# Chapter five

Discrete models and Z-transforms

This chapter is on the theme of discrete time linear models, for example:

y_{k}+a_{1}y_{k-1} + ...+ a_{n}y_{k-n} = b_{1}u_{k-1} + ...+ b_{n}u_{k-n}

where y(t) is the output, u(t) the input and a_{i}, b_{i} are model parameters. The subscript 'k' denotes the sampling index.

Core skills are things such as:

What do I mean by a discrete model/signal and how do I find a discrete 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.

Overview of core results without derivations

## Sections in chapter five

### Section one: Time series models

What is a discrete signal and why/how do these arise? What are simple mathematical tools for handling discrete signals and how do we find the model parameters from measured data?

### Section two: Sampling, Z-Transforms and behaviours

How is sampling expressed mathematically? What are the links between common Z-Transforms and behaviours? Reconstruction of the underlying signal from its Z-transform.

### Section three: Discrete models

How do I represent and model real continuous systems alongside discrete signals? What is a zero-order-hold?

### Section four: Control in discrete time

How do I take a control design which was done using continuous time design methodologies and then apply this in discrete time. What the disadvantages and alternatives?

Tutorial sheets for chapter five

Online quizzes for chapter five