Generalized analytical predictor

Abstract

A generalized analytical predictor (GAP) approach (Wong and Seborg, 1984, 1986) has been proposed to enable the use of any feedback controller with the standard analytical predictor.In this Wong-Seborg GAP (WS-GAP), a disturbance predictor is included to predict future values of the load disturbance effect using the difference between the outputs of the actual process and the process model. The derivation of the disturbance predictor is based on the assumption that the disturbance transfer function is the same as the process transfer function and the process model is known exactly. Wellons and Edgar (1985) have modified the WS-GAP to include a generalized disturbance predictor. The structure of the Wellons-Edgar GAP (WE-GAP) is shown to contain the WS-GAP. In the development of the WE-GAP, a general secondorder disturbance transfer function is used. Assuming a firstorder disturbance transfer function to be the process model and using a deadbeat filter, the WE-GAP reduces to the WS-GAP.In this note, an effort is made to unify the two different derivations of the GAP from the viewpoint of parameter estimation. No assumptions on the disturbance transfer function made in the generalized analytical predictors are used in the present derivation. The assumption made here is believed to be more practical, and has led to the use of autoregressive model for predicting the future effects of load disturbances. Parameter Estimation and Bias TermIn parameter adaptive control, the controlled process is usually approximated as a linear discrete equation with its parameters estimated recursively. The discrete equation for a first-order process isThe u and y are deviation variables, that is, Corrcspndcncc concerning this paper should be addressed to Won-Kym Lee where Ys, and U,, are steady state values. Since u ( k ) and y ( k ) are used in parameter estimation, the steady state values Y, and U,, are required. However, these steady state values are not always available, especially for a nonlinear process experiencing frequent changes in operating conditions. This difficulty can be overcome by including a constant term in the discrete equation. Substituting Eqs. 2 and 3 into Eq. 1 giveswith Thus, with the addition of the bias term d ( k ) , the actual measurements of process input and output can be used directly without knowing the steady state values.Furthermore, this bias term has another important function. In the process industries, unmeasured load disturbances are quite common occurrences. These unmeasured disturbances are actually a part of process inputs that is not accounted for in the discrete process model. When a single-input/single-output (SISO) approach is adopted to obtain a "pure" relation between the desired input and output, a term for the unmeasured load disturbance must be included in the process model to prevent the model parameters from being estimated erroneously due to the additional contribution of the unmeasured disturbance. This can be achieved by assuming that the overall effect of all unrneasured distur...