Barrier function based model predictive control

Abstract

A new formulation of nonlinear model predictive control (MPC) is developed by including a weighted barrier function in the control objective. While the barrier ensures that inequality constraints are strictly satisfied it also provides a smooth transition between points in the interior and those on the boundary of the constraint set. In addition, the resulting optimisation problem, to be solved at each control step, is effectively unconstrained and thus amenable to elegant optimisation techniques. The barrier must satisfy certain conditions in order that the state converges to the origin and we show how to construct such a barrier. Conventional MPC may be seen as a limiting case of the new class as the barrier weighting itself approaches zero. We pay particular attention to the novel approach of fixing the weighting parameter to some positive value—possibly large—and observe that this provides a degree of controller caution near constraint boundaries. We construct an ellipsoidal invariant set by exploiting the geometry of self-concordant functions and show nominal closed-loop stability for this class of controllers under full state feedback.