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Optimal Economic Planning and Control for the Management of Ecosystems Kevin Macksamie

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Optimal Economic Planning and Control for the Management of Ecosystems Kevin Macksamie Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 12-1-2011 Optimal economic planning and control for the management of ecosystems Kevin Macksamie Follow this and additional works at: http://scholarworks.rit.edu/theses Recommended Citation Macksamie, Kevin, "Optimal economic planning and control for the management of ecosystems" (2011). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Optimal Economic Planning and Control for the Management of Ecosystems by Kevin John Macksamie A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Engineering Supervised by Associate Professor Dr. Juan Cockburn Department of Computer Engineering Kate Gleason College of Engineering Rochester Institute of Technology Rochester, New York December 2011 Approved by: Dr. Juan Cockburn, Associate Professor Thesis Advisor, Department of Computer Engineering Dr. Jeffrey Wagner, Associate Professor Committee Member, Department of Economics Dr. Roy Melton, Senior Lecturer Committee Member, Department of Computer Engineering Thesis Release Permission Form Rochester Institute of Technology Kate Gleason College of Engineering Title: Optimal Economic Planning and Control for the Management of Ecosystems I, Kevin John Macksamie, hereby grant permission to the Wallace Memorial Library to reproduce my thesis in whole or part. Kevin John Macksamie Date iii Dedication To my family and friends. iv Acknowledgments I would like to thank my advisor Dr. Cockburn for his support, encouragement and mentorship. I also would like to thank my two committee members Dr. Wagner and Dr. Melton. Dr. Wagner for his tremendous patience and enthusiasm about the thesis research, and Dr. Melton for his patience and willingness to be a part of my thesis committee. In addition to my thesis advisor and committee, there are many professors, which I have had the pleasure of studying under, that have revealed my affinity for control and mathematics. I would like to acknowledge and thank the following professors at RIT. Dr. Hopkins for talking with me and sparking my interest in control systems midway through my academic career. Dr. Mestha for his exceptional teaching of modern control theory and for taking time to talk with me after class about advanced topics. Dr. Mathew for his teaching of optimal control and advice. Dr. Das for introducing me to the world of nonlinear control theory. Dr. Brooks for his generosity and for his incredible instruction on dynamical systems. Finally, Dr. Barth-Hart for his encouragement and inspiration. From outside of RIT, I would like to thank Dr. Rao and his research group at the University of Florida for conversing with me about GPOPS. Additionally, Idaho's Department of Fish and Game for sending me data on the hunting of wolves and elk. Lastly, I sincerely thank all my family and friends for their encouragement through- out my schooling. v Abstract Optimal Economic Planning and Control for the Management of Ecosystems Kevin John Macksamie Supervising Professor: Dr. Juan Cockburn In recent years the interest on sustainable systems has increased significantly. Among the many interested problems, creating and restoring sustainable ecosystems is a chal- lenging and complex problem. One of the fundamental problems within this area is the imbalance between species that have a predator-prey relationship. Solutions in- volving management have become an integral player in many environments. Manage- ment systems typically use ad hoc methods to develop harvesting policies to control the populations of species to desired numbers. In order to amalgamate intelligence and structure, ecological systems require a diverse research effort from three primary fields: ecology, economics, and control theory. In this thesis, all three primary fields aforementioned are researched to develop a theoretical framework that includes an optimal trajectory planning system that exploits an ecosystem to maximize profits for the supporting community, and a ro- bust control system design to track the optimal trajectories subjected to exogenous disturbances. Population ecology is used to select a model that identifies the key characteristics a management system needs to understand the behavior of the natu- ral environment. A bioeconomic model is developed to relate the species populations to revenue. The nonlinear ecosystem is transformed into a linear parameter-varying (LPV) system that is then controlled using H1 synthesis and the gain scheduling methodology. The consequences of the results in this thesis are that optimal trajectories of an ecosystem can be obtained by constructing and solving a nonlinear programming problem (NLP), and the LPV based gain scheduling approach produces a robust con- troller that rejects disturbances and advises quality control policies to the manager an ecosystem. The LPV controller achieves comparable profits with satisfactory tracking performance while minding the induced costs of its high frequency output. Implica- tions of constraining the control effort when designing for robustness are observed. Overall, the theoretical framework provides a solid foundation for future research on the understanding and improvement of ecosystem management. vi Contents Dedication ::::::::::::::::::::::::::::::::::::: iii Acknowledgments :::::::::::::::::::::::::::::::: iv Abstract :::::::::::::::::::::::::::::::::::::: v Glossary :::::::::::::::::::::::::::::::::::::: xi 1 Introduction :::::::::::::::::::::::::::::::::: 1 1.1 Background . .1 1.1.1 Wildlife Management . .1 1.1.2 Feedback Control . .7 1.2 Research Goals . .9 1.2.1 Control Theory Over Alternatives . 10 1.3 Thesis Organization . 13 2 Supporting Work ::::::::::::::::::::::::::::::: 15 2.1 Population Ecology . 16 2.1.1 Population Structures . 16 2.1.2 Predator-Prey Systems . 19 2.1.3 Predator-Prey with Harvesting Dynamics . 20 2.2 Bioeconomics . 22 2.2.1 Optimal Harvest Policies . 23 2.2.2 Literature Survey . 25 2.3 Robust Control Theory . 29 2.3.1 The Robust Control Problem . 30 2.3.2 The H1 Norm . 33 2.3.3 H1 Synthesis . 34 2.4 Linear Parameter-Varying Systems . 37 2.4.1 Formulation . 38 2.4.2 LPV Synthesis . 41 vii 2.4.3 Remarks . 42 3 Bioeconomic Model ::::::::::::::::::::::::::::: 44 3.1 Ecological Model . 44 3.1.1 Lyapunov Stability Analysis . 47 3.1.2 Predator-Prey-Hunter Model . 50 3.2 Economic Model . 50 3.3 Optimal Harvesting Policies . 52 3.3.1 GPOPS . 55 3.4 Case Study . 56 3.4.1 Scenarios . 57 3.5 Discussion . 65 4 LPV Control ::::::::::::::::::::::::::::::::: 66 4.1 Control Oriented Model . 66 4.2 LPV Control System Design . 67 4.2.1 LPV Description . 68 4.2.2 Stability . 69 4.2.3 Scheduling . 70 4.3 Controller Synthesis . 71 4.4 Numerical Results . 72 4.4.1 Quadratic Stability . 73 4.4.2 Weighting Filters . 73 4.4.3 Simulations . 75 4.5 Cost of LPV Control . 85 4.6 Discussion . 87 5 Conclusions :::::::::::::::::::::::::::::::::: 89 5.1 Research Summary . 89 5.2 Future Work . 90 Bibliography ::::::::::::::::::::::::::::::::::: 92 A Optimal Control Theory Fundamentals ::::::::::::::::: 98 A.1 Pontryagin's Maximum Principle . 101 viii List of Tables 3.1 Nominal parameter values for simulations . 56 3.2 Optimal profits for each numerically solved scenario . 65 4.1 Robust performance γ .......................... 75 4.2 Aggressive LPV profits and costs for the decade-long management plan 87 4.3 Relaxed LPV profits and costs for the decade-long management plan 87 ix List of Figures 1.1 Soda Butte Creek . .5 1.2 Feedback loop . .8 1.3 Negative feedback loop of herbivore population . .8 1.4 Positive feedback loop of herbivore population . .9 1.5 Research focus . 10 2.1 Holling's functional responses . 18 2.2 The behavior of the Volterra-Lotka predator-prey model . 19 2.3 Predator-prey with constant harvesting dynamics . 21 2.4 Predator-prey with proportional harvesting dynamics . 22 2.5 Feedback form of the plant G and controller K ............. 30 2.6 Robust control interconnect with uncertainty . 32 2.7 Robust analysis setup . 32 2.8 General system block diagram . 33 2.9 Robust control block diagram . 34 2.10 LPV generalized system with the plant G(ρ) and controller K(ρ)... 38 2.11 Robust LPV control interconnect with uncertainty . 39 3.1 Scenario A simulation results . 62 3.2 Scenario B simulation results . 63 3.3 Scenario C simulation results . 64 4.1 Convex hull of the parameter set . 68 4.2 Robust control interconnect . 71 4.3 The self-scheduled LPV controller controls the nonlinear system . 72 4.4 Disturbances over a decade . 77 4.5 Exogenous disturbances and sensor noise . 78 4.6 Scenario A LPV (aggressive) simulation results . 79 4.7 Scenario B LPV (aggressive) simulation results . 80 4.8 Scenario C LPV (aggressive) simulation results . 81 4.9 Scenario A LPV (relaxed) simulation results . 82 4.10 Scenario B LPV (relaxed) simulation results . 83 x 4.11 Scenario C LPV (relaxed) simulation results . 84 4.12 The control policy (top) contains enough chatter to exemplify the penalty system. The magnitude of the high pass filter is then plotted (bottom) and the peaks are detected as the abrupt changes in policy. Finding a peak is equivalent to the manager losing money to change the policy during the year. 86 xi Glossary B bioeconomics The science determining the socioeconomic activity threshold for which a biological system can be effectively and efficiently utilized with de- stroying the conditions for its regeneration and therefore its sustainability, p. 15. C carrying capacity The maximum population size of a species that an ecosytem can sustain indefinitely, p. 45. control law The mathematical definition evaluated at any instant to determine the control action for a particular system, p. 7. D discount rate A term applied when a future value is to less than its current value, p. 24. dynamical system A way of describing the passage in time of all points of a given space S [1], p.
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