Robust Optimization-Based Affine Abstractions for Uncertain Affine Dynamics

This paper considers affine abstractions for over-approximating uncertain affine discrete-time systems, where the system uncertainties are represented by interval matrices, by a pair of affine functions in the sense of inclusion of all possible trajectories over the entire domain. The affine abstraction problem is a robust optimization problem with nonlinear uncertainties. To make this problem practically solvable, we convert the nonlinear uncertainties into linear uncertainties by exploiting the fact that the system uncertainties are hyperrectangles and thus, we only need to consider the vertices of the hyperrectangles instead of the entire uncertainty sets. Hence, affine abstraction can be solved efficiently by computing its corresponding robust counterpart to obtain a linear programming problem. Finally, we demonstrate the effectiveness of the proposed approach for abstracting uncertain driver intention models in an intersection crossing scenario.