# Chapter two

Modelling and behaviour

## Section eight: linearisation of non-linear models

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 linearisation of nonlinear models.

Linear models are easier to analyse and generic insights and solvers exist.

Despite being non-linear in reality, a linearised approximation which is valid locally is often good enough for design decisions, control and insight.

### 1. What do we mean by a linear model and superposition?

Students often confuse straight lines and linear models. It is important to form a clear distinction and to emphasis the concept of superposition which is a powerful model and simulation device.

Linear models and superposition (PDF, 180 KB)

### 2. Linearisation using Taylor series

Gives a brief introduction to a Taylor series and illustrates its use to linearise non-linear functions.

First order Taylor series and linearisation (PDF, 290 KB)

### 3. Linearisation of models

This note gives a simple algorithm and then some worked examples of developing linear model approximations to nonlinear models.

Linearised modelling case studies (PDF, 274 KB)