Enhanced multiplex binary PIR localization using the Transferable Belief Model

A. HADJ HENNI, R. BEN BACHOUCH, O. BENNIS and N. RAMDANI Abstract-Pyro-electric Infra Red (PIR) sensors have been widely used in different indoor localization applications during the last decade. These sensors are cheap, non-intrusive and nonwearable, nevertheless, the binary PIR sensor detects only the presence of a human motion in its field of view without any other information about the actual location. Therefore, to localize a person in different zones of interest, the use of several PIR sensors with overlapping field of view is necessary. To reduce the number of sensors used, we use multiplex masks with the binary PIR sensors to obtain a compressed overlapping structure. Such a structure induces ambiguity during transitions between zones. In this paper, we show how to circumvent this issue by using a novel localization algorithm based on the transferable belief model TBM. Besides, we show how to tune efficiently the parameters of our algorithm, by choosing an appropriate discounting factor within (TBM). Experiments using standard commercialized sensors equipped with the multiplex masks, emphasize the performance of our novel method Index Terms-Indoor localization, binary PIR sensing, transferable belief model, sensors data fusion, smart homes.

I. INTRODUCTION

N OWADAYS, the human indoor location-tracking using sensor networks has become a necessity for several smart home applications such as anti-intrusion security systems, comfort and energy optimization, and elderly monitoring. A large range of indoor localization systems has been investigated during the last decade, and different sensing technologies were used depending on the considered smart home application. These systems can be mainly divided into two types, the wearable and the non-wearable device systems as shown in a recent survey [START_REF] Ahvar | On analyzing user location discovery methods in smart homes: A taxonomy and survey[END_REF].

In this paper, we are interested in an elderly monitoring application which rather requires non-wearable devices since they are more convenient and more comfortable for the elderly. Among the non-wearable devices, the Pyro Infra-Red PIR sensors are often used alone or with other sensing modalities in smart home applications. In fact, E.Ahvar et al [START_REF] Ahvar | On analyzing user location discovery methods in smart homes: A taxonomy and survey[END_REF] have analyzed some non-wearable localization systems w.r.t their technical and technological features, and discussed their advantages and inconveniences Relying on this analysis, we note that cheap and reliable non-wearable systems often involve PIR sensors. The latter are inexpensive compared to other technologies, are non-intrusive and have low energy consumption. A PIR sensor detects thermal variations occurring in its FOV (Field of View) and translates the variations as an electrical signal known as analog output. The latter can This work was funded by the french CoCaPs project issued from FUI 20. All authors are with the Univ. Orleans, INSA-CVL, PRISME, EA 4229, F45072, Orleans, France. Corresponding author: A. HADJ HENNI (raouf.hadj.henni@gmail.com) be processed either to provide information such as angle and direction of movement, speed and distance from sensor [START_REF] Yun | Human movement detection and identification using pyroelectric infrared sensors[END_REF] [3], or for zoning the moving target as in [START_REF] Monaci | Indoor user zoning and tracking in passive infrared sensing systems[END_REF]. Nevertheless, analog outputs are not widely used for indoor location-tracking applications due to the processing time and the intensive computing resources they require, and also to the saturation of the communication network they induce. Therefore, the analog output is often converted into a binary output to provide only binary data such as presence/absence of a human motion. Related works that use binary PIR outputs for indoor localization can be divided into two categories depending on the sensor network structure.

The first category uses PIR sensor networks without overlapping FOV e.g. each room is covered only by one sensor as in [START_REF] Danancher | Online location tracking of a single inhabitant based on a state estimator[END_REF]. However, in some elderly monitoring applications, several activities can be done within the same room which requires covering each zone of interest with its own sensor to discern the different activities within a same room as in [START_REF] Ballardini | An indoor localization system for telehomecare applications[END_REF]. Unfortunately, covering each zone with its own PIR sensor is not always easy due to sensor placement constraints as induced by the presence of different obstacles [START_REF] Fanti | An integrated framework for binary sensor placement and inhabitants location tracking[END_REF] or constraints related to the smart home architecture. For instance, when several activity zones (i.e. zones of interest) are close to each other within the room, the FOV of the sensors of two neighboring zones may overlap other zone(s) of interest, hence, the FOV's overlapping structure cannot be avoided.

The second category of systems uses binary PIR sensor networks with overlapping FOVs. Such structure suffers from invalid intersections caused by the interference between different PIR FOVs even when tracking a single target. To cope with this issue, Bo.Yang et al proposed several methods in recent years either using classifiers, which require a training phase as in [START_REF] Yang | Distributed multi-human location algorithm using naive bayes classifier for a binary pyroelectric infrared sensor tracking system[END_REF], or without classifiers as in [START_REF] Yang | Credit-based multiple human location for passive binary pyroelectric infrared sensor tracking system: Free from region partition and classifier[END_REF]. These methods use specific sensor-nodes with 6 PIRs per node, however, such sensor nodes may increase the data throughput due to the important number of sensors used. Also, designing such sensor nodes is not always easy in practice since it requires deep technical skills and technological knowledge. Moreover, Bo.yang et al works deal with target position coordinates as in [START_REF] Djuric | Target tracking by particle filtering in binary sensor networks[END_REF]. To reduce the number of PIR sensors used in an indoor localization application, a compressive binary PIR structure was proposed in [START_REF] Lu | Space encoding based compressive multiple human tracking with distributed binary pyroelectric infrared sensor networks[END_REF] [START_REF] Lu | Binary compressive tracking[END_REF] by adding so-called multiplex masks to the PIR sensors. Nevertheless, the application in these studies [START_REF] Lu | Space encoding based compressive multiple human tracking with distributed binary pyroelectric infrared sensor networks[END_REF] [START_REF] Lu | Binary compressive tracking[END_REF] relies on some geometric information in order to enhance the location resolution which is similar to the case of study in [START_REF] Shrivastava | Target tracking with binary proximity sensors: fundamental limits, minimal descriptions, and algorithms[END_REF]. In our case, we are not interested in the precise coordinates of the user's position nor in enhancing the location resolution, we rather aim at estimating the probability of presence over some predefined zones of interest.

Our application falls within the same work as in [START_REF] Ballardini | An indoor localization system for telehomecare applications[END_REF] which addresses human indoor localization within zones of interest using binary PIR sensors for a telehomecare application, that is the monitoring of a person living alone. The localization algorithm in [START_REF] Ballardini | An indoor localization system for telehomecare applications[END_REF] relies on Bayesian filtering, using a probabilistic motion model and past location estimates for prediction, while the likelihoods are specified manually for each zone by the person in charge of the configuration. The main drawback of the method used in [START_REF] Ballardini | An indoor localization system for telehomecare applications[END_REF] resides in the large number of sensors required to cover the zones of interest, since each zone was covered by its own specific sensor. Also, allowing a person specifying a list of boolean expressions for each zone to obtain likelihoods depending on activated sensor(s) is a tedious task, even more when the number of zones is large. To cope with this drawback, we have recently extended this localization approach, first by using a multiplex overlapping structure to reduce the number of needed sensors when covering the zones of interest, and second by automatically computing the likelihoods without using boolean expressions [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF].

Nevertheless, only simulation results were presented as evaluation of the Bayesian filtering in [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF] as it appeared that the overlapping structure of our system induces ambiguity when the person transits between zones. This is because during a transition, the sensors of both neighboring zones may activate leading to the erroneous signature of an other neighboring zone. Such transition issue has already been identified in [START_REF] Ballardini | An indoor localization system for telehomecare applications[END_REF] as the main source explaining large location errors.

In this paper, the first contribution addresses the ambiguity issue induced by the transitions between zones, by using an alternative framework. We propose a prediction-correction filtering algorithm based on the Transferable Belief Model (TBM) framework [START_REF] Smets | The transferable belief model[END_REF] instead of the Bayesian framework since TBM can naturally handle such ambiguity through the power set. The localization is performed by combining two different information sources namely the prediction, which is obtained using past location results and a human motion model, and the observation source which updates the prediction relying on the sensor network measurement data. The trade-off between the two sources depends on whether a transition between zones is occurring or not, a situation that can be distinguished relying on the received measurement data.

The second contribution of this paper addresses an efficient tuning of TBM. We propose a novel way for choosing an appropriate discounting factor, which is a factor used to weight an information source compared to other source(s) within TBM. The appropriate value is chosen by analyzing the behavior of the Dempster-Shafer (DS) normalization function in order to avoid counter-intuitive location results after DS normalization.

Finally, experiments using commercialized sensors, for which we modify the field of view using multiplex masks, show the performance of our novel algorithm compared to the one we presented in [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF]. The rest of this paper is organized as follows: Section II presents some aspects of the TBM framework. Section III describes the sensors specifications and their deployment. Section IV describes our novel localization approach using the TBM. Section V shows the experimental results and section VI concludes the paper.

II. ASPECTS OF THE TRANSFERABLE BELIEF MODEL

This section presents some TBM theoretical aspects which are used next in our localization algorithm.

A. Main definitions:

TBM is a theoretical framework that uses belief functions to represent knowledge about given hypotheses, but also about sets of these hypotheses. It can be considered as an alternative to Dempster-Shafer theory DST [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] with some extensions and additional tools. Among the common aspects between TBM and DST, we have the mass function denoted m and the power set 2 Ω . Let the set of hypotheses Ω = {h 1 , h 2 , ..h n } be the Frame of Discernment (FoD), and let 2 Ω = {{φ}, {h 1 }, ..., {h 1 , h 2 }, ..., {Ω}} be the power set of Ω. A mass function m(H) over a hypothesis (i.e. when H = h i ) or over a set of hypotheses (i.e. when H = {h i , .., h j }), takes values in the range [0, 1]. It is assigned to all elements H of the power set 2 Ω and satisfies:

H∈2 Ω m(H) = 1.
The elements H of 2 Ω that may have m(H) > 0 are called focal elements. The set of all focal elements is denoted ξ. The power set 2 Ω allows us to represent the knowledge over a set of hypotheses which represents the uncertainty when we are not sure about a singleton (i.e. a single element of Ω). This is the main advantage compared to frameworks based on the probability theory such as the Bayes framework. Other common aspects between TBM and DST are the belief function Bel(.) and the plausibility function P l(.) defined for all H k ⊆ Ω as:

Bel(H k ) = Hi|Hi⊆H k m(H i ), P l(H k ) = Hi|Hi∩H k =φ m(H i ).
The belief Bel(H k ) (resp the plausibility P l(H k )) represents the lower bound (resp the upper bound) of the probability of

H k i.e.
Bel(H k ) ≤ P rob(H k ) ≤ P l(H k ).

B. Basic combination rules:

Several combination rules exist within the TBM framework, among them, there is the conjunctive rule and its normalized version which corresponds to the DS rule, and the disjunctive rule known as the prudent rule. Let A, ∈ 2 Ω , and let m 1 , m 2 be mass functions obtained from two independent knowledge sources. The conjunctive rule is then:

∀A ∈ 2 Ω , m 1∩2 (A) = B,C∈2 Ω ,B∩C=A m 1 (C) • m 2 (B). (1)
Then, the normalized conjunctive rule, i.e. DS rule is:

∀A ∈ 2 Ω \ φ, m 1⊕2 (A) = 1 1 -m 1∩2 (φ) • m 1∩2 (A), (2)
and m 1⊕2 (φ) = 0. Finally, the disjunctive combination rule is:

m 1∪2 (A) = m 1 ∪ m 2 (A) = B∪C=A =φ m 1 (C) • m 2 (B). (3)
Note that all these combination rules require that both mass functions are defined over the same frame of discernment and their sources must be independent sources of evidence.

C. Operations over FODs:

This subsection presents only some operations over FoDs which are used in this paper.

1) Refinement and vacuous extension: Let Ω be a frame of discernment and Ω be a refinement of Ω, i.e., each hypothesis h i of Ω has an image in one or in a set of hypotheses of Ω . This means that hypotheses of Ω are more detailed in Ω by splitting at least one hypothesis h i of Ω into several hypotheses in Ω . Refinement operations over FoDs can be generalized to mass functions.

Let m Ω be a mass function over Ω, and let ρ() be the refinement function i.e. Ω = ρ(Ω). The mass m Ω can be transferred to Ω through the vacuous extension m Ω↑Ω as follows:

m Ω↑Ω (h ) = m Ω (h) if h = ρ(h) 0 otherwise (4)
with h ∈ Ω and h ∈ Ω .

2) Coarsening: Coarsening, the dual operation of refinement, reduces the size of a FoD by making it more coarse, i.e. if Ω is a refinement of Ω (Ω = ρ(Ω)) then Ω is a coarsening of Ω . Nevertheless, the refinement is not always surjective i.e. some element(s) of 2 Ω may not have image(s) on 2 Ω . This issue leads to inner and outer reduction as pointed in DST [16, pp 117-118].

3) Inner and outer reduction: Inner reduction ω and outer reduction ω are functions from 2 Ω to 2 Ω that satisfy:

ω(A) = {ω ∈ Ω | ρ({ω}) ⊆ A} (5) ω(A) = {ω ∈ Ω | ρ({ω}) ∩ A = φ} (6)
where A ⊆ Ω and Ω = ρ(Ω).

The inner (resp outer) reduction concept can be extended to mass functions as in [START_REF] Fanti | An integrated framework for binary sensor placement and inhabitants location tracking[END_REF] (resp in [START_REF] Yang | Distributed multi-human location algorithm using naive bayes classifier for a binary pyroelectric infrared sensor tracking system[END_REF] ) :

m Ω (ω) = A⊆Ω , ω(A)=ω m Ω (A), ∀ ω ⊆ Ω (7) m Ω (ω) = A⊆Ω , ω(A)=ω m Ω (A), ∀ ω ⊆ Ω (8)
More details and examples about inner and outer reduction can be found in [START_REF] Denux | Approximating the combination of belief functions using the fast mbius transform in a coarsened frame[END_REF] D. Pignistic transform Within TBM, the pignistic probability BetP can be used to transform the mass function m Ω of a singleton into a probability by transferring the mass functions of an uncertainties as follows:

BetP Ω (C) = 1 1 -m Ω (φ) A⊆Ω, c∈A m Ω (A) | A | (9)
where | A | is the cardinal of A i.e. is the number of elements in A.

III. SENSORS DETAILS AND DEPLOYMENT A. PIR sensors details

In our application, we use wall-mounted 180 • PIR sensors commercialized * by Legrand company and shown in Fig. 2.a Each sensor is provided along with the mask shown in Fig. 2.b that can be used to cover one or several angles of view.

The provided mask is easy to cut with pliers, nevertheless, large errors on the FoV were obtained during experimental tests since the mask is not deep enough. For instance, if we cut a portion of 30 • we will obtain a FoV of around 50 • or more instead of 30 • . To minimize such large error and obtain consistent angles w.r.t desired FoVs, we add a 3D printed shields Fig. 2.c and paste them with the provided mask using standard glue gun to obtain our own multiplex masks as shown in Fig. 2.d. Such a manipulation does not require deep technological knowledge nor high technical skills compared to the manipulations required for designing the specific sensor nodes as in [START_REF] Yang | Distributed multi-human location algorithm using naive bayes classifier for a binary pyroelectric infrared sensor tracking system[END_REF] . Note that the shields used are effective, however, other shields can be investigated.

Since a PIR sensor detects human motions based on a thermal variation, it is also sensitive to any other thermal variation caused by non-human objects (e.g. heater, airflow, ..etc.) which may induce erroneous measurement known as false positive. In addition, such sensors may miss some human motions depending on the distance from the sensor and on the motion's nature (large/small motion, radial/tangential motion w.r.t the sensor, ..etc.) which leads to false negative.

The sensors outputs are sent through a KNX communication network and are acquired using its associated ETS software 1 .

B. The living lab and sensors deployment

Fig. 1.a) illustrates the living lab GIS MADONAH2 which is a nursing home equipped with different sensors and technologies to study senior's behavior in order to improve his/her home support and nursing. For this purpose, and for subsequent decision making studies, we need to localize the person within some predefined zones of interest. The zoning of the living lab is shown in Fig. 1.b) and obtained using seven PIR sensors along with multiplex masks with different FOVs. Each sensor has two FOVs except for purple and green sensors3 , and the uncovered FOVs are marked with discontinuous arcs. The living lab is divided then into thirteen different zones but note that some of these zones are not linked to activities (e.g. Zone3.a, Zone3.b and Zone6). Such zones are obtained due to the overlapping structure and can be considered as passage zones instead of activity zones. Also, very small zones are not considered due to their small size (e.g. zone between Zone6 and Zone9) since if the person moves inside them he/she will activate sensors of their large neighboring zones, hence, they correspond rather to transitions. The BathRoom is not considered and the kitchen is partially considered in this paper. The bed zone is not considered since a bed rest sensor will be used subsequently.

To further investigate the impact of sensors deployment on our novel localization method, we have considered another deployment Deployment B ( where Green, P urple, Y ellow, and Black sensors have been moved to other walls) as shown in Fig. 1.c). Remember that we are interested in covering some specific zones of interest, hence, the Deployment B should cover more or less the same zones as Deployment A.

C. Zone validation:

The validation of each zone is obtained by performing several large motions inside the considered zone and by checking the activation of the sensors covering it. For example, to validate Zone10 we performed several walking motions within and around the sofa and checked that the Blue, Red and P urple sensors will activate. Note that a reasonable uncertainty on the sensor's FOV is not really problematic as long as the zones of interest are still covered by the desired sensors. Indeed, since we are more interested next in the activities and not in the exact position coordinates, we just need to make sure that the person is inside the activity zone regardless of its size variation (e.g. ensure that the person is on the sofa regardless the shape of Zone10), consequently, the FOV's variations are related rather to transitions.

Each of the thirteen zones has a color code corresponding to the sensor(s) covering it. With such a zoning, it is clear that false positive and false negative are critical issues. For example, if the person is moving inside Zone12 and if the red sensor is activated due to a false positive, we will receive a measurement (red + green activated) corresponding to Zone7 in Fig. 1.b). To the contrary, if the person is moving inside Zone7 and red sensor will not detect the motion due to a false negative (caused by a radial motion w.r.t red sensor) we will receive green alone as a measurement which corresponds to Zone12. Such false positive and false negative can be overcome through filtering (i.e. prediction/correction) techniques such as in our previous approach [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF]. Nevertheless, the latter method cannot overcome the ambiguity issues induced by the transitions such as the one between Zone3.b and Zone3.a which sensors activation signature is the same as the signature of Zone4 in Fig. 1.b. This may easily lead to localization errors if we do not handle the transitions between zones.

IV. OUR NOVEL LOCALIZATION APPROACH

A. Problem formulation

The localization problem is formulated as a state estimation problem and solved using a prediction-correction filtering algorithm. Prediction is obtained using the past location estimates and a human motion model. Correction exploits the binary outputs provided by the PIR sensor networks. Since we will combine the prediction with the measurements data within TBM, the first step is to define the Frame of Discernment (FoD) of each source.

Let Ω pre (resp Ω obs ) be the prediction's FoD (resp the observation's FoD). These two FoDs are defined for each Zone(i) as follows:

Ω pre = {Inside, Inside}, (10)
Ω obs = {Inside, N eigh, F ar}. ( 11
)
where N eigh ∪ F ar = Inside. The power sets are

2 Ωpre = {φ, Inside, Inside, Ω pre }, (12)
2

Ω obs = {φ, Inside, N eigh, F ar, Inside ∪ N eigh T ransition , Inside ∪ F ar, N eigh ∪ F ar Inside , Ω obs }. (13)
Therefore, it is clear that Ω obs is a refinement of Ω pre . Ω pre contains two hypotheses which state that a person is either inside or outside a considered Zone(i). In fact, for each Zone(i) we only predict if the person will be inside or outside the zone, because of the motion model used as detailed further in Sect.IV-B. However, Ω obs contains more details since by considering the sensor network data instead of a single sensor data we can distinguish when the person is inside the zone, transiting to a neighbor zone i.e. {Inside ∪ N eigh}, or outside the zone i.e. {N eigh ∪ F ar}. Because of the transition {Inside∪N eigh}, Ω obs is more detailed than Ω pre . The set of focal elements for the observation is then

ξ Ω obs = {Inside, Inside ∪ N eigh, N eigh ∪ F ar, Ω obs }.
One can say that a single binary PIR output can discern only Inside and Inside. However, note that we rather deal with a sensor network, hence, several binary outputs can be received within a time window that may refer to either a zone and/or a transition between zones. In fact, since the prediction requires a motion model which is defined w.r.t a time window, the measurement data will be harvested within the same time window. This can easily lead to receiving several binary PIR outputs from different zones within the same time window. Note that taking Ω = {Zone(1), ....Zone(n)} as a common FoD is not consistent in our case since the prediction must be computed only for each zone separately, i.e. prediction cannot directly discern all zones. Moreover, the cardinal of such FoD increases with the number of zones and may lead to computational issues since the computational cost during the combinations depends on the cardinality of the FoD [START_REF] Yang | A novel approximation of basic probability assignment based on rank-level fusion[END_REF].

B. Mass functions construction 1) Prediction mass functions:

The prediction step relies on the human motion model along with the past location probability results at t-1 to predict the probability of presence in each zone for the current time t. The set of focal elements of Ω pre is ξ Ωpre = {Inside, Inside}. The mass function m Ωpre pre of these focal elements, which is then a Bayesian mass function, is modeled for each Zone(i) as follow:

m Ωpre pre (Inside, i) = (P i,t-1 ) • γ stay + Nj j=1 (P j,t-1 ) • γ move + N k k=1 (P k,t-1 ) • γ jump (14) m Ωpre pre (Inside, i) = 1 -m Ωpre pre (Inside, i), (15)
where N j (resp N k ) is the number of Neighbor (resp Far) zones of Zone(i). Besides, (P i,t-1 ), (P j,t-1 ) and (P k,t-1 ) are respectively the past probability of presence in Zone(i), in its neighbor zones Zone(j) and in its further zones Zone(k). γ stay , γ move and γ jump are the parameters of the motion model and represent respectively the probability of staying within the same zone, moving to a neighbor zone or jumping to a far zone. The probability γ move should generally be smaller than γ stay and the probability γ jump should be far smaller but not null i.e. γ stay > γ move >> γ jump . This rule is inspired from the following reasoning: Firstly, the chances that a person jumps to a far zone within one second time window are null, however, γ jump should be very small but not zero to allow to recover from a failure at next time windows as shown in [START_REF] Ballardini | An indoor localization system for telehomecare applications[END_REF].

Secondly, γ move and γ stay should be equal in theory since if a person is inside a Zone(i) at t -1, then, he/she could either stay in the same zone or move to a neighbor zone at the next time t. Nevertheless, even if the person decides to move into a neighbor zone, the person could be either completely inside the neighbor zone, or still transiting between the two zones due to the uncertainty on the FoV of the PIR sensors. Since a transition is regarded as being inside both zones, then, even if the person decides to move to a neighbor zone at next time, there is a chance that he/she will partially be inside the previous zone (partially stay). In other words, the possibility of transition is included in γ stay . Consequently, the probability γ stay should be higher than γ move in practice. Note that no automatic way exists to estimate these parameters, nevertheless, changing the values of these parameters does not greatly impact the localization results as long as the rule γ stay > γ move >> γ jump is satisfied as shown further in Sect. V-B. Besides, this model is also used in [START_REF] Ballardini | An indoor localization system for telehomecare applications[END_REF] and [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF], therefore, it will be more consistent next to compare the results of the TBM localization algorithm w.r.t the ones of the previous algorithm [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF].

2) Observation mass functions: For each Zone(i), the masses m Ω obs obs of the focal elements are modeled over Ω obs as follow:

• m Ω obs obs over Inside and N eigh ∪ F ar:

m Ω obs obs (Inside, i) = N Act (i) • µ(i) N Cov (i) • N T ot act (16) m Ω obs obs (N eigh ∪ F ar, i) = N N on act (i) • µ(i) N Cov (i) • N T ot act (17)
where N Act (i) (resp N N on act (i) ) represents the number of activated (resp non activated) sensor(s) in Zone(i). N T ot act is the total number of activated sensors of the whole sensor network at time t. N Cov (i) = N Act (i) + N N on act (i) is the number of sensor(s) covering Zone(i).

In fact, the belief of being Inside (resp N eigh ∪ F ar = Inside) a Zone(i) depends on the number of its activated N Act (i) (resp non activated N N on act (i)) sensor(s) at time window t, against both the number of sensor(s) covering Zone(i) i.e. N Cov (i) and the total activated sensors in the whole networks N T ot act at time window t. Nevertheless, to obtain appropriate mass functions, for example, a unit mass m Ω obs obs (Inside, i) in Zone(i) when all its covering sensor(s) are activated and when their number is equal to the number of total activated sensor(s) (i.e. m Ω obs obs (Inside, i) = 1 when N Act (i) = N T ot act ), we add the normalization parameter µ(i). The latter corresponds to N Act (i) since it will be simplified with N T ot act in such example, and omitting it may lead to inconsistent mass functions generation. Indeed, mass normalization is sometime necessary, for example, in [START_REF] Dallil | Sensor fusion and target tracking using evidential data association[END_REF] authors added a normalization factor during mass functions generation to obtain appropriate mass functions. Note that if the number of activated sensors is null in a Zone(i) and if the total number of activated sensors is not null, then the m Ω obs obs (N eigh ∪ F ar, i) is set to one. Also, if there is no measurement at time t, we do not compute observation mass functions since we will keep the previous location results.

• m Ω obs obs for Inside ∪ N eigh during a transition: Transition between zones can be seen as being Inside several zones. In fact, a transition between two zones Inside(i) ∪ N eigh(i) can be seen as Inside(i) ∪ Inside(j) with j related to Zone(j) which is the neighbor zone of Zone(i) involved in the transition. Therefore, constructing mass function for a transition Inside ∪ N eigh is similar to constructing a mass for Inside for several zones. Consequently, the transition mass function depends on the number of activated sensor(s) in each involved zone, against both the number of sensor(s) covering each zone and the total activated number at the time window t. This leads to [START_REF] Yang | A novel approximation of basic probability assignment based on rank-level fusion[END_REF], which shares similar parameters with [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF], as follows:

m Ω obs obs (Inside ∪ N eigh = T ransition, i) = (N Act (i) + N Act (j)) • µ transit (i ∪ j) (N Cov (i) + Cov (j)) • N T ot act (18)
where µ transit (i ∪ j) is the normalization parameter for transitions defined as:

µ transit (i ∪ j) = (N Act (i) + N Act (j)) -N Common
N Common is the number of common activated sensors between zones involved in the transition and we subtract it to have µ transit (i ∪ j) N T ot act (accordingly, to have m Ω obs obs (Inside ∪ N eigh, i) 1). The mass in (18) corresponds to a transition mass function involving two zones, nevertheless, it can be extended if there are additional zones involved. Note that for the zones involved in a transition, their corresponding m Ω obs obs (Inside) and m Ω obs obs (N eigh ∪ F ar) are set to zero, since during a transition we cannot discern between Inside and N eigh ∪ F ar for involved zones.

• m Ω obs obs over Ω obs : In a non-transition case, we have m Ω obs obs (Inside, i) + m Ω obs obs (N eigh ∪ F ar, i) 1. Also, during a transition case we have m Ω obs obs (Inside ∪ N eigh, i) 1. Therefore, to get

H∈2 Ω obs
m Ω obs obs (H) = 1 in both cases we define:

m Ω obs obs (Ω obs , i) = 1 -m Ω obs obs (Inside ∪ N eigh, i)+ m Ω obs obs (N eigh ∪ F ar, i) + m Ω obs obs (Inside, i) (19)
C. Appropriate discounting factor and mass weighting

When fusing data from different sources, it is judicious to weight the provided data w.r.t to the confidence of its corresponding source. In our case, the observation source is less reliable than the prediction source, therefore, the observation mass functions m Ω obs obs are weighted using the discounting defined by Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] as shown in [START_REF] Martin | Comparative study of information fusion methods for sonar images classification[END_REF].

α obs m Ω obs obs (C) = (1 -α obs ) • m Ω obs obs (C), ∀C Ω obs α obs m Ω obs obs (Ω obs ) = (1 -α obs ) • m Ω obs obs (Ω obs ) + α obs (20
) where α obs m Ω obs obs is the weighted mass. α obs is the discounting factor and (1 -α obs ) is the confidence in the observation.

Nevertheless, the choice of an appropriate value for the discounting factor is challenging. This value can be estimated a priori through a learning phase e.g. [START_REF] Martin | Comparative study of information fusion methods for sonar images classification[END_REF] or evaluated a posteriori after treating beliefs of the different sources based on distances [START_REF] Elouedi | Assessing sensor reliability for multisensor data fusion within the transferable belief model[END_REF] and conflict measure [START_REF] Martin | Conflict measure for the discounting operation on belief functions[END_REF].

In our case, the observation source m Ω obs obs is reliable but not at 100% (e.g due to delayed data within a time window, false positive/negative, ... etc.) let say (1 -α obs ) ∈ [0.7, 0.95] i.e. α obs ∈ [0.05, 0.3]. Unfortunately, choosing an inappropriate value of α obs in this range can lead to counter-intuitive results during DS normalization over each zone, hence, leading to inconsistent location results as shown further in Example01 of Appendix B. Note that this counter-intuitive issue in our case is different from the famous Zadeh's counter-intuitive issue [START_REF] Lotfi | Book review: A mathematical theory of evidence[END_REF]. Indeed, Zadeh's issue can occur when a source has a so called dictatorial power as shown in [START_REF] Tchamova | On the behavior of dempster's rule of combination and the foundations of dempster-shafer theory[END_REF], however, in our case the counter-intuitive issue may occur since the belief m pre∩obs (Inside, Z i ) of some zone(s) may increase significantly compared to the other zones during DS normalization since each zone has its own corresponding mass of empty set. Consequently, relying on a priori learning phase [START_REF] Martin | Comparative study of information fusion methods for sonar images classification[END_REF] or on a posteriori evaluation as in [START_REF] Elouedi | Assessing sensor reliability for multisensor data fusion within the transferable belief model[END_REF] [START_REF] Martin | Conflict measure for the discounting operation on belief functions[END_REF] may not necessarily lead to the appropriate discounting, since in our case, this appropriateness does, not only depend on the reliability of the source, but it also depends on the behavior of the DS normalization function as illustrated in Appendix B. Therefore, the discounting factor(s) of information source(s) should satisfy the constraint (21) bellow (See Appendix B for demonstration).

(1 -α 1 ) • .. • (1 -α n ) ≤ max(m φ ) (21)
where: n is the number of sources to be discounted. max(m φ ) is the maximum value that mass of empty set of each zone should not exceed to avoid counter-intuitive results. As a special case, if the n sources are considered equally reliable, this leads to (1 -α) n ≤ max(m φ ) which means that they should be equally discounted with α ≥ 1 -n max(m φ ) to ensure avoiding counter intuitive results. Finally, we show in Appendix B that max(m φ ) ≈ 0.75, and since we discount only the observation source i.e. n = 1, hence, the appropriate α obs for observation should satisfy the following:

1 -α obs max(m φ ) ≈ 0.75 ⇒ α obs 0.25 (22)
We take α obs = 0.25 to maximize beliefs from observation.

D. Combination and state filtering

We should note that prediction and observation are independent sources of evidence since the construction of the prediction mass functions depends only on the human motion model and the past results at t -1 whatever the state of the sensors (i.e. of the observation mass functions) at the current time t.

To combine prediction with observation, the mass functions must be defined over the same FoD. The challenge here is on which FoD should we combine. At first time, we combine prediction and observation masses over Ω obs since otherwise we will loose the transition information. Consequently, the prediction mass functions m Ωpre pre must be redefined over Ω obs using the vacuous extension defined in (4) as follows:

m Ωpre↑Ω obs pre (Inside, i) = m Ωpre pre (Inside, i), m Ωpre↑Ω obs pre (N eigh ∪ F ar, i) = m Ωpre pre (Inside, i).
We use DS combination rule to combine the prediction with the observation. Nevertheless, during a transition the DS normalization may cause a problem since the zones involved in the transition will have the null mass of the empty set (hence no DS normalization) in contrary to the rest of the zones, and this can further impact the location results. To cope with this issue, we switch to the disjunctive rule, known as the prudent rule, during a transition since the disjunctive rule does not induce the mass of the empty set during the combination. Therefore, all zones will be treated judiciously during a transition whether it is involved in the transition or not. The a posteriori combined mass is:

m Ω obs 1,2 = m Ωpre↑Ω obs pre α m Ω obs obs ( 23
)
where dot in is replaced by + when using DS rule, and by ∪ when using the disjunctive rule during a transition.

Indeed, DS rule may lead to counter intuitive results in several cases. The famous case is Zadeh's [START_REF] Lotfi | Book review: A mathematical theory of evidence[END_REF] example which shows counter intuitive results when the mass of the empty set is too high. However, in [START_REF] Tchamova | On the behavior of dempster's rule of combination and the foundations of dempster-shafer theory[END_REF] it is shown that counter-intuitive results can occur at any level of conflict in case where a source has a so-called dictatorial power. Fortunately, such dictatorial power is not present in our case since P l Ω obs pred (C) = 0 and P l Ω obs obs (C) = 0 ∀ C ∈ ξ Ω obs . In fact, since m Ωpre↑Ω obs pre (Inside) is never null neither equal to 1, hence, m Ωpre↑Ω obs pre (N eigh ∪ F ar) is also never null and P l Ω obs pred (C) = 0 ∀ C ∈ ξ Ω obs . Also, since the observation mass functions are discounted, we will always have α m Ω obs obs (Ω obs ) = 0, hence, P l Ω obs obs (C) = 0 ∀ C ∈ ξ Ω obs . Consequently, the dictatorial power issue is not present in both sources and there is no reason to use another combination rule (e.g. P CR5) as an alternative to DS rule during a nontransition.

Nevertheless, the only issue that occurs during transitions is that the a posteriori m Ω obs 1,2 (Inside, i) will increase for the non-involved zones during DS normalization (since their m Ω obs 1∩2 (Inside, i) will be multiplied by the DS normalization factor to remove the induced mass of the empty set) which is not the case for the involved zones. Indeed, a posteriori will never increase for the involved zones since their mass of empty set is null by default (because m Ω obs obs (Inside, i) = 0 = m Ω obs obs (N eigh ∪ F ar, i) as evoked in IV-B) and this is not fair since it may lead to inappropriate results when normalizing (i.e. increasing) the m Ω obs 1∩2 (Inside, j) of the non-involved zones as shown in Appendix A. One can say that we should use the conjunctive rule during transition, unfortunately, this is not possible since we consider that the FoD is exhaustive (i.e. a closed world).

Consequently, the appropriate combination rule for the transition case is the disjunctive rule despite of being less informative than DS rule.

E. Location probability distribution

To have a consistent comparison with our previous work, we need to transform the a posteriori mass function results to a location probability distribution P (with P (Inside) = 1 -P (Inside)). This requires two operations. The first one is the projection of the a posteriori combined mass obtained in [START_REF] Lotfi | Book review: A mathematical theory of evidence[END_REF] over Ω pre as follows:

m Ωpre 1,2 = m Ω obs ↓Ωpre 1,2 (24)
Projection in [START_REF] Tchamova | On the behavior of dempster's rule of combination and the foundations of dempster-shafer theory[END_REF] is obtained using the inner reduction. In fact, during a transition, the a posteriori mass obtained using the disjunctive rule m Ω obs 1∪2 (Inside ∪ N eigh) can be allocated either to m Ωpre 1∪2 (Inside) if using the inner reduction, or to m Ωpre 1∪2 (Inside) if using the outer reduction. However, we have noticed previously in IV-B that a transition Inside ∪ N eigh corresponds rather to Inside in several zones, hence, for each zone it is rather judicious to allocate m Ω obs 1∪2 (Inside ∪ N eigh) to m Ωpre 1∪2 (Inside) and not to m Ωpre 1∪2 (Inside). Also, when there is no transition, inner reduction leads to the same mass as the outer reduction since only m Ω obs 1⊕2 (N eigh) can make a difference (it can be either allocated to m Ωpre 1∪2 (Inside) if outer reduction, or to the empty set if inner reduction). Nevertheless, m Ω obs 1⊕2 (N eigh) is null when there is no transition because it is obtained by m Ωpre 1⊕2 (N eigh ∪ F ar) • m Ω obs 1⊕2 (Inside ∪ N eigh) and m Ω obs 1⊕2 (Inside ∪ N eigh) is null in non-transition case. The second operation transforms the projected mass functions into probabilities using the pignistic transform BetP Ωpre i (Inside) over each Zone(i) as follows:

P (Inside, i) = BetP Ωpre (Inside, i) = 1 1 -m Ωpre ⊕ (φ) • m Ωpre 1,2 (Inside, i) + m Ωpre 1,2 (Ω pre , i) 2 ( 25
)
where

m Ωpre 1,2
is the combined mass and m

Ωpre ⊕ (φ) is the mass of the empty set after combination and projection.

(C, i) = m Ωpre pre (H, i) if C = ρ(H) 0 else ∀ C ∈ ξ Ω obs , ∀ H ∈ ξ Ωpre 3.b) Combine m Ω obs
pre with m Ω obs obs using ( 20):

if Transition measurement case then m Ω obs 1,2 (C, i) = m Ωpre↑Ω obs pre (C, i) ∪ α m Ω obs obs (C, i) else m Ω obs 1,2 (C, i) = m Ωpre↑Ω obs pre (C, i) ⊕ α m Ω obs obs (C, i) end if ∀ C ∈ ξ Ω obs 4) Location probability distribution: 4.a) Redefine m Ω obs
1,2 over Ω pre using ( 22): m end for end for Note that when using DS combination, the mass of the empty set is set to zero after DS normalization in the credal level. Also, when using the disjunctive combination, this mass is also null since it cannot be obtained. Hence, at pignistic level, m Ω obs ⊕ (φ) = 0. Finally, m

Ω obs ↓Ωpre 1,2 (H, i) = C∈ξΩ obs , ρ(H)⊆C m Ω obs 1,2 (C, i)
Ωpre ⊕ (φ) = m Ω obs ⊕ (φ) during the projection, therefore, m Ωpre ⊕ (φ) = 0 at pignistic level. Location probability for each zone P (Inside, i) becomes :

P (Inside, i) = BetP Ωpre (Inside, i) = m Ωpre 1,2 (Inside, i) + m Ωpre 1,2 (Ω pre , i) 2 ( 26
)
To compare the TBM based results with our previous work [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF], we need a normalized location probability distribution which is obtained as follows for all zones (i)

P N orm (Inside, i) = P (Inside, i) Nz k=1 P (Inside, k) (27)
where N z is the total number of zones. Finally, the different steps of the localization algorithm are summarized in the Algorithm 1.

V. EXPERIMENT RESULTS AND DISCUSSION

First of all, to show the interest of the transition handling, we have performed a simple scenario Scenario01 shown in Fig. 4.a) with deployementA. The results of this scenario, which are detailed in V-A, highlights the caution of the TBM method during transitions and shows its interest for subsequent decision-making study. Secondly, we have tested both localization methods for two scenarios with different deployments and configurations as shown in V-B.

The measurements are collected based on a one second time window so that the fusion of the observation with the prediction would be consistent, since if the time window is very large e.g. ten seconds, the person can move to any far zone in the next time window and the prediction defined in [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF] will not be reasonable. Also, to have a fair comparison, the same initialization was taken as equi-probable for both methods. An example of a placed PIR sensor along with its multiplex mask is shown in Fig. 3, and it corresponds to the Red sensor in Fig. 1.b) and in Fig. 1.c).

The proposed experiment scenarios are shown in Fig. 4. In Scenario01, the person has left the bed to reach the table (e.g. to take a glass of water from the table) then went on his/her way to sit down on the chair in the bed room (e.g. to watch TV). To further test our novel method, we have added the Scenario02 which is more complete and solicits almost all the zones of interest. zones and in this situation the TBM based algorithm is prudent since it did not really prefer a zone over the involved zones due to the disjunctive rule which is less informative. Nevertheless, the involved zones have higher probabilities than the noninvolved zones and the maximum probability corresponds to zone3.b which is consistent w.r.t the ground truth. On the other hand, our previous Bayesian based algorithm did not handle the uncertainty induced during the transition since it has preferred zone4 (high difference of probability between zone4 and zone6) which is not consistent w.r.t the ground truth since the person was rather transiting between zone3.b and zone6 and he was not inside zone4.

A. Results highlighting the interest of transition handling

Our previous algorithm may recover from such transition errors depending on the next situations, for example, at t = 22s we can see from Fig. 5.d) that the previous algorithm has recovered from the transition error caused at t = 11s. Unfortunately, such recovering requires time and depends on the next motions and this might not be always good for subsequent decision making study. Indeed, after t = 22s the person left zone9 to go to zone2 and at t = 27s the person was transiting between zone3.a ∪ zone2 to reach the chair. At this t = 27s time window, we can see from Fig. 5.e) that TBM results are consistent since zone3.a and zone2 have the highest probabilities in contrary to the previous algorithm results where the highest probabilities are in zone3.a and zone4. This error in our previous algorithm at t = 27s has induced other errors at the next time window t = 28s where the person has reached the chair in zone2. We can see from 5.f) that at t = 28s the TBM algorithm results are consistent, however, the previous algorithm results are not consistent since probability in zone2 is not the highest yet. This is due to the erroneous results caused by the successive transitions from t = 22s to t = 27s.

From t = 28s to t = 34s no sensor was activated since no sufficient motion has been performed, therefore, the same location results are kept without updating the location results i.e. the probabilities. At t = 35s the person performed a sufficiently large motion in the chair and the Black sensor was activated, at this moment, the previous algorithm started recovering and we can see from 5.g) that zone2 has the highest probability in both algorithms. Nevertheless, the probability of zone2 for the previous algorithm is not significant yet w.r.t zone3.a (small difference of probability between zone2 and zone3.a) and the previous algorithm has to wait for another motion to get more significant probabilities. Hence, for the time interval t = 27s to t = 35s we can say that the person was in zone2 relying on TBM algorithm results, however, we cannot conclude yet if we rely on the previous algorithm results. Consequently, if a decision has to be made within this time interval [START_REF] Delmotte | Target identification based on the transferable belief model interpretation of dempstershafer model[END_REF][START_REF] Ramasso | State filtering and change detection using tbm conflict application to human action recognition in athletics videos[END_REF][29][30][31][32][33][34][35], it can be easily made using TBM results since the person was in the same zone during this interval, however, the location results was not stable and inconsistent when using our previous algorithm for this same time interval. Note that for both algorithms, some location errors may occur (e.g. due to successive false positives, missing data during time window sampling, ...etc.), however, both can recover from such errors once the measurements at next time window(s) are correct.

B. Impact of durable sensors faults, deployment, and setting

To evaluate both methods over different setups and scenarios, we compute the localization error rate as follows:

Error rate = N F alse N all • 100 (28)
where N F alse is the number of measurement time windows when the relatively highest probability is assigned to a wrong zone. N all is the total number of time windows. 1) Impact of durable sensors faults: False positive and false negative are well known faults for PIR sensors. In our case, an additional aspect of these faults is pointed out which depends on the duration of the fault within the sensor network. This aspect concerns fleeting fault(s) and durable fault(s). A fleeting fault is a false positive/negative which lasts for one time window/ Such kind of faults could be overcome through filtering.

A durable fault is a false positive/negative which lasts for several successive time windows. For example, when a person enters Zone09 she will be localized properly as it was the case in Fig. 5.d). However, if the person performs successive small motions that activate only P urple sensor, a durable fault will be induced and the probability will gradually decrease in Zone09 and increase in Zone05. Nevertheless, when the person performs a large motion, she will be localized again in Zone09.

We can see from Table I and II that the error rate of the TBM method is always smaller than the error rate of our previous method. Also, when omitting the errors caused by the durable faults, the error rate of the TBM method becomes much smaller (around 5% or less for almost all cases). This is not the case for the previous method since the error rate is still high because of the additional errors caused during transitions. Finally, the error rates of the TBM method are smaller than the sensors network fault rate4 . Fig. 5: Comparison of the novel TBM algorithm results with the results of our previous algorithm [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF]. Red square corresponds to the person's position inside a zone while the red arrow corresponds to the transition.

2) Impact of sensors deployment: Depending on the case of study, PIR sensors deployment may reduce the sensors fault rate. For example, in [START_REF] Lu | Binary compressive tracking[END_REF] authors showed that an adequate sensors deployment can improve the tracking performance. In [START_REF] Yang | A novel low-cost and small-size human tracking system with pyroelectric infrared sensor mesh network[END_REF], authors showed that different sensor nodes orientations can lead to different tracking accuracy. Nevertheless, in both studies, the location accuracy concerned the error in position coordinates and not the probability of presence in some defined zones of interests. Also, their experiments did not have severe constraints on sensors placement (square areas are considered and no zone of interest constraints).

In our case of study, the sensors can be placed only in some specific places within the smart-home due to architectural constraints (windows, furniture, .. etc.) in one hand, and due to the zones of interest covering in the other hand. Therefore, an optimal sensors deployment is not always easy to implement. Moreover, location errors for a specific sensors deployment may increase or decrease depending on the considered scenario along with the durable sensors faults caused by the nature of human motion during the scenario. In fact, deploymentA shows less location errors compared to deploymentB in the case of scenario01 which is the opposite in the case of scenario02. Consequently, the location errors rate related to a sensors deployment depends also on the scenario and on the nature of the human motions, and these last should be considered when looking for optimal deployment.

3) Impact of parameters setting: In a first a part, we will analyze the impact of the parameters setting within a same scenario to check the consistency of the rule γ stay > γ move >> γ jump . In the second part, we will analyze the impact of the parameters setting for different scenarios.

For the first part, we consider the results of the scenario01 shown in Table .I. We can see from this table that location the error rates increased relatively when γ stay = γ move and that they are still high even when omitting errors caused by durable faults. This is because when γ stay = γ move , the TBM method (and also our previous method) becomes less robust against fleeting faults. For this reason, and as explained in IV-B1, we take the rule γ stay > γ move >> γ jump in practice. Now, let us consider that this rule is satisfied (i.e. by considering only the two settings γ stay = 0.65 and γ stay = 0.85). We can see from Table .I that the best location error rate is obtained with the setting γ stay = 0.85 in the case of scenario01, however, the best error rate is rather obtained with γ stay = 0.65 in the case of scenario02. From this analysis, one can say that the choice of the parameters values depends on the scenario, however, even for a same data set, the parameters setting may increase/decrease the location error rate caused by durable faults. For example, for the scenario02 with deployment B, the location errors caused by the durable faults was 10.5% for γ stay = 0.65 and 9.2% for γ stay = 0.85. In fact, we believe that the parameters values should change w.r.t to the nature of the human motion during a scenario, for example, we should increase γ stay when a small motion is detected, and decrease it when a large motion is detected. We believe that such switching may reduce location errors caused by durable sensor faults, if these faults are detected, which constitutes a track for our next research.

C. Discussion

In V-A, we have shown how the TBM method is prudent during transitions and how its location results could be more significant for subsequent decision-making studies compared to our previous localization algorithm [START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF]. Also, we have shown, with different scenarios and setups in V-B, that the error rates of our previous method are always higher than the error rates of the TBM method. This is caused by the additional errors induced during the transitions between zones that our previous algorithm cannot handle.

We have seen that the TBM algorithm can deal with fleeting sensors network faults, however, it is sensitive to durable sensors faults. Also, we have shown that the localization performance does not only depend on the sensors deployment, but also on the scenario and nature of the human motions during the scenario. Since appropriate sensors deployment cannot always be achieved in practice, one should rather detect and deal with durable faults to increase the location-tracking performances which is the aim of our future work. Finally, the impact of the motion model parameters setting also depends on the scenario along with the nature of the human motions, and we believe that changing the values of the parameters setting depending on the nature of the human motions constitutes the track to deal with durable faults in our future work.

In our long-term project, other sensing modalities will be added and a decision making study will use our data fusion results. Consequently, our long-term studies concerns a state filtering by combining uncertain data from different modalities for a subsequent decision-making study. The TBM seems to be the coherent theoretical framework for our case since it has already shown its advantages in combining uncertain data for both decision making applications such as medical diagnosis [START_REF] Hadj Henni | A transferable belief model decision support tool over complementary clinical conditions[END_REF] or target identification [START_REF] Delmotte | Target identification based on the transferable belief model interpretation of dempstershafer model[END_REF], and also in state filtering [START_REF] Ramasso | State filtering and change detection using tbm conflict application to human action recognition in athletics videos[END_REF].

Finally, note that for multiple persons localization, it is more challenging when using overlapping binary PIR structure since received measurements obtained when multiple persons are moving may easily increase location errors. Indeed, Bo Yang et al pointed out this issue in [START_REF] Yang | A novel low-cost and small-size human tracking system with pyroelectric infrared sensor mesh network[END_REF] and solved it using classifiers with a learning phase [START_REF] Yang | Distributed multi-human location algorithm using naive bayes classifier for a binary pyroelectric infrared sensor tracking system[END_REF] or relying on clustering [START_REF] Yang | Credit-based multiple human location for passive binary pyroelectric infrared sensor tracking system: Free from region partition and classifier[END_REF]. Since other sensing modalities will be further used in our long-term project, hence, we will exploit multi-modal data fusion approach to deal with the multiple person issue without learning phases or clustering requirement.

VI. CONCLUSION

In this paper, we have firstly enhanced our previous multiplex binary localization approach to overcome the ambiguity in the detection of transitions between zones, by using a filtering technique based on the transferable belief model (TBM) instead of the Bayesian framework. Secondly, we have presented a new way for choosing an appropriate discounting factor to weight an information source w.r.t the other source(s), relying on the analysis of the Dempster-Shafer normalization function. The desired zones of interest are covered with a reduced number of sensors thanks to the overlapping multiplex structure. Our novel method is robust against fleeting sensor faults thanks to filtering, and against reasonable uncertainty on sensor's field of view (which is related to transition) thanks to the uncertainty handling using the TBM framework. Finally, experiments using commercialized single PIR sensors, with modified field of view, showed the advantages of the TBM method w.r.t our previous Bayesian method.

Our next work will consist in enhancing the locationtracking results by dealing with durable sensors faults, and addressing the location-tracking of multiple humans. 1-m φ ) w.r.t the mass of the empty set m φ as shown in Fig. 6, we can see that DS norm (m φ ) can be divided into three lines with different slopes which intersect around 0.75 for line A and line B, and around 0.9 for line B and line C. Since each zone will have its own mass of empty set m pre∩obs (φ, Z i ) during the nontransition cases (in transition cases there is no m pre∩obs (φ, Z i ) because the disjunctive rule is used), hence, the combined mass m pre∩obs (Inside, Z i ) of each zone obtained by the conjunctive rule will increase during the DS normalization. Unfortunately, the masses m pre∩obs (Inside, Z i ) will increase differently depending on the m pre∩obs (φ, Z i ) obtained for each zone, especially, this difference becomes more important and problematic when the different m pre∩obs (φ, Z i ) do not belong to the same line A as shown further in the Example01.

APPENDIX A ILLUSTRATIVE EXAMPLE WITH

To minimize this difference during DS normalization accordingly to increase the consistency of localization results, it is necessary to ensure that the mass of the empty set of all zones will belong to the same line A. The limit of this line with line B corresponds to m φ 0.75, hence, if we choose 1 -α obs 0.75 (i.e. α obs 0.25) we will be sure that m φ of all zones will belong to the same line A. We choose only line A since its slope is very small, hence, the mass increasing during the DS normalization will not highly differ from a zone to another. Consequently, it will be unlikely then to obtain counter-intuitive results w.r.t the conjunctive combined masses after DS normalization.

Example01: This example shows the importance of having m 1∩2 (φ) of all zones belonging to the same line A. Let us consider the following scenario (with sensors Deployment A):

At time window t -1, the person is inside zone05. Accordingly, m pre (Inside, Z 8 ) at t is higher than m pre (Inside, Z

Fig. 1 :

 1 Fig. 1: Living lab description and zoning. Uncovered FOVs are marked with discontinuous arcs in (b) and (c).

Fig. 2 :

 2 Fig. 2: Example of a 180 • PIR sensor used in our case.

4

  .b) Presence probability for each Zone(i) using (24): P (Inside, i) = BetP Ωpre (Inside, i) 4.c) Normalize the probability distribution using (25): P N orm (Inside, i) = P (Inside,i) Nz k=1 P (Inside,k)

From Fig. 5 .Fig. 3 :Fig. 4 :

 534 Fig.3: A placed PIR sensor with our multiplex mask

  (a) At t = 5(s) initialization (no motion yet) (b) At t = 10(s) person is in Zone3.b (c) At t = 12(s) the person is transiting between zone3.b to zone6 (d) At t = 22(s) the person is about to quit zone (e) At t = 27(s) the person is transiting between zone3.a and zone2 (f) At t = 28(s) the person entered zone2 (g) At t = 35(s) the person sits on the chair in zone2

  Deployment A This example shows the inappropriate location results induced if we use the DS rule during a transition. Let's consider the transition Z3.a ∪ Z3.b where the person left Z6 to go to Z2. In this case, we will logically have: m Ωpre↑Ω obs pre (Inside, Z3.b) > m Ωpre↑Ω obs pre (Inside, Z3.a) > m Ωpre↑Ω obs pre (Inside, Z4). Let's say their values are: m Ωpre↑Ω obs pre (Inside, Z3.b) = 0.75, m Ωpre↑Ω obs pre (Inside, Z3.a) = 0.6, m Ωpre↑Ω obs pre(Inside, Z4) = 0.3. Also, for the observation we assume that Orange and Y ellow sensors are activated due to this transition, hence:m Ω obs obs (Inside, Z4) = m Ω obs obs (Inside ∪ N eigh, Z3.a) = m Ω obs obs (Inside ∪ N eigh, Z3.b) = 1.After observation and mass discounting, we obtain:α m Ω obs obs (Inside, Z4) = α m Ω obs obs (Inside ∪ N eighZ3.a) = α m Ω obs obs (Inside ∪ N eighZ3.b) = 0.75 α m Ω obs obs (Ω obs , Z4) = α m Ω obs obs (Ω obs , Z3.a) = α m Ω obs obs (Ω obs ,Z3.b) = 0.25 Conjunctive combination results are then: m Ω obs 1∩2 (Inside, Z3.b) = 0.75 > m Ω obs 1∩2 (Inside, Z3.a) = 0.6 > m Ω obs 1∩2 (Inside, Z4) = 0.3 With: m Ω obs 1∩2 (φ, Z3.b) = 0 = m Ω obs 1∩2 (φ, Z3.a). However, m Ω obs 1∩2 (φ, Z4) = 0 = 0.525. The conjunctive combination results are consistent with the proposed scenario. However, the DS normalization leads to: m Ω obs 1⊕2 (Inside, Z3.b) = 0.75. But: m Ω obs 1⊕2 (Inside, Z3.a) = 0.6 < m Ω obs 1⊕2 (Inside, Z4) = 0.63 which is not consistent w.r.t the scenario. Now, if we use the disjunctive combination we obtain: m Ω obs 1∪2 (Inside ∪ N eigh, Z4) = 0.3 m Ω obs 1∪2 (Inside ∪ N eigh, Z3.a) = 0.42 m Ω obs 1∪2 (Inside ∪ N eigh, Z3.b) = 0.6 After inner reduction (24) we will get: m Ωpre 1∪2 (Inside, Z3.b) = 0.6 > m Ωpre 1∪2 (Inside, Z3.a) = 0.42 > m Ωpre 1∪2 (Inside, Z4) = 0.3 We can see that the disjunctive combination results are consistent w.r.t the scenario while considering the closed world property. For such a reason, we use the disjunctive combination instead of the DS combination during transitions. APPENDIX B APPROPRIATE DISCOUNTING FACTOR As evoked in IV-C, the appropriate value of α obs is chosen by analyzing the behavior of the DS normalization function. In fact, if we analyze this function (DS norm (m φ ) = 1

7 )

 7 let's say: m pre (Inside, Z 8 ) = 0.3 > m pre (Inside, Z 7 ) = 0.05. At time window t the person enters zone08 but only Green and Red sensors are activated (P urple sensor is in false negative). With such measurement we obtain:m obs (Inside, Z 7 ) = 1, m obs (N eigh ∪ F ar, Z 7 ) = 0 = m obs (Ω, Z 7 )m obs (Inside, Z 8 ) = 0.66, m obs (N eigh ∪ F ar, Z 8 ) = 0.33. m obs (Ω, Z 8 ) = 0

Fig. 6 :

 6 Fig. 6: DS N orm factor evolution w.r.t m(φ)

  Algorithm 1 Get the location probability distributionRequire: Ω pre = {Inside, Inside} Ω obs = {Inside, N eigh, F ar} ξ Ωpre = {Inside, Inside} ξ Ω obs = {Inside, Inside ∪ N eigh Ωpre using (13)[START_REF] Hadj Henni | A multiplex binary pir sensing approach for a telehome-care application[END_REF].get m Ω obs obs (C, i) ∀ C ∈ ξ Ω obs using (15)-(18). 2) Weight observation mass functions m Ω obs obs : get α m Ω obs obs (C, i) ∀ C ∈ ξ Ω obs using[START_REF] Dallil | Sensor fusion and target tracking using evidential data association[END_REF].Ωpre pre over Ω obs using vacuous extension: m

		, N eigh ∪ F ar	, Ω obs }
	T ransition for each time step t do	Inside
	for each Zone(i) do
	1) Compute mass functions:
	get m
	3) State filtering:
	3.a) Redefine m
	Ωpre↑Ω obs pre

Ωpre

pre (H, i) ∀ H ∈ ξ

TABLE I :

 I Location-tracking error rates for scenario01. Numbers in bold represent the location-tracking error rate when omitting errors caused by durable faults.

	Parameters values for Scenario02	Setup	TBM Meth Error rate %	Previous Meth [14] Error rate %
	  	γstay = 0.85 γmove = 0.14 γ jump = 0.01	A B	18, 4 5,2 14, 4 5,2	32 13,1 27 15,7
	  	γstay = 0.65 γmove = 0.34 γ jump = 0.01	A B	17, 1 3,9 13, 1 2,6	28 14,4 21 10,5
	Sensors network	A		26.7 %
		faults rate	B		22, 3 %
	TABLE II: Location-tracking error rates for scenario02.
	Numbers in bold represent the location-tracking error rate
	when omitting errors caused by durable faults.

http://www.knx.fr/KNX-France-ets5.html

http://www.bourges.univ-orleans.fr/madonah/index.php/experimentations

Colors are used in order to easily distinguish the different sensors and the code of each zone.

Total sensor network faults rate is obtained by checking the consistency of the received measurements of each time window of the data set w.r.t ground truth.


Rihab Hamdi
email: rihab.hamdi2012@gmail.com

Amel Hadri Hamida

Fatima Babaa
email: babaa.fatima@yahoo.fr

Ouafae Bennis
email: ouafae.bennis@univ.orleans.fr

HM-Based SMVC with Adaptive Feedforward Controller Applied to DC-DC Converter

Keywords: Sliding Mode Control, Adaptive control, DC-DC converter, Parallel Multi-cells converter, Robustness

This paper aims to provide a comprehensive study of sliding mode controller (SMC) performance for closedloop voltage control of DC-DC three-cells buck converter based on hysteresis modulation (HM) where an adaptive feedforward technique is adopted. To fix the switching frequency, the approach is to incorporate a feedforward adaptive scheme which is effectively a variant of SM control. Further, the use of an adaptive feedforward control that makes the hysteresis band variable in the hysteresis modulator of the SM controller to restrict the switching frequency variation feeding parameters uncertainties and loads disturbance, in order to overcome the design constraints and to mitigate the undesired transient response. The results obtained under load change, input change and reference change clearly demonstrates a great dynamic response of the proposed technique, as well as provide stability in any operating conditions, the effectiveness is fast with a smooth tracking of the desired output voltage. Simulations studies in MATLAB/Simulink environment have been performed to verify the concept.

I. INTRODUCTION

In the field of strong currents with high switching frequencies, while automating technological processes, there is a tendency to develop and use new structures based on the combination of components. Among them, we find the parallel multicellular converters that have been receiving increased attention recently. The various control methodologies currently being considered to achieve a constant output voltage and have been found to have potential applications in DC-DC converters that require very high performance in dynamical response [START_REF] Said | HIL simulation approach for a multicellular converter controlled by sliding mode[END_REF] - [START_REF] Hadri Hamida | Étude qualitative des convertisseurs multicellulaires entrelacés par la théorie de bifurcation[END_REF].

The various control methodologies currently being considered to achieve a constant output voltage and have been found to have potential applications in DC-DC converters that require very high performance in dynamical response. Theoretically, to achieve a sliding mode control operation in a perfect way, the system should be operated at a high switching frequency so that the controlled variables can follow closely the reference to achieve the desired dynamic response and steady-state operation. Along with this requirement, however, the feasibility of applying SM control to DC-DC converters is questioned [START_REF] Hadri Hamida | Second-Order Sliding Mode Control Scheme with a Non-Linear Phenomenon Analysis of a DC-DC Power Converter Dedicated to Distributed Power Systems[END_REF], [START_REF] Nordstrom | Virtualization of synchronized phasor measurement units within real-time simulators for smart grid applications[END_REF]. This is because extreme high-speed switching in DC-DC converters leads to an excessive switching loss, inductor and transformer core losses, and "electromagnetic interference" noise issues. Hence, their switching frequencies must be constricted within a specifical range in order to make the SM control applicable to DC-DC converters.

Therefore, the design of such converters is a challenge for electrical engineers due to the use requirements and the inherent nonlinear nature. All the converters have few requirements like high density which leads to a smaller size, high efficiency which leads to low losses and robust to any changes in the input or output [START_REF] Thierry | Analysis and Design of Multicell DC/DC Converters using Vectorized Models[END_REF], [START_REF] Utkin | Sliding Mode Control in Electromechanical Systems[END_REF]. For surpassing the problem of high frequency, a new class of power converter appeared: the multicellular converter. The multicellular conversion has inherent advantages that make it an increasingly attractive solution that can be considered for any application. It provides more degrees of freedom to the designer, but it also makes the design more complicated since many options can be considered [START_REF] Yin | Disturbance observer based second order sliding mode control for DC-DC buck converters[END_REF] - [START_REF] Lee | The Chattering Analysis[END_REF]. The HM strategy can be applied to develop fixed-frequency SM controllers and the technique of state-space averaging was incorporated into the controller's modeling [START_REF] Daniel | Improved Design of Sliding-Mode Controllers Based on the Requirements of MPPT Techniques[END_REF], [START_REF] Salimi | Hyper-plane sliding mode control of the DC-DC buck/boost converter in continuous and discontinuous conduction modes of operation[END_REF]. In turn, as known, the limitation of the conventional HM based sliding mode controller consists of the significant variation of the switching frequency in the presence of disturbances and parametric uncertainties. This disadvantage leads to overestimating the properties of the filters in the converter and to impropriate modifications of the regulation. Hence, it is recommended to operate the converter at a constant switching frequency [START_REF] Wang | Extended state observer-based sliding mode control for PWM-based DC-DC buck power converter systems with mismatched disturbances[END_REF].

To surpass this problem, we propose an adaptive controller with a feedforward technique in intent to overcome the imperfect feedback loop that causes a small steady-state error in the output voltage [START_REF] Zhao | A Sliding-Mode Duty-Ratio Controller for DC/DC Buck Converters with Constant Power Loads[END_REF]- [START_REF] Said | Sliding mode control for multicell converters[END_REF]. Unlike the feedback technique, that needs a perception of a past situation, the feedforward technique is illustrated in the formulation of commands to propose solutions for the future. In other words, it consists of the anticipation of the action that follows.

The major contribution of this work is to present thorough a discussion of the problem of switching frequency variation and the effectiveness of the adaptive solution in alleviating the problem through a close examination of the basic structure.

The remainder of the paper is organized into 5 sections:

Section II outlines the system's topology and operating principle. Section III discusses the control scheme and the operating mechanism of the variable hysteresis band. Simulation results are presented and discussed in Section IV; Section V concludes the paper. II. PARALLEL MULTI-CELL CONVERTER DC-DC converters constitute a particular class of nonlinear, time-varying systems. The paralleling of cell switching is accomplished by directly connecting the cells to the voltage source. The multicellular converter is based on the combination of elementary cells of commutation. This signal is equal to 1 when the cell upper switch is conducting and equal to 0 when the lower complementary switch is conducting. These cells are associated in parallel with an RL load and separated by capacitors application to a three-cells buck converter. Fig. 1 depicts a basic topology of the buck converter, in which Vi is the input DC voltage; L is the filter inductor; C is the capacitor; R is the load. Then the system dynamics of the buck converter can be expressed as:

.

L L out i di L i V u V dt     ( 1
)
out out L dv V Ci dt R  ( 2
)
Where iL is the inductor current, Vout is the output DC voltage, u is the control input. The multicellular converter consists of phases, where each cell contains two complementary power electronics components and it is controlled by a binary switch. A 3-phase converter feeds a single load assumed to be resistive. The considered multicellular buck converter is a second-order converter, with Single Input Single Output (SISO) structure, where, the cells are associated in parallel with R load, realized in MATLAB/SIMULINK. Fig. 2 illustrates the configuration adopted of the threecell step-down DC-DC converter associated with parallel realized in the MATLAB/Simulink environment.

ref out out out i out x V V dV V uV V x x dt dt C R L         (3)
For the system modeling and state-space descriptions required for the design of SM voltage control for buck converter operating in CCM, the state-space may be expressed in the following form:

11 22 0 0 1 0 11 ref i xx u V V xx LC RC LC LC                                            (4)
For an SM voltage controller, the switching function u can be determined from the control parameters x1 and x2 using the state trajectory computation:

Where :

    12 ,1 , T J x x x         (6)
This forms the basis for the control law :

1 ' ' 0 ' ' ON When S k u OFF When S k       (7)
To determine asymptotic stability the existence condition, which is derived from Lyapunov's second method, must be obeyed :

0 lim . 0 S SS  (8)
Thus, the condition for SM control to exist is

0 0 0 - 0 Jx for S S Jx for S       ( 9
)
where ξ is an arbitrarily small positive quantity Substituting (3) and ( 5), we find :

1 ()

ref out C S V V i R    (10)
Also, the width of the hysteresis band κ is a fixed parameter that can be determined using :

(1 )

2 ref ref i V V V k fw L   (11)
with fw, Vref, and Vi representing the steady-state switching frequency, the desired output voltage, and the input voltage, respectively.

The above equation is valid only if the sliding coefficient is set as :

1 RC   (12)
Under this configuration, the conditions for SM control to exist are :

1 2 1 2 2 1 11 ( ) 0 11 ( ) 0 ref i ref VV Cxx R L L V Cxx R L L            ( 13
)
The proposed controller-based sliding mode control via hysteresis modulation is applied to buck type DC-DC converter. To keep the switching frequency fixed against line variation, an adaptive feedforward control scheme that varies the hysteresis band in the hysteresis modulator of the SM controller in the event of any change of the line input voltage must be introduced. Fig. 3 shows the basic structure of the adaptive feedforward SMVC buck converter. The operation of the adaptive feedforward variable hysteresis band and the variable hysteresis band scheme are illustrated in Fig. 4 and Fig. 5, which shows the trajectory of S of the steady-state operation for one switching cycle. Here, the terms Soff(min) and Son(min) represent the trajectory of S when the input voltage is minimum for, respectively, the turn-off period Toff(min) and turn-on period Ton(min). Similarly, Soff(max) and Son(max) represent the trajectory of S when the input voltage is maximum, for respectively the turn-off period Toff(max) and turn-on period Ton(max). Also, κmin and κmax represent the required width of the hysteresis band for maintaining the same switching frequency at the minimum and the maximum input voltage.

IV. SIMULATION RESULTS

In order to illustrate the performance of the proposed control, we considered a three-cell converter connected to an R load. The performances of the developed techniques based SM control were verified through simulation using MATLAB software.

Table I show the specifications of the buck converter used in the MATLAB simulation. To test the performance of the proposed controller, a comprehensive simulation under sequence step load change (variation of 50%) with constant input voltage was tested. The reference output voltage was set to 12 V, the load resistance changes from 12Ω to 6Ω. The situation where the load changes suddenly from one value of load resistance to another is considered. This particularly interesting because it is a typical problem for power electronics, where the power supply is supposed to compensate quickly for the load variation.

Robustness test versus the load resistance variation is illustrated by Fig. 6 and Fig. 7, showing that the state variables exhibit a transient but the output voltage converges to the desired reference. According to these results, it can be noticed that the performances of the proposed control for load variation are satisfactory.

Then, the multicellular converter is initially powered by an input voltage 𝑉 𝑖 = 24 𝑉 which varies at instant 𝑡 = 0.4 𝑠 to reach 50 V, to test the robustness of the developed controller, the disturbance is assigned to the level of the supply voltage.

Fig. 8 and Fig. 9 summarizes the results of this test. There is a very brief transient that lasts a few 𝑚𝑠 followed by a steady-state giving voltage values 𝑉 𝑜𝑢𝑡 proportional to the new value of 𝑉 𝑖 .

Finally, the robustness test versus the power reference value changes is illustrated in Fig. 10 and Fig. 11. Indeed, at t = 0.4 s, 𝑉 𝑟𝑒𝑓 is slightly changed from 12V to 14V.

According to these results, it can be noticed that the control strategy objective is fulfilled and the performances are satisfactory. V. CONCLUSION

The HM-based SM controller has improved the dynamic behavior and it is able to cope with input voltage and load variations, as well as provide stability in any operating conditions.

The carried-out simulations show very promising results in terms of reference tracking performances and robustness. They prove the appropriateness of sliding mode control for such kind of system.

Fig. 1 .

 1 Fig. 1. Basic topology of the Buck converter.

Fig. 2 .

 2 Fig. 2. Three-cell Buck converter associated in parallel.III. THE OPERATING MECHANISMThe proposed controller-based SM control via hysteresis modulation is applied to buck type DC/DC converter.The mathematical model of the buck converter is 1

Fig. 3 .

 3 Fig. 3. The basic structure of an adaptive feedforward SMVC buck converter.

Fig. 4 .

 4 Fig. 4. The operating mechanism of the adaptive feedforward variable hysteresis band.

Fig. 5 .

 5 Fig. 5. The variable hysteresis band scheme.

Fig. 6 .

 6 Fig. 6. (a) Simulation results of the Output voltage response with an output load variation. (b) Zoom in.

Fig. 7 .

 7 Fig. 7. (a) Simulation results of the Output load current response with an output load variation. (b) Zoom in.

Fig. 8 .

 8 Fig. 8. (a) Simulation results of the Output voltage response with an input voltage variation. (b) Zoom in.

Fig. 9 .

 9 Fig. 9. (a) Simulation results of the Output load current response with an input voltage variation. (b) Zoom in.

Fig. 10 .

 10 Fig. 10. (a) Simulation results of the Output voltage response with an reference voltage variation. (b) Zoom in.

Fig. 11 .

 11 Fig. 11. (a) Simulation results of the Output load current response with a reference voltage variation. (b) Zoom in.

TABLE I .

 I SPECIFICATION OF THE CONVERTER

Converter Type The Properties Of Each Cell

		Description	Parameter	Nominal value
		Input voltage	Vi	24 V
		Capacitance	C	6e-6 F
	Parallel
	Three-cell	Resistance	R	12 Ω
	Buck Converter	Inductance	L	0.1 H
		Switching frequency	fw	20 kHz
		Desired output voltage	Vref	12 V