Convolution models for output only linear structural system identification and the problem of identifiability

This paper investigates the use of the Gaussian Process Convolution Model (GPCM) as an output only system identification tool for structural systems. The form of the model assumes a priori that the observed data arise as the result of a convolution between an unknown linear filter and an unobserved white noise process, where each of these are modelled as a GP. The GPCM infers both the linear time filter (which is the impulse response function, i.e. Green's function, of the system) and driving white noise process in a Bayesian probabilistic fashion with an approximate variational posterior over both signals. It will be shown that although the model structure is intuitive and sensible priors are applied, the GPCM falls short in recovering the linear impulse response of interest response due to the problem of identifiability. This is an interesting result indicating that physically informed kernel structures alone are not enough to recover the true impulse response in similar non-parametric probabilistic models. Despite this, the avenue of research remains highly promising, and several ideas are proposed to improve the model as a system identification tool.