Identification of Physical Dynamical Processes Via Linear Structure Models (Part 2)

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.