Abstract
Abstract. In this contribution, the limitations of the Carleman linearization approach are presented and discussed. The Carleman linearization transforms an ordinary nonlinear differential equation into an infinite system of linear differential equations. In order to transform the nonlinear differential equation, orthogonal polynomials which represent solutions of a Sturm–Liouville problem are used as basis. The determination of the time derivate of this basis yields an infinite dimensional linear system that depends on the considered nonlinear differential equation. The infinite linear system has the same properties as the nonlinear differential equation such as limit cycles or chaotic behavior. In general, the infinite dimensional linear system cannot be solved. Therefore, the infinite dimensional linear system has to be approximated by a finite dimensional linear system. Due to limitation of dimension the solution of the finite dimensional linear system does not represent the global behavior of the nonlinear differential equation. In fact, the accuracy of the approximation depends on the considered nonlinear system and the initial value. The idea of this contribution is to adapt the range of validity for the Carleman linearization in order to increase the accuracy of the approximation for different ranges of initial values. Instead of truncating the infinite dimensional system after a certain order a Taylor series approach is used to approximate the behavior of the nonlinear differential equation about different equilibrium points. Thus, the adapted finite linear system describes the local behavior of the solution of the nonlinear differential equation.
Highlights
The large signal analysis of nonlinear circuits is an important subject in modern technologies of integrated circuits (Rugh, 1981)
In the dynamical case such circuits are described by nonlinear differential equations
The transformation of the nonlinear differential equation to linear differential equations are usually performed by a Taylor series in an operating point, which is truncated after the first order
Summary
The large signal analysis of nonlinear circuits is an important subject in modern technologies of integrated circuits (Rugh, 1981). The transformation of the nonlinear differential equation to linear differential equations are usually performed by a Taylor series in an operating point, which is truncated after the first order By use of this technique the solution is only valid for small signals about the operating point. Another linearization technique was developed by Carleman in 1932 (Carleman, 1932) He showed that polynomial differential equations can be represented by an infinite dimensional linear system. This approach yields an approximation in the vicinity of the initial value and improves for initial values near to the origin In this contribution an adaption of the range of validity of the Carleman linearization is presented in order to describe the behavior of the system about different equilibrium points.
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