Solving Optimization Problems with the Heuristic Kalman Algorithm

✍ Scribed by Rosario Toscano

• Publisher: Springer Nature Switzerland
• Year: 2024
• Tongue: English
• Leaves: 297
• Series: Springer Optimization and Its Applications 212
• Edition: 1
• Category: Library

✦ Synopsis

This text focuses on simple and easy-to-use design strategies for solving complex engineering problems that arise in several fields of engineering design, namely non-convex optimization problems.

The main optimization tool used in this book to tackle the problem of nonconvexity is the Heuristic Kalman Algorithm (HKA). The main characteristic of HKA is the use of a stochastic search mechanism to solve a given optimization problem. From a computational point of view, the use of a stochastic search procedure appears essential for dealing with non-convex problems.

The topics discussed in this monograph include basic definitions and concepts from the classical optimization theory, the notion of the acceptable solution, machine learning, the concept of preventive maintenance, and more.

The Heuristic Kalman Algorithm discussed in this book applies to many fields such as robust structured control, electrical engineering, mechanical engineering, machine learning, reliability, and preference models. This large coverage of practical optimization problems makes this text very useful to those working on and researching systems design. The intended audience includes industrial engineers, postgraduates, and final-year undergraduates in various fields of systems design.

✦ Table of Contents

Preface
Stochastic Optimization via HKA
Structure of the Book
Contents
Notation and Acronyms
Sets
Relational Operators
Miscellaneous
Matrix Operations
Measure of Size
Acronyms
1 Introduction
1.1 General Formulation of an Optimization Problem
1.1.1 Design Variables, Constraints, and Objective Functions
1.1.1.1 Design Variables
1.1.1.2 Constraints
1.1.1.3 Objective Function
1.1.2 Global and Local Minimum, Descent Direction
1.1.3 Optimality Conditions
1.2 Convex Optimization Problems
1.2.1 Semidefinite Programming
1.2.2 Dual Problem
1.3 Optimization in Engineering Design
1.4 Main Objectives of the Book
1.4.1 Algorithms for Finding an Acceptable Solution
1.5 Notes and References
2 Stochastic Optimization Methods
2.1 Motivations and Basic Notions
2.1.1 Notion of Acceptable Solution
2.1.2 Some Characteristics of Stochastic Methods
2.2 Pure Random Search Methods
2.2.1 Non-localized Search Method
2.2.2 Localized Search Method
2.3 Simulated Annealing
2.3.1 Metropolis Algorithm and Simulated Annealing
2.3.2 Simulated Annealing Algorithm
2.4 Genetic Algorithm
2.4.1 The Main Steps of a Genetic Algorithm
2.4.2 The Standard Genetic Algorithm
2.5 Particle Swarm Optimization
2.5.1 Dynamics of the Particles of a Swarm
2.5.2 The Standard PSO Algorithm
2.6 Robust Optimization
2.6.1 Worst-Case Approach
2.6.2 Average Approach
2.7 Notes and References
3 Heuristic Kalman Algorithm
3.1 Introduction
3.2 Principle of the Algorithm
3.2.1 Gaussian Generator
3.2.2 Measurement Process
3.2.3 Kalman Estimator
3.3 Equation of the Kalman Estimator
3.4 Heuristic Kalman Algorithm and Implementation
3.4.1 Updating Rule of the Gaussian Distribution
3.4.2 Initialization and Parameter Settings
3.4.3 Stopping Rule
3.4.4 The Feasibility Issue
3.5 Numerical Experiments
3.5.1 Comparison with SA, GA, and PSO
3.5.1.1 Formulation of the Control Design Problem
3.5.1.2 Solution of Problem (3.23) with SA, GA, PSO, and HKA
3.5.2 Comparison with Some Other Metaheuristics
3.5.2.1 Unconstrained Case
3.5.2.2 Constrained Case
3.6 Conclusion
3.7 Notes and References
4 Some Notions on System Modeling
4.1 Notion of System
4.2 Mathematical Model of a System
4.3 State Space Models
4.3.1 Linearization of the State Space Model
4.3.2 Transfer Matrix
4.3.3 Parametric Uncertainty
4.3.4 General Representation of an Uncertain Linear System
4.3.4.1 State Space Representation
4.3.4.2 LFT Representation
4.4 Parametric Uncertainty
4.4.1 Affine Parameter-Dependent Model
4.4.2 Polytopic Model
4.4.3 LFT Model
4.5 Experimental Models
4.5.1 Formulation of the Optimization Problem
4.5.2 Common Choice of the Postulated Model
4.6 Examples of Modeling
4.6.1 Active Magnetic Bearing (AMB)
4.6.2 Quadrotor Unmanned Aerial Vehicle (QUAV)
4.7 Examples of Optimal Sizing
4.7.1 Optimal Sizing of an AMB
4.7.2 Optimal Sizing of a Quadrotor
4.8 Notes and References
4.9 Optimal Sizing of the AMB
4.10 Optimal Sizing of the Quadrotor
5 Robust Control of Uncertain Parametric Systems
5.1 Notion of System and Feedback Control
5.2 The Standard Control Problem
5.2.1 The Standard Control Problem as an Optimization Problem
5.2.2 Extended System Model
5.2.3 Controller
5.2.4 Closed-loop System and Stabilizing Controllers
5.2.4.1 Stabilizing Controllers
5.2.4.2 Input-Output Representation of the Closed-loop System
5.3 Performance Specifications
5.4 Structured Controllers
5.4.1 Important Examples of Structured Controllers
5.4.2 Difficulties in Solving the Structured Control Problem
5.4.3 Robustness Issue
5.4.4 Robust Performance Condition
5.4.5 General Stochastic Algorithm
5.5 H2 and Mixed H2/H∞ Design of Structured Controllers
5.6 H2 Design of Structured Controllers
5.6.1 Formulation of the Robust H H H H2 Design Problem
5.6.2 Set of Robustly Stable SOF Controllers
5.6.3 Worst-Case Performance and Average Performance
5.6.4 Guaranteed LQ Cost with Time-Varying Parameters
5.7 Mixed H2/H∞ Design of Structured Controllers
5.7.1 Problem Formulation
5.7.2 Set of Robustly Stable SOF Controllers
5.8 Design Examples
5.8.1 Design Example 1
5.8.2 Design Example 2
5.9 Notes and References
6 Preventive Maintenance
6.1 Problem Statement
6.2 Reliability of a System in the Context of Systematic Preventive Maintenance
6.2.1 Basic Relations Related to the Reliability
6.2.2 Main Reliability Laws Used in Practice
6.2.3 Determination of the Preventive Replacement Time of a Component
6.2.4 Reliability of a System
6.2.5 Optimal Allocation of Reliability
6.3 Fault Detection and Localization in the Context of Conditional Preventive Maintenance
6.3.1 Mathematical Model-Based Diagnosis of a System
6.3.2 Principle Residual Generation
6.3.3 Fault Detection and Localization
6.3.4 Observer-Based Fault Detection and Localization (FDL)
6.3.4.1 Unknown Input Observer with Perfect Decoupling
6.3.4.2 Unknown Input Observer with Optimal Approximate Decoupling
6.3.5 Examples of the Design of Observers for Residual Generation
6.4 Notes and References
6.4.1 Some Historical Aspects of Reliability and Maintenance as Well as Fault Detection and Localization
6.4.2 MatLab Code for the Determination of the Optimal Time of Replacement
6.4.3 MatLab Code for the Optimal Reliability Allocation
6.4.4 MatLab Code for Mixed H2/H∞ Residual Generator
7 Machine Learning
7.1 K-Means Clustering
7.1.1 Numerical Example
7.2 Classification via Gaussian Basis Functions
7.2.1 Modeling of the Classes via Gaussian Basis Functions
7.2.2 Numerical Example
7.3 Nonlinear Regression Based on a Multi-model Approach
7.3.1 Multi-model Representation of a Given Nonlinear System
7.3.2 Global Representation of a System from Local Models
7.3.3 General Procedure for Building a Nonlinear Regression Based Multi-model
7.3.4 Numerical Example 1
7.3.5 Numerical Example 2
7.4 Preference Models
7.4.1 The Notion of Product Space
7.4.2 Basic Principle of the Qualitative Analysis
7.4.3 Preference Value According to an Order on the Qualitative Attributes
7.4.3.1 Determination of an Order on the Qualitative Attributes
7.4.3.2 Preference Order Weights Selection According to the Maximum Entropy Principle
7.4.4 RBF Preference Model
7.4.4.1 Method for Determining the Parameters of the Basis Functions
7.4.5 Numerical Example
7.5 Notes and References
7.5.1 MatLab Code of the K-Means Algorithm
7.5.2 MatLab Code for Building the Multi-model
7.5.3 Learning Database BA and Test Database BT, for the Roughness Prediction
8 Conclusion
8.1 The Necessity of the Uncertain Model
8.1.1 Certainty Prediction
8.1.2 Uncertain Prediction
8.2 Design Procedure
A Signal and System Norms
A.1 Signal Norms
A.1.1 L1-Space and L1-Norm
A.1.2 L2-Space and L2-Norm
A.1.3 L∞-Space and L∞-Norm
A.1.4 Extended Lp-Space
A.1.5 RMS-Value
A.2 LTI Systems
A.2.1 System Stability
A.2.2 Controllability and Observability
A.2.3 Transfer Matrix
A.3 System Norms
A.3.1 Definition of the H2-Norm and H∞-Norm of a System
A.3.2 Singular Values of a Transfer Matrix
A.3.3 Singular Values and H2- and H∞-Norms
A.3.4 Computing Norms from the State Space Equation
A.4 Notes and References
B Convergence Properties of the HKA and Program Code
B.1 Convergence Properties
B.2 Program Code of the Heuristic Kalman Algorithm
B.3 Example of Use of This Program
B.4 MatLab Function Used in Example 5.1
References
Index

✦ Subjects

Stochastic Optimization, Heuristic Kalman Algorithm, Uncertain Parametric Systems, Preventive Maintenance, Machine Learning