Najib Essounbouli

Abdelaziz Hamzaoui

Direct and Indirect Robust Adaptive Fuzzy Controllers for a Class of Nonlinear Systems

Keywords: Adaptive fuzzy control, nonaffine systems, nonlinear systems, sliding mode control

In this paper, we propose direct and indirect adaptive fuzzy sliding mode control approaches for a class of nonaffine nonlinear systems. In the direct case, we use the implicit function theory to prove the existence of an ideal implicit feedback linearization controller, and hence approximate it to attain the desired performances. In the indirect case, we exploit the linear structure of a Takagi-Sugeno fuzzy system with constant conclusion to establish an affine-in-control model, and therefore design an indirect adaptive fuzzy controller. In both cases, the adaptation laws of the adjustable parameters are deduced from the stability analysis, in the sense of Lyapunov, to get a more accurate approximation level. In addition to their robustness, the design of the proposed approaches does not require the upper bounds of both external disturbances and approximation errors. To show the efficiency of the proposed controllers, a simulation example is presented.

INTRODUCTION

Adaptive control schemes for nonlinear systems via feedback linearization concept have been widely employed for decades. The idea of feedback linearization approaches is to transform a nonlinear dynamic system into a linear system through state feedback mechanisms. With such transformations, those well-explored linear controllers can then be applied to meet desired control specifications. Several results and parameter adaptive control schemes have been reported in [START_REF] Isodori | Nonlinear Control Systems[END_REF][START_REF] Sastry | Adaptive Control: Stability, Convergence, and Robustness[END_REF][START_REF] Slotine | Applied Nonlinear Control[END_REF]. However, the performances of these approaches are directly relied to the exact cancellation of nonlinear terms. If these nonlinear terms are uncertain or unknown, the performances can be deteriorated due to non-exact cancellation.

As a model free design method, fuzzy systems have been successfully applied to control complex or illdefined processes whose mathematical models are difficult to obtain [START_REF] King | The application of fuzzy control systems to industrial process[END_REF][START_REF] Takagi | Identification of systems and its applications to modeling and control[END_REF]. The ability of converting linguistic descriptions into automatic control strategy makes it a practical and promising alternative to the classical control scheme for achieving control of complex nonlinear systems. A major drawback of fuzzy control systems is that the fuzzy rules must be previously tuned by trial and error procedures. To overcome this problem, some research has been focused on the Lyapunov synthesis approach to construct stable adaptive fuzzy controllers [START_REF] Spooner | Stable adaptive control of a class of nonlinear systems and neural network[END_REF][START_REF] Sue | Adaptive control of a class of nonlinear systems with fuzzy logic[END_REF][START_REF] Wang | Adaptive Fuzzy Systems and Control[END_REF][START_REF] Wang | Stable adaptive fuzzy controllers with application to inverted pendulum tracking[END_REF][START_REF] Tong | Fuzzy adaptive control for a class of nonlinear systems[END_REF]. The basic idea of most of these works is that with the universal approximation ability of fuzzy systems [START_REF] Wang | Fuzzy systems are universal approximators[END_REF], the plant model is approximated by two adaptive fuzzy systems to construct the control law. To make more accurate the approximation level and hence to improve the tracking performances, the adaptation laws of the adjustable parameters are synthesised from the stability analysis in the sense of Lyapunov. To maintain the performance of fuzzy adaptive control in the presence of external disturbances, some robust schemes based on sliding mode control or H∞ technique are presented in the literature [START_REF] Shigame | Sliding mode controller design based on fuzzy inference for nonlinear systems[END_REF][START_REF] Wang | A Course in Fuzzy Systems And Control[END_REF][START_REF] Hamzaoui | Fuzzy sliding mode control with fuzzy switching function for nonlinear uncertain MIMO systems[END_REF][START_REF] Yoo | Adaptive fuzzy sliding mode control of nonlinear systems[END_REF][START_REF] Wang | Indirect sliding mode control: part I: Fuzzy switching[END_REF][START_REF] Lin | Robust adaptive sliding mode control using fuzzy modeling for a class of uncertain MIMO nonlinear systems[END_REF][START_REF] Hamzaoui | Fuzzy sliding mode control for uncertain SISO systems[END_REF][START_REF] Chen | H∞ tracking design of uncertain nonlinear SISO systems: Adaptive fuzzy approach[END_REF][START_REF] Hamzaoui | Adaptive fuzzy control for uncertain nonlinear systems[END_REF][START_REF] Hamzaoui | Robust adaptive fuzzy control application to uncertain nonlinear systems in robotics[END_REF][START_REF] Chang | Adaptive fuzzy-based tracking control for nonlinear SISO systems via VSS and H∞ approaches[END_REF][START_REF] Essounbouli | A supervisory robust adaptive fuzzy controller[END_REF][START_REF] Essounbouli | A robust adaptive fuzzy control applied to disturbed MIMO systems[END_REF]. However, these approaches are restricted to affine in control plants. To overcome this restriction, some works treating the extension of adaptive control to nonaffine systems have been developed in the literature. Concerning the indirect adaptive control scheme, there are two techniques where the main idea is to synthesise an affine-in-control model of the plant to design the controller. Indeed in [START_REF] Yoon | Adaptive fuzzy control of nonlinear systems using Takagi-Sugeno fuzzy models[END_REF] and [START_REF] Boukezzoula | Fuzzy adaptive linearizing control for nonaffine systems[END_REF], the authors exploit the linear structure of the Takagi-Sugeno systems with triangular membership functions for inputs and constant conclusion, to establish an affine-in-control fuzzy model to describe the dynamic behaviour of the plant. In [START_REF] Park | Robust adaptive fuzzy controller for non-affine nonlinear systems with dynamic rule activation[END_REF], the Taylor series expansion is used to obtain an affine in a control model of the plant. Concerning the direct control scheme presented in [START_REF] Guo | CMAC NN-based adaptive control of non-affine nonlinear systems[END_REF][START_REF] Zhang | Direct adaptive control of non-affine nonlinear systems using multilayer neural network[END_REF], the authors used the implicit theorem and variable structure control to prove the existence of feedback control, which has been approximated using neural networks. To improve the approximation level and hence the tracking performances, an adaptation law is derived from the stability analysis. However, to ensure the stability and the robustness of the closed loop system in these works, an additional control signal is needed to compensate the approximation errors and the external disturbances. The design of this signal depends on the well-known upper bounds of both the approximation errors and the external disturbances, which is a restrictive assumption due to the fact that these bounds are generally unknown.

In this work, we propose direct and indirect adaptive fuzzy controllers for a class of non-affine systems subject to external disturbances. In the direct case, we use an adaptive fuzzy system to approximate the implicit desired feedback control whose existence is proven by the implicit theorem. In the indirect case, we exploit the linear structure of a Takagi-Sugeno fuzzy system to generate a fuzzy affine-in-control model to approximate the dynamic behaviour of the plant. In both cases, we utilize the modified sliding mode control to ensure the robustness of the closed loop system. Indeed, the approaching phase is assured by an attenuation term that allows the chattering phenomenon to be eliminated and the constraint on the knowledge of the upper bounds of both external disturbances and approximation errors to be overcome.

To improve the approximation level, the adaptation laws are derived from the stability analysis in the sense of Lyapunov. To show the efficiency of the proposed approaches, an illustration example is presented.

PROBLEM STATEMENT

Consider a single-input single-output (SISO) nonlinear system described by the following differential equation: The objective is to develop a control law using sliding mode control to ensure the tracking performances and the robustness of the closed loop system. So, forcing the system to track a reference trajectory is equivalent to forcing the plant to attain a sliding surface and to maintain it on it. The Hurwitzian structure of this surface allows the convergence of the plant to the phase plan origin.

( ) ( ) ( )

DIRECT ADAPTIVE FUZZY CONTROLLER

In this section our task is to synthesise a direct adaptive fuzzy controller for the system [START_REF] Isodori | Nonlinear Control Systems[END_REF]. For this, we assume the following:

Assumption A1: ( ) 0 , 0 u f x u b b u ∂ ⎡ ⎤ ⎣ ⎦ = >

≠ ∂

Assumption A2: There exists a smooth function

( ) 0 x β > such that ( ) / u u u u u b b t b x b b β ∂ ∂ = ≤ .
To attain the desired tracking performances, let's consider the sliding surface S or the filtered error given as follows [START_REF] Slotine | Applied Nonlinear Control[END_REF]:

( ) ( ) ( ) ( ) 1 2 1 1 1 1 1 1 ... , n n n n n i i i S e k e k e e ke - - - - - - = = - - -- = - -∑ (2)
where r e y y =denotes the tracking error and r y is a bounded reference trajectory. The gains i k , 1,..., i n =

, are chosen such that the corresponding polynomial is Hurwitzian. Using (1) and ( 2), the time derivative of the switching surface S can be written as:

( ) ( ) ( ) ( ) ( ) 1 1 1 1 , . n n i i i n n i r i i S e k e f x u d y k e - = - = = - - = + - - ∑ ∑ (3)
Lemma 1 [START_REF] Zhang | Direct adaptive control of non-affine nonlinear systems using multilayer neural network[END_REF]: For the system (1) free of external disturbances (d=0) satisfying A1 and A2, there exists a compact set u Φ and an unique ideal input * u such that all ( )

0 u
x ∈ Φ , the equation ( 3) can be expressed as the following form:

( )

S x S β = - , (4)
which allows to obtain lim 0

r x y y →∞ -= .
The previous lemma guarantees only the existence of a control law guaranteeing the convergence of the tracking error toward zero and does not provide the method of constructing it [START_REF] Zhang | Direct adaptive control of non-affine nonlinear systems using multilayer neural network[END_REF]. Based on the fact that a fuzzy system is an universal approximator [START_REF] Wang | Fuzzy systems are universal approximators[END_REF], we use a fuzzy Takagi-Sugeno system to approximate the ideal law * u . To guarantee the stability of the closed loop system, we add a supplementary signal s u . Hence, the proposed control law is given by:

fuzzy s u u u = + , (5)
( )

T fuzzy u x θ = Ψ , ( 6
) 2 s S u ρ = - , ( 7
)
where θ is the vector of the adjustable parameters, ( )

x Ψ the regressive vector, and ρ is a positive constant representing the attenuation level of the effects of both the approximation error and the external disturbances. Note that * u is written as

( ) * * T u x θ δ = Ψ +
, where * θ is the optimal value of θ and δ is the approximation error.

Using the Mean Value Theory [START_REF] Khalil | Nonlinear Systems[END_REF], there exists a positive constant

] [ 0 , 1 λ ∈ such that: ( ) ( ) ( ) * * , , u f x u f x u b u u λ = + - , (8) where ( )
, u u u f x u b u λ λ = ∂ ⎡ ⎤ ⎣ ⎦ = ∂ and ( ) * 1 u u u λ λ λ = + - .
According to the implicit theorem [31, there exists a * u such that ( )

* , 0 v f x u + = where
( ) v x S β = ( ) ( ) 1 1 n n i r i i y ke - = - -∑
. So, (8) can be written as:

( ) ( ) ( ) ( ) ( ) ( ) * 1 * 1 ,
.

u n n i r i u i f x u v b u u x S y k e b u u λ λ β - = = -+ - = - + + + - ∑ (9)
From ( 3) and ( 9), we can obtain:

( ) ( ) * u S x S b u u d λ β = - + - + . ( 10
)
Using [START_REF] Takagi | Identification of systems and its applications to modeling and control[END_REF], equation [START_REF] Tong | Fuzzy adaptive control for a class of nonlinear systems[END_REF] becomes:

( ) ( ) * u fuzzy u s T u u S x S b u b u b x b d λ λ λ λ β θ δ = - + + - Ψ - + (11)
or

( ) ( ) ( ) * 2 , T T u u u u S x S b x b x S b b d λ λ λ λ β θ θ δ ρ = - + Ψ - Ψ - - + (12)
( ) ( )

2 T u u u S S x S b x b b d λ λ λ β θ δ ρ = - + Ψ - - + ,( 13
)
which gives ( ) ( )

1 1 1 2 T u u u S b S b x S x b d λ λ λ β θ δ ρ - - - = - + Ψ - -+ , (14)
where

* θ θ θ = -.
Consider the following Lyapunov function:

1 2 1 1 2 2 T u V b S λ θ θ η - = + . ( 15
)
Differentiating ( 15) along ( 12) yields:

1 1 1 1 1 1 2 2 2 2 T T u u V Sb S Sb S λ λ θ θ θ θ η η - - = + + + . ( 16
)
Using the fact that * θ θ θ θ = -= and since the elements of V are scalars, the equation ( 16) can be rewritten as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( )

1 1 1 2 2 1 2 1 2 2 1 2 1 2 2 1 2 1 1 S 1 1 1 . T u T T u u u T T u u T u u T u V Sb S S b x S x b b d S b x S S x S b d S b x S S b d S x S S b d S x λ λ λ λ λ λ λ λ λ θ θ η β θ δ θθ η ρ β θ δ θθ η ρ β δ θ θ η η ρ δ θ θ η η ρ - - - - - - - - = + ⎛ ⎞ = - + Ψ - - + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ = - + Ψ - - - + = - -- - + + Ψ ≤ -- - + + Ψ (17)
Choosing the following adaptation law:

( ) S x θ η = -Ψ (18)
gives:

( ) ( ) ( ) ( ) 2 1 2 2 2 1 2 2 2 2 2 1 2 1 2 2 2 2 1 2 2 2 u u u u S V S b d S S S b d S S S b d b d λ λ λ λ δ ρ δ ρ ρ δ ρ δ ρ ρ - - - - ≤ - - - ≤ - - - - ⎡ ⎤ ≤ - - + - + - ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ( ) ( ) 2 2 1 2 2 2 1 2 2 . 2 2 u u b d S b d λ λ ρ δ ρ δ ρ - - + - ≤ - + - (19)
Integrating the above inequality from 0 t = and T , we have:

( ) ( ) ( ) 2 2 2 1 2 0 0 0 2 2 T T u S V T V dt b d dt λ ρ δ ρ - - ≤- + - ∫ ∫ , (20) ( ) ( )
( )

2 2 2 1 2 0 0 0 2 2 T T u S dt V V T b d dt λ ρ δ ρ - ≤ - + - ∫ ∫ .( 21
)
Using the fact that ( ) 0 V T ≥ , the above inequality can be simplified as:

( ) ( ) 2 2 2 1 2 0 0 0 2 2 T T u S dt V b d dt λ ρ δ ρ - ≤ + - ∫ ∫ , (22)
(22) guarantees that S L ∞ ∈ . Because all the variables in the right-hand side of ( 14) are bounded, i.e., S L ∞ ∈ . Since the right side of ( 22) are also bounded,

2

S L ∈ [START_REF] Guo | CMAC NN-based adaptive control of non-affine nonlinear systems[END_REF][START_REF] Zhang | Direct adaptive control of non-affine nonlinear systems using multilayer neural network[END_REF]. Using Barbalat Lemma, we have 0 S → when t → ∞ [START_REF] Zhang | Direct adaptive control of non-affine nonlinear systems using multilayer neural network[END_REF]. Therefore, the tracking error converges to the origin, i.e., lim 0

x e →∞ = [3].
To ensure the convergence of the adaptive algorithm, we introduce the projection algorithm defined by [START_REF] Essounbouli | A supervisory robust adaptive fuzzy controller[END_REF].

⎧ ≤ ⎪ -Ψ = ⎪ ⎪ ⎪ Ψ ≥ = ⎨ ⎪ ⎡ ⎤ = ⎪ ⎢ ⎥ - Ψ + Ψ ⎪ Ψ < ⎢ ⎥ ⎪ ⎣ ⎦ ⎩ (23)

INDIRECT ADAPTIVE FUZZY CONTROLLER

In the case where Assumptions A1 and A2 are restrictive, and we cannot satisfy them, we can use an indirect adaptive fuzzy controller. The main idea is to construct an affine model of the plant using a Takagi-Sugeno system. Then we use it to synthesise a robust controller allowing it to ensure the tracking performances and the robustness of the closed loop system.

The plant is constructed from a Takagi-Sugeno system whose inputs are the state variables 1

x , 1,..., i n = , and u . For each variable 1 x , 1,..., i n = , and u , we define i p and M fuzzy sets. Hence, the j th rule can be written in the form:

( )

1

Rule j IF x is A And x is A And u is

B THEN x θ ⎡ ⎤ ⎣ ⎦ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ (24)
where { } Using the singleton fuzzifier, the centre average defuzzification and the product inference engine, the output of the Takagi-Sugeno system can be given by [START_REF] Wang | Adaptive Fuzzy Systems and Control[END_REF]:

( ) ( ) ( ) ( ) ( ) 1 1,..., , 1 1 1 1 1 1 1 ... ... p pn n M j jnm ji i m B A i j jn i p pn n M ji i m B A i j jn i x u n x u x θ μ μ μ μ = = ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ∑ ∑ ∑ ∏ = ∑ ∑ ∑ ∏ , (25)
or on the following vectorial form:

( )

, T n x x u = Θ Ψ , (26) where ( )
, x u Ψ is a 1 ... p pn M × × × dimensional
vector with its ( ) 1,..., , th j jn m element given by:

( ) ( ) ( ) ( ) ( ) 1 1 1 1 1
1,..., , ...

n ji i m B A i i p pn n M ji i m B A i j jn i x u j jnm x u μ μ μ μ = = ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ∏ Ψ = ∑ ∑ ∑ ∏ , (27)
- ⎧ = ⎪ - ⎪ ⎪ - ⎪ = = + ⎨ - ⎪ ⎪ ⎪ ⎪ ⎩ , ( 28
)
where m a is a constant satisfying

( ) 1 m m B a μ = .
For a given value of

{ } 1, 2,..., 1 M α =
-, the control input u exists in the subspace u U α . So, substituting ( 28) in [START_REF] Yoon | Adaptive fuzzy control of nonlinear systems using Takagi-Sugeno fuzzy models[END_REF] gives:

( ) ( ) ( ) ( ) ( ) ( ) ( ) 1
α α α α α α α α θ θ θ θ + + + ⎡ ⎤ Θ = Ψ - ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ Ψ - ⎢ ⎥ ⎣ ⎦ = Φ + Φ ∑ ∑ ∑ ∑ (29)
where ( ) ( ) ( )

1 1 1 1 1 1 ... 1 n ji i A i i p pn n M ji i m B A i j jn i x s x u a a μ α α μ μ = = ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ∏ Ψ = - ∑ ∑ ∑ ∏ .
Therefore, the fuzzy system can be decomposed into M-1 subsystems, which allows to obtain an affine-in-control model of the plant [START_REF] Yoon | Adaptive fuzzy control of nonlinear systems using Takagi-Sugeno fuzzy models[END_REF][START_REF] Boukezzoula | Fuzzy adaptive linearizing control for nonaffine systems[END_REF].

After synthesising the fuzzy model, our next task is to develop a robust controller to ensure the global stability and the robustness of the closed loop system.

the proposed control law is given by:

( ) ( ) ( ) ( ) 1 1 2 1 1 n n i r i s i u x x y ke u α α - - = ⎡ ⎤ ⎡ ⎤ = Φ -Φ + + + ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ∑ , (30)
where s u denotes an additional term guaranteeing the robustness of the closed loop system by attenuating the effects of both the external disturbances and the approximation errors to a prescribed level ρ . To attain this objective, we choose s u as given in (7):

2 s S u ρ = - .
Using [START_REF] Khalil | Nonlinear Systems[END_REF], the time derivative of the sliding surface is given by:

( ) ( ) , , , s S f x u f x u u d = - Θ + + . (31)
If we note by ( )

( ) * * * ˆ, , , T f x u x u Θ = Θ Ψ the optimal value of ( ) ( ) ˆ, , , T f x u x u Θ = Θ Ψ
and by ( )

, w f x u = ( ) * * ˆ, , f x u -
Θ the minimal approximation error, [START_REF] Gho | Model reference control of nonlinear systems via implicit function emulation[END_REF] can be rewritten as:

( ) 2 , T S S w x u d ρ = +Θ Ψ + - , (32)
To determine the adaptation law of the adjustable parameter vector Θ , we consider the following Lyapunov function:

2 1 1 2 2 T V S σ = + Θ Θ , ( 33
)
where σ is a positive constant given by the designer.

Using [START_REF] Ge | Adaptive control of non-affine nonlinear systems using neural networks[END_REF] and the fact that Θ = -Θ , the time derivative of (33) can be written as:

( ) 2 1 , T T S V S w x u d σ ρ ⎡ ⎤ = + Θ Ψ + - -Θ Θ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ , (34) [ ] ( ) 2 2 1 ,
T S V S w d S x u σ σ ρ ⎡ ⎤ = + - -Θ Θ-Ψ ⎣ ⎦ , (35) [ ] ( ) 2 2 2 2 1 , 2 2 T
S S V S w d S x u σ σ ρ ρ ⎡ ⎤ = + - - -Θ Θ-Ψ ⎣ ⎦ , (36) [ ] [ ] [ ] ( )
2 2 2 2 2 2 2 2 1 2 2 1 1 , , 2 2 T
S V w d Sw d S w d S x u ρ ρ ρ σ σ ρ ⎡ ⎤ = - + - + - ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ + + - -Θ Θ-Ψ ⎣ ⎦ (37) [ ] [ ] ( ) 2 2 2 2 2 1 2 2 1 , , 2 T
S V w d w d S S x u ρ ρ ρ σ σ ρ ⎡ ⎤ = - + - + + ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ - -Θ Θ- Ψ ⎣ ⎦ (38)
[ ] ( )

2 2 2 1 1 , . 2 2 T S V w d S x u σ σ ρ ⎡ ⎤ ≤ + - -Θ Θ- Ψ ⎣ ⎦ (39)
If we choose the following adaptation law:

( )

, S x u σ Θ = Ψ (40)
and using the same mathematical tool development presented in the previous section, we can obtain:

( ) [ ] 2 2 2 2 0 0 0 2 2 T T S dt V w d dt ρ ρ ≤ + + ∫ ∫ . ( 41
)
As proven in the previous section, we have 2

S L ∈

and S L ∞ ∈ . Hence, we have 0 S → when t → ∞ . According to the Hurwitzian structure of the sliding surface, the system converges to the origin of the phase plan [START_REF] Slotine | Applied Nonlinear Control[END_REF].

Remark 1: To overcome the singularity problem when ( )

2 0 x α Φ =
without complicating the controller structure, we propose to substitute the term ( )

1 2 x α - ⎡ ⎤ Φ ⎣ ⎦ by ( ) ( ) 2 2 2 x x α α ε ⎡ ⎤ Φ ⎣ ⎦ ⎡ ⎤ + Φ ⎣ ⎦
[33] which yields the new control law:

( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 2 n n i r i s i x u x y k e u x α α α ε - = ⎡ ⎤ Φ ⎡ ⎤ ⎣ ⎦ = -Φ + + + ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎣ ⎦ + Φ ⎣ ⎦ ∑ ,( 42
)
where ε is a small positive constant.

Remark 2:

To ensure the convergence of the adaptive algorithm, we modify the adaptation law by using the projection technique as given by (43).

(

T if f S x u or f and S x u if f S x u S x and S x u σ σ ⎧ Θ ≤ ⎪ Ψ Θ = ⎪ ⎪ ⎪ Ψ ≤ Θ = ⎨ ⎪ ⎡ ⎤ Θ = ΘΘ ⎪ ⎢ ⎥ Ψ - Ψ ⎪ Ψ > ⎢ ⎥ Θ ⎪ ⎣ ⎦ ⎩ (43) ) ( ) ( ) ( ) ( ) max max max 2 : [ ] , [ , 0] : [ , , 0]

AN ILLUSTRATION EXAMPLE

To illustrate the effectiveness of the proposed adaptive controllers, we consider the following system: ( ) ( )

1 2 2 3 2 2 1 2 1 0.1 0.1 1 sin 0.1 . x x x x u x u u y x = ⎧ ⎪ ⎪ = + + + + ⎨ ⎪ = ⎪ ⎩ (44)
In this simulation example, the control objective is to determine u so that the output y follows the desired reference trajectory sin( )

r y t = .
To show the robustness of the proposed approaches, we consider that the system is subject to external disturbances in the form ( )

0.2 sin( ) sin(2 ) d t t = + .
To synthesise the direct adaptive fuzzy controller, we use a fuzzy system with the state variables x₁ and x₂ as inputs and fuzzy u as output. x (solid line) and its reference trajectory (dashed line). x (solid line) and its reference trajectory (dashed line). Fig. 4, we remark that the control signal does not contain any abrupt variation despite the knowledge unavailability of the upper bound of the external disturbances.

To overcome the restrictive assumptions A1 and A2 used to design the control law [START_REF] Takagi | Identification of systems and its applications to modeling and control[END_REF], we propose to use the indirect adaptive fuzzy controller given by [START_REF] Gho | Model reference control of nonlinear systems via implicit function emulation[END_REF]. For this, we must at first define a fuzzy system, which inputs the state variables ( ) For each input variable, we have defined respectively 5, 5, and 6 linguistic sets. Figs. 5 and6 give the evolution of the state variables 1

x and 2

x together with the corresponding reference signals.

Despite using an affine-in-control fuzzy model to synthesise the control law, we can note the good tracking performances and the convergence of the system to the desired trajectories. This can be justified by the good approximation level assured by the proposed approach as illustrated in Fig. 7. The applied control signal to attain our objective is given by Fig. 8. In summary, we can conclude that the proposed (direct and indirect schemes) allow the desired tracking performances to be attained and also ensure the robustness of the closed loop system, without using classical methods of linearization. Furthermore, the design of these controllers does not require any knowledge regarding the disturbances structure or their bounds.

CONCLUSION

In this paper direct and indirect robust adaptive fuzzy controllers for a class of nonaffine nonlinear systems are presented. In the direct case, based on the implicit theorem, a fuzzy adaptive system is used to attain the desired performances. In the indirect case, a Takagi-Sugeno system is used to synthesise an affinein-control model, and hence to design the controller. In the two cases, the robustness of the closed loop system is guaranteed by a modified sliding mode x (solid line) and its reference trajectory (dashed line). x (solid line) and its reference trajectory (dashed line). control where the upper bounds of both the external disturbances and the approximation errors are not required. To improve the approximation accuracy, the adaptation laws of the adjustable parameters are deduced from the stability analysis in the sense of Lyapunov. The simulation results demonstrate both good tracking performances and high efficiency of theses approaches. Current works are focussed on the use of a state observer to overcome the assumption on the availability of the state variables to the measurement.

Fig. 1 .

 1 Fig. 1. The membership functions of u .

Fig. 2 .

 2 Figs. 2, 3 and 4 present the simulation results for the initial sate

Fig. 3 .

 3 Fig. 3. Evolution of the state variable 2x (solid line) and its reference trajectory (dashed line).

Fig. 4 .

 4 Fig. 4. Applied control signal.

2 x

 2 . The universe of discourse of the inputs 1 x , , and u are respectively [ ]

Fig. 5 .

 5 Fig. 5. Evolution of the state variable 1x (solid line) and its reference trajectory (dashed line).

Fig. 6 .

 6 Fig.[START_REF] Spooner | Stable adaptive control of a class of nonlinear systems and neural network[END_REF]. Evolution of the state variable 2x (solid line) and its reference trajectory (dashed line).

Fig. 7 .

 7 Fig. 7. Evolution of the dynamic function ( ) f x,u (dashed line) and its approximation (continuous line).

Fig. 8 .

 8 Fig. 8. Applied control signal.

  International Journal of Control, Automation, and Systems, vol. 4, no. 2, pp. 146-154, April 2006

	__________
	Manuscript received October 1, 2005; accepted January 12,
	2006. Recommended by Editor Zengqi Sun.

Najib Essounbouli and Abdelaziz Hamzaoui are with the Mechanical Engineering Department of the Technical Institute of Troyes, 9, rue de Quebec, B.P. 396-10026 Troyes cedex-France (e-mails: {najib.essounbouli, abdelaziz.hamzaoui}@ univ-reims.fr).


A Ameur

B Mokhtari

N Essounbouli
email: najib.essounbouli@univ-reims.fr.

F Nollet

Modified Direct Torque Control for Permanent Magnet Synchronous Motor Drive Based on Fuzzy Logic Torque Ripple Reduction and Stator Resistance Estimator

Keywords: DTC, PMSM, FLC, Switching table, Torque and flux ripples, Stator resistance estimator. 

The two important disadvantages of Direct Torque and Flux Control (DTC), are torque and flux ripples and stator resistance variation (which affects seriously the estimation stator flux and electromagnetic torque). This paper proposes a fuzzy-logic based DTC (FLC) using modified switching table for permanent magnet synchronous motor (PMSM) and Fuzzy-logic Resistance Estimator. The high-performance of the proposed FLC applied to PMSM drive, are investigated and compared with traditional DTC. The proposed system has proven successful in controlling the instantaneous torque so as not to depend only on the estimation flux, torque and position, but also on variable amplitude torque hysteresis bands and stator resistance (r s ) delivered by fuzzy-logic.

INTRODUCTION

Recently, the areas of industrial application of high performance AC drives are considerably expanded with the development of power electronics, Microelectronics and computer Engineering. Indeed, technological developments have permitted alternative machines, especially the synchronous machine; to acquire the control suppleness and high dynamic performance naturally obtained, [START_REF] Boussak | Implementation and Experimental Investigation of Sensorless Speed Control With Initial Rotor Position Estimation for Interior Permanent Magnet Synchronous Motor Drive[END_REF]. In addition, depth research has led to the development of the new materials, such as permanent magnets based on rare earths, [START_REF] Kadjoudj | Fuzzy-Rule based Model Reference Adaptive Control of PMSM drives[END_REF]. These allow obtaining synchronous machines which are compared to other types of DC machines, synchronous with electrical excitation and asynchronous many advantages, among others, low inertia and high power density and efficiency, [START_REF] Zhong | A Direct Torque Controller for Permanent Magnet Synchronous Motor Drives[END_REF]. They are currently the best placed to meet the requirements of applications such as robotics, machine tools, space technology (actuators aviation, satellites) and general embedded systems.

DTC offer high advantages performances in terms of simplicity control scheme and a very good torque dynamic response, as well as the fact that it does not need the rotor speed or position to realize the torque and flux control. Moreover, except stator resistor, DTC is not sensitive to parameters variations, [START_REF] Takahashi | High-performance direct torque control of an induction motor[END_REF].

In the classic DTC the employment of the hysteresis controllers to regulate the stator magnetic flux and torque is natural to have high torque ripples and variable switching frequency, which is varying with speed, load torque, selected hysteresis bands and difficulty to control torque and flux at very low speed. It also results in higher acoustical noise and in harmonic losses, [START_REF] Wang | novel direct torque control for permanent magnet synchronous motor drive[END_REF].

The variation of r s due to changes in temperature or frequency degrade the performance of DTC controller by introducing errors in the estimated flux linkage magnitude, position and the electromagnetic torque, [START_REF] Haque | Influence of Stator Resistance Variation on Direct Torque Controlled Interior Permanent Magnet Synchronous Motor Drive Performance and Its Compensation[END_REF]. So the compensation for the effect of r s variation then becomes necessary, [START_REF] Ameur | An Improved Sliding Mode Observer for Speed Sensorless Direct Torque Control of PMSM Drive with a Three-Level NPC Inverter Based Speed and Stator Resistance Estimator[END_REF].

In this paper, two fuzzy-logic (FL) are proposed to overcome the problem of torque and flux hysteresis bands with variable amplitude based on switched table, in over hand the effect of r s variation is discussed and the r s is estimated by FL estimator using the error between the reference and actual stator flux linkage. The performance of the controller is examined by extensive simulation studies, which shows that the drive system becomes unstable if the r s value used in the controller differs at real resistance.

PMSM MODEL

The stator and rotor flux equation of PMSM can be written in the reference frame of Park, in the following form:

                                          0 0 0 e sq sd q d q d I I L L    (1)
In this same reference, the stator voltages equations are written in the following form:

0 0 0 0 0                                                                              e

CONVENTIONAL DTC

The methods of DTC as shown in figure 1 consist of directly controlling the turn OFF or turn ON of the inverter switches on calculated values of stator flux and torque from relation [START_REF] Haque | Influence of Stator Resistance Variation on Direct Torque Controlled Interior Permanent Magnet Synchronous Motor Drive Performance and Its Compensation[END_REF]. The changes of state of the switches are linked to the changes in electromagnetic state motor. They are no longer controlled based on voltage and frequency references given to the commutation control of a pulse width voltage modulation inverter, [START_REF] Mokhtari | DTC Applied to Optimize Solar Panel Efficiency[END_REF]. The reference frame related to the stator makes it possible to estimate the stator flux (magnitude and argument) and torque. The aim of the switches control is to give the vector representing the stator flux the direction determined by the reference value.

) (

) ( 0 0              t s s s s t s s s s dt I r v dt I r v         (5)
The DTC is deduced based on the two approximations described by the formulas ( 6) and ( 7): [START_REF] Haque | Influence of Stator Resistance Variation on Direct Torque Controlled Interior Permanent Magnet Synchronous Motor Drive Performance and Its Compensation[END_REF] ( 1) ( ) A two levels classical voltage inverter can achieve seven separate positions in the phase corresponding to the eight sequences of the voltage inverter, [START_REF] Takahashi | High-performance direct torque control of an induction motor[END_REF].

s s s E s s E k k V T V T            ' ' ( ) sin e s r s r T k k         (7) More over:                     
    V 2 =[1 1 0] V 1 =[1 0 0] V 3 =[0 1 0] V 4 =[0 1 1] V 5 =[0 0 1] V 6 =[1 0 1] Sector 1 S 1 S 2 S 4 S 3 S 5 S 6 V 4 (011) 1 2 3 4 5 6 7 Sector 6 V 1 (IF,DT) V 2 (IF,IT) V 4 (DF,IT) V 5 (DF,DT) V 1 (100) V 2 (110) V 3 (010) V 5 (001) V 6 (101) V 0,7 (000)  s * T e * Switching Table  s & T e Estimation  s Position + - + -    s T e
Traditional DTC Modified DTC V 1 -30°  30° Torque undetermined -60°  0° DT , IF V 2 30°  90° IT , IF 0°  60° IT , IF V 3 90°  150° IT , DF 60°  120° Flux undetermined V 4 150°  210° Torque undetermined 120°  180° IT , DF V 5 210°  270° DT , DF 180°  240° DT , DF V 6 270°  330° DT , IF 240°  300° Flux undetermined
Table 3 presents the switching table when the sectors are shifted by 30°. We can see the difference from the conventional DTC in the third and the fourth lines for conventional switching table, [START_REF] Mokhtari | Comparative Experimental Study of Three Switching Tables of a DTC Applied to an Induction Motor for a Tracking System[END_REF]. The basic idea is to replace bands flux and torque controllers hysteresis desired by bands issued by FLC. The value of the change in the speed error can be used for measuring or estimating the torque smoothing, [START_REF] Zidani | Direct Torque Control of Induction Motor with Fuzzy Minimization Torque Ripple[END_REF].

The equation of mechanical motor neglecting viscous friction is:

l e r T - T dt Ω d J  (9)
Replacing the error speed e = Ω r -Ω ref in equation ( 9), we find:

l e ref ref T - T dt de J dt Ω d J dt e Ω d J     ) ( (10)
For a constant reference speed and a constant load, the change of the error speed is related to motor torque by:

J T - T dt de l e  (11)
From ( 11), we can conclude that the change of speed error signal can indeed be a good measure and a good indicator of motor torque ripple.

We will present in the following a fuzzy approach dedicated to reducing torque ripples.

A fuzzy controller seems to be a reasonable choice to evaluate the hysteresis bands of the torque and flux amplitude, [START_REF] Adam | Minimum Torque Ripple Algorithm with Fuzzy Logic Controller for DTC of PMSM[END_REF].

In this approach, the hysteresis bands of torque and flux are not prefixes, but they are determined by two fuzzy controllers. Figure ( 4) shows the two adapted controllers torque and flux hysteresis bands. Based on the proposed analysis and the diagram above, two inputs are chosen, error rate speed and stator current, variations:

) 1 ( ) ( ) ( 1     k - k k e r r (12) 2 ( ) ( )-( 1) s s e k i k i k   (13)
The magnitude of the stator current is deduced from the following relation:

2 2   s s s i i i   .
The net output (with a hysteresis band appropriate) is integrated in such a way that the amplitude of the hysteresis band of the torque and flux is obtained from the following two equations:

) ( ) 1 ( ) ( k b k b k b e e e T T T     ( 14
) ) ( ) 1 ( ) ( k b k b k b s s s        (15)
The fuzzification of input variables and outputs of the two fuzzy controllers is shown in figure (5). For example, for e 1 Positive Big (PB), and a variation e 2 Positive (P), then Δb Te must be Positive (P) as well. In this case, the overshoot in the torque error can touch the upper band which will lead us to select a reverse voltage vector. This will cause a rapid reduction in torque but slow response torque below the hysteresis band, [START_REF] Zidani | Direct Torque Control of Induction Motor with Fuzzy Minimization Torque Ripple[END_REF]. Thus, Δb Te should not be Positive Small (PS) but positive (P), to avoid this situation.

EFFECT OF STATOR RESISTANCE VARIATION

The unknown of r s may have several origins. They may be due to an incorrect identification of the parameters of the machine, change in the load or the temperature of the surroundings. Because an incorrect value of the r s certainly causes an error estimation of the stator flux module its position. It can cause serious problems in the choice of voltage vector to apply, [START_REF] Tang | A Novel PI Stator Resistance Estimator for Direct Torque Controlled Permanent Magnet Synchronous Motor Drive[END_REF]. Several studies have been proposed for this estimate to address the problem of variation of r s , [START_REF] Mir | PI and Fuzzy Estimators for Tuning the Stator Resistance in Direct Torque Control of InductionnMachines[END_REF], . In this section we will propose to illustrate the influence of this variation, and then we present a block estimate that compensates each variation of r s , using a fuzzy-logic controller.

In DTC drive, the stator flux is estimated by integrating the difference between the input voltage and the voltage drop across the r s , as given by equation (5).

) (

ˆ0      t s s s s sa s dt I r v    (16)
The estimated and actual electromagnetic torque are given by

) ( 2 3 ˆ      s s s s e ea e I I p T T T     (17)
Any variation of r s can cause an imbalance in the equations of the stator flux PMSM components, expressed in steady state.

) )( ( (

ˆ0           t s s s s s s s sa dt I I r r v    (18) ] ) )( ( ) )( [( 2 3 ˆ            s s s s s s s s e e ea I I I I p T T T              (19)
The quantities estimated by the controller are given by:

) )( ( ( ˆ0      t s s s s s dt I I r v  (20) ] ) )( ( ) )( [( 2 3 ˆ            s s s s s s s s e I I I I p T           (21)
Errors in the estimation of flux and torque are given by: Equations ( 22) and ( 24) gives the errors of the stator flux and torque are taking into account the variation of the r s during operation. Estimated quantities differ sensibly from actual magnitudes of the machine, so the resistive term can be comparable to the supply voltage V s and the error in the stator flux becomes more important, leading to a dysfunction in the choice of voltage vector applied to the machine, so the DTC will lose performance and may become unstable.

) )( ( ˆ0          t s s s s sa s dt I I r    (22) ] ) )( ( ) )( [( 2 3 ˆ              s s s s s s s s s e ea e I I I I p T T T                (23) ] ) ) (( ) ) [(( 2 3 ˆ00               

FUZZY ESTIMATOR OF STATOR RESISTANCE

A Fuzzy-logic corrector is developed to establish the DTC and to obtain a good estimation of flux and electromagnetic torque. This FL estimator can detects the r s variation by using the error between the reference and actual stator flux linkage The inputs to the r s fuzzy estimator are the normalized error between the reference and estimated stator Flux, e(k) =ϕ s *(k)ϕ s (k), and the normalized change in Flux error e(k)=e(k)e(k-1), whose block diagram are illustrated by the figure [START_REF] Haque | Influence of Stator Resistance Variation on Direct Torque Controlled Interior Permanent Magnet Synchronous Motor Drive Performance and Its Compensation[END_REF]. The selected inputs and output memberships for the fuzzy estimator are given in the figure (5), the centroid defuzzification algorithm is used, in which the output fuzzy variable value is calculated as the centre of gravity of the membership function which is always added to the previously estimated r s (k-1). This updated stator resistance can be used directly in DTC algorithm estimation. In addition, the rule base controlling the defuzzified output according to the fuzzified input values is given in Table (5). In this simulation study, we will carry two cases of the PMSM Control:

2 * 2 * * ) ( ) (      s s s   (12)
8.1 Control without compensation.

We simulated the system drive for a reference speed of 100 (rd / s) load at startup. At t = 0.08(s), the PMSM is tracking load equal to 5 (Nm), then from t = 0.02 (s), we assumed a variation of r s (see Fig. 8).

Figure ( 9), presents the influence of such variation on the actual and estimated electromagnetic torque of PMSM. It is noticed that the couple deviated of its reference. In addition, the actual flux of the PMSM machine shown in figure (10), moves away by approx 0.04(Wb) form its reference value.

Control without compensation.

In this part of simulation, we introduce a fuzzy-logic r s estimator in order to correct the estimate of stator flux and electromagnetic torque using modified switching

+ + r s (k) ) 1 ( ˆ k s r ) ( * k s  + + Z -1 + - e(k) e(k) s   1 1 PI FLC  s   1

CONCLUSIONS

To reduce the ripples torque and stator flux of the PMSM, controlled by conventional DTC, we proposed in this work, firstly a modified switching table which the sectors are shifted by 30° and secondly, two fuzzy-logic controllers adapted hysteresis bands.

The effects of the stator resistance variations have been studied, and to eliminate them, a Fuzzy-logic compensator has applied.

The simulation results suggests that the proposed modifications via conventional DTC of PMSM can achieve a good reduce of ripples, a precise control of the stator flux and torque and a stability to the study system.

NOMENCLATURE dq

Synchronously rotating frame.

Fig. 1 .

 1 Fig. 1. Diagram of DTC control applied for PMSM supplied with a three-phase inverter with PWM.

Fig. 2 .

 2 Fig. 2. Different vectors of stator voltages provided by a two levels inverter. Where: I(D)F : Increase (Decrease) of Flux amplitude. I(D)T : Increase (Decrease) of Torque.

Fig. 4 .

 4 Fig. 4. Adapted fuzzy bands torque and flux hysteresis controller.

FLCFig. 5 .

 5 Fig. 5. Fuzzy controller inputs and outputs membership functions.

Fig. 6 .

 6 Fig. 6. Memberships of the inputs /outputs functions.

Fig. 16 .

 16 Fig. 9. Evolution of motor's electromagnetic torque without compensation of r s .

Table 1 . Vectors voltage localization.

 1

	 s	T e	S 1	S 2	S 3	S 4	S 5	S 6
		1	110	010	011	001	101	100
	1	0	000	000	000	000	000	000
		-1	101	100	110	010	011	001
		1	010	011	001	101	100	110
	0	0	000	000	000	000	000	000
		-1	001	101	100	110	010	011
				4. DTC MODIFIED

The same principle of basic DTC control of the PMSM, which is supplied by a two-level inverter, is applied for the other ST.

All sectors are shifted by an angle of 30° leading to an angle of the first sector between 0° and 60° as shown in figure

3

. Fig. 3. Different vectors of stator voltages, in case of table with sectors shifted by 30°.

The difference between a DTC with a simple switching table and that with sectors shifted is summarized in table 2.

Table 2 . Comparison between the simple table and the table with sectors shifted by 30°.

 2

Table 3 . Switching table of DTC shifting 30°.

 3

	 s	T e	S 1	S 2	S 3	S 4	S 5	S 6
		1	110	010	011	001	101	100
	1	0	000	000	000	000	000	000
		-1	100	110	010	011	001	101
		1	011	001	101	100	110	010
	0	0	000	000	000	000	000	000
		-1	001	101	100	110	010	011
	5.	REDUCTION OF ELECTROMAGNETIC PMSM
	TORQUE RIPPLES USING CONTROLLERS BASED ON
				FUZZY LOGIC

Table 4 . Linguistic rules base for two PI-Type fuzzy-logic controllers.

 4

	e 1 e 2 PB PM	PS	ZE	NS	NM NB
	P	P	P	PS	ZE	NS	NS	N
	ZE	P	P	PS	ZE	NS	N	N
	N	P	NS	PS	ZE	NS	N	N

Table 5 . Rules base of FLC stator resistance estimator.

 5

	e	Δe PB	PS	ZE	NB	NS
	PB	PB	PB	PB	PS	ZE
	PS	PB	PB	PS	ZE	NS
	ZE	PB	PS	ZE	NS	NB
	NB	PS	ZE	NS	NB	NB
	NS	ZE	NS	NB	NB	NB
	Fig.7. Block diagram of fuzzy-logic r s estimator.
		8. RESULTS OF SIMULATION
	Table (6), summarizes the PMSM parameters used in this
	simulation .

Table 6 .

 6 PMSM parameters.

	Pole pairs	3
	Rated power KW (at 50 Hz)	1.5
	Rated voltage (V)	220/380
	Rated Flux (Wb)	0.30
	Rated torque (Nm)	5
	Rs (Ω)	1.4
	Ld ; Lq (H)	0.0066; 0.0058
	Flux magnet (Wb)	0.15
	J(Kg.m²)	0.00176
	fr (Nm/(rad/s))	0.0038


Tanguy Gernot

Patrick Lacharme

Biometric Masterkeys

Keywords: Biometric Authentication, Cancelable Biometric Transformation, Masterkey

Biometric authentication is used to secure digital or physical access. Such an authentication system uses a biometric database, where data are sometimes protected by cancelable transformations. This paper introduces the notion of biometric masterkeys. A masterkey is a feature vector such that the corresponding template matches with a significant number of templates stored in a cancelable biometric database. Such a masterkey is directly researched from a cancelable biometric database, but we also investigate another scenario in which the masterkey is fixed before the creation of the cancelable biometric database, providing additional access rights in the system for the masterkey's owner. Experimental results on the fingerprint database FVC and the face image database LFW show the effectiveness and the efficiency of such masterkeys in both scenarios. In particular, from any given feature vector, we are able to construct a cancelable database, for which the biometric template matches with all the templates of the database.

Introduction

Nowadays, biometrics is more and more used in many applications, from access control on smartphones to payment on the Internet. Biometrics is generally based on physical characteristics such as fingerprints, face, iris or voice. Several ways are used for biometric data protection, as cryptography or cancelable biometric transformations. These transformations should respect several security properties as revocability, unlinkability and non-invertibility while maintaining good performance. They use a secret parameter, called token or seed, but the most used security model considers that this secret can be known by an attacker (stolen token model).

During the enrollment step, one or several biometric data is transformed into a biometric template and stored in a database with biometric templates of other people. During the authentication step, a person sends a biometric data and the seed to the matcher, the corresponding template is computed and compared with templates of the database. If the matching is correct, the person is authenticated.

The performance of the cancelable scheme is expected close to the original scheme without transformations, which means similar biometric data are mapped to similar templates with high probability.

In this paper, we are interested in the creation of feature vectors for which the corresponding templates match with the greatest number of templates of the cancelable database. Such feature vectors, called masterkeys, can be used to attack the system, in the stolen token model, or provide some additional access rights for the system administrator. In this case, the cancelable transformation is used not for the security of biometric data, but for the creation of efficient masterkeys, by considering the secret seed as a public salt. More precisely, masterkeys are computed in the two following scenarios:

• The cancelable database and the corresponding seeds are known. The goal is to create a synthetic feature vector for which the corresponding templates, computed with the seeds, matches with the maximal number of templates of the database.

• The biometric database is known, and a feature vector is fixed. The goal is to find the seeds for which the corresponding templates, computed with the seeds, match with the maximal number of templates, computed with the same seeds.

Experiments use a random projection, with or without a binarization step, for transformation, applied on two biometric databases: the FVC database (fingerprints) and the LFW database (face images).

In the second scenario, a brute force strategy is used for the computation of seeds, providing successful masterkeys for all the biometric database, with no significative performance degradation in these cancelable biometric databases. This second scenario proposes a new paradigm in biometric authentication, with the creation of biometric databases with specific access rights. It can also be seen as a backdoor placed in the system. Seeds are also computed for a set of masterkeys's candidate, providing partial masterkeys for other feature vectors.

Section 2 proposes generic definitions on biometric systems and presents the works close to this paper. Section 3 formalizes the concept of masterkey with several definitions on the effectiveness of such masterkeys. Experiments and results are given in Section 4, whereas conclusion and future works are in Section 5.

Definitions and related works

A biometric authentication scheme is realized into two phases. One (or several) feature vector is extracted from a biometric data during the enrollment process and is stored into a biometric database. During the authentication phase, a second feature vector is extracted from a biometric data and these two vectors are compared. It is expected that feature vectors x and x of the same individual, extracted from two measurements are sufficiently close to verify D A (x, x ) < τ A . This paper uses the Euclidean distance for D A , defined for feature vectors x, x ∈ M A = R N by:

D A (x, x ) = N i=1 (x i -x i ) 2 .
Definition 2.2. Let B = {x i } i=1,...,n be a biometric database, composed of feature vectors. A biometric authentication scheme for B is an algorithm that takes a feature vector x and a threshold τ A as inputs and returns T rue if ∃i between 1 and n such that V (x, x i , τ A ) returns T rue and F alse otherwise. In the first case, x is successfully authenticated by x i .

The quality of a biometric database is estimated with two rates: the False Match Rate (FMR) that provides the percentage of impostors authenticated by the system and the False Non Match Rate (FNMR) that provides the percentage of genuine peoples rejected by the system. These rates depend on τ A and in case of equality these rates correspond to the Equal Error Rate (EER).

A biometric database contains some feature vectors that strongly influence these indicators. Doddington et al. [START_REF] Doddington | Sheep, goats, lambs and wolwes : A statistical analysis of speaker performance in the NIST 1998 speaker recognition evaluation[END_REF] sorted them in 4 categories: sheep, goats, lambs and wolves, where wolves are able to usurp many users, causing false acceptances. These individuals appear to have generic characteristics. Similarly, Yager et al. [START_REF] Yager | The biometric menagerie[END_REF] described 4 categories of animals representing the relationship between genuine and impostor match scores: worms, chameleons, phantoms and doves, where chameleons combine both high genuine and impostor match scores. In these categories, chameleons obtain scores belonging to the first quarter of the best scores.

Ine et al. defined a wolf as a feature vector that has a successful authentication probability greater than the FMR and evaluate the security of a given database with the notion of Wolf Attack Probability (WAP) in [START_REF] Une | Wolf attack probability: A new security measure in biometric authentication systems[END_REF]. Countermeasures against this attack is investigated by Inuma et al. in [START_REF] Inuma | Theoretical framework for constructing matching algorithms in biometric authentication systems[END_REF] and tested on fingerprints by Murakami et al. in [START_REF] Murakami | Towards optimal countermeasures against wolves and lambs in biometrics[END_REF]. Wolf attacks have been realized on the voice by Okhi et al. in [START_REF] Ohki | Evaluation of wolf attack for classified target on speaker verification systems[END_REF], then by Ohki and Otsuka [START_REF] Ohki | Theoretical vulnerability in likelihood-ratio-based biometric verification[END_REF].

Recently, Roy et al. investigated masterprints, partial fingerprints that usurp a large number of users [START_REF] Roy | MasterPrint: Exploring the Vulnerability of Partial Fingerprint-Based Authentication Systems[END_REF] in a multi-enrollment process. They worked with small sensors of smartphones which acquire a lot of partial minutiae-based feature vectors. It was later improved with machine learning based approaches in [START_REF] Roy | Evolutionary Methods for Generating Synthetic MasterPrint Templates: Dictionary Attack in Fingerprint Recognition[END_REF][START_REF] Roy | Masterprint attack resistance: A maximum cover based approach for automatic fingerprint template selection[END_REF][START_REF] Bontrager | DeepMasterPrints: Generating MasterPrints for Dictionary Attacks via Latent Variable Evolution[END_REF] and recently applied on face recognition in [START_REF] Nguyen | Generating master faces for use in performing wolf attacks on face recognition systems[END_REF].

For security reasons, feature vectors are not always directly stored in the database, but are used as input of a cancelable biometric transformation. These transformations were first proposed by Ratha et al. for face recognition [START_REF] Ratha | Enhancing security and privacy in biometrics-based authentication system[END_REF]. There are many works on cancelable transformations, as presented in the surveys [START_REF] Rathgeb | A survey on biometric cryptosystems and cancelable biometrics[END_REF][START_REF] Nandakumar | Biometric template protection: Bridging the performance gap between theory and practice[END_REF] on biometric templates protection. Definition 2.3. Let K be a token (seed) space and (M B , D B ) be a metric space. A cancelable biometric scheme is a set of algorithms (T , V), where

• T takes a secret seed s ∈ K, and a feature vector x ∈ M A as input, and returns a biometric template u = T (s, x) ∈ M B .

• V takes two biometric templates u = T (s, x), u = T (s, x ), and a threshold τ B as input, and returns T rue if D B (u, u ) < τ B , and F alse otherwise.

Definition 2.4. Let D = {u i } i=1,...,n be a cancelable biometric database, composed of biometric templates u i = T (s i , x i ). A cancelable biometric authentication scheme for D is an algorithm that takes a biometric template u = T (s, x) and a threshold τ B as inputs and returns T rue if ∃i between 1 and n such that s = s i and V(u, T (s i , x i ), τ B ) returns T rue and F alse otherwise. In the first case, x is successfully authenticated by x i .

It is expected that the cancelable biometric transformation T does not significantly decrease the performance of the original biometric system, in terms of EER. This paper uses the Hamming distance for D B , defined for biometric templates u, u ∈ M B = {0, 1} M by:

D B (u, u ) = M i=1 u i ⊕ u i ,
where ⊕ is exclusive or (xor) bitwise operator.

Cancelable biometric transformations are generally based on locality sensitive hashing [START_REF] Charikar | Similarity estimation techniques from rounding algorithms[END_REF][START_REF] Andoni | Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions[END_REF][START_REF] Wang | A survey on learning to hash[END_REF] or robust perceptual hashing [START_REF] Voloshynovskiy | Random projections based item authentication[END_REF][START_REF] Voloshynovskiy | Conception and limits of robust perceptual hashing: towards side information assisted hash functions[END_REF], ensuring the performance of the transformation. The transformations used in this paper are based on random projection, possibly with a binarization step. These cancelable transformations were first proposed in [START_REF] Teoh | Personalised cryptographic key generation based on FaceHashing[END_REF][START_REF] Teoh | Cancellable biometrics and annotations on biohash[END_REF] with the biohashing algorithm. Other projections have been later proposed, as in [START_REF] Feng | A hybrid approach for generating secure and discriminating face template[END_REF][START_REF] Pillai | Secure and robust iris recognition using random projections and sparse representations[END_REF][START_REF] Wang | An analysis of random projection for changeable and privacy-preserving biometric verification[END_REF].

Let M s be a real matrix with N rows and M columns, generated pseudo randomly from the random seed s, x a feature vector of size N and D : R M → {0, 1} M defined by D(t 1 , . . . , t M ) = (u 1 , . . . , u M ) where

u i = 0 if t i < 0 1 if t i ≥ 0
In this case, the cancelable transformation T is defined by T (x, s) = D(xM s ). In the biohashing algorithm, the pseudorandom matrix M s is orthonormalized with the Gram-Schmidt algorithm.

The security of a cancelable biometric scheme is evaluated on several attack models, generally in the stolen token attack model, which considers that the token s is known by the attacker. Definition 2.5. Let x ∈ M A be a feature vector, u = T (s, x) ∈ M B is a biometric template, computed from a seed s and τ A is a threshold. A nearby-feature preimage of u with respect to s is a feature vector x * such that V (x, x * , τ A ) = T rue.

There are several methods for the construction of nearby-feature preimages. Genetic algorithms are a generic strategy for this, used in the case of biohashing on fingerprints in [START_REF] Lacharme | Preimage Attack on BioHashing[END_REF], in the iris case in [START_REF] Galbally | Iris image reconstruction from binary templates: An efficient probabilistic approach based on genetic algorithms[END_REF], in the case of minutiae templates in [START_REF] Rozsa | Genetic algorithm attack on minutiaebased fingerprint authentication and protected template fingerprint systems[END_REF] or recently on several transformations in [START_REF] Dong | A Genetic Algorithm Enabled Similarity-Based Attack on Cancellable Biometrics[END_REF]. A genetic algorithm is a set of operators on a population, depending on a fitness function. At each step, the population is updated with three operators: selection, crossing-over and mutation. Selection chooses the best subpart of the population. Crossing-over generates new individuals in the population from the selected population. Mutation modifies the new individuals generated during crossing-over. In the case of cancelable biometric transformations, individuals are candidates for masterkey and the fitness function is defined by the distance D B between the corresponding template and a target template.

Nearby-feature preimages can also be constructed using a specific attack on the transformation. For example, random projection based schemes are vulnerable to linear programming approaches [START_REF] Nagar | Biometric template transformation: a security analysis[END_REF][START_REF] Feng | Masquerade attack on transform-based binary-template protection based on perceptron learning[END_REF][START_REF] Topcu | Practical security and privacy attacks against biometric hashing using sparse recovery[END_REF].

Biometric masterkeys

A biometric masterkey x is a synthetic feature vector such that the corresponding biometric template u is close to several biometric templates in the database. The first part of this section is a generalization of concepts of the previous section (wolves, masterprints) to cancelable biometric database. The second part proposes an alternative by constructing a cancelable biometric database that matches with a given masterkey. In this case, the cancelable transformation is introduced to provide such masterkeys efficiently. Definition 3.1. Let D = {u i } i=1,...,n be a cancelable biometric database and τ B a threshold. x is a masterkey for D, by respect to τ B , if ∀i between 1 and n,

V(T (s i , x), T (s i , x i ), τ B ) = T rue, with T (s i , x i ) = u i .
Obviously, x is a masterkey for the database {T (s, x)}, but the research of a masterkey for a large database is not necessarily an easy task. A less restrictive goal is the research of a minimal number of masterkeys such that all templates of the database are close to at less one template corresponding to one masterkey. Definition 3.2. Let D = {u i } i=1,...,n be a cancelable biometric database and τ B a threshold. D is said covered by a set of r masterkeys {x 1 , . . . x r } by respect to τ B if ∀i between 1 and n, ∃k between 1 and r such that V(T (s i , x k ), T (s i , x i ), τ B ) = T rue. The minimum number R such that D is covered by a set of R masterkeys is called optimal dictionary size.

In the previous definition, each masterkey x i ∈ {x 1 , . . . x r } is a masterkey for a subset D i ⊂ D and we have ∪ r i=1 D i = D. Nevertheless, the number of templates in each subset D i is generally different. Definition 3.3. A cancelable biometric database D = {u i } i=1,...,n is said -covered by a masterkey x, with 0 < ≤ 1 if there exists a subset D of D of n persons such that x is a masterkey for D . The optimal coverage percentage of D is the maximal number E such that D is E-covered by a masterkey x ∈ M A .

Such a set of masterkeys can be used for attacks on the cancelable biometric database, similarly to the work of Roy et al. on minutiae [START_REF] Roy | MasterPrint: Exploring the Vulnerability of Partial Fingerprint-Based Authentication Systems[END_REF], called dictionary attack, using a set of carefully chosen masterkeys. Nevertheless these masterkeys could provide other applications. For example, a masterkey could be used for a system of delegation of biometric authentication, with the possibility for one person to be authenticated as herself and also as other person, similarly to proxy signatures [START_REF] Mambo | Proxy signatures for delegating signing operation[END_REF] or group signatures [START_REF] Chaum | Group signatures[END_REF] in cryptography.

A second approach of the biometric masterkey problem is the construction of a cancelable biometric database from a feature vector x and n feature vectors extracted from n peoples, such that x is a masterkey for the cancelable database. It means that, in this second approach, the seeds used by peoples are constructed, with the knowledge of x, contrary to the previous approach. Definition 3.4. Let x be a feature vector, τ B be a threshold, and B = {x i } i=1,...,n be a biometric database. A cancelable biometric database D is said compliant with x by respect to τ B if there exists s 1 , . . . s n such that D = {T (s i , x i )} and x is a masterkey for D by respect to τ B . This second approach uses not the cancelable biometric transformation for security but uses it for the construction of masterkeys. It makes possible to adjust the choice of seeds to allow super access, i.e., a vector usurping the entire base. This can allow an administrator to have the same access as its users without adding another authentication channel. If an insecure transformation is used, an additional security layer should be added in the template storage.

It can also allow an attacker active in the enrollment phase to build a backdoor in the authentication system, without adding suspicious code or making the database dubious, as in the following definition. Definition 3.5. Let B = {x i } i=1,...,n a biometric database and D = {T (s i , x i )} a cancelable biometric database compliant with a masterkey x. Then x is a backdoor if there exists no polynomial algorithm that distinguishes s 1 , T (x 1 , s 1 ), . . . s n , T (x n , s n ) from s 1 , T (x 1 , s 1 ), . . . s n , T (x n , s n ), where s 1 , . . . s n are randomly generated from the set of seeds (x is unknown).

In this paragraph, we consider the theoretical complexity to find masterkeys. We suppose that the metric space (M B , D B ) is {0, 1} M with the Hamming distance and that the biometric transformation T is pseudorandom. Thus, given a biometric template u, a feature vector x and a threshold τ , the probability p that the distance D B (T (x, s), u) < τ for a random seed

s is p = 1 2 M τ -1 i=0 M i . Let D = {u i } i=1,.
..,n be a cancelable biometric database of n binary templates. A feature vector x, such that D is -covered by x, is expected after 1/p n trials. Let B = {x i } i=1,...,n be a biometric database of n feature vectors and x a feature vector. A set of n seeds s 1 , . . . s n , such that the corresponding cancelable biometric database D is compliant with x (Definition 3.4) is expected after n/p trials. Thus, the research of a masterkey given a biometric database is clearly easier than given a cancelable database.

Experiments

Effectiveness of masterkeys research is investigated on the fingerprint database FVC and the face database LFW. All experiments are processed on a Dell Latitude 7400 laptop with an Intel R Core TM i7-8665U CPU @ 1.90 GHz quad-core processor and 16 Gb of RAM, using python for the first part and C for the second one.

Biometric Databases

The first biometric database is the FVC2002 database [START_REF] Maio | FVC2002: Second Fingerprint Verification Competition[END_REF], which contains t = 8 biometric images of n = 100 peoples obtained from fingerprints. The second one is the sub LFW database [START_REF] Huang | Labeled faces in the wild: A database forStudying face recognition in unconstrained environments[END_REF] used in Dong et al. experiments [START_REF] Dong | A Genetic Algorithm Enabled Similarity-Based Attack on Cancellable Biometrics[END_REF], which contains t = 10 biometric images of n = 158 peoples obtained from face pictures. This database is noted LFW10 in this section. A third database, named LFW8, is extracted from LFW10 by taking the first 100 peoples, with the first 8 biometric images, for comparison with the FVC2002 database.

Feature vectors are extracted from FVC using Gabor filters [START_REF] Belguechi | Comparative study on texture features for fingerprint recognition: application to the BioHashing template protection scheme[END_REF]. Each biometric feature vector is composed of N = 512 real values of 64 bits and the EER of the biometric database is 10% with a threshold τ A = 240.7. Feature vectors extracted from LFW are obtained using the deep network InsightFace [START_REF] Deng | ArcFace: Additive angular margin loss for deep face recognition[END_REF]. Each biometric feature vector is composed of n = 512 real values of 64 bits and the EER of the biometric database is 0.2% with a threshold τ A = 1.227.

Masterkeys from Cancelable Biometric Database

The cancelable biometric transformation used in this subsection is the biohashing algorithm, mapping R N into {0, 1} M where M = 128, described in Section 2. The binary size of the seed is 128 bits. After transformation the EER of the cancelable biometric database is around 16.5% with a threshold τ B = 17 for FVC2002 and around 1.9% with a threshold τ B = 51 for LFW10 and LFW8.

The research of a masterkey, as presented in Definition 3.1, from a cancelable biometric database uses genetic algorithm where the fitness function is not the number of people who are authenticated by the masterkey, because it does not provide the best results. Two other fitness functions have been investigated, the first one based on the average of the minimal Hamming distance obtained for each person and the second one based on the sum of the minimal Hamming distance of not authenticated peoples by the masterkey. The second fitness function (sum) provides the best results on both databases and is used in next experiments.

We only consider the first biometric template for each person of the cancelable biometric database during the evaluation of the best optimal coverage percentage and for the best optimal dictionary size.

Table 1 gives the best optimal coverage percentage (OCP), as described in Definition 3.3, and the best optimal dictionary size (ODS), as described in Definition 3. The genetic algorithm is processed on a population of 200 peoples during 500 iterations. Figure 1 shows that coverage does not increase from 350 iterations. Vulnerabilities of cancelable transformations in the stolen token scenario have been already studied. Typically, nearby-feature preimages have been already constructed on biohashing in this scenario with various methods. Nevertheless, it is the first time that these attacks are extended to the entire database by the construction of masterkeys. The strategy uses a genetic algorithm which is not related to the weaknesses of this transformation. Typically, specific techniques as linear optimization algorithms give efficient results for nearby-feature preimages on biohashing and could be used to improve the performance of our algorithm. Nevertheless, the choice of this algorithm depends on the cancelable transformation, whereas the research of masterkeys should be possible for all transformations.

Masterkeys from Biometric Database

In this second scenario, we have a biometric database and a feature vector, called candidate masterkey. The objective is the generation of seeds for which the feature vector is a masterkey for the corresponding cancelable biometric database as presented in Definition 2.4. In this scenario the transformation is chosen for the efficiency of masterkeys creation, not for data protection. Thus, the beginning of this subsection describes the choice of the transformation and its parameters.

The performance of random projection-based schemes are analyzed using the Johnson-Lindenstauss Lemma [START_REF] Johnson | Extension of Lipschitz mapping into a hilbert space[END_REF][START_REF] Dasgupta | An elementary proof of a theorem of Johnson and Lindenstrauss[END_REF] which establishes that for any 0 < < 1, there exists a map f : R N → R M such that for all x, y in a subset of n points of

R N with M ≥ O( -2 log n) (1 -) x -y 2 2 ≤ f (x) -f (y) 2 2 ≤ (1 + ) x -y 2 2 .
The two following random projection N × M matrices are proposed by Achlioptas in [START_REF] Achlioptas | Database-friendly random projections: Johnson-Lindenstrauss with binary coins[END_REF]. In the first case, called JL1, coefficients of the projection matrix M s are 1/ √ M with a probability 1/2 -1/ √ M with a probability 1/2

In the second case, called JL2, coefficients of M s are    3/M with a probability 1/6 0 with a probability 2/3 -3/M with a probability 1/6

The main advantage of these transformations is the low computational time for matrix generation compared to the Gram-Schmidt algorithm.

The value of in the previous inequality is detailed in figure 2 for N = 512 and M between 32 and 256 using the FVC database (a random database also provides similar curves). Two transformations are used: the map composed of a random orthonormalized matrix (Gram-Schmidt), where decreases from 0.9 to 0.5 and for the two previous transformations (Achlioptas) which provides the similar results ( 0.1 for M ≥ 128). The previous experiments on has very few consequences on the EER. The EER of the cancelable database is close to the EER of the original database in figure 3 for both transformations for M ≥ 128. Even if the value of in the previous experiment is different, in the case of Achlioptas's transformations, the threshold is stable, whereas in the first transformation, the threshold increases. Thus, the research of masterkeys from a biometric database seems more interesting, directly in the Euclidean space, because the EER does not change compared to the original database if M is not too small (for M = 128, around 11% against 10% on FVC and around 1% against 0.2% for LFW). Nevertheless, the computational time for the research of masterkeys is not negligible, particularly for some feature vectors. The EER of the cancelable database obtained with the two previous transformations and a binarization step is a little bit increased compared to the original EER: around 16.5% for FVC and 2.1% for LFW in figure 4 with M = 128. Consequently we investigate the research of masterkeys with transformations T : R N → R M or {0, 1} M , defined by the Achlioptas transformations, ending or not by a binarization step. Random matrices are generated from a seed of 16 bytes as previously. Let B = {x j i } i=1,...,n,j=1...,t be a biometric database, composed of feature vectors (n peoples with t vectors by people). In experiments, the candidate masterkey is a feature vector of a chosen people of a database. The research of seeds for a cancelable biometric database compliant with a masterkey candidate x j i , as presented in Definition 3.4 is realized, taking one by one the (n -1) j th feature vectors of each person, and choice seed, randomly generated from {0, 1} 128 , so that the candidate masterkey is successfully authenticated. Experiments use the following algorithm, and outputs an array of n -1 seeds for each candidate masterkey, with the threshold at EER. The seed research is only limited by a value c max , in the case of the FVC database without binarization of the transformation. If a successful seed is not founded after c max trials, the output of the algorithm is F ailure (in most of experiments, c max is not achieved).

Algorithm 1 Research of seeds for masterkeys

Require: B the database, τ B the threshold, c max SeedT = [] for i = 1 to n and j = 1 to t do The research of seeds for masterkeys is successful for all feature vectors from FVC and LFW10 (called now LFW), for both Achlioptas's transformations (called JL1 and JL2), with the binarization step. Figure 5 presents the number of clock required for each feature vector for both transformations on FVC database (more precisely, it is an average of the t vectors of each person), and the same results for LFW. It shows that the time is more important for some people of the FVC database, compared to the LFW database. It also shows that the second transformation provides slightly better results in both database (but the number of distance comparison is similar). The research of seeds is very efficient and only takes a few seconds for almost masterkey candidates.

SeedL = [] for k = 1 to n, with k = i do cpt ← 0, s k ← rand(1, 2 128 ) while V(T (s k , x j i ), T (s k , x j k ), τ B ) = F alse and cpt < c max do s k ← rand(1, 2 128 ) end while if V(T (s k , x j i ), T (s k , x j k ), τ B ) = T
The average of the EER in these n * t cancelable biometric databases is around 17% for FVC (for both transformations) which is close to the EER of a cancelable database obtained with random seeds (around 16.5% for both transformations). For LFW, the average of the EER is around 2.4 % which is also close to the EER of a cancelable database obtained with random seeds (around 2.1% for both transformations).

If these transformations do not use a binarization step, the time of research increases a lot for the FVC database (more precisely for some people of this database), so we don't detail it. In the case of LFW database, the number of clock is not significantly increased compared to the number of clock required with the binarization step. Figure 6 presents the number of clock during the research of the masterkeys for LFW (more precisely, it is an average of the t vectors of each person). In this case, the second transformation is again more efficient than the first one. The EER of these n * t databases is around 1.3% which is close to the EER of the LFW database obtained with random seeds (around 1% for both transformations without binarization). In this subsection, masterkeys have been efficiently founded for LFW in the Euclidean space. Nevertheless, due to complexity for some vectors of the FVC database, it seems more appropriate to use a transformation with binarization in the case where acquisition of feature vectors has not been perfectly realized. There is a tradeoff between performance of the cancelable database and efficiency of the research of seeds for masterkeys.

Multiple masterkeys

In this subsection, a variant of the previous scenario is investigated. The previous definition ensures for the owner's masterkey that the masterkey is successfully authenticated with enrollment data of other people. Nevertheless it does not ensure that another biometric data of the owner's masterkey is authenticated on this database. This subsection studies the percentage of the cancelable database cov-ered by these other feature vectors. Let B = {x j i } i=1,...,n,j=1...,t be a biometric database, composed of feature vectors (n peoples with t vectors by people). Let x 1 , . . . x t be feature vectors which are candidates for masterkey (x 1 , . . . x t are assumed from the same people, but it doesn't mean that the distance between them is always lower than the used threshold). In a first time seeds are computed for a subset x t1 , . . . x t T , called first set (if such a seed is not founded, the seed minimizing the worst distance is kept). In a second time, we look for the coverage percentage of the database D provided by the t -T other vectors, called second set. The corresponding templates of x 1 , . . . , x t , noted u 1 , . . . , u t , form the first set and the second set of templates, depending to the feature vectors.

Experiments only use the JL2 transformation, with and without binarization, on FVC and LFW, for T = 1 and T = 4. For practical reasons, we used a time limit of 5 minutes in place of c max . The number of templates in D matching with the first set of T templates u t1 , . . . u t T (in blue for T = 1 and green for T = 4) and with the second set of t -T other templates (in orange for T = 1 and red for T = 4) is presented in Figure 7 for FVC the Hamming space and in Figure 8 for LFW the Hamming and Euclidean space.

For example, with T = 1 in FVC, around 60% of templates from the second set of templates match with at less 40 templates of D (orange curve), whereas with T = 4, around 60% of templates from the second set of templates match with at less 75 templates of D (red curve). Moreover we have only one blue point in these figures, because 100% of templates of the first set match with all templates in the database, as seen in the previous subsection (if T = 4 it is not the case because the limit of 5 minutes for each seed computation).

Conclusion and Future Work

This paper has introduced the concept of biometric masterkeys in two scenarios. In the first one, there exists a cancelable biometric database, and we try to find a feature vector for which the corresponding template matches with the greatest number of templates of the database. It is a formalization and a generalization of Wolf Attacks and masterprints on fingerprints, experimented with the biohashing algorithm. In the second scenario, there exists a biometric database, and we try to find a set of seeds for which the corresponding template of a given feature vector matches with the greatest number of templates of the database.

The main contribution of this work is the construction of such a cancelable biometric database covered by one of its feature vectors, with an EER similar to the original cancelable database. The research of masterkeys in this second scenario is completely different from previous works on wolves or masterprints, because it uses the transformation, not for data protection, but for the research of preimages. This work presents new opportunities for a biometric system administrator, to construct a cancelable biometric database which matches with a template obtained from its own feature vector. This masterkey only requires a public seed by template and is very easy to construct.

There are several future works and improvements of this paper, particularly in the research of seeds, providing good coverage percentage of the cancelable database for other acquisitions of feature vectors. Other applications of masterkeys, as biometric systems with fine-grained access controls, will also be investigated in the future. Finally, it would be interesting to use an efficient and secure transformation to ensure the security of templates in this context. Nevertheless, if such a transformation is not available, it is possible to combine masterkeys with any biometric protection scheme.

Definition 2 . 1 .

 21 Let (M A , D A ) be a metric space. A biometric authentication scheme for two biometric data b and b is a pair of algorithms (E, V ), where• E takes biometric data b as input, and returns a feature vector x ∈ M A .• V takes two feature vectors x = E(b), x = E(b ), and a threshold τ A as input, and returns T rue if D A (x, x ) < τ A , and F alse otherwise.

Figure 1 :

 1 Figure 1: Evolution of coverage during iterations

Figure 2 :

 2 Figure 2: Estimation of for different sizes of templates (FVC)

Figure 3 :

 3 Figure 3: EER for FVC (Euclidean distance)

Figure 4 :

 4 Figure 4: EER for FVC (Hamming distance)

Figure 5 :Figure 6 :

 56 Figure 5: Average number of clock (in binary)

Figure 7 :Figure 8 :

 78 Figure 7: Cumulative decreasing percentage of successful masterkeys for T = 1 and 4, on FVC (Binary)

Figure 9 :

 9 Figure 9: Correlation between successful masterkeys and other templates (T = 4 on FVC in Hamming space)

Figure 10 :

 10 Figure 10: Correlation between successful masterkeys and other templates (T = 4 on LFW with Euclidean and Hamming space)

Table 1 :

 1 2. For both cancelable biometric databases, we use thresholds τ B at EER. Optimal coverage percentage / Optimal dictionary size

	db	OCP ODS
	FVC	73	5
	LFW8	21	12
	LFW10 15.2	18

Table 2 :

 2 FMR/OCP corresponding to the thresholds (FVC)