Hi sry for the interruption, I did not understand the following optimization equation for the command governor,
I don't understand how to find the Q on the equation and how the equation is used, hope can get some help and feedback from you all.
You should provide the paper you're reading or at least the reference to the full paper for us to understand your problem.
Luckily I find that paper on IEEE with DOI: 10.1109/ACC.2014.6859176
In this case, J is the cost function and Q is the state cost matrix, just like in Linear quadratic regulator.
LQR
You have to choose the matrix Q, so just start using an identity matrix, then modify it to see what happens.
Related
I am doing a comparison of some alternate linear regression techniques.
Clearly these will be bench-marked relative to OLS (Ordinary Least Squares).
But I just want a pure OLS method, no preconditioning of the data to uncover ill-conditioning in the data as you find when you use regress().
I had hoped to simply use the classic (XX)^-1XY expression? However this would necessitate using the inv() function, but in the MATLAB guide page for inv() it recommends that you use mldivide when doing least squares estimation as it is superior in terms of execution time and numerical accuracy.
However, I'm concerned as to whether it's okay to use mldivide to find the OLS estimates? As an operator it seems I can't see what the function is doing by "stepping-in" in the debugger.
Can I be assume that mldivide will produce the same answers as theoretical OLS under all conditions (including in the presence of) singular/i-ll conditioned matrices)?
If not what is the best way to compute pure OLS answers in MATLAB without any preconditioning of the data?
The short answer is:
When the system A*x = b is overdetermined, both algorithms provide the same answer. When the system is underdetermined, PINV will return the solution x, that has the minimum norm (min NORM(x)). MLDIVIDE will pick the solution with least number of non-zero elements.
As for how mldivide works, MathWorks also posted a description of how the function operates.
However, you might also want to have a look at this answer for the first part of the discussion about mldivide vs. other methods when the matrix A is square.
Depending on the shape and composition of the matrix you would use either Cholesky decomposition for symmetric positive definite, LU decomposition for other square matrix or QR otherwise. Then you can can hold onto the factorization and use linsolve to essentially just do back-substitution for you.
As to whether mldivide is preferable to pinv when A is either not square (overspecified) or is square but singular, the two options will give you two of the infinitely many solutions. According to those docs, both solutions will give you exact solutions:
Both of these are exact solutions in the sense that norm(A*x-b) and norm(A*y-b)are on the order of roundoff error.
According to the help page pinv gives a least squares solution to a system of equations, and so to solve the system Ax=b, just do x=pinv(A)*b.
I read at a few places (in the doc and in this blog post : http://blogs.mathworks.com/loren/2007/05/16/purpose-of-inv/ ) that the use of inv in Matlab is not recommended because it is slow and inaccurate.
I am trying to find the reason of this inaccuracy. As of now, Google did not give m interesting result, so I thought someone here could guide me.
Thanks !
The inaccuracy I mentioned is with the method INV, not MATLAB's implementation of it. You should be using QR, LU, or other methods to solve systems of equations since these methods don't typically require squaring the condition number of the system in question. Using inv typically requires an operation that loses accuracy by squaring the condition number of the original system.
--Loren
I think the point of Loren's blog is not that MATLAB's inv function is particularly slower or more inaccurate than any other numerical implementation of computing a matrix inverse; rather, that in most cases the inverse itself is not needed, and you can proceed by other means (such as solving a linear system using \ - the backslash operator - rather than computing an inverse).
inv() is certainly slower than \ unless you have multiple right hand side vectors to solve for. However, the advice from MathWorks regarding inaccuracy is due to a overly conservative bound in a numerical linear algebra result. In other words, inv() is NOT inaccurate. The link elaborates further : http://arxiv.org/abs/1201.6035
Several widely-used textbooks lead the reader to believe that solving a linear system of equations Ax = b by multiplying the vector b by a computed inverse inv(A) is inaccurate. Virtually all other textbooks on numerical analysis and numerical linear algebra advise against using computed inverses without stating whether this is accurate or not. In fact, under reasonable assumptions on how the inverse is computed, x = inv(A)*b is as accurate as the solution computed by the best backward-stable solvers.
I have just started understanding modeling techniques based on regression models and was going through MATLAB curve fitting toolbox and the SO. I have fundamental doubts and unable to proceed further. I have a single vector set with k=100 data points which I want to fit into an AR model,MA model,ARMA model successively to see which is better suited.Starting with an AR(p) model of the form y(k+1)=a*y(k)+ b*y(k-1)The command
coeff = polyfit(x,y,d)
will fit a polynomial of degree say d=1 with p number of coefficients indicating the order of the model (AR(p)). But I just have 1 set of data which is the recording of the angular moment.So,what will go as the first parameter (x) of the function signature i.e what will be x,y?Then, what if the linear models are not good enough so I may have to select the nonlinear models.Can somebody please guide with code snippets what are the steps in fitting,checking for overfitting,residual calculation etc.
x is likely to be k (index of y). And the whole code:
c =polyfit(1:length(y), y, d).
Matlab has a curve fitting toolbox. You could use it to check different nonlinear fitting in GUI to get some intuition.
If you want steps there's a great Coursera Machine Learning course. The beginning of this course is related to linear regression and I recommend you to spend some hours at least on that beginning.
MATLAB calculate INV wrong sometimes:
See this example
[ 8617412867597445*2^(-25), 5859840749966268*2^(-28)]
[ 5859840749966268*2^(-28), 7969383419954132*2^(-32)]
If you put this in MATLAB it doesn't have inverse but in s calculator it has one.
What is going on?
Please read What every scientist should know about floating point arithmetic
Next, don't compute the inverse anyway. An inverse matrix is almost never necessary, except in textbooks, where it is convenient to write. Sadly, many authors do not appreciate this fact anyway, because they had learned from textbooks by other people who also failed to understand that an inverse matrix is a bad thing to do in general.
Since this matrix is numerically singular in double precision arithmetic, the inverse of that matrix is meaningless.
Use of the matlab backslash operator will be better and faster in general than will inverse. Or use pinv, which will be more robust to problems.
Hi I wanted to comment on Woodchips' answer but since I'm a new user I can't seem to do that, that is one very interesting article and I must read it in more detail when I have the time...
With regards to matrix inversion, you could always use the 'cond' command to calculate the condition number of the matrix, for a non-singular matrix the value should be approaching unity. As Woodchips suggested, 'pinv' does come in handy if you need to find the psuedo-inverse of a non-square matrix.
I read at a few places (in the doc and in this blog post : http://blogs.mathworks.com/loren/2007/05/16/purpose-of-inv/ ) that the use of inv in Matlab is not recommended because it is slow and inaccurate.
I am trying to find the reason of this inaccuracy. As of now, Google did not give m interesting result, so I thought someone here could guide me.
Thanks !
The inaccuracy I mentioned is with the method INV, not MATLAB's implementation of it. You should be using QR, LU, or other methods to solve systems of equations since these methods don't typically require squaring the condition number of the system in question. Using inv typically requires an operation that loses accuracy by squaring the condition number of the original system.
--Loren
I think the point of Loren's blog is not that MATLAB's inv function is particularly slower or more inaccurate than any other numerical implementation of computing a matrix inverse; rather, that in most cases the inverse itself is not needed, and you can proceed by other means (such as solving a linear system using \ - the backslash operator - rather than computing an inverse).
inv() is certainly slower than \ unless you have multiple right hand side vectors to solve for. However, the advice from MathWorks regarding inaccuracy is due to a overly conservative bound in a numerical linear algebra result. In other words, inv() is NOT inaccurate. The link elaborates further : http://arxiv.org/abs/1201.6035
Several widely-used textbooks lead the reader to believe that solving a linear system of equations Ax = b by multiplying the vector b by a computed inverse inv(A) is inaccurate. Virtually all other textbooks on numerical analysis and numerical linear algebra advise against using computed inverses without stating whether this is accurate or not. In fact, under reasonable assumptions on how the inverse is computed, x = inv(A)*b is as accurate as the solution computed by the best backward-stable solvers.