Salnikov M. Modified ellipsoidal estimation method and its applications

The thesis is devoted to the development and justification of the modification of an entire class of ellipsoidal estimation algorithms, ensuring their convergence and robustness with respect to the violation of a priori assumptions about the properties of uncertain quantities when solving problems of guaranteed estimation. An analysis of modern methods of guaranteed estimation showed that the questions of ensuring convergence, robustness, high convergence rate, and computational efficiency are among the most relevant. These questions are largely related to the ellipsoidal estimation method, since by construction, ellipsoidal estimates contain additional domains that do not contain true values. This leads to a decrease in the convergence rate with increase of dimension. The proposed modifications allow to overcome the mentioned drawbacks. In particular, for the problems of estimating linear regression parameters, the additional convergence condition was proposed to be added to ellipsoidal estimation algorithms. The convergence is also proved for the case when there are restrictions on the value of input variables. The method is proposed for generating small changes in these variables that provide an exact solution to the estimation problem, and finite convergence conditions are obtained. The problem of estimating the parameters of a linear regression is analyzed in the presence of bounded noise measurement of regressors. The complexity of its solution is connected with the nonconvexity of the information sets. It is proved that the information sets are a union of a finite number of multidimensional layers, when the set of possible values of measurement errors is a convex polyhedron. The boundary hyperplanes of these sets are determined by the vertices of the polyhedron of the error values. The information sets are convex within each octant when this polyhedron is a multidimensional parallelepiped oriented along the coordinate axes. This property was used in the development of a new efficient algorithm for solution of the linear regression estimation problem based on the modified ellipsoid method. The estimation problems of the state and parameters of linear and some class of nonlinear dynamical systems are solved. The complexity of solving these problems is due to the fact that at the prediction stage it is necessary to approximate the reachability sets by an ellipsoid. The use of the traditional approach based on second–order approximation of a nonlinear mapping leads to strongly overestimated ellipsoidal estimates. It is proposed to use the first–order approximation and an adaptive ellipsoid size increase at the correction stage to ensure a non–empty intersection of the ellipsoidal estimate of the reachable set and the information set associated with measurements. This approach provided a higher rate of convergence at lower computational costs than in case of using the traditional approach. A general approach is formulated and two particular problems of state estimation of dynamic systems with distributed parameters were solved on its basis. For convection-diffusion-reaction processes, as well as processes of magnetic hydrodynamics, a new version of the Petrov-Galerkin finite element method was proposed to obtain their approximating finite-dimensional models, including for three-dimensional domains. These results were used to study the generation of global solar currents. The developed estimation algorithms were used to solve the space weather forecasting problem using an adjusted non-linear model. On the basis of the developed methods for estimating the state and parameters of nonlinear dynamic systems, the problem of determining and compensating unknown displacements in the readings of gyros of strapdown inertial navigation systems was solved.