Nonlinear Systems

What are the state of art of the estimation methods?

I know that the title of question covers a wide area. Currently I am trying to apply Kalman Filter to estimate some parameters of a highly nonlinear system. I took a look at the papers and articles, but I could not see any obvious advances.

If anyone can recommend me a book or a recent review article which includes and explains the state of art estimation methods, I will be glad.

Altan Onat · University of Pardubice

Thank you all who shared his/her opinion and knowledge. I appreciate all the knowledge and experience.

I think I need to be a little more specific. For my Phd thesis I am searching of application of estimation methods to estimate contact conditions between wheelset and rail. I am currently working on wheelset dynamical model, and I know that there exists some fairly new research about it.

In the publication below

"Hussain, I., Mei, T. X., & Ritchings, R. T. (2013). Estimation of wheel–rail contact conditions and adhesion using the multiple model approach. Vehicle System Dynamics, 51(1), 32-53."

They already applied "Kalman Filtering" for estimation of contact conditions based on the dynamic model of a wheelset. However in this work they just linearised the wheelset dynamics around operating point and then applied conventional Kalman Filter. Besides in order to estimate contact conditions Kalman Filter is not enough, so that they used Fuzzy Logic to interpret residuals of Kalman Filter.

I just wanted to know if such kind of thing can be done without linearising the dynamical model and without using post processing of data (i.e. fuzzy logic)?

If someone knows such a method (I think I can do it using an unscented Kalman Filter, but I am not sure) and if he/she can share it with me, it will be greatly appreciated.

Thank you again.

Which tool you prefer for non linear analysis and why?

I just want to know which tool is used more for the fea non linear simulations. And the reason behind why we use that tool.

Miroslav Halas · Slovak University of Technology in Bratislava

I would recommend you either Matlab/SIMULINK or Maple, depends on whether you are more interested in the simulations or in the computations.

If you only need to simulate how nonlinear systems behave, without for example knowing really a solution to some nonlinear equations, then Matlab is better.

If you need to find for example an (analytic/numeric) solution to some nonlinear (differential) equation(s), Maple is more suitable.

Is there any formal definition of superquadratic convergence?

The formal definitions of linear, superlinear  and quadratic convergence order are clear in the context of the convergence of the solution of nonlinear systems of equations. Recently, I have come across some works on superquadratic convergence, but I can't find the formal definition of the convergence order (superquadratic) just like we have in the rest of the existing convergence order.

NB: I need a formal definition.  Definition like the ones given in this text, see the link below:

How to solve a highly nonlinear system of equations numerically with high efficiency? Is there a recommendation on any software package?

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

Cahit Köme · Nevşehir University

Dear @Daniel_Bachrathy, have you ever use your nonlinear solver on the solution of differential equations with an implicit method?. I'm just wondering that is it really better than newton-raphson method while solving nonlinear algebraic system of equtions appearing in implicit methods?

• Hugh Lachlan Kennedy added an answer:
Does anyone have experience with an extended kalman filter without process noise?

I am currently working on an extended kalman filter for a nonlinear system.

And I am very confused.

Assume there is a nonlinear system without process noise but with measurement noise. Then when using EKF to estimate the state of this system, how can I handle Q in EKF.

For example:

x(k+1)= x(k) ; or x(k+1)= x(k) + u(k)

and observer equation
y(k)=x²(k) + v(k)

u(k) is the input and v(k) is the measurement noise.

When EKF is used for the estimate of the state, should I set Q=0?

Hugh Lachlan Kennedy · University of South Australia

Using a small Q matrix  (for nice smooth estimates)  is OK in simulations where your filter model (ideally) matches the process model; though you may need some process noise to help recover from linearization errors (in an EKF). In practice though, using plenty of process noise is a good idea to prevent divergence over time, because your filter models/assumptions are likely to be a poor approximation of the actual measurement and process dynamics.

For a DC3 dynamical system (X, f), is there an uncountable subset S of X such that any two different points of S are a DC3 pair?

Anthony Rosato · New Jersey Institute of Technology

Dear Dr. Li:

While I am unable to answer your question, let me refer you to my colleague (Prof. Blackmore) who is an expert in dynamical systems. His e-mail address is:  denis.l.blackmore@njit.edu

Sincerely,

Tony Rosato

Can one identify complex Four Wave Mixing (FWM) parameters experimentally?

The governing equation of the Manakov model with complex FWM terms is of the following form

1 q1_t + q1_xx  + 2 (a|q1| + c |q2| + b q1q2^* + d q2 q1^*) q1 =0

1 q2_t + q2_xx + 2 (a|q1| + c |q2| + b q1q2^* + d q2 q1^*) q2 =0

Stam Nicolis · University of Tours

The immediate answer is that it's impossible to answer the question, as posed, since the experimental procedure is not defined. If one asks a related question, however, what are the parameters, whose values cannot be rescaled arbitrarily, then dimensional analysis is useful tool and can lead to an answer, that is useful for numerical work, for example and experiment, also.

How can I extract Abaqus data (U or V) from a selected result file (.sel) in Abaqus?

How can I extract Abaqus data (U or V) from selected result file (.sel) in Abaqus? I would like to analyze these data in every time increment when analysis is running in Abaqus/Explicit.

How to check this formula S(r,r0;t)=S(r,r0_bar:t)-(r0-r0_bar)p_s, here p_s can be seen as the initial momentum of S(r,r0_bar:t) with Symplectic maps?

If we consider a classical action from r0 to r in terms of t(time), then for the semiclassical calculation, I can consider a small wavepacket and make an expansion around r0_bar.
Based on classical mechanics, we can have this formula S(r,r0;t)=S(r,r0_bar:t)-(r0-r0_bar)p_s, here p_s can be seen as the initial momentum of the orbit of S(r,r0_bar:t), this is
indeed out of any problem. But I want to check this formula with standard map or kicked top, but I find I can not make sense in this model. As the discrete characteristics, action difference is large as the two orbits should go to the same final position r in the same time. But this formula should fit for the Symplectic maps, so what is wrong with my idea?
I am working on this seemingly very easy job for quite a few days, but now it seems hopeless to find the reason. Maybe I have a lack of some knowledge here. So I ask for your help and very glad to hear some good suggestions from you.  I give the related paper for the background with the formula(15) and (24).

Wenjun Shi · 宁德师范学院学报

Dear Manuel Morales, maybe you can check this formula with your idea, if you have some interest, ok? such as using the very simple model-kicked rotator. I am glad to hear your good news and I am very pleased to see that result. Best!

• Pedro Martín Merino added an answer:
Does the general proof of convergence for Quasi-Newton's matrix exist?

I am wondering whether their is any general proof of the convergence of the Broyden's update matrix. Broyden (1965) discovered a new method for solving nonlinear systems of equations by replacing the Jacobian matrix in Newton's method with an approximate matrix (Broyden's matrix) that can be updated in each iteration.

Dennis and More (1974) gives the characterization of the superlinear convergence of quasi-Newton's method based on the convergences of the sequence generated by quasi-Newton's iterates. There is an assumption for the convergence of the approximate matrix, but to the best of my ability I can't find any attempt in showing that the Broyden's matrix converges to the true jacobian at the solution.

Please I need expert suggestions/answer to my question.

Pedro Martín Merino · Escuela Politécnica Nacional

Perhaps you may also want to have a look at these references

• R.H. Byrd and J. Nocedal, "A tool for the analysis of quasi-Newton methods with application to unconstrained minimization," SIAM Journal on Numerical Analysis 26 (1989) 727-739.
• R.H. Byrd, D.C. Liu and J. Nocedal, "On the behavior of Broyden's class of quasi-newton methods," Report No. NAM 01, Department of Electrical Engineering and Computer Science, Northwestern University (Evanston, IL, 1990).
• How can I calculate interevent intervals (IEI) for a non-autonomous nonlinear system?

This is used to find event rates.

Gopalakrishnan Jayalalitha · RVS College of Engineering and Technology Dindigul-5

using Time series Analysis or Trend Analysis for IEI

Are there apriori conditions to prevent multicollinearity on nonlinear model ?

The issue is: may we find any apriori (necessary and/or sufficient) conditions on a given nonlinear model: y = f(X,β) + e

in order to avoid multicollinearity of the approximated Hessian matrix of the error function (i.e. to get the Newton direction accessible during the iterative procedure of optimization) ?

Deleted · Wasit University

ultivariate analysis of a structural theory, one that stipulates
causal relations among multiple variables. The causal
pattern of intervariable relations within the theory is specified
a priori. The goal is to determine whether a hypothesized
theoretical model is consistent with the data collected
to reflect this theory. The consistency is evaluated through
model-data fit, which indicates the extent to which the postulated
network of relations among variables is plausible.
SEM is a large sample technique (usually N > 200; e.g.,
Kline, 2005, pp. 111, 178) and the sample size required is
somewhat dependent on model complexity, the estimation
method used, and the distributional characteristics of observed
variables (Kline, pp. 14–15). SEM has a number of
synonyms and special cases in the literature including path
analysis, causal modeling, and covariance structure analysis.
In simple terms, SEM involves the evaluation of two models:
a measurement model and a path model. They are described
below.
Path Model
Path analysis is an extension of multiple regression in that it
involves various multiple regression models or equations that
are estimated simultaneously. This provides a more effective
and direct way of modeling mediation, indirect effects, and
other complex relationship among variables. Path analysis
can be considered a special case of SEM in which structural
relations among observed (vs. latent) variables are modeled.
Structural relations are hypotheses about directional influences
or causal relations of multiple variables (e.g., how
independent variables affect dependent variables). Hence,
path analysis (or the more generalized SEM) is sometimes
referred to as causal modeling. Because analyzing interrelations
among variables is a major part of SEM and these interrelations
are hypothesized to generate specific observed
covariance (or correlation) patterns among the variables,
SEM is also sometimes called covariance structure analysis.
In SEM, a variable can serve both as a source variable
(called an exogenous variable, which is analogous to an independent
variable) and a result variable (called an endogenous
variable, which is analogous to a dependent variable)
in a chain of causal hypotheses. This kind of variable is
often called a mediator. As an example, suppose that family
environment has a direct impact on learning motivation
which, in turn, is hypothesized to affect achievement. In this
case motivation is a mediator between family environment
and achievement; it is the source variable for achievement
and the result variable for family environment. Furthermore,
feedback loops among variables (e.g., achievement can in
turn affect family environment in the example) are permissible
in SEM, as are reciprocal effects (e.g., learning
motivation and achievement affect each other).1
In path analyses, observed variables are treated as if they
are measured without error, which is an assumption that
does not likely hold in most social and behavioral sciences.
When observed variables contain error, estimates of path coefficients
may be biased in unpredictable ways, especially for
complex models (e.g., Bollen, 1989, p. 151–178). Estimates
of reliability for the measured variables, if available, can be
incorporated into the model to fix their error variances (e.g.,
squared standard error of measurement via classical test
theory). Alternatively, if multiple observed variables that
are supposed to measure the same latent constructs are
available, then a measurement model can be used to separate
the common variances of the observed variables from
their error variances thus correcting the coefficients in the
model for unreliability.2
Measurement Model
The measurement of latent variables originated from psychometric
theories. Unobserved latent variables cannot be
measured directly but are indicated or inferred by responses
to a number of observable variables (indicators). Latent
constructs such as intelligence or reading ability are often
gauged by responses to a battery of items that are designed
to tap those constructs. Responses of a study participant to
those items are supposed to reflect where the participant
stands on the latent variable. Statistical techniques, such
as factor analysis, exploratory or confirmatory, have been
widely used to examine the number of latent constructs underlying
the observed responses and to evaluate the adequacy
of individual items or variables as indicators for the latent
constructs they are supposed to measure.
The measurement model in SEM is evaluated through con-
firmatory factor analysis (CFA). CFA differs from exploratory
factor analysis (EFA) in that factor structures are hypothesized
a priori and verified empirically rather than derived
from the data. EFA often allows all indicators to load on all
factors and does not permit correlated residuals. Solutions
for different number of factors are often examined in EFA
and the most sensible solution is interpreted. In contrast,
the number of factors in CFA is assumed to be known. In
SEM, these factors correspond to the latent constructs represented
in the model. CFA allows an indicator to load on
multiple factors (if it is believed to measure multiple latent
constructs). It also allows residuals or errors to correlate (if
these indicators are believed to have common causes other
than the latent factors included in the model). Once the measurement
model has been specified, structural relations of
the latent factors are then modeled essentially the same way
as they are in path models. The combination of CFA models
with structural path models on the latent constructs represents
the general SEM framework in analyzing covariance
structures.
Other Models
Current developments in SEM

Is linear analysis in controller design tool (in MATLAB simulink) applicable to unstable systems?

Hi every body,

I have a nonlinear system model in simulink. This system is unstable and I want to design a controller to stabilize it. Due to this, I want to linearize this model and then design a stabilizer controller. To do this, I use the linearization ability of simulnk, but Is linear analysis in controller design (in MATLAB simulnk) applicable to unstable systems ?

Davood Shaghaghi · Khaje Nasir Toosi University of Technology

thanks to all.

What you think about this mathematical question on quasiconvex functions in Morrey sense?
Let $L$ be convex function with superlinear growth and let $L_k$ be an increasing sequence of convex functions with superlinear growth which converges locally uniformly to $L$. Let $F$ be $C^{\infty}$ function with compact support.

Is then this true that $(L_k + F)^{qc} \to (L + F)^{qc}$ pointwisely?

The answer is yes, when $L$ has $p$-growth. However, I am interested in more general cases. Please, let me know what you think about this problem.
Paolo Marcellini · University of Florence

Date: Tue, 20 Aug 2013 12:01:49 +0200
From: Paolo Marcellini <marcellini@math.unifi.it>
Subject: Re: a math question
To: Mikhail Sychev <masychev@math.nsc.ru>

Dear Mikhail,

The $p$-growth condition is fundamental for quasiconvex functions. I do not
have now a final answer to the question, however I presume that some growth
of Orlichz type would be enough, as well as $p$-growth. To my knowledge, a
monotone sequence $L_{k}$ approximating $L$ has been studied in

http://web.math.unifi.it/users/marcellini/lavori/reprints/1985_Marcellini_Manuscripta_Math_Approximation_of_quasiconvex_functions_and_lower_semicontinuity_of_multiple_integrals.pdf

with a big effort to treat perturbations and with a strong use of the
$p$-growth assumption.

With best wishes
Paolo
________________________________________________________________________
Paolo Marcellini
http://www.math.unifi.it/users/marcellini

What should I do if my Fuzzy sliding mode controller does not provide stability to the system?

Fuzzy sliding mode controller does not provide stability to the nonlinear system. how we can solve it, if SMC be stable.

Eric Pigeon · Université de Caen Basse-Normandie

I think the same things that Nadege Kabache

• Hazim Hashim Tahir added an answer:
How could I make a MATLAB code to design a PID controller?

I'm trying to design a PID controller to control a nonlinear system, I finished all the dynamics and error (e) equations, P, I, D; but I couldn't find some useful tips, to start writing my code.

One could found some ready code from internet, but I want to understand how one could be able to make such things; since I'm a bit new to this area!!

Any one can suggest some tutorials, or informative resources?

Hazim Hashim Tahir · Ministry of Science and Technology, Iraq

Well done عفية عليك يا امين; practice makes perfect.

How we can to reduce the output oscillation magnitude in Fuzzy sliding mode controller?

One of the important challenge in intelligent nonlinear control is overshoot. How we can to reduce/eliminate it?

Farzin Piltan · IRANIAN CENTER OF ADVANCED SCIENCE AND TECHNOLOGY (IRAN SSP)

I have about 50 papers related to sliding mode topics, you can follow my research and please don't any hesitate if you have any question.

Regarding to above question if fuzzy type control be PID I think we also can to reduce the output gain updating factor's of PD parts.

Solve y^3 + y''' = 0?
Can anyone suggest an analytical method to solve y^3 + y''' = 0? The solution should satisfy the boundary conditions y tends to plus or minus one as x tends to plus or minus infinity. The solution should also be bounded everywhere. The tanh method doesn't work, but perhaps some variant of it does.
R. Mark Bradley · Colorado State University

Hello everybody---

Thanks for all of your help with this ODE. Your input showed that the correct ODE to look at was somewhat different than the one I asked about here. The paper that came out of this line of research is attached. Hopefully you'll find it interesting.

Best regards,

How can I solve a^2 y = y^3 - y''' - y'?
Can anyone suggest an analytical method to solve the ODE a^2 y = y^3 - y''' - y', where a is a positive constant? The solution should have y(0) = 0. Also y -> a and y' -> 0 as x -> infinity.
R. Mark Bradley · Colorado State University

Hello everybody---

You can find the published version of our paper

How can you solve NMPC objective functions with nonlinear constraint in MATLAB?
NMPC- Nonlinear model predictive control X_dot=f(x,u) Y=C*x objective function: min J = (Y-Ys)^2+du^2+u^2 w.r.t u constraints are : 0
Himour Yassine · Université de Khemis Miliana

Check this funciton "nmpc.m" and examples of use...

http://numerik.mathematik.uni-bayreuth.de/~lgruene/nmpc-book/matlab_nmpc.html

What orthogonal polynomials best fit the Volterra-Wiener nonlinear series?

To reduce the total number of parameters estimated in nonlinear system identification using Volterra-Wiener functions, laguerre polynomials are often used. Is this orthogonal set of functions the optimal choice for parameter compression or are there any better orthogonal functions for this task?

Simone Orcioni · Università Politecnica delle Marche

Polynomial expansions lead to block-structured models, as shown in Fig. 1 of [1]. In [1] you can also find a comparison between LET (Laguerre expansion technique) and FOA (fast orthogonal algorithm). You can see also [2] about Laguerre expansion.

These models (so as Volterrra series) can lead to a matrix formulation that is linear in the coefficients and that can be solved with LMS or RLS methods. In the limit of the approximation given by the expansion, these estimation methods (LMS,RLS) give good results, mostly if you have a reduced number of parameter to be estimated.

But an issue that can affect these estimation methods is the locality.

In simple terms, if you choose an input with high variance you stimulate well high order nonlinearity, resulting in well estimated high order coefficients and pourly estimated lower order (linear part included). The opposite if you use a low energy input.

This problem also affect orthogonal methods like that of Lee-Schetzen, that does not approximate the kernels but have a greater number of coefficients with respect of block-structured methods.

In [3] you can find a proposal for the mitigation of the problem of locality.

The main obstacle to the use of an orthogonal method like that of Lee-Schetzen or that proposed in [3] is given by the system in study. If you can give in input to your system a white noise, the preferred choice is an orthogonal methods, like [3]. If you have a limited and non gaussian inputs for your system, I would choose FOA.

[1] David T. Westwick, Bela Suki, and Kenneth R. Lutchen, “Sensitivity analysis of kernel estimates: Implications in nonlinear physiological system identification,” Annals of Biomedical Engineering, vol. 26, pp. 488–501, 1998.

[2] Vasilis Z. Marmarelis, “Modeling methodology for nonlinear physiological systems,” Annals of Biomedical Engineering, vol. 25, pp. 239–251, 1997.

[3] Orcioni, Simone, "Improving the approximation ability of Volterra series identified with a cross-correlation method", Nonlinear Dynamics, Volume 78, Issue 4, pp 2861-2869, December 2014.

To control any nonlinear system; what is the difference between fuzzy sliding mode controller and sliding mode fuzzy controller?

To design a controller for nonlinear and uncertain systems we have two choice: the first one is fuzzy sliding mode controller and the second one is sliding mode fuzzy controller. I'd like to know that which one is better and why?

Rainer Palm · Örebro universitet

Before I wrote my paper "Slinding mode fuzzy control" I was wondering why a diagonal Mamdani  FC with rules like  IF error=PB and change of error=NS THEN u=NB  work so well and robustly with regard to uncertainties even for a nonlinear system whose parameters are not completely known in advance. The reason is that this kind of diagonal FC works as an SMC with a nonlinear boundary layer.  However the main point is to choose the input scaling factors for error and change of error and the output gain accordingly so that stability, performance and robustness are guaranteed. In the following this SMFC was enhanced by the equivalent control which serves as a kind of compensation term for  well known parts of the system.

In my opinion, now a  controller is called  "FSMC"  for all fuzzy controllers with a sliding mode term included which is also the case for the SMFC. So I can't see a significant difference between the two.

• Flávio Eler De Melo added an answer:
What is the relationship between Moving Horizon Estimation (MHE) and EKF?

Now I want to solve a navigation problem using both MHE and EKF. I found that MHE has a better performance than EKF for highly nonlinear system. Can anyone tell me the reason? What is the relationship between these for estimation method? Are they identical to each other under certain assumptions?

Flávio Eler De Melo · University of Liverpool

The MHE arises as a maximum a posteriori (MAP) probability estimator (maximisation of the log posterior probability density), whereas the EKF is a suboptimal solution in the minimum mean square error (MMSE) sense. If the state and measurement processes are linear Gaussian without constraints, the "one-step" solutions coincide.

The MHE can tackle highly nonlinear problems where the state and uncertainty may have constraints, but at a higher computational cost and some practical difficulties such as approximating the arrival cost. The EKF is computationally simple but it requires both the system nonlinearities to be mild and the variances of the noises to be small (in order to keep the linearisation valid).

Depending on the nature of the system, I would rather consider either the UKF or sequential Monte-Carlo samplers (e.g., particle filters).

• Kheng Ka Tan added an answer:
What is wave-number in Rayleigh Benard instability?

Hi all.

I have just begun with the field of instability and going through texts of Francois Charru.

While going through Rayleigh Benard thermal instability (where, to best of what I have understood, the condition when due to temperature difference the advection will be prominent i.e. movement to fluid will start) I came across the following attached plot. I was unable to get what the wave number represents here?

Like in Taylor stability, it represents the wavelength of the perturbation, but what does k stand for here?

If it stands for perturbation wavelength, then what is perturbation here?

Is it different diameter particles/fluid elements exposed to heat surface?

Kheng Ka Tan · HELP University College

Dear Deewakar,

Pl try to read Lord Rayleigh's 1916 original paper, a classic available free on internet, and also Chandrashekar's book for  more modern treatment, then go back to the definition for the dimensionless wave number and Lord Rayleigh's original derivation, you will have a better understanding of its mathematical and true physical sense.

kk

L. Rayleigh, On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Phil. Mag., 32:529–46, 1916.

Subrahmanyan Chandrasekhar (1982). Hydrodynamic and Hydromagnetic Stability (Dover). ISBN 0-486-64071-X

P.G. Drazin and W.H. Reid (2004). Hydrodynamic Stability, second edition (Cambridge University Press).

How can we define Capabilities of system identification for nonlinear systems?
What does it mean when a nonlinear system can be identified?
What terms do we need to identify a nonlinear system?
Dmitry Kovriguine · Russian Academy of Sciences. Industrial Eng. Institute. Nizhniy Novgorod

Geysers behave periodically, and not almost such, and not so such, - a simple model relates to a linear piecewise point maps with unknown parameters. Let these are identified from time series, then one can set a billboard with a sketch of given dynamical system as Poincare maps, which can be understood almost for everyone, say, in Yellowstone, to show advances in science, in prediction of picturesque beauties in the Nature.

What are the methods for checking stability of zero dynamics of a nonlinear system?

Input-Output feedback linearization method for a nonlinear system requires the check of stability of zero dynamics or internal dynamics.

Zigang Pan · N/A

Dr. Castanos's response is on the mark.  I just have one more suggestion.  After you obtain the zero dynamics, you can apply Lyapunov function analysis, linear approximation to study the local stability of the zero dynamics.  A more powerful local stability analysis tool for nonlinear systems is "center manifold analysis", which you can find in Prof. Khalil's book "Nonlinear Systems".  It is like the linearization except it can deal with cases where the linearization method is indeterminant.

How can we design FPGA-Based sliding mode controller?

One of the important problem to design and implement Sliding Mode controller is dynamical based FPGA-part, so what's your idea about it?

• Mohammed Lamine Moussaoui added an answer:
Any ideas about the stability of two solutions in a nonlinear system?
Suppose a non linear system has at least 2 positive periodic solutions in a bounded domain. If one can show that the system is globally asymptotically stable, then is it a contradiction to the previous statement? If not, what can one say about the other solution? Can anybody suggest me any relevant references?
Mohammed Lamine Moussaoui · University of Science and Technology Houari Boumediene

Dear Santanu Biswas,

To be Convergent the Solution has to be Stable and Consistent (see lax theorem). The choice of the Discretization Step(s) defines a Domain of Stability this gives you several solutions for each choice. But the solution must be unique as it is stated by Fletcher in his book.

• Jose Gonzalez de Durana added an answer:
What are the recent advances, issues to be addressed, and scope of research in the area of stability analysis of multi-agent control systems?
What are the recent advances, issues to be addressed, and scope of research in the area of stability analysis of multi-agent control systems during intermittent and/or permanent sensor faults or loss of observation?
Please shed some light on this area.
In need of help.
Jose Gonzalez de Durana · University of the Basque Country (Spain)

First of all I would like to know what "stability analysis of multi-agent control systems" means (I do not know any study on it).