Imbalance Estimation for Speed-Varying Rigid Rotors Using Time-Varying Observer

Transcription

1 Shiyu Zhou e-ail: Jianjun Shi 1 e-ail: shihang@uich.edu Departent of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI Ibalance Estiation for Speed-Varying Rigid Rotors Using Tie-Varying Observer Rigid rotor dynaic odel is widely used to odel rotating achinery. In this paper, a speed-varying transient rigid rotor odel is developed in the state space for. The states of this odel are augented to include ibalance forces and oents. A tie-varying observer can then be designed for the augented syste by using canonical transforation. After obtaining an estiation of the ibalance forces and oents as the states of the augented syste, the estiated ibalance can be directly calculated. This estiation ethod can be used in the active vibration control or active balancing schees for a rigid rotor. DOI: / Introduction Rotating achines, including achining tools, industrial turboachinery, and aircraft gas turbine engines, are coonly used in industry. Vibration caused by ass ibalance is an iportant factor liiting the perforance and fatigue life of a rotating syste. There are two ajor categories of control ethods for the suppression of excessive ibalance-induced vibration. The first category is balancing. This ethod tries to eliinate the rotor ibalance. Off-line balancing ethods Wowk 1 are widely used in practice, but they are usually tie-consuing and cannot be used if the distribution of ibalance changes during operation. Soe researchers Gosiewski 2,3, Van De Vegte and Lake 4, VanDe Vegte 5 tried to use soe kind of ass redistribution device to actively balance the rotating systes during operation. The ethod directly suppresses the ibalance-induced vibration or the force transitted to the base by using lateral force actuators such as agnetic bearings Knospe et al. 6, Lu et al. 7, Herzog et al. 8. All of the above research concentrated on the constant spin-speed case: the so-called steady-state case. Because of this constant spin-speed assuption, the influence coefficient ethod is used to odel the rotor syste. The whole rotor dynaics are luped into constants: the influence coefficients. The ibalance is ebedded in the influence coefficients. The estiation of the ibalance, which is very iportant in both the balancing and active vibration control schees, is fulfilled by the estiation of influence coefficients. An alternative way to estiate the syste ibalance is provided by Reinig and Desrochers 9 and Zhu et al. 10. In their ethods, the states of the rotor dynaic syste are augented to include the ibalance forces. An observer is then used to estiate the augented states set. Again, their ethods only deal with the constant spin-speed case. Hence, the enlarged syste is a tie invariant linear syste. The traditional Luenberger observer Luenberger 11 can be used to estiate the ibalance forces. Besides the constant rotating speed case, the ibalance vibration control needs to be copleted during speed-varying transient tie to save tie and iprove perforance in soe other cases. For exaple, a achining tool needs to be engaged in cutting as soon as the spindle goes into steady state in high-speed achining. The vibration control has to be active during the acceleration period to reduce the effect on the cutting cycle tie. Although 1 Author to who all correspondence should be addressed. Contributed by the Dynaic Systes and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the Dynaic Systes and Control Division Deceber 12, Associate Editor: S. Fassois. several researchers Knospe et al. 12 have pointed out how to conduct ibalance vibration control during the startup through the critical speed, their basic ethod is to interpolate the influence coefficients between different speeds. This is a quasi-steady strategy because the requireent of a steady-state response is inherited in the influence coefficient ethod. Very little research work has been reported on the balancing and active control for the rotor syste with fast acceleration and low daping ratio. Zhou and Shi 13 found an analytical expression of the ibalanceinduced vibration of a rotor syste during acceleration. Fro this analytical expression, a significant suddenly occurring free vibration coponent can be found in the ibalance-induced vibration if the acceleration is high and the daping is low. The quasisteady-state assuption does not hold under these conditions. Zhou and Shi 14 also proposed a real-tie active balancing schee for fast acceleration case. That schee is based on the least squares estiation of the syste ibalance of the rotor syste. In that schee, a ass redistribution active balancing device is required to excite the syste dynaics and hence ake the estiation converge fast. In this paper, a tie-varying observer is developed to estiate the ibalance force and the ibalance itself of a rigid rotor syste during acceleration. This forulation does not require steadystate response because the rotor is odeled by dynaic differential equations. Moreover, this forulation assues that the dynaic paraeters of rotor syste are known and hence no extra excitation is required. This result can be used in both active balancing and direct vibration control schees for rigid rotors. This paper consists of five sections. In Section 2, a speedvarying transient dynaic odel of a rigid rotor will be presented. It is forulated in a for suitable for ibalance force estiation. This dynaic odel is a tie-varying linear odel due to the gyroscopic effect and the inclusion of the ibalance forces in the states. The tie-varying observer design proble is addressed in Section 3. Section 4 gives soe siulation results to show the effectiveness of this ibalance estiation schee. Conclusions are given in the last section. 2 Speed Varying Dynaic Model of Rigid Rotors The dynaics of rigid rotor are very iportant because the rigid rotor odel has been the odel used for any practical rotors. The geoetric setup is shown in Fig. 1. The basic assuptions used in the odeling procedure are: a The rotor is rigid with circular cross section and the ibalance is represented as a concentrated ass on the shaft. Any static and dynaic ibalance can be represented by the ass and the position of this concentrated ibalance. Journal of Dynaic Systes, Measureent, and Control DECEMBER 2001, Vol. 123 Õ 637 Copyright 2001 by ASME

2 Fig. 1 The geoetric setup of rigid rotor odel b The bearings are odeled as a set of linear springs and dapers in two orthogonal directions with the sae spring rates and daping coefficients. The bearings locate at each end of the shaft and the ass center of the rotor is located at the idpoint between the two bearings. This assuption siplifies the dynaic odel because the translational otion and the conical otion of the rotor are uncoupled under this assuption. The translational otion in the two orthogonal directions X and Z is also uncoupled under this assuption. This siplified rigid rotor odel is used in the following derivation, but the extension to a ore general rigid rotor odel is quite straightforward. c The angular acceleration of the rotational otion is assued a known constant. The linear acceleration speed profile is coon for achining spindles. We also assue that the rotating angle, speed, and acceleration are known. They can be easily easured by attaching an encoder to the spindle. d The lateral vibration otions are assued sall to siplify the dynaics. Two coordinate systes are used in the derivation: the bodyfixed coordinate oxyz and the inertial coordinate OXYZ. The body-fixed y-axis is the rotating axis of the shaft and x- and z- axes are defined by the other two principal inertia axes of the rotor. The origin of xyz is selected as the geoetric center of the shaft. The XYZ coordinate syste is the stationary coordinate and coincides with the xyz coordinate syste when the body is at rest. The elastic and daping forces provided by the bearings are F k 2k R X R Z T, M k kl 2 /2 T /2 F c 2c Ṙ X Ṙ Z T, M c cl 2 /2 T /2. (1) The equation of otion can be obtained by using Newton s law for a rigid body It cl2 /2 kl2 /2 I 2 p I p ud sin cos uy cl 2 /2 kl 2 /2 I p I p u d 2 cos sin u y. (2) R X 2cṘ X 2kR X u d sin 2 cos R Z 2cṘ Z 2kR Z u d 2 sin cos Using the substitution of u x d cos and u z d sin and rewriting Eq. 2 into state space for, we can get the state space odel for rigid rotor, (3) where f 1 cos 2 sin, f 2 sin 2 cos, are known functions of tie. Observing that f 2 and f 1 are the real and iaginary parts of a coplex nuber F ( 2 i )e i actually, F d 2 (e i )/ dt 2 and d F 0 1 dt Ḟ 3i i F, (4) Ḟ we can augent the state space odel and ake the ibalance force be states. In fact, 638 Õ Vol. 123, DECEMBER 2001 Transactions of the ASME

3 uuz u u x u u x u y u u y u z u u x u u z u u y u z u u x u y f 1 f 2 Re uu x I uu x Re uu y u z I uu y u z uu z uu z uu x u y uu x u y F i F i i F i F 1 I F 1 Re F (5) Re F 2 I F 2. Re and I denote the real and iaginary part of a coplex nuber. F 1 and F 2 are F ultiplied by coplex constants as defined in Eq. 5. It is obvious that F 1 and F 2 also satisfy Eq. 4. Equation 4 can be rewritten in real doain as where f r and f i are the real and iaginary part of F. Cobining Eq. 3 through Eq. 6, the augented state space odel of the rigid rotor is d x A t Y1 Y2 0 A 1 t 0 dt 0 0 A 1 t x, y R X R Z T where x R X R Z Ṙ X Ṙ Z f 1r f 1i ḟ 1r ḟ 1i f 2r f 2i ḟ 2r ḟ 2i T, f 1r, f 1i and f 2r, f 2i are the real and iaginary part of F 1 and F 2, respectively. A(t) and A 1 (t) are the atrices shown in Eq. 3 and Eq. 6. Y 1 and Y 2 are defined as Y Y Rearks: The syste ibalance u, u x, u y, and u z are not shown explicitly in Eq. 7. They are iplicitly included in the odel as the initial conditions of f 1r f 1i f 2r f 2i. Since this paper concentrates on the ibalance estiation proble, the control input either lateral forces or balancerinduced forces is not included in the odel. It is straightforward to include these forces. The output of the odel is the displaceent of the ass center and the swinging angle of the rotor about X and Z direction. (6) (7) Since the rotor is rigid, these can be obtained by easuring the otion of any two non-coplanar points on the rotor. The odel is a linear tie variant odel because the rotating speed is included in the dynaic atrices A and A 1. Using this augented odel, the ibalance estiation proble is transfored to a state estiation proble. A linear observer can give the estiation of states fro incoplete states easureents. 3 Tie Varying Observer Design It is well known that the states of a tie-invariant linear syste can be reconstructed by a linear tie-invariant observer. An enorous body of literature has been published on this topic. An excellent review can be found in O Reilly 15. Copared to the tie-invariant observers, relatively few papers Wolovich 16, Yuksel and Bongiorno 17, Nguyen and Lee 18, Shafai and Carroll 19 deal with observers for tie-varying systes, such as the dynaic syste of Eq. 7. In this paper, the design of tie varying observers follows the basic steps in Nguyen and Lee 18. The essential results of their work are that: 1 a linear tievarying syste can be transfored into an observability canonical for under certain conditions; 2 the states of the canonical syste can be estiated by a full order observer; 3 the states of the original syste can be calculated fro the output of the full order observer. A brief review of these results is listed as follows. The tie varying syste is represented by the state space odel ẋ t A t x t B t u t, (8) y t C t x t where x(t), u(t), and y(t) are (n 1), (p 1), and (q 1) vectors and A(t), B(t), and C(t) are atrices with appropriate diensions. For this syste, a full-order observer can be in the for xˆ t F t xˆ t G t y t H t u t. (9) This observer is an asyptotic identity state observer for syste Eq. 8 if H t B t and F A t G t C t (10) F here is a constant atrix with stable eigenvalues. This result can be clearly seen by substituting Eqs. 8 and 10 into Eq. 9, which yields xˆ ẋ F xˆ x. (11) The convergence rate of the observer depends on the eigenvalues of atrix F. Given a general A(t), C(t), and the predeterined eigenvalues of F, it is difficult to find G(t). However, the procedure to find G(t) for an observability canonical for is siple. Under certain conditions, the syste Eq. 8 can be transfored into an observability for Eq. 12 by a Lyapunov transforation. x t Ā t x t B t u t (12) y t C t x t where Ā(t) and C (t) are in the fors where A12 A1q A 21 A 22 A 2q Ā t A11 ] ] ] A q1 A q2 A qq, A ii X ni 1 I ni ni 1, Journal of Dynaic Systes, Measureent, and Control DECEMBER 2001, Vol. 123 Õ 639

4 A ij X ni 10 ni n j 1 for i, j 1,2,...,q and i j, n i and n j are the observability indices associated with the ith and jth row of C(t) atrix, I j is j j identity atrix, 0 i j is i j zero atrix, F12 F1q F 21 F 22 F 2q F F11 ] ] ] F q1 F q2 F qq, where (14) i,0 i,1 F ii... ] 0 1 ni 1, i,ni 1 Ini 1 X represents a nonzero nuber. Nguyen and Lee 10 provide a siple ethod to find Ḡ(t) that satisfies F Ā(t) Ḡ(t)C (t), where F is in observability canonical for with desired eigenvalues. More clearly, F is in the for 0 F 1... (13) ] 0 1 n 1. n 1 In 1 After obtaining the Ḡ(t) atrix, we can get an asyptotic fullorder observer for syste 12. Since Lyapunov transforation is a nonsingular transforation, it is straightforward to estiate the states of the original syste 8 if the estiations of x (t) are available. Soe other observer design issues are not addressed in Nguyen and Lee 18. The first one is the selection of the observer eigenvalues. The decay rate of the observer error is deterined by the eigenvalues of the error dynaic atrix F. Theoretically, we can pick eigenvalues at the far left of the coplex plane for F to ake the observer outputs converge to the true states very fast. However, fast observer poles ay enlarge the effects of sensor noise. The selection of the eigenvalues is a coproise between the fast response speed and good noise soothing capability. Besides the decaying rate, another concern in the observer design is the transient perforance. Noticing the error dynaic equation Eq. 11, the overall error response of the observer is deterined by the F atrix including both the eigenvalues and the structure of F. A large transient error will take ore tie to die out if other conditions are the sae. Moreover, very large transient error could cause nuerical probles in the observer. Hence, sall transient error of the observer is desired. The transient perforance of the observer can be predicted by the fact stated in Chen 20 : For a copanion for dynaic atrix of order n, if all its eigenvalues are distinct, the largest agnitude in transient. is roughly proportional to ( ax ) n 1, where ax is the agnitude of the largest eigenvalue. Hence, in order to have a sall transient error, the order of the largest copanionfor block in the error dynaic atrix F should be kept as sall as possible. For a low diensional dynaic syste, the transient perforance of the observer using the F atrix in Eq. 13 is satisfactory. But for a high diensional dynaic syste, such as the augented rigid rotor odel with diension 16, the transient response of the observer will be too large. For exaple, if the largest agnitude of the eigenvalues of F is 10, then the largest transient response agnitude will be close to the order To deal with this proble, we can break the atrix F into sall copanion-for block atrix. The diension of each block is deterined by the corresponding observability index. The new atrix F has the siilar structure of the Ā(t), F ij 0 ni n j, for i, j 1,2,...,q and i j. The s in Eq. 14 are deterined by the desired eigenvalues. Actually, they are the coefficients of the characteristic equation of each copanion for blocks. To design a full order observer for F, we need to solve the atrix equation Ḡ t C t Ā t F t (15) for Ḡ(t). If all-zero coluns are eliinated fro both left and right sides of Eq. 15 by noting the special structure of F, Ā(t), and C (t), Eq. 15 changes to Ḡ n q t C q q t D n q, (16) where the coluns of D consist of the nonzero coluns of Ā(t) F (t) and the coluns of C consists of the nonzero coluns of C (t). It is clear that C is nonsingular. Equation 16 can be solved by Ḡ t D t C 1 t. (17) The design procedures are suarized as follows: 1 Transfor the original state space odel 8 into observability canonical for 12 by a Lyapunov transforation x(t) P(t)x (t), where P(t) is a nonsingular atrix. 2 Select n distinct desired eigenvalues for the estiator. 3 Group these n values into q groups. Each group has n i (i 1...q) eleents. n i is the observability index of the ith subsyste. Fro these q groups of eigenvalues, get q groups of i,0..ni 1. 4 Obtain F in Eq Obtain Ḡ(t) by the forula of Eq Use the full-order estiator 9 to estiate the states of 12, where F(t) F, H(t) B (t), and G(t) Ḡ(t). 7 Denoting the estiation in step 6 as x ˆ, the estiation of the original syste is xˆ P(t)xR. The rigid rotor syste 7 can be transfored into an observability copanion for. This is confired by the nuerical exaple in Section 4 and also can be easily checked analytically by substituting A(t) and C(t) atrices in 7 into the checking conditions Nguyen and Lee 18. Following the above procedures, we can design a tie-varying observer for the rigid rotor whose estiation error will go to zero asyptotically while the transient perforance is satisfactory. The estiation of ibalance force can be obtained by this observer. Fro the ibalance force estiation, the estiated ibalance itself can be obtained by direct algebraic calculation. If we denote 1 u u z /, 2 u u x /, 3 u u x u y /, 4 u u y u z /, this ethod can be forulized as follows: r f 1 fˆ 1i 3 4 f fˆ (18) 2 fˆ 2r 4 fˆ 2i, 640 Õ Vol. 123, DECEMBER 2001 Transactions of the ASME

5 where f 1 and f 2 are the sae as in Eq. 3, and fˆ 1r fˆ 1i fˆ 2r fˆ 2i T are the estiated ibalance force or oent. Equation 18 can be transfored into f 1 f f 2 f f 1 f f 2 f 1 1 fˆ 1r fˆ 1i (19) fˆ 2r fˆ 2i. f Since the absolute deterinant of 1 f 2 f 2 f is ( 4 2), which is always nonzero, we do not need to worry about the singularities of the calculation during acceleration. The advantage of this ethod is its siplicity. The disadvantage is that the effect of noise on the estiation will be presented in the ibalance estiation directly. The siulation result is presented in the next section to illustrate the above procedures and ethods. 4 Siulation The siulation is done with the paraeters L 0.5, r 0.1, k N/, c 1000 Ns/, u 0.5 kg, u x u y u z 0.08,0.2,0.05 in the body-fixed coordinate syste, and 100 rad s 2. Following the tie-varying observer design procedures, we first obtain the Lyapunov transforation atrix P(t). Then, the equivalent observability canonical for of syste Eq. 7 can be obtained. In this siulation, the four groups of eigenvalues of the F atrix are selected as , , , and The Ḡ(t) atrix can be obtained by using the forula Eq. 17. All these atrices are listed in the Appendix. The original syste Eq. 7 and the observer dynaic syste in step 6 are solved by the Runge-Kutta ethod. The ibalanceinduced translational and conical swinging otions are shown in Fig. 2. Figure 2 a shows the resonant peak of the translational otion is reached at about 400 rad s 1 and Fig. 2 b shows the resonant peak of the conical otion is reached at about 700 rad s 1. The initial states of the observer are all zeros. The output of the observer dynaic syste is ultiplied by P(t) to get the state estiation of the original syste. The error of the ibalance force estiation is shown in Fig. 3. As stated in Section 3, the response of the observer is deterined by the eigenvalues of F and the transient response agnitude. Since the eigenvalues of F are far fro the iaginary axis, the decay rate of the estiation error is high. The transient agnitude is also satisfactory because the block canonical structure is selected for F atrix. Therefore, the Fig. 4 Fig. 3 The observer error The ibalance estiation by solving algebraic equation observer error converges to zero well before the rotating speed hits the critical speeds. This property is desired for active balancing purpose. Fro the ibalance force estiation, the syste ibalance can be estiated using direct algebraic calculation Eq. 19. The ibalance estiation results are shown in Fig. 4. The true values of the ibalance paraeters are shown by dashed lines in Fig. 4. The result shows that the ibalance estiation converges to the true value quickly. Fig. 2 The ibalance-induced vibration of rigid rotor 5 Conclusions This paper presents a new ethod of the ibalance estiation for the rigid rotor during acceleration. The acceleration will excite the dynaics of the rotor. Therefore, the static ethod such as the influence coefficient ethod cannot be used in this case. By viewing the ibalance forces and oents as outputs of a dynaic syste, we can augent the states of the rigid rotor odel to include the ibalance forces and oents. The resulting syste is an autonoous tie-varying linear syste. Under the assuption that the displaceent and the swinging angle of the rigid rotor can be easured, a tie-varying observer can be designed by canonical transforation of the original syste. Transient perforance of the observer is iproved and the upper bound of the agnitude is deterined in this paper by selecting a special structure of the estiation error dynaic atrix. The siulation results show that the estiation error converges to zero quickly and the transient estiation error is kept sall. This estiation ethod can be used in active vibration control or active balancing schees for rigid rotor. Journal of Dynaic Systes, Measureent, and Control DECEMBER 2001, Vol. 123 Õ 641

6 This paper concentrates on the ibalance estiation issues. References and Discussions on other issues of real-tie active balancing of rotating achinery, such as the rotor dynaic odeling, analysis, and control, actuator and sensor layout, etc, can be found in Zhou and Shi 22. Acknowledgents This work was supported by BalaDyne Corporation and the U.S. Departent of Coerce, National Institute of Standards and Technology, Advanced Technology Progra, Cooperative Agreeent Nuber 70NANB7H3029. Noenclature I p, the polar, and the diaetric oents of inertia of the rotor L the length of the rotor OXYZ the stationary coordinate syste R X, R Z the displaceents of the ass center of the rotor in X and Z directions Ṙ X, Ṙ Z the velocities of the ass center of the rotor in X and Z directions R X, R Z the accelerations of the ass center of the rotor in X and Z directions d, the agnitude and angle of the ibalance vector as shown in Fig. 1. u, u x, u y, u z the ass and the position of the ibalance in body-fixed coordinate xyz the ass of the rotor c, k the viscous daping coefficient and the spring rate of the bearings oxyz the body-fixed coordinate syste,, the rotating angle, speed, and acceleration of the rotor, Euler angles to describe the orientation of the body-fixed coordinate in the stationary coordinate.,, fors a body Euler angle set Kane 21 Appendix The explicit expressions of the atrices used in the nuerical study are listed as follows. P t l t t t t t t t t t t t t t t t t t t t t t t t t t t t t C t Ā t l t t t t t t t t t t t t t t t t Õ Vol. 123, DECEMBER 2001 Transactions of the ASME

7 F l Ḡ t l t t t t t t t t t t t t t t t 2100t References 1 Wowk, V., 1995, Machinery Vibration: Balancing, McGraw-Hill, New York. 2 Gosiewski, Z., 1985, Autoatic Balancing of Flexible Rotors, Part 1: Theoretical Background, J. Sound Vib., 100, No. 4, pp Gosiewski, Z., 1987, Autoatic Balancing of Flexible Rotors, Part 2: Synthesis of Syste, J. Sound Vib., 114, No. 1, pp Van De Vegte, J., and Lake, R. T., 1978, Balancing of Rotating Systes During Operation, J. Sound Vib., 57, No. 2, pp Van De Vegte, J., 1981, Balancing of Flexible Rotors During Operation, J. Mech. Eng. Sci., 23, No. 5, pp Knospe, C. R., Hope, R. W., Fedigan, S. J., and Willias, R. D., 1995, Experients in the Control of Ibalance Response Using Magnetic Bearings, Mechanics, 5, No. 4, pp Lu, K. Y., Coppola, V. T., and Bernstein, D. S., 1996, Adaptive Autocentering Control for an Active Magnetic Bearing Supporting a Rotor with Unknown Mass Ibalance, IEEE Trans. Control. Syst. Technol., 4, No. 5, Sept., pp Herzog, R., Buhler, P., Gahler, C., and Larsonneur, R., 1996, Ibalance Copensation Using Generalized Notch Filters in the Multivariable Feedback of Magnetic Bearings, IEEE Control. Syst. Technol., 4, No. 5, Sept., pp Reinig, K. D., and Desrochers, A. A., 1986, Disturbance Accoodating Controllers for Rotating Mechanical Systes, ASME J. Dyn. Syst., Meas., Control, 108, Mar., pp Zhu, W., Castelazo, I., and Nelson, H. D., 1989, An Active Optial Control Strategy of Rotor Vibrations Using External Forces, ASME Design Technical Conference-12th Biennial Conference on Mechanical Vibration and Noise Montreal, Que, Can Luenberger, D. G., 1966, Observers for ultivariable systes, IEEE Trans. Auto. Control, AC-11, Apr., pp Knospe, C. R., Taer, S. M., and Fittro, R., 1997, Rotor Synchronous Response Control: Approaches for Addressing Speed Dependence, J. Vib. Control, 3, No. 4, pp Zhou, S., and Shi, J., 2001, The Analytical Unbalance Response of Jeffcott Rotor During Acceleration, ASME J. Manuf. Sci. Eng., 123, No. 2, pp Zhou, S., and Shi, J., 2000, Supervisory Adaptive Balancing Of Rigid Rotors During Acceleration, Transactions of NAMRI/SME, XXVII, pp O Reilly, J., 1983, Observers for Linear Systes, Acadeic Press, New York. 16 Wolovich, W. A., 1968, On the stabilization of controllable systes, IEEE Journal of Dynaic Systes, Measureent, and Control DECEMBER 2001, Vol. 123 Õ 643

8 Trans. Auto. Control, AC-13, pp Yuksel, Y. O., and Bongiorno, J. J., 1971, Observers for Linear Multivariable Systes with Applications, IEEE Trans. Auto. Control, AC-16, No. 6, pp Nguyen, C., and Lee, T. N., 1985, Design of a State Estiator for a Class of Tie-Varying Multivariable Systes, IEEE Trans. Auto. Control, AC-30, No. 2, Feb., pp Shafai, B., and Carroll, R. L., 1986, Minial-Order Observer Designs for Linear Tie-Varying Multivariable Systes, IEEE Trans. Auto. Control, AC-31, No. 8, Aug., pp Chen, C. T., 1984, Linear Syste Theory and Design, Holt, Rinehart and Winston, New York. 21 Kane, T. R., 1983, Spacecraft Dynaics, McGraw-Hill, New York. 22 Zhou, S., and Shi, J., 2001, Active Balancing and Vibration Control of Rotating Machinery: A Survey, The Shock and Vibration Digest, Sept., 33, No. 5, pp Õ Vol. 123, DECEMBER 2001 Transactions of the ASME