Sliding window assessment for sensor fault model-based diagnosis in inland waterways

Sliding window assessment for sensor fault model-based diagnosis in inland waterways P. Segovia ∗,∗∗,∗∗∗ J. Blesa ∗∗ E. Duviella ∗∗∗ L. Rajaoarisoa ∗∗∗ F. Nejjari ∗V. Puig ∗,∗∗ ∗Automatic Control Department, Technical University of Catalonia (UPC), Terrassa, Spain {pablo.segovia, fatiha.nejjari, vicenc.puig}@upc.edu ∗∗ Institut de Rob`otica i Inform`atica Industrial (CSIC-UPC), Barcelona, Spain {[email protected]} ∗∗∗ URIA, IMT Lille Douai, France {pablo.segovia, eric.duviella, lala.rajaoarisoa}@imt-lille-douai.fr Abstract: The Cuinchy-Fontinettes reach belongs to the inland waterways in the north of France. It is equipped with limnimeters that measure water level data for the management of the water resources. These data can be corrupted by constant or intermittent faults. Hence, it is necessary to detect and localize these faults in order to guarantee efficient management actions. The proposed fault diagnosis method is based on the analysis of the parameters of a grey-box model. These parameters are obtained from available real data by using a sliding window, whose size is determined based on the level of excitation of input signals. Then, several scenarios involving constant and intermittent faults are proposed to discuss the performance of the proposed FDI approach as well as the effect of the sliding window size on the results. Keywords: Large-scale systems, inland waterways, fault diagnosis, grey-box model. 1. INTRODUCTION The Cuinchy-Fontinettes reach (CFr) belongs to the inland waterways in the north of France. It is equipped with controlled gates that are used to regulate the levels by dispatching volumes of water and with locks that make possible the navigation of boats along the waterway. The management condition to be fulfilled aims at keeping the water levels close to the setpoint, which is known as the normal navigation level (NNL). The design and implementation of control algorithms such as those proposed in (Segovia et al., 2017b; Horv´ath et al., 2014c) require the measurement of levels. However, these measurements can be corrupted by several types of faults, which can lead to undesired control actions. Thus, fault detection and isolation (FDI) techniques must be designed to detect and localize the faults. Blesa et al. (2010) proposed a fault detection method based on a linear parametervarying (LPV) model in order to detect faults that occur in open-channel systems. Model-based method and intervals models have been proposed in (Segovia et al., 2017a; Blesa et al., 2014) respectively, for sensor fault detection in inland waterways. A comparison of FDI methods to detect leaks in an irrigation network has been carried out in (Bedjaoui and Weyer, 2011). Nabais et al. (2012) proposed another sensor FDI approach for irrigation canals based on residual generation. An unknown input observer (UIO) design for delayed LPV systems represented in the polytopic framework which has been applied to an open flow canal is proposed by Hassanabadi et al. (2016). FDI techniques ?The authors thank Voies Navigables de France for the data provided, which has made possible this work. based on classification techniques, black-box models or recursive subspace identification method have been proposed in (Pocher et al., 2011, 2012) and (Akhenak et al., 2013), respectively. A combination of a physical model and a classification algorithm has been designed in (Horv´ath et al., 2014a) and developed in (Duviella et al., 2013) by considering the grey-box model detailed in (Horv´ath et al., 2014b). In this paper, the FDI approach is based on the evolution of the grey-box parameters over the time. The parameters of the grey-box model are estimated based on real input/output data using a sliding window. Some rules to tune the length of the time window and guarantee the fitting of the identified model are proposed. The tuning of the time window length is discussed by considering real data from the CFr and three fault scenarios. The paper is organized as follows: the CFr and the management objectives are detailed in Section 2. The grey-box model is presented in Section 3. Section 4 describes the proposed FDI approach. Finally, the modeling and FDI techniques are illustrated in Section 5 by means of a case study. 2. THE CUINCHY-FONTINETTES REACH 2.1 Management objectives The CFr that is located in the north of France plays a crucial role in the transport and the water resources management of the waterways due to its location. During flood periods, the CFr is used to share water volumes between the Aa and the Lys watersheds. During drought periods, it is used to appropriately dispatch the available water resources among the watersheds. The objective is to make navigation possible during the arranged schedule. In order to ensure the navigability of vessels, it is required that the water levels be inside the interval that is defined by considering a tolerance range around the NNL, i.e. lower and higher navigation levels (LNL and HNL, respectively). 2.2 Description The CFr is 42 km long and delimited by the Cuinchy lock upstream and the Fontinettes lock downstream. It is equipped with gates in Cuinchy (beside the lock) and at Aire-sur-la-Lys with the gate of Porte de Garde, located 28 km downstream of Cuinchy (see Fig. 1), and with three limnimeters that allow measuring the water levels at Cuinchy (LC), Aire-sur-la-Lys (LA) and Fontinettes (LF) with a sample time k= 1 min. The controlled discharges are applied by means of the Cuinchy lock and gate (QC), the Porte de Garde gate (QA) and the Fontinettes lock (QF). These magnitudes are measured each 15 min. Cuinchy gate and lock Fontinettes lock Porte de Garde L A LC L FQF QA QC Fig. 1. Schematics of the CFr. The CFr is an open-channel flow characterized by nonlinear dynamics with large time delays. In addition, its dynamics are impacted by strong resonance phenomena due to a negligible slope along its course. Lock operations create waves that travel back and forth along the reach during several hours until their attenuation. 2.3 Fault diagnosis in the CFr In this system, faults mainly occur on sensors. They can be due to an offset error of the measurement during several hours (persistent faults), to transmission problems (intermittent faults) or to a combination of both (transient faults that lead to persistent faults). These faults can have a strong impact on the water level control, where the consideration of wrong data can lead to a system malfunction. Hence, sensor faults must be diagnosed. The main challenge of sensor fault diagnosis lies in the characteristics of the CFr: a large-scale system with nonlinear dynamics and large time delays. Moreover, when real data are considered, it is necessary to deal with strong uncertainties. While classical diagnosis approaches can face some difficulties, data-based approaches seem more suitable to deal with real data, as it was shown in (Duviella et al., 2013). The remaining task consists in detecting intermittent faults. Thus, a grey-box model-based approach is designed to diagnose sensor faults by considering real data of the CFr, from October 30, 2013 (Thursday) to November 17, 2013 (Sunday). Part of these data are used to estimate the parameters of the grey-box model and determine the validity intervals of its main parameters. The remaining data are used to test the designed approach by introducing artificial faults. 3. GREY-BOX MODEL 3.1 Description The structure of the grey-box model is a first order plus time delay for every input/output pair: ˆyk+1 =Ayk|τ+Buk|τ(1) where input variables uk|τ∈Rn2×1correspond to a combination of the components of the input vector uk=[u1 k. . . un k]Twith the corresponding delays, and ul k= Ql k∀l= 1, ..., n are the different discharges along the canal. Similarly, the output variables yk|τ∈Rn2×1correspond to a combination of the components of the output vector yk=[y1 k. . . yn k]Twith the corresponding delays, where yi k=Li k∀i= 1, ..., n are the different level measurements along the canal (see Fig. 2). Matrices A∈Rn×n2and B∈Rn×n2are defined under the assumption that each actuator is equipped with one limnimeter. The matrix τ∈Nn×ngiven in Eq. (2) gathers the time delays between the measurement points Liand Lj, and also between the discharge Qiand the measurement point Lj, respectively. For instance, the value of τi,j ∈Nis the time delay between the measurement points Li(resp. Qi) and Lj. τ=    0τ1,2· · · τ1,n τ2,10· · · τ2,n . . .. . ..... . . τn,1τn,2· · · 0     (2) LLL 12n T1,2 1,n 2,1 2,n n,1 n,2 Q1 Qn Q2 Lock Limnimeter T T T T T Fig. 2. Time delays τi,j between measurement points. The time delays are determined according to the wellknown relations given in (Litrico and Fromion, 2004). The vector yk|τis expressed as: yk|τ= [L1 kL2 k−τ1,2 ··· Ln k−τ1,n L1 k−τ2,1L2 k··· Ln k−τ2,n ··· L1 k−τn,1L2 k−τn,2 ··· Ln k]T(3) Likewise, uk|τreads as follows: uk|τ= [Q1 kQ2 k−τ1,2 ··· Qn k−τ1,n Q1 k−τ2,1Q2 k··· Qn k−τ2,n ··· Q1 k−τn,1Q2 k−τn,2 ··· Qn k]T(4) Matrix Ais defined as the direct sum of vectors Aii,i.e. A=Ln i=1 Aii (Bis expressed similarly): A=    A11 0n0n0n 0nA22 0n0n 0n0n ...0n 0n0n0nAnn     (5) with Aii =ai,1· · · ai,nand 0nthe zero vector of length n.ai,i is the parameter that links ˆyi k+1 to yi kwith no delay. Equation (1) is rewritten as follows: ˆyk+1 =MΦk(6) with M= [A B] and Φk= [yk|τuk|τ]T. Then, Mis the solution of the linear least squares problem. Nsamples of the measured discharges Qi kand levels Li kare considered in its computation: M=YΦT(Φ ΦT)−1(7) with Y= [yχ+1 · · · yN], Φ= [Φχ· · · ΦN−1], and χ= max(τ) + 1, where max(τ) is the maximum entry of matrix τgiven in (2). 3.2 Identification based on a sliding window It is also possible to identify the parameters of the greybox model by using sliding windows. In doing so, a model more reactive to changes and that follows more precisely the dynamics of the reach can be obtained. The value of Mgiven in Eq. (7) is determined by considering a time window of size Nw. Hence, a temporal matrix Mk is computed at every instant k: Mk=YkΦT kΦkΦT k−1 (8) with Yk= [yk+χ+1−Nw· · · yk], Φk= [Φk+χ−Nw· · · Φk−1]. The parameters aj,i kand bj,i k∀i, j = 1...n are obtained from Mk= [AkBk]. The time window Nwis tuned according to persistent excitation indicators in the inputs. The estimation of the parameters is done when the input persistent exciting order is high enough, as it is proposed in section 13.2 in (Ljung, 1999). The Matlab command pexcit is used to determine these indicators. There is no a priori maximum value of Nw. However, the larger Nwis, the lower the re-activity of the grey-box model would be, i.e. the modification time of the parameters’ values would be bigger. 3.3 Validation step The following fit coefficients are used to determine the accuracy of the grey-box model with respect to the measurements. •Pearson product-moment correlation coefficient measures the linear dependence between two variables: Ri= N P k=1 yi k−λyiˆyi k−λˆyi sN P k=1 yi k−λyi2sN P k=1 ˆyi k−λˆyi2 (9) with λyiand λˆyithe mean value of measured and estimated water levels, respectively. This coefficient is bounded between +1 (total positive linear correlation) and -1 (total negative linear correlation), and 0 means that there is no linear correlation. •Nash-Sutcliffe model efficiency coefficient is used to assess the predictive power of hydrological models (Nash and Sutcliffe, 1970): Ei= 1 − N P k=1 yi k−ˆyi k2 N P k=1 yi k−λyi2 (10) Eican range from 1 to −∞, where 1 indicates a perfect match of modeled and observed values, 0 corresponds to the case in which the model predictions are as accurate as the mean of observed data and Ei<0 means that the model predictions are less accurate than the mean of observed data. It can also be expressed in percent when its value is positive. Once the size of the time window Nwis tuned and the proposed model is validated, the coefficients of matrices Akand Bkcan be used to design the FDI strategy. 4. FAULT DIAGNOSIS The fault diagnosis focuses on level sensor faults: Li k=Li,0 k+ ∆i k,∀i= 1, ..., n (11) where Li,0 kdenotes the level iand ∆i kthe fault at time k. As (1) provides the level estimations ˆyk+1, the most straightforward fault detection method consists in evaluating the difference between the level sensor measurements and the estimations: ri k=yi k−ˆyi k,∀i= 1, ..., n (12) where ri kis the temporal residual of the i-th level sensor. The fault detection test can be formulated as follows: φri k=0 if ri k∈[σi, σi]⇒No Fault 1 otherwise (13) with bounds σiand σithe maximum positive and negative deviations of the residual ri kin a fault-free scenario. A similar approach is followed by considering the grey-box model parameters. The main idea is that, under normal conditions, the dynamics of the reach are always similar, and therefore the parameters of the model fluctuate within a narrow interval. Hence, bounds are determined in a faultfree scenario for each parameter of matrix Mk: [aj,i, aj,i] and [bj,i, bj,i]. Conditions on persistent excitation and on the fitting indicators given in Eqs. (9)–(10) must be met. Thus, the parameter fault signals φaj,i kand φbj,i kcan be generated in a similar way as φri kin (13): φaj,i k=(0 if aj,i k∈[aj,i, aj,i]⇒no fault 1 otherwise (14) φbj,i kis computed as φaj,i kbut considering the parameter estimations bj,i kand the bounds [bj,i, bj,i] in (14). A fault is detected when one fault signal φaj,i kor φbj,i kis activated, i.e. its value equals 1. The isolation of the fault benefits from the time delays that characterize the reach dynamics as proposed in (Puig and Blesa, 2013). Indeed, the effect of a fault in the limnimeter Li(∆i k) in the estimation of the level in the limnimeter Lj(ˆyj k) is delayed by τi,j . For instance, when a fault ∆i koccurs, the temporal residual fault signal φri kand the parametric fault signals φaj,i kand φbj,i k∀i= 1,· · · , n should be activated in the first place. In the case of parametric fault signals, the effect of the fault ∆i kis more direct on the estimation of parameter ai,i k. Next, the effect of the fault will be propagated to the nearest measurement point j, which will affect the estimation of the level and the parameters. As the effect of the fault ∆i kis attenuated in the propagation, it might be observed only in the nearest measurement points. The estimation of the grey-box parameters is carried out by considering sliding windows of size Nw. Therefore, an extra delay between the fault occurrence and the effect in the fault signals is often present. By fulfilling the persistent excitation condition, the delay in the fault detection might be reduced using a smaller Nw. However, this improvement might be lost at the expense of the requirement of larger bounds on the parameters. It is thus necessary to reach a satisfactory trade-off between Nwand the detection time. 5. MODELLING AND FAULT DETECTION RESULTS 5.1 Modelling Real data from the limnimeters and discharges are resynchronized by considering a sampling time of 1 minute. Then, the delays between each part of the CFr are estimated according to the characteristics of the system (the values are in minutes): τ="0 78 115 79 0 37 116 37 0 #(15) According to these delays, the input and output vectors uk|τ∈R9and yk|τ∈R9are built by considering the following inputs and outputs: u1 k=QC k,y1 k=LC kfor Cuinchy; u2 k=QA k,y2 k=LA kfor Aire-sur-la-Lys; and u3 k=QF k,y3 k=LF kfor Fontinettes. To determine the minimum required size of the time window Nw, data from five consecutive days starting from October 30, 2013 (Thursday) are considered. Fig. 3 depicts the discharges and the levels in Cuinchy, Aire-sur-la-Lys and Fontinettes. The level of excitation of the input signals are computed for these data and depicted in Fig. 3.e for Nw= 960 (16 hours), Nw= 1020 (17 hours) and Nw= 1440 (24 hours). There is a peak of poor excitation level only for Nw= 960. For Nw= 1020 and Nw= 1440, the excitation levels are suitable and always equal to the maximum value 50. Figure 4 depicts the level of excitation of input signals pexcit according to Nw. It is shown that the minimal length of the window is Nw= 963. Thus, we consider that the minimal length of the window is Nw= 1020 corresponding to 17 hours. That means that at least 40 % of the data of Nwhas to correspond to periods with lock operations. The same data are used to identify a model for each window with Nw= 1020. The levels ˆ Li kare estimated as outputs of the grey-box model. The real measurements Li kand the estimated ˆ Li kare depicted in Fig. 5 in blue and dashed red lines, respectively, for each of the three limnimeters. Notice that these values are relative to the NNL, i.e. Li k=0 corresponds to the NNL in Fig. 5. The Nash-Sutcliffe (Ei) and correlation (Ri) coefficients are computed and their average values for the three -0.2 -0.1 0 (b) L C [m] -0.3 -0.2 -0.1 0 (c) L A [m] -0.4 -0.3 -0.2 -0.1 0 (d) L F [m] -30 -15 0 15 (a) Q [m 3 /s] 0 1 2 3 4 5 0 20 40 60 (e) Pexcit Time [days] Fig. 3. (a) Discharges in Cuinchy (blue), Aire-sur-la-Lys (red) and Fontinettes (magenta). Water levels in: (b) Cuinchy. (c) Aire-sur-la-Lys. (d) Fontinettes. (e) Level of excitation of input signals, for Nw= 960 (blue), Nw= 1020 (red) and for Nw= 1440 (black). 900 920 940 960 980 1000 1020 0 5 10 15 20 25 30 35 40 45 50 N w Pexcit Fig. 4. Minimal level of excitation of input signals pexcit according to the window length Nw. -0.2 -0.1 0 (a) L C [m] -0.3 -0.2 -0.1 0 (b) L A [m] -0.4 -0.3 -0.2 -0.1 0 (c) L F [m] 0 1 2 3 4 5 70 80 90 100 (d) E C , R C [%] Time [days] Fig. 5. Water levels in: (a) Cuinchy. (b) Aire-sur-laLys. (c) Fontinettes. Measured levels: blue solid line; estimated levels: red solid line with Nw= 1020. (d) Fit coefficients for Cuinchy. limnimeters are given in Table 1 by considering the time window sizes Nw= 1020 and Nw= 1440. The quality of the models is confirmed in both cases even if it seems better for the largest time window. A time window of 24 hours (Nw= 1440) covers a whole day of navigation at each step time, which seems to be the main reason of the observed effectiveness of the models. Table 1. Average values of Nash-Sutcliffe (Ei) and correlation (Ri) coefficients Nw= 1020 Nw= 1440 Limnimeter Ei[%] Ri[%] Ei[%] Ri[%] LC88 94 95 95 LA85 92 87 93 LF70 84 76 87 Finally, the identified parameters of the grey-box models for both time window sizes are depicted in Fig. 6 in red and blue lines, respectively. Their minimum and maximum values are computed to determine the intervals of the ai,i kparameters. These values are depicted in dashed lines and are summarized in Table 2. As expected, the larger the time window is, the smaller the intervals are. These intervals will be used to detect and isolate faults. Table 2. Boundaries on the grey-box models Nw= 1020 Nw= 1440 Parameter min max min max a11 0.88 0.95 0.89 0.95 a22 0.76 0.92 0.8 0.89 a33 0.58 0.81 0.61 0.74 0.85 0.9 0.95 1 (a) a 1,1 0.75 0.8 0.85 0.9 0.95 (b) a 2,2 0 1 2 3 4 5 0.5 0.6 0.7 0.8 0.9 (c) a 3,3 Time [days] Fig. 6. Values of the ai,i kparameters and determined thresholds: (a) a1,1. (b) a2,2. (c) a3,3. 5.2 FDI in limnimeters Levels from five consecutive days starting from November 12, 2013 (Tuesday) are considered in order to test the proposed FDI approach. Three faulty scenarios are created by adding artificial faults to the real measurements. Fault f1corresponds to a constant fault of -8 cm on the level LC. Fault f2consists in an intermittent fault that becomes constant over time with a magnitude of 1.5 cm on the level LA. This fault lasts until the end of the considered period of time. Fault f3is an intermittent fault with a magnitude of 5 cm on the level LFduring a bounded time. The features of the three faults are summarized in Table 3. The intermittent faults are generated according to Gaussian random signals, which means that two consecutive scenarios cannot be identical. Hence, 500 simulations based on -0.3 -0.2 -0.1 0 (a) L C [m] -0.3 -0.2 -0.1 0 (b) L A [m] 0 1 2 3 4 5 -0.4 -0.3 -0.2 -0.1 0 (c) L F [m] f 1 f 2 f 3 Time [days] Fig. 7. Water levels in: (a) Cuinchy. (b) Aire-sur-la-Lys. (c) Fontinettes. Measured levels in blue solid line (freefault case), in red dashed line (impacted by faults). the Monte Carlo approach are performed to determine the average, minimum and maximum fault detection delays τD. One of them is depicted in Fig. 7 with the three measured levels in blue solid line (free-fault case) and red dashed line (with faults). The time occurrence and the duration of the faults are indicated with black arrows. Due to the dynamics of the CFr and the magnitude of the faults, their detection is not obvious. Table 3. Considered faults Fault Magn. [cm] Occurrence (dd:hh:mm) Duration [min] f1-8 01:16:44 200 f21.5 02:14:44 2016 f35 00:01:24 300 The FDI approach is performed by considering the ai,i k parameters and the predefined thresholds (see Table 2). These parameters change during the simulation as depicted for the 25 first simulations and Nw= 1020 in Fig. 8. The parameter a1,1crosses the upper threshold after the occurrence of the fault f1, which allows detecting the occurrence of one fault. The isolation is done by considering which parameter is affected by the fault. Here, the fault f1is isolated because the limnimeter LCis impacted. The detection and isolation of faults f2and f3is achieved by considering the parameters a2,2and a3,3, respectively. The faults are detected when these parameters cross one of the bounds. It can be observed that the generation of the intermittent faults based on random signals has an impact on the estimation of the grey-box parameters. The detection delays τDdepend clearly on the dynamics of the intermittent faults. However, the faults f1and f3can be detected after 4 hin average, and only the parameter residual fault signals φaj,i kare activated. The detection delay τDof f2is larger due to its very gradual occurrence. The average, minimum and maximum detection delays τD are computed by considering 500 simulations and the sizes Nw= 1020 and Nw= 1440. They are presented in Table 4. For a constant fault f1, a smaller length of the time window leads to smaller detection delays. For intermittent faults, the time window with length Nw= 1440 leads to a better average performance, even if the minimum time delays are perceptibly worse. Table 4. Detection delay of each fault [min] Nw= 1020 Nw= 1440 Fault λτDmin τDmax τDλτDmin τDmax τD f1217 217 217 247 247 247 f21196 1107 2367 1155 1107 1247 f3243 147 387 221 157 447 The knowledge of the navigation schedule is taken into account and leads to Nw= 1440 (24 hours). All the dynamics of a navigation day are considered. This value allows meeting the required level of excitation of the inputs and the detection of constant and intermittent faults with very small magnitudes considering real data. The performance of the FDI approach remains good even if a smaller time window (Nw= 1020) can improve it in some cases. It is shown that the detection and isolation of constant and intermittent faults can be performed by dealing with the grey-box model parameters. 0.7 0.75 0.8 0.85 0.9 (b) a 2,2 0 1 2 3 4 5 0.3 0.45 0.6 0.75 0.9 (c) a 3,3 0.85 0.9 0.95 1 (a) a 1,1 Time [days] f 1 f 2 f 3 Fig. 8. Value of the ai,i kparameters (blue) and determined thresholds (black) in the faulty case: (a) a1,1, (b) a2,2, (c) a3,3, for the 25 first simulations. 6. CONCLUSION A grey-box model-based diagnosis approach for sensor faults on the Cuinchy-Fontinettes reach is proposed in this paper. It allows detecting and isolating constant and intermittent faults by considering the evolution of the grey-box model parameters on sliding windows. Rules for the determination of the sliding window sizes based on the level of excitation of the input signals are proposed. Then, several fault simulations are realized, confirming the good performance of the proposed FDI approach. REFERENCES Akhenak, A., Duviella, E., Bako, L., and Lecoeuche, S. (2013). Online fault diagnosis using recursive subspace identification: Application to a dam-gallery open channel system. Control Engineering Practice, 21, 797–806. Bedjaoui, N. and Weyer, E. (2011). Algorithms for leak detection, estimation, isolation and localization in open channels. Control Engineering Practice, 19, 564–573. Blesa, J., Horv´ath, K., Duviella, E., Puig, V., Bolea, Y., Rajaoarisoa, L., and Chuquet, K. (2014). Modelbased sensor supervision in inland navigation networks: Cuinchy-Fontinettes case study. Journal of Maritime Research, 11(2), 81–88. Blesa, J., Puig, V., and Bolea, Y. (2010). Fault detection using interval LPV models in an open-flow canal. Control Engineering Practice, 18, 460–470. Duviella, E., Rajaoarisoa, L., Blesa, J., and Chuquet, K. (2013). Fault detection and isolation of inland navigation channel: application to the Cuinchy-Fontinettes reach. 52nd IEEE Conference on Decision and Control, Florence, Firenze, Italy, 10-13 December. Hassanabadi, A.H., Shafiee, M., and Puig, V. (2016). UIO design for singular delayed LPV systems with application to actuator fault detection and isolation. International Journal of Systems Science, 47, 107–121. Horv´ath, K., Blesa, J., Duviella, E., Rajaoarisoa, L., Puig, V., and Chuquet, K. (2014a). Sensor fault diagnosis of inland navigation system using physical model and pattern recognition approach. IFAC WC, Cape Town, South Africa, 24-29 August. Horv´ath, K., Duviella, E., Blesa, J., Rajaoarisoa, L., Bolea, Y., Puig, V., and Chuquet, K. (2014b). Gray-box model of inland navigation channel: application to CuinchyFontinettes reach. Journal of Intelligent Systems. Horv´ath, K., Petreczky, M., Rajaoarisoa, L., Duviella, E., and Chuquet, K. (2014c). MPC of water level in a navigation canal - the Cuinchy-Fontinettes case study. ECC, Strasbourg, France, 24-27 June. Litrico, X. and Fromion, V. (2004). Simplified modeling of irrigation canals for controller design. Journal of Irrigation and Drainage Engineering, 130, 373–383. Ljung, L. (1999). System identification : theory for the user (2nd Edition). Prentice Hall, Upper Saddle River. Nabais, J.L., Mendon¸ca, L.F., and Botto, M.A. (2012). Sensor fault tolerant architecture for irrigation canals. 10th Portuguese Conference on Automatic Control 1618 July 2012, CONTROLO2012, Funchal, Portugal. Nash, J.E. and Sutcliffe, J.V. (1970). River flow forecasting through conceptual models part I: a discussion of principles. Journal of Hydrology, 10(3), 282–290. Pocher, O.L., Duviella, E., Bako, L., and Chuquet, K. (2012). Sensor fault detection of a real undershot/overshot gate based on physical and nonlinear black-box models. SAFEPROCESS’12, Mexico, August 29-31, 8. Pocher, O.L., Duviella, E., and Chuquet, K. (2011). Sensor fault detection in a real hydraulic system using a classification approach. ICINCO’11, Noordwijkerhout, Netherlands, 28-31 July. Puig, V. and Blesa, J. (2013). Limnimeter and rain gauge FDI in sewer networks using an interval parity equations based detection approach and an enhanced isolation scheme. Control Engineering Practice, 21(2), 146–170. Segovia, P., Blesa, J., Horv´ath, K., Rajaoarisoa, L., Nejjari, F., Puig, V., and Duviella, E. (2017a). Fault detection and isolation in flat navigation canals. In 4th International Conference on Control, Decision and Information Technologies, 1–6. Segovia, P., Rajaoarisoa, L., Nejjari, F., Puig, V., and Duviella, E. (2017b). Decentralized control of inland navigation networks with distributaries: application to navigation canals in the north of France. In American Control Conference (ACC), 2017, 3341–3346. IEEE.