The Best Piecewise Linearization of  Nonlinear Functions

Mazarei, Mohammad Mehdi; Behroozpoor, Ali Asghar; Kamyad, Ali Vahidian

doi:10.4236/am.2014.520305

Cited by 4 publications

(2 citation statements)

References 6 publications

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“…Salah satu contoh yang berbentuk persamaan diferensial biasa nonlinier adalah persamaan Van der Pol. Menurut Mazarei, Behroozpoor dan Kamyad (2014), linearisasi pada sistem persamaan nonlinier adalah cara efektif dalam menentukan solusi pendekatan dan analisa system (Mazarei, Behroozpoor, & Kamyad, 2014).…”

Section: A Latar Belakangunclassified

Analisa Kestabilan dan Solusi Pendekatan Pada Persamaan Van der Pol

Yulida

Karim

2019

JTAM

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Abstrak: Di dalam tulisan ini disajikan analisa kestabilan, diselidiki eksistensi dan kestabilan limit cycle, dan ditentukan solusi pendekatan dengan menggunakan metode multiple scale dari persamaan Van der Pol. Penelitian ini dilakukan dalam tiga tahapan metode. Pertama, menganalisa perilaku dinamik persamaan Van der Pol di sekitar ekuilibrium, meliputi transformasi persamaan ke sistem persamaan, analisa kestabilan persamaan melalui linearisasi, dan analisa kemungkinan terjadinya bifukasi pada persamaan. Kedua, membuktikan eksistensi dan kestabilan limit cycle dari persamaan Van der Pol dengan menggunakan teorema Lienard. Ketiga, menentukan solusi pendekatan dari persamaan Van der Pol dengan menggunakan metode multiple scale. Hasil penelitian adalah, berdasarkan variasi nilai parameter kekuatan redaman, daerah kestabilan dari persamaan Van der Pol terbagi menjadi tiga. Untuk parameter kekuatan redaman bernilai positif mengakibatkan ekuilibrium tidak stabil, dan sebaliknya, untuk parameter kekuatan redaman bernilai negatif mengakibatkan ekuilibrium stabil asimtotik, serta tanpa kekuatan redaman mengakibatkan ekuilibrium stabil. Pada kondisi tanpa kekuatan redaman, persamaan Van der Pol memiliki solusi periodik dan mengalami bifurkasi hopf. Selain itu, dengan menggunakan teorema Lienard dapat dibuktikan bahwa solusi periodik dari persamaan Van der Pol berupa limit cycle yang stabil. Pada akhirnya, dengan menggunakan metode multiple scale dan memberikan variasi nilai amplitudo awal dapat ditunjukkan bahwa solusi persamaan Van der Pol konvergen ke solusi periodik dengan periode dua. Abstract: In this paper, the stability analysis is given, the existence and stability of the limit cycle are investigated, and the approach solution is determined using the multiple scale method of the Van der Pol equation. This research was conducted in three stages of method. First, analyzing the dynamic behavior of the equation around the equilibrium, including the transformation of equations into a system of equations, analysis of the stability of equations through linearization, and analysis of the possibility of bifurcation of the equations. Second, the existence and stability of the limit cycle of the equation are proved using the Lienard theorem. Third, the approach solution of the Van der Pol equation is determined using the multiple scale method. Our results, based on variations in the values of the damping strength parameters, the stability region of the Van der Pol equation is divided into three types. For the positive value, it is resulting in unstable equilibrium, and contrary, for the negative value, it is resulting in asymptotic stable equilibrium, and without the damping force, it is resulting in stable equilibrium. In conditions without damping force, the Van der Pol equation has a periodic solution and has hopf bifurcation. In addition, by using the Lienard theorem, it is proven that the periodic solution is a stable limit cycle. Finally, by using the multiple scale method with varying the initial amplitude values, it is shown that the solution of the Van der Pol equation is converge to a periodic solution with a period of two.

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Section: A Latar Belakangunclassified

Analisa Kestabilan dan Solusi Pendekatan Pada Persamaan Van der Pol

Yulida

Karim

2019

JTAM

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“…In [12], the authors introduced a method to find the best piecewise linearization of nonlinear functions based on an optimization problem that is reduced to linear programming. Another algorithm that is used to find the optimal piecewise linearization for a predefined number of linear segments with particle swarm optimization (PSO), without the knowledge of the function and without ideal partitions, is introduced in [13].…”

Section: Introductionmentioning

confidence: 99%

Usage of Selected Swarm Intelligence Algorithms for Piecewise Linearization

2022

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The paper introduces a new approach to enhance optimization algorithms when solving the piecewise linearization problem of a given function. Eight swarm intelligence algorithms were selected to be experimentally compared. The problem is represented by the calculation of the distance between the original function and the estimation from the piecewise linear function. Here, the piecewise linearization of 2D functions is studied. Each of the employed swarm intelligence algorithms is enhanced by a newly proposed automatic detection of the number of piecewise linear parts that determine the discretization points to calculate the distance between the original and piecewise linear function. The original algorithms and their enhanced variants are compared on several examples of piecewise linearization problems. The results show that the enhanced approach performs sufficiently better when it creates a very promising approximation of functions. Moreover, the degree of precision is slightly decreased by the focus on the speed of the optimization process.

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