Automatic Control of Bioprocesses (Control Systems, Robotics and Manufacturing)

Bioprocess Control Bioprocess Control Edited by Denis Dochain First published in France in 2001 by Hermes Science ...

Author: Denis Dochain

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Bioprocess Control

Bioprocess Control

Edited by Denis Dochain

First published in France in 2001 by Hermes Science entitled “Automatique des bioprocédés” First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 6 Fitzroy Square London W1T 5DX UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd, 2008 © Hermes Science, 2001 The rights of Denis Dochain to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Automatique des bioprocédés. English. Bioprocess control / edited by Denis Dochain. p. ; cm. Translation from French. Includes bibliographical references and index. ISBN: 978-1-84821-025-7 1. Biotechnological process control. 2. Biotechnological process monitoring. I. Dochain, D. (Denis), 1956- II. Title. [DNLM: 1. Biomedical Engineering. 2. Bioreactors. 3. Biotechnology. 4. Models, Biological. QT 36 A939 2008a] TP248.25.M65A9813 2008 660.6--dc22 2007046923 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-025-7 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.

Contents

Chapter 1. What are the Challenges for the Control of Bioprocesses? . . . Denis D OCHAIN 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Specific problems of bioprocess control . . . . . . . . . . . . 1.3. A schematic view of monitoring and control of a bioprocess 1.4. Modeling and identification of bioprocesses: some key ideas 1.5. Software sensors: tools for bioprocess monitoring . . . . . . 1.6. Bioprocess control: basic concepts and advanced control . . 1.7. Bioprocess monitoring: the central issue . . . . . . . . . . . . 1.8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 12 12 13 14 15 15 16 16

Chapter 2. Dynamic Models of Biochemical Processes: Properties of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olivier B ERNARD and Isabelle Q UEINNEC

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2.1. Introduction . . . . . . . . . . . . . . . 2.2. Description of biochemical processes 2.2.1. Micro-organisms and their use . 2.2.2. Types of bioreactors . . . . . . . 2.2.3. Three operating modes . . . . . . 2.3. Mass balance modeling . . . . . . . . 2.3.1. Introduction . . . . . . . . . . . . 2.3.2. Reaction scheme . . . . . . . . . 2.3.3. Choice of reactions and variables 2.3.4. Example 1 . . . . . . . . . . . . . 2.4. Mass balance models . . . . . . . . . . 2.4.1. Introduction . . . . . . . . . . . . 2.4.2. Example 2 . . . . . . . . . . . . . 2.4.3. Example 3 . . . . . . . . . . . . .

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2.4.4. Matrix representation . . . . . . . . . . . . . . . . . . . . 2.4.4.1. Example 2 (continuation) . . . . . . . . . . . . . . . 2.4.4.2. Example 1 (continuation) . . . . . . . . . . . . . . . 2.4.5. Gaseous flow . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6. Electroneutrality and affinity constants . . . . . . . . . . 2.4.7. Example 1 (continuation) . . . . . . . . . . . . . . . . . . 2.4.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. Mathematical constraints . . . . . . . . . . . . . . . . . . 2.5.2.1. Positivity of variables . . . . . . . . . . . . . . . . . 2.5.2.2. Variables necessary for the reaction . . . . . . . . . 2.5.2.3. Example 1 (continuation) . . . . . . . . . . . . . . . 2.5.2.4. Phenomenological knowledge . . . . . . . . . . . . 2.5.3. Specific growth rate . . . . . . . . . . . . . . . . . . . . . 2.5.4. Representation of kinetics by means of a neural network 2.6. Validation of the model . . . . . . . . . . . . . . . . . . . . . . 2.6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2. Validation of the reaction scheme . . . . . . . . . . . . . . 2.6.2.1. Mathematical principle . . . . . . . . . . . . . . . . 2.6.2.2. Example 4 . . . . . . . . . . . . . . . . . . . . . . . 2.6.3. Qualitative validation of model . . . . . . . . . . . . . . . 2.6.4. Global validation of the model . . . . . . . . . . . . . . . 2.7. Properties of the models . . . . . . . . . . . . . . . . . . . . . . 2.7.1. Boundedness and positivity of variables . . . . . . . . . . 2.7.2. Equilibrium points and local behavior . . . . . . . . . . . 2.7.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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25 26 26 27 27 28 29 30 30 30 30 31 31 31 32 34 35 35 35 35 36 37 39 39 39 40 40 42 43

Chapter 3. Identification of Bioprocess Models . . . . . . . . . . . . . . . . . Denis D OCHAIN and Peter VANROLLEGHEM

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3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Structural identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Development in Taylor series . . . . . . . . . . . . . . . . . . . . . 3.2.2. Generating series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Examples for the application of the methods of development in series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Some observations on the methods for testing structural identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Practical identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Confidence interval of the estimated parameters . . . . . . . . . .

47 48 49 50 50 51 52 52 54

Contents

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3.3.3. Sensitivity functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Optimum experiment design for parameter estimation (OED/PE) . . . . 3.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Theoretical basis for the OED/PE . . . . . . . . . . . . . . . . . . 3.4.3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Estimation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. Choice of two datasets . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2. Elements of parameter estimation: least squares estimation in the linear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3. Overview of the parameter estimation algorithms . . . . . . . . . . 3.6. A case study: identification of parameters for a process modeled for anaerobic digestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2. Experiment design . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3. Choice of data for calibration and validation . . . . . . . . . . . . 3.6.4. Parameter identification . . . . . . . . . . . . . . . . . . . . . . . . 3.6.5. Analysis of the results . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 57 57 59 61 63 63

Chapter 4. State Estimation for Bioprocesses . . . . . . . . . . . . . . . . . . Olivier B ERNARD and Jean-Luc G OUZÉ

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4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Notions on system observability . . . . . . . . . . . . . . . . . . . . . . 4.2.1. System observability: definitions . . . . . . . . . . . . . . . . . . . 4.2.2. General definition of an observer . . . . . . . . . . . . . . . . . . . 4.2.3. How to manage the uncertainties in the model or in the output . . 4.3. Observers for linear systems . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Luenberger observer . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. The linear case up to an output injection . . . . . . . . . . . . . . . 4.3.3. Local observation of a nonlinear system around an equilibrium point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4. PI observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5. Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6. The extended Kalman filter . . . . . . . . . . . . . . . . . . . . . . 4.4. High gain observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Definitions, hypotheses . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Change of variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3. Fixed gain observer . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4. Variable gain observers (Kalman-like observer) . . . . . . . . . . . 4.4.5. Example: growth of micro-algae . . . . . . . . . . . . . . . . . . . 4.5. Observers for mass balance-based systems . . . . . . . . . . . . . . . . 4.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2. Definitions, hypotheses . . . . . . . . . . . . . . . . . . . . . . . .

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4.5.3. The asymptotic observer . . . . . . . . . . . . . . . 4.5.4. Example . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5. Improvements . . . . . . . . . . . . . . . . . . . . . 4.6. Interval observers . . . . . . . . . . . . . . . . . . . . . . 4.6.1. Principle . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. The linear case up to an output injection . . . . . . 4.6.3. Interval estimator for an activated sludge process . 4.6.4. Bundle of observers . . . . . . . . . . . . . . . . . 4.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Appendix: a comparison theorem . . . . . . . . . . . . . 4.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . .

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96 98 99 101 102 103 105 107 110 111 112

Chapter 5. Recursive Parameter Estimation . . . . . . . . . . . . . . . . . . 115 Denis D OCHAIN 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Parameter estimation based on the structure of the observer . . . . . . . 5.2.1. Example: culture of animal cells . . . . . . . . . . . . . . . . . . . 5.2.2. Estimator based on the structure of the observer . . . . . . . . . . 5.2.3. Example: culture of animal cells (continued) . . . . . . . . . . . . 5.2.4. Calibration of the estimator based on the structure of the observer: theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5. Calibration of the estimator based on the structure of the observer: application to the culture of animal cells . . . . . . . . . . . . . . . 5.2.6. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Recursive least squares estimator . . . . . . . . . . . . . . . . . . . . . . 5.4. Adaptive state observer . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 116 116 117 119 119 124 127 129 133 138 140 141

Chapter 6. Basic Concepts of Bioprocess Control . . . . . . . . . . . . . . . 143 Denis D OCHAIN and Jérôme H ARMAND 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Bioprocess control: basic concepts . . . . . . . . . . . . . . . . . . . . 6.2.1. Biological system dynamics . . . . . . . . . . . . . . . . . . . . . 6.2.2. Sources of uncertainties and disturbances of biological systems 6.3. Stability of biological processes . . . . . . . . . . . . . . . . . . . . . 6.3.1. Basic concept of the stability of a dynamic system . . . . . . . . 6.3.2. Equilibrium point . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Basic concepts of biological process control . . . . . . . . . . . . . . . 6.4.1. Regulation and tracking control . . . . . . . . . . . . . . . . . . . 6.4.2. Strategy selection: direct and indirect control . . . . . . . . . . .

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143 144 144 146 147 147 148 149 150 150 151

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6.4.3. Selection of synthesis method . . . . . . . . . . . . . . . . . . . . . 6.5. Synthesis of biological process control laws . . . . . . . . . . . . . . . . 6.5.1. Representation of systems . . . . . . . . . . . . . . . . . . . . . . . 6.5.2. Structure of control laws . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Advanced control laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1. A nonlinear PI controller . . . . . . . . . . . . . . . . . . . . . . . 6.6.2. Robust control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. Specific approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1. Pulse control: a dialog with bacteria . . . . . . . . . . . . . . . . . 6.7.2. Overall process optimization: towards integrating the control objectives in the initial stage of bioprocess design . . . . . . . . . 6.8. Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . . . 6.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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152 153 153 154 160 160 162 165 165 167 170 170

Chapter 7. Adaptive Linearizing Control and Extremum-Seeking Control of Bioprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Denis D OCHAIN, Martin G UAY, Michel P ERRIER and Mariana T ITICA 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Adaptive linearizing control of bioprocesses . . . . . . . . . . 7.2.1. Design of the adaptive linearizing controller . . . . . . . 7.2.2. Example 1: anaerobic digestion . . . . . . . . . . . . . . . 7.2.2.1. Model order reduction . . . . . . . . . . . . . . . . . 7.2.2.2. Adaptive linearizing control design . . . . . . . . . 7.2.3. Example 2: activated sludge process . . . . . . . . . . . . 7.3. Adaptive extremum-seeking control of bioprocesses . . . . . . 7.3.1. Fed-batch reactor model . . . . . . . . . . . . . . . . . . . 7.3.2. Estimation and controller design . . . . . . . . . . . . . . 7.3.2.1. Estimation equation for the gaseous outflow rate y . 7.3.2.2. Design of the adaptive extremum-seeking controller 7.3.2.3. Stability and convergence analysis . . . . . . . . . . 7.3.2.4. A note on dither signal design . . . . . . . . . . . . 7.3.3. Simulation results . . . . . . . . . . . . . . . . . . . . . . 7.4. Appendix: analysis of the parameter convergence . . . . . . . 7.5. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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173 174 174 176 177 179 183 188 189 191 191 192 195 196 197 202 207

Chapter 8. Tools for Fault Detection and Diagnosis . . . . . . . . . . . . . . 211 Jean-Philippe S TEYER, Antoine G ÉNOVÉSI and Jérôme H ARMAND 8.1. Introduction . . . . . . . . . . . . . . . . 8.2. General definitions . . . . . . . . . . . . 8.2.1. Terminology . . . . . . . . . . . . 8.2.2. Fault types . . . . . . . . . . . . . 8.3. Fault detection and diagnosis . . . . . . 8.3.1. Methods based directly on signals

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8.3.1.1. Hardware redundancy . . . . . . . . . . 8.3.1.2. Specific sensors . . . . . . . . . . . . . 8.3.1.3. Comparison of thresholds . . . . . . . . 8.3.1.4. Spectral analysis . . . . . . . . . . . . . 8.3.1.5. Statistical approaches . . . . . . . . . . 8.3.2. Model-based methods . . . . . . . . . . . . . 8.3.2.1. Parity space . . . . . . . . . . . . . . . . 8.3.2.2. Observers . . . . . . . . . . . . . . . . . 8.3.2.3. Parametric estimation . . . . . . . . . . 8.3.3. Methods based on expertise . . . . . . . . . . 8.3.3.1. AI models . . . . . . . . . . . . . . . . . 8.3.3.2. Artificial neural networks . . . . . . . . 8.3.3.3. Fuzzy inference systems . . . . . . . . . 8.3.4. Choice and combined use of diverse methods 8.4. Application to biological processes . . . . . . . . . 8.4.1. “Simple” biological processes . . . . . . . . 8.4.2. Wastewater treatment processes . . . . . . . 8.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . 8.6. Bibliography . . . . . . . . . . . . . . . . . . . . .

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215 216 217 217 218 218 219 220 221 222 223 224 225 227 227 228 229 231 232

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Chapter 1

What are the Challenges for the Control of Bioprocesses?

1.1. Introduction In simple terms, we can define a fermentation process as the growth of microorganisms (bacteria, yeasts, mushrooms, etc.) resulting from the consumption of substrates or nutrients (sources of carbon, oxygen, nitrogen, phosphorus, etc.). This growth is possible only when favorable “environmental” conditions are present. Environmental conditions refer to physicochemical conditions (pH, temperature, agitation, ventilation, etc.) necessary for good microbial activity. Techniques in the field of biotechnology can be roughly grouped into three major categories: 1. microbiology and genetic engineering; 2. bioprocess engineering; 3. bioprocess control. Microbiology and genetic engineering aim to develop microorganisms, which allow for the production of new products, or aim to choose the best microbial strains so as to obtain certain desired products or product quality. Process engineering chooses the best operating modes or develops processes and/or reactors, which change and improve the output and/or the productivity of bioprocesses. Automatic control aims to increase the output and/or productivity by developing methods of monitoring

Chapter written by Denis D OCHAIN.

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Bioprocess Control

and control, enabling real-time optimization of the bioprocess operation. These approaches are obviously complementary to one another. This book discusses the matter within the context of the final approach. 1.2. Specific problems of bioprocess control Over the past several decades, biotechnological processes have been increasingly used industrially, which is attributed to several reasons (improvement of profitability and quality in production industries, new legislative standards in processing industries, etc.). The problems arising from this industrialization are generally the same as those encountered in any processing industry and we face, in the field of bioprocessing, almost all of the problems that are being tackled in automatic control. Thus, system requirements for supervision, control and monitoring of the processes in order to optimize operation or detect malfunctions are on the increase. However, in reality, very few installations are provided with such systems. Two principal reasons explain this situation: – first of all, biological processes are complex processes involving living organisms whose characteristics are, by nature, very difficult to apprehend. In fact, the modeling of these systems faces two major difficulties. On the one hand, lack of reproducibility of experiments and inaccuracy of measurements result not only in one or several difficulties related to selection of model structure but also in difficulties related to the concepts of structural and practical identifiability at the time of identification of a set of given parameters. On the other hand, difficulties also occur at the time of the validation phase of these models whose sets of parameters could have precisely evolved over course of time. These variations can be the consequence of metabolic changes of biomass or even genetic modifications that could not be foreseen and observed from a macroscopic point of view; – the second major difficulty is the almost systematic absence of sensors providing access to measurements necessary to know the internal functioning of biological processes. The majority of the key variables associated with these systems (concentration of biomass, substrates and products) can be measured only using analyzers on a laboratory scale – where they exist – which are generally very expensive and often require heavy and expensive maintenance. Thus, the majority of the control strategies used in industries are very often limited to indirect control of fermentation processes by control loops of the environmental variables such as dissolved oxygen concentration, temperature, pH, etc. 1.3. A schematic view of monitoring and control of a bioprocess Use of a computer to monitor and control a biological process is represented schematically in Figure 1.1. In the situation outlined, the actuator is the feed rate of the reactor. Its value is the output of the control algorithm, which uses the information of the available process. This information regroups, on the one hand, the state


13

Screen monitor

Set point Computer Keyboard Monitoring algorithm

Software sensors

Control action Output gas

Effluent Concentrated substrate Supply of water

Liquid Sample Bioreactor

Sensors: analyzers and measuring systems

Figure 1.1. Schematic representation of bioprocess control system

of the process to date (i.e. measurements) and, on the other hand, available a priori knowledge (for example, in the form of a “material balance” model type) relative to a dynamic biological process and mutual interactions of different process variables. In certain cases – and in particular, when control objectives directly use variables that could not be measured (certain concentrations of biomass, substrates and/or products) or key parameters of the biological process (growth rate or more generally production rate, yield coefficients, transfer parameters) – information resulting from “in-line” measurements and a priori knowledge will be combined to synthesize “software sensors” or “observers,” whose principles and methods will be presented in Chapters 4 and 5. Thus, according to the available process knowledge and control objectives specified by the user, we will be able to develop and implement more or less complex control algorithms. 1.4. Modeling and identification of bioprocesses: some key ideas The dynamic model concept plays a central role in automatic control. It is in fact on the basis of the time required for the development of the knowledge process that

14

Bioprocess Control

the total design, analysis and implementation of monitoring and control methods are carried out. Within the framework of bioprocesses, the most natural way to determine the models that will enable the characterization of the process dynamics is to consider the material balance (and possibly energy) of major components of the process. It is this approach that we will consider in this work (although certain elements of hybrid modeling, which combines balance equations and neural networks, will be addressed in the chapter on modeling). One of the important aspects of the balance models is that they consist of two types of terms representing, respectively, conversion (i.e. kinetics of various biochemical reactions of the process and conversion yields of various substrates in terms of biomass and products) and the dynamics of transport (which regroups transit of matter within the process in solid, liquid or gaseous form and the transfer phenomena between phases). These models have various properties, which can prove to be interesting for the design of monitoring and control algorithms for bioprocesses, and which will, thus, be reviewed in Chapter 2. Moreover, we will introduce in Chapter 4 on state observers a state transformation that makes it possible to write part of the bioprocess equations in a form independent of the process kinetics. This transformation is largely related to the concept of reaction invariants, which are well known in the literature in chemistry and chemical engineering. An important stage of modeling consists not only of choosing a model suitable and appropriate for describing the bioprocess dynamics studied but also of calibrating the parameters of this model. This stage is far from being understood and therefore no solution has been obtained, given the complexity of models as well as the (frequent) lack of sufficiently numerous and reliable experimental data. Chapter 3 will attempt to introduce the problem of identification of the parameters of the models of the bioprocess (in dealing with questions of structural and practical identifiability as well as experiment design for its identification) and suitable methods to carry out this identification. 1.5. Software sensors: tools for bioprocess monitoring As noted above, sometimes many important variables of the process are not accessible to be measured online. Similarly, many parameters remain unclear and/or are likely to vary with time. There is, thus, a fundamental need to develop a model, which makes it possible to carry out a real-time follow-up of variables and key parameters of the bioprocess. Thus, Chapters 4 and 5 will attempt, respectively, to develop software tools to rebuild the evolution of these parameters and variables in the course of time. Insofar as their design gives reliable values to these parameters and variables, they play the role of sensors and will thus be called “software sensors”. The material is divided between the two chapters on the basis of distinction between state variables (i.e. primarily, component concentrations) whose evolution in time is described by differential equations and parameters (kinetic, conversion and transfer parameters), which are either the functions of process variables (as is typically the case for kinetic parameters


15

such as specific growth rates) or constants (output parameters, transfer parameters)1. For state variables, we will proceed with the design of “software sensors” called state observers (Chapter 4), whereas for estimating the unknown or unclear parameters online, parameter estimators will be used (Chapter 5). Due to space considerations, Chapter 5 will deal exclusively with the estimation of kinetic parameters, which proves to be a more crucial problem to be solved. However, the methods which are developed are also applicable to other parameters. 1.6. Bioprocess control: basic concepts and advanced control An important aspect of bioprocess control is to lay down a stable real time operation, less susceptible to various disturbances, close to a certain state or desired profile compatible with an optimal operating condition. Chapter 6 will attempt to develop the basic concepts of automatic control applied to bioprocesses, particularly the concepts of control and setpoint tracking, feedback, feedforward control and proportional and integral actions. We can also initiate certain control methods specific to bioprocesses. The following chapter will concentrate on the development of more sophisticated control methods with the objective of guaranteeing the best possible bioprocess operation while accounting, in particular, for disturbances and modeling uncertainties. Emphasis will be placed, particularly, on optimal control and adaptive control methods based on the balance model as developed in the chapter on modeling. The objective is clearly to obtain control laws, which seek the best compromise between what is well known in bioprocess dynamics (for example, the reaction scheme and the material balance) and what is less understood (for example, the kinetics). 1.7. Bioprocess monitoring: the central issue With the exception of real-time monitoring of state variables and parameters, there has been little consideration of bioprocess monitoring. In particular, how to manage bioprocesses with respect to various operation problems, which are about malfunctioning or broken down sensors, actuators (valves, pumps, agitators, etc.), or even more basically malfunction of the bioprocess itself, if it starts to deviate from the nominal state (let us not forget that the process implements living organisms, which can possibly undergo certain, at least partial, transformations or changes, which are likely to bring the process to a different state from that expected). This issue is obviously important and cannot be ignored if we wish to guarantee a good real time process operation. This problem calls for all the process information (which is obtained from modeling, physical and software sensors or control). This will be covered in the final chapter.

1. The models used in practice are often so simplified with respect to reality that these parameters can “apparently” undergo certain variations with time. However, it is important to note that these variations are nothing but a reflection of the inaccuracy or inadequacy of the selected model.

16

Bioprocess Control

1.8. Conclusions A certain number of works exist in the literature, which deal with the application of automatic control in bioprocesses. This book is largely based on the following books: [BAS 90, VAN 98]. However, we should also mention other reference works worthy of interest: [MOS 88, PAV 94, PON 92, SCH 00]. Due to lack of space, we have not considered certain topics, which could, however, legitimately have had a place in this book. Initially, the informed reader would have noted that there is no chapter on instrumentation, which is, however, an essential link in monitoring and control. Fortunately, the reader will be able to complement the reading of this work with that (in French) of Boudrant, Corrieu, and Coulet [BOU 94] or that (in English) of Pons [PON 92]. In addition, we did not have the space for approaches such as metabolic engineering, a type of approach, which is already playing a growing role in bioprocess control. We suggest the reader consult the following book on metabolic engineering: [STE 98]. 1.9. Bibliography [BAS 90] G. BASTIN and D. D OCHAIN, On-line Estimation and Adaptive Control of Bioreactors, Elsevier, Amsterdam, 1990. [BOU 94] J. B OUDRANT, G. C ORRIEU and P. C OULET, Capteurs et Mesures en Biotechnologie, Lavoisier, Paris, 1994. [MOS 88] A. M OSER, Bioprocess Technology. Kinetics and Reactors, Springer Verlag, New York, 1988. [PAV 94] A. PAVÉ, Modélisation en Biologie et en Ecologie, Aléas, Lyon, 1994. [PON 92] M.N. P ONS, Bioprocess Monitoring and Control, Hanser, Munich, 1992. [SCH 00] K. S CHÜGERL and K.H. B ELLGARDT, Bioreaction Engineering. Modeling and Control, Springer, Berlin, 2000. [STE 98] G. S TEPHANOPOULOS, J. N IELSEN and A. A RISTIDOU, Metabolic Engineering, Academic Press, Boston, 1998. [VAN 98] J. VAN I MPE, P. VANROLLEGHEM and D. I SERENTANT, Advanced Instrumentation, Data Interpretation and Control of Biotechnological Processes, Kluwer, Amsterdam, 1998.

Chapter 2

Dynamic Models of Biochemical Processes: Properties of Models

2.1. Introduction Modeling biochemical processes is a delicate exercise. Contrary to physics, where there are laws that have been known to man for centuries (Ohm’s law, ideal gas law, Newton’s second law, the principles of thermodynamics, etc.) the majority of the models in biology depend on empirical laws. As it is not possible to base them only on available (and validated) knowledge, it is very important to be able to characterize the reliability of the laws used in the construction of the model. This implies hierarchism in the construction of biochemical models. In this chapter, we will see how to organize knowledge in the model in order to distinguish a reliable part established on the basis of a mass balance, and a more fragile part which will describe the bacterial kinetics. The quality of the model and, above all, its structure must correspond to the objective for which the model was built. In fact, a model can be developed for very different purposes, which will have to be clearly identified from the beginning. Thus, the model could be used to: – reproduce an observed behavior; – explain an observed behavior; – predict the evolution of a system; – help in understanding the mechanisms of the studied system; – estimate variables which are not measured; Chapter written by Olivier B ERNARD and Isabelle Q UEINNEC.

17

18

Bioprocess Control

– estimate parameters of the process; – act on the system to steer and control its variables; – detect an anomaly in the functioning of the process; – etc. The modeling objectives will generally lead to a formalism for designing the model. If we want to explain spatial heterogenity in a fermentor, it will be necessary to resort to a spatialized model (generally described by partial derivative equations). If the objective is to improve the production of a metabolite during the transitional stages, it will be necessary to represent the dynamics of the system. In the same way, the tools that we want to use to achieve the goal will guide us in choosing the model type (continuous/discrete, deterministic/stochastic, etc.). Furthermore, within the limit of these objectives, the model will also have to be adapted to the data available. In fact, a complex model implying a great number of parameters will require a large quantity of data to be identified and validated. Finally, considering the lack of validated laws in biology, the key stage of modeling is the validation of the model. This will be the subject of a special section. It is in fact fundamental to be able to show, on the basis of experimental data, that the model correctly achieves the assigned goals; we invite the reader to refer to [PAV 94] for a thorough reflection on modeling. 2.2. Description of biochemical processes 2.2.1. Micro-organisms and their use Microbial fermentation is a process in which a population of micro-organisms (bacteria, yeasts, moulds, etc.) are grown using certain nutritive elements (nutrients) under favorable surrounding conditions (temperature, pH, agitation, aeration, etc.). It schematically corresponds to the transformation of substances (generally carbonaceous substrates) into products, resulting from the metabolic activity of cells. The main components of the reaction are as follows: – substrates, denoted as Si , which are necessary for the growth of micro-organisms, or even which are precursors of a compound to be produced. These substrates generally contain a source of carbon (glucose, ethanol, etc.) and sometimes nitrogen (NO3 , NH4 , etc.) and phosphorus (PO4 , etc.); – microbial biomasses, denoted as Xi ; – end products, denoted as Pi , for agri-foods (oils, cheese, beer, wines, etc.), chemistry (solvents, enzymes, amino acids, etc.), the pharmaceutical industry (antibiotics, hormones, vitamins, etc.) or for the production of energy (ethanol, biogas, etc.) etc.


19

Mineral salt and vitamins are added to these main components, which, although they seldom appear in the models, are essential for growth. Each type of micro-organism contains some characteristics related to its genetic inheritance and to its regulating systems and fermentation can have various uses: – microbial growth: the primary objective is the growth of the micro-organism itself. This is the case of fermentations aiming to produce baking yeast; – metabolic production: the objective is to synthesize a preferred metabolite (ethanol, penicillin, etc.) using the cell; – substrate consumption: in this case, it is the degradation of the substrate which is desired. In this category, we will primarily find depollution processes (biological treatment of wastewater, breakdown of specific pollutants, etc.); – phenomenologic study: here, the purpose of fermentation is the study of the micro-organism. This can, for example, be to better understand how the micro-organism develops in the natural environment. The majority of biotechnological processes developed at an industrial level use microbial cultures made up of a single species of micro-organism for the possible synthesis of a well defined product (pure culture). However, in certain cases, several species can be made to grow simultaneously, but this is possible only if they are not too competitive. 2.2.2. Types of bioreactors From the view point of mathematical modeling, biological reactors can be divided into two major classes [BAI 86]: – stirred tank reactors (STR) for which the reacting medium is homogenous and the reaction is described by ordinary differential equations; – reactors with a spatial concentration gradient, such as fixed beds, fluidized beds, air lifts, etc., for which the reaction is described by partial derivative equations. In this chapter, we are interested only in the first class of stirred tank reactors, and any reader interested in the second class should see [JAC 96, DOC 94], etc. 2.2.3. Three operating modes Operating modes of bioreactors are generally characterized by liquid exchanges, i.e. by the type of substrate supply of the reactor. We can distinguish three main modes (Figure 2.1). Discontinuous (or batch) mode All the nutritive elements necessary for biological growth are introduced at the beginning of the reaction. Neither supply nor removal (except for some measurements)

20

Bioprocess Control

Batch

Semi-continuous

Continuous

Figure 2.1. Various operating modes of biological processes

is carried out thereafter, and the reaction takes place at constant volume. The only possible actions of the operator relate specifically to the environmental variables (pH, temperature, stirring velocity, aeration, etc.). Thus, few means are necessary for its implementation, which in fact is its attraction, from the industrial point of view. The second advantage is to guarantee the purity of cultures because the risks of culture contamination are less. A disadvantage is the minimal means which make it possible to operate the fermentor to optimize the use of micro-organisms. It also suffers from a major disadvantage: the initial supply of a high quantity of substrate generally inhibits the growth of micro-organisms which consume it, which results in lengthened durations of the process, and limit the acceptable initial load. Semi-continuous (or fed-batch) mode This operating mode is distinguished from the preceding mode by a supply of various nutritive elements as and when there is a need by the micro-organisms. It primarily makes it possible to eliminate the inhibition problems associated with the preceding mode, and to function at specific growth rates close to their maximum value [QUE 99]. From a previously inoculated initial volume the reactor is supplied by a flow controlled in a closed loop. In addition, it is the latter point which strongly limits the use of fedbatch at an industrial scale. Lastly, this operating mode, just like the previous one, is more particularly recommended when the recovery of the products is carried out at intervals (intracellular accumulation for example) or when it is dangerous to release residual toxic matters (which may happen in the continuous mode). Continuous (or chemostat) mode This is the most widely used mode in the field of the biological treatment of water. Characterized by a constant reaction volume, it reaches a state where the extraction of reaction medium equals the nutritive flow rate. The continuous processes work at steady state for fixed supply conditions, by maintaining the system in a stationary state, while avoiding any inhibiting phenomenon owing to the dilution effect due to the supply. Although it generally works in open loop, it is perhaps the richest operating mode from a dynamic point of view, because it makes it possible to study transitory


21

phenomena [GUI 96], the characteristics of a micro-organism over long periods of growth, optimization problems, etc. Moreover, it enables significant productions in small-sized reactors. Sequencing batch reactors (SBR) This is in fact a combination at one time of various operating modes. The idea is to recover the biomass by sedimentation or the supernatant product between two production (or treatment) sequences. The succession of various stages is represented in Figure 2.2.

1–Filling up

2–Reaction

3–Sedimentation

4–Withdrawal of a fraction of the sludge

5–Drain and return to step 1

Figure 2.2. SBR (sequencing batch reactors): time sequence of various stages

In the same way, the SFBR (sequencing fed-batch reactor) differs from the SBR only in the way in which the first filling stage is carried out, as and when there is a need and not in a single action. 2.3. Mass balance modeling 2.3.1. Introduction Modeling biological systems is a delicate task because there are no laws characterizing the evolution of micro-organisms. Nevertheless, these systems, like all physical systems, must comply with rules such as the conservation of mass, electroneutrality of solutions, etc. In this chapter, we will see how to frame the model around these physical laws, so as to guarantee some robustness. 2.3.2. Reaction scheme At the macroscopic level, the reaction scheme of a biochemical process describes the set of the main biological and chemical reactions. For this we adopt [BAS 90] formalism similar to that of chemistry, by simply defining the transformation of two reactants A and B into a product C in the following form: A + B −→ C

22

Bioprocess Control

By convention, contrary to chemistry, we do not consider stoichiometric coefficients in these reactions. In general the reaction rate corresponds to the growing rate of the implied biomass. At the macroscopic level, the reaction scheme is in fact a synthetic way of summarizing all reactions which are supposed to determine the dynamics of the process. Thus, a reaction scheme is generally based on assumptions related to the phenomenological details available. Unlike in chemistry, all the compounds intervening in the reaction are not strictly represented (which is fortunate, because it would be difficult to make a complete balance of compounds (Fe, Pb, F, etc.) necessary for microbial growth). The principal transformation reactions of the mass in a bioprocess are as follows: – growth of micro-organisms and biosynthesis (related to secondary metabolism) S1 + S2 + · · · + Sp −→ X + P1 + · · · + Pk when the growth is aerobic, we find in the substrates of the reaction a source of carbon, nitrogen, phosphorus and mineral salts, as well as oxygen. In the products, there is CO2 ; – synthesis of a product via primary metabolism S1 + S2 + · · · + Sq −→ P1 + · · · + Pl In this case, the manufacture of products is not dependent on the bacterial growth, but generally depends on enzymes produced by micro-organisms; – mortality X −→ Xd where, Xd corresponds to a dead biomass (whereas X is the living biomass). However, this equation can be completed only by adding the reaction rate and stoichiometric coefficients to it. This is why we will prefer to express the reaction scheme in a more complete form: ϕ

kA A + kB B −−→ kC C + X where ϕ is the reaction rate, here corresponding to the rate of biomass formation. The consumption yield of A is kA , that of B, is kB , and kC is the production yield of C. The production rate of C is therefore KC ϕ and the consumption rates of A and B are respectively kA ϕ and kB ϕ. Thereafter, we will assume that the reaction scheme is described by a set of k biological or chemical reactions. We will consider n variables (substrates, products, biomasses, etc.).


23

2.3.3. Choice of reactions and variables The choice of the number of reactions to be considered and components which intervene in these reactions is very important for modeling. It will be carried out based on the knowledge that we have on the process and measurements which could have been carried out. The reaction scheme, as we will see thereafter, will condition the structure of the model. It will thus have to be chosen with parsimony, bearing in mind the objectives of the model and the precision which is expected. The required number of reactions and the reaction scheme can be determined directly from a set of available experimental data [BER 05a, BER 05b]. In addition, in section 2.6 we will present a means of validating the reaction scheme. When we write the mass balance, we make the following assumptions: – the reaction scheme summarizes the distribution of mass and flows between various reactions intervening in the process; – yield coefficients are constant. 2.3.4. Example 1 Here, we will consider the example of anaerobic digestion. This wastewater treatment process uses anaerobic bacteria to degrade organic matter (S1 ). It is in fact a very complex process, in which a great number of bacterial populations intervene [MOS 83, DEL 01]. If the objective is to control this ecosystem, we will need a relatively simple model. We will also limit ourselves to considering two bacterial populations. Thus, we assume that the dynamics of the system can be summarized in two main stages: – acidogenesis stage (at rate r1 (·)), during which substrate S1 is degraded by acidogenic bacteria (X1 ) and is transformed into volatile fatty acids (S2 ) and into CO2 : r1 (·)

k1 S1 −−−−→ X1 + k2 S2 + k4 CO2

(2.1)

– methanogenesis stage (at rate r2 (·)), where the volatile fatty acids (VFA) are degraded into CH4 and CO2 by methanogenic bacteria (X2 ). r2 (·)

k3 S2 −−−−→ X2 + k5 CO2 + k6 CH4

(2.2)

The constants k1 , k2 and k4 respectively represent stoichiometric coefficients associated with consumption of substrate S1 , production of VFA and CO2 in the acidogenesis process. k3 , k5 and k6 respectively represent stoichiometric coefficients in the consumption of VFA and in the production of CO2 and CH4 during the methanogenesis process. It should be noted that this reaction scheme has no biological reality insofar as biomasses X1 and X2 represent a flora of different species. This is the same for substrates S1 and S2 which gather a group of heterogenous compounds. There are

24

Bioprocess Control

many models of this process [HIL 77, MOS 83, MOL 86]. Some authors, according to the modeling objective, chose much finer descriptions of implied processes [COS 91, BAT 97, BAT 02]. 2.4. Mass balance models 2.4.1. Introduction In stirred tank reactors, whatever the operating mode (batch, fed-batch, continuous), the dynamic behavior of the components of the biological reaction arises directly from the expression of mass fluxes, which translates that the variation of the quantity of a compound is equal to the sum of what is produced or supplied, reduced from what is consumed or decanted. In this case, using the reaction scheme, and knowing the hydrodynamics of the fermentor (mass inflow and outflow), we will see that we can directly obtain the mass balance model. 2.4.2. Example 2 Let us consider the growth of a UG5 saccharomyces cerevisiae yeast population on glucose S, and producing ethanol P according to the following reaction scheme: r(·)

k1 S −−−→ X + k2 P Thus, it is represented by the following ordinary differential equations: d(V X) = r(·)V − Qout X (2.3) dt d(V S) = −k1 r(·)V + Qin Sin − Qout S (2.4) dt d(V P ) = k2 r(·)V − Qout P (2.5) dt dV = Qin − Qout (2.6) dt in which X represents the concentration of micro-organisms (g/l), S the concentration of substrate (g/l), P the concentration of formed product (g/l), V the reaction volume (l), Qin the supply flow rate (l/h), Qout the withdrawal rate (l/h) and Sin the substrate influent concentration (g/l). r(·) represents growth rate of micro-organisms (1/h), and k1 and k2 are the coefficients respectively representing the yield of consumption and production. To this basic model, we could add terms of maintenance, mortality or additional equations concerning co-substrates of growth, intermediate products of the reaction, etc.


25

Equations (2.3)–(2.5) are generally re-expressed in terms of concentrations rather than quantities, which gives: dX dt dS dt dP dt dV dt

Qin X V Qin = −k1 r(·) + (Sin − S) V Qin = k2 r(·) − P V

= r(·) −

= Qin − Qout

(2.7) (2.8) (2.9) (2.10)

2.4.3. Example 3 However, growth and production are not strictly linked. We can particularly quote the case of intracellular accumulation of the product during the growing phase, which is then released when growth is completed (because of a substrate limitation for example). This is the case for penicillin production by the growth of the Penicillium fungus on glucose [BAJ 81, LIM 86]. In such a case, the reaction scheme will be divided into two parts: growth of the fungus and production of penicillin: rx (·)

k1 S −−−−→ X rp (·)

k2 S −−−−→ P Thus, the corresponding model is as follows: dX dt dS dt dP dt dV dt

= rx (·) −

Qin X V

= −k1 rx (·) − k2 rp (·) + = rp (·) −

(2.11) Qin (Sin − S) V

Qin P V

= Qin − Qout

(2.12) (2.13) (2.14)

2.4.4. Matrix representation The reaction scheme leads to equations which describe the distribution of the mass in the bioreactor [BAS 90] in an equivalent way: dξ = Kr(·) + D(ξin − ξ) − Q(ξ) + F dt

(2.15)

26

Bioprocess Control

in which ξ represents the process state vector, ξin the influent concentration vector and r(·) the reaction rate vector. K is the matrix containing stoichiometric coefficients (yields). Q(ξ), to which we will return in the following section, represents the terms of gaseous exchange between the gas phase and liquid phase. F is the supply of mass in gaseous form (like for example the mass flow of gaseous oxygen transferred to the reaction medium in aerobic fermentations). Finally, D is the dilution rate, equal to the ratio of supply flow rate Qin on the volume of reactor V . NOTE 2.1. This relatively standard matrix representation is more particularly adapted to the case of continuous fermentations for which volume is constant. On the other hand in the case of fed-batch reactions, the dilution rate varies with volume, which must also appear in the state vector. 2.4.4.1. Example 2 (continuation) If we consider model (2.7)–(2.10) in the case of a continuous reaction (Qout = Qin , D = QVin ), the model can be rewritten in matrix form (2.15) with: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ X 1 0 ξ = ⎣S ⎦ , K = ⎣−k1 ⎦ , ξin = ⎣Sin ⎦ P k2 0 2.4.4.2. Example 1 (continuation) Let us reconsider the example of anaerobic digestion whose reaction scheme is given by (2.1) and (2.2). We will assume that methane is much less soluble, and thus that it passes very quickly in the gas phase. Dissolved CO2 is stored in the liquid phase where it enters the inorganic carbon compartment (C). dX1 dt dX2 dt dS1 dt dS2 dt dC dt

= r1 (·) − DX1

(2.16)

= r2 (·) − DX2

(2.17)

= D(S1in − S1 ) − k1 r1 (·)

(2.18)

= D(S2in − S2 ) + k2 r1 (·) − k3 r2 (·)

(2.19)

= D(Cin − C) − qC (ξ) + k4 r1 (·) + k5 r2 (·)

(2.20)

S1in , S2in and Cin are respectively concentrations of the inflow of substrate, VFA and dissolved inorganic carbon. The term qC (ξ) represents the flow of inorganic carbon (in the form of CO2 ) from liquid phase to gas phase.


27

2.4.5. Gaseous flow At the time of writing the gas balance, we must take into account the species which have a gas phase. These species can thus pass in the gas phase and escape from the fermentor (they can also pass from the gas phase to the liquid phase). It will thus be necessary to add a transfer term with the exterior in the balancing term. For this, Henry’s law is used which expresses the molar flow of compound C from liquid phase to its gas phase: qc = KL a(C − C )

(2.21)

NOTE 2.2. If qc < 0, it means that the gas flow will take place from the gas phase to liquid phase. The transfer coefficient KL a (1/h) strongly depends on the operating conditions and in particular on agitation, pressure and transferring surface between the liquid and gas phases (size of the bubbles) [MER 77, BAI 86]. Modeling of the evolution of these parameters according to the operating conditions can prove very delicate. Quantity C is the concentration of saturation in dissolved C. This quantity is related to partial pressure in gaseous C in gas phase (PC ) using Henry’s constant: C = KH P C

(2.22)

Henry’s constant can also vary, but in a lesser manner, according to the culture medium and temperature. In addition, when several gas species are simultaneously found in the liquid phase, they must follow the ideal gas law, which will result in a constant relationship between molar flow and partial pressure. Thus, for m gas species C1 · · · Cm : Pc1 Pc2 Pcm = = ··· = qc1 qc2 qcm

(2.23)

2.4.6. Electroneutrality and affinity constants The electroneutrality of solutions is the second rule in which biological systems do not derogate: the concentration of anions balanced by the number of charges must equalize the concentration of cations balanced in the same manner. Pure chemical reactions are often well known, and an affinity constant is generally associated with them. This constant is often related to the concentration in protons H + , thus to the pH.

28

Bioprocess Control

2.4.7. Example 1 (continuation) Gas flow Methane flow is directly related to the methanogenesis rate: qM = k6 r2 (·)

(2.24)

qC (ξ) = KL a(CO2 − KH PC )

(2.25)

CO2 flow follows Henry’s law:

where PC is the partial pressure of CO2 . Affinity constants In the example of anaerobic digestion, we will make use of electroneutrality of the solution and chemical balances. We will consider that in the usual range of pH for this type of process (6 ≤ pH ≤ 8) VFA are in their ionized form. Dissolved CO2 is in equilibrium with bicarbonate: CO2 + H2 O ←→ HCO3− + H +

(2.26)

The affinity constant of the reaction is thus Kb =

HCO3− H + CO2

(2.27)

Electroneutrality of the solution Cations (Z) are mainly ions which are not affected by biochemical reactions (Na+ , . . .). Thus, the dynamics of the cations will simply follow, without modification, the concentration of cations Zin in the input, so that: dZ = D(Zin − Z) dt

(2.28)

As for anions they are mainly represented by VFA and bicarbonate, therefore electroneutrality ensures that Z = S2 + HCO3−

(2.29)


29

Matrix representation By adding equation (2.28), the model is finally expressed in matrix form (2.15) with: ⎤ ⎡ ⎤ ⎡ X1 1 0 ⎢X2 ⎥ ⎢ 0 1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢Z⎥ ⎢ r1 (·) 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ (2.30) ξ = ⎢ ⎥ , r(·) = , K=⎢ 0 ⎥ r2 (·) ⎥ ⎢ S1 ⎥ ⎢−k1 ⎣ S2 ⎦ ⎣ k2 −k3 ⎦ C k4 k5 ⎤ 0 ⎢ 0 ⎥ ⎥ ⎢ ⎢ Zin ⎥ ⎥, ⎢ ξin = ⎢ ⎥ ⎢S1in ⎥ ⎣S2in ⎦ Cin ⎡

⎡

⎤ 0 ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥, Q=⎢ ⎥ ⎢ 0 ⎥ ⎣ 0 ⎦ qC (ξ)

(2.31)

An elimination of variables HCO− 3 , CO2 and PC by using equations (2.25), (2.23), (2.27) and (2.29) leads to the following expression for PC (ξ) (see [BER 02]):

φ − φ2 − 4KH PT (C + S2 − Z) PC (ξ) = (2.32) 2KH by showing: φ = C + S2 − Z + KH PT +

k6 kL a r2 (·).

Finally, this gives us: qC (ξ) = kL a(C + S2 − Z − KH PC (ξ))

(2.33)

2.4.8. Conclusion At this stage, we thus arrive at a model based on the following physical and chemical principles: – mass balance; – ionic balance; – affinity constant; – ideal gas law; – Henry’s law. The most important assumption (in terms of reliability) is that which proposes the mass balance. This will have to be verified afterwards.

30

Bioprocess Control

We will see in the following chapters that this model could be used as such in order to monitor the process (observers) or to control it (controllers). Nevertheless, if the objective defined at the beginning includes a simulation phase, it will be necessary to represent the reaction rates ri (·) according to the state of the system and the inputs (forcing environmental variables). This stage is much more delicate, and we will have to make many assumptions that are difficult to verify. 2.5. Kinetics 2.5.1. Introduction In some cases (optimal control, simulation, prediction, etc.) it is necessary to have an expression relating reaction rates to some variables of the system. However, it should be remembered that they are generally approximate relationships based on empirical considerations, and that we thus move away from the physical modeling established in the preceding section. In this chapter, we will see how to treat assumptions made on the kinetics level on a hierarchical basis and thus arrive at a description based on two reliability levels of the kinetics. 2.5.2. Mathematical constraints 2.5.2.1. Positivity of variables A priori, we know a certain number of physical constraints that the model must respect: the variables must remain positive and will be bounded if the mass entering the bioreactor is bounded. These physical constraints will impose constraints on the structure of ri (·). Some quantities (percentages, ratios, etc.) have to remain between two known bounds. To guarantee that the model preserves this property, a sufficient condition is given in the following property: i PROPERTY 2.1 (H1). For each state variable ξi , the field dξ dt on the axis ξi = 0 must point towards the acceptable part of the space. In other words, if the variable ξ ∈ [Lmin , Lmax ], the following property must be verified:

dξi ≥0 dt dξi ≤0 =⇒ dt

ξ = Lmin =⇒ ξ = Lmax

Specific case: in order for a variable ξi to remain positive, it should be ensured that i ξi = 0 ⇒ dξ dt ≥ 0.


31

2.5.2.2. Variables necessary for the reaction The second important constraint that will have to respect the biochemical kinetics is related to the reaction scheme. The reaction cannot take place if one of the reagents necessary for the considered reaction is missing. This explains the following property: PROPERTY 2.2. If ξj is a reactant of reaction i, then ξj can be factorized in ri : ri (ξ, u) = ξj νij (ξ, u) Thus, we easily verify that ξj = 0 ⇒ ri (ξ, u) = 0. In the same way, for the reactions associated with a biomass X, we will also have the same property. Thus, a growth reaction will be written: ri (ξ, u) = μi (ξ, u)X The term μi is then called the specific growth rate. 2.5.2.3. Example 1 (continuation) Let us consider the model given by equations (2.16) to (2.19). Let us apply the positivity principle of variables: X1 = 0 =⇒ r1 (·) ≥ 0

(2.34)

X2 = 0 =⇒ r2 (·) ≥ 0

(2.35)

S1 = 0 =⇒ D(S1in − S1 ) − k1 r1 (·) ≥ 0

(2.36)

S2 = 0 =⇒ D(S2in − S2 ) + k2 r1 (·) − k3 r2 (·) ≥ 0

(2.37)

Equations (2.34) and (2.35) give little information. In order that (2.36) and (2.37) are followed irrespective of experimental conditions, it is necessary that: r1 (·) = S1 φ1 (·)

and

r2 (·) = S2 φ2 (·)

In addition, biomasses X1 and X2 are necessary for reactions 1 and 2 respectively, and from this: r1 (·) = μ1 (·)X1

and

r2 (·) = μ2 (·)X2

Thus, finally we must have: r1 (·) = S1 X1 ν1 (·)

(2.38)

r2 (·) = S2 X1 ν2 (·)

(2.39)

2.5.2.4. Phenomenological knowledge In order to propose an expression for reaction kinetics, we will make use of the phenomenological knowledge available, although this is often highly speculative.

32

Bioprocess Control

First of all, laboratory experiments often make it possible to determine the variables which act on the reaction rates. We saw that among these variables we must find the reactants and possibly the biomass. Then, it is necessary to determine if the reaction rate is activated or inhibited by these variables. It frequently occurs that a variable is activated and that it becomes inhibiting at a very high concentration (toxicity). It now remains to propose an analytical expression which will enable us to consider mathematical constraints and phenomenological details on the process. For this, on the one hand we stress the importance of experimental observations (when they exist), and on the other hand of existing models in the literature. In all cases, we prefer the principle of parsimony, so that the models obtained can be calibrated and validated thereafter. The following sections present the models most often selected to describe the growth of micro-organisms. 2.5.3. Specific growth rate Even if the specific growth rate strongly depends on the operating conditions (temperature, pH, etc.), the reactive medium (concentrations in carbonaceous, nitrogenized, phosphorated compounds, in mineral salt, in oxygen, etc.), the expression most commonly used is Monod’s empirical model [MON 42], which, to describe the bacterial growth, used the law introduced at the beginning of the 20th century by MichaëlisMenten for enzymatic kinetics: μ = μmax

S Ks + S

(2.40)

This expression, in which μmax is the maximum specific growth rate (1/h) and Ks the half-saturation constant (g/l), allows us to describe the phenomenon of limited growth for lack of substrate, and complete breakdown when the substrate is no longer available. Let us note that the analogies with enzymatic kinetics have often been used to determine a growing model [SEG 84, EDE 88]. In addition, the inhibition phenomena due to excess of substrate are generally modeled by Haldane’s expression, introduced in the case of enzymatic reactions, and taken again by Andrews [AND 68] in the case of biological reactions: μ = μmax with Ki inhibition constant (g/l).

S Ks + S +

S2 Ki

(2.41)


33

, Haldane s model 0.4 0.35 μ max = 0.41; K s = 0.002 ; K i = 0.35

Growth rates (1/h)

0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.05

0.1

0.15

0.2 0.25 0.3 Substrate (g/l)

0.35

0.4

0.45

0.5

Figure 2.3. Haldane’s model – μ(S)

It should be noted that many other algebraic relations were established to describe these limitation and/or inhibition phenomena, but their use remains marginal. In the same manner, certain models take into account the influence of concentration of microorganisms, co-metabolite, temperature, pH (see [DOC 86, BAI 86, STE 98] for a list of about 50 models). The case of oxygen is a little different. In fact, in the case of processes functioning in an aerobic environment, oxygen corresponds to a co-substrate of the reaction, and can thus be treated as such, i.e. to intervene in the form of a Monod type term in the expression of a specific growth rate, thus leading to the following expression: μ = μmax

S Ks + S

O2 KO2 + O2

(2.42)

with concentration O2 in dissolved oxygen (g/l) and KO2 the half-saturation constant for oxygen (g/l). However, this expression is often omitted under the assumption that the reactor is sufficiently aerated and that KO2 is very small compared to the concentration of dissolved oxygen present in the reactor under normal conditions.

34

Bioprocess Control

In certain cases, a co-substrate can also act in a competitive way. Let us consider a bacterium such as Pseudomonas putida, which can, under certain conditions, simultaneously use phenol (S1 ) and glucose (S2 ) as sources of energy. Therefore, the specific growth rate can be expressed as the sum of two specific growth rates on each of the substrates: μ = μ1 (S1 , S2 ) + μ2 (S1 , S2 ) The competitive activation/inhibition phenomenon in the bisubstrate mixture can be described by an expression of the form [WAN 96]: μi (Si , Sj ) =

μmax,i Si Ks,i + Si +

Si2 Ki,i

,

+ K3,i Sj + K4,i Si Sj

i, j = 1, 2, i = j (2.43)

In the absence of substrate Sj , μi (Si , 0), is brought to the previously described Andrews expression. By a judicious choice of K3,i , K4,i , i = 1, 2, we can then easily describe the inhibition phenomena whether competitive or not, cross or partial [WAN 96]. 2.5.4. Representation of kinetics by means of a neural network In an alternative way, we can model the kinetics using a neural network (hybrid models). We do not make an assumption a priori on kinetics (or simply the constraints which guarantee that the system trajectories have an acceptable biological meaning) and we directly identify kinetics while learning the network. Nevertheless, it is necessary to decide on variables which influence kinetics, and which will be the inputs of the neural network. A diagrammatic view of the network is represented in Figure 2.4 for a single hidden layer. The expression of the network output according to the inputs is expressed in the following way: m nh

ωk φ υki Si (2.44) μ(S1 , . . . , Sm ) = k=1

i=1

where nh represents the number of neurons in the hidden layer. ωk and υki are respectively the weight of input and output layers. Function φ represents the activation function of the neuron (sigmoid, hyperbolic tangent, Gaussian, etc.). The choice of network type and the number of neurons is a relatively standard question; see [HER 91] for more precise details. Once the structure of the network is chosen, it is then necessary to identify the weight of the network during the learning phase. For more precise details, see [CHE 00].


ν1,1 S1

35

ω1 ω2

S2

ω3 ωk

μ

νm,1 Sm

Input layer

ω nh ν m,n h

Hidden layer

Output layer

Figure 2.4. Diagram of a neural network containing a hidden layer

2.6. Validation of the model 2.6.1. Introduction The last phase of modeling is undoubtedly the most important phase, but also that which is generally neglected. It is all the more important as we saw that it was necessary to make a great number of often speculative assumptions. Before using a model, it is crucial to validate it correctly. This validation stage must be kept on through the identification phase which will be discussed in Chapter 6. The general objective of validation is to verify that the model responds well to the objectives laid down. More precisely, we will see how to individually test the assumptions which were made during the construction of the model. For this we proceed in 3 stages, during which we test: – the reaction scheme; – the qualitative predictions of the model; – the model in its totality (reaction scheme + kinetics + parameters). The validation must be made on the basis of a data file not having been used to construct or identify the model, and for different experimental conditions (if not, we would instead be testing experimental reproducibility). If these conditions are not observed, the model cannot claim to be validated. 2.6.2. Validation of the reaction scheme 2.6.2.1. Mathematical principle The procedure that we propose here relies on an important characteristic, which is a consequence of the mass conservation within the bioreactor. In fact it consists

36

Bioprocess Control

of verifying the consistency of data compared to mass flows induced by the reaction scheme [STE 98, BER 05a]. PROPERTY 2.3. Matrix K is of dimension n × k. We suppose that there are more variables than reactions: n > k. In these conditions, we have a minimum n − k independent vectors vi ∈ Rn such that: viT K = 01×k By convention, we normalize the first component of the vectors vi such that vi1 = 1. Consequence: let us consider the real variable wi = viT ξ, this variable satisfies the following equation: dwi = D(wiin − wi ) − viT Q(ξ) + viT F dt

(2.45)

with wiin = viT ξin . Let us integrate expression (2.45) between two moments t1 and t2 , and develop components vij of vector vi . n

vij φξj (t1 , t2 ) = φξ1 (t1 , t2 )

(2.46)

j=2

by noting φξj (t1 , t2 ) = ξj (t2 ) − ξj (t1 ) −

t2

(D(τ )(ξjin (τ ) − ξj (τ )) − Qj (τ ) + F (τ ))dτ

t1

φξj (t1 , t2 ) can be numerically estimated on the basis of experimental measurements of ξj over the course of time. Expression (2.46) is a linear relation connecting vij to terms φξj (t1 , t2 ). As φξj (t1 , t2 ) can be calculated between different moments t1 and t2 , (2.46) is a linear regression whose validity could be tested in experiments. 2.6.2.2. Example 4 Here, we will consider the simple example of growth of the filamentous fungus Pycnoporus cinnabarinus (X) on two substrates, a glucose source of carbon (C) and an ammonium source of nitrogen (N ). It is thus supposed that the reaction scheme is made up of a reaction: N + C −→ X


37

Therefore, stoichiometric matrix K associated with this reaction is the following T : ξ= X N C K = 1 −k1

−k2

T

ξin = 0

and

Nin

Cin

T

(2.47)

We will consider the following two vectors in the space orthogonal to K: T T 1 1 0 and v2 = 1 0 v1 = 1 k1 k2 We can now define the following quantities: t2 D(τ )X(τ ) φX (t1 , t2 ) = X(t2 ) − X(t1 ) + t1

t2

φN (t1 , t2 ) = N (t2 ) − N (t1 ) − φC (t1 , t2 ) = C(t2 ) − C(t1 ) −

D(τ )(Nin (τ ) − N (τ ))dτ

t1 t2

D(τ )(Cin (τ ) − C(τ ))dτ

t1

which will enable us to write the following two regressions: 1 φN (t1 , t2 ) k1 1 φX (t1 , t2 ) = φC (t1 , t2 ) k2 φX (t1 , t2 ) =

(2.48) (2.49)

It is now easy to verify if relations (2.48) and (2.49) are valid by seeing, on the basis of experimental measurements, if there is a linear relationship between the different φξi (t1 , t2 ). Figure 2.5 shows an example of validation on the basis of a series of experiments. The regression obtained is highly significant, which means that relations (2.48) and (2.49) are valid. Consequently matrix K, which is in the orthogonal of v1 T and v2 is strictly of the type K = 1 −α1 −α2 : the reaction scheme is validated and so is the mass balance. 2.6.3. Qualitative validation of model At this stage, we assume that the reaction scheme, and thus the mass balance model were validated. We will thus consider the simulation model including reaction kinetics. The first thing to do is to verify if the qualitative properties of the model correspond to experimental observations.

38

Bioprocess Control

X estimated from N (g/l)

2.5 A 2 1.5 1 0.5 0.5

1 1.5 2 X measured (g/l)

2.5

1 1.5 2 X measured (g/l)

2.5

X estimated from S (g/l)

2.5 B 2 1.5 1 0.5 0.5

Figure 2.5. Validation of linear relation relating φX and φN (A) and φX and φC (B)

When the inputs of the model are constant, does it predict a steady state, or a more complex behavior (limit cycle, chaos, etc.) in agreement with observations? These qualitative properties are often dependent on the dilution rate of the bioreactor, as well as influent concentrations. Is a qualitative change observed (e.g. a stable equilibrium becomes unstable) when we vary these inputs? In addition, more precise qualitative properties on the nature of transients (oscillation type, extrema, etc.) can sometimes be deduced from the structure of a dynamic system ([JEF 86, SAC 90, GOU 98, BER 95, BER 02]). For example, Hansen and Hubell (1980) [HAN 80] study the competition between two species of bacteria in a chemostat. They show that the result of the competition predicted by the model depends on the dilution rate. For a dilution rate lower than a certain value it is one species which wins, whereas for higher values, it is the other species. These qualitative properties are verified in experiments.


39

2.6.4. Global validation of the model This is the most standard way to validate a model, by quantitatively comparing the results obtained by the model with experimental data. In this approach, the model is considered as a whole. If the adequacy between model and experiment is not good, it will be difficult to determine if the problem is mainly structural (bad assumptions on the reaction scheme), related to mathematical expressions used for kinetics, or if the parameters of the model are not correctly estimated. If the first two stages of the validation are successfully crossed, it can be assumed that the problem is mainly due to incorrect parameter values. In theory, validation of a model consists of verifying statistical assumptions: differences between predictions and measurements must follow a normal law of zero average [WAL 97]. Practically, and considering the difficulty of validating in the strictest sense the biotechnological models, we are satisfied with a good visual adequacy. We can possibly reinforce this subjective criterion by an analysis of correlation between predictions and measurements. 2.7. Properties of the models 2.7.1. Boundedness and positivity of variables We saw in section 2.5.2 that the models must be constructed, in such a manner as to respect constraints as positivity of state variables. We will see here that the models based on mass balance are models of the BIBS type (input bounded state). For this, we use the following property which arises from the principle of mass conservation: HYPOTHESIS 2.1 (H2). There is a vector v + whose components are strictly positive, such as: T

v + K = 01×k T

Consequences: if we set w+ = v + ξ, w satisfies the equation of type (2.45): T T dw+ + = D(win − w+ ) − v + Q(ξ) + v + F dt

(2.50)

We will make an assumption on Q(ξ), which is verified in most cases: HYPOTHESIS 2.2 (H3). Q(ξ) can be compared to a linear expression as follows: T

T

v + Q(ξ) ≥ av + ξ + b where a is a positive real b ∈ Rn .

40

Bioprocess Control

HYPOTHESIS 2.3 (H4). Mass supply in gaseous form F is bounded: F ≤ Fmax , in accordance with physical reality. T

This hypothesis is verified if v + Q(ξ) = 0, or if Q(ξ) is described by Henry’s law. PROPERTY 2.4. If hypothesis (H1), (H2), (H3) and (H4) are verified, then the system is of BIBS type. Proof. The dynamics of w+ can thus be raised by T + Dwin + v + Fmax − b dw+ + ≤ (D + a) −w (2.51) dt D+a T + Dwin +v + Fmax −b . and by applying property 2.1, we deduce w+ ≤ min w+ (0), D+a In other words,

wi+ ξi is bounded. As wi+ > 0, state variables ξi are bounded.

2.7.2. Equilibrium points and local behavior 2.7.2.1. Introduction In this section we recall the principles of the study of model properties; see [KHA 02] for more details. In general, bioreactor models are nonlinear, and often of large dimensions. They include many parameters which also intervene in a nonlinear way. However, passing dimension 3, it becomes mathematically very difficult to characterize the behavior of a dynamic system. However, we show that the models with mass balance have structural properties which can help in understanding the system. In this section, we describe the dynamic system using an equation: dξ = f (ξ, u) dt

(2.52)

However, we remember that f (ξ, u) = Kr(ξ) + D(ξin − ξ) − Q(ξ) + F . We consider the case where u = (D, ξin , F ) is constant. Equilibrium points and local stability dξ Equilibrium points are points such as = 0 when all the inputs are maintained dt as constant.


41

The nonlinear systems differ generically from linear systems by the fact that they can have several equilibrium points. The first stage in model analysis consists of testing if these equilibrium points are locally stable. For this, we consider the Jacobian matrix: J(ξ) =

Df (ξ) Dξ

Equilibrium ξ0 is locally stable if and only if all the eigenvalues of J(ξ0 ) have strictly negative real parts. If there is an eigenvalue with a positive real part, the equilibrium is unstable. We cannot conclude on the stability of the system if the eigenvalues have a non-positive real part and two of them are pure imaginary conjugates. Global behavior The dynamics of a nonlinear system can be very complicated, and behaviors such as limit cycles and chaos can appear in addition to equilibria. It is thus important to find when there is a single locally stable equilibrium, if it is globally stable, i.e. if, irrespective of initial conditions, trajectories will converge towards this equilibrium. The standard method to prove that a steady state is globally stable is based on the Lyapunov functions [KHA 02]. To find a Lyapunov function for a biological system is a difficult task, though; see [LI 98] for constructive methods to find Lyapunov functions in a broad range of growth models. Asymptotic behavior In section 2.6.2 we saw that generally there were n − k vectors vi which make it possible to calculate the wi quantities whose dynamics satisfy equation (2.45). Generally, there exist q independent vectors among the vi , denoted vi0 which verify vi0t Q(ξ) = 0

(2.53)

Moreover, without loss of generality, let us consider the case without mass supply in gas form (F = 0). The associated dynamics of wi0 are thus simple: dwi0 0 = D(wiin − wi0 ) dt

(2.54)

Under the conditions which we consider (i.e. constants D and ξin ), the solutions 0 . This means that the solutions of sysof (2.54) asymptotically converge towards wiin 0 . tem (2.52) will converge towards the hyperplane given by vi0t ξ = wiin The state of the system will thus asymptotically tend towards the vectorial subspace of dimension n − q orthogonal to the q vectors vi0 . This makes it possible to bring back the study of (2.52) of dimension n to dimension n − q.

42

Bioprocess Control

Example 4 (continuation) Let us consider the model given by equation (2.47). Moreover assume that the kinetics follow Monod’s law with respect to the two substrates C and N : r(ξ) = μmax

N C X KC + C KN + N

(2.55)

Let us again take the two vectors v1 and v2 identified in section 2.6.2. These vectors simply verify equation (2.53). Therefore, X +

N k1

−→

Nin k1

and X +

C k2

−→

Cin k2 .

The system study in dimension 3 is thus brought back to that of the following system, of dimension 1: dX Nin − k1 X Cin − k2 X = μmax X − DX dt KC + Cin − k2 X KN + Nin − k1 X

(2.56)

It can be verified that this system has three real equilibrium points (one of them being the trivial equilibrium X = 0). These ordered equilibria are respectively locally stable, unstable and locally stable. Non-trivial equilibria, depending on values of the parameters, are positive (and thus acceptable) or not. For the parametric domains where there is only one strictly positive equilibrium, this equilibrium is then globally stable. 2.8. Conclusion In this chapter, we presented a constructive and systematic method to develop bioprocess models in 4 stages. Let us recall here that bioprocess modeling should be carried out only with the vision of an objective clearly identified at the beginning, and in relation to the quality and quantity of information available to establish and validate the model. The first stage of modeling consists of collecting physical and chemical principles, and making an assumption about a reaction scheme to arrive at a model of mass balance. In the second stage, it is necessary to benefit from the constraints that the model must respect, and make use of empirical relations to determine an expression for kinetics. The third stage of modeling consists of identifying the parameters; these aspects are detailed in Chapter 6. Lastly, we should not neglect the final modeling stage: validation of the model. During this phase, quality of the model must be tested in the most objective way possible, and the operating domains for which it reaches its limits must be identified. In this chapter, we were interested only in the models for processes assumed to be infinitely mixed, and only these spatially homogenous systems will be considered in this book.


43

Finally, let us note that here we restricted ourselves to the knowledge models, resulting from physical considerations. In fact, the black box models of input-output behavior are more concerned with the identification of parameters. 2.9. Bibliography [AND 68] J.F. A NDREWS, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrate”, Biotechnol. & Bioeng., vol. 10, pp. 707–723, 1968. [BAI 86] J.E. BAILEY and D.F. O LLIS, Biochemical Engineering Fundamentals, McGrawHill, New York, 1986. [BAJ 81] R.K. BAJPAI and M. R EUSS, “Evaluation of feeding strategies in carbon-regulated secondary metabolite production through mathematical modelling”, Biotechnol. & Bioeng., vol. 23, pp. 717–738, 1981. [BAS 90] G. BASTIN and D. D OCHAIN, On-line Estimation and Adaptive Control of Bioreactors, Elsevier, Amsterdam, 1990. [BAT 97] D. BATSTONE, J. K ELLERAND, B. N EWELL and M. N EWLAND, “Model development and full scale validation for anaerobic treatment of protein and fat based wastewater”, Water Science and Technology, vol. 36, pp. 423–431, 1997. [BAT 02] D.J. BATSTONE, J. K ELLER, I. A NGELIDAKI, S.V. K ALYUZHNYI, S.G. PAVLOSTATHIS, A. ROZZI, W.T.M. S ANDERS, H. S IEGRIST and V.A. VAVILIN , Anaerobic Digestion Model No.1 (ADM1). IWA Task Group for Mathematical Modelling of Anaerobic Digestion Processes, IWA Publishing, London, 2002. [BER 95] O. B ERNARD and J.-L. G OUZÉ, “Transient behavior of biological loop models, with application to the Droop model”, Mathematical Biosciences, vol. 127, no. 1, pp. 19– 43, 1995. [BER 01] O. B ERNARD, Z. H ADJ -S ADOK, D. D OCHAIN, A. G ENOVESI, and J.-P. S TEYER, “Dynamical model development and parameter identification for an anaerobic wastewater treatment process”, Biotech. Bioeng., vol. 75, pp. 424–438, 2001. [BER 02] O. B ERNARD and J.-L. G OUZÉ, “Global qualitative behavior of a class of nonlinear biological systems: application to the qualitative validation of phytoplankton growth models”, Artif. Intel., vol. 136, no. 1, pp. 29–59, 2002. [BER 05a] O. B ERNARD and G. BASTIN, “Identification of reaction networks for bioprocesses: determination of a partially unknown pseudo-stoichiometric matrix”, Bioprocess Biosyst. Eng., vol. 27, no. 5, pp. 293–301, 2005. [BER 05b] O. B ERNARD and G. BASTIN, “On the estimation of the pseudo-stoichiometric matrix for macroscopic mass balance modelling of biotechnological processes”, Math. Biosciences, vol. 193, no. 1, pp. 51–77, 2005. [CHE 00] L. C HEN, O. B ERNARD, G. BASTIN and P. A NGELOV, “Hybrid modelling of biotechnological processes using neural networks”, Control Eng. Practice, vol. 8, pp. 821– 827, 2000.

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[COS 91] D.J. C OSTELLO, P.F. G REENFIELD and P.L. L EE, “Dynamic modelling of a singlestage high-rate anaerobic reactor - I. Model derivation”, Water Research, vol. 25, pp. 847– 858, 1991. [DEL 01] C. D ELBES, R. M OLETTA and J.-J. G ODON, “Bacterial and archaeal 16s rdna and 16s rrna dynamics during an acetate crisis in an anaerobic digestor ecosystem”, FEMS Microbiology Ecology, vol. 35, pp. 19–26, 2001. [DOC 86] D. D OCHAIN, On line Parameter Estimation, Adaptive State Estimation and Control of Fermentation Processes, PhD thesis, Université Catholique de Louvain, Belgium, 1986. [DOC 94] D. D OCHAIN, Contribution to the Analysis and Control of Distributed Parameter Systems with Application to (Bio) chemical Processes and Robotics, Thèse d’agrégation de l’enseignement supérieur, Université Catholique de Louvain, Belgium, 1994. [EDE 88] L. E DELSTEIN, Mathematical Models in Biology, Random House, New York, 1988. [GOU 98] J.-L. G OUZÉ, “Positive and negative circuits in dynamical systems”, Journal of Biological Systems, vol. 6, no. 1, pp. 11–15, 1998. [GUI 96] V. G UILLOU, I. Q UEINNEC, J.L. U RIBELARREA and A. PAREILLEUX, “Online sensitive lightness measurement of cell-mass in Saccharomyces Cerevisiae culture”, Biotechnology Techniques, vol. 10, no. 1, pp. 19–24, 1996. [HAN 80] S.R. H ANSEN and S.P. H UBBELL, “Single-Nutrient Microbial Competition”, Science, vol. 207, no. 28, pp. 1491–1493, 1980. [HER 91] J. H ERTZ, A. K ROGH and R.G. PALMER, Introduction to the Theory of Neural Computation, Addison-Wesley, 1991. [HIL 77] D.T. H ILL and C.L. BARTH, “A dynamic model for simulation of animal waste digestion”, Journal of the Water Pollution Control Association, vol. 10, pp. 2129–2143, 1977. [JAC 96] J. JACOB, H. P INGAUD, J.M. L E L ANN, S. B OURREL, J.P. BABARY and B. C APDEVILLE, “Dynamic simulation of biofilters”, Simulation – Practice and Theory, vol. 4, pp. 335–348, 1996. [JEF 86] C. J EFFRIES, “Qualitative Stability of Certain Nonlinear systems”, Linear Algebra and its Applications, vol. 75, pp. 133–144, 1986. [KHA 02] H.K. K HALIL, Nonlinear Systems, Prentice-Hall, Englewood Cliffs, 2002. [LI 98] W. L I, “Global asymptotic behavior of the chemostat: general response functions and different removal rates”, SIAM J. Appl. Math., vol. 59, pp. 411–422, 1998. [LIM 86] H.C. L IM, Y.J. TAYEB, J.M. M ODAK and P. B ONTE, “Computational algorithms for optimal feed rates for a class of fed-batch fermentation: numerical results for penicillin and cell mass production”, Biotechnol. & Bioeng., vol. 28, pp. 1408–1420, 1986. [MER 77] J.C. M ERCHUK, “Further considerations on the enhancement factor for oxygen absorption into fermentation broth”, Biotechnol. & Bioeng., vol. 19, pp. 1885–1889, 1977.


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[MOL 86] R. M OLETTA, D. V ERRIER and G. A LBAGNAC, “Dynamic modelling of anaerobic digestion”, Water Research, vol. 20, pp. 427–434, 1986. [MON 42] J. M ONOD, Recherches sur la Croissance des Cultures Bactériennes, Hermann, Paris, 1942. [MOS 83] F.E. M OSEY, “Mathematical modelling of the anaerobic digestion process: regulatory mechanisms for the formation of short-chain volatile acids from glucose”, Water Science and Technology, vol. 15, pp. 209–232, 1983. [PAV 94] A. PAVÉ, Modélisation en Biologie et en Ecologie, Aléas, Lyon, 1994. [QUE 99] I. Q UEINNEC, D. L ÉONARD and C. B EN YOUSSEF, “State observation for fedbatch control of phenol degradation by R alstonia eutropha”, in Proc. of the 4th European Control Conference, Karlsruhe, Germany, 1999. [SAC 90] E. S ACKS, “A dynamic systems perspective on qualitative simulation”, Artif. Intell., vol. 42, pp. 349–362, 1990. [SEG 84] L.A. S EGEL, Modeling Dynamic Phenomena in Molecular and Cellular Biology, Cambridge University Press, Cambridge, 1984. [STE 98] G.N. S TEPHANOPOULOS, A.A. A RISTIDOU and J. N IELSEN, Metabolic Engineering. Principles and Methodologies, Academic Press, San Diego, 1998. [WAL 97] E. WALTER and L. P RONZATO, Identification of Parametric Models from Experimental Data, Springer-Verlag, Heidelberg, 1997. [WAN 96] K.W. WANG, B.C. BALTZIS and G.A. L EWANDOWSKI, “Kinetics of phenol biodegradation in the presence of glucose”, Biotechnol. & Bioeng., vol. 51, pp. 87–94, 1996.

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Chapter 3

Identification of Bioprocess Models

3.1. Introduction The complexity of dynamic bioprocess models, which are usually employed, is actually a reflection of the complexity of the processes themselves as well as of the detailed knowledge about them acquired over the course of the past decades. In the real sense, they comprise a large number of parameters, to which it will be necessary to assign a numerical value either on the basis of prior knowledge about their value or on the basis of experimental data. The quality of the estimation of the parameters will inherently depend on the quantity and quality of the data made available for the calibration (or identification) procedure of the parameters of the model. Apart from these limitations with regard to the available information, another problem emerges with regard to identifying the parameters of the model: the dynamic models of the bioprocesses are intrinsically nonlinear and the parameters of the model can be strongly correlated among themselves. The study into the identifiability of the parameters of the model before their estimation turns out to be an essential task. The central question for analyzing the identifiability can be summarized as follows: let us assume that a certain number of variables of the state of the model are available to the required extent; on the basis of the structure of the model (structural identifiability) or on the basis of the type and quality of the data available (practical identifiability), can we expect to be in a position to attribute a unique value to each parameter of the model by means of parameter estimation? The answer to the analysis of the identifiability cannot usually be expressed in the form of a “yes” or a “no”, but when it gives the results (this is not necessarily evident in the case of nonlinear

Chapter written by Denis D OCHAIN and Peter VANROLLEGHEM.

47

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Bioprocess Control

models), these can very well indicate whether it is possible to identify such subsets or combinations of parameters. The chapter is organized as follows. First, we will deal with the questions of structural identifiability and practical identifiability, before seeing how to design and implement the experiments (experiment design), which will enable us to obtain the best data for the estimation of parameters. We will then review different techniques for parameter identification. We will conclude with a case study: the parameter identification of a model of anaerobic digestion. 3.2. Structural identifiability In the case of structural identifiability, we will examine if, in the (academic) case of ideal measurements (continuous measurements without noise, which are perfectly adjusted with respect to the model), all the parameters of the studied model are identifiable. It may lead to the conclusion that only a combination of parameters is identifiable. If the number of combinations of parameters is less than the number of parameters of the model, or there is no one-to-one relationship between these combinations and parameters, then only prior knowledge of certain parameters enables us to solve the problem of identifiability. Let us consider a simple example: in the model y = ax1 + bx2 + c(x1 + x2 ), only the parameters a + c and b + c are structurally identifiable (and not the three parameters, a, b, c); two parameters (for example, a and b) will be identifiable if the value of the third (here c) is known beforehand. For linear systems, the structural identifiability is very well understood and there are many tests available for this (see, for example [GOD 85]). On the other hand, the problem turns out to be much more complex for nonlinear models and at present only a few methods enable us to test their identifiability, often with the possibility of highlighting only the necessary and/or sufficient conditions of local identifiability (i.e., valid only for certain domains of the model) [WAL 82]. The list below recapitulates the available methods, including the references to applications based on models including Monod kinetics: – transformation of the nonlinear model into a linear model [GOD 85]. Applications: [BOU 98, DOC 95, SPE 00]; – development in series: - Taylor series [GOD 85, POH 78]. Applications: [BOU 98, DOC 95, HOL 82a, JEP 96, SPE 00, PET 00b]; - Generating series [WAL 95]. Applications: [PET 00a]; – similarity transformation or local state isomorphism [WAL 95]. Applications: [JUL 97, JUL 98]; – study of the observability of the nonlinear system [CAS 85]. Applications: [BOU 98].


49

In general, it will be difficult to determine simultaneously for both linear and nonlinear models, the approach that will require the minimum of calculations for a specific example. In most of the cases, we choose the methods of development in series because they are relatively simple to implement and in particular, because their development follows a continuous approach (we can always calculate the higher order terms of the series). As for the Taylor series method, they are generated in the time domain, whereas the generating series are developed in the input domain. These can be viewed as an extension of the Taylor series method in the case where we consider a class of inputs [RAK 85]. In the case of a model without an input variable, the generating series method becomes equivalent to the Taylor series method [VAJ 89]. 3.2.1. Development in Taylor series The approach using the development in Taylor series was initially developed by Pohjanpalo [POH 78]. Let us consider the general model below: ⎧ ⎪ ⎨ dx(t) = f (x(t), u(t), t, θ), x(0) = x0 (θ) dt (3.1) ⎪ ⎩y(t, θ) = g(x(t), θ)

Here, x, u, y and θ represent the state vector, input vector, output vector (measured variables) and parameter vector, respectively. The development in Taylor series of y(t) around t = 0 is equal to: y(t, θ) = y(0) + t

dy t2 dy 2 t k dk y (0) + (0) + · · · + (0), dt 2! dt k! dtk

k = 0, 1, . . . , ∞ (3.2)

This approach assumes that the variable y(t) and its successive derivatives are known (through the value y). Let us write the expressions of the successive derivatives of y(t) (3.2) obtained through calculation by model (3.1) as follows: dk y (0) = ak (θ) dtk

(3.3)

where ak (θ) are (typically nonlinear) functions of the parameters θ. Equation (3.3) is a system of nonlinear equations, where the unknowns are the parameters θ and where k the terms ddtky (0) are known from hypothesis. The method consists of solving this system of equations for a limited number of derivatives of y(t). It is evident that this number should at least be equal to the number of unknown parameters. A sufficient condition of structural identifiability is that there is a unique solution for this equation system.

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Bioprocess Control

3.2.2. Generating series In the case of the generating series, we will consider the general model under the following form [WAL 95]: ⎧ m

⎪ ⎨ dx(t) = f 0 (x(t), θ) + ui (t)f i (x(t), θ), x(0) = x0 (θ) dt (3.4) i=1 ⎪ ⎩ y(t, θ) = g(x(t), θ) where the inputs ui (t) appear in the linear form. The generating series method is based on the output function g(x(t), θ) and its successive Lie derivatives, evaluated at t = 0: Lf i g(x(t), θ)t=0 =

n

f i,k (x, θ)

k=1

∂ g(x(t), θ)t=0 = si (θ) ∂xk

(3.5)

f i,k is the k th component of f i . We also see here a system of equations, where the unknowns are parameters θ (in the functions si (θ)). Therefore, a sufficient condition for structural identifiability will be that there is a unique solution for θ in the above system of equations, similar to the one in the Taylor series development (3.3). 3.2.3. Examples for the application of the methods of development in series The implementation of the two series development methods can follow the following steps: 1) calculation of the successive derivatives of g(x, θ); 2) selection of the set of θ parameters to be identified; 3) evaluation of the successive derivatives by inserting the known quantities and the lower order derivatives; 4) expression of the derivatives as functions of parameter combinations chosen in 2; 5) solution of the equations of step 4; 6) if the solution is unique, the parameters θ are structurally identifiable; otherwise, return to step 2. Let us apply the procedure to a model of the degradation of a substrate S in a batch reactor with Monod kinetics: dS μmax X S =− S(t = 0) = S0 (3.6) dt Y KS + S where the rate of respiration OUR(t) is measured. This is connected to the substrate S using the following equation: OUR = −(1 − Y )

S μmax X Y KS + S

(3.7)


51

Let us calculate the derivatives of OUR(t) at t = 0: OUR(0) =

S0 μmax X(1 − Y ) Y KS + S 0

dOU R μ2 X 2 (1 − Y ) KS S0 (0) = − max 2 dt Y (KS + S0 )3 d2 OUR μ3max X 3 (1 − Y ) KS S0 (KS − 2S0 ) (0) = dt2 Y3 (KS + S0 )5

(3.8) (3.9) (3.10)

There are five parameters to be identified: Y , μmax , X, KS and S0 . If we write i OUR zi = d dt (0), i = 0, 1, 2, . . . , and consider the combinations of the following i parameters: θ1 =

μmax X(1 − Y ) , Y

θ2 = (1 − Y )S0 ,

θ3 = (1 − Y )KS

(3.11)

Equations (3.8), (3.9) and (3.10) can be rewritten in the following (equivalent) form: θ1 θ2 (3.12) z0 = θ2 + θ3 z1 = − z2 =

θ12 θ2 θ3 (θ2 + θ3 )3

θ13 θ2 θ3 (θ3 − 2θ2 ) (θ2 + θ3 )5

(3.13) (3.14)

Consequently, “parameters” θ1 , θ2 and θ3 can be (formally) determined from the values of zi (or otherwise the successive derivatives of OUR(t)) by reversing the expressions above: z0 (z0 z2 − 3z12 ) z0 z2 − z12

(3.15)

θ2 = −

2z02 z1 z0 z2 − 3z12

(3.16)

θ3 = −

(4z02 z13 ) (z0 z2 − 3z12 )(z0 z2 − z12 )

(3.17)

θ1 =

3.2.4. Some observations on the methods for testing structural identifiability From the above example, we can draw some interesting conclusions on the implementation of the methods for testing structural identifiability.

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The advantage of the development methods in series is that it enables us to follow a clearly identified procedure, which is not necessarily the case for the other methods. Having said this, it should also be noted that the maximum number of equations necessary for obtaining a result is unknown. In any case, generally, it is not certain that additional information will be brought in by adding derivatives of a higher order. This absence of an upper limit for the number of terms to be considered means that the result is only a sufficient (but not necessary) condition of identifiability. Furthermore, it will always be difficult to know the combinations of identifiable parameters beforehand. In any case, there is no systematic rule. Having said this, a good basis for an initial choice is frequently either good sense or the structure of terms ) ) in the successive developments. For example, the choice of θ1 (= − μmax X(1−Y Y appears to be obvious from equations (3.8), (3.9) and (3.10). Recently, Petersen [PET 00a] proposed a method, which enables us to suggest potentially identifiable combinations for a class of bioprocess models described by Monod kinetics. 3.3. Practical identifiability Practical identifiability is complementary to structural identifiability, in that we are now interested in the impact of the available experimental data (we are no longer considering ideal measurements!) concerned with the identifiability of the studied model’s parameters. A typical example of the distinction between the two concepts of identifiability is Monod’s model used in a process of simple microbial growth (a substrate, a biomass). We can show that this model is structurally identifiable [ABO 78], but in actual practice, this often turns out to be non-identifiable, given the low quality of available experimental data and their limited number (e.g. [HOL 82a]). This is manifested, for example, in the following form: any variation of a parameter (for example, μmax in the case of Monod’s model) can be almost perfectly compensated by a proportional variation of another parameter (KS in the case of Monod’s model) and yet can still reproduce a goodness of fit between the experimental data and the numerical predictions of the dynamic model. In addition, it may be possible that the algorithms employed for estimating the value of the parameters of a nonlinear model have bad convergence properties or they might turn out not to be well-conditioned numerically. In that case, the estimated values might be very sensitive to the conditions of estimation. We can then have parameter values that are susceptible to variation over large ranges, as a result of which the physical interpretation of their values becomes difficult. The main question is thus as follows: given certain experimental data, what precision can we obtain for the estimated values of parameters, or, in other words, what effect can a small variation in the values of parameters have on the good fit between model and data? 3.3.1. Theoretical framework Let us see how to formulate this question mathematically [MUN 91]. The parameter estimation can, in reality, be formulated as the minimization of the following


53

quadratic criterion using an optimum choice of values for the parameters θ: J(θ) =

N

ˆ − yi )T Qi (yi (θ) ˆ − yi ) (yi (θ)

(3.18)

i=1

ˆ are the vectors of the N values measured and the predictions of the where yi and yi (θ) model at the instants ti (i = 1 to N ) and Qi is a weighted (square) matrix, respectively. The value of J obtained for a set of parameters slightly differing from the optimum values is given by the following expression [MUN 89]: N T N

∂y

∂y T (t (t ) Q ) δθ + tr(Ci Ci−1 ) (3.19) E[J(θ + δθ)] ∼ δθ = i i i ∂θ ∂θ i=1 i=1 where Ci represents the matrix for the covariance of the measurement errors (Qi is often chosen to be equal to Ci−1 ). An important consequence of (3.19) is that for optimizing the practical identifiability (i.e., for maximizing the difference between J(θ + δθ) and Jopt (θ)), it will be necessary to maximize the term enclosed in the brackets [·]. This term is in reality the Fisher information matrix and expresses the information contained in the experimental data [LJU 87]: F =

T ∂y (ti ) Qi (ti ) ∂θ ∂θ

N

∂y i=1

(3.20)

This matrix is the inverse of the covariance matrix of estimation errors of the best unbiased linear estimator [GOD 85]: N T −1

∂y ∂y −1 (ti ) Qi (ti ) (3.21) V =F = ∂θ ∂θ i=1 The terms ∂y ∂θ are the sensitivity functions of the output variables. They quantify the dependence of the model’s predictions with respect to the parameters. Approximation (3.19) of the objective function enables the tracing of the curves of the function for the values given in the space of the parameters. In the case of a model with two parameters (μmax and Ks in the case of Monod’s model, for example), these curves are ellipses. As underlined by Munack [MUN 89], the axes of the ellipses correspond to the characteristic vectors of the Fisher matrix and their length is proportional to the square root of the inverse of the corresponding eigenvalue. Thus, the ratio of the greatest eigenvalue to the smallest one (as absolute value) is a measure for the form of the function close to the optimum estimated values of the parameters. Figure 3.1 (left) shows a very much flattened ellipse corresponding to a very high ratio of eigenvalues, whereas in the figure on the right, the eigenvalues are closely packed.

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Bioprocess Control

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

9 8 7 6 5 4 1

2

3

4

5

6

7

8

9 10 11

3 38

38.5

39

39.5

40

40.5

Figure 3.1. Contour trace of an objective function that is ill-conditioned (left) and well-conditioned (right) in the case of estimation of two parameters [PET 00a]

3.3.2. Confidence interval of the estimated parameters An important result of the practical identifiability is the possibility of calculating the parameter variance, which enables us to calculate a confidence interval for them. If covariance matrix V (3.21) is known and considering Qi to be the inverse of the measurement error covariance matrix we can calculate in an approximate manner the typical deviations for the parameters as follows:

(3.22) σ(θi ) = Vii The confidence intervals are thus obtained in the following manner: θ ± tα;N −p σ(θi )

(3.23)

for a confidence level of 100(1 − α)% and the values of t obtained from Student distributions. On applying equation (3.22), an error occurs because Vii contains the measurement error. In the case where a single measured variable is used for identification, a more realistic estimation of the confidence interval can be obtained by evaluating: s2 =

Jopt (θ) N −p

(3.24)

Here, p is the number of parameters of the model and Jopt (θ) is defined by equation (3.18) with Qi as the identity matrix. In fact, s2 calculated using (3.24) also contains the model error. The standard deviations can thus be calculated in an approximate manner as follows:

(3.25) σ(θi ) = s Vii


55

One method of obtaining the exact confidence intervals for the nonlinear systems is to explore all the values of the objective function systematically for a very large number of parameter combinations. This is done at the cost of several calculations using a computer because the number of evaluations grows exponentially with the number of parameters [LOB 91]. In general, the confidence intervals are described by the hypervolumes limited by hypersurfaces in the space of parameters. In the particular case of a two-parameter model, this corresponds to a 2-dimensional domain in 3-dimensional space generated by (θ1 , θ2 , J(θ)). This enables us to see certain problems of parameter estimation (Figure 3.1). In this space, the objective function is a surface and the confidence interval is delimited by a curve. The confidence interval (1 − α) corresponds to the set of parameter values, for which the objective function is smaller than the limit value [BEA 60]: p Fα;p,N −p Jopt ∗ 1 + (3.26) N −p where N and p are the number of measurements and parameters, respectively, and F is the value of the F -distribution with p and N − p degrees of freedom and a level of confidence α. In Figure 3.1, the contour curves are calculated according to this approach for increasing values of α. 3.3.3. Sensitivity functions As suggested above, the sensitivity functions play a key role in the evaluation of practical identifiability of parameters. An important property is that when the sensitivity functions are proportional, covariance matrix (3.21) is singular. The model is therefore not identifiable [ROB 85]. It turns out that in numerous cases of models used for biological processes, the sensitivity functions are often quasi-linearly dependent. This often gives rise to estimated values of strongly correlated parameters. This can be seen in Figure 3.1 (left) where the functional of the error is widely spread (as a “valley”), thereby indicating that more than one combination of parameters can describe the same data in an equivalent manner. Therefore, one means of studying the practical identifiability of a model involves tracing the sensitivity functions. There are numerous examples of such an approach available in the scientific literature, especially in the case of Monod’s model with biomass and substrate measurements [HOL 82a, HOL 82b, MAR 89, POS 90, ROB 85, VIA 85, PET 00c]. ∂y

Different approaches are possible for determining the sensitivity functions ∂θji . The most precise method is analytical derivation. However, for complex models it

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is necessary to make use of symbolic manipulation software. We can also resort to numerical calculations on the basis of the numerical values of parameters, which deviate slightly from the nominal values. In practice, if we consider a deviation Δθi for parameter θi , the sensitivity of the output variable yj to θi is calculated as follows: ∂yj yj (θi ) − yj (θi + Δθi ) = ∂θi Δθi

(3.27)

To illustrate, let us perform the calculation in the case of Monod’s model (3.6) using the measurements of OUR. The sensitivity of OUR with respect to μmax is equal to: ∂ OUR ∂S ∂ dS d = − (1 − Y ) (3.28) = −(1 − Y ) ∂μmax ∂μmax dt dt ∂μmax is obtained through integration (with zero as In this equation, the sensitivity ∂μ∂S max initial condition) of the following differential equation: d ∂S S ∂ μmax X = − dt ∂μmax ∂μmax Y KS + S (3.29) μmax KS S ∂S X + =− Y KS + S (KS + S)2 ∂μmax where the substrate concentration S is calculated by integrating the dynamic equation of the substrate: μmax X S dS =− dt Y KS + S

(3.30)

The simultaneous solution of differential equations (3.29) and (3.30) enables us to obtain sensitivity functions (3.28). We can proceed in a similar manner for the sensitivity of OUR with respect to KS . We thus obtain the following relationships: ∂ OUR ∂S d = −(1 − Y ) (3.31) ∂KS dt ∂KS ∂S KS ∂S S d μmax X − =− (3.32) dt ∂KS Y (KS + S)2 ∂KS (KS + S)2 We can observe by analyzing the above relationships that the sensitivity functions depend on the value of the model parameters. This is a characteristic of the nonlinear models [DRA 81]. The Fisher matrix will therefore also be a function of the model parameter value: this will have consequences on the optimum design of the experiments (see below).

OUR

Substrate

0

5

10

15

20

25

30

0.250

5.0E−03

Km

0.125

2.5E−03

0.000

0.0E+00

−0.125

–2.5E+03

−0.250

–5.0E+03

μmax

−0.375

–7.5E+03

−0.500 0

Time (min)

5

10

15

20

57

dOUR/dμmax

35 30 25 20 15 10 5 0

dOUR/dKm

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Substrate (mg/l)

OUR (mg O2/l.min)


25

–1.0E+04 30

Time (min)

Figure 3.2. Sensitivity functions (right) for a profile of OUR with Monod’s model (left)

An example of a profile of OUR with the evolutions of the sensitivity functions is shown in Figure 3.2. It can be observed that the sensitivity functions for KS and μmax show an almost linear dependence. Intuitively, the sensitivity functions express the dependence of a variable on a change in the value of a parameter. In the example shown in Figure 3.2, these conditions of quasi-linearity are predominant while the substrate concentration has fallen to a value close to the affinity constant KS . It can be easily understood that a possible first approach for improving the information contained in the data is to choose the sampling instances corresponding to the zones where the parameters exhibit influence, i.e. when the sensitivity is high. 3.4. Optimum experiment design for parameter estimation (OED/PE) 3.4.1. Introduction One method for improving the quality of data for identification, with a view to obtaining the values of reliable parameters, is to plan the experiments, which serve for the collection of data in order to increase the information contained in these data for identification. This task is made particularly difficult in the context of bioprocesses, given the limited experimental resources both from the point of view of the time required for the experiment (this can turn out to be quite long and can extend to many weeks if not months) and the costs involved (the measurements can turn out to be quite cumbersome). The goals pursued by the experiment design in the context of identification may be as follows [VAN 98]: – reliable selection of structures for sufficient mathematical models for describing the process dynamics; – accurate estimation of the model parameters; – combination of the above two goals. Each of these objectives is associated with one or another quantitative function, which will determine the aspects on which the experiments are to be focused. In this

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chapter, we will focus on the experiment design in order to estimate the parameters (second objective). If the reader is interested, they can refer to the chapter in [VAN 98] on aspects of experiment design for the other two goals. Experiment design typically comprises three steps: 1) definition of an objective function of the OED, which is the mathematical translation of the goal pursued by the experiment design; 2) inventory of the degrees of freedom that are available and the experimental constraints; 3) determination of the optimum for the objective function by playing in a suitable manner with the degrees of freedom while at the same time ensuring that the constraints are respected. Basically, the algorithm which carries out the experiment design does so through numerical simulation of experiments so as to quantify the potential effects of the proposed experimental conditions on the objective function. On the basis of the results given by an algorithm of optimization that is capable of taking into account the experimental constraints, we can propose an optimum experiment, which will be subsequently implemented. As the experiment design is largely dependent on the process model, it is evident that a prerequisite for good planning is the existence of a sufficiently reliable process model. The following questions can serve as a guide before the study of the experiment design is implemented. What shall we measure? This question highlights the importance of a good definition of the system (choice of input, output and state variables, etc.). Where shall we measure? This is the problem of sensor location. When shall we measure? Here we are concerned with the choice of sampling time(s). What types of manipulations are involved? This is often the most important question. Once the degrees of freedom are determined, we still have to choose the type of signal to be implemented. This often requires imagination and creativity for designing the signals, which will act on the procedure in such a manner that we can obtain high-quality information. As examples of possible signals, we can have periodic (sinusoidal) signals, pulses, pseudo-random binary signals (PRBS) [LUB 91], etc. What about the treatment of data? Even though most of the time not considered to be an integral part of experiment design, the pretreatment of data is often applied


59

before identifying the model. Typically, this includes the elimination of measurement noise, the detection and elimination of anomalous data and the elimination of dynamics of no interest. The quality of a set of estimated parameter values can be evaluated in diverse ways, for example on the basis of the quality of predictions given by the model. Nevertheless, in most cases, we characterize the quality of parameter estimation on the basis of the estimation errors (confidence intervals as given above (3.23)) or, more generally, on the basis of covariance matrix (3.21). This appears from now on to be clearly linked to the practical identifiability. If the objective of experiment design is to improve the estimation of parameters, it is evident that covariance matrix (3.21) or its elements, or its inverse (Fisher matrix) will constitute a main element thereof. It should be noted that other methods can also turn out to be equally appropriate for treating the experiment design. For example, it is possible that the transformation of the model into a numerically more robust equivalent form would result in a more reliable estimation [RAT 86] (see also the case study at the end of the chapter). We can also think of using prior information (such as the knowledge of the maximum growth rate) in order to impose bounds on the values of parameters and improve in this manner the identifiability of other parameters [MUN 89]. 3.4.2. Theoretical basis for the OED/PE First, let us recall that the variable that describes the reliability of the estimated value of a parameter, i.e. the standard deviation of the estimation error of the parameter, is given by the following relationship (3.22) in the case of a single measured output variable:

σ(θi ) = s Vii (3.33) We can thus see that two terms can be manipulated in order to increase the precision of the estimation of the parameter. The first term (square of the average residue s2 ) is calculated from the sum of errors raised to the power of two between N predictions of the model and N experimental data, Jopt (3.24): s2 =

Jopt (θ) N −p

(3.34)

It is possible to reduce this value by increasing the amount of experimental data N , for example by repeating the experiments. This is particularly useful when p is not negligible with respect to N . When N is very large, the increase in the value of the denominator will be proportional to the increase in the value of the objective function Jopt which is typically a sum N of errors, raised to the power of two.

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Alternatively, we can reduce the covariance of the estimation error V (3.21). We will now concentrate on the methods, which enable the reduction of V . Different approaches have been developed for planning experiments on the basis of covariance matrix V , or equivalently, on the basis of Fisher information matrix F , via the optimization of certain scalar measurements of these matrices [MUN 91]: – A – optimal design criterion: min[tr(F −1 )] – Modified A – optimal design criterion: max[tr(F )] – D – optimal design criterion: max[det(F )] – E – optimal design criterion: max[λmin (F )] (F ) – Modified E – optimal design criterion: min λλmax min (F ) where λmin (F ) and λmax (F ) are the lowest and the highest eigenvalues of the Fisher information matrix, respectively. These different criteria can be interpreted as follows [MUN 91]. Design criteria A and D minimize the arithmetic and geometric means of the identification errors, respectively, whereas criterion E enables the minimization of the largest error. For the two E criteria, the optimization is carried out over the eigenvalues of the Fisher matrix: this ensures the maximization of the distance with respect to the singular (non-informative) case. The modified E criterion should be interpreted on the basis of the objective function’s form. The ratio of the highest eigenvalue to the lowest eigenvalue is an indication of this form. As the aim is to have eigenvalues that are as closely packed to one another as possible, it amounts to having a circle (even so it must be noted that the criterion is not directed at the objective function, i.e. the volume of the confidence interval; in other words, optimizing the modified E criterion can very well lead to a rounder but wider confidence interval). When λmin (F ) is equal to zero, this ratio equals infinity; i.e., there are infinite combinations of parameters, which enable us to describe the experimental data and from then on the experiment becomes non-informative. The Fisher matrix is thus singular. In that case, criteria D and E are equal to zero, while it is impossible to determine the A criterion as the inversion of F is impossible. This case enables us to also highlight problems, which can arise with the modified A criterion: even in the case of a non-informative and thus a non-identifiable experiment, the modified A criterion can always be maximized because one of the other eigenvalues can be very high [GOO 87]. Furthermore, it should be remembered that the elements of the Fisher information matrix depend on the units of the parameters. As a result, the eigenvalues of the Fischer matrix depend on the units; this can therefore form the subject matter of resetting the scale. Further, it is possible that an experiment that is optimum for a certain set of units is not optimum for another. This is true for each of the above criteria, except for the D criterion ([PET 00a]), even though the absolute value of this criterion varies for different units of parameters, the optimum experiment remains the same. Only the optimum experiment generated by the D criterion is independent of the units selected. On the other hand, this dependence of the Fisher information matrix with respect to


61

the units can be used favorably in the case where numerical problems crop up during its inversion. In that case, we can modify the condition number (which is equivalent to the modified E criterion) by simply changing the units and thus retain the reliability of the matrix inversion. This can turn out to be of special interest for a certain number of numerical optimization algorithms used for parameter estimation (see below). Finally, let us mention that other criteria can be proposed, for example the parameter estimation error can be reduced by using the variance of this parameter as the criterion. 3.4.3. Examples Let us now examine some examples of experiment design for parameter estimation. Vialas et al. [VIA 85] proposed a more frequent sampling for certain defined experimental periods. Holmberg [HOL 82a] showed that the identifiability of the parameters for Monod’s model depends very much, for the batch experiments, on the initial value of the substrate concentration; according to Holmberg the initial optimum substrate concentration depends on the noise level and the sampling instant. It is also evident on the basis of her results that the experiment design depends on the value of the parameters, which might induce a modification in the experiment design if we have to take into account the changing character of the process in the course of time. Munack [MUN 89] proposed different modifications of batch experiments and showed that considerable improvements could be achieved in the confidence intervals by means of the experiment design techniques. Vanrolleghem et al. (1995) highlighted the improvement in the accuracy of estimation for the two parameters of a biodegradation model with Monod kinetics. The addition of a substrate pulse (Figure 3.3) enabled the reduction of the confidence interval by 25%. This diminution in the variance was distributed equivalently between the two parameters μmax and KS , which is a typical result of an OED based on the D criterion. A similar study was carried out by [PET 00a] for a real application, where a model for the treatment of wastewater had to be identified. The model contained two submodels, one for the oxidation of nitrogen (nitrification) and the other for the oxidation of carbon. In this case also, the accuracy was improved by adding to the wastewater sample a determined quantity of ammonia. The confidence interval was reduced by 20% for the carbon oxidation submodel and by 50% for the nitrification submodel. Baetens et al. (2000) identified six parameters in a biological phosphate removal process. The experiment design was evaluated by considering more than one degree of freedom. The D criterion could be improved by a factor of at least four. This corresponds to an improvement in the confidence intervals in the estimation of parameters by a factor of two. The evolution of the modified E criterion for changing experimental conditions showed that the longest axis of the confidence ellipsoid could be reduced by a factor of two, as the increase in the modified E criterion by a factor of four could be obtained by increasing the acetate dosage.

Bioprocess Control 0.70

0.70

0.60

0.60

OUR (mg O2/l.min)

OUR (mg O2/l.min)

62

0.50 0.40 0.30 0.20 0.10 0.00

0.50 0.40 0.30 0.20 0.10 0.00

0

5

10

15 Time (min)

20

25

30

0

5

10

15

20

25

30

Time (min)

Figure 3.3. Experimental respirograms obtained for a batch experiment and for a fed-batch experiment with an additional pulse after 14.6 minutes. The parameter variances are 50% less for the fed-batch experiment

Another approach was developed by Petersen et al. [PET 00c] related to the degrees of freedom for the experiment design. The differences in the accuracy of parameter estimation were evaluated when different sets of measured variables were used for identifying the parameters of a nitrification model. The following possibilities were evaluated: – a single measurement of dissolved oxygen SO in an aerated batch reactor; – two measurements of dissolved oxygen SO at the inlet and at the outlet of the respiration chamber; – the respiration rate OURex calculated from these two measurements of oxygen; – the rate of proton production Hp obtained from the pH regulator; – two measurements of dissolved oxygen SO and the rate of proton production Hp . Despite the fact that in this case the mass-transfer parameters KL a and SO,sat should be estimated (in addition to the three nitrification parameters μmax , KN H , and SN H (0)), the estimations of nitrification parameters can be more accurate with the measurements of dissolved oxygen. The confidence intervals are then 10 times smaller (the variances are 100 times smaller). The high(er) level of noise in the respiration rate measurements undoubtedly provides a partial explanation for this phenomenon. This result highlights the importance of an appropriate choice of the measured variables used for the estimation of parameters. Munack [MUN 91] has also shown that the effect of different measurement configurations could lead to datasets having information contents that are significantly different. In his analysis, not only the type of data but also the amount and their position in the core of the reactor play a part. Versyck et al. [VER 97] highlighted a remarkable result with regard to experiment design, in liaison with optimum control: the profile of optimum control for a


63

growth model with Haldane kinetics constitutes an excellent starting point for finding an experiment for which the modified E criterion is optimum (equal to 1). In addition, it is shown in [VER 00] that, in the case of modeling of the temperature’s effect on the kinetics, the uniqueness of the modified E criterion can be attained using an appropriate temperature profile. The authors also show that more than one experiment design enables us to attain this optimum and that it is therefore possible to take into account other selection of criteria, such as practical feasibility or validity of the model (see also [BAL 94]). 3.5. Estimation algorithms Until now we have been concentrating on analyzing the identifiability of parameters and the design of experiments for parameter estimation. Now we move on to the next question. We have constructed and/or selected a model and we have a set of appropriate data for the estimation (obtained ideally after experiment design): how are we going to proceed in order to estimate the values of the parameters of this model? As this chapter gives only an overview of parameter identification, we will not deal with the algorithms used for parameter estimation in a detailed manner. In any case, it could hardly happen that the readers would be asked to develop the estimation algorithm of these parameters by themselves; they would more than certainly have to resort to software available on the market (of which Matlab and its tool boxes are typical examples). We would like to suggest to readers with an interest in the subject to refer to specialist papers, such as [WAL 94]. 3.5.1. Choice of two datasets Once the experimental data are available and before passing on to the estimation of the model parameters, the first essential and fundamental stage is the separation of the data into two distinct datasets: 1) a first set of data used for calibrating the parameters; 2) the remaining data for validating the estimation. The first dataset will be employed for calculating the parameter estimations using an appropriate estimation algorithm. The second dataset will be used for verifying that these parameters reproduce the process dynamics well. The independence between these two datasets is fundamental for validating the identification procedure (while it can be expected that the model obtained on the basis of certain data would reproduce these data, and it will be very important to know whether this model reproduces other data, which have not been used for calibrating its parameters). The separation of the data should be performed in such a manner that the first dataset (calibration) be sufficiently informative and thus covers a sufficient spectrum of experimental conditions (in accordance with an experiment schedule, for example). The second dataset should

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contain sufficient data in order to render the validation as credible as possible. We will avoid having datasets that are very unbalanced in terms of the number of data, even though generally the first set of data (calibration) will have more data than the second set (validation). 3.5.2. Elements of parameter estimation: least squares estimation in the linear case In section 3.3, we stated that the estimation of parameters consisted of basically optimizing a certain error criterion, typically of prediction, such as the objective function (3.18). The concrete form taken by the estimation algorithm will depend largely on the model, especially depending on whether or not the model is linear in the parameters. For example, in the linear case with one equation, the model will be generically written in the following form: yi = φTi θ

(3.35)

where θ is the parameter vector, φi is called the “regressor”, yi contains the independent terms of parameters, and index i corresponds to the temporal index. The choice of a representation of the model in discrete time is natural insofar as the estimation is based on the experimental data themselves available in the form sampled on schedule. To illustrate this, let us consider the balance equation of a substrate S involved in two reactions: growth reaction (where additionally it is assumed that the dependence of the kinetics with respect to the substrate concentration is of the first order) and maintenance reaction. This equation is written as follows: dS = −k1 αSX − km X + DSin − DS dt

(3.36)

Let us assume that neither α (kinetic growth constant) nor km (maintenance coefficient) are known and that we have the data for the concentration of substrate S and biomass X as well as for dilution rate D and of inlet substrate concentration in the feed Sin at time intervals T (i.e., following a sampling period T ). Let us also assume that the yield coefficient k1 is known. Vector θ is thus equal to: α θ= (3.37) km Si+1 −Si By approximating the time derivative dS (where dt with a finite difference T index i indicates the value at instant t = iT ), we can rewrite balance equation (3.36) as follows:

Si+1 − Si − T Di Sin,i + T Di Si = −k1 Si Xi T α + Xi T km

(3.38)


65

In other words, in the formalism of equation (3.35), yi and φi are equal to: −k1 Si Xi T yi = Si+1 − Si − T Di Sin,i + T Di Si , φi = (3.39) Xi T In the simple case of the linear equation in parameters θ, the linear regression techniques, for example an estimator using least squares, can be used. In that case, we consider objective function (3.18) in the following form: J(θ) =

N

ˆ − yi )2 βi (yi (θ)

(3.40)

i=1

The least squares estimator, which minimizes this criterion is written in simple terms as: −1 N N

θˆ = βi φi φTi βi φi yi (3.41) i=1

i=1

When βi in the objective function is not equal to 1, we mean a weighted least squares estimator. A possible choice for βi is to take the inverse of the variance of the measurement yi . 3.5.3. Overview of the parameter estimation algorithms In general, the parameter estimation follows the logic described above for the simple case of the monovariable linear model. It is based on the minimization of a criterion, whose typical model is shown above (3.18)1 J(θ) =

N

ˆ − yi )T Qi (yi (θ) ˆ − yi ) (yi (θ)

(3.42)

i=1

The weights Qi can be different at each sampling instance, which can turn out to be of interest in the case of a measurement quality, which varies with time. A typical choice for matrix Qi consists of taking the inverse of the covariance matrix of measurement errors (as suggested above in the monovariable case).

1. However, use is sometimes made of other objective functions, based for example on the minimization of the absolute value of the prediction error, on the minimization of the maximum absolute error, or on the maximization of the number of changes of sign in the residue sequence.

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Before reviewing the minimization methods, which enable the determination of the values of the selected objective function’s parameters, we can say that in general the problems encountered during parameter estimation using the least-squares type methods are due to three kinds of phenomena. First, it is possible that the residues may not be distributed normally, contrary to the assumptions on the basis of conception of this type of algorithm. Then, for giving correct weighting to the residues, it will be necessary to know the value of the variance of the error at each measurement point. This can be performed precisely only when there is a sufficient number of repeated measurements at a point, which is rarely the case. Finally, the outliers can lead to an inaccurate estimation of the parameters as well as to biased confidence intervals. The search for an optimum for a multivariable function is a common problem in numerous fields of research, and there is substantial expertise on this subject. However, attention should be drawn to the fact that the minimum (the set of parameters minimizing the criterion) can either be local or global (see Figure 3.4): in the first case, it corresponds to a minimum in a region limited in the space, whereas in the second case, it corresponds to the minimum in all of the parameter space. Despite considerable efforts, as of today there is no minimization algorithm, which could ensure the determination of the global minimum for any problem of nonlinear optimization. We see in particular that the convergence toward a (local or global) minimum depends ˆ very much on the initial values selected for the parameters θ(0). We can easily understand this phenomenon on the basis of curve 3.4; by taking the initial values to the left of the local minimum, it is probable that the minimization algorithm will be fixed at the local minimum and will not attain the global minimum, which is attained by starting from the right side of the same. In order to avoid convergence only to a local minimum, it might turn out to be important to select a sufficient number of iterations of the minimization algorithm by taking the sets of initial parameter values that are sufficiently different in order to cover as wide a spectrum of values as possible.

Local minimum Global minimum

Parameter Figure 3.4. Local and global minima for an objective function


67

Schuetze [SCH 98] tried to classify the diverse minimization methods into two groups: the local minimization methods and those of global minimization, even if there is something artificial about this insofar as all the so-called global methods bring into play a local procedure at a certain level. Local minimization There are a large number of procedures for local minimization. Among them, many are based on the derivatives of the objective function to be minimized, information that is thus either considered directly available or numerically calculated. The basic idea of these so-called gradient methods is that a search direction for the minimum is sought in the parameter space. Once this direction is selected, the algorithm chooses the amplitude with which the values of parameters are modified. If we can show the convergence of the method, the convergence can turn out to be very slow [DRA 81]. An alternative involves the use of Newton’s method. Even though it enables a possibly quicker convergence, the disadvantage of the method is that it can diverge or oscillate around the solution. The modification that is probably the best known among the basic methods is the Levenberg-Marquardt algorithm [MAR 63]. It is a compromise between the gradient method and Newton’s method. A suitable choice of the algorithm parameters enables us to combine the advantages of the gradient method (convergence properties) with those of Newton’s method (rapid convergence). In order to improve the convergence, we can make use of the fact that the majority of functions are reasonably well approximated by a quadratic function close to the optimum. The BFGS methods (Broyden, Fletcher, Goldfarb and Shanno) [FLE 87] is probably the best known of this class of methods. [VAN 96] has, however, noted that this type of method, if they converge very quickly, can be sensitive to the local minima. One local minimization method that is not based on the calculation or approximation of the derivatives is the Powell method, with the improvements proposed by Brent [PRE 86, GEG 92]. It is based on a repeated combination of a one-dimensional search along a set of many directions. This probably involves one of the best methods in terms of compromise between convergence speed and insensitivity to local minima. Another well-known local minimization method, which does not require information about the derivatives, was proposed by Nelder and Mead; it is better known as the Simplex method (not to be confused with the simplex method in linear programming!). A simplex is a set of (p + 1) parameter vectors in the parameter space. The method consists of a succession of comparisons of the objective function with these edges, followed by a replacement of the edges having the highest value by another point in the space of the parameters of dimension p. The method is well appreciated

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for its robustness with respect to the local minima, its ease of implementation and its reasonable convergence speed [PRE 86, SCH 98, VAN 96]. Global minimization The global minimization methods can be roughly classified into two groups [SCH 98]. The first group comprises purely deterministic methods, such as the gridding method. It consists of evaluating the objective function for a large number of points predefined over a grid covering the parameter space. If there is a sufficient number of function evaluations, there are chances of attaining the minima. This method is not very effective, unless it is improved by refining the grid after a series of evaluations. The second group of global methods can be called random probing methods, as random decisions are included in the procedure for attaining the optimum. Among them, the adaptive methods take into account the information obtained during the preceding evaluations. For one method (simulated annealing [PRE 86]), the idea is that the search will not always be toward a possible solution (which could just be a local minimum), but may be, from time to time, along another direction. This method can be viewed as preceding the popular methods such as genetic algorithms (GA) [GOL 89]. These algorithms commence with an initial population of prospective solutions (somewhat similar to the edges in the Simplex method) sampled in a random manner in the parameter space. In genetic algorithms, new prospective solutions are obtained by imitating the process of biological evolution of cross breeding, mutation and selection among the parameter “populations”. The definition of the parameters of the algorithm itself is crucial for correct implementation. 3.6. A case study: identification of parameters for a process modeled for anaerobic digestion The anaerobic digestion model (2.16)–(2.29) formed the subject matter for a systematic study to identify parameters on the basis of a fixed bed reactor of LBE - INRA at Narbonne (see [BER 01, BER 00] for a more detailed study). This case study is remarkable in view of many aspects. Firstly, the identification of parameters is riddled with traps that are typical for biological systems: the process is very complex (numerous bacterial populations participate in the process; they can have different behaviors depending on the operating conditions). There are no direct measurements for each of the acidogenic and methanogenic bacterial populations and in general, a limited number of process variables are accessible. The process is slow and can be destabilized easily by the accumulation of fatty acids. These characteristics have significant consequences for the selected model. The model cannot be very complex lest it should turn out to be non-identifiable, whereas the simplified modeling hypotheses can have an impact on its capacity to predict the dynamics of the process. The choice of the structure is therefore a critical stage in view of the fact that the model should necessarily contain elements that are essential for the process dynamics.


69

Moreover, the identification itself contains original elements with respect to the rest of the chapter. Constraints on the process have led us to conceive an experimental plan, not on the basis of the above techniques but motivated by concern for covering as wide a range of operating conditions as possible, while at the same time limiting the duration (here necessarily long) of the experiments as much as possible. In addition, the structure of the reaction system models enables us to separate the parameters into three classes (yield coefficients, kinetic parameters and transfer parameters) and to carry out the identification of each class of parameters in a distinct manner. In this case study, emphasis is laid on parameter estimation using linear regression. Having said this, it will be prudent to draw attention to the fact that this approach cannot turn out to be optimum from a statistical point of view insofar as the statistical conditions on the variables used in the linear regression could not be completely fulfilled in order to enable an estimation that might be statistically accurate and sufficiently reliable. In general, care must be taken during the interpretation of the estimated values of parameters and their standard deviation. Furthermore, these parameter values given by linear regression were suggested to be used as initial values of an estimation on the basis of the nonlinear model. However, such an estimation has not turned out to be useful in the present case. 3.6.1. The model Let us once again proceed from the equations developed in Chapter 2. A variant of these was considered as here we are dealing with a fixed bed reactor, where the bacteria are fixed on supports. Thus, formally there is no dilution term in the balance equations. However, on the other hand, it was noted that a portion of the biomass was detached: it was hypothesized that the rate of detachment was proportional to the rate of dilution, represented by coefficient α. Under these conditions, the model is rewritten as follows: dX1 dt dX2 dt dS1 dt dS2 dt dZ dt dC dt

= μ1 X1 − αDX1

(3.43)

= μ2 X2 − αDX2

(3.44)

= D(S1in − S1 ) − k1 μ1 X1

(3.45)

= D(S2in − S2 ) + k2 μ1 X1 − k3 μ2 X2

(3.46)

= D(Zin − Z)

(3.47)

= D(Cin − C) − qC + k4 μ1 X1 + k5 μ2 X2

(3.48)

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Bioprocess Control

qC = kL a(C − S2 − Z − KH PC )

φ − φ2 − 4KH PT (C + S2 − Z) PC = 2KH φ = C + S2 − Z + KH P T +

k6 μ2 X2 kL a

qM = k6 μ2 X2 C − Z + S2 pH = −log10 Kb Z − S2

(3.49) (3.50) (3.51) (3.52) (3.53)

In addition, growth models were chosen, one Monod model for acidogenesis and one Haldane model for methanization: μmax 1 S1 μ0 S2 , μ2 = (3.54) μ1 = KS1 + S1 KS2 + S2 + S22 /KI2 In the absence of systematic rule, this choice was dictated by a desire to have kinetic models that are sufficiently simple, coherent with those normally used for anaerobic digestion, but also capable of highlighting the potential instability of the process in the presence of the accumulation of volatile fatty acids (explaining the choice of the Haldane model for μ2 ). Knowing that Kb and KH are known chemical and physical constants (Kb = 6.5 10−7 mol/l, KH = 16 mmol/l/atm), we note that the model contains 13 parameters to be identified. In addition, the variables available for the measurement are the dilution rate D, the inlet concentrations, S1in , S2in , Zin , Cin , the gas flow rates qC , qM , the concentrations S1 , S2 , Z, C, and the pH. 3.6.2. Experiment design Given the complexity of the model and the large number of parameters as well as the experimental constraints (long time constants and potential instability of the process), the strategy followed here for experiment design consisted of covering a number of operating points sufficiently representative of how the process works. The experiment design is given in Table 3.1. 3.6.3. Choice of data for calibration and validation In this case, one of the priorities was to obtain a model, which would be capable of correctly reproducing as a priority the equilibrium conditions. This is why the data collected were separated into two categories: the data corresponding to the steadystate conditions for calibrating the parameters and the dynamic data for the validation. The periods corresponding to the values used for the calibration are represented in Figures 3.5 and 3.6 by a bold line on the abscissa.


D(day−1 )

S1in (g/l)

S2in (mmole/l)

pH

0.34

9.5

93.6

5.12

73.68

4.46

38.06

4.49

0.35

10

0.35

4.8

0.36

15.6

0.26

10.6

72.98

4.42

0.51

10.7

71.6

4.47

0.53

9.1

68.78

5.30

112.7

71

4.42

Table 3.1. Characteristics of average feed conditions

250

150

Qch4 (l/h)

Qco2 (l/h)

200 100

150 100

50 50 0

0 10

20

30

40

50

60

70

10

20

30

40

50

60

70

10

20

30 40 50 Time (days)

60

70

7.5

350 300

7 6.5

200

pH

Qtot (l/h)

250

150

6

100 5.5

50 0

10

20

30 40 50 Time (days)

60

70

5

Figure 3.5. Simulated values (fine line) and data values (thick line) of gas flow and pH. The periods used for the calibration are represented in bold on the abscissa

3.6.4. Parameter identification With such a complex model, we can expect to encounter problems of structural and practical identifiability. We expect to be in a position to alleviate, at least partially,

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40

9

35

8

30 VFA (mmol/l)

COD g/l

7 6 5 4 3

20 15 10

2

5

1

0

0 10

20

30

40

50

60

70

80

70

70

60

60

50 C (mmol/l)

Z (mmol/l)

25

50 40 30

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30

40

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20

30

40

50

60

70

40 30 20

20

10

10

0

0 10

20

30

40

Time (days)

50

60

70

Time (days)

Figure 3.6. Simulated values (continuous line) and data values (o) of S1 , S2 , Z and C. The periods used for the calibration are represented in bold on the abscissa

the problems of practical identifiability, thanks to experiment design. However, what about the problems of structural identifiability? Without prejudging the results of an analysis, which could turn out to be complex, it seems to be interesting to take advantage of the structure of the model and its properties for formulating the problem of identification of parameters in a different form. We will see in the next chapter (state observers) that there is a state transformation (4.42)2 which enables us to rewrite a portion of the model in a form independent of the kinetics. This transformation is doubly interesting in our case. On the one hand, it enables us to proceed with the identification of kinetic and other model parameters in a distinct manner. On the other hand, this separation is of particular interest insofar as the kinetic modeling is its weakest link, given that we lack the physical base for the choice of an appropriate model and in addition, it is the chief source of nonlinearities in the model.

2. This transformation is presented in detail and in a more general form in [BAS 90].


73

It is basically this approach that was adopted here. As the identification is carried out on the basis of the steady-state data, static balance equations of biomasses X1 and X2 and the expressions for the specific growth rates, we derive the following expressions: α 1 α 1 = + KS1 ¯ D μ1 max μ1 max S1

(3.55)

1 α 1 α 1 α ¯ S2 = + KS2 + ¯ D μ0 μ0 S2 KI2 μ0

(3.56)

These equations are linear in the parameters μ1 αmax , KS1 μ1 αmax , μα0 , KS2 μα0 and 1 α KI2 μ0 . They can be determined through linear regression. The only problem is that it is not possible to distinguish between μ1 max and α; this is why we have considered a value from the literature ([GHO 74]) for μ1 max . This choice turns out to be even more acceptable, and a sensitivity analysis shows a low sensitivity of μ1 max . We can thus use the equation for the CO2 flow rate for determining kL a, by recalling that qC = kL a(CO2 − KH PC ) and that the reaction for the dissociation of bicarbonate enables the linking of CO2 to C and to pH according to the relationship: CO2 = C

1 1 + Kb 10pH

(3.57)

This enables us to determine the kinetic parameters and the transfer parameters independently of the performance coefficients. The results are summarized in Table 3.2. Parameter

Units

Value

Standard deviation

μ1 max

textDay −1

1.2

(1)

KS1

g/l

7.1

5.0

0.74

0.9

9.28

13.7

μ0

textDay

−1

KS2

mmol

KI2

mmol

α

/

0.5

0.4

kL a

Day−1

19.8

3.5

16

17.9

Table 3.2. Estimated values of kinetic and transfer parameters. (1) taken from [GHO 74]

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The values of the yield coefficients now have to be calculated. The main difficulty results from the absence of measurements of populations X1 and X2 . Without these measurements, it turns out that the yield coefficients are not structurally identifiable. This can be shown by considering, for example, a rescaling for X1 and X2 : X1 = λ1 X1 , X2 = λ2 X2 . This rescaling can be compensated by putting the yield coefficients in scale as follows k1 =

k1 k2 k4 k3 k5 k6 , k2 = , k4 = , k3 = , k5 = , k6 = λ1 λ1 λ1 λ2 λ2 λ2

(3.58)

In order to overcome the difficulty, the identification is carried out in two stages: first, the estimation of the structurally identifiable ratios of the yield coefficients (k2 /k1 , k6 /k3 , k5 /k3 , k4 /k1 ); second, the use of the data of volatile suspended solids (VSS) in order to determine the values of each performance coefficient. The estimation of the ratios of performance coefficients is performed on the basis of balance equations in steady state, which can be rewritten as follows after eliminating X1 and X2 : k4 k2 k5 k5 qC = D(Cin − C) + + D(S1in − S1 ) + D(S2in − S2 ) k1 k1 k3 k3 qM =

k2 k6 k6 D(S1in − S1 ) + D(S2in − S2 ) k1 k3 k3

The ratios kk41 + kk21 kk53 , kk53 , kk21 kk63 , kk63 can thus be determined using linear regression. The result of the calibration is presented in Table 3.3. Ratio

Units

Value

Standard deviation

k2 /k1

mmol/g

2.72

2.16

k6 /k3

/

1.62

0.12

k5 /k3

/

1.28

0.13

k4 /k1

mmol/g

1.18

3.02

Table 3.3. Estimated values of the ratios of performance coefficients

The determination of the coefficients is thus made on the basis of VSS measurements. In fact, the VSS represents an approximate indicator of X1 + X2 . The determination of the VSS distribution between the two bacterial populations has been performed by considering a ratio (= 0.2) between the acidogenic bacteria X1 and the total biomass X1 + X2 taken from [SAN 94]. The results are presented in Table 3.4.


Parameter

Units

Value

Standard deviation

k1

g S1 /g X1

k2

mmol S2 /g X1

116.5

k3

mmol S2 /g X2

268

k4

mmol CO2 /g X1

50.6

143.6

k5

mmol CO2 /g X2

343.6

75.8

k6

mmol CH4 /g X2

453.0

90.9

42.14

75

18.94 113.6 52.31

Table 3.4. Estimated values of performance coefficients

3.6.5. Analysis of the results The first essential stage after the calibration of parameters is their validation for the set of data not used for the calibration. This can be seen in Figures 3.5 and 3.6. We can generally see a good fit between the model and the transient experimental data. This demonstrates the interest in a systematic approach for the identification of parameters, all the more in a case as difficult and complex as this one. However, the reliability of the estimated values of the parameters remains to be evaluated. This corresponds to the standard deviation of each parameter shown in Tables 3.2, 3.3 and 3.4. The increase in this value should be noted while converting the ratios of the performance coefficient to their individual values. This reflects the large uncertainty associated with the determination of biomass concentrations on the basis of VSS data and the assumed ratio of their distribution. We will also note the rather high uncertainty of kinetic parameters, which is not at all surprising, given the fact that the chosen structures are heuristic. Finally, in the ratios of performance coefficients, a worse reliability is found for the parameters involving k1 , which is probably an effect of the variable nature of the concentration of feed S1in . 3.7. Bibliography [ABO 78] S. A BORHEY and D. W ILLIAMSON, “State and parameter estimation of microbial growth processes”, Automatica, vol. 14, pp. 493–498, 1978. [BAL 94] M. BALTES, R. S CHNEIDER, C. S TURM and M. R EUSS, “Optimal experimental design for parameter estimation in unstructured growth models”, Biotechnol. Prog., vol. 10, pp. 480–488, 1994. [BAS 90] G. BASTIN and D. D OCHAIN, On-line Estimation and Adaptive Control of Bioreactors, Elsevier, Amsterdam, 1990.

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[BEA 60] E.M.L. B EALE, “Confidence regions in non-linear estimation”, J. Roy. Statist. Soc. Ser. B, vol. 22, pp. 41–88, 1960. [BER 00] O. B ERNARD, Z. H ADJ -S ADOK and D. D OCHAIN, “Advanced monitoring and control of anaerobic treatment plants: II - Dynamical model development and identification”, in Proc. Watermatex 2000, pp. 3.57–3.64, 2000. [BER 01] O. B ERNARD, Z. H ADJ -S ADOK, D. D OCHAIN, A. G ENOVESI and J.P H . S TEYER, “Dynamical model development and parameter identification for an anaerobic wastewater treatment process”, Biotech. Bioeng., vol. 75, pp. 424–438, 2001. [BOU 98] S.V. B OURREL, J.P. BABARY, S. J ULIEN, M.T. N IHTILÄ and D. D OCHAIN, “Modelling and identification of a fixed-bed denitrification bioreactor”, System Analysis Modelling Simulation (SAMS), vol. 30, pp. 289–309, 1998. [CAS 85] J.L. C ASTI, Nonlinear System Theory, Mathematics in Science and Engineering, vol. 175, Academic Press, Orlando, 1985. [DOC 95] D. D OCHAIN, P.A. VANROLLEGHEM and M. VAN DAELE, “Structural identifiability of biokinetic models of activated sludge respiration”, Wat. Res., vol. 29, pp. 2571–2579, 1995. [DRA 81] N.R. D RAPER and H. S MITH, Applied Regression Analysis, Wiley, New York, 1981. [FLE 87] R. F LETCHER, Practical Methods of Optimization, Wiley, New York, 1987. [GEG 92] K.R. G EGENFURTNER, “PRAXIS: Brent’s algorithm for function minimization”, Behaviour Research Methods, Instruments and Computers, vol. 24, pp. 560–564, 1992. [GHO 74] S. G HOSH and F. P OHLAND, “Kinetics of substrate assimilation and product formation in anaerobic digestion”, J. Wat. Pollut. Control Fed., vol. 46, pp. 748–759, 1974. [GOD 85] K.R. G ODFREY and J.J. D I S TEFANO III, “Identifiability of model parameters”, in Identification and System Parameter Estimation, pp. 89–114, Pergamon Press, Oxford, 1985. [GOL 89] D.E. G OLDBERG, Genetic algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Menlo Park, California, 1989. [GOO 87] G.C. G OODWIN, “Identification: Experiment Design”, in Systems and Control Encyclopedia, M. S INGH (ed.), vol. 4, pp. 2257–2264, Pergamon Press, Oxford, 1987. [HOL 82a] A. H OLMBERG, “On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities”, Math. Biosci., vol. 62, pp. 23–43, 1982. [HOL 82b] A. H OLMBERG and J. R ANTA, “Procedures for parameter and state estimation of microbial growth process models”, Automatica, vol. 18, pp. 181–193, 1982. [JEP 96] U. J EPPSSON, Modelling aspects of wastewater treatment processes, Doctoral thesis, Dep. Ind. Elec. Eng. Aut., Lund Inst. Technol., Sweden, 1996. [JUL 97] S. J ULIEN, Modélisation et estimation pour le contrôle d’un procédé boues activées éliminant l’azote des eaux résiduaires urbaines, Doctoral thesis, LASS-CNRS, Toulouse, Rapport LAAS No. 97499, 1997.


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[JUL 98] S. J ULIEN, J.P. BABARY and P. L ESSARD, “Theoretical and practical identifiability of a reduced order model in an activated sludge process doing nitrification and denitrification”, Wat. Sci. Tech., vol. 37, no. 12, pp. 309–316, 1998. [LJU 87] L. L JUNG, System Identification – Theory for the User, Prentice-Hall, Englewood Cliffs, New Jersey, 1987. [LOB 91] J.R. L OBRY and J.P. F LANDROIS, “Comparison of estimates of Monod’s growth model parameters from the same data set”, Binary, vol. 3, pp. 20–23, 1991. [LUB 91] A. L ÜBBERT, “Characterization of bioreactors”, in Biotechnology, A Multi-volume Comprehensive Treatise, K. S CHÜGERL (ed.), Measuring, Modelling and Control, vol. 4, pp. 107–148, VCH, Weinheim, 1991. [MAR 63] D.W. M ARQUARDT, “An algorithm for least squares estimation of nonlinear parameters”, J. Soc. Ind. Appl. Math., vol. 11, pp. 431–441, 1963. [MAR 89] S. M ARSILI -L IBELLI, “Modelling, identification and control of the activated sludge process”, Adv. Biochem. Eng. Biotechnol., vol. 38, pp. 90–148, 1989. [MUN 89] A. M UNACK, “Optimal feeding strategy for identification of Monod-type models by fed-batch experiments”, in Computer Applications in Fermentation Technology: Modelling and Control of Biotechnological Processes, N. F ISH, R. F OX and N. T HORNHILL (eds.), pp. 195–204, Elsevier, Amsterdam, 1989. [MUN 91] A. M UNACK, “Optimization of sampling”, in Biotechnology, a Multi-volume Comprehensive Treatise, K. S CHÜGERL (ed.), Measuring, Modelling and Control, vol. 4, pp. 251–264, VCH, Weinheim, 1991. [PET 00a] B. P ETERSEN, Calibration, identifiability and optimal experimental design of activated sludge models, PhD thesis, Fac Agr. Appl. Biol. Sci. Ghent University, Belgium, 337 pages, 2000. [PET 00b] B. P ETERSEN, K. G ERNAEY and P.A. VANROLLEGHEM, “Improved theoretical identifiability of model parameters by combined respirometric-titrimetric measurements. A generalisation of results”, in Proc. IMACS 3rd MATHMOD Conference, vol. 2, pp. 639–642, Vienna, Austria, 2000. [PET 00c] B. P ETERSEN, K. G ERNAEY and P.A. VANROLLEGHEM, “Practical identifiability of model parameters by combined respirometric-titrimetric measurements”, in Proc. Watermatex 2000, 2000. [POH 78] H. P OHJANPALO, “System identifiability based on the power series expansion of the solution”, Math. Biosci., vol. 41, pp. 21–33, 1978. [POS 90] C. P OSTEN and A. M UNACK, “On-line application of parameter estimation accuracy to biotechnical processes”, Proc. ACC, pp. 2181–2186, 1990. [PRE 86] W.H. P RESS, B.P. F LANNERY, S.A. T EUKOLSKY and W.T. V ETTERLING, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 1986. [RAK 85] A. R AKSANYI , Y. L ECOURTIER , E. WALTER AND A. V ENOT, “Identifiability and distinguishability testing via computer algebra”, Math. Biosci., vol. 77, pp. 245–266, 1985.

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[RAT 86] D.A. R ATKOWSKY, “A suitable parameterization of the Michaelis-Menten enzyme reaction”, Biochem. J., vol. 240, pp. 357–360, 1986. [ROB 85] J.A. ROBINSON, “Determining microbial parameters using nonlinear regression analysis: Advantages and limitations in microbial ecology”, Adv. Microb. Ecol., vol. 8, pp. 61–114, 1985. [SAN 94] J. S ANCHEZ, S. A RIJO, M. M UNOZ, M. M ORINIGO and J. B ORREGO, “Microbial colonisation of different support materials used to enhance the methanogenic process”, Appl. Microbiol. Biotechnol., vol. 41, pp. 480–486, 1994. [SCH 98] M. S CHUETZE, Integrated simulation and optimum control of the urban wastewater system, PhD thesis, Imperial College, London, UK, 1998. [SPE 00] M. S PÉRANDIO and E. PAUL, “Estimation of wastewater biodegradable COD fractions by combining respirometric experiments in various SO/XO ratios”, Wat. Res., vol. 34, pp. 1233–1246, 2000. [VAJ 89] S. VAJDA, K.R. G ODFREY and H. R ABITZ, “Similarity transformation approach to identifiability analysis of nonlinear compartmental models”, Math. Biosci., vol. 93, pp. 217– 248, 1989. [VAN 96] P.A. VANROLLEGHEM and K.J. K EESMAN, “Identification of biodegradation models under model and data uncertainty”, Wat. Sci. Tech., vol. 33, no. 2, pp. 91–105, 1996. [VAN 98] P.A. VANROLLEGHEM and D. D OCHAIN, “Bioprocess model identification”, in Advanced Instrumentation, Data Interpretation and Control of Biotechnological Processes, J. VAN I MPE, P. VANROLLEGHEM and D. I SERENTANT (eds.), pp. 251–318, Kluwer, Amsterdam, 1998. [VER 97] K.J. V ERSYCK, J.E. C LAES and J.F. VAN I MPE, “Practical identification of unstructured growth kinetics by application of optimal experimental design”, Biotechnol. Prog., vol. 13, pp. 524–531, 1997. [VER 00] K.J. V ERSYCK and J.F. VAN I MPE, “Optimal design of system identification experiments for bioprocesses”, Journal A, vol. 41, no. 2, pp. 25–34, 2000. [VIA 85] C. V IALAS, A. C HERUY and S. G ENTIL, “An experimental approach to improve the Monod model identification”, in Modelling and Control of Biotechnological Processes, A. J OHNSON (ed.), pp. 155–160, Pergamon, Oxford, 1985. [WAL 82] E. WALTER and Y. L ECOURTIER, “Global approaches to identifiability testing for linear and nonlinear state space models”, Math. Computers Sim., vol. 24, pp. 472–482, 1982. [WAL 94] E. WALTER and L. P RONZATO, Identification de Modèles Paramétriques à partir de Données Expérimentales, Masson, Paris, 1994. [WAL 95] E. WALTER and L. P RONZATO, “On the identifiability and distinguishability of nonlinear parametric models”, in Proc. Symp. on Appl. of Mod. and Cont. in Agric. and Bioind., IMACS, V.A.3.1–8, Brussels, Belgium, 1995.

Chapter 4

State Estimation for Bioprocesses

4.1. Introduction One of the main limitations to the improvement of monitoring and optimization of bioreactors is probably due to the difficulty in measuring chemical and biological variables. In fact there are very few sensors which are at the same time cheap and reliable and that can be used for online measurement. The measurement of some biological variables (biomass, cellular quota, etc.) is sometimes very difficult and can necessitate complicated and sophisticated operations. The challenge is to estimate the internal state of a bioreactor when only a few measurements are available. In this chapter we propose methods to build observers which will use the available measurements to estimate non-measured state variables (or at least some of them). The principle of this so-called “software sensor” is to use the process model to asymptotically reconstruct the state on the basis of the available outputs. As will be detailed in this chapter, the system must be observable, or at least detectable, in order to estimate the internal state. There are numerous methods to design an observer. They rely on ideas that can be very different. Thus the best observer must be chosen with respect to the type of problem. The choice will then be strongly connected to the quality and the uncertainties of the model and the data. If the biological kinetics are not precisely known, the mass balance will be the core of the asymptotic observers. If there are bounded uncertainties on the inputs and/or on the parameters, then we will estimate intervals in which the state of the system should lie. If the model has been correctly validated, then we can

Chapter written by Olivier B ERNARD and Jean-Luc G OUZÉ.

79

80

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fully exploit it and – if the outputs are not corrupted with a high level of noise – we can develop a high gain observer. The type of observer to be developed must not be based only on the model quality; it must also take into account the objectives to be achieved. In fact, an observer can have a purpose other than monitoring a bioreactor: it can be developed to apply a control action which needs an estimate of the internal state. It can also be used to determine that there was no failure in the process. 4.2. Notions on system observability We will only recall the main useful notions, we will give references for the more technical parts (see [KAI 80, LUE 79, GAU 01]). The observability notion is fundamental in automatic control. Intuitively, we try to estimate the state variables from the available measurements. If this is possible from a theoretical point of view, the system is said to be observable. Then another question is how to design an observer which is another dynamic system providing a state estimate. Let us mention that the question of observability and of observer design are very different: the observability property does not give any clue on how to build an observer. The theory is extensively developed in the linear case (see next section) and, in the nonlinear case, has been strongly developed during recent years, but for particular classes of models. 4.2.1. System observability: definitions We will consider the general continuous time system: ⎧ ⎨ dx (t) = f x(t), u(t); xt0 = x0 (S) dt ⎩ y(t) = h x(t)

(4.1)

where x ∈ Rn is the state vector, u ∈ Rm is the input vector, y ∈ Rp is the output vector, x0 is the initial condition for initial time t0 , f : Rn × Rm → Rn and h : Rn → Rp . The functions are assumed to be sufficiently smooth in order to avoid problems of existence and solution uniqueness. EXAMPLE 4.1. For the bioreactors described by a mass balance model, we have: f x(t), u(t) = Kr x(t) + D xin (t) − x(t) − Q x(t) where D and xin stand for the input vector.


81

We assume therefore that, for system (S), – the input u(t) is known; – the output y(t) is known; – functions f and h, are known, i.e. the model is known (r(·) is known in the mass balance-based modeling). We want to estimate x(t); the observability is a theoretical notion that states if it is possible. DEFINITION 4.1. Two states x0 and x0 are said to be indiscernible if for any input time function u(t) and for any t ≥ 0, the outputs h(x(t, x0 )) and h(x(t, x0 )) that result are equal. DEFINITION 4.2. The system is said to be observable if it does not have any distinct couples of initial state x0 , x0 that are indiscernible. This means that for any input the initial condition can be uniquely estimated from the output. There exist slightly different definitions, but we will not detail these nuances here. It can be noted that generally for nonlinear systems the observability depends on the input; a system can be observable for some inputs and not observable for others. DEFINITION 4.3. An input is said to be universal if it can distinguish any couple of initial conditions. DEFINITION 4.4. A non-universal input is said to be singular. Even in the case where all the inputs are universal (the system is said to be uniformly observable and can be rewritten under a specific form; see section 4.4), this can be insufficient in practice. We impose then that the universal property persists with time, and we obtain (at least for some systems) the notion of a regularly persisting input (see Hypothesis 4.5, section 4.5.3). For the linear systems things are much simpler (see below). 4.2.2. General definition of an observer Once the system has been proved to be observable, the next step is building the observer in order to estimate state variable x from the inputs, the outputs and the model. The observer principle is presented in Figure 4.1. This is a second dynamic system that will be coupled to the first one thanks to the measured output.

82

Bioprocess Control

u

Process x

Observer z

y

x

Figure 4.1. Observer principle

DEFINITION 4.5. An observer is an auxiliary system coupled with the original system: ⎧ ⎪ ⎨ dz (t) = fˆ z(t), u(t), y(t) ; z t0 = z0 (O) dt (4.2) ⎪ ⎩x ˆ ˆ(t) = h z(t), u(t), y(t) ˆ q × Rm × Rp → Rn such that with z ∈ Rq , fˆRq × Rm × Rp → Rq and hR ˆ(t) = 0 lim x(t) − x

t→∞

(4.3)

It is the classical definition which may be insufficient in some cases. It is stated that the estimation error tends asymptotically toward zero. Moreover we try to regulate the error decreasing rate (convergence rate). We can explain this with a simple n linear example: let us consider the linear system dx dt = Ax + Bu where x ∈ R and let us assume that matrix A is stable. A trivial observer can be obtained with a copy of x x + Bu. In fact, the error e = x − x ˆ follows the same dynamics the system: dˆ dt = Aˆ de dt = Ae and therefore converges towards zero. Let us note that this observer does not necessitate any output. This example shows that the stable internal dynamics are sufficient to estimate the final state. This example highlights a property which will be called detectability for linear systems and which will be the basis of the asymptotic observer (section 4.4) in a different framework. As a consequence, an additional requested property is to be able to regulate the convergence rate of the observer in order to be able to reconstruct the state variables more rapidly than the dynamics of the system. Let us note that the observer variable (z in O) may be larger than the state variable to be estimated x. Another property that we desire is that if the observer is properly initiated, i.e. with the true value x(0), then its estimation remains equal to x(t) for all t. This suggests a peculiar structure for the observer.


83

DEFINITION 4.6. Often, the following observer is taken: ⎧ dˆ x ⎪ ˆ(t), u(t) + k z(t), h(ˆ x(t)) − y(t) ⎨ (t) = f x dt (O) ⎪ ⎩ dz (t) = fˆz(t), u(t), y(t) with k z(t), 0 = 0 dt This is a copy of the system with a correcting term depending on the discrepancy between the true measured outputs and the value of the output calculated using the observer. The correction amplitude is regulated thanks to function k that can be seen as a gain (it is an internal regulation of the observer). In the ideal case, gain k can be regulated in order to obtain a converging rate as large as requested. DEFINITION 4.7. System (O) is said to be an exponential observer if, for any positive λ, gain k can be tuned such that ˆ(t) − x(t) ≤ e−λt x ˆ(0) − x(0). ∀ x0 , x ˆ(0), z0 ∀t > 0, x 4.2.3. How to manage the uncertainties in the model or in the output In real life – and especially in the biological field – we often consider that there are noises either in the output (measurement noise) or in the state equation (model noise). In general, the model noise is assumed to be additive (see section 4.3.5), which is a strong hypothesis (it could be, for example, multiplicative). Another important case which often appears in the bioprocesses is when the model integrates some unknown parts. For example, the biological kinetics in the mass balance models for bioreactors are generally not precisely known [BAS 90]. How do we manage these two problems which have some related aspects?: – Linear filtering, and more specifically Kalman filtering. This is the most popular method. It assumes that the noises are additive and white; it minimizes the error variance (see below). – The L2 , H 2 or H ∞ approach. This consists of assuming that the noises or perturbations w(t) belong to a given class of functions (L2 ) and minimizing their impact on the output using the transfer function. In the H 2 approach, we try to minimize the norm of this transfer function. In the H ∞ approach, we try to minimize the input effect in the worst case (see [BAS 91]). For example, for a γ > 0 and R positive definite matrix, we want observer x ˆ to verify: ∞ 2 2 x ˆ(t) − x(t)R − γ 2 w(t) dt ≤ 0. sup w(·)

0

84

Bioprocess Control

– Disturbance rejection. We try to build observers independently from the unknown perturbation. The disturbance is canceled thanks to linear combinations of variables [DAR 94, KUD 80, HOU 91]. Asymptotic observers belong to this class of systems (see section 4.5). – Bounds on the perturbations and on the uncertainties. We assume that uncertainties are bounded, and try to design interval observers which provide the best possible bounds for the variables to be estimated. For some cases, we try to minimize this bounding (section 4.6). An approach using interval analysis may also be efficient for state or parameter estimation [JAU 01, KIE 04]. – We can also use these bounds to design sliding mode observers which have a correcting term of the type sign(x − x ˆ). Note that the way these observers take the uncertainties into account generates discontinuous dynamics on the sliding manifolds [EDW 94]. – Robustness using LMI. There exist efficient algorithms to solve Linear Matrix Inequality (LMI) problems. It is possible to check if a matrix depending on parameters is positive definite; then it supports a Lyapunov function used to design an observer that is robust with respect to some uncertainties [ARC 01]. NOTE. It is possible to construct examples where a system is observable when the model is known and becomes unobservable when a part of the model is unknown. For such cases the requirement for a classical observer may be relaxed. In particular, we will no longer assume that (4.4) ˆ(t) = 0 lim x(t) − x t→∞

but that the discrepancy tends toward a reasonable value for practical applications. 4.3. Observers for linear systems For single output linear stationary systems we have: ⎧ dx ⎨ (t) = Ax(t) + Bu(t) SL : dt ⎩ y(t) = Cx(t) with A ∈ Mn×n (R) (n ≥ 2), C ∈ M1×n (R). The well known observability criterion is formulated as follows: ⎛ ⎞ C ⎜ CA ⎟ ⎜ ⎟ SL observable ⇐⇒ rank ⎜ .. ⎟ = n. ⎝ . ⎠ CAn−1

(4.5)


85

which relies on the fact that the observability space is generated by the vectors (C, CA, . . . , CAn−1 ). The canonical observability forms, that can be obtained after a linear change of coordinates, highlight the observation structure. They will reappear in the nonlinear case for the high gain observer (section 4.6). THEOREM 4.1. If the pair (A, C) is observable, then there exists an invertible matrix P such that: A0 = P −1 AP with

⎛

−an ⎜−an−1 ⎜ . ⎜ A0 = ⎜ .. ⎜ ⎝ −a2 −a1

1 0 0 1 .. .. . . 0 0 0 0

··· ··· .. . ··· ···

C0 = CP ⎞ 0 0⎟ ⎟ ⎟ ⎟ ⎟ 1⎠ 0

C0 = 1 0

···

0

What happens if the system is not observable? We can rewrite it in two parts, as it is shown in the following theorem. Here A1 and A3 are two square matrices with dimensions corresponding to x1 and x2 . The canonical form clearly shows that x1 cannot be estimated from x2 . THEOREM 4.2. General canonical form: dx1 = A1 x1 + A2 x2 + B1 u dt dx2 = A3 x2 + B2 u dt y = C2 x2 Matrix A1 forces the dynamics of the unobservable part; if they are stable, then the dynamics of the total error will be stable, but the unobservable part will tend toward zero with its own dynamics (given by A1 ); the system is said to be detectable. 4.3.1. Luenberger observer If system (4.5) is observable, a Luenberger observer [LUE 66] can be derived: dˆ x(t) = Aˆ x(t) + Bu(t) + K C x ˆ(t) − y(t) dt where K is a dimension n gain vector, which allows us to regulate the convergence rate of the observer.

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Bioprocess Control

In fact, the dynamics of the observation error e = x − x ˆ is: de = (A + KC)e dt Let us note that these dynamics do not depend on the input. The pole placement theorem sates that the error dynamics can be arbitrarily chosen. THEOREM 4.3. If (A, C) is observable, vector K can be chosen in order to obtain arbitrary linear dynamics of the observation error. In particular, the gain vector K can be chosen in order that the error converges rapidly toward zero. However, the observer will then be very sensitive to perturbations (measurement noise for example). A good compromise must be chosen between stability and precision. The Kalman filter is a way to manage this compromise. 4.3.2. The linear case up to an output injection There is a very simple case for which a linear observer can be designed for a nonlinear system; this is the case where the nonlinearity depends only on output y. ⎧ ⎨ dx (t) = Ax(t) + φt, y(t) + Bu(t) (4.6) (S) : dt ⎩ y(t) = Cx(t) φ is a nonlinear (known) function which takes its values in Rn . The following “Luenberger like” observer generates a linear observation error equation: dˆ x(t) = Aˆ x(t) + φ t, y(t) + Bu(t) + K C x ˆ(t) − y(t) dt The dynamics can be arbitrarily chosen if the pair (A, C) is observable. 4.3.3. Local observation of a nonlinear system around an equilibrium point Let us consider general system (4.1), and let us assume that it admits a single equilibrium point (working point) at (xe , ue ). The system can then be linearized around this point. THEOREM 4.4. The linearized system of (4.1) around (xe , ue ) is ⎧ ⎨ dX (t) = AX + BU (S) dt ⎩ Y (t) = CX with A=

∂f (x, u) ∂x

B=

∂f (x, u) ∂u

C=

∂h(x) ∂x

(4.7)


87

Matrices A, B, C are estimated at xe , ue . Variables X, U, Y are deviations toward equilibrium: U = u − ee ,

X = x − xe ,

Y = y − Cxe

If the pair (A, C) is observable, the nonlinear system is locally observable around the equilibrium. 4.3.4. PI observer The Luenberger observer is based on a correction of the estimations with a term related to the difference between the measured outputs and the predicted outputs. The idea behind the proportional integral observer is to use the integral of this error term. We consider the auxiliary variable w: ˆ t Cx ˆ(τ ) − y(τ ) dτ. w ˆ= 0

The PI observer for system (4.8) will then be rewritten: ⎧ dˆ x ⎪ ⎪ ˆ(t) − y + KP w ˆ ⎨ (t) = Ax(t) + Bu(t) + KI C x dt ⎪ dw ⎪ ⎩ ˆ (t)(t) = C x ˆ−y dt The error equation (ex = x ˆ − x and ew = w) ˆ is then: ⎛ ⎞ dex ⎜ dt (t) ⎟ ex (t) ⎜ ⎟ = F + KI C KP ⎝ dew ⎠ C 0 ew (t) (t) dt

(4.8)

(4.9)

The gains KI and Kp can be chosen so as to ensure stable error dynamics [BEA 89]. The integrator addition provides more robustness to the observer to deal with measurement noise or modeling uncertainties. 4.3.5. Kalman filter The Kalman filter (see [AND 90]) is very famous in the framework of linear systems; it can be seen as a Luenberger observer with a time varying gain; this allows us to minimize the error estimate variance.

88

Bioprocess Control

A stochastic representation can be given by the observable system: ⎧ ⎪ ⎨ dx (t) = Ax(t) + Bu(t) + w(t); x t0 = x0 dt ⎪ ⎩y(t) = Cx(t) + v(t)

(4.10)

where w(t) and v(t) are independent centered white noises (Gaussian perturbations), with respective covariances Q(t) and R(t). Let us also assume that the initial distribution is Gaussian, such that: & T ' = P0 ˆ0 ; E x0 − x ˆ0 x0 − x ˆ0 (4.11) E x0 = x where E represents the expected value and P0 is the initial covariance matrix of the error. The filter is written in several steps: 1. Initialization: ˆ0 ; E x0 = x

& T ' = P0 E x0 − x ˆ0 x0 − x ˆ0

(4.12)

2. Estimation of the state vector: dˆ x (t) = Aˆ x(t) + Bu(t) + K(t) y(t) − C x ˆ(t) ; dt

ˆ0 x ˆ t0 = x

(4.13)

3. Error covariance propagation (Riccati equation): dP (t) = AP (t) + P (t)AT − P (t)C T R(t)−1 CP (t) + Q(t) dt

(4.14)

4. Gain calculation: K(t) = P (t)C T R(t)−1

(4.15)

Some points can be emphasized: – this filter can still be applied when matrices A and C depend on time (the observability must nevertheless be proved); – the estimation of the positive definite matrices R, Q, P0 is often very delicate, especially when the noise properties are not known; – a deterministic interpretation of this observer can be given: it consists of minimizing the integral from 0 to t of the square of the error; – this observer can be extended by adding a term −θP (t) in the Riccati equation. This exponential forgetting factor allows us to consider the cases where Q = 0.


89

4.3.6. The extended Kalman filter This idea consists of linearizing a nonlinear system around its estimated trajectory. Then the problem is equivalent to building a Kalman filter for non-stationary system. Let us consider the system ⎧ ⎨ dx (t) = f x(t) + w(t); xt0 = x0 dt (4.16) ⎩ y(t) = h x(t) + v(t) and the observer is designed as above, with a change in the second step. 2. Estimation of the state vector: dˆ x ˆ0 (t) = f x ˆ(t) + K(t) y(t) − h x ˆ(t) ; x ˆ t0 = x dt and using the matrices of the tangent linearized: ∂f x(t) ∂h x(t) A(t) = C(t) = ∂x(t) x(t)=ˆx(t) ∂x(t) x(t)=ˆx(t)

(4.17)

(4.18)

This extended filter is often used, even if only few theoretical results guarantee its convergence (see section 4.4.4). 4.4. High gain observers 4.4.1. Definitions, hypotheses In this chapter, we will assume that a simulation model of the process is available (i.e. with modeling of the biological kinetics). We also assume that the model has been carefully validated: the high gain observers are dedicated to the nonlinear systems and require high quality models. We will now consider the affine systems with respect to the input, that are described as follows: dξ = f (ξ) + ug(ξ) (4.19) dt We consider here the case where u ∈ R. For bioreactors, the input often corresponds to the dilution rate u = D. In this case f (ξ) = Kr(ξ) − Q(ξ) and g(ξ) = ξin − ξ. Moreover, we assume that the output is a function of the state: y = h(ξ) ∈ R. HYPOTHESIS 4.1. We will state the two following hypotheses: (i) system (4.19) is observable for any input; (ii) there exists a positively invariant compact K, such that for any time t, ξ(t) ∈ K.

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Bioprocess Control

We will denote Lf h(ξ) =

Dh Dξ f (ξ),

which is the Lie derivative of h along the

h(ξ). vector field f . By convention, we will write Lpf h(ξ) = Lf Lp−1 f 4.4.2. Change of variable Let us consider the following change of coordinates, defined on the compact set K: & φ : ξ −→ ζ = h(ξ),

Lf h(ξ),

(n−1)

· · · , Lf

'T h(ξ)

(4.20)

This change of variable consists of considering (in the autonomous case) output y and its n − 1 first derivatives as new coordinates. HYPOTHESIS 4.2. Mapping φ is a global diffeomorphism. We can verify [GAU 81] that under Hypothesis 4.2 φ transforms (4.19) into:

with

⎛

0 ⎜0 ⎜ A=⎜ ⎜· ⎝0 0 ⎛ ⎜ ⎜ ˜ ψ(ζ) =⎜ ⎝

dζ ˜ + ψ(ζ)u ¯ = Aζ + ψ(ζ) dt

(4.21)

y = Cζ

(4.22)

1 0 · 0 0

0 1 · ··· ···

0 .. .

0 Lnf h φ−1 (ζ)

··· ··· · 0 0 ⎞ ⎟ ⎟ ⎟, ⎠

⎞ 0 0⎟ ⎟ ·⎟ ⎟, 1⎠ 0

C = [1, 0, . . . , 0]

⎞ ψ¯1 ζ1 ⎟ ⎜ ψ¯2 ζ1 , ζ2 ⎟ ⎜ ¯ ⎟ ⎜ ψ(ζ) = ⎜ .. ⎟ . ⎠ ⎝ ¯ ψ2 ζ1 , ζ2 , . . . , ζn ⎛

with (i−1) −1 ψ¯i (z) = ψ¯i ζ1 , . . . , ζi = Lg Lf h φ (ζ)

(4.23)

In this canonical form, all the system nonlinearities have been concentrated in ˜ ¯ terms ψ(ζ) and ψ(ζ). We will present the various observers using this canonical form (let us note that this canonical form is very close to the one in section 4.3 for the observer pole assignment).


91

Let us note that an observer in the new basis will provide an estimate ζˆ which will estimate ζ, i.e. the successive output derivatives. The idea consists of writing the observer in this canonical basis i.e. a numerical differentiator of the output. Then, going back to the initial coordinates (applying φ−1 (ζ)), the observer will be expressed in the original basis. To design a high gain observer, we need an additional technical hypothesis. HYPOTHESIS 4.3. Mappings ψ˜ and ψ¯ defined in (4.21) are globally Lipschitz on K. Intuitively, this hypothesis will allow us to dominate the nonlinear part, imposing that the dynamics of the observer can be faster than the system dynamics (this explains the idea of “high gain”). 4.4.3. Fixed gain observer PROPOSITION 4.1 ([GAU 92, BUS 02]). For a sufficiently high gain θ, and under Hypotheses 4.1, 4.2 and 4.3 the following differential system is an exponential observer of (4.19): −1 ∂φ dˆ x = f (ˆ x) + ug(ˆ x) − x) − y (4.24) Sθ−1 C t h(ˆ dt ∂x x=ˆx where Sθ , is the solution of the equation θSθ + At Sθ + Sθ A = C t C. Sθ can be calculated as follows: Sθ (i, j) =

(−1)i+j (i + j − 2)! θi+j−1 (i − 1)!(j − 1)!

(4.25)

For the convergence proof and other details see [GAU 92]. 4.4.4. Variable gain observers (Kalman-like observer) The extended Kalman filter is often used in a framework where its convergence is not guaranteed (see section 4.3.5). We show here how to build a high gain observer very close to the Kalman filter (after a change of variable), whose convergence is guaranteed. PROPOSITION 4.2 ([DEZ 92]). For a gain θ that is sufficiently high, and under Hypotheses 4.1, 4.2 and 4.3 the following differential system is an exponential observer of (4.19): ⎧ −1 ⎪ x 1 ∂φ ⎪ dˆ ⎨ = f (ˆ x) + ug(ˆ x) − x) − y S −1 C t h(ˆ dt r ∂x x=ˆx (4.26) ⎪ dS 1 t ⎪ t ⎩ = −SQθ S − A (ˆ x, u)S − SA (ˆ x, u) + C C dt r

92

Bioprocess Control

with r > 0, Qθ is calculated from the two positive definite symmetric matrices Δθ and Q: Δθ = diag θ, θ2 , . . . , θn (4.27) Qθ = Δθ QΔθ Matrix A can be calculated from diffeomorphism φ: ∂ψ ˆ u) = A + ∂φ A (ξ, +u ∂ζ ζ=φ(ξ) ∂ζ ζ=φ(ξ) ˆ ˆ

(4.28)

(4.29)

See [DEZ 92] for the proof of the convergence of this observer and for more details, especially for the choice of r and matrix Q. It is worth noting that, even if the filtering and noise attenuation performances of this extended Kalman filer are a priori better, this observer is above all a high gain observer; it will therefore present the same generic high sensitivity with respect to the measurement noises and modeling errors. The advantages of Kalman-like high gain observers have a price: this observer is differential equations more difficult to implement. In fact, we have to integrate n(n+3) 2 instead of n equations for the simple high gain observer. 4.4.5. Example: growth of micro-algae We will consider the growth of micro-algae in a continuous photobioreactor. The algal development is limited by a nitrogen source (NO3 ) denoted S, and uses principally inorganic dissolved carbon (C), mainly in the form of CO2 . The algal biomass (X) will then correspond to an amount of particulate nitrogen (N ). In order to simultaneously describe the cellular carbon and nitrogen uptake, we will consider the following reaction scheme r1 (·)

S −−−−→ N r2 (·)

C −−−−→ X Setting ξ = (X, N, S, C)t , the mass balance-based model (4.33) can be written with: ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ 0 0 0 1 ⎜ ⎟ ⎜ ⎟ ⎜1 0 ⎟ ⎟ , ξin = ⎜ 0 ⎟ Q(ξ) = ⎜ 0 ⎟ K=⎜ ⎠ ⎠ ⎝ ⎝ ⎝−1 0 ⎠ Sin 0 Cin Qc (ξ) 0 −k1


93

The units for carbon and nitrogen are the same for biomass and substrate, and moreover the nitrogen uptake yield is assumed to be unitary. The nutrient uptake rate is assumed to follow a Michaelis-Menten law [DUG 67]: S r1 (ξ) = ρmax X S + kS The algal growth from carbon is r2 (ξ) = μ(ξ)X, where the growth rate μ(ξ) is described by the Droop law [DRO 68]: kq μ(ξ) = μ(q) = μ ¯ 1− (4.30) q Variable q represents the internal nitrogen quota defined by the amount of nitrogen N . per biomass unit: q = X We assume that biomass is measured (it is estimated by its total biovolume), and will be used to design a high gain observer to determine S and q. In this case, the nitrate concentration in the renewal medium (Sin ) can be controlled. More precisely, Sin can vary as follows: Sin = sin (1 + u) where u is the control, and sin the nominal concentration, corresponding to u = 0. In the following sections, we will consider only the first three equations of this system, and we will consider the following change of variables: N ; x3 = sSin – x1 = ρsminX ; x2 = Xk q – a1 =

ks sin ;

a2 = μ ¯; a3 =

ρm kq

that leads to the following system: ⎧ ⎨ dx = f (x) + ug(x) dt ⎩y = hx 1 with:

⎞ 1 a − Dx 1 − x 1 1 ⎛ ⎞ ⎟ ⎜ 2 x2 x1 ⎜ ⎟ x3 ⎟ ⎜ ⎝ ⎠ x = x2 , f (x) = ⎜a3 − a2 x2 − 1 ⎟ , ⎟ ⎜ a1 + x3 x3 ⎝ x1 x3 ⎠ D 1 − x3 − a1 + x3 ⎛ ⎞ 0 g(x) = ⎝ 0 ⎠ , h x1 = x1 D

(4.31)

⎛

(4.32)

94

Bioprocess Control

The high gain observer for model (4.31) is then given by: ⎛ ⎞ 3θ ⎜ ⎟ x ˆ x ˆ2 ⎟ D ⎜ ⎜ 3θ 2 1 − 1 − x ˆ2 + 3θ2 2 ⎟ ⎜ x ˆ1 a2 a2 x ˆ1 ⎟ G(ˆ x) = ⎜ ⎟ ⎜ 2 ⎟ ⎜ ⎟ 2 ˆ2 a1 + x ˆ3 ⎝ ⎠ ˆ32 + θ3 x ˆ31 + 3θ2 B 3θB a1 a2 a3 x ˆ1 with: ˆ31 = B

ˆ32 B

ˆ3 1 D2 a3 x + 2a2 + x ˆ22 2a2 − 3D − a1 a3 x ˆ1 a1 + x ˆ3 a2

ˆ3 a3 x D −x ˆ2 2 1− + 4a2 − 4D a1 + x ˆ3 a2 ˆ3 )2 ˆ3 x ˆ2 (a1 + x a3 x = x ˆ2 (2D − 3a2 ) + 4a2 + 2 a1 a2 a3 x ˆ1 a1 + x ˆ3

An experiment where u fluctuates sinusoidally was used to validate the observer. Figure 4.2 proves the observer efficiency when the model is well known. The observer predictions are in agreement with the experimental measurements. For more details on this example, see [BER 98, BER 01]. 4.5. Observers for mass balance-based systems 4.5.1. Introduction In the previous sections we have considered the case where the uncertainties were due to noise on the outputs and, in some cases, were due to modeling noise. We have seen in Chapter 2 that the bioprocess models are often not known very well. In particular, when the model is written on the basis of a mass balance analysis, a term representing the reaction rates appears. This term which represents the biological kinetics with respect to the model state variable is sometimes speculative. Often the modeling of the reaction rate is not reliable enough to base an observer on it. In this section we will use the results for the observers with unknown inputs [KUD 80, HOU 91, DAR 94], whose principle relies on a cancelation of the unknown part after a change of variable in order to build the observer. We will show how to build an observer for a system represented by a mass balance and for which the kinetics would not have been specified. We will see that the main condition to designing such an observer is that enough variables are measured. In particular we will not assume any observability property. This is not particularly


95

12

A

10

9

Biovolume (10 μm3/l)

14

8 6 4 2

Quota (10− 9 μmol/mm3 )

0

0

1

2

3

4

5

10

B

8 6 4 2 0

0

1

2

3

4

5

Nitrate (μmol/l)

0.5

C

0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

Time (days) Figure 4.2. Comparison between direct measurements (•) and observer predictions (—) for model (4.31): (A) biomass estimated from total algal biovolume; (B) internal quota; (C) nitrate concentration

surprising since the observability property relies on its full description (including the kinetics) which is not used to build the mass balance observer. In fact, it is not really an observer in the strict sense, but more precisely a detector, relying on hypothesis that the non-observable part is stable.

96

Bioprocess Control

4.5.2. Definitions, hypotheses In this chapter we will consider the biotechnological processes that are modeled with a mass balance model: dξ = Kr(ξ) − D(t)ξ + D(t)ξin (t) − Q(ξ) (4.33) dt with ξ ∈ Rn

r ∈ Rp

(4.34)

We assume that the set of available measurements y can be decomposed into three vectors: T (4.35) y = y1 y2 y3 where: – y1 is a set of q measured state variables. To simplify the notations, we will order the components of the state so that y1 corresponds to the q first components of ξ; – y2 represents the measured gaseous flow rates: y2 = Q(ξ); – y3 represents the other available measurements (pH, conductivity, etc.) that are related to the state using the following relationship: y3 = h(ξ). Let us rewrite system (4.33) after splitting the measured part (ξ1 = y1 ) from the other part of the state (ξ2 ). dξ1 = K1 r(ξ) − Dξ1 + Dξin1 − Q1 (ξ) dt

(4.36)

dξ2 = K2 r(ξ) − Dξ2 + Dξin2 − Q2 (ξ) dt

(4.37)

Matrices K1 and K2 and vectors ξin1 , ξin2 , Q1 and Q2 are such that ξin1 K1 Q1 K= , ξin = , Q= K2 ξin2 Q2 4.5.3. The asymptotic observer In order to build the asymptotic observers we need the two following technical hypotheses. HYPOTHESIS 4.4. (i) There are more measured quantities than reactions: q ≥ p; (ii) matrix K1 is of full rank. Hypothesis 4.4(ii) means that a non-zero r cannot cancel K1 r (a reaction cannot compensate the other ones with respect to the measured variables).


97

Consequences: under Hypothesis 4.4, the q × p matrix K1 admits a left inverse; there exists a p × q matrix G such that: GK1 = Ip×p

(4.38)

Let us set: A = −K2 G, and let us consider the following linear change of coordinates: ζ1 = ξ1

(4.39)

ζ2 = Aξ1 + ξ2

(4.40)

This change of variable transforms (4.36) and (4.37) into: dζ1 = K1 r(T ζ) − Dζ1 + Dζin1 − Q1 (T ζ) dt dζ2 = −D ζ2 − ζin2 − AQ1 (T ζ) + Q2 (T ζ) dt with

T =

Ip 0p,n−p , −A In−p

M = A In−p

(4.41) (4.42)

(4.43)

and ζin2 = M ξin

(4.44)

The equation of ζ2 can be rewritten using output y2 : dζ2 = −D ζ2 − ζin2 − M y2 dt

(4.45)

NOTE. System (4.45) is a linear system up to an output injection (see section 4.3.2). We can now design an observer for this system by simply copying equations (4.45). However, we must first state an hypothesis to guarantee the observer convergence. HYPOTHESIS 4.5. The positive scalar variable D is a regularly persisting input i.e. there exist positive constants c1 and c2 such that, for all time instant t: t+c2 0 < c1 ≤ D(τ )dτ t

In practice, c2 must be low with respect to the time constant of the system. Moreover, cc12 must be high because it determines the minimal converging rate of the observer.

98

Bioprocess Control

LEMMA 4.1 (see [BAS 90]). Under Hypothesis 4.5, solution ξˆ2 of the following asymptotic observer: dζˆ2 = −D ζˆ2 − ζin2 − M y2 dt ξˆ2 = ζˆ2 − Ay1

(4.46)

converges asymptotically toward solution ξ2 of reduced system (4.37). Proof. It can easily be verified that the estimation error e2 = ξˆ2 − ξ2 = ζˆ2 − ζ2 satisfies: de2 = −De2 . (4.47) dt and converges asymptotically toward ξ2 if Hypothesis 4.5 is fulfilled [BAS 90]. 4.5.4. Example We will consider as an example the growth of the filamentous fungus Pycnoporus cinnabarinus (X) [BER 99]. The fungus uses two substrates to grow: glucose as a carbon source (C) and ammonium as a nitrogen source (N ). The reaction scheme is assumed to be composed by one reaction: N + C −→ X The model is then of the same type as (4.33), with: T T ξ = N C X , K = −k1 −k2 1 , ξin = Nin , Cin , The following measurements are available: y1 = N

C

T

T 0

.

The state partition will then be the following: T ξ1 = N C , ξ2 = X associated with:

K1 = −k1

−k2

T

,

K2 = 1

of left inverses. We will consider two of them: Matrix K1 has an infinite number G1 = − k11 , 0 and G2 = 0, − k12 . These two matrices will naturally lead to two observers. The first one based on the nitrogen measurements: dζˆ21 Nin = −D ζˆ21 − dt k1 (4.48) N ˆ 1 = ζˆ21 − X k1

State Estimation for Bioprocesses 3

3

A

B 2.5

2.5

2 X

X

2 1.5

1.5

1

1

0.5

0.5

0 0

99

1

2 Time (days)

3

0 0

4

1

2 Time (days)

3

4

Figure 4.3. Comparison between direct biomass measurements of Pycnoporus cinnabarinus (o) and observer predictions based on the nitrogen measurement (A) or on the carbon measurement (B)

and the other one based on carbon: dζˆ2

Cin = −D ζˆ22 − dt k2

2

(4.49)

ˆ 2 = ζˆ2 − C X 2 k2 The results of these observers obtained with experimental data are presented in Figure 4.3. In this case, the observer based on the nitrogen measurements is more reliable. 4.5.5. Improvements The asymptotic observers work in open loop. In fact, their estimate relies on the mass balances and are not corrected by a discrepancy between measured and estimated quantities. It assumes that the mass balance model is ideal. Nevertheless, the yield parameters are difficult to estimate properly, and in some cases (wastewater treatment) the mass inputs in the system are not precisely known. In this case it can be dangerous to base the observer only on the mass balance model without taking into account some measurements on the system that reflect its actual state. It can be possible to estimate these unknown parameters online, but we will see here another method aiming at improving the observer robustness with respect to some uncertainties. In this section we will see how to use the available measurements y3 to improve the asymptotic observer performances. We assume here that y3 ∈ R. We define the mapping h: h : ξ1 , ξ2 ∈ Rp × Rn−p −→ y3 = h ξ1 , ξ2 ∈ R We suppose that h satisfies the following hypothesis. HYPOTHESIS 4.6. The mapping h is monotonous with respect to ξ2 , i.e.: fixed sign on the considered domain Ω.

Dh Dξ2

has a

100

Bioprocess Control

EXAMPLE 4.2. In the example detailed hereafter, h(ξ1 , ξ2 ) = αS + βP , and thus: Dh = α Dξ2

β

(4.50)

which has a fixed sign. Of course, h can be nonlinear. PROPOSITION 4.3. Let λ ∈ Rp be a unitary constant vector ( λ = 1), whose signs Dh ), θ is a positive scalar (which can depend are chosen such that sign(λ) = sign( Dξ 2 on time) and zin = M ξin . The following system: dˆ z = −D zˆ − zin − M y2 − θλ h y1 , zˆ − Ay1 − y3 dt

(4.51)

is an asymptotic observer of reduced system (4.45). For the proof of this property, and for more details, see [BER 00]. EXAMPLE 4.3. We will consider a bacterial biomass (X) growing in a bioreactor. The micro-organisms uptake substrate S and metabolize a product P : S −→ X + P The associated model is then: ⎧ dX ⎪ ⎪ = r(ξ) − DX ⎪ ⎪ dt ⎪ ⎪ ⎨ dS = −c1 r(ξ) + D Sin − S ⎪ dt ⎪ ⎪ ⎪ ⎪ dP ⎪ ⎩ = c2 r(ξ) − DP dt

(4.52)

where Sin is the influent substrate, c1 and c2 are the yield coefficients. We assume that the bacterial biomass and the conductivity of the solution can be measured. The conductivity is related to a positive linear combination of the ions in the liquid i.e. S and P . We thus obtain: y1 = X

(4.53)

y2 = (0, 0, 0)t

(4.54)

y3 = αS + βP

(4.55)

We suppose that the substrate concentration in the influent Sin is not precisely known, and we will use an estimate denoted Sîn .


101

Thanks to Proposition 4.3 we can design the following observer (for the sake of clarity we choose λ = 1 0 which satisfies the correct hypotheses) ⎧ dˆ z1 ⎪ = D zîn1 − zˆ1 − θ αSˆ + β Pˆ − y3 ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ dˆ z ⎪ ⎪ = −Dˆ z2 ⎨ dt 2 (4.56) ⎪ y1 ⎪ ⎪ ˆ S = zˆ1 − ⎪ ⎪ ⎪ c1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩Pˆ = zˆ2 + y1 c2 with zîn1 = Sîn . Let us now show the robustness properties when the estimate Sin is false. If S and P represent equilibrium values of S and P , we denote by Sˆ and Pˆ the equilibrium values for the closed loop observer. If the observer is in open loop (θ = 0), using Sîn , a direct calculation provides:

Sˆ = S + Sˆ1in − S1in

(4.57)

The prediction error is thus exactly the error on S1in . With the closed loop observer, the steady state is: Sˆ = S +

D ˆ S1in − S1in θα + D

(4.58)

If gain θ is high, it is easy to see that Sˆ S ; the bias is reduced by the closed loop observer. 4.6. Interval observers The usual observers implicitly assume that the model is a good approximation of the real system. Nevertheless, we have seen that a model of a bioprocess is often poorly known. In this case, the observation principle must be revisited: generally it will no longer be possible to build an exact observer (which would guarantee: e(t) = ˆ x(t) − x(t) → 0 when t → ∞) whose convergence rate could be regulated (such as for example an exponential observer). Therefore, in this case the result must be weaker. We present here a possible method (among others) consisting of bounding the uncertainty on the model. The bound on the variable to be estimated is deduced. To simplify, we will first present the linear (or close to the linear) case (see [GOU 00, RAP 03]).

102

Bioprocess Control

4.6.1. Principle The idea is to use the known dynamic bounds of the uncertainties: the dynamic bounds on the model uncertainties allow us to derive (in the appropriate cases) the dynamic bounds on the state variable to be estimated. Figure 4.4 summarizes the philosophy of the interval estimation. U+

^

X+ X

U

^

X− U−

Figure 4.4. Principle of interval estimation for bounded uncertainties: a priori bounds on the uncertainties U provide bounds on the non-measured state X

Let us consider the general system: ⎧ dx ⎨ (t) = f x(t), u(t), w(t) ; x t0 = x0 dt S0 ⎩ y(t) = h x(t), v(t)

(4.59)

where x ∈ Rn is the state vector y ∈ Rp is the output vector, u ∈ Rm the input vector, x0 the initial condition at t0 , f : Rn × Rm × Rr → Rn and h : Rn × Rs → Rp . The unknown quantities w ∈ Rr and v ∈ Rs are characterized by their upper and lower bounds: w− (t) ≤ w(t) ≤ w+ (t) ∀t ≥ t0

(4.60)

v − (t) ≤ v(t) ≤ v + (t) ∀t ≥ t0

(4.61)

NOTE. The operator ≤ applies to vectors; it corresponds to inequalities between each component. Based on the fixed model structure (S0 ) and the set of known variables, a dynamic auxiliary system can be designed as follows: ⎧ − dz ⎪ ⎪ ⎪ = f − z − , z + , u, y, w− , w+ , v − , v + ; z − t0 = g − x− , x+ 0 0 ⎪ ⎪ dt ⎪ ⎪ ⎪ + ⎨ dz + = f + z − , z + , u, y, w− , w+ , v − , v + ; z + t0 = g + x− O0 0 , x0 dt ⎪ ⎪ ⎪ − ⎪ x = h− z − , z + , u, y, w− , w+ , v − , v + ⎪ ⎪ ⎪ ⎪ ⎩x+ = h+ z − , z + , u, y, w− , w+ , v − , v + (4.62) with z − , z + ∈ Rq , the other functions being defined in the appropriate domains.


103

DEFINITION 4.8 (interval estimator). System (O0 ) is an interval estimator of system + − + (S0 ) if for any pair of initial conditions x− 0 ≤ x0 , there exist bounds z (t0 ), z (t0 ) such that the coupled system (S0 , O0 ) verifies: x− (t) ≤ x(t) ≤ x+ (t);

∀t ≥ t0

(4.63)

The interval estimator is derived from the coupling between two estimators providing each an under-estimate x− (t) and an over-estimate x+ (t) of x(t). The estimator provides a dynamic interval [x− (t), x+ (t)] containing the unknown value x(t) (Figure 4.4). Of course, this interval can be very large and therefore useless. The next step consists of trying to reduce as far as possible this interval and increase the convergence rate toward this interval, for example with an exponential convergence rate. Then, we move back to classical observation problems, with the important difference that we do not require the observation error (the interval amplitude) to tend asymptotically exactly toward zero. 4.6.2. The linear case up to an output injection First, let us take a very simple case. We again consider the following system: ⎧ ⎨ dx (t) = Ax(t) + φ t, y(t) S : dt (4.64) ⎩y(t) = Cx(t) with A ∈ Mn×n (R) (n ≥ 2), C ∈ M1×n (R). If the mapping φ : R+ × R → Rn is known, a Luenberger observer can be designed (section 4.3.1). What happens now if function φ is not well known? We assume that it can be bounded and that the bounds are known. Thus, functions φ− , φ+ : R+ × R → Rn , are known and sufficiently smooth, such that: φ− (t, y) ≤ φ(t, y) ≤ φ+ (t, y),

∀(t, y) ∈ R+ × R

(4.65)

We will then use these bounds to design an upper and a lower estimator: dx + (t) = Ax+ (t) + φ+ t, y(t) + K Cx+ (t) − y(t) dt dx − (t) = Ax− (t) + φ− t, y(t) + K Cx− (t) − y(t) . dt If we now consider the “upper” error e+ (t) = x+ (t) − x(t), we obtain: de + = (A + KC)e+ + b+ (t) dt

(4.66) (4.67)

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Bioprocess Control

with b+ (t) = φ+ t, y(t) − φ t, y(t) . It follows that b+ is positive, and the following lemma can easily be proved: LEMMA 4.2. If the elements of matrix (A + KC) are positive outside the diagonal (the matrix is said to be cooperative), then e+ (0) ≥ 0 implies e+ (t) ≥ 0 for any positive t. Of course, we have the same Lemma for the lower error: e− (t) = x(t) − x− (t) and the total error e(t) = e− (t) + e+ (t). The following theorems can be deduced: THEOREM 4.5. If the gains of vector K can be chosen such that matrix (A + KC) is cooperative, and if we have an initial estimate such that x− (0) ≤ x(0) ≤ x+ (0) then equations (4.66), (4.67) provide an interval estimator for system (4.64). THEOREM 4.6. If hypotheses of the proceeding theorem are verified, if matrix (A + KC) is stable, and if moreover the error on φ can be bounded, i.e. if we have: b(t) = φ+ (t, y) − φ− (t, y) ≤ B where B is a positive constant, then error e(t) converges asymptotically toward an interval smaller (for each component) than the positive vector: emax = −(A + KC)−1 B In particular, if the components of emax are zero, then the corresponding components for e(t) converge towards zero. The proofs are straightforward; the proof of the first theorem follows directly from the above lemma. The proof of the second theorem is due to the differential inequality (A + KC)e + b(t) ≤ (A + KC)e + B which implies (with equal initial conditions) e(t) ≤ em (t), where em (t) is the solution of

de dt m

∀t ≥ 0

= (A + KC)em + B.


105

Some observations: – we use in the observer design the fundamental hypothesis that it is possible to derive inequalities between the variables from inequalities on the left hand side of the differential equations. This hypothesis is connected with the comparison of the solutions of differential equations (see section 4.8). There exist other techniques to estimate the intervals; they are more precise but less explicit [JAU 01]; – we also need the assumption that the initial estimate is valid x− (0) ≤ x(0) ≤ x+ (0); a large estimate can be chosen in practice; – the problem of regulating the convergence rate has not been considered here; is it possible to choose a gain K that will ensure cooperativity, stability and an arbitrary convergence rate? This is a complicated problem; see [RAP 03] for more details. We illustrate this approach with an example of such an estimator for a biochemical process (see also [GOU 00, ALC 02]). 4.6.3. Interval estimator for an activated sludge process We consider a very simplified model of an activated sludge process, used for biological wastewater treatment. The objective is to process wastewater, with an influent flow rate Qin and a pollutant (substrate) concentration sin . We are aiming for a concentration of the effluent that is lower than sout . The process is composed of an aerator (bioreactor) followed by a settler separating the liquid and solid (biomass) phases. Then we recycle a part of the biomass toward the aerator. Let us denote by x, s and xr the three state variables of this simple model, representing respectively the biomass and substrate concentrations in the aerator, and the recycled biomass in the settler. Qin , Qout , Qr and Qw are the flow rates, Va and Vs the volumes (see Figure 4.5). We suppose that the biological reactions only take place in the aerator. 1 μ(·)x s −−−−→ x Y Q in , s in

aerator

x ,s

settler

Va

Qout , sout

Vs Qr , x r

Qw , x r

Figure 4.5. Diagram of an activated sludge process

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Bioprocess Control

Y is a yield coefficient and μ(·) the bacterial growth rate. If we take into account the biomass recycling, we obtain: ⎧ dx ⎪ ⎪ = μ(·)x − (1 + r)D(t)x + rD(t)xr ⎪ ⎪ dt ⎪ ⎪ ⎨ ds (4.68) = − μ(·)x − (1 + r)D(t)s + D(t)sin (t) Y ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dxr = v(1 + r)D(t)x − v(w + r)D(t)xr dt with D(t) =

Qin ; Va

r=

Qr ; Qin

w=

Qw ; Qin

v=

Va Vs

We state the following hypotheses: – we measure only s and we want to estimate x and xr ; – we assume that μ(·) is not known, and we want to design an asymptotic observer (see section 4.5.3); – we know a bounding (even if very loose) of the initial conditions for x and xr ; – the substrate input for sin fluctuates but is not known. However, we know dynamic bounds for sin (t): + s− in (t) ≤ sin (t) ≤ sin (t)

∀t ≥ 0.

These hypotheses correspond to what happens in a urban wastewater treatment plant. The influent varies but is not measured. However, it can be bounded by two periodic functions corresponding to human activities. These bounds will probably evolve with respect to seasons. We will design an asymptotic interval estimator, which will provide bounds for the variables to be estimated. First we carry out a change of variable to eliminate μ(·) (see section 4.5.3): x Y z (4.69) Z=X+ s; Z = 1 ; X = z2 xr 0 and we obtain the 2-dimensional system: dZ x0 + Y s0 = D(t) AZ + B(s, t) ; Z0 = xr 0 dt −(1 + r) r Y sin (t) A= ; B(s, t) = −Y v(1 + r)s v(1 + r) −v(w + r)


107

We can now build two estimators (upper and lower) which use unknown bounds on the influent concentration sin : ⎧ ⎪ Y dZˆ + + ⎪ + + + ˆ ˆ ⎪ = D(t) AZ + B (s, t) ; Z (0) = X0 + s0 ⎪ ⎪ 0 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ˆ− ⎪ Y d Z ⎪ ⎪ ⎪ = D(t) AZˆ − + B − (s, t) ; Zˆ − (0) = X0− + s0 ⎪ ⎨ dt 0 (4.70) ⎪ ⎪ Y ⎪ + + ˆ = Zˆ − ⎪ X s ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y ⎪ ⎪ − − ˆ ˆ ⎪ s ⎩X = Z − 0 with B + (s, t) =

&

s+ in (t) −v(1+r)s

'

Y ; B − (s, t) =

&

s− in (t)

'

−v(1+r)s

Y

In this simple case, the convergence rate is fixed by the system. In Figure 4.6, we have represented the influent concentration and its bounds, the measurement s and the two estimates with the bounds. This very simple observer illustrates how to take into account the knowledge on the dynamic bounds. Under certain hypotheses, it can be shown that this observer can be regulated [GOU 00]. 4.6.4. Bundle of observers Now we will develop a set of regulatable interval observers for a considered system. We benefit from the possibility of regulating the error dynamics in order to run in parallel a broad set of interval observers associated with various gains that all guarantee a bounding of the real state. It is then possible to select the inner envelope of the so-called observer bundle in order to obtain a new observer with a much better convergence rate and much smaller interval predictions [BER 04]. This approach will be illustrated on a very simple example which describes the behavior of the concentrations of a biomass x and a substrate s in a perfectly mixed bioreactor: x˙ = μ(s)x − ux s˙ = u sin − s − kμ(s)x y=s

(4.71)

108

Bioprocess Control s (mg/l)

Sin (mg/l) 200

20

(a)

190

(b)

18

180

16

170

14

160

12

150

10

140

8

130

6 4

120 110

t (h)

2

t (h)

0

100 0

50 sin sin+ sin-

100

150

200

250

300

350

400

450

0

500

50

x (mg/l) 2500

150

200

250

300

350

400

450

500

350

400

450

500

xr (mg/l) 4500

(c)

2250

100

(d)

4000

2000

3500

1750 3000 1500 2500 1250 2000 1000 1500 750 1000

500

500

250

t (h)

t (h) 0

0 0

50 x x+ x-

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

xr xr+ xr-

Figure 4.6. Interval observer: (a) influent bounds, (b) measurements of S, (c) interval estimations of X, (d) interval estimations of Xr

sin corresponds to the concentration of influent substrate, u is the dilution input and k is the conversion yield coefficient. The biological activity of the system are featured by the non-negative growth rate function μ(s) such that μ(0) = 0, known to be highly uncertain. For the sake of simplicity, μ(s) is assumed to be a C 1 function. Our objective is to develop a robust observer based on the interval approach with improved convergence properties in order to estimate the biomass concentration x(t) in a reactor, when monitoring substrate s(t). In the following we assume that the following hypothesis is fulfilled. HYPOTHESIS 4.7. The growth rate is bounded by two known functions μ(s) and μ(s) and a positive constant a > 0: 1. 0 ≤ μ(s) ≤ μ(s) ≤ μ(s) ≤ a, ∀s ≥ 0 2. μ(0) = 0 Let us introduce the variable: z = kx + θ(t)s

(4.72)

where θ(t) ∈ Θ is a gain function. Considering equation (4.71), the dynamics of z can be written as follows: ˙ z˙ = (1 − θ)μ(s)(z − θs) + u θsin − z + θs (4.73)


109

The idea used in [LEM 05] was to implement a hybrid observer combining two types of observers: a bounded error observer with high convergence rate but poor accuracy and an asymptotic observer with fixed convergence rate but high accuracy. HYPOTHESIS 4.8. The influent substrate sin is an unknown input of system (4.71), with known bounds: + s− in (t) ≤ sin (t) ≤ sin (t)

(4.74)

PROPOSITION 4.4. Given x0 , x0 such that x0 ∈ [x0 , x0 ], then for a gain θ, the following systems define interval observers for system (4.71): – for θ(t) < 0 ˙ z˙ θ = (1 − θ) μ(s)z θ − θμ(s)s + u θs− in − z θ + sθ (4.75) ˙ z˙ θ = (1 − θ) μ(s)z θ − θμ(s)s + u θs+ in − z θ + sθ – for 0 ≤ θ(t) < 1 ˙ z˙ θ = (1 − θ) μ(s)z θ − θμ(s)s + u θs+ in − z θ + sθ ˙ z˙θ = (1 − θ) μ(s)z θ − θμ(s)s + u θs− in − z θ + sθ

(4.76)

– for θ(t) ≥ 1 ˙ z˙ θ = (1 − θ) μ(s)z θ − θμ(s)s + u θs+ in − z θ + sθ ˙ z˙ θ = (1 − θ) μ(s)z θ − θμ(s)s + u θs− in − z θ + sθ

(4.77)

with xθ = z θ − θs /k

and

xθ = z θ − θs /k

(4.78)

Proof. See [MOI 07]. One of the key properties related to an interval observer is that we can compare the solutions generated by two or more observers. We thus run several observers in parallel (for different values of gain θ), which all provide guaranteed interval estimates for the state. This is what we then call an observer bundle. Each bundle has an envelope that provides the best bounds i.e., we take the inner envelope from all the sets of estimates generated by different gain values. It is worth noting that we combine the transient behavior of some unstable observers (this may improve transient estimations), with the asymptotic stability of others (this guarantees the boundedness of the envelope).

110

Bioprocess Control

40

biomass (g/l)

30

20

10

0

0

10

20

30

40

time (days) Figure 4.7. Bundle of observer to estimate a biomass from substrate measurements and range of influent substrate

A regular reinitialization of the bundle can be performed to restart all the observers with the best available interval predicted by the bundle. We consider the time interval [tk , tk + Δt ] (we denote by Δt the reinitialization time interval) where the observers run. Then at time instant tk we take the best interval estimates performed by the previous estimation period to reinitialize the whole bundle. The objective behind the regular reinitialization of the interval estimates is to improve the observer efficiency by feeding it with the best available estimate, and thus take the benefit of the transients of some of them. The simulation example in Figure 4.7 performed with model (4.71) when s is measured demonstrates the efficiency of the approach; see [BER 04, MOI 05, MOI 07] for more details on the design of a bundle of observers. 4.7. Conclusion We have seen a set of methods to design an observer for a bioreactor. Other techniques exist and we do not pretend to be exhaustive. Let us mention for example the methods based on neural networks, where the system and its observer are estimated at the same time. The convergence rate of the obtained neural network cannot be regulated.


111

The presented observers assumed that the required model parameters were known and fixed. In some cases these parameters can evolve. Algorithms to estimate the parameters must then be used; they lead to adaptive observers. Of course in that case, the convergence of the full observer-parameter estimator system must be demonstrated. The choice of the type of observer to be developed must be made above all by considering the reliability of the model and the available measurements. A triple tradeoff must then be managed between robustness with respect to modeling uncertainties, robustness with respect to disturbances and convergence rate. Finally, to implement an observer in a computer, a time discretization phase is required. This step will be based on Euler-type algorithms. This step is not difficult, but it requires care. In particular, if the sampling rate is too high with respect to the discretization rate, continuous/discrete observers must be used [PEN 02]. To conclude, we insist that the observers must first be validated before they can be used. For this, their predictions must extensively be compared to direct measurements that were not used during the calibration process. 4.8. Appendix: a comparison theorem We recall here a general theorem in the nonlinear case. It can be useful to apply the interval estimation techniques; see [SMI 95] for details and for a general presentation. DEFINITION 4.9. A nonlinear system in dimension n is said to be cooperative if its Jacobian matrix is positive outside its diagonal on a convex domain. We now give the comparison theorem between x(t) and y(t) defined by the two systems dx = f (x, t); dt

x(0) = x0

dy = g(y, t); dt

y(0) = y0

where f, g : U × R+ → Rn are sufficiently regular on a convex domain U ⊂ Rn . PROPOSITION 4.5. If – ∀z ∈ U , ∀t ≥ 0, f (z, t) ≤ g(z, t) – g is cooperative – x0 ≤ y0 then x(t) ≤ y(t) for t > 0.

112

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The inequalities must be considered component-wise. For a cooperative system, the main property is thus that the order between two solutions is conserved for any time. This property is fundamental for the design of interval estimators.

4.9. Bibliography [ALC 02] V. A LCARAZ -G ONZALEZ, J. H ARMAND, A. R APAPORT, J.-P. S TEYER, A. G ON ZALEZ and C. P ELAYO -O RTIZ , “Software sensors for highly uncertain WWTPs: a new approach based on interval observers”, Wat. Res., vol. 36, pp. 2515–2524, 2002. [AND 90] B.D.O. A NDERSON and J.B. M OORE, Optimal Control. Linear Quadratic Methods., Prentice Hall, Englewood Cliffs, NJ, 1990. [ARC 01] M. A RCAK and P. KOKOTOVI C´ , “Observer-based control of systems with sloperestricted nonlinearities”, IEEE Transactions on Automatic Control, vol. 46, no. 7, pp. 1146–1151, 2001. [BAS 90] G. BASTIN and D. D OCHAIN, On-line Estimation and Adaptive Control of Bioreactors, Elsevier, 1990. [BAS 91] T. BASAR and P. B ERNHARD, H ∞ -Optimal Control and Related Minimal Design Problems: a Dynamic Game Approach, Birkhaüser, Boston, 1991. [BEA 89] S. B EALE and B. S HAFAI, “Robust control system design with a proportional integral observer”, Int. J. Contr., vol. 50, no. 1, pp. 97–111, 1989. [BER 98] O. B ERNARD, G. S ALLET and A. S CIANDRA, “Nonlinear observers for a class of biological systems. Application to validation of a phytoplanktonic growth model”, IEEE Trans. Autom. Contr., vol. 43, pp. 1056–1065, 1998. [BER 99] O. B ERNARD, G. BASTIN, C. S TENTELAIRE, L. L ESAGE -M EESSEN and M. A STHER, “Mass balance modelling of vanillin production from vanillic acid by cultures of the fungus Pycnoporus cinnabarinus in bioreactors”, Biotech. Bioeng, pp. 558–571, 1999. [BER 00] O. B ERNARD, J.-L. G OUZÉ and Z. H ADJ -S ADOK, “Observers for the biotechnological processes with unknown kinetics. Application to wastewater treatment”, in Proceedings of CDC 2000, Sydney, Australia, 11–15 December, 2000. [BER 01] O. B ERNARD, A. S CIANDRA and G. S ALLET, “A non-linear software sensor to monitor the internal nitrogen quota of phytoplanktonic cells”, Oceanologica Acta, vol. 24, pp. 435–442, 2001. [BER 04] O. B ERNARD and J.-L. G OUZÉ, “Closed loop observers bundle for uncertain biotechnological models”, J. Process. Contr., vol. 14:7, pp. 765–774, 2004. [BUS 02] E. B USVELLE and J. G AUTHIER, “High-gain and non-high-gain observers for nonlinear systems”, in Contemporary Trends in Nonlinear Geometric Control Theory and its Applications. International Conference, Mexico City, Mexico, A. A NZALDO -M ENESES, B. B ONNARD, J.P. G AUTHIER and F. M ONROY-P ÉREZ (eds.), World Scientific, 2002.


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[DAR 94] M. DAROUACH, M. Z ASADZINSKI and S.J. X U, “Full-order observers for linear systems with unknown inputs”, IEEE Trans. Automat. Contr., vol. AC-39, pp. 606–609, 1994. [DEZ 92] F. D EZA, E. B USVELLE, J. G AUTHIER and D. R AKOTOPARA, “High gain estimation for nonlinear systems”, System and Control Letters, vol. 18, pp. 292–299, 1992. [DRO 68] M. D ROOP, “Vitamin B12 and marine ecology. IV. The kinetics of uptake growth and inhibition in Monochrysis lutheri”, J. Mar. Biol. Assoc., vol. 48, no. 3, pp. 689–733, 1968. [DUG 67] R. D UGDALE, “Nutrient limitation in the sea: dynamics, identification and significance”, Limnol. Oceanogr., vol. 12, pp. 685–695, 1967. [EDW 94] C. E DWARDS and S.K. S PURGEON, “On the development of discontinuous observers”, Int. J. Control, vol. 59, no. 5, pp. 1211–1229, 1994. [GAU 81] J. G AUTHIER and G. B ORNARD, “Observability for any u(t) of a class of nonlinear systems”, IEEE Trans. Autom. Contr., vol. 26, no. 4, pp. 922–926, 1981. [GAU 92] J. G AUTHIER, H. H AMMOURI and S. OTHMAN, “A simple observer for nonlinear systems applications to bioreactors”, IEEE Trans. Autom. Contr., vol. 37, pp. 875–880, 1992. [GAU 01] J. G AUTHIER and I.A.K. K UPKA, Deterministic Observation Theory and Applications, Cambridge University Press, Cambridge, 2001. [GOU 00] J.-L. G OUZÉ, A. R APAPORT and Z. H ADJ -S ADOK, “Interval observers for uncertain biological systems”, Ecological Modelling, vol. 133, pp. 45–56, 2000. [HOU 91] M. H OU and P. M ÜLLER, “Design of observers for linear systems with unknown inputs”, IEEE Trans. Autom. Contr., vol. AC-37, no. 6, pp. 871–875, 1991. [JAU 01] L. JAULIN, M. K IEFFER,O. D IDRIT and E. WALTER, Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics, SpringerVerlag, London, 2001. [KAI 80] T. K AILATH, Linear Systems, Prentice-Hall, Inc., Englewood Cliffs, N.J., London, 1980. [KIE 04] M. K IEFFER and E. WALTER, “Guaranteed nonlinear state estimator for cooperative systems”, Numerical Algorithms, vol. 37, pp. 187–198, 2004. [KUD 80] P. K UDVA, N. V ISWANADHAM and A. R AMAKRISHNA, “Observers for linear systems with unknown inputs”, IEEE Trans. Autom. Contr., vol. AC-25, no. 1, pp. 113–115, 1980. [LEM 05] V. L EMESLE and J.-L. G OUZÉ, “Hybrid bounded error observers for uncertain bioreactor models”, Bioprocess and Biosystems Engineering, vol. 27, no. 5, pp. 311–318, Springer-Verlag, 2005. [LUE 66] D.G. L UENBERGER, “Observers of multivariables systems”, IEEE Transaction on Automatic Control, vol. 11, pp. 190–197, 1966.

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[LUE 79] D.G. L UENBERGER, Introduction to Dynamic Systems: Theory, Models and Applications, Wiley, 1979. [MOI 05] M. M OISAN and O. B ERNARD, “Interval Observers for non monotone systems. Application to bioprocess models”, in Proceedings of the 16th IFAC World Conference, Prague, Czech Republic, 4th–8th July, 2005. [MOI 07] M. M OISAN, O. B ERNARD and J.-L. G OUZÉ, “Near optimal interval observers bundle for uncertain bioreactors”, in Proceedings of the 9th European Control Conference, Kos, Greece, 2007. [PEN 02] M. P ENGOV, E. R ICHARD and J.-C. V IVALDA, “Continuous-discrete observers for global stabilization of nonlinear systems with applications to bioreactors”, Europ. J. Contr., vol. 8, pp. 465–476, 2002. [RAP 03] A. R APAPORT and J.-L. G OUZÉ, “Parallelotopic and practical observers for nonlinear uncertain systems”, Int. Journal. Control, vol. 76, no. 3, pp. 237–251, 2003. [SMI 95] H.L. S MITH, Monotone Dynamical Dystems: an Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, Rhode Island, 1995.

Chapter 5

Recursive Parameter Estimation

5.1. Introduction In Chapter 4, we dealt with the reconstruction of the evolution of state variables (essentially the concentrations of substrates, products and biomass) over the course of time. In Chapter 3, we calculated the values of the parameters of the model chosen by us for our bioprocess model on the basis of the available experimental data. In this chapter, we deal with a problem that is complementary to the problems described in Chapters 3 and 4: how do we calculate online the value of parameters for our bioprocess model? The motivation for addressing this question is as follows. We have seen earlier that it could be difficult to determine certain parameters with precision. In addition, certain essential parameters such as specific rates often cannot be modeled in a reliable and precise manner. For these reasons, it may turn out to be important, especially in the context of monitoring and control, to be in a position to evaluate over the course of time the value of the most “uncertain” parameters (not only from the point of view of determining the values but also from that of analytical characterization with respect to the process variables) and their variation. The approaches developed here will be substantially different from those considered in Chapter 3 insofar as we want to estimate a parameter at every instant. Therefore, the estimation algorithms in this chapter will be of the recursive type. Here we will be essentially concentrating on two types of approach involving the recursive estimation of parameters: 1) estimator based on the structure of the observer; 2) estimator using recursive least squares. Chapter written by Denis D OCHAIN.

115

116

Bioprocess Control

We will then see how to combine parameter and state estimations in the core of the same estimation algorithm, which we will call an adaptive observer (section 5.4). 5.2. Parameter estimation based on the structure of the observer Let us recall the equations for the general dynamic model for bioreactors: dξ = Kφ(ξ) + F − Q − Dξ dt

(5.1)

where F and Q are the inflow rate of matter and the gaseous outflow rate, respectively [BAS 90]. Let us consider the sufficiently general case where the most uncertain parameters are concentrated in the process kinetics. Let us also assume that the not well-known parameters are linear in the dynamic model. We can, for the requirements of the cause, rewrite the equations for the model as follows: dξ = KG(ξ)θ + F − Q − Dξ dt

(5.2)

Here, θ is the vector of unknown parameters, and G(ξ) a matrix, which is dependent on the state of the process. These equations will serve as a base for designing the estimator. First of all, let us introduce an example that will serve to illustrate the method throughout this section. 5.2.1. Example: culture of animal cells The process with respect to an animal cell, which we are going to consider here, is a culture of human fetus kidney cells (HEK-293) [SIE 99]. It has been shown in [SIE 99] that the process is characterized by the following reaction scheme: k1 S + k2 C −→ X + k3 G

(5.3)

k4 S −→ X + k5 L

(5.4)

Here S, C, X, G and L represent glucose, dissolved oxygen, yeasts, carbon dioxide and lactate, respectively. Reactions (5.3) and (5.4) are, respectively, the oxidation reaction (respiration) on glucose and that of glycolysis (fermentation) on glucose. The dynamics of the process in a stirred tank reactor are given in the matrix format mentioned above (5.2) by the following vectors and matrices: ⎡ ⎤ ⎡ ⎤ −k1 −k4 S ⎢−k2 ⎢C ⎥ 0 ⎥ ⎢ ⎥ ⎢ ⎥ μR X ⎢ ⎥ ⎢ ⎥ 1 ⎥ , G(ξ)θ = ξ = ⎢X ⎥ , K = ⎢ 1 (5.5) μF X ⎣ k3 ⎣G⎦ 0 ⎦ L 0 k5


⎡ ⎤ DSin ⎢ Qin ⎥ ⎢ ⎥ ⎥ F =⎢ ⎢ 0 ⎥, ⎣ 0 ⎦ 0

117

⎡

⎤ 0 ⎢0⎥ ⎢ ⎥ ⎥ Q=⎢ ⎢0⎥ ⎣Q1 ⎦ 0

(5.6)

kj (j = 1 to 5) are the yield coefficients, Sin is the feed concentration of glucose (g/l), Qin is the feed rate of oxygen (g/l/h), Q1 is the outflow rate of CO2 (g/l/h) and μi (i = R, F ) are the specific growth rates (1/h) associated with each growth reaction. Let us now briefly discuss the various possibilities for the distribution of terms in G(x) and θ. The above model suggests the following distribution (this will actually be the one which will be considered below): X 0 μ G(ξ) = , θ= R (5.7) 0 X μF However, there are other possible options, depending on the level of knowledge and on the uncertainty regarding the process kinetics. For example, we can consider: 1) that the whole vector for the reaction rate is unknown: μR X G(ξ) = I, θ = (5.8) μF X 2) or that, depending on the kinetic laws, the reaction rates are explicitly linked to the limiting substrates (the simplest way to express this is to write μR = ρ1 SC and μF = ρ2 S), and to assume that the parameters ρi are unknown; thus we have: SCX 0 ρ G(ξ) = , θ= 1 (5.9) ρ2 0 SX 3) or that we have certain knowledge about the structure of the kinetic models (e.g. Monod’s model) and that the maximum specific growth rate μmax,i (i = 1, 2) is unknown while the affinity constants KS1 , KS2 , KC1 are known; this will lead to the following definitions for G(x) and θ: ⎡ ⎤ SCX 0 μmax,1 ⎢ (KS1 + S)(KC1 + C) ⎥ G(ξ) = ⎣ (5.10) SX ⎦ , θ = μmax,2 0 KS2 + S 5.2.2. Estimator based on the structure of the observer Let us assume the following: – Assumption 1. The p parameters of θ are unknown and possibly time-varying (with bounded time variations: dθ dt < M ). – Assumption 2. Let p state variables be measured online.

118

Bioprocess Control

Starting from Assumption 2, let us define the partition of state as follows: ξ ξ= 1 (5.11) ξ2 Here, ξ1 are the measured variables and ξ2 the unmeasured variables. The dynamic equations can now be rewritten as follows: dξ1 = K1 G(ξ)θ + F1 − Q1 − Dξ1 dt dξ2 = K2 G(ξ)θ + F2 − Q2 − Dξ2 dt

(5.12) (5.13)

Let us make the following assumptions as well: – Assumption 3. G is a diagonal matrix (G = diag{gi }, i = 1 to p) and is full rank for all the admissible values of ξ (in our case, this means essentially that only the positive values of the state variables (i.e. concentrations) are considered). – Assumption 4. K1 is full rank, and its coefficients are known. – Assumption 5. Feed rates F1 (associated with ξ1 ), gas outflows Q1 (associated with ξ1 ) and the dilution rate D are known (via online measurements or choice of the user). Under these assumptions, it is possible to construct an asymptotic observer for variables ξ2 independent of the knowledge about parameters θ (see Chapter 4). In continuation of the section, we will consider that the states included in ξ2 are either accessible to the online measurement or are available via an asymptotic observer or that the dynamics of ξ1 are independent of ξ2 . The concept of the estimator on the basis of the structure of the observer is based on equation (5.12) and actually follows a development similar to that of the Luenberger observer. This gives the following estimation equations: dξˆ1 = K1 G(ξ)θˆ + F1 − Q1 − Dξ1 − Ω(ξ1 − ξˆ1 ) dt dθˆ = [K1 G(ξ)]T Γ(ξ1 − ξˆ1 ) dt

(5.14) (5.15)

The essential motivation for determining the structure of the estimator is as follows. As for any traditional observer, the estimator equations are a combination of the process model equations (K1 G(ξ)θˆ + F1 − Q1 − Dξ1 ) and the terms of correction (−Ω(ξ1 − ξˆ1 ) and [K1 G(ξ)]T Γ(ξ1 − ξˆ1 )) on the measured variables, terms that help in arriving at the estimation. In the estimator based on the structure of the observer,


119

parameters θ have the status of state variables without dynamics. The weighting factor [K1 G(ξ)]T in equation (5.15) is actually the term with which the unknown parameter in the equation of the model is multiplied: its introduction into the equations of the estimator is based on the concept of the traditional estimators (e.g. [GOO 77]) and is generally called a regressor. The estimator based on the structure of the observer was originally developed for estimating specific growth rates [BAS 86] and was subsequently extended to the online estimation of kinetic parameters [BAS 90]. This has found important roles over a large range of bioprocesses occurring in food, pharmaceutical and environmental applications (e.g. [ATR 88, BAS 90, BOU 00, CLA 98, OLI 96, FLA 89, POM 92, POM 95, SUL 97]). NOTE. It is obvious that we can further generalize the approach by considering the case in which certain kinetics are perfectly known and others are not known. The general dynamic model will thus take the following form: dξ = Ki Gi (ξ)θ + Kc φc (ξ) + F − Q − Dξ dt

(5.16)

Here, φc (ξ) is the vector for the known reaction rates and Ki and Kc are the yield coefficient sub-matrices associated with the unknown and known kinetics, respectively. The design of the estimator follows the same approach as that described above. 5.2.3. Example: culture of animal cells (continued) Let us assume that the glucose concentration S and lactate concentration L are measured online and that the aim is to estimate in real time the specific growth rates. Now the vectors and matrices are written in the following specific form: −k1 X −k4 X S , K1 G(ξ) = ξ1 = (5.17) L 0 k5 X DSin μ (5.18) F1 − Q1 = θ= R , μF 0 5.2.4. Calibration of the estimator based on the structure of the observer: theory The theoretical analysis concerning the stability of the estimator based on the observer’s structure is available in the paper [BAS 90]: the essential requirements for this stability are the defined negativity of ΩT Γ + ΓΩ and the persistent excitation of the signal K1 G(ξ), i.e.: 1) negative definiteness: ΩT Γ + ΓΩ < 0;

120

Bioprocess Control

2) persistent excitation: there are positive constants α and β such that αI ≤ ( t+β [K1 G(ξ)]T K1 G(ξ)dτ , for all t ≥ 0. t However, the calibration of the estimator is in practice a difficult problem that has to be solved, which is attributed to the strong interaction among the unknown parameters in the estimator equations and to the dependence of the dynamics on the process variables. This influence of latter constraint can be reduced if the process is made to work around a permanent regime; however, it will be crucial when the system involves a large spectrum of operating conditions (in particular, the case of the batch and fed-batch, reactors, as well as in the case of start-up, shut-down or setpoint change (change in the type of production, for example) of the process) with possibly large variations in the process variables and thus matrix G(ξ). Good performances in terms of tracking the variations of the estimated parameters is particularly important under these circumstances. However, in the above form (5.14)-(5.15) of the estimator, the adjustment of the concept parameters of estimator Γ and Ω may be very conservative and may result in bad performances with respect to tracking the variation in certain operating regions. The objectives of calibration for the estimator based on the structure of the observer mentioned above can be satisfied if we consider the following two stages in the reformulation of the algorithm: 1) a state transformation; 2) the rearrangement of the state vector entries of the estimator. State transformation Let us define the following state transformation: z = K1−1 ξ

(5.19)

Now the dynamic equations of the process can be rewritten as follows: dz = G(z)θ + K1−1 (F1 − Q1 − Dz) dt

(5.20)

With the above transformation, one and only one variable is associated with each unknown parameter θ. In our case, the invertibility of matrix K1 is a result of the independence of the p reactions and p measured variables (see also [BAS 90]). The above transformation has already been proposed in [POM 90, OLI 96]. We can now proceed with a new concept of the estimator based on the structure of the observer on the basis of equation (5.20): dˆ z = G(z)θˆ + K1−1 (F1 − Q1 − Dz) − Ω(z − zˆ) dt

(5.21)


dθˆ = Γ(z − zˆ) dt

121

(5.22)

With transformation (5.19), the estimator based on the structure of the observer has its equations reformulated in a decoupled format for the unknown parameters θi (i = 1 to p). Due to the decoupled formulation of the estimation, an immediate choice for matrices Ω and Γ are diagonal matrices: Ω = diag{−ωi },

Γ = diag{γi },

ωi > 0, γi > 0, i = 1 to p

(5.23)

In this formulation for the estimation algorithm, we eliminated regressor G(z) from the estimation equations of θ (5.22): as one of its main roles consists of transferring explicitly the coupling between the unknown parameters and the measured variables, it is no longer essential here. Rearrangement of the state vector elements of the estimator The final stage before the formulation of the rule for the calibration of the estimator consists of rearranging the equations of the estimator. Let us initially regroup each variable zi with parameter θi to which it is linked and rearrange the elements of the T vector z, θ in the following order in vector ζ: ⎡ ⎤ z1 ⎢ θ1 ⎥ ⎢ ⎥ ⎢ z2 ⎥ ⎢ ⎥ θ ⎥ (5.24) ζ=⎢ ⎢ .2 ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎣z ⎦ p θp Rule for base calibration Let us define the estimation error as follows: e = ζ − ζˆ

(5.25)

The dynamics of the estimation error are obtained immediately from equations (5.20), (5.21) and (5.22): de = Ae + b dt Here, A is a block-diagonal matrix with the dimension blocks 2 × 2: −ωi gi (z) A = diag{Ai }, Ai = , i = 1 to p 0 −γi

(5.26)

(5.27)

122

Bioprocess Control

Here b is equal to: ⎡

⎤ 0 ⎢ dθ1 ⎥ ⎢ ⎥ ⎢ dt ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ dθ2 ⎥ ⎥ b=⎢ ⎢ dt ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎣ ⎦ dθp dt

(5.28)

The characteristic equation of matrix A, det(λI − A), is equal to: det(λI − A) =

p )

(λ2 + ωi λ + γi gi (z))

(5.29)

i=1

The key idea of the calibration rule consists of choosing each γi in an inversely proportional manner to the corresponding term gi (z): γi =

γ¯i , gi (z)

γ¯i > 0, i = 1 to p

(5.30)

With this choice, characteristic equation (5.29) is rewritten as follows: det(λI − A) =

p )

(λ2 + ωi λ + γ¯i )

(5.31)

i=1

and the dynamics of the estimator based on the structure of the observer is now independent of the state variables of the process. Such a choice actually corresponds to a Lyapunov transformation (see [PER 93]). This is obviously valid for the values of gi (z) = 0: this condition is usually easily encountered in bioprocess applications, as will be illustrated in the next section. The values of the calibration parameters can be selected in a way so as to fix the dynamics of the estimator for each unknown parameter θi . As the estimator is reduced, via the transformations, to a set of second order linear systems, one being independent of another, the traditional rules for assigning the dynamics of a set of second order linear systems are immediately applicable here. We therefore suggest to the reader to have a look at standard automatic control textbooks for more detailed information on the subject. Nevertheless, we suggest below the following basic guidelines.


123

An important basic choice consists of considering real poles for the dynamics of the estimator: γi ≥ 0 ωi2 − 4¯

(5.32)

Now the objective is to avoid in the estimation of the parameters inducing oscillations that do not correspond to any physical phenomenon associated with the estimated reaction rates. In this approach, Pomerleau and Perrier [POM 90] suggest the choice of double poles, i.e.: γ¯i =

ωi2 4

(5.33)

In this case, the adjustment of the estimator is reduced to the choice of a single calibration parameter ωi by the estimated parameter. This enables us to have a procedure for adjustment that is both simple (a single parameter for adjustment) and flexible (each estimator of the parameter can be calibrated in different ways if necessary, for example when the variations of the estimated parameters are different). Up to now, we have suggested that it is possible to assign freely the dynamics of the estimator. However, in the presence of noisy data, it appears that a compromise should be made between a rapid convergence of the estimator and a good attenuation of the measurement noise. A detailed analysis is performed in the paper by Bastin and Dochain [BAS 90] (pp. 162–172), which studies the performances of the estimator based on the structure of the observer both in theory and in numerical simulation in the presence of bounded noisy data in the particular case of the estimation of the specific growth rate in a simple microbial growth process. This is characterized by the dynamic material balance equations as follows: dX = μX − DX dt dS = −k1 μX + DSin − DS dt

(5.34) (5.35)

The theoretical analysis of optimization is based on the evaluation of asymptotic properties of the estimator and gives as a result the following optimum value for ω1 : * k1 M1 ω1,opt = 2 , 0

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