system.
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Once you have the ODEs, then submit it to the Mathematica function NDSolve. You can immediately make plots, animations, whatever you wish.
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Hello,
This project aims are to address the theory of dynamic systems from the pedagogy point of view or these intend to study the possibilities for the reformulations, for the re-conceptualizations ... of pedagogy from the perspective of the theory of complex systems?
In any cases, I thing that this project is very interesting and useful too for the knowledge society.
If I misunderstood, please give me some details about the objectives of your proposed project.
Sincerely,
Bogdan Nicolescu
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Thank you, Bogdan!
I am very happy when I have found even one person from the world who understand symstems theory and systems thinking and even the fact that for understanding pedagogy and education - so many intensions and extensions at the same time - sustemic views are needed. Otherwise only a small details ase seen - and it means narrow way of thinking.
sincerely yours,
professor Ulla Härkönen
Finland
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How can I make a optimization code in Matlab for tuned mass damper?
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I suggest you at the first step see the "Vibration Absorber" parts in the classic books for mechanical vibration (for example part 9.11 in Mechanical Vibration by Rao). Then read the following paper:
Sadek, Fahim, et al. "A method of estimating the parameters of tuned mass dampers for seismic applications." Earthquake engineering and structural dynamics 26.6 (1997): 617-636.
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Hi guys,
As you know when predicting multi step ahead if there is even a small error between actual and predicted one at the beginning, this error will be propagated through all predicted values so i want to know if there is any way to reduce this error.
Thanks
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Dear Kazemi,
Please follow the referred paper below:
Boné, R., & Crucianu, M. (2002). Multi-step-ahead prediction with neural networks: a review. 9emes rencontres internationales: Approches Connexionnistes en Sciences, 2, 97-106.
Added with this, you may also follow some researches, like:
[1] Bone, R., & Crucianu, M. (2002, November). An evaluation of constructive algorithms for recurrent networks on multi-step-ahead prediction. In Neural Information Processing, 2002. ICONIP'02. Proceedings of the 9th International Conference on (Vol. 2, pp. 547-551). IEEE.
[2] Bone, R., & Crucianu, M. (2002, November). An evaluation of constructive algorithms for recurrent networks on multi-step-ahead prediction. In Neural Information Processing, 2002. ICONIP'02. Proceedings of the 9th International Conference on (Vol. 2, pp. 547-551). IEEE.
Thanks,
Sobhan
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Hello Guys,
Suppose that we have a data from a chemical process. There are some inputs and one output, the output depends on those inputs and also the time. I am gonna identify this process. So what is the difference between using simple MLP with Lagged output as an input (lagging the output as an input to the model, like 5 step before) and using recurrent neural network such as ELMAN or JORDAN without lagging the output, because i know JORDAN is just feeding back one step before as an input.
Thanks
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Thank you all guys for nice information.
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One of the central themes in Dynamical Systems and Ergodic Theory is that of recurrence, which is a circle of results concerning how points in measurable dynamical systems return close to themselves under iteration. There are several types of recurrent behavior (exact recurrence, Poincaré recurrence, coherent recurrence , ...) for some classes of measurability-preserving discrete time dynamical systems. P. Johnson and A. Sklar in [Recurrence and dispersion under iteration of Čebyšev polynomials. J. Math. Anal. Appl. 54 (1976), no. 3, 752-771] regard the third type („ coherent recurrence” for measurability-preserving transformations) as being of at least equal physical significance, and this type of recurrence fails for Čebyšev polynomials. They also found that there is considerable evidence to support a conjecture that no (strongly) mixing transformation can exhibit coherent recurrence. (This conjecture has been proved by R. E. Rice in [On mixing transformations. Aequationes Math. 17 (1978), no. 1, 104-108].)
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For “the definition of coherent recurrence (for measure/ measurability-preserving transformations) ” see, e.g., in: 1) [P. Johnson and A. Sklar, J. Math. Anal. Appl. 54 (1976), no. 3, 752-771], 2) [R. E. Rice, Aequationes Math. 17 (1978), no. 1, 104-108], 3) H. Fatkić, “O vjerovatnosnim metričkim prostorima i ergodičnim transformacijama (with a summary in English)” on ResarchGate; 4) [ B. Schweizer, A. Sklar, Probabilistic metric spaces, North-Holland Ser. Probab. Appl. Math., North-Holland, New York, 1983; second edition, Dover, Mineola, NY, 2005].
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Time constant is important in control engineering applications. Is there any mathematical concepts that can be used to calculate time constant from system mathematical model?
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Thanks you。
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I am new to membrane dynamics simulation. In recent days i learnt to build a membrane dynamics system, but i doubt whether my system is properly made or not. I checked results of energy minimization, NVT, NPT and solvation layer too, everything is fine. But when i am performing a protein of 300 residues in lysozyme in water, the efficiency of my workstation is it can run 7.5 ns per day. For the same protein, when i am making the membrane simulation system, the efficiency of my workstation becomes it can complete 12 ns per day. Is this a common thing or a strange situation, where i might have commited some mistakes in preparing the system. Hoping for good suggestions and thanks in advance.
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Ya thank you so much, now i understood the reason for longer time and things i should consider for my simulation.
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I am interested in studying the dynamics of a lipid vesicle. Surface Evolver seems to be a great program and quite extensible. However, as I understand it, the program only minimizes the total energy of the surface, and cannot be used to study the evolution (ironically) of the surface over time.
Are there any available programs, packages, or even libraries for programming languages, with similar capabilities as Surface Evolver to model the surface dynamics of systems with various topologies, and types of energies?
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I am interested in simulating the dynamics of membranes. However, the library you had sent me seems to focus on analyzing previously done simulations coming from MD programs.
@Alan I am not necessarily minimizing the energy. I am actually interested in using Monte Carlo simulations for studying dynamics and on the future some MD methods. What I really liked about Surface Evolver is that a surface can be defined and is messed and the energies are easily defined. None of the available MD programs I have seen have a triangulated surface that can be simulated and the energies provided by these programs care not meant for membrane simulations
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How far is it feasible to use an optimally tuned proportional-derivative controller to control a system with fast dynamics?
I understand PD controller suits best for the slow dynamic system.
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PD (or PID) are pretty widely used, because many systems out there already are stable or close to stability, so a constant proportional gain P makes or keeps them stable, an appropriate D gives enough damping to avoid eventual oscillations and an integral term I eliminates the steady state error (as it makes the zero-frequency gain infinite).
However, such systems do not require fast response and so, more complex or mode demanding systems first of all require some more analysis and then require using more complex control design and controllers.
Add to it that the actual parameters could differ or vary from their nominal values that have been used for design and so, in order to maintain performance and even mere stability, this requires using some sort of robust control (with fixed controller parameters) or robust adaptive control (which changes the control parameters on-line, in order to fit them to the specific situation).
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Let's assume we have a dynamic system and we want to identify its unknown parameters in a real time procedure. Observability and identifiability are important for convergence of the estimation process. How can we ensure these two important characteristics?
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I think you have to begin be stating your setting and the your problem more precisely. If I get you correctly, you consider a system with output
dx/dt = f(x,u,p), x(0) = x0
y = h(x)
and parameters p. Now, I understand you want to construct an algorithm that uses knowledge about the input u(t) and the output y(t) to estimate the parameters p. Note: dynamics/processes can be identified, parameters are usually estimated.
Now, to answer your question you need to be more clear what you want:
1. If the exciting input u, the measurement or output map h: x --> y are fixed, one cannot ensure observability or identifiability. All you can do is test for these properties. The conditions depend on the structure of the dynamics and the dependence of the solution x(.; x0, u(.)) on the parameters p. Standard textbooks discuss how to do this for linear and nonlinear settings.
2. If the exciting input u is not given and the measurement or output map h: x --> y and the parameters are fixed, one could try to ensure observability by enforcing that in the vicinity of the resulting trajectory an observability criterion is satisfied. This has, for example, been used in the context of MPC in:
  • Böhm, C.; Findeisen, R. & Allgöwer, F. Avoidance of Poorly Observable Trajectories: A predictive control perspective Proc. 17th IFAC World Congress, Elsevier, 2008, 41, 1952-1957
3. If the exciting input u is not given and the measurement or output map h: x --> y is fixed, you can try to compute trajectories which maximize (loosely speaking) the information contained in the output measurements, see:
  • Körkel, S., et al. "Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes." Optimization Methods and Software 19.3-4 (2004): 327-338.
4. In case that you want to identify parameters p* such that
dx/dt = f(x,u,p*), x(0) = x0
y = h(x)
is observable, you can again try to formulate observability criteria as a constraint of you (optimization-based) parameter estimation procedure, similar to point 2. above.
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Dear all,
I am trying to stabilize an inverted pendulum on cart. I was able to derive the dynamics of the system using Lagrangian mechanics and have a working simulation (I'm using LQR controller for stabilization). However, I have a question when it comes to actually implementing my code using a microcontroller. Here is what I am trying to do:
The laptop implements the LQR controller and passes the control information (u(t)) to the microcontroller (via serial communication). Now, this is where I have a problem. How do I relate the torque information to some appropriate PWM signal so that the microcontroller could pass it to the motor attached to the cart? Am I supposed to do torque control or speed control here? Any help will be much appreciated. Please feel free to ask questions for any clarification.
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Dear G.s. Vasantha Kumar,
Thank you for your response. I really appreciate it.
Dear Elmer,
The papers you mentioned are very useful to me. Thank you for letting me know about them.
Kind regards,
Chaitanya
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The delayed dynamical system is as follows:
dx/dt= -Betta*x(t)+F(x(t))+(Betta-1)F(x(t-d)),
x(t)=Phi(t) for t in[-d, 0] where d is delay and F(x(t)) is a projection function. I have Matlab code (using ODE45) for this delayed dynamical system without delay, but when the delayed added I don't know how I can insert the delay.
I would be very grateful if someone could help me to design Matlab code for such delayed dynamical system?
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Dear Israel Tankam,
Thank you very much for your complete and valuable answer. I will use it.
Kind regards,
Javad.
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The scope of environmental risk assessment has been expanded from the traditional chemical and accident assessment to the inclusion of the potential harm from artificial introduction of species, both natural and genetically modified, into an ecosystem. Thus major categories of environmental hazards include chemical, physical, biological, and/or their combinations. Environmental entities can be more complex with respect to system structure and functions, although the methodologies are largely transplanted from human health risk assessment. Environmental risk assessment processes would involve problem formulation, characterization of exposure and effects, and risk characterization. The assessment can be made retrospective or predicative, depending on the risks involved and management requirements. Thus, is it possible to implemented the dynamic system methods to risk management in order to prevent environmental damage?
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Dear Dino Rimantho
Dynamic system is relation between stimulus and responds. If we know characteristic of the system we can predict the respond, especially related with the stability of the system, stability or chaotic responds. Really if we give input reference to the system we expected the result is the same as input references, if the result is different it is mean the system has a failure. Using risk management method using ISO 31000 or ISO 14971 we can analyze failure mode of your system, effect of this failure (level of severity) , probability of this failure and how to detect.
Salam
Susanto
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If xdot=Ax+Bu is a state space representation, what would a column of zeros in the A matrix depict in particular other than the system being singular in nature.
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Consider the following simple example: A = [0 a1; 0 a2], B = [b1 b2]'
1) det (sI - A)= s (s - a2)  => p1 = 0 and p2 = a2
2) det Co = det ([B | A*B]) = b2 ( b1 * a2 - b2 * a1)
From (1): the system has a pure integrator, but the order is 2 and it cannot be reduced.
From (2): the controllability depend of B, if b2 ≠ 0 and b2 ≠ (a1/a2) b1, the system is completely controllable independently of the zero column. 
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In control theory field, the rank of the controllable matrix is for investigating the control ability of the dynamic systems, and that the matrix is full rank means that the systems is the fully state controllable, and then the bigger the rank is, the stronger the control ability of the systems is. Up to now, no precise concept on the control ability is presented.
In fact, the rank of the controllable matrix and the controllablilty are not good measures on the control ability. For example, for the uncontrollable matrix pair (A,B), a little perturbation on the matrix A or B may lead to that the matrix pair is controllable. So, there isn't exist a distinct line between the controllability and uncontrollability concepts and the two concepts are fuzzy. Considered that the modeling error for the system models of the practical dynamic systems, the rank of the controllable matrix and the controllability aren't appropriate for investigating precisely the control ability of the practical dynamic systems.
In the studying on the precise measure on the control ability of the dynamic systems, the volume value of the controllable region under the unit input energy is propose as a new precise measure on that in my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems). A new theorem on the positive relations among the volume value, the diversity of the control laws (i.e., the size of the solution space of the control laws) in the controller designing, and the performances and robustness of the closed-loop control systems is presented and proven in the paper. That is, the bigger the volume value of the controllable region, the richer the control laws, and then t the closed-loop control systems may be with he better performance and robustness. The volume value is a good measure on the control ability.
How to name the new measure? Which name is appropriate for that, controllable (or controllability) power, controllable (or controllability) abundance, controllable (or controllability) richness, or others? Which colleague gives some helpful suggestions.
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In my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems),a new measure on the state control ability of the linear discrete systems is presented as an improving and promoting concept of the state controllability in the control theory field. The new measure is not only used to distinguish the controllability or uncontrollability, but also can be used to meter precisely the state control ability of the systems, and maybe is a fundamental concept and index to the dynamic systems.
Based on the new measure, many works in the control field and other related fields can be developed and new results will be with the significant value. Some works related to the controllable abundance can be listed as follows.
1. By computing and optimizing on the controllable abundance, the dynamics and kinematics of the dynamic systems can be improved and promoted, such as, enhancing the control ability of the input variable, extending the controllable region in the state space, and so on.
2. By computing and comparing on the controllable abundance, the input variables with the better control ability are determined from the candidate variables in the modeling of the controlled plants and the control efficient of the control systems can be promoted.
3. Based on the relation theorem among the control abundance, the control ability, and the size and richness of the solution set of the control laws in the controller designing, when the controlled plants are optimized on the controllable abundance, the closed-loop control systems designing by the conventional control methods can be with the better performance and robustness.
4. The controllable abundance can provide the theoretical basis and method for determine the control horizon, the optimization horizon, the target state, the reference trajectory in the many control methods , such as, optimal control, adaptive control, predictive control , receding horizon optimal control, and so on.
5. The controllable abundance can be used as the objective function (performance index) or the constrained conditions in the optimal control and robust control, and optimizing the control abundance can be promoting the performance and robustness of the closed-loop systems.
6. As the generalization of the controllable abundance concept, the observable abundance can be proposed to meter the observe ability of the dynamic systems and is used to promote the state filtering systems and communication systems.
More meaningful works about the controllable abundance can be proposed and developed, after deeply understanding the new measure and concept.
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Hello!
As a part of my research on urban transportation, I am using System Dynamics as a tool. If anyone has experience in System Dynamics / Systems Thinking, software tools for System Dynamics or its application on transportation systems, please get in touch with me.
Main Purpose -
1. Make friends from same working background :)
2. I want to engage in discussions with a peer
3. Get some inputs on selecting suitable software package
Thanks,
Madhur Jain
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@Babak Bashirzadeh - Hi Bashir. Thanks for your offer. I see that you use Vensim. Could you please suggest on if Vensim can accept Python or Matlab coding ?
Thanks.
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Following a good tradition of asking for examples of specific systems (non-lin., non-min-phase etc.) I'd like to ask if somebody could give me an example of a simple but still physically relevant LTV system. 
Most textbooks contain examples with terms like t*exp(-2t) and so on, which are clearly artificial. Students are normally not very excited about dealing with such problems (which I find completely reasonable). I thus wonder if there are any examples which stem from real problems, but can be addressed within the framework of a class.
I'd be particularly interested in non-periodic cases, but the periodic ones are also welcome. Any references will be very helpful. 
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Dear Alexander,
1.  You are right:
-- the resistance R is not linearly dependent on V,
-- is time varying.
But this is not the issue. The equation sought should be a linear differential equation for a chosen observed quantity ( i, V , R, C, or whatever else) replacing x in the equation
(*)  d{x(t)}/d{t} = A(t) x(t),   (the LHS denotes the derivative of x with respect to t )
AND  the coefficient A(t) should be independent of anything but t . In your model such a quatity and such an equation  is not distinguished yet.
2. There is also the possibility, that  any other variable is chosen for the independent variable (instead of  t ). Even, one can imagine an equation of the following form d{R(V)}/d{V} = Q(V) R(V) .  I am not developing this point of view.
3. The solution of  (*)  equals   x(t) = x(0) \exp[ \int_0^t \, A(u)\, du ]
4.   Function R(t)=(wt+1)/wC is a solution of the fo