When applying the nonlinear consensus algorithms, most of them are not model-based. Is there any model-based consensus algorithm for a system with arbitrary nonlinear dynamics?
Not that I know of. In fact it would be very surprising if a model based on a consensus could be formulated in a nonlinear dynamic system.
A possible approach would be consider many simultaneous realizations of the same phenomenon, say 10 traders buying and selling the same financial index for instance S&P future index and then consider all the data generated as a multiple dimensional time series and apply an estimation procedure to determine the functional form of the system and the parameters. This would consists in solving an estimation problem, although the estimates might be highly uncertain.
On this line, it is possible to make some products and look through my publications, if you conclude that this approach is in line with what you want from a model based consensus approach.
Yours
Giacomo Patrizi
I am not particularly sure what do you mean by 'model-based consensus approach available for a nonlinear dynamic system'. Please elaborate.
I my group we have been building nonlinear models of for various problems for years.
The affirmation " ... model based consensus ..." was formulated by Ehsan Omidi of the university of Alabama, so you should ask him regarding what he meant.
I interpreted the question as unlikely, which was my first paragraph in my note and discussed a possible approach for a problem indicated in the second paragraph.
To start consider a time series which can be a nonlinear dynamic system and if you want to predict future values you have to obtain a nonlinear dynamic function to carry out predictions. Often this is done by Dynamic chaotic system formulations.
Now consider many realization of the same series, as indicated and the numerous dynamic chaotic systems formulations should be aggregated into one formulation which can be used in trading, but the unification will require an estimation procedure, so this can be interpreted as a model based consensus approach to handle non linear dynamic system, (Q.e.d.)
In my group we have been building nonlinear models for many different types of implementations and are presented in published literature, even in Research gate, and so I am looking forward to scientific discussions on what we do respectively, if it is of interest. The point is that 2 aspects must be analyzed mathematical and statistics and then integrated into one.
Yours Very Sincerely,
Giacomo Patrizi
By the model-based nonlinear consensus approach I mean that for a system such as x_dot(t)=f(x(t), t)+u(t), the consensus protocol of u(t) is also somehow dependent on the definition of f(x(t), t), and not blindly on regular or nonlinear protocols that are commonly being used for linear consensus models.
I do not understand your reply. You are seeking a consensus in multiple realization of a phenomenon to find an optimal control of the aggregate dynamic system. You will find some theoretical and application results in papers authored by myself and published. I hope you will find all you need or tell me more accurately what you need.
I wonder if your question relates to what people call the emergence of a synchronisation manifold. The original model was of a (large) set of oscillators weakly coupled together, and the aim was to find out when they synchronise. One background is that people are interested in macroscale waves in the brain arising from microscale neuron oscillators. I list five articles in my database related to the synchronisation manifolds--maybe one or two may be useful to you.
[1] D. M. Abrams and S. H. Strogatz. Chimera states in a ring of nonlocally coupled oscillators. Int. J. of Bifurcation and Chaos, 16(1):21–37, 2006.
[2] A. C.-L. Chian, R. A. Miranda, E. L. Rempel, Y. Saiki, and M. Yamada. Amplitude-phase synchronization at the onset of permanent spatiotem- poral chaos. Physical Review Letters, 104(254102):1–4, 2010.
[3] I. Chueshov and B. Schmalfuss. Master-slave synchronization and invari- ant manifolds for coupled stochastic systems. Journal of Mathematical Physics, 51(102702):1–23, 2010.
[4] S. J. Moon, R. Ghanem, and I. G. Kevrekidis. Coarse-graining the dy- namics of coupled oscillators. Technical report, http://arXiv.org/abs/ nlin.AO/0509022, 2005.
[5] J. Sun, E. M. Bollt, and T. Nishikawa. Constructing generalized syn- chronization manifolds by manifold equation. Technical report, [http: //arXiv.org/abs/0804.4180v2], Feb. 2009.
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