On-line T-S fuzzy control using Riccati differential equation

2017 ◽  
Vol 33 (6) ◽  
pp. 3871-3881 ◽  
Author(s):  
Jorge Cervantes ◽  
Wen Yu ◽  
Sergio Salazar
Keyword(s):  
Differential Equation ◽  
Fuzzy Control ◽  
Riccati Differential Equation ◽  
On Line

scholarly journals New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1573
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Badah Mohamed Badah
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Tau Method ◽  
Riccati Differential Equation ◽  
Shifted Jacobi Polynomials ◽  
Linearization Problem ◽  
Linearization Coefficients ◽  
New Formula ◽  
Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

scholarly journals Precise Integration Method for Solving Noncooperative LQ Differential Game

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Hai-Jun Peng ◽  
Sheng Zhang ◽  
Zhi-Gang Wu ◽  
Biao-Song Chen
Keyword(s):  
Differential Equation ◽  
Differential Game ◽  
Integration Method ◽  
Transformation Method ◽  
Linear Quadratic ◽  
Riccati Differential Equation ◽  
Precise Integration ◽  
Precise Integration Method ◽  
Direct Integration Method ◽  
Two Parameters

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

Use pH and ORP as fuzzy control parameters of denitrification in SBR process

2002 ◽  
Vol 46 (4-5) ◽  
pp. 131-137 ◽  
Author(s):  
Y.Z. Peng ◽  
J.F. Gao ◽  
S.Y. Wang ◽  
M.H. Sui
Keyword(s):  
Fuzzy Control ◽  
Sequencing Batch Reactor ◽  
Fuzzy Controller ◽  
Batch Reactor ◽  
Limiting Factor ◽  
Control Parameters ◽  
Carbon Addition ◽  
Brewery Wastewater ◽  
Aeration Time ◽  
On Line

In order to achieve fuzzy control of denitrification in a Sequencing Batch Reactor (SBR) brewery wastewater was used as the substrate. The effects of brewery wastewater, sodium acetate, methanol and endogenous carbon source on the relationships between pH, ORP and denitrification were investigated. Also different quantities of brewery wastewater were examined. All the results indicated that the nitrate apex and nitrate knee occurred in the pH and ORP profiles at the end of denitrification. And when carbon was the limiting factor, through comparing the different increasing rate of pH whether the carbon was enough or not could be known, and when the carbon should be added again could be decided. On the basis of this, the fuzzy controller for denitrification in SBR was constructed, and the on-line fuzzy control experiments comparing three methods of carbon addition were carried out. The results showed that continuous carbon addition at a low rate might be the best method, it could not only give higher denitrification rate but also reduce the re-aeration time as much as possible. It appears promising to use pH and ORP as fuzzy control parameters to control the denitrification time and the addition of carbon.

scholarly journals Solution of Scalar Riccati Differential Equation of Fractional Order

2019 ◽  
Vol 22 (4) ◽  
pp. 68-74
Author(s):  
Fadhel Subhi Fadhel ◽  
◽  
Sabreen Hashim Jasim ◽  
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Riccati Differential Equation

A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Hale Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergence Region ◽  
Riccati Differential Equation ◽  
Adomian Polynomials ◽  
Riccati Differential Equations ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

AbstractIn this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.

A note on meromorphic functions with finite order and of bounded l-index

2021 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
Andriy Bandura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Order ◽  
General Problem ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Riccati Differential Equation

We present a generalization of concept of bounded $l$-index for meromorphic functions of finite order. Using known results for entire functions of bounded $l$-index we obtain similar propositions for meromorphic functions. There are presented analogs of Hayman's theorem and logarithmic criterion for this class. The propositions are widely used to investigate $l$-index boundedness of entire solutions of differential equations. Taking this into account we raise a general problem of generalization of some results from theory of entire functions of bounded $l$-index by meromorphic functions of finite order and their applications to meromorphic solutions of differential equations. There are deduced sufficient conditions providing $l$-index boundedness of meromoprhic solutions of finite order for the Riccati differential equation. Also we proved that the Weierstrass $\wp$-function has bounded $l$-index with $l(z)=|z|.$

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