How Simulink handles calls to discrete blocks

I have a Simulink model which includes a Matlab System Block for which I have defined a discrete sample time following this guide.

Indeed, if in Simulink I enable the sample time information (from Debug/Information Overlays panel), I see that the output signals are characterized by that sample time.

Instead, as inputs to this block I have signals with different sample times or multirate.

I am simulating with a ode45 Variable-Step solver.

Let's denote with the simulation end time. The initial time is 0.

If I run the Simulink profiler I see that this Matlab System Block, characerized by a sample time , gets called n times.

Though, I notice that . What I see is that .

Could you help me understand why?

Answer 1

[Fangjun Jiang]

That is right. Read this. For every integration step, the derivative function needs to be called between 4 times (for RK4) and 6 times (for RK5).

https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods

https://maths.cnam.fr/IMG/pdf/RungeKuttaFehlbergProof.pdf

[Paul]

"for which I have defined a discrete sample time"

But if the block has a discrete sample time, shouldn't it only be executed at that sample time?

[Lorenzo]

Thanks @Fangjun Jiang for the reference. I agree that a block should be called more times (between 4 and 6) when using Runge-Kutta methods.

But, as @Paul highlighted, I believe that this holds only for blocks which have continuous states. Instead, for blocks which have discrete dynamics, I think the solver should call them only at time steps which are integer multiples of the discrete blocks sample times.

Maybe I am missing something...

[Fangjun Jiang]

Good question! I have to give it some thoughts and maybe do some experiments later.

1. Assume dy/dt=f(u,y,t)+d(u,y,t) where f(u,y,t) is a continuous function, and you define d(u,y,t) as a "discrete function", or at least, on the surface, you can define the 'sample time' of the implementation of the d(u,y,t) as a discrete sample time. When doing RK45 integration, how do you think we should evaluate dy/dt 4 or 6 times? The u and y in d(u,y,t) are continuous. I don't mean to count questioning but want to trigger some thoughts.
2. In Simulink document, the discrete sample time means "executes the output method of a block". It might be helpful if you can provide more details of your MATLAB System block. Might there be a difference between output method and update method? https://www.mathworks.com/help/simulink/ug/types-of-sample-time.html
3. What I might do, is to construct both f(u,y,t) and d(u,y,t) in a MATLAB Function block, where I can use a persistent variable to count how many times it is called. But this may not reflect your MATLAB System block.

[Fangjun Jiang]

For the mean time, @Lorenzo, can you try different solver and sample time to see how many times it is called?

[Lorenzo]

Thanks @Fangjun Jiang for your reply.

Let me try to answer to your point 1.

So, when I am modeling a discrete system, I am not using differential equations, but finite difference ones.

This means that, for example for a linear system, I have:

So, conversely to continuous systems, there is no derivative. The system state evolves only at discrete time steps. This means that whatever solver I use, I think it should call the block only at multiple integers of the sample time, because there is no integration to perform (which seems to be aligned with the link you shared https://it.mathworks.com/help/simulink/ug/types-of-sample-time.html).

In the meanwhile, I gave a quick read to other help pages (such as https://it.mathworks.com/help/simulink/ug/managing-sample-times-in-systems.html), which might explain why discrete systems could be called more times.

But:

1. I need to carefully evaluate if this is my case.
2. Still, I do not understand why multiple calls are necessary (if not done at multiple steps of the discrete sample time), given that the system state only changes at discrete steps. Also because calling a block could be time consuming (as it is in my case since within the discrete block I am invoking some python code).

Hope it helps to clarify my point.

Thanks for your help

[Paul]

@Fangjun Jiang

According to the linked doc page in your comment:

------------------------------------

Discrete Sample Times

Given a block with a discrete sample time, Simulink® software executes the block output or update method at times .... (emphasis added)

---------------------------------

I realize that a few lines later that same doc page says " executes the output method." Nevertheless, a block with discrete sample time is only executed (update and output methods) at the integer multiples of the sample time (pluss offset if nonzero).

"dy/dt=f(u,y,t)+d(u,y,t) " with u and y continuous.

In this case, dy/dt is computed at the major and minor time steps of the ODE solver, but d(u,y,t) is only executed as its discrete sameple time hits. If using a variable step solver, Simulink ensures that each sample time hit occurs on a major time step. If using a fixed step solver, the step size has to be the sample time of the d(u,y,t) block divided by an ineger >= 1. In either case, the d(u,y,t) block is executed on the major step prior to the start of the integration, i.e., d(u,y,t) is computed at t_n and held constant over all of the dy/dt calculations that are needed to step from t_n to t_n + dt. More precisely, even if t_n + dt = t_(n+1) the d(u,y,t) block is not executed on the final minor time step of the integration. At least that's my understanding based on Simulation Loop Phase

@Lorenzo

What did you see on that linked doc page that "might explain why discrete systems could be called more times."?

Based on your comment above, it sounds like your model is strictly discrete-time, and therefore should be using a discrete-time solver. But your original question mentions using ode45, which is a continuous-time solver? However, even if the solver is set to ode45, Simulink will automatically change it the to the variabie-step discrete-time solver if there are no continuous states. Can you clarify whether or not the model has continuous states? Is the solver is fixed- or variable-step? If the former, what is the step size? In either case, what's the sample time of of the Matlab System Block?