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In this paper, we study the robustness of safety properties of a lineardynamical system with respect to model uncertainties. Our paper involves threeparts. In the first part, we provide symbolic (analytical) and numerical(representation based) techniques for computing the reachable set of uncertainlinear systems. We further prove a relationship between the reachable set of alinear uncertain system and the maximum singular value of the uncertaindynamics matrix. Finally, we propose two heuristics to compute the robustnessthreshold of the system -- the maximum uncertainty that can be introduced tothe system without violating the safety property. We evaluate the reachable setcomputation techniques, effects of singular values, and estimation ofrobustness threshold on two case studies from varied domains, illustrating theapplicability, practicality and scalability of the artifacts, proposed in thispaper, on real-world examples. We further evaluate our artifacts on severallinear dynamical system benchmarks. To the best of the authors' knowledge, thisis the first work to: (i) extend perturbation theory to compute reachable setsof linear uncertain systems, (ii) leverage the relationship between thereachable set of a linear system and the maximum singular values to determinethe effect of uncertainties and (3) estimate the threshold of robustness thatcan be tolerated by the system while remaining safe.

Robustness of Safety for Linear Dynamical Systems: Symbolic and Numerical Approaches