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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_7

KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_7

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1 Chapter 7 Development of Empirical Models From Process Data In some situations it is not feasible to develop a theoretical (physically-based model) due to: 1. Lack of information 2. Model complexity 3. Engineering effort required. An attractive alternative: Develop an empirical dynamic model from input-output data. Advantage: less effort is required Disadvantage: the model is only valid (at best) for the range of data used in its development. i.e., empirical models usually don’t extrapolate very well.

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2 Simple Linear Regression: Steady-State Model As an illustrative example, consider a simple linear model between an output variable y and input variable u , where and are the unknown model parameters to be estimated and ε is a random error. Predictions of y can be made from the regression model, Chapter 7 ˆ y 1 2 β β ε y u = + + 1 β 2 β 1 2 ˆ ˆ ˆ β β (7-3) y u = + where and denote the estimated values of β 1 and β 2 , and denotes the predicted value of y . 1 ˆ β 2 ˆ β 1 2 β β ε (7-1) i i i Y u = + + Let Y denote the measured value of y . Each pair of ( u i , Y i ) observations satisfies:
3 Chapter 7 The Least Squares Approach ( 29 2 2 1 1 2 1 1 ε β β (7-2) N N i i i i S Y u = = = = - - The least squares method is widely used to calculate the values of β 1 and β 2 that minimize the sum of the squares of the errors S for an arbitrary number of data points, N : Replace the unknown values of β 1 and β 2 in (7-2) by their estimates. Then using (7-3), S can be written as: 2 1 where the -th residual, , is defined as, ˆ (7 4) N i i i i i i S e i e e Y y = = = - -

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4 Chapter 7 The least squares solution that minimizes the sum of squared errors, S , is given by: ( 29 1 2 ˆ β (7-5) uu y uy u uu u S S S S NS S - = - ( 29 2 2 ˆ β (7-6) uy u y uu u NS S S NS S - = - where: 2 1 1 N N u i uu i i i S u S u = = 1 1 N N y i uy i i i i S Y S u Y = = The Least Squares Approach (continued)
5 Chapter 7 Least squares estimation can be extended to more general models with: 1. More than one input or output variable. 2. Functionals of the input variables u , such as poly- nomials and exponentials, as long as the unknown parameters appear linearly. A general nonlinear steady-state model which is linear in the parameters has the form, 1 β ε (7-7) p j j j y X = = + Extensions of the Least Squares Approach where each X j is a nonlinear function of u .

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6 Chapter 7 The sum of the squares function analogous to (7-2) is 2 1 1 β (7-8) p N i j ij i j S Y X = = = - ( 29 ( 29 (7-9) T S = - β β Y - X Y X
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