Pipeline Leak Detection Using a Moderate Gain Nonlinear Observer

2020 ◽  
Author(s):  
Sérgio B. Cunha ◽  
Renan M. Baptista
Keyword(s):  
Differential Equations ◽  
Disturbance Observer ◽  
Dynamic Models ◽  
Numerical Solutions ◽  
Leak Detection ◽  
Detection Algorithm ◽  
Nonlinear Observer ◽  
Detection Techniques ◽  
First Order ◽  
Number Of Segments

Abstract Most pipeline control systems use some sort of autonomous leak detection system as a safety feature. Among the pipeline leak detection techniques, state observers stand out as the most sophisticated and promising technique. But its use has been inhibited as the dynamic models employed so far are large and estimating the states of nonlinear systems is not trivial. Pipeline pressure and flow dynamics have been modelled in the literature by means of different numerical solutions to a pair of first order partial differential equations that express mass and linear momentum conservation. The numerical solution requires discretizing the pipeline length in a finite number of segments, resulting in a system of equations with size of twice the number of segments. Although there is nothing wrong with this approach, a smaller system is more convenient if one is concerned exclusively with pressure and flow at the pipeline entrance and exit sections. In this paper, energetic modelling principles are employed to obtain a pair of first order ordinary differential equations representing the dynamics of long liquid pipelines. A recently introduced nonlinear observer enables straightforward use of linear, constant-gain observers with Lipschitz nonlinear dynamics. This observer gives the designer freedom to choose the observer eigenvalues and enables mathematically proven asymptotic stability with low gains. In this paper this observer, using a second-order model to represent the pipeline dynamics, is used as a pipeline leak detection algorithm. Initially the observer was employed directly as a leak detection algorithm, the leak being indicated by a non-transient difference between the measured and the estimated flows. Afterwards the leak was modeled as a disturbance flow and a disturbance observer was designed. Both algorithms were verified by means of computer simulations. It was found that the two methodologies are capable of detecting and estimating very small leaks, but the disturbance observer seems capable of indicating small holes further way from the measuring points.

scholarly journals Everyone can do Differential Equations

2017 ◽  
Vol 5 (4) ◽  
pp. 131-142
Author(s):  
Shinemin Lin ◽  
Domenick Thomas
Keyword(s):  
Differential Equations ◽  
Dynamic Models ◽  
Numerical Solutions ◽  
Real Life ◽  
Original Research ◽  
Initial Value ◽  
First Order ◽  
Life Issues ◽  
Linear Differential ◽  
Predator Model

Abstract: The original research is to make connection between classroom mathematics and real life issues through dynamic models using Excel.  After we completed several sample models such as Prey and Predator model, SIR model, we found we did really solve initial value differential equation problems. We even solve initial values system of linear differential equations numerically and graphically.  Therefore we extend our research to solve initial value differential equations using the same approach as we create dynamic models.  We tested first order and second order differential equations and all got satisfactory numerical solutions.

Aspects of de Wit's Equations Governing the Apparent Spontaneous Yaw of a Ship

1989 ◽  
Vol 42 (1) ◽  
pp. 144-150
Author(s):  
J. O. Flower
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Linearized Equations ◽  
First Order ◽  
Approximate Analytical Solutions

de Wit has produced an analysis of the apparent spontaneous yaw of a ship when undergoing combined rolling and pitching. This analysis produces a set of four first-order simultaneous differential equations which govern the motion. In de Wit the numerical solutions of these equations for a couple of representative examples are given, as well as the corresponding analytical solutions to the linearized equations. In this communication it is shown how two of the four equations can be solved analytically; these solutions can be used to obtain approximate analytical solutions to the remaining two equations.

scholarly journals Numerical solutions of nonstandard first order initial value problems

1992 ◽  
Vol 5 (1) ◽  
pp. 69-82 ◽  
Author(s):  
M. Venkatesulu ◽  
P. D. N. Srinivasu
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Unique Solution ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Initial Value ◽  
First Order ◽  
Physical Phenomena

Differential equations of the form y′=f(t,y,y′), where f is not necessarily linear in its arguments, represent certain physical phenomena and are known for quite some time. The well known Clairut's and Chrystal's equations fall into this category. Earlier, we established the existence of a (unique) solution of the nonstandard initial value problem (NSTD IV P) y′=f(t,y,y′), y(t0)=y0 under certain natural hypotheses on f. In this paper we present some first order convergent numerical methods for finding the approximate solutions of the NST D I V Ps.

scholarly journals SIMULATION BY FIRST-ORDER LINEAR DIFFERENCE EQUATIONS

2021 ◽  
Author(s):  
Valentyna Lisovska ◽  
Tetyana Kudyk ◽  
Dariia Vasylieva
Keyword(s):  
Differential Equations ◽  
Mathematical Models ◽  
Discrete Time ◽  
Difference Equations ◽  
Dynamic Models ◽  
Discrete Models ◽  
Economic Research ◽  
Systems Of Equations ◽  
Linear Difference Equations ◽  
First Order

The article considers economic and mathematical models and studies the socio-economic processes that develop over time, as well as mathematical models that describe them. These are dynamic models. All variables in dynamic models generally depend on the time that acts as an independent variable. In economic research, there are often problems in which variables acquire discrete numerical values. For example, at the end of the month, quarter, year, etc., production results are optimized; accrual of interest on the bank deposit at the end of the month, six months, at the end of the year. In addition, because computers operate only with numbers, so when using computer technology, all continuous processes are reduced to discrete. In this case, from differential equations that describe certain economic processes, we move to difference equations. There are dynamic models with continuous and discrete time, ie continuous and discrete models. Therefore, depending on the type of dynamics of the system under study, dynamic models can be divided into discrete and continuous. In discrete dynamic models, difference equations or systems of difference equations are used; differential equations or systems of differential equations are used in continuous dynamic models. In addition, in some cases there may be systems with mixed dynamics, then differential-equation equations are used to describe them. Difference equations and systems of equations are used successfully in modeling dynamic processes (in economics, banking, etc.). It is when the change of process occurs abruptly, or discretely, that it is convenient and expedient to apply difference equations and systems of equations. The theory of dynamical systems with discrete time, which arose as a result of building mathematical models of real economic and physical processes at the junction of the theory of difference equations and discrete random processes, is currently experiencing a period of rapid development and widespread use in various spheres of human life. In this paper, we investigate the following equations, as well as show their application to solve economic problems. In particular, discrete models described by first-order difference equations are considered. Considerable attention is paid to the analysis of specific models that are meaningful and widely used in economic theory, banking, etc.

scholarly journals An Accurate Implicit Quarter Step First Derivative Block Hybrid Method (AIQSFDBHM) for Solving Ordinary Differential Equations

2019 ◽  
pp. 1-13
Author(s):  
P. Tumba ◽  
J. Sabo ◽  
A. A. Okeke ◽  
D. I. Yakoko
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hybrid Method ◽  
Numerical Experiments ◽  
Stability Region ◽  
Numerical Solutions ◽  
Initial Value ◽  
First Order ◽  
First Derivative ◽  
Stiff Odes

The new accurate implicit quarter step first derivative blocks hybrid method for solving ordinary differential equations have been proposed in this paper via interpolation and collocation method for the solution of stiff ODEs. The analysis of the method was study and it was found to be consistent, convergent, zero-stability, We further compute the region of absolute stability region and it was found to be Aα − stable . It is obvious that, the numerical experiments considered showed that the methods compete favorably with existing ones. Thus, the pair of numerical methods developed in this research is computationally reliable in solving first order initial value problems, as the results from numerical solutions of stiff ODEs shows that this method is superior and best to solve such problems as in tables and figures.

scholarly journals Transformasi Fourier Multiplikatif Dan Aplikasinya Pada Persamaan Diferensial Multiplikatif

2021 ◽  
Vol 8 (2) ◽  
pp. 149-160
Author(s):  
Aurizan Himmi Azhar ◽  
Sugiyanto Sugiyanto ◽  
Muhammad Wakhid Musthofa ◽  
Muhamad Zaki Riyanto
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Research Methods ◽  
Numerical Solutions ◽  
First Order ◽  
Descriptive Research ◽  
Multiplicative Calculus

This research is a development of multiplicative calculus. This study is about the Fourier multiplicative transformation and its application to the multiplicative differential equation. This study aims to determine the Fourier multiplicative transformation as well as the multiplicative differential equation. This study contains numerical simulations to solve the problem of ordinary multiplicative differential equations of the first order. The methods used in this research are descriptive research methods through the study of literature. The results of this study are the application of multiplicative Fourier transformations to multiplicative differential equations and numerical solutions of ordinary multiplicative differential equations with the Adam Bashforth-Moulton multiplicative method.  Keywords: Multiplicative Calculus, Fourier Multiplicative Transformation, Multiplicative Differential Equations, Adams Bashforth Moulton Multiplicative Method

scholarly journals Numerical computation for solving fuzzy differential equations

2019 ◽  
Vol 16 (2) ◽  
pp. 1026
Author(s):  
Fuziyah Ishak ◽  
Najihah Chaini
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Efficient Method ◽  
Dynamic Systems ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Absolute Error ◽  
Initial Value ◽  
First Order

Fuzzy differential equations (FDEs) play important roles in modeling dynamic systems in science, economics and engineering. The modeling roles are important because most problems in nature are indistinct and uncertain. Numerical methods are needed to solve FDEs since it is difficult to obtain exact solutions. Many approaches have been studied and explored by previous researchers to solve FDEs numerically. Most FDEs are solved by adapting numerical solutions of ordinary differential equations. In this study, we propose the extended Trapezoidal method to solve first order initial value problems of FDEs. The computed results are compared to that of Euler and Trapezoidal methods in terms of errors in order to test the accuracy and validity of the proposed method. The results shown that the extended Trapezoidal method is more accurate in terms of absolute error. Since the extended Trapezoidal method has shown to be an efficient method to solve FDEs, this brings an idea for future researchers to explore and improve the existing numerical methods for solving more general FDEs.

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