Applications of Affine Systems of Walsh Type to Generate Smooth Basis

The question on affine Riesz basis of Walsh affine systems is considered. An affine Riesz basis is constructed, generated by a continuous periodic function that belongs to the space on the real line, which has a derivative almost everywhere; in connection with the construction of this example, we note that the functions of the classical Walsh system suffer a discontinuity and their derivatives almost vanish everywhere. A method of regularization (improvement of differential properties) of the generating function of Walsh affine system is proposed, and a criterion for an affine Riesz basis for a regularized generating function that can be represented as a sum of a series in the Rademacher system is obtained.

Equivariant filtering framework for inertial-integrated navigation

This paper proposes an Equivariant Filtering (EqF) framework for the inertial-integrated state estimation. As the kinematic system of the inertial-integrated navigation can be naturally modeled on the matrix Lie group SE2(3), the symmetry of the Lie group can be exploited to design an equivariant filter which extends the invariant extended Kalman filtering on the group-affine system and overcomes the inconsitency issue of the traditional extend Kalman filter. We firstly formulate the inertial-integrated dynamics as the group-affine systems. Then, we prove the left equivariant properties of the inertial-integrated dynamics. Finally, we design an equivariant filtering framework on the earth-centered earth-fixed frame and the local geodetic navigation frame. The experiments show the superiority of the proposed filters when confronting large misalignment angles in Global Navigation Satellite Navigation (GNSS)/Inertial Navigation System (INS) loosely integrated navigation experiments.

The properties of the solution for a class of switched systems with internally forced switching

In this paper, the dynamic behavior of a class of switched systems with internally forced switching (IFS) is investigated. By introducing the definitions of continuous dependence and differentiability, the continuous dependence and differentiability of the solution relative to the control function are obtained. In the past studies, the optimal control problem given by IFS mainly focused on a special class of controlled systems (the piece affine system). Our results lay a good foundation for studying the more general internally forced switching problem.

Statistical Inference for Piecewise Affine System Identification

This short note studies the problem of piecewise affine system identification, being a special nonlinear system based on our previous contribution on it. Two different identification strategies are proposed to achieve our mission, such as centralized identification and distributed identification. More specifically, for centralized identification, the total observed input-output data are used to estimate all unknown parameter vectors simultaneously without any consideration on the classification process. But for distributed identification, after the whole observed input-output data are classified into their own right subregions, then part input-output data, belonging to the same subregion, are applied to estimate the unknown parameter vector. Whatever the centralized identification and distributed identification, the final decision is to determine the unknown parameter vector in one linear form, so the recursive least squares algorithm and its modified form with the dead zone are studied to deal with the statistical noise and bounded noise, respectively. Finally, one simulation example is used to compare the identification accuracy for our considered two identification strategies.

APPLICATION OF CIRCLES FOR CONJUGATION OF FLAT CONTOURS OF THE FIRST ORDER OF SMOOTHNESS

In geometric modeling of contours, especially for conjugation of sections of flat contours of the first order of smoothness, arcs of circles can be applied. The article proposes ways to determine the equations of a circle for two ways of its problem: the problem of a circle with a point and two tangents, none of which contains a given point, and the problem of a circle with three tangents. The equations of the circles were determined in both cases using a projective coordinate system. In the first case, when a circle is given by a point and two tangents, neither of which contains this point, the center of the conjugation circle is defined as the point of intersection of two locus of points - the bisector of the angle between the tangents and the parabola, the focus of which is a given point. given tangents. In the general case, there are 2 conjugation circles for which canonical equations are defined. Parametric equations of conjugate circles, the parameters of which are equal to 0 and ∞ on tangents and equal to one at a given point, with the help of affine and projective coordinates of points of contact are determined first in the projective coordinate system, and then translated into affine system. For the second case, when specifying a circle using three tangent lines, the equation of the second-order curve tangent to these lines is first determined in the projective coordinate system. The tangent lines are taken as the coordinate lines of the projective coordinate system. The unit point of the projective coordinate system is selected in the metacenter of the thus obtained base triangle. The equation of the tangent to the base lines of the second order contains two unknown variables, positive or negative values ​​which determine the location of four possible tangents of the second order. After writing the vector-parametric equation of the tangent curve of the second order in the affine coordinate system, the equation is written to determine the parameters of cyclic points. In order for the equation of the tangent curve of the second order obtained in the projective plane to be an equation of a circle, it must satisfy the coordinates of the cyclic points of the plane, which allows to write the second equation to determine the parameters of cyclic points. By solving a system of two equations, we obtain the required equations of circles tangent to three given lines.

Robust sampled-data control for direct current to direct current converters via switched affine description and error tracking strategy

In this article, a novel sampled-data control method is proposed for direct current to direct current converter. According to its switching property, the direct current to direct current converter is described as a switched affine system. A novel error tracking switching law is designed based on the multi-Lyapunov functions method and sampled-data control strategy with variant sampling intervals. The sufficient condition concerning the state of the constructed switched affine system converging to a finite region is developed, which guarantees the outcome voltage can approach the desired value and achieve the voltage adjustment. Based on it, the condition under uncertain parameters is developed as well, which would be more desirable in applications. The effectiveness of the proposed method is verified by numerical simulations. The proposed method is also applicable to other types of power converters.

Zonotope Parameter Identification for Piecewise Affine System

This paper studies the identification problem for piecewise affine system, which is a special nonlinear system. As the difficulty in identifying piecewise affine system is to determine each separated region and each unknown parameter vector simultaneously, we propose a multi class classification process to determine each separated region. This multi class classification process is similar to the classical data clustering process, and the merit of our strategy is that the first order algorithm of convex optimization can be applied to achieve this classification process. Furthermore to relax the strict probabilistic description on external noise in identifying each unknown parameter vector, zonotope parameter identification algorithm is proposed to computes a set that contains the parameter vector, consistent with the measured output and the given bound of the noise. To guarantee our derived zonotope not growing unbounded with iterations, a sufficient condition for this requirement to hold may be formulated as one linear matrix inequality. Finally a numerical example confirms our theoretical results