MS19.3: Systems with Time Delay

Two mathematical formulations of a model that describes the self-exited oscillations of milling tools are compared: (i) a delay differential equation (DDE) and (ii) a partial differential equation-ordinary differential equation (PDE-ODE) formulation. Although their mathematical expressions appear to be different, the two models should be consistent. Here, we show that the two models are equivalent under specific assumptions. Utilizing the boundary conditions implied by these assumptions, the PDE of the PDE-ODE system is solved analytically and the PDE-ODE formulation is accordingly transformed to a DDE. From this perspective, DDE models can be viewed as a simplifying approximation of PDE-ODE models. As a result, the DDE model is more efficient than the PDE-ODE model in most scenarios but can be less accurate in extreme cases.

Behaviors of city drivers are investigated in heavy traffic scenarios using automatic differentiation-based parameter fitting. Traffic data are collected via video capturing, and the parameters of the optimal velocity model are identified. Then, time-delay and time-delay-like behavior are introduced into the driver model. These extended models are then fitted to the measurement data, and their success and consistency are evaluated.

Delay differential equations (DDEs) have been employed to investigate the El Niño Southern Oscillation (ENSO) phenomenon. This type of conceptual model typically involves a delayed negative feedback term and a nonautonomous periodic forcing. Despite their simplicity, recent studies have revealed that such DDE models exhibit intricate dynamical behavior, especially in cases where delays are not assumed to be constant. Our starting point is an idealized DDE ENSO model in which the negative delayed feedback term is represented by a step function with constant delay. We expand upon previous work on this model by incorporating state-dependent delays, while keeping the step function to enable us to construct explicit solutions. Specifically, we show that the constant delay case serves as an accurate approximation for a family of implicitly defined state-dependent delays.

Spectral submanifolds provide a useful tool for efficient model reduction of nonlinear dynamical systems. Here, we introduce spectral submanifolds for time delay systems which allow us to reduce the dynamics from an infinite-dimensional state space to a few dimensions. We demonstrate that the algorithm allows the direct and accurate approximation of self-excited oscillations.

The nonlinear dynamics of electric scooters is investigated using a spatial mechanical model. The nonlinear governing equations are derived with the help of Kane's method. A hierarchical, linear state feedback controller with two feedback delays is designed in order to balance the e-scooter in a vertical position at zero longitudinal speed. The linear stability charts of the delayed controller are constructed with semi-discretization. The optimal higher-level control gains, corresponding to the fastest decay of the solution, are investigated. The performance of the control algorithm is verified by means of numerical simulations.