# Fault-Tolerant Model Predictive Control Applied to a Sewer Network
## 1 Introduction
Others are based on state estimation, using, for example, a Luenberger observer [34], or determination of normal operating ranges for sensor and actuators [35; 36]. Often, the controller used is an MPC, so it would be a fault-tolerant model predictive control (FTCMPC) [35; 36], but no work has been found in which the PCA technique has been applied to the fault detection and diagnosis in a sewer network. Due to complex physical and chemical processes as well as changing operating conditions and the nonlinearity of the sewer networks, this technique could be applied successfully to this process. The benchmark considered as a case study is described in [37]. The main problem of this system is the high variability of the disturbances (collected flows in each area) that affect the process. Among the whole set of data that the benchmark realistically integrates, there are time intervals of weeks in which rainfall is very low, which means that the control system, even when working properly, has little influence on the performance of the system. In this case, a fault of any sensor or actuator, even a large one, would be virtually undetectable, although it would also have little impact on the system. Something similar occurs in the complete opposite situation: if very heavy and repeated rainfall occurs, which can saturate the sewer network, the control system will not prevent overflows in the network, which can become important even if it works properly, and if a fault occurs in any equipment, it will have little impact on the system and therefore its detection and classification will be more difficult. These same reasons lead us to think that when any type of fault occurs, if it is not very significant, it will most likely go unnoticed. Benchmark simulation tests demonstrate this. Consequently, it is advisable to focus on intermediate situations, i.e., situations after times of moderately high-intensity rainfall, or at longer time intervals when medium-intensity rainfall occurs, but with more continuity. It has been found that it is in these cases that the MPC controller is most useful in reducing overflows at different points in the network and in keeping the inflow to the treatment plant closer to its nominal value. The main contribution of this work consists of the development of a real-time online FTMPC applied to the UDN system considered, consisting of three subsystems: a fault detection system based on an adaptive online PCA moving data window technique, capable of providing a real-time fault monitoring solution for the sewerage system despite the dynamically changing properties of the system; a fault diagnosis system, which will classify the detected fault through statistical calculations that will identify the variable that deviates the most from its normal behavior; and finally, a system for reconfiguring the MPC controller, taking advantage of its constraint handling capability, to try to maintain control over the whole plant, minimizing the effects of the fault. Several case studies with different disturbance profiles will be analyzed. The results have been compared with the behavior of the system without control, with the normal MPC control algorithm and with different fault situations without reconfiguration of the system. This article is structured as follows: after an introduction, the theoretical description of mathematical algorithms that will be used begins, i.e., MPC and PCA. Afterwards, the fault detection, diagnosis and reconfiguration methodology will be detailed. The following section exposes a case study where previous methodology will be applied: first, the sewer system description is presented, and then, the MPC control algorithm and the FTC system that will be applied. Next, the results obtained in each case will be shown to finish with the conclusions of the work.