43 7 System Analysis 7.1 Introduction Models for chemical reactions: ordinary differential equations a System parameters (reaction velocities) remain to be determined. Observe concentrations Y ( j ) for times t i . Statistics: Nonlinear regression with a regression function h given by (numerical) solution of differential eq. Parameters: Reaction constants and initial conditions (=nuisance param.)
44 � � dZ c dt = f Z � t � ; η, u Initial conditions Z � t 0 � = z 0 η : System parameters, reaction constants, u : Experimental conditions, additional explanatory variables. d Y ( j ) Z ( j ) � t i � + E ( j ) = j = 1 , ..., m ≤ q , i i h ( j ) � x i ; θ � Z ( j ) � t i � , = θ = [ η, z 0 ] Example Penicillin e dZ (1) θ 1 Z (1) (1 − Z (1) /θ 2 ) = dt dZ (2) θ 3 Z (1) − θ 4 Z (2) = dt Z (1) Bacteria, Z (2) Penicillin.
45 7.1 Multiple target variables = multivariate regression. f Minimize � � i ( Y ( j ) − h ( j ) � t i � ) 2 j w j i adequate if errors E ( j ) j = var � E ( j ) are independent with σ 2 � = σ 2 /w j . i i → Minimize − � � 2 � � Y ( j ) − h ( j ) � t i ; θ � i σ j j i Remarks: g σ j can be determined from a graph of the Y ( j ) in time. Smooth! • Times t ( j ) may differ for different j . • i Solution as in the univariate case of nonlinear regression. •
46 7.2 Remarks Computational load: Little is known about studying the interplay between opti- a mization and integration – according to my limited enquiry. � Calculate Derivatives ∂η wrt. parameters along with Z s! a � t � := ∂Z � t � b � dZ � � � �� ∂ ∂ = Z � t � ; η, u f ∂η dt ∂η ∂f ∂z a � t � + ∂f da � t � = dt ∂η Differential eq. for a � t � . (Initial condition: a � t 0 � = 0 .) Analogous: Differential eq. for ∂Z � t � /∂z 0 . See Englezos and Kalogerakis (2001).
47 Simple case: All state variables are observed. c Obtain approximate values for Z ( j ) � t i � and dZ ( j ) � t i � /dt by smoothing the observed Y ( j ) � t i � . � � d � Z � + � dt � t i � = f Z � t i � ; η, u E i • Profile traces! d • “Experiment Effekt” E i correlated in time → confidence intervals are invalid! • −
48 7.3 Programs Spezialized programs, e.g. Ferraris and Donati (1971), Ferraris, Donati, Rejna a and Capr` a (1974) Program SimuSolv b User friendly simulation of ordinary differential equations plus estimation of model parameters. Core: Simulation language ACSL. Program EASY-FIT c
49 Literature Literature “parameter estimation” ,“algebraic”and“differential equation models” . Bard (1974): according to Prof. Bonvin“the basic reference” Englezos and Kalogerakis (2001): Application oriented. Contains chapters on biochemical and petrochemical engineering applications, thermodynamics. Based on Fortran programs !!!
38 Literature xx Bard, Y. (1974). Nonlinear parameter estimation, Academic Press, N. Y. Bates, D. M. and Watts, D. G. (1988). Nonlinear Regression Analysis and its Applications, Wiley, N.Y. Englezos, P. and Kalogerakis, N. (2001). Applied parameter estimation for chemical engineers, Marcel Dekker, N. Y. Ferraris, G. B. and Donati, G. (1971). Analysis of the kinetic models for the reaction of synthesis of methanol, Ing. Chim. Ital. 7 : 53–64. Ferraris, G. B., Donati, G., Rejna, F. and Capr` a, S. (1974). An investigation on kinetic models for ammonia synthesis, Chemical Engineering Science 29 : 1621–1627.
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