The problems of the worst-case disturbances acting on multi-mass elastic system

An analytical framework for synthesizing worst-case external disturbances for linear dynamical systems described by ordinary differential equations is presented in the paper. The study is conducted for three classical functional spaces $L_2, L_{\infty}, L_1$ over a fixed time interval, which corresponds to identifying disturbances with bounded energy, bounded amplitude, and bounded impulse, respectively. Linear elastic mechanical systems are chosen as a illustrative object of analysis, thus providing an intuitive interpretation of the results. A unified performance metric is introduced for quantitative assessment of solutions. This metrics is the ratio of a system's target output (e.g., maximum deviation) to the $L_p$-norm of the disturbance (i.e. the normalized system response). Explicit analytical expressions for the worst-case disturbances and their corresponding performance indices are derived. The interrelations between the indices obtained for different disturbance classes are examined. Numerical simulation results are provided for single- and multiple-degree-of-freedom systems, represented as chains of point masses interconnected by elastic and damping elements, and connected to a movable base.