Delft University of Technology. the LS-problem. In [1] the model is parametrized directly. by means of a matrix numerator polynomial and a scalar

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1 Poc. IEEE Confeence on Decision and Contol, pp , Multivaiable Least Squaes Fequency Domain Identication using Polynomial Matix Faction Desciptions R.A. de Callafon z D. de Roove x P.M.J. Van den Hof Mechanical Engineeing Systems and Contol Goup Delft Univesity of Technology Mekelweg 2, 2628 CD Delft, The Nethelands .a.decallafon@wbmt.tudelft.nl Abstact In this pape an appoach is pesented to estimate a linea multivaiable model on the basis of (noisy) fequency domain data via a cuve tting pocedue. The multivaiable model is paametized in eithe a left o a ight polynomial matix faction desciption and the paametes ae computed by using a two-nom minimization of a multivaiable output eo. Additionally, input-output o element-wise based multivaiable fequency weightings can be specied to tune the cuve tting eo in a exible way. The pocedue is demonstated on expeimental data obtained fom a 3 input 3 output Wafe Steppe system. 1 Intoduction Fomulating a pocedue that is able to estimate a model on the basis of fequency domain data has gained consideable attention in the eseach on system identication. Although the clea distinction between time and fequency domain data is geneally oveestimated [12], estimation of models by tting complex fequency domain data has seveal advantages compaed to time domain appoaches. Fistly, epesenting data in the fequency domain domain can yield substantial data eduction [14]. Secondly, compessing a huge amount of time domain data into a nite numbe of fequency points facilitates noise eduction diectly. Both aspects ae used extensively in commecially available sophisticated test equipment fo spectal analysis. Based on Least Squaes (LS) estimation techniques, as used by Levi in [1] and futhe ened by Sanathanan and Koene in [15], multivaiable fequency domain cuve ttes have been fomulated in the liteatue. One is efeed to [11], [4] and the moe ecently intoduced pocedue in [1]. Basically, the pocedues die in the way the multivaiable model is paametized and whethe o not the pocedue allows fo a specication of the model ode and a (multivaiable) weighting on the cuve t eo. As such, in [11] a multivaiable model is found by the composition of scala subsystems, while the ode of the subsequent tansfe functions is detemined by testing the esiduals. A simila z The wok of Raymond de Callafon is nancially suppoted by the Dutch Systems and Contol Theoy Netwok x Reseach is sponsoed by Philips' Reseach Laboatoies, Eindhoven, the Nethelands appoach can be found in [4], wheein a Chebyshev polynomial basis is used to impove numeical conditioning of the LS-poblem. In [1] the model is paametized diectly by means of a matix numeato polynomial and a scala common denominato polynomial, wheeas only a scala fequency dependent weighting on the cuve t eo is allowed. Seveal altenatives to a LS-appoach can also be found in the liteatue. In [13] a subspace based algoithm in the fequency domain is pesented that allows the use to specify an additional fequency weighting. In [9] a fequency domain cuve tte has been developed in which a maximum amplitude of a (weighted) cuve t eo is being consideed. Futhemoe, so-called H 1 -identication pocedues, cuently applicable to scala fequency domain data, can guaantee an uppe bound on the additive eo, see e.g. [8] and the efeences theein. Unfotunately, a maximum amplitude citeion can be highly sensitive to noise, wheeas the available H 1 -identication pocedues might yield high ode models fo modeately damped pocesses [5]. Based on the LS-appoach, this pape pesents a multivaiable fequency domain cuve tte in which the aim is to minimize the two-nom on a (weighted) cuve t eo fo a model having a limited McMillan degee. The multivaiable model is paametized by eithe a left o ight polynomial Matix Faction Desciption (MFD). By use of Konecke calculus it will be shown that both a pe, post o elementwise multivaiable fequency weighting on the cuve t eo can handled elatively easily. Futhemoe, it will be shown that the iteation descibed by [15], denoted by SK-iteation, can be genealized to estimate a polynomial MFD. Due to the subsequent convex optimization steps in the SK-iteation, this appoach suppots the estimation of models with many paametes. Simila to the appoach followed by [1] and suppoted by the wok of [17], the esulting estimate can be used as an initial value fo a Gauss-Newton optimization. Although cumbesome iteations can be avoided by the use of a ealization based algoithm as epoted in [13], the possibility to pespecify the McMillan degee of the model and to intoduce a exible element-wise fequency weighting on the multivaiable data is quite helpful fom a pactical point of view. The pocedue will be illustated by tting amultivaiable model on the fequency esponse obtained fom the positioning mechanism pesent inawafe steppe.

2 2 Poblem fomulation To fomulate the multivaiable fequency domain identication poblem, conside the following set G of noisy complex fequency esponse data obsevations G(! j ), evaluated at N fequency points! j. G := fg(! j ) j G(! j ) 2 C pm ; fo j 2 1;...;Ng (1) The aim of the identication poblem discussed in this pape is to nd a linea time invaiant multivaiable model P of limited complexity, having m inputs and p outputs, that appoximates the data G in (1). To addess the limited complexity, the model P () is paametized by a eithe a left o ight polynomial MFD that depends on a eal valued paamete of limited dimension. The specic paametization of the polynomial MFD of P () is discussed in the next section. The appoximation of the data G by the model P () is addessed by consideing the following additive eo. E a (! j ;):=[G(! j ), P ((! j );)] fo j 2 1;...;N (2) The complex vaiable () in (2) is used to denote the fequency dependency of the model P (). In this way, (! j )=i! j to epesent a continuous time model, wheeas (! j )=e i! j T (shift opeato) o (! j )=(e i!j, 1)=T ( opeato) to epesent a discete time model with sampling time T. To tune the additive eo E a in (2), both an input-output fequency weighted cuve t eo E w with E w (! j ;):=W out (! j )E a (! j ;)W in (! j ) (3) and an element-wise fequency weighted cuve t eo E s with E s (! j ;):=S(! j ): E a (! j ;) (4) will be consideed in this pape. In (4) : is used to denote the Schu poduct; an element-by-element multiplication. Using the notation E to denote the fequency weighted cuve t eo E w in (3) and E s in (4), the deviation of the data G is chaacteized by following the nom function J(). J() := NX i=1 tfe(! j ;)E (! j ;)g = ke()k 2 F (5) In (5) is used to denote the complex conjugate tanspose, tfg is the tace opeato and ke()k F denotes the Fobenius nom opeating on the matix E() = [E(! 1 ;) E(! N ;)]. Consequently, the goal of the pocedue descibed in this pape is to nd a eal valued paamete ^ of limited complexity that can be fomulated by the following minimization. ^ := ag min J() (6) 2 IR 3 Paametization 3.1 Polynomial matix faction desciptions The multivaiable model is epesented by eithe a left o ight polynomial MFD, espectively given by P (;)=A(,1 ;),1 B(,1 ;) (7) P (;)=B(,1 ;)A(,1 ;),1 (8) whee A and B denote paametized polynomial matices in the indeteminate,1. Fo a model having m inputs and p outputs, the the polynomial matix B(,1 ;) is paametized by X d+b,1 B(,1 ;)= B k,k (9) whee B k 2 IR pm, d denotes the numbe of leading zeo matix coecients and b the numbe of non-zeo matix coecients in B(,1 ;). Fo the left MFD in (7), A(,1 ;)is paametized by k=d A(,1 ;)=I pp +,1 ax k=1 A k,k+1 (1) whee A k 2 IR pp and a denotes the numbe of non-zeo matix coecients in the monic polynomial A(,1 ;). The paamete is detemined by the coesponding unknown matix coecients in the polynomials. Hence, = B d B d+b,1 A 1 A a (11) and 2 IR p(mb+pa) fo the left MFD in (7). Dual esults can be fomulated fo the ight MFD in (8). Additionally to the full polynomial paametization pesented hee, so-called stuctual paametes d ij, b ij and a ij with d := minfd ij g, b := maxfb ij g, and a := maxfa ij g can be used to specify a none-full polynomial paametization. In this way, the paamete as given in (11) may contain pespecied zeo enties at specic locations. This may occu in a discete time model with,1 = z,1 whee the value of d ij has a diect connection with the numbe of time delays fom the jth input to the ith output. 3.2 Model ode Due to the indeteminate,1, it can be veied that the MFD of (7) o (8) gives ise to a (stictly) pope tansfe function matix P (;), egadless of the value of the integes d i;j, b i;j o a i;j. Hence, thee ae no estictions on the size of the stuctual paametes, othe than a limitation on the McMillan degee of the esulting model P (; ^). Fo the connection between the stuctual paametes and the McMillan degee of P (;), the following esult can be given. Lemma 3.1 Conside a paamete ^ such that A a 6=and B d+b,1 6=. Dene := maxfa; d + b, 1g (12) and A(; ^) := A(,1 ; ^), B(; ^) := B(,1 ; ^). Let n be used to denote the McMillan degee of the multivaiable tansfe function model P (; ^) obtained by(7) o (8), then n = deg detf A(; ^)g if and only if A(; ^) and B(; ^) ae left copime ove IR[] in case of (7) and ight copime ove IR[] in case of (8). Poof: The poof is given fo (8). With the condition A a 6=, B d+b,1 6=, it follows that A() := A(,1 ) and B() := B(,1 ) ae polynomial matices in. In case of (8), P () = B() A(),1 and a state space ealization [A,B,C,D] fo P ()

3 can be obtained, such that dim A = deg detf A()g and fa,bg contollable, see e.g [3]. Futhemoe, fc,ag is obsevable if and only if A() and B() ae ight copime ove IR[], see theoem 6.1 in [3]. Dually, the esult can be shown fo (7). 2 Unde some mild condition on the polynomials A(,1 ; ^) and B(,1 ; ^) being estimated, lemma 3.1 gives a diect elation between the deg detf A(; ^)g and the McMillan degee of the esulting estimate P (; ^). In case of the left MFD (7), deg detf A(; ^)g geneally will be equal to p. Hence, the stuctual paametes give ise to (an uppe bound) on the McMillan degee of the model being estimated. Fo a moe detailed discussion on the exact elation between the McMillan degee, the ow degee of the polynomial matices A(,1 ;), B(,1 ;) and the obsevability indices of a model computed by a left polynomial MFD, one is efeed to [6] o [16]. Compaed to a paametization of the multivaiable model P (;) using a scala common denominato polynomial d(,1 ;) as pesented in [1], the paametization using a (left) MFD is moe exible, as a scala common denominato esticts A(,1 ;)tobei pp d(,1 ;). A model with one output that is paametized by the left MFD of (7), constitutes a scala common denominato polynomial A(,1 ;). 4 Computational pocedue 4.1 Iteative minimization In this section, the minimization of (6) will be discussed by means of an iteative pocedue of convex optimization steps simila to the SK-iteation of [15]. The attention will be esticted to a paametization of P (;) based on the left MFD (7) as dual esults can be obtained fo a ight MFD. To extend the SK-iteation to the multivaiable case, st conside the (unweighted) additive cuve t eo of (2). Fo a model P (;) paametized by left MFD, (2) can be witten as E a (! j ;)=A((! j ),1 ;),1 ~ E(!j ;) (13) whee ~ E(! j ;) is the equation eo dened by ~E(! j ;):=A((! j ),1 ;)G(! j ), B((! j ),1 ;): (14) Substituting the paametization (7) fo the polynomials A, B, the equation eo in (14) can be epesented by whee is given in (11) and ~E(! j ;)=G(! j ), (! j ) (15) (! j )= 2 64 I mm (! j ),d. I mm (! j ),(d+b,1) G(! j )(! j ),1. G(! j )(! j ),a 3 75 (16) with (! j ) 2 C (mb+pa)m. A matix ~ E() can be fomed by stacking ~ E(! j ;) columnwise fo j 2 1;...;N and this yields ag min 2 IR k E()k ~ 2 F = ag min 2 IR kg, Pk2 F (17) whee G and P ae found by stacking the eal and imaginay pat of espectively G(! j ) and (! j ) fo j 2 1;...;N. Due to the linea appeaance of the paamete, (17) coesponds a standad least squaes poblem that can be solved by numeical eliable tools as e.g a QR-factoization with (patial) pivoting [7]. Due to the fact that A(,1 ;) in (13) also depends on the paamete, the linea appeaance of the paamete in (13) is violated. In ode to facilitate the convexity in minimizing the two-nom on the equation eo in (17), an iteative pocedue simila as in [15] can be used. An estimate ^ t in step t is computed by eplacing A((! j ),1 ;) in (13) by a xed A((! j ),1 ; ^ t,1 ) based on an estimate ^ t,1 obtained fom the pevious step t, 1. In this way the Fobenius nom of an output weighted equation eo ~E w (! j ; ^ t,1 ;)=A((! j ),1 ; ^ t,1 ),1 E(! ~ j ;) needs to be minimized epeatedly accoding to ^ t = ag min 2 IR k ~ E w (^ t,1 ;)k 2 F : This genealizes the SK-iteation to multivaiable models paametized by a left polynomial MFD. A dual appoach can be fomulated fo a ight polynomial MFD. The estimate obtained fom the SK-iteation is not optimal in the sense of (6) in pesence of noise and/o incoect model ode, but it does povide a tool to nd an initial estimate fo a GN-optimization [17]. Futhemoe, the convex optimization to be solved in each step of the multivaiable SK-iteation suppots the estimation of models with many paametes. The computational pocedue to obtain the paamete ^ in case of the (weighted) cuve t eos of (3) and (4) is pesented in the subsequent sections. 4.2 Input-output weighting The input-output weighted cuve t eo of (3) can be ewitten into E w (! j ;)= ~ W out (! j ;) ~ E(! j ;)W in (! j ) (18) whee W ~ out (! j ;):=W out (! j )A((! j ),1 ;),1 and E(! ~ j ;) is given in (14). Using a simila appoach of iteative minimization steps as used in section 4.1, the paamete in W ~ out (! j ;) in (18) is xed to an estimate ^ t,1 obtained fom the pevious step t, 1. Consequently, the weighted equation eo ~E w dened by ~E w (! j ; ^ t,1 ;):= ~ W out (! j ; t ) ~ E(! j ;)W in (! j ) (19) again indicates that the paamete to be estimated appeas linealy in (19). Although the fee paamete appeas linealy in (19), witing down a matix epesentation fo the weighted equation eo ~E w simila to (17) would inevitably lead to additional (lage) spase matices that need to be stoed in ode to compute the least squaes solution. The spase matices aise fom the fequency dependent output (and input) weighting that need to be incopoated [1]. Futhemoe, the paamete might have a stuctue containing zeo enties at pespecied locations if a none-full polynomial paametization is being used. To avoid the computational and memoy stoage issues that aise fom dealing with (lage) spase matices and to be able to take into account the specic stuctue that might

4 be pesent in the paamete, a faily simple and staightfowad computational pocedue based on Konecke calculus is pesented hee. Fo this pupose conside the following denition. Denition 4.1 Conside two matices X 2 C n1n2 and Y 2 C m1m2, then the Konecke vecto vec(x) 2 C n1n21 and the Konecke poduct X Y 2 C n1m1n2m2 ae espectively dened byvec(x) :=[x 1 x n2 ] T and X Y := 2 64 x 1;1Y x 1;n2 Y.. x n1;1y x n1;n2 Y whee x i;j and x j fo i 2 1;...;n 1 and j 2 1;...;n 2 ae used to denote espectively the (i; j)th enty in X and the jth column in X. The Konecke poduct is a well known concept [2] and by stacking the columns of a matix to obtain the coesponding Konecke vecto as mentioned in denition 4.1, the following esult can be obtained. Poposition 4.2 Conside (complex) matices X, Y and Z with appopiate dimensions, such that the matix poduct C := XY Z is well dened. Then vec(c) satises vec(c) =[Z T X]vec(Y ): Poof: The poof can be found in [2]. 2 On the basis of poposition 4.2, the Konecke vecto of the input/output weighted equation eo ~ E w (! j ; ^ t,1 ;)in (19) can be witten as 3 75 vec( ~ E w )=vec( ~ W out GW in ), [[W in ] T ~ W out ]vec() wheein the aguments! j, ^ t,1 and ae left out, to avoid notational issues. As the Fobenius-nom satises kxk 2 F = kvec(x)k 2 F fo an abitay matix X, the Fobenius-nom on ~E w can be chaacteized by a matix epesentation fomed by stacking vec( ~ E w (! j ; ^ t,1 ;)) ow-wise fo j 2 1;...;N. This yields the following estimate ^ = ag min 2 IR kvec( E ~ w (^ t,1 ;))k 2 F = ag min 2 IR kg w, P wk 2 F (2) whee =vec() 2 IR p(mb+pa)1 accoding to (11). Futhemoe, G w 2 IR 2pmN1 and P w 2 IR 2pmNp(mb+pa) ae matices that can be found by ow-wise stacking of the eal and imaginay pat of espectively vec( W ~ out (! j ; ^ t,1 )G(! j )W in (! j )) and vec([(! j )W in (! j )] T W ~ out (! j ; ^ t,1 )) fo j 2 1;...;N. The egession matix P w in (2) does not exhibit any spase matix stuctue as occus e.g. in the method of [1]. In fact, 2pmN p(mb + pa) enties is the smallest dimension of the egession matix P w in ode to compute a least squaes paamete ^ that has p(mb+pa) unknown enties (fo a a left full polynomial paametization) on the basis of N complex fequency domain points of a pm multivaiable system. In this way memoy stoage poblems ae avoided diectly as much as possible. As the paamete is conveted into a column paamete =vec(), any pespecied zeo enties in can be incopoated in the estimation of the paamete elatively easy. This can be done by omitting the columns in the egession matix P w that coespond to the zeo enties in and theeby educing the size of the paamete to be estimated diectly. 4.3 Schu weighting Conside the Schu o element-wise fequency weighted cuve t eo in (4) and ewite this into E s (! j ;)=S(! j ): [A((! j ),1 ;),1 ~ E(!j ;)] (21) whee the equation eo ~E(! j ;)was dened in (14). Using a simila appoach of iteative minimization steps as used in section 4.1, the paamete in A((! j ),1 ;),1 in (21) is xed to an estimate ^ t,1 obtained fom the pevious step t, 1. Consequently, the weighted equation eo E ~ s dened by ~E s (! j ; ^ t,1 ;):=S(! j ): [A((! j ),1 ; ^ t,1 ),1 ~ E(!j ;)] again indicates that the paamete to be estimated appeas linealy. Finally, it can be veied (leaving out the aguments! j, (! j ),1, ^ t,1 and ) that vec( ~ E s ) can be ewitten into vec(s: [A,1 G]), diag(vec(s))[ T A,1 ]vec() (22) by using the esult of poposition 4.2. Hence, stacking vec( ~E s (! j ; ^ t,1 ;)) ow wise fo each j 2 1;...;N will yield a simila expession fo the minimizing agument ^ as given in (2). Howeve, the matix G w in (2) now contains eal and imaginay pat of vec(s(! j ): [A((! j ),1 ; ^ t,1 )G(! j )]), wheeas P w in (2) will consist of the eal and imaginay pat of diag(vec(s(! j )))[(! j ) T A,1 ((! j ),1 ; ^ t,1 )] fo j 2 1;...; N. Hence, the same computational pocedue can be used to incopoate an element-by-element weighted cuve t eo (4) by a slight modication of the matices in (2). 5 Application to expeimental data 5.1 Desciption of the wafe steppe system The multivaiable cuve t pocedue discussed in this pape is illustated by cuve tting expeimental data obtained fom a positioning system of a wafe steppe. 1j 2j 3 j Fig. 1: Schematic view of a wafe stage; 1:wafe chuck, 2:lase intefeometes, 3:linea motos. A wafe steppe is a high accuacy positioning machine, used in chip manufactuing pocesses and a schematic view is depicted in Figue 1. The wafe caies appoximately 8 chips and is placed on a moving table, called the wafe chuck, which needs to be positioned accuately. The position of the wafe chuck on the hoizontal suface of a ganite block is measued by means of thee lase intefeomety measuements, wheeas thee linea motos ae used to position the wafe chuck. In this way, the positioning system is consideed to be a multivaiable system, having thee cuants to the linea motos as inputs and thee position measuements as outputs of the pocess.

5 5.2 Expeimental esults Peiodic andom noise signals of 124 points ae used to excite the system. Using the esulting aveaged time seies, a spectal estimate is computed, esulting in a nite numbe of fequency domain data points that constitutes a suitable stating point fo the subsequent cuve t pocedue. As the esulting model has to be used fo discete time contol design puposes, the aim is to estimate a possibly low ode discete time multivaiable model, that descibes the dynamical behaviou of the positioning system in the fequency domain till appoximately 4 Hz. Fo fequencies smalle than 1 Hz, the positioning system acts like a double integato. To illustate the usage of weighting functions in ode to shape the cuve t eo, an output weighting is used that emphasizes the fequency ange between and 3 Hz and stats to oll o at 3 Hz. The ode of the esulting multivaiable model (without the 3 double integatos) is chosen to be 12, epesented by a full left polynomial matix faction desciption having 81 paametes. The SK-iteation is stated up by st estimating a high ode model to compute an initial value fo the modied output weighting ~ W out in (19). Afte this initialization, the SK-iteation is invoked 8 times. The Bode amplitude plot and phase plot of the 18th ode estimate (including the 3 double integatos) is depicted espectively in Figue 2 and Figue 3. It should be noted that the multivaiable output weighting applied duing the estimation pocedue emphasizes the fequency domain aea of inteest. 6 Conclusions An appoach is pesented to estimate a linea multivaiable model on the basis of noisy fequency domain data using a two-nom minimization of a weighted cuve t eo. The weighting on the cuve t eo can be specied by eithe an input/output o an element-by-element fequency dependent multivaiable weighting function. The multivaiable model is paametized in eithe a left o ight polynomial matix faction desciption wheein stuctual paametes allow the specication of both full polynomial o none-full polynomial desciptions. The computational pocedue is able to estimate complex models by using an iteative pocedue of solving weighted multivaiable least squaes poblems and exploits the stuctue of the least squaes poblem, theeby educing any computation and memoy equiements diectly. The cuve is demonstated on expeimental multivaiable fequency domain data obtained fom a Wafe Steppe system having 3 inputs and 3 outputs. 7 Acknowledgements The authos would like toacknowledge the suppot of a wafe steppe expeimental setup by the Philips' Reseach Laboatoy in Eindhoven, the Nethelands. Futhemoe we like to thank Ewin Walges, fo his contibution to the expeimental pat of this pape. Refeences [1] D.S. Bayad. High ode multivaiable tansfe function cuve tting: algoithms, spase matix methods and expeimental esults. Automatica, Vol. 3, No. 9, pp. 1439{1444, jp (e i! j;)j ( ), jg(! j )j (- -) Fig. 2: Amplitude Bode plot of 18th ode discete time model P (e i! j ; ^) and the data G(! j ) P (e i! j ;) ( ), 6 G(! j )j (- -) Fig. 3: Phase Bode plot of 18th ode discete time model P (e i! j ; ^) and the data G(! j ).

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