A Least Squares Approach to the Subspace Identification Problem

Transcription

1 A Least Squares Approach to the Subspace Identiication Problem Laurent Bako, Guillaume Mercère, Stéphane Lecoeuche To cite this version: Laurent Bako, Guillaume Mercère, Stéphane Lecoeuche A Least Squares Approach to the Subspace Identiication Problem 47th IEEE Conerence on Decision and Control, Dec 2008, Cancun, Mexico pp-6, 2008 <hal > HAL Id: hal Submitted on 4 Nov 2008 HAL is a multi-disciplinary open access archive or the deposit and dissemination o scientiic research documents, whether they are published or not The documents may come rom teaching and research institutions in France or abroad, or rom public or private research centers L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diusion de documents scientiiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche rançais ou étrangers, des laboratoires publics ou privés

2 A Least Squares Approach to the Subspace Identiication Problem L Bako,2, G Mercère 3 and S Lecoeuche,2 Abstract In this paper, we propose a new method or the identiication o linear Multiple Inputs-Multiple Outputs (MIMO) systems By introducing a particular user-deined matrix that does not change the rank o the extended observability matrix when multiplying this latter matrix on the let, the subspace identiication problem is recasted into a simple least squares problem with all regressors available Thereore, the Singular Value Decomposition algorithm which is a customary tool in subspace identiication can be avoided, thus making our method appealing or recursive implementation The technique is such that the state coordinates basis o the estimated matrices is completely determined by the aorementioned userdeined matrix, that is, given such a matrix, the state basis o the identiied matrices does not change with respect to the realization o input-output data I INTRODCTION The identiication o linear dynamical Multiple Input- Multiple Output (MIMO) systems achieved in the last two decades a remarkable development rom the so-called subspace methods [], [2], [3] The main appealing eature o these methods over the more traditional error prediction methods [4], is that they directly provide minimal and not necessarily canonical state space models rom input-output (I/O) data However, the application o subspace identiication methods to the estimation o certain types o systems such as composite systems with linear constituent submodels [5], switched linear systems [6], [7] or recursive identiication [8], [9], sometimes comes with some technical diiculties One issue is related to the multiplicity o possible bases or representing the system equations in the state space For example, in the case o recursive subspace identiication, most o the existing contributions [8], [9] do not guarantee that the state basis remains ixed during the whole recursive identiication procedure This is also a crucial problem when dealing or example with the identiication o multi-models or switched systems Indeed, in these cases, all the dierent submodels o the system must be obtained in the same basis [6] Failing to this requirement may have the consequence o altering the I/O behavior o the system A second issue is related to the SVD algorithm that is generally required in subspace identiication In act, subspace methods involve an SVD step in which one decides arbitrarily on the basis o the range space o the extended Ecole des Mines de Douai, Département Inormatique et Automatique, 59508, Douai, France 2 Laboratoire d Automatique, Génie Inormatique et Signal, MR CNRS 846, niversité des Sciences et Technologies de Lille, 59655, Villeneuve d Ascq, France 3 niversité de Poitiers, Laboratoire d Automatique et d Inormatique Industrielle, 86022, Poitiers, France observability matrix to be estimated An important problem is that the SVD is computationally heavy and technically hard to update recursively, thus making its use in online identiication a astidious task [8] Moreover, updating an SVD neither solves the problem o coordinate basis deviation during the estimation Thereore, an attempt to reconstruct the I/O behavior rom the estimated matrices may result in a shited behavior In this paper, we ocus on developing a new, simple, eicient and SVD-ree identiication method or linear dynamical state space models, that could overcome the aorementioned problems Since a state model holds only up to a similar transormation, one can indeed choose in advance the basis o the model to be identiied This is carried out by introducing a user-deined matrix Λ (notation o the paper), that preserves the rank properties o the extended observability matrix when multiplying this latter matrix on the let We then show that, under the assumption that the considered system is observable, i one draws Λ randomly rom a uniorm distribution or example, we can obtain a consistent realization o the system Being based on a conversion o the subspace estimation problem into a Least Squares problem, the developed method can be regarded as an interesting solution to the problem o subspace tracking requently encountered in signal array processing, recursive system identiication and many other applications [0] The outline o the paper is as ollows In Section II we ormulate the subspace identiication problem A relevant suiciency o excitation concept is deined and used to derive conditions that guarantee consistency o the subspace identiication In Section III, we present a new subspacebased identiication algorithm Some illustrative simulation results shown in Section IV demonstrate the applicability o our method II PROBLEM STATEMENT We consider a Linear Time-Invariant (LTI) system described by a discrete-time model o the orm { x(t + ) = Ax(t) + Bu(t) + w(t) () y(t) = Cx(t) + Du(t) + v(t), where x(t) R n, u(t) R nu, y(t) R ny are respectively the state, the input and output vector and w(t) R n and v(t) R ny symbolize the process noise and measurement noise Here, these noises are assumed to be zero-mean white noise processes A, B, C, D are the system matrices relatively to a certain basis o the state space The identiication problem can then be ormulated as ollows: given I/O data {u(t), y(t)} N t= generated by a system

3 o the orm (), estimate the matrices (A, B, C, D) in any state coordinates To begin with the identiication procedure, let us deine u (t) = [ u(t) u(t + ) ] R n u, (2) with > n In a similar manner as u (t), we deine y (t) R ny, w (t) R n and v (t) R ny Finally, in accordance with these signal vectors, we deine the matrices Γ = [ (C) (CA) (CA ) ] R n y n, D 0 0 CB D 0 H = Rny nu, CA 2 B CB D C 0 0 G = Rny n CA 2 C 0 From the recurrent equation (), one can easily obtain, by successive substitutions over a time horizon [t, t + ], y (t) = Γ x(t) + H u (t) + e (t), t (3) with e (t) = G w (t) + v (t) Let N be an integer such that n < N and deine X t,n = [ x(t) x(t + N ) ] R n N, t,,n = [ u (t) u (t + N ) ] R nu N Similarly to t,,n, we deine also Y t,,n R ny N, E t,,n R ny N Then, based on Eq (3), we can write the ollowing block data equation or t = + Y +,,N = Γ X +,N + H +,,N + E +,,N (4) In the literature o subspace identiication, data matrices o the orm Y,,N and Y +,,N are sometimes respectively reerred to as past and uture data In order to alleviate the notations we will, throughout the paper, denote whenever possible the uture data more simply as Y = Y +,,N, = +,,N, E = E +,,N and X = X +,N In this way, Eq (4) can be written simply as Y = Γ X + H + E (5) Based on this embedded data equation, subspace methods or solving the problem o identiying system () proceed by irst extracting either the state sequence X or the extended observability matrix Γ, making use o geometric projection techniques and rank actorization algorithms such as the SVD [2], [] A second step then consists in computing the system matrices in an arbitrary state coordinates basis However, the use o these methods in or instance a recursive identiication context may be subject to two important issues: ) the SVD actorization in addition o being computationally cumbersome, is technically diicult to update, 2) the state space basis in which the matrices are obtained may change during the identiication procedure or rom one mode to another when dealing with multi-modal systems The method developed in the present paper mainly intends to overcome the aorementioned diiculties But, in a more general ramework, the proposed method turns out to be an interesting Least Squares solution to the well known problem o subspace tracking Consistent estimation o the parameter matrices in (5) that we can re-write as Y = [ Γ H ] [ X ] + E, (6) requires the measured data to have some properties o richness Thereore, we start by introducing such conditions that will be useul throughout the paper We irst make the ollowing deinition o suiciency o excitation o the exogenous input {u(t)} Deinition (suiciently exciting input sequence): The process {u(t)} is said to be suiciently exciting o order at least l i there exist an integer N 0 and a time index t 0 such that rank( t,l,n0 ) = ln u or all t t 0 We then say that {u(t)} is SE(l) Note that Deinition diers rom the classical deinition o persistency o excitation [4] that concerns ininite sequences o signals Here, a inite horizon is considered instead as the number o data available or identiication is generally inite in practice A consequence o Deinition is as ollows Proposition : Assume that the system () is reachable and let the noise terms w(t) and v(t) be identically zero in () Then the ollowing holds I {u(t)} is SE(n + ) in the sense o Deinition with N N 0 + n and + t 0, then ( [ ] X ) rank +,N = n + n u +,,N Proo: Assume that there are α R n and β R nu veriying α X +,N + β +,,N = 0 (7) We then need to show that α and β are both necessarily equal to zero Since N N 0 + n, or any t such that + t + + N N 0, α X t,n0 + β t,,n0 = 0 (8) On the other hand, we can write rom the system equations in (), x(t) = A n x(t n) + n u n (t n), t > n, (9) where n = [ A n B AB B ] R n nnu By making use o the Cayley-Hamilton theorem, we get ater some calculations x(t) = (a I n )x n (t n)+ n (K I nu )u n (t n), (0) where x n (t n) and u n (t n) are deined as in (2), reers to the Kronecker product, a = [ a a n ] is a vector ormed with the coeicients a j, j =,, n o the characteristic polynomial o A, ch A (z) = det (zi A) =

4 a + a 2 z + + a n z n + z n Here, the matrix K is deined as a 2 a 3 a n a n a 3 a 4 a n 0 K = R n n a n For all t > n, Eq (0) reads as x(t) + a n x(t ) + + a x(t n) n (K I nu )u n (t n) = 0 Deine τ = + n + > n Then, Eq () implies X τ,n0 + a n X τ,n0 + + a X τ n,n0 n (K I nu ) τ n,n,n0 = 0 Multiplying the equation (2) on the let by α yields α X τ,n0 + a n α X τ,n0 + + a α X τ n,n0 ᾱ τ n,n,n0 = 0, () (2) (3) with ᾱ = α n (K I nu ) Combining now the oregoing equation with (8) results in β τ,,n0 a n β τ,,n0 a β τ n,,n0 ᾱ τ n,n,n0 = 0 (4) Note that all the matrices o the orm k,,n0, τ n k τ can be expressed as a combination o the rows o τ n,+n,n0 In this way, we have k,,n0 = P k τ n,+n,n0 or τ n k τ, and k,n,n0 = Q k τ n,+n,n0, with [ P τ j = 0nu (n j)nu, I nu, 0 nu jnu] R nu (+n)nu ] Q τ n = [I nnu, 0 nnu nu R nnu (+n)nu, or j = 0,, n sing these notations, we get ( β P τ + a n β P τ + + a β P τ n + ᾱ Q τ n ) τ n,+n,n0 = 0 (5) Since {u(t)} is suiciently o order at least n+ the matrix τ n,+n,n0 = +,+n,n0 is ull row rank Consequently, the term o (5) that is in the parentheses is null I we let β and ᾱ be partitioned as ᾱ = [ ] ᾱ ᾱn, ᾱ j R nu, β = [ ] β β, β j R nu, then, by straightorward calculations, we arrive at a n a a n a 0 0 a n a β β = 0, and a a 2 a 0 a n a n a β β n + ᾱ ᾱ n = 0 It ollows immediately that β = 0 and ᾱ = α n (K I nu ) = 0 Since the system () is reachable, we have rank( n ) = n and so, rom the deinition o K, n (K I nu ) is clearly ull row rank n Consequently, we have also α = 0 and hence, the rows o [ X+,N ] +,,N are linearly independent For uture reerence, we shall also state the ollowing proposition Proposition 2: Assume that the system () is reachable and observable and let w(t) and v(t) be identically null in () Then the ([ ollowing ]) statements are equivalent X ) rank = n + n u 2) rank(xπ ) = n, where Π = I N ( ), (6) I N being the identity matrix o order N ([ ]) Y 3) rank = n + n u 4) rank ( ) Y Π = n Proo: ) 2): Notice that [ ][ ] [ ] X X XX X = X Then, by using the identity [2] [ ] [ ] In X ( ) XX X 0 I nu X [ ] I n 0 ( ) X I nu [ ] XΠ = X 0 0, it is clear that ( [ X rank ] ) ( [ ] XΠ = rank X 0 ) 0 = n u + rank(xπ X ) = n u + rank(xπ ), rom which it can be concluded that ( [ ] X ) rank = n + n u rank(xπ ) = n ) 3): The result ollows directly rom the relation [ [ ] [ ] Y Γ H = X (7) ] 0 I nu }{{} since the underbraced matrix is ull column rank when the system is observable, ie, when rank(γ ) = n

5 3) 4): A proo o this statement is similar to the one derived or ) 2) III A NEW SBSPACE IDENTIFICATION METHOD We develop in this section a new method or the identiication o linear state space models In contrast to most o the existing subspace techniques, our method does not require any SVD Thereore, it can be naturally and straightorwardly extended to recursive identiication [8], [3], [9] Moreover, it allows to set up very simply the state basis o the matrices to be identiied To begin with, assume the order n o the system () to be available Consider a known matrix Λ R n ny that satisies rank(λ Γ ) = n (8) I we multiply Eq (5) on the let by Λ, we get Λ Y = (Λ Γ )X + Λ H + Λ E (9) Since T = Λ Γ R n n is nonsingular, one realizes a state coordinates basis change by setting X X = T X Then, the state sequence can be obtained in the new basis as X = Λ Y Λ H Λ E (20) As a consequence o this change o the coordinates system, the matrices (A, B, C, D) change into (Ā, B, C, D) = (T AT, T B, CT, D) and Γ becomes Γ = Γ T but H remains unchanged as it does not depend on the state basis For the sake o clarity, we shall irst consider the noise-ree case, ie, the case where E in (5) is identically null Then, we shall demonstrate the applicability o the method on noisy data A Deterministic case In the absence o noise in the data, the state equation (20) reduces to X = Λ Y Λ H (2) It is obvious rom this relation that the choice o Λ determines completely the state basis since it directly deines a linear combination o the I/O data that gives the state To obtain Γ, we can report Eq (2) in Eq (5) This results immediately in Y = Γ X + H = Γ Λ Y + (I ny Γ Λ )H = [ ] [ ] Λ Γ Ω Y, (22) where Ω = (I ny Γ Λ )H This shows that the subspace identiication problem can be transormed into an ordinary Least Squares problem by eliminating the unknown state As shown by Propositions and 2, when the input process is Discussion on the concrete determination o such a matrix Λ is deerred to Subsection III-C suiciently exciting o order at least n +, and the model () is minimal, it holds that ( [ ] Y ) rank = n + n u By ollowing a similar procedure as in the proo o Proposition 2 (see Eq (7)), one can easily establish that the ollowing holds also ( [ Λ rank Y ] ) = n + n u and rank(λ Y Π ) = n Thus, we can estimate directly the matrices Γ and Ω rom Eq (22) and then deduce the system matrices Remark : I Λ R n ny obeys the rank property (8), then by denoting im( ) and ker( ) respectively the image and the kernel operators or matrices, we have: ) im(γ ) = im( Γ ) = ker(i ny Γ Λ ) 2) rank(i ny Γ Λ ) = n y n 3) The matrix I ny Γ Λ is a projection matrix that projects the ambiant vector space R ny onto im(i ny Γ Λ ) = ker(λ ) along im(γ ) In this way, Eq (22) can be viewed as a projection o Eq (5) obtained by applying (I ny Γ Λ ) to it Thanks to this remark, we know that I ny Γ Λ is not ull column rank and so H cannot be retrieved directly rom the estimate ˆΩ o Ω in (22) However, by appropriately exploiting the structure o H, it is still possible to recover the matrices B and D rom ˆΩ and ˆ Γ similarly as in [, p 54] Another way to proceed is to remove the term Ω rom (22) by multiplying this latter equation on the right by Π beore computing Γ We then get Γ = Y Π ( Λ Y Π ), = Σ yu Λ ( Λ Σ yu Λ ), (23) where Σ yu = N Y Π Y Here, Σ yu is a correlation matrix while Π deined in (6) is a matrix o projection onto the orthogonal complement o the rows space o The symbol reers here to the Moore-Penrose inverse In practice, the orthogonal projection Π may be implemented by resorting to methods that are known to be numerically robust such as the QR actorization [2] Once Γ is available, an extraction o the matrices Ā and C can be simply achieved by exploiting the so-called A-invariance property o Γ [] Now, given the matrices Ā and C, it remains to determine the matrices B and D This can be achieved by solving a linear regression problem For more details on the concrete procedure, we reer to, or example the papers [2], [8] B Stochastic case Consider now the more realistic case where the data are subject to noise Then, the combination o Eq (5) and Eq (20) gives Y = Γ Λ Y + (I ny Γ Λ )H + (I ny Γ Λ )E

6 As previously, the term Ω = (I ny ΓΛ )H is removed by post-multiplying the equation by Π We get Y Π = Γ Λ Y Π + (I ny Γ Λ )EΠ (24) To deal with the noisy data, we adopt the well known method o instrumental variable (IV) [4] The basic idea o this method is to choose a matrix o instruments Z R nz N, n z n (ormed or example with past input and output [3]) to zero out the eect o the noise while preserving the inormation conveyed by the data By multiplying Eq (24) on the right by Z and dividing by the number o samples N, we obtain N Y Π Z = Γ N Λ Y Π Z + (I ny Γ Λ ) N EZ (I ny Γ Λ ) N E ( N ) N Z (25) nder the assumption that the sequence {u(t)} is ergodic and independent o {w(t)} and {v(t)}, the last term o (25) vanishes when N Thus, Eq (25) reduces to ( lim N N Y Π Z ) = Γ ( lim N N Λ Y Π Z ) + (I ny Γ ( Λ ) lim N N EZ ) (26) In view o the previous equation, we require the instruments matrix Z to veriy the ollowing two conditions ( lim N rank ( lim N N EZ ) = 0, N Λ Y Π Z ) = n (27) When Z ulills the requirements (27), the ollowing asymptotic relation holds lim N N Y Π Z = Γ ( lim N N Λ Y Π Z ) (28) From this equation, we can estimate Γ, then compute as previously Ā and C and subsequently obtain B and D by linear regression as indicated in Subsection III-A Turning now to the question o how to choose the instruments matrix Z, we will ollow [4] where the concatenation o the the past inputs and past outputs, Z = [,,N Y,,N is shown to be a valid set o instruments ] (29) C Setting the matrix Λ The method described above or the identiication o state space models relies on the possibility to ind a matrix Λ R n ny that veriies the rank condition (8) This naturally raises the question o whether such a matrix always exists and more importantly, how to determine it while Γ is unknown The question o the existence is readily answered under the assumption that the system is observable since then, Λ = Γ (or example) veriies (8) But, as we do not know Γ, we cannot set Λ to be equal to Γ However, as stated by Propositions and 2, i {u(t)} is SE(n + ), then rank(λ Γ ) = n i and only i rank(λ Y Π ) = n In view o this remark, Λ can be selected just such that rank(λ Y Π ) = n because contrarily to Γ, Y Π is known But this solution may require an SVD actorization Thereore, an alternative solution is to generate Λ at random as suggested by the ollowing proposition Proposition 3: Assume that the system () is observable Let the above matrix Λ be drawn randomly rom a uniorm distribution Then, it holds with probability one that rank(λ Γ ) = n Proo: Let λ = vec(λ ) R nny, where vec( ) is the vectorization operator Denote Λ = Λ (λ), that is, let { us view Λ as a unction o λ and consider } the set S = λ R nny /P (λ) = det(λ (λ)γ ) = 0 o all λ such that rank(λ (λ)γ ) < n Consider the probability measure Pr associated to the uniorm distribution and deined on a σ- algebra R (containing S) over R nny Clearly, or a given Γ, the polynomial P is not identically null since we have or example P (vec(γ )) 0 Then, the hypersurace S is a subset o the probability space ( R nny, R, Pr ) that is o dimension strictly less than nn y From the measure theory [5], a subset such as S is known to be a null set Thus, the property rank(λ (λ)γ ) = n holds almost everywhere 2 Proposition 3 states in other words that i we draw randomly the matrix Λ rom a uniorm distribution or example, the rank property (8) holds with probability one The system matrices are then correctly estimated by the above method with probability one Remark 2 (On the estimation o the order): In case the order n o the system () is unknown, it would be also interesting to estimate it without having to resort to the SVD To this purpose, let Λ be selected in R ny ny such that 3 or any r n, it holds that rank(λ :r Γ ) = r, where Λ :r = Λ ( : r, :) Then rom the embedded data equation Y = Γ X + H, one can write [ ] [ ] Λ Λ Y Π = Γ XΠ, Λ 2 where Λ = Λ ( : n, :) and Λ 2 = Λ (n + : n y, :) Taking the square o the irst term o this equation yields [ Λ Σ = Σ yu (Λ ) Λ Σ yu(λ 2 ] ) Λ 2 Σ yu(λ ) Λ 2 Σ yu(λ 2, ) with Σ yu = Y Π Y The order is then n = max { r : rank(σ ( : r, : r) ) = r } and can be determined by ollowing a similar procedure as in [7] Λ 2 IV SIMLATION EXAMPLE We shall in this section, evaluate the perormance o our method on a numerical example To this purpose, we consider a linear system o order three with two inputs and two outputs The system matrices are given by 2 (8) is satisied everywhere in R nny except on a set o measure null 3 In act, as in Proposition 3, this is the case almost surely when Λ is generated randomly

7 A = , B = , [ ] C =, D = [ ] To identiy this system, the excitation input is chosen as a zero-mean white Gaussian noise with unit variance The state noise w(t) and output noise v(t) are both set as white noises in a reasonable proportion o 25 db respectively o the state and the output The user-deined parameter is set to be 5 We then carry out a Monte-Carlo simulation with 000 dierent realizations o noise and input Each o these simulations generates 000 I/O data samples With this setting, we perorm two sets o simulations: one with an instrumental variable in the orm o (29) and the second without any instrumental variable The results can be judged through the ollowing graphs In Figure, we present the poles o the actual system and the poles o the estimated model over all the 000 simulations One can notice that when no instrumental variable is introduced, the estimator seems to be biased when the data are corrupted by noise To urther show the perormance o our method in a statistical sense, we represent in Figure 2 the histograms o the relative errors M ˆM / M between the irst ten Markov parameters (which are invariant under state basis change) o the true system together with the Markov parameters o the estimated model All the simulation results presented clearly illustrate the potential o our method as a serious alternative to the classical subspace methods or the identiication o state space models without IV with IV Fig Poles o the system superposed to that o its estimate The big crosses reer here to the poles o the true system V CONCLSION We have proposed a new method or the identiication o MIMO state space models Contrarily to most o the existing subspace identiication schemes, our method is SVD-ree and thereore, lends itsel to a straightorward extension to recursive identiication o multivariable systems Moreover the state space basis o the system matrices to be estimated N s N s without IV Errors with IV Errors Fig 2 Histograms o the relative errors between the Markov parameters o the system on the one hand and that o the model estimated by our method on the other hand N s stands or number o simulations is ixed in advance through a conversion o the subspace problem into a least squares one This is an important eature or example in the ramework o multi-modal systems identiication since in such situation, all the dierent submodels need to be identiied in the same state coordinates basis In uture work we will carry out a comparison study o our algorithms with the standard subspace identiication methods such as MOESP and N4SID in terms o complexity and perormance We will also consider to extend our method to the identiication o multivariable switched linear systems described by state space models REFERENCES [] P Van Overschee and B De Moor, Subspace identiication or linear systems theory, implementation, applications Kluwer Academic Publishers, 996 [2] T Katayama, Subspace methods or system identiication Springer- Verlag, 2005 [3] M Verhaegen and V Verdult, Filtering and System Identiication: A Least Squares Approach Cambridge niversity Press, 2007 [4] L Ljung, System Identiication Theory or the user 2nd ed PTR Prentice Hall pper Saddle River, SA, 999 [5] V Verdult, L Ljung, and M Verhaegen, Identiication o composite local linear state-space models using a projected gradient search, International Journal Control, vol 75, pp , 2002 [6] V Verdult and M Verhaegen, Subspace identiication o piecewise linear systems, in Conerence on Decision and Control, 2004 [7] L Bako, G Mercère, and S Lecoeuche, Online subspace identiication o switching systems with possibly varying orders, in European Control Conerence, 2007 [8] M Lovera, T Gustasson, and M Verhaegen, Recursive subspace identiication o linear and non-linear wiener state-space models, Automatica, vol 36, pp , 2000 [9] H Oku and H Kimura, Recursive 4SID algorithms using gradient type subspace tracking, Automatica, vol 38, pp , 2002 [0] H Krim and M Viberg, Two decades o array signal processing research: the parametric approach, IEEE Signal Processing Magazine, vol 3, pp 67 94, 996 [] S J Qin, An overview o subspace identiication, Computers and Chemical Engineering, vol 30, pp , 2006 [2] T Kailath and A H Sayed, Fast Reliable Algorithms or Matrices with Structure SIAM, Philadelphia, PA, 999 [3] G Mercère, L Bako, and S Lecoeuche, Propagator-based methods or recursive subspace model identiication, Signal Processing, vol 2008, pp , 88 [4] C T Chou and M Verhaegen, Subspace algorithms or the identiication multivariables dynamic error-in-variables models, Automatica, vol 33, pp , 997 [5] P R Halmos, Measure Theory New York: Springer-Verlag, 974