DOCSLIB.ORG

Output Feedback Control of Nonlinear Systems with Unstabilizable/Undetectable Linearization

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Output Feedback Control of Nonlinear Systems with Unstabilizable/Undetectable Linearization 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-
Load more

Recommended publications
  • What Is a Complex Adaptive System?
    PROJECT GUTS What is a Complex Adaptive System? Introduction During the last three decades a leap has been made from the application of computing to help scientists ‘do’ science to the integration of computer science concepts, tools and theorems into the very fabric of science. The modeling of complex adaptive systems (CAS) is an example of such an integration of computer science into the very fabric of science; models of complex systems are used to understand, predict and prevent the most daunting problems we face today; issues such as climate change, loss of biodiversity, energy consumption and virulent disease affect us all. The study of complex adaptive systems, has come to be seen as a scientific frontier, and an increasing ability to interact systematically with highly complex systems that transcend separate disciplines will have a profound affect on future science, engineering and industry as well as in the management of our planet’s resources (Emmott et al., 2006). The name itself, “complex adaptive systems” conjures up images of complicated ideas that might be too difficult for a novice to understand. Instead, the study of CAS does exactly the opposite; it creates a unified method of studying disparate systems that elucidates the processes by which they operate. A complex system is simply a system in which many independent elements or agents interact, leading to emergent outcomes that are often difficult (or impossible) to predict simply by looking at the individual interactions. The “complex” part of CAS refers in fact to the vast interconnectedness of these systems. Using the principles of CAS to study these topics as related disciplines that can be better understood through the application of models, rather than a disparate collection of facts can strengthen learners’ understanding of these topics and prepare them to understand other systems by applying similar methods of analysis (Emmott et al., 2006).
    [Show full text]
  • Environmental Influences: Family Systems Theory
    Environmental Influences: Family Systems Theory Family Systems Theory provides a broad and comprehensive mechanism for understanding the core aspects of the Performance Competence Lifespan Framework — quality of life, member- ship, and a personal sense of competence. It also focuses on the most important component of environmental influences—home and family. From birth, a child’s Quality of Life is directly influ- enced by the kind of care, support, stimulation and education he or she receives from family mem- bers in the home. As infants begin to develop secure attachments with significant others, particu- larly family members, they begin to establish themselves as members of the first and most basic unit of society—the family, which forms the foundation for secure Membership in other groups throughout life. The infant begins to develop a Personal Sense of Competence when his mother responds consistently to his distress, when he takes his first step or says his first word, or when his father praises him for using the toilet. These early beginnings, then, are at the core of what each individual child will come to know and be able to do. As the PC Framework indicates, there are multiple environmental influences on performance and competence, but the family is the first and most important. The influence of family members on one another is not simple, but complex; it is not one-way, but reciprocal. The family, like a mechanical system, is made up of multiple parts that are interdependent. When one part does not function well, all other parts are impacted. Further, the family interacts with other systems, includ- ing those that provide direct services to the child—child care/preschools, schools and community agencies—and each system affects the other.
    [Show full text]
  • Robustness of Multiloop Linear Feedback Systems
    WA1 - 10:45 ROBUSTNESS OF MJLTILOOP LINEAR FEEDBACK SYSTEMS* J. C. Doyle Consultant Honeywell Systems and Research Center 2600 Ridgway Parkway Minneapolis, Minnesota 55413 ABSTRACT critical role in evaluating system robustness. The Bode plot or scalar Nyquist or Inverse Nyquist This paper presents new a approach to the diagram (polar plots of the loop transfer function) frequency-domain analysis of multiloop linear feed-provides a meansof assessing these quantities at back systems. The properties of the return a glance. For multiloop systems, individual loops difference equation are examined using the conceptsmay be analyzed using scalar techniques, but these of singular values, singular vectors and the methods fail to capture the essentially multi- spectral norm of a matrix.A number of new tools variable nature of manysystem. For example, for multiloop systems are developed which are scalar methods may ignore variations which analogous to those for scalar Nyquist and Bode simultaneously affect multiple loops. analysis. These provide a generalization of the scalar frequency-domain notions such as gain, There are a number of other possible ways bandwidth, stability margins and M-circles, and to extend the classical frequency-domain tech- provide considerable insight into system robust- niques. One involves using compensation or feed- ness. back to decouple(or approximately decouple) a multiloop system into a set of scalar systems which may be treated w#th scalar techniques (i.e., 1. Introduction "Diagonal Dominance", bsenbrock[4]). Another A critical property of feedback systems is method uses the eigenvaluesof the loop transfer their robustness; that is, their ability to main-matrix (G(s) in Figure 1) as a function of fre- tain performance in the face of uncertainties.
    [Show full text]
  • Emergence in Psychology: Lessons from the History of Non-Reductionist Science
    Paper Human Development 2002;45:2-28 Human Development Emergence in Psychology: Lessons from the History of Non-Reductionist Science R. Keith Sawyer Washington University, St.louis, Mo., USA KeyWords Cognitive science. Emergence. History. Reductionism. Socioculturalism Abstract Theories of emergence have had a longstanding influence on psychological thought. Emergentism rejects both reductionism and holism; emergentists are scientific materialists, and yet argue that reductionist explanation may not always be scientifically feasible. I begin by summarizing the history of emergence in psy- chology and sociology, from the mid-19th century through the mid-20th century. I then demonstrate several parallels between this history and contemporary psy- chology, focusing on two recent psychological movements: socioculturalism and connectionist cognitive science. Placed in historical context, both sociocultural ism and connectionism are seen to be revivals of 19th and early 20th century emergen- tism. I then draw on this history to identify several unresolved issues facing socio- culturalists and connectionists, and to suggest several promising paths for future theory. Copyright @ 2002 S. Karger AG, Base! 1. Introduction Emergentism is a form of materialism which holds that some complex natural phe- nomena cannot be studied using reductionist methods. My first goal in this paper is to identify the historical roots of two emergentist paradigms in contemporary psychology: sociocultural psychology and cognitive science. My second goal is to draw on this histo- ry to explore several unresolved issues facing these contemporary paradigms. Sociocul- tural psychologists are sociological emergentists; they hold that group behavior is consti- KARG E R <92002 S. Karger AG, Basel R. Keith Sawyer, Washington University .OOI8-716X/02/0451-0()()2$18.50/0 Department of Education, Campus Box 1183 Fax+41613061234 St.
    [Show full text]
  • An Analysis of Power and Stress Using Cybernetic Epistemology Randall Robert Lyle Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1992 An analysis of power and stress using cybernetic epistemology Randall Robert Lyle Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Educational Psychology Commons, Family, Life Course, and Society Commons, and the Philosophy Commons Recommended Citation Lyle, Randall Robert, "An analysis of power and stress using cybernetic epistemology " (1992). Retrospective Theses and Dissertations. 10131. https://lib.dr.iastate.edu/rtd/10131 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely afreet reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from lefr to right in equal sections with small overlaps.
    [Show full text]
  • Adaptation of Elaborated Feedback in E-Learning
    Adaptation of Elaborated Feedback in e-Learning Ekaterina Vasilyeva1,2, Mykola Pechenizkiy2, and Paul De Bra2 1 Department of Computer Science and Information Systems, University of Jyväskylä, P.O. Box 35, 40351 Jyväskylä, Finland 2 Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, the Netherlands {e.vasilyeva,m.pechenizkiy}@tue.nl, [email protected] Abstract. Design of feedback is a critical issue of online assessment develop- ment within Web-based Learning Systems (WBLSs). In our work we demon- strate the possibilities of tailoring the feedback to the students’ learning style (LS), certitude in response and its correctness. We observe in the experimental studies that these factors have a significant influence on the feedback prefer- ences of students and the effectiveness of elaborated feedback (EF), i.e. stu- dents’ performance improvement during the test. These observations helped us to develop a simple EF recommendation approach. Our experimental study shows that (1) many students are eager to follow the recommendations on ne- cessity to read certain EF in the majority of cases; (2) the students more often find the recommended EF to be useful, and (3) the recommended EF helped to answer related questions better. Keywords: feedback authoring, feedback personalization, online assessment. 1 Introduction Online assessment is an important component of modern education. Nowadays it is used not only in e-learning for (self-)evaluation, but also tends to replace or comple- ment traditional methods of formal evaluation of the student’s performance in blended learning. Feedback is usually a significant part of the assessment as students need to be in- formed about the results of their (current and/or overall) performance.
    [Show full text]
  • What Is a Complex System?
    What is a Complex System? James Ladyman, James Lambert Department of Philosophy, University of Bristol, U.K. Karoline Wiesner Department of Mathematics and Centre for Complexity Sciences, University of Bristol, U.K. (Dated: March 8, 2012) Complex systems research is becoming ever more important in both the natural and social sciences. It is commonly implied that there is such a thing as a complex system, different examples of which are studied across many disciplines. However, there is no concise definition of a complex system, let alone a definition on which all scientists agree. We review various attempts to characterize a complex system, and consider a core set of features that are widely associated with complex systems in the literature and by those in the field. We argue that some of these features are neither necessary nor sufficient for complexity, and that some of them are too vague or confused to be of any analytical use. In order to bring mathematical rigour to the issue we then review some standard measures of complexity from the scientific literature, and offer a taxonomy for them, before arguing that the one that best captures the qualitative notion of the order produced by complex systems is that of the Statistical Complexity. Finally, we offer our own list of necessary conditions as a characterization of complexity. These conditions are qualitative and may not be jointly sufficient for complexity. We close with some suggestions for future work. I. INTRODUCTION The idea of complexity is sometimes said to be part of a new unifying framework for science, and a revolution in our understanding of systems the behaviour of which has proved difficult to predict and control thus far, such as the human brain and the world economy.
    [Show full text]
  • Lecture 13: Feedback Linearization1
    Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 13: Feedback Linearization1 Using control authority to transform nonlinear models into linear ones is one of the most commonly used ideas of practical nonlinear control design. Generally, the trick helps one to recognize “simple” nonlinear feedback design tasks. 13.1 Motivation and objectives In this section, we give a motivating example and state technical objectives of theory of feedback linearization. 13.1.1 Example: fully actuated mechanical systems Equations of rather general mechanical systems can be written in the form M (q(t))¨q(t) + F (q(t); q˙(t)) = u(t); (13.1) where q(t) 2 Rk is the position vector, u(t) is the vector of actuation forces and torques, F : Rk £ Rk 7! Rk is a given vector-valued function, and M : Rk 7! Rk£k is a given function taking positive definite symmetric matrix values (the inertia matrix). When u = u(t) is fixed (for example, when u(t) = u0 cos(t) is a harmonic excitation), analysis of (13.1) is usually an extremely difficult task. However, when u(t) is an unrestricted control effort to be chosen, a simple change of control variable u(t) = M (q(t))(v(t) + F (q(t); q˙(t))) (13.2) transforms (13.1) into a linear double integrator model q¨(t) = v(t): (13.3) 1Version of October 29, 2003 2 The transformation from (13.1) to (13.3) is a typical example of feedback linearization, which uses a strong control authority to simplify system equations.
    [Show full text]
  • Complex Adaptive Team Systems (CATS)
    Presented by: John R. Turner, Ph.D. [email protected] Complex Adaptive Team Rose Baker, Ph.D. Systems (CATS) [email protected] Kerry Romine, MS [email protected] University of North Texas College of Information Department of Learning Technologies Systems Open –vs– Closed Systems Systems Theory Complex Adaptive Team System (CATS) Complex Adaptive Systems Complexity Theory Open –vs– Closed Systems Open Systems Closed Systems Linear Non‐Linear Predictable Non‐Predictable Presents Simple Problems Presents Complex & Wicked Problems Systems theory Input – Process – Output Changes in one part of the system effects other parts of the system. The system is the sum of the parts. Changes are generally predictable. Complex Adaptive Systems “Interdependent agents that interact, learn from each other, and adapt their behaviors accordingly.” (Beck & Plowman, 2014, p. 1246) The building block for higher level agents or systems while continuously adapting to environmental changes. (Bovaird, 2008) CAS ‐ Characteristics • Non‐Linearity • Contains many constituents interacting non‐linearly. • Open System • Open in which boundaries permit interaction with environment. • Feedback Loops • Contains Feedback loops. • Scalable • Structure spanning several scales. • Emergence • Emergent behavior leads to new state. Complexity Theory “Targets a sub‐set of all systems; a sub‐set which is abundant and is the basis of all novelty; a sub‐set from which structure emerges…. That is, self‐ organization occurs through the dynamics, interactions and feedback of
    [Show full text]
  • Complex Adaptive Systems
    Evidence scan: Complex adaptive systems August 2010 Identify Innovate Demonstrate Encourage Contents Key messages 3 1. Scope 4 2. Concepts 6 3. Sectors outside of healthcare 10 4. Healthcare 13 5. Practical examples 18 6. Usefulness and lessons learnt 24 References 28 Health Foundation evidence scans provide information to help those involved in improving the quality of healthcare understand what research is available on particular topics. Evidence scans provide a rapid collation of empirical research about a topic relevant to the Health Foundation's work. Although all of the evidence is sourced and compiled systematically, they are not systematic reviews. They do not seek to summarise theoretical literature or to explore in any depth the concepts covered by the scan or those arising from it. This evidence scan was prepared by The Evidence Centre on behalf of the Health Foundation. © 2010 The Health Foundation Previously published as Research scan: Complex adaptive systems Key messages Complex adaptive systems thinking is an approach that challenges simple cause and effect assumptions, and instead sees healthcare and other systems as a dynamic process. One where the interactions and relationships of different components simultaneously affect and are shaped by the system. This research scan collates more than 100 articles The scan suggests that a complex adaptive systems about complex adaptive systems thinking in approach has something to offer when thinking healthcare and other sectors. The purpose is to about leadership and organisational development provide a synopsis of evidence to help inform in healthcare, not least of which because it may discussions and to help identify if there is need for challenge taken for granted assumptions and further research or development in this area.
    [Show full text]
  • Scalability, Resilience, and Complexity Management in Laminar Control of Ultra-Large Scale Systems
    Scalability, Resilience, and Complexity Management in Laminar Control of Ultra-Large Scale Systems Jeffrey Taft, PhD Paul De Martini Connected Energy Networks Resnick Institute Cisco Systems, Inc. California Institute of Technology Abstract Ultra-large scale control systems that are based on layered optimization decomposition and Network Utility Maximization (NUM) have structural properties and operational modes that givesuch control systems useful scalabilityand anti-fragile properties. We denote control architectures based on these principles as Laminar Control and we suggest that the structure so defined has properties such as layer-by-layer segmentation of control signal traffic, abstraction of local state into scalar signal flows, self-similarity of data flow patterns at each layer, and support for islanded operation and re-connection. While the underlying physical networks (such as power systems) and associated communication networks are not necessarily self-similar, the imposition of the Laminar Control paradigm allows the creation such a structure at the application node level. Consequently, instead of having a set of agents with randomly structured logical connections, the optimization nodes become hubs and furthermore,it may be possible to restrict peer-to-peer communication at any level in the hierarchy to delimited specifiable domains. State determination can be similarly partitioned so that domain state need not be shared globally. In combination, these properties can invest distributed control networks with three valuable characteristics: scalability of control system real time network communications, resilience of the logical control network, and complexity bounds. Introduction: Laminar Control Architecture and Data Flows Distributed and hierarchical control methods have been available for decades. In the case of electric power systems, some of these concepts have been used in portions of the grid, but not as an Ultra-Large Scale (ULS) control.
    [Show full text]
  • Feedback Systems: Notes on Linear Systems Theory
    Feedback Systems: Notes on Linear Systems Theory Richard M. Murray Control and Dynamical Systems California Institute of Technology DRAFT – Fall 2019 October 31, 2019 These notes are a supplement for the second edition of Feedback Systems by Astr¨om˚ and Murray (referred to as FBS2e), focused on providing some additional mathematical background and theory for the study of linear systems. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. You are free to copy and redistribute the material and to remix, transform, and build upon the material for any purpose. 2 Contents 1 Signals and Systems 5 1.1 Linear Spaces and Mappings . 5 1.2 Input/OutputDynamicalSystems . 8 1.3 Linear Systems and Transfer Functions . 12 1.4 SystemNorms ....................................... 13 1.5 Exercises .......................................... 15 2 Linear Input/Output Systems 19 2.1 MatrixExponential..................................... 19 2.2 Convolution Equation . 20 2.3 LinearSystemSubspaces ................................. 21 2.4 Input/outputstability ................................... 24 2.5 Time-VaryingSystems ................................... 25 2.6 Exercises .......................................... 27 3 Reachability and Stabilization 31 3.1 ConceptsandDefinitions ................................. 31 3.2 Reachability for Linear State Space Systems . 32 3.3 SystemNorms ....................................... 36 3.4 Stabilization via Linear Feedback . 37 3.5 Exercises .........................................
    [Show full text]