  For systems that are not observable at the very equilibrium of interest to be
stabilized, output-feedback stabilization is considerably challenging. In this
paper we solve this control problem for the case-study of a second-order system
that is bilinear and affine, both in the input and the output, but it is
unobservable at the target equilibrium. The case-study is representative of a
well-studied class of non-uniformly observable systems and stems from
automotive control. Our main contribution is a novel certainty-equivalence
hybrid controller that achieves asymptotic stabilization semiglobally. The
controller relies on a switched observer that estimates the state, provided
that the latter is 'kept away' from the singular equilibrium. To achieve both
competing tasks, stabilization and estimation, the controller also relies on
the keen construction of a piecewise-constant, converging, reference. Our main
results are illustrated via numerical simulations on a meaningful example.
