  Wireless communication offers many benefits for control such as substantially
reduced deployment costs, higher flexibility, as well as easier data access. It
is thus not surprising that smart and wireless sensors and actuators are
increasingly used in industry. With these enhanced possibilities, exciting new
technologies such as Control-as-a-Service arise, where (for example) controller
design or tuning based on input-output-data can be outsourced to a cloud or
mobile device. This implies, however, that sensitive plant information may
become available to service providers or, possibly, attackers.
  Against this background, we focus on privacy-preserving optimal PID tuning
as-a-Service here. In particular, we combine homomorphic encryption with
extremum seeking in order to provide a purely data-driven and confidential
tuning algorithm. The encrypted realization requires several adaptions of
established extremum seekers. These encompass relative parameter updates,
stochastic gradient approximations, and a normalized objective function. As a
result, and as illustrated by various numerical examples, the proposed
encrypted extremum seeker is able to tune PID controllers for a wide variety of
plants without being too conservative.
