# Chapter eight

Predictive control

## section two: Introduction to predictive control and forming system predictions

This section 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

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

### 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.

### 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 second video extends the previous video by introducing compact notation for prediction which enables the easier algebra needed for most MPC algorithms.

Video 1 on Model Predictive Control – prediction with state space models

Video 2 on Model Predictive Control – prediction with state space models (continued)

### 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.

Video 1 on Predictive Control 7 – prediction with Carima models

Video 2 on Predictive Control 7 – prediction with Carima models (continued)

### 6. Prediction with step response models

Many commercial MPC algorithms deploy step response models as these are relative 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.

### 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).

Video on Model Predictive Control – unbiased prediction with CARIMA models

Video on Model Predictive Control – unbiased prediction using steady-state estimates

### 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 naïve 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.