OUTPUT FEEDBACK CONTROL OF NONLINEAR SYSTEMS WITH UNSTABILIZABLE/UNDETECTABLE LINEARIZATION by BO YANG Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Advisor: Dr. Wei Lin Department of Electrical Engineering and Computer Science CASE WESTERN RESERVE UNIVERSITY January, 2006 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of ______________________________________________________ candidate for the Ph.D. degree *. (signed)_______________________________________________ (chair of the committee) ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ (date) _______________________ *We also certify that written approval has been obtained for any proprietary material contained therein. Copyright c 2006 by Bo Yang All rights reserved To my wife and my parents with love and gratitude TABLE OF CONTENTS 1. Introduction ................................ 1 1.1 LiteratureReview............................. 2 1.2 Motivations ................................ 5 1.3 ContributionsofThisDissertation. ... 7 2. Preliminaries ................................ 11 2.1 StabilityTheory.............................. 11 2.2 HomogeneousSystemsTheory . 19 2.2.1 Homogeneous Functions and Vector Fields . 19 2.2.2 Stability of Homogeneous Systems . 21 2.3 UsefulInequalities ............................ 24 3. Smooth Output Feedback Design ................... 26 3.1 Introduction................................ 26 3.2 Output Feedback Design of Linear Systems — A Non-Separation Prin- cipleParadigm .............................. 30 3.3 Smooth Output Feedback Stabilization of Homogeneous Systems. 35 3.4 Global Stabilization of a Class of Non-Homogeneous Systems via Out- putFeedback ............................... 41 3.5 ExampleandDiscussion . .. .. 48 3.6 Summary ................................. 56 4. Robust Control of Uncertain Systems by Smooth Output Feedback 58 iv 4.1 Introduction................................ 58 4.2 Robust Output Feedback Design: the Case of p =1.......... 62 4.3 Robust Output Feedback Design: the p-NormalFormCase . 67 4.4 Output Feedback Stabilization of Uncertain Cascade Systems..... 79 4.5 Summary ................................. 87 5. Global Robust Stabilization by Nonsmooth Output Feedback . 89 5.1 Introduction................................ 89 5.2 Robust Output Feedback Design: A Case Revisit . .. 93 5.3 Nonsmooth Output Feedback Stabilization of Uncertain Nonlinear Sys- tems .................................... 97 5.4 ExtensionandDiscussion. 104 5.5 Summary ................................. 107 6. Global Output Feedback Control with Dynamical Rescaling .... 109 6.1 MainResultandDiscussion . 109 6.2 StateFeedbackDesign . .. .. 113 6.2.1 Dynamic Rescaling of the Original System . 113 6.2.2 StateFeedbackController . 116 6.3 OutputFeedbackDesign . 122 6.3.1 Reduced-OrderObserver . 122 6.3.2 Error Dynamics and Output Feedback Controller . 124 6.3.3 ObserverGainAssignment . 125 6.4 StabilityAnalysis ............................. 130 6.5 Output Feedback Stabilization of Cascade Systems . ..... 131 6.6 Summary ................................. 135 7. Semi-Global Output Feedback Stabilization of Non-Uniformly Ob- servable and Nonsmoothly Stabilizable Systems ........... 137 v 7.1 Introduction................................ 137 7.2 KeyTechnicalLemmas . 141 7.3 ASimplerParadigm ........................... 142 7.4 ProofofMainResult ........................... 149 7.5 Summary ................................. 156 8. Conclusion ................................... 157 8.1 Summary ................................. 157 8.2 FutureWork................................ 158 vi LIST OF FIGURES 3.1 Transient Responses of the closed-loop system (3.57)-(3.59) with x1(0) = 5, x2(0) = −3, xˆ1(0) = 3, xˆ2(0) = −5. .................. 51 3.2 Trajectories of the closed-loop system (3.60)-(3.61) with x1(0) = −2, x2(0) = 5, x3(0) = −3, xˆ2(0) = 3, xˆ3(0) = −5. .................. 52 4.1 Transient responses of the system (4.51)-(4.52) from x1(0) = 2, x2(0) = −10, zˆ2(0) = −15.............................. 78 4.2 Transient responses of the closed-loop system (4.66)-(4.71) with θ =1 and the initial condition (ζ, η1, η2, zˆ2)=(1, 0.3, −6, −5) ........ 88 7.1 The level set Ωx on x-space and the saturation threshold M. ..... 144 7.2 The level set Ω on (x, zˆ)-space. ..................... 148 vii Acknowledgment I would like to first acknowledge my profound debt of gratitude to my advisor Dr. Wei Lin. I could not have asked for a more encouraging and caring mentor than him. Throughout these years, he has suggested many promising projects that I worked on in a very productive way. Without his constructive direction and invaluable advice, this work would not have been completed. I want to extend my gratitude to Dr. Kenneth A. Loparo, Dr. Marc R. Buchner and Dr. M. Cenk Cavusoglu for being on my advisory committee and reviewing my dissertation. I also wish to express my appreciation to Dr. Chunjiang Qian of the University of Texas at San Antonio, Dr. Qi Gong of Naval Postgraduate School, Dr. Radom Pongvuthithum of Chiang Mai University, and Dr. Xianqing Huang for their generous help on either a research level or a personal level. My thanks are also given to our research group members: Jianfeng Wei and Hao Lei with whom I share the research interests and friendship. Hao also helped me a lot complete paperwork at Case and make my defense possible when I was visiting visit Texas Tech University during the semester of Fall 2005. Same thanks to my officemates: Arsit Boonyaprapasorn, Jinbae Choi, Chunrong Dong, Yiming Huang, Ying Wang, and our secretary Marla Radvansky and student affairs coordinator Eliz- abethanne Fuller-Murray for their kind assistance. This career goal of mine would have never materialized had it not been for the sacrifices and support of my family. Words cannot adequately acknowledge their undying support. Finally, special thanks to my wife, Bei, for her unfailingly support and stood behind me in all my endeavors. viii Output Feedback Control of Nonlinear Systems with Unstabilizable/Undetectable Linearization Abstract by BO YANG This dissertation addresses a number of fundamental and challenging output feed- back control problems for a significant class of uncertain nonlinear systems with un- stabilizable/undetectable linearization, including: (1) global asymptotic stabilization via smooth output feedback under a high-order version of Lipschitz-like condition; (2) robust control by smooth or nonsmooth output feedback with dynamic rescal- ing, under appropriate yet restrictive growth conditions; (3) semi-global asymptotic stabilization of non-uniformly observable and nonsmoothly stabilizable systems by nonsmooth output feedback, without imposing any growth condition. First of all, we study the problem of global stabilization by smooth output feed- back, for a class of n-dimensional systems whose Jacobian linearization is neither stabilizable nor detectable. A novel output feedback control scheme is proposed for the explicit design of both high-order observers and controllers in a recursive manner. Its design philosophy is substantially different from that of the traditional “Luen- berger” observer in which the observer gain is determined by observability. Using the new output feedback design method, we then consider the problem of robust smooth or nonsmooth output feedback stabilization for several families of un- certain nonlinear systems with unstabilizable/undetectable linearization. To handle system uncertainties effectively, we introduce a novel rescaling transformation with an appropriate dilation and factor. Depending on the type of growth conditions, the rescaling factor can be either a sufficiently large constant or a time-varying function ix x that needs to be tuned on-line through a Riccati-like differential equation. The con- structions of smooth or nonsmooth state feedback controllers and observers use only the knowledge of the bounding system rather than the uncertain system itself. The robust output feedback design approach thus developed is also extended to uncertain cascade systems beyond a strict-triangular structure. In the last part of this thesis, we show that without imposing any growth condition, semi-global stabilization by nonsmooth output feedback can be achieved for a chain of odd power integrators perturbed by a triangular vector field, which is in general not smoothly stabilizable nor uniformly observable. Chapter 1. INTRODUCTION One of the most important problems in control theory is to make a controlled plant stable at its equilibrium point by state or output feedback in the presence of system uncertainties. The origin of its modern theory dates back to the year of 1868, when J.C. Maxwell analyzed the stability of J. Watt’s flyball governor. One and a half centuries later, extensive research in linear systems revealed the celebrated concepts such as controllability, stabilizability, observability, and detectability and their pow- erful applications. It has been known that for a linear system, a state or output feedback controller can be easily designed as long as the system is stabilizable and detectable. In the past decade, research has been focused on extending the linear output feedback control theory to nonlinear systems. Due to the nature of nonlinear-