  This paper explores the privacy of cloud outsourced Model Predictive Control
(MPC) for a linear system with input constraints. In our cloud-based
architecture, a client sends her private states to the cloud who performs the
MPC computation and returns the control inputs. In order to guarantee that the
cloud can perform this computation without obtaining anything about the
client's private data, we employ a partially homomorphic cryptosystem. We
propose protocols for two cloud-MPC architectures motivated by the current
developments in the Internet of Things: a client-server architecture and a
two-server architecture. In the first case, a control input for the system is
privately computed by the cloud server, with the assistance of the client. In
the second case, the control input is privately computed by two independent,
non-colluding servers, with no additional requirements from the client. We
prove that the proposed protocols preserve the privacy of the client's data and
of the resulting control input. Furthermore, we compute bounds on the errors
introduced by encryption. We present numerical simulations for the two
architectures and discuss the trade-off between communication, MPC performance
and privacy.
