# Chapter 2: Introduction to predictive control and forming system predictions

This chapter discusses briefly why prediction is useful for control design and then focuses directly on how one can form predictions for a number of different model structures. The focus is solely on linear models in line with this book's focus on the basics rather than advanced topics.

- Introduction.
- Main components.
- Modelling assumptions.
- Prediction with state space models.
- Prediction with CARIMA models.
- Prediction with step response models.
- Unbiased prediction.
- Performance indices.

## 1. Introduction

Gives the human or philosophical thinking behind predictive control and explains why this is an intuitively obvious approach to control design.

- A video talk through is available here.

## 2. Main components

Predictive control is a way of thinking not a specific algorithm. This video breaks down the thinking into the different aspects which underpin a well designed algorithm - continued in next video.

## 3. Modelling assumptions

The choice of a model is a fundamental part of MPC. This video gives a brief overview of typical models that have been found to be effective and some of the thinking the user should deploy.

- A video talk through is available here.

## 4. Prediction with state space models

Introduces the procedure of prediction using mathematical models. Prediction is core to the efficacy of MPC and thus good comprehension of how this is done is essential. The 2nd video extends the previous video by introducing compact notation for prediction which enables the easier algebra needed for most MPC algorithms

- Two video talk throughs are available here and here.
- MATLAB demonstrations are available here.

## 5. Prediction with CARIMA models

In the early days and for SISO systems, it may be easier to model with transfer functions rather than state space models. The first video gives an elementary and easy to code mechanism for forming compact n-step ahead predictions. The next video follows on by giving some numerical examples of prediction matrices for a complete prediction horizon, using transfer function models. Includes MATLAB demonstration.

- The videos are available here and here.
- MATLAB demonstrations are available here.

## 6. Prediction with step response models

Many commerical MPC algorithms deploy step response models as these are relativey easy to identify. This video shows how one can form a n-step ahead prediction using step response parameters - viewers will note that this requires a subtlety that might be unexpected.

- A videos is available here.
- MATLAB demonstrations are available here.

## 7. Unbiased Predictions

The previous videos showed how to predict, but gave little attention to the accuracy of these predictions. It transpires that an effective MPC design requires the predictions to be unaffected by uncertainty, in the steady-state; this is called unbiased prediction. One method used is a disturbance estimate. Alternatively, the use of a CARIMA model allows the predictions, in steady-state, to have no errors despite the presence of parameter and other uncertainties. Finally, one can use a disturbance estimate in conjunction with deviation variables relative to the expected steady-state to give unbiased predictions when utilising a state space model (assumes observability).

## 8. Performance indices

Introduces the concept of good and bad performance which is required to distinguish between the predictions you want and those you do not. Noted that in many cases the definition of 'optimal' performance is to some extent arbitrary and not systematic. The next video illustrates how a naive selection of performance index based on the square of tracking errors or inputs can lead to poor performance. Any measure of performance must be unbiased, that is the use of this within control design and decision making should not lead to steady-state offsets. Hence, the final video shows how a performance index can be constructed to make good engineering science and also be unbiased.