Abstract

Explicit MPC can potentially be used for safety critical applications, including applications to systems with fast dynamics. Unfortunately, the off-line calculations at the design stage may be excessively demanding, and the required table size to represent the solution may also be unacceptable for some applications.

Several authors have therefore proposed various approximations to the (exact, optimal) explicit MPC . In approximate explicit MPC one generally accepts some degree of sub-optimality in order to arrive at a simpler solution, requiring fewer regions (and therefore also a smaller look-up table). Key properties of a design procedure for approximate explicit MPC are: i) It should not be necessary to find the exact solution first, and ii) It should be possible to ascertain the closed loop stability of the approximate solution. Most authors guarantee stability by starting from an MPC formulation that guarantees stability for the exact solution, and then ensure that the cost of the approximate solution is ‘close’ to the cost of the exact solution. It can then be shown that the optimal cost function is also a Lyapunov function for the approximate solution.

The talk will focus on alternative approaches to guaranteeing stability of approximate explicit MPC . Two such approaches will be presented: i) Refining the approximate solution until it can be shown that the cost function for the approximate solution is a Lyapunov function. ii) Using an LMI formulation to find a piecewise quadratic Lyapunov function for the approximate solution. For the LMI formulation, a novel relaxation will be proposed. Numerical examples indicate that this new relaxation is superior to the relaxation in common use for finding PWQ Lyapunov functions.