No abstract

Zonotopes are a special subclass of polytopes, which have several favorable properties: They can be represented in a compact way and they are closed under the Minkowski sum as well as under linear transformations. Zonotopes are a popular set representation used e.g. for reachability analysis of dynamic systems, set-based observers and robust control. The complexity of algorithms that work on zonotopes strongly depends on their order (i.e. their number of generators and dimensions), which is often increased by operations like the Minkowski sum. Thus, to keep computations efficient, zonotopes of high orders are often over-approximated as tight as possible by zonotopes of smaller order. This paper has two main contributions: First, we propose new methods based on principle component analysis (PCA), clustering and constrained optimization for tight over-approximation of zonotopes. Second, we provide an overview of the most important known methods for order reduction and compare the performance of new and known methods in low-and high-dimensional spaces.

Abstract-Controlling autonomous vehicles typically has two main components: planning a trajectory and tracking this trajectory using feedback controllers. To benefit from the recent progress in planning algorithms, it is key that the underlying tracking controller is able to follow the planned trajectory as desired. In emergency situations in particular, it is crucial that feedback controllers steer the vehicle as close as possible to the planned trajectory to remain within a safe corridor. While there exists much work on the design of trajectory and path tracking controllers for vehicles, little work has been done to systematically compare different approaches, especially when considering extreme situations, uncertain parameters, and disturbances. In this work, we compare eight tracking controllers in a systematic way, each of them representing a different controller family. By not only considering nominal behavior, but also sensor noise and uncertain parameters, we obtain for the first time a broad comparison of the behavior of different controllers in various situations.

The popularity of model predictive control (MPC) is mainly founded on its easy implementation and its ability to consider state and input constraints. For future applications in safety-critical systems, however, it is necessary to provide formal guarantees of safety despite disturbances and measurement noise. In this paper, we include reachability analysis in an MPC approach to obtain provably safe controllers which are easy to implement. We consider continuous-time, nonlinear systems affected by disturbances and measurement noise. In contrast to most existing techniques, we explicitly consider the computation time and guarantee the satisfaction of state and input constraints despite the previously-mentioned disturbances. We use a novel type of dual mode MPC, which does not require the computation of Lyapunov functions. We demonstrate the applicability of our approach with a numerical example of a chemical reactor, where we show the advantages of our approach compared to existing MPC.

Optimal control finds an optimal input trajectory which steers an initial state to a desired final state while satisfying given state and input constraints. However, most efficient approaches are restricted to a single initial state. In this paper, we present a new approach, which combines reachability analysis with optimal control. This enables us to solve the optimal control problem for a whole set of initial states by optimizing over the set of all possible solutions. At the same time, we are able to provide formal guarantees for the satisfaction of state and input constraints. Taking the effects of sets of disturbances into account ensures that the resulting controller is robust against them, which is a big advantage over many existing approaches. We show the applicability of our approach with a vehicle-platoon example.