On the modeling of time-varying delays

This thesis is an effort to develop generalized dynamic models for systems with time-varying time delays. Unlike the simple time-delay model characterized by a transportation lag in the case of a fixed time delay, time-varying delays exhibit quite different characteristics, making the development of easy to use models a difficult endeavor. First an algorithm is developed to predict the actual input-output behavior when the input signal is directly fed into a device that characterizes the time-varying delay. Input-output behaviour generated with this algorithm serves as the truth model for subsequent approximate model development. Simulation results for different classes of delay and different inputs were obtained using the truth model. The input functions were limited to steps, ramps and sinusoids. This limited class of inputs and delays defines the scope of this thesis and the results are to be interpreted as such. The methodology adopted to identify the basic underpinnings of models was system identification where input-output data came from the truth model. Models for the aforementioned classes of inputs and delays were then derived using elementary system identification tools. These models were then carefully analyzed to extract trends by changing the delay parameter. A satisfactory trend was observed in the case of linearly varying time delay. A generalized model for the linearly varying time delay with step and polynomial inputs was developed. An attempt was also made at developing a generalized model for sinusoidally time-varying time delays. This study proposes a model for linearly time-varying time delay, whose structure is not surprisingly also dependent on the class of inputs. It is shown that the derived model reduces to the well known model in the case of a fixed delay.