Linear difference equation

Abstract: The study of discrete-time integrable systems is currently the focus of an intense activity1, 2, 3, 4.

Abstract: : The following problems are discussed and solved in the paper: Finding computable, necessary and sufficient conditions for complete reachability and complete observability of a linear, time-varying, discrete-time system; Finding sufficient conditions for complete reachability of nonlinear discrete-time systems; Relating reachability to the concept of discrete Pfaffian systems; Obtaining a minimal-dimension difference equation (with possibly variable coefficients) from a given input/output function of a system; Finding necessary and sufficient conditions for Lyapunov stability and finite-time stability of nonlinear difference equations; Giving an algorithm for determining whether a linear difference equation is stable in the finite-time sense. (Author)

Abstract: SUMMARY present a detailed description of a new method of spectral analysis named ‘Sompi’. The basic idea of this method originates in the physical concept of the characteristic property of the linear dynamic system that is described by a linear differential equation. The time series modelling in the Sompi method consists essentially of estimating the governing differential equation of the hypothetical linear dynamic system that has yielded the given time series data. Due to the equivalence of a linear differential equation and a linear difference equation [or an autoregressive (AR) equation], this method takes the form of the familiar AR method. However, our basic concept of the AR model and the exact formulation based on the maximum likelihood principle have led to a model estimation algorithm different from previous AR methods, and further, to spectral estimation with higher resolution and reliability. By the Sompi method, a time series is deconvoluted into a linear combination of coherent oscillations with amplitudes decaying (or growing) exponentially with time, and additional noise. In other words, it yields a line-shaped spectrum in complex frequency space, unlike the traditional harmonic decomposition in real frequency space, and is powerful for the analysis of the decaying characteristics, as well as the periods, of the oscillations. Also, the variances of the spectral estimates by the Sompi method can be given in simple formulae unlike most modern parametric methods. Although some practical problems still remain unresolved, the theory presented here will provide the theoretical prototype for a new discipline of physical spectral analysis.

Abstract: The z-transform is used to solve sampled-data systems which have a periodically time-varying sampling rate, i.e., systems which have a repetitive sampling pattern in which the time duration between the individual samples is not constant. Such systems are described by linear difference equations with periodic coefficients; however, the difference equation which describes the system at sampling instants corresponding to KN, where N is the period of the coefficients of the difference equation, and K = 0, 1, 2, …, is a linear difference equation with constant coefficients. Thus by forming a series of difference equations which individually describe the system at sampling instants corresponding to KN, KN+1, KN+2, …, (K+1)N−1, the time varying features of the system are in essence removed from the analysis and the z-transform can be used to solve the resulting constant coefficient difference equations. Also, the response between the sampling instants can be found using the solutions of these difference equations. The method presented is straightforward and can be used to analyze any linear sampled-data system with a periodic sampling pattern. Such a condition could occur, for example, when a computer is time shared by more than one system or in some telemetering devices which periodically give to control systems information on quantities being monitored but in which the desired information is not available at equally spaced intervals of time. This method can also be used to obtain an approximate solution for the output of any linear system which is excited by a periodic but nonsinusoidal forcing function and, because of the flexibility of the sampling pattern, should give more accurate results than an approximation which uses equally spaced samples. In this analysis, only periodicity of the sampling pattern is assumed, and no relationship between the individual sampling intervals is required. A few examples have been introduced to illustrate the analytical procedure and the features of the response of a system to sinusoidal inputs is indicated in one of the examples.