A sufficient condition for the stability of optimizing controllers with saturating actuators

The quadratic programme that must be solved with certain output–feedback model predictive controllers can be expressed as a continuous sector-bounded nonlinearity together with two linear transformations. Thus, the multivariable circle criterion gives a simple test for stability, with or without model mismatch. In particular, it may be applied if the open-loop plant is stable and the actuators are subject to simple saturation constraints. In the case of single horizon model predictive control, it suffices to check for positive realness a transfer function matrix whose dimension corresponds to the number of inputs. For an arbitrary length receding horizon it suffices to check the poles of a low dimension transfer function matrix and the eigenvalues (over an appropriate range of operator values) of a matrix whose dimension is independent of the horizon length.