Probabilistic bounds for ℓ1ℓ1 uncertainty model validation

This paper is concerned with a mixed deterministic/probabilistic model validation problem, which amounts to determining the probability for a given model to reproduce given experimental data. We consider an additive uncertain model, in which the modelling uncertainty is characterized in time domain by the ℓ1ℓ1 induced system norm. The data available for validation are input–output time series and are assumed to have been corrupted by a random noise. For a given uncertainty norm bound, our objective is to compute the probability for an uncertainty to exist so that it may satisfy the prescribed bound and match the input–output measurements. While the exact computation of this probability poses a formidable task, we derive its upper and lower bounds. This is carried out with respect to noise sequences with Gaussian distribution, and more generally, when only such statistical information as the expectation and covariance of the noise are known.