Well-known methods of joint estimation of the state and parameters (quasilinearization, invariant imbedding, extended Kalman filter and others like them) expand the vector of the state of the system by including equations for parameters in the model. Such a task of joint estimation of the state and parameter is nonlinear even for linear systems. For Linear Structure Models (LSModels), an analytical method is proposed for the transition to an auxiliary model in which the parameter vector is expanded by initial states and the task of identifying parameter and initial states becomes linear. With the help of an auxiliary state vector, the initial dynamic model is reduced to an auxiliary model with residual. In this case, the auxiliary model does not contain derivatives of the measured elements of the initial dynamic model, but contains filtered measured elements. The proof of the identity of solutions according to the initial and auxiliary models is given. An Iterative algorithm of identification of order, parameters and state estimation is proposed. An analytical example of solving the problem of joint estimation of parameters and state for the heat equation is given and its software implementation in the MATLAB is discussed in detail. Next, another auxiliary model is proposed. If the first implies that the order of the differential equation is unknown but only limited by a certain value, then the second model has a given order. Now there can be two types of auxiliary models to it. An example of a nonlinear initial model is given.
| Published in | International Journal of Systems Engineering (Volume 8, Issue 2) |
| DOI | 10.11648/j.ijse.20240802.11 |
| Page(s) | 22-39 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Identification Iterative Algorithm, Linear Structure Models, Auxiliary LSModel, State Estimation
| [1] | Bellman R., Kagiwada H., Kalaba R., Wengert’s Numerical Method for Partial Derivatives, Orbit Determination, and Quasilinearization. The RAND Corporation, RM-4354-PR, November, 1964. |
| [2] | Bellman R., Kalaba R. Quasilinearization and nonlinear boundary-value problems. American Elsevier Publishing Company, Inc. 1965, 218p. |
| [3] | Bellman R., Wing G. M. An introduction to invariant imbedding. SIAM, 1992, 267p. |
| [4] | Bellman R., Kalaba R., Wing G. M., Invariant Imbedding and Mathematical Physics–I: Particle Processes. J. Math. Phys. 1960, 1, pp. 280-308. |
| [5] | Shimizu R., Aoki R., Application of Invariant Imbedding to Reactor Physics, Academic, New York, 1972. |
| [6] | Wall D., Olsson P. Invariant imbedding and hyperbolic heat waves. Technical report CODEN: LUTEDX (TEAT- 7045/1-32/(1996). |
| [7] | Kalman R. E. A New Approach to Linear Filtering and Prediction Problems. J. Basic Engineering, 1960, 82D, pp. 35–45. |
| [8] | Grewal M. S., Andrews A. P. Kalman Filtering: Theory and Practice Using MATLAB. Third Edition. John Wiley & Sons, Inc. 2008. |
| [9] | Eykhoff P. System Identification Parameter and State Estimation. John Wiley & Sons, 1974. |
| [10] | Sage A.P., Melsa J.L. Estimation Theory with Application to Communication and Control. Mc Graw Hill, 1971. |
| [11] | Kopysov O. Yu., Prokopov B. I., Pupkov K. A. Identification of parameters and the state of one class of nonlinear system. Problem of Control and Information Theory, 1982, 11(3), pp. 205–216. |
| [12] | Kopysov O. Yu. Identification via linear structure models. Varna: Publishing house DesCartes Science Center; 2012, 368 p. ISBN 978-954-92807-2-2. |
| [13] | Kopysov O. Yu. Identification of Parameters and Restoring of State Vector of Dynamical Plant. IFAC Proceedings Volumes (IFAC-PapersOnLine), 2013, 46(9), pp. 1494-1499. |
| [14] | Kopysov O. Yu. Elements of the Model Identification. IFAC-PapersOnLine, 2017, 50(1), pp. 2260–2265. 20th IFAC World Congress. |
| [15] | Kopysov O. Yu. Identification of Order, Parameters and State Estimation via Projection onto the Plane of Guarantors. IFAC-PapersOnLine, 2023, 56(2), pp. 7771– 7777. 22th IFAC World Congress. |
APA Style
Kopysov, O. Y. (2024). Identification of Physical Dynamical Processes Via Linear Structure Models (Part 2). International Journal of Systems Engineering, 8(2), 22-39. https://doi.org/10.11648/j.ijse.20240802.11
ACS Style
Kopysov, O. Y. Identification of Physical Dynamical Processes Via Linear Structure Models (Part 2). Int. J. Syst. Eng. 2024, 8(2), 22-39. doi: 10.11648/j.ijse.20240802.11
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Download@article{10.11648/j.ijse.20240802.11,
author = {Oleg Yu. Kopysov},
title = {Identification of Physical Dynamical Processes Via Linear Structure Models (Part 2)},
journal = {International Journal of Systems Engineering},
volume = {8},
number = {2},
pages = {22-39},
doi = {10.11648/j.ijse.20240802.11},
url = {https://doi.org/10.11648/j.ijse.20240802.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijse.20240802.11},
abstract = {Well-known methods of joint estimation of the state and parameters (quasilinearization, invariant imbedding, extended Kalman filter and others like them) expand the vector of the state of the system by including equations for parameters in the model. Such a task of joint estimation of the state and parameter is nonlinear even for linear systems. For Linear Structure Models (LSModels), an analytical method is proposed for the transition to an auxiliary model in which the parameter vector is expanded by initial states and the task of identifying parameter and initial states becomes linear. With the help of an auxiliary state vector, the initial dynamic model is reduced to an auxiliary model with residual. In this case, the auxiliary model does not contain derivatives of the measured elements of the initial dynamic model, but contains filtered measured elements. The proof of the identity of solutions according to the initial and auxiliary models is given. An Iterative algorithm of identification of order, parameters and state estimation is proposed. An analytical example of solving the problem of joint estimation of parameters and state for the heat equation is given and its software implementation in the MATLAB is discussed in detail. Next, another auxiliary model is proposed. If the first implies that the order of the differential equation is unknown but only limited by a certain value, then the second model has a given order. Now there can be two types of auxiliary models to it. An example of a nonlinear initial model is given.},
year = {2024}
}
TY - JOUR T1 - Identification of Physical Dynamical Processes Via Linear Structure Models (Part 2) AU - Oleg Yu. Kopysov Y1 - 2024/10/31 PY - 2024 N1 - https://doi.org/10.11648/j.ijse.20240802.11 DO - 10.11648/j.ijse.20240802.11 T2 - International Journal of Systems Engineering JF - International Journal of Systems Engineering JO - International Journal of Systems Engineering SP - 22 EP - 39 PB - Science Publishing Group SN - 2640-4230 UR - https://doi.org/10.11648/j.ijse.20240802.11 AB - Well-known methods of joint estimation of the state and parameters (quasilinearization, invariant imbedding, extended Kalman filter and others like them) expand the vector of the state of the system by including equations for parameters in the model. Such a task of joint estimation of the state and parameter is nonlinear even for linear systems. For Linear Structure Models (LSModels), an analytical method is proposed for the transition to an auxiliary model in which the parameter vector is expanded by initial states and the task of identifying parameter and initial states becomes linear. With the help of an auxiliary state vector, the initial dynamic model is reduced to an auxiliary model with residual. In this case, the auxiliary model does not contain derivatives of the measured elements of the initial dynamic model, but contains filtered measured elements. The proof of the identity of solutions according to the initial and auxiliary models is given. An Iterative algorithm of identification of order, parameters and state estimation is proposed. An analytical example of solving the problem of joint estimation of parameters and state for the heat equation is given and its software implementation in the MATLAB is discussed in detail. Next, another auxiliary model is proposed. If the first implies that the order of the differential equation is unknown but only limited by a certain value, then the second model has a given order. Now there can be two types of auxiliary models to it. An example of a nonlinear initial model is given. VL - 8 IS - 2 ER -
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