# Chapter 4: Infinite horizon predictive control laws (dual-mode)

Dual-mode approaches build on the understanding of what constitutes a well-posed predictive control optimisation. Initially one can make an argument that one should optimise over predictions which include within the class at least one which is the 'global' unconstrained optimal. Demonstrates, using state space models, how this can be done. Then extends this paradigm to a more general case called dual-mode control and develops the unconstrained MPC control law for this case. Finally gives some discussion to stability and argues that the use of dual-mode prediction eliminates some of the fundamental weakenesses associated to GPC/DMC. Most of the chapter is based on the regulation problem in order to simplify the algebra, so only the last section considers extensions to deal with tracking and disturbance rejection. Also demonstrates that the use of a disturbance estimate will also give offset free tracking in the case of parameter uncertainty.

The focus is on the state-space case because, although possible, the algebra with transfer function and step response models is much more cumbersome without adding to the key concepts.

- Introduction.
- Prediction structures and degrees of freedom.
- The performance index.
- Analysing the performance index.
- Suboptimal dual mode MPC.
- Well-posed recursive decision making.
- Stability with infinite horizons and the tail.
- Numerical examples and MATLAB.
- Tracking and disturbance rejection.
- Disturbance estimates with an independent model.
- Disturbance estimates with an observer.

## 1. Introduction

Gives the human or philosophical thinking behind optimal predictive control and explains why this is an intuitively obvious approach to predictive control algorithm design. Implicitly builds on insights developed in chapter 3.

- A video talk through is available here.

## 2. Prediction structures and degrees of freedom

Shows how the 'global' unconstrained optimal can be embedded within system predictions and also how to add degrees of freedom which can be used for fine tuning. Gives fine details of the associated predictions which are analogous to developments in chapter 1.

## 3. The performance index

Demonstrates how predictions based around the global unconstrained optimal can be substituted into the performance index to give a simple cost function. Introduces the augmented model as a mechanism for keeping the algebra very simple and transparent. Uses Lyapunov to deal with summations over an infinite horizon. Includes MATLAB code for computing the parameters.

- Video talk throughs are available here.

## 4. Analysing the performance index

Analyses each component of the performance index in turn and demonstrates through argument and MATLAB examples how this has a very simple (and well conditioned) structure. Shows the optimised control law is independent of the control horizon.

- A video talk through is available here.

## 5. Suboptimal dual mode MPC

Uses the concept of a dual-mode predictive in OMPC and allows some modification to give a more general dual-mode approach (Subopotimal MPC). Demonstrates how the algebra and computation is almost identical, but the performance index for SOMPC has a slightly more complex structure than OMPC which means the control law is also dependent on the control horizon.

- Videos are available here.

## 6. Well-posed recursive decision making

Introduces concepts of consistent recursive decision making, that is how we ensure the optimisation at the current sample has a synergy with that taken at the previous sample. This is a prerequisiste for the expectation of good behaviour/stability. It is shown how the concept of the tail captures this requirement.

- A video is available here.

## 7. Stability with infinite horizons and the tail

Introduces analysis and anticipation of closed-loop stability. Shows how the use of infinite horizons in conjunction with the tail enables guarantees of stability for the nominal case. Shows how dual-mode approaches, regardless of stabilising terminal mode, can meet these criteria.

- A video is available here.

## 8.Numerical examples and MATLAB

Uses MATLAB to illustrate the OMPC and SOMPC algorithms in the constraint free and regulation case. Both are guaranteed stable, however whereas OMPC gives zero perturbations (well-posed optimisation), SOMPC gives non-zero perturbations and a prediction mismatch. Nevertheless the SOMPC compensator tends towards OMPC and has a small prediction mismatch for high control horizons.

- A video is available here.

## 10. Tracking and disturbance rejection

Extends the earlier videos to include non-zero targets and disturbances. Demonstrates how near identical algebra and optimisation may be applicable by making use of superposition and deviation variables. Makes use of unbiased predictions and performance indices discussed in chapter 1. MATLAB examples are included.

- Videos are available here.

## 11. Disturbance estimates with an independent model

Earlier videos assumed the state and disturbance were known whereas in practice these need to be estimated. This video gives a brief review of he impact of using an independent model as an observer. Critically, although nominal stability results are unaffected, performance can be seriously degraded. Includes MATLAB comparisons.

- Videos are available here.

## 12. Disturbance estimates with an observer

Earlier videos assumed the state and disturbance were known whereas in practice these need to be estimated. This video gives a brief review of he impact of including an observer. Critically, although nominal stability results are unaffected, performance can be seriously degraded. Also includes MATLAB comparisons with the independent model approach.

- A video is available here.