Example: Plain pendulum in index-3 formulation (PENDU_I3) 

The plain pendulum is the classical example to illustrate higher-order differential algebraic equations. Especially, the model is able to generate the drift effect, if modeled as a lower index system.

The Mathematical Model:
The differential state variables are denoted by p1(t) and p2(t) for the coordinates and  v1(t) and v2(t) for the velocities. There is one algebraic variable
l(t). Parameters to be estimated, are the mass m and the length l. The differential equations are

p1(t)t = v1(t) 
t = v2(t)
t = -2 p1(t) l(t) /m
t = (-m g -2 p2(t) l(t))/m

One algebraic equation is needed to enforce that the mass point of the pendulum remains on a cycle,

p1(t)2 + p2(t)2 - l2  =  0

Initial positions and velocities are given, and a consistent initial value can be derived for the algebraic variable. g denotes the gravitational constant. Measurements are generated subject to an error of 1 % between 0 and 10.

Schittkowski (2002): Numerical Data Fitting in Dynamical Systems - A Practical Introduction with Applications and Software, Kluwer Academic Publishers

The complete solution of a data fitting problem is described in six steps:

  1. Define model type and document the experiment
    ... set some informative strings, define the mathematical structure and the variables
  2. Specify details of the model structure
    ... set number of differential and algebraic equations, tolerances, constraints, concentration values, ...
  3. Use editor for declaring variables and for defining functions
    ... the essential part, you have to know the mathematical equations and how to relate them to the format required by
  4. Insert measurement data
    ... the dirty job, can become boring (but you may import data from text files and EXCEL spreadsheets!) 
  5. Select termination tolerances and start a data fitting run
    ... only a few mouse clicks
  6. A separate process is started and all computed data are displayed
    ... MODFIT estimates parameters and performs a statistical analysis

Then you would like to take a look at reports and graphs:
    - parameter values
   - experimental data versus fitting criterion

Documentation and parameters:

  Back to begin of example  

Model structure:

  Back to begin of example  

Model equations (or use your own favorite editor):

  Back to begin of example  

Measurement data (or use import function for text files or Excel):

  Back to begin of example  

Parameters, tolerances, and start of a data fitting run:

  Back to begin of example  

Numerical results (computed by the least squares code DFNLP):

  Back to begin of example  

Report on parameter values, residuals, performance, etc. (or export to Word):

  Back to begin of example  

Experimental data versus fitting criterion (also available for Gnuplot):

  Back to begin of example  

Back to home page Back to EASY-FIT klaus@schittkowski.de