# Chapter two

Modelling and behaviour

## Section six: behaviour characterisation for any order system

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.

This section focuses on developing and applying core analysis tools to a variety of low and high order systems.

For simplicity, all models are expressed using Laplace tools.

How do I describe, analyse and contrast the behaviours of different systems, including those with varying orders?

### 1. What do we mean by behaviours?

An introduction to basic concepts and distinctions between signals, systems and transfer functions.

Characterising signals (PDF, 736 KB)

Transfer functions (PDF, 174 KB)

### 2. Behaviour analysis methodologies

Looks at the links between transfer function poles and system behaviour and thus inference of likely behaviour from poles.

Speed of response and convergence (PDF, 329 KB)

Oscillation with decay (PDF, 238 KB)

### 3. Steady-state analysis

What values do signals and system outputs converge to?

Final value theorem and signals (PDF, 318 KB)

Steady-state gain for systems (PDF, 272 KB)

### 4. Stability and overview of behaviour analysis

Do system outputs converge or diverge? How do I formally compare behaviours of different systems?

Stability (PDF, 228 KB)

Evaluation of system behaviour (PDF, 175 KB)

### 5. Supporting videos

Introduction to LHP and RHP and stability

Links between poles and behaviour