Dynamic analysis and model reduction

Note that Assumption 4.3 effectively captures the need to use a purge stream to prevent the buildup of the impurity in the units of the recycle loop. 

We denote ucg = Ni/NiiS, i = 1. C — 1,1. Also, we define Uj = Fj/FjS to be the scaled (possibly manipulated) inputs that correspond to the flow rates Fj of the material streams. 

Under Assumptions 4.1-4.4, the generic model in Equation (4.15) becomes 

Equation (4.17) can be written in a more compact and mathematically convenient form as 

The mass-transfer rates represented in Equation (4.17) by uCjj are generally functions of the physical parameters of the system, i.e., uCjj = wCjj(x), and cannot be manipulated independently. Hence, for notational convenience, the corresponding terms have been included in the expressions for f(x, u1) and g (x). 

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Previous work using eigenvalue tracking (ET) as a method of spectral association has been successfully applied for the purposes of dynamic analysis and model reduction. ET uses homo-topy methods that transform a system with known eigenvalue-to-state association into the final system and track the eigenvalue associations as the system is transformed. 

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