Active vibration control of a rotor-bearing system based on dynamic stiffness

Transcription

1 Rev. Fac. Ing. Univ. Anioquia N. 55 pp Sepiembre, 21 Acive vibraion conrol of a roor-bearing sysem based on dynamic siffness Conrol acivo de vibraciones en un sisema roor-chumaceras basado en la rigidez dinámica Andrés Blanco Orega 1*, Francisco Belrán Carbajal 2, Gerardo Silva Navarro 3, Marco Anonio Oliver Salazar 1 1 Cenro Nacional de Invesigación y Desarrollo Tecnológico Cenide). Coordinación de Mecarónica. Inerior Inernado Palmira s/n col. Palmira, Cuernavaca Morelos C. P México. 2 Universidad Poliécnica de la Zona Meropoliana de Guadalajara C.P Tlajomulco de Zúñiga, Jalisco, México. 3 CINVESTAV-IPN. Deparameno de Ingeniería Elécrica. Sección de Mecarónica. C.P. 736 México, D.F. Recibido el 13 de Junio de 29. Acepado el 6 de abril de 21) Absrac This paper presens an acive vibraion conrol scheme o reduce unbalanceinduced synchronous vibraion in roor-bearing sysems suppored on wo ball bearings, one of which can be auomaically moved o conrol he effecive roor lengh and, as an immediae consequence, he roor siffness. This dynamic siffness conrol scheme, based on frequency analysis, speed conrol and acceleraion scheduling, is used o avoid resonan vibraion of a roor sysem when i passes run-up or coas down) hrough is firs criical speed. Algebraic idenificaion is used for on-line unbalance esimaion a he same ime ha he roor is aken o he desired operaing speed. Some numerical simulaions and experimens are included o show he unbalance compensaion properies and robusness of he proposed acive vibraion conrol scheme when he roor is sared and operaed over he firs criical speed Keywords: acive vibraion conrol, eccenriciy idenificaion, Jeffco roor * Auor de correspondencia: ex. 15, fax , correo elecrónico: [email protected]. A. Blanco) 125

2 Rev. Fac. Ing. Univ. Anioquia N. 55. Sepiembre 21 Resumen En ese arículo se presena un esquema de conrol acivo de vibraciones para aenuar las ampliudes de vibración síncrona inducidas por el desbalance en un sisema roor-chumaceras; donde una de las chumaceras puede ser desplazada auomáicamene para modificar la longiud efeciva del roor, y como consecuencia, la rigidez del roor. El conrol de la rigidez dinámica se basa en un análisis de la respuesa en frecuencia, conrol de velocidad y en el uso de esquemas de aceleración, para evadir las ampliudes de la vibración en la resonancia mienras el sisema roaorio pasa acelerado o desacelerado) a ravés de una velocidad críica. Se uiliza idenificación algebraica para esimar el desbalance en línea, mienras el roor es llevado a la velocidad de operación deseada. Algunas simulaciones numéricas y resulados experimenales son incluidos para mosrar las propiedades de la compensación del desbalance y la robusez del esquema de conrol acivo de vibraciones propueso, cuando el roor se opera a una velocidad por encima de la primera velocidad críica Palabras clave: conrol acivo de vibraciones, idenificación de excenricidad, roor Jeffco. Inroducion Roaing machinery is commonly used in many mechanical sysems, including elecrical moors, machine ools, compressors, urbo machinery and aircraf gas urbine engines. Typically hese sysems are affeced by exogenous or endogenous vibraions produced by unbalance, misalignmen, resonances, maerial imperfecions and cracks. Vibraion caused by mass unbalance is a common problem in roaing machinery. Roor unbalance occurs when he principal ineria axis of he roor does no coincide wih is geomerical axis and leads o synchronous vibraions and significan undesirable forces ransmied o he mechanical elemens and suppors. Many mehods have been proposed o reduce he unbalance-induced vibraion, where differen devices such as elecromagneic bearings, acive squeeze film dampers, laeral force acuaors, acive balancers and pressurized bearings have been developed [1-5]. Passive and acive balancing echniques are based on he unbalance esimaion o aenuae he unbalance response in he roaing machinery. The Influence Coefficien Mehod has been used o esimae he unbalance while he roaing speed of he roor is consan [6, 7]. This mehod has been used o esimae he unknown dynamics and roor-bearing sysem unbalance during he speedvarying period [8]. On he oher hand, here is a vas lieraure on idenificaion mehods [9-11], which are essenially asympoic, recursive or complex, which generally suffer of poor speed performance. This paper presens an acive vibraion conrol scheme o reduce unbalance-induced synchronous vibraion in roor-bearing sysems suppored on wo ball bearings, one of which can be auomaically moved along he shaf o conrol he effecive roor lengh and, as an immediae consequence, he roor siffness. This dynamic siffness conrol scheme, based on frequency analysis, speed conrol and acceleraion scheduling, is used o avoid resonan vibraion of a roor sysem when i passes run-up or coas down) hrough is firs criical speed. Algebraic idenificaion is used for on-line unbalance esimaion a he same ime ha he roor is aken o he desired operaing speed. The proposed resuls are srongly based on he algebraic approach o parameer idenificaion in linear sysems repored [12], which requires a priori knowledge of he mahemaical model of he sysem. This 126

3 Acive vibraion conrol of a roor-bearing sysem based on dynamic siffness approach has been employed for parameer and signal esimaion in nonlinear and linear vibraing mechanical sysems, where numerical simulaions and experimenal resuls show ha he algebraic idenificaion provides high robusness agains parameer uncerainy, frequency variaions, small measuremen errors and noise [13, 14]. In addiion, algebraic idenificaion is combined wih inegral reconsrucion of ime derivaives of he oupu GPI Conrol) using a simplified mahemaical model of he sysem, where some nonlinear effecs siffness and fricion) were negleced; in spie of ha, he experimenal resuls show ha he esimaed values represen good approximaions of he real parameers and high performance of he proposed acive vibraion conrol scheme, which means ha he algebraic idenificaion and GPI conrol mehodologies could be used for some indusrial applicaions, when a leas a simplified mahemaical model of he sysem is available [14]. Some numerical simulaions and experimens are included o show he unbalance compensaion properies and robusness when he roor is sared and operaed over he firs criical speed. Sysem descripion Mahemaical model The Jeffco roor sysem consiss of a planar and rigid disk of mass m mouned on a flexible shaf of negligible mass and siffness k a he mid-span beween wo symmeric bearing suppors see figure 1a) when a = b). Due o roor unbalance he mass cener is no locaed a he geomeric cener of he disk S bu a he poin G cener of mass of he unbalanced disk); he disance u beween hese wo poins is known as disk eccenriciy or saic unbalance [15, 16]. An end view of he whirling roor is also shown in figure 1b), wih coordinaes ha describe is moion. In our analysis he roor-bearing sysem is modeled as he assembly of a rigid disk, flexible shaf and wo ball bearings. This sysem differs from he classical Jeffco roor because he effecive shaf lengh can be increased or decreased from is nominal value. In fac, his adjusmen is obained by enabling longiudinal moion of one of he bearing suppors righ bearing in figure 1.a) o differen conrolled posiions ino a small inerval by using some servomechanism, which provides he appropriae longiudinal force. Wih his simple approach one can modify he shaf siffness; moreover, one can acually conrol he roor naural frequency, during run-up or coasdown, o evade criical speeds or a leas reduce roor vibraion ampliudes. Our mehodology combines some ideas on variable roor siffness [17] and roor acceleraion scheduling [18] bu compleing he analysis and conrol for he Jeffco-like roor sysem. bearing a a) b disk movable bearing F Z Y y disk G S u Figure 1 Roor-bearing sysem: a) Schemaic diagram of a roor-bearing sysem wih one movable righ) bearing and b) end view of he whirling roor For simpliciy, he following assumpions are considered: flexible shaf wih aached disk, graviy loads negleced insignifican when compared wih he acual dynamic loads), equivalen mass for he base-bearing m b, linear viscous damping c b beween he bearing base and he linear sliding, force acuaor o conrol he shaf siffness F, angular speed d = d conrolled by means of an elecrical moor wih servodrive and local Proporional Inegral PI) conroller o rack he desired speed scheduling in presence of small dynamical disurbances. The mahemaical model of he four degree-offreedom Jeffco-like roor is obained using Newon equaions as follows. x X 127

4 Rev. Fac. Ing. Univ. Anioquia N. 55. Sepiembre mx + cx + kx = m u sin + u 2 cos ) 1) my +.. cy. + ky = m u. 2 sin - u.. cos ) 2) J + mu ) + c = - p = m xu sin - yu cos ) 3) z... mbb + cbb = F 4) where k and c are he siffness and viscous damping of he shaf, J z is he polar momen of ineria of he disk and ) is he applied orque conrol inpu) for roor speed regulaion. In addiion, x and y denoe he orhogonal coordinaes ha describe he disk posiion and. = w is he roor angular velociy. The coordinae b denoes he posiion of he movable righ) bearing, which is conrolled by means of he conrol force F) servomechanism). In our analysis he siffness coefficien for he roor-bearing sysem is given by [19] k = 2 2 3EIla - ab + b ) 5) a 3 b 3 where l = a + b is he oal lengh of he roor beween boh bearings wih b he coordinae o be conrolled, I = D 4 is he momen of ineria of a 64 shaf of diameer D and E is he Young s modulus of elasiciy E = 2.11 x 1 11 N/m 2 ) for AISI 414 seel). The naural frequency of he roor sysem is hen obained as follows [19] k / m n = 6) In such a way ha, conrolling b by means of he conrol force F one is able o manipulae w n o evade appropriaely he criical speeds during roor operaion. The proposed conrol objecive is o reduce as much as possible he roor vibraion ampliude, denoed in adimensional unis by for run-up, coas-down or seady sae operaion of he roor sysem, even in presence of small exogenous or endogenous disurbances. Noe, however, ha his conrol problem is quie difficul because of he 8h order nonlinear model, many couplings erms, underacuaion and unconrollabiliy properies from he wo conrol inpus τ, F). Experimenal resuls Some experimenal resuls were performed in open loop in a Roor-Ki experimenal plaform of Benly Nevada. The posiions of he inerial disk, sensors and bearing suppors in he roorki are shown in figure 2. The experimenal resuls were performed o show how he naural frequency can be modified o differen posiions of he righ bearing. Furhermore, he roor is sared and operaed over he firs criical speed where he speed operaing condiion for he roor is given as j = rad/s 4 rpm) and he acceleraion for he speed ramp was 1.74 rad/s 2. moor k c S1 S2 D-8 S3 S B C Aco. mm Figure 2 Posiions of he inerial disk, sensors and bearing suppors in he roor-bearing sysem The response for he Jeffco roor configuraion righ bearing posiion a poin B, see figure 2) is shown in figure 3 and i was recorded by he sensor S3 see figure 2). The naural frequency for he Jeffco roor configuraion was abou ω n = 22 rpm. R [mils] A x + y R = u 7) [rpm] Figure 3 Unbalance response for a Jeffco Roor righ bearing posiion a poin B) 128

5 Acive vibraion conrol of a roor-bearing sysem based on dynamic siffness R [mils] R [mils] Experimenal resuls for he bearing posiion a poins A l =.4 m) and C l =.5 m), are shown in figures 4 and 5, respecively. Here, he open-loop responses show ha a smaller lengh l =.4 m leads o a higher naural frequency and a bigger lengh l =.5 m leads o a smaller naural frequency. Hence o ge a minimal unbalance response, he roor lengh should sar a l =.4 m and hen abruply change o l =.5 m. This change of he bearing posiion mus occur exacly when he response for l =.4 m crosses he response for l =.5 m [rpm] Figure 4 Roor unbalance response for he righ bearing posiion a poin A [rpm] 3 4 Figure 5 Roor unbalance response for he righ bearing posiion a poin C Speed conrol wih rajecory planning In order o conrol he speed of he Jeffcolike roor sysem, consider ω = z 6 =. equaion 3), under he emporary assumpion ha he eccenriciy u is perfecly known and ha c o simplify he analysis. Then, he following local PI conroller is designed o rack desired reference rajecories of speed ω*) and acceleraion scheduling ω*) for he roor: = Jzv 1 + c + kux sin - kuy cos v = * ) - * )) - - * )) d 8) 1 1 The use of his conroller yields he following closed-loop dynamics for he rajecory racking error e 1 = ω - ω* ):... e + e + e = ) Therefore, selecing he design parameers [ 1, ] so ha he associaed characerisic polynomial for equaion 9) is a Hurwiz polynomial, one guaranees ha he error dynamics is asympoically sable. The prescribed speed and acceleraion scheduling for he planned speed rajecory is given by { * ) = i, i, f) f f < i i < i > f 1) where ω i and ω f are he iniial and final speeds a he imes i and f, respecively, passing hrough he firs criical frequency, and σ, i, f ) is a Bézier polynomials, wih σ, i, f ) = and σ, i, f ) = 1, described by 1-2 [,, ) = - - i f i i f f - - i f i [ i f i - i f i 11) wih g 1 = 252, g 2 = 15, g 3 = 18, g 4 = 1575, g 5 = 7, g 6 = 126, in order o guaranee a sufficienly smooh ransfer beween he iniial and final speeds. The fundamenal problem wih he proposed feedback conrol in equaion 8) is ha he eccenriciy u is no known, excep for he fac ha i is consan. The Algebraic idenificaion mehodology is proposed o on-line esimae he eccenriciy u, which is based on he algebraic approach o parameer idenificaion in linear sysems [12]. 129

6 Rev. Fac. Ing. Univ. Anioquia N. 55. Sepiembre 21 Algebraic idenificaion of eccenriciy Consider equaion 3) wih perfec knowledge of he momen of ineria J z and he shaf siffness k, and ha he posiion coordinaes of he disk x, y) and he conrol inpu τ are available for he idenificaion process of he eccenriciy u. Muliplying equaion 3) by and inegraing by pars he resuling expression once wih respec o ime, one ges 1 z6) d = J z - cz6) d + ku z3 - cos z5- z1 sin z5) d J z 12) Solving for he eccenriciy u in equaion 12) leads o he following on-line algebraic idenifier for he eccenriciy: u = J J z - z c) d, [, )] k y cos - x sin )d 13) where d is a posiive and sufficienly small value. Therefore, when he denominaor of he idenifier of equaion 13) is differen o, a leas for a small ime inerval [O, d] wih d <, one can find from equaion 13) a closed-form expression o on-line idenify he eccenriciy. An adapive-like conroller wih algebraic idenificaion The PI conroller given by equaion 8) can be combined wih he on-line idenificaion of he eccenriciy in equaion 13), resuling he following cerainy equivalence PI conrol law = Jzv 1 + c + kux sin - kuy cos. 14) v = * ) - * )) - - * )) d wih 1 1 u = J J z - z c) d, [, )] k y cos - x sin )d Noe ha, in accordance wih he algebraic idenificaion approach, providing fas idenificaion for he eccenriciy, he proposed conroller 14) resembles an adapive conrol scheme. From a heoreical poin of view, he algebraic idenificaion is insananeous [12]. In pracice, however, here are modeling errors and oher facors ha inhibi he algebraic compuaion. Forunaely, he idenificaion algorihms and closed-loop sysem are robus agains such difficulies [14]. Simulaion resuls Some numerical simulaions were performed using he parameers lised in able 1. Table 1 Roor sysem parameers m r =.9 kg D -.1 m a =.3 m m b =.4 kg r disk =.4 m cϕ = 1.5 x 1-3 Nms/rad c b = 1 Ns/m u = 1 mm z7 =.3 ±.5 m Figure 6 shows he idenificaion process of eccenriciy and he dynamic behavior of he adapive-like PI conroller given by equaion 14), which sars using he nominal value u = mm. One can see ha he idenificaion process is almos insananeous. The conrol objecive is o ake from he res posiion of he roor o he operaing speed ωf = 3 rad/s. Eccenriciy [ m] Roor Speed [rad/s] Torque [N/m] Time [1 s] Time [s] Time [s] Figure 6 Close loop sysem response using he PI conroller: a) idenificaion of eccenriciy, b) roor speed and c) conrol inpu 13

7 Acive vibraion conrol of a roor-bearing sysem based on dynamic siffness The desired speed profile runs up he roor in a very slow and smooh rajecory while passing hrough he firs criical speed. This conrol scheme is appropriae o guaranee sabiliy and racking. The resuling roor vibraion ampliude sysem response when = ) is shown in figure 7, for hree differen and consan posiions of he righ bearing i.e., b =.25 m,.3 m,.35 m), using he PI conroller. Unbalance response [rad/s] Roor Speed [rad/s] Figure 7 Unbalance response R for differen and consan posiions of he movable bearing: dashed line), solid line) and dash-doed line) The purpose of hese simulaions is o illusrae how he posiion of he bearing ruly affecs he roor vibraion ampliudes for he desired speed profile. The nominal lengh of he shaf is l =.6 m. A smaller lengh l =.55 m leads o a higher naural frequency and a bigger lengh l =.65 m leads o a smaller naural frequency see figure 7). Hence o ge a minimal unbalance response, he roor lengh should sar a l =.55 m and hen abruply change o l =.65 m. This change of he bearing posiion mus occur exacly when he response for l =.55 crosses he response for l =.65, in order o evade he resonance condiion, because he roor speed is differen from he naural frequency of he roor-bearing sysem. Posiion conrol of he bearing suppor I is eviden from equaions 5) and 6) ha conrolling he posiion of he movable righ) bearing b applying he conrol force F and according o a pre-specified speed profile ω*) he modificaion of he roor ampliude response o he unbalance is possible. As a maer of fac his mehodology is equivalen o a dynamic siffness conrol for he Jeffco-like roor sysem, enabling smooh changes on coordinae b. To design a conroller for posiion reference racking, consider equaion 4). Then, one can propose he following Generalized Proporional Inegral GPI) conroller for asympoic and robus racking o he desired posiion rajecory b* ) for he bearing posiion and velociy, which employs only posiion measuremens of he bearing. For more deails on GPI conrol see [2]. F = m b v2 - cbb..... v 2 = b* ) - 2 b - b* )) - 1 b - b* )) - b - b* )) d 15) where. b is an inegral reconsrucor of he bearing velociy, which is given by. b = c m b + bb 1 m F ) d 16) b The use of he GPI conroller given yields he following closed-loop dynamics for he rajecory racking error e 2 = b - b* ): 3)... e 2 + 2e e 2 + e 2 = 17) Therefore, selecing he design parameers {b, b 1, b 2,} such ha he associaed characerisic polynomial for equaion 17) be Hurwiz, one guaranees ha he error dynamics be globally asympoically sable. The desired rajecory planning b* ) for he bearing posiion and velociy is also based on Bézier polynomials similar o equaion 1). Resuls and discussion The proposed mehodology for he acive vibraion conrol of he ransien run-up or coasdown of he roor-bearing sysem consiss of he following seps: 131

8 Rev. Fac. Ing. Univ. Anioquia N. 55. Sepiembre Define he rajecory planning for he speed rajecory profile w*) o be asympoically racked by he use of he adapive-like PI conroller wih he algebraic idenifier of he eccenriciy, i.e., lim w*) = w*). 2. Esablish an appropriae smooh swiching on he posiion of he movable bearing b*) o be asympoically racked by he applicaion of he GPI conroller, i.e., lim b) = b*). The swiching ime has o be a he crossing poin leading o minimal unbalance response in figure 7. Figure 8 shows he unbalance response of he roor-bearing sysem when roor speed PI conroller wih algebraic idenificaion of eccenriciy and GPI conrol of he bearing posiion are simulaneously used. Noe ha he swiching of he bearing posiion leads o small ransien oscillaions due o inerial and cenrifugal effecs on he overall roor sysem. Unbalance response [m/m] Roor Speed [rad/s] Figure 8 Roor vibraion ampliude response using acive vibraion conrol solid line) Firs of all, he speed rajecory planning and conrol orque shown in figure 6 are similarly used. The smooh swiching for he bearing posiion is implemened in such a way ha he run-up of he roor sysem sars wih he posiion b i =.25 m i.e., l =.55 m) and changes o b f =.35 m i.e., l =.65 m) exacly a he crossing poin shown in he corresponding response in figure 7. The swiching ime occurs when ω = 17.6 rad/s, ha is, = 23.9 s. The desired posiion of he bearing b) is illusraed in figure 9 ogeher wih he applied conrol force F. A comparison of he open-loop response and he closed-loop response in figure 8 resuls in imporan unbalance reducions abou 64%. Bearning Posiion [m] Force [N] Time [s] Time [s] Figure 9 Response of he bearing suppor using GPI conroller: a) posiion of he movable bearing and b) conrol force Conclusions The acive vibraion conrol of a Jeffco-like roor hrough dynamic siffness conrol and acceleraion scheduling is addressed. The conrol approach consiss of a servomechanism able o move one of he supporing bearings in such a way ha he effecive roor lengh is conrolled. As a consequence, he roor siffness and naural frequency are modified according o an off-line and smooh rajecory planning of he roor speed/ acceleraion in order o reduce he unbalance response when passing hrough he firs criical speed. The vibraion conrol scheme resuls from he combinaion of passive and acive conrol sraegies, leading o robus and sable performance in presence of he synchronous disurbances associaed o he normal operaion of he roor and some small parameer uncerainies. Since his acive vibraion conrol scheme requires informaion of he eccenriciy, a novel algebraic idenificaion approach is proposed 132

9 Acive vibraion conrol of a roor-bearing sysem based on dynamic siffness for on-line esimaion of he eccenriciy. From a heoreical poin of view, he algebraic idenificaion is pracically insananeous and robus wih respec o parameer uncerainy, frequency variaions, small measuremen errors and noise. The proposed acive vibraion conrol scheme, used o reduce unbalance-induced synchronous vibraion, is resriced o use in small roaing machinery e.g., ools machines, moors and generaors). References 1. S. Zhou, J. Shi. Acive balancing and vibraion conrol of roaing machinery: a survey. The Shock and Vibraion Diges. Vol pp G. Sheu, S. Yang, C. Yang. Design of experimens for he conroller of roor sysems wih a magneic bearing. Journal of Vibraion and Acousics. Vol pp Y. Guozhi, Y. F. Fah, C. Guang, M. Guang, F. Tong, Q. Yang. Elecro-rheological muli-layer squeeze film damper and is applicaion o vibraion conrol of roor sysem. Journal of Vibraion and Acousics. Vol pp B. Palazzolo, S. Jagannahan, A. F. Kaskaf, G. T. Monague, L. J. Kiraly. Hybrid acive vibraion conrol of roorbearing sysems using piezoelecric acuaors. Journal of Vibraion and Acousics. Vol pp Q. Jinhao, J. Tani and T. Kwon. Conrol of self-excied vibraion of a roor sysem wih acive gas bearings. Journal of Vibraion and Acousics. Vol pp X. Yu. General influence coefficien algorihm in balancing of roaing machinery.inernaional Journal of Roaing Machinery. Vol pp S. Lee, B. Kim, J. Moon, D. Kim. A sudy on acive balancing for roaing machinery using influence coefficien mehod. Proceedings of Inernaional Symposium on Compuaional Inelligence in Roboics and Auomaion. Espoo. 25. pp S. Zhou, S. Dyer, K. K. Shin, J. Shi, J. Ni. Exended influence coefficien mehod for roor acive balancing during acceleraion. Journal of Dynamics Sysems Measuremens and Conrol. Vol pp L. Ljung. Sysems idenificaion: heory for he user. Ed Prenice-Hall. New Jersey pp S. Sagara, Z. Y. Zhao. Numerical inegraion approach o on-line idenificaion of coninuous sysems. Auomaica. Vol pp S. Sagara, Z. Y. Zhao. Recursive idenificaion of ransfer funcion marix in coninuous sysems via linear inegral filer. Inernaional Journal of Conrol. Vol pp M. Fliess, H. Sira-Ramírez. An algebraic framework for linear idenificaion, ESAIM: Conrol. Opimizaion and Calculus of Variaions. Vol pp F. Belrán-Carbajal, H. Sira-Ramírez, G. Silva- Navarro. Adapive-like acive vibraion supression for a nonlinear mechanical sysem using on-line algebraic idenificaion. Proceedings of he Thireenh Inernaional Congress on Sound and Vibraion. Vienna. July 26. pp F. Belrán-Carbajal, G. Silva-Navarro, H. Sira- Ramírez, J. Quezada Andrade. Acive vibraion conrol using on-line algebraic idenificaion of harmonic vibraions. Proceedings of American Conrol Conference. Porland Oregon). 25. pp J. M. Vance. Roordynamics of Turbomachinery. Ed. John Wiley and Sons. New York pp A. Dimarogonas. Vibraion for Engineers. Ed. Prenice Hall. New Jersey pp Z. Sandler. Roboics: designing he mechanism for auomaed machinery. Ed. Academic Press. San Diego CA) pp K. T. Millsaps, L. Reed. Reducing laeral vibraions of a roor passing hrough criical speeds by acceleraion scheduling. Journal of Engineering for Gas Turbines and Power. Vol pp S. S. Rao. Mechanical Vibraion. Ed. Pearson Educaion. New Jersey. 24. pp M. Fliess, R. Marquez, E. Delaleau, H. Sira Ramírez. Correceurs proporionnels-inegraux generalizes. ESAIM Conrol, Opimisaion and Calculus of Variaions. Vol pp