Projects in mathematical systems and control theory

Mathematical systems and control theory “represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems” [1]. It concerns itself with “the basic theoretical principles underlying the analysis of feedback and the design of control systems [and] differs from the more classical study of dynamical systems in its emphasis on inputs (or controls) and outputs (or measurements)” [2]. 

Control systems and feedback loops are ubiquitous in science and engineering, and arise in traditional areas from aerospace control, manufacturing and process control, through to robotics, electrical power systems, systems biology, and therapeutics. The modern study of control systems traces its roots back to the industrial revolution and the principles of feedback control are at the heart of the flyball governor which was used regulate pressure in steam engines and facilitate their safe operation. Nowadays, the design and deployment of control systems has been fantastically successful, with 31 real-world success stories described in [3]. Novel and emerging applications where control systems play an essential role range from the control of autonomous vehicles, smart grids and devices, through to epidemiology [4].  

Mathematical control theory is broad and makes connections to numerous other mathematical areas, including: 

• analysis and operator theory via the control of infinite-dimensional systems — such as delay- or partial differential- equations from either a PDE or abstract functional analytic perspective [5];
• calculus of variations and optimisation via optimal control theory and, increasingly prevalently, stochastic optimal control theory [6];
• numerical analysis and numerical linear algebra via singular value- or Krylov subspace- methods for model order reduction [7].
• The present project welcomes applications for doctoral research in a wide range of control theoretic settings. Potential PhD projects are offered in themes of expertise of the supervisor, including:
• Topics in nonlinear stability theory — analysing and predicting the long-term behaviour of various classes of control systems specified by nonlinear differential or difference equations. This theme focusses primarily on (controlled) systems of differential equations.
• Regularity properties of classes of linear operators arising commonly in control theory (namely, Hankel and Toeplitz operators). It is well-known that these operators admit representations in terms of the Fourier transform and multiplication by certain classes of complex-valued analytic functions (often called symbols), and this relationship shall be further explored. This theme focusses primarily on functional analysis and operator theory.
• Topics in so-called positive and monotone control systems [8], roughly dynamical systems which evolve on a positive cone, reflecting the typical requirement that state (and possibly other) variables are naturally nonnegative in some sense. This theme is broad but combines elements from linear algebra and dynamical systems theory. There is scope within this theme for applications of control theory to theoretical ecology and mathematical biology; see, for example [9].
• Applications of control theory to energy storage and renewable energy generation. This theme focusses primarily on optimisation and optimal control applications.

The projects offered are mathematical, although there is ample scope for more theoretical or more applications-focussed projects, which may contain a large numerical/computational component, depending on applicant and the overall direction of the research. Further, the second supervisor may change depending on the theme chosen. Please also note that prior experience of mathematical control theory is not a requirement — strong mathematics knowledge and experience is essential, however. Perspective applicants are encouraged to contact the Supervisor before submitting their applications, highlighting which of the above areas are of most interest. Applications should make it clear the project you are applying for and the name of the supervisors. 

Academic qualifications

A first-class honours degree, or a distinction at master level, or equivalent achievements in Mathematics or another cognate discipline

English language requirement

If your first language is not English, comply with the University requirements for research degree programmes in terms of English language.

Application process

Prospective applicants are encouraged to contact the supervisor, Dr Chris Guiver (c.guiver@napier.ac.uk) to discuss the content of the project and the fit with their qualifications and skills before preparing an application. 

Contact details

Should you need more information, please email SCEBERDL@napier.ac.uk.

The application must include: 

Research project outline of 2 pages (list of references excluded). The outline may provide details about

• Background and motivation, explaining the importance of the project, should be supported also by relevant literature. You can also discuss the applications you expect for the project results.
• Research questions or
• Methodology: types of data to be used, approach to data collection, and data analysis methods.
• List of references

The outline must be created solely by the applicant. Supervisors can only offer general discussions about the project idea without providing any additional support.

• Statement no longer than 1 page describing your motivations and fit with the project.
• Recent and complete curriculum vitae. The curriculum must include a declaration regarding the English language qualifications of the candidate.
• Supporting documents will have to be submitted by successful candidates.
• Two academic references (but if you have been out of education for more than three years, you may submit one academic and one professional reference), on the form can be downloaded here.

Applications can be submitted here. To be considered, the application must use:

• “SCEBE1024” as project code.
• the advertised title as project title

Download a copy of the project details here