  This article considers output-feedback control of systems where the function
mapping states to measurements has a set-valued inverse. We show that if the
set has a bounded number of elements, then minimax dual control of such systems
admits finite-dimensional information states. We specialize our results to a
discrete-time integrator with magnitude measurements and derive a surprisingly
simple sub-optimal control policy that ensures finite gain of the closed loop.
The sub-optimal policy is a proportional controller where the magnitude of the
gain is computed offline, but the sign is learned, forgotten, and relearned
online.
  The discrete-time integrator with magnitude measurements captures real-world
applications such as antenna alignment, and despite its simplicity, it defies
established control-design methods. For example, whether a stabilizing linear
time-invariant controller exists for this system is unknown, and we conjecture
that none exists.
