TRADITIONAL system identi cation techniques, 1 3 and more

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1 JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol 23, No 2, Mah Apil 2000 System Identi ation in the Pesene of Completely Unknown Peiodi Distubanes Neil E Goodzeit and Minh Q Phan Pineton Univesity, Pineton, New Jesey A system identi ation method to extat the distubane-fee dynamis and the distubane effet oetly despite the pesene of unknown peiodi distubanes is pesented The distubane fequenies and wavefoms an be ompletely unknown and abitay Only measuements of the exitation input and the distubaneontaminated esponse ae used fo identi ation Initially, the distubanes ae modeled impliitly When the ode of an assumed input-output model exeeds a etain minimum value, the distubane infomation is ompletely absobed in the identi ed model oef ients A speial inteation matix explains the mehanism by whih infomation about the system and the distubanes ae intetwined and moe impotantly, how they an be sepaated uniquely and exatly fo late use in identi ation and ontol Fom the identi ed infomationa feedfowad ontolle an be developed to ejet the unwanted distubanes without equiing the measuement of a sepaate distubane-oelated signal The multi-inputmulti-outputfomulation is st deived in the deteministi setting fo whih the system and distubane identi ation is exat Extensions to handle noise-ontaminated data ae also povided Expeimental esults illustate the method on a exible stutue A ompanion pape addesses the poblem whee the distubane effet is modeled expliitly Intodution TRADITIONAL system identi ation tehniques, 1 3 and moe eentlydevelopedmethods, 4 6 an be used to identifythe system dynamis fom input and output data These methods assume that all of the system inputs ae known, o that any unknown inputs ae white noise whose effets an be aveaged out using suf ient data Sometimes, howeve, not all of the system inputs ae known, and it may not be possible to disable o eliminate these unknown inputs while data is olleted fo system identi ation In addition, these inputs may be deteministi, fo example, smooth o peiodi funtions, and they may be lage enough to dominate the system esponse Thee is no guaantee of pefet identi ation when unknown peiodi distubanesae pesent in the system In this ase, the identi ed model may o may not be auate enough fo ontol design In addition, these methods do not povide the infomation needed to alulate the feedfowad ontol signal to anel the effets of the unknown distubanes Refeene 7 boadens the taditional appoahes by onsideing system identi ation when unknown peiodi inputs at on the system When the ommon distubane peiod is known, Ref 7 shows that the distubane effet and the system input output dynamis an be identi ed exatly The identi ation esults ae then used fo feedbak ontol design and to alulate a feedfowad ontol that anels the effet of the distubane on the system esponse The pesent pape extends the above method by onsideing peiodi distubane inputs whee both the distubane wavefoms and peiods ae unknown By emoving the equied knowledge of the ommon distubane peiod, the pape seeks to answe the fundamental question of whethe it is indeed possible to identify the system distubane-fee dynamis oetly in the pesene of ompletelyunknownpeiodidistubanesbesidesdeteminingthe onditions that ensue pefet identi ation, we also wish to extat the distubane effet and use it to alulate a feedfowad ontol signal that anels the effet of the distubane Fo this pupose, we assume that the only infomation available ae measuements Reeived 11 Mah 1997; evision eeived 2 August 1999; aepted fo publiation 2 Septembe 1999 Copyight 1999 by the Ameian Institute of Aeonautis and Astonautis, In All ights eseved Gaduate Reseah Assistant, Depatment of Mehanial and Aeospae Engineeing; uently Pinipal Enginee Systems, Lokheed Matin Missiles and Spae, Sunnyvale, CA Assistant Pofesso, Depatment of Mehanial and Aeospae Engineeing Membe AIAA of a ontol exitation signal and the system esponse The system esponse is oupted by unmeasuable peiodi distubanes with unknown peiodsthat entethe system at unknown loationsunlike Ref 7, thee is no expliit model of the distubane effet on the output in the uent tehniqueinstead, the distubaneinfomation is initially absobed in the identi ed model oef ients fom whih the system distubane-feedynamis and the distubaneeffet ae eoveed We bie y mention the impliation of the poposedidenti ation tehnique on the distubane-ejetion ontol poblem In addition to the lassial noth lte appoah, the liteatue inludes many methods fo solving distubane-ejetion poblems These methods, some of whih ae apable of handling systems with unknown dynamis and distubanes, inlude state-spae appoahes based on distubane-aommodation ontol (DAC), 8 11 tansfe funtion appoahesthat use adaptive lteing tehniques, and tehniques using neual netwoks Ou pimay goal is to addess the system identi ation poblem in the pesene of unknown distubane inputs One the system input output dynamis and distubane effet ae identi ed, a distubane-ejetionontolle an be designed using these esults In so doing, we avoid some of the assumptions inheent in the efeened tehniques Fo example, unlikedac appoahes,we assumeno knowledgeof the system dynamis Also, athe than modeling and alulatingthe atual distubanes, we identify thei ombined effet on the system esponse instead As a esult, the numbe of distubaneso whee they ente the system is unimpotant Unlike adaptive lteing o neual netwok appoahes, we do not equie a distubane-oelated efeene signal o need to detemine the tansfe funtion elating the distubanes to the system esponse This pape begins by detemining the onditionsfo the existene of an exat model that elates the ontol exitation inputs to the system distubane-ontaminated outputs without an expliit distubane model The unknown peiodi distubanes ae entiely absobedin the oef ients of this model that an be identi ed fom input output data Next, fom the identi ed model, we show that the system (distubane-fee) input output dynamis an still be extated exatly, as if the unknown distubanes wee not pesent We show how the distubane effet an also be extated and then eliminated by a feedfowad ontol signal Finally, we disuss issues elatedto system and distubaneidenti ation using measuements that ae ontaminated by noise Following the mathematial development, expeimental esults illustate the theoy in the identi ation of a exible tuss stutue 251

2 úú úú úú úú 252 GOODZEIT AND PHAN Mathematial Fomulation We now developan inteationmatix fomulationthat shows how infomation about the system and the distubanes ae intetwined and how they an be uniquely sepaated Input Output Models Relating Exitation Input to Distubane-Coupted Output Data In this setion we show that it is possible to onstut a model that an exatly pedit the uent value of the esponse,given only past values of the esponse and ontol inputs, despite the system esponse data being oupted by the effets of unknown peiodi distubaneinputsthe main emphasis of this setion is to detemine the onditions unde whih suh a model exists The system to be identi ed is assumed to be epesentable by a linea disete-time state-spae model x(k + 1) = Ax(k) + Bu(k) + B d d(k), y(k) = Cx(k) (1) whee x(k) is an n 1 state veto, u(k) is the m 1 ontol veto, y(k) is the q 1 output veto, and k is the time step The veto d(k) epesents the unknown peiodi distubane inputs The system maties, A, B, and C have dimensions n n, n m, and q n Both these maties and the distubane input matix B d ae unknown Only measuements of the input u(k) and measuements of the distubane-oupted system esponse y(k) ae available fo identi ation Equation(1) alulatesa one-step-aheadstate peditionx(k + 1) given x(k), u(k), and d(k) By epeatedly evaluating x(k + 1), and then substituting it bak into Eq (1), we geneate a p-step-ahead state veto pedition, x(k + p) = A p x(k) + Cu p (k) + C d d p (k) (2) T d = êê C B d 0 C AB d C B d 0 C A p 2 B d C AB d C B d 0 Note that O is the pq n system obsevability matix The matix Tis a pq pm Toeplitz matix made up of the q m system Makov paametes C B, C AB,, C A p 2 B, whose elements ae the system esponse to a unit pulse applied at eah ontol input To eliminate the effets of the distubanes d p (k) and the initial state x(k) on the system input output mapping, additional degees of feedomae now intoduedinto the model This is aomplished by adding and subtating My p (k) to the ight-hand side of Eq (2) to obtain x(k + p) = A p x(k) + Cu p (k) + C d d p (k) + My p (k) My p (k) (6) whee M is an abitay n pq matix Next, substituting Eq (4) fo y p (k), expession (6) beomes x(k + p) = A p x(k) + Cu p (k) + C d d p (k) + M[Ox(k) + Tu p (k) + T d d p (k)] My p (k) = ( A p + MO)x(k) + (C + MT )u p (k) + (C d + MT d )d p (k) My p (k) (7) As expeted, Eq (7) involves the system state x(k) and the distubane input d p (k), both of whih ae unknown Howeve, these tems an be eliminated fom the equation if thee exists an n pq matix M suh that the following onditions ae satis ed fo all k: (5) whee u p (k) and d p (k) ae vetos of the ontol inputs and distubanes, u p (k) = u(k) u(k + 1) u(k + p 1) The maties Cand C d ae given by, d p (k) = d(k) d(k + 1) d(k + p 1) C = [A p 1 B,, AB, B], C d = [A p 1 B d,, AB d, B d ] (3) Note that C and C d ae ontollability maties assoiated with the ontol exitation and distubane inputs, espetively By epeated appliation of the output equation, the expession fo the pq 1 veto of system outputs y p (k) is y p (k) = Ox(k) + Tu p (k) + T d d p (k) (4) whee y p (k) and the maties O, T, and T d ae given by y p (k) = êê ë T = êê y(k) y(k + 1) y(k + p 1), O = C B 0 C AB C B 0 C A p 2 B C AB C B 0 êê C C A C A p 2 C A p 1 so that Eq (7) beomes A p + MO = 0 (8) (C d + MT d )d p (k) = 0 8 k (9) x(k + p) = (C + MT )u p (k) My p (k) (10) Let us examine Eq (9) moe losely Fo k =1, 2,, N, N + 1,, the onstaint equations imposed by Eq (9) an be gouped togethe as whee (C d + MT d )D = 0 (11) D = [d p (1), d p (2),, d p (N ), d p (N + 1), ] (12) Although it may appea that Eq (11) is a athe lage set of onstaints, not all of these onstaint equations ae linealy independentthe atualnumbeof linealyindependentonstaintequations in Eq (11) is nq, whee n is the ode of the system and q is the ank of D Aoding to Eq (12), the ows of D ae time-shifted sampled histoies of the peiodi distubanes Consequently, its ow ank (fo a suf iently lage N ) is limited by the numbe of distint fequenies pesent the distubanes To see this, onside an example whee the distubane input is a sine wave funtion of a single fequeny, fo example, 1 Hz, and this signal is being sampled with a sampling inteval of less than 05 s Time shifting this 1-Hz signal by one time step will intodue a new (osine) omponent to this signal, thus ausing the time-shifted signal to be linealy independent fom the oiginal sampled signal Additional shiftings will not intodue new linealy independent signals Futhemoe, these statements ae valid even if the peiod of the signal is not an intege multiple of the sampling inteval If the sampling inteval is exatly 05 s, then time shifting it will podue the same signal with the sign evesed, thus eating no new linealy independent signal Thus, evey distint, zeo-mean hamoni omponent of the distubane ontibutes at most two linealy independent ows to

3 úú úú êê úú GOODZEIT AND PHAN 253 D If any one of these distubanes have nonzeo mean, the ank of D will be ineased futhe by one In othe wods, if thee ae f distint fequenies pesent in the distubanes, then q = 2 f o 2 f + 1, depending on whethe any distubane has nonzeo mean Let D f be fomed by q linealy independent olumns of D(a ow ank of a matix is the same as its olumn ank), we an now wite all of the equations that M must satisfy as follows: M[O, T d D f ] = [A p, C d D f ] (13) Equation (13) is a set of n 2 + 2nq linea equations in n qp = nqp unknowns in M Thus, the existene of M is assued povided [O, T d D f ] is full (olumn) ank and p is hosen suh that nqp n 2 + q n When expessed in tems of the distubane fequenies, we have the following ondition fo p: pq n + 2 f + 1 (14) Equation (14) epesents a safe lowe bound fo p beause thee ae ases whee the tue lowe bound fo p is even smalle Fist, Eq (14) assumes the geneal ase whee the distubanes do not have zeo mean Othewise, it beomes pq n + 2 f Seond, it also assumes that the distubanes have no fequeny omponents at exatly the Nyquist fequeny (half the sampling fequeny) As illustatedin the peedingexample, suh sampled signals will ontibute not two but only one linealy independent ow to D Note that all of these exeptions will potentially ause the lowe bound fo p to be even smalle, indiating that Eq (14) is a suf ient ondition It is impotant to ealize that thee is no need to selet p oesponding to its tuly lowe bound beause any lage p an be used In so doing, one an avoid all of the subtletiesdisussedin the peeding explanation Theefoe, fo peiodi distubanes, if p is seleted to be lage enough to satisfy Eq (14), existene of M implies existene of an input output model of the fom y(k + p) = C(C + MT )u p (k) C My p (k) (15) Equation (15) is obtained by pemultiplying both sides of Eq (10) by C Shifting the time index bak by p steps, fo k p this expession beomes whee u p (k p) = êê ë y(k) = C(C + MT )u p (k p) C My p (k p) (16) u(k p) u(k p + 1) u(k 1), y p (k p) = êê ë y(k p) y(k p + 1) y(k 1) Equation (16) shows that, even though the system esponse is oupted by the unknown distubane inputs, a model exists suh that a one-step-ahead esponse pedition y(k) an be alulated exatly fom p past values of the inputs and outputs The distubane infomation is ompletely absobed in suh an input output model, the oef ients of whih ae alulated as will be desibed The speial matix M as deived hee desibes the mehanism by whih the system and distubane infomation is inteelated Fo this eason we use the tem inteation matix to desibe it Calulating Model Coef ients fom Input Output Data Equation (16) expesses y(k) in tems of p past values of the system esponse, y(k 1),, y(k p), and p past values of the ontolinput, u(k 1),, u(k p) This expessionhas the same fom as an autoegessive moving aveage model with exogenous input (ARX) model, y(k) = a 1 y(k 1) + a 2 y(k 2) + + a p y(k p) + b 1u(k 1) + b 2u(k 2) + + b pu(k p) (17) whee a 1, a 2,, a p and b 1, b 2,, b p ae the ARX model oef- ients that ae now known to be elated to the maties in Eq (16) by [a p, a p 1,, a 1] = C M [b p, b p 1,, b 1] = C(C+ MT ) (18) By the assumptionthat measuementsof u(k) and y(k) ae available fo k = 0, 1,, `, the model oef ient maties an be alulated fom [C(C + MT ), C M] = Y V T (V V T ) + (19) whee Y and V ae data maties aanged in the fom Y = [ y( p), y( p + 1),, y(`)] V = [ u p(0) u p (1) u p (` p) y p (0) y p (1) y p (` p)] (20) The + sign in Eq (19) denotes the pseudoinvese opeation that should be pefomed via the singula value deomposition to detet and eliminate numeial ill-onditioningissues if they aise It is assumed that the ontol input is suf iently ih, fo example, andom input, so that the ows ontaining the shifted sequenes of u(k) ae linealy independentand that thee ae suf ient measuements available so that the numbe of equations is at least equal to o geate than the numbe of unknowns As aleady mentioned, in the absene of noise, if p is seleted to be lage enough, Eq (19) will detemine a model that esults in an exat t to the data, despite the peseneof the unknown peiodi distubaneas shown ealie, ineasing p makes available additional degees of feedom so that the distubane effet an be ompletely absobed into the model oef ients In the following setion we show that this absoption ousin suh a spei way that the system pulse esponsesamples (Makov paametes) an still be eoveed exatly fom the alteed oef ients without knowing the inteation matix M The system Makov paametes ompletely desibe the system input output dynamis Reoveing System Makov Paametes fom Model Coef ients One the model oef ients have been deteminedusing Eq (19), any numbe of system Makov paametes an be alulated as follows The ARX model oef ients in Eq (18) ae elated to one anothe by [b p, b p 1,, b 1] = CC [a p, a p 1,, a 1]T = [C A p 1 B,, C AB, C B] [a p, a p 1,, a 1] C B 0 C AB C B 0 C A p 2 B C AB C B 0 (21) By equating of the tems on eah side of this expession, the st p system Makov paametes ae C B = b 1 C AB = b 2 + a 1C B C A 2 B = b 3 + a 2C B + a 1C AB b a a a C A p 1 B = p + p 1C B + p 2C AB + + 1C A p 2 B (22)

4 úú 254 GOODZEIT AND PHAN To detemine additional system Makov paametes, eall that by seletion of the model ode p to be lage enough, a matix M will exist so that Eq (8) is satis ed This ondition an be expessed as A p + M p C A p M 2 C A + M 1 C = 0 (23) whee M 1, M 2,, M p ae the n q patitions of M, that is, M =[M 1, M 2,, M p ] Pemultiplying Eq (23) by C and postmultiplying it by B yield C A p B + (C M p )C A p 1 B + + (C M 2 )C AB + (C M 1 )C B = 0 (24) By the eognitionthat a 1 = C M p, a 2 = C M p 1,, and a p = C M 1, the system Makov paamete C A p B is given by C A p B = a 1C A p 1 B + + a p 1C AB + a pc B (25) Additional Makov paametes an be obtained analogously, C A p + 1 B = a 1C A p B + + a p 1C A 2 B + a pc AB (26) Theefoe, despite the unknown distubanes alteing the ARX model oef ients, the system Makov paametes an still be eoveed exatly as if the distubaneswee not pesent Fom these Makov paametes one an obtain a minimum-ode state-spae ealization that an pedit the distubane-fee esponse to abitay inputs The following setion will show how to onstut this edued-ode model Additionally, note that the fomulas fo the system Makov paametes given ealie tun out to be the same as those deived fom the oef ients of an input output tansfe funtion identi ed fom data that is not ontaminated by any distubanes 4 6 Theefoe, the identi ed model whose oef ients ae thought to be oupted by the unknown distubanesis apable of pediting the system distubane-feeesponse exatly as well Detemining a Minimum-Ode State-Spae Realization Given the system Makov paametes, one an easily obtain a minimum-ode state-spae ealization Fo ompleteness we now povide the key equations involved in the ealization by the eigensystem ealization algoithm (ERA) that is desibed in detail in Ref 4 One st foms the Hankel maties H(0) and H (1) fom the eoveed system Makov paametes, k = 0, 1 H(k) = êê ë C A k B C A k + 1 B C A k + s 1 B C A k + 1 B C A k + 2 B C A k + s B C A k + 1 B C A k + B C A k + s + 2 B (27) Fo suf ientlylage valuesof and s, the ank of the Hankel matix is equal to the ode n of the minimal ealization A minimum-ode state-spae ealization is given by Ā = R n 1 2 R T n H (1)S nr 1 2 n, B = R 1 2 n S T n E m, C = E T q R nr 1 2 n (28) whee the q q matix E T q and the m sm matix E T m ae de ned as Eq T =[I q q O q ( 1)q ] and Em T =[I m m O m (s 1)m ] The singula value deomposition of H(0) is H(0) = RR S T = R n R n Sn T whee n is the ode of the minimum ealization and only n nonzeo singula values of the Hankel maix H(0) ae etained in R n Identifying the Distubane Effet fom Input Output Data In the peeding setions the goal was to identify a model fo the system whee the expliit effet of the unknown distubanes on the input output mapping was ompletely eliminated By ineasing the ARX model ode, we showed that the distubane effet was absobed in the oef ients Moeove, we showed that system Makov paametes ould still be eoveed oetly fom the alteed oef ients and be used to alulatea minimum-ode statespae ealization, as if thee wee no distubanes pesent If the pupose of identi ation is only to podue a oet system model elating the ontol inputs to the system outputs, no futhe steps ae neessay The distubane infomation, howeve, is ontained in the identi ed model oef ients, and if extated, it an be used to alulate a feedfowad ontol that anelsthe distubaneeffet o to detemine the steady-stateeffet of the distubaneon the system esponse In the following we will show that, in addition to exat system identi ation, exat eoveyof the distubaneinfomation is also possible By the use of a simila agument as desibed in Eqs (2 7), a f -step-aheadstate pedition model oespondingto the identi ed minimum-ode state-spae model ealization is x(k + f ) = ( Ā f + MŌ ) x(k) + ( C + M T )u f (k) + ( C d + M T d )d f (k) My f (k) (29) f f whee Ō is the qf n system obsevabilitymatix alulated fom Ā and C, C is the n mf system ontollability matix alulated fom Ā and B, and Tis the qf mf Toeplitz matix alulatedfom Eq (5) using the identi ed system Makov paametes Note that Eq (29) is valid fo any value of and any matix M The tem on the ight-hand side that ontains d (k) is unknown, and in ontast to the pevious development, we wish to extat this distubane infomationathe than making this tem vanish Theefoe, we seek only to eliminate x(k) fom Eq (29) by nding an n qf matix M that satis es the ondition Ā f + MŌ = 0 (30) Beause Ō is full olumn ank (only the system s obsevable subspae an be identi ed), suh an M an be found fom M = ( Ā) f (Ō) + (31) if f is hosen suh that qf n We an even hoose f to be p fo onveniene This hoie of M will eliminate expliit dependene on the state vaiable x(k) in Eq (29), but the distubane effet will appea expliitly as an additive tem in the model Substituting M given in Eq (31) into Eq (29) and eodeing tems yield x(k + f ) = Shifting the time index bak by f Eq (32) by C podue My f (k) + ( C+ M T )u f (k) + ( C d + M T d )d f (k) (32) y(k) = C My f (k f ) + C( C + M T )u f (k f ) whee time steps, and pemultiplying + C( C d + M T d )d f (k f ) (33) y f (k f ) = f f y(k ) y(k + 1) y(k 1) d f (k f ) =, u f (k f ) = f f d(k ) d(k + 1) d(k 1) f f u(k ) u(k + 1) u(k 1) The last tem on the ight-hand side of Eq (33) is a linea ombination of f samples of the peiodi distubane and is, theefoe, also peiodi with a ommon peiod N With this peiodi tem denoted as g (k), Eq (33) beomes y(k) = C My f (k f ) + C( C + M T )u f (k f ) + g (k) (34)

5 GOODZEIT AND PHAN 255 In the fom of an ARX model, we now have y(k) = a 1 y(k + b 2u(k 1) + a 2 y(k 2) + + a f y(k f ) + b 1u(k 1) 2) + + b f u(k f ) + g (k) (35) whee the oef ients ae given by [ a f, a f 1,, a 1] = C M [ b f, b f 1,, b 1] = C( C + M T ) (36) It is impotant to ealize that fo peiodi unknown distubanes both the model given by Eq (17) and the one given by Eq (35) ae equally oet Assuming noise-fee measuements, both models will t the esponse data pefetly In the ase of Eq (17), the distubane effet is entiely embedded in the model oef ients along with the system dynamis In the ase of Eq (35), the model has been patitioned so that the distubane effet appeas as an additive peiodi tem g (k), and the oef ients inlude only the system distubane fee dynamis One Eq (36) is used to detemine the minimum ealization ARX model oef ients a i and b i, i = 1, 2,, f, the distubane effet g (k) fo k f is given by g (k) = y(k) a 1 y(k 1) a 2 y(k 2) a f y(k f ) b 1u(k 1) b 2u(k 2) b f u(k f ) (37) Calulating the Steady-State Distubane Response and Feedfowad Contol Signal One g (k) and the ARX model oef ients a i and b i have been detemined,we have omplete infomationto detemineboth a feedfowad ontol signal that anels the distubane effet and the steady-state esponse of the system to the unknown distubanes This is done as follows With u f (k) denoted as the feedfowad ontol to be applied to the system, the model of Eq (35) beomes y(k) = a 1 y(k 1) + a 2 y(k 2) + + a f y(k f ) + b 1u f (k 1) + b 2u f (k 2) + + b f u f (k f ) + g (k) (38) Fom this expession, the feedfowad ontol signal that makes the steady-state system esponse equal zeo must satisfy b 1u f (k b b f f g 1) + 2u f (k 2) + + u f (k ) + (k) = 0 (39) The needed feedfowad ontol signal is simply the steady-state solution to Eq (39) Vaious ways of alulating the feedfowad ontol signal ae disussed in Ref 7 Whethe a feedfowad ontol signal exists that an exatly anel the distubanes depends on seveal fatos If the distubanesause the output to have a fequeny omponentat whih the ontol input has no in uene (eg, the input-output tansfe funtion equals zeo at this fequeny), then this omponent annot be aneled by feedfowad ontol In addition,povidedthe ontolan in uene the system output,if the numbe of independentinputs is equal to o geate than the numbe of outputs, then a feedfowad ontol exists that exatly anels the distubaneesponse Othewise, a pefet anellationis in geneal not possible Fom Eq (35), in the absene of the input u(k), the system esponse to the unknown distubanes y d (k) is given by y d (k) = a 1 y d (k a a f f g 1) + 2 y d (k 2) + + y d (k ) + (k) (40) Knowing g (k) and a i, i = 1, 2,, f, the estimated distubane esponse y d (k) an be solved fo fom Eq (40), whih will math the atual distubane esponse in the steady state Analysis of Distubane-Coupted Model In the absene of noise, the Hankel matix H(0) will have peisely n nonzeosingulavalues, the same numbe as the tue system ode In the pesene of noise, howeve, model edution by examining the singula values of the Hankel matix H(0) is geneally dif ult beause H(0) tends to appea full ank This limitation an be oveome by eognizing that the identi ed model of Eq (17) an be onveted to a modal state-spae fom fom whih analysis an be aied out to detemine eah espetive mode of the model In geneal, they inlude tue identi able dynami modes of the system, unontollable distubane modes, and unontollable modes due to ovepaameteization In the following we disuss ways by whih one an distinguish these modes fom the identi ed model The distubane modes ompise the unontollable subspae of the ARX model that geneates the distubane effet g (k) In the ideal noise-fee ase, these modes ae easy to identify They ontibute to model s esponse pedition, but annot be exited by the ontol input In addition, beause thei ontibutionto the esponse pedition inludes only sinusoidal omponents at the distubane fequenies, these modes have zeo damping Identifying the distubane modes allows the unknown distubane fequenies to be detemined Ove-paameteization modes esult beause in patie it is neessay to selet a model ode p muh lage than that equied by Eq (14) fo seveal easons This is beause the tue (o effetive) ode of the system is not known exatly and neithe ae the numbe of distint distubane fequenies Theefoe, the model ode must be seleted so that suf ient degees of feedom ae available to absob the identi able dynamis and distubane effet given these unetainties Additionally, fo auate esults, the model ode must be futhe ineased to edue the effets of measuement and poess noise This situation will be illustated late by an example In the peseneof noise, to popelyategoizethe model s modes, it is neessay to obtain identi ation esults fo seveal values of p and, hene, fo seveal diffeent levels of model ovepaameteization As the model ode ineases, the damping of the tue system modes onvege to nonzeo positive values (fo a stable system), wheeas the distubane mode damping dops to nea zeo The distubane mode damping may eithe be positive o negative with equal likelihood Beause of this, it is not unusual fo the distubane-oupted model to ontain unstable modes that estit how it an be used without futhe poessing The ovepaameteization modes also have positive damping, and they geneally onstitute an easily eognizable band of modes distibuted ove the entie fequeny ange Besides modal damping, the othe impotant disimination iteion is the ontibution of a mode to the model s input output mapping o the system pulse esponse Using the method to be desibed enables the modal pulse esponse ontibutionsto be anked in odefom the most signi ant to leastsigni antobsevinghow the ontibution fatos and ankings hange as the model ode p ineases povides a poweful tool fo modal disimination If the system is identi ed when the distubanes ae not pesent the situation is simple The ovepaameteizationmodes have the smallest ontibution fatos, and the dominant dynamis have the lagest As p ineases the ankings hange little, exept some weakly ontollable o weakly obsevable dynami modes may move up as they beome bette esolved The ovepaameteizationmodes an be disaded if a edued-ode model is desied beause they do not ontibute signi antly to the input output mapping Dynami modes that ae weakly ontollable,weakly obsevable,o both have ankings in between and an also be disaded with minimal effet When distubanes ae pesent, the distubane modes may appea to have lage ontibution fatos Howeve, as p ineases they diminish towad zeo In addition, the ontibution fatos fo the dominant system dynamis will in geneal inease and ultimately onvege to some onstant value To ompute the modal ontibution fatos, the identi ed ARX model is onveted to a anonial state-spae fom and then to the modal fom w(k + 1) = K w(k) + C u(k), y(k) = X w(k) (41) whee the dimension of the state veto w(k) is qp 1, whee q is the numbe of outputsand p is the assumed ode of the input output

6 êêêêê úú 256 GOODZEIT AND PHAN model The system matix K is a blok diagonal matix onstuted fom the system eal and omplex eigenvalues, K = diag{ k (1) (2), k,, [ (1) (1) x x (1) (1) ] [, (2) (2) x x (2) (2) ] }, (42) and the output and input in uene maties ae X = [ (1), (2),, (1), (2), ], C = b (1) b (2) ṛ b (1) b (2) (43) whee (i ) and (i ) and b (i ) and b (i) ae the espetiveoutputand input in uene oef ients assoiated with eal eigenvalues k (i ) and the omplex (onjugate) (i) (i) eigenvaluepais jx The modal pulse esponses ae P (i) (k) = (i ) (k (i ) ) k 1 b (i), P (i ) and the total system pulse esponse is P t (k) = n ^ P (i) (k) + i = 1 (k) = (i) [ n ^ P (i) (k) = i = 1 (i ) (i) x x (i) (i ) ] k 1 b (i) (44) n m ^ P (i) (k) (45) i = 1 whee n is the numbe of eal eigenvalues and n is the numbe of omplex-onjugateeigenvaluepais Fo simpliity the two summations ae eplaed with a single one, whee P (i) (k) is the pulse esponse of the ith mode and n m = n + n The matix modal ontibution fatos ae given by S (i) = L ^ k = 1 P t (k) P (i) (k) (46) whee the S (i) ae q m maties and L is the numbe of pulse esponse samples The dot symbol denotes the podutevaluated by multiplying the oespondingelements of eah matix To simplify the disiminationpoess, the sala modal ontibutionfatos s (i) ae detemined by summing the elements of S (i), s (i) = q ^ k = 1 m ^ S (i) (47) kl l = 1 Equations(46) and (47) deteminequantitiesthat ae both a measue of the oelation between the total and individual modal pulse esponsesand the nom of the individualmodal pulse esponsesthemselves Theefoe, modes that have a lage effet on the esponse, and that ae stongly oelated with the total pulse esponse, will have the lagest ontibution fatos One the distubane modes have been identi ed based on thei damping and ontibution fatos, then a distubane-fee model an be deteminedthe edued-odestate-spaemodel is obtained by deleting the states oesponding to the distubane modes, but etaining all othe modes This edued-ode model an then be used to detemine the ARX model given in Eq (35) and to extat the distubane effet Finally, note that in the pesene of noise, the method used to omputeg (k) an be modi ed to takeadvantageof the noise lteing effet of the identi ed model as follows With noise, the identi ed model of Eq (17) will not t the data pefetly The least-squaes solution, howeve, minimizes the Eulidean nom of the eo e (k) between the atual and pedited esponse ove the data eod, whee e (k) = y(k) ˆy(k) (48) ˆy(k) = a 1 y(k 1) + a 2 y(k 2) + + a p y(k p) + b 1u(k 1) + b 2u(k 2) + + b pu(k p) (49) The output ˆy(k) is a lteed vesion of the oiginal output y(k) The lteing ation is povided by the ovepaameteizationof the one-step-ahead esponse model 5,6 Theefoe, to ompute the distubane effet g (k), one should use g (k) = ˆy(k) a 1 y(k 1) a 2 y(k 2) a f y(k f ) b 1u(k 1) b 2u(k 2) b f u(k f ) (50) whee ˆy(k) is used in plae of y(k) and the unontollable modes due to ovepaameteization ae etained in a i and b i to lte the oiginal data y(k 1), y(k 2), u(k 1), u(k 2), et Expeimental Results The exible lightly damped stutue used fo the expeimental study is shown in Fig 1 (Ref 25) Two aeleometesat one end of the stutue ae used as output sensos Loated neaby is a poofmass atuato ating as a distubanesoue Two othe poof-mass atuatosat the fa end of the stutueseve as exitation inputs To ollet data fo system identi ation, andom exitation ae applied to the two input atuatos, and the two aeleomete esponses ae eoded in the pesene of a distubane-ontaining fequeny omponents at 136, 17, 20, and 24 Hz To test the developedtheoy, we use only exitation input and distubane-oupted data in the Fig 1 Flexible tuss stutue

7 GOODZEIT AND PHAN 257 Table 1 System and distubane identi ation esults Refeene values a Model ode 90 Model ode 60 Fequeny, Damping, Contibution Fequeny, Damping, Contibution Fequeny, Damping, Contibution Hz % fato Hz % fato Hz % fato System dynamis b Distubane modes a System dynamis identi ed in the absene of the distubanes with model ode 60 b Ten system modes with the lagest pulse esponse ontibution fatos Fig 2 Refeene pulse esponse Fig 3 Distubane-oupted pulse esponse, p = 60 identi ation, and the distubaneinput is assumed to be unknown The distubane omponents at 136 and 17 Hz ae lose to the tuss stutualmodes, whih makes the identi ationpoblemmoe dif ultfuthemoe,the magnitudeof the distubaneinputis suh that the distubane-oupted esponse is oughly six times lage than the esponse to the exitation input only so that we have a situation whee the identi ation data is signi antly dominated by the unknown distubane Table 1 summaizes the identi ation esults fo assumed ARX model odes p of 60 and 90 Fo ompaison, esults ae also given fo a efeeneo tuth model identi ed in the absene of the distubane In eah ase 3200 data samples (0006-s sampling inteval) ae used fo identi ation Table 1 gives the fequenies, damping atios, and pulse esponse ontibution fatos of the 10 most signi ant system modes (as anked by thei ontibutionfatos) The same infomation is given fo the modes assoiated with the unknown distubane Table 1 eveals among othe things that with noise-ontaminated measuements, models identi ed in the pesene of distubanes an have seious defets that may ende them unusable without additional poessing This is why the mode disiminationtehniquedesibedin the peedingsetion is essential to poduing high-quality models To appeiate the poblem, onside Fig 2, whih shows the efeene model pulse esponse obtained with the distubanetuned off Compae the efeene pulse esponse to the one in Fig 3 identi ed with distubane-oupted data and p = 60 In theoywith noise-feemeasuementsthey should be idential, but in patie they ae not In fat, the pulse esponse obtained with distubane-oupted data is gowing slowly due to the ontibution of two identi ed distubane modes that tun out to be slightly unstable The appeaane of possibly unstable distu- bane modes should not be a supise beause the oet damping fato fo a distubane mode is zeo, and with noise ontaminated data, the identi ed damping atio may tun out to be positive o negative fo unbiasedidenti ation Moeove, the distubanemode at 136 Hz is not only unstable, it has the lagest modal ontibution fato Reall that with noise-fee measuements, the ontibution fatos fo the distubane modes ae zeo Thus, the pesene of these unstable modes endes this model unusable in its uent fom without additional poessing The st step to podue a high-quality model is to disiminate the distubane modes This is best aomplished by examining the modal damping beause distubane modes will have vey low (positive o negative) damping The dispaity between the distubane mode damping and the damping of the othe modes an be seen in Fig 4 fo the ase p = 90 The fou distubanemodes have damping atios that ae moe than two odes of magnitude lowe than those of the system dynamis modes and ovepaameteization modes (lagest positive damping atio of 0002% fo the distubane at 24 Hz and model ode 90) With ineasing p, one obtains a edution in the distubane mode damping atios with ineasing model ode wheeas the system mode damping atios onvege to nonzeo positive values as expeted In the event that the atual system modes have nea-zeo damping, o if the data ae vey noisy, the distubane modes an still be disiminated by examining the modal pulse esponse ontibution fatos In the noise-fee ase, distubane modes have zeo ontibution fatos This is beause, in the input output tansfe funtions, the distubane modes appea along with zeos that anel them pefetly With noisy measuements, the pole-zeo anellations ae impefet, patiulalyif a distubane s effet is lage, that

8 258 GOODZEIT AND PHAN Conlusions We have developed an inteation matix fomulation that shows how it is possible to identify a system input output dynamis oetly in the pesene of unknown peiodi distubanes Povided the ode of an assumed model is suf iently lage, the distubane effetwill be ompletelyabsobedinto its oef ientsfuthemoe, the detailsof theabsoptionae suh thatthesysteminput outputdynamis an still be eoveed oetly fom the alteed oef ients as if thee wee no distubanespesentin additionto identifyingthe system distubane-fee dynamis oetly, the distubane effet an also be extated fom the identi ed model oef ients This infomation an be used to alulate a feedfowad ontol signal and the steady-state esponse to the distubanes The fomulation only equies measuements of the ontol exitation inputs and the system outputs and is geneal enough to handle multi-input/multioutput systems with single o multiple distubane soues Thee is no need to measue the distubanes, o to know thei peiods o po les, o to use steady-state data In addition, the distubane peiods need not be intege multiples of the sampling inteval Requied to be known, howeve, ae uppe bounds on the ode of the system dynamis and the numbe of fequenies pesent in the distubanes When implemented eusively, the developed method an be adapted to handle systems whose dynamis and distubane fequenies ae slowly time vaying Aknowledgments This eseahis suppotedby a gant fom Lokheed-MatinCopoation The omments of the eviewes and the assoiate edito ae gatefully aknowledged Fig 4 Modal damping vs modal fequeny, p = 90 Fig 5 Identi ed and efeene pulse esponse, p = 90 is, when a distubane is oinident with a lightly damped system mode Theefoe, the distubane mode may appea to have a lage and, pehaps, dominant ontibution to the system pulse esponse as illustated in Fig 3 The ontibution fatos fo the distubane modes will, howeve, deease towad zeo as the seleted model ode p ineases This is in ontast to the atual system dynamis, whose modal fatos may inease and ultimately onvege to some nonzeo values with ineasing model ode As seen in Table 1 the ontibution fato fo the distubane mode at 136 Hz dops by almost a fato of ve as the model ode ineases fom 60 to 90 The ontibution fato fo the distubane mode at 17 Hz dops by ove an odeof magnitudein ontast,the ontibutionfato of the tue system mode at 135 Hz nealy doubles Ovepaameteization modes have inonsistentdamping fatos, pulse esponse ontibuting fatos, and fequenies as p hanges; theefoe, they an be easily disiminated Finally, Fig 5 shows the pulse esponse identi ed with distubane-oupted data (with all distubane modes identi ed and emoved) is in lose ageement with the efeene pulse esponse identi ed with distubane-feedata Refeenes 1 Ljung, L, and Södetöm, T, Theoy and Patie of Reusive Identi- ation, MIT Pess, Cambidge, MA, 1983, pp Johnson, C R, J, Reusive Least-Squaes, Letues on Adaptive Paamete Estimation, Pentie Hall, Englewood Cliffs, NJ, 1988, pp Kelle, B, On-Line Physial Paamete Identi ation and Adaptive Contol of a Launh Vehile, PhD Thesis, Dept of Aeonautis and Astonautis, Stanfod Univ, Stanfod, CA, Mah 1993, pp Phan,M, Hota, L G, Juang,J-N, and Longman,R W, Linea System Identi ation Via an Asymptotially Stable Obseve, Jounal of Optimization Theoy and Appliations, Vol 79, No 1, 1993, pp Juang, J-N, Phan, M, Hota, L G, and Longman, R W, Identi ation of Obseve/Kalman Filte Makov Paametes: Theoy and Expeiments, Jounal of Guidane, Contol, and Dynamis, Vol 16, No 2, 1993, pp Phan, M, Hota, L G, Juang, J-N, and Longman, R W, Impovement of Obseve/Kalman Filte Identi ation by Residual Whitening, Jounal of Vibation and Aoustis, Vol 117, No 2, 1995, pp Goodzeit,N E, and Phan,M Q, Systemand DistubaneIdenti ation fo Feedfowad and Feedbak Contol Appliations, Jounal of Guidane, Contol, and Dynamis, Vol 23, No 2, 2000, pp ; 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9 GOODZEIT AND PHAN Hillestom, G, and Stenby, J, Rejetion of Peiodi Distubanes with Unknown Peiod A Fequeny Domain Appoah, Poeedings of the 1994 Ameian Contol Confeene, Vol 2, Inst of Eletial and Eletonis Enginees, New Yok, 1994, pp Saks, A H, Bodson, M, and Messne, W, Advaned Methods fo Repeatable Runout Compensation in Disk Dives, IEEE Tansations on Magnetis, Vol 31, No 2, 1995, pp Hyland, D C, and Davis, L D, A Multiple-Input, Multiple-Output Neual Ahitetue fo Suppession of a Multi-Tone Distubane, 19th Annual AAS Guidane and Contol Conf, Ameian Astonautial Soiety, Pape AAS , Feb Hyland, D C, Davis, L, Das, A, and Yen, G, Autonomous Neual Contol fo Stutue Vibation Suppession, Poeedings of the AIAA Guidane, Navigation, and Contol Confeene, AIAA, Reston, VA, Snyde, S D, and Tanaka, N, A Neual Netwok fo Feedfowad Contolled Smat Stutues, Jounal of Intelligent Mateial Systems and Stutues, Vol 4, No 3, 1993, pp Goodzeit, N E, System and Distubane Identi ation fo Adaptive Distubane-Rejetion Contol, PhD Dissetation, Dept of Mehanial and Aeospae Engineeing, Pineton Univ, Pineton, NJ, June 1998