Tai ngay!!! Ban co the xoa dong chu nay!!! Advances in Industrial Control For further volumes: www.springer.com/series/1412 Se Young Yoon r Zongli Lin r Paul E Allaire Control of Surge in Centrifugal Compressors by Active Magnetic Bearings Theory and Implementation Se Young Yoon Charles L Brown Dpt of El & Comp Eng University of Virginia Charlottesville, USA Paul E Allaire Dept of Mechanical & Aerospace Engin University of Virginia Charlottesville, USA Zongli Lin Charles L Brown Dpt of El & Comp Eng University of Virginia Charlottesville, USA ISSN 1430-9491 ISSN 2193-1577 (electronic) Advances in Industrial Control ISBN 978-1-4471-4239-3 ISBN 978-1-4471-4240-9 (eBook) DOI 10.1007/978-1-4471-4240-9 Springer London Heidelberg New York Dordrecht Library of Congress Control Number: 2012941917 © Springer-Verlag London 2013 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To Our Families Series Editors’ Foreword The series Advances in Industrial Control aims to report and encourage technology transfer in control engineering The rapid development of control technology has an impact on all areas of the control discipline New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies, , new challenges Much of this development work resides in industrial reports, feasibility study papers and the reports of advanced collaborative projects The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination Monographs in the Advances in Industrial Control series can be considered to range in type from the “art of the possible”, a “proof of principle” type and then a “state of the art” type, where the latter often reports on control as it exists in today’s industry For example, some “art of the possible” monographs explore a new theoretical development and demonstrate how it might find application in the control field A good example of this type of monograph is Process Control by J Bao and P.L Lee (ISBN 978-1-84628-892-0, 2007) Other monographs examine the present “state of the art” of control and its technology as found in current industrial practice and look at how better control might enhance efficiency and minimise pollution Recent exemplars of this category are Advanced Control and Supervision of Mineral Processing Plants by D Sbárbaro, R del Villar (ISBN 978-1-84996-105-9, 2010) or the monograph Hydraulic Servo-systems by M Jelali and A Kroll (ISBN 978-185233-692-9, 2002) However, this present, comprehensive Advances in Industrial Control monograph Surge Control of Active-Magnet-Bearing-Suspended Centrifugal Compressors: Theory and Implementation by Se Young Yoon, Zongli Lin and Paul E Allaire is an example of the “proof of concept” monograph It is an excellent addition to the series since its content has broad but complementary contributions from a new technology, from advanced control and from an advanced controller demonstration and assessment using an industrial-standard experimental rig The phenomenon of surge and stall in compressor technology is long standing and when the widespread industrial use of compressors is considered, a successful vii viii Series Editors’ Foreword control strategy that optimally maximises performance and eliminates compressor downtime would be of significant economic benefit to industry This particular control problem has received exposure in the Advances in Industrial Control series previously through the published monograph Compressor Surge and Rotating Stall by J.T Gravdahl and O Egeland (ISBN 978-1-85233-067-5, 1999), a monograph that is often cited in the literature of the compressor control field Some related material can be found in another monograph in the series, namely, Dynamic Modelling of Gas Turbines edited by G.G Kulikov and H.A Thompson (ISBN 978-1-85233-784-1, 2004) However, this monograph by Se Young Yoon, Zongli Lin and Paul E Allaire is distinctive in that it investigates the particular technology of active-magnet-bearingsuspended centrifugal compressors and assesses the authors’ own original advanced control strategies The assessment takes place using “an industrial-size centrifugal compressor test rig designed, built, and commissioned by the Rotating Machinery and Controls Laboratory (ROMAC) at the University of Virginia” (USA) A description of this experimental set-up can be found in Chap of the monograph Access to and use of this industrial-sized test rig is just one of the distinctive features of the research reported in the monograph Another feature is the comprehensiveness of the contents since the authors have taken special care to address the requirements of two readerships, one being readers from the control field, and the second being a more general engineering readership The industrial and academic control community will be interested in the outcome of the linear-quadratic-Gaussian (LQG) and H∞ advanced control trials performed using the experimental rig This group of readers will also find the technical knowledge extracted in terms of models and parameters needed for computer simulation tests before the instrumented control trials of value However, to ensure that the industrial and academic control community can fully comprehend the fundamentals of compressor technology there are invaluable and detailed presentations on the problem of surge and stall (Chap 1), rotor dynamics (Chap 2), magnetic bearings (Chap 3) and on the experimental rig and its associated instrumentation (Chap 4) In addition, to facilitate and ensure a full appreciation of the advanced control developments presented in Chaps and by a more general readership from the mechanical, manufacturing, mechatronics, rotating machinery and other engineering disciplines, the authors have included an introductory chapter on control systems theory (Chap 6) Even readers from the control community might find this chapter useful as a “refresher course” before reading the chapters covering the advanced LQG and H∞ control strategies The original contributions made by the authors in describing the various aspects of the technology, in devising and testing the advanced control strategies and the careful and thorough construction of this monograph make it a very welcome addition to the Advances in Industrial Control series and to the wider literature of compressor technology Industrial Control Centre Glasgow, Scotland, UK M.J Grimble M.A Johnson Preface Compressors are essential machines for a large number of modern manufacturing processes Like the hearts pumping life to the production lines, compressors are vital to the operation of key industrial sectors, such as the petrochemical and the mining industries, which rely on compressors for critical tasks, ranging from temperature control to gas transportation and mixing As a result, there have been continual efforts by the academic and industrial communities to improve the reliability and performance of such turbomachinery as new technologies become available Active magnetic bearing (AMB) is one such enhancing technology that has been gaining strong momentum in recent years Among other benefits, the low maintenance requirements and small parasitic energy losses have made these bearings highly desirable for high performance compressors, particularly those designed to operate in harsh or inaccessible environments Additionally, with their ability to actively change the rotor-dynamic characteristics of the compressor by controlling the bearing parameters in real time, the AMBs can provide a smoother and more reliable operation of the compressor over a wider range of operating conditions Stability is a critical factor that limits the performance of compressors The maximum mass flow output of a compression system is capped by choke, which is generally not a destabilizing phenomenon, and it is caused by the compressed medium reaching sonic conditions At the opposite end, the minimum mass flow is limited by the compressor instabilities known as stall and surge Stall is a localized phenomenon that can be observed in some compression systems, and it is sometimes accompanied by a sudden drop in the average compressor output flow On the other hand, surge is a system-wide instability that is characterized by large amplitude oscillations in the output pressure and mass flow These oscillations can cause extensive damage to the compressor casing and internal components due to high vibrational loads They can even lead to a catastrophic mechanical failure of the compressor if they are not addressed properly A conservative way of dealing with surge is to avoid it, by operating far away from the instability A more efficient way is to implement an active method to stabilize surge and stall, so that the stable operating region of the compression system is extended, resulting in both higher productivity and safer operation ix x Preface Unfortunately, a majority of current compressors operate conservatively to avoid surge In other words, many compressors trade the peak performance at the maximum pressure rise for the stability at the higher mass flow rates The focus in surge avoidance is on guaranteeing the mechanical integrity of the machines and the safety of the work place by keeping a precautionary margin between the operating output flows and the known surge points Additionally, a reset mechanism is built in the system that quickly releases the built-up pressure in the compressor if surge is detected by the different safety triggers An active surge controller, on the other hand, stabilizes the compressor flow during the initiation of surge, effectively extending the operational range of the compressor with no loss in performance The implementation of a control mechanism is much rarer in industrial applications than the surge avoidance strategies for several reasons The main reason is that the modifications to compressors in the field required for the installation of a surge control mechanism are very often complicated and involve very specialized equipments More importantly, there has not been an univocal experimental demonstration of the potential benefits that an effective surge controller could offer to an actual industrial-size compressor Recently, promising results have been presented in the literature on an active surge control scheme that modulates the impeller position to stabilize the flow in an AMB supported single stage centrifugal compressor With the AMB acting as a high bandwidth actuator to regulate the displacement of the impeller, the compressor flow states can be restored to the equilibrium operating point during the early stages of the surge instability, when the amplitude of the limit cycle is relatively small The main advantage of this active surge control scheme is that it can be easily implemented in existing AMB suspended compressors, generally with a simple modification in the control software The purpose of this book is to present the fundamentals on the integration of the AMBs for the suspension of the rotor in compressors, and how this relatively new bearing technology can be employed to actively control and potentially eliminate the compressor surge The material presented here is intended to serve as a comprehensive reference in the areas of compressor surge control and AMB application in turbomachinery For readers who are unfamiliar with compressors, rotor dynamics and magnetic bearings, brief introductions to these topics are presented in the earlier chapters of this book A brief discussion on compressors and compressor instabilities is presented in Chap 1, where the literature on the surge modeling and control is also reviewed Chapter contains a review of the basic theories and tools in the study of rotor dynamics Chapter presents a brief discussion on the operating principles of the AMBs and a summary of the potential benefits that come from the implementation of this bearing technology in compressors Both Chaps and are intended to be a self-contained reference for control engineers In order to develop the theory in a physical context, and to provide experimental validation of the theory developed throughout this book, an industrial-sized AMB suspended compressor system was designed, constructed and commissioned for the study of surge control A thorough description of this compressor test rig is presented in Chap This description includes the integration of the AMBs to the Preface xi compressor for rotor support and for surge control The derivation of the dynamic models for both the AMB/rotor system and the compression system flow, along with their experimental validations, are presented in Chaps and The experimental identification of the system dynamics included in these chapters will demonstrate that the assumptions made in the derivation of the mathematical models are sound These models will serve as the basis on which the AMB levitation controller and the active surge controller are designed, in Chaps and 8, respectively In the design of the AMB levitation controller, performance and robustness specifications that are desirable for AMB suspended compressors are included in the discussion In the design of the surge controller, the performance degradation of the surge controller due to dynamic limitations in the AMB system will be studied For both controllers, the theoretical derivation is accompanied by the experimental data to show their effectiveness in industrial-size compressors Finally, it is important to note that this book is not intended to be reference material for general design and operation of compressors There exists an extensive list of excellent references on the topics of compressor design and flow modeling Instead, this book is intended to serve as a guide for the application of the AMB technology in turbomachinery, and to demonstrate the advantages that this rotor support system can provide in the stabilization of the compressor surge for a particular group of single stage centrifugal compressors Since active magnetic bearings play a central role in the surge control method to be presented in the book, their theory and applications are extensively discussed The stabilization of the compressor surge is mainly discussed from a control theory perspective This book builds on years of work invested by many engineers and scientists from the Rotating Machinery and Controls (ROMAC) Laboratory at the University of Virginia The authors would like to acknowledge those who participated in the different stages of the research presented here The derivation of the theoretical concept for the surge control strategy presented here, as well as the design and the initial preparation of the experimental setup, was executed in the early stages of this project by the team led by Professor Eric Maslen and Dr Dorsa Sanadgol The experience in industrial compressors brought by Kin Tien Lim and the advice of Professor Chris Goyne in experimental fluid dynamic testing came to be of great value during the construction and commissioning of the compressor test rig Finally, the authors would also like to express their appreciation for the generous donations made by Kobe Steel Ltd., Kobe, Japan, and the constant support and funding by the ROMAC Laboratory and its industrial partners around the world Charlottesville, Virginia, USA Se Young Yoon Zongli Lin Paul E Allaire Chapter Conclusions The integration of active magnetic bearings in centrifugal compressors and the use of this bearing technology for the suppression of surge were explored in this book Using the AMBs to perturb the impeller axial tip clearance, a surge controller was developed that is able to extend the stability region of the compression system to a wider range of operating flows The surge point in the compression system, which is normally located at the peak output pressure in the characteristic curve, is effectively moved to a lower flow rate value by the implemented surge controller An important difference between the surge controller proposed here and the more common surge avoidance methods found in many industrial compressors is that the surge controller does not sacrifice system performance in order to achieve stability The controlled compressor is able to operate at the maximum output efficiency, while maintaining a healthy stability margin between the system steady state output and the new surge inception point Additionally, the proposed surge controller is easily implementable in existing AMB suspended compressors, and its implementation generally involves a simple modification in the control software The motivations and objectives of the research effort made in this book were discussed in Chap In this chapter, basic concepts of compression systems and compressor flow instabilities were introduced The mechanisms of the stall instability were briefly described, but the main focus of this book was on the surge instability A review of the different approaches for modeling surge in axial and centrifugal compression systems was presented Additionally, the different strategies used in surge avoidance and surge control were discussed in great detail It was concluded from the review of the literature that the proper selection of the actuator and sensor pair plays an important role in the success of a surge control scheme An introduction to the basic concepts in rotor dynamics that are needed in the study of active magnetic bearings was presented in Chap We first derived the fundamental ideas in the vibration theory of rotor-dynamic systems by analyzing the simplified Jöoppl/Jeffcott rotor on rigid supports We were able to identify in this simple rotor/support system many characteristics that are commonly observed in complex rotating machines Destabilizing dynamics that are found in both centrifugal and axial compressors, such as the gyroscopic moment and the cross-coupled S.Y Yoon et al., Control of Surge in Centrifugal Compressors by Active Magnetic Bearings, Advances in Industrial Control, DOI 10.1007/978-1-4471-4240-9_9, © Springer-Verlag London 2013 261 262 Conclusions stiffness, were briefly introduced The gyroscopic moment and the cross-coupled stiffness are in many cases the main sources of instability in AMB supported systems The API 617 standard, which is widely used for auditing the rotor-dynamic response in industrial compressors, was also presented in this chapter for the analysis of machines with AMBs The chapter concluded with a brief discussion on the finite-element method used for modeling rotors with complex geometries The mathematical model of the rotor is required in the rotor-dynamic analysis specified in API 617, as well as in the design of the AMB levitation controller Chapter presented the fundamentals in the modeling and operation of active magnetic bearings Starting from a brief discussion on magnetic flux and field, we described the operating mechanism of active magnetic bearings and identified the parts that compose the AMB system The open loop gains of the linearized AMB force equation were introduced The linearized AMB force equation is in turn based on a simplified magnetic circuit model of the current/force relationship in electromagnetic actuators The bearing losses limiting the maximum performance of AMB systems were discussed This chapter also presented a summary of the different rotor levitation controllers that have been explored in the literature for AMB supported machines The two most common control methods in industrial AMB applications are the decentralized and the tilt-and-translated PID controllers For high-performance AMB applications, advanced control methods are also explored in the literature Optimal controllers, neural network-based controllers, and selftuning controllers were among the many options that were discussed in the chapter Chapter described an AMB levitated centrifugal compressor test rig This test rig was the platform for the experimental testing of the surge controller presented in this book A detailed description was first provided for all major components of the test rig, including the AMB rotor levitation system Next, the experimental characterization of the steady state flow in the compression system was presented The compressor characteristic curve was measured for different operating speeds in the stable flow region of the test rig Finally, the observed surge instability in the compressor test rig was carefully analyzed The pressure and rotor vibration measurements during the different phases of surge were presented The pressure measurements provided evidence of the existence of two surge modes in the system The first surge component was at 132 rad/s (21 Hz) and was caused by the excitation of the piping acoustic mode The second and larger surge component was at 44 rad/s (7 Hz), which was slightly lower than the system Helmholtz frequency The derivation of a mathematical model for a compression system with variable impeller tip clearance and long exhaust piping was presented in Chap This model was based on the well-known Greitzer compression system model, where the compressor equation was expanded to include the effect of changing the impeller axial position Additionally, the acoustic effects of the pipeline in the compression system were integrated in the final dynamic equations Using the AMBs to perturb the impeller axial position, we obtained the experimental Bode plots of the compressor test rig Good agreement was observed between the simulated and measured Bode plots during the validation of the mathematical model on the compressor test rig An interesting observation from the experimental test results presented in this chapter Conclusions 263 was the correlation between the initiation of surge and the loss in damping of the compression system modes As the compressor operation was pushed towards the surge region by restricting the output flow, the Bode plots showed a rapid loss in the damping of the system modes The first mode to lose its damping was the piping acoustic mode at 132 rad/s (21 Hz), which was quickly followed by the second mode near the Helmholtz frequency at 44 rad/s (7 Hz) A short overview of the classical and modern control theories was presented in Chap This chapter introduced many concepts in control theory that would help to understand the design and implementation of the active surge and rotor levitation controllers presented throughout this book The presentation of the material was divided into the classical control and modern control sections In the discussion of classical control theory, we reviewed the objectives of the feedback controller and the different system parameters that characterize the dynamic response of SISO systems Transfer functions that are important in the auditing of AMB systems were defined, and the basic elements of the PID controller were introduced In the discussion of modern control methods, we introduced the state space representation of linear systems and the different tools available for designing optimal feedback controllers for MIMO systems Optimization-based methods for designing controllers, such as the LQG, H∞ and μ synthesis controllers, were introduced to maximize the robustness and the performance of the closed-loop system Next, the design of the rotor levitation controllers for the AMB system in the compressor test rig was described in Chap The derivation of the mathematical model of the AMB system was first presented This model takes into account some common bearing losses such as the eddy currents and the back-EMF effects From this model, the AMB levitation controllers were designed to satisfy the ISO and API rotor-dynamic standards developed for machines with AMBs The rotor lateral levitation is controlled by an LQG controller, which was selected to minimize the rotor vibration due to the unbalance forces The rotor axial position, on the other hand, is regulated by an H∞ controller that optimizes the rotor axial tracking performance The experimental validation of the open loop and closed-loop transfer functions of the AMB system in the compressor test rig was presented to the end of the chapter Finally, Chap discussed the design, implementation and experimental testing of the active surge controller A linearized compression system model was derived from the dynamic equations presented in Chap The linearization point of the compression system dynamics was selected to be the intersecting point between the compressor characteristic curve and the load curve corresponding to the throttle valve at 17 % opening This puts the equilibrium point in the instability region at the left-hand side of the surge point in the compressor characteristic curve An H∞ control law was designed to suppress the surge instability in the compression system The interaction between the surge controller and the thrust AMB controller was found to affect the effectiveness of the surge suppression method In order to compensate for the uncertainty in the dynamics of the AMB rotor levitation system, a robustness condition was included in the synthesis of the H∞ surge controller The derived controller was implemented and tested in the experimental compressor test rig at the operating speeds of 10,290 rpm, 13,950 rpm and 16,290 rpm 264 Conclusions In all three cases, the surge controller was able to suppress the compressor instability and extend the stable flow range of the compression system At 13,950 rpm, the stability region of the compression system was extended by 20.5 % from the original surge point identified in the uncontrolled case At 16,290 rpm, the stable flow range of the controlled system was extended by 21.3 % The significance of these results are that the surge controller allows the compression system to operate at the peak pressure output, and to so with a large surge margin separating the newly found surge point in the controlled system and the safe compressor operating region Most importantly, the extended stability region is obtained with no loss in the maximum compressor efficiency 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On lq-control of magnetic bearing IEEE Trans Control Syst Technol 8, 344–350 (2000) Index A Active magnetic bearing, ix, 1, 4, 15, 18, 45, 47, 57, 89, 90, 101, 130, 186 air gap, 57, 62, 64, 65, 69, 71, 72, 102 bias current, 65, 82, 104 closed-loop stiffness, 193, 206 coil inductance, 67 coil resistance, 66 decentralized PID, 77 electromagnetic actuator, 60, 67, 68, 102, 107, 196 e-core, 71, 102 heteropolar, 69 homopolar, 70, 83 horseshoe, 70 sectioned stator, 109 force equation, 64, 185 lamination, 72, 98, 102, 198 levitation controller, xi, 16, 18, 80, 81, 185 axial, 114, 213, 214, 231 radial, 110, 199, 201 linearized force equation, 65, 196 Ki , 66, 104, 109 Kx , 66, 104, 109, 193 load capacity, 67, 102, 108 losses back-EMF effect, 185, 198 eddy current, 72, 98, 109, 114, 198 flux fringing, 72 flux leakage, 57, 72 hysteresis, 73 magnetic circuit model, 62, 102 assumptions, 61 modern control, 81 power amplifier, 4, 71, 197 proximity sensor, 4, 71, 102, 197 eddy, 96 variable reluctance, 96 slew rate, 68, 197 tilt and translate PID, 77 vibration compensation, 79 Adaptive control application in AMB, 86 Alford force, see Cross-coupling, force Aliasing, 197 American Petroleum Institute, see API Amplification factor, 39, 208 Annulus, API, 18, 38, 46, 80, 111, 185, 186, 207 Artificial neural network, 84 Auxiliary bearing, 73, 115 damping element, 75, 116 rolling element, 75 soft bushing, 75 B B–H curve, 60 Backup bearing, see Auxiliary bearing Beam theory Bernoulli–Euler, 51 Timoshenko, 51 C Campbell diagram, 99 Characteristic curve, 3, 5, 8, 92, 127, 222, 249, 253 polynomial fitting, 127 Collector, see Plenum Complementary sensitivity function, 113, 114, 154, 214, 218 Compression system, 3, 89, 95, 117, 125, 130, 136, 221, 231, 238, 245, 248, 253 compressor, see Compressor piping, see Piping S.Y Yoon et al., Control of Surge in Centrifugal Compressors by Active Magnetic Bearings, Advances in Industrial Control, DOI 10.1007/978-1-4471-4240-9, © Springer-Verlag London 2013 271 272 Compression system (cont.) plenum, see Plenum throttle valve, see Valve, throttle valve Compressor dynamic compressor, axial, 1, 8, 10, 39 centrifugal, x, 2, 6, 8, 10, 39, 44, 89, 90, 97, 117, 130, 221, 223, 257 performance curve, see Characteristic curve positive displacement compressor, reciprocating, rotary screw, Compressor instability, dynamic instability, stall, see Stall surge, see Surge static instability, Confidence interval network, 84, 86 Control system, 150, 243 closed-loop control, 153 feedback control, 25, 44, 150, 151, 199, 236 open-loop control, 150 Control theory, 11, 149 classical, 57, 149 modern, 57, 149, 167 Controllability, 168, 174 controllability matrix, 174 Critical speed map, 41, 98, 206 Cross-coupling, 36 coefficient, 37 force, 20, 36, 38, 40, 43 stiffness, 17, 37, 44, 46, 189 D Damping critically damped, 162 damping ratio, 22, 161, 208 logarithmic decrement, 43 overdamped, 22, 162 underdamped, 22, 162 Diffuser, 2, 9, 96 vaned, vaneless, 2, 90 Duality property, 177 Ducting system, see Piping E Elastic modulus, 19 Energy momentum wheel, 85 Equilibrium operating point, 5, 230, 231 F Feedback linearization, 83 Final value theorem, 166 Index Finite-element analysis magnetic, 104, 109 rotordynamics, 190 thermal, 94 Föppl/Jeffcott rotor, see Jeffcott rotor G Gas turbine, Gaussian observer, 200 Generalized eigenvalue problem, 191 Gyroscopics, 18, 20, 27, 85, 99, 183, 189, 201 H Helmholtz frequency, 9, 117, 126, 134, 139, 225, 229, 231 Hermite shape function, 50 H∞ control, 181, 213, 214, 234 application in AMB, 82, 85, 114 I Ideal gas law, Impeller, 1, 2, 17, 20, 36, 37, 90, 96, 190 tip clearance, 92, 108, 130, 221, 222, 230, 234, 235, 239 dynamic perturbation test, 133 efficiency, 130 linearized equation, 222 steady state test, 132 Inducer, Instrumentation, 95 flow rate sensor, 96 pressure transducer, 96 rotor displacement sensor, see Active magnetic bearing, proximity sensor temperature sensor, 96 International Organization for Standardization, see ISO ISO, 38, 111, 185, 186, 203, 212, 217, 218 J Jeffcott rotor, 18, 27 Jet propulsion engine, K Kalman filter, 81, 180 L Lagrange’s equation, 51 Laplace transform, 149, 151, 154, 156, 160, 166, 170, 172 Linear fractional transformation, 82 Linear matrix inequality, see LMI Linear quadratic gaussian, 86, 110, 180, 185, 199, 201, 209 application in AMB, 81 Index Linear quadratic regulator, 15, 179 Linear time-invariant, see LTI LMI, 167, 178 Loop shaping control, 83 LTI, 149, 168, 172, 182, 198, 199 LTI system pole, 155, 165, 173, 231 simulation diagram, 169 M Magnetic field, 58 Magnetic flux, 59, 198 flux density, 59, 69 flux linkage, 67 Magnetic permeability, 60 relative permeability, 60 Magnetic reluctance, 60 Magnetomotive force, 58 Mass flow rate observer, 236 convergence, 238 estimation error, 236 Maximum continuous operating speed, see MCOS MCOS, 39, 43, 100 Method of characteristics, 137 MIMO, 81, 167, 199, 201 Model reduction modal, 191 modal damping, 192, 209 Moment of inertia ratio, 29, 33 Motor induction motor, 94 μ synthesis control, 182 application in AMB, 84, 85 structured singular value, 85, 178, 183 Multi-input multi-output, see MIMO N Natural frequency, 27, 39 damped, 22, 162 undamped, 21, 23, 32, 161 Norm H2 norm, 178 H∞ norm, 83, 178, 213 Nyquist frequency, 197 O Observability, 168, 176 observability matrix, 177 Optimal control, 81, 82, 167, 178, 199 objective function, 82, 86, 168, 178, 199 weighting function, 81, 83, 85, 86, 181, 199, 201, 214, 235 273 P Pade delay approximation, 198 PID control, 150, 155, 199, 212 application in AMB, 75 integrator, 209, 218 Pipeline compressor, Piping, 1, 90, 96, 118, 134, 136, 222, 226 acoustic resonance, 117, 229, 231, 236, 238, 245, 248, 256 acoustics, 11, 118, 135, 147, 222, 256 dynamic model, see Transmission line model Plenum, 1, 8, 13, 92, 97, 117, 125, 133, 136, 138, 140, 222, 225, 228, 231, 238, 249, 251, 252 Pump, 4, 39 R Radial basis function network, 84, 86 Riccati equation, 179, 200 Robust control, 85, 182, 233 ROMAC, 89, 193 Rotor mode, 191 backward mode, 21, 32, 101 bending mode, 42, 43, 98, 110, 111, 193, 209 conical mode, 32, 194 critical speed, 17, 98, 195, 203 cylindrical mode, 31 forward mode, 21, 32, 36, 43, 101 free–free mode, 195 mode shape, 41, 80, 100, 191 parallel mode, 194, 207 rigid body mode, 98, 193, 209 Rotordynamic system, 17 bearing, 17, 47 AMB, see Active magnetic bearing fluid film, 5, 36, 186 rolling element, 5, 186 impeller, 46 rotor, 4, 17, 20, 98 journal, 42, 185, 190, 198 thrust disk, 109 seal, 17, 20, 36, 37, 46, 47 Rotordynamics, 1, 5, 16, 17 S Self-tuning control application in AMB, 86 Sensitivity function, 113, 114, 154, 188, 201, 208, 209, 214, 218, 233, 235 ISO stability margin, see Stability analysis Separation margin, 39, 194 Shaft, see Rotordynamic system, rotor 274 Single-input single-output, see SISO SISO, 86, 149, 167, 172 Small gain theorem, 82, 234 Stability, ix asymptotic stability, 155, 180 BIBO stability, 155, 180 external stability, 155, 174 input output stability, 155 internal stability, 155, 174, 180 Stability analysis API level I, 44 API level II, 46 ISO stability margin, 188 Stability margin, 186, 203, 212 gain margin, 189, 201 phase crossover frequency, 167 phase margin, 189, 201 magnitude crossover frequency, 167 Stall, ix, 6, 125 category abrupt, non-rotating blade, progressive, rotating, flow separation, stall cell, Supercritical, 17 Surge, ix, 6, 8, 15, 125, 130, 136 bearing load, 115, 213 category classic, deep, mild, modified, experimental characterization, 116 limit cycle, 9, 14, 117, 128, 238, 244, 245, 254 blowdown period, collapse period, recovery period, surge line, 6, 8, 12, 240 surge point, x, 6, 8, 13, 127, 224, 240, 244, 246, 249, 252 Surge avoidance, 11, 12 ratio control, 12 surge avoidance line, 12 surge margin, 12, 249, 254 Surge control, 12, 13, 16 active control, xi, 1, 11–13, 221, 230 AMB, 15, 95, 221 bleed valve, 15 closed-coupled valve, 14 throttle valve, 14 Index experimental testing at 10,290 rpm, 245 at 13,950 rpm, 247 at 16,290 rpm, 252 implementation, 240 passive control, 12, 13 hydraulic oscillator, 13 movable wall, 13 robustness condition, 235 simulation, 238 Surge model, 10, 125, 221, 252 Greitzer model, 10, 125, 126, 221 Greitzer stability constant, 126 Helmholtz frequency, see Helmholtz frequency non-dimensional mass flow rate, 125 non-dimensional pressure rise, 125 Greitzer model with exhaust pipe, 138 experimental validation, 139, 147 Greitzer model with plenum pipe, 140, 238 experimental validation, 141 model linearization, 229 Moore–Greitzer model, 15 one-dimensional model, 10 two-dimensional model, 10 Surge suppression, see Surge control System representation state space, 168, 171 observer canonical form, 170 transfer function, 89, 133, 149, 229, 232, 234 System response steady state, 160, 165 Bode plots, 111, 114, 165 transient, 160 impulse response, 172 peak time, 163 percentage overshoot, 163, 203 rise time, 163, 203 settling time, 165, 203 step response, 161 time constant, 160 T Taylor expansion, 223 Touch-down bearing, see Auxiliary bearing Transmission line model, 136 frequency dependent friction factor, 137 inviscid characteristic impedance, 137, 226 kinematic viscosity, 144 line dissipation number, 144, 226 modal approximation, 143, 226 two-port transmission line model, 136 Turbine, 17, 36, 39 Index Turbocharger, Turbocompressor, see Compressor Turbomachinery, 4, 10, 11, 125 U Unbalance, 17, 40, 79, 150, 185, 201, 203 eccentricity, 18, 20 unbalance response, 18, 42, 110, 186, 203 Uncertainty, 149, 150, 189, 233 additive, 152 parametric, 85, 86, 183 structured, 85, 183 unstructured, 82, 86 V Valve, bleed valve, 12, 15 275 blow-off valve, 12 closed-coupled valve, 14 recycle valve, 12 throttle valve, 4, 9, 13, 14, 90, 96, 116, 128, 222, 227, 229, 230, 238, 241, 244 Vibration axial, 47, 115 damped free, 22 lateral, 17, 21, 39, 47, 77 subsynchronous, 17, 36 synchronous, 17, 80, 195 torsional, 17 undamped free, 20, 23, 27 whirling, 17, 36