Introductory description

The module gives an introduction to control theory. The emphasis is on theory, i.e. mathematics, rather than application. For applications, best to look at Engineering modules. Nevertheless, it is motivated by paractical questions. Can one control the state of a system to a given place if the control force is only on some components? Can one stabilise it by suitable feedback control? Can one infer the state of the system from observations of only some components? How to achieve a given effect with minimum control cost? We will formulate these questions mathematically and answer them, at least in simple settings. The module will make use of a wide variety of mathematics, some of which you will already know (linear algebra, ordinary differential equations) and some you might not yet have met (optimisation with constraints, Bayesian inference). I’ll give rapid summaries of the things that I expect you to know and more detailed surveys of required topics that are likely to be new to you.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Will include the study of controllability, stabilization, observability, filtering and optimal control. Furthermore connections between these concepts will also be studied. Both linear and nonlinear systems will be considered. The module will comprise six chapters. The necessary background material in linear algebra, differential equations and probability will be developed as part of the course.

1. Introduction to Key Concepts. 2. Background Material. 3. Controllability. 4. Stabilization. 5. Observability and Filtering. 6. Optimal Control.

Learning outcomes

By the end of the module, students should be able to:

• Explain and exploit role of controllability matrix in linear control systems.
• Explain and exploit stabilization for linear control systems.
• Derive and anlayze the Kalman filter.
• Understand linear ODEs and stability theory.
• Understand and manipulate Gaussian probability distributions.
• Understand basic variational calculus for constrained minimization in Hilbert space.

Indicative reading list

E. D. Sontag, Mathematical Control Theory, Texts in Applied Mathematics No 6, Springer Verlag, 1990.

J. Zabczyk, Mathematical Control Theory: An Introduction, Systems and Control, Birkhauser, 1992.

Subject specific skills

Ability to decide whether a linear system is controllable, observable. Ability to design Kalman filters. Ability to design optimal control.

Transferable skills

Understanding of how mathematics relates to real-world engineering.