Linear Systems - Science topic

Yes, I solved the problem. In brief, we should be mindful when assessing the origin of unexpected oscillations in numerical studies, which in fact can be due to numerical reasons (e.g. occurrence of strain discontinuity) but can also represent physical phenomena (especially in more complex nonlinear analysis). The latter is easier to be concluded if the oscillations can be attributed to a particular physical phenomenon but possibly not recognized before in the 'new' context/perspective. (Note here that it is certainly prudent to assume at the very first inspection of the results that any unexpected oscillations in our numerical studies are spurious).

You are welcome to see more on this topic in my publications listed on Researchgate.

Regarding your work, thanks for sharing the link, judging from the abstract of your work (paper not available to my institution), your problem is probably a bit different than mine.

Hi Iman,

For large symmetric matrices, the Conjugate Gradient (CG) and Conjugate Gradient Least Squares (CGLS) methods are very effective, super fast, and require little memory to implement. The entire matrix A need not be loaded into memory, and the method can read columns and rows of matrix A to apply the algorithm. See this excellent attached book on inverse problems (Aster et al.) for more details.

Regards,

Hamzeh

Of course an analytical expression exists-in terms of the minors of A-it’s, just, not useful for any practical calculation. It’s, also, possible to have an expression in terms of the eigenvectors and eigenvalues, that’s more useful.

However what all these, mathematically equivalent, approaches amount to is to provide ways for checking numerical methods.

Where all derivatives are evaluated at the equilibrium point x=x_{\rm e}\ . Its eigenvalues determine linear stability properties of the equilibrium. An equilibrium is asymptotically stable if all eigenvalues have negative real parts; it is unstable if at least one eigenvalue has positive real part.The Jacobian matrix at each equilibrium is J=f'(x)\ . An equilibrium is asymptotically stable when f'(x)<0\ ; that is, the slope of f is negative. It is unstable when f'(x)>0\ . The left two equilibria in the figure are hyperbolic (f'(x) \neq 0), the others are non-hyperbolic because the slope (eigenvalue) is zero.

Can any one share LabVIEW VIs for fractional order systems?

Dear Prof. Ankur Gajjar,

Please refer to my work other than the suggestion of Dear Scientists on the solution of linear equations in the form of AX=B.

Kind regards

The short answer is that there is no "exact" procedure for analyzing non-hyperbolic equilibrium points. However, there are some work using blowing up/down methods. You may look at this methods in a differential equation literature. You can check the way we used in one of our paper. "Bifurcations and global dynamics in a predator–prey model with a strong Allee effect on the prey, and a ratio-dependent functional response." P Aguirre, JD Flores, E González-Olivares - Nonlinear Analysis: Real World Applications, 2014. In this paper we analyzed the stability of the origin using blow up/down techniques.

I thank Victor for pointing me in the right direction.. Vieta's formulas give the easiest proof for the assertion.

Indeed, we suppose that p(x) = x^2 + b x + c as the given polynomial.

Suppose that its roots are: x1 = m + i n, x2 = m - i n (where n = 0 or n > 0).

If p(x) is Hurwitz, it has stable roots. This implies that m < 0.

By Vieta's formulas, x1 + x2 = - b and x1 x2 = c.

This gives: b = - (x1 + x2) = - 2 m > 0 and c = x1 x2 = m^2 + n^2 > 0.

Conversely, suppose that b > 0 and c > 0. We show that p(x) is a Hurwitz polynomial. In other words, we must show that m < 0.

This is quite obvious since 2 m = - b < 0 and so m < 0.

We have proved the claim that p(x) = x^2 + b x + c is a Hurwitz polynomial if and only if b > 0 and c > 0.

This is quite an elegant proof and I thank Victor again.

Dear Delaram Rabiei,

In addition to what is proposed above, i suggest you to see links and attached files on topic.

Best regards

If you are looking for sparse solution to a linear system of arbitrary size m by n, you can apply the iterative method called " Orthogonal Matching Pursuit (OMP) algorithm ".

Also, Lasso Regression model can be used in such cases, which involves a regularization term in a L1 norm. For sparsity in solution, minimization in L1 or Linfinity norm is sought.

OMP algorithm generates a sparse solution based on minimization in L0 norm ( number of non zero entries in the vector)

Dear Chang Che Kuei,

I suggest you to see links and attached files on topic.

Thank you for your help Mr. Zahoor.

First, we cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking at the qualitative dynamics of a system. Equilibrium points– steady states of the system– are an important feature that we look for. Many systems settle into a equilibrium state after some time, so they might tell us about the long-term behavior of the system.

Equilibrium points can be stable or unstable: put loosely, if you start near an equilibrium you might, over time, move closer (stable equilibrium) or away (unstable equilibrium) from the equilibrium. Physicists often draw pictures that look like hills and valleys: if you were to put a ball on a hill top and give it a push, it would roll down either side of the hill. If you were to put a ball at the bottom of a valley and push it, it would fall back to the bottom of the valley.

Agreeing with Pascal Salart, but making it simple. One signal is x(t), the other is y(t), you study the covariance matrix E(x(t+i) y(t+j)), which can be estimated by low pass filtering vectors v(t, i) = (x(t+i-N), ... x(t+i))

and w(t, j) =(y(t+j-N),... y(t+j)) with N a length of observation window,

then the scalar product <v(t, i), w(t, j)> estimates the mean expectation E() above.

The matrix obtained is a Gram matrix, hence it diagonalises with eigenvectors, eigenvalues obtained by Gram-Schmidt algorithm. QED...

Ok ?

Dear Niccolo Lemonis,

Your system is under a state space representation (A,B,C,D), I presume it is in any form it will be necessary to transform it by a change of vector basis in diagonal canonical form if the poles (eigenvalue of matrix A) are distinct case 1 or that of Jordan form if the poles are multiple cases 2.

The state case 1 cannot be controlled if the corresponding row of matrix B is zero, for case 2 it will be the last row of Jordan's block (degree of multiplicity of the pole) and its correspondence of matrix B which must be zero. It is also the same principle applied to the Kalman decomposition.

Best regards

It is difficult to give any hint without knowing the equations. However, at a general level linearization around an equilibrium might not provide a correct behaviour if the linearized system is not exponentially stable, i.e., if some of its eigenvalues are zero. For example, the ODE

y'=-y³

is stable around y=0, but its linearization is

y'=0

for which each initial condition is stationary.

Have a look at

نعم

Dear Joo ...

Check your the step of discretization and linearisation of your simulation.

Regards

Hi, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. You can refer to some papers on this topic:

Best

Dear Matija Perne ,

I think you are not answering the original question. Is linearity assumption useful and actually accurate in many applications? Absolutely. But "do we actually have linear system in nature" or are they "an approximation of nature"?

None of the examples you provided are linear. Let us take your grass and deer example: what about the negative numbers of deer? Or grass? Or even not an integer number of deer (they are quantized, as far as I can tell)? Or even outside of the mathematical definition of linearity, what happens when the number of deer becomes too large to be supported by the meadow of a fixed size - then the 'eating rate' will be capped at some point. Does the postulated proportionality hold at any point in time, or averaged over some period of time?

Ggiven that the absolute majority of "things" in the universe (mass/energy etc) are quantized, the existence of anything "truly linear" seems to be impossible. Maybe one can do something "truly linear" with time or gravity, but then one gets into things like "what happened before the big bang" - or even "what is time any ways"...

Libor Pekař

Dear Libor,

I don’t know if you eventually found the answer to your question. I shall give my answer below.

The output y(t) of a system has two components: the free component y_L(t) generated by the initial conditions and the forced component y_F(t) generated by the input signal.

The forced component is the convolution between the causal impulse response h_C(t) (the inverse Laplace of the transfer function H(s)) and the input signal u(t) and has two sub-components: the transitory component y_T(t) related to the system’s structure (influenced by the signs of the poles) and the permanent component y_P(t) related to the input signal (influenced by the input’s poles).

If the linear system is exponential stable, then the free- and transitory components of the output response vanish as the time increase sufficiently, i.e. y_L(t) -> 0 and y_T(t) -> 0 as t-> infinity. Thus, the only component which remains is y_P(t) and moreover, for exponentially stabile systems the permanent component is of the same shape as the input signal as t-> infinity.

BIBO-stability reflects the input-output stability and it is evaluated on the transfer function, H(s). On the state space model one evaluates Lyapunov, asymptotic, exponential and other kind of stability properties.

Consequently:

- If the linear system is internally Lyapunov exponential stable, then for bounded inputs the output responses are bounded, i.e. THE SYSTEM IS BIBO-STABLE;

- If the linear system is internally unstable, but it is also non-minimal such that all the uncontrollable or the unobservable eigenvalues are those with positive sign (within the open right complex half -plane, inducing instability), then THE SYSTEM IS BIBO-STABLE. Explanation: taking into account that the transfer function obtained from the state-space model is always the irreducible, the unstable roots of the characteristic polynomial (poles of H(s)) will be simplified by the similar zeros of the nominator of H(s); consequently, H(s) will have only negative poles which means that the transitory component vanish as time tends to infinity and for bounded inputs the permanent component of the output response will be also bounded as determined by the poles of the input (the Laplace transform of u(t)).

- Conversely, if the transfer function has all its poles with negative real part, i.e. the system is BIBO stable, we cannot know if it is also Lyapunov exponentially stable (the characteristic polynomial can have uncontrollable or unobservable eigenvalues with positive real parts). From a stable transfer function one obtains a minimal state realization which is thus exponentially stable, but this will not reflect the internal state of the system which can be evaluated only on the system mathematical model of differential equations – i.e. the equations which result directly from the mathematical modeling process.

- A transfer function with all its poles in the open left complex half-plane has the impulse response absolutely integrable, i.e. it is L1-stable and vice-versa.

What I understand from the question is if A is the matrix of your linear system in X basis then one can always get its QR factorization and find its eigenvalues through QR algorithm. Now write your system as per your eigenvalues obtained.

Dear Ádám Domina,

First of all, you need to make sure that the Jacobian matrix evaluated at the equilibrium state is hyperbolic. ( The eigenvalues has non zero real parts) Then you can apply the Hartman-Grobman theorem. That is, the linear system and nonlinear system behave the same in the neighborhood of the equilibrium point. So, the main point is to determine the eigenvalues of the Jacobian matrix which essential to study the stability of your system.

Best regards

Hi, so far, it seems you are on the right path. However, I have some remarks:

1. what's your control goal? I assume is the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. Then that desired position and angle (with respect to your model) giving you the upright position are your controller reference (could be 180° or 0°, it depends which angle you took as the reference to develop your model).

2. Because you are thinking in implementation and as you said the only measurements available will be position and angle, then you would need an observer to estimate the other two states (derivatives). I recommend a Kalman Filter.

3. In order to better understand how to use your linear approximation to control your non-linear model, the following paper could help (which explain how to translate your linear model to the operating point of the nonlinear system):

Regards.

see

Farey sequence in the appearance of subharmonic Shapiro steps

Odavić, Jovan, Mali, Petar, Tekić, JasminaJournal:Physical Review EYear:2015

Mehrdad Babazadeh

Thank you Dr. Babazadeh.

Indeed, I have also faced another problem. The mathematical relation which is available for describing this impedance is a relation between frequency and impedance. On the other hand, this nonlinear element is connected to a nonlinear power-electronic circuit which results in not having any unique or finite frequencies in the voltage which is exerted to this element. I need a time-domain relation for the impedance of this element which depends on the continuous range of frequencies of the voltage which is applied to the element. I do not know how I can do it. Inverse Fourier transform only gives me a function of time. Such an impedance which is only a function of time, does not have any relation with the waveform of the voltage which is applied to it. Since the relation with frequency is nonlinear, I cannot consider all "j2pi*f"s in the relation between voltage and current, equivalent with the derivative of voltage. Maybe the Taylor exception of the frequency-domain relation of impedance can be helpful to find some time-domain relation for impedance as a function of the first, second, and ... derivatives of voltage. But it makes it too much complicated and forces some estimations and errors to the simulation doe to applying finite terms in the Taylor exception. Is it possible to simulate this element's electrical behavior in Simscape in any better way?

sigma = sqrt(18); % standard deviation

mu = 11; % mean

z = randn(100,1); % Sample from Standard Gaussian (mean 0, variance 1)

x = z*sigma + mu; % Sample from Gaussian (mean 11, variance 18)

Thank you very much for your reply Dr Deepak Kumar . Anyhow, I have managed to simulate a TFET inverter circuit using TCAD mixed-mode simulation. Still, there are difficulties with other circuits. If you don't mind, could I contact you in above-given number in future?

Thank you so much.

Thank you everyone for all your time and good explanations. It is clear :)

Dear Parsa Ghofrani ,

According to your assumptions, we have n distinct equations.

To reach your request proceed as the following:

Rearrange the equations as follows:

1). The coefficient of the first unknown in the first equation is not zero.

2).The coefficient of the second unknown in the second equation is not zero.

etc

n).The coefficient of the nth-unknown in the nth-equation is not zero.

The new augmented matrix guaranteed your request.

Otherwise, if your matrix is full of zeros and det(A) =0, then the system has

i). No solution

or

ii) Infinite number of solutions.

Best regards

The Petsc library provide a generic interface to call various solvers, including some direct solvers for square unsymmetric matrices.

A list of available solver is provided at https://www.mcs.anl.gov/petsc/documentation/linearsolvertable.html

Listed direct solver for square unsymmetric sparse matrices (seqaij or aij) are mostly based on LU decomposition : Pastix, MUMPS, superLU, UMFPACK and KLU from SuiteSparse, MATLAB and LUSOL.

Using Petsc makes it easy to change from one solver to another, or try a preconditionner. Direct solvers are provided as preconditionners. For instance, command line options can be:

-pc_type lu -pc_factor_mat_solver_type superlu_dist -ksp_monitor -ksp_view

scipy's scipy.sparse.linalg.spsolve seems to wrap UMFPACK and SuperLU :

Decentralized implies that there is no single point where the choice is made. Each hub settles on a choice for it's very own conduct and the subsequent framework conduct is the total reaction but, a "linear system"deal with all together at once

I'm thinking you might be able to augment your state vector x(t) to incorporate a(t), i.e. use z(t) = [x(t); a(t)], and re-define A, B, C (&D?) for this augmented system. This would also allow you to model aspects of a(t)'s behaviour too.

Hi, Xiang-Ming Gu

LEVINSONG(R,V,N) solves a Hermitian Toeplitz system of equations using the Levinson-Durbin recursion, it is available on http://mathforum.org/kb/message.jspa?messageID=74329

I follow the answer

Thanks dear Ryan Vogt and Debopam Ghosh

Hi

One of the disadvantages of DIgSILENT PowerFactory is the lack of B,C,D matrix in out put. This tool only present the A matrix. Thus for control application it is not suitable.

If you control a system, you often want to operate it around an equilibrium, so it makes sense to use this point. However, there might be cases for which you want shift this operating point via (nonlinear) control and generate a new equilibrium. The simple idea is given here

https://www.cds.caltech.edu/~murray/courses/cds101/fa02/faq/02-10-09_linearization.html.

Then you could linearize about the new equilibrium point of the closed loop.

Generally, for control purposes you also can linearize along a known trajectory. Think of a rocket or a robot arm. In this case your system is time varying. To use well established linear tools, you can check (closed loop) stability on these operating points (i.e. which are not equilibrium points) along the trajectory, or design controllers along the trajectory. This will give you the behavior close to this point. However, stability does not mean that it will return to this point.

In "Incomplete Cholesky factorizations with limited memory" (Lin, More, paragraph above formula (4.1)), it is said: "the vector b is the vector of all ones". Or you can look at https://github.com/vakho10/Sparse-Storage-Formats

Hi,

Use ADRC, see my published papers

The advantages of OMP and MP algorithms for Direction of Arrival estimation:

Applying BS algorithms to a DOA problem enhances resolution

and decreases complexity. Moreover, the knowledge of the number

of signal sources is not required to know in these algorithms. In

addition, they do not need any post-processing to converge to

the ML solution since the output of these algorithms is straightly

the DOAs. ML algorithm compares all feasible directions and then

selects the most likely one. On the other hand, BS algorithms

compare some of the angles and select them in a smart method.

Hence, BS algorithms are much more computationally efficient

in comparison to other algorithms of DOA estimation such as

MUSIC and ESPRIT to approach to the ML solution. Moreover, BS

algorithms converge to the ML solution when the value of SNR

is low, whereas other approaches converge at high SNRs only.

In addition, in other methods for DOA estimation, the number

of estimated DOAs is limited by the number of antennas. The BS

based DOA estimation methods can estimate more DOAs than the

antennas number. Among BS methods, OMP algorithm provides

slightly higher performance than MP algorithm with moderately

higher computational complexity.

By using matlab software, x=pinv(A)*B

Hi Nasser,

Nonlinear systems are complicated because of the high dependency of the system variables on each others. I have to tell you that Most of the Engineers are using linear systems in their analysis. Before start solving any Engineering problem we try do linearization to get a linearized model. That is because, the nonlinear problems are difficult to solve and are so expensive. However, linear problems give very close solution to the nonlinear ones with less cost, time and effort.

I can list you many different examples of solving nonlinear problems with a linearzied model if you want. But I think it might be enough to mention simply why we are perform linearzation to solve nonlinear problems.

Best regards

Mohamed

Yes Houssem, the chosen output could represent the flat output which always has a physical meaning. In this example however, no output is defined.

Dear Ali,

A true system that is not strictly clean is a non-causal system! what i can offer for the moment is to see the file attached this can help you solve your problem.

Best regards

Dear Xu, The question is little confusing since you have asked the asymptotic stability of y=Cx. The question may mean the stability of output signal 'y', or it may mean the stability of output dynamics dy/dt=C dx / dt. I am answering both the cases.

If the stability or boundedness of the output signal ''y" is in question, then the output is stable for any norm bounded matrix C. As the states "x" tends to zero in forward time, the output also asymptotically converges to zero provided the matrix C is finite (If C is time dependent, then also you have to check whether C is norm bounded or not ). However one should be careful about the fact that, depending on the elements of the matrix C, the output may initially peak or oscillate.

The answer to the second case is little complicated. Since

dy/dt=CAx+CBKx (u=Kx), the same K may stabilize the x system, but it does not mean it can always stabilize the y system. Therefore the asymptotic stability of output dynamics (y system) not only depends on x but also depends on the C matrix.

I recommend transforming the system into LTV system. I tried that.you can transform a nonlinear system into a linear time varying one. The uncertainty will be covered by the state and input matrixes.

This is an important question.

A good start on an answer to this question is given in

More to the point, see

and

Dear Ashraf, product of posdef matrix A and negdef matrix B is a negdef matrix:

if u^TAu > 0 for every u, u^TBu < 0 for every u, then u^TABu = u^TAu u^TBu/(uu^T) < 0 because uu^T > 0 .

Gianluca

As a co-author (with H. Gorecki, A. Korytowski and S. Fuksa) I may recommend: Analysis and Synthesis of Time-Dalay Systems. Wiley. Chichester. 1989

@Dimel

Your question belongs to classical  physics or quantum physics . Highlight the same in detail.

B.Rath

Do you have relative degrees higher than one in some of the outputs?

My guess is that you must make sure to use a sparse solver (specifically, the Thomas algorithm) also when solving the linear subsystems associated to each block. It is not clear from your description of the steps you attempted that this was the case.

More generally, a number of specific algorithms are described in the literature, see for example

Heller, Don. "Some aspects of the cyclic reduction algorithm for block tridiagonal linear systems." SIAM Journal on Numerical Analysis 13.4 (1976): 484-496.

and the many papers that refer to it.

Hello. Jose Fernando provided a good answer above. I think I can clarify a bit further.

The "classical" Kalman filter is posed in continuous time for a system of the form dx/dt=Ax(t)+Bu(t)+Gw(t), z(t)=Hx(t)+v(t) where w(t) and v(t) are white Gaussian noise (WGN) with zero mean and known covariance Q and R, respectively, x(0) is Gaussian with known mean and covariance and all three of x(0), v(t) and w(t) are uncorrelated. It is a simple matter to proceed from the constant-coefficient continuous time linear stochastic system to a discrete time equivalent system by "discretizing" the continuous time system via the "variation of constants" formula. This involves integrating the dynamical equation and assuming x(t) is constant between sampling times. The process noise covariance Q(k) of the discrete time system can also be obtained this way. The classical Kalman filter for the discrete time system can then be written down quite easily. In general, the coefficient matrices for the discrete-time KF depend on the time, but this is not a problem.

Now when the continuous time system is not time invariant (in terms of its coefficient matrices), it is not generally possible to discretize the continuous time system to obtain the discrete time equivalent and hence the KF. However, the discrete-time KF is general in the sense that the equations hold even when the coefficient matrices are time varying. You just have to know what they are at each time instant.

So I think this answers your question: if the continuous time system is time varying, you can't easily write down the KF, but you can for a time-varying discrete-time stochastic linear system.

In your opinion what is Frobenius Norm?

You don't just fix the singularity. If the matrix is singular, then you have to work with it by using basic linear algebra. Numerically, its simpler to use, e.g., a QR decomposition of the matrix (instead echelon form) to determine if there are many or no solutions.

One often used solution strategy is to solve it in the least square sense. This gives you a solution which leads to the smallest residual norm in the Euclidean norm which, however, will be nonzero in general.

thank you all to your responses they was really helpfull, and i managed to ''get rid '' of the singularity.

Many thanks for your answers.

Thank you very much. However, I am already having the attached one. I need the original one.

--Vikas

One of the solutions could be using Moore-Penrose pseudoinverse of matrix. This can be obtained using pinv function in Matlab.

I tried with the matrices provided by you and it seems to work and did not result in inf eigenvalue. You can use the following code:

load('M11.mat');load('M12.mat');load('M21.mat');load('M22.mat');

load('K11.mat');load('K12.mat');load('K21.mat');load('K22.mat');

K = [K11 K12; K21 K22];

M = [M11 M12; M21 M22];

[v d] = eig(pinv(M)*K);

diag(d)

Still I would suggest you to have a closer look into the eigenvalues to ensure the correctness of the solution.

Hope this helps.

Dear Alexander,

1.  You are right:

-- the resistance R is not linearly dependent on V,

-- is time varying.

But this is not the issue. The equation sought should be a linear differential equation for a chosen observed quantity ( i, V , R, C, or whatever else) replacing x in the equation

(*)  d{x(t)}/d{t} = A(t) x(t),   (the LHS denotes the derivative of x with respect to t )

AND  the coefficient A(t) should be independent of anything but t . In your model such a quatity and such an equation  is not distinguished yet.

2. There is also the possibility, that  any other variable is chosen for the independent variable (instead of  t ). Even, one can imagine an equation of the following form d{R(V)}/d{V} = Q(V) R(V) .  I am not developing this point of view.

3. The solution of  (*)  equals   x(t) = x(0) \exp[ \int_0^t \, A(u)\, du ]

4.   Function R(t)=(wt+1)/wC is a solution of the following differential linear non-homogeneous time independent equation:

d{R(t)}/d{t} = 1/C, with the initial condition R(0) = 1/wC

On the other hand, its inverse G=1/R = wC/(wt +1) satisfies the following differential linear homogeneous time varying equation:

d{G(t)}/d{t} = - [ w/(wt+1)] G(t), with initial condition G(0) = wC.

5. Other details are omitted for readability of the message:)

Regards, Joachim

Hauke is correct. That particular work was presented at the ASA meeting in 1963. See Section 3 of the attached article. It also mentions some interesting facts about that event.

Piecewise linearisation is the normal technique - take a look at https://www.hindawi.com/journals/mpe/2013/101376/

Your attachment seems unreadable.

Hello Anirudh Nath,

As mentioned, you are able to achieve the complete linearization of your 5th order system, since the order is equal to the relative degree of the system with the selected output. In such case, you can define an input/control signal such that all the nonlinearities are cancelled and the resulting output dynamics (which, in this case, is the whole closed-loop dynamics) turns out to be linear and time-invariant. Therefore, to ensure that the output trajectories converge to the origin, you are able to apply the available theory for linear systems. That is, you can compute the characteristic polynomial of the constant matrix A, which characterize the output dynamics, and apply the Routh-Hurwitz criterion.

However, if you redefine the output signal such that the relative degree of the system is less than its order, then you have to analyse the resulting internal dynamics, in addition to the external dynamics (the output dynamics).

As the Dr. Gómez-Espinosa mentioned, the best books for the study of feedback linearization theory are the ones by Khalil, and Slotine and Li.

I also can recommend some of my papers, where you can find practical applications of the feedback linearization theory to 4th order systems, which are two-degrees-of-freedom underactuated mechanical systems. I'm sure that you can acquire some insight from them in order to solve your problem.

[1] Carlos Aguilar-Avelar and Javier Moreno-Valenzuela, "New Feedback Linearization-Based Control for Arm Trajectory Tracking of the Furuta Pendulum", IEEE/ASME Transactions on Mechatronics, vol. 21, no. 2, pp. 638-648, 2015.

[2] Carlos Aguilar-Avelar and Javier Moreno-Valenzuela, "A Feedback Linearization Controller for Trajectory Tracking of the Furuta Pendulum", Proc. of the 2014 American Control Conference, Portland, Jun. 2014, pp. 4543-4548.

[3] Carlos Aguilar-Avelar and Javier Moreno-Valenzuela, "A composite controller for trajectory tracking applied to the Furuta pendulum", ISA Transactions, Vol. 57, pp. 286-294, 2015.

Best regards,

Carlos Aguilar-Avelar.

If your are looking for source code

Thank you Dr. Chinedu, I will try that. 

I did not notice your message and I am sorry for the late reply.

We calculate without imposing gauge conditions, but no error will occur. Therefore, I can not give any useful advice for you.

I am sorry, I am going to resign the company at the end of this month and I can not use my old program. If you can come up with something, I will comment.