scholarly journals The Second Kummer Function with Matrix Parameters and Its Asymptotic Behaviour

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Georg Wehowar ◽  
Erika Hausenblas
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Closed Form ◽  
Present Article ◽  
Matrix Differential Equation ◽  
Residue Theorem ◽  
Kummer Function ◽  
Matrix Differential

In the present article, we introduce the second Kummer function with matrix parameters and examine its asymptotic behaviour relying on the residue theorem. Further, we provide a closed form of a solution of a Weber matrix differential equation and give a representation using the second Kummer function.

Closed-Form Forced Response of a Damped, Rotating, Multiple Disks/Spindle System

1997 ◽  
Vol 64 (2) ◽  
pp. 343-352 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Differential Equation ◽  
Rigid Body ◽  
Closed Form ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Forced Response ◽  
Vibration Response ◽  
Matrix Differential Equation ◽  
Spindle System ◽  
Matrix Differential

This paper is to study forced vibration response of a rotating disk/spindle system consisting of multiple flexible circular disks clamped to a rigid spindle supported by two flexible bearings. In particular, the disk/spindle system is subjected to prescribed translational base excitations and externally applied loads. Because of the bearing flexibility, the rigid spindle undergoes infinitesimal rigid-body rocking and translation simultaneously. To model real vibration response that has finite resonance amplitudes, the disks and the bearings are assumed to be viscously damped. Equations of motion are then derived through use of Rayleigh dissipation function and Lagrange’s equation. The equations of motion include three sets of matrix differential equations: one for the rigid-body rocking of the spindle and one-nodal-diameter disk modes, one for the axial translation of the spindle and axisymmetric disk modes, and one for disk modes with two or more nodal diameters. Each matrix differential equation contains either a gyroscopic matrix or a damping matrix or both. The causal Green’s function of each matrix differential equation is determined explicitly in closed form through use of matrix inversion and inverse Laplace transforms. Closed-form forced response of the damped rotating disk/spindle system is then obtained from the causal Green’s function and the generalized forces through convolution integrals. Finally, responses of a disk/spindle system subjected to a concentrated sinusoidal load or an impulsive load are demonstrated numerically as an example.

scholarly journals ON CONDITIONING FOR A GENERAL SYSTEM OF FOUR POINT BOUNDARY VALUE PROBLEMS

1996 ◽  
Vol 27 (3) ◽  
pp. 219-225
Author(s):  
M. S. N. MURTY
Keyword(s):  
Differential Equation ◽  
Fundamental Matrix ◽  
Close Relationships ◽  
General System ◽  
Point Boundary ◽  
Matrix Differential Equation ◽  
Growth Behaviour ◽  
First Order ◽  
Matrix Differential ◽  
The Stability

In this paper we investigate the close relationships between the stability constants and the growth behaviour of the fundamental matrix to the general FPBVP'S associated with the general first order matrix differential equation.

Output Control by Linear Time-Varying Systems using Parametric Identification Methods

2020 ◽  
Vol 21 (7) ◽  
pp. 387-393
Author(s):  
V. Q. Dat ◽  
A. A. Bobtsov
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Control Object ◽  
The State ◽  
Time Varying ◽  
State Variables ◽  
Matrix Differential Equation ◽  
Matrix Differential ◽  
Uniform Observability ◽  
Riccati Matrix

In this paper the problem of control for time-varying linear systems by the output (i.e. without measuring the vector of state variables or derivatives of the output signal) was considered. For the control design, the well-known online procedure for solving the Riccati matrix differential equation is chosen. This procedure involves the synthesis of linear static feedbacks on state variables in the case of known parameters of the plant. If state variables are not measured, then for the observer design using the matrix Riccati differential equation, using the dual scheme, which provides for the transposition of the state matrix and the replacement of the input matrix by the output matrix. It is well known that an observer of state variables built on the basis of a solution of the Riccati matrix differential equation ensures the exponential stability of a closed loop system in the case of uniform observability. Despite the fact that this type of observer can be classified as universal, its have a number of significant drawbacks. The main problem of such observers is the need for accurate knowledge of the parameters and the requirement for uniform observability, which in practice cannot always be realized. Thus, the problem of the new methods design for constructing observers of state variables of linear non-stationary systems is still relevant. Some time ago, a number of methods for the adaptive observers design of state variables for nonlinear systems were proposed. The main idea of the synthesis of observers was based on the transformation of the original dynamic system to a linear regression model containing unknown parameters, which in turn were functions of the initial conditions of the state variables of the control object. This approach in the English language literature is called PEBO. This paper, based on the PEBO method, proposes a new approach for the observers design of state for non-stationary systems. This approach provides the possibility of obtaining monotonic convergence estimates with transient time tuning.

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