Control of quasi rational distributed systems with examples on control of cumulative mass fraction of a particle size distribution.

Abstract

The dynamics and control of a class of processes modeled by hyperbolic partial differential equations and having transfer functions of the form $G\\sb{p}(s)$ = $P\\sb1(s) - P\\sb2(s)e\\sp{-st\\sb{d}}\\over Q(s)$ were investigated. Systems with this transfer function are given the name Quasi Rational Distributed Systems or QRDS. Models for the dynamics of particle size distribution in fluidized bed calciners and crystallizers, and heat exchangers are presented as examples. QRSD have been observed to exhibit a wide range of dynamics. This was explained in terms of the location of zeros. QRDS have an infinity of zeros that could lie in the right half-plane, causing large phase lags and being difficult to control. The theory on the asymptotic location of zeros of quasi polynomials was used to characterize the unique nonminimum phase nature of QRDS. Formulas for factoring QRDS, necessary for model based controller design, were presented. A procedure for designing model based controllers for nonminimum phase system, called Generalized Smith Predictor (GSP), was developed. It uses a Smith Predictor (SP) like structure and pole placement to synthesise the main controller. The pole placement controller parameterization yields adjustable parameters that were used to design robust control systems. The GSP procedure overcomes the inability of previous SP procedures to h and ling robustness in a transparent manner. The GSP design procedure was used to design controllers for the cumulative mass fraction (CMF) in fluidized bed calciners and MSMPR crystallizers. As the CMF can be measured easily, it provides an implementable scheme for controlling particle size distribution. However, the CMF has a complicated QRDS model. The performance of GSP controllers was comparable to that of an optimal PID controller, even when the design was based on approximate rational or rational plus delay models. The GSP design procedure required fewer simulations to determine the "optimal" controller parameters. A framework for designing QRDS to be minimum phase and hence easily controlled was also provided. The theory on the asymptotic location of zeros and a root finding program were used to determine limits on parameters to convert a nonminimum phase QRDS to a minimum phase system. The results for the CMF indicate that PSD in the improved minimum phase calciner can be controlled satisfactorily with a PI controller.