ANALYSIS AND DATA PROCESSING SYSTEMS

The Voltaire integral equation of the first kind with a difference kernel is often used as a model of a stationary dynamical system. For such models the problem of non-parametric identification consists in estimating a difference kernel (called the Impulse Response Function) from the measured values of the input and output signals of the identified dynamic system. This task is ill-posed, i.e. the solution may not exist, may not be unique, and may be unstable with respect to errors (measurement noise) of the original data. To obtain a unique and stable (but approximate) solution, various regularization methods in particular the A.N. Tikhonov regularization method are used. The discrete Fourier transform (DFT) forms a computational basis of such an algorithm. It is assumed that an input signal (the core of the integral equation) is specified accurately, and an output signal of the system is recorded with some random error. However, this assumption is seldom implemented in practice, since both the input and output signals of the system are measured and recorded by measuring devices and thus are set with random errors (measurement noise). In this paper, a two-step stable algorithm for nonparametric identification of the Impulse Response Function of a stationary dynamic system is proposed. The algorithm is used in the case when both the input and output signals of the identified system are recorded with random errors. At the first stage, wavelet filtering of the noisy input signal is used. For this, threshold algorithms for processing the coefficients of the wavelet decomposition of a noisy signal are used. Threshold values based on the statistical optimality criterion of the filtering algorithm are calculated to minimize the filtering error. At the second stage, a regularizing algorithm using a discrete Fourier transform is applied to the filtered input signal. To select the optimal value of a regularization parameter a special algorithm is used. The paper analyses the degree of influence of the error levels of the input and output signals on the identification error. The results of the computational experiment are published and discussed. The effectiveness of the proposed approach to the construction of a stable algorithm for non-parametric identification of the Impulse Response Function of a stationary dynamic system with different levels of measurement noise of input and output signals are illustrated.

1. Greblicki W., Pawiak M. Nonparametric system identification. Cambridge, Cambridge University Press, 2008. 400 p.

2. Kondrashin A.V., Khor'kov V.I. Issledovanie i identifikatsiya upravlyaemykh tekhnicheskikh sistem [Research and identification of controllable technical systems]. Moscow, IspoServis Publ., 2000. 220 p.

3. Tikhonov A.N., Arsenin V.Ya. Metody resheniya nekorrektnykh zadach [Methods for solving ill-posed problems]. Moscow, Nauka Publ., 1979. 285 p.

4. Tikhonov A.N., Goncharskii A.V., Stepanov V.V., Yagola A.G. Chislennye metody resheniya nekorrektnykh zadach [Numerical methods for the solution of ill-posed problems]. Moscow, Nauka Publ., 1990. 231 p.

5. Morozov V.A., Grebennikov A.I. Metody resheniya nekorrektno postavlennykh zadach: algoritmicheskii aspekt [Methods for solving ill-posed problems: an algorithmic aspect]. Moscow, Moscow State University Publ., 1992. 319 p.

6. Voskoboinikov Yu.E. Preobrazhenskii N.G., Sedel'nikov A.I. Matematicheskaya obrabotka eksperimenta v molekulyarnoi gazodinamike [Mathematical processing of the experiment in molecular gas dynamics]. Novosibirsk, Nauka Publ., 1984. 238 p.

7. Voskoboinikov Yu.E., Krysov D.A. Neparametricheskaya identifikatsiya dinamicheskoi sistemy pri netochnom vkhodnom signale [Nonparametric identification of a dynamic system with an inaccurate input signal]. Avtomatika i programmnaya inzheneriya – Automatics & Software Enginery, 2017, no. 4 (22), pp. 86–92.

8. Voskoboinikov Yu.E., Krysov D.A. Lokal'nyi regulyariziruyushchii algoritm neparametricheskoi identifikatsii ob"ekta s netochnym vkhodnym signalom [Local regularizing algorithm of nonparametric identification to object with inaccurate input signal]. Nauchnyi vestnik Novosibirskogo gosudarstvennogo tekhnicheskogo universiteta – Science bulletin of the Novosibirsk state technical university, 2018, no. 1 (70), pp. 19–38. DOI: 10.17212/1814-1196-2018-1-19-38.

9. Voskoboinikov Yu.E., Krysov D.A. Algoritm identifikatsiya impul'snoi perekhodnoi funktsii pri vysokom urovne shuma izmereniya vkhodnogo signala sistemy [The algorithm for identifying the impulse response function at a high noise level of measuring the input signal of the system]. Avtomatika i programmnaya inzheneriya – Automatics & Software Enginery, 2018, no. 2 (24), pp. 67–73.

10. Mallat S. Multiresolution approximation and wavelet orthonormal bases of L^2(R). Transactions of the American Mathematical Society, 1989, vol. 315, no. 1, pp. 69–87.

11. Mallat S. А theory of multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1989, vol. 11, no. 7, pp. 674–693.

12. Voskoboinikov Yu.E. Veivlet-fil'tratsiya signalov i izobrazhenii (s primerami v MathCAD) [Wavelet filtering of signals and images: (with examples in MathCAD]. Novosibirsk, NGASU Publ., 2015. 188 p.

13. Voskoboinikov Yu.E., Krysov D.A. Vybor nailuchshei odnoparametricheskoi porogovoi funktsii v algoritmakh veivlet-fil'tratsii [Choose of the best one-parameter threshold function to the wavelet filtering algorithms]. Sbornik nauchnykh trudov Novosibirskogo gosudarstvennogo tekhnicheskogo universiteta – Transaction of scientific papers of the Novosibirsk state technical university, 2016, no. 3 (85), pp. 71–82.

14. Vidakovic B. Statistical modeling by wavelets. Wiley series in probability and statistics. New York, John Wiley & Sons, 1999. 365 p.

15. Voskoboinikov Yu.E. Ustoichivye algoritmy resheniya obratnykh izmeritel'nykh zadach [A stable algorithms for solving inverse measurement problems]. Novosibirsk, NGASU Publ., 2007. 184 p.

16. Urmanov A.M, Gribok A.V., Bozdogan H., Hines J.W., Uhrid R.E. Information complexity-based regularizing parameter selection for solution of ill conditioned inverse problems. Inverse Problems, 2002, vol. 18, no. 2, pp. L1–L9.

17. Vogel C.R. Non-convergence of L-curve regularization parameter selection method. Inverse Problems, 1996, vol. 12, no. 4, pp. 535–547.

18. Lukas M.A. Comparison of parameter choice methods for regularization with discrete noisy data. Inverse Problems, 2000, vol. 14, no. 2, pp. 161–184.

19. Engl H.W., Hanke M., Neubauer F. A regularization of inverse problems. Dordrecht, Boston, Kluwer Academic Publisher, 2000. 383 p.

20. Titarenko V.N., Yagola A.G. Primenenie metoda GVC dlya korrektnykh i nekorrektnykh zadach [Applying the GVC method for correctly posed and ill-posed problems]. Vestnik Moskovskogo Universiteta. Seriya 3, Fizika. Astronomiya – Moscow University Physics Bulletin, 2000, no. 4, pp. 15–18. (In Russian).

21. Morozov V.A. Regulyarnye metody resheniya nekorrektno postavlennykh zadach [Regular methods for solving ill-posed problems]. Moscow, Nauka Publ., 1987. 240 p.

22. Levin M.A., Tatarintsev A.V., Akhkubekov A.E. Metod Laplace-DLTS s vyborom parametra regulyarizatsii po L-krivoi [Laplace-DLTS method with the choice of the regularization parameter on the L-curve]. Fizika i tekhnika poluprovodnikov – Semiconductors, 2009, vol. 43, no. 5, pp. 74–81. (In Russian).

23. Dimaki A.V., Svetlakov A.A. Regulyarizatsiya resheniya zadachi identifikatsii pri is-pol'zovanii algoritma chuvstvitel'nosti [Regularization of the solution to the problem of identification using the sensitivity algorithm]. Izvestiya Tomskogo politekhnicheskogo universiteta – Bulletin of the Tomsk Polytechnic University, 2009, vol. 314, no. 5, pp. 27–31.

24. Voskoboinikov Yu.E., Krysov D.A. Otsenivanie kharakteristik shuma izmereniya v modeli "signal+shum" [Estimation of the Noise Measurement Characteristics in the Model “Signal + Noise”]. Avtomatika i programmnaya inzheneriya – Automatics & Software Enginery, 2018, no. 3 (25), pp. 54–61.