The reachable set and safety guarantee

Author: Huda Ali A Bu Obaid

Bu Obaid, Huda Ali A, 2018 The reachable set and safety guarantee, Flinders University, College of Science and Engineering

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This research focuses on computing the exact reachable set which can be formulated in terms of a Hamilton-Jacobi partial differential equation (PDE). Hamilton-Jacobi (HJ) reachability is a procedure which can provide an accurate analysis of the safety state of the dynamical systems. This procedure is very effective in low-dimensional dynamical systems like aircrafts and quadrotors. It guarantees the safety where there is a potential dangerous scenario. One can get the safety guarantee by computing the reachable set, particularly the backward reachable set (BRS). The system might violate the safety properties in spite of the effort from the system to stay safe. The Hamilton-Jacobi reachability cannot be used for high dimensional systems because the computation of the backward reachable sets is complex and the number of state dimensions scales exponentially. In spite of the fact that; there are many techniques which can be used for approximation and these techniques have the ability to give conservative estimates for the backward reachable set. However, they do not provide an accurate solution and they usually demand limiting assumptions about the system dynamics. Therefore, this project will use a general method to analyze dynamical systems. When the results of subsystems are connected, the high-dimensionality of backward reachable sets can be easily and quickly computed which were previously considered to be intractable or take a long time to do. Also, it gives an exact computation in lower-dimensional subspaces. The method which will be applied in this project, will project the full dimensional of the reachable sets into lower dimensional subsystems. This method is from a decomposition of the true BRS. The theoretical results will be explained through the example of aircraft collision avoidance, 3D Dubins car and a linear system example. The Hamiltonian formulation of mechanics is a stronger formulation than others such as Lagrangian formulation and Newtonian formulation. In the Hamiltonian formulation, we can use coordinates which are much wider class. It expresses the relationship between the conservation and symmetries laws.

Keywords: Reachability, higer-dimensional systems, projections, self-contained subsystems, decomposition
Subject: Mathematical Sciences thesis

Thesis type: Masters
Completed: 2018
School: College of Science and Engineering
Supervisor: Murk Bottema