The aim of this randomized controlled trial is comparing the improvement of diurnal PaCO2 in OHS patients after one month of NIV treatment depending upon the tools used for adjusting settings. In one arm settings will be adjusted using only nocturnal oxygen SaO2 and PaCO2 at awakening whereas in the other arm patients will benefit from a complete polysomnography under NIV.

Initiation of NIV treatment implies technical adjustments (choice of mask and ventilator), but also patient education by the nurses and medical staff. The settings are generally adjusted during the daytime in awake patients. The appropriateness of these settings is generally not optimal during sleep as many respiratory events (hypopnoea, apnea, leaks, desynchronization, and glottic closure) may occur under NIV then reducing the effectiveness of treatment. These nocturnal abnormalities are routinely evaluated by measuring their consequences, i.e. oxygen desaturation and the level of PaCO2 after wake up. Some teams are now proposing to better characterize what happens when using NIV during sleep using polysomnography (PSG). The precise characterization of residual events under treatment may allow optimizing ventilator settings.

• Experimental: patients will benefit from a complete polysomnography under NIV
• Active Comparator: settings will be adjusted using only nocturnal oxygen SaO2 and PaCO2 at awakening whereas

Inclusion Criteria:

• Patient male or female, aged 20 to 75 years
• Patient with a OHS in stable condition, with a BMI> 32 kg/m2 and a PaCO2> 6 kPa in diurnal spontaneous breathing without other cause of alveolar hypoventilation
• Patient affiliated to a social security insurance
• Having given its written informed consent to participate to the study

Exclusion Criteria:

• Patients with inherent chronic obstructive pulmonary disease (COPD) with FEV1 on Forced Vital Capacity (FVC) ratio less than 70%
• Patients suffering from heart failure with periodic breathing
• Associated Neurological Diseases, evolving rapidly, leading to a dependency in daily activities
• Unbalanced Psychiatric Diseases
• Patients with a respiratory decompensation the month preceding the study
• Patients not autonomous in the use of the NIV
• Pacemaker patients, constituting a contraindication to magnetic stimulation
• Sensitive subjects, in accordance with article L 1121-6 of the French Public Health Code
• Patients with long term by steroids or other anti-inflammatory drugs