Conditions for saddle-point equilibria in output-feedback MPC with MHE

A new method for solving output-feedback model predictive control (MPC) and moving horizon estimation (MHE) problems simultaneously as a single min-max optimization problem was recently proposed. This method allows for stability analysis of the joint output-feedback control and estimation problem. In fact, under the main assumption that a saddle-point solution exists for the min-max optimization problem as well as standard observability and controllability assumptions, practical stability can be established in the presence of noise and disturbances. In this paper we derive sufficient conditions for the existence of a saddle-point solution to this min-max optimization problem. For the specialized linear-quadratic case, we show that a saddle-point solution exists if the system is observable and weights in the cost function are chosen appropriately. A numerical example is given to illustrate the effectiveness of this combined control and estimation approach.