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E-raamat: Attractive Ellipsoids in Robust Control

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Attractive Ellipsoids in Robust ControlSuurem pilt
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This monograph introduces a newly developed robust-control design technique for a wide class of continuous-time dynamical systems called the “attractive ellipsoid method.” Along with a coherent introduction to the proposed control design and related topics, the monograph studies nonlinear affine control systems in the presence of uncertainty and presents a constructive and easily implementable control strategy that guarantees certain stability properties. The authors discuss linear-style feedback control synthesis in the context of the above-mentioned systems. The development and physical implementation of high-performance robust-feedback controllers that work in the absence of complete information is addressed, with numerous examples to illustrate how to apply the attractive ellipsoid method to mechanical and electromechanical systems. While theorems are proved systematically, the emphasis is on understanding and applying the theory to real-world situations. Attractive Ellipsoids in Robust Control will appeal to undergraduate and graduate students with a background in modern systems theory as well as researchers in the fields of control engineering and applied mathematics.

Arvustused

The monograph presents a systematically written and organized text focused on understanding and the capability to apply the developed theory of the attractive ellipsoid method to real-world problems. The book will be useful to interested undergraduate and graduate students familiar with sufficient knowledge of modern systems theory as well as to professionals in control engineering and applied mathematics, and also to general readers interested in this emerging subject. (Lubomír Bakule, zbMATH 1314.93006, 2015)

1 Introduction 1(10)
1.1 Complete Information Case: Classical Control Approaches
1(4)
1.1.1 System Description
2(2)
1.1.2 Feasible and Admissible Control
4(1)
1.1.3 Problem Setting in the General Bolza Form
4(1)
1.1.4 Specific Features of Classical Optimal Control
5(1)
1.2 Case of Incomplete Information
5(3)
1.2.1 Robust Tracking Problem Formulation
5(2)
1.2.2 What Is the Effectiveness of a Designed Control in the Case of Incomplete Information9
7(1)
1.3 Ellipsoid-Based Feedback Control Design
8(1)
1.4 Overview of the Book
9(2)
2 Mathematical Background 11(36)
2.1 The Class of Nonlinear Uncertain Models
11(11)
2.1.1 Quasi-Lipschitz Dynamical Systems
11(3)
2.1.2 Examples of Quasi-Lipschitz Systems
14(2)
2.1.3 Differential Inclusions and General Solution Concept
16(3)
2.1.4 The Filippov Regularization Procedure
19(3)
2.2 The Lyapunov Approach to Quasi-Lipschitz Dynamical Systems
22(4)
2.3 Elements of LMIs
26(15)
2.3.1 Main Concepts
26(5)
2.3.2 Existence of Solutions of LMIs
31(7)
2.3.3 Numerical Approaches to LMIs
38(3)
2.4 S-Lemma and Some Useful Mathematical Facts
41(6)
3 Robust State Feedback Control 47(24)
3.1 Introduction
48(1)
3.2 Proportional Feedback Design
49(1)
3.2.1 Model Description
49(1)
3.2.2 Problem Formulation
50(1)
3.3 S -Procedure-Based Approach
50(3)
3.4 Storage Function Method
53(1)
3.5 Minimization of the Attractive Ellipsoid
54(2)
3.6 Practical Stabilization
56(2)
3.7 Other Restrictions on Control and Uncertainties
58(2)
3.8 Illustrative Example
60(3)
3.9 What to Do If We Don't Know the Matrix A?
63(5)
3.9.1 Description of the Dynamic Model in This Case
63(2)
3.9.2 Sufficient Conditions of Attractiveness
65(3)
3.9.3 Optimal Robust Linear Feedback as a Solution of an Optimization Problem with LMI Constraints
68(1)
3.10 Conclusions
68(3)
4 Robust Output Feedback Control 71(26)
4.1 Static Feedback Control
72(6)
4.1.1 System Description and Problem Statement
72(1)
4.1.2 Application of the Attractive Ellipsoids Method
73(3)
4.1.3 Example: Stabilization of a Discontinuous System
76(2)
4.2 Observer-Based Feedback Design
78(14)
4.2.1 State Observer and the Extended Dynamic Model
78(1)
4.2.2 Stabilizing Feedback Gains K and F
79(6)
4.2.3 Numerical Aspects
85(2)
4.2.4 Example: Robust Stabilization of a Spacecraft
87(5)
4.3 Dynamic Regulator
92(4)
4.3.1 Full-Order Linear Dynamic Controllers
92(1)
4.3.2 Main Result on the Attractive Ellipsoid for a Dynamic Controller
93(3)
4.4 Conclusions
96(1)
5 Control with Sample-Data Measurements 97(26)
5.1 Introduction and Motivation
98(1)
5.2 Problem Formulation and Some Preliminaries
99(2)
5.3 Linear Feedback Proportional to a State Estimate Vector
101(12)
5.3.1 Description in Extended Form
101(2)
5.3.2 Lyapunov-Like Analysis
103(7)
5.3.3 Numerical Aspects
110(3)
5.4 Full-Order Robust Linear Dynamic Controller
113(8)
5.4.1 The Structure of a Dynamic Controller
113(5)
5.4.2 The "Minimal-Size" Attractive Ellipsoid and LMI Constrained Optimization
118(2)
5.4.3 On Numerical Realization
120(1)
5.5 Conclusion
121(2)
6 Sample Data and Quantifying Output Control 123(24)
6.1 Introduction
123(2)
6.2 Problem Formulation
125(3)
6.3 A Lyapunov-Krasovskii Functional
128(6)
6.3.1 Main Result
133(1)
6.4 Numerical Aspects
134(6)
6.5 Numerical Examples
140(4)
6.5.1 Example 1
140(2)
6.5.2 Example 2
142(2)
6.6 Conclusions
144(3)
7 Robust Control of Implicit Systems 147(16)
7.1 Introduction
147(2)
7.2 Some Preliminaries
149(5)
7.2.1 Model Description
149(1)
7.2.2 Useful Concepts and Facts
150(1)
7.2.3 Transformation to Differential-Algebraic Form
151(2)
7.2.4 Problem Formulation
153(1)
7.3 Attractive Ellipsoid for Implicit Systems
154(6)
7.3.1 Descriptive Method Application
154(2)
7.3.2 Reduction of Nonlinear Matrix Inequalities to LMIs
156(4)
7.4 Concluding Remarks
160(3)
8 Attractive Ellipsoids in Sliding Mode Control 163(24)
8.1 Minimization of Unmatched Uncertainties Effect in Sliding Mode Control
164(12)
8.1.1 Problem Statement
164(2)
8.1.2 LMI-Based Sliding Mode Control Design
166(2)
8.1.3 Optimal Sliding Surface
168(4)
8.1.4 Numerical Aspects of Sliding Surface Design
172(2)
8.1.5 Numerical Example
174(2)
8.2 Gain Matrix Tuning in Dynamic Actuators
176(9)
8.2.1 Problem Statement
176(2)
8.2.2 Controller Design
178(4)
8.2.3 Example
182(3)
8.3 Conclusion
185(2)
9 Robust Stabilization of Time-Delay Systems 187(38)
9.1 Time-Delay Systems with Known Input Delay
187(20)
9.1.1 Brief Historical Remark
187(1)
9.1.2 System Description and Problem Statement
188(2)
9.1.3 Unavoidable Stabilization Error
190(1)
9.1.4 Minimal Invariant Ellipsoid for the Prediction System
191(6)
9.1.5 Minimal Attractive Ellipsoid of the Original System
197(5)
9.1.6 Computational Aspects
202(3)
9.1.7 Numerical Example
205(2)
9.2 Control of Systems with Unknown Input Delay
207(14)
9.2.1 Introduction
207(1)
9.2.2 Problem Statement
208(2)
9.2.3 Attractive Ellipsoid Method for Time-Delay Systems
210(1)
9.2.4 Predictor-Based Output Feedback Design
210(6)
9.2.5 Adjustment of Control Parameters: Computational Aspects
216(4)
9.2.6 Numerical Example
220(1)
9.3 Conclusion
221(4)
10 Robust Control of Switched Systems 225(42)
10.1 Introduction
226(6)
10.1.1 Some Preliminaries
227(1)
10.1.2 Problem Formulation
228(4)
10.2 Application of the Attractive Ellipsoid Method
232(19)
10.2.1 Practical Stability
233(5)
10.2.2 Intersection of Ellipsoids
238(6)
10.2.3 Bilinear Matrix Inequality Representation
244(3)
10.2.4 Simulation Results
247(4)
10.3 Switched Systems with Quantized and Sampled Output Feedback
251(14)
10.3.1 System Description
251(3)
10.3.2 Lyapunov-Krasovskii-Like Functional
254(3)
10.3.3 On Practical Stability
257(8)
10.4 Conclusions
265(2)
11 Bounded Robust Control 267(28)
11.1 Introduction
268(1)
11.2 The Class of Uncertain Nonlinear Systems and Problem Formulation
268(6)
11.2.1 System Description
268(2)
11.2.2 Basic Assumptions
270(2)
11.2.3 Extended Dynamic Form
272(1)
11.2.4 Problem Formulation
273(1)
11.3 Robust Bounded Output Control Synthesis
274(12)
11.3.1 Storage Function
274(3)
11.3.2 Zone-Convergence Analysis
277(6)
11.3.3 The Attractive Ellipsoid of "Minimal Size"
283(3)
11.4 Numerical Aspects
286(3)
11.4.1 Transformation of BMI Constraints into LMI Constraints
286(2)
11.4.2 Computational Aspects
288(1)
11.5 Illustrative Example
289(4)
11.5.1 Dynamic Model
289(2)
11.5.2 Numerical Results
291(1)
11.5.3 Simulation Results
292(1)
11.6 Conclusion
293(2)
12 Attractive Ellipsoid Method with Adaptation 295(44)
12.1 Introduction
296(1)
12.2 Attractive Ellipsoid Method with KL-Adaptation
297(21)
12.2.1 Basic Assumptions and Constraints
298(1)
12.2.2 System Description and Problem Formulation
298(1)
12.2.3 Main Assumptions
299(1)
12.2.4 Extended Quasilinear Format
300(1)
12.2.5 Problem Formulation
301(1)
12.2.6 Learning Laws, Storage Function Properties, and the "Minimal Size" Ellipsoid
301(4)
12.2.7 Attractive Ellipsoid for Robust Control with KL-Adaptation
305(3)
12.2.8 On the Attractive Ellipsoid in the State Space
308(2)
12.2.9 On the Effectiveness of the Adaptation Process
310(3)
12.2.10 On Transformation BMI Constraints into LMI Constraints
313(3)
12.2.11 Numerical Aspects
316(1)
12.2.12 Illustrative Example
316(2)
12.3 A-Adaptation in the Attractive Ellipsoid Method
318(18)
12.3.1 Quasilinear Model with Adjusted Feedback and Problem Formulation
320(1)
12.3.2 "A"-Adaptation
320(4)
12.3.3 Closed-Loop Representation and Storage Function
324(3)
12.3.4 Stability Analysis
327(5)
12.3.5 On the "Minimal Size" of the Attractive Ellipsoid
332(1)
12.3.6 Numerical Aspects
333(3)
12.4 Conclusion
336(3)
Bibliography 339(8)
Index 347
Alexander S. Poznyak graduated from Moscow Physical Technical Institute (MPhTI) in 1970. He earned PhD and Doctor Degrees from the Institute of Control Sciences of Russian Academy of Sciences in 1978 and 1989, respectively. He is now the head of the Automatic Control Department at CINVESTAV-IPN, and has published more than 120 papers in different international journals and 9 books. Andrey Polyakov is a researcher at NON-A team, Inria Lille-Nord Europe. He has graduated with a PhD in Physics and Mathematics from Voronezh State University, Russia in 2005. His areas of interest lie within control theory and differential equations. Vadim Azhmyakov is a Professor at University of Antonio Nariño, Colombia, in the Department of Electronic and Biomedical Engineering.
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