Carine Jauberthie

Soheib Fergani

Emelyne Saulnier

Fault detection using Upper Bound Interval Kalman Filter for unmaned aerial vehicle

This paper deals with dynamics estimation and fault detection of unmaned aerial vehicles (uav) using the Upper Bound Interval Kalman Filter (UBIKF). An upper bound for all positive semi-definite matrices included in an interval matrix is calculated. This upper bound is used to generate envelopes for the variables to be estimated which are the dynamics of the UAV. It allows to provide a guaranteed estimation envelope for the considered dynamics. Then, the fault detection scheme is used based on a χ 2 test. The faults concern sensors and actuators. Simulations on a discrete uncertain UAV model highlight the efficiency of the proposed filter for both UAV dynamics estimation and fault detection.

I. INTRODUCTION

Using Upper Bound Interval Kalman Filter instead of classical Kalman Filter is mainly motivated by the idea that stochastic and set-membership estimation approaches have specific advantages: they complement each other more than they compete. In fact, the experimental conditions about noise and disturbances, in a stochastic estimation framework, are usually properly modeled through appropriate probability distributions. Yet, other sources of uncertainty are not wellsuited to stochastic modeling, as for example the parameter uncertainties that generally arise from system design tolerances and from process ageing. These uncertainties are often better represented with bounded uncertainties, as intervals. Thus this modeling is described through the set-membership framework.

Combining bounded and stochastic uncertainties opens hence new perspectives for modeling complex systems more accurately.

Thus, motivated by above-mentioned reasons, the filtering issue for discrete time linear models considering bounded uncertainties on parameters and gaussian noise on measurements is studied in this paper. In [START_REF] Chen | Interval Kalman filtering[END_REF], the classical Kalman filter [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF] has been extended to this type of uncertain systems. This proposed solution is not guaranteed, i.e. the solution set may not include all the classical Kalman filter solutions consistent with the bounded uncertainties represented in the system. In [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF], an improved interval Kalman filter has been proposed solving the interval matrix inversion problem with the set inversion algorithm SIVIA (Set Inversion Via Interval Analysis) and constraint satisfaction problems [START_REF] Jaulin | Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics[END_REF]. However, an high computational time is required by the proposed algorithm if the considered system is affected by large uncertainties [START_REF] Tran | Two stochastic filters and their interval extensions[END_REF]. *This work was not supported by any organization 1 The authors are with LAAS-CNRS, Université de Toulouse, UPS, Toulouse, France cjaubert,sfergani,esaulnier@laas.fr Thus the Minimum Upper Bound of Variance Interval Kalman Filter (UBIKF) has been presented in [START_REF] Tran | Interval kalman filter enhanced by positive definite upper bounds[END_REF] with two main goals: minimizing an upper bound for the estimation error covariance and enclosing the set of possible solutions of the filtering problem for interval linear systems. Since the gain matrix handled by UBIKF is punctual, this approach encloses all the estimates consistent with the parameter uncertainties in a much less conservative manner than the iIKF. This filter is used in the following work.

In the proposed work, the fault detection strategy consists in combining the UBIKF filter with the well-known χ 2 test. The χ 2 test is a very useful statistical test providing information about on the significance of any observed differences (with the predicted behaviour) and also detailed information on which categories account for any differences found. Since the χ 2 square is a significance test, coupling it with an accurate estimation strategy can provide satisfying results.

Here, the study case deals with data that are estimated as intervals with no predefined distribution. This choice was also motivated by two main factors: the first is the computation time and the easiness of implementation for both UBIKF filter and the χ 2 test. The second is the compliance of this test with the mixed uncertainties handled by the proposed algorithm. Here, we focus on the efficiency of the filter estimation performances and though the robustness against disturbances is to be developed in on going works.

Interesting cases of study for the proposed strategy are Unmanned Aerial Vehicles (UAVs). Indeed, in the last decade, both industrial and academic communities have been interested by the UAV systems, for civilian and military applications. Due to their high manoeuvrability and flexibility in indoor and outdoor application, a special focus has been given to multirotor UAVs. Therefore, one of the most challenging topics for their autonomous navigation is the vehicle attitude estimation. Also, since most of the localization and control strategies are based on these dynamics estimation, fault detection and isolation on these information is crucial to anticipate system dysfunctions and prevent missions failure. In this work, authors propose an efficient fault detection and estimation strategy based on the Minimum Upper Bound of Variance Interval Kalman Filter. The main objective is estimate UAVs dynamics while supervising its behaviour to detect the eventual actuators and sensors faults.

This paper is organized as follows. The problem formulation is described in Section II. In Section III, the fault detection scheme, based on UBIKF, is described. Then a numerical example has been used to highlight the potential of the proposed approach in Section IV. Some conclusions are presented in Section V.

II. PROBLEM FORMULATION

Interval analysis was developed by [START_REF] Moore | Interval analysis[END_REF] and is useful to deal with bounded uncertainties. Most of the notions of interval analysis can be found in [START_REF] Jaulin | Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics[END_REF]. In this framework, an interval [x] is a closed and connected subset of R:

[x] = {x ∈ R | x ≤ x ≤ x} , (1)
where x and x are respectively the lower and upper bound. The center of [x] is defined by mid([x]) = (x + x)/2 and its radius is rad([x]) = (x -x)/2. The set of all intervals in R is noted as IR.

An interval vector (or matrix) is a vector (or matrix) whose elements are considered as intervals. In this paper, an interval vector and an interval matrix are denoted as In the proposed work, the following class of uncertain linear discrete-time stochastic systems is considered:

x k+1 = A k x k + B k u k + w k , y k = C k x k + v k , (2)
where x k ∈ R nx is the state vector, u k ∈ R nu is the input vector, y k ∈ R ny is the measurement vector and w k , v k are white Gaussian noise sequences with zero mean and covariance matrices Q and R. The initial state x 0 is also Gaussian with mean µ 0 and covariance matrix P 0 . In addition, x 0 , {w 1 , ..., w k } and {v 1 , ..., v k } are assumed to be mutually independent. The structure of the classical Kalman filter for system (2) is represented as follows ( [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]):

xk|k = xk|k-1 + K k (y k -C k xk|k-1 ), (3)
where xk|k is the posterior state estimate, xk|k-1 = A k xk-1|k-1 is the prediction of state a priori and K k is the gain matrix that minimizes the mean square error

E[(x k|k -x k )(x k|k -x k ) T ].
We consider the case where the matrices A k , B k , C k and the covariance matrices Q, R are assumed bounded and are represented by interval matrices, denoted

[A k ], [B k ], [C k ], [Q] and [R]
, containing all possible values of each parameter. Since it is impossible to solve directly the Kalman filtering problem due to parameter uncertainties, our goal is to obtain an upper bound P + k|k such that:

E (x k|k -x k )(x k|k -x k ) T P + k|k , (4)
for the set of all models with parameters bounded by the above interval matrices. The envelope enclosing the set of state estimates corresponding to the gain K is then computed. This idea is similar to [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF] in which the envelopes of the optimal gains and the state estimates given by the Kalman filtering procedure are determined by using interval analysis.

In contrast, the proposed algorithm determines a gain matrix K which minimizes the trace of the upper bound on the error covariance instead of finding a set of gain in [START_REF] Xiong | Fault detection using interval Kalman filtering enhanced by constraint propagation[END_REF].

Since the gain matrix in the UBIKF is punctual, it allows to reduce the conservatism and the computational time of interval operations.

In the following, the UBIKF is combined with a χ 2statistics test for fault detection of an unmaned aerial vehicle.

III. FAULT DETECTION SCHEME

As the conventional Kalman filter, the UBIKF can be designed in two steps: prediction and correction. A description of this filter is given in [START_REF] Tran | Interval kalman filter enhanced by positive definite upper bounds[END_REF] and the main algorithm is recalled below:

A. UBIKF algorithm

The algorithm steps are summarized below:

Algorithm 1 UBIKF Algorithm Input: x0|0 , P + 0|0 , [A], [B], [C], [Q], [R], y k , u k , A k , Q k , R k , k = 1, 2, . . . Output: xk|k , P + k|k 1: for k = 1, 2, . . . do 2:
Prediction step:

xk|k-1 = [A] xk-1|k-1 + [B] u k , P k|k-1 = A k P + k-1|k-1 A T k + Q k , P + k|k-1 P k|k-1 3: Correction step R + k R k K k = (n 0 + 1)P + k|k-1 C T m S -1 k , S k = (n 0 + 1)mid ([C]) P + k|k-1 mid ([C]) T + 2 ny i=1 nx j=1 C (i,j) r P + k|k-1 C (i,j) r T + 1 2 ny i=1 nx j=1 ny m=1 nx l=1 S (m,l) (i,j) + R + k . P + k|k = (n 0 + 1) (I -K k mid ([C])) P + k|k-1 . xk|k = (I -K k [C k ]) xk|k-1 + K k y k . 4: end for

B. Fault detection scheme

In this section, consider the following system:

x k+1 = A k x k + B k u k + w k + f a k , y k = C k x k + v k + f c k (5)
where f a k ∈ R n f a and f c k ∈ R ny are actuators and sensors faults vectors.

The difference between measurement and model output is defined by:

[r k ] = y k -[C] xk|k-1 , (6)
and it contains all values of innovation r k defined by r k = y k -C k xk|k-1 whose the covariance matrix is given by:

[S] = [C] P k|k-1 [C] T + [R] , (7)
with xk|k-1 and P k|k-1 , the predicted state and the covariance matrix of the prediction error respectively. The interval

[β k ] defined by [β k ] = [r k ] T S -1 k [r k ] contains all possible values of β k , ∀r k ∈ [r k ]
where β k is a statistical function detecting a change in the mean of the innovation sequence:

β k = k i=k-W +1 r T i S -1 i r i , (8)
In the literature, using the χ 2 -statistics test for fault detection is a kind of Innovation approach mentioned in [START_REF] Mehra | An innovations approach to fault detection and diagnosis in dynamic systems[END_REF]. In [START_REF] Willsky | Adaptive filtering and self-test methods for failure detection and compensation[END_REF] or [START_REF]Two self-test methods applied to an inertial system problem[END_REF], this method is applied for fault detection problems in which the β k statistic is used where W is a window size (W ≤ k). The authors argue that this statistic is χ 2 -distributed with W n y degrees of freedom. A rule for the fault detection test was established: (H 0 ) β k ≤ δ, no error occurred; (H 1 ) β k > δ, an error occurred, where δ is the threshold determined by P(χ 2 (W n y ) > δ) = α with α a chosen significance level.

Algorithm 2 Fault detection algorithm

Initialization:

[x 0|0 ], P 0|0 , [A], [B], [C], [D], [Q], [R], W , u k , y k , α, k = 1, 2, ..., N . for k = 1, 2, 3, ...N do Implementation: Use Algo.1 to get : [x k|k ], P + k|k , [x k|k-1 ], P k|k-1 . [r k ] = y k -[C][x k|k-1 ] [S k ] = [C]P k|k-1 [C] T + [R] U k = sup{abs([r k ] T [r k ])} Calculate β k Find δ s.t.: P(χ 2 (W n y ) > δ = α Detection signal : π k = I(U k > δ)
end for where I(x) equal 1 if x holds true and null otherwise.

IV. APPLICATION TO UAV

The drone studied in this work is the Parrot ARDRONE 2 which attracted a lot of attention and many works (see for example [START_REF] Hamel | Dynamic modelling and configuration stabilization for an x4-flyer[END_REF] or [START_REF] Mahony | Multirotor aerial vehicles: Modeling, estimation, and control of quadrotor[END_REF]).

The ARDRONE 2, depicted in Fig. 1 is an underactuated system. Indeed, it has 6 degrees of freedom (see Fig. 1), 3 translations ( w.r.t axes (x,y,z)) and 3 rotations ( w.r.t (φ, θ, ψ)). The different movements are obtained by varying the speeds of each one of the 4 motors ( [START_REF] Goerzen | A survey of motion planning algorithms from the perspective of autonomous uav guidance[END_REF] [14]).

A. Estimation of the UAV dynamics

In this work, the commonly used 12-DOF (degree of freedom) linearized model of the UAV is considered (some details can be found in [START_REF] Lecointe | Backstepping control law application to path tracking with an indoor quadrotor[END_REF]). The state vector X is defined as follows: X = (x, y, z, v x , v y , v z , φ, θ, ψ, p, q, r) where x, y, z represent the relative positions of the UAV center in the world frame; v x , v y , v z correspond to the translation speed of the drone in the world frame; φ, θ, ψ correspond to rotation angles of the UAV in the world frame, and p, q, r are the angular velocities in the body frame. Then, the inputs of the system are: u = (u 1 , u 2 , u 3 , u 4 ) where :

u 1 = F 1 + F 2 + F 3 + F 4 , u 2 = (-F 1 + F 2 + F 3 -F 4 )l, u 3 = (-F 1 -F 2 + F 3 + F 4 )l , u 4 = F 1 -F 2 + F 3 -F 4
where l and l are the distance between the center of gravity of the quadrotor and the rotors axis. F i are the vertical forces generated by the rotors. Thus the system can be described by Equation 2. In the following, the attitude dynamics of the UAV are under interest. The drone is hovering at a considered altitude, and then its dynamics are excited by varying the input u 4 (the force differential at each time instant k to study, mainly, the roll and pitch behaviours).

Remark 4.1: It is worth noticing that in this work, the study has been focusing on the roll dynamics (considering that the pitch behaviour is the quit the same on the y-axis). Also, the yaw dynamic estimation is not considered, and usually measured using an OptiTrack capture system.

In Fig 2, the UBIKF filter is used to estimate the UAV states using only four output signals (z, v x , vy , vz ). The objective is to efficiency bound the estimated states in the guaranteed intervals.

One can notice that the UBIK filter provides tight estimated intervals that includes the vehicle dynamics which can be very useful for its behaviour monitoring (especially for UAV pitch and roll dynamics) as in Fig. 3: Also, in Fig. 4 the trace of the upper bound of the covariance matrix is represented.

Thus after few time steps, the trace of this upper bound is constant. The detection threshold is crossed between 30 and 70, which corresponds to the added fault. The χ 2 test detects the bias introduced by the faulty sensor.

B. Fault detection

2) Actuator fault detection: Inspired by [START_REF] Sharifi | Fault tolerant control of a quadrotor uav using sliding mode control[END_REF], the matrices that allow to provide a configuration where one of the 4 rotors of the UAV would fail are used (actuator fault scenario). The inputs of the UAV are scheduled using these matrices to simulate the actuator fault. The considered scenario is the following: The drone is hovering at a considered altitude, and then the attitude dynamics are excited. Also, by simulating a fault on the first rotor (u 1 ) at k = 31. It can be easily noticed that the curves representing the states are affected by this input deterioration. Indeed, one can observe a jump to k ≈ 32 on each of the states.

  [x] and [M ]. The set of n-dimensional interval vectors (or m × n interval matrices) is denoted as IR n (or IR m×n ). Given [M ] ∈ IR m×n , the two functions mid([M]) and rad([M]) provide two m × n real matrices containing the centers and radius of elements of [M ].

Fig. 1 .

 1 Fig. 1. UAV Parrot ARDRONE 2

Fig. 2 Fig. 3 .Fig. 4 .Fig. 5 .

 2345 Fig. 2. State estimation

Fig. 6 .

 6 Fig. 6. Detection test of χ 2 law for a sensor fault

1 )

 1 Sensor fault detection: To test the efficiency of the UBIK filter for estimating the faulty dynamics, the following

							UAV's nominal states estimation
			20
	x 1		0
		-20
			0	2	4	6	8	10	12	14	16	18	20
			20
	x 2		0
		-20
			0	2	4	6	8	10	12	14	16	18	20
	x 3		0
		-20
			0	2	4	6	8	10	12	14	16	18	20
			20
	x 4		0
		-20
			0	2	4	6	8	10	12	14	16	18	20
			20
	x 5		0
		-20
			0	2	4	6	8	10	12	14	16	18	20
		-19
	x 6	-20
		-21
			0	2	4	6	8	10	12	14	16	18	20
			2
	x 7	0
			-2
			0	2	4	6	8	10	12	14	16	18	20
			2
	x 8	0
			-2
			0	2	4	6	8	10	12	14	16	18	20
			2
	x 9	0
			-2
			0	2	4	6	8	10	12	14	16	18	20
								k

Using the χ 2 test, it can be seen in Fig. 9 that at k ≈ 32 the curve crosses the detection threshold. Thus, the test detects the fault correctly.

V. CONCLUSION

An approach to fault detection applied to a UAV model is proposed in this paper. It combines UBIKF and a classical χ 2 test. The proposed filter allows to obtained tight guaranteed estimation intervals of the considered dynamics while the χ 2test allows to detect some abnormal behaviours. Simulation results highlight the efficiency of the proposed fault detection scheme. Beside being efficient, it is easily implemented and can be very useful in real time implementation. Also, the robustness analysis of the filtering considering external disturbances and comparison with the well-known classical estimation approaches represent challenging issues that are been considered by authors in their ongoing works. Furthermore, this study can inspire performance fault tolerant control strategy by taking advantage of the provided guaranteed interval estimations to handle the systems disfunctions (see Fig. 10).


Quoc-Hung Lu

Soheib Fergani

Carine Jauberthie

Active Fault Diagnosis based on Adaptive Degrees of Freedom χ 2 -statistic method

This paper is concerned with an active fault diagnosis method is developed based on an adaptive fault detection approach using χ 2 -statistics [1]. The system under consideration is linear discrete time-variant with sensor faults in the framework of mixed uncertainties (stochastic noises and bounded uncertainty for parameter matrices). The proposed method enhances the fault detection performance and provides the ability of localization and estimation of the detected fault. It provides also an on-line fault diagnosis with no delay in time instant with the computation time depending only on the computer performance.

I. INTRODUCTION

Fault diagnosis is becoming nowadays a functionality of utmost importance for modern systems. Including the fault detection as a substantial task, fault diagnosis is characterized by its ability of fault isolation and identification that allows the system to react in real time.

The fault diagnosis can be organised in two general categories: model-based (analytical model based on a deeper understanding of the system), and non-model based(heuristic knowledge). In the field of model-based fault diagnosis, two approaches can be considered: passive and active fault diagnosis (AFD). The passive approach has been introduced very early in 1970's by [START_REF] Beard | Failure accomodation in linear system through selfreorganization[END_REF], [START_REF] Jones | Failure accomodation in linear system through selfreorganization[END_REF], [START_REF] Mehra | An innovations approach to fault detection and diagnosis in dynamic systems[END_REF] and has been extensively investigated. A general survey on these passive approaches can be found in [START_REF] Chen | Robust Model-Based Fault Diagnosis for Dynamic Systems[END_REF] and [START_REF] Ding | Model-Based Fault Diagnosis Techniques[END_REF] . On the other hand, few works have focused on the active approach that has been developed more recently. In [START_REF] Campbell | Auxiliary Signal design for Failure Detection[END_REF], an approach using auxiliary signal design for failure detection has been introduced, and where the key terminology "active failure detection" (AFD) appears. Also, ( [START_REF] Niemann | Fault tolerant control based on active fault diagnosis[END_REF], [START_REF] Niemann | Active fault diagnosis in closedloop systems[END_REF], [START_REF] Stoustrup | Active fault diagnosis by controller modification[END_REF], [START_REF] Niemann | Active fault detection in MIMO systems[END_REF],...) use the AFD terminology. To the best of our knowledge until now, [START_REF] Puncochar | A survey of active fault diagnosis methods[END_REF] is a rather complete and recent survey dealing with the AFD approach.

A key idea of AFD is to use auxiliary input signals that are injected into the monitored system in order to improve the quality of decision making stage of the fault diagnosis. This technique was investigated in several researches around 1990, well before the release of the AFD terminology ( more deatails in [START_REF] Puncochar | A survey of active fault diagnosis methods[END_REF]). Also, the AFD approach can be classified into groups based on different relevant features as: the deterministic (norm bounded)/probabilistic (based on uncertainty description), the fixed/variable finite time interval or infinite time interval (based on the length of time interval in which auxiliary signals are designed). A list up to date and non exhaustive of related researches can be found in [START_REF] Puncochar | A survey of active fault diagnosis methods[END_REF], [START_REF] Tan | Set-based guaranteed active fault diagnosis for lpv systems with unknown bounded uncertainties[END_REF].

The relevant issue of AFD approach is that the injection of auxiliary signals into the monitored system disturbs the outputs to be controlled both in the fault-free case as well as in the faulty case [START_REF] Niemann | Active fault detection in MIMO systems[END_REF]. It should be ensured that auxiliary signals do not drive the monitored system out of desired control performance specifications. Therefore, there is a tradeoff between the fault diagnosis quality improvement and a minimal disturbance of the controlled outputs.

Very recently, authors have introduced Adaptive degrees of freedom χ 2 -statistic (ADFC) using the optimal upper bound kalman filter and the χ 2 -statistics. It allows to provide a significantly enhance fault detection for systems subject to mixed uncertainties (bounded and stochastic), see [START_REF] Lu | A new scheme for fault detection based on optimal upper bounded interval kalman filter[END_REF]. In this paper, a novel AFD method based on the Adaptive degrees of freedom χ 2 -statistic (ADFC) is presented. The proposed active strategy uses auxiliary signals to enhance fault detection performance of the ADFC method (see Fig. 1). Additionally, it provides the ability of the estimation and localization of the detected fault. It uses the ADFC method as its main fault detector, called ADFC detector in this article, and therefore deals with a dynamical system with mixed uncertainties. More precisely, in this study, a linear discrete time system concerning both stochastic and bounded uncertainties in system parameters and sensor fault context are handled. The main contribution is that the auxiliary signals are not injected into the monitored system but provided only to the diagnoser whenever a fault is detected. Thanks to characteristics of ADFC detector, using these auxiliary signals, the diagnoser can decide whether the detected signal is a false alarm or not and hence reduces false alarms. In the case of positive confirmation, the diagnoser localizes and estimates the detected fault. Then, the estimated fault is returned backwardly to the diagnoser to compensate for the actual fault effect to the next iteration, which also enhances the fault detection. Furthermore, since auxiliary signals are not injected to the monitored systems, the undesired system disturbance due to auxiliary signals is avoided.

The paper is organized as follows: Section I provides an overview and a general framework for the study.In Section II, the considered problem is formulated. Section III presents the main contribution which is the proposed AFD method. Section IV applies the method to the Bicycle vehicle application. Section V is the conclusion.

II. PROBLEM FORMULATION

In this section, the problem formulation for the fault diagnosis purpose is addressed. Based on that, objectives, scope and resolution methodology are determined. In general, a standard fault diagnosis resolution addresses both the fault detection and isolation (FDI) and the fault estimation (FE).

Let's start by presenting briefly the notations used in the whole article (including Algorithms). A real positive semidefinite matrix A is denoted by A 0. For two real square matrices M, N of the same size, M is an upper bound of N , denoted N M , if and only if M -N 0. For a non empty set Ω of square matrices, K is an upper bound of

Ω, denoted Ω K, if and only if M K, ∀M ∈ Ω. A real interval matrix [X] of dimension p × q is a matrix with real interval components [x ij ], i = 1 : p, j = 1 : q. Write X ∈ [X] to indicate a point matrix X = (x ij ) belonging element-wise to [X]. Define: X ≡ sup([X]) = (sup([x ij ])), X ≡ inf([X]) = (inf([x ])), mid([X]) = (X + X)/2, rad([X]) = (X -X)/2 and width([X]) = X -X. Let [X] be an n × n real interval matrix, denote: S([X]) = X ∈ [X] : X = X T , S + ([X]) = {X ∈ S([X]) : X 0}.

Basic interval operators

∈ {+, -, ×, ÷} defined in [START_REF] Jaulin | Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics[END_REF] can be used to compute directly all operations [u] [v] and α [u], for real intervals [u], [v] and α ∈ R, without any further approximation algorithm. More general operators are constructed by meant of inclusion function [START_REF] Jaulin | Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics[END_REF]. In practice, the package Intlab developed for Matlab is used.

Consider the linear discrete time dynamical system:

x k = A k x k-1 + B k u k + w k , y k = C k x k + D k u k + v k + f k , k ∈ N * , (1)
where x k ∈ R nx and y k ∈ R ny represent state variables and measures respectively, u k ∈ R nu inputs, w k ∈ R nx state noises, v k ∈ R ny measurement noises and f k ∈ R ny sensor faults. Each of the f k 's components corresponds to a sensor and f k can be composed by multiple faults (some or all sensors are faulty) or single fault (only one sensor is faulty). Assumptions (A1):

w k , v k are centered Gaussian vectors with covariance matrices Q k and R k . Matrices A k , B k , C k , D k , Q k , R k are unknown,

deterministic and belonging to a given bounded interval matrices

[A], [B], [C], [D], [Q] and [R]
respectively. The initial state x 0 is also Gaussian with mean µ 0 and covariance matrix P 0 . In addition, x 0 , {w 1 , ..., w k } and {v 1 , ..., v k } are assumed to be mutually independent.

Assumption (F1). The fault vector is of the form of single fault, that is f k = b.e j0 ≡ f b k,j0 where b ∈ R and e j0 is the j 0 -th column of the corresponding identity matrix, j 0 ∈ {1, ..., n y }.

Assumption (F2). For each chain j ∈ {1, ..., n y }, there exists a value b * j > 0 so that the detection signal π k satisfies

P π k = 1 |b| ≥ b * j ≥ p * j , P π k = 0 |b| < b * j ≥ p * j ,
with acceptable probabilities p * j , p * j , e.g. p * j = p * j = 0.95. Remark 1. In the notation f b k,j0 , b means for a dependence index of f k and not a power. The assumption (F1) is considered as it is the simplest but indispensable case for a FDI and FE problem. The assumption (F2) raises naturally for any fault detection method. As the fault value b has a large magnitude, the fault is easily detected (π k = 1) and vice versa. Thus, the thresholds b * j 's always exist. The values b * j , p * j and p * j are chosen based on the application and the considered scenario.

Problem. Consider the system (1) with assumptions (A1), (F1) and (F2). P1. For a detected fault, determine (isolate) the chain on which it occurs and estimate its value. P2. Detect (diagnosis if possible) incipient faults, i.e. the ones with small magnitudes (|b| < b * j ), in the early stage. The problem P1 focuses on the diagnosis of large faults (|b| ≥ b * j ) which are detected almost always (with high probabilities) and some moderate ones (|b| < b * j ) which are detected occasionally (with small probabilities). The detected signals (π k = 1) include also false alarms. Beside the ability of isolation and estimation of the real faults, a good fault diagnosis scheme solving P1 might help to reduce false alarms.

Incipient faults are naturally harder to detect. The problem P2 can be seen as solved (or partially solved) by a scheme :

1) reducing thresholds b * j 's as small as possible, 2) solving the problem P1 using reduced thresholds b * j 's with good/acceptable performance, 3) reducing false alarms. A perfect solution detecting all incipient faults might be unrealistic.

III. ACTIVE FAULT DIAGNOSIS SCHEME FOR ADAPTIVE

DEGREES OF FREEDOM χ 2 -STATISTIC METHOD In this section, an AFD scheme applied to a discrete dynamical system based on the ADFC method dealing with sensor additive single fault is developed. This scenario corresponds to the type 2 of error (single fault) presented in [START_REF] Lu | A new scheme for fault detection based on optimal upper bounded interval kalman filter[END_REF]. The scheme focuses to handle the problem P1 and but also solves the problem P2.

A. Motivation

The ADFC method is developed based on Theorems 1-2 of [START_REF] Lu | A new scheme for fault detection based on optimal upper bounded interval kalman filter[END_REF] and summarized by Algorithms 1-2 presented below (also introduced in [START_REF] Lu | A new scheme for fault detection based on optimal upper bounded interval kalman filter[END_REF]). The Algorithms concern the system (1) with assumptions (A1). ADFC method is a fault detection method that can detect multiple and single faults. The fault values can be positive or negative. Consequently, it is compatible also to assumptions (F1) and (F2). In the ADFC method, Algorithm 2, namely OUBIKF which means for Optimal Upper Bound Interval Kalman Filter, is used as a residual generator providing

[r k ] = y k -[ŷ k ]. The whole fault detection Algorithm 1 ADFC METHOD TO LINEAR SYSTEM [1] 1: Initialization: [x 0|0 ], P 0|0 , [A], [B], [C], [D], [Q], [R], α, u k , y k , k = 1, 2, ..., N . 2: for k = 1, 2, 3, ...N do 3:
Use Algorithm 2 to obtain:

[x k|k-1 ], [P k|k-1 ]. 4: [r k ] = y k -[C][x k|k-1 ] -[D]u k 5: [S k ] = [C][P k|k-1 ][C] T + [R] 6: Find a k using Theorem 2 of [1] s.t. : S + ([S k ]) a k I. 7: U k = sup{abs([r k ] T [r k ]/a k )} 8: κ k = ny i=1 width([r k (i)])/n y = mean{width([r k ])} 9: Find δ k s.t.: P(χ 2 (κ k n y ) > δ k ) = α.
Algorithm 2 OUBIKF [1] 1: Initialization: [x 0|0 ], P 0|0 , [A], [B], [C], [D], [Q], [R], u k , y k , k = 1, 2, 3, ..., N . Find n 0 the number of non zero radius of [C]. Find γ such that S + ([R]) γI. 2: for k = 1, 2, 3, ...N do 3: [x k|k-1 ] = [A][x k-1|k-1 ] + [B]u k 4: [P k|k-1 ] = [A]P k-1|k-1 [A] T + [Q] 5: Find α k using Theorem 2 of [1] s.t.: S+([P k|k-1 ]) α k I 6: Choose β k > 0 and σ k > 0 7: S k = mid([C])mid([C]) T + γ α k (1+n0/β k ) I + β k +n0σ k 1+n0/β k Diag rad([C])rad([C]) T 8: K k = mid([C]) T S -1 k 9: [x k|k ] = (I -K k [C])[x k|k-1 ] + K k (y k -[D]u k ) 10: P k|k = (I -K k mid([C]))α k (1 + n 0 /β k ) 11:
end for process of this method is summarized in Algorithm 1 and called in brief the ADFC detector as mentioned so far.

Let k ≥ 1, y k = y 0 k + f k where y 0 k is the fault-free measurement. In the faulty case, f k = b.e j0 with b = 0 and j 0 ∈ {1, ..., n y }. Also, one gets • fk = b 3 e j where j = j 0 ? Assuming that the ADFC detector has a good enough performance, for (Q1), detection signal π k equals 0 with the use of ỹk . Then consider (Q2). In the first case, the fault value has been augmented its magnitude by an additive term of the same sign, so the ADFC detector gives π k = 1 with high probability. In the second case, regarding assumption (F2), one gets with high probability that

[r k ] = [r 0 k ] + f k and [x k|k ] = [x 0 k|k ] + K k f k .
π k = 0 if |b 2 + b| < b * j0 and π k = 1 if |b 2 + b| ≥ b * j0 .
For the last case, it is worthy to note that:

• add a term b 3 to the j-th element of y k is equivalent to add b 3 to the j-th element of [r k ],

• the statistic used in ADFC detector is

U k = sup{abs([r k ] T [r k ]/a k )}.
Since j = j 0 , by assumption, the j-th element of [r k ], says [r k (j)], is fault-free. Therefore, [r k (j)] is centered nearby 0. With the additive term b 3 , whatever it is negative or positive, [r k (j)] deviates more from 0 while its width remains unchanged provided that |b 3 | is not too small. This implies that U k takes a greater value while the threshold δ k determined by

P[χ 2 (κ k n y ) > δ k ] = α is unchanged since κ k
, n y and α are unchanged (Algorithm 1). Thus, π k = 1 also in the last case of (Q2).

B. Methodology and Scheme

In the considered framework, a sensor fault is characterized by a fault value b and the chain j on which it occurs. Denote a detection signal associated to an occurred fault f b k,j using the ADFC detector by π k (f b j ). In addition, since the ADFC detector performs also with the multi faults case, we also use the notation π k (f b j + f b j ), j = j , to designate the detection signal associated to faults occurring on two different chains.

Let ∆ ∈ 0 , min j=1:ny {b * j } and M ∈ N * so that M.∆ is superior to maximum fault magnitude we want to estimate. Generate auxiliary signals, denoted fictive faults, as follows: f q∆ k,j = q.∆.e j , q = -M : M , j = 1 : n y .

If a fault occurs on a chain j 0 ∈ {1, ..., n y }, which means f b k,j0 = b.e j0 = 0, and assuming that 0 < |b| ≤ M.∆, there exists a q * ∈ {-M, ..., M } so that

q * ∆ < b ≤ (q * + 1)∆ . This implies that 0 ≤ |b -(q * + i)∆| ≤ ∆, i ∈ {0, 1}. Therefore, for I = -b * j0 /∆ + 1, ..., b * j0 /∆ , π k f b j0 + f bi j0 = 0, b i = -(q * + i)∆, i ∈ I, (2)
with high probability regarding discussions of (Q1) and (Q2). Also, we have with high probability:

π k f b j0 + f bi j0 = 1 , b i = -(q * + i)∆ , i / ∈ I, π k f b j0 + f q∆ j = 1 , q ∈ {-M : M } , j = j 0 .
Remark 2. The so called "with high probability" mentioned above depends on the probabilities p * j , p * j of assumptions (F2) and the performance of the ADFC detector represented by DR, FAR or EFF indexes defined in [START_REF] Lu | A new scheme for fault detection based on optimal upper bounded interval kalman filter[END_REF] and recalled in Section IV.

The developed AFD scheme is explained via the diagram in Fig. 2 and Algorithm 3. Regarding Fig. 2, the ADFC detector includes OUBIKF as its residual generator and the fault detection part of AFD block. It provides only detection signal π k at each iteration. From the figure, we note that when a fault occurs at a time instant k, it follows the feedback [x k|k ] and affects the residual at next iterations. Thus π k+1 reflects not only the existence of f k+1 but also the effect of previous fault f k . The AFD scheme consists in using fictive faults as auxiliary inputs and providing them only to the AFD block and not to the monitored system. Thanks to these auxiliary inputs, the ADFC detector produces the signature matrix

S k = π k f b j0 + f q∆ j j,q ≡ π q k,j ,
where j = 1 : n y , q = -M : M are row and column indexes respectively. Then the AFD block is equipped with a fault diagnosis part decoding S k to decide whether π k equals 0 or 1, f k is 0 or takes which estimated value. After that, the estimated fault fk is sent backwardly to OUBIKF in order to compensate for the fault effect to [x k|k ] and to the fault diagnosis at the next iteration. Without this fault feedback process, the fault diagnosis has a poor performance. The fault diagnosis part of the AFD block is a signature matrix decoder. Its function is specified in Algorithm 3.

[C] [B]

When S k has no zero element, all yielded detection signals are 1, in particular, [START_REF] Beard | Failure accomodation in linear system through selfreorganization[END_REF]. In other words, it is almost the case that π k (f b j0 ) = 1 for all b nearby and including 0. Therefore, the detected signal π k = 1 (line 3 of Algorithm 3) initially dispatched is a false alarm with high probability.

π f b j0 + f bi j0 1 , b i = -(q * + i)∆ , i ∈ I, which contradicts
When S k has at least one zero element, the error chain is estimated as the one on which S k has the maximum number of zero elements, denoted by ĵ0 . That is because a single fault at a chain j 0 may affect the behavior of residual [r k ] on another chain j = j 0 , however its effects on the chain j 0 is the stronger. Then, the fault is estimated as the additive inverse of the mean of all fictive faults f q∆

Algorithm 3 AFD SCHEME FOR ADFC METHOD

1: Initialization: ∆, M , B * = {b * j , j = 1 : n y }, λ, [x 0|0 ], P 0|0 , [A], [B], [C], [D], [Q], [R], α, {u k , y k } k=1:N 2: for k = 1, 2, 3, ...N do 3:
Use Algorithm 1 to get detection signal: π k 4:

if π k = 1 then 5: q = -M : M ; j = 1 : n y ; ỹq k,j = y k + q.∆.e j ;

6:

Rerun Algorithm 1 with {ỹ q k,j } to obtain S k = (π q k,j ).

7:

if S k has no zero element then end if. 21: end for Note: The parameters in the second line of the initialization are required only for ADFC detector (Algorithm 1).

where r > 0 is a predetermined radius. This condition is also called the r-accuracy.

Next, in order to eliminate more false alarms and reinforce the estimation accuracy, a regularization is performed. Regarding the assumption (F2), ideally, a fault value b ≥ b * j0 is detected and hence b must be at least close to min{b * j , j = 1 : n y }. So, if the estimated value b is such that

| b| ≤ λ. min{b * j , j = 1 : n y }, λ ∈ [0, 1],
we consider that it is not consistent with assumption (F2) and hence b is replaced by 0 and πk = 0 is dispatched. In the case

λ. min{b * j , j = 1 : n y } < | b| ≤ mean{b * j , j = 1 : n y },
we consider that there is something intervening and lessening the estimated value b. So b is replaced by mean{b * j , j = 1 : n y } and πk = 1 is dispatched. The value mean{b * j , j = 1 : n y } is chosen as the replacing value because the actual faulty chain j 0 is not known, otherwise b * j0 could be used. Finally, the estimated fault is fed backwardly to ADFC detector by subtracting the amount K fk to [x k|k ].

IV. APPLICATION

In this section, the proposed method is applied in simulation to the Bicycle vehicle model ( [START_REF] Fergani | Robust multivariable control for vehicle dynamics[END_REF]) which was also used as the simulation application in [START_REF] Lu | A new scheme for fault detection based on optimal upper bounded interval kalman filter[END_REF], that is:

β(t) ψ(t) = -C f mv 1 + µ -lrCr-l f C f mv 2 -lrCr-l f C f Iz -l 2 f C f -l 2 r Cr Izv β(t) ψ(t) + C f mv 0 0 0 l f C f Iz 1 Iz SrRtr 2Iz -SrRtr 2Iz     δ M dz T b rl T brr    
where β(t) is the sideslip angle and ψ(t) is the vehicle yaw which form the model state variables. This is a nonlinear continuous-time model which has been discretized/linearized and thus given under the form [START_REF] Lu | A new scheme for fault detection based on optimal upper bounded interval kalman filter[END_REF], where the state is

x k = (x k (1), x k (2)) T = (β(k), ψ(k)) T .
In the present simulation, the parameters σ k = 1 and β k = 10 -6 are applied inside the Algorithm 2. Other parameter settings and simulation procedure applied here are similar to those used in [START_REF] Lu | A new scheme for fault detection based on optimal upper bounded interval kalman filter[END_REF] without adjustment step.

To evaluate the fault detection performance, indicators introduced in [START_REF] Lu | A new scheme for fault detection based on optimal upper bounded interval kalman filter[END_REF] are used. Assume that system (1) is applied for N iterations among which faults occur in a region R with length l (0 ≤ l ≤ N ). The region R may be a range or union of ranges but is called hereafter an error range for simplicity. Secondly, we continue the simulation to illustrate the AFD scheme performance. Other parameters chosen for Scheme 3 are: ∆ = 5, M = 12, λ = 0.5 and α = 0.03.

Recall that the AFD scheme is dedicated to diagnose faults whose magnitudes beyond thresholds b * j 's. Thus, a fault value b = -25 is fixed. Then, four different faults f k = -25e j , j = 1 : 4, are tested. In each case, an error range of length 50 (time instants) are randomly chosen in which the fault occurs. Then the ADFC detector (Algorithm 1) is performed without and with AFD scheme. Thanks to that we can answer to several questions:

(1) Does the AFD scheme enhance the fault detection of ADFC detector and help to reduce false alarms? (2) How well is the fault diagnosis provided by the AFD scheme ?

The first question is answered positively thanks to Fig. 3 and Tables II-III. Fig. 3 visualizes the reduction of false alarms while Tables II-III show corresponding values of evaluation indexes DR, FAR, EFF. In more details, Tables II-III show that the faults are totally detected in the error range (DR = 100%) by the ADFC detector. In addition, applying the AFD scheme, the false alarm rate (FAR) is reduced from about 8.6% to about 1% and hence the efficiency (EFF) indexes increase to 99%. To deal with the second question above, we measure the diagnosis performance by the accuracy rate A r defined by

A r = k∈Rj I(| bk -b k | ≤ r)I( ĵ = j) |R j | × 100% , (4)
where r > 0 is a predetermined radius, R is the error range corresponding to the fault chain j, |R j | is the length of R j , bk and ĵ are estimates of actual values b k and j, k is the time instant. So, A r is the percentage of fault estimates in R j satisfying (3). The simulation results related to the question (2) are given by Table IV. From the last row of the table, it is shown that all estimated chains are correct and all estimated fault values bk are away from the actual faults b k at most a radius of r = ∆ = 5. The second row of the Apart from the questions discussed above, the fact that AFD scheme using estimated fault to feed backwardly into [x k|k ] increases the estimation performance of the OUBIKF as the results shown in Fig. 4. In two figures in the left, the estimate intervals between two vertical black lines (the error range) deviate from the real states, even no longer contain these states and, in addition, the widths of these estimate intervals increase. In two figures in the right, the estimate intervals still track the real states well with reasonable widths.

V. CONCLUSION

In this paper, an AFD scheme is developed for ADFC method. It can be see as a diagnoser using auxiliary signals that allows to improve greatly the ADFC detector performance as well as FDI and FE of the proposed strategy.

The most relevant characteristic of the scheme is that auxiliary signals are not injected into the monitored system but provided only to the diagnoser. This is also the key difference of the scheme with other AFD methods. This helps to avoid additional disturbances due to auxiliary signals on the monitored system. In addition, auxiliary signals are designed off-line and only injected into the diagnoser once a detected signal (π k = 1) is dispatched at a time instant k. Then, the generated signature matrix is analyzed to provide decisions about the fault candidate without delay of any finite time (instant) interval in which the diagnoser waits reactions of the monitored system being injected. This implies that the developed scheme can provide an on-line fault diagnosis with no delay in time instant and with computation time depending only on the computer performance.

Another important characteristic of the scheme is the compensation for the actual fault effect to the diagnosis at the next iteration by using the estimated fault as a feedback to the diagnoser. It is important for the developed scheme because without it, the diagnosis performance of the scheme degrades severely. It may be also a good additional strategy for several existing AFD and state estimation methods.

Fig. 1 :

 1 Fig. 1: AFD general block diagram.

10 :

 10 Detection signal : π k = I(U k > δ k ). 11: end for ( * ) : I(x) is the indicator function which equal to 1 if the conditions x are true and vanishes otherwise.

  The developed AFD scheme is motivated by following questions: (Q1) Given that the ADFC detector has detected the existence of a fault f k = b.e j0 , what happens if one can add (by chance!) a quantity fk = -b.e j0 to the measurement y k to obtain ỹk then rerun the ADFC detector with ỹk ? (Q2) In the same way, what happens if one adds each of following quantities to y k and rerun the ADFC detector: • fk = b 1 e j0 where b 1 .b > 0, • fk = b 2 e j0 where b 2 .b < 0,

Fig. 2 :

 2 Fig. 2: Active Fault Diagnosis diagram using ADFC detector.

  k, ĵ0 with which π k f b j0 + f q∆ ĵ0 = 0. Denote the estimated fault as fk = b.e ĵ0 and the diagnosed detection signal as πk . The diagnosis is called r-accurate if ĵ0 ≡ j 0 and | b -b| ≤ r ,

Find I 0

 0 ⊆ {-M : M } s.t. π q k, ĵ0 = 0, ∀q ∈ I 0 12: b = -mean {I 0 } .∆ ; fk = b.e ĵ0 ; 13: if | b| ≤ λ. min{B * } then = sign( b). max{| b|, mean(B * )}.e ĵ0 17: [x k|k ] = [x k|k ] -K. fk 18:

  Briefly, + Detection Rate: DR = k∈R I(π k = 1)/l × 100%, + False Alarm Rate: FAR = k ∈R I(π k =1) N -l × 100%, + Efficiency: EFF = DR -FAR. Fistly, we verify the ADFC detector properties as mentioned in Section III-A and determine thresholds b * j 's of assumption (F2). In order to increase the detection rate, the parameter κk = λ k mean{width([r k ])} is applied in Algorithm 1 with λ k = a -1/2 k is a scale parameter (a k is determined in Algorithm 1). The results after 100 times of simulations are shown in the following table.

Fig. 3 :

 3 Fig. 3: Detection signals without (left) and with (right) AFD technique Chain DR (%) FAR (%) EFF (%) 1 100 8.6 91.4 2 100 8.6 91.4 3 100 8.7 91.3 4 100 8.7 91.3

Fig. 4 :

 4 Fig. 4: Active fault diagnosis -State estimates without (left) and with (right) fault estimation feedback.

TABLE I :

 I Detection rate of ADFC detector applied for Bicycle vehicle model. Note that in the case of b = 0, there is in fact no fault, thus the corresponding detection rates shown in the table are actually false alarm rate (FAR(%)) which notation is noted right next to them.

	It is shown that the ADFC method functions with either
	positive or negative fault values, the thresholds mentioned in
	assumptions (F2) are determined as b * 1 = b * 2 = b * 3 = b * 4 = 20
	with probabilities beyond 0.96.

TABLE II :

 II Detection performance without AFD technique

	Chain	DR (%) FAR (%)	EFF (%)
	1	100	0.7	99.3
	2	100	1.0	99.0
	3	100	1.0	99.0
	4	100	1.0	99.0

TABLE III :

 III Detection performance with AFD technique

  Table provides the accuracy percentage corresponding to the radius r = ∆/2.

	Chain	1	2	3	4
	Ar (%)	r = 5/2 r = 5	86 100	100 100 100 100 78 98

TABLE IV :

 IV The A r (%) accuracy rate

LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France. Corresponding author: qhlu@laas.fr