Discontinuous feedbacks, discontinuous optimal controls, and continuous-time model predictive control

It is known that there is a class of nonlinear systems that cannot be stabilized by a continuous time-invariant feedback. This class includes systems with interest in practice, such as nonholonomic systems, frequently appearing in robotics and other areas. Yet, most continuous-time model predictive control (MPC) frameworks had to assume continuity of the resulting feedback law, being unable to address an important class of nonlinear systems. It is also known that the open-loop optimal control problems that are solved in MPC algorithms may not have, in general, a continuous solution. Again, most continuous-time MPC frameworks had to artificially assume continuity of the optimal controls or, alternatively, impose some demanding assumptions on the data of the optimal control problem to achieve the desired continuity. In this work we analyse the reasons why traditional MPC approaches had to impose the continuity assumptions, the difficulties in relaxing these assumptions, and how the concept of 'sampling feedbacks' combines naturally with MPC to overcome these difficulties. A continuous-time MPC framework using a strictly positive inter-sampling time is argued to be appropriate to use with discontinuous optimal controls and discontinuous feedbacks. The essential features for the stability of such MPC framework are reviewed. Copyright (C) 2003 John Wiley Sons, Ltd.