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E-raamat: Multi-parametric Optimization and Control

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Multi-parametric Optimization and ControlSuurem pilt
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Led by a well-known expert in the field, this self-contained book features comprehensive coverage of multi-parametric optimization and control.  The authors provide state-of-the-art coverage of the most recent methodological developments for optimal model-based control via parametric optimization.  Each chapter consists of a theoretical treatment of the topic along with a relevant case study, allowing for the topical complexity to gradually increase throughout.  Each case study describes the needed methods and illustrates real-world applications, aiding readers in gaining a better understanding of the presented material.  Part I presents an overview of the state-of-the-art multi-parametric optimization theory and algorithms in multi-parametric programming.  Introducing the ‘San Francisco to Topeka’ transportation problem case study, this section highlights a conceptual approach and covers the main algorithms.  Part II focuses on multi-parametric model predictive control and emphasizes the connection between multi-parametric programming and model-predictive control, starting from the linear quadratic regulator over hybrid systems to periodic systems and robust control.  Illustrating the natural combination between multi-parametric programming and model predictive control, this part introduces and works through a case study on the continuously stirred tank reactor.  Part III features multi-parametric optimization in process systems engineering.  This section introduces the step-by-step procedure of embedding multi-parametric programming within process system engineering.  This naturally leads to the PAROC framework and software platform, which is an integrated framework and software platform for the optimization and advanced model-based control of process systems.  This section’s case study features a combined heat and power system.  Finally, the book ends with an appendix that includes the history of multi-parametric optimization algorithms as well as the use of the parametric optimization toolbox (POP), which is a comprehensive software tool capable of efficiently solving multi-parametric programming problems, while being easily embedded into other software architectures such as the ones used in the PAROC platform.
Short Bios of the Authors xvii
Preface xxi
1 Introduction
1(18)
1.1 Concepts of Optimization
1(5)
1.1.1 Convex Analysis
1(1)
1.1.1.1 Properties of Convex Sets
2(1)
1.1.1.2 Properties of Convex Functions
2(1)
1.1.2 Optimality Conditions
3(2)
1.1.2.1 Karush-Kuhn-Tucker Necessary Optimality Conditions
5(1)
1.1.2.2 Karun-Kush-Tucker First-Order Sufficient Optimality Conditions
5(1)
1.1.3 Interpretation of Lagrange Multipliers
6(1)
1.2 Concepts of Multi-parametric Programming
6(3)
1.2.1 Basic Sensitivity Theorem
6(3)
1.3 Poly topes
9(7)
1.3.1 Approaches for the Removal of Redundant Constraints
11(1)
1.3.1.1 Lower-Upper Bound Classification
12(1)
1.3.1.2 Solution of Linear Programming Problem
13(1)
1.3.2 Projections
13(1)
1.3.3 Modeling of the Union of Polytopes
14(2)
1.4 Organization of the Book
16(3)
References
16(3)
Part I Multi-parametric Optimization
19(168)
2 Multi-parametric Linear Programming
21(24)
2.1 Solution Properties
22(6)
2.1.1 Local Properties
23(2)
2.1.2 Global Properties
25(3)
2.2 Degeneracy
28(4)
2.2.1 Primal Degeneracy
29(1)
2.2.2 Dual Degeneracy
30(1)
2.2.3 Connections Between Degeneracy and Optimality Conditions
31(1)
2.3 Critical Region Definition
32(1)
2.4 An Example: Chicago to Topeka
33(5)
2.4.1 The Deterministic Solution
34(1)
2.4.2 Considering Demand Uncertainty
35(1)
2.4.3 Interpretation of the Results
36(2)
2.5 Literature Review
38(7)
References
39(6)
3 Multi-Parametric Quadratic Programming
45(22)
3.1 Calculation of the Parametric Solution
47(2)
3.1.1 Solution via the Basic Sensitivity Theorem
47(1)
3.1.2 Solution via the Parametric Solution of the KKT Conditions
48(1)
3.2 Solution Properties
49(6)
3.2.1 Local Properties
49(1)
3.2.2 Global Properties
50(2)
3.2.3 Structural Analysis of the Parametric Solution
52(3)
3.3 Chicago to Topeka with Quadratic Distance Cost
55(6)
3.3.1 Interpretation of the Results
56(5)
3.4 Literature Review
61(6)
References
63(4)
4 Solution Strategies for mp-LP and mp-QP Problems
67(22)
4.1 General Overview
68(2)
4.2 The Geometrical Approach
70(5)
4.2.1 Define A Starting Point 00
70(1)
4.2.2 Fix 90 in Problem (4.1), and Solve the Resulting QP
71(1)
4.2.3 Identify The Active Set for The Solution of The QP Problem
72(1)
4.2.4 Move Outside the Found Critical Region and Explore the Parameter Space
72(3)
4.3 The Combinatorial Approach
75(3)
4.3.1 Pruning Criterion
76(2)
4.4 The Connected-Graph Approach
78(3)
4.5 Discussion
81(2)
4.6 Literature Review
83(6)
References
85(4)
5 Multi-parametric Mixed-integer Linear Programming
89(18)
5.1 Solution Properties
90(2)
5.1.1 From mp-LP to mp-MILP Problems
90(1)
5.1.2 The Properties
91(1)
5.2 Comparing the Solutions from Different mp-LP Problems
92(4)
5.2.1 Identification of Overlapping Critical Regions
93(2)
5.2.2 Performing the Comparison
95(1)
5.2.3 Constraint Reversal for Coverage of Parameter Space
95(1)
5.3 Multi-parametric Integer Linear Programming
96(3)
5.4 Chicago to Topeka Featuring a Purchase Decision
99(3)
5.4.1 Interpretation of the Results
99(3)
5.5 Literature Review
102(5)
References
103(4)
6 Multi-parametric Mixed-integer Quadratic Programming
107(18)
6.1 Solution Properties
109(1)
6.1.1 From mp-QP to mp-MIQP Problems
109(1)
6.1.2 The Properties
109(1)
6.2 Comparing the Solutions from Different mp-QP Problems
110(3)
6.2.1 Identification of overlapping critical regions
112(1)
6.2.2 Performing the Comparison
112(1)
6.3 Envelope of Solutions
113(1)
6.4 Chicago to Topeka Featuring Quadratic Cost and A Purchase Decision
114(5)
6.4.1 Interpretation of the Results
115(4)
6.5 Literature Review
119(6)
References
121(4)
7 Solution Strategies for mp-MILP and mp-MIQP Problems
125(22)
7.1 General Framework
126(1)
7.2 Global Optimization
127(3)
7.2.1 Introducing Suboptimality
129(1)
7.3 Branch-and-Bound
130(3)
1.4 Exhaustive Enumeration
133(1)
7.5 The Comparison Procedure
134(4)
7.5.1 Affine Comparison
135(2)
7.5.2 Exact Comparison
137(1)
7.6 Discussion
138(4)
7.6.1 Integer Handling
138(3)
7.6.2 Comparison Procedure
141(1)
1.1 Literature Review
142(5)
References
144(3)
8 Solving Multi-parametric Programming Problems Using MATLAB®
147(18)
8.1 An Overview over the Functionalities of POP
148(1)
8.2 Problem Solution
148(2)
8.2.1 Solution of mp-QP Problems
148(1)
8.2.2 Solution of mp-MIQP Problems
148(1)
8.2.3 Requirements and Validation
149(1)
8.2.4 Handling of Equality Constraints
149(1)
8.2.5 Solving Problem (7.2)
149(1)
8.3 Problem Generation
150(1)
8.4 Problem Library
151(2)
8.4.1 Merits and Shortcomings of The Problem Library
152(1)
8.5 Graphical User Interface (GUI)
153(1)
8.6 Computational Performance for Test Sets
154(2)
8.6.1 Continuous Problems
154(1)
8.6.2 Mixed-integer Problems
154(2)
8.7 Discussion
156(9)
Acknowledgments
162(1)
References
162(3)
9 Other Developments in Multi-parametric Optimization
165(22)
9.1 Multi-parametric Nonlinear Programming
165(2)
9.1.1 The Convex Case
166(1)
9.1.2 The Non-convex Case
167(1)
9.2 Dynamic Programming via Multi-parametric Programming
167(3)
9.2.1 Direct and Indirect Approaches
169(1)
9.3 Multi-parametric Linear Complementarity Problem
170(1)
9.4 Inverse Multi-parametric Programming
171(1)
9.5 Bilevel Programming Using Multi-parametric Programming
172(1)
9.6 Multi-parametric Multi-objective Optimization
173(14)
References
174(13)
Part II Multi-parametric Model Predictive Control
187(94)
10 Multi-parametric/Explicit Model Predictive Control
189(22)
10.1 Introduction
189(2)
10.2 From Transfer Functions to Discrete Time State-Space Models
191(4)
10.3 From Discrete Time State-Space Models to Multi-parametric Programming
195(5)
10.4 Explicit LQR - An Example of mp-MPC
200(6)
10.4.1 Problem Formulation and Solution
200(2)
10.4.2 Results and Validation
202(4)
10.5 Size of the Solution and Online Computational Effort
206(5)
References
207(4)
11 Extensions to Other Classes of Problems
211(32)
11.1 Hybrid Explicit M PC
211(8)
11.1.1 Explicit Hybrid M PC - An Example of mp-M PC
213(2)
11.1.2 Results and Validation
215(4)
11.2 Disturbance Rejection
219(3)
11.2.1 Explicit Disturbance Rejection - An Example of mp-MPC
220(2)
11.2.2 Results and Validation
222(1)
11.3 Reference Trajectory Tracking
222(10)
11.3.1 Reference Tracking to LQR Reformulation
227(3)
11.3.2 Explicit Reference Tracking - An Example of mp-MPC
230(2)
11.3.3 Results and Validation
232(1)
11.4 Moving Horizon Estimation
232(7)
11.4.1 Multi-parametric Moving Horizon Estimation
232(5)
11.4.1.1 Current State
237(1)
11.4.1.2 Recent Developments
237(1)
11.4.1.3 Future Outlook
238(1)
11.5 Other Developments in Explicit MPC
239(4)
References
240(3)
12 PAROC: PARametric Optimization and Control
243(38)
12.1 Introduction
243(3)
12.2 The PAROC Framework
246(15)
12.2.1 "High Fidelity" Modeling and Analysis
247(1)
12.2.2 Model Approximation
247(1)
12.2.2.1 Model Approximation Algorithms: A User Perspective Within the PAROC Framework
247(10)
12.2.3 Multi-parametric Programming
257(2)
12.2.4 Multi-parametric Moving Horizon Policies
259(1)
12.2.5 Software Implementation and Closed-Loop Validation
259(1)
12.2.5.1 Multi-parametric Programming Software
259(1)
12.2.5.2 Integration of PAROC in gPROMS® ModelBuilder
260(1)
12.3 Case Study: Distillation Column
261(8)
12.3.1 "High Fidelity" Modeling
262(2)
12.3.2 Model Approximation
264(1)
12.3.3 Multi-parametric Programming, Control, and Estimation
265(2)
12.3.4 Closed-Loop Validation
267(1)
12.3.5 Conclusion
268(1)
12.4 Case Study: Simple Buffer Tank
269(1)
12.5 The Tank Example
269(4)
12.5.1 "High Fidelity" Dynamic Modeling
269(1)
12.5.2 Model Approximation
270(1)
12.5.3 Design of the Multi-parametric Model Predictive Controller
271(1)
12.5.4 Closed-Loop Validation
272(1)
12.5.5 Conclusion
273(1)
12.6 Concluding Remarks
273(8)
References
273(8)
A Appendix for the mp-MPC
Chapter 10
281(4)
B Appendix for the mp-MPC
Chapter 11
285(6)
B.1 Matrices for the mp-QP Problem Corresponding to the Example of Section 11.3.2
285(6)
Index 291
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.
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