Range Space or Null Space: Least-Squares Methods for the Realization Problem

Cristian R. Rojas; H\r{a}kan Hjalmarsson; Jiabao He; Yue Ju; Yueyue Xu

REVIEW 3 major objections 7 minor 2 cited by

Range Space or Null Space: Least-Squares Methods for the Realization Problem

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The classic SVD realization is a total least-squares fit, and the null-space method is ordinary least squares; a weighted least-squares estimate has the smallest asymptotic variance.

desk verdict Kung's method as TLS and NUSBR as OLS is a real unification, but the WLS optimality theorem currently has a wrong covariance formula. read the letter →

arxiv 2505.19639 v1 pith:Z5RLJXUS submitted 2025-05-26 eess.SY cs.SY

Jiabao He , Yueyue Xu , Yue Ju , Cristian R. Rojas , Håkan Hjalmarsson This is my paper · ORCID

classification eess.SYcs.SY MSC 93B3093B1593E1062J05

keywords approximaterealizationsubspaceidentificationHankelmatrixtotalleastsquaresordinaryweightedasymptoticnormalitysingularvaluedecomposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper revisits the approximate realization problem, where noisy estimates of the first $n$ Markov parameters are used to build a state-space model. It shows that the two classical approaches are not arbitrary alternatives: the null-space-based method is an ordinary least-squares solution in the characteristic polynomial coefficients, while the SVD-based range-space method is the corresponding total least-squares solution. The paper argues that both are suboptimal because neither noise model matches the Hankel structure, and it proposes a weighted least-squares estimator that it proves has the smallest asymptotic variance. If correct, this gives a unified explanation of when each classical method wins and a concrete route toward statistically efficient subspace identification.

What carries the argument

The central object is the Hankel matrix $H_{n_x p}$ built from the Markov parameters, which factorizes as an observability times controllability matrix; both its range space and left null space encode the system. By the Cayley-Hamilton theorem, the left null space is parameterized by the coefficients $a$ of the characteristic polynomial of $A$, turning realization into the linear relation $aH^+_{n_x p} + H^-_{n_x p} = 0$. The argument then runs through three regression formulations: OLS projects out $H^-$ on $H^+$ (NUSBR), TLS perturbs both blocks and is solved by the SVD (RASBR), and WLS uses the Toeplitz matrix $T(a)$ to account for the Hankel structure of the noise in the Markov parameter estimates.

What would settle it

Run a Monte Carlo study on a known SISO system: estimate the first $n = \beta \log N$ Markov parameters from $N$ samples, compute the three estimators, and compare the empirical covariance of $\sqrt{N}(\hat a_{\mathrm{ols}}-a)$, $\sqrt{N}(\hat a_{\mathrm{tls}}-a)$, and $\sqrt{N}(\hat a_{\mathrm{wls}}-a)$ with (64)--(65); if the WLS covariance is not the smallest, or the OLS and TLS covariances do not coincide, the claims fail.

Watch

Extended reading notes

Core claim

The paper establishes that the null-space-based realization (NUSBR) solves an ordinary least-squares problem for the coefficients $a$ of the characteristic polynomial, while the range-space-based realization (RASBR) solves the matching total least-squares problem, and the $A$-matrix returned by the SVD range-space algorithm is similar to the TLS solution. Because the Hankel structure makes the TLS correction term $\hat{\sigma}_{n_x+1}^2 I$ ineffective and the OLS assumption of a noise-free data matrix unrealistic, both methods are suboptimal and case-dependent. The paper proposes a weighted least-squares estimator $\hat a_{\mathrm{wls}}$ with weighting $W(a) = (T^\top(a) P_g T(a))^{-1}$, where $T(a)$ is a Toeplitz matrix capturing the Hankel-structured noise. Under Assumptions 5.1--5.3, all three estimators are consistent and asymptotically normal, with $P_{a,\mathrm{ols}} = P_{a,\mathrm{tls}} \succeq P_{a,\mathrm{wls}}$, so OLS and TLS are asymptotically equivalent and WLS has the smallest asymptotic variance.

Load-bearing premise

The optimality of the weighted least-squares estimator rests on an imported theorem from earlier work, not re-derived here, which says that replacing the true pole-coefficients in the optimal weighting by any consistent estimate does not change the asymptotic best performance.

Editorial extensions

If this is right

When the ratio $\hat\kappa = \hat\sigma_{n_x}/\hat\sigma^+_{n_x}$ is large, the TLS range-space estimate is expected to beat the OLS null-space estimate; when $\hat\delta = \hat\sigma^+_{n_x} - \hat\sigma_{n_x+1}$ is small, the TLS problem is ill-conditioned and the null-space method is the safer choice.
OLS and TLS realizations are asymptotically equivalent under the stated assumptions, so their finite-sample differences stem from conditioning and noise structure rather than from a fundamental statistical advantage.
The two-step procedure that first obtains an OLS or TLS estimate and then refines it with the weighted least-squares step yields the smallest asymptotic variance among the three methods.
The results explain why many subspace identification algorithms are suboptimal and case-dependent, and they point toward a design principle for asymptotically efficient subspace identification.
The ordering $P_{a,\mathrm{ols}} = P_{a,\mathrm{tls}} \succeq P_{a,\mathrm{wls}}$ shows that exploiting the Hankel structure through the weighting is the key to efficiency.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not stated as a procedure in the paper: the quantities $\hat\kappa$ and $\hat\delta$ could be computed from the noisy Hankel matrix before choosing an algorithm, giving a practical pre-test for whether the SVD range-space method or the null-space method will be more reliable.
The paper's analysis implies that a structured total least-squares estimator that respects the Toeplitz/Hankel structure, rather than the unweighted Frobenius correction used in ordinary TLS, should approach the WLS performance; the paper does not construct such an estimator.
For MIMO systems, the left-null-space parameterization is more involved, but if the block-Hankel extension preserves the weighting form $W(a)$, the variance ordering should carry over; this is a testable extension of the present SISO analysis.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a genuinely useful paper that demystifies a recurring puzzle in subspace identification, but Theorem 5.2's covariance formula is wrong as written and needs fixing before the optimality claim can stand.

What's new and good: The identification of Kung's range-space method as a TLS solution and NUSBR as an OLS solution is clean and, as far as I know, not in earlier literature. Theorem 3.1 is correct and bridges the two prototype algorithms in a way that makes the case-dependent accuracy understandable. The sensitivity analysis with the kappa/delta conditions is a practical contribution: it explains why one method wins depending on conditioning, and the simulations back up the qualitative story. The paper is well-organized and the literature coverage, including structured TLS and WNSF, is honest.

Soft spots, in order of severity. First, Theorem 5.2 is internally inconsistent. The linearization in Appendix IV.C gives

sqrt(N)(a_tls - a) roughly -sqrt(N) g_tilde T(a) H+^T (H+ H+^T)^{-1},

whose covariance is A^{-1} H+ W^{-1} H+^T A^{-1} with A = H+ H+^T. The stated formula (H+)^dagger W^{-1} ((H+)^dagger)^T is a different object, dimensionally (p+1)x(p+1) rather than nx x nx, with H+ and H+^T in the wrong order. The proof of P_ols >= P_wls in the same appendix uses the correct expression, so the theorem statement and its proof contradict each other. The qualitative WLS-optimality inequality may survive after correction since it is essentially a Gauss-Markov statement, but as written the theorem is wrong.

Second, the claim that plugging a consistent estimate into W(a) preserves asymptotic optimality is imported from the authors' own WNSF paper [24], and the conditions are not re-verified in the growing-n setting of Assumption 5.1. That is reliance rather than circularity, but it makes the paper's central claim less self-contained than it appears.

Third, the OLS-versus-TLS preference conditions rest on the approximation ||tilde H|| roughly ||tilde H+|| and on simulations. The approximation is reasonable when f = nx, and the paper is transparent about it, but the boundary between prefer OLS and prefer TLS is not a theorem. Minor: no code or data to reproduce the experiments.

Bottom line: For someone working on subspace identification or approximate realization, this is worth reading and worth citing for the OLS/TLS equivalence and the sensitivity analysis. But the statistical optimality claim should not be taken as established until Theorem 5.2 is corrected. I would send it to review and ask for a revision; the conceptual core is likely right and the fix looks straightforward.

Referee Report

3 major / 7 minor

Summary. The paper studies the approximate state-space realization problem from noisy Markov parameters. It shows that Kung's range-space-based realization (RASBR) is exactly a total least-squares (TLS) solution, while the recently proposed null-space-based realization (NUSBR) is an ordinary least-squares (OLS) solution (Section III, Theorem 3.1). It then analyzes sensitivity to argue when one method outperforms the other, proposes a weighted least-squares (WLS) estimator as the 'optimal' realization (Section IV, Eq. (59)), and provides consistency and asymptotic normality results (Section V, Theorems 5.1 and 5.2), claiming P_a,ols = P_a,tls ≽ P_a,wls. The main statistical claims are backed by proofs in the appendices, with numerical experiments illustrating the finite-sample behavior.

Significance. If the main claims are correct, the paper offers a genuinely useful unification of two prototype realization algorithms, explaining the observed case-dependent performance of subspace methods and pointing toward a principled weighted approach. The explicit equivalence between Kung's method and TLS is valuable, as is the framing of the null-space method as OLS. The paper also makes falsifiable predictions about when TLS or OLS should be preferred, supported by simulations, and provides a statistical analysis including consistency and asymptotic normality. A notable strength is that the TLS proofs are given in detail in the appendices. However, the central variance comparison in Theorem 5.2 is currently marred by a dimensionally inconsistent formula, and the optimality claim for WLS relies on imported results whose conditions are not verified here, so the contribution cannot be accepted in its present form.

major comments (3)

[Section V, Theorem 5.2, Eq. (64a)] The covariance formula in Theorem 5.2 is inconsistent with the paper's own linearization in Appendix IV.C. Since H+_nxp in Eq. (13) is n_x × (p+1) with full row rank, (H+_nxp)^† = H+^T (H+ H+^T)^{-1}, so (H+)^† W^{-1}(a) ((H+)^†)^T is a (p+1) × (p+1) matrix, whereas P_a,ols and P_a,tls must be n_x × n_x. The linearization in Appendix IV.C gives \tilde a_tls ≈ -\tilde g_n T(a) H+^T (H+ H+^T)^{-1}, whose asymptotic covariance is (H+ H+^T)^{-1} H+ W^{-1}(a) H+^T (H+ H+^T)^{-1}; this is exactly the sandwich form used in the proof of Eq. (65) in the same appendix. The theorem statement should therefore use P = ((H+)^†)^T W^{-1}(a) (H+)^†, not the transposed product. Because Theorem 5.2 is the load-bearing statistical claim of the paper, this requires correction.
[Section IV.B and Theorems 5.1–5.2] The proofs of consistency and asymptotic normality for OLS and WLS are delegated to [24] with the statement 'equivalent to the statistical analysis in [24]', and the plug-in property stated in Remark 8 — that replacing a in W(a) by a consistent estimate preserves asymptotic optimality — is also imported from [24] without verification. The conditions of the WNSF theorem in [24] are not checked under the growing-n regime of Assumption 5.1, and the theorem is not stated. Since the optimality of \hat a_wls is a central contribution, the authors should either state the precise result from [24] and verify its hypotheses in this setting, or provide the missing argument.
[Section IV.B and Abstract] The paper calls \hat a_wls 'the optimal realization' (abstract and Section IV.B), but Theorem 5.2 only establishes that its asymptotic covariance is no larger than those of the OLS and TLS estimators. No lower bound over a larger class of estimators is proved, and the regressor \hat H+ in Eq. (59) is itself noisy, so the standard BLUE optimality for fixed regressors does not automatically apply. The authors should either prove asymptotic efficiency within a clearly defined class or qualify the optimality claim to the class of least-squares estimators considered in the paper.

minor comments (7)

[Section III.B, Eqs. (29) and (34)] The summation index in both equations is written as 'i=i'; it should be 'i=1'.
[Section IV.B, Eq. (54)] The vector g_n is defined as [g_1 g_2 ... g_n], but the Markov parameters are indexed from g_0 in Eq. (1); this should be [g_0 g_1 ... g_{n-1}] to be consistent with the Hankel matrix construction.
[Section IV.B, Eq. (59)] The subscripts in \hat H^-_{n×n} and \hat H^+_{n×n} are inconsistent with the H_{nxp} notation used elsewhere; if the intended subscripts are n_x and p, please correct them.
[Experiment 1] '200 Monto Carlo trails' should be '200 Monte Carlo trials'.
[Section III.C, Eq. (44)] The approximation \|\tilde H_nxp\| ≈ \|\tilde H+_nxp\| is introduced without an error bound; a sentence justifying its accuracy relative to the row length would strengthen the heuristic.
[Figure 3] The six panels in Figure 3 are not labeled in the caption; please identify which panel corresponds to which group in Table II.
[Appendix I, proof of Theorem 3.1] The step from the TLS solution (32) satisfying [\hat a_tls 1]\hat O_nx = 0 to the similarity between \hat A_tls and \hat A_R is compressed; please spell out that the unique left null vector of \hat O_nx gives the characteristic coefficients of \hat A_R.

Circularity Check

1 steps flagged · score 4.0 of 10

OLS/TLS equivalence is derived in-paper, but the central WLS-optimality claim is inherited from the authors' own WNSF paper [24] via load-bearing self-citation; no equation-level circularity found.

self citation load bearing [Section IV.B (Eq. (59)) and Section V.B, Proof of Theorem 5.2]
"Although the optimal weighting W(a) depends on the true value of a, as demonstrated in [24], replacing a in W(a) with its estimate aols or atls will not affect the asymptotic optimality of awls. ... The proof for the asymptotic normality of aols and awls is equivalent to the statistical analysis in [24]."

The paper's advertised optimality result (WLS has smallest asymptotic variance, Theorem 5.2) rests on two statements that are not derived here: (i) the plug-in property that replacing a in W(a) preserves asymptotic optimality, and (ii) the asymptotic normality/covariance formulas for the OLS and WLS estimates. Both are referred to [24], a prior paper by three of the authors (Galrinho, Rojas, Hjalmarsson). The paper does not verify the high-level conditions of the imported WNSF theorem for the growing-n Hankel setting beyond Assumptions 5.1-5.3. Thus the central statistical claim is inherited from the authors' own prior work rather than established by the present derivation; this is load-bearing self-citation, though not an equation-level identity with the inputs.

full rationale

The paper's main structural result is self-contained: NUSBR is explicitly posed as the OLS problem (26)-(27), and RASBR is shown to equal the TLS solution (32)-(35) via the paper's own Theorem 3.1 proof and Appendix I, with no input-output circularity. The sensitivity comparisons (Lemmas 1-4) use external results [30], [35], [37]. The WLS estimator itself is constructed from the residual covariance (56)-(58) in a standard way. The circularity risk is concentrated in the statistical optimality claim: the plug-in property and the OLS/WLS asymptotic normality are delegated to [24], which shares authors with this paper. That makes part of the central claim self-citation load-bearing (score 4), but it is reliance on a previously published theorem rather than a fitted parameter renamed as a prediction. The reviewer-flagged inconsistency between Eq. (64a) and the Appendix IV.C linearization is an internal correctness problem, not a circularity pattern; it does not affect this score. No step reduces to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's central claims rest mainly on standard linear systems theory, on classical TLS perturbation bounds, and on the authors' prior WNSF results. No free parameters are fitted to make the theory work, and no new entities are introduced. The most important unproved input is the WNSF optimality theorem imported from [24].

assumptions (6)

domain assumption Assumption 1.1: the true LTI system is stable and minimal (rho(A) < 1, (A,B) controllable, (A,C) observable).
Used throughout to guarantee rank(H_nxp) = nx and the existence of a minimal realization up to similarity.
domain assumption Assumptions 5.1-5.3 on the Markov parameter estimates: consistency, asymptotic normality, and a growth rate n(N) with n^{4+delta}/N -> 0 and vanishing tail.
Theorems 5.1 and 5.2 are stated under these conditions; they are not proved for all estimators but justified for FIR and ARX estimates by standard theory [40].
domain assumption Asymptotic optimality of the WNSF weighting (from [24], with two overlapping authors).
The paper's central 'optimal WLS' claim depends on the WNSF theorem that replacing the true a in W(a) by a consistent estimate preserves asymptotic optimality; this is not proved in the current paper.
standard math Classical TLS sensitivity lemmas from Golub and Van Loan [30] and Van Huffel and Vandewalle [37] (Lemmas 2, 3 and the sin theta inequalities in Lemma 1).
The preference conditions for OLS versus TLS are built on these external perturbation bounds.
standard math Cayley-Hamilton and the Hankel factorization H_nxp = O_nx C_p (standard linear systems theory).
Used to parameterize the left null space by characteristic polynomial coefficients in Eqs. (11)-(14).
domain assumption Generic full-rank and non-degeneracy assumptions: \hat O+ invertible, sigma_hat_nx > sigma_hat_nx+1, and \hat H+ W \hat H+^T invertible.
Theorem 3.1 and the TLS solution (32)-(33) require isolated smallest singular value and invertible sub-blocks; near-degenerate cases are discussed only heuristically.

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Pith. "Pith review of Range Space or Null Space: Least-Squares Methods for the Realization Problem." pith.science (2026). https://pith.science/paper/Z5RLJXUS

@misc{pith2026250519639,
  author       = {Pith},
  title        = {Pith review of: Range Space or Null Space: Least-Squares Methods for the Realization Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z5RLJXUS}},
  note         = {Machine review of arXiv:2505.19639}
}

read the original abstract

This contribution revisits the classical approximate realization problem, which involves determining matrices of a state-space model based on estimates of a truncated series of Markov parameters. A Hankel matrix built up by these Markov parameters plays a fundamental role in this problem, leveraging the fact that both its range space and left null space encode critical information about the state-space model. We examine two prototype realization algorithms based on the Hankel matrix: the classical range-space-based (SVD-based) method and the more recent null-space-based method. It is demonstrated that the range-space-based method corresponds to a total least-squares solution, whereas the null-space-based method corresponds to an ordinary least-squares solution. By analyzing the differences in sensitivity of the two algorithms, we determine the conditions when one or the other realization algorithm is to be preferred, and identify factors that contribute to an ill-conditioned realization problem. Furthermore, recognizing that both methods are suboptimal, we argue that the optimal realization is obtained through a weighted least-squares approach. A statistical analysis of these methods, including their consistency and asymptotic normality is also provided.

Figures

Figures reproduced from arXiv: 2505.19639 by the authors.

**Figure 1.** FITs of impulse responses from 200 Monte Carlo trials: System 1 (◦) and System 2 (N), and the solid line is a bisector line. (2) We argue that the optimal realization is a weighted null space fitting problem, and provide a weighted least-squares (WLS) solution for solving it. (3) We provide a statistical analysis for these methods, including consistency and asymptotic normality. Our results suggest that under mild a… view at source ↗

**Figure 2.** FITs of impulse responses for three experiment groups in Table I: A random system (N), and the solid line is a bisector line. ratio κˆ is mainly influenced by the length and the orientation of H− nxp w.r.t. the singular vectors of H+ nxp . Here we briefly summarize their impact. Before proceeding, we mention that, similar to the SVD of Hˆ+ nxp and Hˆ nxp in (28) and (31), the corresponding SVD of the true matrices H… view at source ↗

**Figure 3.** FITs of impulse responses for six experiment groups in Table II: A random system (N), and the solid line is a bisector line. we evaluate the performance of TLS and OLS using the corrupted Markov parameters of these random systems, where independent Gaussian noise with the variance of 0.5 is added to the true Markov parameters. We conduct six groups of experiments, and run 200 Monte Carlo trials in each group. The p… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FITs of impulse responses from 200 Monte Carlo trials: System 1 (◦, OLS VS WLS) and System 2 (N, TLS VS WLS), and the solid line is a bisector line. of aˆwls. This results in a two-step least squares method: the first step obtains an initial solution using TLS or OLS, …

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Reference graph

Works this paper leans on

45 extracted references · 42 canonical work pages · cited by 2 Pith papers

[24]

Parametr ic identiﬁcation using weighted null-space ﬁtting,

M. Galrinho, C. R. Rojas, and H. Hjalmarsson, “Parametr ic identiﬁcation using weighted null-space ﬁtting,” IEEE Trans. Autom. Control , vol. 64, no. 7, pp. 2798–2813, 2018. 12 IEEE TRANSACTIONS AND JOURNALS TEMPLA TE

work page 2018
[1]

Approximate realization of noisy linear systems: the Hankel and Page matrix approach,

P . van den Hof, “Approximate realization of noisy linear systems: the Hankel and Page matrix approach,” Master’s thesis, the Neth erlands: Eindhoven University of Technology, 1983

work page 1983
[2]

Kailath, Linear Systems

T. Kailath, Linear Systems . Prentice-Hall, 1980

work page 1980
[3]

The four fundamental subspaces: 4 lines,

G. Strang, “The four fundamental subspaces: 4 lines,” Massachusetts Institute of Technology , 2016. [Online]. Available: https://web.mit.edu/18.06/www/Essays/newpaper ver3.pdf

work page 2016
[4]

Effective construction of li near state- variable models from input/output functions,

B. L. Ho and R. E. K´ alm´ an, “Effective construction of li near state- variable models from input/output functions,” Automatisierungstechnik, vol. 14, no. 1-12, pp. 545–548, 1966

work page 1966
[5]

A new identiﬁcation and model reduction alg orithm via singular value decomposition,

S.-Y . Kung, “A new identiﬁcation and model reduction alg orithm via singular value decomposition,” in Proc. Asilomar Conf. Circuits, Syst. Comput., Paciﬁc Grove, USA, 1978

work page 1978
[6]

An eigensystem realization algorithm for modal parameter identiﬁcation and model reduction,

J.-N. Juang and R. S. Pappa, “An eigensystem realization algorithm for modal parameter identiﬁcation and model reduction,” J. Guid. Control Dyn., vol. 8, no. 5, pp. 620–627, 1985

work page 1985
[7]

Canonical variate analysis in identiﬁc ation, ﬁltering and adaptive control,

W. E. Larimore, “Canonical variate analysis in identiﬁc ation, ﬁltering and adaptive control,” in Proc. IEEE Conf. Decis. Control , Honolulu, HI, USA, 1990

work page 1990

Show all 45 references

[8]

N4SID: Subspace algorit hms for the identiﬁcation of combined deterministic-stochast ic systems,

P . V an Overschee and B. De Moor, “N4SID: Subspace algorit hms for the identiﬁcation of combined deterministic-stochast ic systems,” Automatica, vol. 30, no. 1, pp. 75–93, 1994

1994
[9]

Subspace model identiﬁcat ion part I: the output-error state-space model identiﬁcation class of algorithm,

M. V erahegen and P . Dewilde, “Subspace model identiﬁcat ion part I: the output-error state-space model identiﬁcation class of algorithm,” Int. J. Control, vol. 56, pp. 1187–1210, 1992

1992
[10]

A novel subspace identiﬁ cation approach with enforced causal models,

S. J. Qin, W. Lin, and L. Ljung, “A novel subspace identiﬁ cation approach with enforced causal models,” Automatica, vol. 41, no. 12, pp. 2043–2053, 2005

2005
[11]

Subspace identiﬁcation and ARX modeling,

M. Jansson, “Subspace identiﬁcation and ARX modeling, ” in Proc. 13th IF AC Symp. Syst. Identiﬁcation , Netherlands, 2003

2003
[12]

The role of vector autoregressive modeling in predictor- based subspace identiﬁcation,

A. Chiuso, “The role of vector autoregressive modeling in predictor- based subspace identiﬁcation,” Automatica, vol. 43, no. 6, pp. 1034– 1048, 2007

2007
[13]

A subspace based instrumen tal variable method for state-space system identiﬁcation,

B. Ottersten and M. Viberg, “A subspace based instrumen tal variable method for state-space system identiﬁcation,” IF AC Proceedings V ol- umes, vol. 27, no. 8, pp. 449–454, 1994

1994
[14]

Analysis of s tate space system identiﬁcation methods based on instrumental variab les and subspace ﬁtting,

M. Viberg, B. Wahlberg, and B. Ottersten, “Analysis of s tate space system identiﬁcation methods based on instrumental variab les and subspace ﬁtting,” Automatica, vol. 33, no. 9, pp. 1603–1616, 1997

1997
[15]

A subspace ﬁtting method for identiﬁcation of linear state-space models,

A. Swindlehust, R. Roy, B. Ottersten, and T. Kailath, “A subspace ﬁtting method for identiﬁcation of linear state-space models,” IEEE Trans. Auto. Control, vol. 40, no. 2, pp. 311–316, 1995

1995
[16]

A linear regression approa ch to state- space subspace system identiﬁcation,

M. Jansson and B. Wahlberg, “A linear regression approa ch to state- space subspace system identiﬁcation,” Signal Process. , vol. 52, no. 2, pp. 103–129, 1996

1996
[17]

E. J. Hannan and M. Deistler, The Statistical Theory of Linear Systems . SIAM, 2012

2012
[18]

Regalia, Adaptive IIR ﬁltering in signal processing and control

P . Regalia, Adaptive IIR ﬁltering in signal processing and control . Routledge, 2018

2018
[19]

Weighted null space ﬁtting (W NSF): A link between the prediction error method and subspace identiﬁca tion,

J. He and H. Hjalmarsson, “Weighted null space ﬁtting (W NSF): A link between the prediction error method and subspace identiﬁca tion,” arXiv preprint arXiv:2411.00506, 2024

2024 arXiv
[20]

Least squares realization of LTI models is a n eigenvalue problem,

B. De Moor, “Least squares realization of LTI models is a n eigenvalue problem,” in 18th European Control Conf. (ECC) , 2019, pp. 2270–2275

2019
[21]

Least squares optimal realisation of autonomous L TI systems is an eigenvalue problem,

——, “Least squares optimal realisation of autonomous L TI systems is an eigenvalue problem,” Communications in Information and Systems , vol. 20, no. 2, pp. 163–207, 2020

2020
[22]

Realizing LTI models by identifying characteristic param eters using least squares optimization,

T. Nicolai, M. Haring, E. I. Grøtli, J. T. Gravdahl, and J . Reger, “Realizing LTI models by identifying characteristic param eters using least squares optimization,” in European Control Conf. (ECC) , 2023, pp. 1–6

2023
[23]

A new approach t o learning linear dynamical systems,

A. Bakshi, A. Liu, A. Moitra, and M. Y au, “A new approach t o learning linear dynamical systems,” in Proc. 55th Annual ACM Sympo. Theory Computing, 2023, pp. 335–348

2023
[25]

Application of structured total least squares for system i dentiﬁcation and model reduction,

I. Markovsky, J. C. Willems, S. V an Huffel, B. De Moor, an d R. Pintelon, “Application of structured total least squares for system i dentiﬁcation and model reduction,” IEEE Trans. Auto. Control , vol. 50, no. 10, pp. 1490–1500, 2005

2005
[26]

The singular value decomposition and long a nd short spaces of noisy matrices,

B. De Moor, “The singular value decomposition and long a nd short spaces of noisy matrices,” IEEE Trans. Signal Process. , vol. 41, no. 9, pp. 2826–2838, 1993

1993
[27]

Structured total least squares and L2 approximati on problems,

——, “Structured total least squares and L2 approximati on problems,” Linear Algebra Appl. , vol. 188, pp. 163–205, 1993

1993
[28]

Overview of total least -squares methods,

I. Markovsky and S. V an Huffel, “Overview of total least -squares methods,” Signal Process. , vol. 87, no. 10, pp. 2283–2302, 2007

2007
[29]

Total least squares for afﬁnely structured matrices and the noisy realization problem,

B. De Moor, “Total least squares for afﬁnely structured matrices and the noisy realization problem,” IEEE Trans. Signal Process. , vol. 42, no. 11, pp. 3104–3113, 2002

2002
[30]

An analysis of the total le ast squares problem,

G. H. Golub and C. F. V an Loan, “An analysis of the total le ast squares problem,” SIAM J. Numer . Anal. , vol. 17, no. 6, pp. 883–893, 1980

1980
[31]

V an Huffel and J

S. V an Huffel and J. V andewalle, The Total Least Squares Problem: Computational Aspects and Analysis . SIAM, 1991, vol. 9

1991
[32]

G. H. Golub and C. F. V an Loan, Matrix Computations . 3rd Edition, The Johns Hopkins University Press, 2013

2013
[33]

On the role of future horizon in cl osed-loop subspace identiﬁcation,

S. J. Qin and L. Ljung, “On the role of future horizon in cl osed-loop subspace identiﬁcation,” IF AC Proceedings V olumes, vol. 39, no. 1, pp. 1080–1084, 2006

2006
[34]

Some insights on the choice of the future hor izon in closed- loop CCA-type subspace algorithms,

A. Chiuso, “Some insights on the choice of the future hor izon in closed- loop CCA-type subspace algorithms,” in 2007 Am. Control Conf. , 2007, pp. 840–845

2007
[35]

Perturbation bounds in connection with singular value decomposition,

P .- ˚A. Wedin, “Perturbation bounds in connection with singular value decomposition,” BIT Numer . Math., vol. 12, pp. 99–111, 1972

1972
[36]

A critica l view on benchmarks based on randomly generated systems,

C. R. Rojas, P . E. V alenzuela, and R. A. Rojas, “A critica l view on benchmarks based on randomly generated systems,” in Proc. 17th IF AC Symp. Syst. Identiﬁcation , Beijing, China, 2015

2015
[37]

On the accuracy of tota l least squares and least squares techniques in the presence of errors on all data,

S. V an Huffel and J. V andewalle, “On the accuracy of tota l least squares and least squares techniques in the presence of errors on all data,” Automatica, vol. 25, no. 5, pp. 765–769, 1989

1989
[38]

An overview of subspace identiﬁcation,

S. J. Qin, “An overview of subspace identiﬁcation,” Comput. Chem. Eng., vol. 30, no. 10-12, pp. 1502–1513, 2006

2006
[39]

A weighted l east-squares method for parameter estimation in structured models,

M. Galrinho, C. Rojas, and H. Hjalmarsson, “A weighted l east-squares method for parameter estimation in structured models,” in Proc. IEEE Conf. Decis. Control , Los Angeles, California, USA, 2014

2014
[40]

Asymptotic properties of the least-squares method for estimating transfer functions and disturbance s pectra,

L. Ljung and B. Wahlberg, “Asymptotic properties of the least-squares method for estimating transfer functions and disturbance s pectra,” Adv. Appl. Probab., vol. 24, no. 2, pp. 412–440, 1992

1992
[41]

Revisiting Ho-Kalman-based syst em identi- ﬁcation: Robustness and ﬁnite-sample analysis,

S. Oymak and N. Ozay, “Revisiting Ho-Kalman-based syst em identi- ﬁcation: Robustness and ﬁnite-sample analysis,” IEEE Trans. Autom. Control, vol. 67, no. 4, pp. 1914–1928, 2021

1914
[42]

Finite sample analysis of s tochastic system identiﬁcation,

A. Tsiamis and G. J. Pappas, “Finite sample analysis of s tochastic system identiﬁcation,” in IEEE Conf. Decis. Control , 2019, pp. 3648–3654

2019
[43]

Perturbation theory for pseudo-inverses,

P .- ˚A. Wedin, “Perturbation theory for pseudo-inverses,” BIT Numerical Mathematics, vol. 13, pp. 217–232, 1973

1973
[44]

On the distribution of the left singular vectors of a random matrix and its applications,

E. Bura and R. Pfeiffer, “On the distribution of the left singular vectors of a random matrix and its applications,” Stat. Probab. Lett. , vol. 78, no. 15, pp. 2275–2280, 2008. APPENDIX I PROOF OF THEOREM 3.1 Notice that the extended observability matrix ˆOnx in ( 35) satisﬁes...

2008
[45]

a1.png" is available in

satisﬁes [ ˆatls 1 ] ˆOnx = 0 . Moreover, as dim ( K( ˆO ⊤ nx) ) = 1 , the solution to ( I.2) is unique, so we conclude that the solution ( 35) to the equation ( I.1) is the same as the the solution ( 32) to the equation ( I.2), up to a similarity transformation. ■ APPENDIX II...

arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [24]

Parametr ic identiﬁcation using weighted null-space ﬁtting,

M. Galrinho, C. R. Rojas, and H. Hjalmarsson, “Parametr ic identiﬁcation using weighted null-space ﬁtting,” IEEE Trans. Autom. Control , vol. 64, no. 7, pp. 2798–2813, 2018. 12 IEEE TRANSACTIONS AND JOURNALS TEMPLA TE

work page 2018

[2] [1]

Approximate realization of noisy linear systems: the Hankel and Page matrix approach,

P . van den Hof, “Approximate realization of noisy linear systems: the Hankel and Page matrix approach,” Master’s thesis, the Neth erlands: Eindhoven University of Technology, 1983

work page 1983

[3] [2]

Kailath, Linear Systems

T. Kailath, Linear Systems . Prentice-Hall, 1980

work page 1980

[4] [3]

The four fundamental subspaces: 4 lines,

G. Strang, “The four fundamental subspaces: 4 lines,” Massachusetts Institute of Technology , 2016. [Online]. Available: https://web.mit.edu/18.06/www/Essays/newpaper ver3.pdf

work page 2016

[5] [4]

Effective construction of li near state- variable models from input/output functions,

B. L. Ho and R. E. K´ alm´ an, “Effective construction of li near state- variable models from input/output functions,” Automatisierungstechnik, vol. 14, no. 1-12, pp. 545–548, 1966

work page 1966

[6] [5]

A new identiﬁcation and model reduction alg orithm via singular value decomposition,

S.-Y . Kung, “A new identiﬁcation and model reduction alg orithm via singular value decomposition,” in Proc. Asilomar Conf. Circuits, Syst. Comput., Paciﬁc Grove, USA, 1978

work page 1978

[7] [6]

An eigensystem realization algorithm for modal parameter identiﬁcation and model reduction,

J.-N. Juang and R. S. Pappa, “An eigensystem realization algorithm for modal parameter identiﬁcation and model reduction,” J. Guid. Control Dyn., vol. 8, no. 5, pp. 620–627, 1985

work page 1985

[8] [7]

Canonical variate analysis in identiﬁc ation, ﬁltering and adaptive control,

W. E. Larimore, “Canonical variate analysis in identiﬁc ation, ﬁltering and adaptive control,” in Proc. IEEE Conf. Decis. Control , Honolulu, HI, USA, 1990

work page 1990