Multi-parametric Optimization and Control

Recent developments in multi-parametric optimization and control

Multi-Parametric Optimization and Control provides comprehensive coverage of recent methodological developments for optimal model-based control through parametric optimization. It also shares real-world research applications to support deeper understanding of the material.

Researchers and practitioners can use the book as reference. It is also suitable as a primary or a supplementary textbook. Each chapter looks at the theories related to a topic along with a relevant case study. Topic complexity increases gradually as readers progress through the chapters. The first part of the book presents an overview of the state-of-the-art multi-parametric optimization theory and algorithms in multi-parametric programming. The second examines the connection between multi-parametric programming and model-predictive control—from the linear quadratic regulator over hybrid systems to periodic systems and robust control.

The third part of the book addresses multi-parametric optimization in process systems engineering. A step-by-step procedure is introduced for embedding the programming within the system engineering, which leads the reader into the topic of the PAROC framework and software platform. PAROC is an integrated framework and platform for the optimization and advanced model-based control of process systems.

• Uses case studies to illustrate real-world applications for a better understanding of the concepts presented
• Covers the fundamentals of optimization and model predictive control
• Provides information on key topics, such as the basic sensitivity theorem, linear programming, quadratic programming, mixed-integer linear programming, optimal control of continuous systems, and multi-parametric optimal control

An appendix summarizes the history of multi-parametric optimization algorithms. It also covers the use of the parametric optimization toolbox (POP), which is comprehensive software for efficiently solving multi-parametric programming problems.

EFSTRATIOS N. PISTIKOPOULOS is the Director of the Texas A&M Energy Institute and a TEES Eminent Professor in the Artie McFerrin Department of Chemical Engineering at Texas A&M University. He holds a Ph.D. degree from Carnegie Mellon University (1988) and was with Shell Chemicals in Amsterdam before joining Imperial. He has authored or co-authored over 500 major research publications in the areas of modelling, control and optimization of process, energy and systems engineering applications, 15 books and 2 patents.

NIKOLAOS A. DIANGELAKIS is an Optimization Specialist at Octeract Ltd. He holds a PhD and MSc on Advanced Chemical Engineering from Imperial College London and was a member of the Multi-Parametric Optimization and Control group at Imperial and then Texas A&M since 2011. He is the co-author of 16 journal papers, 11 conference papers and 3 book chapters.

RICHARD OBERDIECK is a Technical Account Manager at Gurobi Optimization, LLC. He obtained a bachelor and MSc degrees from ETH Zurich in Switzerland (2009-1013), before pursuing a PhD in Chemical Engineering at Imperial College London, UK, which he completed in 2017. He has published 21 papers and 2 book chapters, has an h-index of 11 and was awarded the FICO Decisions Award 2019 in Optimization, Machine Learning and AI.

Preface v

1 Introduction 1

1.1 Concepts of Optimization 1

1.1.1 Convex Analysis 1

1.1.2 Optimality Conditions 3

1.1.3 Interpretation of Lagrange Multipliers 4

1.2 Concepts of Multiparametric Programming 5

1.2.1 Basic Sensitivity Theorem 5

1.3 Polytopes 8

1.3.1 Approaches for the removal of redundant constraints 10

1.3.2 Projections 11

1.3.3 Modelling of the union of polytopes 12

1.4 Organization of the Book 13

Part I Multi-parametric Optimization 17

2 Multi-parametric linear programming 19

2.1 Solution properties 20

2.1.1 Local properties 20

2.1.2 Global properties 22

2.2 Degeneracy 24

2.3 Critical region definition 27

2.4 An Example: Chicago to Topeka 28

2.4.1 The deterministic solution 29

2.4.2 Considering demand uncertainty 30

2.4.3 Interpretation of the results 32

2.5 Literature review 32

3 Multi-parametric quadratic programming 39

3.1 Calculation of the parametric solution 40

3.1.1 Solution via the Basic Sensitivity Theorem 40

3.1.2 Solution via the parametric solution of the KKT conditions 41

3.2 Solution properties 42

3.2.1 Local properties 42

3.2.2 Global properties 42

3.2.3 Structural analysis of the parametric solution 44

3.3 Chicago to Topeka with quadratic distance cost 47

3.3.1 Interpretation of the results 50

3.4 Literature review 51

4 Solution strategies for mp-LP and mp-QP problems 55

4.1 General overview 56

4.2 The geometrical approach 57

4.2.1 Define a starting point 0 57

4.2.2 Fix 0 in problem (4.1), and solve the resulting QP 58

4.2.3 Identify the active set for the solution of the QP problem 58

4.2.4 Move outside the found critical region and explore the parameter space 59

4.3 The combinatorial approach 62

4.3.1 Pruning criterion 62

4.4 The connected-graph approach 63

4.5 Discussion 66

4.6 Literature Review 67

5 Multi-parametric mixed-integer linear programming 71

5.1 Solution properties 72

5.1.1 From mp-LP to mp-MILP problems 72

5.1.2 The properties 72

5.2 Comparing the solutions from different mp-LP problems 74

5.3 Multi-parametric integer linear programming 76

5.4 Chicago to Topeka featuring a purchase decision 78

5.4.1 Interpretation of the results 79

5.5 Literature review 82

6 Multi-parametric mixed-integer quadratic programming 85

6.1 Solution properties 86

6.1.1 From mp-QP to mp-MIQP problems 86

6.1.2 The properties 86

6.2 Comparing the solutions from different mp-QP problems 88

6.3 Envelope of solutions 90

6.4 Chicago to Topeka featuring quadratic cost and a purchase decision 91

6.4.1 Interpretation of the results 95

6.5 Literature review 95

7 Solution strategies for mp-MILP and mp-MIQP problems 99

7.1 General Framework 99

7.2 Global optimization 100

7.2.1 Introducing suboptimality 102

7.3 Branch-and-bound 103

7.4 Exhaustive enumeration 105

7.5 The comparison procedure 106

7.6 Discussion 111

7.6.1 Integer Handling 111

7.6.2 Comparison procedure 112

7.7 Literature Review 113

8 Solving multi-parametric programming problems using MATLAB® 117

8.1 An overview over the functionalities of POP 117

8.2 Problem solution 118

8.2.1 Solution of mp-QP problems 118

8.2.2 Solution of mp-MIQP problems 118

8.2.3 Requirements and Validation 118

8.2.4 Handling of equality constraints 119

8.2.5 Solving problem (7.2) 119

8.3 Problem generation 119

8.4 Problem library 120

8.4.1 Merits and shortcomings of the problem library 121

8.5 Graphical User Interface (GUI) 123

8.6 Computational performance for test sets 123

8.6.1 Continuous problems 124

8.6.2 Mixed-integer problems 127

8.7 Discussion 130

8.8 Acknowledgments 130

9 Other developments in multi-parametric optimization 133

9.1 Multi-parametric nonlinear programming 133

9.1.1 The convex case 134

9.1.2 The non-convex case 134

9.2 Dynamic programming via multi-parametric programming 135

9.2.1 Direct and indirect approaches 136

9.3 Multi-parametric linear complementarity problem 136

9.4 Inverse multi-parametric programming 137

9.5 Bilevel programming using multi-parametric programming 138

9.6 Multi-parametric multi-objective optimization 139

Part II Multi-parametric Model Predictive Control 147

10 Multi-parametric/explicit Model Predictive Control 149

10.1 Introduction 149

10.2 From transfer functions to discrete time state-space models 151

10.3 From discrete time state-space models to multi-parametric programming 154

10.4 Explicit LQR - an example of mp-MPC 158

10.4.1 Problem formulation and solution 158

10.4.2 Results and validation 159

10.5 Size of the solution and online computational effort 163

11 Extensions to other classes of problems 167

11.1 Hybrid Explicit MPC 167

11.1.1 Explicit Hybrid MPC - an example of mp-MPC 169

11.1.2 Results and validation 170

11.2 Disturbance rejection 174

11.2.1 Explicit disturbance rejection - an example of mp-MPC 175

11.2.2 Results and validation 176

11.3 Reference trajectory tracking 180

11.3.1 Reference tracking to LQR reformulation 181

11.3.2 Explicit reference tracking - an example of mp-MPC 184

11.3.3 Results and validation 186

11.4 Moving Horizon Estimation 190

11.4.1 Multi-parametric Moving Horizon Estimation 190

11.5 Other developments in explicit MPC 192

12 PAROC: PARametric Optimization and Control 197

12.1 Introduction 197

12.2 The PAROC Framework 198

12.2.1 ‘High Fidelity’ Modeling and Analysis 198

12.2.2 Model Approximation 199

12.2.3 Multi-parametric Programming 208

12.2.4 Multi-parametric Moving Horizon Policies 208

12.2.5 Software Implementation and Closed-loop Validation 209

12.3 Case study: Distillation Column 210

12.3.1 ‘High Fidelity’ Modeling 211

12.3.2 Model Approximation 212

12.3.3 Multi-parametric Programming, Control and Estimation 214

12.3.4 Closed loop validation 215

12.3.5 Conclusion 216

12.4 Case study: Simple Buffer Tank 216

12.5 The tank example 216

12.5.1 ‘High Fidelity’ Dynamic Modeling 217

12.5.2 Model Approximation 217

12.5.3 Design of the Multi-parametric Model Predictive Controller 217

12.5.4 Closed-Loop Validation 218

12.5.5 Conclusion 220

12.6 Concluding remarks 220

A Appendix 225

A.1 Appendix for the mp-MPC Chapter 10 225

B Appendix 229

B.1 Appendix for the mp-MPC Chapter 11 229

B.1.1 Matrices for the mp-QP problem corresponding to the example of section 11.3.2 229

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