Using Intersection of Unions to Minimize Multi-directional Linearization Error in Reachability Analysis

In piecewise linearization based reachable set computation, different linear approximations are computed around smaller pieces of the reachable set to reduce the linearization error in reachability analysis. However, this approach suffers from curse of dimensionality because the number of pieces required to restrict the linearization error below a threshold can blow up intractably for high-dimensional systems. Alternatively, we can fix the maximum number of divisions of the reachable set and optimize the division vector to minimize the linearization error. But the functions projecting the linearization error along different directions can be different, which have different optimal solutions for the division vector. Still, we may need to minimize the linearization error along multiple directions to achieve good accuracy along any one direction because the differential equations can be coupled. Therefore, we develop a new method of piecewise linearization based reachable set computation that incorporates different optimized divisions of reachable set for different projections of linearization error to improve accuracy. To do so, we use intersection of unions of sets (IoU) to approximate reachable sets such that different unions in the intersection are obtained from optimized division along different directions and forward propagation. We develop an algorithm to propagate the reachable set of the IoU in a coupled way, such that each intersecting union complements the approximation accuracy of other unions. We validate the advantage of using multiple optimal divisions instead of one optimized division. For this, we compare the performance on high dimensional examples, of the proposed algorithm with a variant of the algorithm which uses only one division vector at each time step. We also draw comparison with state-of-the-art methods and demonstrate that the accuracy of our algorithm is at par or better for the benchmarks.