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Ensuring safety is crucial for the successful deployment of autonomous systems, such as self-driving vehicles, unmanned aerial vehicles, and robots acting close to humans. While there exist many controllers that optimize certain criteria, such as energy consumption, comfort, or low wear, they are usually not able to guarantee safety at all times for constrained nonlinear systems affected by disturbances. Many controllers providing safety guarantees, however, have no optimal performance. The idea of this article is, therefore, to synthesize a formally correct controller that serves as a safety net for an unverified, optimal controller. This way, most of the time, the optimal controller is in charge and leads to a desired, optimal control performance. The safety controller constantly monitors the actions of the optimal controller and takes over if the system would become unsafe. The safety controller utilizes a novel concept of backward reachable set computation, where we avoid the need of computing underapproximations of reachable sets. We have further developed a new approach that analytically describes reachable sets, making it possible to efficiently maximize the size of the backward reachable set. We demonstrate our approach by a numerical example from autonomous driving. [ABSTRACT FROM AUTHOR]

In this article, we present a novel control approach for formally solving reach-avoid problems for constrained and disturbed nonlinear systems by optimizing over reachable sets. Reach-avoid problems arise in many modern control applications of safety-critical systems, such as autonomous driving and human-robot collaboration. They require us to control all states from a set of initial states in finite time into a final set while guaranteeing the satisfaction of state and input constraints despite the presence of disturbances and uncertain measurements. We optimize over reachable sets, thereby, simultaneously improving the control performance and guaranteeing constraint satisfaction for all solutions. Moreover, our new approach involves a novel combination of state-dependent feedforward and feedback control, which leads to better control performance compared to the existing approaches as demonstrated in several numerical examples. Our algorithm is particularly suited for computing motion primitives used in maneuver automata, which realizes fast and efficient online planning. The online applicability is supported by the simple structure of the resulting controller, which is a time-varying linear tracking controller. [ABSTRACT FROM AUTHOR]