Zames-Falb multipliers for quadratic programming

In constrained linear model predictive control a quadratic program must be solved on-line at each control step. If zero is feasible the resultant static nonlinearity is sector bound. We show that the nonlinearity is also monotone nondecreasing and slope restricted; furthermore it may be expressed as the gradient of a convex potential function. Hence we show the existence of Zames-Falb multipliers for such a nonlinearity. For completeness, we construct such multipliers both for the general case of multi-input multi-output static nonlinearities and for the particular case where the nonlinearity arises from a quadratic program. We also express the results in terms of integral quadratic constraints. These multipliers may be used in a general and versatile analysis of the robust stability of constrained model predictive control.