  In this work, we present a new class of models, called uncertain-input
models, that allows us to treat system-identification problems in which a
linear system is subject to a partially unknown input signal. To encode prior
information about the input or the linear system, we use Gaussian-process
models. We estimate the model from data using the empirical Bayes approach: the
input and the impulse responses of the linear system are estimated using the
posterior means of the Gaussian-process models given the data, and the
hyperparameters that characterize the Gaussian-process models are estimated
from the marginal likelihood of the data. We propose an iterative algorithm to
find the hyperparameters that relies on the EM method and results in simple
update steps. In the most general formulation, neither the marginal likelihood
nor the posterior distribution of the unknowns is tractable. Therefore, we
propose two approximation approaches, one based on Markov-chain Monte Carlo
techniques and one based on variational Bayes approximation. We also show
special model structures for which the distributions are treatable exactly.
Through numerical simulations, we study the application of the uncertain-input
model to the identification of Hammerstein systems and cascaded linear systems.
As part of the contribution of the paper, we show that this model structure
encompasses many classical problems in system identification such as classical
PEM, Hammerstein models, errors-in-variables problems, blind system
identification, and cascaded linear systems. This allows us to build a
systematic procedure to apply the algorithms proposed in this work to a wide
class of classical problems.
