Optimal Identification and Estimation of Discrete-Time Transfer Function Models

Abstract

The previous chapter 6 has set the scene for the identification and estimation of transfer function models of stochastic, dynamic systems by revealing the limitations of the linear least squares estimation in this context and showing how instrumental variable modifications to the recursive least squares estimation equations can overcome these limitations in a sub-optimal manner. The reason for these limitations is, of course, that general transfer function models of the kind considered in chapter 6 represent inherently nonlinear and stochastic parameter estimation problems. For reasons that will become obvious as we proceed, the present chapter considers the most interesting of these models, the Box-Jenkins (BJ) model, and shows how the standard, recursive-iterative instrumental variable (SIV) algorithm, as developed and evaluated in chapter 6, can be modified to estimate this model in a statistically optimal manner. For convenience, let us first write down again the details of the Box-Jenkins model: y(k) = B(z −1) A(z −1) u(k − δ) + D(z −1) C(z −1) e(k) e(k) = N(0, σ 2) (7.1) or, in decomposed form: System model : x(k) = B(z −1) A(z −1) u(k − δ) (7.2a) ARMA noise model : ξ (k) = D(z −1) C(z −1) e(k) e(k) = N(0, σ 2) (7.2b) Observation equation : y(k) = x(k) + ξ (k) (7.2c) where, in general, the polynomials A(z −1), B(z −1), C(z −1) and D(z −1) are defined as follows: 197