We consider linear matrix difference equations used in control problems and obtain quantitative characteristics of the local stability of optimal estimates. Bibliography: 25 titles.
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References
A. Björck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA (1996).
M. Aoki and P. C. Yue, “On a priori error estimates of some identification methods,” IEEE Trans. Automat. Control. 15, No. 5, 541–548 (1970).
S. van Huffel and J. Vandewalle, The Total Least Squares Problem, SIAM, Philadelphia, PA (1991).
T. J. Abatzoglou, J. M. Mendel, and G. A. Harada, “The constrained total least squares technique and its applications to harmonic superresolution,” IEEE Trans. Signal Process. 39, No. 5, 1070–1087 (1991).
A. A. Lomov, “Orthoregressive methods for estimating parameters and problems of separating trends in linear systems” [in Russian], Differ. Urav. Proc. Upr. No. 2, 1–86 (2005).
A. A. Lomov, “On quatitative a priori degree of identifiability of parameters of a linear system” [in Russian], In: Proc. VIII Intern. Conf. “Identificaion of systems and control problems”, Moscow, 2009, pp. 479–491. Moscow (2009).
W. A. Fuller, Measurement Error Models, John Wiley & Sons, Chichester (1987).
A. A. Lomov, “Orthoregressive estimates for parameters of systems of linear difference equations” [in Russian], Sib. Zh. Ind. Mat. 8, No. 3, 102–119 (2005); English transl.: J. Appl. Industr. Math. 1, No. 1. 59–76 (2007).
M. R. Osborne, “A class of nonlinear regression problems,” Data Represent., Proc. Semin. Aust. nat. Univ. 94–101. (1970).
A. O. Egorshin, “Numerical closed methods of identification of linear objects” [in Russian], In: Optimal and Self-Adjusting Systems, pp. 40–53, Novosibirsk (1971).
B. Roorda, “Algorithms for global total least squares modelling of finite multivariable time series,” Automatica 31, No. 3, 391–404 (1995).
A. O. Egorshin, “Optimization of parameters of stationary models in a unitary space” [in Russian], Avtom. Telemekh. No. 12, 29–48 (2004); English transl.: Autom. Remote Control 65, No. 12, 1885–1903 (2004).
V. G. Demidenko, “Recovery of parameters of a homogeneous linear model” [in Russian], Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. 8, No. 3, 51–59 (2008).
M. Aoki and P. C. Yue, “On the certain convergence questions in system identification,” SIAM J. Control. 8, No. 2. 239–256 (1970).
B. G. Vorchik, “Identifiability of linear parametric stochastic systems. I. II” [in Russian], Avtom. Telemekh., No. 5, 64–78; No. 7, 96–109 (1985); English transl.: Autom. Remote Control 46, 596–609; 867–878 (1985).
A. A. Lomov, “Identification of linear dynamic systems by short parts of transients with additive measUring disturbances” [in Russian], Izv. Akad. Nauk, Teor. Sist. Upr. No. 3, 20–26 (1997); English transl.: J. Comput. Syst. Sci. Int. 36, No. 3, 346–352 (1997).
A. A. Lomov, “Distinguishability conditions for stationary linear systems” [in Russian], Differ. Uravn. 39, No. 2, 261–266 (2003); English transl.: Differ Equ. 39, No. 2, 283–288 (2003).
A. A. Lomov, “On identifiability of stationary linear systems with coefficients that depend on a parameter” [in Russian], Sib. Zh. Ind. Mat. 6, No. 4 (16), 60–66 (2003).
C. Lanczos, Practical Methods of Applied Analysis [Russian translation], Fiz. Mat. Lit., Moscow (1961).
V. I. Kostin, “On extremum points of one function” [in Russian], Upr. Syst. 24, 35–42 (1984).
G. K. Smyth, “Employing symmetry constraints for improved frequency estimation by eigenanalysis methods” Technometrics 42, 277–289 (2000).
A. A. Lomov, “Minimal descriptions of stationary linear models” [In Russian], Models. Method. Optimiz. 28 91–117 (1994).
L. Schwartz, Analysis. I, II [Russian translation], Mir, Moscow (1972).
P. Lancaster, Theory of Matrices, Academic Press. New York etc. (1969).
L. Mirsky, “Symmetric gauge functions and unitarily invariant norms,” Quart. J. Math. Oxf. II. Ser. 11, 50–59 (1960).
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Translated from Vestnik Novosibirskogo Gosudarstvennogo Universiteta: Seriya Matematika, Mekhanika, Informatika 10, No. 4, 2010, pp. 82–104
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Lomov, A.A. Local stability in the problem of identifying coefficients of a linear difference equation. J Math Sci 188, 410–434 (2013). https://doi.org/10.1007/s10958-012-1138-z
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DOI: https://doi.org/10.1007/s10958-012-1138-z