zbMATH — the first resource for mathematics

On quantitative a priori measures of identifiability of coefficients of linear dynamic systems. (English. Russian original) Zbl 1267.93174
J. Comput. Syst. Sci. Int. 50, No. 1, 1-13 (2011); translation from Izv. Ross. Akad. Nauk, Teor. Sist. Upravl. 2011, No. 1, 3-15 (2011).
Summary: A number of quantitative a-priori measures of identifiability of coefficients of linear dynamic control systems are proposed. The criteria are based on numerical characteristics of the lower bound of the asymptotic variance of estimates of the coefficients, which is approximately calculated using the mean value of the inverse information matrix. Information matrices are calculated in the limit case of small noise amplitude. Additive observation noise is considered. In this case, the optimal estimates of coefficients correspond to the smallest distance between observations of trajectories and the solution set of the system. Numerical examples are given.

93E12 Identification in stochastic control theory
93C05 Linear systems in control theory
93E10 Estimation and detection in stochastic control theory
Full Text: DOI
[1] T. Soderstrom, L. Ljung, and I. Gustavsson, ”Identifiability Conditions for Linear Multivariable Systems Operating Under Feedback,” IEEE Trans. Autom. Control AC-21(6), 837–840 (1976). · Zbl 0344.93023
[2] B. N. Petrov, E. D. Teryaev, and B. M. Shamrikov, ”Conditions of Parametric Identifiability of Control Objects in Open-Loop and Closed-Loop Automatic Systems,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 2, 160–175 (1977). · Zbl 0386.93014
[3] E. Walter, Identifiability of State Space Models (Springer, New York, 1982). · Zbl 0508.93001
[4] B. G. Vorchik, ”Identifiability of Linear Parametric Stochastic Systems,” Avtom. Telemekh., No. 7, 96–109 (1985). · Zbl 0601.93058
[5] T. V. Avdeenko, ”On Planning of the Model Structure in the State Space: Analysis of Structural Identifiability,” Sib. Zh. Industr. Mat. 4(2) (2001). · Zbl 1025.93011
[6] A. A. Lomov, ”Parametric Identifiability of Linear Stochastic Systems by Observations of Intervals with Finite Trajectories,” Izv. Ross. Akad. Nauk, Teor. Sist. Upr., No. 2, 53–58 (2002) [Comp. Syst. Sci. 41 (2), 213–218 (2002)].
[7] Ya. Z. Tsypkin, Foundations of Information Theory of Identifiability (Nauka, Moscow, 1984) [in Russian].
[8] A. A. Lomov, ”Orthogonal Regression Methods of Parameter Estimation and Trend Separation in Linear Systems, in Differential Equations and Control Processes ( http://www.Neva.Ru/Journal ), No. 2, 1–86 (2005).
[9] A. A. Lomov, ”Orthoregressive Estimates of the Parameters of Systems of Linear Difference Equations,” Sib. Zh. Industr. Mat. 8(3(23) (2005). · Zbl 1106.93021
[10] A. A. Lomov, ”Orthoregressive Estimates for the Parameters of Systems of Linear Difference Equations,” J. of Applied and Industrial Mathematics 1(1), 59–76 (2007).
[11] R. E. Maine and K. W. Iliff, ”Formulation and Implementation of a Practical Algorithm for Parameter Estimation with Process and Measurement Noise,” SIAM J. Appl. Math. 41(3), 558–579 (1981). · Zbl 0477.93072
[12] R. L. Kash’yap and A. R. Rao, Construction of Dynamic Stochastic Models Based on Experimental Data (Nauka, Moscow, 1983) [in Russian].
[13] V. I. Denisov, V. M. Chubich, and O. S. Chernikova, ”Active Parametric Identification of Stochastic Linear Discrete Systems in Temporal Domain,” Sib. Zh. Industr. Mat. 6(3) (2003). · Zbl 1075.93543
[14] K. Ostrem and T. Bolin, ”Digital Identification of Linear Dynamic Systems Based on Data on the Normal Operation Regime” in Proceeding of 2nd IFAC Congress on the Theory of Self-Adapting Control Systems (Nauka, Moscow, 1969), pp. 99–116 [in Russian].
[15] L. S. Ljung, System Identification: Theory for the User (Prentice-Hall, Englewood Cliffs, N.J., 1987; Nauka, Moscow, 1991). · Zbl 0615.93004
[16] A. O. Egorshin, ”Least Squares Method and Fast Algorithms in Variational Problems of Identification and Filtering (Method VI),” Avtometriya, No. 1, 30–42 (1988).
[17] A. O. Egorshin, ”Optimization of Parameters of Stationary Models in a Unitary Space,” Avtom. Telemekh., No. 12, 29–48 (2004) [Automat. Remote Control 65 (12), 1885–1903 (2004)]. · Zbl 1095.93006
[18] A. A. Lomov, ”On a Prior Quantitative Measure of Identifiability of Parameters of a Linear System,” in Proceedings of 8th International Conference on ”System Identification and Control Problems” SICPRO’09, IPU RAN, Moscow, Russia, 2009, pp. 479–491 [in Russian].
[19] J. C. Villems, ”From a Time Series to a Linear System,” I, II, III, Automatica 22, 23, 561–580, 675–694, 87–115 (1986, 1987).
[20] A. A. Lomov, ”Distinguishability Conditions for Stationary Linear Systems,” Differ. Uravn. 39(2), 283–288 (2003). · Zbl 1065.93011
[21] A. A. Lomov, ”On Distinguishability of Stationary Linear Systems with Coefficients Depending on Parameters,” Sib. Zh. Industr. Mat. 6 4(16) (2003).
[22] W. A. Fuller, Measurement Error Models (Wiley, New York, 1987). · Zbl 0800.62413
[23] A. A. Lomov, ”Identification of Linear Dynamic Systems by Short Parts of Transients with Additive Measuring Disturbances,” Izv. Ross. Akad. Nauk, Teor. Sist. Upr., No. 3, 20–26 (1997) [Comp. Syst. Sci. 36 (3), 346–352 (1997)]. · Zbl 0905.93014
[24] K. Pearson, ”On Lines and Planes of Closest Fit to Systems of Points in Space,” Philos. Mag. 6(2), 559–572 (1901). · JFM 32.0246.07
[25] H. Cramer, Mathematical Methods of Statistics (Berlin, Springer, 1933; Nauka, Moscow, 1975).
[26] J. Neyman, ”Remarks on a Paper by E.C. Rhodes,” J. Royal Statistical Society 100, 50–57 (1937).
[27] O. Reiersol, ”Identifiability of a Linear Relation between Variables which are Subject to Error,” Econometrica 18, 375–389 (1950). · Zbl 0040.22502
[28] M. Aoki and P. C. Yue, ”On A Priori Error Estimates of Some Identification Methods,” IEEE Trans. on Automat. Control AC-15, 541–548 (1970).
[29] A. A. Lomov, ”Comparison of Methods for Estimating the Parameters of Linear Dynamic Systems by Measurements of Short Sections of Transient Processes,” Avtom. Telemekh., No. 3, 39–47 (2005) [Automat. Remote Control 66 (3), 373–381 (2005)].
[30] A. O. Egorshin, ”Computational Closed Methods of Identification of Linear Objects,” in Optimal and Adaptive Systems (Novosibirsk, 1971), pp. 40–53 [in Russian].
[31] M. R. Osborne, ”A Class of Nonlinear Regression Problems,” in Data Representation, Ed. by R. S. Anderssen and M. R. Osborne (University of Queenland Press, St. Lucia, 1970), pp. 94–101.
[32] E. N. Rozenvasser and R. M. Yusupov, Sensitivity of Control Systems (Nauka, Moscow, 1981) [in Russian]. · Zbl 0206.45204
[33] E. L. Leman, Theory of Point Estimation (Springer, New York, 1988; Nauka, Moscow, 1991).
[34] A. A. Borovkov, Mathematical Statistics (Nauka, Novosibirsk, 1997) [in Russian]. · Zbl 0913.62001
[35] I. G. Zedginidze, Design of Experiment for Investigation of MIMO Systems (Nauka, Moscow, 1976) [in Russian]. · Zbl 0464.62066
[36] M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, 4th ed. (Nauka, Moscow, 1966; Griffin, London, 1977). · Zbl 0416.62001
[37] A. A. Lomov, ”Estimation of Trends and Identification of Time Series Dynamics in Short Observation Sections,” Izv. Ross. Akad. Nauk, Teor. Sist. Upr., No. 1, 25–37 (2009) [Comp. Syst. Sci. 48 (1), 1–13 (2009)]. · Zbl 1194.93203
[38] E. R. Alekseev, E. A. Chesnokova, and E. A. Rudchenko, Scilab: Solution of Engineering and mathematical Problems (ALT Linux, Moscow) [in Russian].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.