Laplace Transformation for Control and Analysis of Nonlinear System
Keywords:Laplace Transformation, Transfer Function, Stability Analysis, Routh-Hurwitz Criterion, Vidale-Wolfe Advertising Model
Nonlinear systems play an important role in modelling real-world problems. In this paper, the Laplace transform is applied to control and analyze nonlinear systems. For this purpose, the nonlinear system is linearized and converted into a system of algebraic equations by applying the Laplace transform. In this way, the transfer function, which indicates the ratio of the output to the input of the system, is defined. Then, the Routh-Hurwitz criterion is performed to examine any sign changes that exist in the first column of the Routh-Hurwitz array. When all the roots in the first column have the same sign, the system is said to be stable. In addition, the transfer function can be expressed in a partial fraction form to provide the solution of the system when the inverse Laplace transform is used. Obviously, the system of differential equations would not be solved directly using the Laplace transform. For illustration, the Vidale-Wolfe advertising model was studied. The results showed that the system is stable. The graphical solutions revealed that the different advertising strategies would provide different market shares, where the respective performance index was enclosed. In conclusion, the application of the Laplace transformation to nonlinear system is properly presented.