Kalman observer design for marine craft models represented in frequency domain

В статье описывается предложенный авторами новый подход к построению стационарного оптимального наблюдателя Калмана по частотной характеристике линейного динамического объекта. Предложенный подход может иметь множество практических применений в задачах управления и анализа состояния морских судов так как различные частные виды движений судов описываются линейными динамическими стационарными моделями представленными изначально именно в частотной области. В частности в работе рассматривается задача оценки состояния системы курсового автопилота, использующего только измерения компаса. Для достижения цели работы – построения соотношений для определения наблюдателя Калмана по частотной характеристике объекта, использовался один из методов поиска решения стабилизирующего решения алгебраического уравнения Риккати – метод резольвенты. Модификация метода резольвенты позволила осуществить целевое построение с привлечением дополнительного условия на свойство линейного динамического объекта, сужающего область применения предложенного подхода до только полностью наблюдаемых объектов. A new approach which constructs a steady-state optimal Kalman observer by frequency response of a linear dynamic object is proposed in the paper. The proposed approach tend to have many practical applications in the area of control and state analysis of sea vessels problems, since various particular ship movements types described by linear dynamic stationary models presented initially in the frequency domain. In particular, the paper considers the problem of assessing the state of the system of the course autopilot using only compass measurements. To achieve the goal of the work - constructing relations for determining the Kalman observer by the object frequency characteristic, one of the algebraic Riccati equation solution method was improved. It was the resolvent method. The proposed resolvent method modification allows reaching the paper goal under one additional auxiliary condition to the linear dynamic object property. The condition narrows the scope of the proposed approach to only completely observable objects.

1/ε problem in resurgence

This paper considers the 1/ε problem, which is the divergent behavior of the ground state energy of asymmetric potential in quantum mechanics, which is calculated with semi-classical expansion and resurgence technique. Using resolvent method, It is shown that including not only one complex bion but multi-complex bion and multi-bounce contributions solves this problem. This result indicates the importance of summing all possible saddle points contribution and also the relationship between exact WKB and path integral formalism.

Resolvent Method for Calculating Dispersion Spectra of the Shear Waves in the Phononic Plates and Waveguides

We propose a new method for calculating dispersion spectra of shear waves in the two-dimensional free phononic plates made of solid matrix with periodically distributed inclusions and in the waveguides composed of a phononic layer between two periodic substrates. The method proceeds from the propagator M which involves exact integration in the depth coordinate. Because the components of M can be very large, the dispersion equation for a free plate is recast in terms of the resolvent of propagator R = (αI - M)-1 (α is a constant) which is numerically stable. The resolvent is the central object of the method. Another key tool, which comes into play in the case of a waveguide, is a projector P expressed as a contour integral of the resolvent of the substrate. The projector allows to extract the "physical" modes decreasing into the depth of the substrates without solving the wave equation. The resulting dispersion equation for a waveguide defined via the projectors for the substrates and the resolvent for the enclosed layer is numerically stable. We provide several options for the calculation of the resolvent and projector. Besides, special attention is given to derivation of the dispersion equations for the uncoupled symmetric and antisymmetric dispersion branches in the case of mirror-symmetric structures.

Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory.