Estimation And Fault Detection of Discrete Event Systems Modeled By Time Interval Petri Nets

This work is interested in studying discrete event systems modeled by time interval Petri nets. Indeed, a new state model describing the time evolution of the system is proposed. The simulation of the proposed state model gives the upper and lower bounds of each element of the whole system state and the system outputs. After that, a new state estimator is proposed allowing estimating the whole system state and inputs. This estimator gives, also, an upper and lower bounds for each elements of the state vector and the input. The state model and the estimator are both proposed along, successively, count and dater approaches. For count approach, the system state is the number of transition ﬁring and for dater approach, it is the dates of transitions ﬁring. The proposed state model and estimator are applied to a model of baking process described by a time interval Petri nets. This application shows the robustness of the state and input estimation. The proposed state model and estimator are used also for fault detection, localization and estimation and it is shown that they give acceptable results.


Introduction
Many processes are described by discrete events systems and can be modeled by various kinds of Petri nets.Generally, time can be added to Petri nets to describe their time evolution [33].Time interval Petri nets are used to model several process in case where transitions firing is allowed only in a particular time interval or the the sojourn time of token is governed by a special time interval [12,14,33].Generally, Petri nets can be used for state estimation [1,17,28], discrete event systems diagnosis [7,10,11,31] and fault detection [5,30,32,36].
State models elaboration is important to study the system time evolution.This Its elaboration must be made thoroughly because it is the base of the system simulation and it gives the initialization states values in the case of state estimation [15].State estimation becomes is very important if the whole system state can not be measured [15,16].
The aim of this work is the state estimation and the fault detection, localization and estimation for discrete events systems described by time interval Petri nets.The considered system state, in this work, is the date of transition firing or their number depending on the followed approach (dater or count).The system state depends on the considered approach.If it is elaborated according to a count approach, this state is the number of firing of each transition.If the dater approach is adopted, the system state is the dates of transitions firing.Two important parts will be presented successively.The first part proposes a new state model giving an upper and lower bounds of the system state.The second focuses on the design of a new estimator able to estimate the upper and lower bounds of the state vector and input.
The first contribution which is presented in this work is to elaborate the elaboration of a new state model outlining the system behavior.This state model is conceived following successively a count and dater approaches.The proposed state model presents two inequalities where the first one leads to compute the upper and lower bonds of the system state and the second one allows computing the upper and lower bounds of the systems outputs.
The second contribution concerns a state estimator elaboration.This elaboration is made also following successively the count approach and the dater approach.The proposed state estimator is also composed of two inequalities where the first one estimates the upper and lower bounds of the state vector and the second one gives an estimation of the upper and lower bounds of the system inputs.So, The the conceived estimator is really a state and input estimator.In a next step, the residuals are calculated computed as the intersection between the intervals given by the simulation of the state model and the estimator in order to test the robustness of the estimation.
The proposed state model and estimator are applied to a baking process modeled by time interval Petri net in order to test the proposed state model and estimator and to apply them for fault detection, localization and estimation.
This paper is organized as following follows.Section 2 summarizes some related works.The third section 3 presents the used notations and variables and recalls some important definitions dealing with Petri nets.A preliminary example is also presented in this section.The section 4 presents an extension of the classical state model to time interval Petri nets following count and dater approaches.This extension is presented in two different forms.Section 5 details the design of a new state model which is proposed following successively the count and the dater approaches.The elaboration of the estimator is the objective of section 6.This estimator is proposed following, also, the count and dater approaches.An application of the proposed state model and estimator to a baking process modeled by a time interval petri net is presented in section 7 in order to show the robustness of the estimation.Their application to the fault estimation, detection and localization is also presented.

Related works
Many works discussed the problem of diagnosis and state estimation of discrete event systems modeled by different classes of Petri nets with regards to the current Petri net marking as the system state.In [7,18], authors tried to make the state estimation under the assumption that the initial marking is known.In [38], authors proposed to bound the estimated marking number for the case where the Petri net contains unobservable transitions sharing the same label.Under the assumption that transitions have non-negative costs, authors in [29] focused on least cost firing sequences.For the class of partial observed Petri nets, reduced observer was used in [8,23] to computes only a subset of the possible current markings, which is different from this work where a complete observer able to estimate the whole state vector is proposed.Under the assumption that Petri net does not contain cycles of unobservable transitions, Ghazel et al. [17] proposed a method of state estimation,.This this hypothesis is not considered in this work.The problem of on line estimation of the current markings probabilities was treated in [6,34] by developing an appropriate recursive algorithm.The results of marking estimation are used for diagnosis in [28,31].
Works cited above focused on the estimation and/or the boundedness of the current marking of the Petri net after each iteration in presence of unobservable transition.The considered system state in these works is the current marking which is different from this work where the considered state is the date of transitions firing in the case of dater approach or their firing number if a count approach is considered.Indeed, the state model and the estimator are proposed in this work following two different approaches : dater and count approaches.
In [15,16], authors proposed an algebraic form allowing describing the whole solution set for a sequence of observed events.The uniqueness and computation of optimal solutions are analyzed in the case of forward/backward Conflict Free Petri nets.The proposed algebraic model is applied after that in [13] using counter approach for the estimation of optimal sequences in partially observable untimed Petri nets.
More related works dealing with some particular classes of Petri nets can be also cited.Indeed, only some works dealing with P-time Petri nets and time interval Petri nets, which are the subject of this work, will be cited in the rest of this section.In [4], author treated the problem of marking estimation for the class of P-Time Petri net using a linear programming problem leading to check the set of firing sequences.The determination of the extremum cycle times was the subject of [24] using a particular extended graph associated with the studied P-time Petri net.In [26,27], authors consider a particular model of P-time Event Graph named Negative Event to analyze the maximum and minimum cycle times by checking the earliest and latest possible regular firing timetables using both standard an max-plus algebra.The liveness is also discussed.In [21], authors analyzed a particular process where time has only a minimum delay and not a maximum duration like the case of P-time Event Graphs.Indeed, authors proposed to consider only one common period to all tasks in order to minimize largest period of the events.Time reachability is considered also in [39].Many works [9,19,22,26] used the technique of linear programming to describe problems relative to discrete event system modeled by Petri nets in order to compute cycle time and model duality principle In [37], authors proposed to add more restriction on each iteration cycle in order to compute bounds in the case of Stochastic Event Graphs.In [2], authors computed the basic quantities depending on a particular experiment interval.
These works are interesting in the analyze of cycle time contrary to this work which the aim is the state estimation and the fault detection and localization of discrete events system described by time interval Petri nets.
Colored Petri nets are also treated recently in [35] and applied for technological facilities and the evolution of systems maintenance.
P-time Petri nets are considered in [4] and recently in [3] of decentralized state estimation in real time and diagnosis.The main difference with this work is that we consider P-time-interval Petri nets.
3 Elementary background on Petri nets

Notations
In this section, notations used along the paper are presented.Main definition related with Petri net formalism will be also presented.
|X| denotes the cardinal of the set X. A place/transition net (a P/T R net) is a structure N = (P, T R, W + , W − ), where P is a set of |P | places and T R is a set of |T R| transitions which are denoted by x (notation t corresponds to the current time and T to the transposition of a matrix).T + l (resp.T − l ) denotes the upper bound (resp.the lower bound) of the token sojourn time of a given place p l ∈ P .
Post-and pre-incidence matrices W + and W − are |P | × |T R| over N where each row l ∈ {1, ..., |P |} corresponds to the weight of the incoming and outgoing arcs of places p l ∈ P respectively.The incidence matrix is computed by In this work the weight of each arc is supposed unitary, i.e., W ij ∈ {−1, 0, 1}.The marking of set P is a vector m ∈ N |P | that assigns to each place of a P/T R net a non-negative integer number of tokens.m l is the marking of place p l with l ∈ {1, ..., |P |}.A net system (N, m) is a net N with an initial marking m.
In this paper, the incidence matrices and the initial marking are supposed known.The internal transitions are denoted x i , u i are the input transitions and y i are the output transitions.The places set P can be decomposed on four sets: -P ux : the set of places between the input and the internal transitions, their upper bounds (resp.lower bounds) of temporizations are denoted T + ux (resp.T + ux ) and their marking is denoted m ux -P xx : the set of places between the internal transitions, their upper bounds (resp.lower bounds) of temporizations are denoted T + xx (resp.T + xx ) and their marking is denoted m xx -P xy : the set of places between the output and the internal transitions, their upper bounds (resp.lower bounds) of temporizations are denoted T + xy (resp.T + xy ) and their marking is denoted m xy -P uy : the set of places which link directly the output and the input, their upper bounds (resp.lower bounds) of temporizations are denoted T + uy (resp.T + uy ) and their marking is denoted m uy

Definitions
Definition 1 [15] Given a Petri net N = (P, T R, W + , W − ): The N Petri net is Forward Conflict Free (FCF), i.e., any two distinct unobservable transitions have no common input place.
Definition 2 [15] The system of linear inequalities Ax ≤ b is sup-monotone if each row of matrix A has one strictly positive element at most.
Definition 3 [15] Let Γ be the solution set of a sup-monotone system Ax ≤ b.Set Γ is a sup-semilattice since each pair of elements has an upper bound.
Moreover, the following lemma guarantees that Γ has an extremum element that is a greatest solution which belongs to Γ .
Lemma 1 [15] Let Γ be the solution set of a sup-monotone system Ax ≤ b.Set Γ has a greatest element if the set is non-empty and has a majorant.

Preliminary example
Example 1 Let us consider the following elementary time interval Petri net given by figure 1.The time evolution of the system trajectory can be described by the equations 1 and 2: In order to estimate the system trajectory, the equations 3, 4 and 5 can be used: The system trajectory presents an upper bound and a lower bound computed using the time interval of the token sojourn time in each place.The main objective of this work is to conceive a general state model allowing computing the upper and lower bound of each element of the system state and to conceive an observer allowing estimating the upper and lower bound of the system trajectory.The obtained result can be used after that for the fault detection.

Introduction
In this section, we present an extension of the the classical state model presented in [25] to time interval petri nets.This extension is proposed following the count approach and dater approach.This state model is presented in two different forms for both.The first proposed form is the "compact form" which uses the whole state vector denoted x, and pre-and post-incidence matrices.The second proposed form gives the equation of each element of the state vector by making explicit the state model.The used state vector x(t) includes the system inputs, outputs and internal transitions.Each element of the state vector x(t) is denoted x i (t) {1,...,n} , the state vector becomes : x(t) = (x 1 (t), x 2 (t), ..., x i (t), ..., x n (t)) T where n is the transitions number.

Compact form
The used state vector x(t) includes the system input, output and internal transitions.Each element of the state vector x(t) is noted x i (t), the state vector becomes : x(t) = (x 1 (t), x 2 (t), ..., x i (t), ..., x n (t)) T where n is the transitions number.
In the case of time interval Petri nets, the system evolution in time is governed by the following inequality: where W + and W − are the incidence matrices, m is the initial marking and T + l an T − l are respectively the upper and lower bounds of the tokens sojourn time in places.The inequality (6) describes only the system trajectory evolution in time.Since the behavior of the system trajectory must be non-decreasing, the inequality ( 8) is added.The global state model allowing the system simulation is: The inequality (7) has the form : x j (t − T l ) is the sum of the output arcs of the place p i , T + l and T − l are respectively the upper and lower bounds of the place temporization interval and m l is the initial marking of the place p l .

Explicit form
The explicit form let to details each term x i (t) of the state vector x.If the studied system is FCF, then it is sup-monotone and the matrix W − presents only one 1 by row, we obtain: where L is the number of column of the matrix W − .Let us denote x + j (t) the greatest state and x − j (t) the smallest state.These particular states are given by the following equations: x + j (t) = min x j (t − T l ) is the sum of the output arcs of the place p i , T + l and T − l are respectively the upper and lower bounds of the place temporization interval and m l is the initial marking of the place p l .

Compact form
The used state vector x(t) includes the system input, output and internal transitions.Each element of the state vector x(t) is noted x i (t), the state vector becomes : x(t) = (x 1 (t), x 2 (t), ..., x i (t), ..., x n (t)) T where n is the transitions number.In the case of time interval Petri nets, the system evolution in time is governed by the following inequality: where W + and W − are the incidence matrices, T + is the upper bound places temporization vector, T − is the lower bound places temporization vector and m l ∈ {0, 1}.The inequality (12) describes the system evolution in time.Since the behavior of the system trajectory must be non-decreasing, the inequality ( 14) is added in order to obtain the following global state model:

Explicit form
The explicit form let to details each term x i (t) of the state vector x.If the studied system is FCF, then it is sup-monotone and the matrix W − presents only one 1 by row, we obtain: where L is the number of column of the matrix W − .Let us denote x + j (t) the greatest state and x − j (t) the smallest state.These particular states are given by the following equations: x  [25].This state model is conceived following successively a count and a dater approaches.

Count approach
The system behavior can be described, for the four sets P ux , P xx , P xy and P uy , by the following inequalities where the matrices xy , W + uy and ❅ −W − uy are elements of the incidence matrix matrices W + and W − .
To guarantee the non decreasing system behavior, the following inequalities are added: Inequalities ( 18) and (19) give the upper and lower bounds of the system state x(t), they can be transform on the matrix inequality 24: Inequality (24) let to compute an upper bound of the system state x(t) denoted x + (t) and a lower bound of the system state x(t) denoted x − (t).
In the same manner, Inequalities (20) and (21) giving the upper and lower bounds of the system output y(t) can be put on the matrix inequality (25): The obtained Inequality (25) let to compute an upper bound of the system output y(t) denoted y + (t) and a lower bound of the system output y(t) denoted y − (t).Inequalities ( 22) and ( 24) can be gathered in the augmented matrix inequality (26): where : SimilarlyUsing the same approach, inequalities ( 23) and ( 25) can be gathered as following: where: Using the iInequalities ( 26) and ( 27), form the state model ( 28) is obtained given as follows : The system (28) describes the obtained state model.It is possible to simulate this model in order to analyze the system's dynamic behavior.This model let to search simultaneously the upper and lower bounds of the system state x(t) and the system output y(t).The upper and lower bounds of x(t) and y(t) are finite.It is possible to conclude that the system state and outputs are bounded and the system trajectory is non-decreasing.

Dater approach
The system behavior can be described, for the four sets P ux , P xx , P xy and P uy , by the following inequalities where the matrices xy , W + uy and ❅ −W − uy are elements of the incidence matrix matrices W + and W − .
To guarantee the non decreasing system behavior, the following inequalities are added: Inequalities ( 29) and (30) give the upper and lower bounds of the system state x(k), they can be transform on the matrix inequality (35): Inequality (35) let to compute an upper bound of the system state x(k) denoted x + (k) and a lower bound of the system state x(k) denoted x − (k).
In the same manner, Inequalities (31) and (32) giving the upper and lower bounds of the system output y(k) can be put on the matrix inequality (36): The obtained Inequality (36) let to compute an upper bound of the system output y(k) denoted y + (k) and a lower bound of the system output y(k) denoted y − (k).Inequalities ( 33) and ( 35) can be gathered in the augmented matrix inequality (37): where : Using the same approach, iInequalities ( 34) and ( 36) can be also gathered as following: Using the iInequalities ( 37) and (38), form the state model ( 39) is obtained: The system (39) describes the obtained state model.It is possible to simulate this model in order to analyze the system's dynamic behavior.This model let to search simultaneously the upper and lower bounds of the system state x(k) and the system output y(k).The upper and lower bounds of x(k) and y(k) are finite.It is possible to conclude that the system state and outputs are bounded and the system trajectory is non-decreasing.Inequalities ( 28) and (39) describe the state models successively using the count and dater approaches.These models allow obtaining an upper and lower bounds of each elements of the state vector.At each moment, state vector must belong in the interval between the upper and the lower bounds.

Numerical example
Example 2 Let us consider the example of Petri net given in figure 1 of section 3.3 For this example, the system input is u = x 1 and the system output is y uy and W − uy are the following: For the system simulation, u = x 1 is assumed to be known.For example we can obtain the following table presenting 5 iterations.6 Design of a state and input estimator

Objective
The aim of this part is to conceive a new estimator able to estimate simultaneously the system state and input.The obtained estimator consists of two inequalities.The first inequality allows estimating obtaining an upper and lower bounds of the system state and the second one let to estimate the system output by computing its upper and lower bounds.The proposed estimator is conceived following successively a count and a dater approaches.

Count approach
To estimate the system state and input, it is supposed that the system output is known.Inequalities ( 18) to ( 21) can be rewritten in the following forms.
Inequalities ( 43) and ( 44) can be gathered in the matrix inequality (47) which permits to estimate the system input by computing a lower bound of u(t) denoted u − (t) and an upper bound denoted u + (t).
In the same manner, inequalities ( 42) and ( 45) can be gathered in the matrix inequality (46) which is able to compute a lower bound of x(t) denoted x − (t) and an upper bound denoted x + (t) : To guarantee the non decreasing behavior of the system trajectory, the following conditions are added: Inequalities ( 47) and ( 48) can be aggregated in the augmented matrix inequality (50): where: 46) and ( 49) can be aggregated in the augmented matrix inequality (51): where: The estimator (52) is obtained using the inequalities (50) and (51): The obtained estimator ( 52) is an algebraic model giving simultaneously an estimation of upper and lower bounds of the system state and input.State and input estimation obtained using this estimator are is generally useful for fault estimation, detection and localization.The upper and lower bounds of the estimation of x(t) and y(t) are finite.The estimation of the system state and inputs are bounded and the system trajectory estimation is non-decreasing.

Dater approach
To estimate the system state and input, inequalities ( 29) to ( 32) can be rewritten in the following form.
Inequalities ( 54) and ( 55) can be gathered in the matrix inequality (57) which permits to estimate the system input by computing a lower bound of u(k) denoted u − (k) and an upper bound denoted u + (k).
In the same manner, inequalities (53) and ( 56) can be gathered in the matrix inequality (58) which is able to compute a lower bound of x(k) denoted x − (k) and an upper bound denoted x + (k) : To guarantee the non decreasing behavior of the system trajectory, the following conditions are added: Inequalities ( 58) and ( 59) can be aggregated in the augmented matrix inequality (61): where: Moreover, inequalities (57) and (60) can be aggregated in the augmented matrix inequality (62): where: The estimator (63) is obtained using the inequalities (61) and (62): The obtained estimator ( 63) is an algebraic model giving simultaneously an upper and lower bounds of the system state and input.State and input estimation obtained using this estimator are generally useful for fault estimation, detection and localization.The upper and lower bounds of the estimation of x(k) and y(k) are finite.The estimation of the system state and inputs are bounded and the system trajectory estimation is non-decreasing.
Inequalities (52) and (63) describe the elaborated estimator using successively the count and dater approaches.These models allow obtaining an estimation of the upper and lower bounds of each elements of the state vector.At each moment, state vector estimation must belong in the interval between the upper and the lower bounds.

Numerical example
Example 3 Let us consider the example of Petri net given in figure 1 of section 3.3 For this example, the system input is u = x 1 and the system output is y uy and W − uy are given in equations ( 40) and (41).For the system simulation, y = x 4 is assumed to be known.For example we can obtain the following table presenting 5 iterations.

Introduction
In this section, the proposed state model (39) and estimator (63) will be applied to a model of baking process.The objective of this application is to validate their efficiency.A dater approach is followed in order to validate the proposed system state and estimator.Indeed, the application is divided into four parts.The first part is an application of the proposed state model (39).The objective of the second part is to apply the proposed estimator.Residuals are computed in the third part in order to validate the proposed state model and estimator (63).In the last fourth part of this application, we propose to use the proposed state model and estimator to the fault detection and localization.
The model of the baking process is proposed in [14,20].It describe two ranges of bread production with two different qualities.The baking process is modeled by a time interval Petri net and its operating is detailed in [14,20].The model of the baking process is given in figure 2 [14].

State estimation
The table 4 gives the estimation result of the system state and input.The symbol (ˆ) means that it is an estimated variable.For the estimation, it is supposed that only the system output y(k) is known.The results given in table 4 are obtained using the estimator proposed in the equation ( 63  Table 4 gives a possible estimation of the state vector at each date k The estimation is based on the interval of y(k).

Residuals computation
Simulation and estimation results give for each variable a lower and upper bound.Residuals are generally computed to know the precision of the state estimation.In this work, residuals are computed as the intersection between the two obtained intervals.If the intersection is different of ∅, it is supposed that the estimation is accepted.Residuals presented in table 5 show that estimation is accepted in more than 89% of cases.Residuals presented in table 5 show that estimation is accepted in more than 89% of cases.Indeed, the estimation is not precise only for few cases.In more than 89% of cases, the estimation is cosidered precise since the intersection between the simulation result and the estimated values is not empty.Residuals computation show that the proposed system state (39) and estimator (63) are sufficiently precise.This result can be used for fault detection and localization and/or fault estimation.

Fault detection and localization
The main objective of this part is to apply the results of estimation and simulation to make fault detection and localization.It is important also to deduce if the fault detection is made at the same date or with a delay.To validate this approach work, it is supposed that the studied system is affected by a fault.In this work, the fault is modeled by an unknown input transition for the place P 11.This transition is denoted α(k).It is supposed that this transition can take, at many dates, very high values.For simulation, it is supposed that the fault α(k) takes the values given in the table 6.The following inequality is added.Table 6 shows that for the dates k = 2, 5 and 7, α(k) takes the value 5000 and the transitions x 11 , x 12 , x 13 and y are immediately (at the same date) affected by the fault α(k).However, the transitions x 3 , x 4 , x 5 and x 6 are affected by the fault with a delay of one iteration.The transitions x 1 , x 2 , x 7 , x 8 , x 9 , x 10 and u are not affected by the fault α(k).
The analysis of the result presented in table 6 let to conclude that the fault detection and localization is made immediately at the same time of the fault appearance or at least after one iteration.Transitions which are not affected by faults are also known.
The state vector presents some elements which are sensibles immediately to faults, other elements are sensibles to this fault with a delay.Some elements are not affected by this fault.
The simulation of the proposed state model let to detect the fault in real time or with a delay and let to know the transitions not affected by the fault

Fault estimation
The objective of this part is to use the proposed system state and estimator to fault estimation.The obtained results are presented in the table 7. The fault estimated α(k) is considered sufficiently precise.The following inequality is added and can be used for fault estimation.According to the obtained fault estimation, it is possible to conclude that this estimation is sufficiently precise since the theoretical values of the fault are belonging in the obtained interval.
The proposed estimator let to estimate the system state and inputs and also the fault affecting the system.

Discussion
The proposed state model (39) and estimator (63) are used successively to simulate and estimate the system state using the values of matrices W + ux , W − ux , W + xx , ❅ −W − xx , W + xy , W − xy , W + uy and W − uy given by equations ( 64) and (65).The system input is also estimated and the obtained results are presented in form of tables.For each element of the state vector, a lower and upper bounds are computed for both cases of simulation and estimation.To analyze the robustness of the estimation, residuals are computed.In this work residuals are computed as the intersection, for each element of the state vector, of both intervals given by simulation and estimation.It is considered that the estimation is acceptable if the intersection between the intervals are different of ∅.The next step of the application is devoted to the fault detection and localization.For this case, it is supposed that an unknown transition is added to the system as an input transition of the place P 11.The application of the proposed state model (39) and estimator (63) show that the fault localization is possible and easy by the analysis of bounds of the state vector.The fault detection is possible can be at the same date of its apparition or with a delay of one iteration at max.The last step of this application is dedicated to the fault estimation.The analysis of the obtained estimation result show that the real value of the unknown input α(k) is included in the obtained interval, so it is supposed that the fault estimation is acceptable.

Conclusion and future works
This work allowed proposing proposes a new state model describing the time evolution in time of discrete events systems modeled by time interval Petri nets.A new estimator is also proposed allowing estimating the whole system state and input.The proposed state model and estimator allow computing an upper and lower bounds for each element of the system state vector.To test the robustness of the state estimation, residuals are computed as the intersection between the two intervals obtained by the estimation and the simulation.The proposed estimator can be used also for fault detection, localization and estimation.The state model and the estimator are proposed following successively a count and a dater approaches.In order to show the efficiency of the proposed state model and estimator, they have been applied, following a dater approach, to a baking process modeled by a time interval Petri net.This The application of the proposed state model and estimator to the baking process shows that the state estimation is acceptable in more than 89% of cases.It shown also that the fault localization is garanteed and its detection is possible at the same date daye of its apparition or, at max, by a delay of one iteration.The fault estimation is also possible and acceptable.
It is possible to extend this work to analyze the optimality of the estimation using the principle of linear programming where we can add a criteria which must be respected in the simulation of the state model and the estimation.

Compliance with Ethical Standards
There is no funding in this study Authors declare that there is no conflict of interest.This article does not contain any studies with human participants or animals performed by any of the authors.

Fig. 1
Fig. 1 Elementary example on time interval Petri net

Table 1
Simulation of example-1

Table 3
System simulation

Table 4
State estimation

Table 5
Residuals computation

Table 6
Fault detection and localization

Table 7
Fault estimation