# Chapter five

Discrete models and Z-transforms

## section four: discrete controller design

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.

This section focuses on the design and implementation of discrete controllers. To what extent can we exploit continuous time design methodologies which are well understood?

### 1. Impact of a ZOH on closed-loop poles

A ZOH means that a feedback law responds intermittently as opposed to continuously, and this in turn effects the closed-loop behaviour.

Impact of ZOH on closed-loop poles (PDF, 718 KB)

### 2. Impact of a ZOH on control and margins

A ZOH has negative phase and this means that the behaviour of a discrete system will be worse than the continuous time system, with an equivalent control law.

Impact of ZOH on Bode diagrams (PDF, 842 KB)

### 3. Simple control design methods

The simplest control design approach is to discretise the continuous time control law using some form of numerical approximation.

Control approximation with simple differencing (PDF, 756 KB)

### 4. Tustin transform for controller approximation

Simple transforms have large phase differences between the discrete and continuous counterparts which affect margins. A tustin transform reduces the approximation errors.

Tustin transform examples (PDF, 310 KB)

### 5. Control design proposal

Given we know the ZOH adds phase lag, the continuous time controller will not be ideal in the discrete domain. Although other alternatives are popular in text books, these are messy so here we show how simple gain reduction is often effective/sufficient.

Control design for a discrete system (PDF, 828 KB)

### 6. Use of MATLAB

Readers will gather than paper based computations are too tedious and slow in general with discrete control, so use a computer. A concise summary of some MATLAB functions is given here.

MATLAB for discrete systems analysis (PDF, 740 KB)

Tutorial sheets for chapter five

Online quizzes for chapter five