Оглавление
Efstratios N. Pistikopoulos. Multi-parametric Optimization and Control
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
Wiley Series in
Multi‐parametric Optimization and Control
Short Bios of the Authors. Efstratios N. Pistikopoulos
Nikolaos A. Diangelakis
Richard Oberdieck
Preface
1 Introduction
1.1 Concepts of Optimization. 1.1.1 Convex Analysis
Definition 1.1 (Line)
Definition 1.2 (Line Segment)
Definition 1.3 (Convex Set)
1.1.1.1 Properties of Convex Sets
Definition 1.4 (Convex Function)
Definition 1.5 (Concave Function)
1.1.1.2 Properties of Convex Functions
1.1.2 Optimality Conditions
Definition 1.6 (Local Minimum)
Definition 1.7 (Global Minimum)
Definition 1.8 (Active Constraints)
Remark 1.1
1.1.2.1 Karush–Kuhn–Tucker Necessary Optimality Conditions
1.1.2.2 Karun–Kush–Tucker First‐Order Sufficient Optimality Conditions
1.1.3 Interpretation of Lagrange Multipliers
1.2 Concepts of Multi‐parametric Programming. 1.2.1 Basic Sensitivity Theorem
Theorem 1.1 (Basic Sensitivity Theorem, [1])
Proof
1.3 Polytopes
Definition 1.9
Definition 1.10
1.3.1 Approaches for the Removal of Redundant Constraints
Theorem 1.2 ([3])
Remark 1.3
Remark 1.4
1.3.1.1 Lower‐Upper Bound Classification
1.3.1.2 Solution of Linear Programming Problem
Remark 1.5
1.3.2 Projections
Definition 1.11 (Projection [7])
Definition 1.12 (Hybrid Projection)
1.3.3 Modeling of the Union of Polytopes
1.4 Organization of the Book
References
Notes
2 Multi‐parametric Linear Programming
Remark 2.1
2.1 Solution Properties. Remark 2.2
2.1.1 Local Properties
Remark 2.3
Lemma 2.1
Proof
Lemma 2.2
Proof
2.1.2 Global Properties
Theorem 2.1 (The Solution of mp‐LP Problems)
Proof
Definition 2.1 (mp‐LP Graph)
Theorem 2.2 (The Connected‐graph Theorem)
Proof
2.2 Degeneracy
2.2.1 Primal Degeneracy
2.2.2 Dual Degeneracy
Remark 2.4
2.2.3 Connections Between Degeneracy and Optimality Conditions
2.3 Critical Region Definition
2.4 An Example: Chicago to Topeka
2.4.1 The Deterministic Solution
2.4.2 Considering Demand Uncertainty
Remark 2.5
2.4.3 Interpretation of the Results
2.5 Literature Review
References
Notes
3 Multi‐Parametric Quadratic Programming
Remark 3.1
3.1 Calculation of the Parametric Solution. Remark 3.2
3.1.1 Solution via the Basic Sensitivity Theorem
Remark 3.3
3.1.2 Solution via the Parametric Solution of the KKT Conditions
3.2 Solution Properties. 3.2.1 Local Properties
Remark 3.4
3.2.2 Global Properties
Theorem 3.1 (The Solution of mp‐QP Problems)
Proof
3.2.3 Structural Analysis of the Parametric Solution
Theorem 3.2 (Active Set of Adjacent Region)
Proof
Lemma 3.1
Proof
Lemma 3.2
Proof
Definition 3.1 (mp‐QP Graph)
Theorem 3.3 (Connected Graph for mp‐QP Problems)
Proof
3.3 Chicago to Topeka with Quadratic Distance Cost
Remark 3.5
3.3.1 Interpretation of the Results
3.4 Literature Review
Remark 3.6
References
Notes
4 Solution Strategies for mp‐LP and mp‐QP Problems
4.1 General Overview
Remark 4.1
4.2 The Geometrical Approach
4.2.1 Define A Starting Point
Remark 4.2
4.2.2 Fix in Problem 4.1, and Solve the Resulting QP
4.2.3 Identify The Active Set for The Solution of The QP Problem
4.2.4 Move Outside the Found Critical Region and Explore the Parameter Space
4.3 The Combinatorial Approach
Remark 4.3
4.3.1 Pruning Criterion
Lemma 4.1
Proof
Remark 4.4
4.4 The Connected‐Graph Approach
Remark 4.6
4.5 Discussion
Remark 4.7
4.6 Literature Review
Remark 4.8
References
Notes
5 Multi‐parametric Mixed‐integer Linear Programming
Remark 5.1
5.1 Solution Properties. 5.1.1 From mp‐LP to mp‐MILP Problems
5.1.2 The Properties
Theorem 5.1 (The solution of mp‐MILP problems)
Proof
5.2 Comparing the Solutions from Different mp‐LP Problems. Remark 5.2
Remark 5.3
5.2.1 Identification of Overlapping Critical Regions
5.2.2 Performing the Comparison
5.2.3 Constraint Reversal for Coverage of Parameter Space
5.3 Multi‐parametric Integer Linear Programming
Remark 5.4
5.4 Chicago to Topeka Featuring a Purchase Decision
5.4.1 Interpretation of the Results
5.5 Literature Review
References
Notes
6 Multi‐parametric Mixed‐integer Quadratic Programming
Remark 6.1
6.1 Solution Properties. 6.1.1 From mp‐QP to mp‐MIQP Problems
6.1.2 The Properties
Theorem 6.1
Proof
Lemma 6.1 (Quadratic Boundaries)
Proof
6.2 Comparing the Solutions from Different mp‐QP Problems. Remark 6.2
Remark 6.3
6.2.1 Identification of overlapping critical regions
6.2.2 Performing the Comparison
6.3 Envelope of Solutions
Definition 6.1 (Envelope of Solutions)
6.4 Chicago to Topeka Featuring Quadratic Cost and A Purchase Decision
Remark 6.4
6.4.1 Interpretation of the Results
6.5 Literature Review
References
Notes
7 Solution Strategies for mp‐MILP and mp‐MIQP Problems
7.1 General Framework
7.2 Global Optimization
7.2.1 Introducing Suboptimality
7.3 Branch‐and‐Bound
Remark 7.1
7.4 Exhaustive Enumeration
7.5 The Comparison Procedure
Remark 7.2
7.5.1 Affine Comparison
7.5.2 Exact Comparison
Remark 7.3
7.6 Discussion
7.6.1 Integer Handling
7.6.2 Comparison Procedure
7.7 Literature Review
References
Notes
8 Solving Multi‐parametric Programming Problems Using MATLAB®
8.1 An Overview over the Functionalities of POP
8.2 Problem Solution. 8.2.1 Solution of mp‐QP Problems
8.2.2 Solution of mp‐MIQP Problems
8.2.3 Requirements and Validation
8.2.4 Handling of Equality Constraints
8.2.5 Solving Problem (7.2)
8.3 Problem Generation
8.4 Problem Library
8.4.1 Merits and Shortcomings of The Problem Library
8.5 Graphical User Interface (GUI)
8.6 Computational Performance for Test Sets. Remark 8.1
8.6.1 Continuous Problems
8.6.2 Mixed‐integer Problems
8.7 Discussion
Acknowledgments
References
Notes
9 Other Developments in Multi‐parametric Optimization
9.1 Multi‐parametric Nonlinear Programming
9.1.1 The Convex Case
9.1.2 The Non‐convex Case
9.2 Dynamic Programming via Multi‐parametric Programming
Remark 9.1
9.2.1 Direct and Indirect Approaches
9.3 Multi‐parametric Linear Complementarity Problem
Remark 9.2
9.4 Inverse Multi‐parametric Programming
9.5 Bilevel Programming Using Multi‐parametric Programming
Remark 9.3
9.6 Multi‐parametric Multi‐objective Optimization
Remark 9.4
References
Notes
10 Multi‐parametric/Explicit Model Predictive Control. 10.1 Introduction
10.2 From Transfer Functions to Discrete Time State‐Space Models
10.3 From Discrete Time State‐Space Models to Multi‐parametric Programming
Remark 10.1
Remark 10.2
Remark 10.3
Remark 10.4
Remark 10.5
Remark 10.6
10.4 Explicit LQR – An Example of mp‐MPC. 10.4.1 Problem Formulation and Solution
10.4.2 Results and Validation
10.5 Size of the Solution and Online Computational Effort
References
Notes
11 Extensions to Other Classes of Problems
11.1 Hybrid Explicit MPC
11.1.1 Explicit Hybrid MPC – An Example of mp‐MPC
11.1.2 Results and Validation
11.2 Disturbance Rejection
11.2.1 Explicit Disturbance Rejection – An Example of mp‐MPC
11.2.2 Results and Validation
11.3 Reference Trajectory Tracking
11.3.1 Reference Tracking to LQR Reformulation
11.3.2 Explicit Reference Tracking – An Example of mp‐MPC
11.3.3 Results and Validation
11.4 Moving Horizon Estimation. 11.4.1 Multi‐parametric Moving Horizon Estimation
11.4.1.1 Current State
11.4.1.2 Recent Developments
11.4.1.3 Future Outlook
11.5 Other Developments in Explicit MPC
References
Notes
12 PAROC: PARametric Optimization and Control. 12.1 Introduction
12.2 The PAROC Framework
12.2.1 “High Fidelity” Modeling and Analysis
12.2.2 Model Approximation
12.2.2.1 Model Approximation Algorithms: A User Perspective Within the PAROC Framework
12.2.2.1.1 Model Approximation: An Outline
Data Collection
Polish and Present Data
Data Filtering
Choosing the Model Structure
The Number of States
Discrete or Continuous Time Representation
The Disturbance Component
The Output Feedthrough
The Focus of the Model
12.2.2.1.2 Model Fitting
12.2.2.1.3 Model Validation and Accepting the Model
12.2.3 Multi‐parametric Programming
12.2.4 Multi‐parametric Moving Horizon Policies
Remark 12.1
12.2.5 Software Implementation and Closed‐Loop Validation. 12.2.5.1 Multi‐parametric Programming Software
12.2.5.2 Integration of PAROC in gPROMS® ModelBuilder
12.3 Case Study: Distillation Column
12.3.1 “High Fidelity” Modeling
12.3.2 Model Approximation
12.3.3 Multi‐parametric Programming, Control, and Estimation
12.3.4 Closed‐Loop Validation
12.3.5 Conclusion
12.4 Case Study: Simple Buffer Tank. 12.5 The Tank Example
12.5.1 “High Fidelity” Dynamic Modeling
12.5.2 Model Approximation
12.5.3 Design of the Multi‐parametric Model Predictive Controller
12.5.4 Closed‐Loop Validation
12.5.5 Conclusion
12.6 Concluding Remarks
References
Notes
A Appendix for the mp‐MPC Chapter 10
Appendix for the mp‐MPC Chapter 11
B.1 Matrices for the mp‐QP Problem Corresponding to the Example of Section 11.3.2
Index
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