Construction of an Uncertainty to Maximize the Gain at Multiple Frequencies

This paper considers the construction of worst- case perturbations for uncertain systems. The uncertain system is modeled as an interconnection of a linear time-invariant (LTI) system and a norm-bounded LTI uncertainty. The worst-case gain (measured in the norm) for the uncertain system can be assessed via skewed-μ analysis. The standard approach is to compute upper and lower bounds for the worst-case gain on a frequency grid. A worst-case LTI perturbation is then constructed to maximize the gain at a single frequency. This perturbation can be used within a high fidelity nonlinear simulation to further explore system robustness. A drawback of this existing approach is that the worst-case perturbation constructed at a single frequency may not necessarily induce poor time-domain performance. It is beneficial to construct a perturbation that maximizes the gain at multiple frequency points, e.g. where the system is most sensitive or where disturbances have large frequency content. This paper provides an algorithm to construct a single perturbation which causes the uncertain system to achieve its largest possible gain at multiple chosen frequency points. This is achieved by interpolating through worst-case samples at the individual frequencies using the boundary Nevanlinna-Pick interpolation. Simple numerical examples are provided to demonstrate the proposed approach.