Algebraic independence of values of exponential and elliptic functions

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations.

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Robust H∞ Loop Shaping Controller Synthesis for SISO Systems by Complex Modular Functions

Mathematical and Computational Applications ◽

10.3390/mca26010021 ◽

2021 ◽

Vol 26 (1) ◽

pp. 21

Author(s):

Ahmad Taher Azar ◽

Fernando E. Serrano ◽

Nashwa Ahmad Kamal

Keyword(s):

Design Methodology ◽

Closed Loop ◽

Complex Analysis ◽

Controller Design ◽

Filter Design ◽

Design Procedure ◽

Elliptic Functions ◽

Loop System ◽

Closed Loop System ◽

Loop Shaping

In this paper, a loop shaping controller design methodology for single input and a single output (SISO) system is proposed. The theoretical background for this approach is based on complex elliptic functions which allow a flexible design of a SISO controller considering that elliptic functions have a double periodicity. The gain and phase margins of the closed-loop system can be selected appropriately with this new loop shaping design procedure. The loop shaping design methodology consists of implementing suitable filters to obtain a desired frequency response of the closed-loop system by selecting appropriate poles and zeros by the Abel theorem that are fundamental in the theory of the elliptic functions. The elliptic function properties are implemented to facilitate the loop shaping controller design along with their fundamental background and contributions from the complex analysis that are very useful in the automatic control field. Finally, apart from the filter design, a PID controller loop shaping synthesis is proposed implementing a similar design procedure as the first part of this study.

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