Modelling and control of nonlinear systems with variable time delays
Lancaster University, Doctor of Philosophy, 2026
Supervisor: C.J. Taylor
Abstract
This thesis concerns the modelling and control of dynamic systems that are subject to uncertain and potentially time-varying time-delays. A pair of HYDROLEK HLK-7W manipulators, each consisting of a 6-degrees-of-freedom (6-DOF) articulated arm, are used to motivate the development of the new algorithms, and to provide experimental data for off-line analysis. Here, a time-varying time-delay refers, for example, to the time in seconds between changing an applied voltage (in open or closed-loop scenarios) and observing a corresponding change in the manipulator movement characteristics e.g. angular velocity or direction of movement.
The first major contribution of the thesis concerns the development of a novel algorithm to estimate the potentially variable time delay of an unknown system from experimental data. The method is a black-box, statistical estimation approach that does not require a dynamic model nor any knowledge of the systematic causes of the time delay variability. The algorithm provides insight into the time delay behaviour and hence guides further modelling and control design decisions. The method is successfully used to estimate the delays observed in the HYDROLEK manipulators, which range from almost zero to over 1.2s, and show how the magnitude of the delay is related to the applied voltage. For example, the time delays are found to be longest when the input signal is close to (but nonetheless outside) the dead-band.
Secondly, a novel scheduled controller that uses a weighting function to intelligently switch between two or more partial models with different (fixed) time delays, is proposed. A conventional, Proportional-Integral-Plus (PIP) controller is used as the foundation for this approach, by providing the control inputs associated with each partial model. The weighting coefficients are calculated via either (i) a Gaussian radial basis function or (ii) linear regression. This approach has potential wide applicability to nonlinear systems and, in the present thesis, it is evaluated in the context of the HYDROLEK manipulators, for which the partial models are chosen to encompass the range of delays identified from experimental data.
The new approach is compared in simulation with two simpler PIP controllers, one that uses a basic, unweighted, switching between each partial model and the other a conventional linear design. The simulation study utilises a previously developed nonlinear model for individual joints of the HYDROLEK manipulators. The model is updated in this thesis for the latest laboratory configuration and experimental data. It is based on a first order, integrating Transfer Function, placed in series with a static non-linear function to represent the dead-band and angular velocity saturation. Finally, the model is augmented here with a variable time delay.
The baseline controller takes the form of the discrete-time, linear PIP algorithm, designed using pole assignment or Linear Quadratic (LQ) optimal control, coupled with an Inverse Dead Zone (IDZ) element. The basic and weighted switching approaches combine the IDZ element with three partial PIP controllers. The Gaussian basis function-based approach shows the most promising results, with an average 100% decrease in Mean Absolute Error (MAE) between the output and the ideal response (based on the chosen closed loop design poles), compared to the simplest linear approach and an average 44% decrease in MAE compared to the basic unweighted switching approach. These figures summarise the results from a range of simulation scenarios, including with consideration of model uncertainty and disturbances.
In this manner, the novel time delay estimation and weighted control algorithms developed in this thesis are complementary and help to improve the closed loop performance. The former provides estimated delays to facilitate development of both the new simulation model and the partial models used for control design. The latter is shown in simulation to yield significantly improved closed-loop performance compared to existing PIP algorithms, presaging its likely utility in future practical applications. The thesis concludes by discussing future research in this area, including experimental work using the HYDROLEKs and the potential for other applications.