Guaranteeing Constraints of Disturbed Nonlinear Systems Using Set-Based Optimal Control in Generator Space

Abstract

We address the problem of finding an optimal solution for a nonlinear system for a set of initial states rather than just for a single initial state. In addition, we consider state and input constraints as well as a set of possible disturbances. While previous optimal control techniques typically ignore the fact that the current state of a system is not exactly known, future safety-critical systems demand that all uncertainties including the initial state are considered; this is required for e.g. automated vehicles, surgical robots, or human-robot interaction. We present a new method that obtains optimal control inputs by finding optimal weights for generators that span the space reachable by the considered system. This solution routine can be used not only for a single initial state but also for a set of initial states - this is not possible using classical optimization techniques. We ensure that all constraints are met by using reachability analysis, which provides formal bounds for all possible system trajectories. We demonstrate the applicability of our approach with an example from automated driving; for this example, the result is obtained within a few seconds and outperforms a classical LQR approach.