Local Decay of Residuals in Dual Gradient Method with Soft State Constraints

摘要

Quadratic programs resulting from a model predictive control problem in real-time control context are solved using a dual gradient method. The projection operator of the method is modified so as to implement soft state constraints with linear and quadratic cost on constraint violation without directly calculating values of slack variables. Evolution of iterates and residuals throughout iterations of the modified method is studied. We notice that in most iterations, the set of the constraints that are active and the ones that are violated does not change. Observing the residuals through multiple iterations in which the active and violated sets do not change leads to interesting results. When the dual residual is transformed into a certain base, its components are decaying independently of each other and at exactly predictable rates. The transformation only depends on the system matrices and on the active and violated sets. Since the matrices are independent of the system state, so is the transformation, and the decay rate of the components stays constant through multiple iterations. The predictions are confirmed by numerical simulations of MPC control, which is shown for the AFTI-16 benchmark example.