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E-raamat: Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays

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Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate DelaysSuurem pilt
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In this brief the authors establish a new frequency-sweeping framework to solve the complete stability problem for time-delay systems with commensurate delays. The text describes an analytic curve perspective which allows a deeper understanding of spectral properties focusing on the asymptotic behavior of the characteristic roots located on the imaginary axis as well as on properties invariant with respect to the delay parameters. This asymptotic behavior is shown to be related by another novel concept, the dual Puiseux series which helps make frequency-sweeping curves useful in the study of general time-delay systems. The comparison of Puiseux and dual Puiseux series leads to three important results:

  • an explicit function of the number of unstable roots simplifying analysis and design of time-delay systems so that to some degree they may be dealt with as finite-dimensional systems;
  • categorization of all time-delay systems into three types according to their ultimate stability properties; and
  • a simple frequency-sweeping criterion allowing asymptotic behavior analysis of critical imaginary roots for all positive critical delays by observation.

Academic researchers and graduate students interested in time-delay systems and practitioners working in a variety of fields – engineering, economics and the life sciences involving transfer of materials, energy or information which are inherently non-instantaneous, will find the results presented here useful in tackling some of the complicated problems posed by delays.

1 Introduction to Complete Stability of Time-Delay Systems
1(16)
1.1 Preliminaries and Prerequisites
1(7)
1.1.1 Basic Concepts
1(3)
1.1.2 Complete Stability Problem
4(2)
1.1.3 τ-Decomposition Idea
6(2)
1.1.4 Frequency-Sweeping Framework of this Book
8(1)
1.2 Detecting Critical Pairs and Frequency-Sweeping Curves
8(5)
1.2.1 Rewriting Characteristic Function
9(1)
1.2.2 Two Useful Indices and Related Remarks
10(1)
1.2.3 Frequency-Sweeping Curves
11(2)
1.3 Asymptotic Behavior of Critical Imaginary Roots
13(2)
1.3.1 Critical Imaginary Root at a Critical Delay
13(1)
1.3.2 Critical Imaginary Root with Infinitely Many Positive Critical Delays
14(1)
1.4 Book Structure
15(2)
2 Introduction to Analytic Curves
17(10)
2.1 Introductory Remarks to Singularities of Analytic Curves
18(2)
2.2 Puiseux Series
20(2)
2.3 Convergence of Puiseux Series
22(1)
2.4 Newton Diagram
22(2)
2.5 A Direct Application of Puiseux Series
24(1)
2.6 Notes and Comments
25(2)
3 Analytic Curve Perspective for Time-Delay Systems
27(8)
3.1 Further Focus on Asymptotic Behavior Analysis of Critical Imaginary Roots
27(6)
3.1.1 A Motivating Example
28(2)
3.1.2 Two-Variable Taylor Expansion
30(1)
3.1.3 Analytic Curves and Asymptotic Behavior of Critical Imaginary Roots
31(2)
3.2 Asymptotic Behavior Analysis of Frequency-Sweeping Curves
33(1)
3.3 Notes and Comments
34(1)
4 Computing Puiseux Series for a Critical Pair
35(12)
4.1 Why Puiseux Series Are a Necessary Tool
35(3)
4.1.1 Introductory Remarks
35(2)
4.1.2 Asymptotic Behavior Must Correspond to Puiseux Series
37(1)
4.2 How to Obtain Puiseux Series
38(5)
4.2.1 An Algorithm for General Case
38(1)
4.2.2 Illustrative Examples
39(4)
4.3 Studying Some Degenerate Cases
43(1)
4.4 Useful Properties for Puiseux Series
44(2)
4.4.1 Conjugacy Class
44(1)
4.4.2 Structure of Puiseux Series
45(1)
4.5 Notes and Comments
46(1)
5 In variance Property: A Unique Idea for Complete Stability Analysis
47(6)
5.1 Infinitesimal Delay Case and Spectral Properties
48(1)
5.2 Quantifying Asymptotic Behavior of a Critical Imaginary Root
49(1)
5.3 Stability Test for Bounded Delay
49(1)
5.4 Limitations
50(1)
5.4.1 Computational Complexity Issues
51(1)
5.4.2 Large Delays and Ultimate Stability Problem
51(1)
5.5 Invariance Property for Some Specific Delay Systems
51(1)
5.6 General Invariance Property Statement
52(1)
5.7 Notes and Comments
52(1)
6 Invariance Property for Critical Imaginary Roots with Index g = 1
53(10)
6.1 Preliminaries
53(1)
6.2 General Expression of Puiseux Series When g = 1
54(1)
6.3 Invariance Property When g = 1
55(3)
6.4 Simple Class of Quasipolynomials
58(1)
6.5 Illustrative Examples
58(4)
6.6 On Some Limitations (Lack of Analyticity)
62(1)
6.7 Notes and Comments
62(1)
7 Invariance Property for Critical Imaginary Roots with Index n = 1
63(10)
7.1 Preliminaries
63(1)
7.2 Embryo of New Frequency-Sweeping Framework
64(3)
7.2.1 Asymptotic Behavior of Simple Critical Imaginary Roots
64(1)
7.2.2 Some New Angles for Frequency-Sweeping Curves
65(2)
7.3 Equivalence Relation Between Two Types of Asymptotic Behavior
67(2)
7.3.1 Nondegenerate Case
67(1)
7.3.2 Degenerate Case
68(1)
7.4 Invariance Property for Simple Critical Imaginary Roots
69(1)
7.5 Illustrative Examples
70(2)
7.6 Notes and Comments
72(1)
8 A New Frequency-Sweeping Framework and Invariance Property in General Case
73(8)
8.1 Preliminaries
73(1)
8.2 Constructing a New Frequency-Sweeping Framework
74(3)
8.3 Proving General Invariance Property
77(1)
8.3.1 Critical Imaginary Roots with One Puiseux Series
77(1)
8.3.2 Critical Imaginary Roots with Multiple Puiseux Series
78(1)
8.4 Illustrative Examples
78(2)
8.5 Notes and Comments
80(1)
9 Complete Stability for Time-Delay Systems: A Unified Approach
81(10)
9.1 Ultimate Stability Property
81(4)
9.1.1 Characterizing Some Limit Cases
82(1)
9.1.2 Classification
83(1)
9.1.3 Delay-Independent Stability (Instability)
83(2)
9.2 A Unified Approach for Complete Stability
85(1)
9.3 Illustrative Examples
86(2)
9.4 Notes and Comments
88(3)
10 Extension to Neutral Time-Delay Systems
91(14)
10.1 Preliminaries
92(2)
10.1.1 Basic Concepts
92(1)
10.1.2 Subtleties of Neutral Time-Delay Systems
93(1)
10.2 Complete Stability Characterization
94(4)
10.2.1 Embedding Stability Condition of Neutral Operator
94(3)
10.2.2 Infinitesimal Delay Case
97(1)
10.2.3 General Invariance Property for Neutral Time-Delay Systems
97(1)
10.2.4 Ultimate Stability Property
97(1)
10.2.5 Frequency-Sweeping Framework: A Unified Approach
98(1)
10.3 Discussions on Neutral Systems with Multiple Delays
98(4)
10.3.1 Multiple Commensurate Delays
99(1)
10.3.2 Multiple Incommensurate Delays
100(2)
10.4 Illustrative Examples
102(2)
10.5 Notes and Comments
104(1)
11 Concluding Remarks and Further Perspectives
105(4)
11.1 Concluding Remarks
105(1)
11.2 Future Perspectives
106(3)
11.2.1 Extra Requirements on Spectra of Time-Delay Systems
106(1)
11.2.2 Design Problem
107(1)
11.2.3 Other Types of Time-Delay Systems
107(1)
11.2.4 Applying Parameter-Sweeping Techniques to Other Problems
108(1)
Appendix A Implicit Function Theorem 109(2)
Appendix B Proof of Theorem 8.3 (One Conjugacy Class) 111(10)
References 121(6)
Series Editors' Biography 127(2)
Index 129
Silviu-Iulian Niculescu was born in Petrosani, Romania in 1968. He received the B.S. degree from the Polytechnical Institute of Bucharest, Romania, the M.Sc. and Ph.D. degrees, both in Automatic Control, from the Institut National Polytechnique de Grenoble, France and the Habilitation à Diriger des Recherches (HDR) in Automatic Control from Université de Technologie de Compiègne, in 1992, 1993, 1996 and 2003, respectively. From 1992 to 1997, he was with the Department of Automatic Control and Computers, University Politehnica Bucharest, Romania. From 1997 to 2006, he was with HEUDIASYC (Heuristics and diagnosis of complex systems) laboratory, Compiègne, France as a Researcher at CNRS (French National Center for Scientific Research). He also held a Post Doctoral position in the Department of Applied Mathematics, ENSTA, Paris, France, from 1996 to 1997. In September 2006, he joined L2S (Laboratory of Signals and Systems), Gif-sur-Yvette, where he is currently Research Director (Senior Researcher) at CNRS and head of the laboratory. He is the single author of two books: delay systems. Qualitative aspects on the stability and stabilization (Diderot: Paris, 1997; in French) and Delay effects on stability. A robust control approach (Springer: Heidelberg, LNCIS, vol. 269, 2001; in English), co-author of one book (with Wim Michiels) entitled Stability and stabilization of time-delay systems. An eigenvalue-based approach (SIAM: Philadelphia, USA, 2007) and co-editor of five volumes (SIAM, 1999; Pergamon Press, 2001; Springer 2004; Springer 2007; Springer 2009). He is author or co-author of more than 300 book chapters or scientific papers. He has been the IPC Chairman of the 3rd IFAC Workshop on Time-Delay Systems (Santa Fe, NM, USA, December 2001) and of the 8th IFAC Workshop on Time-Delay Systems (Sinaia, Romania, September 2009) and the main organizer or co-organizer of the 1st CNRS-NSF Workshop on Time-Delay Systems (Paris: La Défense,January 2003) and of the 23rd European Summer School in Automatic Control (Grenoble, September 2000) devoted to time-delay systems. He was the guest co-editor of five special issues in the area of delay systems (Journal of Mathematical Modelling of Natural Phenomena (MMNP) in 2009 Asian Journal of Control in 2005, IMA Journal of Mathematical Control and Information in 2002 and 2010 and Kybernetika in 2001). Dr. Niculescu has been scientific responsible of 14 international cooperation programs and projects (with Belgium, Mexico, Romania and Eastern European countries, South Korea, Turkey and United States). He is member of IFAC Technical Committee on Linear Systems (since 2002) and of the IPC of 33 International Conferences and he was an Associate Editor of the IEEE Transactions on Automatic Control (2003-2005). Since 2011, He is an Associate Editor of European Journal of Control and IMA Journal of Mathematical Control and Information. He is the responsible of the IFAC Research Group on Time-delay systems since its creation in October 2007. Dr. Niculescu was awarded the CNRS Bronze Medal for scientific research, the Best Paper Presentation Award at American Control Conference, Chicago, IL and the Ph.D. Thesis Award from INPG, Grenoble (France) in 2001, 2000, and 1996, respectively. His research interests include delay systems, robust control, operator theory and numerical methods in optimization and their applications to the design of engineering systems.
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