A Branch-and-Bound Algorithm to Compute the Worst-Case Norm of Uncertain Linear Systems under Inputs with Magnitude and Rate Constraints

This paper extends the worst-case norm (WCN) of linear systems subject to inputs with magnitude and rate bounds to the case of uncertain linear systems. While the WCN for linear systems can be accurately obtained by simply solving a sparse linear programming, the computation of the WCN for uncertain linear systems leads to an NP-hard problem. In this paper, a branch-and-bound algorithm is adopted to calculate the WCN in the presence of uncertainty. Numerical examples demonstrate that computation time of the proposed algorithm is reasonable within certain problem dimensions. An exhaustive search is employed to validate the branch-and-bound algorithm, which later indicates the positive outcome. Finally, we suggest a means to improve the WCN computation for problems with higher dimensions