Set Membership approximation theory for fast implementation of Model Predictive Control laws

In this paper, the use of Set Membership (SM) methodologies is investigated in the approximation of Model Predictive Control (MPC) laws for linear systems. Such approximated MPC laws are derived from a finite number νν of exact control moves computed off-line. Properties, in terms of guaranteed approximation error, closed-loop stability and performance, are derived assuming only the continuity of the exact predictive control law. These results are achieved by means of two main contributions. At first, it will be shown that if the approximating function enjoys two key properties   (i.e. fulfillment of input constraints and explicit evaluation of a bound on the approximation error, which converges to zero as νν increases), then it is possible to guarantee the boundedness of the controlled state trajectories inside a compact set, their convergency to an arbitrary small neighborhood of the origin, and satisfaction of state constraints. Moreover, the guaranteed performance degradation, in terms of maximum state trajectory distance, can be explicitly computed and reduced to an arbitrary small value, by increasing νν. Then, two SM approximations are investigated, both enjoying the above key properties  . The first one minimizes the guaranteed approximation error, but its on-line computational time increases with νν. The second one has higher approximation error, but lower on-line computational time which is constant with νν when the off-line computed moves are suitably chosen. The presented approximation techniques can be systematically employed to obtain an efficient MPC implementation for “fast” processes. The effectiveness of the proposed techniques is tested on two numerical examples.