Time delays are often seen as a source of frustration — missed connections due to delayed flights, disrupted online gaming due to communication lags, or instability of dynamical systems due to tiny control loop delays. But what if we could harness these delays as a force for good?Consider this: your bank delays transactions to prevent fraud, tango dancers pause to suppress inertia and execute precise movements, and thermostats introduce delays to prevent chattering and reduce wear and tear. In this PhD project, we will explore how time delays can be strategically used to enhance the stability and performance of dynamical systems.We will focus on systems described by ordinary differential equations (ODEs) or partial differential equations (PDEs). A key insight is that time delays can approximate derivatives through finite differences. For instance, the proportional-integral-derivative (PID) controller relies on an output derivative, which is not measured directly. Instead, this derivative is approximated by a finite difference, leading to a proportional-integral-delayed controller (where "D" now stands for "Delayed"). With such control implementation, poorly chosen delays can be disastrous, while the right delays can eliminate overshoot and improve robustness.The aim of this PhD project is to develop a comprehensive methodology for implementing derivative-dependent control using time delays. Specifically, the objectives are to:Identify dynamical systems (ODEs or PDEs) that can benefit from a controller that uses output derivatives.Construct and compare different approximations of the output derivatives via delayed measurements.Derive the stability conditions using Lyapunov functionals and linear matrix inequalities.Compare the performance of derivative-dependent and delay-dependent controllers via numerical simulations and experiments.If you're passionate about discovering new ways to enhance system stability and control and intrigued by the unconventional role of time delays, we'd love to hear from you. To learn more about the PhD project or ask any questions, please contact the project supervisor below.Further ReadingA. Selivanov and E. Fridman, “An improved time-delay implementation of derivative-dependent feedback,” Automatica, vol. 98, pp. 269–276, 2018.A. Selivanov and E. Fridman, “Sampled-Data Implementation of Derivative-Dependent Control Using Artificial Delays,” IEEE Transactions on Automatic Control, vol. 63, no. 10, pp. 3594–3600, 2018.E. Fridman and L. Shaikhet, “Stabilisation by using artificial delays: An LMI approach,” Automatica, vol. 81, pp. 429–437, 2017.Candidate RequirementsStrong mathematical background in calculus, linear algebra, and ODEs is essential. Experience in control theory is desirable but not mandatory – a mathematically literate candidate can quickly fill possible gaps. Most importantly, we are looking for candidates passionate about maths and fundamental research in general.Learning EnvironmentThe University of Sheffield is a Russell Group university. It is located in the centre of the UK, right next to the Peak District National Park. The standard duration of a PhD in the UK is 3.5 years. To learn more about student life in Sheffield, visit https://www.sheffield.ac.uk/sheffield-guide.Application ProcessInformal enquiries are encouraged and should be addressed to Dr Anton Selivanov below.You can apply for this project here: https://www.sheffield.ac.uk/postgradapplication/Suitable candidates will be invited for an online interview in December/January.Candidates who pass the interview will be invited to apply for the University of Sheffield PhD scholarship. The application deadline is January 2025.Start date: Autumn Semester 2025.