  This paper is concerned with tube-based model predictive control (MPC) for
both linear and nonlinear, input-affine continuous-time dynamic systems that
are affected by time-varying disturbances. We derive a min-max differential
inequality describing the support function of positive robust forward invariant
tubes, which can be used to construct a variety of tube-based model predictive
controllers. These constructions are conservative, but computationally
tractable and their complexity scales linearly with the length of the
prediction horizon. In contrast to many existing tube-based MPC
implementations, the proposed framework does not involve discretizing the
control policy and, therefore, the conservatism of the predicted tube depends
solely on the accuracy of the set parameterization. The proposed approach is
then used to construct a robust MPC scheme based on tubes with ellipsoidal
cross-sections. This ellipsoidal MPC scheme is based on solving an optimal
control problem under linear matrix inequality constraints. We illustrate these
results with the numerical case study of a spring-mass-damper system.
