A synchronous language with ODEs

Zelus: A Synchronous Language with ODEs

Zélus is a synchronous language extended with Ordinary Differential Equations (ODEs) to program hybrid systems that mix discrete-time and continuous-time models.
An example is a (discrete-time) model of a control software paired with a (continuous-time) model of the plant.
The language shares the basic principles of the synchronous languages Lustre with modularity features from Lucid Synchrone (type inference, hierarchical automata, and higher-order functions).
It is conservatively extended to write continuous-time models expressed by ODEs and zero-crossing events.
The compiler is written in OCaml and is structured as a series of source-to-source and traceable transformations that ultimately yield statically scheduled sequential code.
Continuous-time models are simulated using an off-the-shelf numerical solver (here Sundials CVODE and, for the moment, the two built-in solvers ode23 and ode45).

Getting Started

The easiest way to install Zelus is via Opam, the OCaml package manager.

opam install zelus

You can then test your installation with:

zeluc -version

The manual, examples, and research papers can be found at http://zelus.di.ens.fr

Optional Dependencies

By default Zelus relies on the built-in solvers.
To switch to Sundials CVODE you need to install sundialsml (which requires sundials).
Some examples also depend on the zelus gtk library (which requires gtk2.0)

opam install sundialsml zelus-gtk


We also provide a dockerfile to setup an environment with all the dependencies.
Build the image with (you might need to increase available memory in docker preferences):

make docker-build

Run with:

make docker-run

A Simple Example

Consider the example of a bouncing ball.
The zelus code is the following:

let loose = 0.8
let     g = 9.81
let    x0 = 0.0 
let    y0 = 10.0
let   x'0 = 1.0
let   y'0 = 0.0

let hybrid main () = () where
  rec der x  =  x' init x0
  and der y  =  y' init y0
  and der x' = 0.0 init x'0
  and der y' = -.g init y'0 reset up(-.y) -> -.loose *. last y'
  and present up(-.y) -> local cpt in
    do  cpt = 0 fby cpt + 1
    and  () = print_endline (string_of_int cpt) done

The dynamics of the ball is expressed with four ODEs defining the position (x, y) and the speed (x', y') of the ball given an initial position (x0, y0) and an initial speed (x'0, y'0).
Whenever the ball hits the ground up(-. y) the discrete time code of the body of the present construct is executed (here incrementing a simple counter).


The zeluc compiler takes a zelus file (e.g., bouncing.zls) and compile it to OCaml code (e.g., bouncing.ml).

zeluc bouncing.zls

You can also specify a simulation node.
The compiler then generates an additional file containing the simulation code (e.g., main.ml).

zeluc -s main bouncing.zls

To build an executable, the last thing to do is to compile the OCaml code using the zelus library.

ocamlfind ocamlc -linkpkg -package zelus bouncing.ml main.ml -o bouncing

Other Examples

This repository includes runnable examples demonstrating different aspects of the language.
The source code for several of the examples can be found in the examples directory.
To build most of the examples:

cd examples && make

The executables can be found in each example directory (e.g., horloge/horloge_main.exe).



We use dune to build the compiler, the libraries, and the examples.
To build the project:

dune build

This produces two executables (and some tools in ./tools):

  • compiler/zeluc.exe: native code

  • compiler/zeluc.bc: byte code (can be used with ocamldebug)

Libraries are split in two packages:

  • zelus: the standard libraries

  • zelus-gtk: additional libraries that depend on lablgtk (only built if lablgtk is installed)

The build automatically detects if sundialsml is installed and updates the librairies accordingly.


To run all the tests:

cd test

Tests are split into 3 categories: good, bad, and run.
To launch a single subset (e.g., good):

cd good

To clean generated files:

make clean

Citing Zelus

  author = {Timothy Bourke and Marc Pouzet},
  title = {Zélus: A Synchronous Language with {ODEs}},
  booktitle = {16th International Conference on Hybrid Systems: Computation and Control (HSCC'13)},
  pages = {113--118},
  month = mar,
  year = 2013,
>= "20180523"
>= "2.6"
>= "4.08.1"
Reverse Dependencies
>= "2.2"