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View Example 45 UNIVERSITY OF CINCINNATI _____________ , 20 _____ I,______________________________________________, hereby submit this as part of the requirements for the degree of: ________________________________________________ in: ________________________________________________ It is entitled: ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ Approved by: ________________________ ________________________ ________________________ ________________________ ________________________ Stability Analysis and Design of Servo-Hydraulic Systems - A Bifurcation Study A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of DOCTORATE OF PHILOSOPHY (Ph.D.) in the Department of Mechanical, Industrial and Nuclear Engineering of the College of Engineering 2002 by Amit Shukla B.E.M.E., MNR Engineering College, India 1996 M.S.M.E., University of Cincinnati, 1998 Committee Chair: Dr. David F. Thompson Abstract Design for robust stability is one of the most important issues in nonlinear systems theory. The validity of linear system design in a small neighborhood is not a sufficient criterion for systems that undergo parametric variations and have strong nonlinear characteristics. With rapid growth in the systems theory, the design of nonlinear systems using bifurcation theory- based procedures has been one of the key developments. Servo-hydraulic systems are one of the most commonly used actuation and control devices, due to their force to weight ratio. They also are highly nonlinear in nature and hence provide considerable difficulty in the design and analysis of these systems and their control algorithms. The goal of this dissertation is to tackle some of the issues of the nonlinear systems theory with applications to servo-hydraulic systems. The use of bifurcation theory for the design and analysis of a nonlinear system is illustrated, and a detailed investigation into the dynamics associated with the servo-hydraulic systems is done. Further, the model decomposition/reduction strategy for parametric study in the nonlinear system is suggested. The idea of control-induced bifurcation is introduced and explained in light of servo-hydraulic systems. The servo-hydraulic system nonlinearities are explained and their effects on the robust stability are highlighted. This numerical work is also complemented with the experimental results on the servo-hydraulic circuits. This general procedure for robust stability design and control design, under the influence of nonlinearities, presented in this work can be used for any nonlinear system. The limitations of bifurcation theory based tools are also highlighted. Acknowledgements This work has been possible due to motivation and support of various individuals who gave me sound advice and guidance at numerous occasions throughout my years as a graduate student at the University of Cincinnati. I express my deepest gratitude to my academic advisor Dr. David F. Thompson, whose excellent guidance and critical comments paved the way for this work as well as contributed to my development as an academician. He has been a true guide, a mentor and a friend. I also wish to acknowledge Dr. Randall J. Allemang, Dr. Ronald Huston, and Dr. Edward J. Berger for being on my dissertation committee and for providing valuable help and guidance. All of them have shaped my graduate education and life very significantly by their great teaching and thoughtful concerns. I am also grateful to the members of UC SDRL, specially, Dr. David Brown, Dr. Jay Kim, Dr. Allyn Phillips, Dr. Gregory Kremer, Dr. Doug Adams, Dr. Bill Fladung, Dan Lazor, Srinivas Kowta, Tom Terrell, Jeff Hylok and Bruce Fouts who made my stay at UC most enjoyable. I also would like to thank Rhonda Christman for her exceptional secretarial help during my 6 years at UC. I am also thankful to the National Science Foundation for providing monetary support for this work and Ford Motor Company (Dr. Gregory M. Pietron) for support and fruitful discussions on hydraulic system modeling. This has been a project which would be impossible without unflinching support of my family. I dedicate this work to my parents. They provided an atmosphere and everything else in my upbringing for me to reach this point in my life. The support and understanding that my wife gave me provided extra boost for completion of this work. Her love has been my strength for the past few years. Her parents were also very cooperative and patient regarding my tenure as a graduate student. Table of Contents Abstract List of Figures 4 List of Symbols 9 1. Introduction 11 1.1. Background/Motivation 11 1.2. Applications 16 1.3. Organization of the dissertation 17 2. Servo hydraulic drives 20 2.1. Previous work 20 2.2. Fundamentals of hydraulic system modeling 23 2.2.1. Orifice flows 23 2.2.2. Continuity equations 24 2.2.3. Example 25 2.2.4. Valve modeling 26 2.3. Modeling assumptions 33 2.4. Physically realistic models of servo hydraulic system 34 2.5. Modeling of servo-pump actuator system 37 2.6. Modeling of servo-valve actuator system 39 2.7. Servo-valve actuator model with line dynamics 42 3. Linear and nonlinear analysis of servo-hydraulic systems 45 3.1. Linear analysis of servo-hydraulic system –a review example 45 3.2. Experimental results for servo-hydraulic systems – a review 47 3.3. Nonlinearities in the servo-hydraulic systems and their effect on stability 48 1 3.3.1. Effect of flow nonlinearity on the stability 48 3.3.2. Effect of saturation on the stability 50 3.4. Large scale coupled nonlinear systems 50 3.5. Why nonlinear analysis is needed? 52 4. Nonlinear systems analysis and bifurcation theory 53 4.1. Dynamical systems and equilibrium points 54 4.2. Generic bifurcations 57 4.3. Global bifurcations, jumps and non-local behavior 62 4.4. Stability of forced systems-feedback loop nonlinearities and nonlinear feedback 63 4.5. Nonlinear systems analysis tools 64 5. Nonlinear systems analysis –a bifurcation theory based approach 66 5.1. Previous work 66 5.2. Multi-parameter multi-space bifurcation theory 71 5.2.1. Parameter space, eigen-space and state space 74 5.2.2. Constant velocity solutions 76 5.2.3. Robust bifurcation stability analysis 77 5.3. Decomposition of the system 80 5.3.1. Fastest unstable mode with associated parameters 81 5.3.2. Transformations for parametric model decomposition 82 5.4. Control induced bifurcations 83 5.5. Control of bifurcation instabilities 85 5.6. Measure of nonlinearity 86 5.7. Numerical aspects of bifurcation analysis 87 5.7.1. Data noise and nonlinear effects 89 6. Preliminary bifurcation results: numerical 90 2 6.1. Characterization of nonlinear dynamics- computational results 90 6.2. Results for servo-pump actuator system 90 6.2.1. Simulation studies 91 6.2.2. Parametric bifurcation studies 94 6.3. Results for the servo-valve actuator model 101 6.4. Chapter summary 104 7. Development of experimental apparatus 125 7.1. Development of the test stand 125 7.2. Servo-valve model development and refinement 127 7.2.1. Pressure-flow-voltage nonlinear static characteristics 129 7.2.2. Linear transfer function model of the servo-valve 131 7.2.3. Servo-valve actuator model and real-time control schematic 133 7.3. Effect of accumulators 137 7.4. Chapter summary 139 8. Control Studies- Introduction 140 8.1. PD controller 140 8.2. Effect of PD controller on the servo-valve actuator system 141 8.3. Effect of 5th order linear controller on the servo-valve actuator system 161 8.4. Chapter summary 177 9. Conclusions and suggestions for future research 180 10. Bibliography 184 11. Appendix 198 3 List of Figures 2.1 A sample hydraulic system model 25 2.2 A valve schematic 27 2.3 Port flow area schematic of a spool valve 28 2.4 Open valve port area configuration 29 2.5 Closed valve configuration 30 2.6 Port flow area as a function of the valve displacement 30 2.7 A common four way spool valve 31 2.8 Axial and lateral components of flow forces in a valve port 32 2.9 Model 1, servo-pump actuator system 38 2.10 Block diagram of model 1 38 2.11 Servo-valve actuator model schematic with external loads 41 2.12 Pipe line model schematic 43 3.1 Proportional feedback for the linearized servo-pump actuator model 46 3.2 Root locus for varying the proportional feedback gain 46 3.3 Dead zone and saturation nonlinearity-input and response 50 3.4 Parameter space investigation using simulation 51 4.1 Time response, phase plane and eigenplane 55 4.2 Invariant subspaces and manifolds 56 4.3 Saddle node bifurcation plot representing the behavior of equilibrium point 58 4.4 Trans critical bifurcation 59 4.5 Pitch fork bifurcation 59 4.6 Super critical Hopf bifurcation 60 4.7 Soft and hard generation of limit cycles 60 4.8 Transition from stationary to periodic solution, via Hopf Bifurcations 62 4 4.9 Nonlinear system decomposition- nonlinear feedback 63 5.1 Parameter space, state space and eigenspace 75 5.2 Block diagram of the state space system with proportional output feedback law 78 5.3 Decomposition of system for robust stability analysis 80 5.4 Block diagram of plant and controller depicting extended parameter vector 84 5.5 Control induced bifurcation and distance to bifurcation 84 5.6 Parametric step size: large steps may skip some details 87 6.1 Eigenvalues of the nominal open loop servo-pump actuator model 92 6.2 Open loop simulation response
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