Exponential State Enclosure Techniques for the Implementation of Validated Model Predictive Control

Abstract

The design task of predictive controllers for uncertain systems is commonly formulated on the basis of their kinematic and/or dynamic models. These models are assumed to be expressed as initial value problems (IVPs) for finite-dimensional sets of nonlinear ordinary differential equations (ODEs). If constraints for the admissible state trajectories are formulated, bounds for these trajectories need to be computed by numerical procedures to obtain guaranteed enclosures of all possible states at each time step that contain the solution of the exact IVP-ODEs. Uncertainties in both the initial states and system parameters are considered in this paper by means of bounded interval variables. For this kind of system representation, we apply an exponential enclosure approach to determine guaranteed enclosures of all reachable states. This approach is embedded in a novel manner into the framework of a guaranteed nonlinear model predictive control (NMPC) to acquire optimal and safe control domains along a receding horizon. The NMPC problem is solved at each time step considering several constraints which are crucial for the system's safety and stability, namely, bounds on the state trajectories and the control signals. The capabilities of the combination of the exponential enclosure technique with the set-based NMPC strategy are illustrated through simulations using a nonlinear inverted pendulum.