A New Matrix Formula for the Inverse Laplace Transformation

1967 ◽  
Vol 89 (2) ◽  
pp. 269-272 ◽  
Author(s):  
C. F. Chen ◽  
R. E. Yates
Keyword(s):  
Laplace Transformation ◽  
Transfer Functions ◽  
High Order ◽  
Partial Fraction ◽  
Inverse Laplace Transformation ◽  
Numerical Errors ◽  
Matrix Operations ◽  
Simple Matrix ◽  
Matrix Formula ◽  
Single Matrix

A new matrix formula for the inverse Laplace transformation is established. After substituting the eigenvalues and coefficients and performing some simple matrix operations, one can obtain the inverse Laplace transformation of the function in question. The regular Heaviside techniques involving partial fraction expansions, function differentiations, and so on, are avoided. Since the formula is general, it is particularly advantageous for use on high-order transfer functions; since the formula is exact, the results have no numerical errors. Hundreds of commonly used transform pairs can be replaced by this single matrix formula.

scholarly journals On the Integration Schemes of Retrieving Impulse Response Functions from Transfer Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Kui Fu Chen ◽  
Yan Feng Li
Keyword(s):  
Numerical Integration ◽  
Transfer Functions ◽  
Sampling Interval ◽  
High Order ◽  
Impulse Response Functions ◽  
Inverse Laplace Transformation ◽  
High Order Scheme ◽  
Integration Schemes ◽  
Order Scheme ◽  
Better Than

The numerical inverse Laplace transformation (NILM) makes use of numerical integration. Generally, a high-order scheme of numerical integration renders high accuracy. However, surprisingly, this is not true for the NILM to the transfer function. Numerical examples show that the performance of higher-order schemes is no better than that of the trapezoidal scheme. In particular, the solutions from high-order scheme deviate from the exact one markedly over the rear portion of the period of interest. The underlying essence is examined. The deviation can be reduced by decreasing the frequency-sampling interval.

scholarly journals The Model Order Reduction Method as an Effective Way to Implement GPC Controller for Multidimensional Objects

Algorithms ◽  
2020 ◽  
Vol 13 (8) ◽  
pp. 178
Author(s):  
Sebastian Plamowski ◽  
Richard W Kephart
Keyword(s):  
Predictive Control ◽  
Model Order Reduction ◽  
Order Reduction ◽  
High Order ◽  
Double Precision ◽  
Model Order ◽  
Numerical Errors ◽  
Multiple Input ◽  
Multidimensional Objects ◽  
Order Reduction Method

The paper addresses issues associated with implementing GPC controllers in systems with multiple input signals. Depending on the method of identification, the resulting models may be of a high order and when applied to a control/regulation law, may result in numerical errors due to the limitations of representing values in double-precision floating point numbers. This phenomenon is to be avoided, because even if the model is correct, the resulting numerical errors will lead to poor control performance. An effective way to identify, and at the same time eliminate, this unfavorable feature is to reduce the model order. A method of model order reduction is presented in this paper that effectively mitigates these issues. In this paper, the Generalized Predictive Control (GPC) algorithm is presented, followed by a discussion of the conditions that result in high order models. Examples are included where the discussed problem is demonstrated along with the subsequent results after the reduction. The obtained results and formulated conclusions are valuable for industry practitioners who implement a predictive control in industry.

Fundamental solution and its validation by numerical inverse Laplace transformation and FEM for a damped Timoshenko beam subjected to impact and moving loads

2018 ◽  
Vol 25 (3) ◽  
pp. 593-611
Author(s):  
Xiayang Zhang ◽  
Haoquan Liang ◽  
Meijuan Zhao
Keyword(s):  
Timoshenko Beam ◽  
Laplace Transformation ◽  
Impact Load ◽  
Moving Load ◽  
Numerical Results ◽  
Dynamic Responses ◽  
Superposition Method ◽  
Load Case ◽  
Inverse Laplace Transformation ◽  
Damping Effect

This paper, taking the clamped boundary condition as an example, develops Su and Ma's fundamental solutions of the dynamic responses of a Timoshenko beam subjected to impact load. Based on that, a further extension regarding the general moving load case is also established. Kelvin–Voigt damping, whether proportionally or nonproportionally damped, is incorporated into the model, making it more comprehensive than the model of Su and Ma. Numerical inverse Laplace transformation is introduced to obtain the time-domain solution, where Durbin's formula and the corresponding convergence criteria are utilized in numerical experiments. Further, the real modal superposition method is applied at an analytical level to validate the numerical results by applying a proportionally damped condition. Total comparisons are made between the methods by sufficient case studies. The dynamic responses with and without damping effect are computed with wider slenderness to verify the correctness and effectiveness of the numerical results. Furthermore, parametric studies regarding the damping coefficients are performed to explore the nonproportional damping effect. The results show that the structural damping has significant influences on the dynamic behaviors and is especially stronger at small slender ratios. As the damping decreases the inherent frequencies and excites the low-frequency modal components more actively, a resonant phenomenon appears in high slenderness case when the beam experiences a low-speed moving load. Additionally, the computations in the moving load case indicate that the algorithm convergence is preferable when the number of grids exceeds 1000.

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