DOCSLIB.ORG

Dynamic Ecological System Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dynamic Ecological System Analysis 1 DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 2 HUSEYIN COSKUN∗ 3 A Holistic Analysis of Compartmental Systems 4 Abstract. In this article, a new mathematical method for the dynamic analysis of nonlinear 5 compartmental systems is developed in the context of ecology. The method is based on the novel 6 dynamic system and subsystem partitioning methodologies through which compartmental systems 7 are decomposed to the utmost level. The dynamic system and subsystem partitioning enable the 8 determination of the distribution of environmental inputs and intercompartmental system flows as 9 well as the organization of the associated storages generated by these inputs and flows individu- 10 ally and separately within the system. Moreover, the transient and the dynamic direct, indirect, 11 acyclic, cycling, and transfer (diact) flows and associated storages transmitted along a given flow 12 path or from one compartment, directly or indirectly, to any other are analytically characterized, 13 systematically classified, and mathematically formulated. Major flow- and stock-related concepts 14 and quantities of the current static network analyses are also extended to nonlinear dynamic set- 15 tings and integrated with the proposed dynamic measures and indices within the proposed unifying 16 mathematical framework. This comprehensive methodology enables a holistic view and analysis of ecological systems.17 18 Key words. dynamic ecological network analysis, nonlinear dynamic compartmental systems, 19 dynamic system and subsystem partitioning, transient flows and storages, diact flows and storages 20 AMS subject classifications. 34A34, 70G60, 37N25, 92C42, 92D40 21 1. Introduction. Compartmental systems are mathematical abstractions of net- 22 works that approximate behaviors of continuous physical systems composed of discrete 23 living and nonliving homogeneous components. Based on conservation principles, the 24 system compartments are interconnected through the flow of energy, matter, or cur- 25 rency between them and their environment. Therefore, for the quantification of the 26 compartmental system function, accurate and explicit formulation of flows and as- 27 sociated storages are of paramount importance. Various mathematical aspects of 28 compartmental systems are studied in literature [22,2]. While many fields utilize 29 compartmental modeling, this approach proves particularly well-suited for analysis of 30 ecological systems to address environmental phenomena. 31 Environmental issues have taken center stage in human communities due to cur- 32 rent scientific understandings of population and industrial growth, associated resource 33 demands, and technological advances. Despite this increased attention to the envi- 34 ronment, traditional ecology has an applied nature and is still in the empirical stage 35 of development; a first principles-based formal theory has yet to emerge in its main- 36 stream framework. This narrows the field’s scope of applicability and compromises 37 its ability to deal with complex organism-environment relationships. To that extent, 38 ecology and environmental science are limited in their applied reach by a general in- 39 ability to realistically model and analyze the complex systems of man and nature. 40 Mathematical theories and modeling have significant potential to lead the way to a 41 more formalistic and theoretical ecological science devoted to the discovery of scientific 42 laws. Based on this understanding and prediction, more exact, precise, and incisive 43 environmental applications can be expected to materialize. 44 Sound rationales have been offered in literature for ecological network analysis, 45 but these are for specific cases, such as linear and static models. One such environ- 46 mental system theory known as the environ theory, has been developed over recent ∗Department of Mathematics, University of Georgia, Athens, GA 30602 ([email protected]). 1 This manuscript is for review purposes only. 2 HUSEYIN COSKUN 47 decades for static compartmental models. Building on economic input-output analysis 48 [26, 27] introduced into ecology by [17], the concepts of static flow and storage envi- 49 rons are formulated based on conservation principles [29, 28]. Along parallel research 50 lines, ecological networks and complexity in living systems are also analyzed in the 51 context of information theory and thermodynamics [33, 21, 34, 35] and hierarchy the- 52 ory [1], yet only for static systems. Several software developments computerize these 53 static methods [36,7, 13, 23, 30,5]. 54 Although the steady-state analysis is well-established, dynamic analysis of nonlin- 55 ear systems has remained a long-standing, open problem. For example, Finn's cycling 56 index defined in static network ecology over four decades ago, has still not been made 57 applicable to ecosystem models that change over time [14]. There are earlier dynamic 58 formulations in the literature, but they are essentially designed for the analysis of 59 special cases, such as time dependent linear systems [20] or closed-form abstract for- 60 mulations [16]. There are also computational methods, such as discrete time [31] and 61 individual based techniques [25, 24] that rely on network particle tracking simulations. 62 Today's major environmental problems{human impact, climate change, biodiver- 63 sity loss, etc.{ all involve change, and this makes the need for mathematically dynamic 64 and analytical methods of nonlinear system analysis not only appropriate, but also 65 urgent [6, 19]. In ecosystem ecology, food webs provide a framework to link com- 66 munity structure with flows of energy and material through trophic interactions and, 67 therefore, reconcile biodiversity with ecosystem function. Temporal variation in web 68 architecture and nonlinearity are discussed in the literature, and it is proposed that 69 this dynamic nature is affecting ecosystem attributes [12, 37]. Nonlinearity and dy- 70 namic behavior including extinction in food webs, however, has yet to be addressed 71 methodologically. 72 This is the first manuscript in literature that potentially addresses the mismatch 73 between the current static and computational methods and applied ecological needs. 74 The proposed methodology is a comprehensive approach in the sense that the pro- 75 posed dynamic measures as well as the major flow- and stock-related results of static 76 ecological network analyses are combined and integrated effectively within this novel 77 and unifying mathematical framework. More importantly, the corresponding con- 78 cepts and quantities of ecological network analyses are further extended from static 79 to nonlinear dynamic settings. Therefore, this unifying approach leads to a holistic 80 analysis of ecosystems. The proposed methodology, in effect, brings a novel, formal, 81 deterministic, complex system theory to the service of urgent ecological problems of 82 the day. 83 The proposed method in line with the mathematical theory introduced by [9] is 84 based on the novel dynamic system and subsystem partitioning methodologies. The 85 system partitioning methodology yields the subthroughflow and substorage matrices 86 that represent the flows and storages generated by individual environmental inputs 87 at each compartment separately. Therefore, the system partitioning enables dynami- 88 cally partitioning composite compartmental flows and storages into subcompartmen- 89 tal segments based on their constituent environmental sources. In other words, it 90 enables dynamically tracking the evolution of environmental inputs and associated 91 storages individually and separately within the system. Through the subsystem par- 92 titioning methodology, then, the transient flows and associated storages transmitted 93 along a given subflow path are formulated. Consequently, arbitrary composite inter- 94 compartmental flows and associated storages are dynamically decomposed into the 95 constituent transient subflow and substorage segments along a given set of subflow 96 paths. Therefore, the subsystem partitioning enables dynamically tracking the fate This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 3 97 of arbitrary intercompartmental flows and associated storages within the subsystems. 98 Based on the concept of the transient flow and storage, the dynamic direct, indirect, 99 acyclic, cycling, and transfer (diact) flows and associated storages transmitted from 100 one compartment, directly or indirectly, to any other are also analytically character- 101 ized, systematically classified, and mathematically formulated for the quantification 102 of intercompartmental flow and storage dynamics. 103 In a nutshell, the system and subsystem partitioning methodologies dynamically 104 determine the distribution of environmental inputs and intercompartmental flows as 105 well as the organization of the associated storages generated by these flows indi- 106 vidually and separately within the system. For such system analysis, the dynamic 107 subthroughflows and substorages as well as the transient and diact flows and stor- 108 ages are systematically introduced in the present paper for the first time in literature. 109 Equipped with these measures, the proposed methodology serves as a quantitative 110 platform for testing empirical hypotheses, ecological inferences, and, potentially, the- 111 oretical developments. The method also constructs a base for the development of new 112 dynamic system measures and indices as ecological indicators. Multiple such quanti- 113 tative tools for
Load more

Recommended publications
  • Coping with the Bounds: a Neo-Clausewitzean Primer / Thomas J
    About the CCRP The Command and Control Research Program (CCRP) has the mission of improving DoD’s understanding of the national security implications of the Information Age. Focusing upon improving both the state of the art and the state of the practice of command and control, the CCRP helps DoD take full advantage of the opportunities afforded by emerging technologies. The CCRP pursues a broad program of research and analysis in information superiority, information operations, command and control theory, and associated operational concepts that enable us to leverage shared awareness to improve the effectiveness and efficiency of assigned missions. An important aspect of the CCRP program is its ability to serve as a bridge between the operational, technical, analytical, and educational communities. The CCRP provides leadership for the command and control research community by: • articulating critical research issues; • working to strengthen command and control research infrastructure; • sponsoring a series of workshops and symposia; • serving as a clearing house for command and control related research funding; and • disseminating outreach initiatives that include the CCRP Publication Series. This is a continuation in the series of publications produced by the Center for Advanced Concepts and Technology (ACT), which was created as a “skunk works” with funding provided by the CCRP under the auspices of the Assistant Secretary of Defense (NII). This program has demonstrated the importance of having a research program focused on the national security implications of the Information Age. It develops the theoretical foundations to provide DoD with information superiority and highlights the importance of active outreach and dissemination initiatives designed to acquaint senior military personnel and civilians with these emerging issues.
    [Show full text]
  • Observation of Nonlinear Verticum-Type Systems Applied to Ecological Monitoring
    2nd Reading June 4, 2012 13:53 WSPC S1793-5245 242-IJB 1250051 International Journal of Biomathematics Vol. 5, No. 6 (November 2012) 1250051 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S1793524512500519 OBSERVATION OF NONLINEAR VERTICUM-TYPE SYSTEMS APPLIED TO ECOLOGICAL MONITORING S. MOLNAR´ Institute of Mathematics and Informatics Szent Istv´an University P´ater K. u. 1., H-2103 Godollo, Hungary [email protected] M. GAMEZ´ ∗ and I. LOPEZ´ † Department of Statistics and Applied Mathematics University of Almer´ıa La Ca˜nada de San Urbano 04120 Almer´ıa, Spain ∗[email protected] †[email protected] Received 4 November 2011 Accepted 13 January 2012 Published 7 June 2012 In this paper the concept of a nonlinear verticum-type observation system is introduced. These systems are composed from several “subsystems” connected sequentially in a particular way: a part of the state variables of each “subsystem” also appears in the next “subsystem” as an “exogenous variable” which can also be interpreted as a con- trol generated by an “exosystem”. Therefore, these “subsystems” are not observation systems, but formally can be considered as control-observation systems. The problem of observability of such systems can be reduced to rank conditions on the “subsystems”. Indeed, under the condition of Lyapunov stability of an equilibrium of the “large”, verticum-type system, it is shown that the Kalman rank condition on the linearization of the “subsystems” implies the observability of the original, nonlinear verticum-type system. For an illustration of the above linearization result, a stage-structured fishery model with reserve area is considered.
    [Show full text]
  • Nonlinear System Stability and Behavioral Analysis for Effective
    S S symmetry Article Nonlinear System Stability and Behavioral Analysis for Effective Implementation of Artificial Lower Limb Susmita Das 1 , Dalia Nandi 2, Biswarup Neogi 3 and Biswajit Sarkar 4,* 1 Narula Institute of Technology, Kolkata 700109, India; [email protected] 2 Indian Institute of Information Technology Kalyani, Kalyani 741235, India; [email protected] 3 JIS College of Engineering., Kalyani 741235, India; [email protected] 4 Department of Industrial Engineering, Yonsei University, Seoul 03722, Korea * Correspondence: [email protected] or [email protected] Received: 8 September 2020; Accepted: 14 October 2020; Published: 19 October 2020 Abstract: System performance and efficiency depends on the stability criteria. The lower limb prosthetic model design requires some prerequisites such as hardware design functionality and compatibility of the building block materials. Effective implementation of mathematical model simulation symmetry towards the achievement of hardware design is the focus of the present work. Different postures of lower limb have been considered in this paper to be analyzed for artificial system design of lower limb movement. The generated polynomial equations of the sitting and standing positions of the normal limb are represented with overall system transfer function. The behavioral analysis of the lower limb model shows the nonlinear nature. The Euler-Lagrange method is utilized to describe the nonlinearity in the field of forward dynamics of the artificial system. The stability factor through phase portrait analysis is checked with respect to nonlinear system characteristics of the lower limb. The asymptotic stability has been achieved utilizing the most applicable Lyapunov method for nonlinear systems.
    [Show full text]
  • Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems
    UNIVERSITY of CALIFORNIA Santa Barbara Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering by Ryan M. Mohr Committee in charge: Professor Igor Mezi´c,Chair Professor Bassam Bamieh Professor Jeffrey Moehlis Professor Mihai Putinar June 2014 The dissertation of Ryan M. Mohr is approved: Bassam Bamieh Jeffrey Moehlis Mihai Putinar Igor Mezi´c,Committee Chair May 2014 Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems Copyright c 2014 by Ryan M. Mohr iii To my family and dear friends iv Acknowledgements Above all, I would like to thank my family, my parents Beth and Jim, and brothers Brennan and Gralan. Without their support, encouragement, confidence and love, I would not be the person I am today. I am also grateful to my advisor, Professor Igor Mezi´c,who introduced and guided me through the field of dynamical systems and encouraged my mathematical pursuits, even when I fell down the (many) proverbial (mathematical) rabbit holes. I learned many fascinating things from these wandering forays on a wide range of topics, both contributing to my dissertation and not. Our many interesting discussions on math- ematics, research, and philosophy are among the highlights of my graduate school career. His confidence in my abilities and research has been a constant source of encouragement, especially when I felt uncertain in them myself. His enthusiasm for science and deep, meaningful work has set the tone for the rest of my career and taught me the level of research problems I should consider and the quality I should strive for.
    [Show full text]
  • Rethinking the International System As a Complex Adaptive System
    Munich Personal RePEc Archive A New Taxonomy for International Relations: Rethinking the International System as a Complex Adaptive System Scartozzi, Cesare M. University of Tokyo, Graduate School of Public Policy 2018 Online at https://mpra.ub.uni-muenchen.de/95496/ MPRA Paper No. 95496, posted 12 Aug 2019 10:50 UTC 1SFQVCMJDBUJPOJournal on Policy and Complex Systems • Volume 4, Number 1 • Spring 2018 A New Taxonomy for International Relations: Rethinking the International System as a Complex Adaptive System Cesare Scartozzi Graduate School of Public Policy, University of Tokyo [email protected] 7-3-1, Hongo, Bunkyo, Tokyo 113-0033, Japan ORCID number: 0000-0002-4350-4386. Abstract The international system is a complex adaptive system with emer- gent properties and dynamics of self-organization and informa- tion processing. As such, it is better understood with a multidis- ciplinary approach that borrows methodologies from the field of complexity science and integrates them to the theoretical perspec- tives offered by the field of international relations (IR). This study is set to formalize a complex systems theory approach to the study of international affairs and introduce a new taxonomy for IR with the two-pronged aim of improving interoperability between differ- ent epistemological communities and outlining a formal grammar that set the basis for modeling international politics as a complex adaptive system. Keywords: international politics, international relations theory, complex systems theory, taxonomy, adaptation, fitness, self-orga- nization This is a prepublication version of: Scartozzi, Cesare M. “A New Taxonomy for International Relations: Rethinking the International System as a Complex Adaptive System.” Journal on Policy and Complex Systems 4, no.
    [Show full text]
  • Chapter 8 Nonlinear Systems
    Chapter 8 Nonlinear systems 8.1 Linearization, critical points, and equilibria Note: 1 lecture, §6.1–§6.2 in [EP], §9.2–§9.3 in [BD] Except for a few brief detours in chapter 1, we considered mostly linear equations. Linear equations suffice in many applications, but in reality most phenomena require nonlinear equations. Nonlinear equations, however, are notoriously more difficult to understand than linear ones, and many strange new phenomena appear when we allow our equations to be nonlinear. Not to worry, we did not waste all this time studying linear equations. Nonlinear equations can often be approximated by linear ones if we only need a solution “locally,” for example, only for a short period of time, or only for certain parameters. Understanding linear equations can also give us qualitative understanding about a more general nonlinear problem. The idea is similar to what you did in calculus in trying to approximate a function by a line with the right slope. In § 2.4 we looked at the pendulum of length L. The goal was to solve for the angle θ(t) as a function of the time t. The equation for the setup is the nonlinear equation L g θ�� + sinθ=0. θ L Instead of solving this equation, we solved the rather easier linear equation g θ�� + θ=0. L While the solution to the linear equation is not exactly what we were looking for, it is rather close to the original, as long as the angleθ is small and the time period involved is short. You might ask: Why don’t we just solve the nonlinear problem? Well, it might be very difficult, impractical, or impossible to solve analytically, depending on the equation in question.
    [Show full text]
  • NONLINEARITIES, FEEDBACKS and CRITICAL THRESHOLDS WITHIN the EARTH's CLIMATE SYSTEM 1. Introduction Nonlinear Phenomena Charac
    NONLINEARITIES, FEEDBACKS AND CRITICAL THRESHOLDS WITHIN THE EARTH’S CLIMATE SYSTEM JOSÉ A. RIAL 1,ROGERA.PIELKESR.2, MARTIN BENISTON 3, MARTIN CLAUSSEN 4, JOSEP CANADELL 5, PETER COX 6, HERMANN HELD 4, NATHALIE DE NOBLET-DUCOUDRÉ 7, RONALD PRINN 8, JAMES F. REYNOLDS 9 and JOSÉ D. SALAS 10 1Wave Propagation Laboratory, Department of Geological Sciences CB#3315, University of North Carolina, Chapel Hill, NC 27599-3315, U.S.A. E-mail: [email protected] 2Atmospheric Science Dept., Colorado State University, Fort Collins, CO 80523, U.S.A. 3Dept. of Geosciences, Geography, Univ. of Fribourg, Pérolles, Ch-1700 Fribourg, Switzerland 4Potsdam Institute for Climate Impact Research, Telegrafenberg C4, 14473 Potsdam, P.O. Box 601203, Potsdam, Germany 5GCP-IPO, Earth Observation Centre, CSIRO, GPO Box 3023, Canberra, ACT 2601, Australia 6Met Office Hadley Centre, London Road, Bracknell, Berkshire RG12 2SY, U.K. 7DSM/LSCE, Laboratoire des Sciences du Climat et de l’Environnement, Unité mixte de Recherche CEA-CNRS, Bat. 709 Orme des Merisiers, 91191 Gif-sur-Yvette, France 8Dept. of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, U.S.A. 9Department of Biology and Nicholas School of the Environmental and Earth Sciences, Phytotron Bldg., Science Dr., Box 90340, Duke University, Durham, NC 27708, U.S.A. 10Dept. of Civil Engineering, Colorado State University, Fort Collins, CO 80523, U.S.A. Abstract. The Earth’s climate system is highly nonlinear: inputs and outputs are not proportional, change is often episodic and abrupt, rather than slow and gradual, and multiple equilibria are the norm.
    [Show full text]
  • Modeling Vascular Homeostasis and Improving Data Filtering Methods for Model Calibration
    Modeling Vascular Homeostasis and Improving Data Filtering Methods for Model Calibration by Jiacheng Wu A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering − Mechanical Engineering and the Designated Emphasis in Computational Science and Engineering in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Shawn C. Shadden, Chair Professor Grace D. O'Connell Professor Peter L. Bartlett Spring 2019 Modeling Vascular Homeostasis and Improving Data Filtering Methods for Model Calibration Copyright 2019 by Jiacheng Wu 1 Abstract Modeling Vascular Homeostasis and Improving Data Filtering Methods for Model Calibration by Jiacheng Wu Doctor of Philosophy in Engineering − Mechanical Engineering and the Designated Emphasis in Computational Science and Engineering University of California, Berkeley Professor Shawn C. Shadden, Chair Vascular homeostasis is the preferred state that blood vessels try to maintain against external mechanical and chemical stimuli. The vascular adaptive behavior around the homeostatic state is closely related to cardiovascular disease progressions such as arterial aneurysms. In this work, we develop a multi-physics computational framework that couples vascular growth & remodeling (G&R), wall mechanics and hemodynamics to describe the overall vascular adaptive behavior. The coupled simulation is implemented in patient-specific geometries to predict aneurysm progression. Lyapunov stability analysis of the governing
    [Show full text]
  • Nonlinear Control Systems
    1. Introduction to Nonlinear Systems Nonlinear Control Systems Ant´onioPedro Aguiar [email protected] 1. Introduction to Nonlinear Systems IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7Epedro/NCS2012/ 2011/2012 1 1. Introduction to Nonlinear Systems Objective The main goal of this course is to provide to the students a solid background in analysis and design of nonlinear control systems Why analysis? (and not only simulation) • Every day computers are becoming more and more powerful to simulate complex systems • Simulation combined with good intuition can provide useful insight into system's behavior Nevertheless • It is not feasible to rely only on simulations when trying to obtain guarantees of stability and performance of nonlinear systems, since crucial cases may be missed • Analysis tools provide the means to obtain formal mathematical proofs (certificates) about the system's behavior • results may be surprising, i.e, something we had not thought to simulate. 2 1. Introduction to Nonlinear Systems Why study nonlinear systems? Nonlinear versus linear systems • Huge body of work in analysis and control of linear systems • most models currently available are linear (but most real systems are nonlinear...) However • dynamics of linear systems are not rich enough to describe many commonly observed phenomena 3 1. Introduction to Nonlinear Systems Examples of essentially nonlinear phenomena • Finite escape time, i.e, the state can go to infinity in finite time (while this is impossible to happen for linear systems) • Multiple isolated equilibria, while linear systems can only have one isolated equilibrium point, that is, one steady state operating point • Limit cycles (oscillation of fixed amplitude and frequency, irrespective of the initial state) • Subharmonic, harmonic or almost-periodic oscillations; • A stable linear system under a periodic input produces an output of the same frequency.
    [Show full text]
  • Application of Symmetry and Symmetry Analyses to Systems of First-Order Equations Arising from Mathematical Modelling in Epidemiology
    Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 1, 159–169 Application of Symmetry and Symmetry Analyses to Systems of First-Order Equations Arising from Mathematical Modelling in Epidemiology P.G.L. LEACH † and K. ANDRIOPOULOS ‡ † School of Mathematical and Statistical Sciences, University of Natal, Durban 4041, South Africa E-mail: [email protected] ‡ Department of Mathematics, National and Capodistrian University of Athens, Panepistimioupolis, Ilisia, 15771 Athens, Greece E-mail: [email protected] We examine the integrability, in terms of the Painlev´e singularity analysis and of the Lie symmetry analysis, of systems of nonlinear first-order ordinary differential equations that arise in the particular area of Mathematical Modelling known as Rational Epidemiology. These analyses are presented as being complementary to the standard analysis using the methods of Dynamical Systems. The importance to obtain complete understanding of evo- lution of epidemics – one need think only of the potential for devastation by HIV-AIDS in African countries or the more recent threat posed by SARS – demands that all possi- ble approaches of analysis be used. The concept of decomposed systems is introduced as illustrating some rather attractive mathematical properties of certain classes of systems of equations arising in Mathematical Modelling. Such systems allow the mathematician some opportunity to enjoy Mathematics even in the context of quite strict applications. 1 Introduction There are four standard approaches to the analysis of a system of nonlinear first-order ordi- nary differential equations of the type which arise frequently in Mathematical Modelling. The approaches comprise numerical computation, dynamical systems, singularity analysis and sym- metry analysis, all of which possess extensive literatures.
    [Show full text]
  • Nonlinear Control - an Overview
    1 NONLINEAR CONTROL - AN OVERVIEW Fernando Lobo Pereira, fl[email protected] Porto University, Portugal Nonlinear Control, 2008, FEUP References 1 M. Vidyasagar, Nonlinear Systems Analysis, Prentice-Hall, Englewood Cli®s, NJ, 1978. 2 A. Isidori, Nonlinear Control Systems, Springer-Verlag, 199X. 3 E. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer-Verlag, 1998. 4 C. T. Chen, Introduction to Linear Systems Theory, Van Nostrand Reinhold Company, New York, NY, 1970. 5 C. A. Desoer, Notes for a Second Course on Linear Systems, Holt Rhinehart, Winston, 1970. 6 K. Ogata, State Space Analysis of Control Systems, Prentice-Hall, Englewood Cli®s, NJ, 1965. 7 J. M. Carvalho, Dynamical Systems and Automatic Control, Prentice-Hall, New York, NY, 1994. Fernando Lobo Pereira, fl[email protected] 1 Nonlinear Control, 2008, FEUP Introduction Problem description - system components, objectives and control issues. Example 1: telescope mirrors control PICTURE Most of real systems involve nonlinearities in one way or another. However, many concepts for linear systems play an important role since some of the techniques to deal with nonlinear systems are: ² Change of variable in phase space so that the resulting system is linear. ² Nonlinearities may be cancelled by picking up appropriate feedback control laws. ² Pervasive analogy ... Fernando Lobo Pereira, fl[email protected] 2 Nonlinear Control, 2008, FEUP Introduction Example 2: Pendulum Modeling PICTURE Dynamics (Newton's law for rotating objects): mθÄ(t) + mgsin(θ(t)) = u(t) State variables: θ; θ_ Control/Input variable: u (external applied torque) Let us assume m = 1; l = 1; g = 1. Fernando Lobo Pereira, fl[email protected] 3 Nonlinear Control, 2008, FEUP Introduction Control Design ½ (0; 0) stable equilibrium Stationary positions: (1) (¼; 0) unstable equilibrium.
    [Show full text]
  • Viewpointviewpoint Complexity in Biology
    viewpointviewpoint Complexity in biology Exceeding the limits of reductionism and determinism using complexity theory Fulvio Mazzocchi he ultimate aim of scientific research scientific knowledge has to provide an objec- due course, reductionism proved to be an is to understand the natural world. In tive representation of the external world. The extremely powerful analytical methodology Torder to achieve this goal, Western world’s apparent complexity can be resolved and it enabled scientists to analyse many science has relied on different cognitive by analysis and reducing phenomena to basic molecular and cellular processes. strategies, including simplification, both in their simplest components. “Once you have terms of analysis and explanation. As the done that, [the evolution of phenomena] Complex systems exist at British natural philosopher Sir Isaac Newton will turn out to be perfectly regular, revers- (1643–1727) put it, “Truth is ever to be ible and predictable, while the knowledge different levels of organization found in the simplicity, and not in the multi- you gained will merely be a reflection of that that range from the subatomic plicity and confusion of things.” In a way, pre-existing order” (Heylighen et al, 2007). realm to individual organisms to examples of simplification include using Ever since Newton formulated the first whole populations and beyond idealized models, such as a ‘perfect sphere laws of gravity, the conceptual model of the rolling down a smooth plane in a vacuum’; physical world had successfully described conducting experiments in a strictly con- the shape, movements and actions of the Nonetheless, biologists might be reaching trolled environment such as the laboratory; objects within it.
    [Show full text]