# Chapter two

Modelling and behaviour

This chapter is on the theme of linear models. For example:

A d^{3}x/dt^{3} + B d^{2}x/dt^{2} + C dx/dt + D x = K u

where x(t) is the state, u(t) the input and A, B, C, D, K are model parameters.

Core skills are things such as:

How do I find a mathematical model representation of a real physical system?

How do such systems behave and how does the behaviour link to the model parameters?

Are there generic analysis tools that help with understanding?

It is implicit that students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials and Laplace.

### Rapid summary

Relatively quick overview videos and summary notes which introduce the core topics.

Modelling concepts and analogies video and notes (PDF, 942KB). The basics of first principles modelling and links between systems.

First order modelling video and notes (PDF, 903KB). Examples of several first order models and analogies between them.

First order responses video, notes (PDF, 936KB) and related problem solving. How do first order models behave and why? Concepts of time constant and gain.

Second order modelling video and notes (PDF, 885KB). Examples of several second order models and analogies between them.

Second order responses video and notes (PDF, 783KB). How do second order models behave and why? Concepts of damping and oscillation.

Generic behaviours video and notes (PDF, 803KB). An overview of characterisation of system behaviours, stability, LHP and RHP.

## Sections in chapter two

### Section one: Modelling principles

How do we do physical modelling? Are there common concepts we can exploit? What are the analogies between different disciplines? Derive models for example scenarios.

### Section two: Modelling first order systems

Define a number of engineering scenarios which lead to first order models. Demonstrate the modelling from first principles and illustrate analogies.

### Section three: Responses of first order systems

How do first order systems behave? Are there efficient and insightful ways of defining and illustrating behaviour? How do we choose system parameters to achieve the desired behaviour?

### Section four: Modelling second order systems

Define a number of engineering scenarios which lead to second order models. Demonstrate the modelling from first principles and illustrate analogies.

### Section five: Responses of second order systems

How do second order systems behave? Are there efficient and insightful ways of defining and illustrating behaviour? How do we choose system parameters to achieve the desired behaviour?

### Section six: Behaviour characterisation for any order systems

Discussion of how to characterise behaviour in general, including for higher order systems.

### Section seven: Case studies on modelling and behaviours

Examples of a variety of engineering scenarios and modelling from first principles, leading to models with different orders and attributes.

### Section eight: Linearisation of nonlinear models

Most real models included nonlinear components and relationships. However, they can be approximated well enough locally by a linear model.