Searcharxiv⌕ Search

arXiv subjects

Robert Baier

Publications and source records attributed to Robert Baier.

7 recordsLinked to original sources
arXiv2019

Computation of avoidance regions for driver assistance systems by using a Hamilton-Jacobi approach

We consider the problem of computing safety regions, modeled as nonconvex backward reachable sets, for a nonlinear car collision avoidance model with time-dependent obstacles. The Hamilton-Jacobi-Bellman framework is used. A new formulation of level set functions for obstacle avoidance is given and sufficient conditions for granting the obstacle avoidance on the whole time interval are obtained, even though the conditions are checked only at discrete times. Different scenarios including various road configurations, different geometry of vehicle and obstacles, as well as fixed or moving obstacles, are then studied and computed. Computations involve solving nonlinear partial differential equations of up to five space dimensions plus time with nonsmooth obstacle representations, and an efficient solver is used to this end. A comparison with a direct optimal control approach is also done for one of the examples.

math.OC↗
arXiv2018

Construction of the Minimum Time Function for Linear Systems Via Higher-Order Set-Valued Methods

The paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Higher-order discretization of the reachable set of the linear control problem can balance missing regularity (e.g., Holder continuity) of the minimum time function for smoother problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e., we study cases in which the minimum time function is Holder continuous or even discontinuous.

math.OC↗
arXiv2015

Construction of the Minimum Time Function Via Reachable Sets of Linear Control Systems. Part I: Error Estimates

The first part of this paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Numerical examples are included in the second part.

math.OC↗
arXiv2015

Construction of the Minimum Time Function Via Reachable Sets of Linear Control Systems. Part 2: Numerical Computations

In the first part of this paper we introduced an algorithm that uses reachable set approximation to approximate the minimum time function of linear control problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e. we study cases in which the minimum time function is Hölder continuous or even discontinuous.

math.OC↗
arXiv2013

Directed Subdifferentiable Functions and the Directed Subdifferential without Delta-Convex Structure

We show that the directed subdifferential introduced for differences of convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from the directional derivative without using any information on the DC structure of the function. The new definition extends to a more general class of functions, which includes Lipschitz functions definable on o-minimal structure and quasidifferentiable functions.

math.OC↗