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MACROBAL: A Program for Robust Data Reconciliation and Gross Error Detection
Copyright © IFAC Modeling and Control of Biotechnical Processes, Colorado, USA, 1992
MACROBAL: A PROGRAM FOR ROBUST DATA RECONCILIATION AND GROSS ERROR DETECTION C. Hellinga and
n. Romeln
Department of Biochemical Engineering, Delft University of Technology, lulianalaan 67, 2628 BC, Delft, The Netherlands
Abstract. A set of linear constraint equations provides the mathematical basis for relating conversion rates in (bio-)chemical processes. In the computer program MACROBAL, running on any type IBM-PC, such equations can be defined by summing up all relevant compounds and the conserved quantities in an 'elemental' composition matrix. MACROBAL uses the concept of the redundancy matrix for analyzing the system of equations. If possible, measured rates will be balanced and non-measured rates will be calculated. A Chi-square test is performed for detecting errors in the system description and/or measured rates. The serial elimination method can be used to trace the error location.
Keywords. Classification; computability; computer software; error analysis; fermentation process; matrix algebra; redundancy; statistics.
INTRODUCTION
Heijden (1991) for detailed information on the calculations). The redundancy matrix could also conveniently be used in the subsequent calculations. MACROBAL uses this method, and has proven to be an easy-to-use program in industry, education and research.
The concept of using linear constraint relation for data reconciliation and error diagnosis of fermentation processes gained attention some decade ago (de Kok and Roels, 1980; Wang and Stephanopoulos,1983). Linear constraint equations can be derived relatively easy by summing up 1) conserved quantities like chemical elements, charge, enthalpy and Gibbs energy, and by specifying 2) the chemical compounds that play the major role in the turnover of these quantities. In doing so, the conversion rates of these compounds are related so that non-measured rates can sometimes be calculated and possibly measured rates can be balanced. Hence, in principle, the accuracy of the measured rates can be improved. The matrix equations that are used to perform all necessary computations are not so easy to solve. Often, a so- called pathological set of equations occurs. The calculability and balanceability of conversion rates depend on the division between measured an non-measured rates and on the location of these rates in the equations. Singularities can make calculations cumbersome. Some measured rates may appear to be balanceable, while others are not, etc. The concept of the redundancy matrix has been worked out to deal with the classification of the conversion rates into calculable and balanceable ones (see van der
In order to perform balancing and error diagnosis (a subset) of the system of equations must be redundant. The degree of redundancy can be increased (or the need for performing measurements decreased) by adding knowledge about metabolic pathways to the black box description. Noorman and coworkers (1991) show that such knowledge can be represented in an analogous matrix form . MACROBAL can then again be used to investigate the possible gain in redundancy and for performing all subsequent calculations.
MAIN PROGRAM FUNCTIONS The program consists of four sections. Figure 1 shows the page where the elemental composition matrix can be defined (up to 16 conserved quantities and 16 compounds). In this example, the heat balance is included as well. In Fig. 2 the "calculation sheet" is presented where the user defines which conversion rates have been 459
REFERENCES
measured, the values of these rates and their standard deviations. MACROBAL returns which rates are calculable and which are balanceable (a "C" or a "B" in the second column), and calculates the appropriate values along with their standard deviations. On the lower lines the results of the Chi-square consistency test are summarized. The third and fourth section are for storing and retrieving data from disk and for an automatic serial elimination procedure respectively.
van der Heijden, R.T.J.M. (1991). State Estimation and Error Diagnosis for Biotechnological Processes, PhD thesis, Delft, The Netherlands. de Kok, H.E. and J.A Roels (1980). Method for the statistical treatment of elemental and energy balances with application to steady state continuous culture. Biotechnol. Bioeng.22, 1097-1104. Noorman, H.J., J.J. Heijnen and K.Ch.AM. Luyben (1991). Linear relations in microbial reaction systems: A general overview of their origin, form, and use. Biotechnol. Bioeng. 38, 603-618. Wang, N.S. and G. Stephanopoulos (1983). Application of macroscopic balances to the identification of gross measurement errors. Biotechnol. Bioeng. 25, 2177-2208.
HARDWARE REQUIREMENTS & DISTRIBUTION MACROBAL runs on an any IBM-PC (XT or above). It requires no mathematical co-processor, no hard disk and runs in alpha-numerical mode only. The program can be obtained by remission of a certified check of $50 to the attention of Ir. A Braat at the above mentioned address.
C water Carbon Dioxide Glucose Biomass Ammonia Oxygen Ethanol HeatProd
I
Fl: add col
• • • H
0.0000 1.0000 6.0000 1.0000 0.0000 0.0000 2.0000 0.0000
2.0000 0.0000 12.000 1.8300 3.0000 0.0000 6.0000 0.0000
F2: del col
I
N
0
1.0000 2.0000 6.0000 0.5600 0.0000 2.0000 1.0000 0.0000
F3: add row
0.0000 0.0000 0.0000 0.1700 1.0000 0.0000 0.0000 0.0000
I
•
Enth -285.000 -393.000 -1260.000 -90.700 -79.400 0.0000 -276.000 1.0000
F4: del row
I
FIO: quit
Fig. 1. MACROBAL screen for defining the set of constraint equations. net cony. rate of
measured st. dev
Water Carbon Dioxide Glucose Biomass Ammonia oxygen Ethanol HeatProd
* * * *
C B B C C B C B
******** 7.50E-02 1.60E-02 ******** ******** 3.60E-02 ******** 3.10E+01
measured cony rate
estimated cony rate
st. dev
********* 1.500E+00 -3.200E-Ol ********* ********* -7.250E-Ol ********* 3.100E+02
2.187E-Ol 1.469E+00 -3.216E-Ol -1.213E+00 2.063E-Ol -6.920E-Ol 8.372E-Ol 3.717E+02
1.25E-Ol 7.37E-02 1.60E-02 2.83E-Ol 4.81E-02 3.28E-02 9.64E-02 1. 41E+Ol
Degree of redundancy: 1. Confidence level: 75 90 95 97.5 Test h = 4.98 against Chi square (1) : 1.32 2.71 3.84 5.02 Measurement error(s) or wrong system definition (with 95 % confidence). Fl: calculate
I
F2: gross error
I
F3: show matrices
Fig. 2. MACROBAL calculation sheet. 460
I
99% 6.63
FlO: quit
MACROBAL: A PROGRAM FOR ROBUST DATA RECONCILIATION AND GROSS ERROR DETECTION C. Hellinga and
n. Romeln
Department of Biochemical Engineering, Delft University of Technology, lulianalaan 67, 2628 BC, Delft, The Netherlands
Abstract. A set of linear constraint equations provides the mathematical basis for relating conversion rates in (bio-)chemical processes. In the computer program MACROBAL, running on any type IBM-PC, such equations can be defined by summing up all relevant compounds and the conserved quantities in an 'elemental' composition matrix. MACROBAL uses the concept of the redundancy matrix for analyzing the system of equations. If possible, measured rates will be balanced and non-measured rates will be calculated. A Chi-square test is performed for detecting errors in the system description and/or measured rates. The serial elimination method can be used to trace the error location.
Keywords. Classification; computability; computer software; error analysis; fermentation process; matrix algebra; redundancy; statistics.
INTRODUCTION
Heijden (1991) for detailed information on the calculations). The redundancy matrix could also conveniently be used in the subsequent calculations. MACROBAL uses this method, and has proven to be an easy-to-use program in industry, education and research.
The concept of using linear constraint relation for data reconciliation and error diagnosis of fermentation processes gained attention some decade ago (de Kok and Roels, 1980; Wang and Stephanopoulos,1983). Linear constraint equations can be derived relatively easy by summing up 1) conserved quantities like chemical elements, charge, enthalpy and Gibbs energy, and by specifying 2) the chemical compounds that play the major role in the turnover of these quantities. In doing so, the conversion rates of these compounds are related so that non-measured rates can sometimes be calculated and possibly measured rates can be balanced. Hence, in principle, the accuracy of the measured rates can be improved. The matrix equations that are used to perform all necessary computations are not so easy to solve. Often, a so- called pathological set of equations occurs. The calculability and balanceability of conversion rates depend on the division between measured an non-measured rates and on the location of these rates in the equations. Singularities can make calculations cumbersome. Some measured rates may appear to be balanceable, while others are not, etc. The concept of the redundancy matrix has been worked out to deal with the classification of the conversion rates into calculable and balanceable ones (see van der
In order to perform balancing and error diagnosis (a subset) of the system of equations must be redundant. The degree of redundancy can be increased (or the need for performing measurements decreased) by adding knowledge about metabolic pathways to the black box description. Noorman and coworkers (1991) show that such knowledge can be represented in an analogous matrix form . MACROBAL can then again be used to investigate the possible gain in redundancy and for performing all subsequent calculations.
MAIN PROGRAM FUNCTIONS The program consists of four sections. Figure 1 shows the page where the elemental composition matrix can be defined (up to 16 conserved quantities and 16 compounds). In this example, the heat balance is included as well. In Fig. 2 the "calculation sheet" is presented where the user defines which conversion rates have been 459
REFERENCES
measured, the values of these rates and their standard deviations. MACROBAL returns which rates are calculable and which are balanceable (a "C" or a "B" in the second column), and calculates the appropriate values along with their standard deviations. On the lower lines the results of the Chi-square consistency test are summarized. The third and fourth section are for storing and retrieving data from disk and for an automatic serial elimination procedure respectively.
van der Heijden, R.T.J.M. (1991). State Estimation and Error Diagnosis for Biotechnological Processes, PhD thesis, Delft, The Netherlands. de Kok, H.E. and J.A Roels (1980). Method for the statistical treatment of elemental and energy balances with application to steady state continuous culture. Biotechnol. Bioeng.22, 1097-1104. Noorman, H.J., J.J. Heijnen and K.Ch.AM. Luyben (1991). Linear relations in microbial reaction systems: A general overview of their origin, form, and use. Biotechnol. Bioeng. 38, 603-618. Wang, N.S. and G. Stephanopoulos (1983). Application of macroscopic balances to the identification of gross measurement errors. Biotechnol. Bioeng. 25, 2177-2208.
HARDWARE REQUIREMENTS & DISTRIBUTION MACROBAL runs on an any IBM-PC (XT or above). It requires no mathematical co-processor, no hard disk and runs in alpha-numerical mode only. The program can be obtained by remission of a certified check of $50 to the attention of Ir. A Braat at the above mentioned address.
C water Carbon Dioxide Glucose Biomass Ammonia Oxygen Ethanol HeatProd
I
Fl: add col
• • • H
0.0000 1.0000 6.0000 1.0000 0.0000 0.0000 2.0000 0.0000
2.0000 0.0000 12.000 1.8300 3.0000 0.0000 6.0000 0.0000
F2: del col
I
N
0
1.0000 2.0000 6.0000 0.5600 0.0000 2.0000 1.0000 0.0000
F3: add row
0.0000 0.0000 0.0000 0.1700 1.0000 0.0000 0.0000 0.0000
I
•
Enth -285.000 -393.000 -1260.000 -90.700 -79.400 0.0000 -276.000 1.0000
F4: del row
I
FIO: quit
Fig. 1. MACROBAL screen for defining the set of constraint equations. net cony. rate of
measured st. dev
Water Carbon Dioxide Glucose Biomass Ammonia oxygen Ethanol HeatProd
* * * *
C B B C C B C B
******** 7.50E-02 1.60E-02 ******** ******** 3.60E-02 ******** 3.10E+01
measured cony rate
estimated cony rate
st. dev
********* 1.500E+00 -3.200E-Ol ********* ********* -7.250E-Ol ********* 3.100E+02
2.187E-Ol 1.469E+00 -3.216E-Ol -1.213E+00 2.063E-Ol -6.920E-Ol 8.372E-Ol 3.717E+02
1.25E-Ol 7.37E-02 1.60E-02 2.83E-Ol 4.81E-02 3.28E-02 9.64E-02 1. 41E+Ol
Degree of redundancy: 1. Confidence level: 75 90 95 97.5 Test h = 4.98 against Chi square (1) : 1.32 2.71 3.84 5.02 Measurement error(s) or wrong system definition (with 95 % confidence). Fl: calculate
I
F2: gross error
I
F3: show matrices
Fig. 2. MACROBAL calculation sheet. 460
I
99% 6.63
FlO: quit