Subspace Methods for System Identification

Communications and Control Engineering

Volumn , Issue 9781852339814, 2005, Pages 1-392

  Katayama, Tohru ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

CONTROL ENGINEERING; ENGINEERING;

SUB-SPACE METHODS;

INDUSTRIAL ENGINEERING;

EID: 85040751730     PISSN: 01785354     EISSN: 21977119     Source Type: Book Series    
DOI: None     Document Type: Chapter

References (186)

1. H. Akaike, “Stochastic theory of minimal realization,” IEEE Trans. Automatic Control, vol. AC-19, no. 6, pp. 667–674, 1974.
2. H. Akaike, “Markovian representation of stochastic processes by canonical variables,” SIAM J. Control, vol. 13, no. 1, pp. 162–173, 1975.
3. H. Akaike, “Canonical correlation analysis of time series and the use of an information criterion,” In System Identification: Advances and Case Studies (R. Mehra and D. Lainiotis, eds.), Academic, pp. 27–96, 1976.
4. H. Akaike, “Comments on ‘On model structure testing in system identification’,” Int. J. Control, vol. 27, no. 2, pp. 323–324, 1978.
5. U. M. Al-Saggaf and G. F. Franklin, “An error bound for a discrete reduced order model of a linear multivariable system,” IEEE Trans. Automatic Control, vol. 32, no. 9, pp. 815–819, 1987.
6. U. M. Al-Saggaf and G. F. Franklin, “Model reduction via balanced realizations: An extension and frequency weighting techniques,” IEEE Trans. Automatic Control, vol. 33, no. 7, pp. 687–692, 1988.
7. B. D. O. Anderson, “A system theory criterion for positive real matrices,” SIAM J. Control, vol. 5, no. 2, pp. 171–182, 1967.
8. B. D. O. Anderson, “An algebraic solution to the spectral factorization problem,” IEEE Trans. Automatic Control, vol. AC-12, no. 4, pp. 410–414, 1967.
9. B. D. O. Anderson, “The inverse problem of stationary covariance generation,” J. Statistical Physics, vol. 1, no. 1, pp. 133–147, 1969.
10. B. D. O. Anderson, K. L. Hitz and N. D. Diem, “Recursive algorithm for spectral fac-torization,” IEEE Trans. Circuits Systems, vol. CAS-21, no. 6, pp. 742–750, 1974.
11. B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, 1979.
12. B. D. O. Anderson and M. R. Gevers, “Identifiability of linear stochastic systems operating under linear feedback,” Automatica, vol. 18, no. 2, pp. 195–213, 1982.
13. T. W. Anderson, The Statistical Analysis of Time Series, Wiley, 1971.
14. T. W. Anderson, An Introduction to Multivariable Statistical Analysis (2nd ed.), Wiley, 1984.
15. M. Aoki, State Space Modeling of Time Series (2nd ed.), Springer, 1990.
16. M. Aoki and A. M. Havenner (eds.), Applications of Computer Aided Time Series Modeling (Lecture Notes in Statistics 119), Springer, 1997
17. K. J. Åström and T. Bohlin, “Numerical identification of linear dynamic systems for normal operating records,” Proc. 2nd IFAC Symp. Theory of Self-Adaptive Systems, Ted-dington, pp. 96–111, 1965.
18. K. S. Arun and S. Y. Kung, “Balanced approximation of stochastic systems,” SIAM J. Matrix Analysis and Applications, vol. 11, no. 1, pp. 42–68, 1990.
19. D. Bauer and M. Jansson, “Analysis of the asymptotic properties of the MOESP type of subspace algorithms,” Automatica, vol. 36, no. 4, pp. 497–509, 2000.
20. S. Bittanti, A. J. Laub and J. C. Willems (eds.), The Riccati Equation, Springer, 1991.
21. A. Björck and G. H. Golub, “Numerical methods for computing angles between linear subspaces,” Mathematics of Computation, vol. 27, pp. 579–594, July 1973.
22. G. E. P. Box and G. M. Jenkins, Time Series Analysis - Forecasting and Control, Holden-Day, 1970.
23. R. S. Bucy, Lectures on Discrete Time Filtering, Springer, 1994.
24. P. E. Caines and C. W. Chan, “Feedback between stationary stochastic processes,” IEEE Trans. Automatic Control, vol. AC-20, no. 4, pp. 498–508, 1975.
25. P. E. Caines and C. W. Chan, “Estimation, identification and feedback,” In System Identification: Advances and Case Studies (R. Mehra and D. Lainiotis, eds.), Academic, pp. 349–405, 1976.
26. P. E. Caines, “Weak and strong feedback free processes,” IEEE Trans. Automatic Control, vol. AC-21, no. 5, pp. 737–739, 1976.
27. C. T. Chen, Linear System Theory and Design, Holt-Saunders, 1984.
28. H. -F. Chen, P. K. Kumar and J. H. van Schuppen, “On Kalman filtering for conditionally Gaussian systems with random matrices,” Systems and Control Letters, vol. 13, pp. 397–404, 1989.
29. A. Chiuso and G. Picci, “Subspace identification by orthogonal decomposition,” Proc. 14th IFAC World Congress, Beijing, vol. H, pp. 241–246, 1999.
30. A. Chiuso and G. Picci, “Some algorithmic aspects of subspace identification with in-puts,” Int. J. Applied Math. and Computer Science, vol. 11, no. 1, pp. 55–75, 2001.
31. A. Chiuso and G. Picci, “On the ill-conditioning of subspace identification with inputs,” Automatica, vol. 40, no. 4, pp. 575–589, 2004.
32. A. Chiuso and G. Picci, “Numerical conditioning and asymptotic variance of subspace estimates,” Automatica, vol. 40, no. 4, pp. 677–683, 2004.
33. T. Chonavel, Statistical Signal Processing, Springer, 2002.
34. C. T. Chou and M. Verhaegen, “Closed-loop identification using canonical correlation analysis,” Proc. 5th European Control Conference, Karlsruhe, F-162, 1999.
35. N. L. C. Chui and J. M. Maciejowski, “Realization of stable models with subspace meth-ods,” Automatica, vol. 32, no. 11, pp. 1587–1595, 1996.
36. R. V. Churchill and J. W. Brown, Complex Variables and Applications (4th ed.), McGraw-Hill, 1984.
37. B. Codrons, B. D. O. Anderson and M. Gevers, “Closed-loop identification with an unstable or nonminimum phase controller,” Automatica, vol. 38, no. 12, pp. 2127–2137, 2002.
38. A. Dahlén, A. Lindquist and J. Mari, “Experimental evidence showing that stochastic subspace identification methods may fail,” Systems and Control Letters, vol. 34, no. 2, pp. 302–312, 1998.
39. K. De Cock, Principal Angles in System Theory, Information Theory and Signal Processing, Ph.D. Thesis, Katholieke Universiteit Lueven, Belgium, 2002.
40. B. De Moor, “The singular value decomposition and long and short spaces of noisy matrices,” IEEE Trans. Signal Processing, vol. 41, no. 9, pp. 2826–2838, 1993.
41. B. De Moor, M. Moonen, L. Vandenberghe and J. Vandewalle, “Identification of linear state space models with SVD using canonical correlation analysis,” In Singular Value Decomposition and Signal Processing (E. Deprettere, ed.), North-Holland, pp. 161–169, 1988.
42. U. B. Desai and D. Pal, “A transformation approach to stochastic model reduction,” IEEE Trans. Automatic Control, vol. AC-29, no. 12, pp. 1097–1100, 1984.
43. U. B. Desai, D. Pal and R. D. Kirkpatrick, “A realization approach to stochastic model reduction,” Int. J. Control, vol. 42, no. 4, pp. 821–838, 1985.
44. J. L. Doob, Stochastic Processes, Wiley, 1953.
45. P. Faurre, “Réalisations markoviennes de processus stationnaires,” Technical Report No. 13, INRIA, March 1973.
46. P. L. Faurre, “Stochastic realization algorithms,” In System Identification: Advances and Case Studies (R. Mehra and D. Lainiotis, eds.), Academic, pp. 1–25, 1976.
47. P. Faurre, M. Clerget and F. Germain, Opérateurs Rationnels Positifs, Dunod, 1979.
48. U. Forssell and L. Ljung, “Closed-loop identification revisited,” Automatica, vol. 35, no. 7, pp. 1215–1241, 1999.
49. U. Forssell and L. Ljung, “A projection method for closed-loop identification,” IEEE Trans. Automatic Control, vol. 45, no. 11, pp. 2101–2106, 2000.
50. S. Fujishige, H. Nagai and Y. Sawaragi, “System-theoretic approach to model reduction and system-order determination,” Int. J. Control, vol. 22, no. 6, pp. 807–819, 1975.
51. F. R. Gantmacher, The Theory of Matrices (2 vols.), Chelsea, 1959.
52. I. M. Gel’fand and A. M. Yaglom, “Calculation of the amount of information about a random function contained in another such function,” American Math. Soc. Transl., vol. 12, pp. 199–246, 1959.
53. M. Gevers and B. D. O. Anderson, “On joint stationary feedback-free stochastic pro-cesses,” IEEE Trans. Automatic Control, vol. AC-27, no. 2, pp. 431–436, 1982.
54. M. Gevers and V. Wertz, “Uniquely identifiable state-space and ARMA parametrizations for multivariable linear systems,” Automatica, vol. 20, no. 3, pp. 333–347, 1984.
55. M. Gevers, “A personal view on the development of system identification,” Proc. 13th IFAC Symp. System Identification, Rotterdam, pp. 773–784, 2003.
56. E. G. Gilbert, “Controllability and observability in multivariable control systems,” SIAM J. Control, vol. 1, no. 2, pp. 128–151, 1963.
57. M. Glover and J. C. Willems, “Parametrizations of linear dynamical systems: Canonical forms and identifiability,” IEEE Trans. Automatic Control, vol. AC-19, no. 6, pp. 640–646, 1974.
58. I. Goethals, T. Van Gestel, J. Suykens, P. Van Dooren and B. De Moor, “Identification of positive real models in subspace identification by using regularization,” IEEE Trans. Automatic Control, vol. 48, no. 10, pp. 1843–1847, 2003.
59. G. H. Golub and C. F. van Loan, Matrix Computations (3rd ed.), The Johns Hopkins University Press, 1996.
60. G. H. Golub and H. A. van der Vorst, “Eigenvalue computation in the 20th century,” J. Computational and Applied Mathematics, vol.123, pp. 35–65, 2000.
61. G. C. Goodwin and R. L. Payne, Dynamic System Identification: Experiment Design and Data Analysis, Academic, 1977.
62. B. Gopinath, “On the identification of linear time-invariant systems from input-output data,” Bell Systems Technical Journal, vol. 48, no. 5, pp. 1101–1113, 1969.
63. C. W. J. Granger, “Economic processes involving feedback,” Information and Control, vol. 6, pp. 28–48, 1963.
64. M. Green, “All-pass matrices, the positive real lemma, and unit canonical correlations between future and past,” J. Multivariate Analysis, vol. 24, pp. 143–154, 1988.
65. U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, Chelsea, 1958.
66. M. S. Grewal and A. P. Andrews, Kalman Filtering – Theory and Practice, Prentice-Hall, 1993.
67. R. Guidorzi, “Canonical structures in the identification of multivariable systems,” Automatica, vol. 11, no. 4, pp. 361–374, 1975.
68. E. J. Hannan and M. Deistler, The Statistical Theory of Linear Systems, Wiley, 1988.
69. E. J. Hannan and D. S. Poskit, “Unit canonical correlations between future and past,” Annals of Statistics, vol. 16, pp. 784–790, 1988.
70. F. R. Hansen, G. F. Franklin and R. Kosut, “Closed-loop identification via the fractional representation: Experiment design,” Proc. 1989 American Control Conference, Pittsburgh, pp. 1422–1427, 1989.
71. D. Hinrichsen and A. J. Prichard, “An improved error estimate for reduced-order models of discrete-time systems,” IEEE Trans. Automatic Control, vol. AC-35, no. 3, pp. 317–320, 1990.
72. B. L. Ho and R. E. Kalman, “Effective construction of linear state-variable models from input/output functions,” Regelungstechnik, vol. 14, no. 12, pp. 545–548, 1966.
73. R. A. Horn and C. A. Johnson, Matrix Analysis, Cambridge University Press, 1985.
74. H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educational Psychology, vol. 24, pp. 417–441 and pp. 498–520, 1933.
75. H. Hotelling, “Relations between two sets of variates,” Biometrica, vol. 28, pp. 321–377, 1936.
76. M. Jansson and B. Wahlberg, “On consistency of subspace methods for system identifi-cation,” Automatica, vol. 34, no. 12, pp. 1507–1519, 1998.
77. E. A. Jonckheere and J. W. Helton, “Power spectrum reduction by optimal Hankel norm approximation of the phase of the outer spectral factor,” IEEE Trans. Automatic Control, vol. AC-30, no. 12, pp. 1192–1201, 1985.
78. T. Kailath, “A view of three decades of linear filtering theory,” IEEE Trans. Information Theory, vol. IT-20, no. 2, pp. 146–181, 1974.
79. T. Kailath, Lectures on Linear Least-Squares Estimation, Springer, 1976.
80. T. Kailath, Linear Systems, Prentice-Hall, 1980
81. R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME, J. Basic Engineering, vol. 82D, no.1, pp. 34–45, 1960.
82. R. E. Kalman, “Mathematical description of linear dynamical systems,” SIAM J. Control, vol. 1, no. 2, pp. 152–192, 1963.
83. R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” In Generalized Inverses and Applications (Z. Nashed, ed.), pp. 111–121, Academic, 1976.
84. R. E. Kalman and R. S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME, J. Basic Engineering, vol. 83D, no.1, pp. 95–108, 1961.
85. R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, 1969.
86. T. Katayama, “Subspace-based system identification – A view from realization theory,” J. Systems, Control and Information Engineers, vol. 41, no. 9, pp. 380–387, 1997 (in Japanese).
87. T. Katayama, H. Kawauchi and G. Picci, “A subspace identification of deterministic part of state space model operating in closed loop,” Proc. 6th European Control Conference, Porto, pp. 2505–2510, 2001.
88. T. Katayama, H. Kawauchi and G. Picci, “Subspace identification of closed loop systems by stochastic realization,” CD-ROM Preprints 15th IFAC World Congress, Barcelona, Paper # T-Mo-M02-2, 2002.
89. T. Katayama, H. Kawauchi and G. Picci, “Subspace identification of closed loop systems by the orthogonal decomposition method,” Automatica, vol. 41, no. 5, pp. 863–872, 2005.
90. T. Katayama and G. Picci, “Realization of stochastic systems with exogenous inputs and subspace identification methods,” Automatica, vol. 35, no. 10, pp. 1635–1652, 1999.
91. T. Katayama and S. Sugimoto (eds.), Statistical Methods in Control and Signal Processing, Marcel Dekker, 1997.
92. T. Katayama, H. Tanaka and T. Enomoto, “A simple subspace identification method of closed-loop systems using orthogonal decomposition,” Preprints 16th IFAC World Congress, Prague, July 2005.
93. H. Kawauchi, A. Chiuso, T. Katayama and G. Picci, “Comparison of two subspace identification methods for combined deterministic-stochastic systems,” Proc. 31st ISCIE Symp. Stochastic Systems Theory and Applications, Yokohama, pp. 7–12, 1999.
94. V. Klema and A. J. Laub, “The singular value decomposition: Its computation and some applications,” IEEE Trans. Automatic Control, vol. AC-25, no. 2, pp. 164–176, 1980.
95. L. H. Koopmans, The Spectral Analysis of Time Series, Academic, 1974.
96. T. C. Koopmans (ed.), Statistical Inference in Dynamic Economic Models, Wiley, 1950.
97. V. Kucèra, “The discrete Riccati equation of optimal control,” Kybernetika, vol. 8, no. 5, pp. 430–447, 1972.
98. H. Kwakernaak and R. Sivan, Modern Signals and Systems, Prentice-Hall, 1991.
99. P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford Science Publications, 1995.
100. W. E. Larimore, “System identification, reduced-order filtering and modeling via canonical variate analysis,” Proc. 1983 American Control Conference, San Francisco, pp. 445–451, 1983.
101. W. E. Larimore, “Canonical variate analysis in identification, filtering, and adaptive control,” Proc. 29th IEEE Conference on Decision and Control, Honolulu, pp. 596–604, 1990.
102. W. E. Larimore, “The ADAPTx software for automated and real-time multivariable system identification,” Proc. 13th IFAC Symp. System Identification, Rotterdam, pp. 1496–1501, 2003.
103. A. J. Laub, “A Schur method for solving algebraic Riccati equations,” IEEE Trans. Automatic Control, vol. AC-24, no. 6, pp. 913–921, 1979.
104. M. J. Levin, “Estimation of a system pulse transfer function in the presence of noise,” IEEE Trans. Automatic Control, vol. 9, no. 2, pp. 229–235, 1964.
105. A. Lindquist and G. Picci, “A geometric approach to modelling and estimation of linear stochastic systems,” J. Math. Systems, Estimation and Control, vol. 1, no. 3, pp. 241–333, 1991.
106. A. Lindquist and G. Picci, “Canonical correlation analysis, approximate covariance extension, and identification of stationary time series,” Automatica, vol. 32, no. 5, pp. 709–733, 1996.
107. A. Lindquist and G. Picci, “Geometric methods for state space identification,” In Identification, Adaptation, Learning: The Science of Learning Models from Data (S. Bittanti and G. Picci, eds.), Springer, pp. 1–69, 1996.
108. Y. Liu and B. D. O. Anderson, “Singular perturbation approximation of balanced sys-tems,” Int. J. Control, vol. 50, no. 4, pp. 1379–1405, 1989.
109. L. Ljung, System Identification - Theory for The User (2nd ed.), Prentice-Hall, 1999.
110. L. Ljung and T. McKelvey, “Subspace identification from closed loop data,” Signal Processing, vol. 52, no. 2, pp. 209–216, 1996.
111. D. Luenberger, Optimization by Vector Space Methods, Wiley, 1969.
112. J. M. Maciejowski, “Parameter estimation of multivariable systems using balanced real-izations,” In Identification, Adaptation, Learning: The Science of Learning Models from Data (S. Bittanti and G. Picci, eds.), Springer, pp. 70–119, 1996.
113. T. McKelvey, H. Ackay and L. Ljung, “Subspace-based multivariable identification from frequency response data,” IEEE Trans. Automatic Control, vol. AC-41, no. 7, pp. 960–979, 1996.
114. T. McKelvey, A. Helmersson and T. Ribarits, “Data driven local coordinates for multivariable linear systems and their application to system identification,” Automatica, vol. 40, no. 9, pp. 1629–1635, 2004.
115. P. Masani, “The prediction theory of multivariable stochastic processes - III,” Acta Mathematica, vol. 104, pp. 141–162, 1960.
116. R. K. Mehra and C. T. Leondes (eds.), System Identification - Advances and Case Studies, Academic, 1976.
117. B. P. Molinari, “The stabilizing solution of the discrete algebraic Riccati equation,” IEEE Trans. Automatic Control, vol. AC-20, no. 3, pp. 396–399, 1975.
118. M. Moonen, B. De Moor, L. Vandenberghe and J. Vandewalle, “On- and off-line identification of linear state-space models,” Int. J. Control, vol. 49, no. 1, pp. 219–232, 1989.
119. M. Moonen and J. Vandewalle, “QSVD approach to on- and off-line state-space identi-fication,” Int. J. Control, vol. 51, no. 5, pp. 1133–1146, 1990.
120. A. Ohsumi, K. Kameyama and K. Yamaguchi, “Subspace identification for continuous-time stochastic systems via distribution-based approach,” Automatica, vol. 38, no. 1, pp. 63–79, 2002.
121. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing, Prentice-Hall, 1975.
122. C. C. Paige, “Properties of numerical algorithms related to computing controllability,” IEEE Trans. Automatic Control, vol. AC-26, no. 1, pp. 130–138, 1981.
123. A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes (4th ed.), McGraw-Hill, 2002.
124. T. Pappas, A. J. Laub and N. R. Sandell, “On the numerical solution of the discrete time algebraic Riccati equation,” IEEE Trans. Automatic Control, vol. AC-25, no. 4, pp. 631–641, 1980.
125. R. Patel, A. J. Laub and P. M. Van Dooren (eds.), Numerical Linear Algebra Techniques for Systems and Control, IEEE Press, 1994.
126. M. Pavon, “Canonical correlations of past inputs and future outputs for linear stochastic systems,” Systems and Control Letters, vol. 4, pp. 209–215, 1984.
127. L. Pernebo and L. M. Silverman, “Model reduction via balanced state space representa-tions,” IEEE Trans. Automatic Control, vol. AC-27, no. 2, pp. 382–387, 1982.
128. K. Peternell, W. Scherrer and M. Deistler, “Statistical analysis of novel subspace identification methods,” Signal Processing, vol. 52, no. 2, pp. 161–177, 1996.
129. G. Picci, “Stochastic realization and system identification,” In Statistical Methods in Control and Signal Processing (T. Katayama and S. Sugimoto, eds.), Marcel Dekker, pp. 1–36, 1997.
130. G. Picci and T. Katayama, “Stochastic realization with exogenous inputs and ‘subspace methods’ identification,” Signal Processing, vol. 52, no. 2, pp. 145–160, 1996.
131. G. Picci and T. Katayama, “A simple ‘subspace’ identification method with exogenous inputs,” Proc. IFAC 13th World Congress, San Francisco, Vol. I, pp. 175–180, 1996.
132. R. Pintelon and J. Schoukens, System Identification – A Frequency Domain Approach, IEEE Press, 2001.
133. V. F. Pisarenko, “The retrieval of harmonics from a covariance function,” Geophysical J. Royal Astronomical Society, vol. 33, pp. 347–366, 1973.
134. M. B. Priestley, Spectral Analysis and Time Series, vol. 1: Univariate Series; vol. 2: Multivariate Series, Prediction and Control, Academic, 1981.
135. A. Rantzer, “On the Kalman-Yakubovich-Popov lemma,” Systems and Control Letters, vol. 28, no. 1, pp. 7–10, 1996.
136. C. R. Rao, Linear Statistical Inference and Its Applications (2nd ed.), Wiley, 1973.
137. J. Rissanen, “Recursive identification of linear systems,” SIAM J. Control, vol. 9, no. 3, pp. 420–430, 1971.
138. Yu. A. Rozanov, Stationary Random Processes, Holden-Day, 1967.
139. A. H. Sayed, Fundamentals of Adaptive Filtering, Wiley, 2003.
140. R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas and Propagation, vol. 34, no. 3, pp. 276–280, 1986.
141. R. Schrama, “Control oriented approximate closed loop identification via fractional rep-resentation,” Proc. 1991 American Control Conference, Boston, pp. 719–720, 1991.
142. R. H. Shumway, Applied Statistical Time Series Analysis, Prentice-Hall, 1988.
143. L. M. Silverman, “Realization of linear dynamical systems,” IEEE Trans. Automatic Control, vol. AC-16, no. 6, pp. 554–567, 1971.
144. T. Söderström, Discrete-time Stochastic Systems, Springer, 2002.
145. T. Söderström and P. Stoica, System Identification, Prentice-Hall, 1989.
146. L. H. Son and B. D. O. Anderson, “Design of Kalman filters using signal-model output statistics,” Proc. IEE, vol. 120, no. 2, pp. 312–318, 1973.
147. E. D. Sontag, Mathematical Control Theory, Springer, 1990.
148. G. W. Stewart, “On the early history of the singular value decomposition,” SIAM Review, vol. 35, no. 4, pp. 551–566, 1993.
149. S. M. Stigler, The History of Statistics: The Measurement of Uncertainty before 1900, The Belknap Press of Harvard University Press, 1986.
150. P. Stoica and R. Moses, Introduction to Spectral Analysis, Prentice-Hall, 1997.
151. H. Tanaka and T. Katayama, “A stochastic realization in a Hilbert space based on ‘LQ decomposition’ with application to subspace identification,” Proc. 13th IFAC Symp. System Identification, Rotterdam, pp. 899–904, 2003.
152. H. Tanaka and T. Katayama, “Stochastic realization on a finite interval via ‘LQ decom-position’ in Hilbert space,” Proc. 7th European Control Conference, Cambridge, Paper No. SI-2-2, 2003.
153. H. Tanaka and T. Katayama, “A stochastically balanced realization on a finite interval,” Proc. 16th Int. Symp. on Mathematical Theory of Network and Systems, THP8 (Identifi-cation), Leuven, 2004.
154. H. Tanaka and T. Katayama, “Stochastic subspace identification algorithm guarnteeing stability and minimum phase,” Preprints 16th IFAC World Congress, Prague, July 2005.
155. A. J. Tether, “Construction of minimal linear state-variable models from finite input-output data,” IEEE Trans. Automatic Control, vol. 15, no. 4, pp. 427–436, 1970.
156. L. Tong and S. Perreau, “Multichannel blind identification: From subspace to maximum likelihood methods,” Proc. IEEE, vol. 86, no. 10, pp. 1951–1968, 1998.
157. L. N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, 1997.
158. P. M. J. Van den Hof, “Closed-loop issues in system identification,” Proc. 11th IFAC Symp. System Identification, Kitakyushu, Japan, pp. 1651–1664, 1997.
159. P. M. J. Van den Hof and R. A. de Callafon, “Multivariable closed-loop identification: From indirect identification to dual-Youla parametrization,” Proc. 35th IEEE Conference on Decision and Control, Kobe, Japan, pp. 1397–1402, 1996.
160. P. M. J. Van den Hof and R. J. P. Schrama, “An indirect method for transfer function estimation from closed loop data,” Automatica, vol. 29, no. 6, pp. 1523–1527, 1993.
161. A. C. Van der Klauw, M. Verhaegen and P. P. J. Van den Bosch, “State space identification of closed loop systems,” Proc. 30th IEEE Conference on Decision and Control, Brighton, pp. 1327–1332, 1991.
162. A. J. Van der Veen, E. F. Deprettere and A. Swindlehurst, “Subspace-based signal analysis using singular value decomposition,” Proc. IEEE, vol. 81, no. 9, pp. 1277–1308, 1993.
163. P. Van Overschee and B. De Moor, “Subspace algorithms for the stochastic identification problem,” Automatica, vol. 29, no. 3, pp. 649–660, 1993.
164. P. Van Overschee and B. De Moor, “N4SID - Subspace algorithms for the identification of combined deterministic - stochastic systems,” Automatica, vol. 30, no. 1, pp. 75–93, 1994.
165. P. Van Overschee and B. De Moor, Subspace Identification for Linear Systems, Kluwer Academic Pub., 1996.
166. P. Van Overschee, B. De Moor, W. Dehandschutter and J. Swevers, “A subspace algorithm for the identification of discrete-time frequency domain power spectra,” Automatica, vol. 33, no. 12, pp. 2147–2157, 1997.
167. P. Van Overschee and B. De Moor, “Closed loop subspace systems identification,” Proc. 36th IEEE Conference on Decision and Control, San Diego, pp. 1848–1853, 1997.
168. A. Varga, “On balancing and order reduction of unstable periodic systems,” Preprints 1st IFAC Workshop on Periodic Control Systems, Como, pp. 177–182, 2001.
169. M. Verhaegen, “Subspace model identification, Part 3: Analysis of the ordinary output-error state-space model identification algorithm,” Int. J. Control, vol. 58, no. 3, pp. 555–586, 1993.
170. M. Verhaegen, “Application of a subspace model identification technique to identify LTI systems operating in closed loop,” Automatica, vol. 29, no. 4, pp. 1027–1040, 1993.
171. M. Verhaegen, “Identification of the deterministic part of MIMO state space models given in innovations form from input-output data,” Automatica, vol. 30, no. 1, pp. 61–74, 1994.
172. M. Verhaegen and P. Dewilde, “Subspace model identification, Part 1: The output-error state-space model identification class of algorithms,” Int. J. Control, vol. 56, no. 5, pp. 1187–1210, 1992.
173. M. Verhaegen and P. Dewilde, “Subspace model identification, Part 2: Analysis of the elementary output-error state space model identification algorithm,” Int. J. Control, vol. 56, no. 5, pp. 1211–1241, 1992.
174. M. Viberg, B. Ottersten, B. Wahlberg and L. Ljung, “A statistical perspective on state-space modeling using subspace methods,” Proc. 30th IEEE Conference on Decision and Control, Brighton, pp. 1337–1342, 1991.
175. M. Viberg, “Subspace-based methods for identification of linear time-invariant systems,” Automatica, vol. 31, no. 12, pp. 1835–1851, 1995.
176. M. Viberg and P. Stoica, “Editorial note,” Signal Processing, Special Issue on Subspace Methods, Part I: Array Processing and Subspace Computations, vol. 50, nos. 1-2, pp. 1–3, 1996; Part II: System Identification, vol. 52, no. 2, pp. 99–101, 1996.
177. K. Wada, “What’s the subspace identification methods?” J. Society Instrument and Control Engineers, vol. 36, no. 8, pp. 569–574, 1997 (in Japanese).
178. P. Whittle, Prediction and Regulation by Linear Least-Square Methods (2nd ed.), Basil Blackwell Publisher, 1984.
179. N. Wiener and P. Masani, “The prediction theory of multivariable stochastic processes -I,” Acta Mathematica, vol. 98, pp. 111–150, 1957.
180. N. Wiener and P. Masani, “The prediction theory of multivariable stochastic processes -II,” Acta Mathematica, vol. 99, pp. 93–137, 1958.
181. J. Willems, I. Markovsky, P. Rapisarda and B. De Moor, “A note on persistency of exci-tation,” Technical Report 04-54, Department of Electrical Engineering, Katholieke Universiteit Lueven, 2004.
182. W. M. Wonham, “On a matrix Riccati equation of stochastic control,” SIAM J. Control, vol. 6, no. 4, pp. 681–698, 1968.
183. N. Young, An Introduction to Hilbert Space, Cambridge University Press, 1988.
184. H. P. Zeiger and A. J. McEwen, “Approximate linear realization of given dimension via Ho’s algorithm,” IEEE Trans. Automatic Control, vol. AC-19, no. 2, p. 153, 1974.
185. K. Zhou, J. C. Doyle and K. Glover, Robust Optimal Control, Prentice-Hall, 1996.
186. K. Zhou, G. Salomon and E. Wu, “Balanced realization and model reduction for unstable systems,” Int. J. Robust and Nonlinear Control, vol. 9, no. 2, pp. 183–198, 1999.