Dr Florent Arregle
email: florent.arregle@ap-hm.fr

Nicolas Iline

MD Roch Giorgi

PhD Mary Philip

Sandrine Hubert

Frederique Gouriet

MD Jean-Paul Casalta

MD Frédéric Collart

Alberto Riberi

MD Hélène Martel

MD Sébastien Renard

MD Laurence Camoin

PhD Anne Claire Casalta

MD Hubert Lepidi

PhD Michel Drancourt

PhD Gilbert Habib

Influence of the health-care pathway on the outcome of patients with infective endocarditis

Keywords: endocarditis, mortality, health-care pathway, referral center Methods endocarditis, mortality, health-care pathway, referral center

Aims: To determine the prognosis of patients treated for infective endocarditis (IE) according to their health-care pathway. To assess how the ESC guidelines are implemented concerning the performance of transesophageal echocardiography (TEE), the use of antibiotic therapy, and the performance of valve surgery; and to compare the epidemiological profile of IE according to the type of centers in which the patients are hospitalized.

Methods and results:

In a prospective multicentric study including 22 hospitals in the South-East of France, 342 patients were classified in three groups according to their health-care pathway: 119 patients diagnosed and taken care entirely in a reference center or hospital with cardiac surgery (Referral Center (RC) group), 111 patients diagnosed and initially taken care in a non-referral center (NRC) , then referred in a center including cardiac surgery (Transferred to Referral Center (TRC) group) and 112 patients totally taken care in NRC (NRC group). Oneyear mortality was 26% (88 deaths) and was not significantly different between groups 1 and 2 (20% vs 21%, p = 0.83). Patients in NRC group had a higher mortality (37%) compared to patients in RC and TRC groups (p <0.001). ESC guidelines were not implemented similarly depending on the health-care pathway (p=0.04). Patients in NRC group were significantly older (p<0.001) and had more comorbidities (p<0.001) than patients treated in referral centers.

Conclusion:

Prognosis of patients with IE is influenced by their health-care pathway. Patients treated exclusively in NRC have a worse prognosis than patients treated in referral or surgical centers.

Epidemiologic profile of infective endocarditis (IE) has changed over the past twenty years, 1 2 3 but mortality remains high 4 5 6 despite diagnostic improvement and access to surgery in up to 50% of cases. [START_REF] Thuny | The timing of surgery influences mortality and morbidity in adults with severe complicated infective endocarditis: a propensity analysis[END_REF] This persistent severity has justified the development of new diagnostic and therapeutic strategies to improve prognosis. 8 9 Management by a specialized multidisciplinary team within a referent medical-surgical center seems to be a key factor in improving patients prognosis, 10 11 and is recommended by the ESC guidelines. 8 9 However, few studies have compared the prognosis and management of patients with IE according to their health-care pathway. Indeed, three situations are usually observed: patients for whom the diagnosis and management are carried out entirely in a referral center and/or having a cardiac surgery department, patients secondarily referred to a referral hospital or surgical center by another care center, patients for whom the entire care is provided within a non-referral care center. In addition, application of European guidelines remains difficult to assess in daily clinical practice outside the major referral centers. Published epidemiological data only represent part of the IE, most often not taking into account patients hospitalized in non-referral centers and therefore introducing a significant bias in the knowledge of this pathology. 12 Finally, although the ESC guidelines suggest that patients with uncomplicated IE can be managed in a non-reference center 9 , if all patients with IE 13 or only the sickest patients 14 should be referred to reference centers is still a matter of debate.

The main objective of this study was to determine the prognosis of patients treated for IE according to their health-care pathway. The secondary objectives were to assess how the ESC guidelines are implemented concerning the performance of transesophageal echocardiography (TEE), the use of antibiotic therapy, and the performance of valve surgery, and to compare the epidemiological profile of IE according to the type of centers in which the patients are hospitalized.

-Transferred to Referral Center group (TRC), including patients diagnosed and initially taken care in a non-referral center, then referred in a center including cardiac surgery -Non-Referral Center group (NRC), including patients totally taken care in non-referral centers (18 centers)

Follow-up

Patients had a one-year follow-up with a clinical evaluation or if not possible, phone contact.

Primary and secondary endpoints

Primary endpoint was death from all cause at one-year follow-up. Secondary endpoints were:

-the evaluation of the implementation of European guidelines 8 9 in the three patients groups regarding: the use of TEE, the choice and duration of antibiotic therapy, and the performance of valvular surgery and/or extraction of pacemaker or implantable automatic defibrillator when indicated -comparison of demographical and microbiological data among the three groups.

Statistical analysis

Continuous variables were described by their mean, standard deviation, minimum and maximum value, 1st and 3rd quartile. Continuous variables were compared using the Mann-Whitney non parametric U test or Student's t test, depending on the application conditions.

Categorical variables were compared using the chi-square test or the Fisher test, depending on the application conditions. The event-free survival distributions were estimated with the Kaplan-Meier method. The log-rank test was used to compare survival distributions for categorical variables. The effect of continuous variables on the risk of an event occurring was estimated and tested using a Cox model. The surgery variable, which is a time dependent variable, was estimated using a Poisson approach.

Multivariate analyzes were tested using a Cox model. Multivariate analyzes were tested using a Cox model. Inclusion of variables in the fitted model was carried out according to the following criteria : result in univariate analysis with statistical significance p<0.20, number of missing values <10 % for this variable, effective presence in both categories of a bivariate variable > 10 %, as well as according to their clinical relevance. The final models were selected using the Akaike Information Criteria (AIC).

The comparison tests of group 1 to group 2, group 1 to group 3, group 1 + 2 to group 3 were performed in bilateral situation using Bonferroni correction and were considered statistically significant for p <0.017. For all other tests, they were performed bilaterally and were considered statistically significant for p <0.05. Statistical analysis was performed with R software (version 3.6.2) and RStudio (v 1.2.5).

Results

Patient population

Between January 2014 and June 2017, a total of 342 patients with definite IE were included. Among them, 119 were diagnosed and entirely managed in a reference center or hospital with cardiac surgery (RC group), 111 were diagnosed and initially managed in a nonreferral center, then referred in a center including cardiac surgery (TRC group) and 112 were totally managed in non-referral centers (NRC group). Clinical, demographical, biological, and microbiological characteristics of patients are presented in was not significantly different between RC and TRC groups (p=0.83), however it was significantly higher in NRC group compared to RC and TRC groups (p<0.001).

Characteristics of patients only treated in non-referral centers

During the study period, 112 patients were managed only in a NRC. Among them, 28 patients (25%) had indication for valvular surgery or intracardiac device removal. These patients had a mean age of 76+/-11 years and were at high surgical risk (mean Euroscore 17+/-16%). Reasons for absence of transfer and surgery included: death before surgery for 16 patients risk (comorbidities) for 16 patients (14%). 6 patients (5%) had intracardiac device removal (pacemaker lead or catheter) in the NRC. Among these 28 patients, one year mortality was 50% (14 deaths). 84 patients (75%) had indication for medical treatment of IE. One year mortality among them was 29% (24 deaths).

Prognostic factors of mortality

Results of univariate analysis of prognostic factors of one year mortality are shown in Table S1.

Evaluation of European guidelines application according to health-care pathway

Table 3 shows the implementation of ESC guidelines in the three groups regarding use of TEE, valvular surgery or intracardiac device removal and antibiotic therapy. In overall population, ESC guidelines were correctly implemented in 52% of patients. TEE was performed in most IE cases (92%); however it was significantly less frequently used in NRC group as compared to the two other groups (p<0.001). Guidelines concerning surgery or intracardiac device extraction were correctly implemented in 78% of cases and there was no significant difference between the three groups. Antibiotic therapy was in accordance with guidelines in 72% of cases and there was no significant difference between patients taken care entirely or partially in RC (p=0.66). However, treatment was less frequently in agreement with guidelines in patients in NRC group compared to the others (p<0.001).

Discussion

This prospective study carried out in 22 hospitals in the South-East of France including 342 IE cases divided into three groups according to their health-care pathway shows that:

-The prognosis of patients is influenced by their health-care pathway -The number of patients managed in non-referral centers is high. They are older, have more comorbidities, and less severe cardiac lesions than the others.

-Surgery or device removal is less frequently performed in those patients -European recommendations on the management of these patients are not followed similarly depending on their health-care pathway -Both in-hospital and long-term mortalities are higher in non-referral centers.

General population

By including patients treated in NRC, our study has the advantage of providing global epidemiological information concerning the cases of IE within the south-eastern region, reflecting the management of this pathology in "real life" and not only in RC. Almost a third of the patients treated for IE during the inclusion period were treated in a NRC.

Consistent with known data 19 20 , there was a predominance of IE cases in males (69%).

The mean age of diagnosis of IE was 68 years in our study. This age seemed more advanced in comparison to other French (average age of 59 +/-17 years in 1999 2 , 62 +/-16 years in 2008 21 ) or European studies (57 +/-16 years in 2001 3 , 59 +/-18 years in the Euro-Endo registry 20 ).

Concerning microbiological data, S. aureus was the predominant germ (26% of IE), which was in agreement with previous studies 21 22 . There was a majority of left-sided IE (85%). An embolic event was observed in 38% of cases. This rate was comparable to other studies that describe the occurrence of an embolic event in 20 to 50% of cases 21 23 24 In our cohort, hospital mortality was 13% and one-year mortality was 26%. Few studies have compared the prognosis of patients with IE according to their health-care pathway. A study by Fernandez-Hidalgo et al. 25 , compared the hospital mortality of 223 patients treated in a RC with cardiac surgery with 144 patients taken care in a NRC and secondarily referred to the RC. This study did not show any significant difference in terms of hospital mortality between the two groups. Likewise, an international multicenter study 12 carried out in RC in the management of IE did not show difference in hospital mortality between 1,164 patients transferred from NRC and 1,596 patients treated directly within the RC. However, these studies have some limitations: by considering only patients treated in RC they are not representative of epidemiology of patients treated in NRC. The originality of our study is to compare three groups, including a group of patients treated exclusively in NRC, these patients usually being not included in previous studies. Our main findings is a significantly higher hospital (20%) and one year mortality (37%) in patients treated only in NRC. Belonging to NRC group was an independent risk factor in multivariate analysis of one-year mortality in the total cohort of patients. This excess of mortality observed in NRC group could be explained by various factors: epidemiological differences compared to other groups, recommendations on the management of IE that are less implemented and a higher absence of surgery rate in patients with a theoretical surgical indication.

Implementation of European guidelines

In our study, we observed a more frequent use of TEE (92% of patients) than in other multicenter studies (60-75% 3 20 ). However, within NRC group, TEE was significantly less frequently performed compared to the other groups. TEE has a key role in the diagnostic process of IE and in particular in the detection of peri-annular complications and should be performed systematically, in accordance with current recommendations 8 9 26 . Thus, it is possible that these complications, often requiring surgery, have not been diagnosed in some patients, especially in NRC group.

Regarding valve surgery or the extraction of intracardiac material, there was no significant difference in application of recommendations between the three groups. However, we noted within each group patients with a theoretical indication for surgery for who surgery was finally not performed. Within NRC group, 25% of patients presented a theoretical indication and only 5% received interventional management. Conversely, in RC and TRC groups, 65% of patients with theoretical surgical indication benefited from intervention.

Different studies have already shown the poor prognosis of patients with a theoretical indication for surgery and not operated 20 27 , which may in part explain the higher mortality of patients in NRC group. In addition, the main multicenter studies carried out over the last few years highlight a surgery rate close to 50% (52% in Euro Heart Survey 3 , 51% in the EuroEndo register 20 , 48% in the ICE register 4 ). Among the patients treated exclusively in RC, the rate of surgery or extraction of intracardiac material was 44%, close to previous published data. On the other hand, if we considered all the patients for whom the diagnosis of IE was made in a non-referral center (TRC and NRC groups), the overall rate of valve surgery or extraction of intracardiac material was only 30 %. Several factors could explain this discrepancy with the pre-existing studies. Indeed, most studies on IE focused on patients treated in RC with a cardiac surgery department. It is possible that a referral bias may cause an overestimation of the rate of cardiac surgery in this pathology, by not considering many IE of "medical" management and treated exclusively in NRC. On the other hand, it is also possible that among the patients diagnosed in a NRC, surgical indications were sometimes not identified, especially in NRC group in which, unlike TRC group, it was not possible to re-evaluate the surgical indications by a multidisciplinary team in a reference center. In a same way, a Spanish study by Lopez-establishment of a multidisciplinary team including a cardiac surgeon was associated with an increase in the rate of surgical indication identified (14.5% vs 34.5% ). Finally, the European guidelines concerning the antibiotic treatment were significantly less applied among patients in group 3 with 35% of inappropriate antibiotic therapy. Here again, few studies concerning the antibiotic treatment used in non-referral centers are available. Fernandez-Hidalgo et al. 25 showed that among the patients diagnosed in a NRC and referred secondarily to a surgical center, the antibiotic treatment initiated in the non-referral center was inadequate in 54% of cases. In this study, inadequate treatment was associated with an increased risk of in-hospital mortality in multivariate analysis. The goal of our study was to evaluate the implementation of rtant to highlight that in some cases, there may be logical reasons why patients do not undergo an indicated procedure such as TEE, for instance when a patient is not operable, or surgery.

Epidemiological characteristics of IE according to health-care pathway

The epidemiological profile of patients in RC and TRC groups was comparable. These were patients with more severe IE than in NRC group with more hemodynamic complications, a higher rate of vegetations, more peri-annular complications. These elements partly explain the higher surgical indication rate in these two groups. The patients in NRC group were significantly older, presented more comorbidities, and a greater operative risk. Some studies focusing on IE cases in the elderly have shown, including during treatment in a RC, limited access to surgery and significant mortality in these patients when surgery, although indicated, was not performed 29 30 .

Limitations

Some data could not be recorded, such as the reason for the transfer of patients to a referral center, the time between diagnosis and the transfer of patients, and some factors who could have influenced transfer decision (such as cirrhosis or poor nutritional status). Due to the large number of participating non-referral centers, the reasons for transferring patients may have differed from one center to another. Moreover, modified Duke criteria were used for diagnosis of IE and have limited sensitivity. This study reflects the epidemiology and practices used in the south-eastern region of France and cannot be generalized to all the other centers.

Finally, the study design was meant to compare prognosis of patients according to the healthcare pathway but reasons explaining the different prognosis observed could only be hypothesized.

Conclusion

Prognosis of patients with IE is influenced by their health-care pathway. Patients treated exclusively in NRC have a worse prognosis than patients treated in RC. These patients are often fragile and their management difficult. Given the difference in mortality between patients managed in referral vs non-referral centers, the current recommendations of treating patients with non-complicated IE in non-referral centers should be questioned. IE is a deadly disease with a better prognosis when managed in RC.

Supplementary data

Supplementary material is available online. Epidemiologic profile of infective endocarditis (IE) has changed over the past twenty years, 1 2 3 but mortality remains high 4 5 6 despite diagnostic improvement and access to surgery in up to 50% of cases. [START_REF] Thuny | The timing of surgery influences mortality and morbidity in adults with severe complicated infective endocarditis: a propensity analysis[END_REF] This persistent severity has justified the development of new diagnostic and therapeutic strategies to improve prognosis. 8 9 Management by a specialized multidisciplinary team within a referent medical-surgical center seems to be a key factor in improving patients prognosis, 10 11 and is recommended by the ESC guidelines. 8 9 However, few studies have compared the prognosis and management of patients with IE according to their health-care pathway. Indeed, three situations are usually observed: patients for whom the diagnosis and management are carried out entirely in a referral center and/or having a cardiac surgery department, patients secondarily referred to a referral hospital or surgical center by another care center, patients for whom the entire care is provided within a non-referral care center. In addition, application of European guidelines remains difficult to assess in daily clinical practice outside the major referral centers. Published epidemiological data only represent part of the IE, most often not taking into account patients hospitalized in non-referral centers and therefore introducing a significant bias in the knowledge of this pathology. 12 Finally, although the ESC guidelines suggest that patients with uncomplicated IE can be managed in a non-reference center 9 , if all patients with IE 13 or only the sickest patients 14 should be referred to reference centers is still a matter of debate.

Funding

The main objective of this study was to determine the prognosis of patients treated for IE according to their health-care pathway. The secondary objectives were to assess how the ESC guidelines are implemented concerning the performance of transesophageal echocardiography (TEE), the use of antibiotic therapy, and the performance of valve surgery, and to compare the epidemiological profile of IE according to the type of centers in which the patients are hospitalized.

Biological data were collected, including hemoglobin level, platelet count, leukocytes count, C-Reactive Protein serum level, creatinine level, rheumatoid factor. Microbiological data included: positive or negative blood cultures, type of microorganism and antibiogram.

Echocardiographic data collected were: presence of vegetation and its maximal length, presence of periannular lesion (defined as the presence of abscess and/or pseudoaneurysm and/or fistula), presence of valvular regurgitation, quantified according to current recommendations. 16 17 Type and date of the following complications were collected: congestive heart failure, peripheral embolism, neurological complications (including symptomatic and silent), mycotic aneurysm, acute kidney failure (>26µmol/L increase of serum cretinine in 48 hours or >50% increase in 7 days) , glomerulonephritis, uncontrolled infection (fever and/or positive blood cultures after 7 days of appropriate antibiotic therapy and unlinked to an extracardiac cause).

Moreover, type, dose and length of antibiotic therapy was reported. Antibiotic therapy was considered appropriate if drugs used and full duration of treatment were in accordance with ESC guidelines (taking into account patients terminating antibiotics prematurely due to death).

Analysis of antibiotic treatment was conducted retrospectively by the Endocarditis team of the investigator center (CHU La Timone, Marseille).

During hospitalization or follow-up the indication (according to ESC guidelines) and date of surgery was also collected. Finally, surgical risk was evaluated using the Euroscore II. 18 Classification of patients according to health-care pathway Patients were classified in three groups according to their health-care pathway:

-Referral Center group (RC), including patients diagnosed and taken care entirely in a reference center or hospital with cardiac surgery (4 hospitals). These centers were the only surgical centers of this region of France.

-Transferred to Referral Center group (TRC), including patients diagnosed and initially taken care in a non-referral center, then referred in a center including cardiac surgery -Non-Referral Center group (NRC), including patients totally taken care in non-referral centers (18 centers)

Follow-up

Patients had a one-year follow-up with a clinical evaluation or if not possible, phone contact.

Primary and secondary endpoints

Primary endpoint was death from all cause at one-year follow-up. Secondary endpoints were:

-the evaluation of the implementation of European guidelines 8 9 in the three patients groups regarding: the use of TEE, the choice and duration of antibiotic therapy, and the performance of valvular surgery and/or extraction of pacemaker or implantable automatic defibrillator when indicated -comparison of demographical and microbiological data among the three groups.

Statistical analysis

Continuous variables were described by their mean, standard deviation, minimum and maximum value, 1st and 3rd quartile. Continuous variables were compared using the Mann-Whitney non parametric U test or Student's t test, depending on the application conditions.

Categorical variables were compared using the chi-square test or the Fisher test, depending on the application conditions. The event-free survival distributions were estimated with the Kaplan-Meier method. The log-rank test was used to compare survival distributions for categorical variables. The effect of continuous variables on the risk of an event occurring was estimated and tested using a Cox model. The surgery variable, which is a time dependent variable, was estimated using a Poisson approach.

Multivariate analyzes were tested using a Cox model. Multivariate analyzes were tested using a Cox model. Inclusion of variables in the fitted model was carried out according to the following criteria : result in univariate analysis with statistical significance p<0.20, number of missing values <10 % for this variable, effective presence in both categories of a bivariate variable > 10 %, as well as according to their clinical relevance. The final models were selected using the Akaike Information Criteria (AIC).

The comparison tests of group 1 to group 2, group 1 to group 3, group 1 + 2 to group 3 were performed in bilateral situation using Bonferroni correction and were considered statistically significant for p <0.017. For all other tests, they were performed bilaterally and were considered statistically significant for p <0.05. Statistical analysis was performed with R software (version 3.6.2) and RStudio (v 1.2.5).

Results

Patient population

Between January 2014 and June 2017, a total of 342 patients with definite IE were included. Among them, 119 were diagnosed and entirely managed in a reference center or hospital with cardiac surgery (RC group), 111 were diagnosed and initially managed in a nonreferral center, then referred in a center including cardiac surgery (TRC group) and 112 were totally managed in non-referral centers (NRC group). Clinical, demographical, biological, and microbiological characteristics of patients are presented in Table 1.

In the overall population, mean age was 68 +/-15 years. IE were mainly communityacquired (83% of cases), S. aureus was the most frequent bacteria involved (26% of cases).

Negative blood culture IE represented 17% of cases. Left-sided IE counted for 85% of all cases and aortic (43%) and mitral (42%) localization were almost equally represented. Perivalvular complications were observed in 16% of cases. 33% of cases were prosthetic valve IE. More than half IE cases (58%) showed theoretical indication for valvular surgery or intracardiac device removal but only 35% had such interventions.

Characteristics according to health-care pathway (Table 1)

Patients in NRC group were significantly older and had higher Charlson index. Sources of infection did not differ between the three groups and microbiological profile of IE cases was not significantly different. Echocardiographic findings showed more severe lesions in RC group and TRC group: longer vegetations, more peri-valvular complications, and more severe valvular regurgitation. Concerning IE complications, there was no significant difference between groups in embolic events, however more silent embolisms were diagnosed in RC group and TRC group. Spondylodiscitis was more frequent in NRC group.

Patients in RC and TRC groups had more frequent surgical indication, but surgery or intracardiac device removal was performed in only 52 patients (44%) in RC group, 60 patients (55%) in TRC group and 6 patients (5%) in NRC group, p<0.001. In RC group , 81 patients had an indication for intervention and 29 were not operated and in TRC group , 91 patients had an indication for intervention and 31 were not operated. Reasons for absence of intervention were: deaths before surgery (6 patients in RC group and 2 patients in TRC group RC group and 1 patient in TRC group) and decision of the endocarditis team due to comorbidities (21 patients in RC group and 28 patients in TRC risk (comorbidities) for 16 patients (14%). 6 patients (5%) had intracardiac device removal (pacemaker lead or catheter) in the NRC. Among these 28 patients, one year mortality was 50% (14 deaths). 84 patients (75%) had indication for medical treatment of IE. One year mortality among them was 29% (24 deaths).

Prognostic factors of mortality

Results of univariate analysis of prognostic factors of one year mortality are shown in Table S1.

Evaluation of European guidelines application according to health-care pathway

Table 3 shows the implementation of ESC guidelines in the three groups regarding use of TEE, valvular surgery or intracardiac device removal and antibiotic therapy. In overall population, ESC guidelines were correctly implemented in 52% of patients. TEE was performed in most IE cases (92%); however it was significantly less frequently used in NRC group as compared to the two other groups (p<0.001). Guidelines concerning surgery or intracardiac device extraction were correctly implemented in 78% of cases and there was no significant difference between the three groups. Antibiotic therapy was in accordance with guidelines in 72% of cases and there was no significant difference between patients taken care entirely or partially in RC (p=0.66). However, treatment was less frequently in agreement with guidelines in patients in NRC group compared to the others (p<0.001).

Discussion

This prospective study carried out in 22 hospitals in the South-East of France including 342 IE cases divided into three groups according to their health-care pathway shows that:

-The prognosis of patients is influenced by their health-care pathway -The number of patients managed in non-referral centers is high. They are older, have more comorbidities, and less severe cardiac lesions than the others.

-Surgery or device removal is less frequently performed in those patients -European recommendations on the management of these patients are not followed similarly depending on their health-care pathway -Both in-hospital and long-term mortalities are higher in non-referral centers.

General population

By including patients treated in NRC, our study has the advantage of providing global epidemiological information concerning the cases of IE within the south-eastern region, reflecting the management of this pathology in "real life" and not only in RC. Almost a third of the patients treated for IE during the inclusion period were treated in a NRC.

Consistent with known data 19 20 , there was a predominance of IE cases in males (69%).

The mean age of diagnosis of IE was 68 years in our study. This age seemed more advanced in comparison to other French (average age of 59 +/-17 years in 1999 2 , 62 +/-16 years in 2008 21 ) or European studies (57 +/-16 years in 2001 3 , 59 +/-18 years in the Euro-Endo registry 20 ).

Concerning microbiological data, S. aureus was the predominant germ (26% of IE), which was in agreement with previous studies 21 22 . There was a majority of left-sided IE (85%). An embolic event was observed in 38% of cases. This rate was comparable to other studies that describe the occurrence of an embolic event in 20 to 50% of cases 21 23 24 In our cohort, hospital mortality was 13% and one-year mortality was 26%. Few studies have compared the prognosis of patients with IE according to their health-care pathway. A study by Fernandez-Hidalgo et al. 25 , compared the hospital mortality of 223 patients treated in a RC with cardiac surgery with 144 patients taken care in a NRC and secondarily referred to the RC. This study did not show any significant difference in terms of hospital mortality between the two groups. Likewise, an international multicenter study 12 carried out in RC in the management of IE did not show difference in hospital mortality between 1,164 patients transferred from NRC and 1,596 patients treated directly within the RC. However, these studies have some limitations: by considering only patients treated in RC they are not representative of epidemiology of patients treated in NRC. The originality of our study is to compare three groups, including a group of patients treated exclusively in NRC, these patients usually being not included in previous studies. Our main findings is a significantly higher hospital (20%) and one year mortality (37%) in patients treated only in NRC. Belonging to NRC group was an independent risk factor in multivariate analysis of one-year mortality in the total cohort of patients. This excess of mortality observed in NRC group could be explained by various factors: epidemiological differences compared to other groups, recommendations on the management of IE that are less implemented and a higher absence of surgery rate in patients with a theoretical surgical indication.

Implementation of European guidelines

In our study, we observed a more frequent use of TEE (92% of patients) than in other multicenter studies (60-75% 3 20 ). However, within NRC group, TEE was significantly less frequently performed compared to the other groups. TEE has a key role in the diagnostic process of IE and in particular in the detection of peri-annular complications and should be performed systematically, in accordance with current recommendations 8 9 26 . Thus, it is possible that these complications, often requiring surgery, have not been diagnosed in some patients, especially in NRC group.

Regarding valve surgery or the extraction of intracardiac material, there was no significant difference in application of recommendations between the three groups. However, we noted within each group patients with a theoretical indication for surgery for who surgery was finally not performed. Within NRC group, 25% of patients presented a theoretical indication and only 5% received interventional management. Conversely, in RC and TRC groups, 65% of patients with theoretical surgical indication benefited from intervention.

Different studies have already shown the poor prognosis of patients with a theoretical indication for surgery and not operated 20 27 , which may in part explain the higher mortality of patients in NRC group. In addition, the main multicenter studies carried out over the last few years highlight a surgery rate close to 50% (52% in Euro Heart Survey 3 , 51% in the EuroEndo register 20 , 48% in the ICE register 4 ). Among the patients treated exclusively in RC, the rate of surgery or extraction of intracardiac material was 44%, close to previous published data. On the other hand, if we considered all the patients for whom the diagnosis of IE was made in a non-referral center (TRC and NRC groups), the overall rate of valve surgery or extraction of intracardiac material was only 30 %. Several factors could explain this discrepancy with the pre-existing studies. Indeed, most studies on IE focused on patients treated in RC with a cardiac surgery department. It is possible that a referral bias may cause an overestimation of the rate of cardiac surgery in this pathology, by not considering many IE of "medical" management and treated exclusively in NRC. On the other hand, it is also possible that among the patients diagnosed in a NRC, surgical indications were sometimes not identified, especially in NRC group in which, unlike TRC group, it was not possible to re-evaluate the surgical indications by a multidisciplinary team in a reference center. In a same way, a Spanish study by Lopez-establishment of a multidisciplinary team including a cardiac surgeon was associated with an increase in the rate of surgical indication identified (14.5% vs 34.5% ). Finally, the European guidelines concerning the antibiotic treatment were significantly less applied among patients in group 3 with 35% of inappropriate antibiotic therapy. Here again, few studies concerning the antibiotic treatment used in non-referral centers are available. Fernandez-Hidalgo et al. 25 showed that among the patients diagnosed in a NRC and referred secondarily to a surgical center, the antibiotic treatment initiated in the non-referral center was inadequate in 54% of cases. In this study, inadequate treatment was associated with an increased risk of in-hospital mortality in multivariate analysis. The goal of our study was to evaluate the implementation of e may be logical reasons why patients do not undergo an indicated procedure such as TEE, for instance when a patient is not operable, or surgery.

Epidemiological characteristics of IE according to health-care pathway

The epidemiological profile of patients in RC and TRC groups was comparable. These were patients with more severe IE than in NRC group with more hemodynamic complications, a higher rate of vegetations, more peri-annular complications. These elements partly explain the higher surgical indication rate in these two groups. The patients in NRC group were significantly older, presented more comorbidities, and a greater operative risk. Some studies focusing on IE cases in the elderly have shown, including during treatment in a RC, limited access to surgery and significant mortality in these patients when surgery, although indicated, was not performed 29 30 .

Some data could not be recorded, such as the reason for the transfer of patients to a referral center, the time between diagnosis and the transfer of patients, and some factors who could have influenced transfer decision (such as cirrhosis or poor nutritional status). Due to the large number of participating non-referral centers, the reasons for transferring patients may have differed from one center to another. Moreover, modified Duke criteria were used for diagnosis of IE and have limited sensitivity. This study reflects the epidemiology and practices used in the south-eastern region of France and cannot be generalized to all the other centers.

Finally, the study design was meant to compare prognosis of patients according to the healthcare pathway but reasons explaining the different prognosis observed could only be hypothesized.

Conclusion

Prognosis of patients with IE is influenced by their health-care pathway. Patients treated exclusively in NRC have a worse prognosis than patients treated in RC. These patients are often fragile and their management difficult. Given the difference in mortality between patients managed in referral vs non-referral centers, the current recommendations of treating patients with non-complicated IE in non-referral centers should be questioned. IE is a deadly disease with a better prognosis when managed in RC.

Supplementary data

Supplementary material is available online.

Funding

8 .

 8 Investigators. Clinical presentation, etiology, and outcome of infective endocarditis in the 21st century: the International Collaboration on Endocarditis-Prospective Cohort Study. Arch Intern Med 2009;169:463 473. 5. Gálvez-Acebal J, Rodríguez-Baño J, Martínez-Marcos FJ, Reguera JM, Plata A, Ruiz J, Marquez M, Lomas JM, Torre-Lima J de la, Hidalgo-Tenorio C, Alarcón A de, Grupo para el Estudio de las Infecciones Cardiovasculares de la Sociedad Andaluza de Enfermedades Infecciosas (SAEI). Prognostic factors in left-sided endocarditis: results from the Andalusian multicenter cohort. BMC Infect Dis 2010;10:17. 6. Thuny F, Disalvo G, Belliard O, Avierinos J-F, Pergola V, Rosenberg V, Casalta J-P, Gouvernet J, Derumeaux G, Iarussi D, Ambrosi P, Calabro R, Riberi A, Collart F, Metras D, Lepidi H, Raoult D, Harle J-R, Weiller P-J, Cohen A, Habib G. Risk of Embolism and Death in Infective Endocarditis: Prognostic Value of Echocardiography: A Prospective Multicenter Study. Circulation 2005;112:69 75. 7. Thuny F, Beurtheret S, Mancini J, Gariboldi V, Casalta J-P, Riberi A, Giorgi R, Gouriet F, Tafanelli L, Avierinos J-F, Renard S, Collart F, Raoult D, Habib G. The timing of surgery influences mortality and morbidity in adults with severe complicated infective endocarditis: a propensity analysis. Eur Heart J 2011;32:2027 2033. Endorsed by the European Society of Clinical Microbiology and Infectious Diseases (ESCMID) and by the International Society of Chemotherapy (ISC) for Infection and Cancer, Authors/Task Force Members, Habib G, Hoen B, Tornos P, Thuny F, Prendergast B, Vilacosta I, Moreillon P, Jesus Antunes M de, Thilen U, Lekakis J, Lengyel M, Muller L, Naber CK, Nihoyannopoulos P, Moritz A, Zamorano JL, ESC Committee for Practice Guidelines (CPG), Vahanian A, Auricchio A, Bax J, Ceconi C, Dean V, Filippatos G, Funck-Brentano C, Hobbs R, Kearney P, McDonagh T, McGregor K, et al. Guidelines on L, Avierinos J-F, Maysou L-A, Riberi A, Grisoli D, Casalta A-C, Collart F, Raoult D, Habib G. Infective endocarditis in octogenarians. Heart 2017;103:1602 1609. 30. Ramírez-Duque N, García-Cabrera E, Ivanova-Georgieva R, Noureddine M, Lomas JM, Hidalgo-Tenorio C, Plata A, Gálvez-Acebal J, Ruíz-Morales J, Torre-Lima J de la, Reguera JM, Martínez-Marcos FJ, Alarcón A de. Surgical treatment for infective endocarditis in elderly patients. J Infect 2011;63:131 138.

Figure 1 :

 1 Figure 1 : Probability of survival according to health-care pathway

Figure 1 :

 1 Figure 1 : Probability of survival according to health-care pathway

Table

.

  In the overall population, mean age was 68 +/-15 years. IE were mainly communityacquired (83% of cases), S. aureus was the most frequent bacteria involved (26% of cases).

	younger (69.4 +/-13 years vs
	62.4 +/-12 years, p=0.01), had lower Charlson index (3.9 +/-1.9 vs 2.8 +/-2.7, p=0.007),
	had more frequently vegetations (86 patients (73%) vs 95 patients (85%), p=0.002) and severe
	Negative blood culture IE represented 17% of cases. Left-sided IE counted for 85% of all cases valvular regurgitation (41 patients (35%) vs 68 patients (61%), p<0.001) but less renal failure
	and aortic (43%) and mitral (42%) localization were almost equally represented. Perivalvular (29 patients (25%) vs 5 patients (4%), p<0.001) and diabetes (39 patients (33%) vs 20 patients
	complications were observed in 16% of cases. 33% of cases were prosthetic valve IE. More (18%), p=0.01).
	than half IE cases (58%) showed theoretical indication for valvular surgery or intracardiac Time from diagnosis to intervention did not significantly differ between RC group and
	device removal but only 35% had such interventions. TRC group (mean time 30+/-32 days vs 22+/-29 days, p=0.36). In our cohort, mean time for
	right-sided IE was 17+/-18 days and 25+/-28 days for left-sided IE.
	Prognosis of patients according to their health-care pathway
	Among the 342 patients, one year mortality was 26% (88 deaths), hospital mortality was
	13% (44 deaths) and one year recurrence rate was 4% (13 patients). Figure 1 shows survival
	probability according to health-care pathway.
	Hospital mortality was 11% (13 deaths) in RC group, 8% (9 patients) in TRC group and
	20% (22 patients) in NRC group, p=0.03. One year mortality was 20% (24 deaths) in RC group,
	21% (23 deaths) in TRC group and 37% (41 deaths) in NRC group, p=0.003. One year mortality

Characteristics according to health-care pathway (Table

1

)

Patients in NRC group were significantly older and had higher Charlson index. Sources of infection did not differ between the three groups and microbiological profile of IE cases was not significantly different. Echocardiographic findings showed more severe lesions in RC group and TRC group: longer vegetations, more peri-valvular complications, and more severe valvular regurgitation. Concerning IE complications, there was no significant difference between groups in embolic events, however more silent embolisms were diagnosed in RC group and TRC group. Spondylodiscitis was more frequent in NRC group.

Patients in RC and TRC groups had more frequent surgical indication, but surgery or intracardiac device removal was performed in only 52 patients (44%) in RC group, 60 patients (55%) in TRC group and 6 patients (5%) in NRC group, p<0.001. In RC group , 81 patients had an indication for intervention and 29 were not operated and in TRC group , 91 patients had an indication for intervention and 31 were not operated. Reasons for absence of intervention were: deaths before surgery (6 patients in RC group and 2 patients in TRC group endocarditis team due to comorbidities (21 patients in RC group and 28 patients in TRC group). Patients who benefited from surgery were significantly

Table 2

 2

	shows prognostic factors of mortality among the 342 patients in multivariate
	analysis. Belonging to NRC group was associated with higher risk of death (HR 2.56; 95% CI
	1.44-4.55, p=0.001). Heart failure (HR 2.4; 95% CI 1.52-3.78, p<0.001) and peri-valvular
	involvement (HR 3.25; 95% CI 1.86-5.68, p<0.001) were IE complications predictive of
	mortality.

Table 2

 2 shows prognostic factors of mortality among the 342 patients in multivariate

	analysis. Belonging to NRC group was associated with higher risk of death (HR 2.56; 95% CI
	1.44-4.55, p=0.001). Heart failure (HR 2.4; 95% CI 1.52-3.78, p<0.001) and peri-valvular
	involvement (HR 3.25; 95% CI 1.86-5.68, p<0.001) were IE complications predictive of
	mortality.

This study was supported by a grant from: the PHRCI (Projet Hospitalier de Recherche Clinique Inter-régional 2012)

Franciscaines, Nîmes: Vlad Ciobataru. MD, Hôpital de Fréjus: Xavier Lamit. MD, Hôpital de Bastia: Aurelia Tho, MD, Thierry James, MD, Abdelkader Bensalah. MD.

Data availability

The data underlying this article will be shared on reasonable request to the corresponding author.

Data availability

The data underlying this article will be shared on reasonable request to the corresponding author.

Conflict of interest

None declared

Methods

Study design and patients

This was a prospective multicentric study including patients consecutively admitted for suspicion of IE in 22 hospitals in South-East of France between January 2014 and June 2017. These hospitals were both tertiary centers with cardiac surgery and second-level community hospitals. Inclusion criteria were a diagnosis of definite IE based on Duke modified criteria 15 on native, prosthetic or pacemaker/implantable automatic defibrillator. Non-inclusion criteria were age < 18 years and patients already included in the study and presenting with a new IE diagnosis. Patients were screened in the echocardiography lab of each institution and were prospectively included by local investigators.

Patient and Public involvement

All patients hospitalized for suspected IE were accepted on admission to participate in this research protocol. However, they were not involved in the design, conduct, reporting or dissemination plans of our research. The study complies with the Declaration of Helsinki.

Written consent was waived by La Timone Institutional Review Board, which gave its approval for this study.

Data collected at the time of inclusion and during hospitalization

The following clinical data were collected: age, sex, previous cardiac and non-cardiac diseases, Charlson index score, date of first clinical signs, body temperature, presence of cardiac murmur, congestive heart failure, neurological complications, septic or cariogenic shock or atrio-ventricular block on electrocardiogram at time of admission and during hospitalization.

Conflict of interest

None declared

Franciscaines, Nîmes: Vlad Ciobataru. MD, Hôpital de Fréjus: Xavier Lamit. MD, Hôpital de Bastia: Aurelia Tho, MD, Thierry James, MD, Abdelkader Bensalah. MD.


Kutlu Can

Yuksel

Silviu-Iulian Niculescu

Tomáš Vyhlídal

A Spectrum-based Filter Design for Periodic Output Regulation of Systems with Dead-Time ⋆

Keywords: delay, filter design, IMC, vibration control

This paper presents a filter structure that allows the user to place desired poles and the necessary zeros to achieve periodic regulation to the closed-loop sensitivity through Internal Model Control framework. The periodic regulation is aimed at stable SISO linear systems suffering from input dead-time and having no closed right-half plane zeros. The negative effects of the dead time on tracking/rejection are compensated by introducing another delay which is tuned independently from the filter thanks to its structure. The main focus is given to constructing the filter with polynomials. However, it can be readily extended to quasi-polynomials. The findings are verified with a computer simulation of a vibration control task.

INTRODUCTION

Periodic output regulation aims to ensure that the particular system under control performs in the desired manner when subjected to periodic exogenous signals. If this exogenous signal is received from the reference, the controller's objective becomes to follow the reference without lag or imperfection. If the signal acts on the system output as a disturbance, then the goal is also to cancel the effects caused by this disturbance. From a systems theory perspective, these two goals are identical and the celebrated Internal Model Principle (IMP) stated by Francis and Wonham (1976) forms a roadmap to achieve them.

The IMP states that perfect tracking/rejection of a signal can be achieved when the close-loop control system encapsulates a model of the signal and is guaranteed to be stable. This translates to placing the poles of the signal model as the zeros of a stable sensitivity function and hence the perfect tracking/rejection. The renowned repetitive control strategy for periodic regulation, introduced in [START_REF] Inoue | High accuracy control of a proton synchrotron magnet power supply[END_REF], is a particular realization of this principle derived by employing a time-delay model of the periodic signal expressed by the transfer function

G signal (s) = 1 1 -F (s)e -sT d , (1)
where F (s) is a low-pass filter and T d is the period of the signal. It is easy to observe that the ideal model that captures every possible signal with period T d is achieved when F (s) = 1. However, employing such a model leads to a closed-loop sensitivity whose dynamics can be described by delay-differential equations of neutral type, and, as shown in [START_REF] Hara | Repetitive control system: A new type servo system for periodic exogenous signals[END_REF], is only BIBO stabilizable for systems with relative degree zero. Therefore, with the expense of a degraded signal model, the associated lowpass filter, with sufficient bandwidth, is utilized to make the overall setup applicable to (strictly) proper systems with achievable stability. As expected, the degradation in the signal model is due to the fact that the filter alters the ideal harmonic pole distribution. In such a case, the filter bandwidth is what determines the number of finitely targeted harmonic poles; those harmonic frequencies that lie outside the bandwidth diverge from their ideal position drastically.

Inspired by this approach, there have been many generalizations or alternatives of the signal model (1) for reasons to improve the transient performance, robustness, and applicability of the strategy to more sophisticated systems. Among these, one particular generalization is obtained by replacing the lumped delay e -sT d with an exponential polynomial over R, as in

G signal (s) = 1 1 -F (s)W (s) , (2)
where W (s) = m i=1 w i e -msT d with w i ∈ R. Such configurations with model (2) have been referred to as the high-order repetitive control, and tuning of the weights w i to improve performance and robustness was studied in [START_REF] Inoue | Practical repetitive control system design[END_REF]; [START_REF] Chang | Analysis and design of two types of digital repetitive systems[END_REF]; [START_REF] Steinbuch | Design of noise and period-time robust high-order repetitive control, with application to optical storage[END_REF]; [START_REF] Pipeleers | Robust high-order repetitive control: Optimal performance trade-offs[END_REF] mainly for discrete-time systems. Model (2) is of particular interest because it marks an important observation: despite the known negative impacts of delays on system stability, the presence of more delays in the model can lead to greater flexibility to optimize with respect to a desired property. Intuitively, the increased capability for further improvement can be reasoned by the fact that model (2) describes a signal based on multiple past values in contrast to Model (1) where only one past information is utilized. With similar motivation, an alternative transport equation representation of the signal was proposed in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF].

Nevertheless, all these signal models can prove their functionality only if the closed-loop system is stable, and it is a known fact that such a property can be easily lost due to uncounted additional delays that can occur in the loop through true or approximation of the controlled system. Thankfully, many notions of stability analysis for finite dimensional systems, mainly eigenvalue-based and Lyapunov-based approaches, carry over to infinite dimensional systems under which many time-delay systems are categorized.

This paper proposes a controller design that can simultaneously achieve periodic regulation and input-delay compensation as in [START_REF] Vyhlídal | Control system design based on a universal first order model with time delays[END_REF]; Yuksel et al. (2021a,b). The design steps for periodic regulation differ from the approach seen in repetitive control: rather than a time-delay signal model, a model of the system is employed in the loop, and the required zeros for regulation are introduced one by one to the sensitivity with a special filter in contrast to first placing infinitely many zeros through the signal model and then drastically drifting them away from the imaginary axis except those which are of interest. The proposed individual treatment of zeros ensures the harmonic zeros to stay on the imaginary axis and prevents unwanted zeros from occurring between targeted frequencies. Other features of the design are that it does not require the system model to be of first-order as was the case in the previous works of the authors, and the controller is derived directly from a reference sensitivity that specifies the desired spectrum. Therefore, stability is assured from the very beginning for the ideal case, and furthermore, all possible controllers fulfilling the demands are parameterized by the location of the stable poles of the sensitivity. Clearly, having controllers meeting the demands parameterized makes the approach suitable for optimization, though discussion on this topic is not in the scope of the paper, and only some remarks are given in the subsequent sections.

The aim of the paper is to introduce the reader to a straightforward yet effective design concept and to contribute to the previous works of the authors. It is organized in the following manner: Section 2 covers the controller design and introduces the Internal Model Control scheme, which forms the framework for the design. The section also elaborates on the conditions that need to be satisfied by the filter within the controller and the additional delay for regulation/compensation and proceeds to the state-space formulation due to the convenience it brings to implement and impose the conditions. Section 3 gives remarks about the proposed controller, including the generalization of the filter via quasi-polynomials and methods to improve the performance of the control system to delay variations. Section 4 numerically validates the effectiveness of the controller structure introduced in Section 2 via a simulation of a vibration control applied to a fourth-order system with input delay. Finally, the paper is concluded with Section 5.

CONTROLLER DESIGN

The controller design begins by specifying a stable sensitivity with the desired zeros for periodic regulation and deduces the controller structure and the conditions based on Internal Model Control scheme.

We consider a stable SISO plant G(s) of the form

G(s) = Y (s) U (s) = s α + a α-1 s α-1 + ... + a 1 s + a 0 b β s β + b β-1 s β-1 + ... + b 1 s + b 0 e -sτp ,
(3) where the polynomials in the numerator and the denominator are Hurwitz with β ≥ α and τ p > 0.

The exogenous signal v(t) to be ideally tracked/rejected is assumed to have a finite Fourier series expansion i.e.

v(t) = v 0 + k i=-k v i e jωit , (4)
where frequencies ω i are commensurate i.e. integer multiples of a base frequency ω b , v i is the corresponding Fourier coefficient and k ∈ N \ {∞}. In accordance with IMP, this implies that the sensitivity transfer function S(s) of the closed-loop needs to be stable and have zeros at s = 0 and s = ±jω i for i = 1, ..., k i.e.

S(s) = z(s) p(s) , (5)
where z(s) and p(s) are the numerator and denominator of the sensitivity, respectively, and such that z(0) = z(jω i ) = 0 and p(s) is Hurwitz.

More specifically, if z(s) and p(s) happen to be polynomials, the desired sensitivity has the form

S(s) = s k i=1 (s 2 + ω 2 i ) m (s -z m ) n (s -p n ) , (6)
where p n ∈ C -and z m denote the poles and the additional zeros, respectively.

Internal Model Control

Internal Model Control (IMC) is the control arrangement depicted in Figure 1, in which the model of the controlled plant is employed in the control loop to compare the actual output of the plant with the estimated one. The features of the IMC scheme were extensively studied by [START_REF] Garcia | Internal model control. a unifying review and some new results[END_REF]. Further adaptive extensions of this scheme was proposed in [START_REF] Hahn | Adaptive imc control for drug infusion for biological systems[END_REF] The sensitivity transfer function S(s) of the IMC scheme in Fig. 1 is given by τm+θ) ) ,

S(s) = 1 -Q(s)G m (s)e -s(τm+θ) 1 + Q(s)(G p (s)e -s(τp+θ) -G m (s)e -s(
(7) and it is a retarded time-delay system with infinitely many zeros and poles. Assuming that the ideal case holds i.e.

G p (s) = G m (s) = G(s) and τ p = τ m = τ , it reduces to S(s) = 1 -Q(s)G m (s)e -s(τm+θ) . (8)
Based on the proposed design by [START_REF] Vyhlídal | Control system design based on a universal first order model with time delays[END_REF], if the controller structure is constructed as

Q(s) = 1 G m (s) F (s), (9)
where F (s) is a low-pass filter, then reflecting this to the ideal sensitivity in Equation (8) yields

S(s) = 1 -F (s)e -s(τ +θ) , (10)
Notice that, the inverse of G m (s) can be utilized in (9) thanks to having no non-minimum phase zeros. The delay θ associated with the controller Q(s) is used to compensate for the negative effects of the other delay.

Imposing the conditions on zeros of the sensitivity given by ( 5) and studying Equation (10) in terms of its modulus and argument yield the conditions need to be satisfied by F (s) and θ:

lim s→0 F (s) = 1 (11) |F (jω i )| = 1 (12) ∠F (jω i ) -ω i (τ + θ) = 2πh ( 13
)
where h ∈ Z.

Ideal Filter Structure

Suppose initially that there are no delays in the loop, i.e. τ = θ = 0. Then based on Equation (10), the filter that results in the desired reference sensitivity expressed via (5) can be found as

F (s) = p(s) -z(s) p(s) , (14)
which is referred to as the ideal filter. Notice that, it satisfies condition ( 11)-( 12) and has no phase shift at ω = ω i i.e. ∠F (jω i ) = 0 since

F (jω i ) = p(jω i ) -z(jω i ) p(jω i ) = p(jω i ) p(jω i ) = 1. (15)
However, in order to make the controller Q(s) proper, the filter needs to be proper with a relative degree n r ≥ β -α.

Throughout the paper, we consider filters that are strictly proper i.e. n r ̸ = 0, for being generally applicable to all systems expressed by (3). For a filter to be strictly proper, the terms with the highest power from p(s) and z(s) must match. Hence, if the filter is derived through a sensitivity based on monic polynomials as in ( 6) then the condition n = 2k + m + 1 (16) must hold. Notice that, for each introduced additional zeros, we need to place one additional stable pole and vice versa.

State-Space Formulation

Even though the transfer function representation of the ideal filter is straightforward, its direct implementation is likely to lead to numerical problems. To overcome this issue for filters with polynomials, the sensitivity and, later on, the filter is expressed in a state-space form, which allows employing numerically more reliable methods. In addition, the conditions to achieve desired relative degree for the filter can be posed as linear constraints, which can subsequently be used to solve for the unknown system matrices.

Let [A, B, C, D] denote the standard matrices of the statespace representation of the sensitivity (6). Then from ( 14), it simply follows that the filter is [A, B, -C, (1 -D)].

The system matrix A is such that its characteristic equation corresponds to p(s) i.e det(λI -A) = p(s). Without loss of generality, set C = [1 1 ... 1] 1×n and D = 1. The harmonic zeros z ω ∈ {-jω k , ..., 0, ..., jω k } of the sensitivity must satisfy

C(z ω I -A) -1 B = -D, (17)
which gives 2k + 1 equation:

        CA -1 ℜ(C(jω 1 I -A) -1 ) ℑ(C(jω 1 I -A) -1 ) . . . ℜ(C(jω k I -A) -1 ) ℑ(C(jω k I -A) -1 )         B n×1 =         -D -D 0 . . . -D 0         (18)
The other n -2k -1 equations required to have a solution for B come from the conditions for the relative degree.

Notice that the required number of equations is equal to the number of additional zeros m. It can be shown that the relative degree of the filter in state space form can be determined by the number of the terms of the sequence (1 -D), -CB, -CAB, -CA 2 B, . . . that are equal to zero starting from the first one. The first term happens to be zero thanks to setting D = 1, therefore the filter is at least of relative degree one. For n r ≥ 2, the following set of equations must be satisfied:

    C CA . . . CA (nr-2)     B =     0 0 . . . 0     (19)
Hence, the relationship between the number of additional zeros and the relative filter order corresponds to m = n r -1.

(20) Combining Equations ( 18) and ( 19) gives in total n equations from which B can be uniquely found.

Subsequently, the inverse model dynamics is added to the filter to give the state space model of the controller Q(s):

A Q = Ã = A 0 -B α C A α ( 21
)
B Q = B = B 0 (22) C Q = [b β b β-1 . . . b 1 b 0 ]      C Ãβ C Ãβ-1 . . . C      (23) D Q = b β C Ã(β-1) B ( 24
)
where

[A α , B α , C α , D α ] is a minimal state-space realiza- tion of G α (s) = (s α + a α-1 s α-1 + ... + a 1 s + a 0 ) -1 and C = [-D α C C α ]. Note that, [ Ã, B, C, 0] is obtained when the filter is combined with G α (s)
and it has a relative order not less than β. Using this fact yields the controller.

Delay Adjustment

Compensation for the negative effects of system delay τ on periodic tracking/rejection is achieved by tuning the userdefinable delay θ to satisfy the argument condition (13).

Since the phase-shift induced by the filter ( 14) is zero at harmonic frequencies, the condition reduces to

ω i (τ + θ) = 2πl, ( 25
)
where l = -h. The assumption that all frequencies ω i are commensurate implies that there exists a frequency, so-called the base frequency ω b , such that ω b ≤ ω i and ωi ω b = γ ∈ Z. Utilizing this base frequency yields an equation for the smallest θ satisfying (25) which is valid for all frequencies ω i :

θ = 2l b π ω b -τ ≥ 0, (26)
where

l b = τ ω b 2π + 1. Note that, if ω b (τ + θ) is an integer multiple of 2π then so as ω i (τ + θ) = γω b (τ + θ).
It is clear from Equation (8) that the resulting sensitivity in the presence of delays is different than the sensitivity taken as a reference to form the ideal filter. In fact, it attains infinitely many zeros since the numerator corresponds to a quasi-polynomial. However, when θ is chosen according to Equation ( 26) the resulting sensitivity encapsulates the harmonic zeros from the reference sensitivity and has identical poles.

FURTHER REMARKS

To this point, the only condition on the poles has been for them to lie on the left half plane of the complex space. Even though this is sufficient to guarantee the stability of the close loop for the ideal case, the real-life application of the controller requires the close-loop to be robust to inevitable mismatches between the plant and the model. Furthermore, not every pole distribution leads to a desirable response performance. For this reason, the placing of the poles must be carried out with respect to a chosen robustness and/or performance measure, such as the H ∞ -norm of the sensitivity. Such an operation can be done either analytically or numerically by posing an optimization problem. However, in doing so, one must make sure that the computations comply with the nature of time-delay systems. For instance, a method to compute the H ∞ norm of SISO time-delay systems was proposed by [START_REF] Gumussoy | Computation of extremum singular values and the strong h-infinity norm of siso time-delay systems[END_REF].

The proposed filter is also suitable for applying delay adaptive methods thanks to Equation (25). For instance, a wavelet-based algorithm for θ to compensate for varying plant delays was recently proposed in [START_REF] Peichl | Two-harmonic periodic disturbance compensation for dead-time processes with online identification of input delay[END_REF].

Additionally, the filter does not necessarily have to consist of polynomials. In fact, if the utilization of quasipolynomials is permitted, one can attain dynamics otherwise not achievable and can lower the order of the overall filter. For instance, the distributed delay approach studied by [START_REF] Yuksel | Internal model control with distributed-delay-compensator to attenuate multiharmonic periodic disturbance of time-delay system[END_REF] for first-order systems with input-delay, corresponds to choosing z(s) to be the quasipolynomial of the form

z(s) = s - k a k e -sθ k , (27)
with p(s) = s. The zeros are placed by tuning the coefficients a k associated with fixed delays. Such configuration was shown to lead to a pole-less sensitivity which made the response dynamics governed by a difference equation.

ILLUSTRATIVE EXAMPLE

The proposed controller model is tested on a computer simulation of a fourth-order model corresponding to the serially connected mass-spring-damper with two actuators depicted in Figure 2 where m 1 = 1.1 kg, m 2 = 0.514 kg,

k 1 = 1768 N m -1 , k 2 = 424 N m -1 , c 1 = 4.43 N s m -1 , c 2 = 2.
41 N s m -1 and is described by,

m 1 ẍ1 (t) + (c 1 + c 2 ) ẋ1 (t) + (k 1 + k 2 )x 1 (t) = c 2 ẋ2 (t) + k 2 x 2 (t) + u(t -τ ) -d(t), (28) m 2 ẍ2 (t)+c 2 ẋ2 (t)+k 2 x 2 (t) = c 2 ẋ1 (t)+k 2 x 1 (t)+d(t). ( 29
)
The goal is to control the position of mass m 1 by actuator u with output feedback despite the oscillations caused by actuator d and artificially added input delay of τ =0.2 s. The position response of mass m 1 to a unit step from input u is as shown in Figure 3.

The disturbance is caused by forcing the masses in a sawtooth manner with frequency f d = 2Hz, resulting in m 1 being exposed to oscillations with spectrum shown in Figure 4.

The controller is designed to cancel the first 8 harmonics of the disturbance and contains an all-polynomial filter Figure 6 shows the pole-zero distribution, obtained via QPmR by [START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF], of the ideal sensitivity S(s) and the non-ideal sensitivity S per (s) of the form (7) with the system subjected to the perturbation, G p (s) = 0.9 0.05s+1 G m (s). As can be seen, the harmonic zeros are placed and preserved, and the close loop remains stable despite the perturbation. Note that, in the ideal case, the close-loop has infinitely many zeros but finitely many poles. However, in the non-ideal case, both the zeros and the poles form chains that tend to each other at infinity. Notice also that, after perturbation, the poles moved dangerously close to the imaginary axis. 8, respectively. The filter has a relatively slow rejection performance with residual oscillation caused by the unaccounted higher harmonics and a relatively fast tracking performance with undesirable spiky behavior, indicating the necessity of further optimization for the pole locations.

CONCLUSION

Alternative to the repetitive control, an analytical design method is proposed to derive a controller that can achieve both periodic regulation and input time-delay compensation for stable systems with dead-time and no closed righthalf plane zeros. The controller design does not necessarily require a first-order model approximation of the plant, as it was the case in the authors' previous works. This is achieved by utilizing the so-called ideal filter in the IMC framework. The filter poles ensure the stability of the close-loop. Nevertheless, the decision on the pole locations is an open question. Additionally, the proposed filter structure can be further extended with quasi-polynomials, forming a new perspective for employing delays in the control loop. Future works will focus on this together with experimentation and the placing of poles with respect to a performance/robustness measure.

⋆

  This work was supported by the Czech Science Foundation project No. 21-07321S -Persistent problems of repetitive control. The work of the first and second authors was also supported by a public grant overseen by the French National research Agency (ANR) as part of the Investissements d'Avenir program, through the "ADI 2020" project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02. The first author was also supported by the Student Grant Agency of the Czech Technical University in Prague, student grant No. SGS23/157/OHK2/3T/12.

Fig. 1 .

 1 Fig.1. The internal model control scheme removing periodic disturbance from systems approximated by first-order models with dead-time was proposed by[START_REF] Vyhlídal | Control system design based on a universal first order model with time delays[END_REF] andYuksel et al. (2021a).

Fig. 2 .Fig. 4 .

 24 Fig. 2. Diagram of the controlled system

Figure 5

 5 Figure5shows the Bode magnitude plot of the resulting ideal sensitivity in the form (10) with filter (14). It is clear from the figure that the chosen pole distribution gives a favorable ||S(jω)|| ∞ < 2 for robustness.

Fig. 5 .

 5 Fig. 5. Bode Magnitude Plot of the Resulting Sensitivity

Fig. 6 .

 6 Fig. 6. Pole Zero Distribution of the ideal sensitivity of the form (6) and the perturbed sensitivity S per (s) of the form (7)(Top) Overall spectrum of S(s) and S per (s) (Bottom) Zoomed view showing the placed harmonic zeros. The transient rejection and tracking performance of the ideal control system are demonstrated in Figures 7 and

Fig. 7 .

 7 Fig. 7. Disturbance rejection performance of the proposed controller with specified poles. The activation moment t = 1s of the controller is indicated by the vertical dashed line.