thÈse - inria · evolución dinámica de sus ariablesv de importancia. los sensores existentes son...

Université de Nice SophiaAntipolis UFR Sciences

École DoctoraleSciences et Technologies de l'Information et de la Communication

THÈSE

pour obtenir le grade de

Docteur de l'Université de Nice SophiaAntipolis

Mention

Sciences de l'Ingenieur

présentée par

MarceloMOISAN

Titre de la thèse

Synthèse d'Observateurs par Intervalles

pour des Systèmes Biologiques Mal Connus

Thèse préparée dans le projet Comore, INRIA SophiaAntipolis

Dirigée par Olivier Bernard

Soutenue le 6 décembre 2007 devant le jury composé de:

Examinateurs M. Georges Bastin

M. Olivier Bernard

M. Jacques Blum

M. Jean-Luc Gouzé

M. Michel Kieffer

Rapporteurs M. Jaime Moreno

M. Alain Rapaport

Acknowledgements

A mi familia: Victor, Magaly,

Romina, Patricio y Soa

I have had the privilege of being supervised by one of the brightest and most exceptional persons

I have ever met. I would like to thank Olivier Bernard, for his understanding and constant support.

Working with Olivier I learned to take on new challenges with enthusiasm and to do my best to achieve

the objectives.

I would also like to express my gratitude to the COMORE research team. Particularly to Jean-Luc

Gouzé and Frédéric Grognard. Thanks for always being ready to give valuable advise and help. Also,

Stéphanie Sorres, France Limouzis and Christine Riehl deserve all my recognition.

Many thanks to the members of my committee for their constructive corrections, fruitful technical

discussions and remarks on my work: Professor Georges Bastin, president of the committee, Doctor

Alain Rapaport, Professor Jaime Moreno, Professor Jacques Blum and Doctor Michel Kieer.

I would like to thank the French National Institute for Research in Computer Science and Control

INRIA, and the Chilean Council of Research CONICyT, for the nancial support granted.

My friends played an important role in my life during these last four years. I would like to say

gracias to Sapna Nundloll, for always being there, ready to share a word and give her appreciation

about photography, which I appreciated very much. Many thanks to the Chilean community at INRIA,

for your encouragements and unconditional friendship. Among others I would express my gratitude

to Ángela, MaríaJosé, Marcela, Carlos, Gonzalo, Marcos, Mario and los colocadores: Antonio, Diego,

Patricio and Adán. I wish you all the best.

Résumé Étendu

Les bioprocédés sont des systèmes plus complexes que les systèmes industriels classiques construits par

des ingénieurs dans la mesure où ils impliquent des organismes vivants. Les bioprocédés sont de plus

en plus étudiés dans le monde, et ils sont utilisés dans divers domaines, comme le traitement des eaux

usées, la production pharmaceutique, l'industrie alimentaire, etc.

Les systèmes vivants sont diciles à gérer car les organismes sont capables d'adaptation (leurs

propriétés évoluent au cours du temps), et peuvent être très sensibles à des changements minimes

de leur environnement. En conséquence, la modélisation mathématique des bioprocédés est délicate,

souvent caractérisée par des dynamiques non linéaires et incertaines, ou par une grande variabilité des

paramètres.

Il est donc indispensable de développer des stratégies de surveillance des procédés biologiques.

Cependant, l'une des dicultés majeures au niveau industriel est le manque de capteurs permettant

de suivre dans le détail lévolution dynamique de leurs variables clé. Les capteurs sont souvent onéreux

et nécessitent une expertise technique. De plus, la plupart des mesures sont généralement obtenues

horsligne, après une analyse en laboratoire. Enn, notamment pour les bioprocédés consacrés à la

dépollution des eaux useés, il arrive souvent que les inuents, qui varient au cours du temps, ne soient

pas régulièrement mesurés.

L'automatique vise à concevoir des stratégies de contrôle et des schémas d'estimation d'état. À ce

titre, elle joue un rôle central dans l'etude des bioprocédés et permet des améliorations substantielles

de leurs performances. Elle fournit des outils pour améliorer l'ecacité des procédés et pour garantir

la qualité de la production. Ceci conduit à optimiser le fonctionnement du système par rapport à

diérents critères (maximiser le rendement de production dans l'industrie pharmaceutique, du taux

de biodégradation pour la dépollution, ...). La production industrielle cherche souvent à satisfaire des

normes an d'assurer une qualité constante du produit. Parfois, des contraintes de sécurité doivent

également être prises en compte, par exemple lorsqu'il existe des points d'opération instables (ce qui

peut conduire, par exemple, au lessivage de la biomasse d'un bioréacteur).

Les outils classiques de l'automatique, mis au point pour des modèles bien identiés, ne sont souvent

pas adaptés aux bioprocédés avec leur dynamique incertaine. Des méthodes dédiées de surveillance et

de contrôle doivent alors être développées, an de garantir la robustesse aux incertitudes et au manque

de mesures.

Cette thèse traite du problème de l'estimation de l'état des systèmes biologiques. Les techniques

d'estimation d'état nonlinéaires ont suscité une attention croissante au cours des dernières décennies.

Depuis l'observateur de Luenberger (Luenberger, 1966) et le ltre de Kalman (Kalman and Bucy, 1961),

les méthodes d'estimation d'état ont évolué vers des méthodes non linéaires et robustes, étendant leur

champ d'application à une large gamme de systèmes de la vie réelle. Nous nous concentrons sur un

iii

iv Résumé Étendu

cadre déterministe et nonlinéaire, et nous proposons des méthodes d'estimation garantie. Pour ces

méthodes, l'objectif d'estimation est de déterminer une région de l'espace d'état contenant les variables

inconnues. Les méthodes ellipsoidales (Kurzhanski and Vályi, 1997) et l'estimation avec l'analyse par

intervalles (Jaulin et al., 2001) sont les exemples les plus populaires de méthodes d'estimation d'état

garantie.

Nous fondons notre approche sur la notion d'observateurs par intervalles (Gouzé et al., 2000). Ce

schéma d'estimation consiste à obtenir des bornes supérieure et inférieure garanties des variables incon-

nues d'un système dynamique, en considerant que l'on connaît des bornes pour les quantités incertaines.

La synthèse des observateurs par intervalles est basée sur la théorie des systèmes dynamiques positifs

et des systèmes coopératifs.

Après un examen de la recherche actuelle associée à ces observateurs, il a été possible d'identier

certaines limitations. Tout d'abord, le développement de ces observateurs est limité à des systèmes

dynamiques coopératifs (Alcaraz-González, 2001; Hadj-Sadok, 1999), c'est-à-dire des systèmes dont

la matrice jacobienne a des valeurs positives en dehors de la diagonale (Smith, 1995). Ces systèmes

préservent la relation d'ordre partiel dans l'espace d'état. Ceci réduit donc le champ d'application à une

certaine classe de systèmes biologiques. Les approches actuelles conduisent souvent à des observateurs

dont les taux de convergence ne sont pas réglables. En eet, divers observateurs par intervalles pour

des modèles de bioprocédés ont été dévelopés sur la base d'observateurs asymptotiques (Bastin and

Dochain, 1990). En conséquence, le taux de convergence est limité par les conditions d'opération

du système (le taux de dilution du bioréacteur doit être susamment grand). Enn, la dégradation

des performances dûe aux grandes incertitudes est une forte limite de ces observateurs. Parfois, ils

produisent de larges intervalles et ne fournissent donc pas d'information utile.

Le principal objectif de cette thèse est de proposer des idées nouvelles et originales sur la conception

de ces estimateurs, en surmontant les limites précédement évoquées.

La première partie de cette thèse est consacrée à la modélisation de bioprocédés et à un examen

de l'état de l'art des méthodes d'estimation. Dans le Chapitre 1, nous exposons les modes opératoires

des bioréacteurs, puis nous nous concentrons sur la digestion anaérobie. Nous présentons alors un

modèle bidimensionnel largement utilisé et eectuons une analyse qualitative de son comportement.

Un exemple industriel réel de la digestion anaérobie est présenté (qui sera dans la suite notre cas

d'application principal). Dans le Chapitre 2, nous présentons l'etat de l'art concernant des méthodes

d'estimation robuste avec des applications aux bioprocédés. Nous présentons tout d'abord les méthodes

d'estimation classiques, puis nous mettons l'accent sur les innovations récentes. Ce chapitre propose

également le cadre théorique des systèmes dynamiques monotones et positifs.

La seconde partie de ce mémoire présente plus particulièrement notre contribution et est organisée

en quatre chapitres. Dans le Chapitre 3 de nouveaux concepts sur les observateurs par intervalles

sont introduits. En particulier, nous en donnons une nouvelle dénition, en introduisant le concept d'

encadreur et de faisceau (Bernard and Gouzé, 2004). Nous lançons en parallèle diverses estimations par

intervalles puis nous prenons l'enveloppe intérieure, en protant de la garantie des bornes supérieures

et inférieures obtenues. Un exemple d'application non classique à un système chaotique est présenté.

La conception des observateurs par intervalles pour des systèmes non monotones est étudiée dans le

Chapitre 4. Nous considérons la partie nonlinéaire (et incertaine) de la dynamique et, sous certaines

hypothèses techniques, nous l'écrivons comme la diérence de deux fonctions croissantes. Ceci nous

permet d'obtenir une bornitude de la partie non linéaire, et de formuler des observateurs avec une

dynamique d'erreur monotone. Cette approche est illustrée par deux observateurs par intervalles d'un

Résumé Étendu v

procédé de digestion anaérobie.

Dans le Chapitre 5, nous développons un critère d'optimalité (qui peut être appliqué à un obser-

vateur par intervalles quelconque), an de trouver les gains qui fournissent les meilleures estimations.

Ensuite, nous introduisons un nouveau schéma d'estimation qui améliore l'estimation de l'intervalle

des conditions initiales, en construisant des observateurs pour le système en temps inverse, et en ef-

fectuant des marches en arrière. Ces idées sont illustrées sur l'observateur proposé par Lemesle and

Gouzé (2005). Nous ecrivons alors un observateur par intervalles dont les propriétés de convergence

sont améliorées, mais qui produit une estimation discontinues des intervalles d'état.

Enn, nous proposons au Chapitre 6 des observateurs par intervalles dédiés à l'estimation des entrées

incertaines. Ces observateurs sont élaborés dans le cas d'une entrée constante, puis variable par rapport

au temps. Dans le premier cas, nous montrons que l'estimation par intervalles de l'entrée dépend

directement de l'estimation eectuée sur l'état du système (estimation d'état et d'entrée simultanée).

Pour le deuxième cas, on obtient des estimations par intervalles en construisant des bornes sur l'erreur

d'estimation d'un observateur de l'entrée de type grand gain. Ces observateurs sont testés et validés

sur des données réelles issues d'un fermenteur anaérobie industriel.

Resumen Extendido

Los bioprocesos son sistemas más complejos que los sistemas industriales clásicos construidos por los

ingenieros, en la medida de que involucran organismos vivientes. Los bioprocesos son cada vez más

estudiados en el mundo, siendo útiles en diversos campos, como el tratamiento de aguas contaminadas,

la producción de farmaceuticos, la industria alimentaria, etc.

Los sistemas vivientes son difíciles de tratar ya que los organismos son capaces de adaptarse (sus

propiedades evolucionan en el tiempo), y pueden ser muy sensibles a cambios pequeños en su medio

ambiente. Como consecuencia, el modelamiento matemático de bioprocesos es una tarea delicada. Los

modelos obtenidos son con frecuencia caracterizados por dinámicas no lineales e inciertas, o por una

gran variabilidad de sus parámetros.

Es entonces indispensable desarrollar estrategias de monitoreo de bioprocesos. Sin embargo, una

de las mayores dicultades a nivel industrial es la falta de sensores que permitan seguir en detalle la

evolución dinámica de sus variables de importancia. Los sensores existentes son usualmente costosos y

necesitan de una expertiz técnica. Adicionalmente, una gran parte de las mediciones son generalmente

obtenidas fuera de línea, después de un análisis de laboratorio. Finalmente, y sobre todo en bioprocesos

dedicados a la descontaminación de aguas, con frecuencia se encuentra que las concentraciones de

entrada, que varían en el curso del tiempo, no son medidas de manera regular.

La automática tiene como objetivo concebir estrategias de control y esquemas de estimación de

estado. Esta juega un papel central en el estudio de bioprocesos y permite mejoras substanciales en

su rendimiento. Además, provee herramientas para mejorar la ecacia de los procesos y garantizar

la calidad de la producción. Esto conduce a la optimización de un sistema con respecto a diferentes

criterios (maximizar el rendimiento de producción de farmacéuticos la tasa de biodegradación en pro-

cesos de descontaminación, ). La producción industrial busca con frecuencia satisfacer normas con el

n de asegurar una calidad constante del producto. Normalmente, condiciones de seguridad deben ser

también tomadas en cuenta, por ejemplo, cuando existen puntos de operación inestables (que pueden

conducir, por ejemplo, al vaciamiento de la biomasa de un birreactor).

Las herramientas clásicas de la automática, ajustadas para modelos bien identicados, no se adaptan

de manera óptima a modelos de bioprocesos con dinámica incierta. Entonces, métodos dedicados a

su monitoreo y control deben ser desarrollados, con el n de garantizar la robustez con respecto a las

incertidumbres y falta de mediciones.

Esta tesis trata sobre el problema de estimación de estado de sistemas biológicos. Las técnicas de

estimación de estado no lineales han suscitado una atención creciente en el curso de las últimas décadas.

Desde el observador de Luenberger (Luenberger, 1966) y el ltro de Kalman (Kalman and Bucy, 1961),

los métodos de estimación de estado han evolucionado hacia métodos no lineales y robustos, extendiendo

su campo de aplicación a una gran gama de sistemas reales. En particular, nos concentraremos en un

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viii Resumen Extendido

cuadro determinista y no lineal, proponiendo métodos de estimación garantizada. Para estos métodos,

el objetivo de estimación es de determinar una región del espacio de estado que contenga las variables

desconocidas. Los métodos elipsoidales (Kurzhanski and Vályi, 1997) y la estimación a través del

análisis por intervalos (Jaulin et al., 2001) son los ejemplos más populares de métodos de estimación

de estado garantizada.

Nosotros fundamos nuestro enfoque sobre la noción de observadores por intervalos (Gouzé et al.,

2000). Este esquema de estimación consiste en obtener cotas superiores e inferiores garantizadas de las

variables desconocidas de un sistema dinámico, considerando que se conocen cotas para las cantidades

inciertas. El diseño de observadores por intervalos esta basado en la teoría de sistemas dinámicos

positivos y sistemas cooperativos.

Después de realizar un examen de la investigación actual asociada a este tipo de observadores,

ha sido posible identicar ciertas limitaciones. En primer lugar, el desarrollo de estos observadores

es limitado a sistema dinámicos cooperativos (Alcaraz-González, 2001; Hadj-Sadok, 1999), es decir

sistemas cuya matriz jacobiana tiene elementos no negativos fuera de su diagonal (Smith, 1995). Estos

sistemas preservan la relación de orden parcial en el espacio de estado. Esto reduce el campo de

aplicación a una clase reducida de sistemas biológicos. Los enfoques actuales conducen usualmente a

observadores cuya razón de convergencia no es ajustable. En efecto, diversos observadores por intervalos

para modelos de bioprocesos han sido desarrollados sobre la base de observadores asintóticos (Bastin

and Dochain, 1990). Como consecuencia, la razón de convergencia esta limitada a las condiciones de

operación del sistema (la dilución de un bioreactor debe ser lo sucientemente grande). Finalmente, la

degradación del rendimiento de observadores por intervalos debido a incertidumbres de gran escala es

una fuerte limitante de estos observadores. A veces, ellos producen grandes intervalos que no entregan

ninguna información útil.

El objetivo principal de esta tesis es de proponer nuevas y originales ideas sobre la concepción de

este tipo de estimadores, estableciendo mejoras sobre las limitaciones mencionadas anteriormente.

La primera parte de esta tesis esta dedicada al modelamiento de los bioprocesos y a un examen

del estado del arte de los métodos de estimación de estado. En el Capítulo 1, exponemos los modos

de operación de bioreactores y luego nos concentramos en procesos de digestión anaeróbica. Entonces

presentamos un modelo bidimensional ampliamente usado y efectuamos un análisis cualitativo de su

comportamiento. Un ejemplo real asociado a un proceso industrial de la digestión anaeróbica es

presentado (proceso que será en lo que sigue nuestro principal caso de aplicación). En el Capítulo

2, presentamos el estado del arte de métodos de estimación robusta de estado, con aplicaciones a

bioprocesos. En primer lugar presentamos los métodos de estimación clásicos, para luego hacer hincapié

sobre las innovaciones recientes. Este capitulo igualmente propone el cuadro teórico de los sistemas

dinámicos monótonos y positivos.

La segunda parte de esta memoria presenta nuestra contribución y es organizada en cuatro capítulos.

En el Capítulo 3 introducimos nuevos conceptos relacionados con la construcción de observadores por

intervalos. En particular entregamos una nueva denición, introduciendo el concepto de acotador y pila

de observadores (Bernard and Gouzé, 2004). Corremos en paralelo diversas estimaciones por intervalos

y luego tomamos la envoltura interna, tomando ventaja de la garantía de las cotas superiores e inferiores

obtenidas. Un ejemplo no clásico de aplicación, considerando un sistema caótico es presentado.

La concepción de observadores por intervalos para sistemas dinámicos no monótonos es estudiada

en el Capítulo 4. Nos concentramos en la parte no lineal (e incierta) del sistema y, bajo ciertas

hipótesis técnicas, la escribimos como la diferencia de dos funciones crecientes. Esto nos permite

Resumen Extendido ix

obtener cotas de la parte no lineal y así establecer observadores cuyo error es caracterizado por una

dinámica monótona. Nuestra propuesta es ilustrada por dos observadores por intervalos diseñados

para un proceso de digestión anaeróbica.

En el Capítulo 5, desarrollamos un criterio de optimización (que puede ser aplicado a cualquier

observador por intervalos), con el objetivo de encontrar las ganancias que proveen las mejores esti-

maciones. En seguida, introducimos un nuevo esquema de estimación que mejora la estimación del

intervalo de condiciones iniciales, por medio de la construcción de observadores para el sistema en

tiempo inverso, efectuando estimaciones hacia atrás. Estas ideas son ilustradas considerando el obser-

vador a error acotado propuesto por Lemesle and Gouzé (2005). Entonces, a partir de este observador

proponemos un observador por intervalos con propiedades de convergencia mejoradas, pero que produce

una estimación caracterizada por intervalos discontinuos.

Finalmente, en el Capítulo 6 proponemos observadores por intervalos para la estimación de entradas

inciertas. Estos observadores son elaborados para los casos de entradas constantes y variables en

el tiempo. En el primer caso, mostramos que la estimación por intervalos de la entrada depende

directamente de la estimación efectuada para el estado del sistema (estimación simultanea de entradas

y estado). Para el segundo caso, las estimaciones por intervalos son obtenidas construyendo cotas para

el error de estimación de un observador de la entrada de tipo gran ganancia.

Contents

Acknowledgements i

Résumé Étendu iii

Resumen Extendido vii

Contents xi

Introduction 1

1 Modelling of biotechnological processes 5

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 The main types of bioreactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Working of a bioreactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Mass balance Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Properties of the reaction rate r(ξ) . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 The Anaerobic Digestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Models for anaerobic digestion processes . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.2 Uncertainty and model complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.3 Monitoring of wastewater treatment plants . . . . . . . . . . . . . . . . . . . . . 15

1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 A review on robust state estimation methods 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Denition of observability and observers . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Observability of nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Observability of linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Denition of an observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Classical state estimation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Luenberger observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.3 High Gain observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.4 LMI approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Estimation methods dedicated to uncertain systems . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Observers for linear systems with unknown inputs . . . . . . . . . . . . . . . . . 28

2.4.2 Guaranteed state estimation methods . . . . . . . . . . . . . . . . . . . . . . . . 30

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xii Contents

2.4.3 Monotone and positive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.4 Interval observers based on positive systems theory . . . . . . . . . . . . . . . . . 33

2.4.5 Estimation through Interval Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.6 Ellipsoidal methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4.7 Interval Kalman ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Bundle of observers 45

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Observers bundle: key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 Observers bundle and reinitialisation . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Application to chaotic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.1 Chua's chaotic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.2 Interval observers with perfect knowledge . . . . . . . . . . . . . . . . . . . . . . 49

3.3.3 Interval observer with uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Interval observers for nonmonotone systems 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Some properties of non-monotone mappings . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.1 Example: Haldane growth rate function . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Closed loop interval observer for nonmonotone systems . . . . . . . . . . . . . . . . . . 61

4.3.1 Application to the AMH1 model . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.2 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 An interval observer using measurements of the nonlinearity . . . . . . . . . . . . . . . . 66

4.4.1 Class of systems and example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.2 Observer Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.3 Application to an industrial anaerobic digestion process . . . . . . . . . . . . . . 69

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Optimality criterion and reverse time observers 71

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Optimality of interval observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.1 Denitions and Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.2 A bounded error interval observer . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.3 Computing the optimal gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.4 Biased output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Reverse time interval observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.1 Estimation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.2 Nearly optimal gain in reverse time . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.3 Biased output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.4 Application to the Agralco plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Contents xiii

6 Interval estimation of unknown inputs 89

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Interval estimation of a piecewise constant unknown input . . . . . . . . . . . . . . . . . 89

6.2.1 A closed loop interval observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2.2 Interval estimation of inuent concentrations . . . . . . . . . . . . . . . . . . . . 93

6.2.3 Application to the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3 Interval estimation of a time varying input . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3.1 Assumptions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3.2 High Gain observer for the input . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.3 An interval observer for the input . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3.4 Change of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3.5 An interval observer with perfect knowledge . . . . . . . . . . . . . . . . . . . . . 99

6.3.6 Interval observer with uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3.7 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Conclusions and perspectives 103

Publications 107

xiv Contents

Introduction

Bioprocesses are more complex than the classical industrial systems which have been constructed by

engineers. They distinguish from these systems since they involve living organisms. Bioprocesses

are more and more engineered around the world, and they are used in various elds, like wastewater

treatment, pharmaceutical production, food industry, etc.

Living systems are dicult to manage since organisms are susceptible to adapt (their properties

evolve along time), or can be highly sensitive to a small change in their environment. As a consequence,

the mathematical modelling of bioprocesses is delicate and often characterized by highly nonlinear and

uncertain dynamics, or by a high parameter variability.

It is therefore crucial to monitor such adapting biological processes. However, at the industrial level,

there is a lack of sensors to follow their dynamical evolution. Sensors are often expensive, require a high

expertise and a strong maintenance. Moreover, most of the measurements are generally obtained o

line, after laboratory analysis. Finally, especially for the bioprocesses dedicated to pollution removal,

it often occurs that inuent concentrations, which may vary along time, are not measured.

Automatic control aims at designing control strategies and monitoring schemes (state estimation).

Automatic control can play a central role in order to achieve substantial improvements in bioprocess

performance. It provides tools to operate processes in a protable manner and to ensure the quality

of the production. The prot target leads to optimize the operation of the system under dierent

performance criteria (maximizing product yield in pharmaceutical industry, bioproduction rate for

food, ...). An industrial production often tries to satisfy a quality standard in order to ensure a

constant quality of the product. Sometimes, safety is also required, for example when there exist

unstable operation points (which can lead, for instance, to the washout of biomass of a reactor).

However the classical tools of automatic control, developed for well known and identied models

are often not suitable for bioprocesses with uncertain dynamics. Special monitoring and control ap-

proaches must then be developed that are robust toward uncertainties and lack of measurements.

This thesis deals with the state estimation problem of biological systems. Nonlinear state esti-

mation techniques have attracted increasing attention in the last decades. From the already classical

Luenberger observer (Luenberger, 1966) and Kalman lter (Kalman and Bucy, 1961), state estimation

methods have evolved toward both nonlinear and robust methods, extending their application in a

wide range of real life systems. We focus on a deterministic and nonlinear framework, dealing with

guaranteed state estimation methods. For these methods, the estimation objective is to compute a

region of the state space where the unknown variables are sure to lie. Ellipsoidal methods (Kurzhanski

and Vályi, 1997) and estimation through interval analysis (Jaulin et al., 2001) are the most popular

examples of guaranteed state estimation methods.

1

2 Introduction

We base our approach on the concept of the socalled interval observers (Gouzé et al., 2000). This

estimation scheme consists in obtaining guaranteed upper and lower bounds of the unknown variables

of a dynamical system, considering that known bounds of the uncertain terms are available. Interval

observers are designed considering the theory of positive dynamic systems and dierential inequalities.

After an examination of the current research associated with these observers, it has been possible

to identify some limitations. In particular, the development of these observers have been restricted

to cooperative dynamic systems (Alcaraz-González, 2001; Hadj-Sadok, 1999). Cooperative systems

preserve the partial order relationship in the state space. They have nonnegative odiagonal entries

in their Jacobian matrix (Smith, 1995). This restricts the range of application to a reduced class

of biological systems. The convergence rate is another limitation of the current approaches. Several

interval observers for bioprocess models have been designed on the basis of asymptotic observers (Bastin

and Dochain, 1990). Therefore, the convergence rate is limited to the operation conditions of the system

(the dilution rate of the bioreactor must be high enough). The degraded performance with respect to

large uncertainties is a third strong limit of these observers. Sometimes these observers produce large

intervals which hardly provide useful information about the unknown state.

The main objective of this thesis is to propose new and original ideas on the design of these

estimators, overcoming the presented limitations.

The rst introductory part of this thesis is devoted to the modelling of bioprocesses and to a review

of state estimation methods. In Chapter 1, we describe the main working modes of bioreactors and

then we focus on anaerobic digestion processes. We present a widely used model in dimension two and

we derive a qualitative analysis of its behavior. A real life example of anaerobic digestion is presented

(which will be in the sequel our main case of application). Then, in Chapter 2, a stateoftheart

of the robust state estimation methods with applications in bioprocesses is presented. We present

wellknown and classical methods and then we focus on recent innovations. This chapter also gives a

theoretical framework on monotone and positive dierential systems, which are used for the statement

of our results.

The second part of this thesis presents our contribution, which is organized in four chapters. In

Chapter 3 new concepts about interval observers are introduced. In particular, we give a new meaning

to the denition of interval observer, by introducing the concept of framers and bundle of observers

(Bernard and Gouzé, 2004). We run in parallel various interval estimates and then we take the inner

envelope, taking advantage of the guaranteed upper and lower bounds. A nonclassical application to

a chaotic system is presented.

The design of interval observers for nonmonotone systems is considered in Chapter 4. We focus

on the nonlinear (and uncertain) part of the dynamics and then, under some technical assumptions,

we write it as the dierence of two monotone increasing functions. This allows us to obtain a bounding

of the nonlinear part and then formulate observers with monotone error dynamics. This approach is

illustrated with two observers designed for an anaerobic process.

In Chapter 5, we develop an optimality criterion (which can be applied to any interval observer) in

order to nd the gains that provide the best estimates. Then we introduce a new estimation scheme

considering interval observers running in reverse time. For the developments of these ideas we consider

a particular observer structure proposed in Lemesle and Gouzé (2005), and then we write an interval

observer with improved convergence properties.

Finally, interval observers dedicated to the estimation of uncertain inputs are introduced in Chapter

6. These observers are developed for two cases: piecewise constant input and time varying input. For

Introduction 3

the rst case, we show that the estimates on the input depend directly on the estimates performed on

the system state (joint state-input estimation). For the second case, we obtain interval estimates by

bounding the estimation error of a high gain input observer.

4 Introduction

Chapter 1

Modelling of biotechnological processes

1.1 Introduction

Biotechnological processes play an important role in the nowadays industry. A wide range of applica-

tions of biotechnology can be found e.g. in: food industry (beer, wine, cheese, etc.), pharmaceutical

(production of antibiotics, vaccines, hormones, etc.), energetic (fuel production) and depollution pro-

cesses (wastewater treatment) (Stephanopoulos et al., 1998). In contrast with other kinds of systems

that are perfectly described by physical laws -like mechanical or electrical systems- biotechnological

processes deal with living organisms. As a consequence, their mathematical models are uncertain and

are known to have a lower aptitude to accurately match experimental results.

Mathematical models for biotechnological systems can be constructed for various objectives. For

example to predict the behavior of the system, to estimate key parameters, to develop and test a

control strategy or to estimate variables through observers. The aim of this chapter is to describe bio-

processes and modelling issues using ordinary dierential equations, which are suitable for the design

and application of automatic control tools.

This chapter is organized as follows. After presenting the classes of bioreactors and their working

modes, we present a general model structure based on mass balance. Then, we focus on a particular

biotechnological process related to the treatment of wastewater, called anaerobic digestion. We present

a simple model of one biomass and one substrate describing this process, studying its main properties

(this simple model will be in the sequel our main example of study). A real life example of anaerobic

digestion plant is also presented.

1.2 The main types of bioreactors

A bioreactor is mainly a culture vessel of volume V where microorganisms grow. A pipe feeds the

vessel with an inuent medium (with ow rate Qin) and another one withdraws the culture medium

with a ow rate Qout.

Depending on the type of microorganisms growing in the bioreactor, they need a support to settle

or can be free in the liquid. They can resist to more or less intense shearing constraints which will

5

6 Modelling of biotechnological processes

implicate a specic steering system. These two main requirements determine the type of bioreactor.

Two classes can be identied (Bailey and Ollis, 1986):

stirred tank reactors (CSTR) in which the medium is homogeneous and each element of volume

will represent the concentrations in the whole fermenter.

bioreactors with non homogenous concentration along space. In particular the bioreactors for

microorganisms needing a support to grow (called bed) are in this category.

When the medium is homogeneous, it can be modeled by ordinary dierential equations (our case

of study). When a strong spatial distribution must be taken into account, a model based on partial

dierential equations is more appropriate.

1.2.1 Working of a bioreactor

Depending on the way the bioreactor is fed and withdrawn, three basic working modes can be identied

(see g. 1.1).

Batch mode

In batch mode the system has a constant volume, since no feeding or withdrawal are performed during

the fermentation (Qin = Qout = 0). An inoculum of micro-organisms is introduced at the initial time

with all the nutrients and substrates. The biomass or the nal product are recovered at the end of

the fermentation. The advantage of this approach is that it avoids the contaminations with other

bacteria which may occur in an open system. The drawback is the limited means of action to act on

the fermentation (pH, temperature, aeration...). This is the most used working modes in the industry.

Fedbatch mode

As for the batch mode the duration of a fedbatch is nite. But here the fermenter is fed and starts

from a volume V0 to reach a volume Vf at the end of the fermentation (Qin > 0 and Qout = 0). Thismode allows a better control of the growth and biotransformation process along the fermentation. The

fedbatch processes are often in closed loop. This operating mode is particularly used when the product

to be recovered necessitates to empty the bioreactor like e.g. for intracellular components.

The continuous mode (chemostat)

This is the most popular mode in the eld of wastewater treatment. The volume of the bioreactor is

constant since the inuent ow rate is equal to the euent ow rate (Qin = Qout > 0). This mode

provides the richest dynamics, and therefore presents the more latitude to optimise the process. It is

also often used in laboratories to study the physiology of a microorganism. The advantage is also that

it allows important productions in small size reactors.

A Sequencing Batch Reactor (SBR) can be considered as a combination of the 3 main working

modes. The idea is to recover the biomass before emptying the bioreactor. For this, the agitation is

stopped to let the biomass settle. In the same way, the SFBR (sequencing fedbatch reactor) is a SBR

with a stage of lling that follows a fedbatch mode.

1.3. Mass balance Modelling 7

Figure 1.1: Working modes of a CSTR. From Bernard and Queinnec (2002).

1.3 Mass balance Modelling

Bioprocesses deal with living organisms and therefore, are characterised by the biotransformations or

reactions that take place in the process. The reaction scheme of a biochemical process is a macroscopic

description of the set of biological and chemical reactions which represent the main mass transfer within

the reactor.

We are interested in reactions associated to growth of microorganisms. This can be represented by the

following formalism:

k1s1 + k2s2 + ... + kpsprk−→ k′1p1 + ... + k′qpq + xk (1.1)

where a set of substrates si are transformed into products pi and a biomass xk. rk denotes the

production rate of the biomass, and ki, k′i are respectively the stoichiometric coecient related to the

consumption or production of substrates and products.

A reaction scheme is a concise way to summarise at the macroscopic level a set of reactions that are

assumed to determine the process dynamics. The reaction scheme is therefore based on the available

phenomenological knowledge of the process, and represents the main mass and ow repartitions that

take place in the process.

In the sequel we will assume that the reaction scheme is composed of a set of p biological or chemical

reactions.

Mass balance modelling has been widely used for biological and chemical reactors. It is based on

the fundamental principle of conservation of mass, which states that matter can not disappear or be

created. This leads to mathematical models where physical and biological dynamics can be clearly

identied. In mathematical terms, we consider the following general model:

ξ = −DHξ +Kr(ξ) +Dξin −Q(ξ) (1.2)

In this model:

ξ = [ξ1, ξ2, . . . , ξn]T is the vector of all the process concentrations (biomasses, substrates and

products).


The matrix K contains the stoichiometric coecients, also known as yield coecients of the

model. We will assume that these coecients remain constant.

The vector r(ξ) = [r1(ξ), r2(ξ), . . . , rp(ξ)]T is a vector of reaction rates (or conversion rates)

representing the microbial activity for the p considered reactions.

The diagonal matrix H = diag(αi), with αi ∈ [0, 1] is the heterogeneity matrix and stands for

the fraction of biomasses, substrates or products in the liquid phase: αi = 0 corresponds to an

ideal xedbed reactor, and αi = 1 corresponds to an ideal CSTR, i = 1, . . . , n (Bernard et

al., 2001).

The inuent feeding concentration is represented by the positive vector ξin.

The dilution rate D is the ratio of the inuent ow rate and of the volume of the fermenter. The

gaseous exchange with the outside of the fermenter is represented by the term Q(ξ).

Note that model (1.2) diers slightly from the mass balance model introduced in Bastin and Dochain

(1990), because of matrix H.

1.3.1 Properties of the reaction rate r(ξ)

The reaction rate r(ξ) describes the biological dynamics inside the bioreactor. According to the reaction

scheme (1.1) and the general model (1.2), the reaction generates biomass and products from substrates,

then it is necessarily a nonnegative function. In the case where no biomass neither substrate are present

in the bioreactor, the reaction does not take place. These facts are summarized in the following

hypothesis:

Hypothesis 1.1 The reaction rate r(ξ) ∈ C1 fulls:

∀ξ ≥ 0, r(ξ) ≥ 0.

∀i ∈ 1, . . . , p, si = 0 ⇒ rk(ξ|si=0) = 0.

x = 0 ⇒ rk(ξ|xi=0) = 0.

1.4 The Anaerobic Digestion

Anaerobic digestion has been on the rise in recent years because of its applications in pollution removal.

The wide type of wastewater that can be treated with this technology has made anaerobic digestion

processes attractive in rural, municipal and industrial installations (Malina and Pohland, 1992).

The anaerobic digestion is a complex multistep process of biological fermentation, for the treatment

of wastewater. It takes place in an oxygen free environment reducing organic matter (waste) into

methane and carbonic gas.

The anaerobic community is made of several taxonomic groups, whose diversity can be very high. The

anaerobic community evolves dynamically and involves more than 140 bacterial species (Delbès et al.,

2001). Microorganisms can be classied in trophic groups, which play a well identied role in the

following four steps that characterise the process:

1.4. The Anaerobic Digestion 9

Hydrolysis: In this rst step, substrate is transformed into soluble molecules (proteins, lipids,

etc.) and then into soluble monomers (amino acids, etc.).

Acidogenesis: Soluble monomers are transformed into volatile fatty acids (VFA). In the models,

acidogenesis and hydrolysis are often grouped in one step, as they are produced by the same

bacterial populations.

Acetogenesis: Formation of carbonic gas, hydrogen and acetate from long chained VFA generated

in the acidogenesis.

Methanogenesis: Methane is produced in two ways from a mixture of carbonic gas and hydrogen

and from the decarboxylation of acetate.

Anaerobic digestion presents several advantages:

The energy required by anaerobic digestion processes is minimal compared to the energy required

by other aerobic processes. The methane obtained can be used to generate electricity, which can

be used on site (giving energetic autonomy to the plant), or even integrate a local electrical

distribution network, in the case of large scale facilities.

Low production of biological sludge, which can be 10 times lower than the quantity produced by

aerobic reactions, for the same quantity of substrate.

Despite these advantages, engineers have been reluctant to fully utilize the potential and eectiveness

of anaerobic technologies, mainly because of the high instability of the process. Anaerobic digestion

can be very sensitive to operational conditions. The accumulation of acid compounds can lead to the

Figure 1.2: Anaerobic digestion trophic groups and metabolic pathways. From Malina and Pohland(1992)


process acidication with complete loss of biomass (Fripiat et al., 1984). This point motivated the

development of algorithms for monitoring and control.

1.4.1 Models for anaerobic digestion processes

The ecosystem complexity is one of the main drawbacks when modelling anaerobic digestion processes.

Various models have been developed considering dierent number of trophic groups. For example, the

ADM1 model (which stands for Anaerobic Digestion Model 1) considers 7 distinct trophic groups and

11 substrates and products. This model covers a wide range of wastes (animal, vegetal, lipidic, etc.)

and can be used when it is desired to follow the evolution of a high number of intermediate products

in the fermentation process. The high quantity of variables and parameters to be tuned is one of the

main drawbacks of this model.

As it has been shown in Bernard et al. (2004), simpler models can be proposed, obtaining a good

qualitative representation for an available set of data.

A generic bioreactor model

This macroscopic model reduces the behavior to one biomass and one substrate in a bioreactor. The

state vector is then denoted by ξ = [x s]T . The organic matter s is degraded in one step into methane

and carbonic gas, according to the following pathway:

k1sr−→ x+ k4CO2 + k6CH4 (1.3)

The equations are the following: x = r(ξ)− αDx

s = D(sin − s)− k1r(ξ)(1.4)

where sin is the inuent substrate, k1, k4 and k6 are stoichiometric coecients. The methane ow rate

from the liquid phase to the gaseous phase is given by the quantity qM = k6r(ξ).This model corresponds to a slightly modied version of the classical Monod model (Monod, 1942).

Equation (1.4) has the form of system (1.2) with:

H =

[α 00 1

], K =

[1k1

], ξin =

[0sin

]and Q =

[00

](1.5)

The reaction rate r(ξ) is often expressed as the product of a growth rate µ(s) and biomass: r(ξ) = µ(s)x,being µ(s) the growth rate function of the biomass x. The following hypothesis is assumed for the

growth rate function.

Hypothesis 1.2 In the sequel we will consider that µ(s) is a C1(R) function such that:

µ(0) = 0 and µ(s) ≥ 0.

∃ a ∈ R+ : µ(s) ≤ a, ∀s ≥ 0.

Property 1.1 For any positive initial condition, trajectories of system (1.4) remain positive and

bounded for any positive time.


Proof. Positivity of the system is veried considering that Hypothesis 1.1 (or equivalently Hypothesis

1.2) is fullled. Indeed, it is straightforward to verify that:

x|x=0 = 0s|s=0 = Dsin ≥ 0

(1.6)

then, x = 0 remains invariant and s = 0 repulsive, which implies that the variables remain positive

considering their positive initialization.

In order to prove the boundedness of system (1.4), consider the variable z = s + k1x. It is clear

from the computation of its dynamics that:

D(sin − z) ≤ z ≤ D(sin − αz), ∀α ∈]0, 1] (1.7)

Since x and s are positive variables, from (1.7) it holds that:

min (z(0), sin) ≤ z ≤ max (z(0), sin/α)

Moreover, from the substrate dynamics it holds that:

s ≤ D(sin − s), ∀α ∈]0, 1] (1.8)

which implies that: 0 ≤ s ≤ max (s(0), sin)

0 ≤ x ≤ max (z(0), sin/α)k1

2

For a more detailed study of system (1.4), a specic structure for the growth rate is required. Several

models have been introduced for the function µ(s) (see Bastin and Dochain (1990), for an exhaustive

list). In particular, we will focus on the Monod and Haldane kinetics.

Monod kinetics

Also called the Michaelis-Menten law, it was originally introduced in Monod (1942). It could be

considered as the most known and used growth rate model. Its analytical expression is given by:

µM (s) = µms

s+ ks(1.9)

This is a monotone function where µm corresponds to the maximum growth rate and ks is a saturation

positive constant. When the substrate is equal to ks, the growth rate value reaches the half of the

maximum value: µM (ks) = µm/2.

Haldane kinetics

A more complex non-monotone growth rate model is dened by the Haldane equation (Andrews, 1968):

µH(s) = µhs

s+ ks + s2/ki(1.10)


where ks and ki are respectively the saturation and inhibition constants. The maximum growth rate

value is reached at µMH = µH(

√kski). Haldane function presents a positive derivative for substrate

values lower than µMH .For higher substrate values its derivative is negative. This growth rate function

introduces two non trivial equilibrium points in the bioreactor model, as it is shown hereafter. From

equation (1.10), it is easy to see that, when ki → ∞, there is no inhibitory eect and the Haldane

model is reduced to a Monod model.

Equation (1.4) with a Haldane kinetics is called Anaerobic Digestion Model Haldane 1 (AMH1).

Equilibrium points and stability analysis

For any bacterial growth rate function µ(s), system (1.4) has the following equilibrium points:

ξ?1 =

[0sin

]and ξ?

2 =

[x?

s?

](1.11)

where µ(s?) = αD and k1µ(s?)x? = D(sin−s?). A local stability analysis of these points can be easily

carried out, looking at the sign of the determinant and trace of the Jacobian matrix J (x, s) of system(1.4), which is written as:

J (x, s) =

[µ(s)− αD xµ′(s)−k1µ(s) −D − k1xµ

′(s)

](1.12)

where µ′(s) =∂µ

∂s(s).

Stability analysis of point ξ?1 . J (x, s) evaluated in ξ?

1 leads to:

J (ξ?1) =

[µ(sin)− αD 0−k1µ(sin) −D

](1.13)

and directly det(J (ξ?1)) = αD2−Dµ(sin) and tr(J (ξ?

1)) = µ(sin)−D(1+α). A sucient and necessary

local stability condition of ξ?1 is then given by the inequality:

αD > µ(sin) (1.14)

This last condition can be interpreted as a washout of the biomass in the bioreactor (extinction of

living organism). As the equilibrium point ξ?1 does not depend on the growth rate function µ(s) the

stability of this point is always conditioned by inequality (1.14).

Stability analysis of point ξ?2 . This corresponds to an inner point of operation of the bioreactor,

that is, the equilibrium point veries: 0 < s?2 < sin and x?

2 > 0 . J (x, s) evaluated in ξ?2 leads to:

J (ξ?2) =

[0 x?µ′(s?)

−αk1D −D − k1x?µ′(s?)

](1.15)

and then det(J (ξ?2)) = αk1Dx

?µ′(s?) and tr(J (ξ?2)) = −D − k1x

?µ′(s?). Now, the stability analysis

depends on the growth rate function µ(s). A specic analysis for Monod and Haldane growth rate

functions must now be carried out.


Figure 1.3: Monod and Haldane equilibrium points.

• Monod Model.

The existence of the inner equilibrium point ξ?2 is guaranteed for a xed D, such that αD ≤ µ(sin).

The Monod growth rate function given by equation (1.9) fullls µ′M (·) > 0, then det(J (ξ?2)) > 0 and

tr(J (ξ?2)) < 0. For this case the equilibrium point ξ?

2 is then locally stable. Global stability for this

point can also be shown. The reader is referred to Hess and Bernard (2007); Smith and Waltman

(1995) to see more details.

(1.16)

• Haldane Model.

Let us focus on the case where√kiks < sin (otherwise µ(s) is monotone increasing in the interval

]0, sin]) and also for a xed D, such that αD ∈ [µ(sin), µMH ]. The dynamics of the system under the

equilibrium point satises µH(s?) = αD. From the Haldane equation (1.10), it follows that:

µhs?

s? + ks + s?2/ki= αD ⇒ s? + ki(1− µh/(αD))s? + kski = 0 (1.17)

It is possible to obtain the equilibrium s? as the solution of the previous quadratic equation:

s?± =

ki(µh/(αD)− 1)±√k2

i (1− µh/(αD))2 − 4kski

2(1.18)

then the Haldane model has two inner equilibrium points ξ?− and ξ?

+. Consider s?− < s?

+. As

αD ≤ µMH , then straightforwardly µ′H(s?

−) > 0 and µ′H(s?+) < 0.

With this last result, it holds that det(J (ξ?2−)) = αk1Dx

?−µ

′H(s?

−) > 0 and tr(J(ξ?2−)) = −D −

k1x?−µ

′H(s?

−) < 0 then ξ?2− is locally stable.

For the second point, we have det(J (ξ?2+)) = αk1Dx

?+µ

′H(s?

+) < 0 which implies that one of the

eigenvalues of the Jacobian matrix J (ξ?2+) is positive and then ξ?

2+ is unstable. This implies the

partition of the state space into two attraction basins: one for the stable inner equilibrium and one for

the washout.

Fig. 1.3 shows a phase portrait for the Monod and Haldane models. More details about the

behavior of the Haldane model can be found in Hess and Bernard (2007).


The AM2 model

This model (Bernard et al., 2001), considers a two step degradation of the organic matter, representing

the acidogenesis and methanogenesis steps of an anaerobic digester. In the acidogenesis step a substrate

s1 is degraded by acidogenic bacteria x1 (with a rate r1), generating volatile fatty acids (VFA, s2) and

CO2. In the methanogenesis step, volatile fatty acids are degraded by methanogenic bacteria x2 (with

a rate r2), into CH4 and CO2. The reaction schemes are the following:

acidogenesis: k1s1r1−→ x1 + k2s2 + k4CO2

methanogenesis: k3s2r2−→ x2 + k5CO2 + k6CH4

(1.19)

The state vector is then ξ = [x1 s1 x2 s2]T , and the associated model is described by the following

system: x1 = r1(ξ)− αDx1

s1 = D(s1in − s1)− k1r1(ξ)x2 = r2(ξ)− αDx2

s2 = D(s2in − s2) + k2r1(ξ)− k3r2(ξ)

(1.20)

where k2 and k3 are stoichiometric coecients. The methane ow rate is given by the quantity

qM = k6r2(ξ). The reactions rates can be written as r1(x1, s1) = x1µ1(s1) and r2(x2, s2) = x2µ2(s2),where µ1(s1) and µ2(s2) are respectively Monod and Haldane equations (Bernard et al., 2001).

Equilibrium points

The acidogenic step, described by variables x1 and s1, corresponds to a simple Monod bioreactor

model. Its equilibrium is then given by equation (1.11). The methanogenic step, described by

variables x2 and s2 should be understood as a simple Haldane bioreactor with an inuent sin =s2in +k2µ1(s1)x1/D. Considering that the acidogenic step has reached its equilibrium, the equilibrium

points of the methanogenic step can be expressed by:

x?2± =

1αk3

(s2in − s?

2± +k2

k1(s1in − s?

1))

(1.21)

where the values s?2± are obtained from the quadratic equation (1.17) and s?

1 corresponds to the inner

equilibrium point of the substrate of the acidogenic step. The stability analysis of the equilibrium

points can be extended from the analysis carried out for the simple bioreactor model (1.4).

More details about model (1.20), considering qualitative properties and calibration using real data

values from an anaerobic digester can be found in Bernard et al. (2001).

1.4.2 Uncertainty and model complexity

Real life processes are rarely accurately represented by any mathematical model. Above all in modelling

biological systems, badly modelled dynamics and parametric uncertainty have to be taken into account.

In the case of wastewater treatment plants, it can also occur that inuent concentrations are not

accurately known.

In Hadj-Sadok (1999), the following classication of the uncertainties of a bioprocess is given:


Structural uncertainties: They are related to the lack of bio(chemical) knowledge on the con-

sidered process. A typical example is the uncertainty related to the growth rate function of

microorganism in a reactor.

Environmental uncertainties: They are determined by the interaction of the bioprocess and

exogenous conditions.

Measurements uncertainties: They correspond to the errors generated by the measurement system

(hardware) or procedure used to monitor variables.

Another important issue related to modelling is the level of details desired. Models that are tar-

geting a high detail representation of a process will straightly increase in number of variables and

parameters. Parameter identication, and automatic control design or observers development may

become cumbersome for nonlinear models of high dimension. Table 1.4.2 shows some features of the

cited model related to anaerobic digestion (Bernard et al., 2004).

Table 1.1: Features of dierent models of anaerobic digestion.

AMH1 AM2 ADM1state variables 2 6 26biomasses 1 2 7

number of reactions 1 2 19parameters 5 13 86outputs 3 18 32

1.4.3 Monitoring of wastewater treatment plants

The state of the art related to the instrumentation of wastewater treatment plants is rapidly changing

(Olsson, 2006). Monitoring this kind of processes is a key point toward their optimization, and a limited

number of measurements are available. Sensors for wastewater treatment plants can be classied in

two basic types (Vanrolleghem and Lee, 2003): reliable and low maintenance sensors, and advanced,

high maintenance sensors. The complexity involved by the measurement of a key variable in anaerobic

digestion can lead to very expensive sensors. This is in particular the case of sensors for suspended

solids and chemical oxygen demand. In contrast, measurements in the gaseous phase at the digester

exhaust appear more reliable and less expensive, but they only provide indirect and global information.

AGRALCO industrial wastewater treatment plant

The example presented is a real industrial anaerobic digester 1. Available data from this plant have

been used to test the contributions of this thesis.

This plant is owned by the Agralco company located in Stella, Spain. It corresponds to a real

industrial anaerobic digestion wastewater treatment plant processing raw industrial vinasses with a

volume of 2000m3. Available data from the plant are:

Dilution rate D [1/day].

1part of the TELEMAC project (European commission, Information Society Technologies program, Key action ISystems & Services for the Citizen, contract TELEMAC number IST-2000- 28256)


0 10 20 30 40 50 60 700.02

0.04

0.06

0.08

time [days]

dilu

tion

[day

s−1 ]

0 10 20 30 40 50 60 70

47

37

27

time [days]

Infl

uent

CO

D [

g/l]

0 10 20 30 40 50 60 7020

40

60

80

100

time [days]

Qch

4 [l

/h]

Figure 1.4: Example of data from AGRALCO plant: Dilution rate, inuent COD and methane owrate.

0 10 20 30 40 50 60 70

4

6

8

time [days]

CO

D [

g/l]

0 10 20 30 40 50 60 70

20

30

40

time [days]

TSS

[g/

l]

Figure 1.5: Experimental data from AGRALCO plant: COD and Total Suspended Solids.

Methane ow rate Qch4 [1/h].

Oline samples of inuent Chemical Oxygen Demand CODinf [g/l], which corresponds to the

inuent substrate sin.

Chemical Oxygen Demand COD [g/l], which corresponds to the substrate s.

The COD is an index of organic matter concentration and corresponds to the oxygen quantity

required for the chemical degradation of the residuals contained in the wastewater.

Oline samples of Total Suspended Solids TSS [g/l].

This corresponds to an index of the total matter suspended in the wastewater obtained by

ltration or centrifugation. Due to the high solubility of the substrate treated in this plant, this

measurement can be compared to the total biomass x within the reactor.

These data are shown in g. 1.4 and 1.5. Considering the available data, model (1.4) has been

adjusted, assuming a Haldane model for the biological dynamics. It is worth to remark that parameter

µh of the Haldane equation (1.10) is not accurately known. Parameters values are summarized in Table

1.2.

More details about the Agralco plant can be found in Bernard et al. (2005).

1.5 Conclusions

The basis of biotechnological processes modelling has been presented in this chapter. A special emphasis

was made to present the anaerobic digestion process, that will illustrate our developments along this

thesis. Several models have been proposed for this process, with increasing complexity when a more

1.5. Conclusions 17

Table 1.2: System parameters.

parameter meaning value unitsµh maximal growth rate [0.72,1.08] day−1

ks saturation constant 40 g/lki inhibition constant 50 g/lk1 biomass yield conversion 19.5 -k6 methane yield conversion 25 mmol/gα heterogeneity constant 1 -

detailed representation of the system is required. There exists a trade o between model complexity

and the nal objective for which a model is used.

When an analytical approach must be deployed to implement some algorithms (automatic con-

trollers, observers, diagnosis) a simple model that can be mathematically handled is preferred. How-

ever, if the objective is to forecast the process evolution with a simulator, a complex model closer to

the real system is more relevant.

Finally, there exists a lack of sensors in industrial bioprocesses, specially in the case of wastewater

treatment plants. This is one of the main constraints when optimizing this kind of plants, justifying

the development of robust state estimation methods.

Chapter 2

A review on robust state estimation

methods

2.1 Introduction

The objective of this chapter is to review the robust state estimation methods with a special empha-

sis on developments related to biotechnological processes. The methods presented here cover a wide

range of techniques. We rst present the techniques that do not take uncertainty into account. Then

we present the estimation methods for uncertain systems. We focus on continuous time models in a

deterministic framework.

The challenge of the research associated with state estimation is to adapt to industrial constraints:

uncertain dynamics, unknown inputs, and noise in the measurements. There does not exist any

standard procedure when designing an observer: it rst depends on the nature of available models

(stochastic, deterministic), representation in time (discrete, continuous) and on how uncertainties are

considered in the design. Some observers are more suitable than others, in terms of robustness, and

the price to pay is often a less rapid convergence.

The observer design and analysis followed an important evolution: the classical methods of ob-

server design (asymptotic observers, highgain observers, ...) have been combined with sophisticated

techniques (like interval analysis, linear matrix inequalities, etc.) giving place to a second generation of

observers, extending classical observers to a robust framework. Examples of this trend are illustrated

in this chapter.

This chapter is organized as follows. Section 2.2 gives a general introduction on observability of

linear and nonlinear systems, recalling concepts that are essential for the design of observers. Then we

present several estimation methods, which are grouped in two families: classical observers, presented

in section 2.3, and estimation methods for uncertain systems, presented in section 2.4

19

20 A review on robust state estimation methods

2.2 Denition of observability and observers

In this section we recall some concepts related with observability of linear and nonlinear dynamic

systems. We consider the dynamic system:x(t) = f(x(t), u(t)); x(0) = x0

y(t) = h(x(t))(2.1)

where x ∈ Ω ⊂ Rn is the state vector, f : Rn × Rm → Rn and h : Rn × Rm → Rp are nonlinear

functions, y ∈ Rp is the vector of measured variables and u ∈ Rm is the system control input.

Observability states whether the unknown variables can be reconstructed from the measured ones.

2.2.1 Observability of nonlinear systems

Observability properties of nonlinear systems have been studied by several authors (Gauthier and

Kupka, 2001; Sontag, 1984). In this section we consider observability as a result of the denition of

discernability. This concept has been introduced by Hermann and Krener (1977), where the following

denitions are given:

Denition 2.1 Two states x0 and x′0 are indiscernible if for any input u the outputs y(t, x0) and

y(t, x′0) are equal for any t ≥ 0.

Denition 2.2 A system is observable if it does not have any distinct couple of initial state x0, x′0

that are indiscernible. In other words, the mapping x0 7→ y(t)t≥0 is injective.

These denitions establish a link between the system output and the system initialisation: if the

system is observable, then the initial condition can be uniquely determined from the system output.

It is worth noting that the observability of a nonlinear system usually depends on its inputs. The

following denitions are then necessary:

Denition 2.3 An input is said to be universal if it can distinguish any couple of initial conditions.

Denition 2.4 A non universal input is said to be singular.

A more sophisticated condition to test observability can be proposed, using the denition of obser-

vation space (Hermann and Krener, 1977; Nijmeijer and Van Der Schaft, 1990).

Denition 2.5 The observation space of system (2.1) is the space containing the measurements y(t)and its successive derivatives with respect to the vector eld f .

L = spanh(x), Lfh(x), . . . , Lkfh(x), ... (2.2)

where Lfh(x) = DhDxf(x) is the Lie derivative of h with respect to the eld f (with L0

fh(x) = h(x) andLk

fh(x) = Lf (Lk−1f h(x)), for k ≥ 1)

Denition 2.6 System (2.1) is said to be observable if its observation space L separates the points of

Ω.

This denition leads straightforwardly to the criterion of r-observability (Gauthier and Kupka, 2001).

2.2. Denition of observability and observers 21

Denition 2.7 System (2.1) is said to be r-observable if its r-observability map, with r ≥ n:

qr : x 7→ [h(x), Lfh(x), . . . , Lr−1f h(x)]T (2.3)

is globally injective.

Many nonlinear observer design is carried out assuming r = n. In Gauthier et al. (1991) and Vargas

et al. (2002), the case when r > n has been analyzed.

2.2.2 Observability of linear systems

Consider the linear system expressed by the equations:x(t) = Ax(t) +Bu(t)y(t) = Cx(t)

(2.4)

where A ∈ Rn×n, B ∈ Rn×m and C ∈ Rp×n.

Denition 2.5 leads directly to the well known observability matrix O for linear systems:

O =

C

CA...

CAn−1

(2.5)

Then the linear system (2.4) is said to be observable, or equivalently, the pair A,C is said to be

observable if rank(O) = n.

If the system is not observable, it is possible to consider a canonical decomposition of the system in

two parts: an observable part and an unobservable one. Then, the system is said to be detectable if

and only if its unobservable part is stable. In this case, the estimation convergence can not be tuned as

desired, and the convergence time will be given by the time constants of the dynamics (Sontag, 1990).

2.2.3 Denition of an observer

An observer is an auxiliary system whose objective is to reconstruct the unmeasured variables x of a

dynamic system, using the available information such as the mathematical model (2.1) and the online

measured outputs y.

A classical denition is the following (Luenberger, 1979):

Denition 2.8 An observer of system (2.1) is a dynamic system of the form:˙z = g(z(t), u(t), y(t)); z(0) = z0

x = h(z(t), u(t), y(t))(2.6)

such that

limt→∞

‖x(t)− x(t)‖ = 0 (2.7)

this principle is illustrated in g. 2.1. Note that in the previous denition the observer converges

asymptotically towards the state.

A more specic and classical observer structure is given by the following denition:


Denition 2.9 A closed loop observer is an observer with the form:˙x(t) = f(x(t), u(t)) + k (z(t), h(x(t))− y(t)) , x(0) = x0

z(t) = g(z(t), u(t), y(t))(2.8)

which is characterized by the comparison between the observed and measured variables and a tunable

gain k, with k(z(t), 0) = 0, which is used to obtain a desired convergence response of the observer.

Denition 2.10 System (2.8) is an exponential observer of system (2.1), if for any λ > 0 we can

adjust the gain k(t), such that:

‖x(t)− x(t)‖ ≤ e−λt‖x(0)− x(0)‖, ∀t ≥ 0 (2.9)

Considering an uncertain framework, robustness and exponential convergence properties of estima-

tion methods may not hold. Denition 2.9 can be revisited into a practical convergence in a bounded

error context. We adopt the denitions given in Rapaport and Gouzé (2003):

Denition 2.11 System (2.8) is a weak practical observer of system (2.1), if the gain k(t) can be

adjusted such that:

∃ε > 0, ∃T > 0 : ∀t > T, ‖x(t)− x(t)‖ ≤ ε (2.10)

Denition 2.12 System (2.8) is a strong practical observer of system (2.1), if the gain k(t) can be

adjusted such that:

∀ε > 0, ∃λ(ε) > 0 : ‖x(t)− x(t)‖ ≤ e−λ(ε)(‖x(0)− x(0)‖ − ε) + ε (2.11)

with λ(ε) → +∞ when ε→ 0.

The following denition gives us another notion of convergence towards a bounded error. In this

case, the observer copes with an uncertain dynamic of the system, and then a loose approximation f

is used in the observer design (Lemesle and Gouzé, 2005).

Figure 2.1: State observer principle.

2.3. Classical state estimation methods 23

Denition 2.13 A bounded error observer of system (2.1) is a dynamic system˙z = f(z, u, y)ξ = g(z, u, y)

(2.12)

such that:

limt→∞

||x(t)− x(t)|| ≤ r (2.13)

where r is positive real constant.

2.3 Classical state estimation methods

We recall the basis of some classical approaches for continuous time linear systems such as the Luen-

berger observer and the well known Kalman Filter, in a stochastic framework. We present then the

High Gain observer for nonlinear systems.

2.3.1 Luenberger observer

One of the pioneer state estimation approaches is the one introduced in (Luenberger, 1966). Let us

consider the linear system:

x(t) = Ax(t) +Bu(t), x(0) = x0

y(t) = Cx(t)(2.14)

where x ∈ Rn. If the system (2.14) is observable, then there exists a matrix K ∈ Rn×p such that

A+KC is a stable matrix. The following system:

˙x = Ax(t) +Bu(t) +K(Cx(t)− y(t)), x(0) = x0 (2.15)

is then a state observer of system (2.14), with exponential convergence rate. Indeed, the error dynamics

are governed by the system e = (A +KC)e, with e = x − x. Therefore, the convergence rate can be

adjusted solving the eigenvalues placement problem (Luenberger, 1979) associated to matrix A+KC

by means of the gain K.

2.3.2 Kalman Filtering

The Kalman lter (Kalman and Bucy, 1961; Anderson and Moore, 1990) is very famous in the frame-

work of linear systems; it can be seen as a Luenberger observer with a time varying gain; this allows

to minimise the error estimate variance.

Kalman lter for linear systems

A stochastic representation can be given by the system:x(t) = Ax(t) +Bu(t) + w(t), x(0) = x0

y(t) = Cx(t) + v(t)(2.16)


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 system is observable and that the initial

distribution is Gaussian, such that:

E[x0] = x0, E[(x0 − x0)(x0 − x0)T ] = P0 (2.17)

where E represents the expected value and P0 is the initial covariance matrix of the error. The lter

is written in four steps (Gelb et al., 1992):

1. Initialisation:

E[x0] = x0, E[(x0 − x0)(x0 − x0)T ] = P0 (2.18)

2. Estimation of the state vector:

˙x(t) = A x(t) +Bu(t) +K(t) [y(t)− C x(t)], x(0) = x0 (2.19)

3. Error covariance propagation (Riccati equation):

P (t) = A P (t) + P (t) AT − P (t)CTR(t)−1CP (t) +Q(t) (2.20)

4. Gain computation:

K(t) = P (t) CTR(t)−1 (2.21)

Some points can be emphasised:

This lter can still be applied when matrices A and C depend on time (the observability must

nevertheless be proven).

The estimation of the positive denite matrices R,Q,P0 is often very delicate, especially when

the noise properties are not known.

This observer consists in minimising 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 exponentialforgetting factor allows to consider the cases where Q = 0.

Extended Kalman lter

In order to design an observer for nonlinear systems, the idea consists in considering the linearized

tangent around its estimated trajectory. Then the problem is equivalent to build a Kalman lter for

non stationary system. Let us consider the systemx(t) = f(x(t)) + w(t), x(0) = x0

y(t) = h(x(t)) + v(t)(2.22)

and the observer is designed as above, with a change in the second step:

2.3. Classical state estimation methods 25

2. Estimation of the state vector:

˙x(t) = f(x(t)) +K(t)(y(t)− h(x(t))), x(0) = x0 (2.23)

and using the matrices of the tangent linearised:

A(t) =∂f(x(t))∂x(t)

∣∣∣∣∣x(t)=x(t)

C(t) =∂h(x(t))∂x(t)

∣∣∣∣∣x(t)=x(t)

(2.24)

This extended lter is often used, even if only few theoretical results guarantee its convergence.

The Kalman lter is very popular and has been subject to increasing theoretical and applied research.

See for example Zhao and Kümmel (1995), Lukasse et al. (1999) and Bogaerts (1999) for applications

of this technique in bioprocesses.

Note that an analogous version of the Kalman lter for discrete time systems has been proposed, with

similar characteristics. The reader is invited to refer to (Chen et al., 1995) for more details.

2.3.3 High Gain observers

This approach has been proposed by Gauthier et al. (1992). Let us consider the dynamic system:x = f(x, u), x(0) = x0

y = h(x)(2.25)

where x ∈ Rn and u ∈ Rp is a system input.

Consider the following transformation:

φ : x 7→ z = [h(x), Lfh(x), . . . , Ln−1f h(x)] (2.26)

where Lfh(x) is the Lie derivative of h with respect to the eld f .

If φ is a global dieomorphism, transformation (2.26) allows to write system (2.25) in the canonic

form (Gauthier and Kupka, 2001): z = Az + ϕ(z, u)y = Cz

(2.27)

with A =

0 1 0...

. . .... 10 . . . . . . 0

, C = [1, 0, . . . , 0] and ϕi(z, u) = ϕi(z1, . . . , zi, u).

Hypothesis 2.1 The following hypotheses are considered:

System (2.25) is observable for any input u ∈ Rp

The nonlinear terms ϕi are globally Lipschitz functions with respect to z.

φ is a global dieomorphism.


Under the previous conditions, the High Gain observer can be written as follows.

Proposition 2.1 The following dynamic system is an exponential observer of system (2.27):

˙z = Az + ϕ(z, u)− S−1θ CT (Cz − y) (2.28)

where θ > 0 is a constant which must be chosen suciently large and Sθ is a symmetric positive denite

matrix, solution of the algebraic Lyapunov equation:

θSθ +ATSθ + SθA = CTC (2.29)

Proof: See Gauthier et al. (1992). 2

A well known drawback of this observer is the socalled peaking phenomena, in which the quantity

e(t) = z(t) − z(t) exhibits a high overshoot at the beginning of the estimates. In El Yaagoubi et al.

(2004), some modications to the original high gain observer are introduced in order to reduce this

drawback.

Despite the sensitivity of this method to noise in the measurements (which are amplied by the

high values of the gain), high gain observers have been successfully applied in the estimation of the

state of bioprocesses (Gauthier et al., 1992) and also in the online estimation of reaction rates (Farza et

al., 1998). In Bernard et al. (1998), high gain observers are used in the validation of phytoplanktonic

growth models using real data values. In Nadri (2001) the application of high gain observers to

wastewater treatment plants is carried out. In Deza et al. (1992) an observer with a structure similar

to the Kalman lter is proposed, from the basis of a High Gain lter.

2.3.4 LMI approaches

Linear Matrix Inequalities have become a powerful tool for the stability analysis of dynamic systems

(Boyd et al., 1994) and for the design of observers (Arcak and Kokotovi¢, 2001).

In order to illustrate these approaches, we consider here an extension of Luenberger observers (Ze-

mouche, 2007), for systems with sloperestricted nonlinearities.

Consider the system: x(t) = Ax(t) +Gγ(Hx(t)) + %(y(t), u(t))y(t) = Cx(t)

(2.30)

where % : Rp × Rm → Rn is a known function. The nonlinear term γ is a nondecreasing real function.

A bound on the slope of this nonlinearity is known:

0 ≤ γ(v)− γ(w)v − w

≤ b, ∀ v, w ∈ R (2.31)

The proposed observer has the following structure:

˙x = Ax+ L(Cx− y) +Gγ(Hx+K(Cx− y)) + %(y, u) (2.32)

2.4. Estimation methods dedicated to uncertain systems 27

whose error dynamics e = x− x satisfy the equation:e = (A+ LC)e+Gδ(t)ηη = (H +KC)e

(2.33)

where δ(t) is a timevarying gain taking values in the interval [0, b], according to the condition (2.31).

The objective now is to nd K and L to render system (2.33) asymptotically stable. For this, a

quadratic Lyapunov function V = eTPe is used, giving place to the following result.

Theorem 2.1 If a matrix P = PT > 0 and a constant ν can be found such that:[(A+ LC)TP + P (A+ LC) + νI PG+ (H +KC)T

GTP + (H +KC) −2/b

]≤ 0 (2.34)

then the observer error system (2.33) converges exponentially to 0.

Proof: See Arcak and Kokotovi¢ (2001). 2

We invite the reader to consult Boyd et al. (1994) for linear matrix inequalities theory.

Application of linear matrix inequalities can be found in Osorio and Moreno (2006), for the design

of observers for nonlinear systems based on dissipativity. In Zemouche (2007), LMI are used to the

formulation of observers covering discrete time and delayed systems, with applications to observer

based synchronization.

2.4 Estimation methods dedicated to uncertain systems

The role played by uncertainty in automatic control issues has become a central research subject

(Herzallah, 2005). Observability when considering uncertainties in the system is a delicate subject and

there exist few results in a general context. Most of the studies are dedicated to particular class of

systems (Külmiz and Göknar, 1996; Darouach et al., 2003) or nding conditions for which an observer

candidate can deal with the uncertain terms (Floquet and Barbot, 2006).

Several estimation methods can be found for uncertain systems. Most of them rely on classical

approaches like the robust Kalman ltering (Xie et al., 1994), H∞ or H2 ltering methods (Geromel

et al., 2000). Also in a stochastic context, the socalled Particle Filters (Salmon et al., 1993; Goaux

and Vande Wouwer, 2005) is a recent contribution.

We will focus on a deterministic framework assuming that the only information available about un-

certainties is that they are bounded by upper and lower known values (these values can be functions

of time), giving place to the socalled guaranteed estimation techniques. In this eld, we will focus

on interval observers based on positive dierential systems (which are the main subject of this thesis),

guaranteed state estimation using interval analysis (Jaulin et al., 2001) and approximation of reachable

trajectories by means of convex sets, in particular using ellipsoidal methods (Kurzhanski and Vályi,

1997). All these methods provide a guaranteed outer envelope of the unknown state and they can be

even combined. Finally, we present an improvement of the Kalman lter by integrating an interval

analysis.

We begin this section presenting some particular results about observers for systems with unknown

inputs and then we focus on the guaranteed state estimation methods.


2.4.1 Observers for linear systems with unknown inputs

Results about observability of linear systems with unknown inputs are the basis of the design of

unknown input observers (Kudva et al., 1980; O'Reilly, 1983; Darouach, 1994). In Darouach et al.

(1994), necessary and sucient conditions for the existence of an observer are given. We briey recall

these results herein. Consider the linear system:x(t) = Ax(t) +Bu(t) +Dv(t)y(t) = Cx(t)

(2.35)

where u(t) ∈ Rm and v(t) ∈ Rd are the known and unknown inputs, respectively. The proposed

observer is written as follows: z(t) = Nz(t) + Ly(t) +Gu(t)x(t) = z(t)− Ey(t)

(2.36)

where z, x ∈ Rn and N,L,G and E are matrices of appropriate dimensions, to be determined such

that the error e = x − x converges asymptotically to zero. From (2.35) and (2.36) the error dynamic

is:

e(t) = Ne(t) + (NP + LC − PA)x(t) + (G− PB)u(t)− PDv(t) (2.37)

with P = In + EC.

The following theorem states necessary and sucient conditions for observability of linear systems with

unknown inputs.

Theorem 2.2 The observer (2.36) for the system (2.35) exists if and only if:

1. rank(CD) = d

2. rank

[sIn − PA

C

]= n

Proof: See Darouach et al. (1994). 2

The rst condition of this theorem is necessary for the existence of the observer (2.36), assuring

that the uncertain input v(t) can be eliminated from the error equation (2.37). The second condition

stands for the observer stability. It guarantees that a gain K = L+NE can be adjusted such that the

observer is asymptotically stable.

Asymptotic Observers

Asymptotic observers for bioprocess models (Bastin and Dochain, 1990) are a particular application

of unknown input obervers. Consider a biotechnological process modeled by the mass balance model

(1.2) with the uncertain reaction rate vector r(ξ) (considered as an unknown input). We assume that

the set of available measurements y can be decomposed into y = [y1 y2]T , 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 rst components of ξ.

y2 represents the measured gaseous ow rates: y2 = Q(ξ).


Let us rewrite system (1.2) after splitting the measured part (ξ1 = y1) from the other part of the state

(ξ2).ξ1 = K1r(ξ)−DH1ξ1 +Dξ1in −Q1(ξ)ξ2 = K2r(ξ)−DH2ξ2 +Dξ2in −Q2(ξ)

(2.38)

Matrices K1, K2, H1 and H2, and vectors ξ1in, ξ2in, Q1 and Q2 are such that

K =

[K1

K2

], H =

[H1 0

0 H2

], ξin =

[ξ1inξ2in

], Q =

[Q1

Q2

]

Hypothesis 2.2 The following hypotheses are needed:

There are more measured quantities than reactions: q ≥ p.

Matrix K1 is of full rank.

Under Hypothesis 2.2, the q× p matrix K1 admits a left inverse, that is, there exists a p× q matrix G

such that:

GK1 = Ip×p (2.39)

Let us set A = −K2G, and let us consider the following linear change of coordinates:

ζ1 = ξ1

ζ2 = Aξ1 + ξ2(2.40)

this change of variable transforms (2.38) into:

ζ1 = K1r(Tζ)−DH1ζ1 +Dζ1in −Q1(Tζ)

ζ2 = D(ζ2in −Rζ1 −H2ζ2)− (AQ1(Tζ) +Q2(Tζ))

(2.41)

where R = AH1 −H2A, ζ2in = Mξin and:

T =

[Ip 0p,n−p

−A In−p

], M = [A In−p] (2.42)

The equation of ζ2 can be rewritten using the output y1 and y2 as follows:

ζ2 = D(ζ2in −Ry1 −H2ζ2)−My2 (2.43)

Remark 2.1 Note that in the case of a CSTR, the heterogeneity matrix H = In×n and then matrix

R = 0.

We can now design an observer for system (2.43), based on its detectability properties. In order to

guarantee the observer convergence let us rst state the following hypothesis:

Hypothesis 2.3 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:

0 < c1 ≤∫ t+c2

t

D(τ)dτ


In practice, c2 must be low with respect to the time constant of the system. Moreover c1/c2 must be

high because it determines the minimal converging rate of the observer.

Lemma 2.1 Considering Hypothesis 2.3 and that −H2 is stable, then the solution ξ2 of the following

asymptotic observer:˙ζ2 = D(ζ2

in −Ry1 −H2ζ2)−My2

ξ2 = ζ2 −Ay1(2.44)

converges asymptotically toward solution ξ2 of system (2.38).

Proof. It can be easily veried that the estimation error e2 = ξ2 − ξ2 = ζ2 − ζ2 satises:

e2 = −DH2e2. (2.45)

and converges asymptotically toward ξ2 if hypothesis 2.3 is fullled (Bastin and Dochain, 1990). 2

2.4.2 Guaranteed state estimation methods

Before presenting the estimation methods, let us introduce some useful denitions and properties of

monotone systems.

Denition 2.14 Let us denote [x] = [x, x] a closed interval, subset of R. The width of [x] is denedas w(x) = x− x.

Arithmetical operations over intervals can be dened. See Jaulin et al. (2001) for more details.

Denition 2.15 An interval vector or box is a vector with interval components [x] = ([x1], . . . , [xn])T

or equivalently, the cartesian product of scalar intervals: [x] = [x1] × . . . × [xn]. The width of a box

w([x]) is the maximum of the widths of its components. The union of nonoverlapping boxes is called

a subpaving.

Denition 2.16 An inclusion function [f ](.) for a function f ∈ D ⊂ R is such that the image of an

interval by this function is an interval, guaranteed to contain the image of the same interval by the

original function:

∀[x] ⊂ D, f([x]) ⊂ [f ]([x]) (2.46)

An inclusion function is said to be convergent if limw([x])→0 w([f ][x]) = 0 and inclusion monotonic if

for all [x] ⊂ [y] ⇒ [f ]([x]) ⊂ [f ]([y]). If the inclusion in (2.46) becomes an equality, then the inclusion

function is said to be minimal.

2.4.3 Monotone and positive systems

Let us present some theoretical arguments about monotone and positive systems, that are important

for the formulation of guaranteed estimation methods.

Positive systems

A positive system is a system in which the state variables are always positive. This kind of system

appears in real-life cases where negative values are meaningless, like in biological systems, whose vari-

ables describe concentrations, number of individuals, etc. that must be nonnegative.


For x, y ∈ Rn, we denote x ≥ y when for all the components xi ≥ yi. This means, the operator

≥ is understood as a set of inequalities applied component by component. The same concept can be

extended to inequalities between matrices.

Therefore, a vector x ∈ Rn is said to be nonnegative if x ≥ 0, and positive if x > 0.

Denition 2.17 The closed positive orthant of real vectors is dened by Rn+ = x ∈ Rn, x ≥ 0.

Denition 2.18 A square matrix A is said to be cooperative if its odiagonal terms are nonnegative:

aij ≥ 0, ∀i 6= j.

Theorem 2.3 (Luenberger, 1979) Let A be a cooperative matrix and b a positive vector. Matrix Ahas all of its eigenvalues strictly within the left half of the complex plane if and only if:

∃x > 0 : Ax+ b = 0 (2.47)

Theorem 2.4 (Luenberger, 1979) Let A be a cooperative matrix. Then −A−1 exists and is a positive

matrix if and only if A has all of its eigenvalues strictly within the left half of the complex plane (the

matrix A is stable).

Now, let us consider the positivity property from a dynamical point of view. Consider the system:

x = f(x), x(0) = x0 (2.48)

with x ∈ Rn and f ∈ C1. System (2.48) is positive if all the trajectories x(t, x0) generated by (2.48),

with intitialisation x0 > 0 remain positive, for all t ≥ 0. This means that the positive orthant Rn+ is

invariant.

Property 2.1 System (2.48) is positive if and only if:

∀i ∈ 1, . . . , n, x0 > 0, fi(x1 ≥ 0, . . . , xi = 0, . . . , xn ≥ 0) ≥ 0, (2.49)

This means that each face of the positive orthant Rn+ (i.e. the hyperplanes xi = 0) is repulsive and

then, considering a positive initialisation of system (2.48), it will remain positive along time.

Nonlinear positive systems

In this section we consider a special class of nonlinear systems, whose main feature is the preservation

of the partial order between trajectories. These systems, called monotone have been studied in Müller

(1926), Kamke (1932) and Smith (1995).

Consider the system:

x = f(t, x), x(0) = x0 (2.50)

dened in the open domain D ⊂ Rn. Note that system (2.50) can be non-autonomous, with t ≥ 0. Letφt(x0) denote the solution of (2.50) starting at x0 at time t = 0.

Denition 2.19 f is said to be of type K in D if for each i, fi(a) ≤ fi(b), for any couple of points a

and b in D satisfying a ≤ b and ai = bi.


The following result asserts that the type K condition, otherwise called KamkeMüller condition, is

sucient and necessary for the order preserving property.

Proposition 2.2 Let f be of type K on D and x0, y0 ∈ D. If x0 ≤ y0 then φt(x0) ≤ φt(y0).

Proof. See Smith (1995) for a detailed proof. 2

The type K condition is most easily identiable from the sign structure of the Jacobian matrix of

the vector eld f .

If D is a convex subset of Rn and

∀i, ∀j 6= i, ∀x ∈ D, ∂fi

∂xj(x) ≥ 0 (2.51)

(in other words, its Jacobian matrix is cooperative), then it holds that f is of type K in D. In fact, if

f fulls (2.51) and for any a, b ∈ D such that a ≤ b and ai = bi, then:

fi(b)− fi(a) =∫ 1

0

∑i 6=j

∂fi

∂xj(a+ σ(b− a))(bj − aj)dσ ≥ 0 (2.52)

Denition 2.20 The system (2.50) is said to be cooperative if the jacobian matrix of the function f

is a cooperative matrix.

The following theorem exploits Denition 2.20, stating a result for two dierential systems.

Theorem 2.5 Consider the dynamic systems x = f(t, x) and z = h(t, z) dened in an open convex

domain D ⊂ Rn. The following conditions are fullled:

x = f(t, x) is a cooperative system.

∀a ∈ D and t ≥ 0, f(t, a) ≤ h(t, a).

x0 ≤ z0.

then x(t, x0) ≤ z(t, z0) ∀t ≥ 0.

Proof. See Smith and Waltman (1995).

Theorem 2.6 The cooperative dynamic system (2.50) is a positive system if and only if f(t, 0) ≥ 0.

Proof. A detailed proof of can be found in Mailleret (2004). 2

More properties and theoretical aspects about cooperative systems can be found in Smith (1995).

Linear positive systems

Some useful results have been obtained for linear positive systems. Property 2.1 of positive systems

characterizes the structure of linear positive systems. Consider the linear dynamics in Rn described

by:

x = Ax+ b, x(0) = x0 (2.53)

the following results (Luenberger, 1979) hold.


Theorem 2.7 The linear system (2.53) is a positive system if and only if A is a cooperative matrix

(see Denition 2.18) and b ≥ 0.

Proof. First consider x = 0, then, from Property 2.1 it is necessary that b ≥ 0. Now consider xi = 0(with x ≥ 0), then xi = bi +

∑i 6=j aijxj , for bi ≥ 0 and xi ≥ 0. In order to guarantee that xi ≥ 0 for

any xj , it is necessary that ∀i, ∀j 6= i, aij ≥ 0, which implies that A is a cooperative matrix. 2

The reciprocal of this theorem is straightforward.

Theorem 2.8 Consider system (2.53) with matrix A cooperative and stable, and b ≥ 0. Then system

(2.53) admits a unique positive equilibrium point x?.

Proof. The proof is a direct consequence of Theorem 2.4. Given matrix A cooperative and stable, and

b ≥ 0, then −A−1 > 0 and the equilibrium x? = −A−1b is positive and unique. 2

Theorem 2.8 is in particular used to look for asymptotic bounds on the error of interval observers,

as it is shown hereafter.

2.4.4 Interval observers based on positive systems theory

Interval observers (Gouzé et al., 2000; Rapaport and Dochain, 2005), are formulated using properties

of positive dynamic systems. They consist in an auxiliary dynamic system that provides guaranteed

bounds of the state to be estimated. Interval observers incorporate uncertainty into the observer design,

dealing with unknown dynamics, inputs and outputs. Nonetheless, known bounds on the uncertain

terms are required. Consider the system:x(t) = f(x(t), u(t), w(t)), x(0) = x0

y(t) = h(x(t), v(t))(2.54)

where w ∈ Rr and v ∈ Rq are uncertain terms related respectively to the state and output equa-

tions. Initial condition x0 is also assumed to be unknown. These uncertainties are bounded by known

quantities such that:

w−(t) ≤ w(t) ≤ w+(t), v−(t) ≤ v(t) ≤ v+(t) and x−0 ≤ x0 ≤ x+0 (2.55)

Then, an interval observer for system (2.54) can be dened. We start from a general framework,

proposing rst the denition of a framer.

Denition 2.21 A framer for system (3.1) is a dynamical system of the form:˙z+ = f+(z−, z+, w−, w+, v−, v+, θ, u, y), z+

0 = g+(x+0 , x

−0 , θ)

˙z− = f−(z−, z+, w−, w+, v−, v+, θ, u, y), z−0 = g−(x+0 , x

−0 , θ)

x+ = h+(z−, z+, w+, v−, v+, θ, u, y)x− = h−(z−, z+, w+, v−, v+, θ, u, y)

(2.56)

with z−, z+ ∈ Rl and functions f−, f+, h−, h+, g−, g+ dened in appropriate domains, such that, for

any initial condition verifying:

x−0 ≤ x0 ≤ x+0 we have x−(t) ≤ x(t) ≤ x+(t)


Figure 2.2: Interval observer principle.

Note that a framer is parameterized by a gain θ. Denition 2.21 is rather general, highlighting the fact

that a framer is conceived to provide an upper and a lower bound of the unknown state. Stability can

be considered as an additional feature: if it can be shown that the obtained interval estimate is error

bounded (see Denition 2.13), then the framer becomes an interval observer.

Denition 2.22 An interval observer is a bounded error framer.

∃T, ∃M ≥ 0 : ‖x+(t)− x−(t)‖ ≤M, ∀t > T (2.57)

Now, the theorems about monotone and positive systems presented above become useful in order

to formulate interval observers. This can be achieved in several ways:

If the original system is monotone (cooperative), then Denition 2.19 leads directly to an interval

observer.

If the original system is not monotone, then the main objective will be to construct an observer

candidate and nd conditions for which the error dynamics are a positive system.

It is worth to remark that monotony and positivity properties of dynamic systems are coordinate

dependent. If the original system is not monotone, a change of base where the new dynamics are

monotone may lead to interval observers.

The following examples show some interval observers developed using these concepts.


Asymptotic interval observer. In Hadj-Sadok (1999) and Alcaraz-González et al. (2001), interval

observers have been developed as an extension of asymptotic observers applied to bioprocesses models.

Let us consider the framework for the design of asymptotic observers presented in section 2.4.1, and

add the following hypothesis.

Hypothesis 2.4 The inuent concentrations vector is unknown, but bounded by two known (possibly

time varying) vectors:

ξ−in(t) ≤ ξin(t) ≤ ξ+in(t).

The objective is to write an interval observer for the unknown part of the vector ξ (denoted by ξ2),

which does not depend on the reaction rate vector r(ξ). The following interval asymptotic observer is

then proposed.

Proposition 2.3 Given ξ−2 (0) and ξ+2 (0) such that ξ2(0) ∈ [ξ−2 (0), ξ+2 (0)], then the following pair of

dynamic systems is an interval observer of system (2.38):

ζ−2 = D(ζ−in −Ry1 −H2ζ−2 )−My2

ζ−2 (0) = ξ−2 (0) +Ay1(0)ξ−2 = ζ−2 −Ay1

ζ+2 = D(ζ+

in −Ry1 −H2ζ+2 )−My2

ζ+2 (0) = ξ+2 (0) +Ay1(0)ξ+2 = ζ+

2 −Ay1

(2.58)

with ζ−in and ζ+in such that ζ−in ≤Mξin ≤ ζ+

in.

Proof. The error dynamics have the following form:

e? = −DH2e? + ν?(t) (2.59)

where e? stands for the upper error e+(t) = ξ+2 (t)− ξ2(t) or the lower error e−(t) = ξ2(t)− ξ−2 (t). ν?(t)is a nonnegative vector that depends on the input uncertainty and its bounds. Note that from the

denition of the heterogeneity matrix H in equation (1.2), it holds that −H2 is cooperative. Therefore,

from Theorem 2.7 it holds that the error e?(t) is a nonnegative variable if it is initialised such that

e?(0) ≥ 0 and then it is guaranteed that ξ−2 (t) ≤ ξ2(t) ≤ ξ+2 (t). 2

This observer inherits the good characteristics of classical asymptotic observers: it does not depend

on the reaction rate of the system, moreover, it is able to deal with the uncertainty of the inuent

concentrations using the known bounds ξ−in(t) and ξ+in(t). Nevertheless, the convergence rate of this

observer is again determined by the operational conditions of the system.

Asymptotic interval observers have been applied to anaerobic digestion. Results can be seen in Hadj-

Sadok (1999) and Alcaraz-González et al. (2001).

Linear case up to an output injection. In Gouzé et al. (2000) an interval observer is developed

for systems of the form: x(t) = Ax(t) + φ(t, y(t))y(t) = Cx(t)

(2.60)


with A ∈ Rn×n (n ≥ 2), C ∈ R1×n. If the mapping φ : R+ ×R → Rn is known, a Luenberger observer

can be designed.

Let us assume that φ is badly known and that it can be bounded by two known bounds. Thus, there

exist known functions φ−, φ+ : R+ × R → Rn, suciently smooth, such that:

φ−(t, y) ≤ φ(t, y) ≤ φ+(t, y), ∀(t, y) ∈ R+ × R (2.61)

Theorem 2.9 If there exists a gain vector K such that matrix A+KC is cooperative, then the following

equations:

x+(t) = Ax+(t) + φ+(t, y(t)) +K(Cx+(t)− y(t)) (2.62)

x−(t) = Ax−(t) + φ−(t, y(t)) +K(Cx−(t)− y(t)) (2.63)

provide an interval estimator for system (2.60), that is:

x−(0) ≤ x(0) ≤ x+(0) ⇒ x−(t) ≤ x(t) ≤ x+(t)

Proof. The proof follows directly from Theorem 2.7. 2

Let us now consider the upper error e+(t) = x+(t)− x(t), its dynamics are given by:

e+ = (A+KC)e+ + b+(t) (2.64)

with b+(t) = φ+(t, y(t))− φ(t, y(t)) ≥ 0.

Theorem 2.10 If hypotheses of Theorem 2.9 are veried, with a stable matrix A+KC, 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 the error e(t) is ultimately bounded by the positive vector:

emax = −(A+KC)−1B

In particular, if the components of emax are zero, then the corresponding components for e(t) convergetoward zero.

Proof. The proof is due to the inequality:

(A+KC)e+ b(t) ≤ (A+KC)e+B

which implies (with equal initial conditions):

e(t) ≤ em(t), ∀t ≥ 0

where em(t) is the solution of em = (A+KC)em +B. 2


In Moisan and Bernard (2005), the extension where φ = φ(x, y) is analyzed, when it is a a non

monotone function with respect to the unknown state variables. This makes the design of an interval

observer less straightforward. See chapter 4 for more details.

Additional remarks

A property of interval observers is that they work in a guaranteed estimation context, and there-

fore, the solutions of dierent observers can be compared in order to obtain an improved estimation

interval. This is the basic idea behind the concept of bundle of observers, introduced in Bernard and

Gouzé (2004). This is deeply studied in the following chapters.

Interval observers have also been used working jointly with classical observers. In the framework of

linear systems with a parametric isolated uncertainty, see Sauvage et al. (2007), where estimates gen-

erated by the interval observer are used by a second observer, which guarantees asymptotic convergence.

Applications of interval observers in an observercontroller loop have also been studied, leading to

robust control schemes. See for example Rapaport and Harmand (2002).

2.4.5 Estimation through Interval Analysis

Interval analysis is a tool that allows to cope with uncertainties in several engineering problems,

like robotics, aeronautics and computeraided design (Grandón, 2007). These techniques have been

successfully applied to state and parameter estimation.

Problem statement

Let us consider the following continuous time dynamic model, with discrete time output samples:

x(t) = f(x(t),p,v(t), t), x(t0) = x0

ym(p, tk) = h(x(tk),p,wk, tk)(2.65)

where x ∈ Rnx is the state vector, p ∈ Rnp is a constant parameter vector. v(t) and w(t) are state

perturbation and output noise respectively. Output is sampled at the time instant tk with k = 1, . . . , N .

It is assumed that the state initial condition, system parameters, state and output disturbances are

uncertain, but bounded by known quantities: x0 ∈ [x0] = [x0,x0], p0 ∈ [p0] = [p0,p0], v(t) ∈ [v(t)] =

[v(t),v(t)] for all t ∈ [t0, tk] and w(tk) ∈ [w(tk)] = [w(tk),w(tk)].The available information at any time instant t ∈ [t0, tk] is then expressed by the set:

I(t) = [x0], [p0], [v(τ)τ∈[t0,t]], y(tk), [w(tk)]Mk=1 (2.66)

where tM is such that tM ≤ t ≤ tM+1. Therefore, the state estimation problem deals with the

characterization of the set Xt|t of all values of x(t) that are consistent with I, for all t ≥ t0.


Estimation procedure

The following algorithm is proposed (Jaulin and Walter, 1993), to perform the state estimates. Let us

denote:

ϕ(x, t, tk,p, v(τ)τ∈[tk,t]) (2.67)

the ow at time t associated to (2.65). The estimation procedure has two steps (Jaulin, 2002; Kieer

and Walter, 2004):

Prediction: For a given x ∈ Xtk|tk, the predicted set Xk+1|k can be written as:

Xtk+1|tk=

ϕ(x, ttk+1 , tk,p, v(τ)τ∈[tk+1,tk])|p ∈ [p0],v(τ) ∈ [v(τ)],x ∈ Xtk|tk

, τ ∈ [tk+1, tk] (2.68)

By construction x(tk+1) ∈ Xtk+1|tk.

Correction: The information contained in the system output y((tk+1)) is incorporated to the

algorithm. The objective is to nd x(tk+1) ∈ Xtk+1|tkthat are consistent with the measurements

at tk+1:

Xtk+1|tk+1 =x(tk+1) ∈ Xtk+1|tk

|y(tk+1) ∈ h(x, [p0], [w(tk+1)], tk+1)

(2.69)

The main diculty of this estimation procedure lies in determining Xtk+1|tkin the prediction step,

which may be viewed as the direct image of a set by the function ϕ: an inclusion function associated

to ϕ can be very dicult to obtain. This problem has been faced using dierent tools:

Guaranteed numerical integrations of ODEs: These methods are mainly based on numerical

integration using a Taylor expansion associated to system (2.65) and then bounding the remainder

in a guaranteed way (Corliss and Rihm, 1996; Neldiakov and Jackson, 2001). These approaches

can become very inecient when dealing with uncertain parameters and bounded perturbations

as far as the bounds on the remainder can become extremely large. In Raïssi et al. (2005), a

moving horizon state estimator for continuous time systems is developed using this principle, and

interval constraint propagation techniques are used in order to improve results.

Dierential inequalities theorems: In Kieer and Walter (2004) the cooperativity properties of

dynamic systems are exploited in order to obtain an inclusion of function ϕ. In Kieer and

Walter (2006), an adaptation of the KamkeMüller's condition (Denition 2.19) is used with the

same objective.

Once an inclusion function for ϕ is obtained, then a direct image evaluation is performed. This

corresponds to a characterization of the set Y = f(X) when X and f are known. In (Kieer et al., 2002)

the algorithm ImageSp is proposed to perform this task, and is described in box 2.1. This algorithm

is based on interval analysis and works building a subpaving Y such that Y ⊂ Y when X is itself a

subpaving and an inclusion function [f] is available.


Box 2.1 ImageSp algorithm:

Mincing: All boxes X are bisected, obtaining a subpaving X′ such that all the boxes

have a width less than a prespecied precision parameter εi.

Evaluation: The images of all boxes of X′ are evaluated using the inclusion function

[f] and stored in a list of image boxes Y.

Regularisation: All boxes in Y are merged into a new subpaving Y.

Then, Y is guaranteed to contain Y. The precision of the approximation is controlled by

the parameter εi.

The characterization of the set consistent with the output is carried out using an inverse image

evaluation procedure. The Sivia algorithm (Set Inverter Via Interval Analysis, see box 2.2) (Jaulin

and Walter, 1993) is intended to the characterization of set dened by X = h−1(Y) and applies to any

function h for which a prior box [x](0) and an inclusion function [h] are available (note that [h] canbe easily obtained from system (2.65)).

Box 2.2 Sivia algorithm:

initialisation: k = 0, S = ∅, K = ∅, K = ∅.

iteration:

step 1. If [h]([x](k)) ⊂ Y then K = K ∪ [x](k). Go to step 4.

step 2. If [h]([x](k)) ∩ Y = ∅, then go to step 4.

step 3. If w([x](k)) ≤ εs then K = K ∪ [x](k),else bisect [x](k) and store in S the two resulting boxes.

step 4. If S is not empty, then assign [x](k + 1) = S. Go to step 1.

end.

The parameter εs is used to to denote the accuracy of the subpaving. Upon completion of

the algorithm, all boxes will have a width lower than or equal than εs.

This estimation procedure has been applied to state estimation of an activated sludge process (Kief-

fer and Walter, 2004), and to the joint estimation of the state and parameters of a twocompartmental

model, with MichaelisMenten nonlinearities (Kieer and Walter, 2006).

2.4.6 Ellipsoidal methods

Ellipsoidal techniques have been extensively applied to state estimation and control problems (Chernousko,

1985; Kurzhanski and Vályi, 1997).


Let us consider the dynamic system:x = f(x, u, t), x(t0) ∈ X0, t ≥ t0

y = h(x, v, t)(2.70)

where x ∈ Rn, u ∈ U , v ∈ V are the set of state and output uncertainties and X0 is a closed set in Rn

which contains the initial condition x0.

Ellipsoidal sets in Rn are characterised by a ratio a ∈ Rn and a positive denite shape matrixQ ∈ Rn×n.

The mathematical expression of an ellipsoid E(a,Q) is:

E(a,Q) = x ∈ Rn : (x− a)TQ−1(x− a) ≤ 1

Ellipsoidal methods approximate, by means of ellipsoids, the so called attainable set of a (uncertain)

dierential system (see g. 2.3).

Denition 2.23 The attainable set D(t, t0, X0) of system (2.70) is the set of all the trajectories x(t)that veries the dierential inclusion:

x ∈ f(x,U , t), x(t0) ∈ X0, t ≥ t0

Attainable sets have the following evolutional property:

D(t, t0, X0) = D(t, τ,D(τ, t0, X0)), ∀τ ∈ [t0, t] (2.71)

Let us assume that system (2.70) can be written in discrete time version using the following expression:xk+1 = (In + σA)xk + σψu

k

yk = Hxk + ψvk

(2.72)

Figure 2.3: Ellipsoidal estimation principle.


where σ is a small positive real number, A stands for the linear part of system (2.70) and ψuk = ψ(xk, u)

concentrates uncertainties and nonlinearities. Analogously, the output equation is characterised by the

known matrix H and the uncertain term ψvk = ψ(xk, v).

If xk and ψk belong to some ellipsoids, then in can be seen from equation (2.72) that, in order to

obtain an ellipsoid for xk+1, the following operation with ellipsoids are essential: linear transformation,

summing and intersection of two ellipsoids. Intersection of ellipsoids will be performed once an ellipsoid

for xk consistent with the output yk has been computed. Hence, it is aimed to obtain a one step forward

outer approximation of the attainability domain of the system and then, guaranteed bounds for the

state.

One of the advantages of this method is that the operation between ellipsoids mentioned above

are well dened, allowing the computation of the outer set immediately using equation (2.72) and its

implementation can lead to ecient algorithms (Maksarov and Norton, 2002). For details the reader

is referred to Kurzhanski and Vályi (1997).

2.4.7 Interval Kalman ltering

An interval version of the Kalman lter has been introduced in Chen et al. (1997), in order to deal

with uncertainties of linear systems. The key idea is to propose an interval version of the statistical

expectation of random variables, and then, using the tools of interval analysis, derive an interval

Kalman lter.

For a real Lipschitz function f and its associated inclusion function F , consider the real numbers a

and b, and the partition into N subintervals of the interval [a, b]:

[x1] = [x1, x1], . . . , [xN ] = [xN , xN ] : a = x1 < x1 = x2 < x2 = . . . = xN < xN = b (2.73)

dene:

SN (F ; [a, b]) =b− a

N

N∑i=1

F ([xi]) (2.74)

then one has: ∫ b

a

f(z)dz =∞⋂

N=1

SN (F ; [a, b]) = limN→∞

SN (F ; [a, b]) (2.75)

This can be also written in the recursive form:y1 = S1

yk+1 = Sk+1 ∩ yk, k = 1, 2, . . .(2.76)

where Sk = Sk(F ; [a, b]), then yk is a nested sequence of intervals that converges to the exact value of

the integral∫ b

af(z)dz. Now let us consider x an interval of realvalued random variables and

f(x) =1√

2πσx

exp−(x− µx)2

2σ2x

, x ∈ x (2.77)


be an ordinary Gaussian density function with known µx and σx > 0.Then, f(x) is associated to an interval Lipchitz inclusion function, so that the interval expectation

E(x) =∫ +∞

−∞xf(x)dx =

∫ +∞

−∞

x√2πσx

exp−(x− µx)2

2σ2x

dx, x ∈ x (2.78)

is well dened, based on the nite integral dened above with a → −∞ and b → +∞. This leads

straightforwardly to the denition of interval variance, V(x). The conditional interval expectation and

conditional variance are also well dened and are expressed by:

E(x|y ∈ y) = E(x) + σ2xy[y − E(y)]/σ2

y (2.79)

V(x|y ∈ y) = V(x)− σ2xyσ

2yx/σ

2y (2.80)

with σ2xy = σ2

yx = E(xy)− E(x)E(y) (Chen et al., 1995).

Let us consider the following interval linear system, described by:xk+1 = AI

kxk +BIkξk,

yk = CIk + ηk

(2.81)

where xk ∈ Rn, yk ∈ Rm and AIk = [Ak, Ak], BI

k = [Bk, Bk] and CIk = [Ck, Ck] are interval matrices of

appropriate dimension and ξk and ηk are mutually independent zeromean Gaussian noise sequences

with known covariance matrices Qk and Rk. Note that the interval arithmetic operations between

matrices are based on basic interval arithmetic operations, and therefore, they are well dened.

The interval Kalman lter is given by the following algorithm:

xI0 = ExI

0, (given) (2.82)

xIk = AI

k−1xIk−1 +GI

k[yIk − CI

kAIk−1x

Ik−1] (2.83)

P I0 = VxI

0, (given) (2.84)

M Ik−1 = AI

k−1PIk−1[A

Ik−1]

T +BIk−1Qk−1[BI

k−1]T (2.85)

GIk = M I

k−1[CIk ]T [CI

kMIk−1[C

Ik ]T +Rk]−1 (2.86)

P Ik = [I −GI

kCIk ]M I

k−1[I −GIkC

Ik ]T +GI

kRk[GIk]T (2.87)

The ltering result produced by the interval Kalman lter is a sequence of interval estimates given

by the interval conditional expectations, that contains all possible optimal states x that the intervalsystem (2.81) may generate.

Other improvements of the Kalman lter have been introduced, in order to have a more robust version

of the lter. See for example Xie et al. (1994).

2.5. Conclusions 43

2.5 Conclusions

This chapter has presented dierent estimations methods, considering classical approaches and esti-

mation methods that deal with uncertainties. It is possible to remark that the performance of any

estimation method is directly related to the information available on the system. In a general context,

if there exists an accurate mathematical model representing the real life system, then several methods

can guarantee exponential convergence toward the unknown state variables. However, when uncer-

tainties and bad modeled dynamics are present, estimation methods in a guaranteed bounded error

context can be a wiser choice. For this last case, we presented several estimation techniques. All of

them are based on dierent philosophies, however they provide guaranteed upper and lower bounds

for the unknown state. Therefore, it is possible to intersect their estimations, generating an improved

envelope. This characteristic is unique in the framework of state estimation of uncertain systems and is

exploited in the next chapter, from the point of view of interval observers based on positive systems.

Chapter 3

Bundle of observers

3.1 Introduction

Interval observers provide guaranteed bounds of the state to be estimated. In this chapter we exploit

this fundamental characteristic in order to run multiple estimates in parallel, (changing a tunable gain

of the observer) and then taking the best generated estimate.

Interval observers were introduced in a general context in Chapter 2 by introducing the concept

of framer, that is, a sole pair of estimates able to provide bounds of the unknown state (without

considering any stability constraint). An interval observer is then a bounded error framer. In this

chapter we revisit this denition: an interval observer is the result of comparing multiple framers

running in parallel. The socalled bundle of observers, will generate an interval observer under the

assumption that at least one of the framers provides a bounded error interval estimate.

Taking again advantage of the guaranteed estimation framework, regular reinitialisation of the ob-

server bundle is proposed, as a way to exploit the dierent stable and unstable behaviors provided by

the set of the considered framers. These concepts are complemented by the design of a convergence

index, which simply let us online assess convergence properties of the interval estimate.

In this chapter we formalise these concepts, which will be in the sequel the base of further develop-

ment. We illustrate these concepts with a detailed example of an interval observer design for a chaotic

system (Moisan and Bernard, 2006b).

3.2 Observers bundle: key concepts

Let us recall the uncertain dynamical system:x(t) = f(x(t), u(t), w(t)), x(0) = x0

y(t) = h(x(t), v(t))(3.1)

where x ∈ Rn and y ∈ Rp and uncertainties x0, v(t) and w(t) are bounded by known quantities (see

equation (2.55)).

According to Denition 2.21, a framer is basically a dynamical system able to provide bounds for

45

46 Bundle of observers

Figure 3.1: Bundle of observers and best estimate.

the state of system (3.1), parameterized by a tunable gain θ, which can be a xed gain or a time

function.

This denition, still general, does not provide information about the stability of a framer.

3.2.1 Observers bundle and reinitialisation

One of the key properties is that we can compare the solutions generated by two or more framers.

Consider that there exist p framers associated with p dierent θi, with θi ∈ GΘ. We run therefore

several framers in parallel, which all provide guaranteed interval estimates.

Denition 3.1 An observer bundle is a set of interval estimates generated by a nite set of p framers.

B(t) = x+θi

(t) : θi ∈ GΘ and B(t) = x−θi(t) : θi ∈ GΘ, ∀i ∈ 1, . . . , p (3.2)

B and B are respectively the lower and upper bundle.

Remark 3.1 The upper and lower bundle can be generated by dierent gains θ+, θ− ∈ GΘ, which are

gathered in θ in order to lighten notation.

Each bundle has an envelope that corresponds to the best bounds:

B+inf(t) = minB(t) and B−sup(t) = maxB(t) (3.3)

That is, we take the inner envelope from the set of framers.

Property 3.1 If there exists one framer [x−θj, x+

θk], j, k ∈ 1, . . . , p belonging to the bundle (3.2),

such that ‖x+θk

(t)− x−θj(t)‖ ≤M , M ≥ 0, then the envelope [B−sup(t),B+

inf(t)] is an interval observer of

system (3.1).

Proof. The proof becomes clear considering equation (3.3) and Denition 2.22 (interval observer). 2

It is worth noting that we combine transient behavior of some unstable framers (that may improve

transient estimations), with the asymptotic stability of others (that guarantees the boundedness of the

3.2. Observers bundle: key concepts 47

Figure 3.2: Reinitialisation and best estimate.

envelope).

It is easy to verify the following properties:

Property 3.2 For an observer bundle given by equation (3.2), the following properties hold:

At least one stable framer guarantees the boundedness of the envelope.

A positive lower framer guarantees positivity of B−sup(t).

A regular reinitialisation of the bundle can be performed to restart all the framers with the best

available interval predicted by Bsup and Binf. We consider the time interval [tk, tk + ∆t] (we denote by∆t the reinitialisation time interval) where the framers run. Then at the time instant tk we take the

best interval estimates performed by the previous estimation period to reinitialize the whole bundle.

[x−0θ (tk), x+0

θ (tk)] = [B−sup(tk),B+inf(tk)] (3.4)

The objective behind the regular reinitialisation of the interval estimates is to improve the framer

eciency by feeding them with the best available estimate, and thus take benet of the transients of

some of them.

Convergence index

One of the advantages of obtaining interval estimates is that we can assess the observer convergence

by comparing upper and lower bound estimates. This leads to a convergence index ϑ, which can be

designed in several ways. Here we propose the following expression, associated to the estimation of the

kth unknown variable:

ϑk(t) =B+inf,k(t)− B−sup,k(t)

|x+0,k|+ |x−0,k|

(3.5)

which compares estimation result of an observer bundle with respect to the initial condition.

In the following section, a complete example of design of an interval observer for a chaotic system

(the well known Chua's system) is presented.


3.3 Application to chaotic systems

We consider a chaotic dynamic system that can be written in the form:x(t) = A(κ)x(t) + ψ(x, κ), x(0) = x0

y(t) = Cx(t)(3.6)

where x ∈ Ω ⊂ Rn, A ∈ Rn×n. The mapping ψ(x) concentrates the non linearities of the system. We

will assume in the sequel that ψ(x) is a Lipschitz function.

System (3.6) is also characterized by a set of parameters κ. This last point is treated in details here-

after, when considering uncertainties in the observer formulation.

Furthermore, we consider that a linear combination of the state vector is measured, with C ∈ R1×n+ .

For the sake of simplicity we focus here on the single output case, but the same principle of observer

design straightforwardly applies to an output of dimension p.

For the formulation and analysis of the observers, we consider the following hypotheses on system

(3.6).

Hypothesis 3.1 The trajectories of (3.6) are bounded for all time: ∃ xmax > 0 such that |x(t, x0)| ≤xmax, ∀x0 ∈ Ω.

Hypothesis 3.2 There exists a function ψ(xa, xb) : Ω×Ω 7→ Rn that fulls the following requirements:

ψ(x, x) = ψ(x).

x− ≤ x ≤ x+ ⇒ ψ(x−, x+) ≤ ψ(x) ≤ ψ(x+, x−)

This hypothesis is developed with more details in Chapter 4, where it is explained how to construct

function ψ on the basis of monotonicity considerations.

3.3.1 Chua's chaotic system

Chua's system is one of the most cited and studied paradigms of chaotic behavior. The double scroll

attractor that features this system is generated from a simple electrical circuit. The dimensionless

−4−1014

−0.30

0.3

−5

−3

0

3

5

x1x2

x3

−3 −1 0 1 3−4

−2

0

2

4

s

g(s)

Figure 3.3: Chua's attractor and nonlinearity g(s).

3.3. Application to chaotic systems 49

equations that represent Chua's system are the following:x1 = α[x2 − x1(1 + b)− g(x1)]x2 = x1 − x2 + x3

x3 = −βx2 − γx3

(3.7)

The nonlinear feature is given by the function:

g(s) =12(a− b)[|s+ 1| − |s− 1|] (3.8)

where α, β, γ ∈ R+ and a, b ∈ R− are system parameters.

Chua's system (3.7) has the form of system (3.6) considering x = [x1 x2 x3]T ∈ R3 and:

A =

−α(1 + b) α 01 −1 10 −β −γ

, ψ(x) =

−αg(x1)00

(3.9)

For the forthcoming analysis, we will consider that variable x2 is measured, therefore C = [0 1 0].

Property 3.3 The solutions of system (3.7) remain bounded for all time.

Proof. See for example Leonov et al. (1993) and Strogatz (1994), where a qualitative study of chaotic

dynamics can be found. 2

We develop the interval observer in two steps: rst, we consider the case of perfect knowledge

of system (3.6). Then, we consider the case when system parameters are not exactly known and

measurements are biased. We show how to obtain a guaranteed stable and error bounded estimation

of the chaotic state.

3.3.2 Interval observers with perfect knowledge

The observer design is similar to the interval observer for a linear system up to an output injection

presented in Chapter 2.

A perfect knowledge interval observer

Let us consider the following system:x+ = Ax+ + ψ(x+, x−) + Θ1(y+ − y)x− = Ax− + ψ(x−, x+)−Θ2(y − y−)x+(0) = x+

0 , x−(0) = x−0

(3.10)

where Θi = (θi1, ..., θ

in)t ∈ Rn are two gain vectors, x− and x+ ∈ Rn and y+ = Cx+, y− = Cx−.

Proposition 3.1 If there exist Θ1 and Θ2 such that the matrices A1 = A+ Θ1C and A2 = A+ Θ2C

are cooperative, then system (3.10) is a framer of system (3.6).


Proof. We write the error dynamics by comparing the equations (3.10) and (3.6). This leads to the

following system: e+ = A1e

+ + ψ(x+, x−)− ψ(x)e− = A2e

− + ψ(x)− ψ(x−, x+)(3.11)

Considering the error dynamics e =

[e+

e−

], it can be rewritten in the compact form e = Ae +

φ(x+, x−, x), where:

A =

[A1 0

0 A2

]and φ(.) =

[ψ(x+, x−)− ψ(x, x)ψ(x, x)− ψ(x−, x+)

](3.12)

By initial condition hypothesis, e(0) ≥ 0.Let us consider the rst time instant t? when one of the component of vector e is equal to zero. Let

us e.g. assume that it is the kth component of ek. We have for this error component:

ek|t=t? =2n∑i 6=k

akiei + φk(x+, x−, x) ≥ 0 (3.13)

where the components of matrix A are denoted aij . As a consequence ek will stay nonnegative and

nally e(t) will remain nonnegative for any time t. 2

Application to the example

Let us consider Θ1 = Θ2 = Θ. For Chua's equation (3.7) we have:

A+ ΘC =

−α(1 + b) α+ θ1 01 −1 + θ2 10 −β + θ3 −γ

(3.14)

The nonlinear part is perfectly known, then:

ψ1(x−1 ) ≤ ψ1(x1) ≤ ψ1(x

+1 ) (3.15)

with:

ψ1(s) =−α2

(a− b)[|s+ 1| − |s− 1|] (3.16)

For Chua's equation, we have considered α = 11.85, β = 14.9, γ = 0.29, a = −1.14 and b = −0.71(Strogatz, 1994) and the initial condition x0 = [−0.1 0.2 0.05]. We run the framer provided by equation

(3.10) considering x+0 = [3 3 3]t and x−0 = [−3 − 3 − 3]t .

The gain value Θ = [−α 0 β]t makes the matrix A + ΘC cooperative. Fig. 3.4 shows the framer

associated to this gain value.

Remark 3.2 It is worth noting that when x1 ∈ [−1, 1] equation (3.10) does not provide stable esti-


0 5 10 15 20

−3

0

3

time

x1

0 5 10 15 20

−3

0

3

time

x3

Figure 3.4: Interval estimation of variables x1 and x3.

mates. Indeed, equation (3.7) can be represented by the linear system x = Lx, with:

L =

−α(1 + a) α 01 −1 10 −β −γ

(3.17)

In this case, the pair L,C is not observable and the origin is an unstable point. In other words, the

output y = x2 does not allow to assign all the eigenvalues of L + ΘC in the left hand complex plane.

However, convergence of the estimates can still be reached, as it is discussed hereafter.

3.3.3 Interval observer with uncertainties

Uncertain parameters

Let us consider equation (3.6) in the case where κ is uncertain, but known to live in the intervals

κi ∈ [κi, κi]. Under this consideration, matrix A is not known anymore. However, there exist two

nonnegative matrices M1 and M2 ∈ Rn×n+ such that:

A+M1 = A and A−M2 = A (3.18)

whereA andA ∈ Rn×n are known matrices. Therefore, the uncertain matrix A is bounded: A ≤ A ≤ A.

For a simpler notation, we dene R = A−A = M1 +M2 which is a non-negative matrix.

We will show that an interval observer for this class of system will depend on the sign of the estimates.

For this purpose let us introduce the following matrices:

σ+n×n = diag[pos(x+

i )] and σ−n×n = diag[pos(x−i )] (3.19)

where:

pos(x) =

1 if x ≥ 00 otherwise

(3.20)

In other words, σ+(x+) denotes a diagonal matrix where the element σk will be 1 if the kth component

of the vector x+ is non-negative and 0 if not. In an analogous way, the diagonal matrix σ−(x−) is

constructed over the sign of the components of the vector x−.


Now we write a candidate framer equation as follows:x+ = Bx+ + ψ(x+, x−, κ, κ) + Θ1(y+ − y)x− = Bx− + ψ(x−, x+, κ, κ)−Θ2(y − y−)x+(0) = x+

0 , x−(0) = x−0

(3.21)

where:

B = Aσ+ +A(I − σ+) and B = Aσ− +A(I − σ−) (3.22)

Proposition 3.2 If there exist Θ1 and Θ2 such that the matrices A1 = A+ Θ1C and A2 = A+ Θ2C

are cooperative, then system (3.21) is a framer of system (3.6).

Proof. Let us check the error dynamics. After some algebraic arrangements, equation (3.22) can be

written as:

B = [A−M2 +Rσ+] and B = [A+M1 −Rσ−] (3.23)

then the error dynamics hold:e+ = (A+ Θ1C)e+ + φ+(.) + (Rσ+ −M2)x+

e− = (A+ Θ2C)e− + φ−(.) + (Rσ− −M1)x−(3.24)

which allows the following compact form:

e = Ae+ φ(x+, x−, x) +H

[x+

x−

](3.25)

where A and φ(x+, x−, x) remain the same as in equation (3.12) and

H =

[Rσ+ −M2 0

0 Rσ− −M1

](3.26)

The term H is related to the parametric uncertainty. Considering that rij = mij1 +mij

2 we have:

[Rσ+ −M2]ij =

m1

ij if pos(x+j ) = 1

−m2ij otherwise

(3.27)

Then it is clear that the kth column of Rσ+−M2 has the same sign as the kth component of x+. The

same statement applies to matrix Rσ− −M1 with respect to x−. This implies that:

[Rσ+ −M2]ijx+i ≥ 0 and [Rσ− −M1]ijx−i ≥ 0 (3.28)

and nally H

[x+

x−

]= Hu ≥ 0.

To nish the proof we write again the derivative of the rst component ek of the vector error e becoming


equal to zero, for the rst time t0:

ek|t=t0=

2n∑i 6=k

akiei + hkiui+ φk(x+, x−, x) ≥ 0 (3.29)

Then it is shown that the error will stay nonnegative, using the same arguments for the proof of Propo-

sition (3.1). 2

For the example, guaranteed estimates can be obtained considering:

A+ ΘC =

−α(1 + b) α+ θ1 01 −1 + θ2 10 −β + θ3 −γ

(3.30)

For the nonlinear part, we have to take into account the sign of the estimates. This leads to:

if x+1 ≥ 0 : ψ1(x

+1 ) = −1

2(αa− αb)[|x+

1 + 1| − |x+1 − 1|] (3.31)

if x+1 < 0 : ψ1(x

+1 ) = −1

2(αa− αb)[|x+

1 + 1| − |x+1 − 1|] (3.32)

if x−1 ≥ 0 : ψ1(x−1 ) = −1

2(αa− αb)[|x−1 + 1| − |x−1 − 1|] (3.33)

if x−1 < 0 : ψ1(x−1 ) = −1

2(αa− αb)[|x−1 + 1| − |x−1 − 1|] (3.34)

Remark 3.3 In general, the choice of κi or κi to construct bounds of ψ(x, κ) depends exclusively

on the nonlinearity structure. Without loss of generality, in the sequel we will assume that the term

φ(x+, x−, x) remains bounded by a positive constant vector φ = [φ1 φ2]T .

Biased output

In practice, measurements are corrupted with noise. We assume that this noise is bounded:

y = Cx(1 + δ), |δ| ≤ ∆ < 1 (3.35)

In other words, the original measured set of variables Cx is corrupted by a bounded multiplicative

perturbation δ.

Depending on the sign of the measurements y, we can bound the actually unknown quantity Cx

considering:

y =y

1 + ε∆≤ Cx ≤ y

1− ε∆= y (3.36)

where ε =

1 if y ≥ 0−1 otherwise

.


These bounds are known quantities that allow to establish a new framer equation.x+ = Bx+ + ψ(.) + Θ1(y+ − y) + F1(x+ − x−)x− = Bx− + ψ(.)−Θ2(y − y−)− F2(x+ − x−)x+(0) = x+

0 , x−(0) = x−0

(3.37)

Proposition 3.3 If there exist Θ1, Θ2, F1 and F2 such that the matrices A1 = A + Θ1C + F1 and

A2 = A + Θ2C + F2 are cooperative, with Θ1 and Θ2 negative vectors and F1 and F2 nonnegative

matrices, then system (3.37) is a framer of system (3.6).

Proof. Let us focus on the correction terms. For the upper bound now we have:

y+ − y =[Cx+ − Cx

1 + δ

1− ε∆

]

= Ce+ − Cxε∆ + δ

1− ε∆

(3.38)

Similarly, the lower bound correction can be written as:

y − y− = Ce− − Cxε∆− δ

1 + ε∆(3.39)

Now we compute the error dynamics.e+ = (A+ Θ1C)e+ + (Rσ+ −M2)x++

−η1Θ1Cx+ φ+(.) + F1(e+ + e−)e− = (A+ Θ2C)e− + (Rσ− −M1)x−+

−η2Θ2Cx+ φ−(.) + F2(e+ + e−)

(3.40)

where:

η1 =ε∆ + δ

1− ε∆and η2 =

ε∆− δ

1 + ε∆(3.41)

Again, we use the compact notation of equation (3.25) to write the error dynamics. Then we have:

A =

[A+ Θ1C + F1 F1

F2 A+ Θ2C + F2

](3.42)

and

H =

[Rσ+ −M2 0

0 Rσ− −M1

][x+

x−

]+

[−η1Θ1C 0

0 −η2Θ2C

][x

x

](3.43)

We already proved that the vectors (Rσ+ −M2)x+ and (Rσ− −M1)x− remain by construction non-

negative for all time. Now, provided negative gains vectors Θ1 and Θ2, and considering equations

(3.36) and (3.41) it is possible to verify that:

sign(y) = sign(η1) = sign(η2) (3.44)


Moreover, from equation (3.35) it follows that the output y and Cx have the same sign, then:

sign(η1) = sign(η2) = sign(Cx)

⇒

[−η1Θ1C 0

0 −η2Θ2C

][x

x

]≥ 0, ∀Θi < 0

(3.45)

With this last result the vector H is positive for bounded uncertainties in the system parameters and

biased measurements.

To complete the proof, we just need to verify that matrix A is cooperative. Indeed, if matrices

A+Θ1C+F1 and A+Θ2C+F2 are cooperative, then it is straightforward that A is also cooperative,

under the sign condition for Θi and Fi of Proposition 3.3. 2

Observer boundedness

Until this point we have obtained guaranteed upper and lower bounds of system (3.6), because of the

positivity of the error systems. Now let us analyze the stability of the system (3.40), with A and H as

dened in (3.42) and (3.43).

For our purposes, Theorem 2.8 of Chapter 2 will be useful.

Lemma 3.1 If conditions of Proposition (3.3) and theorem (2.8) are fullled, then the estimation

error is ultimately bounded by:

eeq = −N−1

[(R− η1Θ1C)xmax + φ1

(R− η2Θ2C)xmax + φ2

](3.46)

where

N =

[A+ Θ1C +Rσ+ + F1 F1

F2 A+ Θ2C +Rσ− + F2

](3.47)

Proof. From equation (3.43) and considering that x+ = x+ e+ and x− = x− e−, the term H can be

written as:

H =

[Rσ+ −M2 0

0 Rσ− −M1

][e+

e−

]+

[(Rσ+ −M2 − η1Θ1C)x(Rσ− −M1 − η2Θ2C)x

](3.48)

then system (3.40) can be written in the form of (2.53) considering:

b =

[(Rσ+ −M2 − η1Θ1C)x+ φ+(.)(Rσ− −M1 − η2Θ2C)x+ φ−(.)

](3.49)

and matrix N as dened in equation (3.47). From Proposition 3.3, matrix N is also cooperative.

Indeed, matrices G, σ+ and σ− are nonnegatives.

On the other hand, the term b is bounded by a positive vector. Using Hypothesis (3.1) and considering

the following bounds:

Rσ+ −M2 ≤ R and Rσ− −M1 ≤ R

η1 ≤2∆

1−∆= η1 and η2 ≤

2∆1 + ∆

= η2


0 5 10 15 20

−3

0

3

time

x1

0 5 10 15 20

−3

0

3

time

x3

Figure 3.5: Bundle of observers for variables x1 and x3.

we arrive to the error bound of equation (3.46). 2

Because of Remark 3.2, the bound (3.46) is not valid for Chua's system when x1 ∈ [−1, 1].The idea here is to use the positive entries of Fi in order to obtain A cooperative, while using Θi to

obtain matrix N stable and therefore a bounded interval estimation.

Application to Chua's example

All the parameters of Chua's system are supposed to be uncertain, with known upper and lower bounds:

α ∈ [11.49, 12.21], β ∈ [14.45, 15.35], γ ∈ [0.29, 0.31] and a ∈ [−1.17,−1.1], b ∈ [−0.73 − 0.69] whichcorresponds to a ±3% with respect to their real values. For this case we have considered the framer

given by equation (3.37). In order to have matrix A+ ΘC + Fi cooperative, we consider the matrices

F = F1 = F2 of the form:

F =

0 f12 00 0 00 f32 0

(3.50)

Then, in order to full the conditions of Proposition (3.3), we adjust f12 and f32 such that:

fi2 =

|ai2 + θi| if ai2 + θi < 0

0 otherwise, i = 1, 3 (3.51)

We have considered an uniformly distributed noise signal characterized by ∆ = 0.1 and the two

following cases of simulation:

An observers bundle considering 30 framers running in parallel with gains θ1 ∈ [−200 0], θ2 ∈[−500 0] and θ3 ∈ [−100 0]. See g. 3.5.

A simple interval estimate generated by the framer with gain value Θ = [−α − 200 0]T . See g.3.6.

Even though the origin is an unstable point, as it was pointed out in Remark (3.2), the observer

can reproduce the behavior of Chua's system. This is because the error dynamics remain stable when

x ∈]−∞,−1[⋃

]1,+∞[ and when x1 passes into the inner unstable region the error has not enough

time to diverge and the interval observer remains bounded. This is clear in g. 3.4 and g. 3.6:

convergence of both upper and lower bounds of the variable x1 is reached when the real variable is out

of the interval [−1, 1].

3.4. Conclusions 57

0 5 10 15 20

−3

0

3

time

x1

0 5 10 15 20

−3

0

3

time

x3

Figure 3.6: A simple estimate (dotted lines) and best estimate (solid lines) for variables x1 and x3.

0 5 10 15 200

0.5

1

time

conv

erge

nce

inde

x

0 5 10 15 200

0.5

1

time

x3 c

onve

rgen

ce in

dex

Figure 3.7: Index of convergence for variables x1 and x3. Solid line: best estimate, dotted line: simpleestimate.

For the case of parametric uncertainties and noise in the measurement of variable x2, the interval

observer provides acceptable bounds of variables x1 and x3.

3.4 Conclusions

In this chapter we have introduced the concept of observers bundle, as a way to capitalize the guaranteed

estimates provided for various framers. These concepts represent an important improvement with

respect to the initial eorts (see some examples in Chapter 2), where the interval estimate is computed

from an observer with a xed gain.

The example based on a chaotic system illustrates the observers bundle performance, with a regular

reinitialisation. The observer design was developed considering a perfect knowledge framework, and

then we incorporated progressively uncertainties in parameters and noise in measurements. This let

us verify that the original space admissible gains G(θ) becomes more restricted, as the uncertainties

are taken into account.

Chapter 4

Interval observers for nonmonotone

systems

4.1 Introduction

As we have seen in Chapter 2, a central feature of monotone dynamic systems is the preservation of

the partial order of trajectories. This property can be used to construct interval observers, as a useful

tool in order to manage uncertainty of systems. The earlier works of Müller (1926) and Kamke (1932)

introduced theoretical aspects of monotone ows based on dierential inequalities, giving the base of

cooperative systems (Smith, 1995).

In this chapter we deal with the problem of obtaining interval estimates for nonmonotone systems.

We exploit some properties of Lipschitz functions in order to obtain a monotone bounding, allowing

the design of interval observers.

Nonmonotone dynamics can appear in bioprocess modelling. An example is the inhibitory eect on

the growth rate of microorganisms at high substrate concentrations. In particular, we focus on the

design of interval observers for bioprocesses with Haldane kinetics (see Chapter 1 for details).

This chapter begins giving some properties of nonmonotone mappings, leading to two classes of

interval observers for nonmonotone systems. In section 4.3 we present a class of closed loop interval

observer, based on Moisan and Bernard (2005). Section 4.4 is based on Moisan and Bernard (2007b)

and is devoted to the design of a class of open loop interval observer, which considers that a function

of the state variables is measured. This observer is applied to the Agralco industrial anaerobic plant.

4.2 Some properties of non-monotone mappings

Let us focus on a nonlinear non-monotone Lipschitz function ψ(ξ) ∈ C1 : Ω 7→ Rq, where Ω is a compact

subset of Rn.

Property 4.1 The Lipschitz function ψ(ξ) can be written as the dierence of f(ξ) and g(ξ) which are

two increasing functions of ξ:

ψ(ξ) = f(ξ)− g(ξ)

59

60 Interval observers for nonmonotone systems

Proof. Let us consider fj(ξ) = γ∑n

i ξi and gj(ξ) = fj(ξ)−ψj(ξ), where γ ≥ 0 is the Lipschitz constant

of function ψ and j = 1, . . . , q. It is clear that gj(ξ) is increasing since:

∂gj

∂ξi= γ − ∂ψj

∂ξi≥ 0, ∀i = 1, . . . , n

then Property 4.1 holds. 2

Property 4.2 For any Lipschitz function ψ(ξ), there exists a function ψ(ξa, ξb) : Ω × Ω 7→ Rq, such

that:

ψ(ξ, ξ) = ψ(ξ)

[∂ψ

∂ξa

]≥ 0 and

[∂ψ

∂ξb

]≤ 0

Proof. From Property 4.1, it holds that ψ(ξ) = f(ξ) − g(ξ) and therefore, mapping ψ can be written

as ψ(ξa, ξb) = f(ξa)− g(ξb), with f and g monotone increasing. 2

The main consequence of this property is that we have, for the bounds of the argument ξ:

ξ− ≤ ξ ≤ ξ+ ⇒ ψ(ξ−, ξ+) ≤ ψ(ξ, ξ) ≤ ψ(ξ+, ξ−) (4.1)

Moreover, using the generalized Taylor formula, the upper dierence can be written as follows:

ψ(ξ+, ξ−)− ψ(ξ, ξ) =∫ 1

0

∂ψ

∂v(τv + (1− τ)u)dτ (v − u)

=[∫ 1

0

∂f

∂ξ(τξ+ + (1− τ)ξ)dτ −

∫ 1

0

∂g

∂ξ(τξ + (1− τ)ξ−)dτ

](v − u)

= [N1(ξ+, ξ) −N2(ξ, ξ−)](v − u)

(4.2)

where v =

[ξ+

ξ−

]and u =

[ξ

ξ

]. Analogously for the lower bound we have:

ψ(ξ, ξ)− ψ(ξ−, ξ+) =∫ 1

0

∂ψ

∂v(τv + (1− τ)u)dτ (v − u)

=[∫ 1

0

∂f

∂ξ(τξ + (1− τ)ξ−)dτ −

∫ 1

0

∂g

∂ξ(τξ+ + (1− τ)ξ)dτ

](v − u)

= [N3(ξ, ξ−) −N4(ξ+, ξ)](v − u)

(4.3)

with Ni ∈ Rq×n+ , i = 1, ..., 4. This let us introduce the following property.

Property 4.3 If the Jacobian matrix∂ψ

∂v∈ Rq×2n with v =

[ξa

ξb

]is bounded then there exist

4.3. Closed loop interval observer for nonmonotone systems 61

matrices Ni ∈ Rq×n+ , i = 1, ..., 4 such that:

ψ(ξ+, ξ−)− ψ(ξ, ξ) ≤ N1e+ +N2e

−

ψ(ξ, ξ)− ψ(ξ−, ξ+) ≤ N3e+ +N4e

− (4.4)

where e+ = ξ+ − ξ and e− = ξ − ξ−.

Proof. The proof becomes trivial considering equations (4.2) and (4.3). 2

Hypothesis 4.1 We assume that the function ψ can be bounded by a lower and an upper (known)

Lipschitz function. As a consequence, from Property 4.1 there exist two known functions ψ+(ξ1, ξ2, y)and ψ−(ξ1, ξ2, y), increasing with respect to ξ1 and decreasing with respect to ξ2 such that ∀(ξ1, ξ2, y) ∈Ω× Ω× R:

ψ−(ξ1, ξ1, y) ≤ ψ(ξ1, y) ≤ ψ+(ξ1, ξ1, y)

and

ξ1 ≤ ξ ≤ ξ2 ⇒ ψ−(ξ1, ξ2, y) ≤ ψ(ξ, y) ≤ ψ+(ξ2, ξ1, y) (4.5)

4.2.1 Example: Haldane growth rate function

As it was pointed out in Chapter 1, the Haldane equation is a classical nonmonotone growth rate

model. This equation models the inhibitory eect on the biomass growth at high substrate concentra-

tions. We propose to bound the Haldane equation with a function of two variables, which is monotone

with respect to the two variables. Several mathematical expressions for the upper and lower bounds

could be proposed. We propose an expression which is not too conservative in order to keep a good

accuracy for the design of interval observers:

∀s−, s+ : s− ≤ s ≤ s+, µhρ(s−, s+) ≤ µ(s) ≤ µhρ(s+, s−) (4.6)

where

ρ(s−, s+) =s−

s− + ks + s−s+/ki(4.7)

Fig. 4.1 shows a graph of function ρ(s−, s+). It is easy to see that function ρ(s−, s+) remains bounded

for any couple (s−, s+) ∈ R2+.

On this basis we can design interval observers for bioprocesses models with Haldane kinetics. In

what follows we propose two observers.

4.3 Closed loop interval observer for nonmonotone systems

Consider the following class of systems:ξ(t) = Aξ(t) + ψ(ξ, y), ξ(0) = ξ0

y(t) = Cξ(t)(4.8)

where ξ ∈ Ω ⊂ Rn is the state vector of the system, A ∈ Rn×n, C ∈ R1×n and y ∈ R is the system

output. This specic mathematical structure is composed by a linear term and an uncertain nonlinear

function ψ(ξ, y). We will assume in the sequel that ψ(ξ, y) is a C1 Lipschitz function.


ρ(s−

,s+)

s− s+

Figure 4.1: Mapping ρ(s−, s+) for Haldane model.

In the same spirit of the interval observer of Proposition 3.1, a closed loop observer can be written

for system (4.8), with nonmonotone nonlinearity ψ(ξ, y).

Proposition 4.1 If there exist gain vectors Θi = (θi1, ..., θ

in)T , i = 1, 2 such that matrices A + ΘiC

are cooperative, then the following system is a framer of system (4.8):

ζ =

[A+ Θ1C 0

0 A+ Θ2C

]ζ −

[Θ1

Θ2

]y + ψ(·) (4.9)

where ζ = [ξ+ ξ−]T .

The gains Θ1 and Θ2 are the correcting gains between the observer predictions and the real system

output. The nonlinear part ψ(ξ−, ξ+, y) = [ψ+(ξ+, ξ−, y) ψ−(ξ−, ξ+, y)]T is written considering Hy-

pothesis 4.1.

Proof. Let us denote P =

[A+ Θ1C 0

0 A+ Θ2C

], and the error vector e = [e+ e−]T , where

e+(t) = ξ+(t)− ξ(t) and e−(t) = ξ(t)− ξ−(t).We have thus the following dynamics for the error:

e = Pe+ φ(ξ+, ξ−, y) (4.10)

Function φ(ξ+, ξ−, y) is dened as follows.

φ(ξ+, ξ−, y) =

[ψ+(ξ+, ξ−, y)− ψ(ξ, y)ψ(ξ, y)− ψ−(ξ−, ξ+, y)

](4.11)

which is a nonnegative vector. The rest of the proof is the same as the proof of Proposition 3.1. 2

Now, considering the bounds of equation (4.4) we can write a dierential inequality of the form e ≤ Pe,


where:

P =

[A+ Θ1C +N1 N2

N3 A+ Θ2C +N4

](4.12)

Proposition 4.2 If matrix P is stable and condition of Proposition (4.1) are fullled, then the framer

(4.9) converges asymptotically towards the solution of (4.8).

Proof. It follows that P is also cooperative because each Ni is nonnegative. On the other hand

Pe + φ(ξ+, ξ−, y) ≤ Pe, which implies that e?(t) ≥ e(t) ≥ 0, where e? is the solution of the system

e? = Pe?. 2

4.3.1 Application to the AMH1 model

The general mass balance bioprocess model of equation (1.2) can be rewritten in the form of system

(4.8) considering:

A = −DH and ψ(ξ) = Kr(ξ) +Dξin −Q(ξ)

with A = −DH a diagonal matrix. Therefore the choice Θ = 0 provides a matrix P which is both stable

and cooperative. Since bioprocess models are bounded for bounded inputs, this proves the boundness

of the trivial open loop framer for Θ = 0.

Consider the two dimensional bioprocess model with ξ = [x s]T . We assume that the biomass is

measured: y(t) = x(t). It is worth noting that this case, although a bit theoretical, is much more

interesting to illustrate our method than the classical case when the substrate is measured.

The associated model is then the following:x = µ(s)x− αDx

s = D(sin − s)− k1µ(s)xy = x

(4.13)

Hypothesis 4.2 The following hypotheses are considered:

The growth rate function µ(s) is modeled by a Haldane equation.

The parameter µh of the Haldane kinetic is uncertain but bounded by known values:

µ−h ≤ µh ≤ µ+h

The inuent substrate concentration is not accurately measured but is bounded but two known

functions of time:

s−in(t) ≤ sin(t) ≤ s+in(t)

.

For the two dimensional bioreactor model (4.13), matrix A and function ψ(ξ, y) are given by:

A =

[−αD 0

0 −D

]and ψ(s, x) =

[µ(s)x

Dsin − kµ(s)x

](4.14)


Observer design

Considering the bounding of the Haldane equation provided by equation (4.7), we can derive ψ+(s+, s−, x)and ψ−(s−, s+, x) as follows:

ψ+(s+, s−, x) =

[µ+

h ρ(s+, s−)x

−µ−h ρ(s−, s+)x+Ds+in

]

ψ−(s+, s−, x) =

[µ−h ρ(s

−, s+)x−µ+

h ρ(s+, s−)x+Ds−in

] (4.15)

The observer gain is Θ = [θ1 θ2]T and since the biomass is measured C = [1 0]T .The matrix A+ ΘC is expressed by:

A+ ΘC =

[−αD + θ1 0

θ2 −D

]

Cooperativity condition of matrix P is fullled for θ2 ≥ 0. The eigenvalues of matrix A+ ΘC are

λ1 = −αD + θ1 and λ2 = −D.

Since ψ(ξ, y) is bounded for a bounded sin, the boundness of the observer is guaranteed for θ1 < αD.

A framer for the state vector of system (4.13) can be then written as follows:

s+ = D(s+in − s+) +−kµ−h s−x

s− + ks + s+s−/ki+ θ2(x+ − x)

s− = D(s−in − s−) +−kµ+

h s+x

s+ + ks + s+s−/ki+ θ2(x− − x)

x+ = −αDx+ +−kµ−h s−x

s− + ks + s+s−/ki+ θ1(x+ − x)

x− = −αDx− +−kµ+

h s+x

s+ + ks + s+s−/ki+ θ1(x− − x)

(4.16)

Observer positivity

Now we will slightly modify our observer in order to keep a positive lower bound of the estimate.

Indeed, the variables of the original system (4.13) stay positive, and therefore a natural trivial bound

is ξ− = 0. As we can see from the observer equation for s− = 0:

s−∣∣s−=0

= Ds−in −kµhs

+x

s+ + ks+ θ2(x− − x)

which can be of any sign, showing thus that the face s− = 0 is not repulsive. Assuring positive values

of s− will be useful to avoid division by zero in the upper observer equation and also to improve the

nal interval of estimation (because of the positivity of s(t)). To do this, we rst remark that there

exists a positive real ε such that s is always larger than ε. Indeed, we can nd ε such that s|s=ε is

as close as wanted to Dsin (because µh(0) = 0), proving that we will never go under the value ε (we

assume that the initial condition verify s(0) > ε).

We could then take ε into account in the set of lower observer when computing the upper envelope.


0 5 10 15 200

0.5

1

1.5

2

time

Dilu

tion

0 5 10 15 20

72

80

88

time

inpu

t sub

stra

te

Figure 4.2: Dilution and inuent COD.

However it is even more convenient to have a meaningful behaviour of each lower observer, and therefore

we introduce the term:

g(s) =s

s+ ε

in the equation of the lower bound, obtaining then the following expression:

s− = D(s−in − s−) + g(s−)[

−kµ+h s

+x

s+ + ks + s+s−/ki+ θ2(x− − x)

](4.17)

This new equation does exactly match the previous observer for s− ε since g(s−) ' 1. When s− → 0then it ensures that s− stays positive (since g(0) = 0).

Application

The considered bundle of observers is made of 77 framers working with various gains. Specically, we

run the framers considering −10 ≤ θ1 ≤ 2 and 0 ≤ θ2 ≤ 100.Fig. 4.2 shows the dilution input and the inuent substrate with the considered uncertainties. Bounds

for sin and µh have been xed in a ±15% of their real values. Observer initial conditions have been

xed in a large interval in order to be sure that they include the real initial condition (s+(0) = 100and s−(0) = 0).Fig. 4.3 presents the estimates for the substrate. Parameters of the system considered in this applica-

tion are summarized in Table 4.1. The framers were reinitialised more frequently (each 0.5 days) after

a change of value of the dilution input, reinitialising every 2.5 days in steady state conditions.

Table 4.1: System parameters.

parameter meaning value unitsµh maximal growth rate [0.72,1.08] days−1

ks saturation constant 9.28 mmol/lki inhibition constant 256 mmol/lk1 biomass yield conversion 42.14 -α heterogeneity constant 0.5 -


0 5 10 15 200

25

50

75

95

time

subs

trat

e

0 5 10 15 200

50

95

time

subs

trat

e

Figure 4.3: Observers bundle and nal envelope for the substrate.

4.3.2 Some remarks

It is worth noting that function ψ is computed using both values of the upper and lower observers.

This way, the observer involves a strong coupling between the upper bound and the lower bound. The

performance of the observer may then rapidly be degraded if one of these bounds turns out to be too

loose. This is the reason why considering the observer bundle with regular reinitialisation maintains

both bounds in a reasonable range.

Note that we could consider a bundle of observers issued from several observers based on various

bounding functions ψ+ and ψ−. Thus we take benet from the fact that in some regions some bounding

are betters than others, and that it may change with respect to the regions.

4.4 An interval observer using measurements of the nonlinearity

4.4.1 Class of systems and example

We consider a general class of nonlinear systems:ξ = Aξ +Br(ξ) + d, ξ(0) = ξ0

y = r(ξ)(4.18)

where ξ ∈ Ω ⊂ Rn, A ∈ Rn×n is a diagonal and stable matrix, B ∈ Rn is a vector of (at least) known

sign entries and d ∈ Rn is a system input. For the sake of simplicity, we restrict the analysis to a

nonlinear Lipschitz function r(ξ) such that r(ξ) ∈ C1 : Rn 7→ R.The mass balance model (1.2) can be rewritten in the form of system (4.18) considering:

A = −DH, B = K and d = Dξin −Q(ξ)

Our objective is to derive an interval observer for the class of systems (4.18) considering uncertainties

in the measurements, in the function r(ξ), as well as uncertainties in the vector d.

4.4.2 Observer Formulation

Considering that r(ξ) is a Lipschitz function, Properties 4.1 and 4.2 hold. Therefore, r(ξ) can be

written as the dierence of two monotone increasing functions f and g, and there exists a mapping

4.4. An interval observer using measurements of the nonlinearity 67

r(ξa, ξb) : Ω× Ω 7→ Rn, r(ξa, ξb) = f(ξa)− g(ξb) such that:r(ξ+, ξ−)− r(ξ, ξ) = N1(ξ+, ξ)e+ +N2(ξ, ξ−)e−

r(ξ, ξ)− r(ξ−, ξ+) = N3(ξ, ξ−)e+ +N4(ξ+, ξ)e−(4.19)

where e+ = ξ+ − ξ, e− = ξ − ξ−, and matrices Ni ∈ R1×n+ , i = 1, . . . , 4 computed according to

equations (4.2) and (4.3).

Perfect knowledge framework

We consider the following system associated to equation (4.18).ξ+ = Aξ+ + (I − Γ1)By + Γ1Br(ξ+, ξ−) + d

ξ− = Aξ− + (I − Γ2)By + Γ2Br(ξ−, ξ+) + d(4.20)

where Γ1,Γ2 ∈ Rn×n are the observers gains matrices to be tuned.

Let us write the dynamic system associated with the dierential comparison e = [ξ+ − ξ; ξ − ξ−]T .Considering equation (4.19) and after some algebraic manipulations we obtain a system of the type

e = L(ξ+, ξ−, ξ)e where matrix L ∈ R2n×2n is of the form:

L(ξ+, ξ−, ξ) =

A+ Γ1BN1 Γ1BN2

Γ2BN3 A+ Γ2BN4

(4.21)

Remark 4.1 Components of matrix L are not exactly known: matrices Ni, i = 1, . . . , 4, depend on

the unknown state ξ. However the terms BNk have known signs.

Direct cooperative observer.

A rst interval observer is derived, choosing matrices Γk such that matrix L becomes cooperative.

Proposition 4.3 If there exist matrices Γ1 and Γ2 such that matrix L(ξ+, ξ−, ξ) is cooperative, then

system (4.20) is a framer of system (4.18).

Proof. Positivity of the error dynamics is deduced from the cooperativity of matrix L (Smith, 1995).

Considering that B = [bi] andN = [ni]. Note that the construction of Γ such that ΓBNk is nonnegative

is straightforward. It suces to take a matrix Γ whose (xed) sign for the jth column is the sign of bj .

Then ΓB is a positive vector. 2

Remark 4.2 If we choose Γ = 0 we obtain a (stable) asymptotic interval observer with xed conver-

gence rate given by matrix A .

Indirect cooperative observer.

Now let us focus on the terms Fk = ΓkBNk. When their entries are nonnegative (to get a cooperative

matrix L), the stability of matrix L can be aected. We propose a second design, that guarantees the

stability of matrix L through a simple change of base.


The idea consists in choosing a matrix Γ1 (or Γ2) containing zeros everywhere except on the kth

row. This has the opposite signs of B, i.e. γk,jbj ≤ 0. This leads to a matrix L of the form:

L =

a11 . . . 0. . .

...

(−)k1 . . . (−)kk . . . (−)k×2n

.... . .

0 . . . ann

(4.22)

this means, the kth line of L is negative and the rest of the matrix has a diagonal form whose entries

are the same entries of matrix A.

We can now propose the following interval observer.

Proposition 4.4 Choosing Γ1 and Γ2 such that matrix L(ξ+, ξ−, ξ) has the form (4.22), and moreover

if ξ−i (0) ≤ ξi(0) ≤ ξ+i (0) for i 6= k and ξ+k (0) ≤ ξk(0) ≤ ξ−k (0) then system (4.20) is an interval observer

of system (4.18): ∀t ≥ 0, ξ−i (t) ≤ ξi(t) ≤ ξ+i (t) for i 6= k and ξ+k (t) ≤ ξk(t) ≤ ξ−k (t).

Proof. Assume k ≤ n and consider a change of variable where the kth and (k + n)th variables of the

error system are multiplied by (−1):

[e1 . . . ek . . . ek+n . . . e2n]T = [e1 . . .− ek . . .− ek+n . . . e2n]T

This leads to the error dynamics ˙e = L(ξ+, ξ−, ξ)e, where e(0) ≥ 0 and L is a cooperative matrix (after

this variable change, the kth line of matrix L becomes positive, except for the diagonal element lkk).

The rest of the proof is the same as for Proposition 4.3. 2

Remark 4.3 The observers based on Proposition 4.4 lead to a class of observers for which the matrix

L is cooperative and stable. The observers based on Proposition 4.3 may lead to unstable estimates.

Uncertainty framework

Now we consider the case where the function r(ξ), the input d and the available measurement y are

not perfectly known but upper and lower bounded by two known functions. This is formalized in the

following hypothesis:

Hypothesis 4.3 Function r(ξ1, ξ2) is assumed to be bounded by two known functions r+(ξ1, ξ2) and

r−(ξ1, ξ2).

Hypothesis 4.4 A bounded noise δ such that δ ≤ ∆ < 1, perturbs the system output. We assume that

this noise is of multiplicative nature.

y

1 + ∆≤ r(ξ) ≤ y

1−∆(4.23)

Hypothesis 4.5 The input vector d is unknown but bounded: d ≤ d ≤ d.

Now the observer is reformulated considering the bounds on the uncertain terms. Consider the following

observer candidate: ˙ξ+ = Aξ+ + y+(I − Γ1)B + r+(ξ+, ξ−)Γ1B + d+

˙ξ− = Aξ− + y−(I − Γ2)B + r−(ξ−, ξ+)Γ2B + d−(4.24)

4.4. An interval observer using measurements of the nonlinearity 69

where y+, y−, r+(ξ+, ξ−), r−(ξ−, ξ+), d+ and d− ∈ Rn are vectors constructed using the known

bounds of the uncertainties and taking into account the signs of Γi. The choice of these bounds is

straightforward: the objective is to obtain error dynamics of the form e = Le+ φ, where matrix L is,

as in equation (4.21), either a cooperative matrix (direct cooperativity) or a diagonal matrix plus a k

negative row (Equation (4.22)). Vector φ is a residual vector generated be the comparison of system

(4.24) and system (4.18). It is either a nonnegative vector (direct cooperativity), or nonnegative with

the k nonpositive row (indirect cooperativity).

4.4.3 Application to an industrial anaerobic digestion process

We apply framer equation (4.24) to the interval estimation of the total biomass and substrate of an

industrial anaerobic digestion process. We use the data from Agralco plant, presented in Chapter 1.

Let us recall the two dimensional model (4.13) with ξ = [x s]T and assume that the methane gaseous

ow is measured: y = k2r(ξ). x = r(ξ)−Dx

s = D(sin − s)− k1r(ξ)y = k2r(ξ)

(4.25)

Equation (4.25) has the form (4.18) with:

A =

[−1 00 −1

], B =

[1−k1

]and d =

[0sin

](4.26)

The reaction rate is written as r(ξ) = xµ(s), where µ(s) is a Haldane equation.

The critical part of the observer design correspond to the formulation of bounds for the function

r(ξ). Let us consider the bounds for the Haldane model provided by equation (4.7) and also an

uncertainty on the parameter µh. The nal bounds on the reaction rate function r(ξ) are:

r+(x+, s+, s−) = xµ+

h s+

s+ + ks + s−s+/ki

r−(x−, s−, s+) = x−µ−h s

−

s− + ks + s−s+/ki

The gains are chosen from the signs of B = K (see Equation (4.26)): b11 > 0 and b21 < 0.

The following sign congurations for the gains Γ1 = Γ2 = Γ have been considered:

γ11 > 0, γ22 < 0, γ12 < 0 and γ21 > 0.

That makes L cooperative, fullling conditions of Proposition 4.3.

γ11 < 0 and γ12 > 0 with γ22 = 0 and γ21 = 0.

γ21 < 0 and γ22 > 0 with γ11 = 0 and γ12 = 0.

That transforms matrix L into the particular structure (4.22).

We have run a broad set of framers including Γ = 0 (which guarantees the existence of an interval

observer, according to Property 3.1), and 40 direct/indirect cooperative observers with gain values


0 10 20 30 400

20

40

60

time [days]

biom

ass

[g/l]

0 10 20 30 40

0

10

20

30

40

50

time [days]

CO

D [

g/l]

Figure 4.4: Total biomass and substrate interval estimates.

varying in the interval γij ∈ [−24, 24].A 3% multiplicative noise on the measurement has been assumed in order to derive the bounds.

Bounds for the unknown inuent substrate sin are known to uctuate around ±30% of the real value.

Fig. 4.4 shows the estimates performed by the proposed set of observers (in solid lines, only the

bundle envelope is presented) for the biomass and the substrate in the reactor. It can be seen that

the convergence is much more rapid than for the asymptotic observer (dashed lines), especially for the

substrate estimation.

Remark 4.4 The combination of the two types of framers (direct and undirect cooperativity) associated

with various gain values, allowed a strong improvement of the observer performances. The method can

be straightforwardly extended to more complex systems with a known vector function r(ξ).

4.5 Conclusions

A new class of interval observers has been proposed, managing a wide uncertainty framework of a

class of nonlinear systems. The presented observers are designed in order to full particular monotone

conditions for the error dynamics, considering that the original system dynamics are nonmonotone.

It is worth noting that we have transformed the n dimensional nonmonotone system into a monotone

system in dimension 2n.The developed approaches were illustrated with a model of an anaerobic digestion process where the

Haldane function represents growth rate inhibition at high concentration, and for which the design of

a classical observer is still an open problem.

Two interval observers were proposed. The rst one illustrated in simulation the principle of interval

estimation for nonmonotone systems, with a rather theoretical example. The second observer was

tested considering an industrial setup. The good convergence properties of the observer illustrate the

method eciency, and its potentiality of enhancing the convergence rate, especially compared with

classical asymptotic observers (see chapter 2).

Chapter 5

Optimality criterion and reverse time

observers

5.1 Introduction

In the previous chapters we have introduced the concept of bundle of observers. This allows to improve

the interval estimation result, by comparing various framers. Until now, no hint about how to chose

(or tune) the framers gains have been given: the best estimate could be obtained by exploring the

admissible space of gains, and compare as much as possible framers belonging to this space. One of the

objectives of this chapter is to propose an optimality criterion, leading to the denition of an optimal

observer. We design a new interval observer which is based on a bounded error observer, as proposed in

Lemesle and Gouzé (2005), that makes use of a loose approximation of the growth rate. We show that

the proposed optimality criterion (that applies to any interval observer) provides a gain set containing

the best framers, increasing the eciency of a bundle of observers.

A second part of this chapter is devoted to the design of reverse time interval observers. Running

observers in reverse time has been rst proposed in Auroux and Blum (2005) for data assimilation. We

show that running interval observers in reverse time can be very ecient under specic assumptions.

Reverse time observers improves dramatically the convergence rate leading to a discontinuous interval

estimation procedure.

This chapter is organized as follows. In section 5.2, we present a new interval observer for a

biotechnological model. This is followed by the proposition of an optimality criterion (Moisan et al.,

2007). Section 5.3 is based on Moisan and Bernard (2007a) and it is an extension of the proposed

methodology, where reverse time observer are designed. The method is illustrated considering its

application to an industrial biotechnological process.

71

72 Optimality and reverse time observers

5.2 Optimality of interval observers

5.2.1 Denitions and Hypotheses

Let us consider the mathematical model of a perfectly mixed bioreactor, where the substrate s is online

measured:x = µ(s)x− ux

s = D(sin − s)− kµ(s)xy = s

(5.1)

In the sequel we assume that the following hypothesis is fullled.

Hypothesis 5.1 We assume that the growth rate is bounded by two known functions µ(s) and µ(s)and a positive constant a:

1. 0 ≤ µ(s) ≤ µ(s) ≤ µ(s) ≤ a, ∀s ≥ 0

2. µ(0) = 0

Our objective is to develop a robust observer based on interval approach with improved convergence

properties in order to estimate the biomass concentration x(t) in a bioreactor, when monitoring the

substrate s(t).

5.2.2 A bounded error interval observer

In the sequel we will consider the class of observers introduced in (Lemesle and Gouzé, 2005). These

observers do not converge to zero but to a bounded error (see Denition 2.13).

Bounded error observers

A bounded error observer for the variable x(t) of system (5.1) can be derived, considering that the

system dynamics (growth rate) are poorly known. Let us introduce the variable:

z = kx+ θ(t)s (5.2)

where θ(t) ∈ C1(R) is a gain that will be discussed later on. Considering equation (5.1), the dynamics

of z can be written as follows:

z = (1− θ)µ(s)(z − θs) +D(θsin − z) + θs (5.3)

Since the inuent substrate sin is well known, the following bounded error observer can be derived:

Proposition 5.1 The following system is a bounded error observer of (5.1)

˙z = (1− θ)µ(s)(z − θs) +D(θsin − z) + θs

x = (z − θs)/k(5.4)

where the function µ(s) is such that |µ(s)− µ(s)| < a, and a is a positive real.

5.2. Optimality of interval observers 73

Proof. See (Lemesle and Gouzé, 2005). 2

One can note that function µ(s) can be e.g. any of the known functions µ(s) or µ(s) introduced in

Hypothesis 5.1, fullling the previously mentioned condition.

Remark: I f θ = 1 then equation (5.4) becomes the classical asymptotic observer (see (Bastin and

Dochain, 1990)).

The idea used in (Lemesle and Gouzé, 2005) was to implement a hybrid observer based on equation

(5.4). The estimations start with a high and positive value of θ which decreases to one along time.

From a methodological point of view, the hybrid approach combines two type of observers: a bounded

error observer with high convergence rate but poor accuracy and an asymptotic observer with xed

convergence rate but high accuracy. Even though this observer demonstrated to be more ecient than

asymptotic observers, adjusting the gain θ seems rather complicated. This last issue can be overcome

using properties of interval estimates, as explained in the next sections.

In the sequel we consider the following hypothesis.

Hypothesis 5.2 The inuent substrate sin is an unknown bounded input of system (5.1). However,

we know bounds for this input such that:

s−in(t) ≤ sin(t) ≤ s+in(t) (5.5)

We denote e−in = sin − s−in, e+in = s+in − sin and ein = s+in − s−in.

Interval observers

Let us propose analytical expressions for framers of the variable x of system (5.1). These expressions

are deduced from equation (5.4).

Proposition 5.2 For a gain θ ∈ C1(R), the following system denes a framer for system (5.1). The

framer depends on the value of θ(t) as follows:

• for θ(t) < 0zθ = (1− θ)(µ(s)zθ − θµ(s)s) +D(θs−in − zθ) + sθ

zθ = (1− θ)(µ(s)zθ − θµ(s)s) +D(θs+in − zθ) + sθ(5.6)

• for 0 ≤ θ(t) < 1zθ = (1− θ)(µ(s)zθ − θµ(s)s) +D(θs+in − zθ) + sθ

zθ = (1− θ)(µ(s)zθ − θµ(s)s) +D(θs−in − zθ) + sθ(5.7)

• for θ(t) ≥ 1zθ = (1− θ)(µ(s)zθ − θµ(s)s) +D(θs+in − zθ) + sθ

zθ = (1− θ)(µ(s)zθ − θµ(s)s) +D(θs−in − zθ) + sθ(5.8)

with

xθ = (zθ − θs)/k and xθ = (zθ − θs)/k (5.9)


Proof. The proof is carried out for θ ≥ 1 (the same arguments hold for the other framers). Let us

consider the dierence e between the upper candidate estimate and the unknown state, this is:

e = xθ − x = (zθ − z)/k (5.10)

Its dynamics are:

e = (1− θ)((µ(s)zθ − µ(s)z)− θ(µ(s)− µ(s))s

)+D(θe+in − ke) (5.11)

Now we show that e stays nonnegative after its nonnegative initialisation. We consider the rst time

instant t? when e = 0, i.e. x = x and zθ = z, then:

e(t?) = (1− θ)((µ(s)− µ(s))z − θ(µ(s)− µ(s))s

)+Dθe+in (5.12)

Considering that z is positive (because of the positivity of the state and θ ≥ 1) and Hypothesis 5.1,

we have:

e(t?) ≥ 0 (5.13)

which guarantees that the error will stay positive after t?.

Using similar arguments it is possible to show that e(t) ≥ 0 and therefore the interval that contains

the state is positive:

e = xθ − xθ = e+ e ≥ 0 (5.14)

2

Let us detail two cases of specic interest.

The cases θ = 0 and θ = 1

The framer for θ = 1 is the interval version of the classical asymptotic observer (see chapter 2): it does

not depend on the biological kinetics, however it has a xed convergence rate given by D.

Property 5.1 The framer [xθ=1(t), xθ=1(t)] provides a bounded interval estimation of variable x(t).

Proof. This property is straightforward and relies on the boundedness of sin. 2

Property 5.2 The framer xθ=0 is nonnegative, if xθ=0(0) ≥ 0.

Proof. The proof is trivial. 2

Note that the framer for θ = 0 may be unstable.

As a consequence, the framers obtained for θ = 0 and θ = 1 will provide a guaranteed upper and

nonnegative lower bound for the state.

Property 5.3 The interval I = [Bsup,Binf] is bounded if θ(t) ≡ 1 ∈ Θ.

Proof. First, it is worth to point out that if one framer is bounded, then Bsup and Binf are bounded

too. Boundedness of the interval I is a direct consequence of Property 5.1. 2

Property 5.4 The lower bound best value Bsup is positive if θ ≡ 0 ∈ Θ.

Proof. This is a straight consequence of Property 5.2, which provides a lower positive framer ∀t. 2


5.2.3 Computing the optimal gain

Running in parallel a dense enough bundle (i.e. with a large number of framers) in order to obtain the

best interval [Bsup,Binf] can be time consuming without guarantee of optimal estimation. Therefore,

a characterization of the gain set that can generate the best solutions is proposed.

Denition of an optimality criterion

Let us assume that at time t the best estimate [x0θ(t), x

0θ(t)] is provided. The best upper [resp. lower]

interval observer will be given by the value of θ(t) that minimizes [resp. maximizes] x [resp. x]. The

proposed criterion consists then in nding a pair of gains θ(t) and θ(t) which respectively minimizes and

maximizes x and x at any time instant t (see g. 5.1). More formally, the idea consists in optimizing

the following functional:

J(θ, x, x, y) = xθ

J(θ, x, x, y) = xθ

(5.15)

The optimal gains θ and θ verify:

J(θ(s, x, x)) = minθJ(θ, s, x, x)

J(θ(s, x, x)) = maxθJ(θ, s, x, x)

(5.16)

For the proposed observer, based on the change of variables (5.3), the functional (5.15) is expressed

by:J(θ, x, x, y) = (zθ − θ(t)s− θ(t)s)/kJ(θ, x, x, y) = (zθ − θ(t)s− θ(t)s)/k

(5.17)

where the mappings θ 7→ zθ and θ 7→ zθ are piecewise continuous with respect to θ, determined by

equations (5.6), (5.7) and (5.8). Using these expressions, we can write J(x, x, s, θ) and J(x, x, s, θ) as

Figure 5.1: Concept associated to the optimality criterion.


polynomials with respect to the gain θ:

J i(x, x, s, θ) = ai(x, x, s)θ2 + bi(x, x, s)θ + ci(x, x, s)J i(x, x, s, θ) = ai(x, x, s)θ2 + bi(x, x, s)θ + ci(x, x, s)

(5.18)

where i = 1 for θ < 0, i = 2 for 0 ≤ θ < 1 and i = 3 for θ ≥ 1. See Box 5.1 for a detailed computation

of the coecients of equation (5.18). J is drawn in g. 5.2 with respect to θ. It can be easily veried

that for θ ≥ 1 there exists a global nontrivial solution that minimizes J [resp. maximizes J ] denoted β

[resp. β]. On the other hand, for θ < 1, the optimal gain value is θ = 0. Thus, there are two candidatesto solve the optimization problem (5.16) for θ ∈ R:

θ(t) ∈0,max1, β

(5.19)

θ(t) ∈0,max1, β

(5.20)

where

β(t) =s∆µ −De−in + k(µ(s)x− µ(s)x)

2s∆µ

β(t) =s∆µ −De+in − k(µ(s)x− µ(s)x)

2s∆µ, ∆µ = µ(s)− µ(s)

(5.21)

Remark 5.1 All the framers based on gains which belong to the set A = θ ∈ (−∞, 1) − 0 will

never provide the best estimate. As a consequence we eliminate framers (5.6) and consider framers

(5.7) only for the case θ ≡ 0 to compute the interval observer.

It is very important to note that the optimal solution θ and θ cannot be computed because they

depend on the unknown state x(t). We must therefore try to locate the region where the optimal

values for θ lie, and run as many framers as possible in this region. Indeed, intervals that contain

the nontrivial optimal solution β and β can be determined on the basis of the present state esti-

mate. Let us focus on the set θ ≥ 1 (where β and β live) to dene the bounds of this interval.

Figure 5.2: Portrait of J(θ, x, x, y) with respect to θ. θ ∈0,max1, β

.


Box 5.1 Coecients of polynomial (5.18).

• for θ < 0a1 = 0, a1 = 0b1 = µ(s)x− µ(s)x−De−in/k

b1 = µ(s)x− µ(s)x+De+in/k

c1 = (µ(s)−D)xc1 = (µ(s)−D)x

• for θ ∈ [0, 1[a2 = −s∆µ/k, a2 = s∆µ/k

b2 = µ(s)x− µ(s)x+ (s∆µ +De+in)/kb2 = µ(s)x− µ(s)x− (s∆µ +De−in)/kc2 = (µ(s)−D)xc2 = (µ(s)−D)x

• for θ ≥ 1a3 = s∆µ/k, a3 = −s∆µ/k

b3

= µ(s)x− µ(s)x− (s∆µ −De+in)/kb3 = µ(s)x− µ(s)x+ (s∆µ −De−in)/kc3 = (µ(s)−D)xc3 = (µ(s)−D)x

It is possible to verify that:

1. J1(0) = J2(0) and J2(1) = J3(1). Then, the piecewise function J(θ) is acontinuous function. The same statement holds for the function J(θ).

2. For any θ < 0 we have that J1(θ) and J1(θ) are straight lines with positive and

negative slopes respectively. Indeed:

b1 = µ(s)x− µ(s)x−De−in/k ≤ 0b1 = µ(s)x− µ(s)x+De+in/k ≥ 0

3. For any θ ∈ [0, 1[, J2(θ) and J2(θ) are convex and concave parabolas respectively.

4. For any θ ≥ 1, J3(θ) and J3(θ) are concave and convex parabolas respectively.

Bounding of the optimal gain

The objective is to compute the sets that contains the unknown optimal gain values, and run an

observer bundle considering these values, to get as close as possible to the optimal framer.

Bounding at time t

The following hypothesis is not mandatory, but it simplies signicantly the computation of the optimal

interval.

Hypothesis 5.3 The unknown function µ(s) can be written as µ(s) = γρ(s), where ρ(s) is a known


C1 function and γ = [γ, γ].

Proposition 1 At any time t, the optimal nontrivial gains β(t) and β(t) are in the following intervals:

β(t) ∈ [1, ϕ(t)] and β(t) ∈ [1, ϕ(t)] (5.22)

with

ϕ(t) =12

+kγ∆B(t)2s(γ − γ)

and ϕ(t) =12

+kγ∆B(t)2s(γ − γ)

(5.23)

where ∆B(t) = xθ=1(t)− xθ=1(t) and s is the solution of the equation

s = u(s−in − s)− kγρ(s)xθ=1 (5.24)

reinitialized at tk such that s(tk) = s(tk). In the same way xθ=1(tk) = Binf(tk) and xθ=1(tk) = Bsup(tk).

Proof. First it is clear from Property 5.1 that x(t) ∈ [xθ=1(t), xθ=1(t)]. It follows that s ≥ s, and since

s(tk) = s(tk), we have s ≥ s. Finally β(t) ≤ ϕ(t) and β(t) ≤ ϕ(t). 2

From a practical point of view, our strategy consists in splitting the intervals (5.22) into N parts

in order to get N + 1 gain values for each lower and upper bundles.

θi =i+ ϕ(t)(N − i)

Nand θi =

i+ ϕ(t)(N − i)N

(5.25)

for i ∈ 0, .., N.Remark: A better upper bound could be proposed if s is kept in equation (5.23). However the

computation of θ in equation (5.3) would then involve an estimate of s. This is avoided with the

proposed method (it can be checked that ∆B does not need the computation of s).

5.2.4 Biased output

Now we consider a bounded noise aecting the system output.

Hypothesis 5.4 Online measurement s(t) is perturbed by a noise δ(t). We assume that this pertur-

bation is of multiplicative nature: y(t) = s(t)(1 + δ(t)).

Moreover, the noise is bounded by ∆ ∈ R+ such that |δ| ≤ ∆ < 1.

Considering that s(t) is a positive variable, Hypothesis 5.4 implies that we know two bounds for this

quantity:y(t)

(1 + ∆)≤ s(t) ≤ y(t)

(1−∆)(5.26)

We rewrite the framer equations (5.8) and (5.7) (see remark 5.1) taking into account the output

uncertainty:

Proposition 5.3 The following system is a framer of system (5.1):


• for θ ≥ 1

zθ = (1− θ)(ν(y, y)zθ − θν(y, y)y) +D(θs+in − zθ) + θ(εy + (1− ε)y)

zθ = (1− θ)(ν(y, y)zθ − θν(y, y)y) +D(θs−in − zθ) + θ(εy + (1− ε)y)

xθ = (zθ − θy)/k and xθ = (zθ − θy)/k

(5.27)

• for θ = 0

xθ=0 = (ν(y, y)− u)xθ=0 and xθ=0 = (ν(y, y)− u)xθ=0 (5.28)

This means that if x0 ≤ x0 ≤ x0 then x(t) ≤ x(t) ≤ x(t) for any time. The functions ν(.) and ν(.) aredened such that:

ν(y, y) = minq∈[y,y]

µ(q) and ν(y, y) = maxq∈[y,y]

µ(q) (5.29)

and ε =

1 if θ ≥ 00 otherwise

, uses the sign of θ in order to provide the correct bounding.

Proof. The same arguments as in the proof of Proposition 5.2 can be applied. Considering the

upper bound error equation for θ ≥ 1 at the time instant t? where e = 0 we have:

e(t?) = uθe+in + θ(εy + (1− ε)y − s) + (1− θ)((ν(y, y)− µ(s))z? + θ(ν(y, y)y − µ(s)s)

)(5.30)

which under the stated assumptions veries e(t?) ≥ 0 and therefore e(t) stays positive after t? for θ ≥ 1(the proof for θ = 0 is trivial). 2

The same principles to design an observer bundle as in Section 5.2.3 can now be applied. The already

computed bounds for the optimal gain keep the same expressions, except for the reinitialisation of s in

Proposition 1: s(tk) = y(tk)(1+∆) .

5.2.5 Application

We present two applications of the developed method: a simulation for a bioreactor working in batch

mode (and thus an asymptotic observer does not converge) and the application to an industrial anaer-

obic digestion plant.

Bounds for the unknown inuent substrate sin are known to uctuate in a ±30% range (see g. 5.3).

For both cases we have considered a reinitialisation time ∆t = 2 [days]. The observer initial condition

x0 = 0, x0 = 100 has been chosen very large in order to assess the convergence properties of the

method. Gains values were selected considering a homogeneous partition of 8 framers on the gain

interval (in open loop) of equation 5.22, including the gains θ = 0 and θ = 1 to guarantee stability and

positivity of the predicted intervals.


0 10 20 30 40 50 60 70

0

0.02

0.04

0.06

0.08

time [days]

dilu

tion

[1/h

]

0 10 20 30 40 50 60 70

25

35

45

time [days]

Infl

uent

CO

D [

mg/

l]

Figure 5.3: Dilution (batch mode) and inuent COD.

0 10 20 30 40 50 60 700

25

50

75

100

time [days]

biom

ass

[g/l]

0 10 20 30 40 50 60 700

0.25

0.5

0.75

1

time [days]

conv

erge

nce

inde

x

Figure 5.4: Biomass estimates and convergence index for a bioreactor in batch mode.

Simulation in batch mode

It is well known that the classical asymptotic observer (θ = 1) does not converge when the bioreactor is

working in batch mode. In fact, one can take D = 0 and θ = 1 in equation (5.11) and verify that e = 0.In g. 5.4 we demonstrate the eciency of our observer, even in this unfavourable condition. For

simulation purposes, we run model (5.1) considering a dilution input D[1/h] with two batch periods

(see g. 5.3). Biomass interval estimates and observer convergence index can be seen in g. (5.4). The

convergence properties of the observer bundle (solid lines) are clearly demonstrated, despite the batch

phase. The obtained results are compared with an interval asymptotic observer (dashed lines).

Application to the Agralco plant

We considered a noise aecting the output featured by ∆ = 0.03 (see equation (5.26)). Fig. 5.5 shows

the observer performance. The estimation performed for the bundle of observers are compared with the

classical asymptotic interval observer. The convergence properties of the bundle of observers, provides

estimates that become accurate much more rapidly than the asymptotic observer. The lower and upper

gain sets are shown in g. 5.6, evolving discontinuously because of the update of their values at each

reinitialisation time instant tk. At the end of the estimations, it is worth noting that the upper bounds

of the gain intervals progressively reduce to one, following thus a framework similar to (Lemesle and

Gouzé, 2005).

5.3. Reverse time interval observers 81

0 10 20 30 40 50 60 700

25

50

75

100

time [days]

biom

ass

[g/l]

0 10 20 30 40 50 60 700

0.25

0.5

0.75

1

time [days]

conv

erge

nce

inde

x

Figure 5.5: Biomass estimates and convergence index for AGRALCO plant.

0 10 20 30 40 50 60 700

100

200

300

400

time [days]

low

er b

undl

e ga

ins

0 10 20 30 40 50 60 700

100

200

300

400

time [days]

uppe

r bu

ndle

gai

ns

Figure 5.6: Gain sets for lower and upper framers.

5.3 Reverse time interval observers

A new interval estimation concept is obtained when considering reverse time dynamics of the original

system. Designing interval observers for these new dynamics provides a new set of interval estimates,

complementing the estimates performed by the observers running in direct time. Let us focus on the

time interval [tk, tk+1]. The objective is then to improve the estimates at the time instant tk+1 using

direct and reverse time interval observers.

Consider reverse time dynamics of system (5.1) in the time interval [tk, tk+1] and the time change

of coordinates tr = 2tk+1 − t. Denoting by q(tr) the reverse time dynamics of variable z(t), they can

be directly deduced from equation (5.3):

q = (γ − 1)µ(s)(q − γs) +D(q − γsin) + γs (5.31)

where ´ denotes the derivative with respect to tr. The following interval observer can be derived,

provided the reverse time dynamics of system (5.1).

Proposition 5.4 Let us denote xf = x(tk+1) and assume that an upper and lower bounds for xf

are known (they are indeed the result of the direct bundle at tk+1): xf ∈ [xf , xf ]. Then for a gain

γ ∈ C1(R), the following system denes an interval observer for system (5.1). The interval estimate

depends on the value of γ as follows:

• for γ < 0qγ = (γ − 1)(µ(s)qγ − γµ(s)s) +D(qγ − γs+in) + sγ

qγ

= (γ − 1)(µ(s)qγ− γµ(s)s) +D(q

γ− γs−in) + sγ

(5.32)


• for 0 ≤ γ < 1qγ = (γ − 1)(µ(s)qγ − γµ(s)s) +D(qγ − γs−in) + sγ

qγ

= (γ − 1)(µ(s)qγ− γµ(s)s) +D(q

γ− γs+in) + sγ

(5.33)

• for γ ≥ 1qγ = (γ − 1)(µ(s)qγ − γµ(s)s) +D(qγ − γs−in) + sγ

qγ

= (γ − 1)(µ(s)qγ− γµ(s)s) +D(q

γ− γs+in) + sγ

(5.34)

with

xγ(tr) = (q(tr)− γs(tr))/k and xγ(tr) = (q(tr)− γs(tr))/k (5.35)

Proof. The proof can be carried out considering the same arguments as in the proof of Proposition 5.2.

2

5.3.1 Estimation procedure

The procedure consists rst in providing a standard interval estimation using the envelope of an observer

bundle on time interval [tk, tk+1]. Then, the best estimate [xa(tk+1), xa(tk+1)] is used to initialize a

reverse time observer in order to improve the state interval estimate at tk: [xb(tk), xb(tk)]. If this

improvement can be reached, the direct time observer initialized with [xb(tk), xb(tk)] will provide

an improved estimate at time tk+1: [xb(tk+1), xb(tk+1)]. The procedure is repeated until no more

improvement at time instant tk+1 is obtained. Then the same procedure is applied on [tk+1, tk+2]. Theidea is summarized on g. 5.7.

We use the following property to guarantee that an improvement of the initial condition will improve

the estimates of an (improved) direct time bundle of observers.

Property 5.5 Given two sets of initial conditions xak, x

ak and xb

k, xbk such that xa

k ≥ xbk and xa

k ≤ xbk

and gain sets Θ1 and Θ2 such that Θ1 ⊂ Θ2, then:

Binf(t, xak,Θ1) ≥ Binf(t, xb

k,Θ2)Bsup(t, x

ak,Θ1) ≤ Bsup(t, x

bk,Θ2)

(5.36)

Proof: The proof is straightforward. Let us denote θ?(t) ∈ Θ1 the gain sequence that generates the best

estimates Binf(t, xak) and Bsup(t, x

ak). Since θ?(t) is also belonging to Θ2, then it will provide a better

estimate since the bundle of observers envelope keep the partial order with respect to it initialization.

Finally the estimates of the other observers within the bundle Θ2 may still improve this estimate. 2

As a consequence, each time we run a direct time observer bundle on [tk, tk+1], we keep the previ-

ously computed optimal gain in order to ensure that the interval estimates will be improved.

5.3.2 Nearly optimal gain in reverse time

We adapt the optimality criterion (5.16) to the reverse time observer in order to nd the value for the

gain γ that provides the best estimate. Functions γ 7→ qγand γ 7→ qγ are piecewise continuous with


respect to the gain γ. Using expressions (5.32), (5.33) and (5.34), we can write xγ = g(x, x, s, γ) andxγ = g(x, x, s, γ) as polynomials:

gi(x, x, s, γ) = u(x, x, s)iγ2 + vi(x, x, s)γ + wi(x, x, s)

gi(x, x, s, γ) = ui(x, x, s)γ2 + vi(x, x, s)γ + wi(x, x, s)

(5.37)

where i = 1 for θ < 0, i = 2 for 0 ≤ θ < 1 and i = 3 for θ ≥ 1. See Box 5.2 for a detailed computation

of the coecients of equation (5.37). The graph of function g is represented in g. 5.8.

Box 5.2 Coecients of polynomial (5.37)

• for γ < 0u1 = 0, u1 = 0v1 = −µ(s)x+ µ(s)x−De+in/k

v1 = −µ(s)x+ µ(s)x+De−in/k

w1 = (D − µ(s))xw1 = (D − µ(s))x

(5.38)

• for γ ∈ [0, 1[u2 = −s∆µ/k, u2 = s∆µ/k

v2 = −µ(s)x+ µ(s)x+ (s∆µ +De−in)/kv2 = −µ(s)x+ µ(s)x− (s∆µ +De+in)/kw2 = (D − µ(s))xw2 = (D − µ(s))x

(5.39)

• for γ ≥ 1u3 = s∆µ/k, u3 = −s∆µ/k

v3 = −µ(s)x+ µ(s)x− (s∆µ −De−in)/kv3 = −µ(s)x+ µ(s)x+ (s∆µ −De+in)/kw3 = (D − µ(s))xw3 = (D − µ(s))x

(5.40)

It is possible to verify that:

1. g1(0) = g2(0) and g2(1) = g3(1).Then, the piecewise function g(γ) is a continuous function for all γ. The same

statement holds for the function g(γ).

2. For any γ ∈ [0, 1[, g2(γ) and g2(γ) are convex and concave parabolas respectively.

3. For any γ ≥ 1, g3(γ) and g3(γ) are concave and convex parabolas respectively.

Moreover, the maximum and minimum of these parabolas are obtained for gains lower

than 1/2, in fact:

minγg3(γ)

=12

+D(s−in − sin) + k(µ(s)x− µ(s)x)

2s∆µ≤ 1

2(5.41)

The same holds analogously for the lower bound.


Figure 5.7: Estimation procedure with reverse time framers.

It is straightforward to verify that:

For γ ≥ 1 the minimum of the associated parabola is obtained for a value γ? < 1.

For γ ∈ [0, 1[, g2(γ) and g2(γ) are convex and concave parabolas respectively.

For γ < 0 the function g [resp. g ] is a straight line whose slope depends on the uncertainty on

the input sin and on the real state. That is:

v1 = −µ(s)x+ µ(s)x−De+in/k

v1 = −µ(s)x+ µ(s)x+De−in/k(5.42)

The following proposition allows to compare solutions of observers running in direct and reverse

time. It proposes a criterion to evaluate if the reverse time observer will improve (at time tk+1) the

interval estimate.

Proposition 5.5 The reverse time interval estimates improve the direct time best estimates if at the

time instant tk+1:

−Binf − xγ |t=tk+1 ≥ 0−Bsup − xγ |t=tk+1 ≤ 0

(5.43)

Proof. Let us dene the following variable:

∆(t′) = Binf(2tk+1 − t′)− xγ(t′) (5.44)

This system is initialized for t′ = tk+1 such that ∆(tk+1) = 0, i.e. Binf(tk+1) = xγ(tk+1). Since by

denition, ∆(tk+1) ≥ 0, it implies that ∆ stays positive for a nonempty time interval: Binf(tr) ≥ xγ(tr).The same can be obtained analogously for the lower bound.

Now the following proposition holds:

Proposition 5.6 Based on the sign of v1 and v1, the optimal value for the gain γ can be tuned:

a. If v1(tk+1) > 0 [resp. v1(tk+1) < 0] then a reverse time observer with a high enough negative gain


Figure 5.8: Piecewise function g(γ).

value γ improves the estimates performed by a direct time observer bundle.

b. If v1(t) ≤ 0 [resp. v1(t) ≥ 0] ∀t ∈ [tk, tk+1], then reverse time observers do not improve the estimates

performed by a direct time observer bundle.

Proof.

a. The reverse time estimates xγ [resp. xγ ] can be as decreasing [resp. increasing] as desired at tk+1

selecting high enough negative gain values γ. This leads to the improvement in reverse time of the

best estimates performed in direct time.

b. In this case the gain values γ = 0, 1 are the only candidates to minimize [resp. maximize] the

function g [resp. g] (see g. 5.8). Let us show that the reverse time estimates generated by these gains

do not improve the direct time estimates according to Proposition (5.5).

Let us consider γ = 0 and then consider variable ∆ as in (5.44) with ∆(tk+1) = 0, i.e. Binf(tk+1) =xγ=0(tk+1). Let us show that ∆(tk+1) ≤ 0. As Binf has been generated by a set of framers xθ with

gain values θ = 0 ∪ [1, ϕ] (according equation (5.22)), at t = tk+1 one has the following two cases:

• θ = 0

−xθ=0 − xγ=0 = −x∆µ|t=tk+1 ≤ 0 (5.45)

• θ ∈ [1, β]

−xθ − xγ=0 = s∆µθ(1− θ)/k − uθe+in/k − θ(µ(s)x− µ(s)x)|t=tk+1(5.46)

as v1 < 0 from equation (5.42), one has that −k(µ(s)x− µ(s)x) ≤ De+in, then:

−xθ − xγ=0 ≤ 0 (5.47)

at time tk+1. Then Binf(tr) ≤ xγ=0(tr) ∀tr ∈ [tk, tk+1] and no improvement can be expected. The

same can be demonstrated analogously for γ = 1. The same proof holds for the lower bounds. 2

5.3.3 Biased output

We extend the optimality analysis applied to reverse time framers and write the useful framer (5.32)

considering a bounded noise disturbing the output. Considering that Hypothesis 5.4 holds, bounds


0 10 20 30 40 50 60 700

25

50

75

100

time [days]

biom

ass

[g/l]

0 10 20 30 40 50 60 700

0.25

0.5

0.75

1

time [days]

conv

erge

nce

inde

x

Figure 5.9: Biomass estimation for AGRALCO plant. Direct time and reverse time observers andconvergence indexes.

for the measured substrate are obtained using equation (5.26). The following reverse time interval

observer is proposed:

Proposition 5.7 For a xed γ < 0, the following system is a reverse time framer of system (5.1):

qγ = (1− γ)(ν(y, y)qγ − γν(y, y)y) +D(qθ − γs+in)q

γ= (1− γ)(ν(y, y)q

γ− γν(y, y)y) +D(q

θ− γs−in)

(5.48)

xγ(tr) = (q(tr)− γy(tr))/k and xγ(tr) = (q(tr)− γy(tr))/k (5.49)

where functions ν and ν are dened in equation (5.29).

Proof. The proof is carried out as the proof of Proposition 5.3.

5.3.4 Application to the Agralco plant

We have chosen γ = −10 for the gain of the reverse time observer. Simulation results can be seen in

g. 5.9 (solid lines). Reverse time observers have signicantly improved the knowledge of the initial

state and consequently the bounds on the biomass at day 2. This allows to perform new estimates for

the next estimation period (between days 2 and 4) with an improved initialization. The discontinuity

of the estimation scheme can be observed at dierent reinitialisation time instants, see for example

days 2, 4, 36 and 58.

The method has been compared with the result obtained by the observer bundle working exclusively

in direct time (presented in section 5.2). Fig. 5.9 shows the interval predicted for the biomass for

the direct time bundle (dashed lines). The convergence index let us verify a great improvement using

reverse time observers.

5.4 Conclusions

This chapter has been devoted to the analysis of an optimality criterion for interval observers. This

criterion was applied to a particular observer structure, dedicated to the biomass estimation of un-

certain bioreactors. The proposed optimality criterion allowed to identify a region of the admissible

gain space, containing the framers that provide the best estimates. We run a bundle of observers in

this reduced space of gains, improving the eciency of the method in terms of computation time and

accuracy.

5.4. Conclusions 87

This chapter also presented a new estimation concept based on observers running in reverse time. This

scheme leads to possibly discontinuous estimates (at each reinitialisation time instants), leading to a

dramatic improvement of the convergence rate. We applied the proposed optimality criterion to the

reverse time observer, letting us characterise these observers by a single valued high gain framer.

The application of the scheme to the estimation of the total biomass in a real bioreactor let us conclude

about its good performance.

Chapter 6

Interval estimation of unknown inputs

6.1 Introduction

Unknown input estimation is a delicate issue. This problem has been solved with observers that are

constructed to simultaneously estimate the state and the inputs. Most of these methods have been for-

mulated for linear models (Kim and Goodall, 2005). Some approaches apply to nonlinear models where

nonlinearities are perfectly known, and cancel when writing error dynamics equations. In a nonlinear

framework, Liu and Peng (2002) consider an a priori modeling of the disturbance estimates according

to the available system knowledge. Another approach can be found in Corless and Tu (1998), where

the disturbance is treated as a nonlinear non autonomous function. This method is characterized by

Lyapunov functionals for the stability analysis of the estimations.

In the case of biotechnological processes (in particular in depollution processes), it can often occur

that inuent concentrations are unknown. Aubrun et al. (2001) present a method based on a stochastic

approach in order to obtain estimates of the inuent substrate. In Theilliol et al. (2003) a method that

uses online derivatives of available outputs is provided.

This chapter is devoted to the design of two interval observers associated to the unknown inuent

concentration of a bioreactor model. The rst approach is based on Moisan and Bernard (2006a) and

uses a closed loop interval observer working in direct and reverse time and assumes a piecewise constant

input. The second approach, based on Moisan et al. (2008), works generating guaranteed bounds on

the error of a conventional observer for the input and allows the estimation of a time varying inuent

concentration.

6.2 Interval estimation of a piecewise constant unknown input

This approach is based on the particular closed loop interval observer proposed in Bernard and Gouzé

(2004), which takes benet of additional measurements to introduce a correction term in the observer.

Consider the general bioprocess model (1.2) and assume that a part ξ1 ∈ Rq of the state ξ ∈ Rn can

be measured on-line. The part of the state which is not measured is denoted ξ2. We assume moreover

89

90 Interval estimation of unknown inputs

that the output y of the system can be split into 3 parts:

y =

y1

y2

y3

with

y1 = ξ1 ∈ Rq

y2 = Q(ξ) ∈ Rn

y3 = h(ξ1, ξ2) ∈ Rm

(6.1)

In other words we suppose that we can measure the gaseous ow rates Q(ξ) and also a set of

functions of the state h(ξ1, ξ2). They can be physical variables (pH, partial pressure of a gas, etc.)

related to the state ξ.

We consider the following example, with one biomass, substrate and product with state ξ = [x s p]T :

x = r(ξ)−Dx

s = −k1r(ξ) +D(sin − s)p = k2r(ξ)−D(pin − p)

(6.2)

where sin and pin are the unknown inuent substrate and product concentrations, and k1 and k2 are

yield coecients. This model has the form presented in (1.2) with:

K = [1 − k1 k2]T , ξin = [0 sin pin]T and Q(ξ) = [0 0 0]T (6.3)

We assume that the bacterial biomass can be measured. Moreover we suppose that we can estimate

the proportion of s(t) in the medium with respect to p(t). These relative measurements are often much

easier (and also less expensive) to obtain than the absolute values.

We have therefore: y1 = x

y2 = [0 0 0]T

y3 =s

s+ p

(6.4)

Our aim is to estimate the unknown states s and p and the inuent concentrations sin and pin,

without using the unknown kinetics r(ξ).

6.2.1 A closed loop interval observer

This observer is based on asymptotic observers (see Chapter 2 for more details). We separate the

measured state variables ξ1 = x and the unmeasured state variables ξ2 = [s p]T . Considering the linearchange of variables (2.40), we obtain a new system of order n− l (with l the number of reactions) thatdoes not depends on the reaction rates:

z = D(zin −Ry1 −H2z)−My2 (6.5)

where M = [A In−l], z = Mξ and zin = Mξin. The asymptotic observer is then:

˙z = D(zin −Ry1 −H2z)−My2

ξ2 = z2 −Ay1(6.6)

6.2. Interval estimation of a piecewise constant unknown input 91

For the mentioned example (6.2) we obtain the following open loop observer:˙z1 = D(zin1 − z1), s = z1 − y1k1

˙z2 = D(zin2 − z2), p = z2 + y1k2

(6.7)

with z1 = s, z2 = p, zin1 = sin and zin2 = pin.

This observer has nevertheless several defects. It is in open loop and therefore it is assumed that

the model (except the unknown reaction rates) is perfectly known and that there is no error in the

measurements, in the estimations of the feeding inputs and in the estimates of the yield coecient

matrix K. Moreover this asymptotic observer does not take benet of all the available information (y2and y3).

We take advantage of additional (linear or non linear) outputs, to build closedloop interval ob-

servers based on asymptotic observers.

We suppose here that y3 ∈ R. The mapping h is thus dened as follows:

h : (ξ1, ξ2) ∈ (Rn−p × Rp) −→ y3 = h(ξ1, ξ2)

Hypothesis 6.1 The mapping h is monotone with respect to ξ2, i.e.∂h

∂ξ2is of xed signs on Ω.

In our example h(ξ1, ξ2) =s

s+ p, and thus:

∂h

∂ξ2=[

p

(s+ p)2− s

(s+ p)2

](6.8)

which is of xed signs.

In order to deal with uncertainties in the system, interval observers give a suitable way to obtain

guaranteed estimates. We will suppose in our example that the inuent concentrations are uncertain

but bounded. We build interval observers, focusing on the time interval [tk, tk+1].

Hypothesis 6.2 We suppose that we know upper and lower bounds on the inputs:

z−in(t) ≤ zin ≤ z+in(t) (6.9)

Additionally, we consider that zin is constant on the considered time interval [t0, tN ].

To simplify the presentation, we focus on the example in dimension two (z = [z1 z2]T ), but the resultand the proofs are the same in dimension p.

Proposition 6.1 The following system is a framer of system (6.2):z+1 = D(z+

in1 − z+1 )− λ1(h(z+

1 , z+2 )− y3), s+ = z+

1 − y1k1

z+2 = D(z+

in2 − z+2 )− λ2(h(z+

1 , z+2 )− y3), p+ = z+

2 + y1k2

z−1 = D(z−in1 − z−1 )− λ1(h(z−1 , z−2 )− y3), s− = z−1 − y1k1

z−2 = D(z−in2 − z−2 )− λ2(h(z−1 , z−2 )− y3), p− = z−2 + y1k2

(6.10)

Vector λ = [λ1 λ2]T is chosen such that λi∂h

∂ξ2i≥ 0.


In the considered example, the function h is supposed to be increasing with respect to the rst

component of z and decreasing with respect to the second one. Therefore λ1 ≥ 0 and λ2 ≤ 0.Proof. The vector e = [e+ e−]T = [z+(t) − z(t) z(t) − z−(t)]T is positive at time tk. Now let us

consider the rst time instant t? ≥ tk where one of the component of e is equal to zero. For the sake

of simplicity, we consider the rst component e1. Then we have:

e1(t?) = z+1 (t?)− z1(t?) = D(z+

in1 − zin1)− λ1(h(z+1 , z

+2 )− h(z+

1 , z2))|t=t? (6.11)

In one hand, it is clear that D(z+in1 − zin1) ≥ 0, on the other hand (h(z+

1 , z+2 ) − h(z+

1 , z2))|t=t? ≤ 0because the function h is decreasing with respect to the second variable and z+

2 (t?) ≥ z2(t?). Since

λ2 ≤ 0, e1(t?) ≥ 0 which implies that e1(t?) is increasing , and therefore e(t) ≥ 0, ∀t ≥ tk. 2

Remark 6.1 In Bernard and Gouzé (2004) the boundedness of the interval associated to framer equa-

tion (6.10) is demonstrated. However, this property is not required in the sequel, and the framer can

be unstable. It suces to take the framer λ = 0 in the bundle to show the existence of an interval

observer (see Property 3.1).

Now we propose the formulation of interval observers running in reverse time in the time interval

[tk, tk+1]. For this, we consider reverse time dynamics of system (6.2) and then build observers under

the procedure explained in Chapter 5. Variable q = q(tr) will denote the reverse time evolution of

system (6.5), where tr = 2tk+1 − t. Its dynamics are represented by the system:

q = D(q − zin) +My2, ( ´ denotes the derivative with respect to tr) (6.12)

Proposition 6.2 The initialization of an interval reverse time observer will be the best estimate per-

formed by a direct time bundle at time tk+1, that is:

[q−(tk+1), q+(tk+1)] = [z−d (tk+1), z+d (tk+1)]

Proposition 6.3 The following system is a reverse time framer of system (6.2):q+1 = D(q+1 − z−in1)− γ1(h(q+1 , q

+2 )− y3), s+ = q+1 − y1k1

q+2 = D(q+2 − z−in2)− γ2(h(q+1 , q+2 )− y3), p+ = q+2 + y1k2

q−1 = D(q−1 − z−in1)− γ1(h(q−1 , q−2 )− y3), s− = q−1 − y1k1

q−2 = D(q−2 − z−in2)− γ2(h(q−1 , q−2 )− y3), p− = q−2 + y1k2

(6.13)

Vector γ = [γ1 γ2]T veries γi∂h

∂ξ2i≥ 0.

Proof. The proof is the same as the one of Proposition 6.1. 2

Remark 6.2 It is worth noting that system (6.12) is unstable. Then, we use the correction term

γ(h(q+1 , q+2 ) − y3) in the observer (6.13), in order to obtain stable estimations in reverse time, in the

direction of γ.

6.2. Interval estimation of a piecewise constant unknown input 93

Direct and reverse time observers bundle

We have obtained new guaranteed estimates of the state running in reverse time. Considering dierent

values for gain γ, we obtain a new bundle of observers working in reverse time. Now we compare these

estimates with the direct time bundle best value [z−d (t), z+d (t)] in order to obtain a new best value:

[z−r (t), z+r (t)] = [max

γz−d (t), q−γ (2tk+1 − t),min

γz+

d (t), q+γ (2tk+1 − t)]

where z−r , z+r are the best lower and upper envelopes, from direct and reverse time bundles.

6.2.2 Interval estimation of inuent concentrations

We exploit the simplicity of the dynamics of total mass described by equation (6.5), which is indeed a

linear system up to an output injection y2.

Let us consider equation (6.5) with y2 = 0 as in the example (computations without this hypothesis

are obviously possible). System (6.5) is thus expressed as:

zi = D(ziin − zi) (6.14)

Considering D =∫ t

tkD(τ)dτ and assumption (6.2), we can write the explicit solution of the previous

equation and deduce:

zi(t) = (zi(tk)− ziin)e−D + zi

in ⇒ ziin =

zi(t)− zi(tk)e−D

1− e−D(6.15)

Now we consider the best estimates performed by a direct and reverse time bundle [z−r , z+r ] to obtain

the following inequality:

z−in(i) = maxt

(z−r (t)− z+

r (ti)e−D

1− e−D

)≤ zi

in ≤ mint

(z+r (t)− z−r (ti)e−D

1− e−D

)= z+

in(i) (6.16)

Therefore the bounds for the inuent concentrations are deduced from the bounds of the state z.

This way, the performance of the input estimator is directly related with the performance of the state

observer.

Iterative estimations

At this point, it should be quite logical to consider an iterative procedure in order to perform sequential

estimates to progressively improve initial condition and input estimate. Taking into account that a

rst step of observation performed by equations (6.10) and (6.13) gives us bound of the state, in a

second step we use these bounds to obtain new (possibly better) bounds of the uncertain inuent

concentrations. A new iteration will perform a new estimation of the state considering the new bounds

of the inuent concentrations and so on. The stability of the process is guaranteed by the stability of

the direct time bundle of observer.


0 4 8 12 16 200

10

30

50

time

S

0 4 8 12 16 200

5

15

25

time

p

Figure 6.1: Substrate and product interval estimates.

0 4 8 12 16 20

35

45

55

65

time

s in

0 4 8 12 16 20

2

4

6

8

time

p in

Figure 6.2: Estimation of unknown inputs sin and pin.

6.2.3 Application to the example

We have applied the observers dened in equations (6.10) and (6.13) to system (6.2). For simulation

purposes, we consider that the bacterial growth rate (which is of course supposed to be not known), is

a saturated function of the substrate and is inhibited by the product:

r(ξ) = a1s

s+ a2

a3

a3 + px

Simulations have been performed considering parameter values in Table 6.1 and a time varying

dilution input D(t) = 0.4 + 0.3sin(t/2)[days−1]. We apply the presented observers considering a 30%uncertainty for sin and a 15% uncertainty for pin. The reinitialised observers bundle working in direct

and reverse time has been obtained considering various framers with gains in the following intervals:

0 ≤ λ1 ≤ 1000, 0 ≤ γ1 ≤ 1000, −1000 ≤ λ2 ≤ 0 and −1000 ≤ γ1 ≤ 0. Reinitialisation has been applied

Table 6.1: System parameters and considered uncertainty.

parameter meaning valuek1 yield coecient 2k2 yield coecient 0.5a1 maximal growth rate 1.2a2 half saturation constant 0.5a3 inhibition constant 15sin unknown inuent substrate 50pin unknown inuent product 5

6.3. Interval estimation of a time varying input 95

each two days. The framers are initialised at s−0 = 0[g/l], s+0 = 20[g/l], p−0 = 0[g/l] and p+0 = 30[g/l].

The observer predictions are presented on g. (6.1) for the unknown states, and g. (6.2) for the

unknown inputs. It is worth noting that reverse time observers have a direct impact in the improvement

of the initial knowledge: uncertainty in the initial condition of both variables is signicantly reduced.

Also, reverse time observers nally improve the convergence rate of the bundle. This let us run several

times the direct and reverse observers in each sub period of simulation. From assumption (6.2), we

can take: z+in(i) = min(z+

in(i)) and z−in(i) = max(z+in(i)).

6.3 Interval estimation of a time varying input

This development consists in obtaining guaranteed bounds of the error from a conventional input

observer. In particular, we consider the input observer presented in Mazenc (2007), which guarantees

convergence for a high gain value, and then we derive an interval observer (for any gain value) for its

error. After considering some technical details, we construct an interval observer through a change of

coordinates that guarantees stable interval estimates.

6.3.1 Assumptions and notations

Consider the simple bioreactor model in dimension two:x = r(ξ)− ux

s = u(v − s)− k1r(ξ)(6.17)

with ξ = [x s]T . We assume that there is no model available for the reaction rate r(ξ). The time

dependant function v = v(t) ∈ C1(R) corresponds to the unknown concentration of inuent substrate.

Consider that the system output can be written as:

y =

y1 = s

y2 = k6r(ξ)(6.18)

using equation (6.18), system (6.17) can be written as a linear system plus a term related to the

uncertain input.

The following hypotheses are needed for the statement of our result.

Hypothesis 6.3 There exist a constant γ ∈ R+ such that:∫ t

0

u(σ)dσ ≥ γt

Hypothesis 6.4 There exists a known constant β ∈ R+ such that:

|v(t)| ≤ β

Hypothesis 6.5 There exist known intervals containing the initial condition of the state of system

(6.17) and the unknown initial condition of the input v:

x0 ∈ [x−0 , x+0 ], s0 ∈ [s−0 , s

+0 ], v0 ∈ [v−0 , v

+0 ]


Our objective is to obtain an interval estimate of the unknown input v(t), using the available informa-

tion provided in equation (6.18).

In a rst step, a conventional observer with convergence guaranteed for high gain values is consid-

ered. Then, on the basis of a bounded error estimation scheme, an interval observer for the input is

constructed.

6.3.2 High Gain observer for the input

Let us consider the following input observer candidate for system (6.17):

˙s = u(v − y1)− k1k6y2 + u(θ + θ2)(y1 − s)

˙v = uθ3(y1 − s)(6.19)

where θ ∈ R is a tunable gain (for the sake of simplicity, θ will be considered constant). System (6.19)

is initialised considering:

s0 = s0 and v0 ∈ [v−0 , v+0 ] (6.20)

Equation (6.19) corresponds to a slightly modied equation of the input observer proposed by Mazenc

(2007), following a scheme similar to the observer introduced in Gauthier et al. (1992).

Let us denote es = s− s and ev = v − v. The error dynamics are then expressed by the system:

es = −u(θ + θ2)es + uev

ev = −uθ3es − v(t)(6.21)

Proposition 6.4 If Hypotheses 6.3 and 6.4 are veried, then for a gain θ > 1 there exists a bound

B(θ) ∈ R+ such that |ev(t)| ≤ B(θ), with:

B(θ) =2βθ − 1

+θ

θ − 1(v+

0 − v−0 )e−γθt, ∀t ≥ 0 (6.22)

Proof. See the proof in Box 6.1. See also Mazenc (2007) for more details. 2

From equation (6.22), it is clearly seen that for a high gain θ, the bound B(θ) can be made as small

as desired with a convergence rate arbitrary fast. Note that if β = 0 (the case where the input is a

constant) then asymptotic exponential convergence is reached.

Remark 6.3 The bound provided by equation (6.22) is valid for well known stoichiometric coecients

k1 and k6, and noise free measurements y1 and y2.

6.3.3 An interval observer for the input

Let us note that, as there is no model available for the unknown input v(t), proposing an interval

observer is not straightforward.

We develop an interval observer for the input, assuming known bounds on the initial conditions.

The idea is to dynamically compute guaranteed upper and lower bounds for the error system (6.21),

and then to obtain the bounds on the input considering:

v(t)− e+v (t) ≤ v(t) ≤ v(t)− e−v (t) (6.23)


We propose an interval observer considering a similar framework of the closed loop observers pro-

posed in Chapters 3 and 4.

Consider the error dynamics given by equation (6.21). Denoting e = [es ev]T , it can be rewritten

in the following form: e(t) = uAe(t) + b(t)z = Ce

(6.24)

with:

A =

[−(θ + θ2) 1−θ3 0

], b(t) =

[0

−v(t)

]and C = [1 0] (6.25)

Box 6.1 Proof of Proposition 6.4.

Consider the time parametrization:

τ =∫ t

0

u(σ)dσ ≥ γt, γ, t ≥ 0

then system (6.21) can be rewritten as:

ddτ

[es

ev

]= A

[es

ev

]+

[0

−ϕ(τ)

]

where

ψ(τ) =dvdt (τ)u(τ)

∈[−βγ,β

γ

]considering the change of variables ζ1 = θes − ev and ζ2 = −θ2es + ev, it follows that:

ddτζ1 = −θζ1 + ψ(τ) and

ddτζ2 = −θ2ζ2 − ψ(τ)

Bounds for the term ψ(τ) in equation (6.1), let us write:

|ζi(τ)| ≤β

γθi+ |ζi(0)|e−θiτ , i = 1, 2

Now we can come back to the original coordinates:

|v(τ)− v(τ)| ≤ (θ + θ−1)β/γ + θ2(|ζ1(0)|+ |ζ2(0)|)e−θτ

θ2 − θ

and nally considering equation (6.1):

|v(t)− v(t)| ≤ 2βθ − 1

+θ

θ − 1(v+

0 − v−0 )e−γθt, ∀t ≥ 0

Then Proposition 6.4 holds. 2


It is clear that, from Hypothesis 6.4, the term b(t) is bounded by:

b−(t) =

[0−β

]and b+(t) =

[0β

](6.26)

It is not dicult to verify that the interval observer structure given by the pair of systems:e+(t) = Ae+(t) + b+ + Γ(Ce+(t)− z)e−(t) = Ae−(t) + b− − Γ(Ce−(t)− z)

(6.27)

where Γ = [γ1, . . . , γn]T does not provide at the same time stability and cooperativity. Indeed, if we

consider the error e? = [e+ − e, e− e−]T :

e?(t) =

[A+ ΓC 0

0 A+ ΓC

]e?(t) +

[b+(t)− b(t)b(t)− b−(t)

](6.28)

considering equation (6.24), it is possible to check that matrix A+ ΓC can be written as:

A+ ΓC =

[−(θ + θ2) + γ1 1−θ3 + γ2 0

](6.29)

which is cooperative if γ2 ≥ θ3, however it is not stable.

To overcome this problem we propose a change of coordinates of system (6.21) in which stable estimates

can be obtained.

6.3.4 Change of coordinates

Matrix A is stable, with real and strictly negative eigenvalues given by λ1 = −θ and λ2 = −θ2.Therefore, it can be diagonalized:

A = P∆P−1 (6.30)

where:

∆ =

[−θ 00 −θ2

], P−1 =

1θ − 1

[−1 1/θθ2 −1

](6.31)

Consider the change of variables:

ζ = P−1e (6.32)

then, in the new coordinates system (6.24) is expressed by:

ζ(t) = u∆ζ(t) + P−1b(t) (6.33)

Taking advantage of the diagonal (and thus cooperative) structure of matrix ∆, plus its stability for a

gain θ ∈ R+−0, 1, an interval observer can be obtained in the ζ-coordinates, considering the bounds

on the term b(t) provided by Hypothesis (6.4).


6.3.5 An interval observer with perfect knowledge

Consider the following system:[ζ+(t)ζ−(t)

]= u

[∆(θ) 0

0 ∆(θ)

][ζ+(t)ζ−(t)

]+M(θ)

[b+

b−

][e+(t)e−(t)

]= G(θ)

[ζ+(t)ζ−(t)

] (6.34)

where M(θ) and G(θ) are linear transformations that full the following properties:

Property 6.1 For e ∈ [e, e] ⊂ Rn and z = P−1e:

M(θ)

[e

e

]=

[z

z

]such that z ≤ z ≤ z

Property 6.2 For ζ ∈ [ζ, ζ] ⊂ Rn and ε = Pζ:

G(θ)

[ζ

ζ

]=

[ε

ε

]such that ε ≤ ε ≤ ε

See Box 6.2 for more details about M(θ) and G(θ).

Proposition 6.5 For a gain θ ∈ R+−0, 1, then system (6.34) is a framer of system (6.24), leading

to bounds on the input v(t) by equation (6.23).

Proof. We need to prove that the error generated by comparing systems (6.33) and (6.34) is a positive

system. The error dynamics are expressed by the equation:[e+ζ (t)e−ζ (t)

]= u

[∆(θ) 0

0 ∆(θ)

][e+ζ (t)e−ζ (t)

]+ B(θ, b(t), b−, b+) (6.35)

it can be shown that the residual term B(.) ∈ R2n is nonnegative (see Box 6.2 for details), and then

Proposition 6.5 holds. 2

Remark 6.4 This observer inherits the convergence characteristic of the conventional observer (6.19).

6.3.6 Interval observer with uncertainties

Consider now that the stoichiometric coecients k1 and k6 and the measurements are uncertain quan-

tities. The following hypotheses are considered:

Hypothesis 6.6 The stoichiometric coecients k1 and k6 are unknown but bounded by known positive

values.

k1 ∈ [k−1 , k+1 ] and k6 ∈ [k−6 , k

+6 ] (6.36)

Hypothesis 6.7 Online measurements y1(t) and y2(t) are perturbed respectively by noises δs(t) and

δq(t). We assume that these perturbations are of multiplicative nature:

y1(t) = s(t)(1 + δs(t)) and y2(t) = k6r(1 + δq(t)) (6.37)


Box 6.2 Proof of Proposition 6.5.

We need to specify transformations M and G, and then verify that B is a nonnegative

vector.

Transformations M and G provide a link between ecoordinates and ζcoordinates.

Considering the change of variables ζ = P−1e, we specify M as follows:

M =

[P−1

+ P−1−

P−1− P−1

+

]

Denote P−1 = pij, ∀ i, j = 1, . . . , n. Matrices P−1− and P−1

+ ∈ Rn×n are dened as:

P−1+ =

pij if pij ≥ 00 otherwise

and P−1− = P−1 − P−1

+

that is, the positive and negative parts of matrix P−1 have been separated.

Vector B can be written as:

B =

[P−1

+ b+ + P−1− b− − P−1b

P−1b− P−1− b+ − P−1

+ b−

]

Considering b(t) ∈ [b−, b+] = [b(t)− ε−, b(t) + ε+] with ε−, ε+ ∈ Rn+, it follows that:

B =

[P−1

+ ε+ − P−1− ε−

P−1+ ε− − P−1

− ε+

]

it is easy to see that B ≥ 0. Matrix G is obtained in an analogous way (considering matrix

P ), assuring the bounding in the original coordinates. 2

Moreover, these noise signals are bounded such that |δs(t)| ≤ ∆s < 1 and |δq(t)| ≤ ∆q < 1.

Considering Hypotheses 6.6 and 6.7, observer equation (6.19) leads to error dynamics with the same

structure as equation (6.24), except for the vector b(t) which now depends on the uncertain values and

on the gain θ. Vector b(t) is then expressed by:

b(t) =

[usδs(θ + θ2 − 1)− y2

(k1

k6(1+δq) −k1

k6

)uθ3sδs − v(t)

](6.38)

where k1 ∈ [k−1 , k+1 ] and k6 ∈ [k−6 , k

+6 ]. From the previously introduced hypotheses it is easy to see

that b(t) ∈ [b−(t), b+(t)], with:• for θ ∈]0, θ?]:

b±(t) =

∓us∆s(θ + θ2 − 1) + y2

(k±1

k∓6 (1∓∆q)− k∓1

k±6

)±uθ3sδs ± β

(6.39)


0 10 20 30 40 50 60 700.02

0.03

0.04

0.05

0.06

0.07

time

dilu

tion

0 10 20 30 40 50 60 70

5

10

15

time

y 1 (su

bstr

ate)

0 10 20 30 40 50 60 70

20

40

60

time

y 2 (m

etha

ne f

low

rat

e)

Figure 6.3: Dilution u, substrate s and methane ow qM .

0 5 10 15 20 25

40

50

time

infl

uent

sub

stra

te

0 10 20 30 40 50 60 7010

30

50

70

time

infl

uent

sub

stra

te

Figure 6.4: Input interval estimation: perfect knowledge (left) and with uncertainties (right).

• for θ > θ?, θ 6= 1:

b±(t) =

±us∆s(θ + θ2 − 1) + y2

(k±1

k∓6 (1∓∆q)− k∓1

k±6

)±uθ3sδs ± β

(6.40)

where θ? is the positive solution of the equation θ + θ2 − 1 = 0.

Remark 6.5 The residual vector b(t) and its bounds depend on the gain θ which may dramatically

amplify the uncertainties when using high values of θ.

The observer candidate also keeps the same structure as equation (6.34), then the following proposition

holds.

Proposition 6.6 For a gain θ ∈ R+−0, 1, then system (6.34) (for bounds b−(t) and b+(t) expressedby equation (6.39) and (6.40)) is a framer of system (6.24), leading to bounds on the input v(t) by

equation (6.23).

Proof. The proof is the same as the one of Proposition 6.5. 2

6.3.7 Application

A simple application of the estimation scheme has been performed, inspired from anaerobic digestion

processes. In this kind of processes, the output y2 corresponds to the methane ow rate and is usually

online monitored. System parameters can be seen in Table 6.2. Dilution and available outputs y1(t)(substrate) and y2(t) (methane ow rate) are shown in g. 6.3.

We have run the proposed interval observer considering a perfect knowledge of the system and noise

free measurements. Simulation results for a single interval estimate considering θ = 100 are shown


Table 6.2: System parameters and considered uncertainty.

parameter meaning valuek1 biomass production yield coecient 1.28k6 methane production yield coecient 25∆s y1 multiplicative noise upper bound 0.03∆q y2 multiplicative noise upper bound 0.03

in g. 6.4. Fast convergence of the interval estimates toward the inuent substrate unknown value is

veried.

A bundle of interval estimates has been run under an uncertain framework. We considered a ±10%uncertainty for the stoichiometric coecients and a multiplicative noise on the measurements up to a

3%. A bundle of observers with 30 framers running in parallel, with θ ∈ [0.1, 50] is then shown in g.

6.4.

6.4 Conclusions

Two methods have been presented for the estimation of the unknown input of a bioprocess model.

Both schemes use interval observers introduced in the previous chapters, extending the results to the

estimation of the input. The rst approach considered direct and reverse time interval observers. The

input (considered to be constant) is estimated under a cascade observer structure. This allowed to im-

plement an iterative procedure in order to improve both steps (state estimation and input estimation).

The second input estimation approach considered a time varying unknown inuent concentration. The

interval estimates are obtained by bounding the error of a conventional observer (Mazenc, 2007), and

then extended to an uncertain framework (where the conventional observer does not oer any guarantee

of convergence). A bundle of observers allowed us to relax the high gain requirement of the proposed

conventional observer, constructing interval estimates with both low or high gain values.

Conclusions and perspectives

This thesis presented new results in the eld of robust state estimation through the design of interval

observers. Our contribution is mainly focused on the estimation of the state variables of bioprocesses.

The lack of sensors and reliable procedures to online monitor bioprocesses is an actual problem. Our

contribution therefore, aims at improving and facilitating bioprocesses operation, proposing a robust

estimation scheme.

In this thesis, the theoretical developments improving interval estimations have been assessed using

real bioprocesses. Indeed, this is the rst time that such observers are applied to an industrial plant

(anaerobic wastewater treatment plant). It demonstrates their eciency considering high levels of un-

certainties on the inuent concentrations, kinetics and noise measurements. Our results signicantly

improved the predictions of asymptotic interval observers.

Interval observers belong to a specic class of estimators called guaranteed state estimation methods.

The strength of these methods is that they provide a region of the state space where the unknown

variables are sure to lie. This allows the intersection of any set of predictions coming from estimators

based on dierent principles. As a key consequence, it leads to a drastic improvement of the nal

estimate.

This property motivated the rst contribution of this thesis: we reformulated the concept of in-

terval observers as a result of multiple interval estimates running in parallel. The socalled bundle

of observers, allowed to improve the nal estimate by comparing various framers. A framer provides

guaranteed upper and lower bounds of the unknown state, based on arguments of positive dierential

systems. Then a requirement is that at least one framer in the bundle is error bounded, guaranteeing

the existence of an interval observer.

The improvement obtained using a bundle of observers, in terms of convergence rate, was demon-

strated through the design of several interval observers. In our examples, most of the time the framer

associated to the asymptotic observer was used to guaranty the boundedness of the nal envelope.

A solution for the problem of generating interval observers for nonmonotone systems was given.

This consists in identifying monotone increasing and decreasing parts of a nonmonotone mapping.

This led to the design of full order interval observers with monotone error dynamics. Hence, mono-

tonicity properties of the original system are not needed, allowing to extend the formulation of these

observers to a wider range of systems. The Haldane model, was studied to illustrate this point. A

monotone representation of the Haldane equation in two variables was used in the observer design. It

is worth to remark that this representation is not unique, and more interval estimates can be obtained

by systematically running dierent monotone representations of the Haldane equation.

103

104 Conclusions and perspectives

A second example that used a function of the state as measurement was given, considering a Haldane

bioreactor. Guaranteed interval estimates were obtained with two interval observers: a direct interval

observer, based on cooperative error dynamics, and an undirect interval observer which achieves co-

operative error dynamics after a simple change of variables of the error dynamics. This method was

illustrated considering an industrial wastewater treatment plant, showing its eciency when estimating

the total biomass and substrate using available measurements of the methane ow rate. It is worth

to remark that the gaseous ow rate is a measurement that can be easily obtained in most of the

industrial or experimental setups. Therefore, this observer becomes specially attractive as a ecient

way to obtain biomass and substrate estimates.

An optimality criterion was proposed afterwards. This allowed to signicantly improve the per-

formance of a bundle of observers in terms of accuracy of the nal estimate and computing time.

The criterion was applied to a particular interval observer for the total biomass, based on a simple

parameterizable change of coordinates. The criterion led to a gain set that provides the best framers.

This result allowed us to run a bundle of observers, using few framers, aiming at the estimation of the

biomass of an industrial wastewater treatment plant. The good convergence properties were achieved

with a short computation time.

In order to complement the previous results, reverse time interval observers were then introduced.

The idea corresponds to a non conventional estimation scheme, where after running interval estimates

in direct time, interval estimates are run considering reverse time dynamics of the system. The objective

is to improve the initial condition knowledge leading to a discontinuous evolution of the estimates.

We determined conditions for which this kind of interval estimates can be useful (under the pro-

posed optimality criterion), allowing a dramatic improvement of the convergence rate. The application

of reverse time observers to the wastewater treatment plant, let us conrm a substantial improvement

of the estimates performed in direct time, with extremely fast convergence rates.

The last part of our contribution corresponds to the interval estimation of the unknown inputs.

This subject is rather new, with few contributions in the current literature. We considered the problem

from the framework of bioprocess models. It is worth to note that uncertain inuent concentrations

appeared in all the interval observers proposed in this thesis. Therefore, an improved estimate of

this uncertain quantity will improve straightforwardly the state observers performance. First, the

interval estimation of a piecewise constant input was considered. This case was treated considering a

jointly state and input estimation, using a directreverse time observer bundle. A second case was the

interval estimation of an unknown time varying input. This was carried out bounding the estimate of

a conventional (highgain) input observer. Simulation results showed the eectiveness of the proposed

observers for both cases.

Future works

Interval observers are fertile ground for further research. They can provide useful estimates for robust

control loop, where the control strategy and the interval estimates complement each other in order to

deal with system uncertainties. They can also be combined with classical observers. This idea can be

seen from two points of view. The rst one consists in feeding a standard observer with the interval

predicted by an interval observer, through a correction term and then nd conditions in order to obtain

Conclusions and perspectives 105

an improved performance of the standard scheme. The second one corresponds to nd bounds of the

error of any classical observer, in order to generate the interval. The second application presented

for the estimation of the input in Chapter 6 gives some hints on how to achieve this. This can be in

particular useful to assess the convergence of classical approaches and also to deal with some drawbacks

(attenuating the peaking eect of high gain observers, for example).

From the point of view of the observer performance, the design of sensitivity indexes with respect to

the uncertainty can provide useful information. In this thesis we provided a convergence index which

let us assess the observer performance by comparing the actual values of upper and lower bounds.

This concept can be extended, in order to monitor the rate of change of the predicted interval with

respect to the total system uncertainty. Such an index can be useful to compare dierent classes of

interval observers, aiming at obtaining the best estimates from a set of heterogeneous interval observers.

Another point of fruitful research can be the proposition of a new and ecient change of variables.

A basic idea behind interval observers is the fact that the design can be carried out in a new suitable

base (for instance, a base where the original system becomes monotone). A new base can provide a way

to minimize the eect of the uncertainty on the original coordinates. This can be studied in two ways:

the rst is to work with a parameterizable change of base and then apply the proposed optimality

criterion in order to nd the best estimates (this follows the same principle as the observer proposed

in Chapter 5). The second is running in parallel several interval observers, based on dierent changes

of bases, where each observer is designed to adapt to a particular kind of uncertainty.

Even though this thesis dealt with the problem of state estimation of bioprocesses, an example

considering the estimation of chaotic systems was also presented. It corresponds to a rst attempt of

application of this technique to chaotic systems. Observer based synchronization of systems has received

great attention in the eld of secure communications. A robust observer can deal with disturbances in

communication channels and moreover, provide online diagnostic of communication conditions. Appli-

cations in protective relaying of power electrical systems can also be expected. A backup protection

system based on interval observers can provide a simple and low cost alternative protection. These

systems are not as uncertain as biological systems, however, an extremely fast reaction is required when

faults arise. For such cases, the online assessment of the convergence of interval observers becomes a

fundamental property which is not provided by classical observers.

106 Conclusions and perspectives

Publications

List of publications

M. Moisan, O. Bernard and J.-L. Gouzé, Near optimal interval observers bundle for un-

certain bioreactors, submitted to Automatica.

M. Moisan and O. Bernard and J.-L. Gouzé,A high/low gain interval observer: application

to the input estimation of a bioreactor, 17th IFAC World conference, Seoul, Korea, 2008.

M. Moisan and O. Bernard, Robust estimation using directreverse time interval ob-

servers: application to an industrial bioreactor, in: Proceedings of the STIC et Environ-

nement 2007 conference, Lyon, France, 2007.

M. Moisan, O. Bernard and J.-L. Gouzé, A directreverse time interval observer. Appli-

cation to biotechnological models, in: Proceedings of the 7th IFAC Symposium on Nonlinear

Control Systems, Pretoria, South Africa, 2007.

M. Moisan, O. Bernard and J.-L. Gouzé, Near optimal interval observers bundle for un-

certain bioreactors, in: Proceedings of the 9th European Control Conference, Kos, Greece,

2007.

M. Moisan, O. Bernard, An interval observer for non-monotone systems: Application to

an industrial anaerobic digestion process, in: Proceedings of the 10th Computer Applications

in Biotechnology conference, Cancún, México, 2007.

M. Moisan and O. Bernard, Robust interval observers for uncertain chaotic systems, in:

Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, USA, 2006.

M. Moisan and O. Bernard, Guaranteed state estimation of a biotechnological process

by means of cooperative and ellipsoidal techniques, in: Proceedings of the 5th IFAC

Symposium on Robust Control Design, Toulouse, France, 2006.

J.-L. Gouzé, M. Moisan and O. Bernard, A simple improvement of interval asymptotic

observers for biotechnological processes, in: Proceedings of the 5th IFAC Symposium on

Robust Control Design, Toulouse, France, 2006.

107

108 Publications

M. Moisan and O. Bernard, Direct and reverse time observer bundle: Application to the

state and input estimation of a bioprocess, in: Proceedings of the STIC et Environnement

2006 conference, Narbonne, France, 2006.

J.-L. Gouzé, M. Moisan and O. Bernard, Simple interval observers for biotechnological

processes, in: Proceedings of the ISCCSP 2006 conference, Marrakech, Moroco, 2006.

M. Moisan and O. Bernard, Interval Observers for non monotone systems. Applica-

tion to bioprocess models, in: Proceedings of the 16th IFAC World Congress, Prague, Czech

Republic, 2005.

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Synthèse d'Observateurs par Intervalles

pour des Systèmes Biologiques Mal Connus

Cette thèse présente de nouveaux résultats sur l'estimation robuste d'état, avec des applications

aux systèmes biologiques représentés par des modèles approximatifs. Les modèles mathématiques en

biologie sont connus pour être fortement incertains. Nous développons des observateurs par intervalles,

basés sur la théorie des systèmes diérentiels positifs. L'objectif d'un observateur par intervalles est

d'obtenir des bornes supérieures et inférieures garanties pour les variables non mesurées du système.

Ceci est réalisé en considérant que des bornes sur les incertitudes sont connues. Le principe des ob-

servateurs que nous développons repose sur un faisceau d'estimateurs par intervalle : nous lançons en

parallèle plusieurs estimations par intervalles, puis nous en prenons l'enveloppe supérieure et inferieure

pour obtenir l'estimation nale. Nous étendons tout d'abord les résultats existants aux systèmes non

monotones. Nous proposons ensuite une stratégie pour localiser les gains optimaux pour le faisceau

d'observateur. Nous améliorons également les estimations en construisant des observateurs pour le

système en temps inversé. Enn ces principes sont étendus a l'estimation des entrées inconnues. Les

observateurs par intervalles orent de nombreux avantages en comparaison aux méthodes classiques

d'estimation d'état. Tout d'abord ils permettent de gérer l'incertitude dans un cadre déterministe. Ils

peuvent être comparés à d'autres estimations par intervalles, et nalement être améliorés en prenant

l'intersection des prédictions. En outre, il est possible d'évaluer en ligne la performance de l'estimateur

en suivant la largeur de l'intervalle prédit. Les contributions présentées dans cette thèse sont illustrées

par des applications à divers bioprocédés, notamment en utilisant des modèles de croissance de mi-

croorganismes sur un substrat. Des essais sur un procédé industriel d'épuration de l'eau valident ces

observateurs dans un cadre réel. Finalement des applications à des systèmes chaotiques sont également

étudiées.

Design of Interval Observers

for Uncertain Biological systems

This thesis presents new results in the eld of robust state estimation, with applications to un-

certain biotechnological systems. Mathematical models in biology are known to be highly uncertain.

Therefore, we develop interval observers, based on the theory of positive dierential systems.

The objective of an interval observer is to obtain guaranteed upper and lower bounds for the unmea-

sured variables of the system. This is achieved considering that bounds on the uncertainties are known.

Our observer design is featured by the socalled bundle of observers: we run in parallel several interval

estimates and then we take the best inner envelope. We design interval observers for nonmonotone

systems. We develop an optimality criterion associated to interval observers, allowing to nd the op-

timal gains for an observer bundle. We construct reverse time interval observers in order to improve

convergence rate and we extend these principles to the estimation of unknown inputs.

Some of the main advantages of interval observers are that they oer a way to manage uncertainty,

considering a deterministic framework. They can be compared with other guaranteed state estimation

methods, allowing improvements when taking the intersection of the predictions. They also allow to

online assess the convergence of the estimates.

The contributions presented in this thesis are illustrated through their application to biotechnologi-

cal systems, namely in models of microorganisms consuming a substrate. The application to a real

industrial wastewater treatment plant let us validate the proposed methods. Finally, we also studied

applications to the estimation of uncertain chaotic systems.

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