A Fast Computational Approach in Optimal Distributed-Parameter State Estimation

1983 ◽  
Vol 105 (1) ◽  
pp. 1-10 ◽  
Author(s):  
K. Watanabe ◽  
M. Iwasaki
Keyword(s):  
Steady State ◽  
Riccati Equation ◽  
Computation Time ◽  
Computational Approach ◽  
Distributed Parameter ◽  
Kutta Method ◽  
Step Size ◽  
Runge Kutta Method ◽  
Operator Riccati Equation ◽  
Runge Kutta

A fast computational approach is considered for solving of a time-invariant operator Riccati equation accompanied with the optimal steady-state filtering problem of a distributed-parameter system. The partitioned filter with the effective initialization is briefly explained and some relationships between its filter and the well-known Kalman-type filter are shown in terms of the Meditch-type fixed-point smoother in Hilbert spaces. Then, with the aid of these results the time doubling algorithm is proposed to solve the steady-state solution of the operator Riccati equation. Some numerical examples are included and a comparison of the computation time required by the proposed method is made with other algorithms—the distributed partitioned numerical algorithm, and the Runge-Kutta method. It is found that the proposed algorithm is approximately from 40 to 50 times faster than the classical Runge-Kutta method with constant step-size for the case of 9th order mode Fourier expansion.

scholarly journals The Runge–Kutta method in geometric multiplicative calculus

2015 ◽  
Vol 18 (1) ◽  
pp. 539-554 ◽  
Author(s):  
Mustafa Riza ◽  
Hatice Aktöre
Keyword(s):  
Relative Error ◽  
Computation Time ◽  
Kutta Method ◽  
Stability Analyses ◽  
Runge Kutta Method ◽  
Multiplicative Calculus ◽  
Runge Kutta ◽  
General Advantage

This paper illuminates the derivation, applicability, and efficiency of the multiplicative Runge–Kutta method, derived in the framework of geometric multiplicative calculus. The removal of the restrictions of geometric multiplicative calculus on positive-valued functions of real variables and the fact that the multiplicative derivative does not exist at the roots of the function are presented explicitly to ensure that the proposed method is universally applicable. The error and stability analyses are also carried out explicitly in the framework of geometric multiplicative calculus. The method presented is applied to various problems and the results are compared to those obtained from the ordinary Runge–Kutta method. Moreover, for one example, a comparison of the computation time against relative error is worked out to illustrate the general advantage of the proposed method.

scholarly journals Study of electrical circuits using Runge Kutta method of order 4

2021 ◽  
Vol 5 (2) ◽  
pp. 109-120
Author(s):  
Malarvizhi M ◽  
Karunanithi S
Keyword(s):  
Matrix Representation ◽  
Testing Procedure ◽  
Kutta Method ◽  
Step Size ◽  
Integration Interval ◽  
Electrical Circuits ◽  
Voltage Coefficient ◽  
Runge Kutta Method ◽  
Dependent Variables ◽  
Runge Kutta

In this paper, Runge Kutta method of order 4 is used to study the electrical circuits designs through past, intermediate and present voltages. When integrating differential equations with Runge Kutta method of order 4, a constant step size (ℎ) is used until a testing procedure confirms that the discontinuity occurs in the present integration interval. This step size function calculations would take place at the end of the functional calculations, but before the dependent variables were updated. Runge Kutta methods along with convolution are given by array interpretation (Butcher matrix) representation, this leads to identify the equilibrium state. The input parameters indicate the voltage coefficient controlled by current sources and measures it a random periodic time. The output parameters provide stable independent values and calculated from past voltage and current values. Finally solutions are compared with exact values and RK method of order 4 along with Heun, Midpoint and Taylors’s method with various ℎ values.

scholarly journals Numerical Solutions of Nonlinear Ordinary Differential Equations by Using Adaptive Runge-Kutta Method

2019 ◽  
Vol 17 ◽  
pp. 147-154
Author(s):  
Abhinandan Chowdhury ◽  
Sammie Clayton ◽  
Mulatu Lemma
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Adaptive Methods ◽  
Integration Step ◽  
Nonlinear Ordinary Differential Equations ◽  
Kutta Method ◽  
Step Size ◽  
Runge Kutta Method ◽  
Runge Kutta

We present a study on numerical solutions of nonlinear ordinary differential equations by applying Runge-Kutta-Fehlberg (RKF) method, a well-known adaptive Runge-kutta method. The adaptive Runge-kutta methods use embedded integration formulas which appear in pairs. Typically adaptive methods monitor the truncation error at each integration step and automatically adjust the step size to keep the error within prescribed limit. Numerical solutions to different nonlinear initial value problems (IVPs) attained by RKF method are compared with corresponding classical Runge-Kutta (RK4) approximations in order to investigate the computational superiority of the former. The resulting gain in efficiency is compatible with the theoretical prediction. Moreover, with the aid of a suitable time-stepping scheme, we show that the RKF method invariably requires less number of steps to arrive at the right endpoint of the finite interval where the IVP is being considered.

scholarly journals Eficiencia y efectividad del método de continuación aplicado al diseño biaxial de columnas cortas de concreto reforzado

Respuestas ◽  
2013 ◽  
Vol 18 (2) ◽  
pp. 6-15
Author(s):  
Álvaro Ortega-Sierra ◽  
Breiner Reynaldo Sierra-Santos
Keyword(s):  
Initial Value Problem ◽  
Continuation Method ◽  
Kutta Method ◽  
Biaxial Bending ◽  
Initial Value ◽  
Step Size ◽  
Runge Kutta Method ◽  
Quasi Experimental ◽  
Runge Kutta ◽  
The One

 La investigación se realizó con 178 diseños correspondientes a columnas de sección cuadrada, por medio de un estudio descriptivo y cuasi experimental. Para dar solución al sistema de tres ecuaciones no lineales que gobierna el diseño se aplicó el Método de Continuación. De acuerdo con éste método, una solución equivalente a la del sistema anterior está dada por la que se obtiene cuando t=1 en un problema de valor inicial en el intervalo 0 ≤ t ≤ 1, el cual fue resuelto implementando el método de Runge-Kutta de orden 4 con diferentes tamaños de paso. Para explorar la eficiencia se investigó acerca de la existencia de diferencias significativas entre promedios del error numérico al final de dos iteraciones consecutivas, considerando un máximo de cinco. Respecto a la efectividad, ésta se cuantificó para diferentes tolerancias del error en función del tamaño de paso y del número de iteraciones.Palabras clave: flexión biaxial, eficiencia, efectividad, problema de valor inicial, Método de Runge-Kutta, tamaño de paso, iteración. ABSTRACTThe research was conducted with 178 designs for square columns, using a quasi-experimental descriptive study. To solve the system of three nonlinear equations that govern the design was applied continuation method. According to this method, an equivalent solution to the above system is given by the one obtained when an initial value problem in the interval 0 d t d 1, which was resolved by implementing the Runge-Kutta method of order 4 with different sizes step. To explore the efficiency was investigated on the existence of significant differences between means of numerical error at the end of two consecutive iterations, considering a maximum of five. With regard to effectiveness, this was quantified for different tolerances of the error depending on the step size and number of iterations.Keywords: biaxial bending, efficiency, effectiveness, initial value problem, Runge-Kutta method, step size, iteration.  

Simulation and Optimization of an Auto-Thermal Ammonia Synthesis Reactor

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Rakesh Angira
Keyword(s):  
Ammonia Synthesis ◽  
Coupled Model ◽  
Kutta Method ◽  
Step Size ◽  
Simulation And Optimization ◽  
Independent Variables ◽  
Runge Kutta Method ◽  
Reactor Length ◽  
Runge Kutta ◽  
Model Equations

This paper deals with the simulation and optimization of an auto-thermal ammonia synthesis reactor. In the present study, three methods, viz., Euler’s method, Runge-Kutta method (4th order with fixed step size) and NAG subroutine (D02EJF) in MATLAB, are used for simulation. A software package POYMATH is also used to solve three coupled model equations. The effect of top temperature on the optimal reactor length is discussed and results are compared with that reported in literature. Also, the Differential Evolution (DE) in combination with Runge-Kutta method is used for optimization, considering both reactor length and top temperature simultaneously as independent variables. The new results obtained for optimal reactor length and top temperature are presented.

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