Tensor Formulations for the Modelling of Discrete-Time Nonlinear and Multidimensional Systems

摘要

The modelling of nonlinear and multidimensional systems from input and/or output measurements is considered. Tensor concepts are used to reformulate old results and develop several new ones. These results are verified through non-trivial computer simulations. A generalized tensor formulation for the modelling of discrete-time stationary nonlinear systems is presented. Tensor equivalents of the normal equations are derived and several efficient methods for their solution are discussed. Conditions are established that ensure a diagonal correlation tensor so that a solution can be obtained directly without matrix inversion. Using a tensor formulation, a new proof of the Generalized Lattice Theory is obtained. Tensor extensions of the Levinson and Schur algorithms are presented. New two-dimensional lattice parameter models are derived. Using the tensor form of the Generalized Lattice Theory the 2-D multi-point error order-updates are decomposed into 0(N2) single point updates. 2-D extensions of the Levinson and Schur algorithms are given. The quarter plane lattice is considered in detail, first in a general form, then in forms which reduce the computational complexity by assuming shift-invariance. Based on the 2-D lattice, a new nonlinear lattice model is developed. The model is capable of updates in the nonlinear as well as time order. (Author)