# Fault-Tolerant Model Predictive Control Applied to a Sewer Network
## 1 Introduction  
Urban drainage networks (UDN) collect and carry both urban wastewater and that which comes from precipitation to wastewater treatment plants (WWTPs) for treatment before being discharged into the environment, constituting a combined urban drainage system (CUDS). During periods of heavy rain, the residual water resulting from the mixing can overload the urban system and produce overflows (combined system overflows, CSOs) that can be harmful to the environment. To avoid CSOs, the current UDNs have retention systems capable of storing the water that reaches the network in times of intense rain and later releasing the stored volume at lower flows suitable for treatment by the WWTPs. Adequate real-time control (RTC) of the volume of water stored in the tanks can significantly improve the operation of the network to minimize the impact of CSOs [1; 2; 3; 4]. Among the techniques used for optimal control of these systems, those that use a simplified model of the process to predict its behavior stand out. This is how Model Predictive Control (MPC) works [5]. MPCs are part of a control methodology that uses a process prediction model to calculate the manipulated variables over a future horizon to optimize a certain cost function. It is an algorithm that has been successfully implemented for several decades and has also been applied to UDNs with great success [6; 7; 8; 9]. On the other hand, urban water treatment systems (UWS), which integrate both UDNs and WWTPs, have a high degree of interconnection and their proper functioning depends on the reliability of the equipment used, such as sensors (flowmeters, level sensors) and actuators (pumping stations, gates, valves) and communication systems. Theenvironmental conditions surrounding this equipment can cause its deterioration and malfunction. For this reason, it is necessary to develop Fault Tolerant Control Systems (FTCS) to maintain safe and efficient operation. In this way, a Fault-Tolerant Controller (FTC) is one that can achieve control objectives even though faults may exist, which can lead to a reduction in system performance [10; 11; 12]. Fault-tolerant control takes advantage of the physical and analytical redundancies of the system to increase its performance when an element suffers a malfunction. Furthermore, rapid detection and identification of a fault can help avoid serious and even dangerous breakdowns. Generally, FTCS can be classified into two types: passive (PFTCS) and active (AFTCS). AFTCS react to system component faults by actively reconfiguring control actions so that stability and performance are acceptable, even if performance has degraded [11; 13]. Normally, AFTCS consist of four subsystems: (1) a reconfigurable controller, (2) a fault detection and diagnosis scheme (FDD), (3) a controller reconfiguration mechanism, and (4) a command/reference governor. Existing FDD approaches can be generally classified into two categories: (1) model-based scheme and (2) data-driven (model-free) scheme [14; 15]. Data-driven schemes are divided mainly into two approaches: the multivariate statistical process control methods (MSPC) and machine learning (ML) methodologies. In the first case, the most applied methodology is the principal component analysis (PCA) [16; 17; 18; 19; 20]. The second approach considered is the machine learning or artificial intelligence techniques [21; 22; 23; 24; 25]. Furthermore, deep learning strategies have become increasingly popular in the face of complex nonlinearity and can be used for modeling, control, or management of WWTPs as can be seen in [26; 27; 28; 29; 30; 31]; however, very few studies have addressed the fault detection problems in fewer networks. For sewer systems, some methods are based on data analysis [32]. Furthermore, there are methods using closed-circuit television (CCTV) inspections and artificial intelligence to classify defects automatically [33].