Delay-induced stabilisation of dynamical systems

   Department of Automatic Control and Systems Engineering

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  Dr Anton Selivanov  Applications accepted all year round  Competition Funded PhD Project (Students Worldwide)

About the Project

Time delays are often a source of system instability. There are many examples of nonlinear or distributed parameter systems where an arbitrarily small input or output delay destabilises the system. In some cases, however, an artificially introduced time delay can significantly improve the system performance. One of the reasons for this mysterious behaviour is that time delays help approximate output derivatives. The knowledge of the output derivative allows for better controllers that stabilise otherwise unstabilisable systems. For example, the PID controller uses an output derivative, which may not be available directly. Instead, the derivative is approximated by a finite difference, which gives rise to a proportional-integral-delayed controller (now “D” in PID stands for “Delayed”). Adding more delays, one can approximate higher-order output derivatives, making the controller even more capable. Observers or filters can also approximate output derivatives, but they are more challenging to implement than the time-delay feedback.

This PhD project aims to take advantage of the available history of measurements to approximate output derivatives and use this additional information to derive improved controllers. That is, we will investigate the time-delay implementation of derivative-dependent control. The objectives are:

1. Identify dynamical systems (ODEs or PDEs) that can benefit from a controller that uses output derivatives.

2. Construct and compare different approximations of the output derivatives via delayed measurements.

3. Derive the stability conditions using Lyapunov functionals and linear matrix inequalities.

4. Compare the performance of derivative-dependent and delay-dependent controllers via numerical simulations.

If you would like to learn about the PhD project more or have any questions, please, feel free to contact the project supervisor, Dr Anton Selivanov, at [Email Address Removed].

Candidate Requirements

Strong 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 fulfil possible gaps. Most importantly, we are looking for candidates passionate about maths and fundamental research in general.

Applicants are required to hold a BSc/MSc degree in mathematics or engineering. If the degree is not from an English-speaking country, the applicant needs an overall IELTS grade of 6.5 with a minimum of 6.0 in each component (or equivalent). For further details, visit

Learning Environment

The 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 Department of Automatic Control and Systems Engineering (ACSE) is the only department in the UK dedicated to Control Engineering. The standard duration of a PhD in the UK is 3.5 years. To learn more about student life in Sheffield, visit

Application Process

Informal enquiries are encouraged and should be addressed to Dr Anton Selivanov at [Email Address Removed].

  1. You can apply for this project here:
  2. Suitable candidates will be invited for an online interview in December/January.
  3. Candidates who pass the interview will be invited to apply for the University of Sheffield PhD scholarship.
  4. Start date: Autumn Semester 2024.
Engineering (12) Mathematics (25)


1. A. Selivanov and E. Fridman, “An improved time-delay implementation of derivative-dependent feedback,” Automatica, vol. 98, pp. 269–276, 2018.
2. 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.
3. E. Fridman and L. Shaikhet, “Stabilisation by using artificial delays: An LMI approach,” Automatica, vol. 81, pp. 429–437, 2017.

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 About the Project