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# Linear Systems - Science topic

Explore the latest questions and answers in Linear Systems, and find Linear Systems experts.

Questions related to Linear Systems

Dear all,

I am currently dealing with a problem of spurious oscillations in wave propagation in elastoplastic bodies.

The problem in question is a 1D column subjected to a 10Hz sine wave acceleration input motion at the bottom of the mesh. Abaqus/Standard is used for the dynamic analysis.

Generally, the stress strain behavior looks fine, the same with displacements and velocities. The oscillations are most pronouncedly experienced in the calculated accelerations. They appear in time when the element/Gauss point is subjected to a sharp change in stiffness, e.g. from elastic branch into plastic one when no smooth hyperbolic relationship is used. The oscillations are of much higher frequency than the input motion [say around 100Hz].

I am wondering where these oscillations come from. I have tried to include numerical damping in the HHT direct time integration scheme however this did not influence the observed oscillations. I am wondering if further “play” with alpha, beta, gamma parameters in the HHT method could result in damping out the oscillation I experience. So far I have only used set of parameters as suggested in Abaqus Manual for “application=moderate dissipation” option.

I have also tried the effects of time and space discretization, non of the two was effective [for space discretization going for much finer solution than the minimum of 10 nodes per wave length as typically advised]. The problem remains and is insensitive to mesh or time step refinement. The only way of removing the oscillations is applying the Rayleigh damping, however this seems to be artificial way of removing the problem since the constitutive model is elastoplastic and deemed to be capable of accounting for material damping.

Generally, Abaqus manual says

*“The principal advantage of these operators is that they are unconditionally stable for linear systems; there is no mathematical limit on the size of the time increment that can be used to integrate a linear system”*so I understand that maybe this scheme is not stable or inaccurate for the nonlinear dynamic problems or a family of nonlinear dynamic problems. Would some another commonly used direct time integration scheme, such as lets say Bathe be more accurate here?Anyone has experienced maybe a similar problem? Where is it I can look for the reason of the oscillations?

Thanks in advance for any help and advice on that.

Suppose there is a linear system of equation Ax =b, x is required. A is a large symmetric matrix.

Which method is faster to compute x?

1. If I solve it through the calculation of inverse of A and compute x = A

^{-1}b.2. If I solve the linear equations through some iterative method (like Gauss-elimination, Gauss-seidel).

I am working on signal inversion. I am facing a matrix inversion problem. I have a linear system where, the signal and data are known. I am computing the inverse by simple matrix inversion. However, I am appending a vector with both data and signal matrix. I want to compute the inverse without going for simple inversion again. I want an analytical formula of the data coefficient matrix in terms of data, signal, previous coefficient matrix and the appended vector.

The response function is other than ratio-dependent.

I am trying to model a Chaotic fractional order system in LabVIEW. I can do it in matlab but i want to create the model in LabVIEW

Dear all,

I would like to study regarding control of linear and nonlinear systems in detail. So, please suggest me some books which can provide in-depth knowledge regarding it. Thank you.

Linear stability analysis fails to determine the local stability property of a non-hyperbolic equilibrium point as there is a emergence of a centre subspace (other than stable and unstable subspace) of the linearized system corresponding to the eigenvalue whose real part is zero. Centre manifold of the corresponding nonlinear system may not be unique. So, what is the exact procedure to analyze such kind of situation?

In control theory, using Routh array test, it can be established that a quadratic polynomial

p(x) = a x^2 + b x + c (where a > 0)

is a Hurwitz polynomial (i.e. it has roots with negative real part) if and only if a > 0, b > 0 and c > 0. In an equivalent way, this can be proved using Hurwitz determinants.

I am looking for a simple proof for this fact. Without loss of generality, we can assume a = 1.

For p(x) = x^2 + b x + c, we can use the root finder formula and discuss various cases.

Is there any simple proof? I welcome your ideas and suggestions. Thank you!

I have torques and angular positions data (p) to model a second-order linear model T=Is2p+Bsp+kp(s=j*2*pi*f). So first I converted my data( torque, angular position ) from the time domain into the frequency domain. next, frequency domain derivative is done from angular positions to obtain velocity and acceleration data. finally, a least square command lsqminnorm(MATLAB) used to predict its coefficients, I expect to have a linear relation but the results showed very low R2 (<30%), and my coefficient not positive always!

filtering data :

angular displacements: moving average

torques: low pass Butterworth cutoff frequency(4 HZ) sampling (130 Hz )

velocities and accelerations: only pass frequency between [-5 5] to decrease noise

Could anyone help me out with this?

what Can I do to get a better estimation?

here is part of my codes

%%

angle_Data_p = movmean(angle_Data,5);

%% derivative

N=2^nextpow2(length(angle_Data_p ));

df = 1/(N*dt); %Fs/K

Nyq = 1/(2*dt); %Fs/2

A = fft(angle_Data_p );

A = fftshift(A);

f=-Nyq : df : Nyq-df;

A(f>5)=0+0i;

A(f<-5)=0+0i;

iomega_array = 1i*2*pi*(-Nyq : df : Nyq-df); %-FS/2:Fs/N:FS/2

iomega_exp =1 % 1 for velocity and 2 for acceleration

for j = 1 : N

if iomega_array(j) ~= 0

A(j) = A(j) * (iomega_array(j) ^ iomega_exp); % *iw or *-w2

else

A(j) = complex(0.0,0.0);

end

end

A = ifftshift(A);

velocity_freq_p=A; %% including both part (real + imaginary ) in least square

Velocity_time=real( ifft(A));

%%

[b2,a2] = butter(4,fc/(Fs/2));

torque=filter(b2,a2,S(5).data.torque);

T = fft(torque);

T = fftshift(T);

f=-Nyq : df : Nyq-df;

A(f>7)=0+0i;

A(f<-7)=0+0i;

torque_freq=ifftshift(T);

% same procedure for fft of angular frequency data --> angle_freqData_p

phi_P=[accele_freq_p(1:end) velocity_freq_p(1:end) angle_freqData_p(1:end)];

TorqueP_freqData=(torque_freq(1:end));

Theta = lsqminnorm((phi_P),(TorqueP_freqData))

stimatedT2=phi_P*Theta ;

Rsq2_S = 1 - sum((TorqueP_freqData - stimatedT2).^2)/sum((TorqueP_freqData - mean(TorqueP_freqData)).^2)

As is well known, iterative methods for solving linear systems such as Successive Over Relaxation and the like, are very attractive for solving many problems such as sparse matrices. These methods, in general, are formulated in the context of determined system in which the number of equations is equal to the unknowns. Now and for sake of simplicity, let us assume that we have one additional observation and We need to update the previous solution. In other words, now we have an over determined system with the provision of this additional observation. The question is how to include this observation to the to update the previously computed parameters. Indeed, the theory of parameter estimation provides a lot of guidelines to handle this task to get an optimal solution in the sense of least squares. But let us assume that we need to stick to the iterative approach for parameters. Then with assumption how we could handle the additional observation for parameters update and what kind of errors that we need to minimize.

When I read papers in computer vision field, I saw many energy functions or objective functions. I can understand the paper from high level, but when I try to implement the paper I stuck on solving the energy function.

I am implementing DynamicFusion: Reconstruction and Tracking of Non-rigid Scenes in Real-Time, published in 2015 CVPR. The energy function is that:

E(Wt,V,Dt,E)=Data(Wt,V,Dt)+λReg(Wt,E)

The paper use Gauss-Newton to solve the parameters. Should I convert the energy function into a linear system like

**? If so, how to combine Data term and Reg term to be***Ax+b***?***Ax+b*And why papers usually don't give the deviation of the energy function?

Dear all, I have some questions concerning using pole placement techniques (state feedback controller)with model predictive control for linear systems? To be precise, will be possible using pole placement to stabilise or ameliorate the system's stability and then apply Model predictive Control to the closedloop stabilised system.

Any ideas will be apreciated.

Thank you.

I faced such a problem. I have a nonlinear system for control synthesis and I should compare not only my controllers but also a linear version of my system to describe the legitimacy of this linearization. But it never occurred to me how to compare it in numerical. We often do it in a frequency domain for linear systems comparing bandwidth, gain, or phase margin. And we have a numerical result. But I can't do the Laplace transform (for example) because of no superposition principle being. I heard about nonlinear Fourier transform but I doubt that it could help me

A signal is split into two parts and one of them is going through a filter (say, with a transfer function H(f)) and the other part stays unchanged. Then I want to know how to calculate their cross correlation function. My guess is, given the spectral density function S(f), it will be the ordinary Wiener-Khinchine theorem with an addition of the transfer function: R=Integral{S(f)H(f)*exp(i*2*pi*f*t)df}

Hi all, I have a question for Control system development specialists.

I compute controllability of linear system through MATLAB function ctrb, and I know that system have 1 uncontrollable state. How to define which state is uncontrollable?

I have a model that consist of two ODEs and one PDE which are all coupled and non-linear. When I linearize the system around the steady state and simulate the system with initial values that are different from steady state I get a stationary deviation in one of the solutions. Due to non-linearities I expect a quite different behavior for the linear system. However, if I multiply some the elements of the system matrix with a constant (relaxation or dampning coefficient?), the linear system now resembles the non-linear system much better in terms of transient behavior and that I no longer have a stationary deviation which I had before. Is this method for fitting the linear model allowed and have some theory that can justify the use of such relaxation constants? Or is it just a 'cheat' which is not valid as the system does change due to the use of such constant?

Dear collegs,

Let us have a linear system:

Ax=b, (1)

A is a matrix of (N,N) shape, b, x are vectors of N shape. We decrease number of equations to form a underdetermined system:

A'x=b', (2)

A' is a matrix of (M,N) shape, b' is a vector of M shape, M < N.

Can we find any formula for a difference between normal pseudo-solution of the system (2) and exact solution of the system (1)?

Can we find how many equations must we use to estimate exact solution by normal pseudo-solution with the known precision?

Can we determine a convergence rate of the normal pseudo-solution to the exact one in dependence of matrix A parameters (condition number, singular values, etc.)?

Hello dear Dr.Krack!

I have the mass and stiffness matrices of the finite element model of a free structure. Of course the linear system made by these matrices have 3 rigid modes and the first three natural frequencies are zero! When trying to compute the linear FRF of this system using your codes, I notice that the results are not close to the test structure's FRF results. I think some additional modifications need to be made to the system or your codes to account for the free-free boundary condition of the system. What do you think those modifications would be?

I thank you in advance for your attention!

Dear community,

I am trying to build a model of a Furuta pendulum in Simulink/Simscape. Unfortunately, when I try to linearize my model with the integrated Model Linearizer, I get an unexpected result. The evaluation of the linearized system shows, that it is only poorly controllable, although a classic Furuta pendulum should be fully controllable according to literature. Therefore I assume, that there must be something wrong with my model or the way I linearized it but I can´t figure out what it is....
I´d highly appreciate any help on that, as this is bothing me for quite some time now. The model is attached to this post. Furthermore I have attached a screenshot of the linearized system.

My controllability matrix (ctrb(A,B)) then looks like this with rank = 1, which I believe can´t be right...

Controllability matrix =

1.0e+26 *

0 0.0000 0.0000 -0.0000 0.0000

0.0000 0.0000 -0.0000 0.0000 -0.0007

0 0.0000 0.0000 -0.0000 0.0000

0.0000 0.0000 -0.0000 0.0000 -0.0015

0.0000 -0.0000 0.0000 -0.0000 9.2972

Thank you and best regards,
Joo

I am teaching an holistic course about ecosystem and use of natural recurses.

During the course we discuss how ecosystems are complex systems and intervention may brig very unpredictable consequences.

...I can´t help and wonder: are all nature interactions part of complex systems, leaving liner systems to be an approximation of nature to try and describe parts of it, or do we actually have linear system in nature?

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 want to transform a linear system matrix from x basis to z basis where Z=T

_{r}X. T_{r}is obtained through QR factorization. The linear system is in state-space form. Furthermore, I can use the Transformation to obtain regular form matrix for Eigenvalue placement design or Linear-Quadratic minimization feedback sliding mode control.I have a nonlinear system, I linearized it using the Jacobian method and got the A, B, C, D matrices for state-space representation. How could I take into account the change of the equilibrium point - i.e. the point where the system is linearized - during a simulation? I tried to solve it by subtracting u0 from the input and adding y0 to the output of the system - please see the attached picture -, but the results are incorrect because the operating point where the system is linearized is changing.

Dear friends and colleagues! I have designed a Simulink model of an inverted pendulum control system. In order to study the LQR I have also linearised the model and obtained the K gain matrix for the controller, which works well for the linear system, now I am hoping to apply it to the non-linear system as well. However, I do not quite understand the reference for the control system.

The system has 4 states: position of the cart, pendulum angle and their derivatives. I am modeling a data acquisition system, so I assume I can only have encoders measuring the position and angle, and then I model software derivatives I could perform on a computer. Then I have a block to get u=K1*x1+ K2*x2+K3*x3+K4*x4. That seems to be my input in the system. I, however, assume there must be some comparison block, but I can not seem to wrap my head around it. In the diagram I have attached to this question you can see the reference is set to zero, but I did it merely to have some refernece in general. But what is this zero? What if my desired states are not zero? What if i want to move my reference to some other point during execution pf the program? Which one do I move? If I still want to stabilise the pendulum but not at x=0, but somewhere at x=ref?

a) Can subharmonics ever appear in linear systems?

b) Is it possible for subharmonics to appear in p.e. 2-DOF nonlinear systems? What are the conditions for this to happen? For example, this does not happen in the periodically forced Duffing oscillator (typical nonlinear system).

Thank you in advance!

I have an element with an impedance that has a complicated relationship with frequency. I have defined the mathematical relation for the impedance as a MATLAB function and used MATLAB function block to simulate it in Simulink. The problem is that I could not find any variable impedance element which can be controlled by a signal in Simulink. The only element which could be used is variable resistance that neglects the imaginary part of the impedance. On the other hand, impedance block is only useful for defining linear systems with zeros and poles and polynomial type relations for impedance. The impedance relation with angular frequency is attached.

Thank you in advance.

Let say i want to write a program/script on MATLAB, and condition is that MATLAB code should generate a noise signal from gauss distribution of mean "11" and variance="18". But i am confused about which formula/expression i will use here? formula in which i will put values of mean and variance and then i will get noise signal generated from gauss distribution

I m doing mixed mode simulation for TFETs inverter in Sentaurus .Its showing linear system cannot be solve and didnt converge. can anyone know this error or any other ways to do the same.

For extended Kalman filter, we need to do matrix differentiation for converting the non linear system to a linear system using first order Taylor approximation

How does K (The controller Gain) affect J (the objective function) in the LQR Control? How do I explain this a bit theoritically?

Hello,

I am using ILU factorization as the preconditioner of a Bi-CGSTAB solver for solving a linear system of equations Ax=b. The preconditioner matrix M=(LU)^-1 is calculated by backward substitution method, solving Ly=c and Us=y. However, when A has zero diagonal elements (e.g. A(2,2) = 0) U will also have zero diagonal elements ( U(2,2 )=0) ,which makes M impossible to be solved by the backward substitution method.

How could I reorder my system of equation in order to tackle this problem?

I'm looking for an algorithm for solving (and representation of non-zero elements) linear systems with large sparse unsymmetric matrices (not diagonally dominant).

I'm interested in ONLY DIRECT method.

(The matrices are obtained from FDM for hyperbolic PDE).

Generally speaking, a linear system is solvable and a decentralized system can be solved more efficiently.

Your kind help is required.

Consider the state-space representation of a linear system as described by

x(t) = Ax(t)+Bu(t)

y(t) =Cx(t)

state estimation of above system is possible with Luenberger observer .

BUT IF

x(t) = Ax(t)+Bu(t)

y(t) =Cx(t)+a(t)

Question is that if an attack vector is added , then how can you estimate the states keeping that in mind there is no noise and disturbance in the system.

Hi, everyone,

Do you know some good MATLAB subroutines (direct solvers) for solving

**nonsymmetric**Toeplitz linear systems ? I note that most direct solvers are only suitable for symmetric Toeplitz matrix, but recently, during the implementation of my numerical simulations, I need to solve a series of Nonsymmetric Toeplitz linear systems, i.e., A*x_i = b_i. i = 1:s. the size of A is from 8 to 256. I feel that this size is not large, so I do not want to consider the iterative solvers, such as gmres with circulant preconditioners. I think that if the subroutines can be used like the following form, it will be great:>> y = nonsymToeplitz(c,r,x) % c is the first column of Toeplitz matrix A, and r is the first row of A, x is the rhs, i.e., y = A^{-1} x;

If you know some information for this, please tell me.

for linear control systems x_dot=Ax+Bu the reachability set can be calculated using the Image of the controllability matrix, i.e

R=([B AB A^2B,....,]) and reachability set=Im(R)

when rank(R)=n, and we do not have any control constraint the reachibilaty set of linear system is R^n (n is the dimension of states)

if we have a non-linear affine-control system

x_dot=f(x)+g(x)*u

R can be calculated using Lie algebra

R=[g1,g2,[f,g1],[f,g2],...]

my question is, in this case reachability set is again Im(R)?

and if reachability set=Im(R), how can we compute the rachability set, because here R will be a matrix with function arrays (function of states)

Dear friends

I have a serious problem in order to regenerate the results of the attached paper. I do exactly mentioned method (FEM) in the paper but I don't get correct results.

This paper about gas face seal and compressible zeroth and first order perturbed Reynolds equation. I Solved zeroth order with FEM and FDM correctly and I got same results that bring out in the paper (Opening force for various pressure ratio) and I solve first order equation with 1 pressure ratio (figure 5 and 6) (I am thankful to you if look at the paper), but in other figure I can't get the paper's results. I try many method to solve linear system of equation (Ax=b) such as direct method, iterative method (i.e. gauss-sidel, cgs, gmres, bicg, pcg ...) but I failed, also I try many network grid and there is no achievement.

So what should I do?

I really know my question is general, but I really don't know about the errors.

Hello

anybody knows how we can get the A,B,C and D state-space matrices from PowerFactory?

you can get ABCD matrices from PowerFactory which are MATLAB readable file but they arenot the A,B,C and D state-space matrices.

Thank you so much

In control system, if we have to analysze a onlinear system, we linearize the system around its equilibrium point. I would like to ask why we consider only the equilibrium points?? Suppose, if we consider some other random point for linearization, what shall be the effect in the analysis?

Hello People!

I have a question regarding how researchers use the matrices from the sparse matrix marketplace(https://sparse.tamu.edu/) for solving linear systems of equations.

So given that we have to solve for x in : Ax = b

The matrices from the market are the A for the above equation. But what about the vector b? How do we decide on that? Do we just pre-select some random vector x and multiply A with it to give us the vector b, and use that vector b to recalculate x using some method (that's the focus of the researcher)? Or is it something else.

I've come across some papers where people mention the name of the matrices taken for testing and their initial guess vector x0. But I am confused as to how to they select a vector b? Is there a general practice? Or does it depend from author to author

There must be something I am missing, might be very silly, so I apologize in advance!

Some Papers for reference and example (with page numbers) :

Page 6-7 of:

Page 848 -849 of :

Thanks !

I am wondering to enforce state dissipation to stabilize nonlinear or linear systems. Assume a nonlinear control system as xdot=-x

^{3}+u. Then to forcefully dissipate x as exponentially by setting: x = x0*exp(-a*t), where a is dissipation rate, so xdot = -a*x0*exp(-a*t), hence from the evolution dynamics; xdot=-x^{3}+u, we have the control variable as u= xdot+x^{3}= -a*x0*exp(-a*t) + (x0*exp(-a*t))^{3},or in state feedback format; u(x)=-a*x+x

^{3}.This is a time-varying open-loop control and in other format a state feedback strategy. So what is your idea? How do you think about that? Does it worth as a new control methodology?!

what are the advantages and disadvantages of Matching Pursuit Algorithms for Sparse Approximation? and if there are alternative method better than Matching Pursuit

Hello,

I have the following system of equation Ax = B where A is a 2x4 matrix, B is a 2x1 and x is a 4x1 matrix.

A = [ a b 0 0]

0 0 c d]

I am looking for X can I use a pseudo inverse matrix here , if no? may you tell me how to proceed or what other method to use.

Thank you in advance .

Most Engineer use nonlinear system than linear system, but the mathematical methods is difficult for nonlinear.

x

_{1}*=x_{12}^{ }+x_{2}+ux

_{2* =-2u}_{* represents time derivative.}

In most of the literature for Generalized MPC, it is assumed that D matrix in state space model of the system is zero, I have a system with non-zero D (not a strictly proper system), what should I do?

Consider linear system dx/dt=Ax+Bu, y=Cx. Assume that there exists u=Kx such that the system is asymptotically stable , i.e. x is asymptotically stable. Does this mean that y=cx is also asymptotically stable ?

Thank you!

Dear friends,

Is it a correct way to use a linear system with uncertainties instead of the main nonlinear system?why?

I am trying to perform a simulation of polymer brushes based on PNIPAM with GROMACS code.

Generating a coordinate file for the system of interest is not problematic, but creating a topology of this non-linear case is complicated (default gromacs tools are dedicated to linear systems like peptide).

Could you recommend the best tool for this problem (topology generation)?

Is the multiplication of positive definite and negative definite matrix is a positive definite matrix even if they do not commute.

The literature on the time delay systems is reach with many good books. I would like to know if there any introductory book on this subject that is suitable for undergraduate level.

I want to know if in the literature respect to non-minimum phase nonlinear systems there are methods for tracking control. I researched a lot and I found some methods as Byrnes Isidori regulartor(use exosystem to generate trajectories) and other as Method of Devasisa, Chen . But the problems is that i not sure if those methods have been tested since I have only seen cases for regulation or control from one fixed point to another one (solving the iternal dynamics using Bounday Value Problem) but not for tracking explicitly.

Regards

I have a nonlinear MIMO system

$\dot{x}=f(x)+g(x)u$, also the sliding surface for my outputs happens to be a vector S=0. Now I can write $\dot{S}=F(x)+G(x)U$. Can I not extract U from this relationship by enforcing attraction condition. i.e $\dot{V}< -\eta*|s|*-k*s$?

or is it necessary to convert the system into normal form first?

Hi everyone, I'm trying to solve a large sparse block tridiagonal linear equations (about 10^5 X 10^5), and when I use block LU decomposition or backslash in Matlab, it is very time consuming. So could you please advice me how to solve it fast?

Hi

please, i want to know if i can use a classic kalman filter to estimate the states of a linear system with a variable state matrix A.

Thank you

Consider the Lyapunov equation given by A'P+PA+I=0, where I is the identity matrix, A is Hurwitz, and P is a positive definite and symmetric n by n matrix. How can we find an upper bound to the Frobenius norm of P, i.e., ||P||_F, using the eigenvalues of A ?

I was able to find a relationship with eigenvalues of P(see attached picture) but I need to find out relation with eigenvalues of A.

Any idea how should I proceed?

Hi Dears, I have a problem in solving system of linear equations involving singular matrix. How we can fix the singularity of the matrix?

Hello,

i wish to know if there is any algorithme to get rid of matrix singularity.

thank you.

Is there a definition of convex optimization problem for COMLEX-VALUED MATRIX VARIABLES where objective and constraint functions are real-valued ?

Are KKT conditions true for convex optimization problem with COMPLEX-VALUED MATRIX VARIABLES where objective and constraint functions are real-valued ?

They are often seen in MIMO communication problems.

Thank you very much.

I have currently used the Krylov and Multifrontal methods in order to solve disperse linear systems. Next figure shows the characteristic curves

*number of nodes vs CPU time*of the Krylov and Multifrontal methods. Does anyone know about direct or iterative methods for decrementing the computation time in the solution of large sparse linear systems? Which one?HAUTUS, M. L. J., Controllability and observability conditions of linear autonomous systems. Nederl. Akad. Wetensch., Proc., Ser. A 72, 443-448 (1969).

Thanks in advance.

Suppose we want to solve following attached general eigenvalue problem in which the matrices K11, M11 are symmetric, positive definite and very ill conditioned as well as K12.

What method do you recommend.

I attached my matrices in Matlab, It will be so appreciable if someone could help me solve it with Matlab.

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.

I know it's in Hamilton's Nonlinear Acoustics, but what's the original source?

I am trying to linearize two nonlinear functions which I attached. Please check the attachment there I clearly explained the formulations.

Thank you for your comments

I have

**5th order****nonlinear**system of**relative degree 5**. That is, for feedback linearization this system is completely**feedback linearizable**where the input-output feedback linearization is equivalent to input-state feedback linearization.Can any one suggest me relevant theory behind

**Asymptotic Output tracking**problem to track a desired step-signal for this system which is based on feedback linearization?Furthermore, it will be helpful enough if this theory of Asymptotic Output Regulation problem can be extended to the case where

**only the output**(assume, the first state) is only available for measurement.could anyone provide the source code to plot basin of attraction of a given nonlinear ODE system?

I'm using the Machine Learning Algorithms (i.e. Linear Regression) for doing a prediction model for the arrival time of CMEs based on the CME initial characteristics and the interplanetary state.

I was wondering how to perform the Multivariable Linear Regression algorithm for Multioutput in Matlab.

Do you use ungauged or gauged formulation for the MVP A using edge element discretization? Which kind of linear solver did you use for solving the resultant linear system? Thanks.

Solve the continuous-time nonconvex optimal control problems as the attached file drawed with high efficiency. If you have a good idea, can you provide me with the example code?