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Nonlinear control of flexible rotating system with varying velocity

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Nonlinear control of flexible rotating system with varying velocity

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Content NONLINEAR CONTROL OF FLEXIBLE ROTATING SYSTEM WITH VARYING VELOCITY by Yeh Lin A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) May 2016 Copyright 2016 Yeh Lin Dedication I dedicate this do cumen t to m y wife and m y paren ts for their patience, whole- hearted supp ort, and unconditional lo v e. i Acknowledgments First and foremost, I w ould lik e to thank m y paren ts for ev erything they ha v e done for me. Without their lo v e and supp ort, I could not ha v e made it so far in m y life. My sincere thanks also go es to m y wife, for her trust and consideration. She has inspired me in ev ery w a y and c hanged m y life for b etter. I w ould lik e to express m y gratitude to m y advisor Dr. Flashner for the guidance of m y Ph.D researc h, for his patience, understanding and encouragemen t. He help ed me in all the time of this researc h and thesis writing. Without his advices, I w ouldn’t b e able to accomplish this w ork. Besides m y advisor, I w ould lik e to thank the rest of m y thesis committee: Prof. Shiett, Prof. Y ang, Prof. Jonc kheere and Prof. Domaradzki, who k eep me on the righ t researc h direction. Also, I w ould lik e to thank Dr. Udw adia for his teac hing, who help ed me to establish the foundation of the kno wledge in m y researc h eld. Last, I w ould lik e to thank all m y USC friends: Sumo, Harsh, Bo, Chang, Da yung, K orkut, Kevin and Shalini, for the sleepless nigh ts w e w ere w orking and preparing the exams together, for the hard time w e w ere sharing to eac h others, and for all the fun w e ha v e had in the past few y ears. ii Contents 1 In tro duction 1 2 Preliminaries 5 2.1 Mathematical Bac kground . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Quadratic F orms . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.3 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.4 Lipsc hitz Con tin uit y . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Ly apuno v Stabilit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Stabilit y Denition . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Stabilit y Theorem of Equilibrium P oin t . . . . . . . . . . . . . 8 2.2.3 Asymptotic Stabilit y of Linear System . . . . . . . . . . . . . 10 2.3 Boundedness and Ultimate Boundedness . . . . . . . . . . . . . . . . 11 3 System Mo deling 13 3.1 Go v erning Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 Energy F unctions . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.2 Vibration and T orque Equation . . . . . . . . . . . . . . . . . 14 3.1.3 A Rotating System with Linear Elastic Deformation . . . . . 15 3.2 Jecott Rotor Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.2 Dimensionless Equations of Motion . . . . . . . . . . . . . . . 21 3.3 Magnetic Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 Dynamic Beha vior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4.1 Jump Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4.2 Eect of Equation Decoupling . . . . . . . . . . . . . . . . . . 28 4 Linearized Rotor Mo del with PI T orque Con trol 35 4.1 System Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1.1 State-Space Represen tation . . . . . . . . . . . . . . . . . . . 36 4.2 Stabilit y Analysis of Linearized System with T orque PI con trol . . . . 38 4.3 Eect of PI con trol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5 Con trol Dev elopmen t using Ly apuno v Redesign Metho d 54 5.1 Ly apuno v Redesign Metho d . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Equations of Nominal System and A ctual System . . . . . . . . . . . 58 iii 5.2.1 Nominal System . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2.2 P erturb ed System . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3 Con trol La w Deriv ation . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.4 Sim ulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.4.1 System with Nominal Con trol . . . . . . . . . . . . . . . . . . 74 5.4.2 System with Ly apuno v Redesign Con trol (Robustness) . . . . 78 6 Con trol Dev elopmen t using Sliding Mo de Con trol 84 6.1 Sliding Mo de Con trol . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2 Con trol La w Deriv ation . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2.1 Sliding Mo de Con trol . . . . . . . . . . . . . . . . . . . . . . . 87 6.2.2 Sliding Mo de with PI torque Con trol . . . . . . . . . . . . . . 95 6.3 Sim ulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3.1 Sliding Mo de Con trol . . . . . . . . . . . . . . . . . . . . . . . 98 6.3.2 Sliding Mo de Con trol with T orque PI Con trol . . . . . . . . . 101 7 F uture Researc h T opics 105 A Routh Criterion 110 B Stabilit y Pro of of Ly apuno v Redesign 111 C Linear Matrix Inequalit y 113 iv List of Figures 3.1 Rotating Bo dy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 2DOF deection mo del . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 2DOF rotor system for lateral vibration . . . . . . . . . . . . . . . . . 18 3.4 2DOF deection mo del (p olar co ordinates) . . . . . . . . . . . . . . . 20 3.5 The rigid rotor equipp ed with b earing magnets and sensors . . . . . . 23 3.6 Decen tralize con trol structure with PID con trol . . . . . . . . . . . . 25 3.7 Jump phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.8 Time resp onse of the rotor system during jump phenomena . . . . . 28 3.9 Stabilit y Region of decoupled system . . . . . . . . . . . . . . . . . . 30 3.10 Time resp onse of decoupled system . . . . . . . . . . . . . . . . . . . 31 3.11 Stabilit y Region of coupled system, = 0 . . . . . . . . . . . . . . . . 32 3.12 Stabilit y Region of coupled system, > 0 . . . . . . . . . . . . . . . . 33 3.13 Time resp onse of coupled system (P oin t A) . . . . . . . . . . . . . . 33 3.14 Time resp onse of coupled system (P oin t B) . . . . . . . . . . . . . . 34 4.1 F eedbac k con trol mo del with PI con troller . . . . . . . . . . . . . . . 39 4.2 Stabilit y conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3 Stabilit y conditions comparison . . . . . . . . . . . . . . . . . . . . . 40 4.4 Stabilit y region of nominal system . . . . . . . . . . . . . . . . . . . . 41 4.5 Time resp onse of the linearized system with PI con trol (stable case) 41 4.6 Time resp onse of the linearized system with PI con trol (unstable case) 42 4.7 Comparison of stabilit y region with dieren t rotating sp eeds . . . . . 43 4.8 Comparison of stabilit y region with dieren t damping . . . . . . . . 43 4.9 Comparison of stabilit y region with dieren t r . . . . . . . . . . . . . 44 4.10 Stabilit y condition for e = 0:4 . . . . . . . . . . . . . . . . . . . . . . 45 4.11 Stabilit y condition for e = 1 . . . . . . . . . . . . . . . . . . . . . . . 45 4.12 Stabilit y Region (e = 0:4) . . . . . . . . . . . . . . . . . . . . . . . . 46 4.13 Stabilit y Region (e=1) . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.14 Critical eccen tricit y e c (case 1) . . . . . . . . . . . . . . . . . . . . . . 47 4.15 Critical eccen tricit y e c (case 2) . . . . . . . . . . . . . . . . . . . . . . 47 4.16 Stabilit y region for critical eccen tricit y c hart (case 2) . . . . . . . . . 48 4.18 Time resp onse for critical eccen tricit y c hart (Unstable case) . . . . . . 49 4.17 Time resp onse for critical eccen tricit y c hart (Stable case) . . . . . . . 49 4.19 Time resp onse of the linearized system with no in tegral con troller . . 50 4.20 Comparison of stabilit y region with dieren t PI con trol gains . . . . . 51 4.21 Critical eccen tricit y e c with PI con troller (case 1) . . . . . . . . . . . 52 4.22 Critical eccen tricit y e c with PI con troller (case 2) . . . . . . . . . . . 52 4.23 Critical eccen tricit y e c with lo w K P gain and high K I gain (case 3) . 53 4.24 Critical eccen tricit y e c with high K P gain and lo w K I gain (case 4) . 53 5.1 Ly apuno v redesign sim ulation mo del . . . . . . . . . . . . . . . . . . 74 5.2 Time resp onse of the system with nominal con troller (w c = 1.2) . . . 75 5.3 Con trol eort of the system with nominal con troller (w c = 1.2) . . . . 76 5.4 Time resp onse of the system with nominal con troller (w c = 1.8) . . . 76 5.5 Con trol eort of the system with nominal con troller (w c = 1.8) . . . . 77 v 5.6 Time resp onse of the system with nominal con trol (high PI gain) . . . 77 5.7 Con trol eort of the system with nominal con trol (high PI gain) . . . 78 5.8 Time resp onse of the system with Ly apuno v redesign con trol . . . . . 79 5.9 Con trol eort of the system with Ly apuno v redesign con trol . . . . . 79 5.10 Con trol eort of the system with Ly apuno v redesign con trol (zo om in) 80 5.11 Time resp onse of the system with Ly apuno v redesign con trol (Ultimate b oundness) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.12 Con trol eort of the system with Ly apuno v redesign con trol (Ultimate b oundness) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.13 Con trol eort of the system with Ly apuno v redesign con trol b y using ultimate b oundness (zo om in) . . . . . . . . . . . . . . . . . . . . . . 82 5.14 Time resp onse of the system with Ly apuno v redesign con trol (Ultimate b oundness, e = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.15 Con trol eort of the system with Ly apuno v redesign con trol (Ultimate b oundness, e = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.16 Con trol eort of the system with Ly apuno v redesign con trol b y using ultimate b oundness, e = 1 (zo om in) . . . . . . . . . . . . . . . . . . 83 6.1 Sliding Mo de Con trol Sim ulation Mo del . . . . . . . . . . . . . . . . 98 6.2 Time resp onse of the system with sliding mo de con trol . . . . . . . . 99 6.3 Con trol eort of the system with sliding mo de con trol . . . . . . . . . 99 6.4 Con trol eort of the system with sliding mo de con trol (zo om in) . . . 100 6.5 Time resp onse of the system with sliding mo de con trol (Saturation) . 101 6.6 Con trol eort of the system with sliding mo de con trol (Saturation) . 101 6.7 Sliding Mo de Con trol Sim ulation Mo del (with PI con trol) . . . . . . . 102 6.8 Time resp onse of the system with sliding mo de con trol and PI torque con trol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.9 Con trol eort of the system with sliding mo de con trol and PI torque con trol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.10 Time resp onse of the system with sliding mo de con trol and PI torque con trol (Saturation on con trol eort) . . . . . . . . . . . . . . . . . . 104 6.11 Con trol eort of the system with sliding mo de con trol and PI torque con trol (Saturation on con trol eort) . . . . . . . . . . . . . . . . . . 104 vi Abstract Rotordynamic systems are commonly used in mec hanical elds suc h as mac hining to ols, ywheels, and turbine engines. Some studies ha v e sho w ed that the h ysteretic jump phenomena o ccurs in the rotating sp eed and the p o w er consumption of the rotor during the pro cess of acceleration around the natural frequency due to the vibration caused b y the mass im balance. This sudden jump of the rotating sp eed results in a signican t deterioration in the system’s p erformance. Presen ted here are the feedbac k con trol design approac hes for the exible rotating system with v arying v elo cit y and the uncertain ties of the system parameters consid- ered in the design pro cess. First, it is sho wn that the decoupling from the driving torque equation and the exible motion of the system b y assuming constan t rotat- ing sp eed not only limit the application of mec hanical mo dels, but also lead to an inadequate stabilit y estimation of the actual system. Second, the dynamics and b eha vior of linearized rotor system with torque PI con troller has b een prop osed. The stabilit y and conditions of the linearized system are studied and sim ulated with dieren t com binations of system parameters. The results sho w that the larger eccen tricit y of the rotating disk causes larger instabilit y region and b y adding PI torque con trol the p erformance of the linearized system can b e impro v ed. Last, to reduce the eect from the jump phenomena, the nonlinear feedbac k con- trol la ws are dev elop ed b y using Ly apuno v redesign metho d and sliding mo de con trol. Both theoretical analysis and sim ulation results sho w that the radius of the rotating shaft and the rotating sp eed of the disk settle at the desire equilibrium p oin t during the acceleration pro cess around the system’s natural frequency with the nonlinear feedbac k con trol la ws applied. vii 1 Introduction Rotating mac hinery is commonly used in mec hanical elds suc h as mac hining to ols, ywheels, turbine engines and information storage devices. [51] Due to the cen trifugal force from eccen tricit y of the rotating disk, vibration caused b y mass im balance results in a signican t deterioration in system’s p erformance.[48] Cen trifugal forces due to mass im balance gro w with the rotating sp eed and the curren t trend of the applications of the rotor system aims on higher op erational sp eeds. In some systems, the rotating sp eed is ab o v e the ma jor critical sp eed, whic h is the angular v elo cit y that excites at the natural frequency of a rotating ob ject. F or example, a steam turbine generator can reac h ab out t wice the rst critical sp eed, and an aircraft gas turbine is op erated in the sp eed range ab o v e the second or third critical sp eed.[47] In suc h rotating mac hines, the vibration tends to b e larger in the pro cess of acceleration and causes the uctuation of the sp eed around the system’s natural frequencies. The dynamic b eha vior of the whirling motion has b een studied with the Jecott rotor mo del[19, 3, 2, 20]. In [15], it is sho wn that the h ysteretic jump phenomena exist in the rotating sp eed and the p o w er consumption of the rotor due to the whirling motion around the natural frequency . The jump of rotating sp eed means that a sudden c hange of the rotating sp eed o ccurs under mec hanical resonance. This jump not only results the v ariation of the p o w er consumption, but also damages the system’s structure sometimes. Therefore, the vibration con trol of the exible rotating system is essen tial in impro ving mac hining surface nish, ac hieving longer to ol and b earing life in high-sp eed mac hine to ols. The vibration con trol of rotating mac hinery can b e categorized in to passiv e con trol and activ e con trol. In passiv e con trol metho ds, mass, stiness and damping are xed at the design stage; while dynamic forces are applied to the system to minimize the vibration in activ e con trol. Although an activ e con trol system migh t b e more com- plicated than a system with passiv e con trol la w, activ e con trol has man y adv an tages suc h as that it can b e adjusted according to the vibration c haracteristic during the op eration. Th us, the activ e vibration tec hnique is m uc h more exible than passiv e vibration.[51] There are t w o ma jor categories in activ e vibration con trol tec hniques for rotat- ing mac hinery: activ e balancing tec hniques whic h adjust the mass distribution b y a mass redistribution actuator and direct activ e vibration con trol tec hniques apply- ing a lateral con trol force to the rotor. Balancing approac hes include o-line meth- o ds that use either mo dal balancing metho d[31, 28, 12] or the inuence co ecien t metho d[40, 29, 7]. Mo dal balancing metho d are easy to implemen t, but it can only b e 1 applied to the rigid-shaft system or lo w-sp eed rotors. The inuence co ecien t metho d is an en tirely exp erimen tal pro cedure dealing with the exible rotating systems, but eac h exp erimen t result can only b e used for a xed rotating sp eed. Another balanc- ing tec hnique, real-time activ e balancing metho d, uses serv omotors that redistribute balancing masses during the system’s op eration and the vibrations are measured at the supp orting b earings[44, 45, 46, 13, 14, 23]. It should b e noted that in dev eloping these balancing tec hnique, it is assumed that the system op erates at constan t sp eed and the system c haracteristics, suc h as inuence co ecien ts, are kno wn for dieren t ranges of rotating sp eed and are stored in con troller’s memory . A v ariet y of adaptiv e con trol tec hniques w as prop osed to p erform the balancing when the sp eed of rotation v aries[33, 49, 50]. Although these balancing tec hniques adapt to v arying rotational sp eeds, they ignore the nonlinear coupling b et w een the vibrations and the driving torque is required. Moreo v er, the abilit y of the torque actuator, e.g electric motor, to reac h and main tain the commanded sp eed is not part of the analysis. This issue is critically imp ortan t for the tasks where the op erating sp eed v aries and the designs of high p erformance rotating systems.[51] As discussed ab o v e, the inheren t c haracteristic of exible rotating systems with a v arying rotational sp eed is the nonlinear coupling b et w een the con trol torque and vibration motion[1, 10, 16, 27, 41, 20, 47, 8]. In activ e vibration con trol eld, man y studies of rotating system are based on the assumption of a constan t rotating sp eed to simplify the analysis, whic h also implies that a driving torque of a rotating system is decoupled from its dynamic b eha vior[8, 30, 21, 22, 11]. This assumption leads to the decoupling of the system’s exible motion from rotational degree of freedom. Ho w ev er, in man y applications the assumption of constan t rotating sp eed is not v alid. F or example, when a large turbine starts up, it passes through sev eral critical sp eeds at whic h the vibration tends to gro w. In the case of jet gh ters, the rotating sp eed of the turb o engine is v arying due to fast maneuv ers during op eration. This decoupling of the system equation has limited the application of the con trol metho d. Th us, this pap er is mainly fo cusing on a con trol design metho d of exible systems undergoing v arying rotating sp eed. A recen t researc h ab out a torque-based con trol metho d of whirling motion in a rotating electric mac hineries dev elop ed b y K. Inoue, S. Y amamoto et al. in 2003[15]. The h ysteretic jump phenomena of the rotating system when passing through the critical sp eed is studied b y their exp erimen t. An activ e con trol design metho d for the nonlinear rotating system to comp ensate the vibration with a v arying rotating sp eed is also prop osed. The results of b oth sim ulations and exp erimen ts sho w that 2 the vibration from whirling motion can b e reduced eectiv ely . Ho w ev er, the con trol torque presen ted in the pap er is miscalculated with unmatc hed units in the pap er. Besides, the con trol metho d is required to kno w the exact system parameters and the uncertain ties of the system ha v e not b een discussed in the researc h y et. Presen ted here is a nonlinear con trol design approac h for the exible rotating system with v arying v elo cit y . Also, the uncertain ties of the system parameters are considered in the design pro cedure. It is sho wn that the decoupling of the driving torque equation and the system’s exible motions b y assuming constan t rotating sp eed not only limit the application of mec hanical mo dels, but also lead to a dieren t stabilit y conditions from the actual system. W e found that some terms app earing in the driving torque equation suc h as the damping of the rotating disk w ould reduce the eect of the vibrations from the system’s exible motions through the coupling of the equations. Moreo v er, the inheren t coupling b et w een the rotational and exible motions can b e exploited to stabilize the system b y con trolling the rotational sp eed of the system. In this researc h, a prop ortional-in tegral feedbac k con troller is applied to the nominal driving torque equation to suppress the vibrations from exible motions and guaran tee that the rotating sp eed reac hes the input v alue under some conditions. T o accoun t for the uncertain ties existing in the actual system whic h is usually from the aging, disturbance and mo deling errors, an additional con trol is designed b y using Ly apuno v Redesign concepts and sliding mo de con trol metho d. In Ly apuno v redesign metho d, the con trol la w deriv ation con tains t w o parts: con trol la ws for nominal system and for p erturb ed system. The rst requiremen t for Ly apuno v redesign is to design the con trol la w whic h stabilizes the nominal system and pro v e the stabilit y b y Ly apuno v function, whic h will b e used in the con trol design of p erturb ed system. Ho w ev er, it could b e dicult to c ho ose Ly apuno v function candidate directly sometimes, w e add addtional con trol terms in the nominal con troller whic h shap e the system to a desire system so that the Ly apuno v function can b e found.[4, 6, 5, 24, 25] In sliding mo de con trol design, the sliding manifold S = 0 is c hosen at b eginning and the reduced-order system is required to b e stable. Similar to Ly apuno v redesign metho d, the sliding mo de con trol la w also con tains t w o parts - nominal and p erturb ed system con trollers, whic h ensures that the tra jectory on the manifold S = 0 will con v erge asymptotically to the origin. In addition, all tra jectories are guaran teed to reac h the manifold S = 0 in nite time and sta y on it for all future time. Both theoretical stabilit y analysis and sim ulation results sho w that the vibration o ccurred during the acceleration of rotating sp eed around the system’s natural frequency can b e reduced b y applying the robust con trols b y these t w o design metho ds. Whic h the 3 radius of the rotating shaft and the rotating sp eed of the disk settle at the desire equilibrium p oin ts within a nite time.[38, 36, 39, 42, 43, 35, 34] In the follo wing sections, the nonlinear con trol design pro cedure is prop osed for the nonlinear rotating system with uncertain ties b y com bining feedbac k con trol theory and Ly apuno v Redesign metho d to stabilize the system from vibrations and uncertain- ties. In Chapter 2, the mathematical bac kground and the con trol stabilit y theorem are in tro duced. In Chapter 3, the mo del of rotor system is deriv ed and the system b eha vior has b een studied. The system dynamics analysis of the rotor system with a PI torque con trol is studied in Chapter 4. The con trol la w dev elopmen t of Ly apuno v redesign is presen ted in Chapter 5 and the con trol design using sliding mo de metho d is sho wn in Chapter 6. Last, the future w ork topics are discussed in Chapter 7. 4 2 Preliminaries The follo wing in tro ductions of mathematical bac kground and stabilit y theorems are from [18, 26, 37]. 2.1 Mathematical Background 2.1.1 Norm In this w ork, the norm is dened as the follo wing: Consider the ve ctor sp ac e R n . F or e ach, 1p1, the functionkk p , known as the p-norm in ,wher e kxk p def = (jx 1 j p + +jx n j p ) 1=p : In p articular kxk 1 def =jx 1 j + +jx n j: kxk 2 def = q jx 1 j 2 + +jx n j 2 : The1-norm is dene d as fol lows: kxk p def = max i jx i j: 2.1.2 Quadratic F orms L et A2R nn b e a symmetric matrix and let x2R n . Theorem 1. Each of the fol lowing tests is ne c essary and sucient for the r e al symmetric A to b e p ositiv e denite : 1. x T Ax> 08x6= 0. 2. Al l the eigenvalues of A, i (A) satisfy i (A)> 0; i = 1:::n . 3. Al l the princip al minors of A, det(A i )> 0; i = 1;:::n. 4. Ther e exist a matrix W , which is nonsingular such that A =W T W . 5 Theorem 2. Each of the fol lowing tests is ne c essary and sucient for the r e al symmetric A to b e p ositiv e semi-denite : 1. x T Ax 08x6= 0. 2. Al l the eigenvalues of A, i (A) satisfy i (A) 0;8i = 1; 2n . 3. Al l the princip al minors of A, det(A i ) 0;8i = 1; 2;n. 4. Ther e exist a matrix W , p ossibly singular such that A =W T W . Theorem 3. Each of the fol lowing tests is ne c essary and sucient for the r e al symmetric A to b e negativ e denite : 1. x T Ax> 08x6= 0. 2. Al l the eigenvalues of A, i (A) satisfy i (A)< 0;8i = 1; 2n . 3. The princip al minors of A, have alternation for i = 1; 2n starting with a 11 < 0. 4. The matrixA is p ositiv e The matrix A is said to b e indenite if and only if it has p ositiv e and negativ e eigen v alues. Theorem 4. (Ra yleigh Inequalit y) Consider a nonsingular symmetric matrix Q2 R nn , and let min and max b e r esp e ctively the minimum and maximum eigenvalues of Q. Under these c onditions, for any x2R n , min (Q)kxk 2 x T Qx max (Q)kxk 2 : 2.1.3 Sets Consider a set AR n . 6 Denition 1. (Neigh b orho o d) A neighb orho o d of a p oint p2 A R n is the set B r (p) dene d as fol lows: B r (p) = x2R n :kxpk<r : Denition 2. (Op en Set) A set AR n is said to b e op en if for every p2A one c an nd a neighb orho o d B r (p)A. Denition 3. (Bounded set) A set AR n is said to b e b ounde d if ther e exists a r e al numb er M > 0 such that kxk<M 8x2A: Denition 4. (Compact set) A set A R n is said to b e c omp act if it is close d and b ounde d. 2.1.4 Lipsc hitz Con tin uit y Denition 5. A function f(x) : R n !R m is said to b e lo c al ly Lipschitz on D if every p oint of D has a neighb orho o d D 0 R n over which the r estriction of f with domain D 1 satises kf(x 1 )f(x 2 )kLkx 1 x 2 k: (2.1) It is said to b e Lipschitz on an op en set DR n if it satises (2.1) for al l x 1 ;x 2 2D with the same Lipschitz c onstant. Final ly, f is said to b e glob al ly Lipschitz if it satises (2.1) with D =R n . Theorem 2. If a function f :R n R m is c ontinuously dier entiable on an op en set DR n ,then it is lo c al ly Lipschitz on D . 2.2 Lyapunov Stability 2.2.1 Stabilit y Denition Denition 7. Consider the autonomous system _ x =f(x) f :D!R n (2.2) 7 wher e D is an op en and c onne cte d subset of R n and f is a lo c al ly Lipschitz map fr om D into R n . The e quilibrium p oint x =x e of the system (2.2) is said to b e stable if for e ach > 0,9 =()> 0 kx(0)x e k< ) lim x!1 x(t) =x e 8tt 0 Otherwise, the e quilibrium p oint is said to b e unstable. Denition 8. The e quilibrium p oint x =x e of the system (2.2) is said to b e c onver- gent if ther e exists 1 > 0: kx(0)x e k< ) lim x!1 x(t) =x e : Denition 9. The e quilibrium p oint x =x e of the system (2.2) is said to b e asymp- totic al ly stable if it is b oth stable and c onver gent. Denition 10. A function V :D!R is said to b e p ositive semi denite in D if it satises the fol lowing c onditions: 1. 02D and V (0) = 0. 2. V (x) 0; 8x in Df0g. V :D!R is said to b e p ositive denite in D if c ondition 2 is r eplac e d by the fol lowing c ondition V (x)> 0 in Df0g: Final ly, V : D! R is said to b e ne gative denite (semi denite) in D ifV is p ositive denite (semi denite). 2.2.2 Stabilit y Theorem of Equilibrium P oin t Theorem 3. (Ly apuno v Stabilit y Theorem) L et x = 0 b e an e quilibrium p oint of _ x =f(x), f :D!R n , and let V :D!R b e a c ontinuously dier entiable function such that 1. V (0) = 0, 8 2. V (x)> 0 in Df0g, 3. _ V (x) 0 in Df0g, thus x = 0 is stable. Theorem 4. (Asymptotic Stabilit y Theorem) Under the c onditions of The or em 3 , if V () is such that 1. V (0) = 0, 2. V (x)> 0 in Df0g, 3. _ V (x)< 0 in Df0g, thus x = 0 is asymptotic al ly stable. Denition 11. A set M is said to b e an invariant set with r esp e ct to the dynamic al system _ x =f(x) if: x(0)2M ) x(t)2M 8t2R + : In other wor ds, M is the set of p oints such that if a solution of _ x = f(x) b elongs to M at t = 0, then it stays in M for al l futur e time. Theorem 5. (LaSalle’s In v ariance Principle) L et V :D!R b e a c ontinuously dier entiable function and assume that 1. MD is a c omp act set, invariant with r esp e ct to the solutions of (2.2). 2. _ V (x)< 0 in M . 3. E :fx : x2 M and _ V = 0g; that is, E is the set of al l p oints of M such that _ V = 0. 4. N is the lar gest invariant set in E . Then every solution starting in M appr o aches N as t!1. Theorem 6. (Barbashin-Kraso vski-LaSalle Theorem) L et x = 0 b e an e quilib- rium p oint for (2.2). L et V :D!R b e a c ontinuously dier entiable p ositive denite function on a domain D c ontaining the origin x = 0, such that _ V (x) 0 in D . L et S =fx2Dj _ V (x) = 0g and supp ose that no solution c an stay identic al ly in S , other than the trivial solution x(t) 0. Then, the origin is asymptotic al ly stable. 9 2.2.3 Asymptotic Stabilit y of Linear System F or the linear time-in v arian t system _ x =Ax (2.3) has an equilibrium p oin t at the origin. Theorem 7. The e quilibrium p oint x = 0 of _ x = Ax is stable if and only if al l eigenvalues of A satisfy Re i 0 and for every eigenvalue Re i = 0 with and alge- br aic multiplicity q i 2, rank(A i I) =nq i , wher e n is the dimension of x. The e quilibrium p oint x = 0 is (glob al ly) asymptotic al ly stable if and only if al l eigenvalues of A satisfy Re i < 0. Theorem 8. The e quilibrium p oint x = 0 of _ x = Ax is asymptotic al ly stable if and only if for any given p ositive denite symmetric matrix Q ther e exists a p ositive denite symmetric matrix P that satises the Lyapunov e quation PA +A T P =Q (2.4) If A is Hurwitz (i.e. Re i < 0 for all eigen v alues of A), then P is the unique solution of (2.4). Theorem 9. L et x = 0 b e an e quilibrium p oint for the nonline ar system _ x =f(x) Wher e f : D ! R n is c ontinuously dier entiable and D is a neighb orho o d of the origin. L et A = @f @x (x)j x=0 then, 1. The origin is asymptotic al ly stable if Re i < 0 for al l eigenvalues of A. 2. The origin is unstable if Re i > 0 for one or mor e of the eigenvalues of A. 10 2.3 Boundedness and Ultimate Boundedness Denition 12. A c ontinuous function : [0;)! [0;1) is said to b elong to classK if it is strictly incr e asing and (0) = 0. It is said to b elong to class K 1 if =1and (r)!1 as r!1. Denition 13. A c ontinuous function : [0;) [0;1)! [0;1) is said to b elong to classKL if, for e ach xe d s the mapping (r;s) b elongs to classK with r esp e ct to r and for e ach xe d r , the mapping (r;s) is de cr e asing with r esp e ct to s and(r;s)! 0 as s!1. Denition 14. The solutions to 2.2 ar e uniformly b ounde d if ther e exists a p ositive c onstant c, indep endent of t 0 0, and for every 2 (0;c), ther e is =()> 0, indep endent of t 0 such that kx(t 0 )k)kx(t)k; 8tt 0 (2.5) glob al ly uniformly b ounde d if 2.5 holds for arbitr arily lar ge a; uniformly ultimately b ounde d with ultimate b ound b if ther e exist p ositive c onstants b and c, indep endent of t 0 0, and for every 2 (0;c), ther e is T =T (a;b) 0, indep endent of t 0 such that kx(t 0 )k)kx(t)kb; 8tt 0 +T (2.6) glob al ly uniformly ultimately b ounde d if 2.6 holds for arbitr arily lar ge a. Consider the follo wing system _ x =f(t;x) +G(t;x)[u;(t;x;u)] where x2R n is the state and u2R p is the con trol input. The function f , G, and are dened for (t;x;u)2 [0;1)DR n , where DR n is a domain that con tains the origin. The function f andG are kno wn precisely , while is a b ounded but kno wn function that com bines the v arious uncertain terms. 11 Theorem 10. (Ultimate Boundedness) L et D R n b e a domain that c ontains the origin and V :D!R b e a c ontinuously dier entiable function such that 1. 1 (kxk)V (x) 2 kxk 2. @v @x f(x)W 3 (x) 8 kxk>> 0 for al l x2 D wher e 1 and 2 ar e class K functions and W 3 (x) is a c ontinuous p ositive denite function. T ake r> 0 such that B r D and supp ose that < 1 2 ( 1 (r)) Then ther e exists a class KL function and for every initial state x(t 0 ), satisfying kx(t 0 )k, ther e is T > 0 (dep end on x(t 0 ) and) such that the solution of _ x =f(x) satises 1.kx(t)k(kx(t 0 )k;tt 0 ) 8t 0 <t<t 0 +T 2.kx(t)kr = 1 1 ( 2 ()) 8tt 0 +T Thus, the solution x(t) is uniformly b ounde d 8tt 0 and uniformly ultimately b ounde d with the ultimate b ound r = 1 1 ( 2 ()). Mor e over, if D = R n and 1 b elongs to class K 1 , then these c onditions hold for any initial state x(t 0 ) with no r estrictions on how lar ge c an b e. 12 3 System Modeling 3.1 Governing Equations The follo wing deriv ation of the go v erning equations is from [48]. In Fig. 3.1, a b o dy (a rigid-b o dy system supp orted b y springs or an elastic con tin uum) is driv en b y a torque T 0 (t), and rotates ab out the X -axis of a xed co ordinate system OXYZ , where line CC 0 is either the geometrically cen tral line of a rigid-b o dy system or the neutral line of an elastic deformation. The dynamic resp onse of suc h a rotating system can b e describ ed b y a set of n generalized co ordinates q 1 (t);q 2 (t); ;q n (t), whic h c haracterize the vibration of the b o dy due to its elastic deformation, and the rotation (t) of the b o dy ab out line CC 0 . Selection of q 1 (t);q 2 (t); ;q n (t) rela ys either on the n um b er of degrees of freedom for a rigid-b o dy system, or on the discretization pro cedure used for an elastic con tin uum. In this section, the go v erning equations of the rotating system are deriv ed b y Lagrange’s equation. Figure 3.1: Rotating Bo dy 3.1.1 Energy F unctions The kinetic energy of the rotating system is written as T = 1 2 f _ qg T [M 11 (fqg;)]f _ qg + 1 2 _ 2 [M 22 (fqg;)] + _ f _ qg T [M 12 (fqg;)] (3.1) wherefqg is the v ector of generalized co ordinates fq(t)g = (q 1 (t);q 2 (t); ;q n (t)) T (3.2) 13 f _ qg = dfqg =dt, [M 11 ]2 R nxn , [M 21 ]2 R nx1 , and M 22 is a scalar. The [M ] are suc h that the mass matrix " [M 11 ] [M 12 ] [M 12 ] T [M 22 ] # is symmetric and p ositiv e denite for an y fqg and . The p oten tial energy of the b o dy is a function offqg and : V =(fqg;) (3.3) Denote the damping forces applied in the direction of fqg b y a v ector F (d) q = n q (fqg;f _ qg;; _ ) o (3.4) and the damping torque in the direction of b y F (d) = (fqg;f _ qg;; _ ): (3.5) The external force in the direction of generalized co ordinates are represen ted b y a v ector n F (e) q (t) o ; the external force in the direction of rotation is the driving torque T 0 (t) (see Fig. 3.1). 3.1.2 Vibration and T orque Equation By Lagrange’s equation, the equation of motion of the b o dy in the direction of fqg is d dt @T @f _ qg @T @fqg + @ @fqg +f q g = F (e) q (t) (3.6) and the equation of motion of the b o dy in the direction of is d dt @T @ n _ o @T @fg + @ @fg +f g =T 0 (t) (3.7) Eq. (3.6) shall b e called vibration equation b ecause it describ es the vibration of the rotating system. Eq. (3.7) shall b e referred to as the torque equation as it go v erns the rotation of the b o dy driv en b y a torque. substitute Eqs. (3.1), (3.3) and (3.4) in to Eq. (3.6), to obtain the vibration equation of the rotating system 14 [M 11 ]f qg + @[M 11 ] @ _ + @[M 12 ] @q _ @[M 12 ] T @q _ + [A] 1 2 [B] f _ qg + [M 12 ] + @[M 12 ] @ 1 2 @[M 22 ] @q _ 2 + @ @fqg +f q g = F (e) q (t) (3.8) where [A] = n X k=1 _ q k @[M 11 ] @q k ; [B] = [ @[M 11 ] @q 1 f _ qg @[M 11 ] @q n f _ qg] T @[M 12 ] @q = [ @[M 12 ] @q 1 @[M 12 ] @q n ] @[M 22 ] @q = ( @[M 22 ] @q 1 @[M 22 ] @q 1 ) T ; @ @fqg = ( @ @q 1 @ @q n ) T Similarly , plugging Eqs. (3.1), (3.3) and (3.5) in to Eq. (3.7) giv es the torque equation of the rotating system [M 22 ] + [M 12 ] T f qg + _ @[M 22 ] @q f _ qg + 1 2 @[M 22 ] @ _ 2 +f _ qg T @[M 12 ] @fqg 1 2 @[M 11 ] @ f _ qg + @ @ +f g =fT 0 (t)g (3.9) Th us, the vibration and rotation of the b o dy sho wn in Fig. 3.1is go v erned b y Eqs. (3.8) and (3.9). These equations are coupled nonlinear dieren tial equations, whic h in general are solv ed n umerically . 3.1.3 A Rotating System with Linear Elastic Deformation Assume that a rotating b o dy undergo es linear elastic deformation, whic h meets the follo wing three co ordinates. (C1) The kinetic energy of the b o dy is quadratic inf _ qg, _ and _ fqg . In particular, [M 11 ] = [M] [M 12 ] = [R]fqg + [ ^ M 12 ()] [M 22 ] =fqg T [S]fqg +m 22 15 where [M] and [S] are symmetric constan t matrices, [R] is a constan t matrix, m 22 is a constan t, and v ector [ ^ M 12 ()] is a function of . (C2) The p oten tial energy of the b o dy has the form = 1 2 fqg T [K]fqg where the stiness matrix [K] is a constan t matrix. (C3) The damping force in the direction offqg are of the linear form f q g = [D 1 ]f _ qg + _ [D 2 ]fqg where [D1] and [D2] are constan t matrices describing b oth non-rotating damping and rotating damping. (The damping force in the direction of can b e nonlinear.) With condition (C1) to (C3), the go v erning equations are reduced to [M 11 ]f qg + _ [R] [R] T f _ qg + [R]fqg + [ ^ M 12 ] + _ 2 ( d d [ ^ M 12 ] [S]fqg) + [K]fqg + [D 1 ]f _ qg + _ [D 2 ]fqg = F (e) q (t) (3.10) [M 22 ] + [M 12 ] T f qg + 2 fqg T [S]f _ qg +f _ qg T [R]f _ qg +f g =T 0 (t) (3.11) Note that the vibration equation (3.10) is nonlinear ev en though the b o dy has linear deformation. Let b o dy b e rotating at a constan t sp eed , namely , = t, _ = and = 0. The go v erning equations (3.10) and (3.11) are reduced to [M]f qg + ([D 1 ] + [G( )])f _ qg + ([K] 2 [S] + [D 2 ])fqg = n F (e) q (t) o 2 d dt [ ^ M 12 ( t)] (3.12) [M 12 ] T f qg + 2 fqg T [S]f _ qg +f _ qg T [R]f _ qg +f g =T 0 (t) (3.13) where [G( )] = ([R] [R] T ) is a sk ew-symmetric matrix. Eq. (3.12) is a linear equation that is commonly used in mo deling of exible rotating system s in linear vibration. The torque equation (3.13) correlates the driving torque to the vibration of 16 the rotating b o dy . This torque-resp onse decoupling b y assuming constan t rotating sp eed is further in v estigated in Section 3.4.2. 3.2 Jecott Rotor Model In the vibration analysis of rotating mac hinery , a lump ed-parameter system in whic h a rigid disk is moun ted on a massless elastic shaft is often adopted as a mathematical mo del. This simplied mo del that can b e used to study the exual b eha vior of rotors consists of a p oin t mass attac hed to a massless shaft sho wn in Fig. 3.2. As its dynamic b eha vior w as deeply studied in a pap er published b y Jecott in 1919 [11, 17], it is often referred to as Jec ott r otor. Although the Jecott rotor mo del is a simplication of real-w orld rotors, it retains some basic c haracteristics and allo w us to gain a qualitativ e insigh t in to imp ortan t phenomena t ypical of rotor dynamics. The mo deling of the Jecott rotor system is sho wn as the follo wing and it is used in Chapter 4,5 and 6 with con trol metho ds applied. Figure 3.2: 2DOF deection mo del 3.2.1 Equations of Motion The dynamical b eha vior of whirling motion has b een studied with the Jecott rotor. The follo wing equations are dev elop ed in [47] and [11]. Fig. 3.2 sho ws a v ertical rotor in lateral whirling motion and a co ordinate system. The gra vit y force is not considered. The z -axis of the co ordinate systemOxyz coincides with the b earing cen terline, whic h connects the cen ters of the upp er and lo w er b earings. The cen terline of an elastic shaft passes through the geometrical cen ter M(x;y). The cen ter of the gra vit y G(x G ;y G ) is separated b y eccen tricit y e from p oin t M . The rotational angle of the rotor is denoted b y the angle , whic h is the angle b et w een line MG and the x-axis and the rotational angle of the rotating shaft is denoted b y the angle , whic h is the angle b et w een line OM and the x-axis. 17 Figure 3.3: 2DOF rotor system for lateral vibration Let the mass of the disk b e m, the p olar momen t of inertia b e J P . Note that a driving torque is applied to the rotor b y the shaft and that, although the Jecott rotor is considered to b e a p oin t mass, a p olar momen t of inertia J P is asso ciated with it. This is needed to study the acceleration of the rotor once the external torque is imp osed. The v ariables, x andy are the displacemen ts of the rotating shaft from the origin along x and y axes. By follo wing the step of c ho osing as generalized co ordinates those of p oin t C . The p osition and v elo cit y of p oin t P can b e expressed as GO = ( x G y G ) = ( x +e cos() y +e sin() ) V G = ( _ x G _ y G ) = ( _ xe _ sin() _ y +e _ cos() ) The kinetic energy T and p oten tial energy U are, resp ectiv ely , T = 1 2 m( _ x 2 G + _ y 2 G ) + 1 2 J P _ 2 = 1 2 mf _ x 2 + _ y 2 +e 2 _ 2 + 2e _ [ _ x sin() + _ y cos()]g + 1 2 J P _ 2 U = 1 2 k(x 2 +y 2 ) Assuming that an external force acts on p oin t G inxy -plane, to obtain the gener- alized force Q i to b e in tro duced in to the equation of motion, the virtual displacemen t 18 of p oin t G can b e written ( x G y G ) = ( xe sin() y +e cos() ) The virtual w ork of the force F with comp onen ts F x and F y acting on p oin t G and of the input momen t T d is L =F x x +F y y + [T d F x e sin() +F y e cos()] Where the forces F x and F y can b e considered as forces from damping terms, F x =d _ x andF y =d _ y , where d is damping co ecien t in x,y directions suc h that L =d _ xx +d _ yy + [T d +d _ xe sin()d _ ye cos()] The equation of motion can then b e obtained from the Lagrange equation, written in the form d dt ( @(TU) @ _ q i ) @(TU) @q i =Q i (3.14) The generalized forces are Q 1 =d _ x, Q 2 =d _ y and Q 3 = [T d +d _ xe sin() d _ ye cos()]. By substituting Q i in to equations the horizon tal displacemen t x of the rotating shaft is presen ted: d dt ( @(TU) @ _ x ) @(TU) @x = Q 1 m xme[ _ 2 cos() + sin()] +kx =d _ x (3.15) The equation of motion for the v ertical displacemen t of the rotating shaft y is : d dt ( @(TU) @ _ y ) @(TU) @y = Q 2 m yme[ _ 2 sin() cos()] +ky =d _ y (3.16) The equation of motion for the angular displacemen t of the rotating disk is : d dt ( @(TU) @ _ ) @(TU) @ = Q 3 19 (J P +me 2 ) +me[ x sin() + y cos()] = [T d +d _ xe sin()d _ ye cos()] By substituting x and y in to the equation ab o v e, it can b e sho wn in simplied form J P +d r _ +ke[x sin()ycos()] =T d (3.17) where d r is the damping co ecien t of the rotation and T d is the input torque. The displacemen t along x and y axis can also b e represen ted b y the radius of the rotating shaft r and the rotating angle of the shaft in the follo wing gure. The reason of this transformation is that the equilibrium p oin t of the system go es to a constan t for the radius r , but the displacemen t x and y sho ws harmonic oscillation in steady-state. When Ly apuno v stabilit y theory is applied to determine the system stabilit y , a constan t equilibrium p oin t is required. The system dynamic b eha vior is discussed in Section 3.4. Figure 3.4: 2DOF deection mo del (p olar co ordinates) By using the transformations x =r cos ; y =r sin these equations are represen ted b y the p olar co ordinates (r; ) as follo ws: m rmr _ 2 +d _ r +kr = me _ 2 cos( )me sin( ) (3.18) mr + 2m _ r _ +dr _ =me cos( )me _ 2 sin( ) (3.19) 20 J p +d r _ ker sin( ) =T d (3.20) 3.2.2 Dimensionless Equations of Motion The equation of motion can b e rewritten in dimensionless form in order to normalized the parameters suc h as natural frequency , damping co ecien t and spring stiness. F rom Chain rule in calculus, d dt = d d ( d dt ) Let =! n t, where ! n is normalized natural frequency whic h resonance o ccurs at ! n = 1. Suc h that d dt =! n ( d d ); ( d dt ) 2 =! 2 n ( d d ) 2 Here, w e dene the follo wing dimensionless quan tities using the eccen tricit y e as a reference quan tit y: =t p k=m r =r 0 =r g where r 0 represen ts the radius of the rotating shaft with dimension and r g is the radius of gyration. The system parameters can also b e written as: d m = 2! n ; k m =! 2 n d r J p = 2 r ! J ; kr 2 g J p =! 2 J where is the damping ratio of the rotating shaft and r is the damping ratio of rotating disk. Moreo v er, w e dene the follo wing parameters as: J P =mr 2 g and e = ( e 0 r g ); similarly , e 0 stands for the eccen tricit y of the rotating disk with dimension. suc h that 21 ! 2 J ! 2 n = kr 2 g mr 2 g k m = 1 By substituting the parameters ab o v e and the dimensionless quan tities in to the Eqs. (3.19) - (3.20), the dimensionless equations of motion are giv en as follo ws r =r _ 2 2 _ rr +e _ 2 cos( )e sin( ) (3.21) r =2 _ r _ 2r _ e cos( )e _ 2 sin( ) (3.22) =2 r _ +er sin( ) +T d (3.23) 3.3 Magnetic Bearings A ctiv e magnetic b earings (AMB) generate forces through magnetic elds. There is no con tact b et w een b earing and rotor, and this p ermits op eration with no lubrication and no mec hanical w ear. A sp ecial adv an tage of these unique b earings is that the rotor dynamics can b e con trolled activ ely through the b earings.[32] Although the dynamics of magnetic b earing is not considered in this researc h, it pro vides an idea of generating forces in x and y direction of the rotating shaft and mak es the con trol of radius of the rotating shaft p ossible in Chapter 5 and 6. The follo wing materials are referred to [32] to illustrate ho w activ e magnetic b ear- ing generate forces on x and y direction. Fig. 3.5 displa ys a rigid rotor together with the b earing magnets and the p osition sensors. This setup basically corresp onds to a practical and most straigh tforw ard implemen tation of suc h a system. Dieren tly from the one DOF system its output signals, i.e. the measured rotor displacemen ts x seA and x seB , are comprised in the output v ector y , yielding the follo wing expressions: M q +G _ q = Bu f (3.24) y = Cq (3.25) 22 M = 2 6 6 6 6 4 I x 0 0 0 0 m 0 0 0 0 I y 0 0 0 0 m 3 7 7 7 7 5 ; G = 2 6 6 6 6 4 0 0 I z 0 0 0 0 0 I z 0 0 0 0 0 0 0 3 7 7 7 7 5 ; (3.26) B = 2 6 6 6 6 4 a b 0 0 1 1 0 0 0 0 a b 0 0 1 1 3 7 7 7 7 5 (3.27) q = (;x;;y) T ; u f = (f xA ;f xB ;f yA ;f yB ) T (3.28) C = 2 6 6 6 6 4 c 1 0 0 d 1 0 0 0 0 c 1 0 0 d 1 3 7 7 7 7 5 ; y = (x seA ;x seB ;y seA ;y seB ) T (3.29) Figure 3.5: The rigid rotor equipp ed with b earing magnets and sensors By closing the con trol lo op, the magnetic b earing force u f can b e describ ed as a 23 linearized function of the rotor displacemen ts in the b earing and the coil curren ts, in v olving the for c e/curr ent factork i and the for c e/displac ement factork s . In general, these constan ts are not equal in eac h b earing, ho w ev er, they are equal in b oth x and y directions, since the b earing is usually symmetric. Hence, the follo wing relationship results for the force v ector u f used in (3.24): u f = 2 6 6 6 6 4 f xA f xB f yA f yB 3 7 7 7 7 5 = 2 6 6 6 6 4 k sA 0 0 0 0 k sB 0 0 0 0 k sA 0 0 0 0 k sB 3 7 7 7 7 5 2 6 6 6 6 4 x bA x bB y bA y bB 3 7 7 7 7 5 + 2 6 6 6 6 4 k iA 0 0 0 0 k iB 0 0 0 0 k iA 0 0 0 0 k iB 3 7 7 7 7 5 2 6 6 6 6 4 i xA i xB i yA i yB 3 7 7 7 7 5 = K s q b +K i i (3.30) The v ector q b = (x bA; x bB ;y bA ;y bB ) T in tro duced in (3.30) comprises the rotor dis- placemen ts within the magnetic b earings, whereas the v ector i = (i xA ;i xB ;i yA ;i yB ) T con tains the individual coil con trol curren ts of all four b earing magnets. By com bin- ing the rotor mo del (3.24 to 3.29) and the linearized b earing force description (3.30) w e obtain the follo wing basic matrix dieren tial equation of motion for the rigid rotor to b e levitated b y AMBs: M q +G _ q = B(K s q b +K i i) y = Cq Also, the Fig. 3.6 sho ws the structure of com bining magnetic b earings and PID con trol. The dynamic of the system is not discussed here. But this leads a future researc h direction that the mo del of magnetic b earings and the forces generated from the actuators can b e com bined with the nonlinear con trol design la w discussed in Chapter 5 and 6 as a system. 24 Figure 3.6: Decen tralize con trol structure with PID con trol 3.4 Dynamic Behavior In this section the dynamic b eha vior of the ab o v e rotor system is in v estigated. The topics of the jumping phenomenon and the eect of the equation decoupling are discussed in the follo wing subsections. 3.4.1 Jump Phenomenon The jump phenomenon of a nonlinear rotor system o ccurs in the pro cess of the sp eed alteration around the natural frequency . It means that a con tin uous c hange of rotating sp eed can not b e ac hiev ed. The follo wing deriv ation of the jump phenomenon is dev elop ed in [15]. The rotor system giv en in (3.15) - (3.17) is used as an example to illustrate the eect of the jump phenomena. The input torque T d in the Eq. (3.17) can b e c hosen as hV i , where h is a prop ortional factor and V i is the input v oltage to the driving motor. Let the angular acceleration of the disk, = 0 in the Eq. (3.17), the equations in the steady-state of the rotating sp eed are deriv ed as follo ws m x +d _ x +kx =me _ 2 cos() (3.31) m y +d _ y +ky =me _ 2 sin() (3.32) d r _ +ke[x sin()ycos()] =hV i (3.33) 25 The dieren tial equations (3.31) and (3.32) can b e solv ed with resp ect to x and y , resp ectiv ely: x = me _ 2 (km _ 2 ) 2 + (d _ ) 2 [(km _ 2 )cos() +d _ sin()] (3.34) y = me _ 2 (km _ 2 ) 2 + (d _ ) 2 [(km _ 2 )sin()d _ cos()] (3.35) By substituting (3.34) and (3.35) in to (3.33), the relationship b et w een input v olt- age V i and rotating sp eed of the disk _ is giv en as V i = 1 h (d r _ + mdke 2 _ 3 (km _ 2 ) 2 + (d _ ) 2 ) (3.36) T able 1: Rotor system parameters setup The relationship b et w een V i and _ giv en in the Eq. (3.36) is plotted in Fig. (3.7) with the parameters setup in T able (1), whic h the parameters w ere established in [15]. There are t w o folds that cause the jump phenomena. Eq. (3.36) has t w o solutions at the p oin ts a and b, and three solutions exist in the in terv al a<V i <b where the middle solution is unstable and the others are stable. In the pro cess of acceleration, the rotating sp eed tends to gro w slo wly and settle at the natural frequency (220 rad/s) as the input v oltage increasing in the zone b et w een the p oin ts a and b. When the input v oltage is larger than p oin t b, the rotating sp eed suddenly jumps to a higher sp eed around 360 rad/s. The same situation happ ens in the pro cess of deceleration. The rotating sp eed drops at the p oin t a. Hence, the bifurcations at the p oin ts a and b are a saddle-no de bifurcation whic h cause the jump phenomena. 26 Figure 3.7: Jump phenomena W e c ho ose a input v alue(1.6 v olt) from the zone b et w een p oin ts a andb in Fig. 3.7 to illustrate the b eha vior that the rotor system could settle at dieren t rotating sp eeds in steady state with the same input v oltage. The resp onses of t w o cases (acceleration and deceleration) are sho wn in Fig. 3.8. The b ottom curv e is the sim ulation of the time resp onse of rotating sp eed in acceleration b y c ho osing zero initial v elo cit y . The rotating sp eed reac hes the nal v alue at 220 (rad/s). While the top one is the resp onse for the deceleration pro cess, in whic h the initial v elo cit y is set at 350 rad/s. F rom the plot, the rotating sp eed settles at 303 (rad/s), whic h is m uc h higher than the v alue at Point 1 . It is sho wn that the con tin uous acceleration of the rotating sp eed around the natural frequency is not ac hiev able in the system without a closed-lo op con trol eort. The sudden jump of the sp eed not only causes the instabilit y of the system, but also increases the p o w er consumption. In order to main tain the system running at precise rotating sp eed and reduce the large uctuation of the whirling amplitude, some appropriate con trol metho ds are required to suppress the whirling motion. 27 Figure 3.8: Time resp onse of the rotor system during jump phenomena 3.4.2 Eect of Equation Decoupling In the rotor system giv en in (3.21) - (3.33), w e can nd that the rst t w o equations are coupled with the torque equation (3.23) due to the term . Also, the torque equation (3.23) con tains the term er sin( ), whic h means that the torque is aected b y the radius r and the rotating angle of the shaft . In some rotor system studies [8, 30, 21, 22, 11], a constan t rotating sp eed is assumed to simplify the analysis, whic h implies that the motion of the exible rotating shaft is decoupled from its driving torque and the transien t resp onse is not considered in the pro cess of acceleration. Ho w ev er, this assumption ma y not b e applied in the mec hanical systems with frequen t c hange in sp eed or high rotating v elo cit y o v er the natural frequencies, suc h as turb o engines of air crafts. Moreo v er, w e nd that the decoupling w ould lead to a dieren t result of stabilit y prediction and migh t cause the distortion for the stabilit y analysis. The study of the decoupling b y b oth analytical and n umerical metho ds are presen ted in the follo wing. Assume that the rotating sp eed of the disk _ is equal to a constan t sp eed w c in the system (3.21) - (3.33) suc h that the rst t w o equations are completely decoupled from the torque equation. 28 r = r _ 2 2 _ rr +ew 2 c cos( w c t) (3.37) r = 2 _ r _ 2r _ ew 2 c sin( w c t) (3.38) T d = 2 r w c +er sin( w c t) (3.39) Cho ose the states x 1 =r , x 2 = , x 3 = _ r and x 4 = _ . The state-space equations of the decoupled system (3.37) and (3.38) can b e written as _ x 1 = x 3 _ x 2 = x 4 _ x 3 = x 1 (x 2 4 1) 2x 3 +ew 2 c cos(x 2 w c t) _ x 4 = 2x 3 x 4 x 1 2x 4 ew 2 c x 1 sin(x 2 w c t) Denote that x i ; i = 1; 2:::4 are the equilibrium p oin ts and the v alues can b e found b y setting _ X = 0, where X = [x 1 ; x 2 ; x 3 ; x 4 ] T . The equilibrium p oin ts of the states are sho wn as x 1 = r 0 = ew 2 c p (1w 2 c ) 2 + (2w c ) 2 x 2 = w c t; = arctan( 2w c 1w 2 c ) x 3 = 0 x 4 = w c By using the equilibrium p oin ts, the Jac obian matrix A can b e found from the 29 linearization @ @x _ Xj x=x =AX , where A = 2 6 6 6 6 4 0; 0; 1; 0 0; 0; 0; 1 w 2 c 1; w 2 c sin; 2; 2r 0 w c w c 2 r 2 0 sin; w c 2 r 0 cos; 2wc r 0 ; 2 3 7 7 7 7 5 (3.40) The stabilit y around the equilibrium p oin ts x i is in v estigated b y the R outhHurwitz stability criterion . It is sho wn in App endix A that all the co ecien ts of the c harac- teristic p olynomial and all terms in Routh arra y are p ositiv e for > 0. Th us, the decoupled system with damping is stable at all times from the stabilit y criterion. In the case of the system without an y damping ( = 0 ), some terms in the Routh arra y turn to b e zero. In other w ords, the system w ould k eep oscillating and do es not settle do wn at the equilibrium p oin ts. The result can b e v eried b y the eigen v alues of matrix A as w ell. In the Fig. 3.9, the stabilit y region of the rotating sp eed w c is sho wn. The y-axis is just the ag set in the Matlab co de so that the stabilit y region of the rotating sp eed w c onx- axis w ould b e easier to c hec k. The stabilit y region plot is generated according to the eigen v alues of the system and the result demonstrates that the eigen v alues of the decoupled system for all w c ha v e negativ e real parts, so the the system is globally stable for > 0. Figure 3.9: Stabilit y Region of decoupled system In addition, the mo del is built in Sim uLink to sim ulate the time resp onse. In 30 Figure 3.10: Time resp onse of decoupled system Fig. 3.10, the time resp onse for w c = 1:5 and = 0:1 is presen ted as an example of the decoupled system b eha vior. The radius of the exible shaft con v erges to its equilibrium p oin t r 0 and the rotating sp eed of the shaft also settles at the input v alue w c = 1:5 in the steady state. The stabilit y of the decoupled system can b e easily sho wn b y the criterion. Ho w- ev er, the transien t resp onse of the system is concealed b y the decoupling metho d. The stabilit y of the actual system migh t b e distinct from the decoupled one with the v arying rotating sp eed and the torque equation considered. The stabilit y for the system (3.21) - (3.23) is studied. First, w e start with the case when the damping ratio = 0. The Routh arra y of the coupled equations is pro vided in App endix A. It is sho wn that the coupled system is stable for w c < 1 and unstable for w c 1, where w c = 1 is the natural frequency in the dimensionless equation. The stabilit y region of the coupled system is presen ted in the Fig. 3.11. It can b e found that the stabilit y regions of b oth coupled and decoupled cases are matc hed when the the systems are undamp ed. 31 Figure 3.11: Stabilit y Region of coupled system, = 0 Last, the case discussed here is when the damping > 0. F rom the terms in the Routh arra y , w e found that the stabilit y conditions are the functions of the system parameters e, , r andw c . Though the conditions can b e acquired from the criterion, there are to o man y terms to deriv e for the analytical solution. Matlab sim ulations are used as the n umerical results to nd the stabilit y region sho wn in the Fig. 3.12. A ccording to the plot, there is an unstable area b et w een w c = 1 1:5 (around the natural frequency), where the stabilit y region is v arying with the system parameters. W e can clearly see that the system is no longer globally stable as the decoupled case sho wn in Fig. 3.9. This stabilit y region can also b e v eried b y the sim ulations. The system at p oin t A Fig. 3.12 is unstable and the time resp onse of the radius and the rotating sp eed are sho wn in Fig. 3.13 with the system parameters e = 0:55, = 0:1, r = 0:1 andw c = 1:2. Another stable case at P oin t B is sho wn in Fig. 3.4.2 with the input sp eed w c = 2. The result implies that when > 0 and w c > 1, the decoupling of the system c hanges the stabilit y conditions and leads to a inaccurate prediction of the system motion. Th us, more sophisticated con trol design approac hes of the nonlinear system with v arying rotating sp eed are needed for the vibration reduction studies. The con trol design metho d and the analysis of the stabilit y condition are in v estigated in Chapter 4. 32 Figure 3.12: Stabilit y Region of coupled system, > 0 Figure 3.13: Time resp onse of coupled system (P oin t A) 33 Figure 3.14: Time resp onse of coupled system (P oin t B) 34 4 Linearized Rotor Model with PI T orque Control 4.1 System Linearization The Jecott rotor system (3.21) - (3.23) is used as an example to study the linearized system dynamics with PI torque con trol. A PI con trol is added in the torque equation T d =K P ( _ w c )K I ( _ w c ), where K P and K I are the con trol gains and w c represen ts the input rotating sp eed. The system with the con trol is sho wn as the follo wing r = r _ 2 2 _ rr +e _ 2 cos( )e sin( ) r = 2 _ r _ 2r _ e cos( )e _ 2 sin( ) = 2 r _ +er sin( )K P ( _ w c )K I ( _ w c ) Assume the equilibrium p oin ts r = r 0 , _ = w c , _ = w c , and =( ). The equations in steady state can b e deriv ed b y setting the acceleration as zero sho wn b elo w r = 0 ) 0 = r 0 w 2 c r 0 +ew 2 c cos (4.1) = 0 ) 0 = 2r 0 w c +ew 2 c sin (4.2) = 0 ) 0 = 2 r w c er 0 sinK I C (4.3) where r 0 represen ts the equilibrium of the radius r . =( ) is the phase dierence b et w een the rotating shaft and the disk in steady state. C is the constan t of the in tegration, K I ( _ w c ) =K I ( _ w c )t +C . F rom the the Eqs. (4.1) - (4.3), w e can obtain cos = r 0 (1w 2 c ) ew 2 c ; sin = 2r 0 w c ew 2 c (4.4) C = 2 r w c K I er 0 sin K I (4.5) 35 tan = sin cos = 2w c 1w 2 c ; ) = arctan( 2w c 1w 2 c ) Since sec 2 tan 2 = 1) tan 2 + 1 = sec 2 = 1 cos 2 , cos = 1 p 1+tan 2 . By substituting tan = 2wc 1w 2 c yields, cos = 1 r 1+( 2wc 1w 2 c ) 2 = 1w 2 c p (1w 2 c ) 2 + (2w c ) 2 (4.6) ) r 0 = ew 2 c 1w 2 c cos = ew 2 c p (1w 2 c ) 2 + (2w c ) 2 (4.7) 4.1.1 State-Space Represen tation By using the equilibrium p oin ts giv en in (4.4) - (4.7), the states can b e c hosen as the follo wing, so that all the states will settle at the origin in the steady-state resp onse. x 1 =rr 0 ; x 2 = ( _ w c )dtC +; x 3 = ( _ w c )dtC x 4 = _ r; x 5 = _ w c ; x 6 = _ w c The system (3.21) - (3.23) with the ab o v e states can b e written in the state-space form giv en in the follo wing 36 _ x 1 = x 4 (4.8) _ x 2 = x 5 (4.9) _ x 3 = x 6 (4.10) _ x 4 = (x 1 +r 0 )(x 5 +w c ) 2 2x 4 (x 1 +r 0 ) +e(x 6 +w c ) 2 cos(x 2 x 3 ) + 2 r e(x 6 +w c ) sin(x 2 x 3 ) e 2 (x 1 +r 0 ) sin 2 (x 2 x 3 ) +K P ex 6 sin(x 2 x 3 ) +K I e(x 3 +C) sin(x 2 x 3 ) (4.11) _ x 5 = 2x 4 (x 5 +w c ) (x 1 +r 0 ) 2(x 5 +w c ) e(x 6 +w c ) 2 (x 1 +r 0 ) sin(x 2 x 3 ) + 2 r e(x 6 +w c ) (x 1 +r 0 ) cos(x 2 x 3 ) e 2 sin(x 2 x 3 ) cos(x 2 x 3 ) + K P ex 6 (x 1 +r 0 ) cos(x 2 x 3 ) + K I e(x 3 +C) (x 1 +r 0 ) cos(x 2 x 3 ) (4.12) _ x 6 = 2 r (x 6 +w c ) +e(x 1 +r 0 ) sin(x 2 x 3 )K P x 6 K I (x 3 +C) (4.13) The general form of the state-space equation can b e written as _ x =F (x) and the linearized system is giv en as @ @x [F (x)]j x=0 = A c X where 37 A c = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 [ 0; 0; 0; 1; 0; 0 ] [ 0; 0; 0; 0; 1; 0 ] [ 0; 0; 0; 0; 0; 1 ] [w 2 c 1e 2 sin; ew 2 c sin +e 2 r 0 cos sin; ew 2 c sin (er 0 cos +K I )e sin; 2; 2r 0 w c ; 2ew c cos (2 r +K P )e sin] [ ew 2 c sin r 2 0 + e 2 sincos r 0 ; ew 2 c cos r 0 e 2 cos 2 ; ew 2 c cos r 0 + (er 0 cos+K I )ecos r 0 ; 2wc r 0 ;2; 2ewcsin r 0 + (2r+K P ) r 0 e cos] [e sin; er 0 cos;er 0 cosK I ; 0; 0;2 r K P ] 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (4.14) 4.2 Stability Analysis of Linearized System with T orque PI control In this section, the stabilit y of the linearized system is prob ed b y using R outh-Hurwitz criterion for the damping ratio > 0. The stabilit y conditions can b e found from the Routh arra y of the c haracteristic equation det(IA c ) = 0, where all the co ecien ts in the rst column ha v e to b e p ositiv e for the stabilit y . Ho w ev er, there are to o man y terms in the co ecien ts to determine the sym b olic solutions. Instead, the n umerical analysis b y eigen v alues, Routh criterion and Matlab sim ulations are presen ted as the result. The sim ulation mo del of the feedbac k con trol system is established in Matlab Sim ulink sho wn in Fig. 4.1. 38 Figure 4.1: F eedbac k con trol mo del with PI con troller F rom the Routh arra y of the linearized system (4.14), it can b e seen clearly that the co ecien ts of 6 , 5 and 0 are alw a ys p ositiv e for all sets of the system parameters (con trol gains6= 0 ). The other four stabilit y conditions in Routh arra y are plotted in Fig. 4.2 for comparison. It is sho wn that the curv e of condition 4 (co ecien t of 1 ) is closer to zero line than other three stabilit y conditions and crosses to the unstable area (negativ e sign in Roth criterion) as the rotating sp eed w c gro ws on the b ottom plot. In another gure 4.3, w e can see that the unstable zone of the fourth condition co v ers the other three conditions and this result can b e rep eated in other com binations of system parameters ( , r , e and w c ). Whic h implies that condition 4 dominates the system stabilit y , or sa y , if the co ecien t of 1 in Routh arra y is p ositiv e, the nominal system is stable. The result can also b e v eried b y c hec king the eigen v alues of matrix A c . In Fig. 4.4, the stabilit y region based on the eigen v alues is plotted and it can b e seen that the unstable zone (around w c = 1 1:7 ) matc hes the negativ e part of condition 4 in Fig. 4.2. F urthermore, the time resp onses of the case in Fig. 4.2 ( = 0:1, r = 0:3 and w c = 1:06 ) are sim ulated b y Matlab. F rom the Fig. 4.2, it clearly sho ws that the system is stable only when e < 0:05. W e sim ulate the same system with the eccen tricit y e = 0:04 and e = 0:08 in Fig. 4.5 and Fig. 4.6. As can b e seen the system is stable for in the case e = 0:05, whic h v eries the stabilit y condition. 39 Figure 4.2: Stabilit y conditions Figure 4.3: Stabilit y conditions comparison 40 Figure 4.4: Stabilit y region of nominal system Figure 4.5: Time resp onse of the linearized system with PI con trol (stable case) 41 Figure 4.6: Time resp onse of the linearized system with PI con trol (unstable case) T o in v estigate the system stabilit y more, the next topic is to kno w ho w do the system parameters aect the stabilit y conditions. F rom the jump phenomenon of the system in tro duced in Chapter 3, the con tin uous c hange of the rotating sp eed can not b e ac hiev ed around its natural frequency . The result can b e examined b y the Routh arra y sho wn in Fig. 4.7. The curv e of dominated stabilit y condition starts turning do wn and en ters to the negativ e area when w c > 1. Also, in man y stabilit y region tests b y the eigen v alues w e can nd that the unstable resp onse lo cate b et w een the rotating sp eed w c = 1 2:3 in this mo del, whic h satises the bifurcation area in jump phenomenon in Fig. 3.7. 42 Figure 4.7: Comparison of stabilit y region with dieren t rotating sp eeds F or the eect from the damp ers, the stabilit y condition is computed with a set of v alues of damping ratio and r displa y ed in Fig. 4.8 and Fig. 4.9. F rom the plots, it is ob vious that the system b ears more eccen tricit y e with either larger damping ratio or damping of the rotating shaft r . The stabilit y region can b e expanded b y increasing the damping terms. Figure 4.8: Comparison of stabilit y region with dieren t damping 43 Figure 4.9: Comparison of stabilit y region with dieren t r Last, the system parameter related to the stabilit y is the eccen tricit y e, whic h is the main factor to cause the vibration in the whirling motion. If there is no eccen tricit y , the equation of motion of the rotor system (3.18) - (3.20) can b e decoupled and the system will alw a ys b e stable if the damping is not zero. On the con trary , when eccen tricit y increases, the unstable area gro ws, or sa y , the larger damping terms or con trol gains are needed for the stabilit y . In Fig. 4.10 and Fig. 4.11, the stabilit y condition are plotted with t w o dieren t eccen tricit y v alues ( e = 0:4 and e = 1 ) and the corresp onding stabilit y region c harts sho w in Fig. 4.12 and Fig. 4.13, in whic h the system has larger unstable zone with higher eccen tricit y v alue. 44 Figure 4.10: Stabilit y condition for e = 0:4 Figure 4.11: Stabilit y condition for e = 1 45 Figure 4.12: Stabilit y Region (e = 0:4) Figure 4.13: Stabilit y Region (e=1) F rom the n umerical analysis of dieren t sets of system parameters, w e also found that the dominated stabilit y condition (the co ecien t of 1 in Routh arra y > 0 ) can b e con v erted to a critical v alue of eccen tricit y . If the stabilit y condition is plotted with a v arying eccen tricit y and xed damping v alues, a critical p oin t of e, named 46 e c , can b e found on the plot suc h that the system is stable when the eccen tricit y is under this p oin t. This criterion w ould b e helpful for the con trol design, b ecause w e only need to c hec k one system parameter e for the stabilit y and e c b ecomes the only b ound for c ho osing the nominal system. Figure 4.14: Critical eccen tricit y e c (case 1) Figure 4.15: Critical eccen tricit y e c (case 2) 47 Figure 4.16: Stabilit y region for critical eccen tricit y c hart (case 2) In Fig. 4.15 and Fig. 4.14, the critical p oin t e c with t w o dieren t sets of the system parameters are sho wn. The curv es in the plots represen t the b oundaries of the stabilit y region. The system is alw a ys stable under e c and is unstable when the rotating sp eed lo cates within the b o wl-shap e curv e when e>e c . The stabilit y region plot in Fig. 4.16 v eries the critical p oin t c hart in Fig. 4.15, where e = 0:2 and the unstable region of t w o plots are matc hed. W e also sim ulate the time resp onses of the case in Fig. 4.15 , where the critic p oin t e c = 0:028. In Fig. 4.17, the eccen tricit y v alue is selected as e < e c and Fig. 4.18 sho ws the case of e>e c . The result of the sim ulations conrms that the critical p oin t e c determines the stabilit y of the system. 48 Figure 4.18: Time resp onse for critical eccen tricit y c hart (Unstable case) Figure 4.17: Time resp onse for critical eccen tricit y c hart (Stable case) 4.3 Eect of PI control In linear system con trol, a PI con troller is added in the torque equation to bring the rotating sp eed _ to the reference input sp eed w c in steady state. The resp onse of the 49 system without con trol is sim ulated in Fig. 4.19, in whic h the input rotating sp eed is set as w c = 1:06. As sho wn in the gure, the rotating sp eed on the b ottom plot do es not con v erge to the input v alue in steady state. Th us, the system w ould p erform as the jump phenomenon plot discussed in Chapter 3 (Fig. 3.7) without feedbac k con trol applied. Figure 4.19: Time resp onse of the linearized system with no in tegral con troller In Fig. 4.20, it is sho wn that the stable region of the system increases with higher con trol gain K P and K I . In other w ords, with the feedbac k con trol, the system can b ear larger eccen tricit y . The results of dieren t con trol gain trials are sho wn in T able 2. Also, the critical p oin t c harts tested with sev eral com binations of PI gains are displa y ed in Fig. 4.21 and Fig. 4.24. F rom the plots, w e notice the critical p oin t e c raises with P con trol gain K P . Though the in tegral con trol gain K I can eliminate the steady state error of the rotating sp eed, it do es not ob viously impro v e the b ound of the critical eccen tricit y . Ev en when system has high K I gain, the unstable region of the case e > e c tend to b e larger. The example can b e seen b y comparing the Fig. 4.23 and Fig. 4.24 at eccen tricit y e = 0:5. With high K P and lo w K I gain in Fig. 4.24, the unstable region of rotating sp eed is around w c = 1 1:45, while in the case of high K I and lo w K P gain in Fig. 4.23, w c = 1:05 2:05 is the unstable zone. 50 Figure 4.20: Comparison of stabilit y region with dieren t PI con trol gains System parameters T esting v alues Damping ratio 0.1 0.1 0.1 0.1 0.1 0.1 Damping ratio(shaft) r 0.1 0.1 0.1 0.1 0.1 0.1 P con trol gain K P 0.01 0.05 0.1 0.01 0.5 0.5 I con trol gain K I 0.01 0.05 0.1 0.5 0.01 0.5 Critical eccen tricit y e c 0.2694 0.2910 0.3153 0.2802 0.3667 0.4973 T able 2: Critical eccen tricit y e c with PI gains 51 Figure 4.21: Critical eccen tricit y e c with PI con troller (case 1) Figure 4.22: Critical eccen tricit y e c with PI con troller (case 2) 52 Figure 4.23: Critical eccen tricit y e c with lo w K P gain and high K I gain (case 3) Figure 4.24: Critical eccen tricit y e c with high K P gain and lo w K I gain (case 4) 53 5 Control Development using Lyapunov Redesign Method The dynamic b eha vior of rotor system is presen ted in Chapter 3. Though the equation decoupling b y assuming constan t rotating sp eed migh t reduce some load in the con trol design pro cess, the usage of this tec hnique is limited when the system is op erating in a high sp eed range or during fast maneuv ers of the rotating sp eed, suc h as the case of the turbine engines for the jet gh ters. Besides, the stabilit y conditions of the system migh t b e aected b y the decoupling. It is sho wn in the previous c hapter that the decoupled system results a dieren t stabilit y region and lose the observ ation of the transien t resp onse from the original system. A new con trol design approac h presen ted here is to impro v e the p erformance of the rotor system with v arying rotating v elo cit y . Moreo v er, the p erturbations or un- certain ties in the parameter v alues are considered to main tain the robustness of the con trol. The con trol la w dev elopmen t of Ly apuno v redesign metho d can b e catego- rized in to three steps: the nominal system with the prop ortional-in tegral feedbac k con trol, the nonlinear cancellation and the p erturb ed system with the additional con- troller, v . In the nominal system, PI con troller is used in the torque equation to ensure that the nal v alue of rotating sp eed con v erges to the reference input and the nonlinear cancellation term (t;x) is added to mo dify the nominal equations. F or the p erturb ed system con trol design, Lyapunov r e design metho d is applied to reduce the eect b y the uncertain ties. The con trol v in the p erturb ed system can b e determined on the base of the nominal con trol design. In Section 5.1, w e in tro duce the idea of Ly apuno v redesign. The nominal and actual system equations are listed in Section 5.2 and the con trol la w deriv ations are sho w ed in Section 5.3. Last, the sim ulation studies are discussed in Section 5.4. 5.1 Lyapunov Redesign Method The follo wing denitions and theorems of Ly apuno v redesign are from [18]. Consider the system _ x = ^ F (t;x) + ^ G(t;x)[u +(t;x;u)] (5.1) where x2R n is the state and u2R p is the con trol input. The functions ^ F , ^ G, and are dened for (t;x;u)2 [0;1)DR p , where D R n is a domain that con tains the origin. 54 W e assume that ^ F , ^ G and are piecewise con tin uous in t and lo cally Lip c hitz in x and u. The functions ^ F and ^ G are kno wn precisely , while the function is an unkno wn function that lumps together v arious uncertain terms due to mo del sim- plications, parameter uncertain t y , and so on. The uncertain term satises the matc hing condition; that is , when uncertain terms en ter the state equation at the same p oin t as the con trol input. A nominal mo del of the system can b e tak en as _ x = ^ F (t;x) + ^ G(t;x)u (5.2) W e pro ceed to design a stabilizing state feedbac k con troller b y using this nominal mo del. Supp ose w e ha v e succeeded to design a feedbac k con trol la w u = (t;x) suc h that the origin of the nominal closed-lo op system _ x = ^ F (t;x) + ^ G(t;x) (t;x) (5.3) is uniformly asymptotically stable. Supp ose further that w e kno w a Ly apuno v function for (5.3); that is, w e ha v e a con tin uously dieren tiable function V (t;x) that satises the inequalities 1 (kxk) V (t;x) 2 (kxk) (5.4) @V @t + @V @x [ ^ F (t;x) + ^ G(t;x) (t;x)] 3 (kxk) (5.5) for all (t;x)2 [0;1)D , where 1 , 2 and 3 are classK functions. Assume that, with u = (t;x) +v , the uncertain term satises the inequalit y k(t;x; (t;x) +v)k (t;x) + 0 kvk; 0 0 < 1 (5.6) where : [0;1)D!R is a nonnegativ e con tin uous function. The estimate (5.6) is the only information w e need to kno w ab out the uncertain term . The function is a measure of the size of the uncertain t y . It is imp ortan t to emphasize that w e will not require to b e small. W e will only require it to b e kno wn. With the kno wledge 55 of Ly apuno v function V , the function , and the constan t 0 in (5.6) , an additional feedbac k con trol v can b e designed suc h that the o v erall con trol u = (t;x) +v is called Ly apuno v redesign. The system (5.1) with the con trol u = (t;x) +v applied is sho wn as _ x = ^ F (t;x) + ^ G(t;x) (t;x) + ^ G(t;x)[v +(t;x; (t;x) +v)] (5.7) The closed-lo op system is a p erturbation of the nominal closed-lo op system (5.3). The deriv ativ e of V (t;x) along the tra jectories of (5.7) can b e written as _ V = @V @t + @V @x ( ^ F + ^ G ) + @V @x ^ G(v +) 3 (kxk) + @V @x ^ G(v +) Set w T = [@V=@x]G and rewrite the inequalit y ab o v e as _ V 3 (kxk) +w T v +w T The rst term on the righ t-hand side is due to the nominal closed-lo op system. The second and the third term represen t , resp ectiv ely , the eect of the con trol v and the uncertain t y term on _ V . Due to the matc hing condition, the uncertain t y term app ears on the righ t-hand side exactly at the same p oin t where v app ears. Consequen tly , it is p ossible to c ho ose v to cancel the (destabilizing) eect of on _ V . W e will no w explore t w o dieren t metho ds for c ho osing v so that w T v +w T 0. Supp ose inequalit y (5.6) is satised with kk 2 ; that is, k(t;x; (t;x) +v)k 2 (t;x) + 0 kvk 2 ; 0 0 < 1 (5.8) W e ha v e w T v +w T w T v +kwk 2 kk 2 w T v +kwk 2 [(t;x) + 0 kvk 2 ] T aking 56 v = (t;x) w kwk 2 (5.9) with a nonnegativ e function , w e obtain w T v +w T kwk 2 +kwk 2 + 0 kwk 2 =(1 0 )kwk 2 +kwk 2 Cho osing (t;x)(t;x)=(1 0 ) for all (t;x)2 [0;1)D yields, w T v +w T kwk 2 +kwk 2 = 0 Hence, with the con trol (5.9), the deriv ativ e of V (t;x) along the tra jectories of the closed-lo op system (5.7) is negativ e denite. As an alternativ e idea, Cho ose v = (t;x)sgn(w) (5.10) where(t;x)(t;x)=(1 0 ) for all (t;x)2 [0;1)D andsgn(w) is a p-dimensional v ector whose ith comp onen t is sgn(w i ). With the con trol (5.10), the deriv ativ e of V (t;x) along the tra jectories of the closed-lo op system (5.7) is negativ e denite. The stabilit y of the system with the con trol la w (5.10) is pro ofed in App endix B. The con trol la ws giv en b y (5.9) and (5.10) are discon tin uous functions of the state x. This discon tin uit y causes some theoretical as w ell as practical problems. Theoret- ically , w e ha v e to c hange the denition of the con trol la w to a v oid division b y zero. W e also ha v e to examine the question of existence and uniqueness of solutions more carefully , since the feedbac k functions are not lo cally Lipsc hitz in x. Practically , the implemen tation of suc h discon tin uous con trollers is c haracterized b y the phenomenon of c hattering, where, due to imp erfections in switc hing devices or computational de- la ys, the con trol has fast switc hing uctuations across the switc hing surface. Instead of trying to w ork out all these problems, w e will c ho ose the easy and more practi- cal route of appro ximating the discon tin uous con trol la w b y a con tin uous one. The dev elopmen t of suc h appro ximation is similar for b oth con trol la ws. 57 Consider the feedbac k con trol la w v = 8 > > > < > > > : (t;x)(w=kwk 2 ); if(t;x)kwk 2 (t;x)(w=) if(t;x)kwk 2 < (5.11) With (5.11), the deriv ativ e of V along the tra jectories of the closed-lo op system (5.7) will b e negativ e denite whenev er (t;x)kwk 2 . The pro of of the stabilit y in the case (t;x)kwk 2 < is presen ted in App endix B. Where _ V 3 (kxk 2 ) + (1 0 ) 4 is satised irresp ectiv e of the v alue of (t;x)kwk 2 . T ak e r> 0 suc h that B r D , c ho ose < 2 3 ( 1 2 ( 1 (r)))=(1 0 ), and set = 1 3 ((1 0 )=2) < 1 2 ( 1 (r)). Then, _ V 3 (kxk 2 ); 8kxk 2 <r Whic h sho ws that the solution of the closed-lo op system are uniformly ultimately b ounded b y a classK function of . 5.2 Equations of Nominal System and Actual System In man y con trol design tasks, the system parameters are not kno wn exactly due to the error from system iden tication or other disturbances. F or these kind of cases, robust con trol metho ds suc h as Ly apuno v redesign or sliding mo de con trol can b e used to o v ercome the aect of the error b et w een the prediction and the actual system. Before w e pro ceed to the con trol la w deriv ation, the equations of the nominal and actual rotor system are in tro duced in the follo wing sections. 5.2.1 Nominal System The system with kno wn parameters is called Nominal system . It is a prediction of the real system. In the follo wing deriv ation, the notation ^ () is used to represen t a kno wn function or a system parameter. F or example, ^ e means the estimation of the actual eccen tricit y and ^ e is a kno wn v alue. 58 Consider the Jecott rotor system (3.21) - (3.23) with the input torque c hosen as the linear PI con troller,K P ( _ w c )K I ( _ w c ). Assume that all the parameters are kno wn in the nominal system, suc h that the system equations are sho wn as r = r _ 2 2 ^ _ rr + ^ e _ 2 cos( ) ^ e sin( ) (5.12) r = 2 _ r _ 2 ^ r _ ^ e cos( ) ^ e _ 2 sin( ) (5.13) = 2 ^ r _ + ^ er sin( )K P ( _ w c )K I ( _ w c ) (5.14) The system parameters are in tro duced in previous c hapters and w e c ho ose the states as follo ws x 1 =r ^ r 0 ; x 2 = ( _ w c )dt ^ C + ^ ; x 3 = ( _ w c )dt ^ C x 4 = _ r; x 5 =r _ ^ r 0 w c ; x 6 = _ w c where ^ C is a constan t and ^ is the phase dierence from the rotating shaft and the disk, whic h can b e calculated from the system equilibrium p oin t. ^ C = 2 ^ r w c K I ^ e ^ r 0 sin ^ K I ; ^ = arctan( 2 ^ w c 1w 2 c ) The state-space equation can b e presen ted as 59 _ x 1 = x 4 (5.15) _ x 2 = x 5 w c x 1 (x 1 + ^ r 0 ) (5.16) _ x 3 = x 6 (5.17) _ x 4 = (x 5 + ^ r 0 w c ) 2 (x 1 + ^ r 0 ) 2 ^ x 4 (x 1 + ^ r 0 ) + ^ e(x 6 +w c ) 2 cos(x 2 x 3 ^ ) +2 ^ r ^ e(x 6 +w c ) sin(x 2 x 3 ^ ) ^ e 2 (x 1 + ^ r 0 ) sin 2 (x 2 x 3 ^ ) +K P ^ ex 6 sin(x 2 x 3 ^ ) +K I ^ e(x 3 + ^ C) sin(x 2 x 3 ^ ) (5.18) _ x 5 = (x 5 + ^ r 0 w c )x 4 (x 1 + ^ r 0 ) 2 ^ (x 5 + ^ r 0 w c ) ^ e(x 6 +w c ) 2 sin(x 2 x 3 ^ ) +2 ^ r ^ e(x 6 +w c ) cos(x 2 x 3 ^ ) ^ e 2 (x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) cos(x 2 x 3 ^ ) +K P ^ ex 6 cos(x 2 x 3 ^ ) +K I ^ e(x 3 + ^ C) cos(x 2 x 3 ^ ) (5.19) _ x 6 = 2 ^ r (x 6 +w c ) + ^ e(x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) K P x 6 K I (x 3 + ^ C) (5.20) The nominal system can b e presen ted as _ X = ^ F (x) + ^ G(x) (x) (5.21) where (x) represen ts the nominal system con trol and the functions are dened b elo w. ^ F (x) 61 = " ^ f 1 (x) 31 ^ f 2 (x) 31 # ; ^ G(x) 66 = " 0 33 ^ g(x) 33 # ; where 60 ^ f 1 (x) 31 = 2 6 4 x 4 x 5 wcx 1 (x 1 +^ r 0 ) x 6 3 7 5 (5.22) ^ f 2 (x) 31 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 (x 5 +^ r 0 wc) 2 (x 1 +^ r 0 ) 2 ^ x 4 (x 1 + ^ r 0 ) +^ e(x 6 +w c ) 2 cos(x 2 x 3 ^ ) +2 ^ r ^ e(x 6 +w c ) sin(x 2 x 3 ^ ) ^ e 2 (x 1 + ^ r 0 ) sin 2 (x 2 x 3 ^ ) (x 5 +^ r 0 wc)x 4 (x 1 +^ r 0 ) 2 ^ (x 5 + ^ r 0 w c ) ^ e(x 6 +w c ) 2 sin(x 2 x 3 ^ ) +2 ^ r ^ e(x 6 +w c ) cos(x 2 x 3 ^ ) ^ e 2 (x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) cos(x 2 x 3 ^ ) 2 ^ r (x 6 +w c ) + ^ e(x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (5.23) ^ g(x) 33 = 2 6 4 1 0 ^ e sin(x 2 x 3 ^ ) 0 1 ^ e cos(x 2 x 3 ^ ) 0 0 1 3 7 5 (5.24) and ^ g 1 (x) = 2 6 4 1 0 ^ e sin(x 2 x 3 ^ ) 0 1 ^ e cos(x 2 x 3 ^ ) 0 0 1 3 7 5 (5.25) 5.2.2 P erturb ed System In p erturb ed system, the parameters are uncertain. A ccording the the dimensionless equations in Section 3.2.2, the system equations are divided b y the kno wn natural frequency , whic h is one of the dimensionless quan tit y . W e mark it as ^ w n here to distinguish from the actual system parameter. In the deriv ation of the dimensionless equations for the actual system, w e need to dene another dimensionless system 61 parameter, that is w n ^ w n = suc h that the dimensionless equations are r = r _ 2 2 _ r 2 r +e _ 2 cos( )e sin( ) (5.26) r = 2 _ r _ 2r _ e cos( )e _ 2 sin( ) (5.27) = 2 r _ + 2 er sin( ) +u (5.28) where u is the con trol input. By c ho osing the same states as the nominal system, the state-space equation are presen ted as 62 _ x 1 = x 4 (5.29) _ x 2 = x 5 w c x 1 (x 1 + ^ r 0 ) (5.30) _ x 3 = x 6 (5.31) _ x 4 = (x 5 + ^ r 0 w c ) 2 (x 1 + ^ r 0 ) 2x 4 2 (x 1 + ^ r 0 ) +e(x 6 +w c ) 2 cos(x 2 x 3 ^ ) +2 r e(x 6 +w c ) sin(x 2 x 3 ^ ) 2 e 2 (x 1 + ^ r 0 ) sin 2 (x 2 x 3 ^ ) e sin(x 2 x 3 ^ )u (5.32) _ x 5 = (x 5 + ^ r 0 w c )x 4 (x 1 + ^ r 0 ) 2(x 5 + ^ r 0 w c ) e(x 6 +w c ) 2 sin(x 2 x 3 ^ ) +2 r e(x 6 +w c ) cos(x 2 x 3 ^ ) 2 e 2 (x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) cos(x 2 x 3 ^ ) e cos(x 2 x 3 ^ )u (5.33) _ x 6 = 2 r (x 6 +w c ) + 2 e(x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) +u (5.34) Let the actual system b e _ x = F (x) +G(x)u (5.35) The functions in the equation are dened as F (x) 61 = " f 1 (x) 31 f 2 (x) 31 # ; G(x) 66 = " 0 33 g(x) 33 # (5.36) 63 f 1 (x) 31 = 2 6 4 x 4 x 5 wcx 1 (x 1 +^ r 0 ) x 6 3 7 5 (5.37) f 2 (x) 31 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 (x 5 +^ r 0 wc) 2 (x 1 +^ r 0 ) 2x 4 2 (x 1 + ^ r 0 ) +e(x 6 +w c ) 2 cos(x 2 x 3 ^ ) +2 r e(x 6 +w c ) sin(x 2 x 3 ^ ) 2 e 2 (x 1 + ^ r 0 ) sin 2 (x 2 x 3 ^ ) (x 5 +^ r 0 wc)x 4 (x 1 +^ r 0 ) 2(x 5 + ^ r 0 w c ) e(x 6 +w c ) 2 sin(x 2 x 3 ^ ) +2 r e(x 6 +w c ) cos(x 2 x 3 ^ ) 2 e 2 (x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) cos(x 2 x 3 ^ ) 2 r (x 6 +w c ) + 2 e(x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (5.38) g(x) 33 = 2 6 4 1 0 e sin(x 2 x 3 ^ ) 0 1 e cos(x 2 x 3 ^ ) 0 0 1 3 7 5 (5.39) and g 1 (x) = 2 6 4 1 0 e sin(x 2 x 3 ^ ) 0 1 e cos(x 2 x 3 ^ ) 0 0 1 3 7 5 (5.40) 5.3 Control Law Deriv ation Prop osition. Consider the r otor dynamic system given in (5.35) and the functions F (x) and G(x) as dene d in (5.36) - (5.39). If the line ar c ontr ol gain K P satises the c ondition K P > ^ ^ e 2 4 ^ r 2 (5.41) 64 and the functions (x) and 0 ar e given as (x) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 2 ^ jx 4 j + 2(x 1 + ^ r 0 ) + (x 6 +w c ) 2 +2 r (x 6 +w c ) + 2 ^ r w c +jx 1 j +K P jx 6 j +K I jx 3 j 2 ^ (x 5 + ^ r 0 w c ) + (x 6 +w c ) 2 +2 r (x 6 +w c ) + (x 1 + ^ r 0 ) + 2 ^ r w c +jx 1 j +K P jx 6 j +K I jx 3 j 2 ^ r (x 6 +w c ) + (x 1 + ^ r 0 ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 (5.42) and 0 = j(e ^ e)j (5.43) Then the r otor system given in (5.35) is stabilize d by the c ontr ol u = (x) +v given as the fol lowing 65 (x) = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 x 5 2 (x 1 +^ r 0 ) (x 5 +^ r 0 wc) 2 (x 1 +^ r 0 ) + ^ r 0 + ^ ex 6 2 cos(x e ) ^ e(x 6 +w c ) 2 cos(x e ^ ) +2 ^ r ^ ex 6 [sin(x e ) sin(x e ^ )] ^ e 2 x 1 sin(x e )[sin(x e ) sin(x e ^ )] +K P ^ ex 6 [sin(x e ) sin(x e ^ )] +K I ^ ex 3 [sin(x e ) sin(x e ^ )] +^ ew c x 1 x 6 (x 1 +^ r 0 ) cos(x e ) ^ r 0 wcx 4 (x 1 +^ r 0 ) + 2 ^ ^ r 0 w c ^ ex 6 2 sin(x e ) +^ e(x 6 +w c ) 2 sin(x e ^ ) +2 ^ r ^ ex 6 [cos(x e ) cos(x e ^ )] ^ e 2 x 1 sin(x e )[cos(x e ) cos(x e ^ )] +K P ^ ex 6 [cos(x e ) cos(x e ^ )] +K I ^ ex 3 [cos(x e ) cos(x e ^ )] ^ ew c x 1 x 6 (x 1 +^ r 0 ) sin(x e ) 2 ^ r w c + ^ e[x 1 sin(x e ) (x 1 + ^ r 0 ) sin(x e ^ )] K P x 6 K I x 3 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (5.44) wher e x e =x 2 x 3 . In addition, the c ontr ol v is chosen as v = (x) w kwk 2 (5.45) or v = 8 > > > < > > > : (x)(w=kwk 2 ); if(x)kwk 2 (x)(w=) if(x)kwk 2 < (5.46) wher e is a p ositive c onstant. The functions (x) and w ar e given as 66 (x) (x)=(1 0 ) and w T = [@V=@x]G wher e G is dene d in (5.36) and V is the Lyapunov function given by V = 1 2 x 2 4 + 1 2 x 2 5 + 1 2 ^ e 2 x 2 6 + ^ ex 4 x 6 sin(x ) + ^ ex 5 x 6 cos(x ) + 1 2 x 2 6 + 1 2 x 2 1 + 1 2 K I x 2 3 (5.47) Pr o of. The rst step is to sho w that the nominal system (5.15) - (5.20) is stabilized b y the con trol (x) giv en in (5.44). Since all the parameters are kno wn in the nominal system, w e can design the con troller to form the nominal system in to the desired system suc h that the Ly apuno v function candidate can b e found. The nominal con trol (x) consists of three parts (x) = PI (x) + ^ g(x) 1 ( c (x) + a (x)) (5.48) where PI (x) represen ts linear PI con troller, c (x) is the con trol to shap e the nominal system equations and a (x) is the additional con trol to ensure that the Ly apuno v function satises all the stabilit y conditions. PI con trol la w PI (x) can b e written as PI (x) = 2 6 4 0 0 K P x 6 K I (x 3 + ^ C) 3 7 5 and the additional con trollers are giv en b y c (x) = ^ f 2 (x) ( ^ f 2 (x) + ^ g(x) PI (x)) 67 a (x) = 2 6 4 u 1 u 2 0 3 7 5 where u 1 = ^ ew c x 1 x 6 (x 1 + ^ r 0 ) cos(x e ) u 2 = ^ ew c x 1 x 6 (x 1 + ^ r 0 ) sin(x e ) Substituting the con trol giv en in (5:48) in to the nominal system (5.48) yields _ X = " ^ f 1 (x) ^ f 2 (x) + ^ g(x) (x) # = " ^ f 1 (x) ^ f 2 (x) + a (x) # where the function ^ f 2 (x) is dened in the follo wing. Cho ose the desire nominal system as _ x 1 = x 4 (5.49) _ x 2 = x 5 w c x 1 (x 1 + ^ r 0 ) (5.50) _ x 3 = x 6 (5.51) _ x 4 = x 5 2 (x 1 + ^ r 0 ) 2 ^ x 4 x 1 + ^ ex 6 2 cos(x e ) + 2 ^ r ^ ex 6 sin(x e ) ^ e 2 x 1 sin 2 (x e ) +K P ^ ex 6 sin(x e ) +K I ^ ex 3 sin(x e ) +u 1 (5.52) _ x 5 = x 4 x 5 (x 1 + ^ r 0 ) 2 ^ x 5 ^ ex 6 2 sin(x e ) + 2 ^ r ^ ex 6 cos(x e ) ^ e 2 x 1 sin(x e ) cos(x e ) +K P ^ ex 6 cos(x e ) +K I ^ ex 3 cos(x e ) +u 2 (5.53) _ x 6 = 2 ^ r x 6 + ^ ex 1 sin(x e )K P x 6 K I x 3 (5.54) 68 and ^ f 2 (x) = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 x 5 2 (x 1 +^ r 0 ) 2 ^ x 4 x 1 + ^ ex 6 2 cos(x e ) + 2 ^ r ^ ex 6 sin(x e ) ^ e 2 x 1 sin 2 (x e ) +K P ^ ex 6 sin(x e ) +K I ^ ex 3 sin(x e ) x 4 x 5 (x 1 +^ r 0 ) 2 ^ x 5 ^ ex 6 2 sin(x e ) + 2 ^ r ^ ex 6 cos(x e ) ^ e 2 x 1 sin(x e ) cos(x e ) +K P ^ ex 6 cos(x e ) +K I ^ ex 3 cos(x e ) 2 ^ r x 6 + ^ ex 1 sin(x e )K P x 6 K I x 3 3 7 7 7 7 7 7 7 7 7 7 7 7 5 (5.55) where x e =x 2 x 3 . Let the Ly apuno v function candidate b e V = 1 2 x 2 4 + 1 2 x 2 5 + 1 2 ^ e 2 x 2 6 + ^ ex 4 x 6 sin(x e ) + ^ ex 5 x 6 cos(x e ) + 1 2 x 2 6 + 1 2 x 2 1 + 1 2 K I x 2 3 (5.56) Note thatV is a con tin uously dieren tiable function and is p ositiv e denite. T o pro v e the stabilit y of the nominal system with the con trol la w (x), ev aluate _ V : _ V = x 4 _ x 4 +x 5 _ x 5 + ^ e 2 x 6 _ x 6 + ^ e _ x 4 x 6 sin(x e ) + ^ ex 4 _ x 6 sin(x e ) +^ e _ x 2 x 4 x 6 cos(x e ) ^ e _ x 3 x 4 x 6 cos(x e ) + ^ e _ x 5 x 6 cos(x e ) + ^ ex 5 _ x 6 cos(x e ) ^ e _ x 2 x 5 x 6 sin(x e ) + ^ e _ x 3 x 5 x 6 sin(x e ) +x 6 _ x 6 +x 1 _ x 1 +K I x 3 _ x 3 Substituting Eqs. (5.15) - (5.20) in to _ V yields, 69 _ V = 2 ^ x 2 4 2 ^ x 2 5 2 ^ ^ ex 4 x 6 sin(x e ) ^ ew c x 1 x 4 x 6 (x 1 + ^ r 0 ) cos(x e ) 2 ^ ^ ex 5 x 6 cos(x e ) +^ ew c x 1 x 5 x 6 (x 1 + ^ r 0 ) sin(x e ) 2 ^ r x 6 2 K P x 2 6 + ^ ew c x 1 x 4 x 6 (x 1 + ^ r 0 ) cos(x e ) +^ e 2 w c x 1 x 2 6 (x 1 + ^ r 0 ) sin(x e ) cos(x e ) ^ ew c x 1 x 5 x 6 (x 1 + ^ r 0 ) sin(x e ) ^ e 2 w c x 1 x 2 6 (x 1 + ^ r 0 ) sin(x e ) cos(x e ) (5.57) = 2 ^ x 2 4 2 ^ x 2 5 2 ^ ^ ex 4 x 6 sin(x e ) 2 ^ ^ ex 5 x 6 cos(x e ) 2 ^ r x 6 2 K P x 2 6 = h x 4 x 5 x 6 i 2 6 4 2 ^ 0 ^ ^ e sin(x e ) 0 2 ^ ^ ^ e cos(x e ) ^ ^ e sin(x e ) ^ ^ e cos(x e ) 2 ^ r +K P 3 7 5 2 6 4 x 4 x 5 x 6 3 7 5 (5.58) Let P = 2 6 4 2 ^ 0 ^ ^ e sin(x e ) 0 2 ^ ^ ^ e cos(x e ) ^ ^ e sin(x e ) ^ ^ e cos(x e ) 2 ^ r +K P 3 7 5 T o c hec k if the matrix P is p ositiv e denite, leading principal minors are calculated as follo ws 1 = 2 ^ > 0 2 = 4 ^ 2 > 0 3 = 4 ^ 2 (2 ^ r +K P ) 2 ^ ( ^ ^ e sin(x e )) 2 2 ^ ( ^ ^ e cos(x e )) 2 = 4 ^ 2 (2 ^ r +K P ) 2 ^ 3 ^ e 2 If K P > ^ ^ e 2 4 ^ r 2 3 > 0, whic h implies the matrix P is p ositiv e denite and _ V 0. 70 Ho w ev er, in Eq. (5.58) only the states x 4 ,x 5 andx 6 are equal to zero when _ V = 0. A ccording to LaSalle’s In v ariance Principle giv en in Section 2.2.2, w e need to v erify if the equilibrium p oin t is the largest in v arian t set for _ V = 0 in order to establish the stabilit y . By substituting x 4 =x 5 =x 6 = 0 in to the state equations (5.49) - (5.55) yields, _ x 1 = x 4 = 0 ) x 1 =C 1 _ x 3 = x 6 = 0 ) x 3 =C 3 where C 1 and C 3 are constan ts. Since x 6 = 0, _ x 6 = 0. F rom the last state equation, _ x 6 = ^ eC 1 sin(x e )K I C 3 = 0 ) C 3 = ^ eC 1 sin(x e ) K I By substituting C 3 in to _ x 4 = 0 yields, _ x 4 = C 1 ^ e 2 C 1 sin 2 (x e ) + ^ e 2 C 1 sin 2 (x e ) = 0 ) C 1 = 0 Also, C 3 = ^ eC 1 sin(xe) K I = 0 Last, from the state equation _ x 2 _ x 2 = x 5 w c x 1 (x 1 + ^ r 0 ) = 0 whic h implies x 2 is a constan t and it is the only solution when all other states go to the origin for _ V = 0. 71 Hence, the tra jectory settles at the stable equilibrium p oin t for t!1 with the nominal con troller (x). Next is to sho w that with the functions (x) and 0 found in (5.42) and (5.43), the additional con trol v stabilizes the system with uncertain ties. The actual system with uncertain system parameters can b e written as _ x = F (x) +G(x)u = ^ F (x) + ^ G(x)u + [F (x) ^ F (x)] + [G(x) ^ G(x)]u = " ^ f 1 (x) ^ f 2 (x) + ^ g(x)u # + " 0 f 2 (x) ^ f 2 (x) # + " 0 33 g(x) ^ g(x) # u = " ^ f 1 (x) ^ f 2 (x) + ^ g(x)u + ^ g(x)[^ g 1 (x)(f 2 (x) ^ f 2 (x)) + ^ g 1 (x) (g(x) ^ g(x) )u] # (5.59) where the functions ^ f 1 , ^ f 2 , f 1 , f 2 , ^ g and g are dened in Section 5.2. By substituting u = (x) +v in to the equation yields _ x = " ^ f 1 (x) ^ f 2 (x) + ^ g(x) (x) + ^ g(x)[(x; (x) +v) +v] # The function(x; (x) +v) is the uncertain t y term and (x; (x) +v) = ^ g 1 (x)(f 2 (x) ^ f 2 (x)) + ^ g 1 (x) (g(x) ^ g(x) ) ( (x) +v) Rearranging the equation yields, (x;v) = ^ g 1 (x)(f 2 (x) ^ f 2 (x)) + ^ g 1 (x) (g(x) ^ g(x) ) (x) + ^ g 1 (x) (g(x) ^ g(x) )v and k(x;v)k 2 = ^ g 1 (x)(f 2 (x) ^ f 2 (x)) + ^ g 1 (x) (g(x) ^ g(x) ) (x) 2 + ^ g 1 (x) (g(x) ^ g(x) )v 2 (x) + 0 kvk 2 72 where ^ g 1 (x)(f 2 (x) ^ f 2 (x)) + ^ g 1 (x) (g(x) ^ g(x) ) (x) 2 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 2( + ^ )x 4 + ( 2 + 1)(x 1 + ^ r 0 ) +(e ^ e)(x 6 +w c ) 2 cos(x 2 x 3 ^ ) +2 r (e ^ e)(x 6 +w c ) sin(x 2 x 3 ^ ) 2 e(e ^ e)(x 1 + ^ r 0 ) sin 2 (x 2 x 3 ^ ) 2 ^ r w c (e ^ e) sin(x 2 x 3 ^ ) ^ e(e ^ e)[x 1 sin(x e ) (x 1 + ^ r 0 ) sin(x 2 x 3 ^ )] sin(x 2 x 3 ^ ) +K P (e ^ e)x 6 sin(x 2 x 3 ^ ) +K I (e ^ e)x 3 sin(x 2 x 3 ^ ) 2( + ^ )(x 5 + ^ r 0 w c ) (e ^ e)(x 6 +w c ) 2 sin(x 2 x 3 ^ ) +2 r (e ^ e)(x 6 +w c ) cos(x 2 x 3 ^ ) 2 e(e ^ e)(x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) cos(x 2 x 3 ^ ) 2 ^ r w c (e ^ e) cos(x 2 x 3 ^ ) ^ e(e ^ e)[x 1 sin(x e ) (x 1 + ^ r 0 ) sin(x 2 x 3 ^ )] cos(x 2 x 3 ^ ) +K P (e ^ e)x 6 cos(x 2 x 3 ^ ) +K I (e ^ e)x 3 cos(x 2 x 3 ^ ) 2( r + ^ r )(x 6 +w c ) + ( 2 e ^ e)(x 1 + ^ r 0 ) sin(x 2 x 3 ^ ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 2 2 6 6 6 6 6 6 6 6 6 6 6 4 2 ^ jx 4 j + 2(x 1 + ^ r 0 ) + (x 6 +w c ) 2 + 2 r (x 6 +w c ) +2 ^ r w c +jx 1 j +K P jx 6 j +K I jx 3 j 2 ^ (x 5 + ^ r 0 w c ) + (x 6 +w c ) 2 + 2 r (x 6 +w c ) + (x 1 + ^ r 0 ) +2 ^ r w c +jx 1 j +K P jx 6 j +K I jx 3 j 2 ^ r (x 6 +w c ) + (x 1 + ^ r 0 ) 3 7 7 7 7 7 7 7 7 7 7 7 5 2 (x) and 0 = ^ g 1 (x) (g(x) ^ g(x) ) 2 = j(e ^ e)j Assume that the b ound of the eccen tricit y e is kno wn and e min e e max and the 73 largest dimensionless eccen tricit y e max 1. Cho ose the nominal eccen tricit y ^ e as ^ e = (e min +e max )=2 suc h that 0 satises the condition 0 0 < 1. Last step is to design the additional con trol in the form presen ted in (5.45) or (5.46) and w T = [@V=@x] 16 G 63 where V is Ly apuno v function of nominal system listed in (5.56). The pro of of stabilit y is discussed in App endix B and the deriv ation sho ws that the rotor system with uncertain ties can b e stabilized b y the con trol v . 5.4 Simulation Results The sim ulation mo del of the feedbac k con trol system using Ly apuno v redesign metho d w as p erformed using Sim ulink, see the diagram sho wn in Fig. 5.1. T o sho w that the robust con trol v o v ercomes the p erturbation from the system parameters, w e set the additional con trol v = 0 in Section 5.3.1, whic h the system con tains only nominal con trol design giv en in (5.44). The sim ulation studies of the system with full Ly apuno v redesign con troller ( (x) +v ) are presen ted in Section 5.4.2. Figure 5.1: Ly apuno v redesign sim ulation mo del 5.4.1 System with Nominal Con trol In Chapter 4, the stem stabilit y conditions are in v estigated and the unstable zone during the acceleration pro cess are around the system’s natural frequency . T o test the p erformance the nonlinear con troller, the case lo cates in the unstable region in 74 Section 4.2. is used for comparison. In Fig. 5.2, the time resp onse of the radius of the shaft and the rotating sp eed of the disk are sho wn, in whic h the dimensionless system parameters are set as input rotating sp eedw c = 1:2, actual eccen tricit y e = 0:8, actual damping ratio of the shaft = 0:1, actual damping ratio of the disk r = 0:1, actual natural frequency w n = 1:05, PI torque con trol gain K p = 0:5, K I = 0:1 and the desire radius of the shaft r 0 = 0. The uncertain ties are considered in this case, so the nominal system parameters are set dieren tly as ^ e = 0:5, ^ = 0:15, ^ r = 0:15 and ^ w n = 1:05. The corresp onding con trol eort is giv en in Fig. 5.3. The result sho ws that the radius r and the rotating sp eed w do not con v erge to the target (r 0 = 0 and w c = 1:2). In other case w c = 1:8 sho wn in Fig. 5.4 and Fig. 5.5, whic h is the case outside the unstable region of the linear system, the system can still settle at equilibrium p oin t (r 0 = 0:4, w c = 1:8) with nominal con troller. In Fig. 5.6 and Fig. 5.7, the rotating sp eed is set as w c = 1:2, but with higher PI con trol gain (K p = 5, K I = 1). The results demonstrate that the system with the con trol is stable. F rom ab o v e three cases, the nominal con troller can not guaran tee the closed-lo op stabilit y for all com binations of the system parameters. Although the system migh t still main tain the stabilit y around the natural frequency with high PI gain, but the p erformance of the radius r is far from the target r 0 = 0. T o impro v e the system p erformance, additional con trol v is applied to the system to k eep the robustness to the uncertain ties in the follo wing subsection. Figure 5.2: Time resp onse of the system with nominal con troller (w c = 1.2) 75 Figure 5.3: Con trol eort of the system with nominal con troller (w c = 1.2) Figure 5.4: Time resp onse of the system with nominal con troller (w c = 1.8) 76 Figure 5.5: Con trol eort of the system with nominal con troller (w c = 1.8) Figure 5.6: Time resp onse of the system with nominal con trol (high PI gain) 77 Figure 5.7: Con trol eort of the system with nominal con trol (high PI gain) 5.4.2 System with Ly apuno v Redesign Con trol (Robustness) T o compare with the results ab o v e, the system parameters are c hosen as the same with K p = 5 and K I = 1. In Fig. 5.8, w e can nd that the radius of the shaft settle atr 0 = 0:0016 in a nite time. The radius do es not con v erge to r 0 = 0 b ecause of the dela y of the con trol and the limitation of the sampling time of the computer. The corresp onding con trol eort are sho wn in Fig. 5.9 and Fig. 5.10. The c hattering of the con trol eort o ccurs due to fast switc hing on the con troller ( v =(x) w kwk 2 ). 78 Figure 5.8: Time resp onse of the system with Ly apuno v redesign con trol Figure 5.9: Con trol eort of the system with Ly apuno v redesign con trol 79 Figure 5.10: Con trol eort of the system with Ly apuno v redesign con trol (zo om in) T o x the c hattering on the con trol eort , the con trol v can b e designed b y using ultimate b oundness giv en in (5.11). Fig. 5.11 sho ws the time resp onse of the system with Ly apuno v redesign con trol using ultimate b oundness giv en in (5.46), where = 0:1. The radius r settle at r 0 = 0:0014 in steady-state. The con trol eort of this case are sho wn in Fig. 5.12 and Fig. 5.13. W e can nd from the Fig. 5.13 that the switc h time of the con troller slo ws do wn a lot from the previous case in Section 5.4.1. Last, w e set up = 1 for the ultimate b oundness con trol design giv en in (5.46) and k eep all other parameters the same as the previous case. The results of time resp onse and con trol eort are giv en in Fig. 5.14 - Fig. 5.16. It is sho wn that with slo w er switc hing on the con troller, the system has lesser c hattering. But due to the slo w resp onse to the vibration, the radius r settles ten times further from the target. (r 0 = 0:014). 80 Figure 5.11: Time resp onse of the system with Ly apuno v redesign con trol (Ultimate b oundness) Figure 5.12: Con trol eort of the system with Ly apuno v redesign con trol (Ultimate b oundness) 81 Figure 5.13: Con trol eort of the system with Ly apuno v redesign con trol b y using ultimate b oundness (zo om in) Figure 5.14: Time resp onse of the system with Ly apuno v redesign con trol (Ultimate b oundness, e = 1) 82 Figure 5.15: Con trol eort of the system with Ly apuno v redesign con trol (Ultimate b oundness, e = 1) Figure 5.16: Con trol eort of the system with Ly apuno v redesign con trol b y using ultimate b oundness, e = 1 (zo om in) 83 6 Control Development using Sliding Mode Control 6.1 Sliding Mode Control The follo wing theorems and equation deriv ations of sliding mo de con trol are from [18]. Consider the system _ x = f(x) +B(x)[G(x)E(x)u +(t;x;u)] (6.1) where x2 R n is the state, u = R p is the con trol input, and f , B , G, and E are sucien tly smo oth functions in a domain DR n that con tains the origin. The function is piecewise con tin uous in t and sucien t smo oth in (x;u) for (t;x;u)2 [0;1)DR p . W e assume that f , B , and E are kno wn, while G and could b e uncertain. F urthermore, w e assume that E(x) is a nonsingular matrix and G(x) is a diagonal matrix whose elemen ts are p ositiv e and b ounded a w a y from zero; that is g i (x) g 0 > 0, for all x2 D 3 . Supp ose f(0) = 0 so that, in the absence of , the origin is an op en-lo op equilibrium p oin t. Our goal is to design a state feedbac k con trol la w to stabilize the origin for all uncertain ties in G and . Let T :D!R n b e a dieomorphism suc h that @T @x B(x) = " 0 I # (6.2) where I is the pp iden tit y matrix. The c hange of v ariables " # = T (x); 2R np ; 2R p (6.3) transforms the system in to the form _ = f a (;) _ = f b (;) +G(x)E(x)u +(t;x;u) (6.4) The form (6.4) is usually referred to as the regular form. T o design the sliding 84 mo de con trol, w e start b y designing the sliding manifold s =() = 0 suc h that, when motion is restricted to the manifold, the reduced-order mo del _ = f a (;()) (6.5) has an asymptotically stable equilibrium p oin t at the origin. The design of () amoun ts to solving a stabilization problem for the system _ = f a (;) with view ed as the con trol input. This stabilization problem ma y b e solv ed b y using the tec hniques of linearization or feedbac k linearization. W e assume that w e can nd a stabilizing con tin uously dieren tiable function () with (0) = 0. Next, w e design u to bring s to zero in nite time and main tain it there for all future time. T o w ard that end, let us write the _ s-equation: _ s = f b (;) @ @ f a (;) +G(x)E(x)u +(t;x;u) (6.6) W e can design u as a pure switc hing comp onen t or it ma y con tain an additional con tin uous comp onen t that cancels the kno wn terms on the righ t-hand side of (6.6). If ^ G(x) is a nominal mo del ofG(x), the con tin uous comp onen t of u will b eE 1 ^ G 1 [f b @ @ f a ]. In the absence of uncertain t y; that is when = 0 and G is kno wn, taking u =E 1 ^ G 1 [f b @ @ f a ] results in _ s = 0, whic h ensures that the condition s = 0 can b e main tained for all future time. T o analyze b oth cases sim ultaneously , w e write the con trol u as u = E 1 (x)fL(x)[f b (;) @ @ f a (;)] +vg (6.7) where L(x) = ^ G 1 (x), if the kno wn terms are canceled, and L = 0, otherwise. Substituting (6.7) in to (6.6) yields, _ s i = g i (x)v i +4 i (t;x;u); 1ip (6.8) 85 where4 i is the ith comp onen t of 4(x;v) = (t;x;E 1 (x)L(x)(f b (;) @ @ f a (;)) +E 1 (x)v) +[IG(x)L(x)][f b (;) @ @ f a (;)] (6.9) and g i is the ith diagonal elemen t of G. W e assume that the ratio4 i =g i satises the inequalit y 4(x;v) g i (x) (x) + 0 kvk 1 ; 8(t;x;u)2 [0;1)DR p ;81ip (6.10) where (x) > 0 (a con tin uous function) and 0 2 [0; 1) are kno wn. Using the estimate (6.10), w e pro ceed to design v to forces to w ard the manifold s = 0. Utilizing V i = (1=2)s 2 i as a Ly apuno v function candidate for (6.9), w e obtain _ V i = S i _ S i =S i g i (x)v i +S i 4 i (x;v) g i (x)fS i v i +jS i j ((x) + 0 kvk 1 )g T ak e v i =(x)sign(S i ); 1ip (6.11) where (x) (x) 1 0 + 0 ; 8x2D (6.12) and 0 > 0. Then, 86 _ V i g i (x)[(x) +(x) + 0 (x)]jS i j = g i (x)[(1 0 )(x) +(x)]jS i j g i (x)[(x) (1 0 ) 0 +(x)]jS i j = g 0 0 (1 0 )jS i j (6.13) The inequalit y _ V i g 0 0 (1 0 )jS i j ensures that all tra jectories starting o the manifold s = 0 reac h it in nite time and those on the manifold cannot lea v e it. The sliding mo de con troller con tains the discon tin uous sign um function, sgn (s i ), whic h raises some theoretical as w ell as practical issues. Theoretical issues lik e exis- tence and uniqueness of solutions and v alidit y of Ly apuno v analysis will ha v e to b e examined in a framew ork that do es not require the state equation to ha v e lo cally Lip- sc hitz righ t-hand-side functions. there is also the practical issue of c hattering due to imp erfections in switc hing devices and dela ys. T o eliminate c hattering, w e use a con- tin uous appro ximation of the sign um function. By using a con tin uous appro ximation, w e also a v oid the theoretical diculties asso ciated with discon tin uous con trollers. W e appro ximate the sign um function sgn (s i ) b y the high-slop e function sat(s i ="); that is, v i = (x)sat( s i " ); 1ip where (x) satises (6.12). 6.2 Control Law Deriv ation 6.2.1 Sliding Mo de Con trol Prop osition. Consider the r otor dynamic system given by _ X 1 = f 1 (x) (6.14) _ X 2 = f 2 (x) +g(x)u (6.15) 87 wher e the functions f 1 (x), f 2 (x) and g(x) ar e dene d in the e quations (5.37) - (5.39) and denotes u is the fe e db ack c ontr ol law. Dene the sliding manifold as S =X 2 (X 1 ) = 0, wher e X 2 = 2 6 4 x 4 x 5 x 6 3 7 5 ; (X 1 ) = 2 6 4 x 1 x 2 (x 1 + ^ r 0 ) +w c x 1 x 3 3 7 5 (6.16) and > 0. The c ontr ol u is sele cte d as u = ^ g(x) 1 ( ^ f 2 (x) @ @X 1 _ X 1 ) + ^ g(x) 1 v = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 (x 5 +^ r 0 wc) 2 (x 1 +^ r 0 ) + (2 ^ )x 4 + (x 1 + ^ r 0 ) ^ e(x 6 +w c ) 2 cos(x ) ^ ex 6 sin(x ) (x 5 +^ r 0 wc)x 4 (x 1 +^ r 0 ) + (2 ^ )x 5 + 2 ^ ^ r 0 w c +^ e(x 6 +w c ) 2 sin(x ) + (w c x 2 )x 4 +w c x 1 ^ ex 6 cos(x ) 2 ^ r (x 6 +w c ) ^ e(x 1 + ^ r 0 ) sin(x )x 6 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 + ^ g(x) 1 v (6.17) where x =x 2 x 3 ^ and v is given by v = [ v 1 v 2 v 3 ] T , v i = (x)sign(S i ) i = 1; 2; 3 (6.18) with (x) dene d as (x) (x) 1 0 + 0 ; 0 > 0 (6.19) and 88 (x) 2 6 6 6 6 6 6 6 6 6 6 6 4 2jx 4 j + 2j(x 1 + ^ r 0 )j + (x 6 +w c ) 2 +2j(x 6 +w c )j +jx 6 j 2j(x 5 + ^ r 0 w c )j (x 6 +w c ) 2 +2j(x 6 +w c )j +j(x 1 + ^ r 0 )jjx 6 j 2j(x 6 +w c )j +j(x 1 + ^ r 0 )j 3 7 7 7 7 7 7 7 7 7 7 7 5 1 (6.20) Then the tra jectory on the manifold S = 0 will con v erge asymptotically to the origin. In addition, all tra jectories are guaran teed to reac h the manifold S = 0 in nite time and sta y on it for all future time. Pr o of. The sliding manifold S = X 2 (X 1 ) = 0 ) X 2 = (X 1 ), where the functions are dened in (6.16). By substituting the manifold in to Eq. (5.37) yields _ X 1 = 2 6 4 _ x 1 _ x 2 _ x 3 3 7 5 = 2 6 4 x 4 x5wcx 1 (x 1 +^ r 0 ) x 6 3 7 5 = 2 6 4 x 1 x 2 x 3 3 7 5 (6.21) This system is called reduced-order system, since the state equation _ X 1 is decoupled from _ X 2 . The stabilit y of the reduced-order system can b e c hec k ed b y the Ly apuno v function candidate V = 1 2 x 2 1 + 1 2 x 2 2 + 1 2 x 2 3 , whic h satises V (0) = 0 and V > 0. ) _ V =x 1 _ x 1 +x 2 _ x 2 +x 3 _ x 3 =x 2 1 x 2 2 x 2 3 < 0 The reduced-order system satises Ly apuno v stabilit y conditions. In other w ords, the rotor system giv en in (6.14) - (6.15) is stable or sa y all the states settle at origin in a nite time if the manifold S = 0 is reac hed. Next, w e need to pro of that once the tra jectory reac hes the manifold s = 0 , the con trol u main tains it there for all future time, whic h is _ S = 0. 89 The Eqs (6.14) - (6.15) can also b e written as _ X 1 = f 1 (x) = ^ f 1 (x) (6.22) _ X 2 = f 2 (x) +g(x)u = ^ f 2 (x) + ^ g(x)u + (f 2 (x) ^ f 2 (x)) + (g(x) ^ g(x))u (6.23) Where ^ f 1 (x), ^ f 2 (x) and ^ g(x) are nominal system functions dened in (5.22), (5.23), and (5.24) in Section 5.2 Let (f 2 (x) ^ f 2 (x)) + (g(x) ^ g(x))u =(x), suc h that _ X 2 = ^ f 2 (x) + ^ g(x)u +(x) (6.24) F rom the manifold S =X 2 (X 1 ), the deriv ativ e _ S can b e found as _ S = _ X 2 @ @X 1 _ X 1 = ^ f 2 (x) @ @X 1 _ X 1 + ^ g(x)u +(x) (6.25) In nominal case, whic h is all the system parameters are kno wn and the uncertain t y (x) and the additional con trol v equal to zero. By substituting the con trol u dened in (6.17) in to (6.25) results in _ s = 0. Whic h k eep the manifold s = 0 for all future time. If the uncertain t y is not zero or sa y the tra jectory are o the manifold, an addi- tional con trol v is needed to guaran tee that _ s = 0 is satised. F or (x)6= 0, _ S = v + [(f 2 (x) ^ f 2 (x)) (g(x) ^ g(x)) ^ g(x) 1 ( ^ f 2 (x) @ @X 1 _ X 1 ) +(g(x) ^ g(x)) ^ g(x) 1 v] Let 4(x;v) = [(f 2 (x) ^ f 2 (x)) (g(x) ^ g(x)) ^ g(x) 1 ( ^ f 2 (x) @ @X 1 _ X 1 ) +(g(x) ^ g(x)) ^ g(x) 1 v] th us _ S = v +4(x;v) 90 where4(x;v) satises the condition j4(x;v)j (x) + 0 kvk 1 (6.26) and the function(x) can b e found as (x) (f 2 (x) ^ f 2 (x)) (g(x) ^ g(x)) ^ g(x) 1 ( ^ f 2 (x) @ @X 1 _ X 1 ) 1 (6.27) By substituting functions f 2 (x), ^ f 2 (x), g(x) and ^ g(x) in to (6.27) yields, (x) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 2( + ^ )x 4 + ( 2 + 1)(x 1 + ^ r 0 ) +(e ^ e)(x 6 +w c ) 2 cos(x ) +2e( r ^ r )(x 6 +w c ) sin(x ) +( 2 e 2 + ^ ee)(x 1 + ^ r 0 ) sin 2 (x ) (^ ee)x 6 sin(x ) 2( + ^ )(x 5 + ^ r 0 w c ) (e ^ e)(x 6 +w c ) 2 sin(x ) +2e( r ^ r )(x 6 +w c ) cos(x ) +( 2 e 2 + ^ ee)(x 1 + ^ r 0 ) sin(x ) cos(x ) (^ ee)x 6 cos(x ) 2( r + ^ r )(x 6 +w c ) + ( 2 e ^ e)(x 1 + ^ r 0 ) sin(x ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1 T o simplify the expression, w e can c ho ose (x) b y putting the b ound on eac h term in the v ector, suc h that (x) 2 6 6 6 6 6 6 4 2jx 4 j + 2j(x 1 + ^ r 0 )j + (x 6 +w c ) 2 + 2j(x 6 +w c )j +jx 6 j 2j(x 5 + ^ r 0 w c )j (x 6 +w c ) 2 + 2j(x 6 +w c )j +j(x 1 + ^ r 0 )jjx 6 j 2j(x 6 +w c )j +j(x 1 + ^ r 0 )j 3 7 7 7 7 7 7 5 1 In Eq. (6.26), 0 needs to b e b ounded in the range 0 0 < 1 and the v erication is 91 sho wn b elo w 0 = (g(x) ^ g(x)) ^ g(x) 1 1 = 2 6 4 0 0 (^ ee) sin(x ) 0 0 (^ ee) cos(x ) 0 0 0 3 7 5 2 6 4 1 0 ^ e sin(x ) 0 1 ^ e cos(x ) 0 0 1 3 7 5 1 = 2 6 4 0 0 (^ ee) sin(x ) 0 0 (^ ee) cos(x ) 0 0 0 3 7 5 1 j(^ ee)j Assume that the b ound of the eccen tricit y e is kno wn, whic he min <e<e max . Cho ose the nominal eccen tricit y ^ e = emax+e min 2 suc h thatj(^ ee)j< 1. Th us, 0 < 1. Last, to bring S to the manifold S = 0, the con trol v is c hosen as v i = (x)sign(S i ) i = 1; 2; 3 where (x) (x) 1 0 + 0 and 0 > 0. T o sho w that the tra jectories outside the manifold S = 0 reac h it in nite time and sta ys on it for all future time, the Ly apuno v function candidate is selected as follo ws V i = (1=2)s 2 i and the pro of of _ V < 0 is sho wn in (6.13). F rom (6.13), _ V i jS i j where is a p ositiv e constan t. _ V i = S i _ S i jS i j 92 If S i > 0, d dt S i Let t r b e the time required to reac h the surface S = 0. In tegrating b et w een t = 0 and t =t r yields, S(t r )S(0) = 0S(0) (t r 0) )t r S(0) Similarly , for S i < 0, t r jS(0)j Th us, S = 0 will b e reac hed in a nite time. This ab o v e pro of is referred to [37]. Similar to Ly apuno v redesign metho d, to reduce the c hattering of fast switc hing in the con troller, con trol v can also b e selected b y using saturation function sho wn as follo ws, whic h is discussed in Section 6.1. v i = (x)sat( s i " ); 1ip T o simplify the expression of the con trol u in (6.17), w e pro vide a dieren t c hoice of the nominal system and the functions ^ f2 and ^ g(x) are sho wn b elo w ^ f 2 (x) 31 = 2 6 6 6 6 6 6 4 2 ^ x 4 2 ^ (x 5 + ^ r 0 w c ) 2 ^ r (x 6 +w c ) 3 7 7 7 7 7 7 5 ; ^ g(x) =I 33 = 2 6 4 1 0 0 0 1 0 0 0 1 3 7 5 (6.28) 93 Cho ose the con trol u as u = ( ^ f 2 (x) @ @X 1 _ X 1 ) +v = 2 6 4 2 ^ x 4 x 4 +v 1 2 ^ (x 5 + ^ r 0 w c ) + (w c x 2 )x 4 (x 5 w c x 1 ) +v 2 2 ^ r (x 6 +w c )x 6 +v 3 3 7 5 By using the Eqs. (6.26) and (6.27), the functions (x) and 0 are found as (x) 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 (x 5 +^ r 0 wc) 2 (x 1 +^ r 0 ) + 2( + ^ )x 4 2 (x 1 + ^ r 0 ) +e(x 6 +w c ) 2 cos(x ) +2e( r ^ r )(x 6 +w c ) sin(x ) 2 e 2 (x 1 + ^ r 0 ) sin 2 (x ) +ex 6 sin(x ) (x 5 +^ r 0 wc)x 4 (x 1 +^ r 0 ) + 2( + ^ )(x 5 + ^ r 0 w c )e(x 6 +w c ) 2 sin(x ) +2e( r ^ r )(x 6 +w c ) cos(x ) 2 e 2 (x 1 + ^ r 0 ) sin(x ) cos(x ) +ex 6 cos(x ) 2( r + ^ r )(x 6 +w c ) + 2 e(x 1 + ^ r 0 ) sin(x ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1 and 0 = kg 0 (x)k 1 = 2 6 4 0 0 e sin(x ) 0 0 e cos(x ) 0 0 0 3 7 5 1 = je sin(x )j e e is dimensionless eccen tricit y and is dened as e 0 rg . Where e 0 is the eccen tricit y with dimension in the equations (3.18) - (3.20) and r g is the radius of gyration dis- cussed in 3.2.2. Assume e 0 is b ounded as e 0 min < e < e 0 max and e 0 max < r g or sa y e amax < R p 2 if the rotor is a round disk and R is the radius of the rotating disk, then 0 < 1. Usually , the eccen tricit y is quite small in the rotor system. e 0 max < R p 2 means 94 the largest eccen tricit y is less than 70% of the radius of the rotating disk. The b ound is more than sucien t in ph ysical cases. Similarly , the additional con trol v can b e calculated b y the same Eqs. (6.18) - (6.19) giv en in the previous case. 6.2.2 Sliding Mo de with PI torque Con trol In this case, PI con troller in the torque equation is added in the sliding mo de con trol design pro cess to impro v e the system p erformance. The total con trol u con tained t w o parts are presen ted as the follo wing with the states giv en in Section 5.2.1, u = u PI +u s = 2 6 4 0 0 K P x 6 K I (x 3 + ^ C) 3 7 5 + 2 6 4 u 1 u 2 u 3 3 7 5 (6.29) where u PI represen ts PI con trol and u s is the con trol designed b y using sliding mo de metho d sho wn in the cases in Section 6.2.1. The rotor system Eqs. (6.14) and (6.15) can b e written as _ X 1 = f 1 (x) = ^ f 1 (x) _ X 2 = f 2 (x) +g(x)u = f 2 (x) +g(x)u PI +g(x)u s = ^ f 2 (x) + ^ g(x)u PI +u s + (f 2 (x) ^ f 2 (x)) + (g(x) ^ g(x))u PI + (g(x)I 33 )u s = ^ F PI (x) +u s +(x) where ^ F PI (x) = ^ f 2 (x) + ^ g(x)u PI . The functions ^ f 2 (x) and ^ g(x) are dened in (6.28). Similar to the deriv ation sho wn in Section 6.2.1, the sliding mo de con trol u s can b e obtained from 95 u s = ( ^ F PI (x) @ @X 1 _ X 1 ) +v = 2 6 6 6 6 6 6 6 6 6 4 2 ^ x 4 x 4 K P ^ ex 6 K I ^ e(x 3 + ^ C) +v 1 2 ^ (x 5 + ^ r 0 w c ) + (w c x 2 )x 4 (x 5 w c x 1 ) K P ^ ex 6 K I ^ e(x 3 + ^ C) +v 2 2 ^ r (x 6 +w c ) +K P x 6 +K I (x 3 + ^ C)x 6 +v 3 3 7 7 7 7 7 7 7 7 7 5 (6.30) The functions (x) and 0 are giv en as (x) (f 2 (x) ^ f 2 (x)) + (g(x) ^ g(x))u PI g 0 (x)( ^ F PI (x) @ @X 1 _ X 1 ) 1 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 (x 5 +^ r 0 wc) 2 (x 1 +^ r 0 ) + 2( + ^ )x 4 2 (x 1 + ^ r 0 ) +e(x 6 +w c ) 2 cos(x ) +2e( r ^ r )(x 6 +w c ) sin(x ) 2 e 2 (x 1 + ^ r 0 ) sin 2 (x ) +ex 6 sin(x ) K P ^ ex 6 K I ^ e(x 3 + ^ C) (x 5 +^ r 0 wc)x 4 (x 1 +^ r 0 ) + 2( + ^ )(x 5 + ^ r 0 w c ) e(x 6 +w c ) 2 sin(x ) +2e( r ^ r )(x 6 +w c ) cos(x ) 2 e 2 (x 1 + ^ r 0 ) sin(x ) cos(x ) K P ^ ex 6 K I ^ e(x 3 + ^ C) +ex 6 cos(x ) 2( r + ^ r )(x 6 +w c ) + 2 e(x 1 + ^ r 0 ) sin(x ) 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1 (6.31) and 96 0 = kg 0 (x)k 1 = 2 6 4 0 0 e sin(x ) 0 0 e cos(x ) 0 0 0 3 7 5 1 = je sin(x )j e Assume the maxim um b ound of eccen tricit y is smaller than 1, then 0 < 1, whic h is discussed in Section 6.2.1. A ccording to the pro of sho wn in Section 6.2.1, the additional con trol v can b e obtained b y the Eqs. (6.18) - (6.19). The results of the sliding mo de con trol with and without the PI con trol are presen ted in Section 6.3. 97 6.3 Simulation Results 6.3.1 Sliding Mo de Con trol The sim ulation mo del of the feedbac k con trol system using sliding mo de is established in Matlab Sim ulink sho wn in Fig. 6.1. Figure 6.1: Sliding Mo de Con trol Sim ulation Mo del T o test the p erformance of the sliding mo de con trol, w e pic k the case lo cates in the unstable region of the system with only PI torque con trol analyzed in Section 4.2. Fig. 6.2 sho ws the time resp onse of the radius of the shaft and the rotating sp eed of the disk, in whic h the dimensionless system parameters are set as input rotating sp eedw c = 1:2, eccen tricit y e = 0:8, damping ratio of the shaft = 0:1, damping ratio of the disk r = 0:1, natural frequency w n = 1:05 and the desire radius of the shaft r 0 = 0. The uncertain ties are considered in this case, so the nominal system parameters are set dieren tly as ^ e = 0:5, ^ = 0:15, ^ r = 0:15 and ^ w n = 1:05 for testing the robustness of the con troller. F rom the plot, w e can nd that the rotating sp eed con v erge to the target w c = 1:2 in ab out 3 sec and the radius of the shaft tend to settle at zero. Due to the dela y of the con troller and the sampling time if the computer, the radius can only approac h to the target, but not the exact v alue(r 0 = 0:001, the target 98 is zero). The con trol eort is sho wn in Fig. 6.3 and Fig. 6.4. Since the sign function is used in the con trol v =(x)sign(s), the c hattering due to the fast switc hing of the con trol o ccurs in the case. This c hattering can b e seen in Fig. 6.4, where the time in terv als are small(switc hing in 0.001 sec). Figure 6.2: Time resp onse of the system with sliding mo de con trol Figure 6.3: Con trol eort of the system with sliding mo de con trol 99 Figure 6.4: Con trol eort of the system with sliding mo de con trol (zo om in) T o x the c hattering in con trol eort, the sign function can b e replaced b y the saturation function. In this case, v ==(x)sat( s " )," = 0:08 and all the parameters are the same as previous case for comparison. The time resp onse and con trol eort are sho wn in Fig. 6.5 and 6.6. The con trol eort runs smo othly without c hattering, but due to slo w er switc hing in the con trol the equilibrium of the radius is larger than the previous case (r 0 = 0:016). 100 Figure 6.5: Time resp onse of the system with sliding mo de con trol (Saturation) Figure 6.6: Con trol eort of the system with sliding mo de con trol (Saturation) 6.3.2 Sliding Mo de Con trol with T orque PI Con trol T o impro v e the system p erformance (k eep radius r 0 as small as p ossible), the PI torque con trol is used and the sim ulation conguration is presen ted in Fig. 6.7. 101 Figure 6.7: Sliding Mo de Con trol Sim ulation Mo del (with PI con trol) W e use the same system parameters giv en in Section 6.3.1 and the con trol v is designed b y using saturation function. With K p = 5 and K I = 1, the resp onse and the con trol eort are sho wn in Fig. 6.8 and Fig. 6.9. The radius of the shaft r 0 settles at 0:004 in the steady-state, whic h is ab out 25% of the radius in Fig. 6.6. But due to the extra term (PI) in the con trol, the con trol eort is larger at starting p oin t (t = 0). T o reduce the p o w er consumption at t = 0, w e add the b ound (saturation) on the con trol eort. The results in Fig. 6.10 and Fig. 6.11 sho w that the resp onse settle at the targetw c = 1:2 and r 0 = 0:004 with lo w er con trol eort. 102 Figure 6.8: Time resp onse of the system with sliding mo de con trol and PI torque con trol Figure 6.9: Con trol eort of the system with sliding mo de con trol and PI torque con trol 103 Figure 6.10: Time resp onse of the system with sliding mo de con trol and PI torque con trol (Saturation on con trol eort) Figure 6.11: Con trol eort of the system with sliding mo de con trol and PI torque con trol (Saturation on con trol eort) 104 7 F uture Research T opics 1. Vibration studies of con tin uous rotor system In this researc h, the rotor system is mo deled as a Jecott rotor system. The same con- trol design o w can b e applied to a more complicated dynamic system or a con tin uous rotor system mo deled b y nite elemen t metho d. 2. Rotor dynamic system with v arying parameters In industry , the damping and the spring stiness k sometimes are considers as func- tions of rotating sp eed w . The system parameters or the c haracters of b earings are v arying with the rotating sp eed, whic h mak es the con trol design tasks more c halleng- ing. The system parameters could b e replaced b y the functions(w) and k(w) as an extension of the researc h. 3. Rotor Mo deling including Magnetic Bearing Dynamics The idea of magnetic b earing is in tro duced in Chapter 3. F rom the studies, the magnetic b earings pro vide forces on x and y directions of the rotating shaft, whic h implies that the radius of the exible rotating shaft can b e con trolled through the actuator. In this researc h, w e fo cus on the nonlinear con trol design and the stabilit y studies. 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Complex mo dal balancing of exible rotors including residual b o w. Journal of Pr opulsion , V ol. 4, No.3:241251, 1988. [29] W. D. Pilk ey , 1. Bailey , and P . D. Simith. A computational tec hnique for opti- mizing correction w eigh ts and axial lo cation of balance planes of rotating shafts. T r ansactions of ASME, Journal of Vibr ation, A c oustics, Str ess, and R eliability in Design , V ol.105:9093, 1983. [30] S. Rao. R otor Dynamics, 3r d e d . New Age In ternational Publishers, India, 1996. [31] S. Saito and T. Azuma. Balancing of exible rotors b y the complex mo dal metho d. T r ansactions of ASME, Journal of Vibr ation, A c oustics, Str ess, and R eliability in Design , V ol.105:94100, 1983. [32] Gerhard Sc h w eitzer and Eric H. Maslen. Magnetic Be arings . Springer, 2009. [33] K. Shin and J. Ni. A daptiv e con trol of activ e balancing systems for sp eed-v arying rotors using feedforwrd gain adaptation tec hnique. ASME Journal of Dynamic Systems, Me asur ement and Contr ol, Septemb er , V ol.123, 2001. [34] Jean-Jacques E. Slotine. The robust con trol of rob ot manipulators. Int. J. R ob otics R ese ar ch . [35] Jean-Jacques E. Slotine. Sliding con troller design for nonlinear systems. Int. J. Contr ol , page 40. [36] Jean-Jacques E. Slotine and W eiping Li. On the adaptiv e con trol of rob ot ma- nipulators. Int. J. Contr. , V ol. 38:465492, 1983. [37] Jean-Jacques E. Slotine and W eiping Li. Applie d Nonline ar Contr ol . P earson Education, 1991. [38] Serdar So ylu, Bradley J. Buc kham, and Ron P . P o dhoro deski. Mimo sliding- mo de and h-innit y con troller design for dynamic coupling reduction in underw ater-manipulator systems. T r ansactions of the Canadian So ciety for Me- chanic al Engine ering , V ol. 33, No. 4:731743, 2009. 108 [39] M. W. Sp ong and M. Vidy asagar. R ob ot Dynamics and Contr ol . Wiley , New Y ork, 1989. [40] I. M. T essarzik, R. H. Badgley , and W. 1. Anderson. Flexible rotor balancing b y the exact p oin t-sp eed inuence co ecien t metho d. Journal of Engine ering for Industry, T r ansactions of ASME, F ebruary , pages 148158, 1972. [41] K. T suc hiy a. P assage of a rotor through a critical sp eed. T r ansc ations of the Americ an So ciety of Me chanic al Engine ers; ournal of Me chanic al Design , 104:370374, 1982. [42] V. Utkin, J. Guldner, and J. Shi. Sliding Mo de Contr ol in Ele ctr ome chanic al Systems . T a ylor R F rancis, London, 1999. [43] V. I. Utkin. Sliding Mo des in Optimization and Contr ol . Springer-V erlag, New Y ork, 1992. [44] J. V an De V egte. Con tin uous automatic balancing of rotating systems. Journal of Me chanic al Engine ering Scienc e , V ol.6:264269, 1964. [45] J. V an De V egte. Balancing of exible rotors during op eration. Journal of Me chanic al Engine ering Scienc e , V ol.23:257261, 1981. [46] J. V an De V egte and Lak e R. T. Balancing of rotating systems during op eration. Journal of Sound and Vibr aiton , pages 225235, 1978. [47] T oshio Y amamoto and Y ukio Ishida. Line ar and Nonline ar R otor dynamics . JOHN WILEY & SONS, 2001. [48] Bingen. Y ang and Henryk. Flashner. Dynamics and con trol of exible rotating systems. Go v ening Equations. [49] S. Zhou and J. Shi. Optimal one-plane activ e balancing of rigid rotor during acceleration. Journal of Sound and Vibr aiton , 2001. [50] S. Zhou and J. Shi. Un balance estimation for sp eed-v arying rigid rotors us- ing time-v arying observ er. ASME T r ansc ations, Journal of Dynamic Systems, Me asur ement and c ontr ol , 2001. [51] Shiyu Zhou and Jianjun Shi. A ctiv e balancing and vibration con trol of rotating mac hinery: A surv ey . The sho ck and Vibr ation Digest , pages 361371, 2001. 109 A Routh Criterion The stabilit y can b e in v estigated b y R outhHurwitz stability criterion . F rom det(IA) = 0, the c haracteristic equation 4 + 4 3 + (2w 2 c + 2 + 4 2 ) 2 + (4w 2 c + 4) + (w 2 c 1) 2 + 4 2 w 2 c = 0 . All the co ecien ts of the c haracteristic p olynomial are greater than zero for > 0. The Routh arra y of the Jacobian matrix of the decoupled system (3.40) is sho wn as the follo wing. 4 : 1 3 : 4 2 : 2w 2 c + 2 + 4 2 1 4 4w 2 c + 4 =w 2 c + 4 2 + 1 1 : 4w 2 c + 4 4(w 4 c 2w 2 c + 1 + 4 2 w 2 c ) 2w 2 c + 2 + 4 2 1 4 4w 2 c +4 = 4 w 2 c + 4 2 + 1 (4w 2 c + 4 2 ) 0 : w 4 c 2w 2 c + 1 + 4 2 w 2 c = (w 2 c 1) 2 + 4 2 w 2 c When > 0, all the terms in the Routh arra y are greater than zero. Th us, the decoupled system is stable. In the case = 0, the system is marginally stable, but do esn’t settle at equilibrium p oin t. Consider the coupled system (3.21) - (3.23) without the damping ( = 0), where the torque con trol T d = K P ( _ w c ) K I ( _ w c ). The stabilit y around its 110 equilibrium p oin t is in v estigated b y the Routh arra y 6 : 1 5 : 2 r +K P 4 : er 0 +K I +e 2 3 : e 2 w 2 c (2 r +K P )(r 0 +e) r 0 (er 0 +K I +e 2 ) 2 : 4w 2 c r 0 (er 0 +K I +e 2 ) +K I ew 2 c r 0 1 : 4ew 2 c (2 r +K P )(r 0 +e)(1w 2 c ) 4r 0 (er 0 +K I +e 2 ) +K I e 0 : K I (1w 2 c )w 2 c r 0 When the rotating sp eed w c < 1, all the terms in Routh arra y are p ositiv e. While in the case of w c 1, the last t w o terms ( 1 and 0 ) in the Routh arra y ha v e negativ e sign due to the term (1w 2 c ). Th us, the coupled system without the damping is unstable for an y rotating sp eed higher than its natural frequency (w c = 1). B Stability Proof of Lyapunov Redesign The follo wing pro of is from [18]. In Ly apuno v redesign, the additional con trol v can b e selected as couple of dieren t t yp es. F or the case v =(t;x)sgn(w), the stabilit y of the en tire system is pro ofed as the follo wing. Supp ose (5.6) is satised with kk 1 ; that is, k(t;x; (t;x) +v)k 1 (t;x) + 0 kvk 1 ; 0 0 < 1 W e ha v e w T v +w T w T v +kwk 1 kk 1 w T v +kwk 1 [(t;x) + 0 kvk 1 ] 111 Cho ose v = (t;x)sgn(w) (B.1) where (t;x) (t;x)=(1 0 ) for all (t;x)2 [0;1)D and sgn(w) is a p- dimensional v ector whose ith comp onen t is sgn(w i ). Then w T v +w T kwk 1 +kwk 1 + 0 kwk 1 = (1 0 )kwk 1 +kwk 1 kwk 1 +kwk 1 = 0 Hence, with the con trol (B.1), the deriv ativ e of V (t;x) along the tra jectories of the closed-lo op system (5.7) is negativ e denite. Another con trol la w is c hosen as v = 8 > > > < > > > : (t;x)(w=kwk 2 ); if(t;x)kwk 2 (t;x)(w=) if(t;x)kwk 2 < (B.2) With (B.2), the deriv ativ e of V along the tra jectories of the closed-lo op system (5.7) will b e negativ e denite whenev er (t;x)kwk 2 . W e only need to c hec k _ V when (t;x)kwk 2 < . In this case, _ V 3 (kxk 2 ) +w T [ 2 w +] 3 (kxk 2 ) 2 kwk 2 2 +kwk 2 + 0 kwk 2 kvk 2 3 (kxk 2 ) 2 kwk 2 2 +kwk 2 + 0 2 kwk 2 2 3 (kxk 2 ) + (1 0 )( 2 kwk 2 2 +kwk 2 ) The term 112 2 kwk 2 2 + kwk 2 attains a maxim um v alue =4 at kwk 2 ==2. Therefore, _ V 3 (kxk 2 ) + "(1 0 ) 4 whenev er (t;x)kwk 2 < . On the other hand, when (t;x)kwk 2 , _ V satises _ V 3 (kxk 2 ) 3 (kxk 2 ) + (1 0 ) 4 Th us, the inequalit y _ V 3 (kxk 2 ) + (1 0 ) 4 is satised irresp ectiv e of the v alue of (t;x)kwk 2 . C Linear Matrix Inequality The follo wing corollaries and theorems regarding linear matrix inequalit y are from [9] Hurwitz Stability Consider a con tin uous-time linear system in the form of _ x(t) =Ax(t) (C.1) with A2 R nxn . It is said to b e stable if all the eigen v alues of the matrix A ha v e non-p ositiv e real parts; in this case, w e also sa y that the matrix A is Hurwitz critic al ly stable , and it is said to b e asymptotically stable if all eigen v alues of the matrix A ha v e negativ e real parts, and in this case w e also sa y that the matrix A is Hurwitz stable . Theorem. (Hurwitz Stability) 113 The c ontinuous-time system _ x(t) = Ax(t) is Hurwitz stable if and only if ther e exists a matrix P2S n , such that 8 < : P > 0 A T P +PA< 0 Quadratic Stability Consider the follo wing uncertain system x(t) =A((t))x(t) (C.2) with b eing c hosen as in one of the follo wing t w o case: Case1. represen ts the dieren tial op erator. In this case, the system is a con tin uous- time one, and p ossesses the follo wing form: _ x(t) =A((t))x(t) and here t is a con tin uous v ariable. Case2.represen ts the one step forw ard shift op erator. In this case, the system is a discrete-time one, and p ossesses the follo wing form: x(t + 1) =A((t))x(t) and here t is a con tin uous v ariable. The system co ecien t matrix tak es the form of A((t)) =A 0 +4A((t)) (C.3) whereA 0 2R nxn is a kno wn matrix, whic h represen ts the nominal system matrix, while 4A((t)) = 1 (t)A 1 + 2 (t)A 2 +::: + k (t)A k (C.4) is the system matrix p erturbation, where A i 2R nxn ;i = 1; 2;:::k; are kno wn matrices, whic h represen ts the p erturbation directions. 114 i (t);i = 1; 2;:::;k; are arbitrary time functions, whic h represen t the uncertain parameters in the system. (t) = [ 1 (t) 2 (t)::: k (t)] T is the uncertain parameter v ector, whic h is often assumed to b e within a certain compact and con v ex set 4, that is (t) = [ 1 (t) 2 (t)::: k (t)]24; (C.5) and the set4 is called the set of p erturbation parameters. In practical applications, often t w o t yp es of p erturbation parameters sets are used. One is the follo wing regular p olyhedron: 4 I =f(t)j i (t)2 [ i ; + i ]; i = 1; 2;:::kg (C.6) In this case the system through is called a t yp e of in terv al systems. The other t yp e of p erturbation parameter set is the follo wing p olytop e 4 P =f(t)j k X i=1 i (t) = 1; i (t) 0; i = 1; 2;:::;kg; (C.7) whic h is clearly a regular p olyhedron in R k . In this case, the system through is called a t yp e of p olytopic systems. Denition. (Quadratic Hurwitz Stabilit y) The system (C.2) thr ough (C.4), with 8(t)24, is c al le d quadr atic al ly Hurwitz stable if ther e exist a symmetric p ositive denite matrix P , such that A T ((t))P +PA((t))< 0; 8(t)24 Theorem 1. L et the system (C.2) thr ough (C.4), with 8(t)24, b e quadr atic al ly Hurwitz stable. Then, the system (C.2) thr ough (C.4) is uniformly and asymptotic al ly stable in the Lyapunov sense for every (t)24. 115 Theorem 2. The system (C.2) thr ough (C.4) is quadr atic al ly Hurwitz stable if and only if ther e exists a symmetric denite matrix P , such that A T ()P +PA()< 0; 824 E wher e4 E r epr esents the set of al l the extr eme p oints of 4. Corollary 1. The system(C.2) thr ough (C.4) , with 4 =4 I given by (C.6), is quadr atic al ly Hurwitz stable if and only if ther e exists a symmetric p ositive denite matrix P , such that A T ()P +PA()< 0; 8 i = i or + i ; i = 1; 2;:::k Corollary 2. The system (C.2) thr ough (C.4) , with 4 =4 I given by (C.7), is quadr atic al ly Hurwitz stable if and only if ther e exists a symmetric p ositive denite matrix P , such that (A 0 +A i ) T P +P (A 0 +A i )< 0; i = 1; 2;:::k 116

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Creator Lin, Yeh (author)

Core Title Nonlinear control of flexible rotating system with varying velocity

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 04/05/2016

Defense Date 11/23/2015

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Tag active control,control design,control system,dynamic system,eccentricity,feedback control,flexible rotating system,Jeffcott rotor,jump phenomenon,linearization,linearized system,Lyapunov redesign,mass imbalance,nonlinear,nonlinear control,nonlinear system,OAI-PMH Harvest,PID control,rotor,rotor system,simulation,sliding mode control,stability,system modeling,varying velocity,vibration,vibration reduction

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Advisor Flashner, Henryk (committee chair), Jonckheere, Edmond (committee member), Shiflett, Geoffrey (committee member)

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