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Computational tools for eigenvalue perturbation problems associated with linear time-delay systems
The dynamic behavior and stability properties of linear time-invariant systems with delays are related to the solutions of a nonlinear eigenvalue problem.We consider delay systems prone to uncertainty on the coefficient matrices and discuss methods for solving associated eigenvalue perturbation problems, such as the computation of pseudospectral abscissa and the distance to instability.
From an application point of view, it is important to exploit structure on the perturbations at various levels. First, we wish to exploit the structure of the delay equation, i.e., we consider combined perturbation on the individual coefficient matrices rather than unstructured perturbations on an equivalent linear (operator) eigenvalue problem.
Second, we wish to exploit the fact that physical perturbations are real valued. Finally, we wish to have the flexibility to impose additional structure on the perturbations on the coefficient matrices. The latter is, in particular, important in the analysis of interconnections of (sub)systems and controllers, which we model by a system of delay differential algebraic equations (DDAEs).
The aim of the talk is to present on overview of recent results on solving distance problems in the context of time-delay systems, focusing on the main ideas. In the presentation gradually more structure will be imposed on the perturbations, thereby pointing out the additional challenges and solution concepts, and emphasizing similarities and differences with the standard eigenvalue problem. All these methods essentially boil down to solving an optimization problem. Both local and global methods are discussed, as well as the applicability to large-scale problems (by incorporating subspace restrictions and/or exploiting the presence of critical perturbations with low rank).
One of the numerical examples stems from vibration control, where the delay in the model is due to the presence of a distributed delay input shaper (a so-called DZV shaper) in the control loop.