Science topics: Applied MathematicsStability

Science topic

# Stability - Science topic

Explore the latest questions and answers in Stability, and find Stability experts.

Questions related to Stability

I have some mutants that have produced atypical melt curves via just standard DSF with Sypro orange then nanoDSF looking at intrinsic fluorescence. Ideas of further experiments that could be used to probe this further and elucidate details of stability and what's going on with these proteins?

I am currently focusing on 3D geomechanical modeling. And in the future, I want to extend it to a 4D model.
During my recent studies, I realized that most of the 4D geomechanical modeling that has been done has not properly updated the elastic properties such as Young's modulus, bulk modulus, Poisson ratio, etc.
If a 3D static model is extended to a dynamic model, or a two-way or one-way coupling is performed, it is necessary to consider all material behaviors in a time-dependent manner.
Please share if you have useful information in this regard or if you have a suggestion, I would be grateful if you could comment.

**Dear researchers**

**As we know, recently a new type of derivatives have been introduced which depend two parameters such as fractional order and fractal dimension. These derivatives are called fractal-fractional derivatives and divided into three categories with respect to kernel: power-law type kernel, exponential decay-type kernel, generalized Mittag-Leffler type kernel.**

**The power and accuracy of these operators in simulations have motivated many researchers for using them in modeling of different diseases and processes.**

**Is there any researchers working on these operators for working on equilibrium points, sensitivity analysis and local and global stability?**

**If you would like to collaborate with me, please contact me by the following:**

**Thank you very much.**

**Best regards**

**Sina Etemad, PhD**

Hello, can someone help me in understanding how the variation in angle of attack effect the features of the transonic flow? I have read books on transonic flow, but they explain the effect of variation in mach number with transonic flow. How does tansonic features over a wing evolve with angle of attack ? What constitutes a good transonic wing ?

Hello,

Would you have an idea on how to model stable regular networks in a phase field model ?

For instance a square network like in the joined picture (the picture was made artificially) ? I know that the Swift-Hohenberg equation allows to create some regularity but I would like to control more the size of the network and the distance between the branches, and maybe the type of network.

Thank you in advance :)

I am looking for alkali stable radical scavengers.

I have a glycol solution of 80:20 ethyl diglycol:butyl glycol.

And want to add 10% KOH (so a waterless alkaline).

I have tried spiking the glycol solution with up to 2% BHT (butylated hydroxy toluene) before adding the KOH. As BHT is commonly used as antioxidant (radical scavenger)

However, it still seems unstable.

I was wondering if there are any Alkali stable radical scavengers or anything to supress discoloration and further (aldol) condensation of the present glycols.

Dear researchers

As you know, nowadays optimal control is one of the important criteria in studying the qualitative properties of a given fractional dynamical system. What is the best source and paper for introducing optimal control analysis for fractional dynamical models?

Any suggestions on analytical and numerical solutions? Any available codes to solve Mathieu equation available other than in math work websites ?

hello dear colleagues

I was working on a problem where I decide to merge sliding mode controller with homogeneous one. I was planning to define the sliding manifold based on homogeneous system of integrators.

Has anybody tried it out? are there any advantages?

best regards

I am trying to evaluate MWCNT nanofluid (NF) stability by measuring absorbance by UV vis, however since the concentration I am interested in leads to an opaque solution, I get absorbance higher than the max recommended for UV measurements (I get around 5). Is there is any why to come around this without changing concentration? , would decreasing cuvette path length solve the problem?

concentration: 1and 2 %w

Solvent: deionized water

Storage time: 1 or a few days

Also a layer of mold can be see on it.

The Van der Pol oscillator can be give in state model form as follows:

dx/dt = y

dy/dt = mu (1 - x^2) y - x,

where mu is a scalar parameter.

When mu = 0, the Van der Pol oscillator has simple harmonic motion. Its behavior is well-known.

When mu > 0, the Van der Pol oscillator has a stable limit cycle (with Hopf bifurcation).

While we can show the existence of a stable limit cycle with a MATLAB / SCILAB plot with some initial conditions and some positive value for mu like mu = 0.1 or 0.5 (for simulation), I like to know if there is a smart analytical proof (without any simulation) showing the existence of a limit cycle.

Specifically I like to know - is there any energy function V having time-derivative equal to zero along the trajectories of Van der Pol oscillator? Is there some smart calculation showing the existence of a stable limit cycle..

I am interested in knowing this - your help on my query is most welcome. Thanks!

I have a real stable system. However, when I try to reconstruct the state-space matrices of my system by using the subspace identification, it resulted in an unstable A matrix where its eigenvalues are located outside the unit circle.

I know that there are some ways to forced the A matrix to be stable. But it tends to give us a biased result since the stability is forced, not naturally identify as a stable system.

Please suggest if any specific software is used.

Hi everyone,

As known, the buoyancy frequency is defined as N^2=g/rho*drho/dz. Normally, the positive or negative characteristics of N^2 is the same as the gradient of water density (drho/dz), depended on the water temperature vertical profiles. In summer, the temperature of surface water is always higher than the lower zone, so drho/dz is positive. However, in autumn and winter, the surface temperature decreases, the temperature of surface water becomes lower than the lower zone, so drho/dz should be negative. The paradox is N^2 is a square number and should be always positive. I'd like to know how to explain such paradox of negative density gradient (drho/dz) and always positive buoyancy frequency (N^2)? Thanks a lot.

Hello

Why do we see the deposition of nanoparticles by adding different surfactants (cationic, anionic and non-ionic) with different concentrations to cobalt ferrite nanofluids?

Cobalt ferrite nanoparticles were synthesized by co-precipitation method.

Thank you all in advance

In the case of water splitting the water oxidation (OER) is the bottleneck for the process and because of the harsh conditions of the WO reaction, molecular catalysts with high TOFs are not stable in the matrix and quickly decompose. On the other hand heterogeneous catalysts are more stable but they often do not have high TOF quantities. Now what is your opinion about the future catalyst type? Will the ideal catalysts fabricated by turning the molecular catalysts to heterogeneous catalysts ( by loading the molecular catalysts on the solid surfaces ) or it will be from the single atom heterogeneous catalyst type?

Hi! I am trying to find some studies (or be educated) on the stability of the following groups in highly acidic (pH ~ 1-3) aqueous solutions of either an oxidizing acid (sulfuric or nitric) or reducing acid (HBr):

-The urethane linkage itself

-Polycarbonate diol

-Polyacrylate (say, HEA or IBOA)

I have a UV curable coating with the above components (polyurethane prepolymer dissolved in an acrylate monomer) that will be exposed to various acidic solutions for up to 30 days at RT.

I am trying to determine where any reactions would take place and understand if the polymer would degrade. I don't think it will based off the backbone, but I am curious to learn the reasons why / why not.

Thanks!

Dear all,

I am trying to couple a simulation with two codes, say code A and code B (different discretizations). For code A we solve simply the conduction equation (solid), while in B we solve the NS + temperature convection/diffusion (fluid).

The coupling is explicit, while the time scheme for each code is implicit.

at each iteration, code A gives code B the heat flux to impose at the boundary of domain B, while code B gives code A the temperature to impose at the boundary of domain A. So in short, the coupling is done via a Neumann/Dirichlet BCs.

I am facing stability issues that don't allow me to finish correctly the transient solution.

Let us focus only on domain A (code A). When code A receives the temperature values from code B to apply a Dirichlet BC, the values are stocked in a layer of cells; so called the ghost cells. We denote these values as T_wall.

Now, we need to compute the heat flux at the boundary to give to code B. There are two possibilities to compute such a flux :

1- use T_wall and one inner temperature value

2- use 2 or 3 inner temperature values to calculate the flux, and then extrapolate it to the boundary.

I am getting serious instabilities if I calculate the flux via the first approach (T_wall and 1 inner temperature value). Otherwise, with the second approach (only inner temperature values), the simulation is stable and a steady-state is reached.

Can I have please your opinion on this subject. Have you ever faced such kind of problems ?

Is it really that computing the flux with the second approach leads to a stable solution, but to a non-consevative scheme ?

Best regards

Elie Saikali

I need to measure the CCT from rotor angle graph >> is it possible ? HOW?

it will be highly appreciated, if you advice any method to calculate the CCT

I need guidance on the analytical model and calculation for shape (cross-section) optimization of Euler Bernoulli cantilever column under buckling with a point load. It may be using variational methods, with the use of Rayleigh Quotient.

For it, I found a similar case in "Haftka, Raphael & Gurdal, Zafer & Kamat, Manohar. Elements of structural optimization. 2nd revised ed." but with different boundary conditions, I have attached the scans of the procedure used in the text for reference.

Dear all,

I am trying to couple a simulation with two codes, say code A and code B (different discretizations). For code A we solve simply the conduction equation (solid), while in B we solve the NS + temperature convection/diffusion (fluid).

The coupling is explicit, while the time scheme for each code is implicit.

at each iteration, code A gives code B the heat flux to impose at the boundary of domain B, while code B gives code A the temperature to impose at the boundary of domain A. So in short, the coupling is done via a Neumann/Dirichlet BCs.

I am facing stability issues that don't allow me to finish correctly the transient solution.

Let us focus only on domain A (code A). When code A receives the temperature values from code B to apply a Dirichlet BC, the values are stocked in a layer of cells; so called the ghost cells. We denote these values as T_wall.

Now, we need to compute the heat flux at the boundary to give to code B. There are two possibilities to compute such a flux :

1- use T_wall and one inner temperature value

2- use 2 or 3 inner temperature values to calculate the flux, and then extrapolate it to the boundary.

I am getting serious instabilities if I calculate the flux via the first approach (T_wall and 1 inner temperature value). Otherwise, with the second approach (only inner temperature values), the simulation is stable and a steady-state is reached.

Can I have please your opinion on this subject. Have you ever faced such kind of problems ?

Is it really true that computing the flux with the second approach leads to a stable solution, but to a non-consevative scheme ?

Best regards

Elie Saikali

I have a question regarding STB in cadence should I set the operating point with STB analysis or not important? also, another question when I get the loop gain I found it below 0 dB what does it mean does that mean that there is no oscillation and the system is stable?
… Read more

I have the water dispersion of WO

_{3}and Ni(OH)_{2}. Both the dispersions containing 2D nanosheets are stable for more than a month. Dispersions of the bulk materials are not stable though. There are no other additives/surfactants in the dispersion. Measured Zeta potential value is -2mV. How the dispersion is stable then? Is it because of the less mass of the nanosheet compared to bulk Or any other reason?We know that these elements are non-carbide formers. Therefore, C is more likely to be present in the solution which should make the austenite more stable. By this logic, adding these elements should make austenite stable up to an even lower temperature.

I am still unsure about the relationship between BIBO and Lyapunov stability of simple undelayed LTI SISO systems.

Basic facts:

1) The system is STABLE if it has all system poles (eigenvalues) in the open left-half plane (LHP) or even single poles on the imaginary axis.

2) The system is ASYMPTOTICALLY or EXPONENTIALLY STABLE if it has all system poles (eigenvalues) in the open left-half plane (LHP).

3) The system is BIBO STABLE if it has all system poles (eigenvalues) in the open left-half plane. Or, the system is BIBO STABLE if its impulse function is absolutely integrable (i.e., it is L1-stable).

4) Btw., it is a fact that LTI SISO systems with DELAYS can have infinitely many poles in the LHP except for the complex infinity. Such systems are EXPONENTIALLY stable but they can/cannot be ASYMPTOTICALLY, Hinf or BIBO stable. Here, moreover, BIBO implies Hinf stability.

Notes:

- Some authors consider BIBO stability as a feature of the TRANSFER FUNCTION, not the SYSTEM itself. That is, there may exist unstable modes that cannot be seen at the output in the system. Therefore, every asymptotically Lyapunov stable system is BIBO, not vice-versa.

- I found also the idea in the literature that BIBO is stronger than asymptotical Lyapunov stability – however, I mean that this is incorrect.

Could anyone clearly explain me whether it exist any general relationship (inclusion, implication,…) between BIBO and (asymptotic) Lyapunov stability for SISO LTI delay-free systems, please?

I would like to store biological samples containing biotinylated proteins at -80 °C for several weeks, before continuing protein extraction and streptavidin affinity purification. I was wondering whether the biotin-protein bond is stable for long periods at -80 °C? Thank you for your advice.

If an operating region is divided into a number of sectors and if it is shown that the system is stable in equilibrium point corresponding to each of the sectors, can it be said that the system is asymptotically stable in the entire operating region consisting of each of these sectors? Is there any theorem that can be cited to reinforce the premise?

I'm using the Turbiscan Lab Instrument to assess shelf life and stability properties of my dispersions at different temperatures.

However, once analysing my data, I can only export the graphs into a pdf format.

I couldn’t find a method helping me access my raw data.

Are you aware of any tricks on the software to attain my raw data as a table enabling me to compare and analyse them?

I am working on postural stability during static standing position in condition of using mobile phone and without mobile.

I need literature an suggestion for methods.

I'm currently do my research about stability for gait analysis by using Lyapunov Exponent. Can I calculate the Lyapunov exponent based on the ankle angle and ankle moment? Currently I'm still confusing with the the code steps on how to calculate the Lyapunov exponent. Can someone guide me.

*Thank you so much.*I have seen unit cell designs that only perform at normal incidence while there are some designs that perform for a wide range of incidence angles. what is the reason or explanation behind this. what makes them angularly stable.

As MRT express the decay rate (average time that biochar can persist from decaying) and this rate decreases with time. Therefore, the incubation period becomes pivotal for the MRT calculation because shorter duration may lead to higher estimated mineralization rate and shorter MRT. So, can we consider other means of getting justifiable stability period?

please suggest a methodology for above.

In a diverse panel of genotypes, why some genotypes have high stability in certain traits (i.e. flowering time and yield) across multiple environments? In other words, why certain genotypes have consistent or similar phynotypic performance across different environments? Why other genotypes have low stability? Thanks for your help in advance!

What if the pendulum angle set-point of a inverted pendulum (IP) is not zero? Have you ever tried it?

If the result is the same with the zero set-point , Does it mean that the system is controllable but not accessible?

As attached in the picture, my emulsion contains Tween80 or food emulsifier occasionally. They both show very opposite but large magnitude of zeta potential. Can i consider this as stable emulsion? Or does anybody know how i could improve.

Tried concentration from 4% to 30% oil

I'm currently looking into the existing methods of estimating the lightship weight. In particular the estmation of outfitting weight is of interest to me, since it is very dependend on the type of vessel. Ultimately I would like to find out if it is possible to parameterize the outfitting weight for different vessel types.

I would very much like to hear your suggestions on literatur etc. or just your opinions.

I want to use NiCo2O4 as a substrate in 0.5 M H2SO4 solution. Just wondering will it be chemically stable in strong acidic solutions?

Hey everyone,

I'm reading the "Stability- and performance-robustness tradeoffs: MIMO mixed.." and I need to simulate everything.

I would like to know if there is a space/place where the Matlab code has been shared, so that I can simulate the reported results as I need.

Is the paper code available anywhere?

If anyone could help, I would be immensely grateful!

Link to the paper openly available -> http://welcome.isr.tecnico.ulisboa.pt/wp-content/uploads/2015/05/2142_JI1.pdf

Hi all.

I am looking for clarifications on the hexahedral mixed displacement-pressure elements.

1.) What exactly is the Taylor-Hood hexahedral element? Is it hex element with 20 nodes for displacement and 8 nodes for pressure or 27 nodes for displacement and 8 nodes for pressure? (Considering continuous interpolation for pressure.)

2.) Are both 20/8 or 27/8 elements inf-sup stable? If not, which one is inf-sup stable and which is not? Is there any good reference on this topic?

Thanks.

If I have a MIMO system with a number of subsystems with moderate amount of coupling sensitivity and if those individual subsystems are BIBO stable then under what conditions the whole system can be considered to be BIBO stable ? Is there any necessary and sufficient conditions that needs to be satisfied ? An answer from the control theoretic point of view would be most appreciated.

We know that the balancing of load and generation cannot be exactly achieved in a power system. This means that the swing equation cannot be in the steady-state conditions. So in such conditions are the frequencies of a small generating unit and a large generating unit equal?

Dears all

I would like to know what are the relationships among watershed health, watershed sustainability and watershed stability!

Please share your experiences with me.

Thanks

PEDOT:PSS is hygroscopic in nature and absorbs moisture from sorrounding environment. One more problem with PEDOT:PSS is it's acidic nature which affects ITO/HTL interface.These problems lead to degradation of perovskite material and hence stability of PSCs reduced. Now question arises, what are the best solutions to enhance the stability of PEDOT:PSS based PSCs?

The independence property is defined by Shelah. I am looking for other versions or generalization of the definition in which some circumstances are changed.

Main reference (p.316, Def. 4.1):

Shelah, S. "Stability, the f.c.p. and superstability." Ann. Math. Logic 3, 271–362 (1971)

Another reference: (First Def. in introduction)

Gurevich, Yuri, and Peter H. Schmitt. "The theory of ordered abelian groups does not have the independence property."

*Transactions of the American Mathematical Society*284.1 (1984): 171-182.Generally it is and industry practice to conduct the accelerated shelf life study of foods products at a temperature of 40+/-5 degree Celsius and 70+/-5% RH. Is there any authentic document which we can refer for this?

Spectral methods are one of the good tools to find the solution of the PDEs,

I have a parabolic PDE like

dP/dt=f(P, dP/dx1, dP/dx2, d^2P/dx1^2,..)

where f could be

f=a(x1,x2)*dP/dx1 +b (x1,x2)* d^P/dx2+c(x1,x2)*P

I formulated a 2D PDE, but the dimensions could be higher than 2.

a formal way in spectral method to solve the PDE is to solve the eigenvalue problem for operator f like:

f(phi_j)=lambda_j * phi_j

where phi_j is eigenfunctions (preferably orthogonal) and lamda_j are corresponding eigenvalues.

this way we can find the solution of parabolic PDE as well as analyze the stability of the solution based on the eigenvalues.

I should note that the boundary condition might be in Dirichlet , Neumman or Robbin form.

is there any general method to solve this eigenvalue problem and find the eigenfunctions and eignevalues of this general operator?

I need literature how set-up for markers for 3D analysis of upper body stability with Vicon ( 9 cameras stereophotogrammetric system). Suggestion and advice please.

I am trying to test some algebraic theory using many different Runge-Kutta schemes, with varying order, number of stages, stability, etc. Does anybody have a reference with a coherent list of lots of established RK schemes and their stability functions? I have pieced together a good number from various books and literature, but an encyclopedia of sorts would be very useful.

Hi,

Let us say that a plant model of the following form is given:

\dot{q} = A*q + B*u,

y = C*q.

The output of the model (y) is its position (x) and velocity (v).

Using this output, I am going to construct a PI controller (u) of the following form:

u = K

_{1}*x + K_{2}*v.Then, the task is to find the optimal gains (K

_{1}) and (K_{2}) such that some given cost function is minimized.Without any constraints, this problem can be solved easily using the MATLAB command 'fminsearch' or 'fminunc.'

But I'd like to add a stability constraint such that the closed-loop system with PI control is always stable.

Mathematically, it is realized by constraining the eigenvalues of the closed-loop system in the left-hand plane.

But I'm wondering how to add this constraint using MATLAB commands.

I tried to use the command 'eig' in the constraint function (nonlcon) of 'fmincon' but MATLAB says that command is not allowed to be used in the fmincon constraint function.

Any comments will be greatly appreciated.

Thank you.

The bifurcation curve generated in MatCont is displayed in the same color(blue). The stability of the equilibrium or limit cycle can only be tracked in the layout of "numeric" in "Window" by Eigenvalues or Multipliers. Does anyone know how to display the stability of the curve graphically by color or any other ways(e.g. line style)?

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.

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?

in most cases for continuous time stochastic systems which are modeled by SDE, the Lyapunov stability conditions can guarantee the stochastic stability of the system,

another definition In stochastic literature is detailed balance which guarantee the convergence of the probability of the states of SDE to a stationary probability density.

I want to know which one is more strong stability condition?

In the theory of the stability of the differential operators, one could prove the stability results based on spectra of an operator, (all eigenvalues must be negative for example).

one problem with the above method is that not all linear operators are self-adjoint (for examples operators in convection diffusion form) and their corresponding eigenvalue problem can not be solved analytically, hence spectra of the operator can not be calculated analytically. On the other hand there is a definition related to spectra, which is called pseudo-spectra, that somehow evaluates the approximated spectrum , even for non self-adjoint operators.

I want to know is it possible to establish stability results for a differential operator based on pseudo-spectra?

Is there any scientific data avaibale on the AAV vector stability in 15-20% Tween 20 solution? Also any information on the AAV9 aggregation behaviour at 6E+13 concentration range?

Could there be nonlinear ODEs that could not even be solved numerically?

I am working on a third-order nonlinear problem which is giving accurate results for a given set of initial conditions (ICs) but for other ICs the matlab ODE solver (checked 4-5 algorithms) do not converge.

Hi,

I was wondering if anyone has experience with the stability of lyophilised human hemoglobin after being dissolved in solution (e.g., MilliQ water, tap water or buffer) and stored at 4

^{o}C, -20^{o}C or -81^{o}C for weeks/months?Thank you.

I am trying to study the stability of lattice Boltzmann method. The the eigenvalues of the evolution matrix in wave number space need to be studied carefully.

I solved the eigenvalues using Matlab.

I thought that the fluctuation is an artifact resulting from using Matlab, since it shifts the order of the eigenvalues each time I am calculating them. However, this is not the case.

To prove that, I derived the characteristic polynomial and found analytical solutions for each mode. The analytical solution is correct, however some modes fluctuate between it self and other mode (see the picture). Note: there is another mode that complement this mode, so that they together form complete two modes.

what is the reason for this fluctuation? and what is its impact on stability and accuracy?

Here, by element-wise bound, I mean bounds on each of the elements of A, B, and P matrices.

i.e., |a

_{ij}| <= a0_{ij }>=0, etc...Also, P is the solution to the Algebraic Riccati Equation.

for deterministic systems, with defining proper terminal constraint , terminal cost and local controller we can prove the recursive feasibility and stability of nonlinear system under model predictive control. For stochastic nonlinear system it is impossible to do that since we do not have bounded sets for states.

what is the framework for establishing the recursive feasibility and stability of MPC for stochastic nonlinear systems?

Please describe your opinion on this topic and suggest some literature.

Effects and mechanism of control during

*chewing gum*in quite standing position.We are working on some projects related to networks of coupled nonlinear oscillators. We want to obtain “Master Stability Function” (MSF) in our network.

1. Is there any sample code available in the internet for it (we searched, but we couldn’t find any)?

2. Can someone please introduce us some papers who have obtained MSF for simple examples?

3. Is there any major difference between 2D systems and 3D systems in the procedure of calculating MSF?

Let me say everything in an easier way. Suppose that we have only 2 identical Lorenz systems, which are coupled through variable “x” with the coupling strength “d”. How can I obtain MSF when I change “d”? Is there any available code for it? If you have any answer for this question, is it applicable and meaningful for two identical Van der Pol systems (which are 2D) which are coupled through variable “x” with the coupling strength “d”?

Many thanks in advance,

Sajad

I would like to know which assays can be performed to study the stability of antibodies.

Is it possible to generalize chemical stability of group of materials versus the others? For instance, can we arrange oxides, fluorides and haloids in an order from most to least chemically stable ?

The term "Chemically stable" sounds confusing too. When we say a material is chemically stable, are we including factors such as Temperature, Pressure, pH of medium, etc all at the same time ? I'm asking this question because I saw in a reference that they said Oxides are more chemically stable than Fluorides. While, in another paper the exact opposite claim was mentioned without any condition explanation.

Is there anybody with an insightful note or explanation on how to establish the stability of the solution of ODE (bvp) using Poincare-Lyapunov Theorem?

Reduction of the bvp to initial values problem and generation of the Lyapunov function do not incorporate the boundary conditions. We believe these are not acceptable.

Do you know how to include the Neumann boundary condition into Jacobian matrix or how to generate Lyapunov function suitable to establish the stability of the solution of ODE bvp; see the attached problem.

I am using the DSATools software to do a simulation of four generators eleven buses. Firstly, I suppose to do a simple simulation with single machine single line. However, I don't know how to use the PSAT and how to calculate the parameters of the load.

I have to design a fractional order PID controller for a maglev plant. The maglev plant is an open-loop unstable system. Its transfer function has two real poles, one on the RHS of s-plane and one on the LHS of s-plane, G(s)=-K/(s

^{2}- p).The controller parameters are tuned using an evolutionary algorithm.For a particular set of the controller gains I achieve good closed loop response.I have attached the figure of the system response. However, frequency domain analysis (bode,nyquist and nichols-chart) of the system, using MATLAB, shows negative Gain Margin and positive Phase Margin. I have attached the Nichols Chart obtained from MATLAB.

It shows that the gain margin is negative. But says "Yes" to "Closed loop stable?"

How do I correlate these facts? The system exhibits stable response. But the Gain margin is negative! How so?

Is the system actually closed loop system? If so, then how?

Please clarify. I am perplexed!

for ODEs we can use the definition of stability,

for example when the ODE is linear, by calculating the

Eigenvalues or poles of the system the stability characteristic of the system can be achieved. or in nonlinear case the Lyapunov theory of stability can be used.

do such definitions (equilibrium point, stability, Lyaponov theory) exist for PDEs?