System Identification of Static Nonlinear Elements: A Unified Approach
of Active Learning, Over-fit Avoidance, and Model Structure
Determination
Abstract
Systems containing linear first-order dynamics and static nonlinear
elements (i.e., nonlinear elements whose outputs depend only on the
present value of inputs) are often encountered; for example, certain
automobile engine subsystems. Therefore, system identification of static
nonlinear elements becomes a crucial component that underpins the
success of the overall identification of such dynamical systems. In
relation to identifying such systems, we are often required to identify
models in differential equation form, and consequently, we are required
to describe static nonlinear elements in the form of functions in time
domain. Identification of such functions describing static elements is
often a black-box identification exercise; although the inputs and
outputs are known, correct mathematical models describing the static
nonlinear elements may be unknown. Therefore, with the aim of obtaining
computationally efficient models, calibrating polynomial models for such
static elements is often attempted. With that approach comes several
issues, such as long time requirements to collect adequate amounts of
measurements to calibrate models, having to test different models to
pick the best one, and having to avoid models over-fitting to noisy
measurements. Given that premise, this paper proposes an approach to
tackle some of those issues. The approach involves collecting
measurements based on an uncertainty-driven Active Learning scheme to
reduce time spent on measurements, and simultaneously fitting smooth
models using Gaussian Process (GP) regression to avoid over-fitting, and
subsequently picking best fitting polynomial models using GP-regressed
smooth models. The principles for the single-input-single-output (SISO)
static nonlinear element case are demonstrated in this paper through
simulation. These principles can easily be extended to MISO systems.