Mahjoub El Mountassir
email: m.elmountassir@isgroupe.com

Gilles Mourot
email: gilles.mourot@univ-lorraine.fr

Slah Yaacoubi
email: s.yaacoubi@isgroupe.com

Didier Maquin
email: didier.maquin@univ-lorraine.fr

Damage Detection and Localization in Pipeline Using Sparse Estimation of Ultrasonic Guided Waves Signals

Keywords: Sparse estimation, damage detection and localization, structural health monitoring, environmental and operational conditions. 

Damage detection and localization in pipeline using sparse estimation of ultrasonic guided waves signals

Mahjoub El Mountassir, Gilles Mourot, Slah Yaacoubi, Didier Maquin

INTRODUCTION

Pipelines are very critical structures, especially those used for transporting oil, gas and other chemical substances. To ensure better working conditions of these structures, they must be continuously monitored. Structural health monitoring (SHM) systems were proposed to tackle this issue [START_REF] Farrar | An introduction to structural health monitoring[END_REF]. They aim at detecting, localizing and estimating the degree of severity of damage in the structure. Damage detection is performed by comparing the reference signals obtained from the healthy state and the current signal. For pipeline structure, the signals are generally acquired using Ultrasonic Guided Waves (UGW) technique [START_REF] Lowe | Defect detection in pipes using guided waves[END_REF]. UGW are stress waves which propagate through a medium and are guided by the structure boundary. These waves can travel in all directions ensuring hence a volumetric coverage and they interact with the structure discontinuities (damage, welds, structure ends, etc.).

The task of comparison between the reference signal and the current signal is not easy to achieve because the healthy state of the structure could vary due to the changes in the environmental and operational conditions (EOCs) (e.g. temperature, humidity, vibration loads, etc.) [START_REF] Sohn | Effects of environmental and operational variability on structural health monitoring[END_REF]. The effects of these EOCs could be similar to those produced by damage. This would result in false warnings. The differentiation between the aforementioned types of changes is a challenging task. Some methods were proposed to compensate the effects of these EOCs namely BSS (Baseline Signal Stretch) and OBS (Optimal Baseline Selection) [START_REF] Croxford | Efficient temperature compensation strategies for guided wave structural health monitoring[END_REF]. However, damage detection with these methods is based on a simple subtraction between the reference signal and current signal which is not reliable. Data-driven methods were also proposed to tackle this issue. For example, [START_REF] Eybpoosh | An energybased sparse representation of ultrasonic guided-waves for online damage detection of pipelines under varying environmental and operational conditions[END_REF] have proposed a supervised method based on sparse representation of UGW signals which can discriminate between damage and variation in EOCs. However, this method requires the use of signals from damage state which are not usually available. Besides, the fact of having reference signals with limited range of EOCs in the training step does not guarantee the robustness of the proposed method in the case of abrupt variation in EOCS. [START_REF] Liu | Robust ultrasonic damage detection under complex environmental conditions using singular value decomposition[END_REF] have proposed an unsupervised damage detection method using Singular Value Decomposition (SVD) of the matrix of collected signals. It was developed on the premise that the effect of damage and the effect of EOCs will be represented in different singular vectors. In this case, damage is detected by observing a jump of the mean in the right singular vectors. But the question here is in which singular vector the jump can be observed and how can we automatically detect this jump. Also, this can be only done in the case where the following hypotheses are fulfilled. Firstly, it supposes that damage occurs abruptly, which is not always true because in real cases, damage may develop progressively during a long period of time. Secondly, EOCs should be constantly changing so a jump cannot be observed in their associated singular vectors, otherwise it will be considered as indication of presence of damage. In the present case, temperature changes, for example, could have a clear trend, so the latter hypothesis might be not verified.

In this study, since signals from healthy state are the only available information which can be provided in the training step, the proposed method for damage detection is based on a novelty detection technique. To deal with variation in EOCs, we consider a learning method in non-stationary environment. This method consists in estimating the current signal using only few signals among the reference database. Physically, if the current signal is from a healthy state it will be estimated using few reference signals with similar or very close EOCs, the others being discarded. As all identification methods, the proposed approach is based on the assumption that the database of reference signal contains sufficient variation of EOCs. Otherwise, an update of the reference database is necessary to ensure viable damage detection. Such development is not implemented in the present study. To localize the damage, we suggest applying the sparse estimation on a sliding window over the damaged signal. This is motivated by the fact that when dealing with UGW the effect of damage is only local on the signal.

In the next section, the proposed methodology for damage detection and localization in pipeline structure is presented. In section 3, the procedure for collecting the database of healthy and damaged signals is explained. Also, the preprocessing of these signals is exposed. Section 4 is devoted to discussions on the obtained results. Finally, section 5 concludes the paper.

METHODOLOGY

Sparse estimation

The idea behind using sparse estimation of the current signal is that an undamaged signal should be well estimated by reference signals which have similar EOCs. Initially, an estimation of the current signal 𝒙 ∈ ℝ 𝑚 by the matrix of reference signals 𝐶 ∈ ℝ 𝑚×𝑛 could be provided by minimizing the quadric error given below:

J(𝜽) = ‖𝐶𝜽 -𝒙‖ 2 2 (1) 𝜽 ̂= arg min 𝜽 J(𝜽) (2)
where 𝜽 ∈ ℝ 𝑛 is the vector of regression coefficients, 𝑚 and 𝑛 are the number of samples and the number of reference signals respectively.

The optimal regression coefficients 𝜽 ̂ provide an estimation of the current signal by all the reference signals. But, the current signal is measured in specific EOCs while the reference signals are generally acquired in a wide range of EOCs. Hence, such estimation could give rise to an overfitting. This will jeopardize damage detectability because the regression model will tend to minimize the estimation error for a damage signal which should be very high compared to that of a reference signal. Besides, the coefficients should be positive to avoid the compensation between the reference signals. To overcome this issue, sparsity on the regression coefficients must be defined. It can be obtained by adding a regularisation term to equation (2) as following:

𝜽 ̂= arg min 𝜽 (J(𝜽) + λ ‖𝜽‖ 1 ) (3) subject to 𝜽 ≥ 𝟎
This optimization problem can be solved using the lasso method with non-negative constraint [START_REF] Efron | Least angle regression[END_REF]. The optimal solution will select from the reference signals a subset which will be used to estimate the current signal and assign zero to the others. The tuning parameter λ controls the power of regularization and it must be chosen in the training step. Generally, it is very difficult to determine a value of this parameter. It requires the use of cross validation technique. In this context, some studies have shown that in the case of positively correlated signals, least squares method with nonnegative constraint is an efficient regularization technique [START_REF] Meinshausen | Sign-constrained least squares estimation for high-dimensional regression[END_REF][START_REF] Slawski | Non-negative least squares for high-dimensional linear models: Consistency and sparse recovery without regularization[END_REF]. In other words, we can say that the method has a self-regularization property which means that it automatically generates a regularization term. Thus, there is no need to determine the tuning parameter λ.

𝑆 = 1 𝑛 𝐶 𝑇 𝐶 (5)
It must be noted here that the farther the parameter σ is to zero the higher the self-regularizing effect.

The non-negative least squares (NNLS) problem is defined as:

𝜽 ̂= arg min 𝜽 J(𝜽) [START_REF] Bertsekas | Nonlinear programming[END_REF] subject to 𝜽 ≥ 0

To get an optimal solution of this problem, the Karush-Kuhn-Tucker (KKT) conditions must be satisfied. These conditions are defined as follows [START_REF] Bertsekas | Nonlinear programming[END_REF]:

𝜽 ̂ ≥ 0 (7) ∇J(𝜽 ̂) ≥ 0 (8) ∇J(𝜽 ̂) 𝑇 𝜽 ̂= 0 (9)
where ∇ denotes the gradient of J(𝜽 ̂). Its expression is given by:

∇J(𝜽 ̂) = 2𝐶 𝑇 (𝐶𝜽 ̂-𝒙) ( 10
)
The constrained minimization problem described in equation ( 6) can be solved using different numerical approaches. The most common ones are: active set method, projected quasi Newton approach, principal block pivoting method and interior point method [START_REF] Chen | Nonnegativity constraints in numerical analysis[END_REF]. A comparison between these approaches is beyond the scope of this study. However, we have chosen to use the active set method because a recursive version of this method can be easily implemented. This will be used later for the purpose of localization of damage. Active set method divides the constraints into active and passive ones. Actually, active constraint refers to zero regression coefficient, otherwise it is considered as passive. At each iteration, the algorithm calculates the solution by the least squares method on the passive set. Then, it tests if the new regression coefficients satisfy the KKT conditions. Afterthat, it updates the set of active coefficients until a final set is found. [START_REF] Lawson | Solving least squares problems[END_REF] have proposed an algorithm to solve NNLS problem which is an active set method. Let us denote by 𝐶 𝑃 the matrix associated with the constraints in the passive set 𝑃. Similarly, let 𝒔 be a vector of same length as 𝜽. Let 𝒔 𝑷 denote the subvector with indexes from 𝑃 and 𝒔 𝑹 the subvector with indexes from the active constraints 𝑅. The algorithm is described hereafter:

Algorithm 1: Non Negative Least Squares Inputs: 𝐶, 𝒙 and ϵ a real value fixing the tolerance for stopping criterion.

Initialization: 𝑃 = ∅, 𝑅 = {1,2, … , 𝑛}, 𝜽 ̂= 0, 𝒘 = 1 2 ∇J(𝜽 ̂) While: 𝑅 ≠ ∅ and (𝑚𝑎𝑥 𝑖∈𝑅 (𝑤 𝑖 ) > ϵ) do 𝑗 = arg 𝑚𝑎𝑥 𝑖∈𝑅 (𝑤 𝑖 )
Include the index 𝑗 in 𝑃 and remove it from 𝑅

𝒔 𝑷 = [(𝐶 𝑃 ) 𝑇 𝐶 𝑃 ] -1 (𝐶 𝑃 ) 𝑇 𝒙, 𝒔 𝑹 = 0 While: min (𝒔 𝑷 ) ≤ 0 do 𝛼 = 𝑚𝑖𝑛 𝑖∈𝑃 [ 𝜃 ̂𝑖 (𝜃 ̂𝑖-𝑠 𝑖 )
] where 𝑠 𝑖 ≤ 0

𝜽 ̂= 𝜽 ̂+ 𝛼(𝒔 -𝜽 ̂)
Move to 𝑅 all indices in 𝑃 such that 𝜽 ̂𝑗 = 0

𝒔 𝑷 = [(𝐶 𝑃 ) 𝑇 𝐶 𝑃 ] -1 (𝐶 𝑃 ) 𝑇 𝒙, 𝒔 𝑹 = 0 end while 𝜽 ̂= 𝒔, 𝒘 = 1 2 ∇J(𝜽 ̂)
end while

It was proved that this algorithm converges within a finite number of iterations. However, an exact number of these iterations cannot be known in advance and it could be very high.

Damage indicators

To detect the presence of damage, an indicator must be derived from the sparse estimation of the current signal. If the signal is damaged, we expect that the estimation error will be very high compared to that of an undamaged signal. Hence, a damage indicator could be chosen as the quadratic estimation error J(𝜽 ̂) given by the solution of NNLS problem described in equation ( 6):

The sparsity of the regression coefficients (𝑆𝑟) could be also used as damage indicator [START_REF] Eybpoosh | Sparse representation of ultrasonic guided waves for robust damage detection in pipelines under varying environmental and operational conditions[END_REF]. It is defined as the ratio of the number of zero coefficients in 𝜽 ̂ to the total number of reference signals:

𝑆𝑟 = 𝑛 -card(𝑃) 𝑛 × 100 (11)
where card(𝑃) denotes the cardinal of the final set of the passive constraints 𝑃. In fact, a damaged signal will be characterized by a very small number of passive coefficients.

Hence, the value of 𝑆𝑟 will converge to unity. While, for a reference signal, the value of 𝑆𝑟 will be theoretically a little far from unity. Furthermore, this sparsity will increases as the damage size increases. Thus, the damage severity can be assessed using this factor.

A joint damage indicator 𝐼 that account for both the quadratic estimation error and the sparsity ratio could be established to maximize the chances for damage detectability as proposed in [START_REF] Boracchi | Novelty detection in images by sparse representations[END_REF]. It is defined by the vector:

𝐼 = (J(𝜽 ̂), 𝑆𝑟) (12)
To ensure automatic damage detection, a threshold must be defined. For this purpose, in the case of one dimensional damage indicator (i.e. J(𝜽 ̂) or 𝑆𝑟), an empirical distribution for damage indicators, calculated from reference signals, has to be established. Afterthat, the threshold can be chosen by fixing the confidence limit of the distribution. The current signal is considered as damage if its damage indicator exceeds the value of the specified confidence limit. When using a bivariate indicator as defined in equation ( 12), the signals will be represented in a two-dimensional space. In this case, the threshold can be chosen by defining a metric to measure how close the damage indicator vector of the current signal to those of the reference signals. This metric can be calculated using multivariate statistical tools such as the Mahalanobis Square Distance [START_REF] Johnson | Applied multivariate statistical analysis[END_REF].

Damage localization using recursive NNLS

To get the position of damage, we suggest calculating the sparse estimation of the damaged signal on a sliding window over the matrix of reference signals. The already found regression coefficients, obtained for the entire signal, are no longer suitable. Hence, the solution of NNLS problem must be obtained for every sliding window. Taking into consideration the fact that adding and removing one sample at a time do not change significantly the regression coefficients found in the previous window, it is of interest to apply a recursive version for the solution of the NNLS.

In the case of classical least squares method (unconstrained), a method to update the regression coefficients already exists [START_REF] Björck | Numerical methods for least squares problems[END_REF]. It is known as Recursive Least Squares (RLS). Since the NNLS solving method requires the use of least squares on the passive set, the RLS could be applied to update NNLS solution. However, the non-negativity of the new regression coefficients is not guaranteed. In other words, the KKT conditions could be violated. Thus, to get an optimal solution of the recursive NNLS problem, these KKT conditions must be verified. Afterthat, a pivoting exchange between the passive set and the active set could be eventually needed [START_REF] Mosesov | Adaptive Non-negative Least Squares with Applications to Non-Negative Matrix Factorization[END_REF]. Under the assumption of minimal set changes, we expect that only a single pivoting exchange is made between the active set and the passive set. Hence, block pivoting is not necessary in this case.

The calculation of the RLS on a sliding window includes two successive steps: updating and downdating [START_REF] Zhang | Some implementation aspects of sliding window least squares algorithms[END_REF]. which not does represent a difficult issue. However, when this matrix is not invertible, QR decomposition might be used to enforce the stability of the proposed recursive algorithm [START_REF] Björck | Numerical methods for least squares problems[END_REF]. The QR decomposition could be also used to calculate recursively the vector 𝒔 𝑷 in the case of adding and/or removing a column from the matrix 𝐶 𝑘+𝐻-1 𝑃 which results from the single pivoting exchange made between the active set and the passive set.

DATA COLLECTION AND PREPROCESSING

Data collection

The specimen considered in this study consists of a tube with 6.4 m length. It was placed in laboratory conditions where temperature fluctuates between 19°C and 26°C during the monitoring period. This variation is due to the seasonal temperature changes. The used sensor can excite two separate guided waves modes which are: torsion and flexion at five different frequencies: 14, 18, 24, 30 and 37 kHz. For the sake of brevity, only torsional mode with an excitation frequency of 14 kHz is considered in the present study. Figure 1 illustrates the pipeline configuration. The excitation signal has been removed by the acquisition system (portion in the middle of the signal). The three echoes with the highest amplitude represent multiple reflections from the end of the pipe. These echoes have to be removed also from the original signal because they are not useful for dam-age detection. Besides, their amplitudes are very high, hence they can mask small changes induced by the presence of damage.

Damage was created by removing material from the inside of the pipeline in six increasing steps in order to simulate corrosion growing within the structure. Figure 1 in bottom left shows the defect in the last step. The dimensions and the form of the damage were set randomly as in the case of real corrosion which results from a natural aging of the pipeline.

At the end of the monitoring period, a total of 236 signals were collected where 207 ones were undamaged and 29 signals were acquired from a damage state. Table 1 summarizes information related to the collected database.

Temperature effect

The analysis of the collected signals in the time domain shows that the reference signals underwent drastic changes in both amplitude and phase. Since temperature is the only environmental factor that varied during the measurements, the observed changes in the collected signals are due to the variation of this factor. This can be clearly noticed in Figure 3 which illustrates a zoom over a part of two reference signals acquired at different temperature.

Damage effect

Figure 4 illustrates an example of a damage signal and a reference one. Note that, to clearly see the differences between the two signals, a zoom was displayed at the position of damage. This figure shows that the effect of damage is drift in the amplitude with a significant change in the phase.

These results seem to be very similar to those observed in the case of temperature variation. However, the changes in the damaged signal are not uniform as in the case of temperature variation. We can clearly observe in the beginning that the two signals are almost identical while at the end, the amplitude of damaged signal drops suddenly and the time arrival of the two signals changes. This confirms the hypothesis that the effect of damage is local on the signal while the temperature variation impacts the entire signal. Even though, damage could not be detected using classical damage sensitive features proposed by [START_REF] Rizzo | Wavelet-based outlier analysis for guided wave structural monitoring: application to multi-wire strands[END_REF]. Let us consider for example the RMS as a damage sensitive feature. As a reminder, the RMS of a signal 𝒙 = {𝑥 𝑖 } , 𝒙 ∈ ℝ 𝑛 is given by:

RMS(𝒙) = √ 1 𝑛 ∑ 𝑥 𝑖 2 𝑛 𝑖=1 (19)
Result in Figure 5 shows that we cannot discriminate between damage and reference signals using the RMS. The variation of temperature is responsible of the great dispersion of RMS values. The dispersion is mainly associated with the amplitude of the signals since RMS uses only amplitude values at each signal sample.

RESULTS AND DISCUSSIONS

Damage detection

One hundred and forty signals were used as a database of reference signals and the others served for the test of the proposed method. Due to the seasonal changes in temperature, the reference signals used for the test could be obtained with different variation of temperature which might be not included in the database of reference signals. Each new signal (𝒙) is estimated by sparse model 𝜽 ̂ obtained by solving the constrained minimization problem described in equation ( 6). Here, the sparse estimation is calculated on the entire signal. Let us verify that the reference signals are positively correlated which is the condition of using NNLS rather than the nonnegative lasso. In this study, the minimum of the covariance matrix was calculated for the reference signals. It is equal to 0.0414 which is strictly positive. Thus, the condition of positively correlated signals is fulfilled as stated in inequality (4). To confirm this, Principal Component Analysis (PCA) was firstly performed on the matrix of reference signals [START_REF] Jolliffe | Principal Component Analysis and Factor Analysis[END_REF]. Afterthat, the correlation coefficient between the first two representative principal components and the reference signals was calculated. The result is presented in Figure 6.This result shows that the reference signals are well represented by the first principal component (the correlation coefficient exceeds 0.88). This can be explained by the fact that there is a strong positive correlation between the reference signals. The variation of the correlation between the reference signals and the second principal component is probably due to the changes in EOCs namely the temperature. Since the condition of positively correlated signals is now strongly fulfilled, an estimation of a new measured signal can be provided using NNLS. Figure 7 shows an example of an original undamaged signal and its estimation. It can be clearly noticed that the two signals are almost the same which proves that the sparse estimation provides a very good estimation of the undamaged signal. Figure 8 illustrates the error calculated for the original signal showed in Figure 7. This error presents a random behaviour, except in the zones where the echoes from the end of pipe and excitation signal were deleted. When a defect is present, the error will tend to deviate from the random behaviour to a deterministic one. Also, a damaged signal will be badly estimated by the reference signal; in this case the error will be very high.  The quadratic estimation error J(𝜽 ̂) of the reference signals presents very low variation if we compared them with the result of RMS shown in Figure 7.

 The values of J(𝜽 ̂) increase as the size of damage increases. Thus, J(𝜽 ̂) can be used to assess the severity of damage.

Fig. 10. Quadratic estimation error J(𝜽 ̂) of undamaged and damaged signals.

As described in section 2.2 the sparsity ratio (𝑆𝑟) can be also used for damage detection. Figure 11 shows the 𝑆𝑟 for the monitoring signals including damaged and reference signals.

The variation of the 𝑆𝑟 for the reference signals is important while the 𝑆𝑟 of damaged signals presents a relatively low variation. Besides, the values of 𝑆𝑟 increase as the size of damage increases. However, a threshold cannot be defined to ensure automatic damage detection without triggering false alarms. Thus, J(𝜽 ̂) outperforms 𝑆𝑟 in terms of damage detectability. In this case, the use of a bivariate indicator which was defined in equation ( 12) will be useless. The quadratic estimation error J(𝜽 ̂) is sufficient to ensure reliable damage detection.

Fig. 11. Sparsity ratio of undamaged and damaged signals.

Damage localization

To determine the position of damage, the idea which is described in section 2.3, is to apply the sparse estimation on a sliding window over the signal (i.e. samples). Thus, by moving the window through the signal, each part of it will be examined separately. In this case, the window width may influence the final result. It can be determined based on the desired precision of localization. In the present study, it was set initially at forty samples. Note that larger window width could be tested to investigate its influence on the result of localization. This window was moved by one sample per step. At each step, we calculate the quadratic estimation error J(𝜽 ̂) using recursive NNLS algorithm. At the end, we obtain a value of J(𝜽 ̂) at each moving window over the damaged signal.

Figure 12 shows the found position of the created defect using the proposed method for damage localization. To avoid interpretation of false damage position, the damage signal was truncated between the first two arrivals of the end of pipe echoes. This Figure shows that the position of damage which corresponds to the maximum of J(𝜽 ̂) is 3 m. The real position of damage, as shown in Figure 1, is 2.6 m. Thus, we have an error of localization of 0.4 m. This error can be explained by the fact that the window width (𝐻) induces an error of localization. This error can be calculated by knowing the propagation velocity (V) of the waves in the pipeline and the sampling frequency (F). Knowing that this error has been minimized by affecting the value of J(𝜽 ̂) to the position of the half of the window. The error of localization 𝐸 is given as follows:

𝐸 = 𝑉 𝐹 × ( 𝐻 2 ) ( 20
)
The propagation velocity of the waves in the pipeline is equal to 3200 m/s. The sampling frequency is fixed by the acquisition system at 195 kHz. Finally, the position of damage 𝐷 is: 𝐷 = 3 ± 0.41 𝑚 (21) This result is coherent with the value of the real position of damage indicated in Figure 1. Finally, it is worth noting that the result of localisation could be optimized by assigning the position of damage to a value of J(𝜽 ̂) where we estimate that the error is significant and not necessarily to the maximum of J(𝜽 ̂).

CONCLUSION

In this paper, a method for damage detection and localization in pipeline structures was proposed. It is based on sparse estimation of the measured signals by the reference signals. A simplified form of this estimation using the non-negative least squares is investigated. It is based on the fact that the acquired UGW signals are highly correlated. The sparsity helps to enhance damage detectability because a damaged signal will have a high estimation error compared to that of a healthy signal. Besides, it can face the problem of variation in EOCs provided that the database of reference signals contains large variations of these EOCs.

The detection of defect was ensured by calculating the quadratic estimation error J(𝜽 ̂) on the entire current signal. While the localization of damage was established by implementing a recursive version of the sparse estimation on a sliding window over the damaged signal.

As a perspective of this work, an update of the database of reference signals could be considered in the case where these signals present limited variation in EOCS. This can be achieved by adding to this database new healthy signals with unknown variation in EOCs. Also, the proposed method has to be validated on operational pipeline serves in different EOCs.

Algorithm 2 :

 2 The former adds a new sample to the window and the latter removes a sample which is excluded from the sliding window. Let us denotes by 𝐻 the window width and by 𝑘 the index of the sample. The algorithm starts first with a window which comprises samples from 𝑘 to 𝑘 + 𝐻 -1. This window corresponds to the matrix of reference signals 𝐶 𝑘+𝐻-1 and the current signal 𝒙 𝑘+𝐻-1 . The solution in this case is denoted as 𝜽 ̂𝑘+𝐻-1 .When adding a new sample (𝒄 𝑘+𝐻 , 𝑥 𝑘+𝐻 ) where 𝐶 𝑘+𝐻 = ( 𝐶 𝑘+𝐻-𝜽 ̂𝑘+𝐻-1 + 𝜷 𝑘+𝐻 (𝑥 𝑘+𝐻 -𝒄 𝑘+H 𝜽 ̂𝑘+𝐻-1 ) (15) These values will be then used to perform the downdating operation by removing the sample (𝒄 𝑘 , 𝑥 𝑘 ). It is expressed as follows: 1 = 𝜽 ̂𝑘+𝐻 -𝜷 𝑘+H-1 (𝑥 𝑘 -𝒄 𝑘 𝜽 ̂𝑘+𝐻 ) (18) Finally, the algorithm for recursive NNLS can be described hereafter: Recursive Non Negative Least Squares Inputs: data 𝐶 𝑘+𝐻-1 and 𝒙 𝑘+𝐻-1 Update 𝐬 𝐏 = arg min 𝜃 ‖𝐶 𝑘+𝐻-1 𝑃 𝜽 -𝒙 𝑘+𝐻-1 ‖ 2 2 using RLS by adding sample (𝒄 𝑘+𝐻 , 𝑥 𝑘+𝐻 ) as in equation (15) and removing (𝒄 𝑘 , 𝑥 𝑘 ) as in equation (18). If (𝑚𝑖𝑛 (𝒔 𝑷 ) ≤ 0) then remove index 𝑗 = 𝑎𝑟𝑔 𝑚𝑖𝑛 𝑖 ( 𝑠 𝑖 ) from 𝑃 and add it to 𝑅. ̂), if (𝑚𝑎𝑥 (𝒘) ≥ 0) then move index 𝑗 = arg 𝑚𝑎𝑥 𝑖 ( 𝑤 𝑖 ) into 𝑃 While (KKT conditions are not fulfilled) do Calculate 𝒔 𝑷 = [(𝐶 𝑘+𝐻-(𝒔 𝑷 ) ≤ 0) then remove index 𝑗 = 𝑎𝑟𝑔 𝑚𝑖𝑛 𝑖 ( 𝑠 𝑖 ) from 𝑃 and add it to 𝑅. ̂), if (𝑚𝑎𝑥 (𝒘) ≥ 0) then move index 𝑗 = 𝑎𝑟𝑔 𝑚𝑎𝑥 𝑖 ( 𝑤 𝑖 ) into 𝑃 end whileNotice that the window width (𝐻) must be chosen in order to ensure the invertibility of the matrix ((𝐶 𝑘+𝐻-

Fig. 1 .

 1 Fig. 1. Pipeline configuration showing the used sensor and the created damage. The specimen has been monitored during a period of almost 3 months. Each week, multiple measurements were scheduled. At each measurement, five signals were acquired in the morning and at the evening in order to capture temperature changes during the day and to investigate its effects on the collected signals. An example of an acquired signal is illustrated in Figure 2.

Fig. 2 .

 2 Fig. 2. Example of an ultrasonic guided wave signal excited with torsion mode (frequency: 14 kHz).

Fig. 3 .

 3 Fig. 3. Zoom over two reference signals acquired at different temperatures.

Fig. 4 .

 4 Fig. 4. Zoom at the position of damage for a reference signal and a damaged signal.

Fig. 5 .

 5 Fig. 5. RMS of the monitoring signals, the damaged signals are coloured in red with a circle marker.

Fig. 6 .

 6 Fig. 6. Correlation coefficients of the reference signals and the first two principal components.

Fig. 7 .

 7 Fig. 7. Original undamaged signal and its estimation.

Fig. 8 .

 8 Fig. 8. Estimation error of the reference signal shown in Fig 7.

Figure 9

 9 Figure9illustrates an example of the estimation error of a damaged signal. It can be noticed that orders of the magnitude of the estimation error for a damaged signal are far from that of a reference signal.

Fig. 12 .

 12 Fig. 12. Quadratic estimation error 𝐉(𝜽 ̂) of a damaged signal showing the position damage.

Table 1 . Characteristics of the collected data

 1

	Monitoring period	3 months
	Reference state	207 signals
	Damage state	6 increasing defects
		(29 signals)
	Temperature	19 °C  26 °C


César Tlakaélel

Martínez Villegas

Didier Theilliol
email: didier.theilliol@univ-lorraine.fr

Flor Lizeth

Torres Ortiz
email: ftorreso@iingen.unam.mx

César T Martínez-Villegas
email: cesar-tlakaelel.martinez-villegas@univ-lorraine.fr

Lizeth Torres

Review of two non-centralized observer-based diagnosis schemes for interconnected systems

Keywords: Fault detection and isolation, Interconnected systems, Large-scale systems, Observer-based design

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The term "interconnection" is related somehow to information exchange between two or more entities. Depending on the particular context considered, these entities can be very different, ranging from biological organisms to electronic devices or even abstract concepts like equations.

With the advance of science and technology, we are now able to build and control complex interconnected systems. In fact, our modern society relies heavily on those systems every day with important tasks such as water and electricity distribution, transportation, communication, etc. All of these systems are comprised within what is known as "critical infrastructure systems" (Goetz and Shenoi, 2008).

Interconnected systems are prone to faults or failures that can modify their internal dynamics in an undesirable way. These faults can be viewed as input signals injected to either actuators, sensors or the process itself, but also as signals affecting the communication channels or the interconnection variables among subsystems [START_REF] Teixeira | Distributed fault detection and isolation resilient to network model uncertainties[END_REF].

Modern technological systems are equipped with robust control schemes to ensure their correct operation as well as to make them tolerant to disturbances, noise and/or faults [START_REF] Noura | Fault -tolerant Control Systems: Design and Practical Applications[END_REF]. Considering particularly the case of tolerance to faults or failures, these controllers are known as "fault-tolerant". One common strategy to achieve fault tolerance considers the fulfillment of two tasks: fault diagnosis and control reconfiguration [START_REF] Blanke | Diagnosis and Fault -Tolerant Control[END_REF]. Fault diagnosis concern is to obtain as much information about the fault as possible while control reconfiguration involves certain adaptation of the overall control scheme considering the information gathered during fault diagnosis step. An efficient fault diagnosis system is very important to achieve fault-tolerance since control reconfiguration strategies tend to get better with more information about faults affecting the system.

Control and fault diagnosis of large-scale interconnected networks are tasks with some design challenges due to the particular characteristics of each network. An engineer trying to design a control or fault diagnosis scheme for an interconnected network should take into account the order of the overall network (number of variables involved in the process), the complexity within dynamics of the components (nonlinearities, interconnection dynamics) and the availability of information (measurements, sensor location, communication capabilities). In addition, it is always a good idea to consider also a certain amount of scalability to ensure that with little extra effort, the overall system properties are maintained even if it is required to add more components to the network.

Typical fault diagnosis schemes known as centralized schemes assume that a single processing unit executes the fault diagnosis task and all the information related to the system is transmitted to it. The characteristics of largescale interconnected networks make these assumptions hard or even impossible to verify. Moreover, the communication and computational cost are high and the scalability of the overall system poor (the effort to include structural changes in the network suppose a complete redesign of the diagnosis scheme). Ways to deal with these difficulties can be found on non-centralized schemes, where the diagnosis task is shared among a number of diagnosis agents, and the information is not supposed to be available at a single point.

Quantitative observer-based non-centralized fault diagnosis schemes for large-scale interconnected systems has been under study in the last decade. A pioneering contribution was the work of [START_REF] Ferrari | A fault detection scheme for distributed nonlinear uncertain systems[END_REF], where a distributed fault detection scheme was proposed. This work was later improved to use overlapping state sets among subsystems by adding a consensus type term to the estimation equations [START_REF] Ferrari | Distributed fault diagnosis with overlapping decompositions: An adaptive approximation approach[END_REF]. Complete distributed fault diagnosis schemes can be found in [START_REF] Boem | Distributed fault detection and isolation of continuous-time non-linear systems[END_REF] and [START_REF] Ferrari | Distributed fault detection and isolation of large-scale discrete-time nonlinear systems: An adaptive approximation approach[END_REF] for continuous and discrete-time systems respectively. A common aspect of all these works is that all states are assumed to be measured. This assumption was dropped in [START_REF] Boem | Distributed fault diagnosis for continuous-time nonlinear systems: The input-output case[END_REF], where only partial state measurements were considered. A more recent paper covers also the problem of time delays to further improve the applicability of the scheme [START_REF] Boem | A distributed networked approach for fault detection of large-scale systems[END_REF]. All these works consider adaptive approximation of the interconnection function and dynamic fault decision thresholds to improve the performance of the diagnosis system.

An alternative to the adaptive approximation is the use of unknown input observers (UIOs). The fault diagnosis algorithm, in this case, uses a bank of UIOs in a "generalized observer scheme" (GOS) fashion [START_REF] Chen | Robust Model-Based Fault Diagnosis for Dynamic Systems[END_REF]. The application of UIOs for non-centralized schemes considers also the interconnection function decoupling, with the advantage that it does not need any assumption over such interconnection function. These ideas were successfully used by using monolithic systems to diagnose state faults [START_REF] Shames | Distributed fault detection for interconnected secondorder systems[END_REF] and later extended by [START_REF] Teixeira | Distributed fault detection and isolation resilient to network model uncertainties[END_REF] to cover sensing and communication channel type of faults and using local models, thus improving the scalability of the scheme. This paper is organized as follows. Section 2 presents a description of a general class of interconnected systems; in particular, interconnected double-integrator networks are described. Section 3 presents a review of different architectures used for fault diagnosis with particular emphasis on those based on the use of observers, either using monolithic or non-monolithic system models. Section 4 includes a qualitative comparison including the advantages and disadvantages of using different architectures. Section 5 presents some conclusions.

INTERCONNECTED SYSTEM DESCRIPTION

The system under study, in general, is a non-linear multiinput multi-output dynamic system described by a set of differential equations in state-space form. This system referred as monolithic or global system

Σ : ẋ(t) = f (x(t), u(t), d(t), t), y(t) = g(x(t), u(t), d(t), t), (1)
where x(t) ∈ R n , u(t) ∈ R m , y(t) ∈ R p are the state, input and outputs vectors of the system and d(t) ∈ R q is a vector representing some external unknown signals such as disturbances or noise.1 It is important to note that this model is a general one since no assumptions are made over functions f and g, nonetheless, each FDI scheme makes some assumptions to simplify its task. A common assumption used on FDI schemes for large-scale distributed interconnected systems is that system (1) can be modeled also with a set of N models known as local subsystems or local models [START_REF] Blanke | Diagnosis and Fault -Tolerant Control[END_REF] Σ I : While for linear time-invariant systems there are some techniques to decompose the overall system into purely local and purely interconnection dynamics (refer to [START_REF] Langbort | Distributed control design for systems interconnected over an arbitrary graph[END_REF]), this decomposition for the general nonlinear case is usually not possible or very difficult to accomplish. So, it is common that FDI schemes designed for nonlinear models consider another type of decomposition detailed later in this section.

   ẋI (t) = f loc I (x I (t), u I (t), d(t), t) +f int I (x I (t), ūI (t), d(t), t), y I (t) = g I (x(t), u(t), d(t), t),
Within general description (2), it can be also considered the case of a collection of N interacting subsystems, for example, robots or electronic devices; where each one has its own set of actuators, sensors, and local dynamics (see Fig. 1). A control scheme is also considered (either centralized, decentralized or distributed). It is worth noting that the interaction between subsystems can be physical (where subsystems are physically connected to others) or induced by the control scheme, for example, in flocking of multi-agent systems, where a collection of mobile agents are supposed to move all together in a given direction while avoiding contact with other robots and some possible obstacles in their way [START_REF] Olfati-Saber | Flocking for multi-agent dynamic systems: Algorithms and theory[END_REF].

From a fault diagnosis point of view, one particular case of special interest is presented in [START_REF] Boem | Distributed fault diagnosis for continuous-time nonlinear systems: The input-output case[END_REF]. Authors of this work have considered a MIMO uncertain system whose dynamics are given by Σ :

ẋ = A x + f (x, u) + η x (x, u, t), y = C x + η y (x, u, t), (3)
where x ∈ R n , u ∈ R m , y ∈ R p are the state, input and outputs vectors, the term A x + f (x, u) represents the known dynamics with A being an n × n matrix and

f : R n × R m → R n .
C is a p × n matrix representing the nominal output. η x and η y are unknown functions representing model uncertainties, perturbations and/or noise that might affect the dynamics of the system.

The distributed FDI scheme proposed by [START_REF] Boem | Distributed fault diagnosis for continuous-time nonlinear systems: The input-output case[END_REF] assumes that system (3) is decomposed into N subsystems Σ I with the following structure

Σ I : ẋI = A I x I + f I (x I , u I ) + g I (C I x I , u I , z I ), y I = C I x I + η y, I (x I , u I , t), (4)
where x I ∈ R n I , u I ∈ R m I and y I ∈ R p I are the local states, inputs and outputs respectively, z I ∈ R q I is a vector comprising the interconnection variables. A practical way to represent subsystems (2) or (4) and their interactions is through the mathematical concept of graph [START_REF] Mesbahi | Graph Theoretic Methods in Multiagent Networks[END_REF]. A graph G(V, E) is a mathematical entity that is formed by two sets: a set of nodes V and a set of edges E (see Fig. 2). Two ways to use this concept in the design of non-centralized diagnosis schemes are trough what are called as structural and fundamental graphs. A structural graph is a graph in which nodes represent the variables of the monolithic system and the edges represent interactions among such variables. In a fundamental graph, the nodes represent subsystems (2) and the edges the interaction between them [START_REF] Ferrari | Distributed fault detection and isolation of large-scale discrete-time nonlinear systems: An adaptive approximation approach[END_REF].

Interconnected Double-Integrator Systems

A class of interconnected systems that is of particular interest for control design and therefore for fault diagnosis is the one comprising the so-called interconnected secondorder or interconnected double-integrator systems. Taking as a starting point a fundamental graph such as Fig. 2. Each node represents a subsystem whose dynamics can be modeled by a second-order state-space model

Σ I : ẋ1, I = x 2, I , ẋ2, I = u I + v I , (5)
where I = 1, . . . , N , x 1, I and x 2, I are the local states of subsystem I, v I : R → R is a function of time representing the action of an external input on subsystem I. u I is a function representing the interaction between subsystem I and its neighborhood ν I (comprising the subsystems that share a link with subsystem I following the fundamental graph of the network). A general expression for functions u I can be found in [START_REF] Shames | Distributed fault detection for interconnected secondorder systems[END_REF].

u I = -κ I x 2, I + J∈ν I w IJ [(x 1, J -x 1, I ) + γ(x 2, J -x 2,I )], (6)
where κ I , γ ∈ R + and w IJ > 0 is a weight parameter which somehow represents the "strength" of the interaction between nodes I and J.

The corresponding monolithic system computed from subsystems (5) and local inputs( 6) is

ẋ = A x + B v, (7)
where

x = [x 1, 1 , . . . , x 1, N , x 2, 1 , . . . , x 2,N ] T is the overall state, and v = [v 1 , . . . , v I ]
T is a vector including all external inputs v I . A and B are 2N × 2N and 2N × N matrices respectively, with the following structure

A = 0 N ×N I N -L -γL -K , B = 0 N ×N I N ,
where 0 N ×N is a N × N matrix with all zero entries, I N is the N × N identity matrix, K = diag([κ 1 , . . . , κ I ]) and L is the weighted Laplacian matrix of the network whose entries are L IJ = -w IJ ∀I = J and L II = I =J w IJ , I, J = 1, . . . , N .

The monolithic system ( 7) is useful to represent a number of real world networks such as a collection of mobile robots, where the local states x 1, I and x 2, I can be interpreted as the position and velocity of each one of them and the interaction is not a physical one but induced through control law (6); and also an electrical power grid where each node represents a bus of the network, local states are the phase angle and its derivative and the external inputs v I represents power injected or drawn from bus I. In the case of a power grid, the interaction between subsystems (buses) is a physical one since they are supposed to be interconnected following an interconnection graph or diagram [START_REF] Shames | Distributed fault detection for interconnected secondorder systems[END_REF].

ARCHITECTURES FOR FAULT DETECTION AND ISOLATION

According to [START_REF] Boem | Distributed fault detection and isolation of continuous-time non-linear systems[END_REF], an architecture is "a combination of hardware and software used to implement and execute the control or estimation task". This definition refers to the physical system used to compute the elements of fault detection (in the case of quantitative fault diagnosis schemes), the communication network or devices used to transmit valuable information obtained from the system altogether with a design strategy that fulfills the requirements of the overall system.

Control, estimation and fault diagnosis architectures can be classified into two general classes: centralized architectures and non-centralized architectures:

• In a centralized architecture, the overall control, estimation or fault diagnosis is performed by a single entity with all the available knowledge (inputs, outputs, model). Related to model-based strategies to perform such tasks, a single model that represents the dynamics of the system is assumed. Fig. 3 shows a pictorial representation of different architectures used in fault diagnosis schemes. An important thing to note is that the layout of each architecture is independent of the system to be diagnosed. One can design a fault diagnosis scheme based on a centralized architecture even if the system to be diagnosed is represented as a distributed or interconnected one, a way to do so taking as a starting point a collection of state-space systems is to form an overall state and after doing so, use algebraic relations to form an overall set of differential equations. Also, non-centralized architectures can be used on monolithic systems, this is done by performing a decomposition over the monolithic model as commented previously.

Another type of architecture that is not covered by the classification we have done so far is one called hierarchical architecture. This type of architecture is not different from the other ones in terms of available information and system decomposition. Hierarchical architectures are just mixtures of other architectures on different layers of diagnosis, so, first, a non-centralized architecture might be used to achieve some "local" fault diagnosis decision which can be communicated later to another fault diagnosis agent that is designed to process all local fault diagnosis decisions to compute a "global" one.

Observer-based non-centralized FDI methods for interconnected systems

As we have pointed out, system decomposition is a useful tool to overcome the complexity challenge that is caused by the number of interactions in a large-scale interconnected system. In principle, with a good decomposition we can reduce the computational burden of each computing unit running a part of the overall fault diagnosis algorithm ("distributing" computational effort over a number of machines), nonetheless the design procedure should take care some other restrictions, first of all, the decomposition should be made in such a way that some properties for each subsystem are ensured, for example, detectability or observability properties.

A reasonable assumption is that the structural graph of the monolithic system under consideration should be weakly connected, this implies that every variable influence or is influenced by another one. If this assumption is dropped then there are groups of variables naturally decoupled from others thus forming perfectly decoupled subsystems, in the sense that one is completely independent of others.

In this case, one can simply solve the problem of FDI independently for each subsystem (using any architecture if its particular conditions are satisfied). One straightforward consequence of this assumption is that after decomposition, there will be some subsystems whose dynamics will be influenced by variables of other subsystems (interconnection variables), in a more precise way and following equation (4), g I (C I , x I , u I , z I ) ≡ 0 for at least one I ∈ {1, . . . , N }. This fact result being a challenge for observer design since the interconnection function acts as an input to a particular subsystem and is not assumed to be known, so it should be taken into account either in the observer design itself or by a suitable threshold definition as explained in the following sections.

3.2 Observer-based non-centralized FDI using monolithic models.

One way to deal with the existence of unknown interconnection functions g I in observer-design is to leave aside the decomposition itself.

The fault diagnosis scheme is designed taking as a starting point a monolithic model of the system with the only difference compared with centralized architectures being that in the non-centralized one, the fault diagnosis task is divided among N fault diagnosis agents. The noncentralized nature of this scheme is that the available system information (inputs and outputs) is assumed to be different for each agent. Also, it is considered that no agent can have access to all inputs and outputs from the monolithic system.

An example of this idea applied to a network of secondorder interconnected systems (5) can be found on [START_REF] Shames | Distributed fault detection for interconnected secondorder systems[END_REF] article. The authors of this work have taken as a starting point the monolithic system (7) to prove that a bank of Unknown Input Observers (UIO) can be locally designed to perform fault diagnosis.

Each fault diagnosis agent is composed of 2 (|N I |+1) UIOs (where |N I | is the cardinality of the set containing all the nodes that share a link with node I, this is, nodes J ∈ {J = 1, . . . , N }|w IJ = 0 and is called neighborhood set of I or just neighborhood of I), each of one decoupled from a fault signal affecting one state of one subsystem in the neighborhood of I (following the fundamental graph).

The model used by each fault diagnosis agent is

ẋ = A x + B v + E i, I f i, I (t), y I = C I x, (8)
where y I is the output information available an node I which is suppose to include independent measurements from both states of node I and its neighborhood, so matrix C I should be a 2 (|N I |+1)×2 N full row rank matrix, f i (t) is a fault function while matrix E f, I is a column vector called fault distribution vector (or matrix if applicable)

and it has all 0 entries but one 1 at a position related to the state from which it is expected for the observer to be decoupled, for example to decouple a fault in the first state of the first subsystem and following the same overall state definition as in (6), vector E fx 1, 1 = [1, 0, . . . , 0] T .

Each UIO from a local diagnosis agent I is based on the well-known LTI structure (Chen and Patton, 2012

) żi = F i z i + T i B v + K i y i xi = z i + H i y I (9)
where the design matrices F i , T i , K i , H i are chosen so that the following conditions are satisfied.

F i = (A -H i C i A -K 1, i C I ), T i = (I -H i C I ), K i = K 1, i + F i H i , 0 = (H i C I -I) E i, I
K 1, i should be chosen so that the all the eigenvalues of F i lie on the left-hand side of the complex plane.

The fault decision logic is made following a generalized observer scheme (GOS) in which every residual in the bank is compared to a threshold. A fault f i (t) is detected and isolated if the following conditions are fulfilled:

r i (t) < Θ fi r j (t) ≥ Θ fj ( 10
)
where the residual signals are

r i (t) = y I -C I xi (11)
It is worth noting that even when the models used by each local fault diagnosis agent are based on the monolithic system, the strategy presented in this section can be considered as a decentralized one since the authors did not assume that all the information is available at a single place and there is no information exchange among different local fault diagnosis agents (LFDAs). It is also important to note that by a proper choice of matrices E i, I this scheme can be extended to diagnose sensing faults or communication channel type of faults modeled as a changes in parameters w IJ in (6) [START_REF] Teixeira | Distributed fault detection and isolation resilient to network model uncertainties[END_REF].

In spite of the reach of this scheme, two desired characteristics of FDI for large-scale interconnected systems aren't achieved. This scheme still has applicability problems due to its high need for computational resources since each LFDA should estimate 2 N (|N I | + 1) states (for example the 7th LFDA of the fundamental graph on Fig. 2 should estimate 156 states since N = 13 and |N i | = 6). Scalability is also an issue since if it is needed to add more nodes to the network, the model at each LFDA agent should be modified to include the dynamics of these new systems, making it difficult to scale much further.

A way to overcome the high computational burden issue can be to reduce the number of agents supervising each node. If we take a close look at the architecture used, each node is supervised by the agents related to every node on its neighborhood and the LFDA related to himself. This fact has the added advantage that the overall FDI scheme has a lot of redundancy making it robust to failures on LFDAs. A trade-off can be made by sacrificing this redundancy to reduce the total number of states that need to be estimated. As pointed out by [START_REF] Teixeira | Distributed fault detection and isolation resilient to network model uncertainties[END_REF] "the objective is to select a minimum number of observer nodes that covers all the nodes in the network". Solutions to this already studied problem called a "set cover problem" can be found in [START_REF] Grandoni | A note on the complexity of minimum dominating set[END_REF].

Another way to reduce complexity and enhance scalability is to decompose the monolithic system and use local models at each FDI agent, such idea has been studied by [START_REF] Teixeira | Distributed fault detection and isolation resilient to network model uncertainties[END_REF] and applied on second-order interconnected networks and also on a more general class of continuous and discrete time nonlinear interconnected systems [START_REF] Boem | Distributed fault diagnosis for continuous-time nonlinear systems: The input-output case[END_REF] and [START_REF] Ferrari | Distributed fault detection and isolation of large-scale discrete-time nonlinear systems: An adaptive approximation approach[END_REF] respectively). where G(V, E) is a fundamental graph of a second-order interconnected system. The dynamic behavior of a subsystem I associated with subgraph ĜI ( VI , ÊI ) is given by:

FDI using distributed models

φ ĜI (t) = A ĜI φ ĜI (t) + B ĜI v ĜI (t) + ψ ĜI (t), ( 12
)
where φ ĜI is the local state related to the subsystem induced by subgraph ĜI formed by the states of the nodes V ĜI arranged first the first states of all nodes and then the second states (φ

ĜI = [x 1, 1 , . . . , x 1, | VI | , x 2, 1 , . . .], x 2,| V| ).
A ĜI and B ĜI are matrices with the same structure as in ( 7) but related to the subgraph. As one can see, the only structural difference between the system (12) and the monolithic system ( 7) is that the local one has an extra term ψ ĜI which act as an external signal and represents, in this case, the interconnection function of subsystem I. This interconnection function is not assumed to be known due to either lack of knowledge of the function itself or lack of knowledge about the arguments (interconnection states) of the function.

The fault diagnosis scheme is similar to the one presented by [START_REF] Shames | Distributed fault detection for interconnected secondorder systems[END_REF] but based on local models ( 12) making the assumption that the interconnection function ψ ĜI has the following structure

ψ ĜI = E ĜI ψ I (t), ( 13
)
where E ĜI is a full column rank matrix where each column has all zero entries but one 1 at the position where the node affected by the external signal is located. Considering the effect of faults in local dynamics, the dynamics of each local subsystem are given by

φ ĜI (t) = A ĜI φ ĜI (t) + B ĜI v ĜI (t) + E ĜI E i, I ψ I (t) f i, I (t) . ( 14
)
From this expression it is easy to verify that an UIO structure ( 9) is applicable considering E e i, I = E ĜI E i, I as an extended unknown input distribution matrix. The existence of such UIO, in this case, depends on the information that we suppose to be available at subsystem I. As in the case of monolithic models, this information is related to local input v ĜI and output signals y I = C I φ ĜI which are both states of a subset of nodes V s ⊆ VI .

A question that remains open is how to chose the decomposition of the monolithic system into local subsystems (12).

To give an answer to this question it is convenient to start thinking in the smallest order subsystem possible, that is to consider the local model of only one node I (eqn.

(5)). In this case φ ĜI = [x 1, I , x 2, I ] and the system model based on structure ( 12) is

φ ĜI (t) = A ĜI φ ĜI (t) + E ĜI ψ I (t)b (15)
where

A ĜI = 0 1 0 0 , E ĜI = 0 1 ,
and ψ I (t) = u I (t) defined on eqn. ( 6) and considering a fundamental graph G such that ĜI ⊆ G. The fault diagnosis scheme proposed by [START_REF] Shames | Distributed fault detection for interconnected secondorder systems[END_REF] and extended by [START_REF] Teixeira | Distributed fault detection and isolation resilient to network model uncertainties[END_REF] would assume in this case that an LFDA with two observers should be designed to detect and isolate faults in the local subsystem I4 . The fault distribution matrices related to state faults are E 1, I = [ 1 0 ] related to a fault in the first state of node I and E 2, I = [ 0 1 ] to diagnose faults appearing in the second state. It is simple to see that the observer designed to be insensitive to faults with distribution matrix E 1, I will in any case be insensitive also to faults appearing in the second state since E 2, I = E ĜI and E e 1, I = 1 0 0 1 E e 2, I = 0 0 1 1 so, one cannot apply a decision logic such as in (10) to detect and particularly to isolate faults. This impossibility arises from the use of over-simplified local models (since in the previous section we discussed the problem of fault diagnosis using monolithic systems, which is a completely opposite problem from decomposition perspective). To overcome this problem we have to incorporate more nodes and interactions to the local models, for example, nodes in the vicinity of a given node I. In that case, the problem will be that some observers will decouple at the same time faults in the second state and interconnection functions, thus making impossible to isolate such faults. To be able to diagnose the same faults as in the monolithic system case, we have to add to local models, nodes from all the vicinity of a given node I but also nodes that are linked to those nodes in the vicinity (2-hop neighbors of I), and in any case the measurements related to them also.

The authors of the work taken as a base to this section ( [START_REF] Teixeira | Distributed fault detection and isolation resilient to network model uncertainties[END_REF]), had proved that the graph that minimizes both measurement transmission cost and computational burden, given that the same weight is assigned to both measurement transmission and computational burden, is one called proximity graph of node I. The proximity graph P I ⊆ G includes on its node set, nodes from the vicinity of a given node I and every node that shares a link with at least one of them, and its edge set is composed of all the edges linking nodes in the vicinity of node I plus the edges incident to at least one of the nodes in the vicinity.

3.4 FDI using distributed models and adaptive techniques to approximate the interconnection function g I An alternative to the use of UIOs to decouple the effects of the interconnection function in residual dynamics is the use of adaptive approximation techniques to learn or approximate these effects. This work has been developed and extended by different authors starting with the work of [START_REF] Zhang | A robust detection and isolation scheme for abrupt and incipient faults in nonlinear systems[END_REF] that was applied to distributed networks to diagnose faults appearing in state equations [START_REF] Boem | Distributed fault diagnosis for continuous-time nonlinear systems: The input-output case[END_REF]. This scheme also has been successfully used to diagnose sensor faults following a distributed architecture [START_REF] Reppa | Distributed sensor fault diagnosis for a network of interconnected cyber-physical systems[END_REF].

The fault diagnosis scheme, in this case, is designed using a hierarchical architecture with two layers: a first layer to obtain some local fault diagnosis decisions and a second layer to determine a global fault diagnosis decision as well as to coordinate the actions of agents in the first layer [START_REF] Boem | Distributed fault diagnosis for continuous-time nonlinear systems: The input-output case[END_REF].

The first layer is composed by N local fault diagnosers each of one considering one local subsystem (4) after a decomposition of a monolithic system. Local fault diagnosis on each of these fault diagnosers is achieved by implementing two algorithms with different objectives:

I = J W (I, J) s A (s J ) J xJ + f (s J ) J (x J , u J ) +ĝ (s J ) J (y J , u J , z J , φJ ) + L (S J ) J (y J -ŷJ ) (18)
• An algorithm is formed by a bank of N F I observers called Fault Isolation Estimators (FIE), where N F I is the cardinality of a predefined fault set containing all faults considered for fault diagnosis.

• Each observer on the bank will be designed considering the effect of one particular fault in the fault set, so, if we consider the difference between output estimates and the real output of the system as residual, we will have each residual being insensitive to just one fault. The bank of observers acts following a generalized observer scheme.

It is worth saying that this architecture considers noise and uncertainties. Also, the adaptive approximation of the interconnection function takes some time to adapt its parameters and it can still be imprecise. So, each residual will not be completely zero if it is supposed to be insensitive to a particular fault. This fact adds some com-plexity to fault decision tasks, encouraging the necessity of designing some adaptive thresholds to ensure that no false-positive alarms appear. These thresholds are designed by assuming some boundaries in the uncertainty, noise, and interconnection functions.

• The Global Fault Diagnoser has two tasks to perform:

to coordinate the actions of each local fault diagnoser (to enable either fault detection or fault isolation), and to compute a general fault decision based on local ones.

ABOUT THE CHARACTERISTICS OF FDI ARCHITECTURES.

Recalling that due to the characteristics of large-scale interconnected networks, a "good" FDI scheme should be able to achieve fault diagnosis with some degree of performance, but also deal with three particular challenges: computational burden, communication costs, and scalability. Thus the general performance of each FDI scheme is easily related to its ability to do fault diagnosis with the least amount of computational and communication cost and as easily scalable as possible.

As has been briefly covered in this paper, each FDI architecture presents its own design challenges and has different conditions to be fulfilled by the system to be viable for an application. These challenges become even more demanding if we consider also the restrictions that we pointed out in the previous paragraph.

The first work that we reviewed in this paper uses a monolithic model for observer design. The distributed nature comes from the decomposition of the output vector, reducing the communication cost. Another advantage of this architecture is that since the model has more explicit information about the relations between variables of the system, it is easier to fulfill design requirements. Also, FDI design does not have the necessity to deal with unknown interconnection functions. On the other hand, this architecture requires more or less the same amount of computational effort of a centralized architecture since each LFD need to estimate all the states of the network. Another consequence of the use of monolithic models is that it does not scale well. To keep its diagnosis capabilities, one should change the model on every LFD in the face of changes on the structural graph of the network.

To both improve scalability and reduce the amount of computational effort needed to perform fault diagnosis, local models can be used. As we pointed out, system decomposition always induces the appearance of interconnection functions not assumed to be known a priori. In order to perform fault diagnosis, we have to deal with the interconnection functions. Two ways to deal with this problems are the use of UIOs to fully decouple the effects of this interconnection networks and adaptive approximation to, as the name suggests approximate such function.

Since UIOs decouples completely the effect of the interconnection function, the UIO based scheme needs only to assume that the interconnection function has the form E ĜI ψ I (t), so we only have to assume that we know a priori which variables are directly affected by the interconnection function. Once the fault is decoupled, a GOS is deployed to both detect and isolate faults with the same bank of observers. Another important feature of this scheme is that it does not need a global fault diagnoser to coordinate local ones. Compared to the use of UIOs, the use of adaptive approximation is more demanding in terms of computational requirements because every LFDA needs to run the approximation algorithm, the general observer to deal with fault detection and the bank of observers to perform fault isolation. The main advantage of the use of nonlinear observer with adaptive approximation is that it has more flexibility in terms of applicability, this scheme is suited to a more general class of interconnected networks affected by model uncertainties and noise. The use of UIOs has more restrictive design restrictions, for example, it is suited to be applied on a class of LTI systems, with some observability (detectability) properties. A way to cope with these design restrictions, UIOs require relies on the use of more transmitted measurements.

As an additional remark, it is important to note that both schemes presented uses overlapping decomposition. The scheme based on UIOs does not relies on shared variables in any way, nonetheless this overlapping decomposition gives some degree of robustness of the overall FDI in the face of unreliable local fault diagnosis. The work developed over the use of nonlinear observers with adaptive approximation uses directly this decomposition by leveraging on the redundant estimation of shared variables by more than one LFDA. The local state estimation of at each LFDA is based on a consensus algorithm over such estimated shared variables.

CONCLUDING REMARKS AND PERSPECTIVES

Two different alternatives to board non-centralized fault diagnosis schemes for interconnected networks have been reviewed. These techniques are based on the use of unknown input observers and nonlinear observers with an adaptive approximation to deal with the effects of the interconnection function on the state estimation errors.

It has been discussed also some diagnosis challenges imposed by the nature of large-scale interconnected networks: availability of transmitted information, computational burden, and scalability. The most promising architecture to overcome these difficulties is one based on local models with the least amount of transmitted information. Nonetheless, diagnosis scheme design should consider that there is a compromise between information included in one model and the diagnosing capacities of an FDI based on such model. Thus model over-simplification should be avoided.

As future perspectives on the subject covered, there are still some open problems such as time delays, and system decomposition on nonlinear systems. Another question that remains open is regarding the applicability of these techniques on dynamical structure networks instead of fixed structure ones. For example, if we consider that the interconnection parameters are functions of time and/or internal states.

  of only local variables x I and local inputs u I and f int I is a function representing the interaction of local system I with other elements of the system not considered by local dynamics. Subsystems (2) are induced by partitions of vectors u, y, x 2 into N ≥ 1 sub-vectors. These partitions are {u I , I = 1, . . . , N }, {y I , I = 1, . . . , N } and {x I , I = 1, . . . , N }, where xI and ūI are the components of x and u not considered in x I and u I respectively.

Fig. 1 .

 1 Fig. 1. General distributed system.

3

  Terms A I x I + f I (x I , u I ) and C I x I represents the nominal dynamics and nominal output respectively while the term η y, I represents the uncertainty in the output equation (noise). Function g I comprises both interconnection dynamics and the corresponding local model uncertainty (a part of model uncertainty η x ).

Fig. 2 .

 2 Fig. 2. Directed graph.

Fig. 3 .

 3 Fig. 3. Architectures used for fault diagnosis.

  [START_REF] Teixeira | Distributed fault detection and isolation resilient to network model uncertainties[END_REF] have extended the results of[START_REF] Shames | Distributed fault detection for interconnected secondorder systems[END_REF] using UIOs to decouple both the effects of the faults and the effects of the interconnection signals (g I (C x I , u I , z I ) for a more general model) on state estimation errors. The authors used a subgraph ĜI ⊆ G,

Following the notation used in[START_REF] Boem | Distributed fault diagnosis for continuous-time nonlinear systems: The input-output case[END_REF], bold letters are used to denote variables related with the monolithic system (1), while non-bold letters are related to subsystems variables.

One should remember that local and global states, inputs and outputs are always functions of time, here and in the following, its explicit relation has been dropped for sake of clarity and brevity

The interconnection variables z I are the states of neighboring subsystems that influence the dynamics of subsystem I trough the function g I .

For simplicity, we have assumed that the faults to be diagnosed are only node faults, but there might also be edge faults affecting the system

The authors gratefully acknowledge LabCOM PHM-Factory and the financial support provided by the Research National Agency (RNA) of France. This work was supported by Consejo Nacional de Ciencia y Tecnología, México (CONACYT) through a scholarship jointly with French government.


Marcin Witczak
email: m.witczak@issi.uz.zgora.pl

Ralf Stetter
email: stetter@hs-weingarten.de

Mariusz Buciakowski
email: m.buciakowski@issi.uz.zgora.pl

Didier Theilliol
email: didier.theilliol@univ-lorraine.fr

Norbert Kukurowski

Design of diagnostic estimators for an automated guided vehicle

Keywords: Fault diagnosis, observers, automated guided vehicles, input estimation, force balance

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The main reason behind such a situation is that their application can lead to significant production effectiveness improvements. They allow to relief the personnel from time-consuming routine work and they can increase the efficiency and accuracy of production and service activities. AGVs can be used in hazardous environments and can reduce the risk of explosions and contaminations in certain environments such as coal mines or pharmaceutical production. In service applications, AGVs can enhance the availability, safety and security of industrial sites and infrastructure systems. However, in many potential applications AGVs are not yet used or the their possible potential in terms of flexibility and efficiency is not yet realized exploited. The causes for this non-satisfying situation are manifold. One of them is the limited maneuvering capability of AGVs with conventional steering systems such as classical Ackermann steering. Another one is the high cost of sensors on AGVs. Further causes also arise from the sensors of an AGV: very often sensor data filtering, sensor data plausibility assessment and sensor fusion are important challenges for developers and operators of AGVs. Finally, causes can also be a consequence of a less than sufficient flexibility, availability and reliability of current AGV designs.

This paper combines two main approaches to address this causes: the innovative AGV design described allows unlimited maneuvering capabilities and the virtual diagnostic sensor design enhance reliable sensor information without additional expenditures. The design of virtual sensors is one of the emerging streams in the area on Fault-Tolerant Control (FTC) and Fault Diagnosis (FD) [START_REF] Mahmoud | Active Fault Tolerant Control Systems: Stochastic Analysis and Synthesis[END_REF]; [START_REF] Witczak | Fault diagnosis and fault-tolerant control strategies for non-linear systems[END_REF]; [START_REF] Ding | Model-based fault diagnosis techniques: design schemes, algorithms and tools[END_REF]; [START_REF] Chen | Robust Model-based Fault Diagnosis for Dynamic Systems[END_REF]; [START_REF] Seybold | Towards robust predictive fault-tolerant control for a battery assembly system[END_REF]; [START_REF] Pazera | Robust multiple sensor fault-tolerant control for dynamic non-linear systems: application to the aerodynamical twin-rotor system[END_REF]. Many approaches have been reported to design virtual diagnostic sensor: observer-based [START_REF] Aouaouda | Discrete-time H / H-inf sensor fault detection observer design for nonlinear systems with parameter uncertainty[END_REF], Kalman filter-based [START_REF] Foo | A sensor fault detection and isolation method in interior permanent-magnet synchronous motor drives based on an extended Kalman filter[END_REF]; [START_REF] Pourbabaee | Sensor fault detection, isolation, and identification using multiple-model-based hybrid Kalman filter for gas turbine engines[END_REF], parameter identification-based [START_REF] Cai | Modelbased fault detection, estimation, and prediction for a class of linear distributed parameter systems[END_REF].

This paper proposes a novel approach to virtual sensor design of longitudinal forces as well as torques acting onto an AGV using a quadratic boundedness (QB) [START_REF] Alessandri | Design of state estimators for uncertain linear systems using quadratic boundedness[END_REF] approach, which makes it possible to include bounded disturbances acting onto the AGV. Another advantage is that it avoids unnecessary state estimation, which is fully available via measurements. From the mathematical AGV description viewpoint, the main profit is that the proposed approach makes it possible to exclude sophisticated tire models, which usually impair the performance of the approaches presented in the literature [START_REF] Kiencke | Automotive control systems[END_REF]; [START_REF] Rajamani | Algorithms for real-time estimation of individual wheel tire-road friction coefficients[END_REF]. Finally, the design procedure of the proposed approach is expressed in the form of Linear Matrix inequalities (LMIs), which can be solved using widely available computational packages.

The paper is organized as follows. Section 2 introduces essential preliminaries, which are necessary to undertake the problem being investigated. Section 3 describes AGV, which will be used in this the work. Section 3.1 presents its mathematical. Section 4 proposes a strategy for design virtual diagnostic sensor. Section presents application of the proposed strategy to AGV. Finally, the last section concludes the paper.

PRELIMINARIES

Let us consider the following discrete-time system

x k+1 = Ax k + Bu k + Bd k + W w k , (1) where x k ∈ X ⊂ R n is the state vector, u k ∈ U ⊂ R r stands for the input, d k ∈ R n
d is an unknown input, and w k ∈ E ⊂ R nw is an exogenous disturbance vector. Let V k = x T k P x k denote a Lyaponov candidate function. For an unforced u k = 0 and unknown input-free d k = 0 system (1), the following definitions are recalled: Definition 1. The system (1) is strictly quadratically bounded for all allowable

w k ∈ E, k ≥ 0, if V k > 1 ⇒ V k+1 -V k < 0, (2)
for any w k ∈ E. Definition 2. A set E x is a robust invariant set for the system (1) for all allowable w k ∈ E if

x k ∈ E x ⇒ x k+1 ∈ E x ,
(3) for any w k ∈ E.

It should be pointed out that the strict quadratic boundedness enables decreasing the value of Lyapunov function V k , i.e, it means that V k+1 < V k for any 1) is quadratically bounded as well as there exists at least one vector W w k = 0, then such quadratic boundedness is always strict [START_REF] Alessandri | Design of state estimators for uncertain linear systems using quadratic boundedness[END_REF]. Moreover, the notation of the quadratic boundedness can be expressed using the theory of invariant sets [START_REF] Alessandri | Design of state estimators for uncertain linear systems using quadratic boundedness[END_REF].

w k ∈ E when V k > 1. If (
The set E x = {x :

x T P x ≤ 1} is an invariant set for any w k ∈ E iff x k ∈ E x implies x k+1 ∈ E x . Thus, if V k > 1 then x k is outside an invariant set, and hence, by Definition 1 V k+1 < V k . This means that V k decreases until x k is outside E x .

ROBOT DESCRIPTION

This section describes the AGV developed at the University of Applied Sciences Ravensburg-Weingarten, which will be used in this the work (Fig. 1). It consists of four drive motors which are fastened on arms that may freely rotate. Each arm has an angular velocity encoder of each wheel, which allows independent use of each motor. The ability to use four independent motors results from a steering system that is already registered as a patent and is based on the concept to use the torque differences between wheels to steer four independent axles of a vehicle.

The developed AGV is able to turn around its own center and to drive directly in any direction without time-and space-consuming turning maneuvers. This characteristic is very important from the practical viewpoint while operating in small spaces. The semi-autonomous vehicle is controlled by a small PC-ALIX that it is connected to 4 EPOS motor controllers, one per wheel. For the communication between the controllers it uses the CANopen protocol. Each of the wheel contains a suspension to absorb possible different heights of the terrain and the impacts against the ground. The movement from the motor to the wheel is transmitted using two two belt driven pulleys. This transmission is a reduction system, because the torque required by the vehicle is more than the torque produced by the motor. The control of the vehicle occurs on different levels: manual driving, manual driving, autonomous driving mode. The control system implemented in AVG send the speed command to the drive motors as well as desired angles for the wheels to move AVG for to the desired position. Apart from the above mentioned angular velocity sensors, the robot is equipped with acceleration, velocity and yaw rate sensors, which are available via Bluetooth protocol. The principal steering system of AGV

F x = cos(δ f )(F x,f l + F x,f r ) + cos(δ r )(F x,rl + F x,rr )+ -sin(δ f )F y,f -sin(δ r )F y,r . (4)
Analogously, the lateral forces can be anlysed:

F y = sin(δ f )(F x,f l + F x,f r ) + sin(δ r )(F x,rl + F x,rr )+ + cos(δ f )F y,f + cos(δ r )F y,r , (5)
where the longitudinal wheel forces obey:

I xw ωfl = p f,l T -F x,f l R e , (6)
I xw ωfr = p f,r T -F x,f r R e , (7)
I xw ωrl = p r,l T -F x,rl R e , (8)
I xw ωrr = p r,r T -F x,rr R e , (9)
with the torque distribution coefficients p i,j ≥ 0 satisfying:

p f,l + p f,r + p r,l + p r,r = 1. ( 10
)
It should be also noticed that p i,j are assumed to be known parameters. Finally, the yaw rate dynamics is given by

I z ṙ = L f (sin(δ f )(F x,f l + F x,f r ) + cos(δ f )F y,f )+ + L a cos(δ f )(F x,f r -F x,f l ) + L r (sin(δ r )(F x,rl + F x,rr ) -cos(δ r )F y,r ) + L a cos(δ r )(F x,rr -F x,rl ). ( 11
)
Having a mathematical description of AGV, the objective of the subsequent part of this paper is to develop a set of virtual sensors enabling estimation of F x,f r , F x,f l , F x,rr , F x,rl and T based on the available measurement vector:

y = [r, ω f l , ω f r , ω rl , ω rr ] T , (12)
as well as lateral and longitudinal accelerations a x and a y .

VIRTUAL SENSOR DESIGN

The proposed virtual sensor design strategy starts with extracting the lateral forces F y,f and F y,r from ( 5) and ( 11), which yields

F y,f = 1 cos(δ f )(L f + L r ) (I z ṙ + p f,f l F x,f l + +p f,f r F x,f r + p f,rl F x,rl + p f,rr F x,rr + +mL r a y ) , (13)
F y,r = 1 cos(δ f )(L f + L r ) (-I z ṙ + p r,f l F x,f l + +p r,f r F x,f r + p r,rl F x,rl + p r,rr F x,rr +mL r a y ) , (14)
where

p f,f l = -sin(δ f )L f -sin(δ f )L r + L a cos(δ f ), (15) p f,f r = -sin(δ f )L f -sin(δ f )L r -L a cos(δ f ), (16) p f,rl = -2 sin(δ r )L r + L a cos(δ r ), ( 17
) p f,rr = -2 sin(δ r )L r -L a cos(δ r ), ( 18
) p r,f l = L a cos(δ f ), (19) p r,f r = L a cos(δ f ) (20) p r,rl = sin(δ r )L f -sin(δ r )L r + L a cos(δ r ), (21) p r,rr = sin(δ r )L f -sin(δ r )L r -L a cos(δ r ).
(22) Bearing in mind the fact that F x = ma x and then substituting ( 13) and ( 14) into (3.1) yield:

p r ṙ = F x,f l p x,f l + F x,f r p x,f r + + F x,rl p x,rl + F x,rr p x,rr + a x p x + a y p y (23)
where 30) Finally, the state space model of AGV is is given by is given by ( 6)-( 9) and ( 13)-( 14)

p r = I z sin(δ f -δ r ), (24)
p x,f l = - 1 2 L a sin(-δ r + 2δ f ) + 1 2 sin(δ r )L a + + cos(δ r )(L f + L r ), (25)
p x,f r = 1 2 L a sin(-δ r + 2δ f ) - 1 2 sin(δ r )L a + + cos(δ r )(L f + L r ), (26)
p x,rl = - 1 2 L a sin(δ f -2δ r ) - 1 2 sin(δ f )L a + L r cos(δ f -2δ r ) + cos(δ f )L f , (27)
p x,rr = 1 2 L a sin(δ f -2δ r ) + 1 2 sin(δ f )L a + + L r cos(δ f -2δ r ) + cos(δ f )L f , (28)
p x = -(L f + L r ) cos(δ f ) cos(δ r )m, ( 29
) p y = -m(sin(δ f ) cos(δ r )L r + cos(δ f ) sin(δ r )L f ). (
G ẋ = Bu + Ed, (31)
where

x = [r, ω f l , ω f r , ω rl , ω rr ] T , (32) with u = [a x , a y ] T .
(33) Finally, the unknown input, which has to be estimated by the virtual sensor is given by:

d = [F x,f l , F x,f r , F x,rl , F x,rr , T ] T ,
(34) Additionally, the system matrices are:

B =      p x p y 0 0 0 0 0 0 0 0      , ( 35
) E =      0 -p x,f l -p x,f r -p x,rl -p x,rr -p f,l R e 0 0 0 -p f,r 0 R e 0 0 -p r,l 0 0 R e 0 -p r,r 0 0 0 R e      . (36) G = diag(p r , 1, 1, 1, 1). ( 37
)
Finally, while all state variables [r, ω f l , ω f r , ω rl , ω rr ] T are measured, the output equation is given by

y = Cx, (38)
with C = I. To facilitate the implementation on an onboard device, the system (31) was discretized with the sampling time T s = 0.01[s] using the Euler methods, which leads to:

G k x k+1 = G k x k + B k u k + E k d k + W w k , (39)
with

G k = G(δ f,k , δ r,k ), B k = T s B(δ f,k , δ r,k ), ( 40
) E k = T s E(δ f,k , δ r,k ) (41)
where w k stands for an exogenous disturbance vector (which includes the discretization error) with known distribution matrix W , while

G(δ f,k , δ r,k ), B(δ f,k , δ r,k
) and E(δ f,k , δ r,k ) are obtained by substituting δ f,k and δ r,k into ( 35)-( 37), respectively.

For the purpose of further deliberations, it is necessary to underline the fact that all state variables of (39) are measured. Thus, contrarily to the approaches present in the literature (see, e.g., [START_REF] Witczak | An lmi approach to robust fault estimation for a class of nonlinear systems[END_REF] and the references therein), the attention is focused on estimating d k only. Indeed, as the estimation of the state vector is unnecessary, it will simplify the proposed design procedure.

Finally, to tackle the virtual sensor design problem, the following novel adaptive estimator structure is proposed:

dk+1 = dk + L(G k x k+1 -G k x k -B k u k -E k dk ), ( 42
)
where dk stands for an estimate of d k and L is the estimator gain matrix. Substituting (39) into (42) yields:

dk+1 = dk + L(E k e d,k -W w k ), ( 43
)
where e d,k = d k -dk is an unknown input estimation error. While its dynamics is governed by:

e d,k+1 = d k+1 -d k + d k -dk -LE k e d,k -LW w k = (I -LE k ) e d,k + [I -LW ] wk , ( 44
)
where 44) is transformed into a compact form:

ε k = d k+1 -d k and wk = ε k w k . Finally, equa- tion (
e d,k+1 = X k e d,k + Z wk , , (45)
with

X k = I -LE k and Z = [I -LW ].
To make further deliberations tractable, it is assumed that wk is bounded within an ellipsoid

wk ∈ E w , E w = { w : wT Q w w ≤ 1}, ( 46
)
This allows formulating the following theorem, which constitute the main result of this section. Theorem 1. The system (45) is strictly quadratically bounded for all E k and all allowable wk ∈ E w if there exist N , P 0 and 0 < α < 1, such the following conditions are satisfied

  -P + αP 0 P -E T k N T 0 -αQ w R T P -N E k R -P   ≺ 0, k = 0, 1, . . . (47) with R = [P -N W ].
Proof 1. Using Definition 1 and the fact that wT k Q wk ≤ 1 (cf. ( 46)) it can be concluded that

wT k Q wk < e T f,k P e f,k ⇒ e T f,k+1 P e f,k+1 -e T f,k P e f,k < 0. (48)
Consequently, using (45) and defining v k = e f,k wk it can be shown that

v T k X T P X X T P X Z T P X Z T P X v k < 0. (49)
From ( 48) it is evident that for any α > 0

αv T k -P 0 0 Q w v k < 0. ( 50
)
Thus, applying the S-procedure to ( 49) and ( 50) gives

v T k X T k P X k -P + αP X T k P Z Z T P X k Z T P Z -αQ w v k < 0, (51)
which by implementing Schur complement yields

  -P + αP 0 X T k P 0 -αQ w Z T P P X k P Z -P   ≺ 0. (52)
Finally, substituting 52) gives (47), which completes the proof.

P X k = P (I -LE k ) = P -P LE k = P -N E k (53) P Z = P [I -LW ] = [P -P LB] = [P -N W ] (54) into (
In spite of the incontestable appeal of the approach summarized by Theorem 1, it is impossible to use to obtain a solution of (47), which will be feasible for all k = 0, 1, . . .. To settle the design problem, the system (45) is transformed into Linear Parameter-Varying (LPV) form:

e d,k+1 = i={f,r},j={l,r} p x,i,j X i,j e d,k + Z wk , (55)
where

X i,j = I -LE i,j (56)
with

E f,l = Ts    0 -1 0 0 0 -p f,l Re 0 0 0 -p f,r 0 Re 0 0 -p r,l 0 0 Re 0 -pr,r 0 0 0 Re    , E f,r = Ts   0 0 -1 0 0 -p f,l Re 0 0 0 -p f,r 0 Re 0 0 -p r,l 0 0 Re 0 -pr,r 0 0 0 Re   , E r,l = Ts    0 0 0 -1 0 -p f,l Re 0 0 0 -p f,r 0 Re 0 0 -p r,l 0 0 Re 0 -pr,r 0 0 0 Re    , E r,r = Ts   0 0 0 0 -1 -p f,l Re 0 0 0 -p f,r 0 Re 0 0 -p r,l 0 0 Re 0 -pr,r 0 0 0 Re   .
Thus, Theorem (1), can be reformulated in the following fashion: Theorem 2. The system (55) is strictly quadratically bounded for all E k and all allowable wk ∈ E w if there exist N , P 0 and 0 < α < 1, such the following conditions are satisfied

  -P + αP 0 P -(E i,j ) T N T 0 -αQ w R T P -N E i,j R -P   ≺ 0, i = {f, r}, j = {l, r}, (57)
with R = [P -N W ].

Finally, the design procedure of virtual sensors boils down to:

Off-line:

(1) Select Q w in ( 46).

(2) Select 0 < α < 1 and obtain the gain matrix L of (42) by solving (57) and then substituting L = P -1 N . On-line:

(1) Set d0 and k = 0.

(2) Obtain dk+1 with ( 42).

(3) Set k = k + 1 and go to Step 1.

Diagnostic principles

As it was already mentioned in the introductory part of this paper, the main objective was to develop virtual sensors providing: dk = [ Fx,fl,k , Fx,fr , Fx,rl , Fx,rr , T ] T .

(58) Having the above estimates the primary residual signal is formed:

z T,k = T k -Tk , (59)
which is used as a source of knowledge concerning desired torque distribution within AGV. The remaining set of residuals concerns longitudinal forces. The general idea starts with defining the longitudinal sleep ratio, which for all wheels is given by [START_REF] Rajamani | Algorithms for real-time estimation of individual wheel tire-road friction coefficients[END_REF]:

σ i,j = ω i,j R e v x -1. ( 60
)
As can be be observed Fig. 1, apart from the fact that the wheels are identical they are made with metal and sealed with a thin gum strip. Moreover, it is assumed that the AGV is operating on a level stiff surface. Thus, without loss of generality it is possible to assume that σ i,j,k = 0. This leads to the following relation:

ω i,j = v x R e (61)
Since the actual v x of AGV is available, it is to us (61) to calculate desired ω i,j . Subsequently, the desired ω i,j along with T are employed to calculate reference longitudinal forces F x,i,j,k using ( 6)-( 9). Note that both ω i,j and F x,i,j,k are perceived as fault-free as they are generated solely with the AGV model. While real AGV is exposed to various faults of mechanical nature as well as unexpected working conditions like sliding surface, which are also perceived as faults. As a result, the following set of residuals is formed z F,x,i,j,k = F x,i,j,k -Fx,i,j,k .

(62)

EXPERIMENTAL RESULTS

The objective of this section is to provide experimental results regarding application of the developed AGV virtual sensors and their application to fault diagnosis according to the principles detailed in Section 4.1. The experimental validation starts with describing the driving scenario: are consistent with the longitudinal forces obtained by the model and the forces are equally distributed among the wheels. Indeed, Figure 8 presents the residual z x,f l , which clearly indicates that it is close to zero for the faultfree case. However, it is not centered around zero, and hence, some bias in the estimates can be observed. The reason behind such a situation is manifold, e.g. sensor biases and/or modeling inaccuracies. The same situation occurs for the remaining wheels as well as the torque. However, due to the space limits they are omitted. If the Fig. 7. Residual z x,f l for the fault-free case virtual sensors operate properly in the fault-free case then their performance can be evaluated in the fault case. For that purpose, AGV was steered to the two overlapping surfaces in such a way as one of the wheels is hanging in the air. In particular, the front right wheel has no contact with the surface, and hence, it does not generate the longitudinal force appropriately. This unappealing phenomenon is immediately indicated by residual z x,f r which is shown in Fig. The main objective of this paper was to propose a set of diagnostics virtual sensors enabling continuous monitoring of longitudinal tire forces and torque. The proposed strategy was implemented on a prototype AGV, which was recently developed at the University of Applied Sciences Ravensburg-Weingarten. The main advantage of the proposed approach is that it avoids utilization of sophisticated and unreliable tire models, which impair the performance of the approaches presented in the literature and make the whole estimation problem a nonlinear one.

Fig. 1 .

 1 Fig. 1. Automated Guided Vehicle with all the forces considered is shown in Fig. 2. While its parameters are given in Tab. 2.

  Figures 3-6 present the longitudinal forces calculated with the model (nominal case) and their counterparts obtained with based on a set of measurements available from the AGV sensors.It can be observed that the estimates

Fig. 6 .

 6 Fig. 6. Nominal and estimated F x,rr

Fig. 8 .

 8 Fig. 8. Residual z x,f r for the faulty case

Table 2 .

 2 AGV parameters

	δf	δf
	Fx,f l		Fx,f r
			x0
	Fy,f		Fy,f
		La		Lf
	y0	δr	COG	δr
	Fx,r l		Fx,rr	Lr
	Fy,r	La	Fy,r
	Fig. 2. Steering system
	3.1 Modeling AGV
	From Fig. 2, it can be derived that the force causing
	longitudinal motion is given by