This uncertainty analysis problem is motivated by various industrial initiatives for Quality by Design (QbD) (Juran 1992), which have the objective of designing a manufacturing process so as to ensure that the product achieves its desired quality qualifications despite perturbations in the manufacturing process. The QbD literature refers to the set of allowable parameters as the design space (Juran 1992), which in industrial practice is constructed from the data collected from a large number of time-consuming and expensive experiments (e.g., see Garcia-Munoz et al, 2010, for a description of the state-of-the-art in industrial practice and Togkalidou et al, 2001, for an example of experimental design for generating models for pharmaceutical processes). This presentation considers the alternative construction of the design space based on a first-principles, grey-box, or black-box model of the manufacturing process, as fewer experiments are required to construct a model than to directly construct a design space from experiments.
In the control literature, this design space is typically characterized in terms of a robustness margin (Safonov and Athans 1977). Numerous publications quantify the effects of a specified norm-bounded perturbation in a system model on a signal norm on the system output (Wang et al 1992; Zhou et al, 1995) or on outer bounds for state or output trajectories (Durieu et al 2001; Kishida and Braatz 2010, 2011; Kishida et al 2011; Scott and Barton, 2009). The structured singular value and quadratic stability margin quantify the amount of uncertainty allowed to maintain closed-loop stability or a signal norm characterizing output performance (Ma et al 2001, 2002; Hovd et al 2003; Zhou et al 1995; Boyd et al 1997), for linear time-invariant or time-varying systems or linear systems coupled with bounded nonlinear perturbations. This presentation considers nonlinear systems and defines allowable variations in parameters for instantaneous specified bounds on the states or outputs to be satisfied, as this characterization of output performance is used in QbD applications (Juran 1992). The overall approach applies to both open- and closed-loop systems. As the exact generation of a design space is a challenging computational problem for a large number of parameters (Braatz et al 1994), tight upper bounds are determined.
The system output is approximated by a polynomial function (Taylor series expansion) or a rational function (Pad´e approximant) followed by application of the skewed structured singular value to obtain a set of allowable real parameters that ensures that the system output will be in a target set. A target set described by upper and lower bounds on each output is considered here; the same analysis can be used for an ellipsoidal target set by slight modification of the skewed structured singular value analysis (Smith 1990). Both polynomial and rational approximants can be written in the form of a linear fractional transformation, which is used in the next step of the analysis of applying the skewed structured singular value to quantify the allowable uncertainty set. Algorithms are also discussed that generate the algebraic constructions needed by the method as well as model reductions to reduce computational cost (Russell et al 1997; Russell and Braatz 1998).
Several design spaces are constructed for models reported by pharmaceutical companies, including a crystallization operation and a nasal spray.
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