Convert continuous time (ODESystem) to discrete time system?

sdwfrost · May 22, 2024, 11:58pm

I’d like to take a continuous time ODE, and convert it to a discrete time system, along the lines of c2d in ControlSystems.jl.

Here’s the ODE version:

github.com

epirecipes/sir-julia/blob/master/markdown/ode/ode.md

# Ordinary differential equation model
Simon Frost (@sdwfrost), 2020-04-27

## Introduction

The classical ODE version of the SIR model is:

- Deterministic
- Continuous in time
- Continuous in state

## Libraries

```julia
using DifferentialEquations
using SimpleDiffEq
using Tables
using DataFrames
using StatsPlots
using BenchmarkTools

This file has been truncated. show original

Here’s the discrete-time equivalent:

github.com

epirecipes/sir-julia/blob/master/markdown/function_map/function_map.md

# Function map
Simon Frost (@sdwfrost), 2020-04-27

## Introduction

The function map approach taken here is:

- Deterministic
- Discrete in time
- Continuous in state

## Libraries

```julia
using DifferentialEquations
using SimpleDiffEq
using DataFrames
using StatsPlots
using BenchmarkTools
```

This file has been truncated. show original

I suspect that this is easy using ModelingToolkit (now that it has DiscreteProblem support), and that there is something out-of-the-box to do this, but I can’t find it.

A slightly harder version is to take a ReactionSystem and convert it to a discrete time Markov process - any pointers here would also be great!

baggepinnen · May 23, 2024, 6:08am

I’m not sure exactly what you would like the result to be, a discrete-time simulation, or the discrete-time symbolic equations?

In either case, there is no support for this at the moment. For general MTK models, this “conversion” implies integration, so what you’re asking for would require the ability to choose the integration method etc. Looking at the discrete-time equations is pretty meaningless already for simple integrators like RK4, so I assume that you are interested in a numerical simulation with a particular integrator? You can of course already achieve this by adaptive=false and dt=δt when you call solve

sdwfrost · May 23, 2024, 4:21pm

To clarify, f you look at the specific examples I gave, there is an integration of the rates in the ODE to give proportions in the map; my question was whether this can be done symbolically in the current MTK. I’ll think about it some more (c2d does much the same thing in a different setting…)

baggepinnen · May 23, 2024, 4:30pm

I don’t recognize the integration scheme used in your code.

There is no feature to carry out this transformation at the moment, but if you describe it as an algorithm it can certainly be automated.

ChrisRackauckas · May 23, 2024, 7:46pm

The example there is just a discrete time model? the new way to do this is to use clocks:

There isn’t an automatic conversion of an ODESystem into a discrete time clocked system at this time, but if it’s just for simulating then Euler with the right step size is the same thing.