Discrete Models

Abstract

The models described in Chapter 3 are continuous time models. The time variable is a continuous function and the models are expressed in terms of differential equations of the form \(L[{\text{u(t)}}] = f(t)\) where u(t) is the response of the system to the excitation f (t), and L[u(t)] is some differential operator acting on u(t). In some applications, for example digital system control, the dynamics of the system is sampled with sampling rate ∆t. The response of the system is presented in terms of a discrete function, ie. a finite vector u = (u 1, u 2,..., u m)T where u k = u(k∆t), k = 1,2,...,m. The input may be also discrete f = (f 1, f 2,…, f m )T where f k = f (k∆t) is the input force applied at t = k∆t. If there exists a linear relation between the exciting force f and the response u then the input-output relation can be expressed in terms of a linear difference equation of order n \( \sum\limits_{j = 0}^n {{a_j}(k){u_{k + j}} = {f_k},k = 1,2,...,m - j,} \) where a j (k) are the coeffients of the difference equation. The finite difference method allows us to replace the differential equations associated with the continuous time model by difference equations representing the dynamics of the discrete time system. An introduction to finite difference approximation is presented in Section 4.2. The solution to initial value linear difference equations may be simulated easily by the computer. In some cases we may obtain an analytical closed form solution of the discrete model. A method for determining analytical solutions to linear difference equations and systems of difference equations with constant coefficients is presented in Section 4.3.