University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 12-2006 The Stability Analysis of Linear Dynamical Systems with Time- Delays Ajeet Ganesh Kamath University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Mechanical Engineering Commons Recommended Citation Kamath, Ajeet Ganesh, "The Stability Analysis of Linear Dynamical Systems with Time-Delays. " PhD diss., University of Tennessee, 2006. https://trace.tennessee.edu/utk_graddiss/1985 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Ajeet Ganesh Kamath entitled "The Stability Analysis of Linear Dynamical Systems with Time-Delays." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mechanical Engineering. VijaySekhar Chellaboina, Major Professor We have read this dissertation and recommend its acceptance: William R. Hamel, Dongjun Lee, Seddik M. Djouadi Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a dissertation written by Ajeet Ganesh Kamath entitled \The Stability Analysis of Linear Dynamical Systems with Time-Delays". I have examined the ¯nal electronic copy of this dissertation for form and content and rec- ommend that it be accepted in partial ful¯llment of the requirements for the degree of Doctor of Philosophy, with a major in Mechanical Engineering. VijaySekhar Chellaboina Major Professor We have read this dissertation and recommend its acceptance: William R. Hamel Dongjun Lee Seddik M. Djouadi Accepted for the Council: Linda Painter Interim Dean of Graduate Studies (Original signatures are on ¯le with o±cial student records.) The Stability Analysis of Linear Dynamical Systems with Time-Delays A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Ajeet Ganesh Kamath December 2006 Copyright °c 2006 by Ajeet Ganesh Kamath. All rights reserved. ii Dedication This dissertation is dedicated to the members of my family, who nurtured me, endured me and humored me all my life. My father, Ganesh Kamath, my mother Suman Kamath and my brother, Manju (Prashant) Kamath, thank you for all your love, support, guidance and prayers. To my ¯anc¶ee,Aparna Kher, your presence makes me strong. iii Acknowledgments First, and foremost, I am deeply indebted to my advisor, Dr. VijaySekhar Chellaboina for his guidance and advice throughout the course of my graduate education. This work is a result of his encouragement, support, ideas and constructive criticism. His willingness to support me and his guidance during my studies and research has allowed me to mature and develop as a researcher and as a person. I would also like to thank Dr. Wassim Haddad at GeorgiaTech for his support in the work that went into this research. I am also thankful to the members of my doctoral dissertation committee, Dr. Bill Hamel, Dr. Seddik Djouadi and Dr. Dongjun Lee, for their suggestions and input in this dissertation. I am also deeply thankful for all the advice and support from Dr. Gary Smith during my time at the University of Tennessee. Also, many thanks to Dr. Satish Nair and Dr. Carmen Chicone at the University of Missouri, who served on my M.S. thesis committee and with whom I took many courses. I would also like to thank all the members of my family, my father Ganesh Kamath, my mother Suman Kamath and my brother Manju (Prashant) Kamath for their love, guidance and encouragement, which have played a de¯ning role in my life. Many thanks to my ¯anc¶eeand my best friend, Aparna Kher, for being the shining light in my life, and for her support and fortitude. I look forward to spending the rest of my living years with her. In our research group, I am grateful to my colleagues and friends Jayanthy Ra- makrishnan and Alex Melin for their help and input, and for putting up with me over the years. Also deserving a special mention are the other present and past members of the research group, Hancao Li, Mithun Ranga and Sushma Kalavagunta. To all my teachers, present and past, I am deeply indebted to you for the role you have played in taking me to where I stand today. A big \thank you" to all of you. If I have omitted your names, it is only for a lack of space, and most de¯nitely not for a lack of appreciation for your e®orts. iv Abstract Time-delay systems, which are also sometimes known as hereditary systems or sys- tems with memory, aftere®ects or time-lag, represent a class of in¯nite-dimensional systems, and are used to describe, among other types of systems, propagation and transport phenomena, population dynamics, economic systems, communication net- works and neural network models. A key method for the stability analysis of time- delay dynamical systems is the Lyapunov's second method, applied to functional di®erential equations. Speci¯cally, stability of a given linear time-delay dynamical system is typically shown using a Lyapunov-Krasovskii functional, which involves a quadratic part and an integral part. The quadratic part is usually associated with the stability of the forward delay-independent part of the retarded dynamical system, but the integral part of the functional is less understood. We present a concrete method of arriving at the Lyapunov-Krasovskii functional using dissipativity theory. The stability analysis of time-delay systems has been mainly classi¯ed into two cate- gories: delay-dependent and delay-independent analysis. Delay-independent stability criteria provide su±cient conditions for stability of time-delay dynamical systems in- dependent of the amount of delay, whereas delay-dependent stability criteria provide su±cient conditions that are dependent on an upper bound of the delay. In systems where the time delay is known to be bounded, delay-dependent criteria usually give far less conservative stability predictions as compared to delay-independent results. Hence, for such systems it is of paramount importance to derive the sharpest possible delay-dependent stability margins. We show how the stability criteria may also be in- terpreted in the frequency domain in terms of a feedback interconnection of a matrix transfer function and a phase uncertainty block. We develop and present the general framework for a robust stability analysis method to account for phase uncertainties in linear systems. We present new robust stability results for time-delay systems based on pure phase information, and then, using this approach, we derive new and improved time-domain delay-dependent stability criteria for stability analysis of both retarded and neutral type time-delay systems, which we then compare with existing results in the literature. v Contents 1 Introduction 1 1.1. Motivation and Historical Overview . 1 1.2. Applications and Examples of Time-Delay Systems . 2 1.2.1. Transport and Communication Delays . 3 1.2.2. Biology and Population Dynamics . 4 1.2.3. Propagation Phenomena . 4 1.3. The Problem Statement . 5 1.4. Outline of the Dissertation . 8 1.4.1. Dissipativity Approach to Time-Delay Systems . 8 1.4.2. Dynamic Dissipativity Theory . 8 1.4.3. Structured Phase Margin . 9 1.4.4. Neutral Delay Systems . 9 1.4.5. Conclusions and Future Work . 9 2 A Dissipative Dynamical Systems Approach to the Stability Analysis of Time-Delay Systems 10 2.1. Introduction . 10 2.2. Mathematical Preliminaries . 11 2.3. Stability Theory for Continuous-Time Time-Delay Dynamical Systems using Dissipativity Theory . 14 2.4. Stability Theory for Discrete-Time Time-Delay Dynamical Systems us- ing Dissipativity Theory . 18 2.5. Conclusion . 24 3 Stability Analysis of Time Delay Systems using Dynamic Dissipa- tivity Theory 26 3.1. Introduction . 26 3.2. Mathematical Preliminaries . 27 3.3. Stability Theory for Time-Delay Dynamical Systems using Dissipativ- ity Theory . 33 3.4. Illustrative Numerical Example . 40 3.5. Conclusion . 40 vi 4 Structured Phase Margin for Stability Analysis of Time-Delay Sys- tems 42 4.1. Introduction . 42 4.2. Mathematical Preliminaries . 47 4.3. The Structured Phase Margin of a Complex Matrix . 50 4.4. A Computable Lower Bound for the Structured Phase Margin . 52 4.5. Connections between the Structured Phase Margin and the Structured Singular Value . 56 4.6. Stability of Linear Dynamical Systems with Structured Phase Uncer- tainties . 59 4.7. Stability Theory for Time Delay Dynamical Systems . 61 4.8. Time-Domain Conditions for Stability Analysis of Time-Delay Systems 67 4.9. Illustrative Numerical Examples . 70 4.10. Conclusion . 74 5 Su±cient Conditions for Stability of Neutral Time-Delay Systems using the Structured Phase Margin 75 5.1. Introduction . 75 5.2. Frequency-Domain Stability Conditions for Neutral Time-Delay Dy- namical Systems . 77 5.3. Time-Domain Test for Stability Analysis of Linear Neutral Time-Delay Systems . 81 5.4. Illustrative Numerical Examples . 82 6 Conclusions and Future Research 85 6.1. Contributions . 85 References 88 Vita 94 vii List of Tables 5.1 Maximum allowable delay prediction for Example 5.4.2 . 84 viii List of Figures 1.1 Metal strip with two Lp cells (three capacitive cells dashed) . 4 1.2 Small PEEC model for metal strip .