Computation of the periodic steady-state response of nonlinear networks by extrapolation methods
Computation of the periodic steady-state response of nonlinear networks by extrapolation methods
复制标题
通过外推法计算非线性网络的周期稳态响应
DOI：
10.1109/tcs.1980.1084794
复制
发表时间：
1980
影响因子：
2.1
通讯作者：
S. Skelboe
中科院分区：
文献类型：
--
作者：
S. Skelboe
计算周期稳态响应的问题可以用公式表示为求解形式为z = F（z）的非线性方程，其中F（z）是从初始向量z积分一个周期后的非线性网络的解向量。由Y_{r+1} = F（y_r）生成的序列y_0，y_1，\cdots的收敛性可以用外推法加速。本文对标量外推法、向量外推法和最小多项式外推法三种外推方法进行了统一分析。本文的主要结果是给出外推法二次收敛条件的定理。为了得到这个结果的方法进行了研究的线性问题（其中F是一个线性函数）和误差传播特性进行了研究。对于自治系统，可以定义类似于F的称为G的函数。为了从外推方法获得二次收敛，F和G的导数必须是Lipschitz连续的。附录给出了Lipschitz连续的充分条件。讨论的实际问题有关的实施外推方法的收敛定理和误差分析的基础上。外推方法的性能进行了演示，并与其他方法相比，稳态分析的四个例子，两个自治和两个非自治。外推法是非常容易实现的，他们是有效的非线性电路的稳态分析与几个电抗元件引起缓慢衰减的瞬态。
The problem of computing the periodic steady-state response can be formulated as solving a nonlinear equation of the form z = F(z) where F(z) Is the solution vector for the nonlinear network after one period of integration from the initial vector z . The convergence of the sequence y_0 , y_1 , \cdots generated by Y_{r+1} = F(y_r) can be accelerated by extrapolation methods. This paper presents a unified analysis of three extrapolation methods: the scalar and vector \epsilon -algorithms and the minimum polynomial extrapolation algorithm. The main result of the paper is the theorem giving conditions for quadratic convergence of the extrapolation methods. To obtain this result the methods are studied for linear problems (where F is a linear function) and the error propagation properties are investigated. For autonomous systems a function called G similar to F can be defined. In order to obtain quadratic convergence from the extrapolation methods, the derivatives of F and G must be Lipschitz continuous. The appendixes give sufficient conditions for the Lipschitz continuity. A discussion of practical problems related to the implementation of the extrapolation methods is based on the convergence theorem and the error analysis. The performance of the extrapolation methods is demonstrated and compared with other methods for steady-state analysis by four examples, two autonomous and two nonautonomous. Extrapolation methods are very easy to implement, and they are efficient for the steady-state analysis of nonlinear circuits with few reactive elements giving rise to slowly decaying transients.