We show that the evolution of a dynamical system driven by controls obtained by the solution of an embedded optimization problem (as done in MPC) can be cast as a differential variational inequality (DVI). The DVI abstraction reveals that standard sampled-data MPC implementations (in which the control law is computed using states that are sampled at predefined sampling intervals) corresponds to an explicit Euler time-stepping scheme applied to the DVI. As expected, this explicit scheme induces large approximation errors (with respect to the optimal solution manifold of the DVI) when long sampling intervals are used. Motivated by this observation, we propose to use an implicit Euler scheme which allow us to use much longer sampling intervals, but note that this scheme requires the solution of a complicated variational inequality at every sampling time. We thus propose to use a predictor- corrector scheme, which can be implemented with off-the-shelf optimization tools and which can significantly reduce approximation errors incurred by standard sampled-data MPC.
View MPC as a DVI: Implications on Sampling Rates and Accuracy