Background:
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 p_{1}(t) and
p_{2}(t) for the coordinates and v_{1}(t)
and v_{2}(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
p_{1}(t)_{t}
= v_{1}(t)
p_{2}(t)_{t}
= v_{2}(t)
v_{1}(t)_{t}
= -2 p_{1}(t) l(t) /m
v_{2}(t)_{t}
= (-m g -2 p_{2}(t)
l(t))/m
One algebraic equation is needed to enforce that the mass point of the pendulum remains on a cycle,
p_{1}(t)^{2} + p_{2}(t)^{2} - l^{2}^{ } = 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.
Literature:
Schittkowski (2002):
Numerical Data Fitting in Dynamical Systems - A Practical Introduction with
Applications and Software,
Kluwer
Academic Publishers
Implementation:
The complete solution of a data fitting problem is described
in six
steps:
Results:
Then you would like to take a look at reports and graphs:
- parameter values
- experimental data versus fitting criterion
Model equations (or use your own favorite editor):
Measurement data (or use import function for text files or Excel):
Parameters, tolerances, and start of a data fitting run:
Numerical results (computed by the least squares code DFNLP):
Report on parameter values, residuals, performance, etc. (or export to Word):
Experimental data versus fitting criterion (also available for Gnuplot):