  Recent developments in communication technologies, such as 5G, together with
innovative computing paradigms, such as edge computing, provide further
possibilities for the implementation of real-time networked control systems.
However, privacy and cyber-security concerns arise when sharing private data
between sensors, agents and a third-party computing facility. In this paper, a
secure version of the distributed formation control is presented, analyzed and
simulated, where gradient-based formation control law is implemented in the
edge, with sensor and actuator information being secured by fully homomorphic
encryption method based on learning with error (FHE-LWE) combined with a
proposed mixed uniform-logarithmic quantizer (MULQ). The novel quantizer is
shown to be suitable for realizing secure control systems with FHE-LWE where
the critical real-time information can be quantized into a prescribed bounded
space of plaintext while satisfying a sector bound condition whose lower and
upper-bound can be made sufficiently close to an identity. An absolute
stability analysis is presented, that shows the asymptotic stability of the
closed-loop secure control system.
