Estimation of flexible space tether state based on end measurement by finite element Kalman filter state estimator

Model predictive control for electrodynamic tether geometric profile in orbital maneuvering with finite element state estimator

“…where the over dot denotes time derivative,    and 2  being damping coefficients [12,16]. It is noted the mass matrix m is a constant matrix, which is advantageous in numerical integration [16,17].…”

“…, EX  are the Kronecker delta and expectation functions, respectively, the subscripts i and j represents the i and j component of a vector, Equation ( 30) is numerically solved by the Lagrangian multiplier method directly [12]. However, the solution 2 obs Y is not unique if   11 6n q q  , i.e., the dimension of the non-measurable state vector 2 obs  Y is greater than that of the measurable state vector 1 obs Y at the boundary  .…”

“…Then, the extended Kalman filter is applied to propagate the full state (measured and non-measurable) in the time domain to avoid the singularity in case the degrees of freedom of non-measurable state are greater than that of measured state. Random noises are introduced in the measurement model to account for the sensor noises, unmodeled dynamics and model uncertainties associated with the finite element method [12]. Once the full state estimation is obtained, the model predictive control (MPC) method is applied to control the geometric profile of EDT by minimizing the difference between the real and virtual straight tethers as shown in Fig.…”