# Mathematics

## MTH3011 - Nonlinear Systems and Control (2019)

MODULE TITLE CREDIT VALUE Nonlinear Systems and Control 15 MTH3011 Dr Prathyush P Menon (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 weeks 0
 Number of Students Taking Module (anticipated) 11
DESCRIPTION - summary of the module content

Lyapunov theory is a landmark in the stability of dynamical systems and differential equations which has profoundly influenced both significant mathematical results and important applications. This theory is built around a study of energy-like Lyapunov functions of a system. Such functions are first motivated via simple examples and then developed into a powerful tool for analysis and control of nonlinear systems. We study stability types of equilibria and basins of attraction. Rate of change of energy is manipulated via feedback control to force equilibria to have the desired qualitative properties. Applications to mechanical, bio-chemical and economic systems will be developed. Feedback design techniques such as recursive back-stepping and adaptation are studied.

Energy-like functions play a key role in the qualitative study of the dynamical behaviour of nonlinear systems, replacing algebraic tools like eigenvalues so important for linear systems. Mechanical systems and electrical circuits have naturally defined energy. Energy can be manipulated via external control and especially feedback control. The module will develop a conceptual framework interwoven with several case studies. The emphasis for the mathematics is the application of the theory, not so much in the development of the theory. Case studies will include examples such as inverted pendula, rotating bodies, bio-reactors, etc.

On this module, there is ample opportunity for use of computer software (e.g. Maple and similar packages). You will find out how the need to choose suitable Lyapunov functions or stabilising feedbacks lends itself for developing creative mathematical processes and intuition.

Pre-requisite: MTH2003 Differential Equations, MTH2005 Modelling: Theory & Practice or equivalent

AIMS - intentions of the module

The aims of the modules include helping you to understand the nonlinear models and nonlinear phenomena, and the qualitative behavior of second order linear systems and near equilibrium points. On this module, you will learn how to locally linearise a nonlinear system and study the qualitative behavior and discover Lyapunov methods for studying stability of nonlinear systems. Furthermore, you will study the stability of the perturbed systems, the small gain theorem, controllability condition, the principles of Lyapunov based feedback design techniques and a set of mechanical, biological examples. Finally, the course will give you a good understanding of stability and analysis techniques for nonlinear systems, and prepare you to carry out basic feedback design techniques based on Lyapunov theory such as recursive back-stepping.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 use the Lyapunov stability method to study stability of equilibria of simple differential equations;

2 understand the subtleties of stability in its various forms;

3 utilise the Lyapunov method as a framework for feedback stabilisation.

Discipline Specific Skills and Knowledge:

4 appreciate that Lyapunov theory is both qualitative and quantitative and the mathematical arguments and methods built around it have broader implications in applied mathematics, to some extent in pure mathematics and even in economics and engineering.

Personal and Key Transferable/ Employment Skills and  Knowledge:

5 demonstrate further logical reasoning.

SYLLABUS PLAN - summary of the structure and academic content of the module

- motivating examples: linear systems, mechanical systems, bio-reactors, economic systems, free systems and forced (controlled) systems, models and preliminaries, case studies and applications;

- stability method of Lyapunov: Lyapunov functions and stability criteria, stability of equilibria of conservative and gradient systems, Lyapunov functions for linear systems and linearisation;

- further concepts to include: invariance principles, Barbalat's Lemma, converse Lyapunov results, basins of attraction;

- case study applications of Lyapunov methods: simple pendulum, rotating bodies, recurrent neural networks, Walrasian equilibria in economic models, bio-reactors;

- feedback design: Lyapunov design, adaptive control, recursive back-stepping, feedback control of coupled systems, case studies for recursive control design.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 28 Lectures Scheduled learning and teaching activities 5 Example classes/group discussion Guided independent study 117 Lecture and assessment preparation; private study

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework: problem sheets 10 hours, 4-6 questions per problem sheet (2 sheets) All Written

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework – based on questions submitted for assessment 20 1 assignment, 24 hours total All Annotated script and written/verbal feedback
Written Exam – closed book 80 2 hours All Written/verbal on request, SRS

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written Exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener