Diplôme De L'etat

Paul Jennings These

Soutenue Publiquement

A La

Faculte De Medecine De Grenoble

Monsieur Le Professeur

Patrice Francois Membres

Thomas Decaens Monsieur Le Docteur

Arnaud Seigneurin

Madame Le Docteur

Catherine Exbrayat

THESE PRESENTEE POUR L'OBTENTION DU DOCTORAT EN MEDECINE

N° 12 year study of the prevalence, risk factors and characteristics of Interval colorectal cancers after negative colonoscopy performed for a positive gFOBT in a screening programme in Isère (France) from 2002 -2013

A M. le Professeur Thomas DECAENS, membre du jury Je vous remercie d'avoir accepté de juger ce travail. Veuillez trouver ici l'expression de mon profond respect.

A M. le Docteur Arnaud SEIGNEURIN, directeur de thèse Un grand merci d'avoir accepté la direction de ce travail de thèse. Merci pour ta disponibilité, ton professionnalisme, ton soutien et ton écoute attentive tout au long de ce travail. Soyez assuré de tout mon respect.

A Mme le Docteur Catherine EXBRAYAT, membre du jury Merci de m'avoir proposé un tel travail. Merci aussi pour ton encadrement, tes commentaires constructifs, tes relectures patientes et minutieuses.

Sois assurée de tout mon respect et de ma reconnaissance.

Je souhaite tout particulièrement dédier cette thèse; À toutes les personnes sans qui ce travail n'aurait pu voir le jour : Florence PONCET (Registre Du Cancer de l' Isère), Jocelyne CHEVALIER ( !) , Dr Anne GARNIER, Géraldine FERREIRA, Isabelle ROVIRA et toute l'équipe de l'ODLC, merci pour votre implication de près ou de loin dans ce projet.

A Dr Xavier COURTOIS pour tout ce que tu as fait pour moi depuis mon arrivé en France, ta générosité, ta patience, tes conseils, et ton amitié.

A Dr Jérôme JUND, Dr Pierre « non, tu ne peux pas me demander une question, tu peux me poser une question » CASEZ, Dr Christoph STEFFEN, Cécile RICARD et toute l'équipe du SIEM d' Annecy pour votre disponibilité, votre aide, et tous ces temps conviviaux qu'on a passé ensemble.

A Dr Martine WALTER, Dr Caroline DELARUE, Dr Sandra L'EPLATTENIER, Dr Elise GUILLET, Eve CAILLOT, Maud PIROT, Audrey BRIAND, Virginie ROSSET, Emilie RIVOLLET, Silvia DA COSTA, Louisa CHEVALEYRE, Caroline BUTRUILLE, Marielle BERNE, et Séverine GANDOIS, merci à vous toutes pour mon semestre préféré de l'internat ou j'ai trouvé ma voie, vous avez toutes joué un rôle important.

A mes co-internes de santé publique, pour tous les moments conviviaux qu'on a passé ensemble. Je vous souhaite le meilleur pour l'avenir.

A tous les internes que j'avais le bonheur de rencontrer à l'internat d'Annecy qui m'ont fait sentir « at home » ! To Mum and Dad for sacrificing so much for me growing up, and always encouraging and believing in me and for letting me follow my dreams.

To Keith, for all the stuff you taught me growing up (like that Santa didn't exist at the tender age of 3!), the intriguing conversations and your refreshing open mindedness.

To Kevin, for all the laughs (initially the best ones were at your expense, now mostly shared!), for all your advice and for having chosen such an original and nourishing path in life, which revitalises me also.

To Juliet, for all your patience and love, for always sticking by me (at times "like a turkey that votes for Christmas"!!), your support has made a huge difference to me, and made me happier than I've been for years. To lots more voyages, many more incredible new experiences and loads more evenings where we put the world to rights !! A Marc-Antoine pour m'avoir supporté pendant les moments hauts (le Lac Jovet), les moments bas L'étiologie des PCCRC était: 41% cancers ratés, 29% nouveaux cancers, 24% échecs de détection à la biopsie et 6% prélèvement incomplet.

Les TDAs (taux de détection des adénomes) étaient hétérogènes et variaient de 11% à 49% (moy : 30 %). On observait une tendance linéaire inverse entre des TDA et le nombre de PCCRs qui n'était pas statistiquement significative.

Conclusion

La prévalence des PCCRCs et le taux d'incidence étaient bas par rapport à la littérature. Une forte hétérogénéité était présente. Une homogénéisation des TDAs par l'établissement d'un seuil national de Colonoscopy is widely used as part of colorectal cancer screening [START_REF] Lieberman | Use of colonoscopy to screen asymptomatic adults for colorectal cancer. Veterans Affairs Cooperative Study Group 380[END_REF][START_REF] Schoenfeld | Colonoscopic screening of average-risk women for colorectal neoplasia[END_REF][START_REF] Zauber | Colonoscopic polypectomy and long-term prevention of colorectal-cancer deaths[END_REF] . Colonoscopy can be used directly as a first line screening tool, or second line, after a positive gFOBT (guaiac positive faecal occult blood test) or FIT (faecal immunochemical test). Colonoscopy is not however an infallible test with incidence of post-colonoscopy colorectal cancer (PCCRC) ranging from 0.2/1000 person-years [START_REF] Kaminski | Quality indicators for colonoscopy and the risk of interval cancer[END_REF] to 2.2 cases / 1000 person years 7 and 1 in 27 colorectal cancers being diagnosed between 6 and 36 months of colonoscopy [START_REF] Singh | risk factors, and outcomes of interval colorectal cancers: a systematic review and meta-analysis[END_REF] .

Following an endoscopist working group, an interval cancer was defined as "a colorectal cancer diagnosed after a screening or surveillance exam in which no cancer is detected, and before the date of the next recommended exam" [START_REF] Sanduleanu | Definition and taxonomy of interval colorectal cancers: a proposal for standardising nomenclature[END_REF] .

For the purposes of our research we focused solely on Interval Colorectal Cancer after Colonoscopy performed following a positive Guaiac Faecal Occult Blood Test (gFOBT) in the Isère regional screening programme from 2002 -2013.

Significant research has been done on Interval Colorectal Cancer. Several recent studies have estimated the risk of PCCRCs. There is a wide range of heterogeneity relating to study setting, study design, patient characteristics, age, comorbidities, and definition of PCCRC, with most being performed outside of a screening context (see appendix 2). Those performed in a screening context either used colonoscopy as a primary screening tool or else used FIT as opposed to gFOBT as the primary screening tool. In the Isere region, the sensitivity of gFOBT has already been studied 10 . To date however; there has been no study that has examined the risk of PCCRC following a positive gFOBT in a colorectal cancer screening programme.

Describing the characteristics of the PCCRCs in the region, would enable us to identify certain particularities that may pertain to the region, and how the screening programme could be modified in the future to better handle such particularities.

To better understand how we could decrease the risk of PCCRCs in the future in our screening programme, we analysed the characteristics of each of the 17 cases and divided them up by aetiology. When analysing the characteristics of the PCCRCs in isolation we used the 2002-2014 data ( 17PCCRCs). We used the 2002-2009 period when calculating the incidence rate and the number of PCCRCs per colonoscopy, to allow possible PCCRCs to be diagnosed up until 2014) 3. The results of all the patients that underwent colonoscopy after a positive gFOBT are communicated to the ODLC by general practitioners, patients and the gastroenterologists that carry out the colonoscopies in the screening programme in the region. By cross-referencing this information with the Isère Cancer Registry we were able to identify all persons who were diagnosed with an invasive colorectal cancer (In-situ cancers and anal cancers were not included) diagnosed between 6 months and 60 months after their colonoscopy, who could potentially be defined as having a PCCRC.

Colorectal cancers diagnosed between 0 and 6 months post-colonoscopy are considered as screened cancers.

4. The list of potential PCCRCs is compared where possible with data from the relevant healthcare providers in order to collate as much information as possible. This enabled us to: 1) identify that the The initial six months were subtracted and people were observed up to a maximum of 4.5 years, (as patients were only at risk of PCCRC after the first six months following their initial colonoscopy until five years later). We included all patients who had their index colonoscopy up to the end of 2009, and correspondingly all PCCRCs diagnosed until the end of 2014.

A Kaplan Meier survival analysis was used to analyse the time post colonoscopy until diagnosis of the PCCRC.

A poisson regression model was utilised to analyse the relationship between a gastroenterologist's number of PCCRCs (independent variable), and their ADR, controlling for the number of colonoscopies they had performed. We examined other possible confounding factors variables including percentage of satisfactory bowel preparations and caecum intubation rate.

All analyses were conducted using STATA data analysis and statistical software (Stata Corp. LP, College Station, TX, USA.

Results:

The 10,557 negative colonoscopies from January 2002 to December 2013, came from a cohort of 590,912 gFOBT tests carried out by participants between the ages of 50-74 who underwent a gFOBT as part of the Isère regional screening programme. Participation rates varied over time, from 23% in 2002 to 45% in 2009. 12,281 (2.1%) positive gFOBT were recorded, which led to 11,177 (90.5%) colonoscopies being performed. 620 participants who underwent colonoscopy were diagnosed with a cancer (Screening Cancer). The 10,557 negative colonoscopies (findings: normal, hyperplastic polyp, adenoma or carcinoma in situ), were followed for an interval of five years (up to a maximum of 31 st December 2013). During this follow up period 15 PCCRCs arose from this group. Amongst the 578,631 negative gFOBT tests that were recorded, 711 interval cancers were diagnosed in the following two years (IC gFOBT). All the patient populations showed a predominance of male patients. PCCRCs had the highest male proportion of 73.3%. Colorectal cancers were categorised as proximal (caecum/appendix, right colon and transverse colon) and distal (left colon, sigmoid colon and rectum) [START_REF] Singh | risk factors, and outcomes of interval colorectal cancers: a systematic review and meta-analysis[END_REF] .

All the groups of cancers showed a predominance of distal versus proximal location. PCCRCs had an almost equal distal / proximal distribution (53% / 47%), the SCs had the most extreme distal / proximal distribution (73% / 27%). Index colonoscopy not completed due to fixed macro diverticulosis and voluminous sigmoid polyp. Barium enema confirmed this, therefore surgical indication for sigmoidectomy, which was performed and found a carcinoma in-situ. 6 weeks later a control colonoscopy was performed which was normal. 9 months later another colonoscopy was performed which found a 3cm adenocarcinoma at the anastomosis site. Index colonoscopy found a carcinoma in situ in the sigmoid, resected via colonoscopy 3 weeks later with clear margins. 1.5 years later a control colonoscopy found a Lieberkuhnian adenocarcinoma in the sigmoid Index colonoscopy found 3 hyperplastic polyps of the sigmoid colon. During a routine work-up for an abdominal eventration, 0.8 years later a voluminous lieberkuhnian adenocarcinoma of right colon complicated with peritoneal carcinomatosis and metastic lung lesions were found. Index colonoscopy found an adenoma in the sigmoid. 4.2 years later colonoscopy finds a rectal adenocarcinoma, stage I Index colonoscopy normal. 2.2 years later, colonoscopy finds sigmoid adenocarcinoma, 3cm and stage IIIb Index colonoscopy found hyperplastic polyp in caecum. The aetiology of the 17 PCCRCs in our study were as follows: 7 (41%) missed cancers, 5 (29%) new cancers, 4 (24%) failed biopsy detection, 1 (6%) incomplete removal.

In theory 47% (incomplete removal and missed cancer) of our PCCRCs could have been avoided.

Time to PCCRC diagnosis post index colonoscopy

The time to diagnosis of PCCRC was as follows. (By definition there were no PCCRCs in the first 6 months). Endoscopist performance related PCCRC Risk factors.

Finally we investigated endoscopist performance-related PCCRC risk factors. We paid particular attention to the adenoma detection rate ADR (a metric defined as the proportion of colonoscopy examinations that detects ≥ 1 adenomas [START_REF] Van Rijn | Polyp miss rate determined by tandem colonoscopy: a systematic review[END_REF] ). There was an average ADR of 30.1% for our entire group of endoscopists. For the endoscopists who had performed the index colonoscopy for zero PCCRCs there was an average ADR of 31.0%, for the endoscopists who had performed the index colonoscopy for one PCCRC there was an average ADR of 28.2%, the endoscopist who had performed the index colonoscopy for two PCCRCs had an ADR of 22.8%, and the endoscopist who had performed the index colonoscopy for three PCCRCs had an ADR of 18.1%.

A large degree of variability was noted in the ADRs, ranging from 11% to 49% with a standard deviation of 8.2%.

A poisson regression model was used to analyse the relationship between the number of PCCRCs per gastroenterologist (dependent variable), and ADR (independent variable). ADR was found to be statistically significant (p=0.03). However when the model was adjusted to take into account the number of colonoscopies per gastroenterologist, ADR was no longer statistically significant explanatory variable (p=0.109).

We [START_REF] Kaminski | Quality indicators for colonoscopy and the risk of interval cancer[END_REF] where the participants were aged between 40-66 and colonoscopy was used as a primary screening tool, both of these factors would significantly lower the incidence rate.

The lower level of risk may be attributable to the fact that all our colonoscopies are carried out by gastroenterologists. Colonoscopies performed by gastroenterologists are more effective at colorectal cancer prevention [START_REF] Baxter | Association between colonoscopy and colorectal cancer mortality in a US cohort according to site of cancer and colonoscopist specialty[END_REF] , and thereby decrease the number of PCCRCs.

Secondly participants in our study had a maximum age of 75, whereas in many other studies patients of all ages were included. The annual transition rate from adenoma to carcinoma may not be constant over a lifetime, may increase with age and be as high as 5% per year in men over the age of 70 [START_REF] Brenner | Natural History of Colorectal Adenomas: Birth Cohort Analysis Among 3.6 Million Participants of Screening Colonoscopy[END_REF] .

Finally, the fact that our population relates to a screening population that had an uptake rate of between 23-45% may bias our results. People who agree to participate in a screening programme may be more health conscious and therefore in better health, with less comorbidities than those who had a colonoscopy performed for symptomatic reasons. Green et al [START_REF] Green | Nonparticipation in a Population-Based Trial to Increase Colorectal Cancer Screening[END_REF] showed that non-participants in Colorectal Cancer Screening programs had behaviours that can negatively affect cancer outcomes (less continuity of care with one physician, fewer preventive visits, less use of recommended screening tests, and were current or past smokers.)

Characteristics of the PCCRC Colorectal Cancers in the region

Age

Our study showed patients diagnosed with PCCRCs to be older (average age = 67.1) than those with screened CRCs (average age = 65.2), however younger than those who did not participate in the screening programme (average age 72.0). Our patients tended to be younger than those in the literature, Singh et al showed that older patients (>65-70) had a higher risk of PCCRCs than younger individuals (<65-70), adjusted OR = 1.15. As referenced above, the annual transition rate from adenoma to carcinoma may not be constant over a lifetime, may increase with age and be as high as 5% per year in men over the age of 70 [START_REF] Brenner | Natural History of Colorectal Adenomas: Birth Cohort Analysis Among 3.6 Million Participants of Screening Colonoscopy[END_REF] .

The reason the patients who developed a PCCRC in our study were younger than the literature is most likely due to the maximum age limit of 75 in our screening programme.

Our patients' ages had a large standard deviation (1.5 years) relating to the small number of cases.

Gender

Our study found that out of the 17 PCCRCs, 12 of them were male. Meta-analysis has not shown a clear relationship between gender and PCCRCs, however some individual studies found a higher prevalence of PCCRCs in females as compared with males [START_REF] Samadder | Characteristics of Missed or Interval Colorectal Cancer and Patient Survival: A Population-Based Study[END_REF][START_REF] Brenner | Interval cancers after negative colonoscopy: population-based case-control study[END_REF][START_REF] Bressler | Rates of New or Missed Colorectal Cancers After Colonoscopy and Their Risk Factors: A Population-Based Analysis[END_REF][START_REF] Erichsen | Characteristics and Survival of Interval and Sporadic Colorectal Cancer Patients: A Nationwide Population-Based Cohort Study[END_REF] . This is thought to be due to the female colon being longer and more redundant, leading to more discomfort, longer insertion times and lower completion rates [START_REF] Chutkan | Colonoscopy Issues Related to Women[END_REF] . The gender distribution in our study may be due to the small number of cases.

interval colorectal cancer [START_REF] Kaminski | Quality indicators for colonoscopy and the risk of interval cancer[END_REF] and colorectal cancer deaths [START_REF] Corley | Adenoma Detection Rate and Risk of Colorectal Cancer and Death[END_REF] . ADR is therefore a key quality indicator and has been identified as a target for intervention in terms of quality control [START_REF] Coe | An endoscopic quality improvement program improves detection of colorectal adenomas[END_REF][START_REF] Brenner | Trends in Adenoma Detection Rates During the First 10 Years of the German Screening Colonoscopy Program[END_REF][START_REF] Kaminski | Leadership training to improve adenoma detection rate in screening colonoscopy: a randomised trial[END_REF] Although not statistically significant, our data showed a similar trend that as ADR decreased the number of PCCRCs increased.

There was an average ADR of 30% for our entire group of endoscopists. Our data did however come from a screening population who had a positive gFOBT, which would be at greater risk of adenomas than a general population. Participants presenting for colonoscopy following a positive guiac-based FOBT have a 35% risk of detection of adenoma and a nearly 11 % risk of cancer [START_REF] Uk Colorectal | Results of the first round of a demonstration pilot of screening for colorectal cancer in the United Kingdom[END_REF] . This compares with a 15%-25% risk of detection of benign adenoma by ad hoc case finding [START_REF] Lieberman | Use of colonoscopy to screen asymptomatic adults for colorectal cancer. Veterans Affairs Cooperative Study Group 380[END_REF][START_REF] Schoenfeld | Colonoscopic screening of average-risk women for colorectal neoplasia[END_REF][START_REF] Regula | Colonoscopy in Colorectal-Cancer Screening for Detection of Advanced Neoplasia[END_REF] . In the UK NHS bowel cancer screening progamme the benchmark ADR rate is 35%. Within the Isere region; significant levels of heterogeneity within the ADRs were noted. To reduce this heterogeneity and thereby reduce the number of PCCRCs a benchmark ADR rate should be included in the French Colorectal Cancer Screening Guidelines and gastroenterologists should be informed of their current ADR. Kaminski et al [START_REF] Kaminski | Leadership training to improve adenoma detection rate in screening colonoscopy: a randomised trial[END_REF] showed that performance feedback improved ADR by 1.8%. Training programs could be offered to those who have ADRs below the benchmark level.

Studies have shown that ADR can be improved by further training [START_REF] Coe | An endoscopic quality improvement program improves detection of colorectal adenomas[END_REF][START_REF] Kaminski | Leadership training to improve adenoma detection rate in screening colonoscopy: a randomised trial[END_REF] and these improvements in ADR are associated with a reduced risk of interval colorectal cancer and death [START_REF] Kaminski | Increased Rate of Adenoma Detection Associates With Reduced Risk of Colorectal Cancer and Death[END_REF] . Kaminski et al [START_REF] Kaminski | Leadership training to improve adenoma detection rate in screening colonoscopy: a randomised trial[END_REF] demonstrated that training programmes had a greater effect on ADR than performance feedback, with an absolute improvement of 3.9% (OR 1.25) over performance feedback alone.

Caecal intubation rate was a very satisfactory 96.4%. Current US guidelines recommend a caecal intubation rate of 95% for routine screening exams [START_REF] Rex | Quality indicators for colonoscopy[END_REF] , whereas EU guidelines [START_REF] Rembacken | Quality in screening colonoscopy: position statement of the European Society of Gastrointestinal Endoscopy (ESGE)[END_REF] do not differentiate between screening and non-screening colonoscopies and recommend a caecal intubation rate of 90%.

Out of the 62 gastroenterologists who had performed > 30 colonoscopies, 87.5% of all the colonoscopies had a preparation level that ranged from sufficient to good. Amongst our cases of PCCRC, 94.1% (16/17) had between good to sufficient bowel preparation, with one case unknown.

Inadequate colonic cleansing has been shown in the literature to account for a significant proportion of missed lesions and often results in longer, more difficult, and incomplete exams [START_REF] Froehlich | Impact of colonic cleansing on quality and diagnostic yield of colonoscopy: the European Panel of Appropriateness of Gastrointestinal Endoscopy European multicenter study[END_REF][START_REF] Lebwohl | The impact of suboptimal bowel preparation on adenoma miss rates and the factors associated with early repeat colonoscopy[END_REF] .

Limitations and Strengths of our Study

Limitations:

We did not have access to information regarding certain variables which would have enabled a more complete description of the PCCRCs in the region, such as family history of colorectal cancer, presence of diverticular disease, comorbidities, smoker status, and molecular phenotype of PCCRC.

Our results may slightly underestimate the incidence of PCCRCs in Isère due to people either entering or leaving the region. It was not possible to identify PCCRCs amongst the patient population who had positive gFOBT followed by negative colonoscopies but who then left the region. Neither was it possible to identify PCCRCs in the patient population who moved to the region following a positive gFOBT as these people were not included in our database either. It was not possible to address these two sources of bias, but we assume the effect to be quite small. Out of all the regions in France, Isère has one of the smallest levels of migration among this age bracket, ranked 15 th lowest out of the 96

French regions with a net migration rate of -0.90 per 1000 people per annum [START_REF] Baccaini | Recensement de la population de 2006 : Les migrations entre départements : le Sud et l'Ouest toujours très attractifs[END_REF] .

Another source of bias in our study is that we were unable to definitively identify all PCCRCs for patients who had positive gFOBT near the end of the study period of 31 December 2013, as we were unable to follow the patients for the required 5 years. We identified all the PCCRCs which occurred up to and including the 31 December 2014.

Strengths:

We had complete, exhaustive data on all cancers over a lengthy study period due to the input of data from the Régistre Du Cancer de l' Isere. Interval cancers were able to be identified by crossreferencing all the data from the ODLC with data from the Régistre Du Cancer de l' Isere.

(

  les catacombes) et tous les moments rigolos, et pour ta manière de penser, pleine de compassion et de sérénité. (Et pour tous les mails que tu m'as traduits bien évidemment !!!) « Cher »Matthieu pour ta « confraternité » et ta « sagesse » mais au-delà de ça ton amitié et tous les supers instants qu'on a partagé! A Clément pour ta générosité, ta joie de vivre et ta disponibilité permanente pour prendre une bière ! A Angèle, Alex et Sophie, Vince et Caro, Manu et Marianne, Stéphane, Chloé, Flo et Valentina, Stéphane et Vanessa, Olivier, Delphine, Nico et Désirée, Laeti et Jé, Ludo et Delphine, Ludo et Véronique, Julien et Sophie d'avoir été là. Résumé Introduction L'objectif de cette étude est de décrire les caractéristiques des PCCRCs (cancer d'intervalle post coloscopie) et les facteurs de risques liés aux performances de l'endoscopiste. Méthode L'étude a porté sur les coloscopies faites après un test de dépistage positif entre 2002 et 2013 dans le département de l'Isère. Les PCCRCs étaient définis comme des adénocarcinomes diagnostiqués entre six et soixante mois après coloscopie. Les facteurs de risque liés à la performance de l'endoscopiste ont été analysés parmi les 62 gastroentérologues ayant réalisé au moins 30 coloscopies. Résultat 10 557 coloscopies négatives ont été réalisées. 15 PCCRCs ont été diagnostiqué entre 2002-2013, l'âge moyen était 67,1 ans, 73% ont touché les hommes, 53% étaient localisés dans le colon distal.1.1% (0.7%, 1.8%) des cancers diagnostiqués en dépistage étaient des PCCRCs avec un taux d'incidence de 0.41 (0.21, 0.71) pour 1 000 personnes-années.

L

  'augmentation du risque de PCCRC avec une diminution des TDAs a été montrée dans la littérature, le TDA étant un indicateur de qualité reconnu des coloscopies. Le faible effectif des cancers d'intervalle post coloscopie n'a pas permis de retrouver de tendance significative dans notre département. L'établissement d'un seuil de TDA dans les recommandations du programme de dépistage français et l'information des praticiens sur leurs résultats devrait permettre d'homogénéiser les TDAs et diminuer ainsi le nombre de cancers d'intervalle. Des formations pourraient être organisées et proposées aux gastroentérologues ayant des résultats inférieurs au seuil déterminé. Une étude plus large incluant d'autres départements avec des Registres de cancer pourrait permettre de consolider ces résultats dans le programme français de dépistage des cancers colorectaux. Mots clés : Cancer colorectal; Programme de Dépistage, cancer d'intervalle, Test Hémoccult, cancer (CRC) is the third most commonly diagnosed malignancy in males and the second in females with an estimated 1.4 million new cases and 693,000 deaths occurring in 2012 1 . In 2012, in France; Colorectal Cancer was the fourth most commonly diagnosed cancer behind, Prostate, Breast and Lung. In 2012 40,825 new cases were diagnosed, and 17,148 people died as a result of colorectal cancer in France 2 .

  Finally we investigated endoscopist related PCCRC risk factors with particular attention paid to adenoma detection rate, and what modifications could be made to minimise these risks.ObjectivesThe objectives of our study were:Calculate the risk of PCCRC in the Isère region, and to compare it to the levels of risk of PCCRC in other regions.  Describe and compare the characteristics of the PCCRCs to all the colorectal cancers in the region  Define the PCCRC by type (Incomplete Removal, Failed Biopsy Detection, Missed Cancer, New Cancer) according to the algorithm proposed by Pabby et al 7 ,  Investigate Endoscopist performance-related risk factors relating to these PCCRCs Methods: Study Design: This is a population-based retrospective cohort study. Screening Procedures The Colorectal cancer screening programme exists in the Isère region since 2002. The screening programme was based on a biennial Guaiac Faecal Occult Blood Test (gFOBT), targeting men and women from age 50 to 74, with a colonoscopy for positive gFOBT cases. Since 2015 the gFOBT has been replaced by a Faecal Immunochemical Test (FIT). Data Collection: 1. Data relating to the PCCRC was obtained from the Office de Lutte contre le Cancer (ODLC) screening programme database. Data relating to all the colorectal cancers in the Isère region was obtained from the InVS Régistre des cancers de l'Isère. Inclusion Criteria  All subjects between the ages of 50-74 living in the Isère department who had a colonoscopy between the dates of 1 st January 2002 to 31 st December 2013 following a positive Guaiac Faecal Occult Blood Test screening test. Exclusion Criterion  All colonoscopies outside the period of 1 st January 2002 to 31 st December 2013 following a positive Guaiac Faecal Occult Blood Test .  All colonoscopies which found an invasive colorectal cancer. 2. Three different time periods were used in our analysis 2002-2013, 2002-2014 and 2002-2009. Data from the Isère Cancer Registry identifying all cancers diagnosed in the region was only available up to 2013. Data relating to all endoscopists pertained to colonoscopies performed from 2002-2013. Meanwhile using the data from the ODLC we were able to identify all PCCRCs diagnosed up to 2014. Therefore in analysing the performance of the endoscopists or comparing our results with results from the Isère Cancer Registry we used data up to 2013 (15 PCCRCs).

  cases were indeed confirmed PCCRCs ( by analysing; circumstances of the diagnosis of the CRC, Colonoscopy report, Surgical Operation Report, Pathology report, Location, Stage pTNM of the CRC and Index Colonoscopy Report following the positive Guaiac Faecal Occult Blood Test test.) and 2) identify the PCCRC by type (Incomplete Removal, Failed Biopsy Detection, Missed Cancer, New Cancer) according to the algorithm proposed by Pabby et al 7 Incomplete colonoscopies were classified as Missed Lesions.5. The adenoma detection rate (ADR) and all other data relating to each gastroenterologist are recorded and calculated by the ODLC. The ADR is based on all colonoscopies performed as a result of a positive gFOBT in the period from 2002-2013 by a given gastroenterologist. The ADR is defined as the proportion of screening colonoscopies that detect at least one adenoma. All colonoscopies where the gastroenterologist had performed less than thirty colonoscopies for gFOBT positive patients during the study period were excluded for the purpose of ADR analysis. They were however included for the purposes of the calculation of the incidence rate of PCCRC. To analyse the ADR between 2002-2007 and 2008-2013, we again included only those gastroenterologists who had performed > 30 in the period. Variables of Interest: Patient: Sex, Age. Colonoscopy: time interval between positive gFOBT and index Colonoscopy, the number of previous gFOBT performed, result of their index colonoscopy (type of lesion, location of lesion, caecal intubation achieved), result of diagnostic colonoscopy (date of colonoscopy, location of lesion, staging of lesion, size of lesion) Gastroenterologist: number of colonoscopies performed in the context of the screening programme in the study period, ADR, bowel preparations and caecum intubation rate. PCCRC / Screened Cancers / Non-Screened Cancers: Location of the cancer: recto sigmoid, descending colon, transverse colon, ascending colon and caecum/appendix.Statistical analysisThe incidence rate was calculated as: 𝑡𝑖𝑚𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑙𝑜𝑛𝑜𝑠𝑐𝑜𝑝𝑦 𝑎𝑛𝑑 𝑃𝐶𝐶𝑅𝐶 -0.5) 𝑜𝑟 min(𝑝𝑒𝑟𝑖𝑜𝑑 𝑓𝑟𝑜𝑚 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐𝑜𝑙𝑜𝑛𝑜𝑠𝑐𝑜𝑝𝑦 𝑎𝑛𝑑 31 𝐷𝑒𝑐 2003 ; 5 𝑦𝑒𝑎𝑟𝑠) -0. )

Figure 1 :

 1 Figure 1: Flowchart of study participants 2002-2013

Figure 2 :

 2 Figure 2: Cancer location by type of Colorectal Cancer diagnosed in the Isère region from 1 January 2002 to 31 December 2013

4 . 2

 42 years later in a work-up for GI motility disorder, a Stage IIIb caecal adenocarcinoma is found. Index colonoscopy: 10 polyps in transverse and sigmoid colon, 9 of which were hyperplastic, one showed severe dysplasia (location not mentioned), resected with clear margins. 4.8 years later, during a work-up for an anal fistula a rectal adenocarcinoma was found. Stage IIa, 3.5cm. Index colonoscopy: 12 polyps removed, 1 sessile polyp in the right colon 4mm resected by cold excision, right transverse colon: 3 sessile polyps 5-6mm each resected by hot excision, left colon: 3 sessile polyps 8-10mm polypectomy, Sigmoid: 3 polyps resected by polypectomy. Pathology found low grade dysplasia in the 2 lesions in the right colon (i.e. the pathology report interpreted one of the transverse polyps as a right colon polyp, does not mention clear margins), 1 transverse left, 1 left and 3 sigmoid. Control 3mths later never performed. 2.8 years later in the work-up for an Iron deficiency Anaemia: Lieberkuhnian adenocarcinoma of the right colon, stage I, 4cm.

Figure 3 .

 3 Figure 3. Kaplan Meier curve; Time to Interval Colorectal Cancer diagnosis following negative colonoscopy from 2002-2014.

32 Figure 4 :

 324 Figure 4: Relationship of ADR to Number of PCCRCs per endoscopist diagnosed between 2002-2014

  TDA, l'information des praticiens sur leurs résultats et des formations, devraient homogénéiser les TDAs et diminuer le nombre de cancer d'intervalle dans la région. Mots clés : Cancer colorectal; Programme de Dépistage, cancer d'intervalle, Test Hemoccult, En France, le dépistage du cancer colorectal est basé sur un test de recherche de saignement occulte dans les selles et une coloscopie est réalisée en présence d'un test positif. La coloscopie n'a pas une sensibilité de 100% et le taux d'incidence de cancer d'intervalle post coloscopie (PCCRC) varie entre 0,22 et 2,2 cas pour 1 000 personnes-années et un sur 27 cancers colorectaux étant diagnostiqué entre 6 et 36 mois après une coloscopie. Les objectifs de cette étude sont :  De décrire les caractéristiques des PCCRCs. De calculer l'incidence de PCCRC en Isère, et de la comparer à l'incidence des autres régions.

	Résumé Détaillé 117 gastroentérologues ont réalisé 10557 coloscopies négatives. Les facteurs de risque liés à Comparativement à la littérature, les PCCRCs du département de l'Isère étaient plus jeunes, touchaient
	l'endoscopiste ont été analysés parmi les 62 gastroentérologues ayant réalisé au moins 30 coloscopies majoritairement des hommes et présentaient une distribution égale des lésions entre proximal et distal.
	Introduction pendant la période. Les méta-analyses ont montré que les patients âgés sont plus touchés par les PCCRCs, sans différence
	Le cancer colorectal est le troisième cancer le plus souvent diagnostiqué chez les hommes et le entre les sexes, et avec une fréquence 2,4 fois supérieure des cancers proximaux.
	Coloscopie; Diagnostique deuxième chez les femmes, avec 1.4 million de nouveaux cas et 693 000 décès en 2012 dans le Résultats
	monde. Au total, 10 557 coloscopies négatives ont été réalisées. La répartition des cancers colorectaux au
	cours de cette période était la suivante 6 249 NPC (cancer hors dépistage), 620 SC (cancer dépisté),
	15 PCCRC, 711 CI gFOBT. Parmi les PCCRCs l'âge moyen était 67,1 ans, 73% ont touché les
	hommes, 53% étaient localisés dans le colon distal.
	1.1% (IC 95% 0.7% à 1.8%) de tous les cancers diagnostiqués dans le programme de dépistage étaient
	des PCCRCs avec un taux d'incidence de 0.41 (IC 95% 0.21 à 0.71) pour 1 000 personnes-années. Il y
	avait 1.8 (IC 95% 1.0-3.2) PCCRCs pour 1 000 coloscopies.
	L'étiologie des PCCRCs était la suivante : 41% cancers ratés, 29% nouveaux cancers, 24% échecs de
	détection à la biopsie et 6% prélèvement incomplet.
	 De classer les PCCRCs par étiologie (prélèvement incomplet, échec de détection par biopsie, Les TDAs (taux de détection des adénomes) étaient très hétérogènes et variaient de 11% à 49% (moy :
	cancer raté, nouveau cancer) 30 %). On observait une diminution des TDAs quand le nombre de PCCRCs augmentait mais cette
	 D'étudier les facteurs de risques de PCCRC liés à l'endoscopiste. tendance n'était pas statistiquement significative.
	Méthodes
	L'étude a porté sur les coloscopies faites après un test de dépistage positif entre 2002 et 2013 dans le Discussion et Conclusion
	département de l'Isère. Les PCCRCs étaient définis comme des adénocarcinomes diagnostiqués entre La prévalence des PCCRC (1,1 %) et le taux d'incidence (0,41 pour 1 000 personnes-années) sont très
	six et soixante mois après coloscopie. Le croisement des participants au programme départemental de bas contre respectivement : 1.8% à 9% et 0.2 à .2.2 cas pour 1 000 personnes-années dans la
	dépistage des cancers colorectaux entre 1 er janvier 2002 et 31 décembre 2013 avec la base de données littérature. Ce résultat semble indiquer un bon niveau de qualité des endoscopies réalisées dans notre
	du Registre du cancer de l'Isère a permis d'identifier les PCCRCs et de comparer leurs caractéristiques département. La comparaison reste cependant difficile compte tenu de l'hétérogénéité des études
	avec : les cancers d'intervalle post hemoccult (CI gFOBT), les cancers dépistés (SC) et les cancers publiées.
	hors dépistage (NPC) pour cette période.

Table 1 :

 1 Distribution of type of colorectal cancer by age category from 1 January 2002 to 31 December 2013 *>70 years old Our study showed patients diagnosed with PCCRCs to be older (average age = 67.1) than those with Screened CRCs (average age = 65.2), however younger than those who did not participate in the screening programme (NPC) (average age 72.0). The age of those diagnosed with PCCRCs had a larger standard deviation (1.5 years).

	Characteristics of the Colorectal Cancers in the region 2002-2013
		NPC	SC		IC gFOBT		PCCRC
		n	%	n	%	n	%	n	%
	50-54	737	11.8%	48	7.7%	50	7.0%	0	0.0%
	55-59	447	7.2%	83	13.4%	100	14.1%	1	6.7%
	60-64	523	8.4%	136	21.9%	151	21.2%	5	33.3%
	65-69	602	9.6%	159	25.6%	160	22.5%	4	26.7%
	70-74	3,940*	63.1%	194	31.3%	250	35.2%	5	33.3%
	Total	6,249	100%	620	100%	711	100%	15	100%

Table 2 :

 2 Comparison of Gender distribution by PCCRC and all other Colorectal Cancers in the Isère region from 2002-2013.

Table 4 :

 4 Summary of Colonoscopy findings and cancer characteristics of the 17 PCCRCs diagnosed in Isère Screening Programme from 2002-2014 Index colonoscopy found a hyperplastic polyp in rectum. 4.8 years later stage I cancer found in Caecum Appendix Colonoscopy performed, difficult to visualise sigmoid due to insufficient bowel preparation, CT scan performed which was normal. Hospitalised 1.33 years later for occlusive syndrome, stage IIa sigmoid tumour found. Index colonoscopy found advanced polyp in right colon. Colonoscopy repeated 3 years later, stage IIIc cancer of left colon. Index colonoscopy was normal, 1.8 years later stage IIIb cancer found in caecum.

		70%
		60%
	% of Cancers	30% 40% 50%			Rectosigmoid Descending Colon Transverse Colon
		20%
					Ascending Colon / Caecum /
		10%			Appendix
		0%
		NPC	SC	IC gFOBT	PCCRC
			Detection Context

Index colonoscopy found an advanced polyp in right colon and rectum, Colonoscopy 1 month later: sigmoid adenoma (not fully resected as difficult position) and Carcinoma in-situ of rectum, resected with clear margins. Rectal Cancer (Lieberkuhnian adenocarcinoma) diagnosed July 2010 Index Colonoscopy: 2 hyperplastic polyps. 2.2 years later: 6mm, Stage IIa cancer found in right colon Index colonoscopy normal. 2.0 years later 2.5cm stage IIIb tumour found in transverse colon Index colonoscopy : adenoma in the sigmoid removed with diathermy , NOT clear margins, colonoscopy repeated 2 months later showed the remnant of the sigmoid sessile polyp which was resected with clear margins. Colonoscopy 7.5 months later: stage I sigmoid adenocarcinoma (Thermal Artefact?) Index colonoscopy found sigmoid adenoma, resected with clear margins. 2.6 years later, a neuroendocrine tumour of the sigmoid.

  compared the ADR of endoscopists from the period 2002-2007 to 2008-2013. The average ADRs were 29.1% compared to 31.4%, with no statistical difference between the periods.It is difficult to compare these results with other studies due to the wide range of heterogeneity relating to patient characteristics, age, comorbidities, study setting, study design, definition of PCCRC (see appendix 2). Our study has a prevalence and incidence rate that are low relative to other studies. Other studies had prevalence rates ranging from 1.8% -9% and an incidence rate ranging from 0.2-2.2 per 1000 person years. Amongst all the literature it had the lowest prevalence rate and the second lowest incidence rate, however the study with the lowest incident rate was in a study carried out by Kaminski et al

	0.5 Caecal intubation rate was a very satisfactory 96.4%.
	0.4
	0.3 Discussion:
	ADR
	0.2 Risk of PCCRC
	0.1 PCCRC had a prevalene of 1.1%, an incident rate of 0.41 cases per 1,000 person-years and a rate of
	0.0
	0	1	2	3
		Number of PCCRC per Endoscopist

  . Kaminski et al[START_REF] Kaminski | Quality indicators for colonoscopy and the risk of interval cancer[END_REF] in a screening programme setting demonstrated that endoscopists with an ADR <20% significantly increased the risk of PCCRC as compared with those with ADR>20%. Corley et al[START_REF] Corley | Adenoma Detection Rate and Risk of Colorectal Cancer and Death[END_REF] in a study of all colonoscopies in patients aged > 50, showed that the risk of advanced stage PCCRC and fatal PCCRC were reduced by 57% and 62% respectively, with ADRs >33% compared with ADRs < 19%.

Remerciements Aux membres du jury :

A M. le Professeur Patrice FRANCOIS, président du jury Vous me faites l'honneur de présider ce jury. Merci de votre disponibilité et de votre soutien dans ce travail. Je vous remercie pour vos conseils et votre disponibilité tout au long de mon internat. Veuillez trouver ici l'expression de mon profond respect et de ma sincère gratitude.

Specific PCCRC case analysis

To better understand how we could decrease the risk of PCCRCs in the future in our screening programme, we analysed the characteristics of each of the 17 PCCRCs diagnosed between 2002 and 2014 and divided them up by aetiology.

PCCRCs have been categorized as (i) those arising because of missed lesions, either owing to inadequate examination (poor bowel preparation, incomplete colonoscopy) or because of incomplete visualisation (or failure to recognize a polyp despite visualisation of bowel segment), (ii) arising from incomplete resection of a recognized dysplastic polyp, (iii) arising de novo , as new, rapidly forming cancers, and (iv) arising as a result of failure of biopsy to identify a lesion as a CRC [START_REF] Singh | risk factors, and outcomes of interval colorectal cancers: a systematic review and meta-analysis[END_REF] . By dividing them by aetiology, special attention could be paid to the procedural factors (missed or incompletely resected lesions) as these are potentially avoidable [START_REF] Le Clercq | Postcolonoscopy colorectal cancers are preventable: a populationbased study[END_REF] .

Below are the key pieces of information of each of the PCCRCs, along with the type of PCCRC according to the Pabby algorithm (see appendix. A). Commentary for each case is included below. Table 3 describes the characteristics of these 17 PCCRCs. Preparation was "good" in 64.7% 11/17 of colonoscopies, "sufficient" in 29.4% (5/17), and unknown in one case. The caecum was intubated in all but two cases. In these two cases CT colonography was performed in one instance and in the other, a barium enema confirmed an indication for a sigmoidectomy due to a "fixed macro diverticulosis", which was performed. Hyperplastic polyps were found in 4/17 (23.5%) index colonoscopies, 4/17 (23.5%) found no lesion, and 9/17 (52.9%) colonoscopies found adenomas. Out of the 9 adenomas 6/9 (66.6%) were in the same location as the PCCRC, 2/9 (22.2%) were in a different location, and 1/9 (11.1%) was unknown. The site of PCCRC in our population shows an almost equal distribution between distal (53%) and proximal (47%). Our result is therefore quite unusual.

The hypotheses for the greater frequency of proximal versus distal PCCRCs are as follows: failure of endoscopists to intubate the cecum, even when they believe they have reached the cecum, or the likelihood that inadequate bowel preparations affect the right side more than the rest of the colon [START_REF] Singh | Risk of Developing Colorectal Cancer Following a Negative Colonoscopy Examination: Evidence for a 10-Year Interval Between Colonoscopies[END_REF] .

Furthermore, there are differences in gene expression and molecular characteristics between rightsided and left-sided CRCs [START_REF] Birkenkamp-Demtroder | Differential gene expression in colon cancer of the caecum versus the sigmoid and rectosigmoid[END_REF] with microsatellite instability and CpG island methylator phenotypes being more common in right-sided tumours [START_REF] Iacopetta | Are there two sides to colorectal cancer?[END_REF][START_REF] Chirieac | Phenotype of microsatellite-stable colorectal carcinomas with CpG island methylation[END_REF] . These characteristics have been linked with more rapid carcinogenesis [START_REF] Huang | The clinical significance of serrated polyps[END_REF] . Moreover, depressed adenomatous lesions, which are more difficult to detect endoscopically, may occur more often in the right side of the colon, [START_REF] Konishi | Clinicopathological differences between colonic and rectal carcinomas: are they based on the same mechanism of carcinogenesis?[END_REF][START_REF] Okamoto | Flat-type early colorectal cancer preferentially develops in right-sided colon in older patients[END_REF] and these lesions may be more likely to rapidly progress to adenocarcinoma [START_REF] Soetikno | Nonpolypoid (Flat and Depressed) Colorectal Neoplasms[END_REF] .

Possible explanations for the difference in distribution of lesions in our study include the following:

Research has shown that colonoscopies performed by gastroenterologists are more effective at colorectal cancer prevention [START_REF] Baxter | Association between colonoscopy and colorectal cancer mortality in a US cohort according to site of cancer and colonoscopist specialty[END_REF] . As only gastroenterologists perform colonoscopies, there should be a decreased risk of "failure of endoscopists to intubate the cecum, even when they believe they have reached the cecum". This greater proficiency theory was supported by the fewer number of "missed cases" in our study compared to other studies.

The difference in distribution may also just be a statistical anomaly due to the small number of PCCRCs in our study.

Definition of PCCRC by type

The aetiology of PCCRC in our study was divided up as follows: Missed Cancer 7 (41%), New Cancer 5 (29%), Failed Biopsy Detection 4 (24%), and Incomplete Removal 1 (6%).

We compared our results with the results of Roberston et al [START_REF] Robertson | Colorectal cancers soon after colonoscopy: a pooled multicohort analysis[END_REF] and Le Clerq [START_REF] Le Clercq | Postcolonoscopy colorectal cancers are preventable: a populationbased study[END_REF] . As previously mentioned only gastroenterologists performed the colonoscopies in our study, their great proficiency [START_REF] Baxter | Association between colonoscopy and colorectal cancer mortality in a US cohort according to site of cancer and colonoscopist specialty[END_REF] may result in a decreased percentage of missed cancers and incomplete removals.

Endoscopist performance related PCCRC Risk factors.

Endoscopists also have an important role to play in avoiding


Qinghua Zhang
email: qinghua.zhang@inria.fr

On Stability of the Kalman Filter for Discrete Time Output Error Systems

Keywords: Kalman filter, discrete time output error system, time varying system, stability

The stability of the Kalman filter is classically ensured by the uniform complete controllability regarding the process noise and the uniform complete observability of linear time varying systems. This paper studies the case of discrete time output error (OE) systems, in which the process noise is totally absent. The classical stability analysis assuming the controllability regarding the process noise is thus not applicable. It is shown in this paper that the uniform complete observability is sufficient to ensure the stability of the Kalman filter applied to time varying OE systems, regardless of the stability of the OE systems. Though the continuous time case has been studied recently, the results on continuous time systems cannot be directly transposed to discrete time systems, because of a difficulty related to the observability of the discrete time filter error dynamics system.

Introduction

The well known Kalman filter has been extensively studied and is being applied in many different fields [START_REF] Anderson | Optimal Filtering[END_REF][START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF][START_REF] Zarchan | Fundamentals of Kalman filtering: a practical approach[END_REF][START_REF] Kim | Kalman Filter for Beginners: with Matlab Examples[END_REF][START_REF] Grewal | Kalman Filtering: Theory and Practice with Matlab[END_REF]. The purpose of the present paper is to study the stability of the Kalman filter in a particular case not yet covered in the literature: the absence of process noise in the state equation of a discrete time linear time varying (LTV) system. Such systems are known as output error (OE) systems in the literature on system identification. The recent studies on continuous time OE systems in (Ni andZhang, 2015, 2016) have been mainly motivated by applications where state equations originate from physical laws that are believed sufficiently accurate. For discrete time systems considered in this paper, the motivation is mainly for OE system identification [START_REF] Goodman | An output error model and algorithm for electromegnetic system identification[END_REF][START_REF] Forssell | Identification of unstable systems using output error and boxjenkins model structures[END_REF][START_REF] Wang | A robust output error identifier for continuous-time systems[END_REF]. In control applications, the use of OE models has the advantage of focusing system identification on the dynamics of the controlled plant, rather than on noise properties [START_REF] Forssell | Identification of unstable systems using output error and boxjenkins model structures[END_REF]. The result presented in this paper ensures the stability of the Kalman filter applied to LTV OE systems. This result is particularly useful for linear parameter varying (LPV) system identification based on prediction error minimization, as it ensures stable predictions, regardless of the stability of the estimated LPV models during the iterations of prediction error minimization.

While the optimal properties of the Kalman filter are frequently recalled, its stability properties are less often mentioned in the recent literature. The classical stability analysis is based on both the uniform complete controllability regarding the process noise and the uniform complete observability of LTV systems [START_REF] Kalman | New methods in Wiener filtering theory[END_REF][START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF]. In the case of OE systems, there is no process noise at all in the state equation, hence the controllability regarding the process noise cannot be fulfilled.

The stability of the Kalman filter for continuous time OE systems has been recently studied in (Ni andZhang, 2015, 2016). It is often straightforward to transpose theoretic results from continuous time systems to discrete time systems, and vice versa, but there are exceptions. For the stability problem studied in this paper, there are two main extra difficulties for discrete time systems.

First, in the continuous time case, the observability of an OE system induces the observability of its Kalman filter error dynamics system, and this observability plays an important role in the stability analysis of the error dynamics. In the discrete time case, however, the observability of the Kalman filter error dynamics system cannot be induced in a similar way, as explained in Section 5.

Second, in the continuous time case, in the first step of the proof of the Kalman filter asymptotic stability, the Lyapunov stability is naturally proved within a few lines in [START_REF] Ni | Stability of the Kalman filter for continuous time output error systems[END_REF]. In the discrete time case, however, by indirectly analyzing the Kalman filter error dynamics, due to the complexity related to the separation between the prediction step and the update step (no such separation exists in the continuous time case), the proof of the Lyapunov stability takes more than one page (see the proof of Theorem 1 in this paper), with non trivial choices of appropriate equalities involved in the discrete time case only.

The classical optimality results of the Kalman filter are also valid in the case of OE systems (Jazwinski, 1970, chapter 7). However, it is necessary to complete the stability analysis, as the classical results are not applicable in this case.

As the main contribution of this paper, it will be shown that the uniform complete observability is sufficient to guarantee the stability of the dynamics of the Kalman filter applied to a discrete time LTV OE system, regardless of the stability of the OE system itself. The boundedness of the state estimate covariance, as well as the boundedness of the Kalman gain, will also be proved under the same condition. These results complete the classical results [START_REF] Kalman | New methods in Wiener filtering theory[END_REF][START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF], which do not cover the case of OE systems.

Preliminary results of this study have been submitted to the upcoming IFAC Word Congress [START_REF] Zhang | On stability of the Kalman filter for discrete time output error systems[END_REF]. The present paper enriches these preliminary results with technical details, notably the relationship between the Kalman filter error dynamics and the Kalman predictor error dynamics, and also with numerical examples.

The rest of this paper is organized as follows. Some preliminary elements are introduced in Section 2. The problem considered in this paper is formulated in Section 3. The boundedness of the Kalman filter for OE systems is analyzed in Section 4, and the asymptotic stability of the Kalman filter is established in Section 5. Numerical examples are presented in Sections 6. Finally, concluding remarks are drawn in Section 7.

Definitions and basic facts

Let m and n be any two positive integers. For a vector x ∈ R n , x denotes its Euclidean norm. For a matrix A ∈ R m×n , A denotes the matrix norm induced by the Euclidean vector norm, which is equal to the largest singular value of A. Then Ax ≤ A x for all A ∈ R m×n and all x ∈ R n . For two real square symmetric positive definite matrices A and B, A > B means A -B is positive definite.

Let A(k) ∈ R m×n be a sequence of matrices for k = 0, 1, 2, . . . . It is said (upper) bounded if A(k) is bounded.

Consider the homogeneous discrete time LTV system

x(k) = A(k)x(k-1) (1)
with x(k) ∈ R n and A(k) ∈ R n×n , and with the associated state transition matrix defined as

Φ(k, k) = I n (2) Φ(k, l) = A(k)A(k-1) • • • A(l + 1) (3) with I n denoting the n × n identity matrix. Then x(k) = Φ(k, l)x(l).
Definition 1. System (1) is Lyapunov stable if there exists a positive constant γ such that, for all integers k, k 0 satisfying k ≥ k 0 , the following inequality holds

Φ(k, k 0 ) ≤ γ. ( 4
)
Definition 2. System (1) is asymptotically stable if it is Lyapunov stable and if the following limiting behavior holds

lim k→+∞ x(k) = 0 (5)
for any initial state x(0) ∈ R n .

The following uniform complete observability 1 definition for LTV systems follows [START_REF] Kalman | New methods in Wiener filtering theory[END_REF].

Definition 3. The matrix pair {A(k), C(k)} with A(k) ∈ R n×n and C(k) ∈ R m×n is uniformly completely observable if there exist positive constants ρ 1 , ρ 2 and a positive integer h such that, for all k ≥ h, the following inequalities hold

ρ 1 I n ≤ k s=k-h Φ T (s, k)C T (s)R -1 (s)C(s)Φ(s, k) (6) ≤ ρ 2 I n (7)
with some bounded symmetric positive definite matrix R(s) ∈ R m×m (typically the covariance matrix of the output noise in a stochastic state space system).

Problem formulation and assumptions

In this section the considered OE system and its Kalman filter are first formulated, before the statement of the assumptions for stability analysis.

Output error system and Kalman filter

The discrete time output error (OE) systems considered in this paper are in the form of

x(k) = A(k)x(k-1) + B(k)u(k) (8a) y(k) = C(k)x(k) + R 1 2 (k)v(k) (8b)
where k = 0, 1, 2, . . . represents the discrete time index, 1 Some variants of the definition of the uniform complete observability exist in the literature. The definition recalled here follows [START_REF] Kalman | New methods in Wiener filtering theory[END_REF].

x(k) ∈ R n is the state vector, u(k) ∈ R l the input, y(k) ∈ R m the output, v(k) ∈ R m a
square root of R(k). The initial state x(0) ∈ R n is a random vector following the Gaussian distribution x(0) ∼ N (x 0 , P 0 ) with x 0 ∈ R n and P 0 ∈ R n×n .

Compared to the general discrete time state equation

x(k) = A(k)x(k-1) + B(k)u(k) + Q 1 2 (k)w(k) (9)
with the process noise w(k) and the covariance matrix Q(k), an OE system corresponds to the particular case of Q(k) ≡ 0.

After the initialization with P (0|0) = P 0 and x(0|0) = x 0 , the Kalman filter for the OE system (8) consists of the following recursions for k = 1, 2, . . . ,

P (k|k-1) = A(k)P (k-1|k-1)A T (k) (10a) Σ(k) = C(k)P (k|k-1)C T (k) + R(k) (10b) K(k) = P (k|k-1)C T (k)Σ -1 (k) (10c) P (k|k) = [I n -K(k)C(k)]P (k|k-1) (10d) x(k|k-1) = A(k)x(k-1|k-1) + B(k)u(k) (10e) ỹ(k) = y(k) -C(k)x(k|k-1) (10f) x(k|k) = x(k|k-1) + K(k)ỹ(k). (10g)
For general LTV systems with a process noise as in the state equation ( 9), the first equation of the Kalman filter would be

P (k|k-1) = A(k)P (k-1|k-1)A T (k) + Q(k) (11)
with an extra term Q(k) representing the process noise covariance matrix. Therefore, the OE system Kalman filter corresponds to the particular case with Q(k) ≡ 0 of the general LTV system Kalman filter.

It is known that the dynamics of the general LTV system Kalman filter is stable, provided the matrix pair {A(k), Q 1 2 (k)} is uniformly completely controllable and the matrix pair {A(k), C(k)} is uniformly completely observable [START_REF] Kalman | New methods in Wiener filtering theory[END_REF][START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF]. As OE systems correspond to the case with Q(k) ≡ 0, the controllability condition cannot be satisfied. Consequently, the classical results on the stability of the Kalman filter cannot be applied here. The main purpose of the present paper is to study the Kalman filter stability in this particular case.

It will be shown that the error dynamics of the Kalman filter ( 10) is asymptotically stable, and that its iteratively computed variables are all bounded.

Note that the classical optimality results of the Kalman filter remain valid in the case of OE systems (Jazwinski, 1970, chapter 7).

Assumptions

The assumptions stated here are required in all the following sections of this paper.

The considered OE system ( 8) is defined with bounded real matrices A(k), B(k), C(k), R(k) of appropriate sizes. For all k ≥ 0, the square matrix A(k) is assumed invertible, so that the system dynamics is reversible, i.e., the equality x(k) = Φ(k, l)x(l) holds (with an appropriate state transition matrix Φ(k, l)) for all integers k, l ≥ 0, including the case k < l. The noise covariance matrix R(k) is symmetric positive definite and has a strictly positive lower bound.

The initial state x(0) ∈ R n is a random vector following the Gaussian distribution

x(0) ∼ N (x 0 , P 0 ) (12)
with some mean vector x 0 ∈ R n and a symmetric positive definite covariance matrix P 0 ∈ R n×n .

It is further assumed that the matrix pair {A(k), C(k)} is uniformly completely observable (see Definition 3).

Boundedness of the covariance and gain matrices

The purpose of this section is to show that the OE system Kalman filter covariance matrices P (k|k), P (k|k -1) and the Kalman gain K(k) are all bounded under the assumptions stated in Section 3.2.

It will be shown that the covariance of the state estimate P (k|k) is closely related to the matrix sequence Ω(k) recursively defined by

Ω(0) = P -1 (0|0) (13a) Ω(k) = Φ T (k-1, k)Ω(k-1)Φ(k-1, k) + Λ(k), (13b)
where

Λ(k) C T (k)R -1 (k)C (k), (14)
so that the properties of P (k|k) can be analyzed through those of Ω(k).

Proposition 1. Under the assumptions stated in Section 3.2, the matrix sequence Ω(k) ∈ R n×n defined in (13) is symmetric positive definite for all k ≥ 0 and has a strictly positive lower bound.

Proof. Repeatedly applying the recursion (13b) yields

Ω(k) = Φ T (0, k)Ω(0)Φ(0, k) + k s=1 Φ T (s, k)Λ(s)Φ(s, k). ( 15
)
It is assumed that A(k) is invertible for all k ≥ 0, hence Φ(0, k) is also invertible. The initial condition Ω(0) = P -1 (0|0) > 0 then implies that Ω(k) > 0 for all k ≥ 0. When k < h (h is the integer introduced in ( 6)),

Ω(k) ≥ min 0≤k<h Φ T (0, k)Ω(0)Φ(0, k) ≥ ρ 0 I n ( 16
)
where ρ 0 > 0 is the smallest eigenvalue of Φ T (0, k)Ω(0)Φ(0, k) for 0 ≤ k < h. When k ≥ h, because of the assumed uniforme complete observability (see inequality (6) in Definition 3) and the definition of Λ(k) in equation ( 14)),

Ω(k) ≥ k s=k-h Φ T (s, k)Λ(s)Φ(s, k) ≥ ρ 1 I n > 0 (17)
where ρ 1 is the constant in inequality (6). Therefore, min(ρ 0 , ρ 1 )I n > 0 is a lower bound of Ω(k) for all k ≥ 0.

Proposition 2. Under the assumptions stated in Section 3.2, for all k ≥ 0, the Kalman filter covariance matrix P (k|k), iteratively computed through (10a)-(10d) with the initial condition P (0|0) = P 0 > 0, is symmetric positive definite and is upper bounded.

Proof. Recall that Φ(k, k-1) = A(k) and Φ(k-1, k) = A -1 (k). Rewrite (13b) as Ω(k) = A -T (k)Ω(k-1)A -1 (k) + Λ(k). ( 18
)
Take the matrix inverses of both sides of (18) while applying the matrix equality

(G + V T HV ) -1 = G -1 -G -1 V T (H -1 + V G -1 V T ) -1 V G -1 with G = A -T (k)Ω(k-1)A -1 (k), H = R -1 (k) and V = C (k), then Ω -1 (k) = A(k)Ω -1 (k-1)A T (k) -A(k)Ω -1 (k-1)A T (k)C T (k) R(k) + C (k)A(k)Ω -1 (k-1)A T (k)C T (k) -1 • C (k)A(k)Ω -1 (k-1)A T (k) (19)
Define the new notations

P (k|k) Ω -1 (k) (20) P (k|k-1) A(k) P (k-1|k-1)A T (k) (21) Σ(k) C (k) P (k|k-1)C T (k) + R(k) (22) K (k) P (k|k-1)C T (k) Σ-1 (k), (23) 7
then the recursion of P (k|k) = Ω -1 (k) in ( 19) is rewritten as P (k|k) = P (k|k-1) -P (k|k-1)C T (k) R(k)

+ C (k) P (k|k-1)C T (k) -1 • C (k) P (k|k-1) = P (k|k-1) -P (k|k-1)C T (k) Σ-1 (k) • C (k) P (k|k-1), hence P (k|k) = I n -K (k)C (k) P (k|k-1). ( 24
)
It turns out that, the recursions ( 21)-( 24) coincide exactly with those of (10a)-(10d). Moreover, P (0|0) = Ω -1 (0) = P (0|0), therefore, P (k|k) = Ω -1 (k) = P (k|k) for all k ≥ 0.

The results of Proposition 1 then imply that P (k|k) = Ω -1 (k) is positive definite and upper bounded.

Corollary 1. Under the assumptions stated in Section 3.2, for all k ≥ 0, the prediction error covariance matrix P (k|k -1) computed with (10a) is symmetric positive definite and upper bounded, and the gain matrix K(k) computed with (10c) is bounded.

The proof of this corollary is trivial: based on Proposition 2 and on the assumption that A(k) is bounded and invertible, the positive definiteness and the boundedness of P (k|k -1) follow from (10a); while the boundedness of K(k) is due to the boundedness of P (k|k -1) and of C(k) in (10c), and also to the assumed strictly positive lower bound of R(k) in (10b) (see Section 3.2 for the assumptions).

Asymptotic stability of the Kalman filter

It has been shown in Proposition 2 and Corollary 1 that the covariance matrices P (k|k), P (k|k-1) and the Kalman gain K(k) are all bounded. These are obviously important "stability" properties of a recursive algorithm. The purpose of this section is to study the asymptotic stability of the Kalman filter error dynamics, in the sense of Definition 2.

Kalman filter error dynamics

The computations in the steps (10e)-(10g) can be combined into a single step for state estimation:

x(k|k) = A(k)x(k-1|k-1) + B(k)u(k) + K(k) y(k) -C(k) A(k)x(k-1|k-1) + B(k)u(k) . ( 25
)
The homogeneous part of this iterative equation (from x(k-1|k-1) to x(k|k)) is

z(k|k) = [I n -K(k)C(k)]A(k)z(k-1|k-1), (26)
which corresponds to the dynamics of the mathematical expectation of the state estimation error, i.e.,

z(k|k) = E[x(k) -x(k|k)] ( 27
)
with the initial condition z(0|0) = Ex(0) -x(0|0). For this reason, equation ( 26) will be referred to as the Kalman filter error dynamics equation. Like in [START_REF] Kalman | New methods in Wiener filtering theory[END_REF][START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF], the Kalman filter stability analysis consists in analyzing the stability of the error dynamics (26), which reflects the intrinsic stability property of the Kalman filter dynamics as expressed in (25).

For this stability analysis, it seems natural to follow the approach adopted in the continuous time case as studied in [START_REF] Ni | Stability of the Kalman filter for output error systems[END_REF]. In this approach, an important step would be to establish the uniform complete observability of the error dynamics equation ( 26) adjoined with the "output equation"2

ȳ(k) = C(k)z(k|k). (28)
In [START_REF] Ni | Stability of the Kalman filter for output error systems[END_REF], the continuous time counterpart of ( 26) and ( 28) is

dz(t)/dt = [A(t) -K(t)C(t)]z(t) (29) ȳ(t) = C(t)z(t). (30)
In ( 29), the term K(t)C(t)z(t) = K(t)ȳ(t) is known from the "output" ȳ(t) and can be viewed as an "input" term, hence intuitively the pair {[A(t) -K(t)C(t)], C(t)} is observable if the pair {A(t), C(t)} is observable. Unfortunately, in the discrete time case, such a reasoning cannot be applied to ( 26) and ( 28), since the corresponding term in (26), namely K(k)C(k)A(k)z(k-1|k-1), cannot be written as K(k)ȳ(k) with ȳ(k) as in ( 28), because of the matrix A(k) between C(k) and z(k-1|k-1), and also because of the different indexes k and k-1.

In order to bypass this difficulty, let us first analyze the stability of the Kalman predictor dynamics. The stability of the filter dynamics will then be established through a relationship between the filter dynamics and the predictor dynamics.

Stability of the Kalman predictor

By combining the computations in the steps (10e)-(10g), the Kalman predictor dynamics is summarized as

x(k|k-1) = A(k)x(k-1|k-2) + A(k)K(k-1)[y(k-1) -C(k-1)x(k-1|k-2)] + B(k)u(k).
The homogeneous part of this iterative equation (from x(k-1|k-2) to x(k|k-1)) is

z(k|k-1) = A(k)[I n -K(k-1)C(k-1)]z(k-1|k-2), (31)
which corresponds to the dynamics of the mathematical expectation of the state prediction error, i.e.,

z(k|k-1) = E[x(k) -x(k|k-1)] ( 32
)
with the initial condition z(1|0) = Ex(1) -x(1|0). For this reason, equation ( 31) will be referred to as the Kalman predictor error dynamics equation.

The stability of predictor error dynamics (31) will be analyzed in this subsection.

In what follows, the shorter notation

z(k) z(k|k-1) (33)
will be adopted, thus the predictor error dynamics equation ( 31) is rewritten as

z(k) = A(k)[I n -K(k-1)C(k-1)]z(k-1). ( 34
)
The observability of this "state equation", when adjoined with the "output equation"3

ȳ(k) = C(k)z(k), (35)
will play a key role in the stability analysis of (34). The uniform complete observability of the pair {A(k), C(k)} is assumed in this paper (see Section 3.2). Notice that the two equations ( 34) and ( 35) can be rewritten in the equivalent form

z(k) = A(k)z(k-1) -A(k)K(k-1)ȳ(k-1) (36a) ȳ(k) = C(k)z(k). ( 36b
)
By viewing A(k)K(k-1)ȳ(k-1) as a (known) input term, this system is clearly observable, since the pair {A(k), C(k)} is assumed observable. Equivalently, the system defined in ( 34) and ( 35) is also observable. Nevertheless, this intuitive reasoning is not sufficient for the following stability analysis, which requires the uniform complete observability of the pair {A(k)[I n -K(k-1)C(k-1)], C(k)} (see Definition 3), that is established in the following lemma.

Lemma 1. Let Φ(k, l) denote the state transition matrix of the homogeneous iterative equation (34), such that z(k) = Φ(k, l)z(l). Under the assumptions stated in Section 3.2, in particular the uniform complete observability of the pair {A(k), C(k)} as defined in Definition 3, there exists a constant ρ 3 > 0 such that

ρ 3 I n ≤ k s=k-h ΦT (s, k)Λ(s) Φ(s, k) (37)
for all k ≥ h, with the same positive integer h as in (6), and

Λ(s) = C T (s)R -1 (s)C(s) as defined in (14).
A proof of this lemma is given in Appendix 1. The continuous time counterpart of this result is known as part of the lemma on observability conservation by output feedback [START_REF] Anderson | Stability of adaptive systems: passivity and averaging analysis[END_REF][START_REF] Sastry | Adaptive Control: Stability, Convergence and Robustness[END_REF][START_REF] Ioannou | Robust Adaptive Control[END_REF][START_REF] Zhang | Observability conservation by output feedback and observability Gramian bounds[END_REF]. However, for discrete time systems, this result seems not reported in the literature.

Theorem 1. Under the assumptions stated in Section 3.2, the Kalman predictor error dynamics equation (31), which is also expressed in the shorter form (34), is asymptotically stable.

Proof. Define

V (z(k), k) z T (k)P -1 (k|k-1)z(k), (38)
which cannot be a Lyapunov function in the classical sense, because P (k|k-1) does not have a strictly positive lower bound. The stability analysis based on V (z(k), k) will not exactly follow the classical Lyapunov stability theory. Substitute z(k) with ( 34), then

V (z(k), k) = z T (k-1)Ξ(k)z(k-1) (39) with Ξ(k) [I n -K (k-1)C (k-1)] T A T (k)P -1 (k|k-1) • A(k)[I n -K (k-1)C (k-1)]. ( 40
)
Rewrite (10a) as

A T (k)P -1 (k|k-1)A(k) = P -1 (k-1|k-1), ( 41
) then Ξ(k) =[I n -K (k-1)C (k-1)] T P -1 (k-1|k-1) • [I n -K (k-1)C (k-1)]. ( 42
)
It follows from ( 10d) that

[I n -K (k-1)C (k-1)] = P (k-1|k-1)P -1 (k-1|k-2), ( 43
) then Ξ(k) =[I n -K (k-1)C (k-1)] T P -1 (k-1|k-1) • P (k-1|k-1)P -1 (k-1|k-2) (44) =[I n -K (k-1)C (k-1)] T P -1 (k-1|k-2) (45) =P -1 (k-1|k-2) -C T (k-1)K T (k-1)P -1 (k-1|k-2). (46)
Rearrange (10c) as

K T (k-1)P -1 (k-1|k-2) = Σ -1 (k-1)C (k-1), (47) then
Ξ(k) =P -1 (k-1|k-2) -C T (k-1)Σ -1 (k-1)C (k-1) (48)
It is then derived from ( 39) and (38) that

V (z(k), k) -V (z(k-1), k-1) = z T (k-1)Ξ(k)z(k-1) -z T (k-1)P -1 (k-1|k-2)z(k-1). (49)
Applying (48) yields

V (z(k), k) -V (z(k-1), k-1) = -z T (k-1)C T (k-1)Σ -1 (k-1)C (k-1)z(k-1). ( 50
)
It has been shown that P (k|k-1) is an upper bounded positive definite matrix (see Corollary 1), therefore C (k)P (k|k-1)C T (k) is positive semidefinite and upper bounded. It is assumed that the noise covariance matrix R(k) has a strictly positive lower bound (See Section 3.2). Let σ 1 > 0 be an upper bound of the largest eigenvalue of C (k)P (k|k -1)C T (k) for all k ≥ 0, and σ 2 > 0 be a lower bound of the smallest eigenvalue of R(k) for all k ≥ 0. It then follows from (10b) that

Σ(k) = C (k)P (k|k-1)C T (k) + R(k) (51) ≤ σ 1 σ 2 R(k) + R(k) (52) = σ 3 R(k) (53)
with the positive constant σ 3 σ 1 /σ 2 + 1.

It is then derived from (50) that

V (z(k), k) -V (z(k-1), k-1) ≤ - 1 σ 3 z T (k-1)C T (k-1)R -1 (k-1)C (k-1)z(k-1) (54) = - 1 σ 3 z T (k-1)Λ(k-1)z(k-1) (55) ≤ 0, (56)
where Λ(k-1) is as defined in ( 14). This result implies that V (z(k), k) cannot increase with k, hence V (z(k), k) = z T (k)P -1 (k|k-1)z(k) is upper bounded. Moreover, P (k|k-1) is upper bounded (see Corollary 1), therefore z(k) is also upper bounded. The iterative equation (34) is thus Lyapunov stable.

It still remains to show that z(k) → 0 for asymptotic stability.

Repeatedly applying inequality (55) yields

V (z(k+1), k+1) -V (z(k-h), k-h) ≤ - 1 σ 3 k s=k-h z T (s)Λ(s)z(s) (57) = - 1 σ 3 z T (k) k s=k-h ΦT (s, k)Λ(s) Φ(s, k)z(k). (58)
where Φ(s, k) is the state transition matrix of (34) such that z(s) = Φ(s, k)z(k), as introduced in Lemma 1.

Applying Lemma 1 then yields

V (z(k+1), k+1) -V (z(k-h), k-h) ≤ - ρ 3 σ 3 z(k) 2 . ( 59
)
In order to prove, by contradiction, the asymptotic stability of the iterative equation ( 34), assume that z(k) does not tend to zero when k → ∞. This assumption implies that there exists a constant ε > 0 such that for any (arbitrarily large) integer κ, there exists k > κ such that z(k) > ε, hence

V (z(k+1), k+1) -V (z(k-h), k-h) ≤ - ρ 3 σ 3 ε 2 . ( 60
)
Because κ can be arbitrarily large, there exist infinitely many such values of k > κ, Moreover, it was already shown that the value of V (z(k), k) cannot increase with the time index k. Inequality (60) then says that V (z(k), k) is repeatedly decreased by ρ 3 ε 2 /σ 3 > 0 for larger and larger values of k, each being distant of at least h from the previous k. Consequently, V (z(k), k) will become negative for sufficiently large values of k. This is in contradiction with the definition of V (z(k), k) in (38) which is positive definite. Therefore, it is proved that z(k) tends to zero when k → +∞.

The asymptotic stability of the Kalman predictor error dynamics (34), or equivalently (31), is then proved.

Back to the Kalman filter error dynamics

Now let us consider the stability of the Kalman filter error dynamics as expressed in (26). Given the sequence z(k|k) satisfying the recursive equation ( 26), let ζ(k) be another sequence such that

z(k|k) = [I n -K(k)C(k)]ζ(k). ( 61
)
According to Proposition 2 and Corollary 1, both P (k|k) and P (k|k -1) are positive definite, it then follows from (10d) that the matrix [I n -K(k)C(k)] is invertible. Insert (61) into (26), then

[I n -K(k)C(k)]ζ(k) = [I n -K(k)C(k)]A(k)[I n -K(k-1)C(k-1)]ζ(k-1). ( 62
)
Remove the invertible factor [I n -K(k)C(k)] from both sides, then

ζ(k) = A(k)[I n -K(k-1)C(k-1)]ζ(k-1). ( 63
)
This recursive equation is exactly the same as the predictor error dynamics equation (34), whose asymptotic stability has been shown in Theorem 1. Therefore, the recursive equation ( 63 Corollary 2. Under the assumptions stated in Section 3.2, the Kalman filter error dynamics equation ( 26) is asymptotically stable.

Numerical examples

Two numerical examples are presented in this section to illustrate the asymptotic stability of the Kalman filter error dynamics and the boundedness of the covariance matrix.

Example 1

Consider a piecewise constant system with

A(k) = 0.3 0.4 0 1.2 , C(k) = -0.5 1 , R(k) = 0.01, for k < 10 (64)
and

A(k) = -0.15 1 -0.2 -0.15 , C(k) = -0.6 1 , R(k) = 0.01, for k ≥ 10. ( 65
)
At k = 10, the system switches from an unstable to a stable model. The error dynamics of z(k|k) as defined in (26) and the covariance matrix P (k|k) as computed through (10d) are fully characterized by A(k), C(k), R(k) and the initial conditions z(0|0) = [1, 1] T , P (0|0) = I 2 . The trajectories of z(k|k) and of the two eigenvalues of P (k|k) at every instant k are respectively illustrated in Figure 1 and Figure 2. These results confirm that the error z(k|k) converge to zero and the covariance P (k|k) is bounded.

Example 2 Instead of the piecewise constant system in the previous example, a system varying at every time instant is considered in this example, with

A(k) = 1.2 1 + 0.5 cos(k) 0 0.5 + 0.2 * sin(k) , C(k) = 1 0 , R(k) = 0.01. ( 66
)
The error dynamics of z(k|k) as defined in (26) and the covariance matrix P (k|k) as computed through (10d) are fully characterized by A(k), C(k), R(k) and the initial conditions z(0|0) = [1, 1] T , P (0|0) = I 2 . The trajectories of z(k|k) and of the two eigenvalues of P (k|k) at every instant k are respectively illustrated in Figure 3 and Figure 4. Again these results confirm that the error z(k|k) converge to zero and the covariance P (k|k) is bounded.

Conclusion

For any recursive algorithm running in real time, the boundedness of all the involved variables is obviously an important property. It is established in this paper that, when the Kalman filter is applied to discrete time OE systems, all the iteratively computed variables are bounded, essentially under an observability condition. It is further shown that the Kalman filter error dynamics is asymptotically stable. Further analysis on convergence rates will be reported elsewhere, due to space limitation of the present paper.

Since the zero covariance matrix Q(k) = 0 of the process noise may be seen as a singularity, in practice it is possible to "regularize" by replacing the zero matrix with some chosen (small) matrix. The classical stability results are then applicable to the "regularized" Kalman filter. However, because a wrong covariance matrix Q(k) is used, the resulting state estimation is no longer optimal in the sense of minimum variance, and the innovation sequence of the Kalman filter is no longer a white noise. The results of this paper show that there is no need to "regularize" the zero covariance matrix, and the optimal Kalman filter for OE systems is indeed stable under the uniform complete observability condition. 16 (A.7)

In this equality, the usual notation convention of the sum for p = k+1, k+1, . . . , s assumes that k < s. The same equality holds also when k ≥ s, but the sum is for p = s, s+1, . . . , k+1, and the equality is accordingly written as In what follows, the indexes s and p are restricted such that |p -s| ≤ h, with the integer h as in (6), then Φ(s, p) with such restricted indexes is bounded. Left multiply both sides of (A.9) by R -1 2 (s)C (s) and simplify the writing of the result as

  white Gaussian noise with identity covariance matrix, A(k), B(k), C(k), R(k) are real matrices of appropriate sizes. The noise covariance matrix R(k) is symmetric positive definite. The notation R 1 2 (k) denotes the symmetric positive definite matrix

  ) characterizing ζ(k) is asymptotically stable, i.e., ζ(k) is bounded and tends to zero when k → ∞. The asymptotic stability of the recursive equation (26) satisfied by z(k|k) is then established through z(k|k) = [I n -K(k)C(k)]ζ(k), with bounded K(k) (Corollary 1) and bounded C(k). This result is summarized as follows.

Figure 1 :Figure 2 :

 12 Figure 1: Trajectory of the two components of the filter error mean z(k|k) = E[x(k) -x(k|k)].

Figure 3 :Figure 4 :

 34 Figure 3: Trajectory of the two components of the filter error mean z(k|k) = E[x(k) -x(k|k)].

  and treat -A(s)K (s-1)C (s-1)z(s-1) as an input term inserted into (A.1), then z(s) =Φ(s, k)z(k) p)A(p)K (p-1)C (p-1)z(p-1).

  z(s) =Φ(s, k)z(k) -k+1 p=s Φ(s, p)A(p)K (p-1)C (p-1)z(p-1). (A.8)This form will be used in what follows, by assuming s ≤ k.Because of (A.3) and (A.4), Φ(s, k)z(k) = x(s), it then follows from (A.8

  Ψ(p) ∈ R m×m , which are bounded due to the boundedness of the involved matrices R -1 2 (s), C (s), Φ(s, p), A(p) and K (p-1inequality (foru(p) ∈ R, v(p) ∈ R) p) = 1 and v(p) = Ψ(p)C (p)z(p) , then R -1 2 (s)C (s)x(s) 2 ≤ (k -s + 1) ) 2 • C (p)z(p) 2 ≤ (k -s + 1)c 1 k p=s C (p)z(p) 2where c 1 is the upper bound of Ψ(p) c 2 = h + 1 is to account for the fact that in the double sum each term C (p)z(p) 2 appears not more than h + 1 times. By inserting upper bounded R(s) in the right hand side, there exists a constant c 3 > 0 such that substituting with x(s) = Φ(s, k)x(k) and z(p) = Φ(p, k)z(k) x T (k) k s=k-h Φ T (s, k)C T (s)R -1 (s)C (s)Φ(s, k)x(k) ≤ c 3 z T (k)k p=k-h ΦT (p, k)C T (p)R -1 (p)C (p) Φ(p, k)z(k)

The fictive "output" ȳ(k) is introduced for stability analysis only. There is no need to know or to measure it in a real Kalman filter implementation.

Again the fictive "output" ȳ(k) is introduced for stability analysis only. There is no need to know or to measure it in a real Kalman filter implementation.

Appendix A. Proof of Lemma 1

Consider two discrete time state equations

with their states synchronized at instant k, i.e.,

) and Φ(s, k) denote respectively the state transition matrices of these two systems, such that

This last inequality holds for any x(k) = z(k) ∈ R n , hence by recalling inequality (6),

(A.17)

Lemma 1 is then established with ρ 3 = ρ 1 /c 3 .