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## MTH3011 - Nonlinear Systems and Control (2019)

MODULE TITLE | Nonlinear Systems and Control | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | MTH3011 | MODULE CONVENER | Dr Prathyush P Menon (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 0 | 11 weeks | 0 |

Number of Students Taking Module (anticipated) | 11 |
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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

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.

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.

- 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.

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad | 0.00 |
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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 |

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 |

Coursework | 20 | Written Exams | 80 | Practical Exams | 0 |
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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 |

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 |

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.

information that you are expected to consult. Further guidance will be provided by the Module Convener

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Khalil, H.K. | Nonlinear Systems | Prentice-Hall | 2000 | 000-0-132-28024-8 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
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PRE-REQUISITE MODULES | MTH2003, MTH2005 |
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CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 6 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Tuesday 10 July 2018 | LAST REVISION DATE | Friday 30 August 2019 |

KEY WORDS SEARCH | None Defined |
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