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New ODE and DAE Solver


John Hedengren (view profile)


Differential and algebraic equation solver for high-index, stiff, and mixed-integer systems.

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This example demonstrates the numerical integration of simple Differtial Algebraic Equations (DAEs). Similar to ODE integrators in MATLAB, it can handle ODE stiff systems as well as high index (Index 2+) DAE systems. See the following YouTube video for a step-by-step tutorial.

The directory contains a folder (apm) that contains the library of functions for working with APM. The model file (demo.apm) contains the following text:

   tau = 5
   K = 3
  End Parameters

   x = 0
   y = 0
  End Variables

   tau * $x + x = K * u
   y = 2 * x
  End Equations
End Model

The data file (demo.csv) specifies the time points and any inputs to the model. In this case, the input 'u' is specified at the following time intervals:

time, u
0, 0
0.5, 0
1, 1
2, 1
3, 1
5, 1
8, 1
12, 1
15, 1
18, 1
21, 1
22, 1
25, 1
28, 1
30, 1

The function 'apm_solve' receives an input of the application name (in this case 'demo') and returns a structure with the results of the simulation. In this case, the solution is returned into a structure named 'z'.

z = apm_solve('demo');

The structure contains all of the parameters and variables defined in the model file as well as the time points. A plot of 'time' and 'x' is provided with the demo and these are referenced as 'z.time' and 'z.x'.

The APMonitor Modeling Language ( is optimization software for ODEs and DAEs. It is a full-featured modeling language with interfaces to MATLAB and Python. It is coupled with large-scale nonlinear programming solvers (including APOPT and IPOPT) for parameter estimation, nonlinear optimization, simulation, and model predictive control. There is a discussion group as well as regular webinars for those interested in dynamic modeling and simulation.


Optimization, Nonlinear Control, And Estimation Toolbox, Minlp: Mixed Integer Nonlinear Programming, and Model Predictive Control inspired this file.

Required Products MATLAB
MATLAB release MATLAB 7.13 (R2011b)
Other requirements Internet connection to run software as a web-service
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Comments and Ratings (1)
08 Jan 2013 Vasily Kozhevnikov  

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