  We model and study the problem of localizing a set of sparse forcing inputs
for linear dynamical systems from noisy measurements when the initial state is
unknown. This problem is of particular relevance to detecting forced
oscillations in electric power networks. We express measurements as an additive
model comprising the initial state and inputs grouped over time, both expanded
in terms of the basis functions (i.e., impulse response coefficients). Using
this model, with probabilistic guarantees, we recover the locations and
simultaneously estimate the initial state and forcing inputs using a variant of
the group LASSO (linear absolute shrinkage and selection operator) method.
Specifically, we provide a tight upper bound on: (i) the probability that the
group LASSO estimator wrongly identifies the source locations, and (ii) the
$\ell_2$-norm of the estimation error. Our bounds explicitly depend upon the
length of the measurement horizon, the noise statistics, the number of inputs
and sensors, and the singular values of impulse response matrices. Our
theoretical analysis is one of the first to provide a complete treatment for
the group LASSO estimator for linear dynamical systems under input-to-output
delay assumptions. Finally, we validate our results on synthetic models and the
IEEE 68-bus, 16-machine system.
