Application of modern estimation and identification techniques to chemical processes

Abstract

This paper describes a new application of modern estimation theory to nonlinear chemical processes. A particularly convenient form of the extended Kalman esti-mator is presented and its applications are discussed. T h e method described may be used to compute nonmeasureable process states and system parameters in real time with an on-line process control computer. An application of the extended Kalman filter to a six-dimensional nonlinear well-stirred reactor is discussed in detail. T h e results clearly indicate the feasibility of on-line application of this technique. In some cases these computations could eliminate the requirement for an on-I ine ona ly s i s . In modern control applications, a description of the physical processes i s required. Often in the chemical processing industries, lack of an appropriate mathematical model of the process prevents effective control. Some of the difficulties associated with model building and parameter identification in multivariable systems may be overcome through the use of extended Kalrnan estimation (2 ). Since the complete nonlinear structure of the model i s retained in this technique and a measure of the accuracy of the models i s directly obtainable, estimation theory lends itself readily to development of models.The extended Kalman estimation technique i s well established and has been applied to many defense and aerospace systems ( 2 to 4 ) . Recent applications in the electric power industry (5) and in supercritical shipboard boilers (6) and tubular reactors (7) indicate a trend toward process industry applications.Section I presents a summary of the linear and extended Kalman equations. Section I1 develops sets of normalized estimation and identification equations for use with the extended Kalman filter. A characteristic set of reactor parameters and typical instrumentation system accuracy i s selected. Reactor responses and simulated observations are also computed. The application and results are presented in Section 111. A summary which includes the conclusions and recommendations for future stndy is presented in Section IV. I. THE EXTENDED KALMAN FILTER Linear Kalman FilterThe problem of finite-time optimal filtering and prediction for linear systems with nonstationary random noise was first solved by Kalman (8). Additional results for application of this technique to discrete time systems ((?,lo) led to easy implementation on a digital computer.The discrete Kalman filter equations specify an optimal estimate of the state of a linear, time-varying, dynamic system observed sequentially in the presence of additive white Gaussian noise. The estimate obtained at each time i s the maximum likelihood estimate conditioned on all observations up to that time. This i s equivalent to the leastsquares estimate of the state, since disturbance noises are Gaussian. The vector difference equationx k + l = @k x k + r k W k describes such a system where k denotes a particular in-

A hybrid neural network‐first principles approach to process modeling

“…3-5) were discretized using Euler's method. The discretization error can be accounted for with a judicious choice of the process state noise wk, as noted by Wells (1971). We did not assume any prior parameterization of the growth rate, which was considered constant within each sampling time and was the parameterp to be directly estimated, given measurements of the process variables: biomass concentration X and substrate concentration S. Measurement noise and process state noise were both assumed white Gaussian, with Rk = N(0, 0.01) and Qk = N(0, 0.001) respectively; these matrices were diagonal, and the choice of values reported here gave the best filter performance.…”

Comparison of two approaches to on-line parameter and state estimation of nonlinear systems

“…The method has been used by a number of authors to study process systems (Wells, 1971;Seinfeld, 1970;Hamilton et al 1973;Wismer and Wells, 1972). The method has been used by a number of authors to study process systems (Wells, 1971;Seinfeld, 1970;Hamilton et al 1973;Wismer and Wells, 1972).…”

An experimental evaluation of Kalman filtering

“…Simulation studies include applications to continuous stirred tank reactors by Seinfeld et al (1969), Seinfeld (1970), andWells (1971) ; to tubular and packed bed reactors by Gavalas and Seinfeld (1969), Joffe and Sargent (1972), McGreavy andVago (1972), andVakil et al (1972) ; to heat exchangers by Coggan and Noton (1970), and Coggan and Wilson (1971b) ; and to a basic oxygen furnace by Wells (1970). In several of these studies, such as those by Seinfeld (1970) and Wells and Larson (1970) the filters were implemented as part of a feedback control scheme and resulted in significantly better control.…”