Interpolated Mapping System Identification for Free and Forced Systems

Abstract

In this paper, we discuss a method of system identification based on Interpolated Mapping (IM). The method assumes the presence of sampled values of system states, based on which a map is defined that takes a set of initial conditions to their respective images. Using this map with time step T, maps with time steps T/2, T/4, [[ellipsis]], are subsequently found. Ultimately, when the time step has become small enough, difference quotients are used to estimate the state derivatives. A function fit can then be applied to these derivatives in order to obtain the equations of motion of the system. The methodology of IM is used to approximate the images of states that are not on the primary grid. The above scheme works for autonomous systems because a single map can be used to follow the progress of the system at all times; the vector field of the system is invariant with time. However, the procedure breaks down when the system is forced. In this case, time enters the equations of motion explicitly, and different vector fields (and, hence, different maps) exist at different times. We show in this paper that the necessary modifications to the basic methodology are relatively minor. The work and storage requirements can increase by as little as a factor of two, regardless of the dimension of the state space. This is essentially a result of the fact that IM may be used to perform interpolation in the time direction, as well. However, when the forcing is such that its variation over time is entirely unknown, more data would be required to correctly observe its progress over the time period of interest. Functions of time that are flat for small times also necessitate additional effort for proper identification.