Laguerre based design of fractional-order PD controller

Mohammad. Tabatabaei,Romina. Salehi 제어로봇시스템학회 2016 제어로봇시스템학회 국제학술대회 논문집 Vol.2016 No.10

In this paper, a fractional-order PD controller based on Laguerre orthonormal functions is constructed for commensurate fractional order systems. The main idea is to expand the transfer functions of the fractional order plant, the desired open loop gain, and the fractional order PD controller in terms of their Laguerre basis functions. The fractional order PD controller coefficients are obtained by matching the first two coefficients of the open loop gain Laguerre series with the desired one. The optimum value of the Laguerre basis function pole is appropriately selected to minimize an integral square error performance index. The numerical simulations demonstrate the performance of the proposed Laguerre based fractional order PD controller, as well.

EXTENSIONS OF MULTIPLE LAURICELLA AND HUMBERT'S CONFLUENT HYPERGEOMETRIC FUNCTIONS THROUGH A HIGHLY GENERALIZED POCHHAMMER SYMBOL AND THEIR RELATED PROPERTIES

Ritu Agarwal,Junesang Choi,Naveen Kumar,Rakesh K. Parmar Korean Mathematical Society 2023 대한수학회보 Vol.60 No.3

Motivated by several generalizations of the Pochhammer symbol and their associated families of hypergeometric functions and hypergeometric polynomials, by choosing to use a very generalized Pochhammer symbol, we aim to introduce certain extensions of the generalized Lauricella function F⁽ⁿ⁾_A and the Humbert's confluent hypergeometric function Ψ⁽ⁿ⁾ of n variables with, as their respective particular cases, the second Appell hypergeometric function F₂ and the generalized Humbert's confluent hypergeometric functions Ψ₂ and investigate their several properties including, for example, various integral representations, finite summation formulas with an s-fold sum and integral representations involving the Laguerre polynomials, the incomplete gamma functions, and the Bessel and modified Bessel functions. Also, pertinent links between the major identities discussed in this article and different (existing or novel) findings are revealed.

Bounds for radii of convexity of some q-Bessel functions

Ibrahim Aktas,Halit Orhan 대한수학회 2020 대한수학회보 Vol.57 No.2

In the present investigation, by applying two different normalizations of the Jackson's second and third $q$-Bessel functions tight lower and upper bounds for the radii of convexity of the same functions are obtained. In addition, it was shown that these radii obtained are solutions of some transcendental equations. The known Euler-Rayleigh inequalities are intensively used in the proof of main results. Also, the Laguerre-P\'olya class of real entire functions plays an important role in this work.

BOUNDS FOR RADII OF CONVEXITY OF SOME q-BESSEL FUNCTIONS

Aktas, Ibrahim,Orhan, Halit Korean Mathematical Society 2020 대한수학회보 Vol.57 No.2

In the present investigation, by applying two different normalizations of the Jackson's second and third q-Bessel functions tight lower and upper bounds for the radii of convexity of the same functions are obtained. In addition, it was shown that these radii obtained are solutions of some transcendental equations. The known Euler-Rayleigh inequalities are intensively used in the proof of main results. Also, the Laguerre-Pólya class of real entire functions plays an important role in this work.

Closed Loop Modelling Method for Non-linear System Using Laguerre Polynomials

Yusuke Hirama,Hiroto Hamane,Fujio Hiroki 제어로봇시스템학회 2010 제어로봇시스템학회 국제학술대회 논문집 Vol.2010 No.10

This paper presents a modelling method for noisy response data of a closed loop with a PI controller. A general pre-￡ltering procedure is not required in this method. A three-step procedure for estimating Laplace transfer function of a process is proposed. The true closed loop response is estimated from noisy response data, exploiting orthonormal properties of Laguerre functions. Then the closed loop transfer function model (called the Laguerre model) is represented by Laplace transforms of Laguerre polynomials approximated to a true response. Lastly, the process transfer function model is computed from the Laguerre model and the PI controller. PI parameters are given by gain constant, time constant and dead time of process approximated to ￡rst-order lag element plus dead-time system. Using this algorithm, the process model is estimated only by the settling time of response data. Simulation and experiment results show that the proposed method is effective for non-linear systems in modelling.

라게르 함수의 조합을 이용한 안테나의 시간영역 해삭

정백호(Baek Ho Jung) 호서대학교 공업기술연구소 2014 공업기술연구 논문집 Vol.33 No.1

전자파 수치해석 문제에서 시간영역 적분방정식을 풀기 위한 (marching-on-in-degree) 기법이 제안된 바가 있다. 이 방법에서는 과도 함수로 전개하여 최종 계산식에서 시간 변수를 해석적으로 제거하였다. 라게르 함수를 시간영역 기저함 수로 사용하면,라게르 함수의 미분은 바로 그 아래 차수까지의 합으로 구성되었다. 본 논문에서는 도선 안테나의 과도 해석을 위하여 새로운 MOD 기법의 공식화를 수행한다. 시간 영역의 기저함수는 라게르 함수의 장점을 그대로 가지면서, 그 미분의 형태는 라게르 함수의 합이 아닌 조합으로 구성된다. In the computational electromagnetics, the marching-on-in-degree (MOD) method has been presented earlier for solving time domain integral equations in a stable fashion. This is accomplished by expanding the transient responses by a complete set of Laguerre functions, which helps one to analytically integrate out the time variable from the final computations. The final equations that are conventionally used will contain a large number of summations. Therefore, it is proposed to use a new basis function set, which is a combination of Laguerre functions. This new basis function set will retain all the advantage of the Laguerre functions while its derivative will now be another combination of polynomials instead of a summation. Using this methodology, we solve time domain electric field integral equation for dipole and loop antennas. Representative numerical results are presented and compared with frequency domain data.

A GENERALIZATION OF THE LAGUERRE POLYNOMIALS

Ali, Asad Korean Mathematical Society 2021 대한수학회논문집 Vol.36 No.2

The main aim of this paper is to introduce and study the generalized Laguerre polynomials and prove that these polynomials are characterized by the generalized hypergeometric function. Also we investigate some properties and formulas for these polynomials such as explicit representations, generating functions, recurrence relations, differential equation, Rodrigues formula, and orthogonality.

GENERALIZATION OF LAGUERRE MATRIX POLYNOMIALS FOR TWO VARIABLES

( Asad Ali ),( Muhammad ),( Zafar Iqbal ) 호남수학회 2021 호남수학학술지 Vol.43 No.1

The main object of the present paper is to introduce the generalized Laguerre matrix polynomials for two variables. We prove that these matrix polynomials are characterized by the generalized hypergeometric matrix function. An explicit representation, generating functions and some recurrence relations are obtained here. Moreover, these matrix polynomials appear as solution of a differential equation.

라게르 힘수를 이용한 일반적인 분산 매질의 시간영역 해석

정백호(Baek-Ho Jung) 호서대학교 공업기술연구소 2011 공업기술연구 논문집 Vol.30 No.1

본 논문에서는 시간영역 유한차분법 (finite difference time-domain,FDTD)을 사용한 일반적인 분산 매질의 해석시, 효과적이고 정확한 해를 얻기 위한 MOD (marching-on in degree) 기법을 제안한다. 기존의 FDTD 방법에서는 시간에 대한 미분항을 유한차분으로^ 시간 영역의 상승적을 합의 항으로 근사시키는 단점이 있었다. 본 논문에서는 전자장과 유전율 및 투자율 등의 모든 시간 변수를 라게르 함수로 전개하고, 라게르 함수의 특성을 이용하여 시간 영역의 상승적을 해석적으로 처리하였다. 대표적인 드바이, 드루드 및 로렌츠 분산 매질에 대한 전자기 과도 응답을 수치예로 보인다. In tms paper, we illustrate how the marching-on in degree (MOD) method can be used fca* efficient and accurate solution of time domain problems when using the finite difference time-domain (FDTD) technique fca* the solution in a general dispasive media. Traditiaial FDTD methods in media dispersion have disadvantages because they approximate time domain derivatives by differences and time domain convolutions by summations. H^e we provide the compact formulations for goi^al dispersive medium models using the property of the convolution between Laguerre basis functions. The basic idea here is that we fit the fields, permittivity and permeability with a series of Laguerre functions in the time domain. Representative numerical examples are presented for transient wave propagation in general Debye, Drude, and a Lorentz dispersive medium.

CERTAIN INTEGRALS INVOLVING ₂F₁, KAMPÉDE FÉRIET FUNCTION AND SRIVASTAVA POLYNOMIALS

Agarwal, Praveen,Chand, Mehar,Choi, Junesang Korean Mathematical Society 2016 대한수학회논문집 Vol.31 No.2

A remarkably large number of integrals whose integrands are associated, in particular, with a variety of special functions, for example, the hypergeometric and generalized hypergeometric functions have been recorded. Here we aim at presenting certain (presumably) new and (potentially) useful integral formulas whose integrands are involved in a product of $_2F_1$, Srivastava polynomials, and $Kamp{\acute{e}}$ de $F{\acute{e}}riet$ functions. The main results are derived with the help of some known definite integrals obtained earlier by Qureshi et al. [4]. Some interesting special cases of our main results are also considered.