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 denotes the Lagrangian multiplier and g the gravitational constant then the Lagrangian is
Now substituting , 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 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 are measured (but the values of the velocities u(t), v(t) are not). This gives 11 times ,..., 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 and . These weights will be used as weights 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 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 .
The ''true'' value of g is .