MS19.2: Systems with Time Delay

A well-established approach for analyzing the stability of delayed differential equations (DDE) is the semi-discretisation method, which approximates the monodromy operator and the characteristic multipliers through linear mapping. While powerful, it can be computationally expensive and hard to implement. An alternative method, the implicit subspace iteration (ISSI), builds upon the same principles but instead of a linear mapping based on a monodromy matrix, it relies only on numerical simulation. By integrating the initial state over the time period and performing pseudo-inverse calculations, the dominant multipliers can be iteratively computed. The advantage of this approach is its flexibility: any numerical solver can be used. Furthermore, it is applicable to difficult problems involving state-dependent delays (SDD), neutral, non-smooth or stiff systems, as long as the system can be numerically simulated. In our current work, we present the extended ISSI method to handle SDD systems. We compare the results of the ISSI with a linearised solution in a case study related to turning operations.

We present a systematic control design method for achieving non-collocated vibration suppression, with a focus on multi-frequency suppression. The objective is to suppress vibrations of various frequencies at a specific location while maximizing the closed-loop stability margin. This objective is translated into an optimization problem of minimizing the spectral abscissa of the closed-loop, subject to zero-location constraints. A simple output feedback controller is insufficient for multiple frequencies due to limited free parameters in the non-collocated setting. To address this, we propose a dynamic feedback controller with intentional delays applied to the measured output signal. By incorporating multiple delays and increasing the order of the dynamic controller, we aim to increase the number of free parameters for control. The design methodology involves remodelling the system of equations as a delay-differential algebraic equation (DDAE) and minimizing the spectral abscissa subject to zero-location constraints. Constraint elimination is used to impose the zero-location constraints, which is non-trivial in the case of dynamic feedback. We demonstrate that with the additional delays and dynamic feedback, we can achieve multi-frequency vibration suppression and closed-loop stabilization with a sufficient stability margin, even in cases where simple static output feedback is not feasible.