Research Article
Identification of Physical Dynamical Processes Via Linear Structure Models (Part 2)
Oleg Yu. Kopysov*
Issue:
Volume 8, Issue 2, December 2024
Pages:
22-39
Received:
26 July 2024
Accepted:
24 September 2024
Published:
31 October 2024
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.
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. Fo...
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Research Article
Identification of Physical Dynamical Processes via Linear Structure Models (Part 1)
Oleg Yu. Kopysov*
Issue:
Volume 8, Issue 2, December 2024
Pages:
40-65
Received:
26 July 2024
Accepted:
24 September 2024
Published:
18 December 2024
Abstract: The article discusses methods for identifying parameters of partial differential equations. Identification problems are often called poorly conditioned. However, the reason is the ambiguity of the solution, even at the point of minimality of the criterion. In particular, the article discusses: (1) An analysis of singular values for identify the unambiguous solution. The basis of these methods is a singular value decomposition of the matrix of experimental data, what makes it possible to abandon the inversion of matrices and, as a consequence, translate the problem of ill-conditioned problems into the problem of ambiguity of the solution. (2) The issues of anomalous measurements and combination of various experiments. (3) A universal optimization method for identifying parameters by their complete simple enumeration. The method is based on fast calculation of points on a multidimensional sphere. (4) The issues of identifiability of linear structure models and construction of experiments guaranteeing identification. (5) A method for identifying parameters via projecting linear structure model elements onto the plane of guarantors. (6) An approach to constructing histograms of unknown parameters of dynamic systems before calculating them using any algorithm. The approach is based on linear structure models with parameters on a sphere and a rather unexpected application of singular value decomposition. (7) The methods are accompanied by examples of heat equations. The Appendix containsalgorithmsintheMATLABlanguageforallexamples. (8)The presented optimization, projection and statistical methods based on the concept of linear structure models allow solving the same identification problem in fundamentally different ways, which significantly increases the reliability of the results obtained.
Abstract: The article discusses methods for identifying parameters of partial differential equations. Identification problems are often called poorly conditioned. However, the reason is the ambiguity of the solution, even at the point of minimality of the criterion. In particular, the article discusses: (1) An analysis of singular values for identify the una...
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