Relative Gain Array Analysis for Uncertain Process Models

Dan Chen

Ph.D. Candidate  

Department of Chemical Engineering

University of California, Santa Barbara, CA, 93106

danchen@engineering.ucsb.edu

Abstract

The relative-gain array (RGA) has been widely used as a measure of process interactions and as a tool for control structure selection for decentralized (multiloop) control systems. Many fundamental closed-loop system properties, such as stability, decentralized integral controllability and integrity, have been developed based on the open-loop RGA. Furthermore, because plants with large RGA elements are very sensitive to modeling errors, the RGA can be used as a sensitivity measure with respect to model uncertainty. Additional information about the control system, such as decentralized integral controllability, can also be obtained from the RGA.

In practice there are many factors that can contribute to model uncertainty such as plant/model mismatch, the accuracy of the identification method, changes of operating conditions, drift of physical parameters, etc. Thus process models are never perfect. For uncertain plant models, the nominal RGA analysis may provide misleading information about an appropriate controller pairing. Although the process control literature is replete with analyses of the properties and applications of RGA, the effect of model uncertainty on RGA analysis has received little attention. The objective of this research is to develop RGA uncertainty bounds for various uncertainty models and to explore their use in controller design and tuning.

First, analytical expressions for RGA uncertainty bounds are derived for 2 x 2  control problems. Two types of model uncertainty are considered: worst-case bounds (hard limits) and statistical descriptions of uncertainty bounds. The RGA uncertainty bounds for 2 x 2 control problems are then extended to general n x n  control problems. The new RGA uncertainty bounds provide useful information concerning the recommended control structure and the uncertainty associated with the recommendation. Furthermore, RGA uncertainty analysis can be used to estimate the maximum degree of uncertainty in the plant model that will not affect the recommended controller pairing. This useful information provides insight concerning the required accuracy of the plant model and how much the operating condition can change before the controller pairing decision becomes ambiguous.

1.

Chen, D. and D. E. Seborg, “Relative Gain Array (RGA) Analysis for Uncertain Process Models”, paper presented at the AIChE Annual Meeting, Dallas, TX, 1999.