Marco Fantuzzi
email: marco.fantuzzi3@unibo.it

Alessandra Costanzo
email: alessandra.costanzo@unibo.it

Smail Tedjini
email: smail.tedjini@grenoble-inp.fr

Pierre Lemaitre-Auger
email: pierre.lemaitre-auger@grenoble-inp.fr

Low-cost UHF near-field power transmission for RFID applications

Keywords: wireless power transfer (WPT), near-field, RFID, ultra-high frequency (UHF) communication, defected ground structure (DGS)

This paper presents a low-cost contactless power transfer structure for planar configuration working at UHF frequencies. The theory of defected ground structure (DGS) is applied to unconventional materials for power transmission between two closely spaced microstrip lines. Such configuration is studied and described in details, for an easy implementation employing very low-cost materials, characteristics of deemed importance in radio frequency identification (RFID) applications. Influence of nearby material is analyzed and theoretical results are experimentally validated. A possible application for RFID readers is finally proposed.

I. INTRODUCTION

RFID technology is a very well-known technique, which has been deeply investigated since its pervasive distribution in the last decades [START_REF] Rao | Antenna design for UHF RFID tags: a review and a practical application[END_REF][START_REF] Marrocco | The art of UHF RFID antenna design: impedancematching and size-reduction techniques[END_REF][START_REF] Perret | Design of Antennas for UHF RFID Tags[END_REF]. Its applications span a high number of different scenarios, such as logistics and supply chain, itemlevel inventory tracking, access control and real time location systems (RTLS). A key feature demanded in the vast majority of cases is the necessity for very low-cost materials and manufacturing techniques, both deemed important requirements for the effective mass deployment of RFID technology in market applications [START_REF] Lolli | Electromagnetic Characterization of Paper-Glue Compound for System-in-Package on Paper (SiPoP) Future Developments[END_REF]. In such a context, the possibility to replace physical connections between hardware components with contactless techniques using inexpensive materials and featuring high repeatability is very attractive and enabling for a practical implementation.

The concept of near-field WPT is extensively studied for different applications, from powering of small devices to wireless charging electric vehicles [START_REF] Cannon | Magnetic Resonant Coupling As a Potential Means for Wireless Power Transfer to Multiple Small Receivers[END_REF][START_REF] Low | Design and Test of a High-Power High-Efficiency Loosely Coupled Planar Wireless Power Transfer System[END_REF][START_REF] Shin | Design and Implementation of Shaped Magnetic-Resonance-Based Wireless Power Transfer System for Roadway-Powered Moving Electric Vehicles[END_REF]. In almost all cases though, the operating frequency is in the range of the LF or HF, while very few solutions have been proposed at UHF and at microwave frequencies [START_REF] Trevisan | A UHF Near-Field Link for Passive Sensing in Industrial Wireless Power Transfer Systems[END_REF][START_REF] Kim | Wireless Power Transfer to Miniature Implants: Transmitter Optimization[END_REF]. From the RFID point of view, the simplest and less expensive choice would be realizing nearfield coupling at the same frequency of operation of the RFID communication, with no need for additional frequency conversion modules.

Recently a method for near-field coupling at UHF has been proposed [START_REF] Hekal | Strong resonant coupling for short-range wireless power transfer applications using defected ground structures[END_REF][START_REF] Hekal | A Novel Technique for Compact Size Wireless Power Transfer Applications Using Defected Ground Structures[END_REF]. This solution exploits the resonant behavior of an optimized DGS to create a near-field coupling between two closely spaced resonators facing each other. This method is here further extended and proposed for low-cost implementation envisioned for RFID applications in the European UHF 868 MHz band. The contactless power transmission between two closely spaced microstrip lines, realized on unconventional low-cost material, is investigated for different application scenarios. Experimental validation of the expected theoretical results is reported.

The paper is organized as follow: in section II, the adopted material and its configuration for the microstrip lines are described; in section III, a brief description of the resonant coupling principles, together with the related parametric investigations, are introduced. Results for quasi real-scenario implementations are provided in section IV, while, in section V, a possible application for RFID readers is proposed. Finally, the main results obtained are summarized in the conclusions.

II. MATERIAL CHOICE AND CHOSEN SUBSTRATE

One of the most challenging requirements of RFID systems is a low-cost implementation. A second feature that is of increasing importance in many emerging applications is a flexibility of the structure, in order to be conformable with nonplanar surfaces. The choice of material should be based on Low-Cost UHF Near-Field Power Transmission for RFID Applications these two requirements. For this purpose thin PET, which is flexible, low-cost and widely adopted in RFID systems especially for the tags realization, is a good candidate. Nevertheless, it is suitable as a supporting material for printing antennas, but it is usually too thin to be adopted as the sole substrate for microstrip lines and antennas. In order to provide the convenient thickness, linoleum, a very cheap material widely used for floor covering, tough not directly suitable for RF applications, can be adopted as a further supporting medium besides the PET. The combination of these two materials offers a very cheap solution that can be effectively adopted for large-scale applications. The electrical properties of the two materials are measured by means of a standard resonant cavity method and are found to be ε r = 3.2, tan(δ) = 0.003 (@ 1 GHz) and εr = 1.89, tan(δ) = 0.011 (@ 1 GHz) for PET and linoleum, respectively. A schematic representation of a possible multilayer low-cost substrate adopted for the present design is reported in Fig. 1. Printed aluminum on PET is employed for the metallic layers. The three layers can be simply superimposed on top of each other, without any extra glue or fixing material.

III. CONTACTLESS RESONATOR DESIGN

As described in [START_REF] Hekal | A Novel Technique for Compact Size Wireless Power Transfer Applications Using Defected Ground Structures[END_REF], the resonant coupling circuit can be obtained by a proper design of an H-shaped DGS, which can be modelled as a parallel LC resonator, where the inductive contribution comes from the two rectangular removals in the ground plane and the capacitive one is realized by introducing a lumped SMD capacitor that provides the needed value of capacitance. In the present design, SMD components are avoided and an interdigital capacitor is adopted to simplify the manufacturing and to lower the overall cost. Therefore, all metallic structures are planar and can be directly printed on a PET substrate.

For an effective implementation in real scenarios, the evaluation of the impact of surrounding materials which are placed nearby the resonating structure are carried out. The strip line of the structure reported in [START_REF] Hekal | A Novel Technique for Compact Size Wireless Power Transfer Applications Using Defected Ground Structures[END_REF] is facing the supporting material. So, placing the whole structure onto any material other than air will deeply affect its behavior. Indeed, either the effective dielectric constant of the bottom microstrip will change or it will become a stripline if it is placed on top of a metallic surface. To overcome this limitation, the bottom microstrip line is flip over such that the ground plane is facing the supporting material. The effect of the latter is restricted to the area of the DGS due to the partial metal removals in the ground plane. This way, the effect of different materials used as support has to be evaluated only in this region.

The multilayer substrate described in the previous section is chosen for the investigation of the near-field coupling performance between the two closely spaced resonators. The proposed structure is depicted in Fig. 2, along with the corresponding dimensions. By tuning the dimensions of the structure, a parametric study is conducted with electromagnetic (EM) simulations in order to provide the desired coupling behavior at the European 868 MHz RFID band. As a first step, the dimensions of the interdigital capacitor are found in order to provide a fixed capacitance of approximately 1 pF [START_REF] Withayachumnankul | Compact electric-LC resonators for metamaterials[END_REF]. Secondly, the other dimensions reported in Fig. 2 are tuned in order to highlight their impact on the inductive behavior of the DGS and therefore on the operating frequency. Fig. 3 illustrates some results reporting the impact of the parameters L L, WL and WC on the resonant frequency. One parameter is changed at a time while keeping the other two fixed. From these results, it is possible to notice that LL has a smaller impact, it produces a shift of approximately 10-15 MHz/mm, and it is therefore suitable for fine tuning of the resonant frequency. resonant WPT applications that moving the two resonators from the optimal distance produces a frequency separation of the two resonances [START_REF] Costanzo | Rigorous modeling of mid-range wireless power transfer systems based on royer oscillators[END_REF]. Such phenomenon is graphically described in Fig. 4, where the simulated performance of the near-field coupling is reported for different values of distance.

For a real physical implementation of the structure, the material used as spacer between the two resonators cannot simply be free space but necessarily has to be a physical medium. Hence, to comply with the low-cost requirements, the same linoleum material already used is employed as spacer.

It is interesting to notice from the results of Fig. 4 that, even though the optimum distance provides the best operating bandwidth, the maximum transmission value is obtained for K=1, which is therefore chosen for the present design. The 0.5-1 dB drop of the transmission coefficient for higher values of K can be explained by the increased quantity of lossy materials inserted between the two resonators. Even if it is reduced, the operating bandwidth obtained is still well suitable for applications in the European 868 MHz band. Obviously, in case a higher bandwidth would be required, for example for coverage of other RFID frequency bands, the optimum distance could be chosen, at the expense of a slight reduction of the transmission coefficient.

IV. IMPLEMENTATION AND MEASURED RESULTS

Based on the previously described considerations, two prototypes of the proposed system were realized and measured. For a practical implementation, it is extremely important to evaluate the effect of the materials surrounding the DGS section, specifically the physical support on which the coupling structure is placed during operation. In this section, the effects of plastic and wood materials are first considered as they have the same influence on the coupling structure. In a second step, metallic material is also assessed, estimated as the most influencing yet widely adopted medium for a vast number of applications. The pictures of the two measurement setups are reported in Fig. 5. In both cases, a single additional layer of linoleum is placed between the coupling structure and the physical support.

A. Non-metallic Environmenet

The dimensions of the DGS are first optimized for the structure to be put over a plastic or wood material. For that, the first of the two resonances is shifted in the UHF band. The values of the parameters are L L = 26 mm, WL = 6 mm, WC = 7 mm and Lstub = 18 mm. Fig. 6 reports the behavior of the structure in terms of reflection and transmission coefficients. A good agreement is observed between measured and simulated data. The slight discrepancies are mainly due to realization tolerances and to small airgaps inevitably present between the different layers of the adopted substrate, depicted in Fig. 1. Moreover, it is worth noting that a slight influence on the performance is also introduced by the bending of the second microstrip line, because of the thickness of the linoleum layer introduced as spacer between the two resonators.

B. Metallic Environment

In a large number of applications, the RFID operation has to take place in close proximity of metallic objects. This represents by far the worst possible scenario and it is therefore chosen as the second case study. Once again, the dimensions of the structure were optimized by EM simulations. It is found that placing the resonating structure on top of a metallic surface exerts a significant influence on the behavior of the coupling. In particular, the presence of the metallic surface appears to have a detrimental effect mainly on the first of the two resonances of Fig. 6, canceling it, while the second resonance is only slightly affected. This effect is explained by the fact that the first resonance is mainly attributed to the DGS closer to the metallic surface. The second resonance, conversely, mainly attributed to the further DGS, features only a slight frequency shift and a small degradation in the transmission coefficient. It is therefore chosen for the present design. A second prototype was realized by shifting the frequency of the second resonance to be centered at 868 MHz. The corresponding dimensions are in this case: LL = 27 mm, WL = 10 mm, WC = 11 mm and Lstub = 14 mm. Comparison between the measured and simulated data is provided in Fig. 7.

V. RFID READER APPLICATION

The present design can be applied to a large number of applications, where the possibility of providing contactless coupling between two planar microstrip lines that need to operate at UHF sounds attractive. A possible implementation can be that of wireless connection between an RFID reader unit and its antenna. Indeed, there are a several number of cases in which the radiating element of an RFID reader needs to be replaced or modified in a certain way. Since these kinds of operations typically involve a very high number of units at a time, a contactless connection can result in a significant cost reduction with respect to a solution requiring physical connections. Moreover, its high repeatability can easily speed up the entire process.

As a proof of concept, the microstrip line connected to the second DGS, terminated by port 2 of Fig. 2, is connected to a patch antenna, designed on the same multilayer substrate reported in Fig. 1. The designed patch antenna is fed with an inset of 34 mm for impedance matching and it has a width and length of 145 and 122 mm respectively. The dimensions of the DGS are the same as those employed for the prototype of section IV-A, except for the stubs which are in this case tuned to different lengths for matching purposes, since the loads of the two DGSs are different in the present case. In particular, the length of the stub fed by port 1 is fixed to 16 mm, while the length of the second stub is chosen equal to 20 mm. The simulated performance of the system show a maximum realized gain equal to 4.9 dBi and a radiation efficiency of 45%, values mainly limited by the lossy substrate employed.

VI. CONCLUSIONS

A novel low-cost implementation for contactless near-field coupling between two resonating structure at UHF is demonstrated for European 868 MHz RFID applications. The effects of different nearby low-cost materials used as support for the resonant structure were evaluated and the EM design was confirmed by measurement results. The substrate used in the present study entitles the proposed system for wide deployment in mass market applications thanks to its very low cost and high repeatability. Finally, a possible application envisioned for antenna wireless connection in RFID readers is proposed.

Fig. 1 .

 1 Fig. 1. Multilayer substrate representation. Two PET layers with printed conductive aluminum are employed as top and bottom metallization.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Representation of the proposed structure and relative dimensions. (a) Perspective view of the two microstrip lines ending/beginning with the DGS section; one layer of linoleum is placed as spacer between the two DGSs. (b) Top view of the DGS section. (c) Bottom view of the DGS section.

Fig. 4 .Fig. 5 .Fig.

 45 Fig. 4. Effect of the number of linoleum layers, K. DGS dimensions are LL=27 mm, WL=8 mm, WC=8 mm and Lstub=18 mm. (a) Reflection coefficient. (b) Transmission coefficients.

Fig. 7 .

 7 Fig. 7. Comparison between simulated and measured reflection and transmission coefficients for metallic environment.

ACKNOWLEDGMENT

This work was partially supported by the COST Action IC1301 -Wireless Power Transmission for Sustainable Electronics (WiPE).


Jean-Frédéric Brun

Emmanuelle Varlet-Marie

E Raynaud

Christine Fedou

Marion Pollatz

Hemorheologic effects of low intensity endurance training in type 2 diabetic patients: A pilot study


Keywords: Non-linear flat system, flat output, fault detection and isolation, three-tank system

This paper presents a rigorous definition of the isolability of a defect in a flat system whose flat outputs are measured by sensors that are subject to faults. Particularly, we show that the isolation of higher-dimensional defects can be attained under a certain condition pertaining to the relation between the flat outputs. Accordingly, a detailed characterization of this relation is presented in a mathematical framework. Finally, the validity of the results is demonstrated using the three-tank system.

On the Choice of Multiple Flat Outputs for Fault Detection and Isolation of a Flat System

Rim RAMMAL * , Tudor-Bogdan AIRIMITOAIE * , Franck CAZAURANG * , Jean LEVINE * * , Pierre MELCHIOR *

INTRODUCTION

The fault detection and isolation (FDI) problem has been introduced in automatic control as a paradigm for designing algorithms able to detect the outbreak of faults and isolate their causes. Various FDI techniques have been developed and can be found in survey papers (see e.g. [START_REF] Zhou | Overview of fault detection and identification for non-linear dynamic systems[END_REF], [START_REF] Thirumarimurugan | Comparison of fault detection and isolation methods: A review[END_REF]).

The first proposed method is the hardware redundancy in which multiple sensors and actuators are used to measure and control a particular variable [START_REF] Chen | A hybrid data-driven modeling method on sensor condition monitoring and fault diagnosis for power plants[END_REF]). The drawbacks of this method are the extra equipment, maintenance cost and additional space required to accommodate the equipment. Later, this method has been replaced by the analytical redundancy, which is based on the notion of generating residual signals. These residues are defined as the difference between the measured variables and the estimated ones. In the case of no defect, the value of the residue is close to zero and it is different than zero otherwise. There exist several methods to generate residues, such as the observer-based approach [START_REF] Tousi | Robust observerbased fault diagnosis for an unmanned aerial vehicle[END_REF]), the parity-space approach [START_REF] Diversi | Robust residual generation for dynamic processes using decoupling technique[END_REF]) or the Kalman-based approach [START_REF] Izadian | Application of kalman filters in model-based fault diagnosis of a dc-dc boost converter[END_REF]).

Recently, the flatness property has been introduced into the repertoire of FDI techniques ( [START_REF] Suryawan | Fault detection, isolation, and recovery using spline tools and differential flatness with application to a magnetic levitation system[END_REF]), [START_REF] Martínez-Torres | Flatnessbased fault tolerant control[END_REF])). Here, residues are calculated using the differential flatness property. Roughly speaking, a system is said to be flat if all the state and input variables can be expressed as function of a particular variable, called flat output, and a finite number of its successive derivatives. The method presented in [START_REF] Suryawan | Fault detection, isolation, and recovery using spline tools and differential flatness with application to a magnetic levitation system[END_REF]) is dedicated to linear flat systems and it uses B-splines parametrisation to estimate the time derivatives of the flat output, which may not be defined because of the presence of noises. The disadvantage of this method is that the derivative estimation could take time and might delay the reconfiguration process. This issue has been overcome later in (Martínez- [START_REF] Martínez-Torres | Flatnessbased fault tolerant control[END_REF]), where a high-gain observer is used to evaluate the time derivative of noisy signals and a low-pass filter is synthesized to improve its performance. In addition, this method can be applied to both, linear and non-linear flat systems.

In the flatness-based FDI approach, residues betweeen the measured state and input variables and their expression using the measured flat output are computed online. Then, the fault detection algorithm is similar to that of other approaches: if a residue exceeds its threshold then a fault is detected. Therefore the problem of the isolability of a fault is directly related to the dependence of the generated residues with respect to the state variables [START_REF] Kóscielny | Fault isolability with different forms of the faults-symptoms relation[END_REF]). Moreover, if several flat outputs are required, these flat outputs must be such that, if a fault affects one flat output, the others are not totally affected (Martínez- [START_REF] Martínez-Torres | Flatnessbased fault tolerant control[END_REF]).

In the present paper, a rigorous definition of the isolability of defects is presented as well as a characterization of the relations between the flat outputs used. The latter flat outputs may be computed by the method of unimodular completion of polynomial matrices [START_REF] Franke | On the computation of flat outputs for nonlinear control systems[END_REF]) that, in some particular cases, directly provides the flat outputs [START_REF] Fritzsche | Unimodular completion and direct flat representation in the context of differential flatness[END_REF]).

The main contributions of this paper are the above mentioned rigorous definition of the isolability of defects and the characterization of the flat outputs to be used in the defect isolation.

This paper is organised as follows: section 2 introduces the basic concepts of FDI for non-linear differentially flat systems and their definitions. After a brief recall in section 3 of the direct flat output computation method, section 4 discusses the relations that exist between flat outputs. Section 5 presents the application of this FDI approach to the three-tank system. Finally, section 6 concludes the paper.

FLATNESS-BASED FDI

Differentially Flat System

Consider the following non-linear system ẋ = f (x, u)

(1) where x, the vector of states, evolves in a n-dimensional manifold X, u ∈ R m is the vector of inputs, m ≤ n and rank( ∂f ∂u ) = m. Let (x, u) (x, u, u, ü, . . .) be a prolongation of the coordinates (x, u) to the manifold of jets of infinite order X X × R m ∞ (Levine, 2009, Chapter 5).

The system (1) is flat at a point (x 0 , u 0 ) ∈ X if and only if there exist a vector y = (y 1 , . . . , y m ) ∈ R m , two integers r and s and mappings ψ defined on a neighbourhood V of (x 0 , u 0 ) in X and φ defined on a neighbourhood W ⊂ ψ(V) of y 0 (y 0 , ẏ0 , ÿ0 , . . .) ψ(x 0 , u 0 ) in R m ∞ such that: (1) y = ψ(x, u, u, . . . , u (s) ) ∈ W (2) y 1 , . . . , y m and their successive derivatives are linearly independent in W (3) (x, u) = ϕ(y) = (ϕ 0 (y, ẏ, . . . , y (r) , ϕ 1 (y, ẏ, . . . , y (r+1) )) ∈ pr X×R m (V) where pr X×R m (V) is the projection of V in X × R m (4) The differential equation φ0 (y) = f (ϕ 0 (y), ϕ 1 (y)) is identically satisfied in W.

The vector y is called flat output of the system. The mappings ψ and ϕ are called isomorphisms of Lie-Bäcklund and are inverse of one another.

After elimination of the input u in equation (1), the implicit system associated with the system (1) is given by: F (x, ẋ) = 0 (2) where F is supposed to be meromorphic [START_REF] Levine | Analysis and control of nonlinear systems: A flatness-based approach[END_REF]) and rank( ∂F ∂ ẋ ) = n -m. Integral curves of the systems (1) and (2) coincide on the set X 0 defined by

X 0 = x ∈ X | d k dt k F (x, ẋ) = 0, ∀k ∈ N \ {x ∈ X | ∃ u ∈ R m s.t. ẋ -f (x, u) = 0}.
Remark 1. The property of flatness is not defined globally. The Lie-Bäcklund isomorphisms ψ and φ are only locally defined. Thus, there might exist points in X 0 where no such isomorphisms exist or, otherwise stated, where the system is not flat. It has been proven in [START_REF] Kaminski | Intrinsic and apparent singularities in differentially flat systems, and application to global motion planning[END_REF]) that the set of intrinsic singularities contains the set of equilibrium points of the system that are not first order controllable.

Fault Detection and Isolation

We recall the residue design method of (Martínez-Torres et al. ( 2014)), using the expressions of the state and input variables in function of the flat output. Suppose that the system (1) is flat with y = ψ(x, u, u, . . . , u (s) ) a flat output, then the full state and input read:

x = ϕ 0 (y, ẏ, . . . , y (r) ) (2a) u = ϕ 1 (y, ẏ, . . . , y (r+1) ). ( 2b
)
Definition 1. (Martínez- [START_REF] Martínez-Torres | Flatnessbased fault tolerant control[END_REF]. The k th -state (resp. l th -input) residue r x k (resp. r u l ) is defined by the difference between the state (resp. input) measurement ξ k (resp. ν l ) and the one calculated by (2a) (resp. (2b)) using the measured flat output y:

r x k = ξ k -x k , , k = 1, . . . , n r u l = ν l -u l , l = 1, . . . , m, (3)
with x k = ϕ 0,k (y) and u l = ϕ 0,l (y).

The components (y, ẏ, ÿ, . . .) of the flat output take their values from the sensors and actuators. However, due to the presence of noises on sensors and actuators, the derivatives of the flat outputs may not be defined. For this purpose, a high-gain observer is used to estimate these derivatives and a low-pass filter is synthesized to improve its performance.

Conditions of robustness of this method are detailed in (Martínez- [START_REF] Martínez-Torres | Flatnessbased fault tolerant control[END_REF]).

Also, due to the existence of noises, a threshold is fixed for each residue. In the case of no defects, the calculated states x k (resp. inputs u l ) and their measurements ξ k (resp. ν l ) have about the same values and their residues do not exceed their thresholds. In contrast, if, at least, one of the calculated residues exceeds its threshold then a fault is detected on the corresponding sensor or actuator. However, for an arbitrary flat output, that may depend on all the system variables, several thresholds may be exceeded simultaneously and the possibility of isolating a defect thus highly depends on the choice of flat output. The definition of the isolability for the structured residual approach in the framework of general polynomial systems has been introduced in [START_REF] Staroswiecki | Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems[END_REF]). We pose the following rigorous fault isolability definition in the flatness context:

Definition 2. (Isolability) A fault on the state x i is isolable if it verifies the following conditions:

(1)

∂r xi ∂x i = 0 ;
(2)

∂r xj ∂x i = 0 ∀ j = i and ∂r u l ∂x i = 0 ∀ l ∈ {1, . . . , m}.
An isolable fault on the input u l is defined in the same way. This definition of isolability reflects the fact that a fault on a state (resp. input) is isolable if this state is not a component of the flat output y.

Hypothesis: From now on, we assume that there is only one fault at a time affecting the sensors or actuators.

Application on the Three-Tank System

The three-tank system represents the dynamics of three cylindrical tanks of cross-sectional area S, connected to each other by means of cylindrical pipes of section S n , and two pumps P 1 and P 2 that supply tanks T 1 and T 3 . These three tanks are also connected to a central reservoir through pipes (see Fig. 1).

The explicit system of equations of the three-tank model is given by:

S ẋ1 = -Q 10 (x 1 ) -Q 13 (x 1 , x 3 ) + u 1 (4) S ẋ2 = -Q 20 (x 2 ) + Q 32 (x 2 , x 3 ) + u 2 (5) S ẋ3 = Q 13 (x 1 , x 3 ) -Q 32 (x 2 , x 3 ) -Q 30 (x 3 ) (6)
where the state variables x i , i = 1, 2, 3 represent the water level of each tank, Q i0 , i = 1, 2, 3 the outflow between each tank and the central reservoir, Q 13 is the outflow between tank 1 and tank 3 and Q 32 the outflow between tank 3 and tank 2, u 1 and u 2 are the input variables, namely the incoming flow of each pump. Hypothesis: The following configuration is considered to avoid singularities1 :

x 1 > x 3 > x 2 .
We consider that the valves connecting tanks 1 and 3 with the central reservoir are closed, i.e. Q 10 ≡ 0 and Q 30 ≡ 0.

The expressions of Q 13 , Q 32 and Q 20 are given by:

Q 13 (x 1 , x 3 ) = a z1 S n 2g(x 1 -x 3 ) (7) Q 20 (x 2 ) = a z2 S n 2g(x 2 ) (8) Q 32 (x 2 , x 3 ) = a z3 S n 2g(x 3 -x 2 ) (9)
where a zr , r = 1, 2, 3, is the flow coefficient and g the gravitational force.

This system is flat with y = (x 1 , x 3 ) T = (y 1 , y 2 ) T as flat output. The state and input variables of the system are then constructed using (2a) and (2b) as follows:

x 1 = y 1 x 2 = y 2 - 1 2g a z1 S n 2g(y 1 -y 2 ) -S ẏ2 a z3 S n 2 x 3 = y 2 u 1 = S ẏ1 + a z1 S n 2g(y 1 -y 2 ) u 2 = S ẋ2 -a z3 S n 2g(y 2 -x 2 ) + a z2 S n 2gx 2 .
Since the flat outputs take their values from the sensors then the variables x 1 and x 3 being measured as flat output, they are equal to the measurements ξ 1 and ξ 3 . Hence r x1 ≡ 0 and r x3 ≡ 0, so that they can be eliminated from the vector of residues, that thus reads:

r x2 r u1 r u2 = ξ 2 ν 1 ν 2 - x 2 u 1 u 2 . ( 10
)
In this system, the state x 2 is not a component of the flat output y, then, according to the Definition 2, only the residue r x2 is triggered by a fault on the sensor x 2 . As a consequence, a fault on the sensor x 2 can be detected and isolated. But it can be easily verified that if a fault affects one of the components of the flat output y then all the residues will be affected and the fault cannot be isolated at this stage.

In [START_REF] Nagy | State estimation of the three-tank system using a multiple model[END_REF], it has been shown that the system of three-tank is observable with only the state variable x 1 . This means that the other state variables x 2 and x 3 can be estimated using x 1 while given the measurements of u 1 and u 2 . In this case, two additional residues r x2 = ξ 2 -x 2 and r x3 = ξ 3 -x 3 are added to (10), where x 2 and x 3 are the estimated values of x 2 and x 3 , respectively. If a fault affects the component of the flat output x 3 , then all the residues, except r x2 which is independent of x 3 , will be triggered and, consequently, this fault is detected and isolated. Finally, if all the above residues exceed their thresholds then we conclude that a fault appears on the sensor x 1 . Table 1 summarizes the residues triggered by each fault.

Fault rx 2 ru 1 ru 2 r x 2 r x 3 Fx 1 1 1 1 1 1 Fx 2 1 0 0 1 0 Fx 3 1 1 1 0 1 Fu 1 0 1 0 1 1 Fu 2 0 0 1 1 1 Table 1. Faults signatures
Due to the difficulty of estimating x 2 and x 3 as functions of x 1 , and the need to know the measurements of u 1 and u 2 , this method can be replaced by calculating another flat output of the considered flat system. In this case, the measurements of the inputs are not need to be known. This approach is available only if these flat outputs verify the condition of isolability represented above, so that they are algebraically independent in the sense that if a fault affects one of them the others will not be totally affected.

In the following, the concept of algebraically independent flat outputs is defined. Moreover, we show that this feature is valid for the class of direct flat systems that are introduced in the next section.

DIRECT FLAT SYSTEM

Unimodular Completion Algorithm

The variation of the implicit system (2) gives the following tangent system

0 = dF (x, ẋ) = P d dt dx, (11)
with

P d dt =       ∂F 1 ∂ x1 + ∂F 1 ∂ ẋ1 d dt . . . ∂F 1 ∂ xn + ∂F 1 ∂ ẋn d dt . . . . . . ∂F n-m ∂ x1 + ∂F n-m ∂ ẋ1 d dt . . . ∂F n-m ∂ xn + ∂F n-m ∂ ẋn d dt       (
P d dt U = I (n-m) 0 (n-m)×m . ( 13
)
The main property of the flatness is given by the following proposition: Proposition 1. [START_REF] Kaminski | Intrinsic and apparent singularities in differentially flat systems, and application to global motion planning[END_REF]). If the system ( 2) is flat at a point x 0 ∈ X 0 , then there exists a neighbourhood

V of x 0 where P d dt is hyper-regular. Let U = U 0 (n-m)×m I m
and ω = (ω 1 , . . . , ω m ) a vector of m independent 1-forms defined by

ω = U † dx, (14) with U † (0 m×(n-m) I m )U -1 , ( 15
)
the pseudo-inverse of U . The vector ω is a flat output of the variational system (11) or simply a tangent flat output.

Remark 2. The pseudo-inverse U † is not unique, then the tangent flat output ω, associated to U † , is not unique too.

The next definition is borrowed from (Fritzsche et al. (2016a)) and [START_REF] Fritzsche | Unimodular completion and direct flat representation in the context of differential flatness[END_REF]):

Definition 4. Given a hyper-regular matrix M ∈ M p,q d dt with p ≤ q, we say that N ∈ M (q-p)×q d dt is a unimodular completion of M if and only if

M N ∈ U q d dt . (16)
Proposition 2. The matrix U † is a unimodular completion of P d dt .

Proof.

Let U ∈ U n [ d dt ] such that P d dt U = I (n-m) 0 (n-m)×m , (17) then
P d dt = I n-m 0 (n-m)×m U -1 constitutes the first n -m rows of the matrix U -1 , i.e. U -1 = P d dt W (18)
with

W ∈ M m×n [ d dt ]
. Moreover, according to (12), the matrix U † constitutes the last m rows of U -1 , hence

U -1 =   P d dt U †   , (19)
which proves that U † is a unimodular completion of

P d dt .
An algorithm of computation of a unimodular completion of the matrix P d dt is developed in (Fritzsche et al. (2016a)). The flat output y = (y 1 , . . . , y m ) of the system (2) is given by dy = ω (20) provided that ω is integrable, i.e. dω = 0. The conditions of integrability are detailed in [START_REF] Levine | Analysis and control of nonlinear systems: A flatness-based approach[END_REF], [START_REF] Lévine | On necessary and sufficient conditions for differential flatness[END_REF]) and they require the resolution of PDEs whose complexity depends on the system itself.

Direct Flat Representation

Definition 5. [START_REF] Pomet | On dynamic feedback linearization of four-dimensional affine control systems with two inputs[END_REF]). Let (1) be a flat system, it is called (-1)-flat or x-flat if and only if there is a flat output y such that y depends only on x, i.e. y = ψ(x).

Consider a subclass of (-1)-flat systems called direct flat systems defined as follows:

Definition 6. [START_REF] Fritzsche | Unimodular completion and direct flat representation in the context of differential flatness[END_REF]). We say that a (-1)-flat system is a direct flat system if there exists a permutation σ : {1, . . . , n} -→ {σ(1), . . . , σ(n)} such that there exists a flat output given by y = (x σ(1) , . . . , x σ(m) ). Such flat output is called direct flat output.

Proposition 3. [START_REF] Fritzsche | Unimodular completion and direct flat representation in the context of differential flatness[END_REF]). Let P d dt defined by (12) be hyper-regular. Assume that there exists Π, a column permutation matrix such that Remark 3. A direct flat representation is indeed not unique.

P d dt P d dt Π = (A B) (22)

ALGEBRAICALLY INDEPENDENT FLAT OUTPUTS

As mentioned in [START_REF] Torres | Fault detection and isolation on a three tank system using differential flatness[END_REF]), in the aim to provide a total isolation of defects on a systems sensors or actuators, we need to increase the number of residues by calculating several flat outputs. These flat outputs must be algebraically independent in the sense that a fault that affect one of them, the others will not be totally affected. In this section, we present a way to characterize the relation between different flat outputs.

Proposition 4. Let Ω x be the set of all tangent flat outputs at x of a flat system. Then, for all ω 1 and ω 2 ∈ Ω x , there exists a unimodular matrix

K ∈ U m [ d dt ] such that ω 1 = K ω 2 . ( 24
)
Proof. Since the matrix P ( d dt ) is supposed to be hyperregular, there exists a hyper-regular matrix U such that

ω = U † dx (25)
and, indeed, according to (14) dx = U ω.

Let ω 1 = U † 1 dx and ω 2 = U † 2 dx be two different tangent flat outputs at x, then

ω 1 = U † 1 U 2 ω 2 K ω 2 (26) with K = U † 1 U 2 . ( 27
)
Let us prove that K is unimodular. U † 1 and U 2 are hyperregular by construction, then if ξ is a vector such that U † 1 U 2 ξ = 0 with ξ = 0 then ζ = U 2 ξ is also = 0 and U † 1 ζ = 0 which contradicts the hyper-regularity of U † 1 . Hence, K is hyper-regular and square, which proves the unimodularity.

From here two types of relations between the tangent flat outputs are introduced:

(1) if deg(K) = 0 then the tangent flat outputs ω 1 and ω 2 are said to be algebraically dependent; (2) if deg(K) ≥ 1 then the tangent flat outputs ω 1 and ω 2 are said to be algebraically independent.

It is easy to see that, according to the implicit form (2) of the system, at least one state variable is an integral of order larger than or equal to 1 of the other state variables.

As a consequence, we have the following corollary:

Corollary 1. Two different direct tangent flat outputs of a direct flat system are algebraically independent.

Remark 4. A direct flat output is given by dy = ω, then consequently, two direct flat outputs are algebraically independent.

APPLICATION ON THE THREE TANK SYSTEM

For the reasons explained in section 2.3, and in order to detect and isolate faults on sensors and actuators of the three-tank system, we need to calculate another flat output that will be algebraically independent of y = (x 1 , x 3 ) T . In this section, we show that this system is a directly flat with two direct flat outputs. After that we will show how these two flat outputs are useful for the flatness-based FDI.

The implicit equation of this system is calculated as follows: equation ( 6) of the explicit system is free of inputs, so after elimination of equations ( 4) and ( 5) and replacing ( 7) and ( 9) in ( 6), we obtain:

F (x, ẋ) = ẋ3 -a z1 S 2g(x 1 -x 3 ) + a z3 S 2g(x 3 -x 2 ) = 0. (28) with S = S n /S.
According to (12), the matrix P d dt , associated to (28), is given by:

P d dt = - a z1 Sg 2g(x 1 -x 3 ) - a z3 Sg 2g(x 3 -x 2 ) a z1 Sg 2g(x 1 -x 3 ) + a z3 Sg 2g(x 3 -x 2 ) + d dt . (29)
The matrix P and it is integrable, i.e. dω = 0. Then the system (28) is a direct flat system with y = x 2 x 3 (32) a direct flat output. One can also find the flat output y = (x 1 , x 3 ) T using the direct flat representation.

In the following, we denote by y 1 = (x 1 , x 3 ) T and ω 1 = (dx 1 , dx 3 ) T the corresponding tangent flat output, y 2 = (x 2 , x 3 ) T and ω 2 = (dx 2 , dx 3 ) T . In fact, the direct flat outputs y 1 and y 2 are algebraically independent: let According to the Definition 2 of the isolability, the state x 1 is not a component of the flat output y 2 = (x 2 , x 3 ) T , then a fault on the sensor x 1 can be detected and isolated. Finally, by associating the two flat outputs y 1 and y 2 , if a fault is detected but can not be isolated on x 1 and x 2 , then x 3 is inevitably faulty, which allows the complete isolation of defects.

K = γ η + κ

CONCLUSION

The current paper introduced a novel and rigorous definition of the isolability of faults affecting a system's sensors and actuators, using the flatness-based FDI approach. The described condition of isolability provided an efficient way to select a handful of flat outputs useful for fault isolation from an infinite number of possible flat outputs. Our results were tested and validated using the threetank system. Future work needs to focus on providing new definitions of residue signals in order to overcome the reliance on a high-gain observer when estimating the time derivatives of flat outputs.

Fig. 1 .

 1 Fig. 1. Three Tank System, Source: Noura et al. (2009)

  by H = 0 m×(n-m) I m and H = HΠ T , which are constant matrices, H (resp. H) is a unimodular comtangent flat output ω is given by ω = Hdx and always satisfies the integrability condition dω = 0. Hence, a (direct) flat output y of the non linear system is given by y = Hx. Proposition 3, is called a direct flat representation, for which y Hx is a direct flat output.

  3 -x 2 ) and deg(K) = 1. With respect to the implicit form (28), one can easily verify thatω 1 = K ω 2 ,(35)which proves the algebraic independence.

  12)and dx = (dx 1 , . . . , dx n ).

	The matrix P nomial matrices in the operator d d dt ∈ M (n-m)×n d dt dt with meromorphic , the ring of poly-coefficients (Levine (2009)). An invertible matrix U in
	M p×p unimodular. The set of unimodular matrices is denoted by d dt whose inverse is also in M p×p d is called dt
	U p	d dt	. The degree in d dt of a matrix K ∈ M p×q	d dt	,
	Definition 3. (Lévine and Nguyen (2003)). The matrix
	P a unimodular matrix U ∈ U n d is said to be hyper-regular if and only if there exists dt d dt such that

denoted by deg(K), is defined by: deg(K) = max{deg row (K i ), i = 1, . . . , p}, where deg row (K i ) = max{deg(K i,j ), j = 1, . . . , q}.

According to the Remark 1, the point x ∈ X 0 s.t. x 1 = x

= x

is an equilibrium point which is not first order controllable, then it is a point of intrinsic singularity.