In this paper, we propose a new method for ensuring formally that a controlled trajectory stay inside a given safety set S for a given duration T. Using a finite gridding X of S, we first synthesize, for a subset of initial nodes x of X , an admissible control for which the Euler-based approximate trajectories lie in S at t @math [0,T]. We then give sufficient conditions which ensure that the exact trajectories, under the same control, also lie in S for t @math [0,T], when starting at initial points 'close' to nodes x. The statement of such conditions relies on results giving estimates of the deviation of Euler-based approximate trajectories, using one-sided Lipschitz constants. We illustrate the interest of the method on several examples, including a stochastic one.
In @cite_26 , the authors also provide a formal guarantee of safety property. Contrarily to @cite_11 @cite_25 , they do use SL-schemes. They also focus to (periodically) sampled systems as we do. However, they still perform a form of reachability analysis similar to @cite_11 @cite_25 , using convex polytopes as state set representations. Their growth bound are not based on OSL constants as here, but rather on overapproximations of Lagrange remainders in Taylor series.
@cite_26 @cite_25 @cite_11
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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. 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.