This paper presents a method which combines the modelling and simulation of a nonlinear network together. The main idea is to divide the rectangle of interest into simplices by imposing a fixed ordering. It turns out that the creation of a piecewise-linear model and the computation of an approximate solution are greatly simplified hecause of the use of simplices. Some examples show that it is possible to apply the method to 'fixed point' problems and other types of problems.
1. INTRODUCHON
Most electrical engineers are familiar with the technique and physical interpretation of 'linearization' around an 'operating point' of a nonlinear network (or device). This method is to approximate the behaviour of a nonlinear network (or device) around an operating point by an affineS function. Consequently, the analysis of a nonlinear network around the operating point becomes the analysis of a network which is described by an affine function. The operating point is determined by the input to the network. If the 'fixed' input is perturbed by a small amount, then a solution of the linearized system will be closed to the operating point and is an approximate solution of the nonlinear network. Otherwise a new operating point has to be found in order to obtain satisfactory accuracy.
This simple and effective idea has been greatly studied and generalized during the past decade.'-" The domain of a nonlinear network function is divided into a finite number of operating regions. Within an operating region, the network is described by an affine function, i.e. an operating region is similar to the vicinity of an operating point. Whenever the input is so changed that the solution is not in the present operating region, a new region is chosen to compute an approximate solution of the original nonlinear network. Therefore the analysis of a nonlinear network and the computation of an appropriate operating point (operating region) are reduced to the analysis of a sequence of networks described by affine functions. This not only simplifies the analysis, but also gives considerable insight to the characteristics of the network.
It also becomes apparent, from the results which have been derived, that piecewise-linear analysis is especially suitable for analyzing large-scale networks. First of all a large-scale network usually contains so many elements (subnetworks) and so much information that the very detailed behaviour of most of the elements (subnetworks) is not concerned. Instead, we are interested only in the global performance of the connection of these elements (subnetworks). Therefore it is often adequate to use piecewise-linear models, i.e. to describe elements by tables of sample data. Besides, the behaviour of some elements might be too cumbersome to be described by continuously differentiable functions which is required by many iterative methods.
This paper considers a nonlinear network which can be described by