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Parameter identification of a planar pendulum


Consider a pendulum of length l and mass m which is fixed at the origin of a coordinate system and moves in the x-y plane.


We consider the formulation of the dynamics of motion as a system of differential-algebraic equations. If tex2html_wrap_inline2033 denotes the Lagrangian multiplier and g the gravitational constant then the Lagrangian is


Now substituting tex2html_wrap_inline2037 , tex2html_wrap_inline2039 for the velocities and applying index reduction techniques, the equations of motions read as a system of differential-algebraic equations of index 1


Whereas the two additional entry conditions to be satisfied at tex2html_wrap_inline1301 due to the index reduction are


Now the dynamic model has the dimensions

NY = DIMY = 4, NV = DIMV = 1, NU = DIMU = 0, NP = NPAR = 1, NRB = NRB = 2, NZB = NZB = 0.

We assume that the gravitational constant g is unknown and has to be identified by using the results of an experiment:
The pendulum moves for two seconds. After every 0.2 seconds the values of x(t), y(t), and tex2html_wrap_inline2061 are measured (but the values of the velocities u(t), v(t) are not). This gives 11 times tex2html_wrap_inline2067 ,..., tex2html_wrap_inline2069 of measurements and 33 measurement values in total.

It is assumed that the measurements are not fully precise but with measurement errors of standard deviations of tex2html_wrap_inline2071 and tex2html_wrap_inline2073 . These weights will be used as weights tex2html_wrap_inline1253 in the nonlinear least squares objective of Equation (5).

A multiple shooting node is selected at every time of measurement. As initial estimates of the state variables x(t), y(t) and tex2html_wrap_inline2061 at the multiple shooting nodes we simply use the given measurements. Initial estimates of the velocities u(t), v(t) can be computed by local interpolation of the measurements of x(t), y(t).

As initial estimate of the parameter g we use tex2html_wrap_inline2093 .

The ''true'' value of g is tex2html_wrap_inline2097 .

next up previous contents
Next: A minimum energy problem Up: Examples Previous: Examples

Oskar von Stryk
Tue Feb 1 13:50:42 CET 2000