LINEAR system identification has been the subject of

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL., NO., 1 Set-Memberhip error-in-variable identification through convex relaxation technique Vito Cerone, Member, IEEE, Dario Piga, Member, IEEE, and Diego Regruto, Member, IEEE Abtract In thi paper the et memberhip error-in-variable identification problem i conidered, that i the identification of linear dynamic ytem when both output and input meaurement are corrupted by bounded noie. A new approach for the computation of parameter uncertainty interval i preented. Firt, the identification problem i formulated in term of nonconvex optimization. Then, relaxation technique baed on linear matrix inequalitie are employed to evaluate parameter bound by mean of convex optimization. The inherent tructured parity of the original identification problem i exploited to reduce the computational complexity of the relaxed problem. Finally, convergence propertie and complexity of the propoed procedure are dicued. Advantage of the preented technique with repect to previouly publihed reult are dicued and hown by mean of two imulated example. Index Term Error-in-Variable, LMI relaxation, Setmemberhip identification, Spare polynomial optimization. I. INTRODUCTION LINEAR ytem identification ha been the ubject of extenive tudie over the lat decade. A good deal of well aeed methodologie for the olution of modeling problem in the time or in the frequency domain through either recurive or batch cheme are preented in [1], [2]. Mot of the propoed identification method rely on the aumption that the input ignal i exactly known. Either the output error or the equation error tructure i uually conidered. In the output error framework output meaurement are affected by additive noie, while a ingle error term added to the difference equation i ued to generically take into account all poible ource of uncertainty in the equation error tructure. However, in many practical problem input and output data equence are experimentally collected, thu the aumption of noie free input i not a realitic one in uch ituation. Identification problem where both input and output ignal are affected by noie are referred to a error-in-variable (EIV) problem. A detailed review of the main contribution on the identification of linear dynamic ytem when both input and output equence are corrupted by tochatic meaurement noie can be found in the urvey paper by Södertröm [3]. A poible alternative to the tochatic decription of meaurement noie i the bounded error characterization, which lead to the etmemberhip identification approach (ee, e.g., the urvey paper [4] and the pecial iue [5] for detail). Only few work on the EIV identification problem with bounded error can be found The author are with the Dipartimento di Automatica e Informatica, Politecnico di Torino, Torino, Italy. (vito.cerone@polito.it; dario.piga@polito.it; diego.regruto@polito.it). Tel: ; Fax: in the literature. A far a the cae of tatic model i conidered, two different technique to compute tight parameter bound can be found in [6] and [7]. The algorithm preented in [6] require the olution to a et of linear programming (LP) problem, while the procedure in [7] i baed on the reult about interval linear algebraic equation reported in [8]. A imilar problem in the context of data reconciliation i alo conidered in [9]. A to the identification of linear dynamic ytem when both input and output ignal are corrupted by bounded noie, firt inight on the problem are given in [10], [11] in the context of ARMAX model identification. In uch paper it i hown that exact feaible parameter region for dynamic EIV model i decribed by nonconvex bound and the ue of either polytopic or ellipoidal outer approximation i uggeted. An alternative outer approximation of the true feaible parameter et i derived in [12] by exploiting reult in [6]. Such an outer-bounding et i ued to compute parameter uncertainty interval (PUI) through the olution to a number of LP problem. Actually, PUI obtained in [12] are not tight and their degree of conervativene i, in general, not eay to quantify. In thi paper we preent an alternative approach for the computation of PUI through the olution to a et of polynomial optimization problem, which are relaxed by mean of linear matrix inequality (LMI) relaxation technique. The particular tructure of the addreed identification problem i exploited to reduce the computational complexity of the relaxed problem. The paper i organized a follow. Section II i devoted to the formulation of the identification problem. In Section III we how that the computation of tight parameter bound require the olution to nonconvex optimization problem. Relaxation technique for et-memberhip EIV identification of linear dynamic ytem are preented. Firt, the approach propoed in [12] i briefly reviewed in Section IV, then, the new LMI relaxation-baed technique, together with a detailed analyi of it propertie, i preented in Section V. The effectivene of the new preented approach i hown in Section VI by mean of two imulated example. In the appendix, background reult on the relaxation of emialgebraic optimization problem through LMI-relaxation technique are preented. II. PROBLEM SETTING Conider a ingle-input ingle-output (SISO) linear-timeinvariant (LTI) ytem which tranform the noie-free input x t into the noie-free output w t according to w t + a i w t i = b j x t j. (1)

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL., NO., 2 Both input and output data equence are corrupted by additive noie, ξ t and η t repectively, i.e. u t = x t + ξ t, y t = w t + η t. (2) The noie ample ξ t and η t are bounded by given ξ t and η t repectively, that i: ξ t ξ t, η t η t. (3) The unknown parameter vector θ R n θ i defined a θ = [a 1... a na b 0 b 1... b nb ] T, with n θ = na + nb + 1. In thi work we addre the problem of evaluating uncertainty interval on each parameter θ j. III. EVALUATION OF TIGHT PARAMETERS UNCERTAINTY INTERVALS The et D of all parameter θ and noie ample ξ t and η t conitent with the meaurement data equence, the aumed model tructure and error bound i decribed by (1)-(3), i.e. D = { (θ, ξ, η) R nθ+2n : (y t η t ) + a i (y t i η t i ) = = b j (u t j ξ t j ), t = na + 1,..., N, ξ k ξ k, η k η k, k = 1,..., N }, where ξ = [ξ 1, ξ 2,..., ξ N ] T, η = [η 1, η 2,..., η N ] T and N i the meaurement number. Therefore, for all j = 1,..., n θ, the tight parameter uncertainty interval (P UI j ) on θ j, conitent with meaurement, error bound and the aumed model tructure, i P UI j = [ ] θ j ; θ j, with θ j = (4) min θ,ξ,η D θ j, θ j = max θ,ξ,η D θ j. (5) In order to guarantee well-poedne of problem (5), we aume that D i bounded, i.e, for ome contant G > 0 arbitrarily large, the following condition on θ hold: θ G. (6) Becaue of nonlinear contraint (y t η t ) + a i (y t i η t i ) = b j (u t j ξ t j ) defining the feaible et D, optimization problem (5) are nonconvex. Thu, tandard nonlinear optimization tool (e.g. gradient method, Newton method, etc.) can not be ued becaue they can trap in local minima/maxima, leading to uncertainty interval that are not guaranteed to contain the true unknown parameter, which i a key requirement of any et-memberhip identification method. A poible olution to overcome uch a problem i to relax (5) into convex problem in order to efficiently compute a lower and an upper bound of θ j and θ j, repectively. A relaxation technique (called tatic EIV) which provide an outer approximation of the FPS D i propoed in [12] and briefly decribed in Section IV for elf-conitency of the paper. In thi work, we preent a new approach to compute parameter bound through LMI relaxation technique. IV. STATIC EIV RELAXATION The main idea of the tatic EIV approach propoed in [12] i to relax the original problem (5) by auming that the uncertain regreor decribing the feaible et D are uncorrelated. In uch a way, an outer approximation D of D i obtained. The et D i defined by piecewie linear contraint and, although generally nonconvex, D i the union of at mot 2 n θ convex et D i, that i: D = 2 p Di, where Di i defined by linear contraint and it i the interection between D and the i-th orthant of the parameter pace R n θ. Let u define ] the relaxed parameter uncertainty interval P UIj [θ = j, θ j, where θ j = min,...,2 θ ji, θ p j = max,...,2 θ ji, (7) p θ ji = min θ j, θ Di θ ji = max θ j. (8) θ Di Since D i an outer approximation of D, then θ j θ j and θ j θ j for all j = 1,..., n θ, that i P UI j P UIj for all j = 1,..., n θ. Computational complexity The evaluation of P UIj for j = 1,..., n θ require the computation of θ j and θ j, which in turn require the olution to 2n θ 2 n θ LP problem. In fact, problem (8) mut be olved for each of the n θ parameter θ j and in each of the 2 n θ et Di. V. LMI-BASED EIV RELAXATION In thi ection we decribe a new technique, which will be referred to a LMI-baed EIV relaxation, to compute guaranteed parameter uncertainty interval. Firt, note that problem (5) are emialgebraic optimization problem where the number of optimization variable i n θ +2N (i.e. n θ ytem parameter, N noie ample ξ on the input ignal and N noie ample η on the output ignal) and the feaible region D i defined by N na bilinear polynomial equalitie of the kind (y t η t ) + a i (y t i η t i ) = b j (u t j ξ t j ) and 4N linear inequalitie of the kind ξ t ξ t or η t η t. Since (5) are emialgebraic optimization problem, they can be relaxed through a direct implementation of the LMI-relaxation technique baed on the theory of moment and propoed by Laerre in [13]. Such a procedure i baed on the idea of relaxing a polynomial optimization problem by a hierarchy equence of emidefinite programming (SDP) problem with increaing dimenion, whoe optima are guaranteed to converge monotonically to the global optimum of the original polynomial problem. Although the method i guaranteed to converge a far a the length of the number of ucceive SDP problem (relaxation order) goe to infinity, exact global optima can be obtained in practice with a reaonably low relaxation order (ee [14] for detail). In particular, for a given relaxation order 1, relaxing problem (5) through the approach in [13] lead to ( SDP problem where ) the number of nθ + 2N + optimization variable i and the feaible

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL., NO., 3 ( ) nθ + 2N + region i decribed by LMI of ize + ( ) nθ + 2N + 1 (2(N na) + 4N). Due to high computational burden and memory torage requirement, the ue 1 of uch an LMI-relaxation technique to relax (5) i limited, in practice, to identification problem with a mall number N of meaurement (roughly not greater than 10), that i certainly an unuual and unfavorable condition in the identification framework. A poible way to overcome thi retriction i to relax (5) through the procedure preented in [15] in the pirit of [16]. Such an approach exploit the parity in the original polynomial problem to formulate a pare verion of the Laerre relaxation [13], in order to extend the applicability of uch a methodology to medium and large cale problem. In order to apply the pare LMI-relaxation given in [15], the tructure of problem (5) will be analyzed. To thi aim, let u rewrite the FPS D given in (4) a D = {(θ, ξ, η) R nθ+2n. : g t = (y t i η t i )a i + + (u t j ξ t j )b j +η t y t 0, g t+n (. = (y t i η t i )a i + + (u t j ξ t j )b j +η t y t ) 0, g k+2n. = ξk ξ k 0, g k+3n. = ξk + ξ k 0,.. g k+4n = ηk η k 0, g k+5n = ηk + η k 0, } t = na + 1,..., N; k = 1,..., N. The decription of D in (9) i traightforwardly obtained from (4) by rewriting each equality contraint (y t η t ) + a i (y t i η t i ) = b j (u t j ξ t j ) a two inequality contraint, i.e. (y t η t ) + a i (y t i η t i ) b j (u t j ξ t j ) and na nb (y t η t ) + a i (y t i η t i ) b j (u t j ξ t j ). The inherent parity of problem (5) i decribed by Property 1. Property 1: Structure of identification problem (5) Problem (5) enjoy the following feature: P 1.1: the functional depend only on the variable θ j, P 1.2: for all t = na + 1,..., N, the polynomial contraint g t 0 g t+n 0 defining D in (9) depend only on n θ +na+nb+2 = 2n θ +1 variable, i.e.: unknown parameter θ j, j = 1,..., n θ ; input ample noie ξ k, k = t, t 1,..., t nb; output ample noie η k, k = t, t 1,..., t na. (9) P 1.3: for all t = 1,..., N, the contraint g t+2n 0, g t+3n 0, g t+4n 0 and g t+5n 0 defining D in (9) depend only on the variable ξ t or η t. Thank to the parity of problem (5) decribed by Property 1, pare SDP-relaxed problem for (5) can be formulated a decribed in the following. Let X R n θ+2n be the collection of the deciion variable of (5), that i X = [θ T, ξ T, η T ] T, and let the index et I r and S r be defined a I r = {1, 2,..., n θ, n θ + r + na nb, n θ + r + na nb + 1,..., n θ + r + na, n θ + N + r, n θ + N + r + 1,..., n θ + N + r + na}, for r = 1,..., N na; S r = {na + 1, N + na + 1, 2N + 1, 2N + 2,..., 2N + na + 1, 3N + 1, 3N + 2,..., 3N + na + 1, 4N + 1, 4N + 2,..., 4N + na + 1, 5N + 1, 5N + 2,..., 5N + na + 1} for r = 1; S r = {na + r, N + na + r, 2N + na + r, 3N + na + r, 4N + na + r, 5N + na + r}, (10) (11) for r = 2,..., N na. (12) The index et I r and S r are contructed on the bai of the pare tructure of identification problem (5) highlighted by Property 1. More preciely, the et I r and S r are defined o that, for all S r, all the polynomial contraint g 0 in the definition of D γ depend only on the variable X i, with i I r. The index et I r and S r defined in (10)-(12) enjoy the feature highlighted by Property 2, which will play a crucial role in proving key propertie of the propoed identification method (ee Property 3 below). Property 2: For all r = 1,..., N na, index et I r and S r are uch that: P 2.1: The et of the variable indexe I 0 = {1, 2,..., n θ + 2N} i the union of the et I r, that i N na I 0 = I r. P 2.2: The et of the contraint indexe S 0 = {na + 1,..., N, N + na + 1,..., 2N, 2N + 1,..., 6N} defining D N na i the union of the et S r, that i S 0 = S r. P 2.3: The et S r are mutually dijoint. P 2.4: For all S r and r = 1,..., N na, the polynomial contraint g 0 defining D depend only on the variable X(I r ) = {X i : i I r }. P 2.5: The objective function θ j for identification problem N na (5) can be written a θ j = f r, where f r i a polynomial involving only the variable X(I r ) = {X i : i I r }, for all r = 1,..., N na. P 2.6: For all r = 1,..., N na 1, the et I r+1 i uch that: I r+1 r j=1 I j I r.

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL., NO., 4 For a given { relaxation order 1, let u define the et A n θ+2n = α N n } θ+2n nθ +2N 0 : α i, where α i i the i-th component of the vector α and N n θ+2n 0 denote the et of n θ + 2N-dimenional nonnegative integer vector. Let p = {p α } n α A θ +2N be the equence of moment up to order aociated with the variable X of a probability meaure µ on R nθ+2n, i.e. p α = X α µ(dx), where X α = X α1 1 Xα2 2 X α n θ +2N n θ +2N. Let u conider the following SDP problem, obtained from the application of the pare LMI-relaxation propoed in [15] to problem (5): θ j = min p D p α j, θ j = max p D p α j, (13) where α j i a n θ + 2N-dimenional vector uch that { α j 1 if i = j i = 0 otherwie and the feaible region D i a convex et defined a D = {p : M (p, I r ) 0, M 1 (g p, I r ) 0, S r, r = 1,..., N na }, (14) (15) where M (p, I r ) i the moment matrix of order aociated with the variable X(I r ) and M 1 (g p, I r ) i the localizing matrix aociated with the variable X(I r ), which take into account the contraint g 0 defining the emialgebraic region D. See the appendix for detail. Property 3: For a given 1, let u [ define ] the - relaxed uncertainty interval a P UIj = θ j; θ j. For all j = 1,..., n θ, interval P UIj atify the following propertie. P 3.1: Guaranteed relaxed uncertainty interval. For any 1, the interval P UIj i guaranteed to contain the true parameter θ j, i.e. θ j P UIj. P 3.2: Increaing accuracy in relaxed uncertainty interval evaluation. For any 1, the interval P UIj become tighter a the relaxation order increae, that i P UI +1 j P UIj. P 3.3: Convergence to tight uncertainty interval. The interval P UIj converge to the tight interval P UI j a the LMI relaxation order goe to infinity, that i: lim θ j = θ j, lim θ j = θ j. Proof: Since the index et I r and S r defined in (10)- (12) atify Propertie P , the firt part of Propoition 1 (in appendix) can be applied to problem (5) and to the correponding SDP-relaxed problem (13). Therefore, θ j θ +1 j θ j, θ j θ +1 j θ j for all j = 1,..., n θ. (16) Then, from the definition of the interval P UI j and P UIj and eq. (16), we get θ j P UI +1 j P UI j P UI j for all j = 1,..., n θ a tated in Propertie P 3.1 and P 3.2. Beide, from condition (3) and (6), the infinity norm X of the variable vector X i alway bounded and, ince the index et I r and S r atify alo Property P 2.6 for all r = 1,..., N na 1, from the econd part of Propoition 1 convergence condition given by Property P 3.3 follow. Indeed, one of the the main advantage of the LMI-baed EIV over the tatic EIV relaxation i the convergence of the relaxed interval P UI j to the tight interval P UI j, a tated in Property P 3.3. Thu, provided that the relaxation order goe to infinity, LMI-baed EIV provide parameter bound tighter than thoe obtained via the tatic EIV approach, that i: θ j lim θ j = θ j, θ j lim θ j = θ j. (17) Although condition (17) are guaranteed a goe to infinity, the LMI-baed relaxation procedure provide P UIj tighter than P UIj alo for mall value of relaxation order (i.e. = 2). Computational complexity The evaluation of P UIj require the olution to SDP problem (13), where number of optimization variable i ( ) ( ) 2nθ nθ 1 + (N na) (N na 1) and feaible region D i decribed( by N na) moment 2nθ matrixe, each one of ize, and 2(N ( na) ) + 4N localizing matrixe, each one of ize 2nθ +. 1 Remark 1: The number of optimization variable involved in the tatic EIV relaxation doe not depend on the number of meaurement N, it i contant and equal to n θ, ince only the unknown parameter θ j are treated a variable. On the contrary, in the LMI-baed EIV relaxation, where alo the ample noie ξ t and η t are optimization variable of problem (5), the number of optimization variable grow linearly with N. On the other hand, tatic EIV require the olution to 2n θ 2 n θ LP optimization problem in order to compute parameter bound, while, by uing the LMI-baed EIV relaxation, only 2n θ SDP problem have to be olved. VI. SIMULATED EXAMPLES AND DISCUSSION Example 1. Conider a trictly-proper econd-order ytem with true parameter vector θ T = [a 1 a 2 b 1 b 2 ] = [ ]. Parameter bound are evaluated for three imulated data et of divere length N, i.e. N = 30, N = 200 and N = Numerical computation ha been carried out with Matlab 7.4 run on a 2.40-GHz Intel Pentium IV with 3 GB of RAM. The ytem i excited by a random input equence x t uniformly ditributed between [ 1, +1]. Both input x t and output equence w t are corrupted by random additive noie ξ t and η t, uniformly ditributed between [ ξ t, + ξ t ] and [ η t, + η t ], repectively. The choen error bound ξ t and η t are uch that the both the ignal to noie ratio on the input { N SNR x = 10 log { t=1 N SNR w = 10 log t=1 x 2 t w 2 t / N / t=1 N t=1 ξ 2 t η 2 t } } and on the output are between 20 db and 22 db in both the imulation. The central etimate

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL., NO., 5 N SN Rx SN Rw True [db] [db] Value θjc θj θjc, θj θjc = (θj + θj )/2 and the parameter uncertainty bound θj = (θj θj )/2 obtained through the tatic EIV approach, a well a θjc, = (θj + θj )/2 and θj = (θj θj )/2 obtained with the LMI-baed EIV procedure for a relaxation order = 2, are reported in Table I. The CPU elaped time to compute a ingle parameter bound (θj or θj ) i between 0.4 and 0.6 irrepective of the value of N, while the CPU time taken by the SDP olver SeDuMi in olving problem (13) to compute θj or θj i between 30 and 37 when N = 30, between 267 and 382 when N = 200 and between 2876 and 3503 when N = Reult in Table I how that, even for a low relaxation order, the LMI-baed EIV approach provide parameter bound tighter than the one obtained by uing the tatic EIV procedure. In fact, for all choen value of N, the width θj of P U Ij i ignificantly maller than the width of P U Ij. For intance, in the cae N = 1000, the uncertainty bound θj on the firt parameter θ1 = a1 i about 10 time maller than θj. Futhermore, the θj computed by mean of the LMI-baed relaxation for N = 30 are maller than the θj computed through the tatic relaxation for N = Beide, the propoed LMI-baed procedure provide atifactory parameter bound alo for a mall and medium data et (i.e. N = 30 and N = 200), with parameter bound to central value ratio θj /θjc, maller than 50% when N = 30, and maller than 35% when N = 200. We point out that, in the cae N = 1000, the number of optimization variable in (13) i 1, 042, 580, while the feaible region i defined by 998 moment matrixe of ize 715 and 5, 996 localizing matrixe of ize 10. On the other hand, if the LMI-relaxation procedure given in [13] wa directly applied to relax problem (5) without taking into account their tructured parity, then the number of variable of the correponding SDP-relaxed problem would be about 675 billion and the feaible region would be decribed by a moment matrix of ize 2, 011, 015 and 5, 996 localizing matrixe of ize 2, 005, leading to an untractable optimization problem. It i worth remarking that, although taking into account parity of the identification problem (5) ignificantly reduce the computational complexity of the correponding LMI-relaxed problem, it i preently difficult to apply the dicued procedure to the identification of ytem of order higher than 5 becaue of the high computational time required by the olver SeDuMi to compute the olution of SDP problem (13). However, the development of more efficient algorithm to olve SDP problem i growing rapidly. For example, the interior point algorithm propoed in [17] eem to be a promiing method to efficiently olve SDP problem (13). Example 2. We now conider a firt-order ytem with two parameter in order to plot the exact feaible parameter et D together with it outer approximation obtained both through the tatic EIV approach D and through the LMIbaed relaxation D. The true parameter vector of the ytem i θt = [a1 b0 ] = [ ]. The length of the imulated data N i 50 and the value of the SN Rx and the SN Rw are 21 db and 20 db, repectively. The projection of the exact FPS D on the parameter plane (a1, b0 ) i reported in Fig. 1, together with the projection of relaxed et D and D, for = 2. Fig. 1 how that the et D i ignificantly more conervative than the et D, which, in turn, i quite cloe to the convex hull of the exact FPS D. VII. C ONCLUSIONS Identification of linear dynamic ytem when both input and output meaurement are corrupted by bounded noie i addreed in the paper. Firt, we how that the et of parameter and noie ample conitent with meaurement, error bound and the aumed model tructure i a nonconvex et decribed by polynomial inequalitie. Then, the problem of computing parameter uncertainty interval i recat in term of olution to a collection of nonconvex emialgebraic optimization problem whoe global optima are approximated by contructing a hierarchy of convex SDP relaxed problem. Unfortunately, becaue of a large number of variable and contraint involved in the SDP relaxed problem, identification problem are practically intractable when more than 10 meaurement are conidered. In order to overcome thi ignificant limitation, the peculiar parity tructure of the identification problem i exploited to ignificantly reduce the computational burden of the propoed approach while preerving monotone convergence to tight parameter bound. The capabilitie of the preented procedure to provide parameter bound tighter than thoe obtained with previouly publihed tatic EIV relaxation i hown both theoretically and by mean of two numerical parameter b0 TABLE I PARAMETER CENTRAL ESTIMATES (θjc, θjc, ) AND UNCERTAINTY c, BOUNDS ( θj, θj ) AGAINST NUMBER OF DATA N. VALUES OF θj AND θj ARE COMPUTED FOR RELAXATION ORDER = parameter a 1 Fig. 1. Exact feaible parameter et D (gray region), approximate feaible parameter et D (region inide thick line) and approximate feaible parameter et D (region inide thin line) projected on the parameter plane (a1, b0 ).

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL., NO., 6 example. It i worth remarking that, although taking into account parity of the identification problem ignificantly reduce the computational burden of the correponding relaxed problem, it i preently difficult to apply the preented approach to the identification of ytem of order higher than 5 becaue of the high computational time required by the available general purpoe SDP olver. Neverthele, poible direction for further reearch are related to the development of efficient algorithm for SDP problem ariing from the LMIrelaxation of polynomial problem. APPENDIX LMI-RELAXATION FOR SPARSE POLYNOMIAL OPTIMIZATION PROBLEMS A brief review on the relaxation of pare polynomial optimization problem propoed by Laerre in [15] in the pirit of the work of Waki et al [16] i preented here. Let u conider the contrained optimization problem f = min f(x), (18) x S where S R n i a emialgebraic et defined a S = {x R n : g (x) 0, = 1,..., Ξ} with g a real-valued polynomial in the variable x = [x 1, x 2,..., x n ] T R n of degree d = deg(g ), and f Pm[x], n with Pm[x] n denoting the pace of real-valued polynomial of the degree at mot m in the variable x R n. By defining the et A n m = {α N n 0 : n i α i m}, where α i i the i-th component of the vector α and N n 0 denote the et of n-dimenional nonnegative integer vector, the canonical bai h n m = [ 1 x 1 ] x 2 x n x 2 1 x 1 x 2 x 1 x n x 2 2 x 2 x 3 x 2 n x 3 1 x m T n of the pace Pm[x] n can be written a h n m = {x α } α A n m, where x α = x α 1 1 xα 2 2 xα n n. Let p = {p α } α A n m be the equence of moment (up to order m) of a probability meaure µ on R n, i.e. p α = x α µ(dx) and I 0 = {1,..., n} be the union of a collection of R et R I r {1,..., n}, that i {1,..., n} = I r. Further, let u partition the index et S 0 = {1,..., Ξ} into R dijoint et S r, r = 1,..., R. Let h n r m (I r ) be the canonical bai of the polynomial Pm nr [x(i r )], where n r i the cardinality of the et I r and x(i r ) = {x i i I r }. The truncated moment matrix M m (p, I r ) aociated with the moment e- quence p and the variable x(i r ) i defined a M m (p, I r ) = h n r m (I r )h nr m (I r ) T µ(dx). By denoting with p β(i,j)(ir ) the entry (i, j) of the matrix M m (p, I r ), the localizing matrix M m (g p, I r ) aociated with the moment equence p and the polynomial g (x) i defined a M m (g p, I r )(i, j) = α A n g r αp m {β(i,j)(ir )+α}, where g α i the coefficient of the term x α in the polynomial g (x) and M m (g p, I r )(i, j) denote the entry (i, j) of the matrix M m (g p, I r ). For a given integer uch that max{m, max d }, let u conider the SDP problem f = min f α p α p α A n.t. M (p, I r ) 0, M d (g p, I r ) 0, S r, r = 1,..., R where d i the mallet integer greater or equal than d 2 and f = {f α } α A n i the equence of coefficient of the polynomial f in the canonical bai h n = {xα } α A n of the pace P n [x], i.e. f(x) = α A f n α x α. Let u conider the following aumption. Aumption 1: For all r = 1,..., R and for all S r, the contraint g (x) 0 defining the feaible et S in problem (18), depend only on the variable x(i r ) = {x i i I r }. Aumption 2: The objective function f can be written a R f = f r, with f r P n r m [x(i r )], for all r = 1,..., R. Aumption 3: There exit a value G > 0 uch that x G for all x S. Aumption 4: For all r = 1,..., R 1, I r+1 i uch that: r I r+1 I j I q, for ome q r. j=1 The following reult hold. Propoition 1: Under Aumption 1 and 2 we have f f +1 f. Furthermore, if alo Aumption 3 and 4 are atified, then lim f = f. REFERENCES [1] L. Ljung, Sytem Identification, Theory for the Uer. Upper Saddle River: Prentince Hall, [2] L. 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