scholarly journals Two Simultaneous Leak Diagnosis in Pipelines Based on Input–Output Numerical Differentiation

Sensors ◽  
2021 ◽  
Vol 21 (23) ◽  
pp. 8035
Author(s):  
Adrián Navarro-Díaz ◽  
Jorge-Alejandro Delgado-Aguiñaga ◽  
Ofelia Begovich ◽  
Gildas Besançon
Keyword(s):  
State Vector ◽  
Pressure Head ◽  
Numerical Differentiation ◽  
State Variables ◽  
Test Bed ◽  
Water Shortages ◽  
Third Order ◽  
Large Cities ◽  
Noise Rejection ◽  
Derivatives Of

This paper addresses the two simultaneous leak diagnosis problem in pipelines based on a state vector reconstruction as a strategy to improve water shortages in large cities by only considering the availability of the flow rate and pressure head measurements at both ends of the pipeline. The proposed algorithm considers the parameters of both leaks as new state variables with constant dynamics, which results in an extended state representation. By applying a suitable persistent input, an invertible mapping in x can be obtained as a function of the input and output, including their time derivatives of the third-order. The state vector can then be reconstructed by means of an algebraic-like observer through the computation of time derivatives using a Numerical Differentiation with Annihilatorsconsidering its inherent noise rejection properties. Experimental results showed that leak parameters were reconstructed with accuracy using a test bed plant built at Cinvestav Guadalajara.

scholarly journals Evolutionary Observer Ensemble for Leak Diagnosis in Water Pipelines

Processes ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 913 ◽  
Author(s):  
A. Navarro ◽  
J. A. Delgado-Aguiñaga ◽  
J. D. Sánchez-Torres ◽  
O. Begovich ◽  
G. Besançon
Keyword(s):  
Heuristic Method ◽  
Pressure Head ◽  
Search Space ◽  
Observation Error ◽  
State Variables ◽  
Test Bed ◽  
Interior Node ◽  
Water Pipelines ◽  
Exponentially Stable ◽  
Integral Observation

This work deals with the Leak Detection and Isolation (LDI) problem in water pipelines based on some heuristic method and assuming only flow rate and pressure head measurements at both ends of the duct. By considering the single leak case at an interior node of the pipeline, it has been shown that observability is indeed satisfied in this case, which allows designing an observer for the unmeasurable state variables, i.e., the pressure head at leak position. Relying on the fact that the origin of the observation error is exponentially stable if all parameters (including the leak coefficients) are known and uniformly ultimately bounded otherwise, the authors propose a bank of observers as follows: taking into account that the physical pipeline parameters are well-known, and there is only uncertainty about leak coefficients (position and magnitude), a pair of such coefficients is taken from a search space and is assigned to an observer. Then, a Genetic Algorithm (GA) is exploited to minimize the integration of the square observation error. The minimum integral observation error will be reached in the observer where the estimated leak parameters match the real ones. Finally, some results are presented by using real-noisy databases coming from a test bed plant built at Cinvestav-Guadalajara, aiming to show the potentiality of this method.

Switching Strategy for Optimal Adaptive Third Order Control Systems

1970 ◽  
Vol 3 (10) ◽  
pp. T168-T172
Author(s):  
Satish C. Mohleji ◽  
Paul A. V. Thomas
Keyword(s):  
Control Systems ◽  
State Variables ◽  
Optimal Switching ◽  
Optimal Control Strategy ◽  
Actual System ◽  
Third Order ◽  
Time Optimal ◽  
The Difference ◽  
Bang Bang Control ◽  
Derivatives Of

The conventional design of bang bang control systems for time optimal performance involves an accurate mathematical model of the system and complex computations to determine the optimal switching conditions. Such a design is normally based on the assumption that the system parameters remain fairly constant. Recomputation of optimal trajectories becomes necessary for any change in the parameters. This paper describes a simple adaptive-optimal control strategy for systems representable in phase space and can be easily adopted on a digital controller. Projections of optimal curves are obtained by storing sections of actual system trajectories in the form of tables in the computer memory. Optimal switching instants are determined by shifting and comparing the stored trajectory with the actual trajectory. Since no manipulations are required and the switching curves are obtained by constantly storing and updating the system trajectories, the control remains optimum even when there is a change in parameters. The proposed strategy requires only measurement of error in the system, derivatives of error are formed by using the difference equations, a method well suited for digital formulation of state variables.

scholarly journals Third derivatives of the integrable part of an asteroid Hamiltonian

2007 ◽  
pp. 53-60 ◽  
Author(s):  
R. Pavlovic
Keyword(s):  
Third Order ◽  
Geometrical Condition ◽  
The Third ◽  
Derivatives Of ◽  
Quasi Convexity ◽  
Explicit Expressions

To apply the theorem of Nekhoroshev (1977) to asteroids, one first has to check whether a necessary geometrical condition is fulfilled: either convexity, or quasi-convexity, or only a 3-jet non-degeneracy. This requires computation of the derivatives of the integrable part of the corresponding Hamiltonian up to the third order over actions and a thorough analysis of their properties. In this paper we describe in detail the procedure of derivation and we give explicit expressions for the obtained derivatives. .

scholarly journals Numerical differentiation on scattered data through multivariate polynomial interpolation

2021 ◽  
Author(s):  
Francesco Dell’Accio ◽  
Filomena Di Tommaso ◽  
Najoua Siar ◽  
Marco Vianello
Keyword(s):  
Polynomial Interpolation ◽  
Numerical Differentiation ◽  
Scattered Data ◽  
Monomial Basis ◽  
Multivariate Polynomial Interpolation ◽  
Differentiation Formula ◽  
Local Polynomial ◽  
Leja Points ◽  
Derivatives Of ◽  
Numerical Differentiation Formula

AbstractWe discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

scholarly journals On accurate high-order numerical derivatives computations for quantum chemistry purposes

2018 ◽  
Keyword(s):  
Quantum Chemistry ◽  
Finite Field ◽  
Numerical Differentiation ◽  
Field Method ◽  
High Order ◽  
Parameter Choice ◽  
Derivatives Of ◽  
Finite Field Method ◽  
Account Electron

Various molecular parameters in quantum chemistry could be computed as derivatives of energy over different arguments. Unfortunately, it is quite complicated to obtain analytical expression for characteristics that are of interest in the framework of methods that account electron correlation. Especially it relates to the coupled cluster (CC) theory. In such cases, numerical differentiation comes to rescue. This approach, like any other numerical method has empirical parameters and restrictions that require investigation. Current work is called to clarify the details of Finite-Field method usage for high-order derivatives calculation in CC approaches. General approach to the parameter choice and corresponding recommendations about numerical steadiness verification are proposed. As an example of Finite-Field approach implementation characterization of optical properties of fullerene passing process through the aperture of carbon nanotorus is given.

Identification of principal screws of two- and three-screw systems

2007 ◽  
Vol 221 (12) ◽  
pp. 1701-1715 ◽  
Author(s):  
J-S Zhao ◽  
F Chu ◽  
Z-J Feng
Keyword(s):  
Screw Theory ◽  
Coordinate Systems ◽  
Third Order ◽  
Analytical Methodology ◽  
Partial Derivatives ◽  
Spatial Mechanisms ◽  
Analysis Theory ◽  
Definition Of ◽  
Derivatives Of ◽  
Homogeneous Linear

The current paper proposes a unified analytical methodology to identify the principal screws of two- and three-screw systems. Based on the definition of the pitch of a screw, it first obtains an identical homogeneous quadric equation. According to functional analysis theory, it is known that the partial derivatives of an identical quadric equation with respect to its variables must be zero. Therefore, the paper deduces a set of linear homogeneous equations that are made up of the partial derivatives of the quadric equation. With the existing criteria of non-zero solutions for homogeneous linear algebra equations, it ultimately obtains the formulas of the principal pitches and the associated principal screws of the system. The most outstanding contribution of this methodology is that it proposes a unified analytical approach to identify the principal pitches and the principal coordinate systems of the second-order and the third-order screw systems. This should be a new contribution to the screw theory and will boost its applications to the kinematics analysis of robots and spatial mechanisms.

scholarly journals The Elastic Properties of β-Mg2SiO4 Containing 0.73 wt.% of H2O to 10 GPa and 600 K by Ultrasonic Interferometry with Synchrotron X-Radiation

Minerals ◽  
2020 ◽  
Vol 10 (3) ◽  
pp. 209
Author(s):  
Gabriel D. Gwanmesia ◽  
Matthew L. Whitaker ◽  
Lidong Dai ◽  
Alwin James ◽  
Haiyan Chen ◽  
...  
Keyword(s):  
Finite Strain ◽  
Seismic Velocity ◽  
Strain Analysis ◽  
Data Sets ◽  
Third Order ◽  
Data Set ◽  
Earth’S Mantle ◽  
Ultrasonic Interferometry ◽  
Linear Fit ◽  
Derivatives Of

We measured the elastic velocities of a synthetic polycrystalline β-Mg2SiO4 containing 0.73 wt.% H2O to 10 GPa and 600 K using ultrasonic interferometry combined with synchrotron X-radiation. Third-order Eulerian finite strain analysis of the high P and T data set yielded Kso = 161.5(2) GPa, Go = 101.6(1) GPa, and (∂Ks/∂P)T = 4.84(4), (∂G/∂P)T = 1.68(2) indistinguishable from Kso = 161.1(3) GPa, Go = 101.4(1) GPa, and (∂Ks/∂P)T = 4.93(4), (∂G/∂P)T = 1.73(2) from the linear fit. The hydration of the wadsleyite by 0.73 wt.% decreases Ks and G moduli by 5.3% and 8.6%, respectively, but no measurable effect was noted for (∂Ks/∂P)T and (∂G/∂P)T. The temperature derivatives of the Ks and G moduli from the finite strain analysis (∂KS/∂T)P = −0.013(2) GPaK−1, (∂G/∂T)P = −0.015(0.4) GPaK−1, and the linear fit (∂KS/∂T)P = −0.015(1) GPaK−1, (∂G/∂T)P = −0.016(1) GPaK−1 are in agreement, and both data sets indicating the |(∂G/∂T)P| to be greater than |(∂KS/∂T)P|. Calculations yield ∆Vp(α-β) = 9.88% and ∆VS(α-β) = 8.70% for the hydrous β-Mg2SiO4 and hydrous α-Mg2SiO4, implying 46–52% olivine volume content in the Earth’s mantle to satisfy the seismic velocity contrast ∆Vs = ∆VP = 4.6% at the 410 km depth.

scholarly journals Formulae for Numerical Differentiation

1941 ◽  
Vol 25 (263) ◽  
pp. 19-27 ◽  
Author(s):  
W G. Bickley
Keyword(s):  
Finite Difference ◽  
Numerical Integration ◽  
Numerical Differentiation ◽  
Companion Paper ◽  
Limited Range ◽  
Approximate Methods ◽  
Derivatives Of

In a recent paper (1) formulae were given for the numerical integration of a function in terms of its values at a set of arguments at equal intervals. In this companion paper, formulae for numerical differentiation, using the same data, are collected. Their utility in enabling derivatives of a function given numerically at such a set of arguments to be computed is obvious, the need arises in several approximate methods which are coming more and more into use (2), (3). The formulae avoid the labour of preliminary differencing, and are indeed more convenient than the finite difference formulae when the derivative is required at all the points of subdivision of a limited range.

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