On single-basis online asymptotic trajectory decomposition for control applications

In this paper, the problem of decomposing a trajectory online by using single basis function (and its time-shifted copies) for control applications is considered. Trajectory decomposition has been explored in control areas including fuzzy and neural-network control, system identification, and learning control, where a given trajectory is decomposed (approximated) by using a set of basis functions. This work aims to achieve online asymptotic trajectory approximation (where the trajectory to be decomposed is only partially known) by using only single basis function and its time-shifted copies without truncation. First, we consider the problem as a least-square minimization problem, then the issue of truncating basis functions in decomposition is resolved through a zero-period extension of the trajectory (i.e., extending the beginning of the trajectory at its initial value for a finite period). It is shown that the coefficients of the basis functions at the beginning portion of the extended period approach to zero as the extended period increases. Finally, a sectional interactive decomposition algorithm is proposed for online trajectory decomposition. Numerical example by using B-spline as the basis function is presented to demonstrate the proposed decomposition approach.