Impact of Model Uncertainty Descriptions for High-Purity Distillation Control zy The ways in which modeling uncertainties are described for a particu- lar process critically affects the results obtained in robustness studies. In this paper, four multivariable robust stability methodologies are used to characterize and analyze the effects of model inaccuracy due to non- linearity in high-purity distillation processes. The unstructured and structured singular value, numerical range, and a mapping of det(/ + G,GJ are compared in terms of their ability to predict the stabil- ity of the dual-composition control system over a wide composition range. The importance of using uncertainty descriptions that include a realistic representation of the phase-magnitude relationship as well as the correlations between uncertainties in each element of the model is demonstrated. The conservatism associated with norm-bounded uncer- tainty descriptions reveals itself by the extent of detuning needed to insure stability and the subsequent degradation in control perfor- mance. zyxwvutsrqp Karen A. McDonald Ahmet Palazoglu B. Wayne Bequette Department of Chemical Engineering University of California Davis, CA 95616 introduction zyxwvutsrqp Control systems for chemical processes are typically designed using an approximate, linear, time-invariant model of the plant. The actual plant dynamics may differ from the nominal model due to many sources of uncertainty, such as nonlinearity, the selection of low-order models to represent a plant with inher- ently high-order dynamics, inaccurate identification of model parameters due to poor measurements or incomplete knowledge, and uncertainty in the manipulative variables and/or time-vary- ing phenomena. In light of the differences between the actual plant and the nominal model, it is necessary to insure that the control system will be stable (and meet some predetermined per- formance criteria) when applied to the actual plant. One of the most difficult steps in analyzing the robust stabil- ity and performance of any control system is the specification of an estimate of the uncertainty associated with the nominal pro- cess model. It is a critical step because an overestimation of the model inaccuracy will lead to excessively poor control perfor- mance and an underestimation may lead to instability. Several papers discuss ways in which model inaccuracy can be described and methods that can be used for assessing robust stability. The most common multivariable approaches that use singular values Correspondence concerning this paper may zyxwvutsrqpo be addressed to K. A. McDonald or A. Palam. The current address of B.W. Beguette is Department of Chemical Engineering, Rensselaer glu. Polytechnic Institute, Troy, NY 12181. (Doyle and Stein, 1981; Arkun et al., 1984) and structured sin- gular values (Doyle, 1982) assume that the actual plant can be described by a norm-bounded perturbation matrix in the fre- quency domain; Figure 1 shows a single-input/single-output (SISO) representation. The structured singular value (SSV) approach provides necessary and sufficient conditions for robust stability and performance for the situation in which uncertainty occurs simultaneously and independently in various parts of the overall control system (e.g., output and input uncertainty) but the perturbation matrix is still norm-bounded. Other ap- proaches that do not require norm-bounded uncertainty descrip- tions are region mapping techniques, such as the methods used by Horowitz and Breiner (1981), Laughlin et al. (1986), and Saeki (1986), and the numerical range approach (Owens, 1984; Palazoglu, 1987). Horowitz uses arbitrarily shaped uncertainty regions on the complex plane to represent uncertain, nonlinear plants and presents a mapping technique to synthesize control- lers. Laughlin utilizes a mapping technique to design SISO IMC controllers for systems characterized by arbitrary uncer- tainty sets. Saeki presents multivariable robust stability criteria for systems with arbitrarily shaped uncertainties. The numerical range approach introduces an effective way of expressing the magnitude-phase characteristics of the process perturbations. In chemical process control, nonlinearity is one of the most significant sources of model inaccuracy. We usually have some knowledge about the structure of model inaccuracy due to non- z 1996 December 1988 Vol. 34, No. 12 AIChE Journal