Introductory description

In many areas of science and engineering, systems are modelled using differential equations that describe how the system dynamics evolve over time. These equations help us model systems like robots and cars, electrical circuits, and even how populations of animals interact in nature. These systems can often be influenced through actuators using control inputs. At the same time, we might only have access to limited measurements from which we must infer the internal state of the system.
In this course we will focus on the mathematical tools that can be used to model, analyze, and control such systems. Emphasis will be given to linear, time-invariant systems in state-space form. We will study how to determine whether a system can be controlled or observed, and how to design strategies to stabilise or guide the system to a desired behavior.
Topics include:

• the analysis of controllability, observability, stabilisability, and detectability.
• the design of controllers via feedback, pole placement, and Lyapunov-based methods.
• the design of optimal control using the linear-quadratic regulator (LQR).
• illustration of these concepts through examples and numerical simulations, using programming tools such as MATLAB or Python.

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.

Learning outcomes

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

• Understand how complex systems can be described using mathematical equations, and how to actually control their behaviour.
• Understand of the concepts of controllability, observability, and stabilizability, and know why they matters.
• Learn how to design controllers using feedback control and pole placement methods.
• Be able to analyse systems’ stability using Lyapunov’s method.
• learn how to design optimal controllers that meet specific goals like minimizing error or energy use using ideas like the Linear Quadratic Regulator.
• Recognise practical problems where control methods can be used effectively.

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.

Interdisciplinary

The course draws concepts from multiple disciplines including mathematics, engineering, physics, and computer science.

Subject specific skills

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

Transferable skills

Understanding of how mathematics relates to real-world engineering.