linear 1st order ODEs sytem with time-dependent coefficients solver

I need to solve many times the following ODEs system:

F'(t) = A(t)*F(t) + B(t)*U(t)
F(0) = F0

where matrices A(t), B(t) and vector control function U(t) contains time-dependent coefficients or functions, which are represented in my case by discrete time-series (fixed sampling period ~ 1sec, 1e3-1e5 samples). Typical dimension of problem is ~ 2 - 6.

Proper interpolation (smooth and fast enough) of discrete time-series is the crucial requirement.

This ODEs systems will be solved many times (~ 1e2-1e4 times) for different A(t),B(t), U(t) and F0 , so solver should be fast enough + parallelizеd (multithreaded).

The MATLAB offers a set of generic ODE solvers, but I am looking for a specific problem solver (linear 1st order ODEs system with time-dependent coefficients), which will be fast and reliable. So far I found only one solver - LDE.

My questions are:

1. Is there any other special solver (implemented in MATLAB) suitable to solve my specific problem effectively and reliably?
2. Is there any reason to expect, that some special solver would be better than standard MATLAB ODE solvers in my specific problem?
3. Is there any suitable or recomended strategy how to solve this specific problem?

[Paul]

Is there any insight as to how A(t), B(t), and U(t) should behave between times n*T and (n+1)*T ?

[Michal]

Smoothly ... continuous up to the 1st derivative.

[Paul]

Is linear interpolation between the samples of A(t), B(t), and U(t) considered to be "proper"?

[Michal]

Did you read my previous comment?

[Paul]

I did. I interpreted "up to the 1st derivative" as meaning the functions A(t) etc. are continuous, but their first derivatives might not be. Perhaps I misundertood and that means the first derivatives are also continuous but the second derivatives might not be. But the basic question still stands, which is that if a solver like (if not excactly) ode45 is going to be used, what should the solver assume to evaluate A(t) et., al. at all values of t, not just at the sample times defined by the sampling period?

[Michal]

See this paper as a theoretical background of LDE solver: http://vixra.org/abs/1611.0002

Still possible to use integral form of general solution, but numerical more complicated approach.

[Michal]

Due to the @Star Strider chaotic actions here (removing of his own answer and consequent comments) at this thread is a bit difficult or confusing to understand who asked (@Torsten) for mathematical background info of the LDE solver.

[Star Strider]

@Michal —

Don’t worry — I won’t deign to respond to any of your future posts. Being insulted once is more than enough. (Single-trial learning.)

[Michal]

@Star Strider My last comment was not mentioned as insult at all. Your future answers and comments are still very welcome, really!!! The only objection is the fact, that you deleted your answer together with all relevant comments. This is defnitely non-standard behavior here!

[Star Strider]

It was obvious that my answer did not solve your problem, so I deleted it. (with very rare exceptions, I delete my answers that are not accepted or that I believe are not likely to be accepted.) You had a comment that included a citation, so before deleting my answer, I moved that comment to a comment to your original question (I checked to be certain that it had been moved before deleting my answer), although it now seems to have disappeared.

[Michal]

OK ... I understand your habits here, but as I said before, now is really not clear, that my comment with citation was requested by @Torsten. This is (of course) a negligible problem, and it was certainly not meant as an insult to you!

[Bjorn Gustavsson]

I recall reading the answers to this question and gaining some insight to this type of problem. Now, what remains is nothing more than the cold ashes. That's a pity.

Answer 1

[Michal]

This problem is now (starting from R2023a) fully covered by LTV models implemented at Control System Toolbox. These LTV models are based on former LTVTools toolbox, but from now fully implemented and supported by TMW Control System Toolbox.

So far, are not implemented all dicrete methods available for LTI models (FOH, for example), but this is probably the best way how to solve effectivelly linear time-varying systems by MATLAB.

Answer 2

[Michal]

There are two available toolboxes for time-dependent linear 1st order ODEs:

LTVtools: https://www.mathworks.com/matlabcentral/fileexchange/69563-ltvtools

LDE: https://www.mathworks.com/matlabcentral/fileexchange/60475-linear-differential-equation-solver-lde-m

[Paul]

As I understand it, all of the examples in LDE used closed form expressions for the differential equations. But in your problem you only have the A(t), B(t) and U(t) as discrete time series at a fixed sampling period. How did you use those discrete time series to implement the D and C functions for lde?

[Michal]

As I said before by proper interpolation method via anonymous function handle.

[Paul]

I'm not familiar with the term "proper interpolation." Is "proper interpolation" a specfic way to interpolate between time samples?

[Michal]

Again, did you read my question???

Proper interpolation = smooth and fast enough

Try to focus on my main questions and not on unimportant details.

[Paul]

Yes, I've read your question, comments, and answer. Which interpolation method did you use in the anonymous function handles input to LDE? Just asking out of curiosity at this point because it seems you have a satisfactory answer to your question.