Kamel Menighed
email: k.menighed@univ-skikda.dz

Ahmed Chemori
email: ahmed.chemori@lirmm.fr

Boumedyen Boussaid
email: boumedyen.boussaid@gmail.com

Joseph-Julien Yamé

Networked Cooperation-Based Distributed Model Predictive Control using Laguerre Functions for Large-Scale Systems

Keywords: large-scale interconnected systems, distributed model predictive control, Laguerre functions

This paper proposes a novel cooperative distributed control system architecture based on unsupervised and independent Model Predictive Control (MPC) using discretetime Laguerre functions to improve the performance of the whole system. In this distributed framework, local MPCs algorithms might exchange and require information from other sub-controllers via the communication network to achieve their task in a cooperative way. In order to reduce the computational burden in the local rolling optimization with a sufficiently large prediction horizon, the orthonormal Laguerre functions are used to approximate the predicted control trajectory. Simulation results show that the proposed architecture could guarantee satisfactory global performance even under strong interactions among the subsystems.

I. INTRODUCTION

During the past two decades a lot of interest has been given to the control of a class of large-scale systems which are composed by many physically or geographically divided subsystems. However, these subsystems are interconnected and characterized by significant interactions. Each subsystem interacts with the other subsystems through their states and their inputs. At the same time, due to the high performance requirements, modern control systems are becoming more and more complex. For these processes, different control solutions have be developed to achieve some global performance of the whole system.

In centralized MPC architecture, a control agent is able to acquire the information of the global system and all the input sequences are optimized with respect to one given performance index in a single optimization problem. However, when the number of the state variables and inputs of the system becomes larger, the computation burden of the centralized optimization problem may increase significantly. Therefore, it is often impractical to apply the centralized MPC to largescale systems due to their size and their higher risk of failure of the MPC controller due to its centralized nature. However, the centralized control scheme in these industrial process has gradually given way to decentralized and distributed control strategies for their remarkable decrease in system dimensionality using the local MPC control [START_REF] Christofides | Distributed Model Predictive control: A Tutorial Review and Future Research Directions[END_REF], [START_REF] Negenborn | Distributed Model Predictive Control: An Overview and Roadmap of Future Research Opportunities[END_REF].

The main idea in both decentralized and distributed control in large interconnected systems is to approximate the overall system with several separated subsystem models, where each subsystem is controlled by a local controller. Indeed, each local controller uses the so-called local performance index. On the one hand, these control architectures have the advantages of being flexible to system structure, error-tolerance, less computational efforts and no global information requirements [START_REF] Vaccarini | Unconstrained Networked Decentralized Model Predictive Control[END_REF], [START_REF] Zheng | Distributed Model Predictive Control Over Network Information Exchange for Large-Scale Systems[END_REF], [START_REF] Zheng | Networked Coordination-Based Distributed Model Predictive Control for Large-Scale System[END_REF]. The only difference between them is that distributed control schemes give due consideration to the external state and/or control signal information communicated between subcontrollers from other subsystems while decentralized ones follow a completely decoupled approach [START_REF] Ikeda | Overlapping Decentralized Control with Input State and Output Inclusion[END_REF], [START_REF] Lunze | Feedback Control of Large-Scale Systems[END_REF], [START_REF] Zečević | A New Approach to Control Design with Overlapping Information Structure Constraints[END_REF].

On the other hand, the development of DCS (Distributed Control Systems), field-bus, communication network technologies in process industries allows the control methodologies to utilize their potentials for improving control performance. In this case, the improvement of the global control performance of the entire system using the network information exchange for large-scale systems is a valuable open problem.

In order to fulfill the global objective for the global system, cooperation between the controllers through a digital communication network might be necessary. Thanks to the digital network, the required cooperation can be achieved by means of a proper information exchange between the subcontrollers. From the control point of view, it is well known that MPC allows to deal with linear, nonlinear, multivariable and constrained systems [START_REF] Jia | Distributed Model Predictive Control[END_REF]. Moreover, MPC technique benefits greatly from both advances in communication technology and advances in computational resources.

Recalling that the MPC strategy is based on an on-line optimization problem and uses a process model to predict the effect of the control sequence on the behavior of the plant. Typically, MPC is implemented in a centralized way. The complete system is modeled and all the control inputs are computed in one optimization problem. However, for large interconnected systems, it may be necessary to have a distributed control scheme as mentioned above, where local control inputs are computed using local measurements and small order models of the local dynamics. But with information exchange between the controllers, the objective is to achieve some degree of cooperation between sub-controllers that are solving MPC problem with locally relevant variables, costs and constraints.

Previous works on distributed MPC are reported in [START_REF] Menighed | Distributed State Estimation and Model Predictive Control : Application to Fault Tolerant Control[END_REF], [START_REF] Vaccarini | State Space Analysis of Unconstrained Decentralized Model Predictive Control Systems[END_REF]- [START_REF] Venkat | Stability and Optimality of Distributed Model Predictive Control[END_REF]. A preliminary analysis of the control performance of distributed MPC has been addressed in [START_REF] Vaccarini | State Space Analysis of Unconstrained Decentralized Model Predictive Control Systems[END_REF]. Menighed et al [START_REF] Menighed | Distributed State Estimation and Model Predictive Control : Application to Fault Tolerant Control[END_REF] and Razavinasab et al [START_REF] Razavinasab | State Estimationbased Distributed Model Predictive Control of Large-Scale Networked Systems with Communication Delays[END_REF] proposed a distributed state estimation strategy, developed for supporting distributed state feedback MPC for large-scale interconnected systems. In [START_REF] Venkat | Stability and Optimality of Distributed Model Predictive Control[END_REF], two approaches for a coordination between sub-controllers are proposed leading to the so-called communication and cooperation based MPC. In the cooperation-based MPC, each sub-controller knows the global objective in order to improve optimality and stability and makes the decentralized strategy very close to the centralized one. When only the local objectives are known, a hierarchical decentralized control architecture uses a supervisor to compute the global optimum and to coordinate the sub-controllers, in communication-based MPC [START_REF] Patton | Challenges of Networked Control Systems: Autonomy, Reconfiguration and Plug and Play[END_REF].

In this paper, we deal with unconstrained distributed model predictive control of large-scale interconnected systems to achieve global performance based on the use of a cooperative strategy between the sub-controllers. Thanks to the flexibility and the on-line optimization process inherent to MPC algorithms, we propose a distributed framework based on MPC with Laguerre functions. In order to reduce the computational burden in the local rolling optimization with a sufficiently large prediction horizon, the orthonormal Laguerre functions are utilized to approximate the predicted control trajectory. It is worth to emphasize that the MPC/Laguerre approach is becoming an important topic in modern control theory and practice [START_REF] Wang | Model Predictive Control System Design and Implementation Using MATLAB[END_REF].

II. COMPLEX PROCESS INTERACTION MODELING

We consider the overall system represented as a discrete, linear time-invariant (LTI) model of the form

S x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) (1)
where x(k) ∈ nx , u(k) ∈ nu and y(k) ∈ ny are the state, the control input and the output, respectively with their corresponding dimensions, n x , n u , n y .

In the decentralized modeling framework, it is assumed that subsystem-subsystem interactions have a negligible effect on system variables, i.e., we suppose that the previous overall model S is composed of N subsystems S ii , each subsystem is represented by the following state-space equations

S ii x i (k + 1) = A ii x i (k) + B ii u i (k) y i (k) = C ii x i (k) i = 1, 2, . . . , N (2)
where x i (k) ∈ nx i , u i (k) ∈ nu i and y i (k) ∈ ny i are the local state, control input and output, respectively and

n x = i n xi , n u = i n ui and n y = i n yi .
Frequently, the components of the interconnected system are tightly coupled due the material/energy and/or information flow between them. In such cases, the decentralized assumption leads to a loss in achievable control performance. It is natural to view the previous overall model S composed of N subsystems S i interacting with each other through linear interconnections. Each subsystem is represented by the following state-space equations

S i x i (k + 1) = A ii x i (k) + B ii u i (k) + w i (k) y i (k) = C ii x i (k) + v i (k) i = 1, 2, . . . , N (3)
where the state and output interaction vectors w i and v i are given by

w i (k) N j=1;j =i A ij x j (k) + N j=1;j =i B ij u j (k) v i (k) N j=1;j =i C ij x j (k) (4)
These vectors represent the interaction of subsystem j = i on subsystem i. The proposed distributed control architecture fosters implementation of cooperation-based strategy for several interacting processes (3)-( 4) in order to emulate the benefits achievable with centralized control.

III. PROBLEM STATEMENT

By means of ( 3) and ( 4), the future state and output prediction over a horizon p are given by

xi (k + l|k) = A l ii xi (k|k) + l s=1 A s-1 ii B ii u i (k + l -s|k) (5a)
+ l s=1 A s-1 ii ŵi (k + l -s|k -1) ŷi (k + l|k) = C ii xi (k + l|k) + vi (k + l|k -1) (5b)
Given the overall system S composed by N interactive subsystems S i , i = 1, 2, . . . , N , the unconstrained Distributed Model Predictive Control (DMPC) problem with prediction horizon p > 0 and control horizon m > 0 two integer values, with m ≤ p consists of finding, at time k, a set of independent sub-controllers C i such that each C i minimizes the local cost function J i .

J i = p l=1 y d i (k+l|k)-ŷ i (k+l|k) 2 Qi + m l=1 ∆u i (k+l-1|k) 2 Ri for i = 1, . . . , N (6)
subject, for l = 1, . . . , p, to model constraints given by( 5) initial condition : xi (k|k)=x i (k) where α 2 Λ α T Λαis the norm of vector α induced by matrixΛ.

y d i (k + l|k): the desired output. ∆u i (k + l -1|k): future control increment at time k, with u i (k) = u i (k -1) + ∆u i (k|k).
Q i : a symmetric and positive semi-definite matrix of appropriate dimension (Q i ≥ 0). R i : a symmetric and positive definite matrix of appropriate dimension (R i > 0).

It should be noted that ∆u j (k + l -1|k) is excluded in the performance index, since it is independent of the future control sequence of S j .

IV. COOPERATION BASED DISTRIBUTED MPC PROBLEM

In order to find an explicit solution to the first version of DMPC problem, each sub-controller C i is decomposed into three connected function blocks: an optimizer, a state predictor and an interaction predictor.

Assumptions 1

The following assumptions are considered:

• the prediction and control horizons are the same for all sub-controllers, i.e., m i = m j =, p i = p j = p, ∀i, j = 1, 2, . . . , N, j = i; • all the sub-controllers are synchronous; • all the sub-controllers communicate only once within a interval; • all the communication channel introduces a delay of one sampling period.

To simplify the mathematical expressions, the following notations are adopted

• 0 a×b is the a × b null matrix ; • I a (0 a )
is the a × a identity (null) matrix;

• diag a {A} is a diagonal block matrix made by a blocks equal to A;

1) Interaction prediction: Under assumptions 1, at step k, the predictions of the interaction vectors are given by

Ŵi (k, p|k -1) = Ãi X(k, p|k -1) + Bi Γi U (k -1, m|k -1) Vi (k, p|k -1) = Ci X(k, p|k -1) (7)
Hi

diag p {H i,1 } . . . diag p {H i,i-1 } 0 diag p {H i,i+1 } . . . diag p {H i,N }
where Hi ∈ Ãi , Bi , Ci

2) State predictor: Under assumptions 1, at step k, the local state prediction for the sub-controller

C i is expressed by xi (k + l|k) = A l ii xi (k|k) + l s=1 A s-1 ii B ii u i (k + l -s|k) + l s=1 A s-1 ii ŵi (k + l -s|k -1) (8)
3) Optimal Control Sequence: Under assumptions 1, at step k, based on the exchanged information, the interaction prediction together with the local measurement is used by the optimizer to solve the MPC optimization problem (without constraints). Once computed the optimal control sequence ∆u i (k|k), . . . , ∆u i (k + m -1|k) , which minimize the local cost function [START_REF] Menighed | Distributed State Estimation and Model Predictive Control : Application to Fault Tolerant Control[END_REF], only the first element of the optimal sequence ∆u i (k|k) is selected and the control action u i (k) = u i (k -1) + ∆u i (k|k) is computed and applied as control action to the subsystem S i , following the receding horizon strategy. Generally, the parameters of the above optimization problem are p and m; these parameters are directly affecting computational load in MPC. One of the MPC formulation is the classic approach presented in [START_REF] Maciejowski | Predictive control with constraints[END_REF]. In this approach, for the case of rapid sampling, complicated process dynamics and/or high demands on closed-loop performance, satisfactory approximation of the control signal ∆u(k) may require a very large number of parameters (large m), leading to poorly numerically conditioned solutions and heavy computational load when implemented on-line. Instead, a more appropriate technique would be to use Laguerre network in the design of MPC presented in [START_REF] Qian | Model Predictive Controller Using Laguerre Functions for Dynamic Positioning System[END_REF], [START_REF] Wang | Model Predictive Control System Design and Implementation Using MATLAB[END_REF].

V. DESCRIPTION OF THE CONTROL SIGNAL TRAJECTORY

The Z-transforms of the discrete-time Laguerre networks are written as follows

Γ n (z) = Γ n-1 (z) z -1 -a 1 -az -1 n = 2, 3, . . . , M (9)
with

Γ 1 (z) = √ 1 -a 2 1 -az -1
where M is the number of Laguerre functions in the network and a is the pole of the Laguerre network. The scaling factor a is required to be selected by the user, where 0 ≤ a < 1 for the stability of the network. Note that the Laguerre networks are well known for their orthonormality.

With the relation ( 9), the Laguerre network is illustrated in Fig. The discrete-time Laguerre functions are obtained through the inverse Z-transform of the Laguerre network. The set of Laguerre functions can be expressed as

L(k) = l 1 (k) l 2 (k) . . . l M (k)
T , where l i (k) denotes the inverse Z-transform of Γ i (z, a). Taking advantage of the network realization [START_REF] Qian | Model Predictive Controller Using Laguerre Functions for Dynamic Positioning System[END_REF], the set of discrete-time Laguerre functions satisfies the following difference equation

L(k + 1) = A l L(k) (10)
where A l ∈ M ×M and is a function of the parameters a and β = (1a 2 ), and the initial condition is given by

L(0) T = β 1 -a a 2 -a 3 . . . (-1) M -1 a M -1
The orthonormality can be expressed by

∞ k=0 l i (k)l j (k) = 0 for i = j (11a) ∞ k=0 l i (k)l j (k) = 1 for i = j (11b)
The orthonormal property of the Laguerre functions will be used in the design of MPC. The key idea in MPC based on Laguerre functions lies in the approximating member of control sequence by a set of Laguerre functions as

               ∆u(k + l|k) = M n=1 l n (l)c n (k) = L(l) T η(k) for l = 0, 1, . . . , m -1 where ∆u(k|k) = u(k|k) -u(k -1|k) (12)
with k being the initial time of the moving horizon window and l being the future sampling. The parameter vector η comprises M Laguerre coefficients: η = [c 1 c 2 . . . c M ] T and L(l) T is the transposed Laguerre functions vector as defined in the difference equation [START_REF] Razavinasab | State Estimationbased Distributed Model Predictive Control of Large-Scale Networked Systems with Communication Delays[END_REF].

By using this approximation, the optimization problem (6) can be expressed in terms of coefficient vector η, instead of ∆u(k) as in the classic approach. Thus, the coefficient vector η will be optimized and computed in the design. With this design framework, the control horizon m from the classical MPC approach has vanished. Instead, the number of terms M (M < m) is used to describe the complexity of the trajectory in conjunction with the free parameter a. Furthermore, a long control horizon m can be achieved without using a large number of parameters, leading to low computational burden and memory storage. In this paper, the MPC based on Laguerre functions is used in the proposed DMPC scheme.

VI. USE OF LAGUERRE FUNCTIONS IN DMPC DESIGN

A. Modified-State predictor 1) For SISO subsystem S i , i = 1, 2, . . . , N : Under assumptions 1, at step k, the future state prediction over the horizon p are given by

xi (k + l|k) = A l ii xi (k|k) + l s=1 A s-1 ii B ii u i (k + l -s|k) + l s=1 A s-1 ii ŵi (k + l -s|k -1) (13) let us denote u i (k) = u i (k -1) + ∆u i (k) and u i (k + l -s) = u i (k -1) + l-s r=0 ∆u i (k + r) for l = 1, 2, . . . , p.
Then, the prediction of the future state variables at time l becomes

xi (k + l|k) = A l ii xi (k|k) + l s=1 A s-1 ii B ii u i (k -1) + l s=1 l h=s A l-h ii B ii L i (s-1) T η i + l s=1 A s-1
ii ŵi (k+l-s|k-1)

for l = 1, 2, . . . , p (14)
where the function ∆u i (k+s-1) is replaced by L i (s-1) T η i for s = 1, 2, . . . , l. The prediction of the plant output will be

ŷi (k + l|k) = C ii A l ii xi (k|k) + C ii l s=1 A s-1 ii B ii u i (k -1) + C ii l s=1 l h=s A l-h ii B ii L i (s -1) T η i + C ii l s=1 A s-1 ii ŵi (k + l -s|k -1) + vi (k + l|k -1) (15)
With this formulation, both predictions of state and output variables are expressed in terms of the coefficient vector η i of the Laguerre network, instead of u i as in the classical approach. Thus, the coefficient vector η i will be optimized and computed in the control design. To compute the prediction, the convolution sum

S ci (l) = l s=1 l h=s A l-h ii B ii L i (s -1) T (16)
needs to be computed. To this end, note that

S ci (1) = B ii L i (0) T S ci (2) = A ii B ii + B ii L i (0) T + B ii L i (1) T = A ii + I B ii L i (0) T + B ii L i (0) T A T li = S ci (1) + A ii S ci (1) + S ci (1)A T li S ci (3) = S ci (2) + A 2 ii S ci (1) + A ii S ci (1)A T li + S ci (1)(A 2 li ) T S ci (4) = S ci (3) + A 3 ii S ci (1) + A 2 ii S ci (1)A T li + A ii S ci (1)(A 2 li ) T + S ci (1)(A 3 li ) T (17)
Continuing the recursion in [START_REF] Zečević | A New Approach to Control Design with Overlapping Information Structure Constraints[END_REF] reveals that for l = 2, 3, . . . , p

       S ci (l) = S ci (l -1) + l h=1 A h-1 ii S ci (1) A (l-h) li T with S ci (1) = B ii L i (0) T (18)
The difference equation L i (k + 1) = A li L i (k) is used for generating the set of Laguerre functions, where A li ∈ Mi×Mi and is a function of the parameters a i and β i = (1a 2 i ). 2) For MIMO subsystem S i , i = 1, 2, . . . , N : To extend the description to MIMO subsystem, with full flexibility in the choice of a i and M i parameters, let ∆u i (k) be

∆u i (k) = ∆u 1 i (k) ∆u 2 i (k) • • • ∆u nu i i (k)
T and the input matrix be partitioned to

B ii = B 1 ii B 2 ii • • • B nu i ii
where n ui is the number of inputs and B g ii for g = 1, 2, . . . , n ui is the g th column of the B ii matrix. We express the g th control signal ∆u g i (k) by choosing a scaling factor a g i and order M g i , where a g i and M g i are selected for this particular input, such that

∆u g i (k) = L g i (k) T η g i for g = 1, 2, . . . , n ui
where η g i and L g i (k) are the Laguerre network description of the g th control, namely

L g i (k) T = l g i(1) (k) l g i(2) (k) . . . l g i(M g i ) (k)
Based on the partition of the input matrix and given the state variable information at xi (k), the prediction of the future state at time l can be written as

xi (k + l|k)= A l ii xi (k|k) + Ψ i (l)u i (k -1) + Φ i (l) T η i + Ω i (l) (19)
with

Ψ i (l) = l s=1 A s-1 ii B ii , Φ i (l) T = l s=1 l h=s A l-h ii B ii L i (s -1) T Ω i (l) = l s=1 A s-1 ii ŵi (k + l -s|k -1)
where the parameter vector η i and the data matrix Φ i (l) T consist of the following individual coefficient vectors

η T i = [η 1 i η 2 i . . . η nu i i ] T Φ i (l) T = l s=1 l h=s A l-h ii B 1 ii L 1 i (s) T B 2 ii L 2 i (s) T . . . B nu i ii L nu i i (s) T
for i = 1, 2, . . . , N and g = 1, 2, . . . , n ui . Note that the k th block matrix

Φ k i (l) T = l s=1 l h=s A l-h ii B k ii L k i (s -1) T
has an identical structure as the single-input case defined by S ci (l), thus it can be computed recursively using [START_REF] Zheng | Distributed Model Predictive Control Over Network Information Exchange for Large-Scale Systems[END_REF]. From here on, the convolution sum in each multi-input subsystem S i is decomposed into computing the subsubsystems, and the computed results are put together block by block to form the multi-input structure. It is worth stressing that in the formulation of the multivariable problem, the scaling factors a g i and the number of terms M g i can be chosen independently for each input signal of subsystem S i .

B. Modified-Optimizer

Under assumptions 1, since the discrete Laguerre functions are orthonormal for a sufficiently large prediction horizon p, the cost function ( 6) is equivalent to the modified local cost function defined as

J i = p l=1 y d i (k + l|k) -ŷi (k + l|k) 2 Qi +η T i R Li η i (20)
where R Li ∈ Mi×Mi is a diagonal matrix with the weighting matrix R i on its diagonal. Therefore, the objective is to find the coefficient vector η i that minimizes J i . Without any constraints, by substituting ( 15) into (20), the optimal solution of the parameter vector ηi is

ηi = η opt i = p l=1 Φ i (l)Q i Φ i (l) T + R Li -1 × p l=1 Φ i (l)Q i y d i (k + l|k) -Θ i (l)x i (k|k) -Ψ i (l)u i (k -1)
-

Ω i (l) -vi (k + l|k -1) (21)
with

Θ i (l) = C ii A l ii , Ψ i (l) = C ii Ψ i (l) Φ i (l) T = C ii Φ i (l) T , Ω i (l) = C ii Ω i (l)
Upon obtaining the optimal parameter vector ηi , the control increment at time k is as follows

∆u i (k) =      L 1 i (0) T 0 T 2 . . . 0 T nu i 0 T 1 L 2 i (0) T . . . 0 T nu i . . . . . . . . . . . . 0 T 1 0 T 2 . . . L nu i i (0) T      ηi (22)
where 0 T r , for r = 1, 2, . . . , n ui , represents a zero block row vector with identical dimension of L r i (0) T . Consequently, the control signal u i (k) can be calculated as

u i (k) = u i (k -1) + ∆u i (k) (23)
1) Modified Interaction prediction: Under assumptions 1, at step k, the predictions of the new interaction vectors are given by Ŵi

(k, |k -1) = Ãi X(k|k -1) + Bi Ũ (k -1|k -1) Vi (k, |k -1) = Ci X(k|k -1) (24)
Fi F i,1 . . . F i,i-1 0 F i,i+1 . . . F i,N
where Fi ∈ Ãi , Bi , Ci

VII. NC-MPC ALGORITHM

For the i th sub-controller C i , where the desired output y d i (k + l|k) is provided by a proper reference generator, the algorithm for the novel distributed MPC is outlined in detail:

1) Set k = 1 2) Acquire by network the predicted future state trajectories Xj (k|k -1) and control inputs U j (k -1|k -1) from sub-controllers C j .

3) Build X(k|k -1) and U (k-1|k -1) by combining the local state trajectory xi (k|k -1) and control input u i (k -1|k -1) with the acquired information, and compute the corresponding predictions of the interactions according to (24). 4) Acquire the measures xi (k) and the desired trajectory y d i (k + l|k) over the horizon p. 5) Compute the optimal control u i (k) and broadcast it by network to sub-controllers C j , cf. ( 21), ( 22) and (23). 6) Apply the control input u i (k) to subsystem S i . 7) Compute the future state trajectory of subsystem S i over the horizon p and broadcast it by network to subcontrollers C j , cf. [START_REF] Zheng | Networked Coordination-Based Distributed Model Predictive Control for Large-Scale System[END_REF]. 8) Increment the sample time index k ← k + 1 and iterates by going to step 2.

VIII. SIMULATION AND RESULTS

In this section, the performance of the proposed NC-DMPC is investigated and compared to the centralized MPC. Consider the following unstable-non minimum phase plant S introduced in [START_REF] Vaccarini | Stability Analysis Tool for Tuning Unconstrained Decentralized Model Predictive Controllers[END_REF] In the proposed test, the stability performances depend on the choice of the tuning parameters α and p. The control performance of the resulting closed-loop system is plotted in Figs 2, and3, where the black lines correspond to the desired outputs, the red solid lines correspond to the system outputs using Centralized MPC (C-MPC) based on Laguerre functions, and the blue solid lines represent the system outputs using the proposed NC-DMPC. In To sum up, for the given example, the NC-DMPC can achieve a satisfactory global performance even whether the interactions among subsystems are strong or not. Furthermore, the cost of computation is very small as compared with the classical centralized controller.

IX. CONCLUSION

In the present study, a formulation of a novel distributed model predictive control for a class of large-scale systems is proposed, in which the whole system is divided into many small scale subsystems interacting with each other by both their states and inputs. The NC-DMPC solution, was proposed for improving the global performance of closed-loop system; it is based on Laguerre functions used in MPC formulation. The proposed methodology is demonstrated on an example for a set-point tracking. Compared to the C-MPC scheme, the proposed NC-DMPC allowed to achieve an improved performance of the whole system through using a local index in optimization.

1

 1

Fig. 1 .

 1 Fig. 1. Illustration of a discrete Laguerre network

C 11 =Fig. 2 .Fig. 3 .

 1123 Fig. 2. Control performance with α = 1, p =

  Fig 2, the performance of ND-DMPC is comparable to the one of C-MPC. However, for strong interactions with α = 3.5, Fig 3 shows that the ND-DMPC strategy can achieve a better global performance of the closed-loop system than C-MPC.

  discretized with a sampling time T s = 0.2 s. A statespace realization of S has the form (1), with the following matricesThe examination of the process model S leads to decomposition into two interconnected subsystems S 1 and S 2 . The corresponding state-space realizations of S 1 and S 2 have the form (2), with matrices {A 11 , B 11 , C 11 } and {A 22 , B 22 , C 22 }, respectively. The constant parameter α is used to study the effect of the interactions between S 1 and S 2 .

	A=	A 11 0	0 A 22	, B=	B 11 0	0 B 22	, C=	C 11 C 12 C 21 C 22

z -1 -a 1-az -1 √ 1-a 2 1-az -1 z -1 -a 1-az -1 1 Γ 1 (z) Γ 2 (z) Γ M (z)


Jérôme Euzière

Régis Guinvarc'
email: regis.guinvarch@centralesupelec.fr

Israel Hinostroza Sáenz

Raphaël Gillard

Bernard Uguen

Time Modulated Array -A Database Approach

Keywords: TMA, radar, rejection, search space, database

Time Modulated Array offers a simple way to synthesize radiation patterns with low sidelobe levels, in timeaverage, using an optimized strategy of switching ON and OFF the elements of the array. In this work we show that the complexity of the optimization strategy can be reduced by first shrinking the search space to a subset, or database, of useful solutions over which the optimization will be launched. This reduction is obtained by applying the constraints in a progressive and hierarchical order. In this way, this problem of exponential growth is reduced to a polynomial growth, with respect to the size of the antenna array. We present the case of designing the time modulated array for radar applications where constant directivity and ability of intereference rejection is needed.

I. INTRODUCTION

Time Modulated Array (TMA) [START_REF] Kummer | Ultra-Low Sidelobes from Time-Modulated Arrays[END_REF] offers a simple way to synthesize low sidelobe radiation patterns. It relies on averaging several radiation patterns. These radiation patterns are obtained by switching (ON and OFF) the elements in the array during variable time intervals so that the average pattern has the desired characteristics. In this way, TMA offers a lot of possible degrees of freedom to which the community has been applying different optimization techniques. In order to make the optimization process acceptable, the community has focused on the time domain [START_REF] Manica | Almost Time-Independent Performance in Time-Modulated Linear Arrays[END_REF] [3] [4] [START_REF] Yang | Design of a Uniform Amplitude Time Modulated Linear Array With Optimized Time Sequences[END_REF] [START_REF] Yang | An effective hybrid optimization algorithm for the synthesis of 4-D linear antenna arrays[END_REF]. In this work, we reintroduce the spatial domain (weighting on the elements of the array) as a way to reduce the search space by a progressive, hierarchical application of the constraints. By discretizing the weights (amplitude and phase) on the antennas, we obtain a finite number of possible solutions (possibly large). From this, by progressively applying a sequence of filters (constraints), the search space is progressively reduced to obtain a database containing the possible solutions. In this way, this sequence of filters permits to address a large number of different problems sharing some common constraints.

In section II we recall the concepts and definitions of TMA. Next, in section III, it is shown how the filters can be applied to obtain a database of useful instantaneous patterns. Later, in section IV, we present an example of the technique considering the design of a linear antenna array for radar applications. Finally, we present some conclusions.

II. TIME MODULATED ARRAY

The original TMA technique introduced by Kummer [START_REF] Kummer | Ultra-Low Sidelobes from Time-Modulated Arrays[END_REF] uses a linear array which elements are switched ON or OFF. This switching is in fact a weigthing on the elements of the antenna array as a function of time that repeats itself after a period T . In this way, each switching combination corresponds to an instantaneous array factor, among a total of P possible ones, over a certain duration ∆t p , p ∈ {1, 2, . . . , P }. Next equation shows the instantaneous pattern p for a uniform linear antenna array of M elements.

AF p (θ) = M m=1 w mp • e j•(m-1)k•d•cos(θ) ( 1
)
where k is the wave number, d is the constant distance between elements, w mp ∈ C is the weight applied to the m-th element of the array (m ∈ {1, 2, . . . , M }) during the ∆t p time duration. Note that in [START_REF] Kummer | Ultra-Low Sidelobes from Time-Modulated Arrays[END_REF] w mp ∈ {0, 1}, whereas, in general, the weight w mp is defined by its magnitude a mp and phase ϕ mp . Then, the time-average pattern AF av is:

AF av (θ) = 1 T P p=1 AF p (θ) • ∆t p (2)
We can see that the final time-average pattern AF av depends on the weighting w mp of the elements and durations ∆t p . Many authors have already worked on the optimization of the latter time dimension. On the former, specially for radar, we can mention the use of sparse arrays (STMA [START_REF] Euzière | Optimization of Sparse Time-Modulated Array by Genetic Algorithm for Radar Applications[END_REF]) and Phase Only Synthesis solution (POSTMA [START_REF] Euzière | Phase Only Synthesis in Time Modulated Array[END_REF]). For all these cases, the number of degrees of freedom can be large (multiple amplitudes and/or phases for w mp , different durations ∆t p ), which gives better results, but longer optimization time. Hence, heuristic methods need to be used.

III. DATABASE STMA

Due to the large number of degrees of freedom, the search space also becomes very large. In this section we propose a technique to reduce the size of the search space. To do so, for simplicity, we consider the case of STMA (w mp ∈ {0, 1} and constant time step ∆t p ), hence optimization only applies to the weights of the elements.

For an antenna array of M elements having given positions and weightings, there are N s = 2 M -1 possible instantaneous array factors. Hence, one possible average pattern can be obtained by selecting P instantaneous array factors among N s , without repetition and indifferent order of selection. Note that the case of selecting with repetition is a way to consider optimization along the dimension of time.

Then, for our case, the total number of possible time-average patterns N av patt is:

N av patt = N s P = N s ! P !(N s -P )! (3)
We can see that, as M → ∞, N av patt = O(2 M •P ), which is an exponential growth with respect to the size of the antenna array. However, many of the possible instantaneous array factors may not be useful and can be discarded. The idea is then to define filters to select only useful instantaneous patterns according to the specifications. In this way we obtain a new reduced number of possible instantaneous array factors N r s , hence a reduction of the search space size is achieved. This database of N r s useful patterns is the input for the optimization process.

As an example of how this database can be built let us consider the case of radar and interference suppression using STMA.

A. Case of radar and interference suppression 1) Directivity filter: A radar antenna needs to have an average pattern with low sidelobe level (SLL) with respect to main lobe and a prescribed directivity. Additionally, during the integration time, the directivity needs to be constant when switching from one pattern to the next one. In order to assure constant directivity, the optimization needs to be restricted to a constant number of elements of the array that are turned ON. Hence, the database, under constant directivity, is composed of N r,direct s solutions:

N r,direct s = M k OF F ( 4
)
where k OF F are the number of elements of the array that are turned OFF. This means that, by considering N r,direct s instead of N s in (3), the new reduced total number of possible time-average patterns is N r,direct av patt = O(M k OF F •P ) instead of N av patt = O(2 M •P ), as M → ∞. Hence the optimization problem with exponential growth has been transformed into a one with polynomial growth, with respect to the size of the array.

2) Interference rejection filter: In [START_REF] Haupt | Time Modulated Receive Arrays[END_REF], Haupt noted that, if the original TMA [START_REF] Kummer | Ultra-Low Sidelobes from Time-Modulated Arrays[END_REF] is used for radar applications, an interference can be damaging even if the average rejection of interference is large. The problem comes from the fact that the rejection for some of the instantaneous patterns may be much higher. As an illustration, we consider the case of a linear array with N elements and d = λ 0 /2 interelement spacing (λ 0 is the working wavelength). We assume the array is radiating at broadside (θ max = 90 • ) and 2 elements out of N are switched OFF. Note that this scenario will be used all along this paper.

For example, if a minimum rejection of 25 dB is needed, there is no solution for an array of 10 elements at θ = 50 • . In fact, the maximum rejection is 22 dB. As expected, this constraint is not a problem at other angles and for larger arrays. For example, for 80 elements, the maximum rejection is better than 47 dB for all angles.

Hence, while building the database of useful patterns, adding an additional filter to take into account the minimum rejection will further reduce the search space. Fig. 2 shows the number of useful radiation patterns that comply with the constraint of minimum rejection of 25 dB.

From Fig. 2 we can see that the number of useful patterns is very low (no more than 22 at θ = 10 • ) for the case of antenna arrays of 10 elements. As expected, for larger arrays this number is larger. For example, for antenna arrays of 40 elements the number of the useful patterns is at least 461 for all angles. It is interesting to note that, globally, the number of useful patterns is quite stable for all angles, once the number of elements is set.

As an example of how much reduction is obtained after applying the second filter, we can consider different values of minimum rejection at θ = 55 • as in Fig. 3, which shows the ratio between the output of the second filter and the output of the first filter versus array length. Fig. 3 also shows how effective is the second filter considering the constraints and array size. It can be seen that, eventually, the number of solutions after applying the second filter (rejection) approaches the number of solutions after applying only the first filter (directivity) as long as the size of the antenna array is large enough. In the previous section it has been shown how the number of useful possibles solutions is reduced by applying different filters. Let us now consider the case of finding the best TMA configuration considering an antenna array of 12 elements under the constraints specified in Tab. I. Fig. 4 shows how a database, for later optimization, is built for this case.

Note, in Fig. 4, how the original number of possible instantaneous patterns is reduced to less than a hundredth thanks to the application of the filters. Fig. 5 shows the 10 instantaneous patterns (due to 10 different weightings) of the database. Note that the superimposition of amplitude levels of the patterns shows only 4 different curves, but their phases are different.

We then launch an optimization to choose 4 different instantaneous patterns (out of the 10 patterns from the database) in order to obtain, in time-average, minimum SLL and maximum interference rejection at θ = 55 • . Fig. 6 shows the result of the optimization. The time-average SLL of the optimized solution is -15.44 dB and the interference rejection at θ = 55 • is 40 dB. Note that if we allow repetition in the choice of instantaneous patterns, the optimization then includes the time dimension.

V. CONCLUSIONS

It has been shown that the search space for the optimization can be strongly reduced (from exponential to polynomial growth relative to the size of the antenna array) when a progressive and hierarchical introduction of filters (constraints on the instantanenous patterns) is carried out. A database of useful instantaneous patterns can be built from which an optimal average pattern can be obtained through optimization techniques. An example has been shown on how a database is built for the case of radar and interference rejection, followed by the final choice (through optimization) of optimum instantaneous patterns which also gives the desired average pattern.

Fig. 1 .

 1 Fig. 1. Opposite of maximum rejection of interference in [dB] (relative to the main lobe level) versus angle (θ ∈ [10 • , 70 • ], main beam at θmax = 90 • ) for different size of antenna array, blue for 8 elements out of 10, green for 18 out of 20, red for 38 out of 40 and black for 78 out of 80.

Fig. 2 .

 2 Fig. 2. Number of useful radiation patterns with minimum rejection of 25 dB versus angle (θ ∈ [10 • , 70 • ], main beam at θmax = 90 • ) and for different size of antenna array, blue for 8 elements out of 10, green for 18 out of 20, red for 28 out of 30 and black for 38 out of 40.

Fig. 3 .

 3 Fig. 3. Ratio of number of final useful radiation patterns (after rejection filter) over number of radiation patterns after applying directivity filter versus antenna array length, per constraint of minimum rejection at θ = 55 • : blue for rejection of 25 dB, green for 30 dB, red for 35 dB and black for 40 dB.

Fig. 4 .

 4 Fig. 4. Example of how a database is built by applying filters. Considering an antenna array of 12 elements: constant directivity of 10 elements ON, minimum rejection of 25 dB at θ = 55 • .

Fig. 5 .Fig. 6 .

 56 Fig. 5. The 10 instantaneous patterns (not all of them seen since they have same amplitude but different phases) of the database STMA after applying the filters (directivity, rejection) and under the constraints of Tab. I. Note that SLL at θ = 55 • is barely lower than -25 dB.


Christian Deffo Tassak

Louis Aimé

Christian Deffo Tassak
email: dtassak@yahoo.com

Louis Aimé Fono

Jules Sadefo Kamdem

Dominance: An application for poverty Measurement »

Keywords: Credibility measure, Fuzzy variable, Fuzzy lower partial moment, Mean-Risk dominance, Poverty index

A more general concept of risk in economics consists on the chance of getting an income or a return less than a threshold one. Risk has been studied and generalized more earlier by Fishburn [8] through Mean Partial Lower Moment specially when income can be described by a random variable. In this paper, we present a new concept of partial moment, namely Fuzzy Lower Partial Moment (FLPM) based on credibility measure, to quantify risk of getting a return described by a fuzzy variable and we study its properties. Based on FLPM, we introduce mean risk dominance for fuzzy variables, we characterize the dominance for some specific cases and we determine some of its properties. Furthermore, we study the consistency of mean-risk models with respect to first and second order dominances. We display one application of FLPM by introducing a new poverty index for poverty measurement in the context of fuzzy environment and we examine some of its properties.

Introduction

The concept of downside risk has been for a long time a meaningful topic in portfolio selection and management. Since the seminal work of Markowitz [START_REF] Markowitz | Portfolio Selection[END_REF] where he proposed below-mean semi-variance and more generally belowtarget semi-variance as risk measures which take into account investors'attitude towards risk, downside risk has been developed more later by Bawa [1] and Fishburn [START_REF] Fishburn | Mean-risk analysis with risk associated with belowtarget returns[END_REF] in their respective frameworks based on Lower Partial Moment (LPM) as below-target risk. The first proved that LPM measure is mathematically related to stochastic dominance for risk tolerance values equal to one or two and the second proved the equivalence of the LPM measure to stochastic dominance for all risk tolerance values as strictly nonnegative integers. Kaplan and Knowles [START_REF] Kaplan | A generalised downside-risk performance measure[END_REF], Brogan and Stidham [3], Sunoj and Maya [START_REF] Sunoj | The role of lower partial moments in stochastic modeling[END_REF] also contributed to develop downside risk measure in their recent frameworks. Let us notice that all precede results rely on the fact that assets' returns are described by random variables. However, according to the recent litterature (Zadeh [26], Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF], Sadefo et al. [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF], Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]), some situations in real life are related to fuzzy phenomena which induces imprecise and vague concepts. For instance, assets' returns can be expressed as follows: "around 15 F", "between 20F and 30 F ", etc..., and fuzzy variables are used to deal with those concepts. Thus, an open question is to study downside risk for fuzzy variable and its related concepts and applications. The aim of this paper is three folds: introduce and study the Fuzzy Lower Partial Moment (FLPM) based on credibility measure, define and study the meanrisk dominance for two fuzzy variables based on FLPM and, apply FLPM to analyze poverty in the case where source of fuzziness for poverty is due to imprecise estimation of income. Before we display the roadmap of our article, let us review some frameworks on poverty in order to specify the guideline of our third specific objective. In fact, literature's review on poverty started with Sen [START_REF] Sen | Poverty: an ordinal approach to measurement[END_REF] who argued that the first issue when measuring poverty is to identify the poor among the total population by using a poverty line which allows to distinguish poor and non poor, and the second issue is the construction of an index to measure poverty's level. Other scholars (Foster [9], Hagenaars [START_REF] Hagenaars | A class of poverty indices[END_REF], Seidls [START_REF] Seidl | Poverty measurement: a survey, Welfare and Efficiency in Public Economics[END_REF], Chakravarty [START_REF] Chakravarty | Ethical social index numbers[END_REF] , Ravallion [START_REF] Ravallion | Poverty comparisons[END_REF] and Zhen ( [START_REF] Zheng | Agregate poverty measures[END_REF], [START_REF] Zheng | Poverty orderings[END_REF], [START_REF] Zheng | Fuzzy poverty measurement and crisp dominance[END_REF]) brought a useful contribution with their respective reviews on poverty measure construction issue. More recently, Gallardo [START_REF] Gallardo | Identifying vulnerability to poverty: a critical survey[END_REF] offered an analysis of the state of the art in the conceptualization of vulnerability to poverty through an approach based on the mean-risk dominance criterion which used the downside mean semideviation as the risk parameter. However, poverty line research has longer been discussed in poverty measurement literature and led to introduce fuzzi-ness by the fact people don't always agree on an individual's poverty status. Following that, Zheng [START_REF] Zheng | Fuzzy poverty measurement and crisp dominance[END_REF] indicated three sources of fuzziness for poverty which have been analyzed in literature: fuzziness due to imprecise estimation of income; fuzziness due to the fact that poverty is essentially a multidimensional concept and, fuzziness due to fact that different people may have different ideas about the poverty line which makes an individual poor or non poor. Those sources of fuzziness have been studied by Chakravarty [START_REF] Chakravarty | An axiomatic approach to multidimensional poverty measurement via fuzzy sets, Fuzzy Set Approach to Multidimensional Poverty Measurement[END_REF], Shorrocks and Subramanian [START_REF] Shorrocks | Fuzzy Poverty indices[END_REF], Beliakov [2] and Dombi [START_REF] Dombi | Membership functions as an evaluation[END_REF], Qizilbah [START_REF] Qizilbash | Philosophical accounts of vagueness, fuzzy poverty measures and multidimensionality[END_REF], Cerioli and zani [4], Zheng [START_REF] Zheng | Fuzzy poverty measurement and crisp dominance[END_REF]. Recently, Zheng [START_REF] Zheng | Fuzzy poverty measurement and crisp dominance[END_REF] explored the third source of fuzziness by introducing a meaningful "density function" for a poverty membership function and he proposed axioms for the characterization of additively separable fuzzy poverty measures. In this paper, we do not consider this approach and we focus only on the first source of fuzziness. This can be justified by the fact that in some societies, people have many jobs in order to improve welfare and overcome bad conditions life. More concretely, it has been observed in many african societies that one individual may have several economic activities besides his official and recognized activity with respect to his qualification or competence. Thus, some unknown incomes are undeclared and it is not often possible to know the exact income of one individual. The paper is organized as follows. Section 2 gives some preliminaries on fuzzy variable and its characteristics. In Section 3, we define Fuzzy Lower Partial Moment (FLPM) based on credibility measure for measuring investment risk of returns described by fuzzy variables. We characterize the FLPM for a particular class of fuzzy variables whose distribution functions belong to family of exponential functions. Based on FLPM, we define the mean-risk dominance for two fuzzy variables and characterize it in some particular cases. Some properties of the new dominance are established and we justify that it is not complete binary relation on fuzzy variables. Moreover, we study the consistency of mean-risk models and justify their optimal portfolios belong to the set of best portfolios with respect to the first or second order dominances based on credibility measure developped earlier by Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]. Section 4 presents an application of FLPM for poverty measurement. Section 5 gives some concluding remarks.

Preliminaries

Let ξ be a fuzzy variable with membership function µ where for any x ∈ R, µ(x) represents the possibility that ξ takes value x. ξ is normal if ∃x 0 /µ(x 0 ) = 1. ξ is non-negative fuzzy variable if ∀x < 0, µ(x) = 0. The support of ξ is the crisp set defined by Supp(ξ) = {x ∈ R, µ(x) > 0}.

Throughout this paper, we will assume that all fuzzy variables are normal, that is, ∃x 0 ∈ R, µ(x 0 ) = 1.

Note that for ξ taking values in B, Zadeh [START_REF] Zadeh | Fuzzy set as a basis of theory of possibility[END_REF] has defined the possibility measure of B by P os({ξ ∈ B}) = sup x∈B µ(x) and the necessity measure of ξ by N ec({ξ ∈ B}) = 1 -sup x∈B c µ(x) respectively. But neither of these measures are self-dual. Therefore, Liu and Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] introduced the credibility measure as the average of possibility measure and necessity measure as follows: for any set B,

Cr({ξ ∈ B}) = 1 2 sup x∈B µ(x) -sup x∈B c µ(x) + 1 . ( 1
)
Credibility measure is self-dual, that is,

Cr({ξ ∈ B}) + Cr({ξ ∈ B c }) = 1.
Liu and Liu [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] introduced the expected value of ξ defined as follows

E[ξ] = e = +∞ 0 Cr{ξ ≥ r} dr - 0 -∞ Cr{ξ ≤ r} dr (2)
provided that at least one of the above integrals is finite. Note that, expected value is one of the most important concept of fuzzy variable which gives the center of its distribution. Liu [START_REF] Liu | Uncertainty Theory: An introduction to its axiomatics foundations[END_REF] also defined the credibility distribution Φ : R → [0, 1] of a fuzzy variable ξ as follows:

∀t ∈ R, Φ(t) = Cr{ξ ≤ t} = 1 2 [1 + sup x∈]-∞;t] µ(x) -sup x∈]t;+∞[ µ(x)]. (3)
When Φ is absolutely continuous, we have the credibility density function

φ : R → [0, ∞[ such that ∀t ∈ R, Φ(t) = t -∞ φ(u)du. (4)
Obviously, we have

+∞ -∞ φ(u)du = 1.
φ belongs to exponential family if there exists γ > 0, K(•) an arbitrary function and D(•) a differentiable function such that φ(x) = e γ x+K(x)+D(γ) ,

x ∈ (0, ∞).

(

) 5
Notice that, Φ(x -) = sup{Φ(y), y < x} is the credibility of getting a value of ξ not exceeding x.

A distribution function Φ of ξ is a non-degenerate distribution function if ∀t ∈ R, t = E(ξ) ⇒ Φ(t) = 0.
The distribution function Φ of a fuzzy number ξ = (a, b, c, d) is defined by:

∀r ∈ R, Φ(r) =            0 if r < a 1 2 µ(r) if a ≤ r < b 1 2 if b ≤ r < c 1 -1 2 µ(r) if c ≤ r < d 1 if d ≤ r . ( 6
)
It is an increasing function, that means, if

x ∈ [a, b], y ∈ [b, c], and z ∈ [c, d], then Φ(x) ≤ Φ(y) ≤ Φ(z).
In the next Section, we will introduce and study the lower partial moment of a fuzzy variable ξ, namely, the Fuzzy Lower Partial Moment (FLPM) of ξ and the Mean risk dominance between fuzzy variables.

FLPM n,τ [ξ] = E[max(τ -ξ, 0) n ] (7)
where the expectation operator E is defined by (2).

Throughout this paper, we will simply say "FLPM of ξ" instead of "The Fuzzy Lower Partial Moment (FLPM) of ξ with order n and target value τ."

Let us deduce some values of a target value or of the order for which the FLPM of ξ becomes some well-known statistics parameters of ξ and some new fuzzy counterparts of well-known notions of the Probability Theory. In addition, we express the FLPM of a fuzzy variable by means of its credibility distribution function.

Remark 1. Let ξ be a fuzzy variable, τ ∈ R and n ∈ N * .

1. If the target value τ = E [ξ], then we obtain the following notions: [ξ] is the semi-moment of order n of ξ introduced earlier by Sadefo et al. [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF].

• For n ∈ 2N -{0}, FLPM n,τ
• For n = 4, FLPM n,τ [ξ] becomes the fuzzy semi-kurtosis of ξ introduced by Sadefo et al. [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF].

• For n = 2, FLPM n,τ [ξ] becomes the fuzzy semi-variance of ξ introduced by Huang [START_REF] Huang | Mean-semivariance models for fuzzy portfolio selection[END_REF].

2. For some specific values of n, we obtain fuzzy counterparts of some well-known notions of the downside risk under probability theory:

• We introduce the so-called credibility of loss equals to the 0 th order FLPM 0,τ of ξ.

• In the case where n = 1, FLPM 1,τ = E[max(τ -ξ, 0)] is called the expected loss of ξ. Here the constant target value τ can be considered as the threshold point separating returns in two parts: downside returns and upside returns depending of the threshold.

• For n = 2, FLPM 2,τ = E[max(τ -ξ, 0) 2 ]
is the Credibilistic Fuzzy Target Semi-Variance (CFTS).

• By setting n = 3 (resp. n = 4), we obtain FLPM 3,τ (resp. FLPM 4,τ ) which is the Credibilistic Fuzzy Target Semi-Skewness (CFTSS) (resp. the Credibilistic Fuzzy Target Semi-Kurtosis (CFTSV)).

3. The expression of FLPM n,τ with respect to credibility distribution is:

FLPM n,τ [ξ] = +∞ 0 Cr{max(τ -ξ, 0) n ≥ r} dr = n τ -∞ (τ -u) n-1 Φ(u)du. (8)
4. If Φ has a derivative and ξ has a lower bounded support then:

FLPM n,τ [ξ] = τ -∞ (τ -u) n dΦ(u) (9) = τ -∞ (τ -u) n φ(u). ( 10
)
The previous definition of FLPM n,τ justifies that it is a function of the credibility distribution function and, it is a non-decreasing function of its target τ return. As τ increases, FLPM n,τ also increases.

We end this Section with the expressions of FLPM of trapezoidal and triangular fuzzy variables.

Corollary 1.

The FLPM of the trapezoidal fuzzy variable

ξ = (a, b, c, d) is: FLPM n,τ [ξ] =                0 if τ < a (τ -a) n+1 2(α+1)(b-a) if a ≤ τ < b [(τ -a) n+1 -(τ -b) n+1 ] 2(n+1)(b-a) if b ≤ τ < c [(τ -a) n+1 -(τ -b) n+1 ] 2(n+1)(b-a) + (τ -c) n+1 2(n+1)(d-c) if c ≤ τ < d [(τ -a) n+1 -(τ -b) n+1 ] 2(n+1)(b-a) + [(τ -c) n+1 -(τ -d) n+1 ] 2(n+1)(d-c) if τ ≥ d . ( 11
)
2. The FLPM of the triangular fuzzy number ξ = (a, b, d) is:

FLPM n,τ [ξ] =            0 if τ < a (τ -a) n+1 2(n+1)(b-a) if a ≤ τ < b [(τ -a) n+1 -(τ -b) n+1 ] 2(n+1)(b-a) + (τ -b) n+1 2(n+1)(d-b) if b ≤ τ < d [(τ -a) n+1 -(τ -b) n+1 ] 2(n+1)(b-a) + [(τ -b) n+1 -(τ -d) n+1 ] 2(n+1)(d-b) if τ ≥ d . (12)
In the next Subsection, we establish some properties of the F LP M of a fuzzy variable ξ based on its absolutely continuous credibility distribution function.

Some results on FLPM n,τ

The following result determines the credibility distribution function Φ of ξ in term of derivatives of its F LP M when Φ has a compact support. More precisely, it suggests that we can determine the credibility distribution Φ(τ ) uniquely given FLPM n,τ with n ∈ N * . Proposition 1. The credibility distribution function Φ of a fuzzy variable ξ with a lower bounded support satisfies the following relation:

d n dτ n FLPM n,τ = n! Φ(τ ), that is, Φ(τ ) = 1 n! d n dτ n FLPM n,τ . (13)
Proof : Let Φ be the credibility distribution function with compact support of the fuzzy variable ξ. We have:

d n dτ n FLPM n,τ = d n dτ n [ τ -∞ (τ -u) n dΦ(u)] = τ -∞ d n dτ n [(τ -u) n dΦ(u)]. It is easy to check that ∀n ∈ N * , d n dτ n (τ -u) n = n!
and finally, we have:

d n dτ n FLPM n,τ = τ -∞ n!dΦ(u) = n! τ -∞ dΦ(u) = n![Φ(τ ) -lim u→-∞ Φ(u)] = n![Φ(τ ) -0] = n!Φ(τ ).
Hence the result.

The following result determines necessary and sufficient conditions on a FLPM under which the density function φ of ξ, satisfying a particular inequality, belongs to exponential family. Proposition 2. Let φ γ be the credibility density function of a nonnegative fuzzy variable ξ satisfying the following condition:

d dγ φ γ (u) ≥ φ γ (u)(u + D (γ)), ∀u ∈ (0, ∞) (14)
where D (γ) is the derivative of D(•) with respect to γ. φ γ belongs to exponential family, that means,

φ γ (u) = e γ u+K(u)+D(γ) , u ∈ (0, ∞), γ > 0, (15)
where K(•) is an arbitrary function, if and only if, its FLPM α,τ satisfy the following recurrence relationship:

FLPM n+1,τ = (τ + D (γ))FLPM n,τ - d dγ FLPM n,τ . (16)
Proof : (⇒) Assume that the credibility density function φ γ is defined by: φ γ (u) = e γ u+K(u)+D(γ) where u ∈ (0, ∞), γ > 0, K and D two arbitrary functions. By computing the derivative of φ γ given by relation [START_REF] Liu | Expected value of fuzzy variable and fuzzy expected value models[END_REF] with respect to γ, one can easily check that φ γ satisfies relation [START_REF] Liu | Uncertainty Theory: An introduction to its axiomatics foundations[END_REF].

Let us prove that FLPM n+1,τ = (τ + D (γ))FLPM n,τ -d dγ FLPM n,τ .
We have:

d dγ FLPM n,τ = d dγ [ τ 0 (τ -u) α e γ u+K(u)+D(γ) du] = τ 0 (τ -u) n d dγ e γ u+K(u)+D(γ) du = τ 0 (τ -u) n (u + D (γ))e γ u+K(u)+D(α) du = τ 0 u(τ -u) n e γ u+K(u)+D(γ) du + τ 0 D (γ)(τ -u) n e γ u+K(u)+D(γ) du = τ 0 (u -τ + τ )(τ -u) n e γ u+K(u)+D(γ) du + D (γ)FLPM n,τ = - τ 0 (τ -u) n+1 e γ u+K(u)+D(γ) du + τ τ 0 (τ -u) n e γ u+K(u)+D(γ) du + D (γ)FLPM n,τ = -FLPM n+1,τ + (τ + D (γ))FLPM n,τ .
Hence the result.

(⇐) Now we prove the sufficient condition. By means of relation (??), relation ( 16) can be expressed as follows:

τ 0 (τ -u) α+1 φ γ (u) du = (τ +D (γ)) τ 0 (τ -u) n φ γ (u) du-d dγ τ 0 (τ -u) α+1 φ γ (u) du which implies that: τ 0 [(τ -u) n φ(u)(-u -D (γ)) + d dγ ((τ -u) n φ γ (u))] du = 0 ( 17
)
By using the fact that φ γ satisfies relation [START_REF] Liu | Uncertainty Theory: An introduction to its axiomatics foundations[END_REF], relation [START_REF] Ogryczak | On consistency of stochastic dominance and mean-semideviation models[END_REF] traduces the nullity of the integration of a positive function. Thus, we obtain:

d dγ ((τ -u) n φ γ (u)) = (τ -u) n φ γ (u)(u + D (γ)) (18)
Finally, by integrating each side of relation [START_REF] Qizilbash | Philosophical accounts of vagueness, fuzzy poverty measures and multidimensionality[END_REF] with respect to γ, we obtain:

(τ -u) n φ γ (u) = ke γu+D(γ) , k > 0, which leads to: φ γ (u) = e γu+D(γ)+K(u) with K(u) = ln( k ((τ -u) n ) ), u ∈]0; τ [. It suffices to consider the function φ γ defined as: φ γ (u) = e γu+D(γ)+K(u) with K(u) = ln |( k ((τ -u) n ) |, u ∈ (0; +∞) \ {τ } and K(τ ) = 0.
We end this Subsection with the following useful result establishing necessary and sufficient condition under which the FLPM of ξ is null. In addition, it gives expression of the expected loss of ξ (its FLPM for n = 1) by means of the distribution function of ξ.

Proposition 3. Let ξ be a fuzzy variable, Φ its credibility distribution function, n ∈ N * and τ ∈ R.

1. FLPM n,τ [ξ] = 0 ⇔ Φ(τ -) = 0. ( 19
)
2. If n = 1, then Φ(τ ) = FLPM n,τ [ξ]. ( 20
)
Proof : 1) (⇒) Assume that FLPM n,τ [ξ] = 0, then (8) implies ∀r ∈ R, r < τ =⇒ Φ(r) = 0, that means, Φ(τ -) = sup{Φ(r), r < τ } = 0. (⇐) If Φ(τ -) = 0, then the inequality Φ(r) ≥ 0 implies ∀r ∈ R, r < τ =⇒ Φ(r) = 0. According to the relation (8), the previous implication FLPM n,τ [ξ] = 0.
2) Assume that n = 1. According to the relation ( 8), we have:

FLPM n,τ [ξ] = n τ -∞ φ(u)du = nΦ(τ ) = Φ(τ ).
In the following Subsection, we introduce and study Mean Risk dominance of fuzzy variables and the consistency of mean-risk models.

Mean-risk dominance based on FLPM n,τ

Definition and properties

We now define the fuzzy mean-risk dominance relation based on FLPM. Definition 2. Let n ∈ N * and τ ∈ R. The fuzzy mean-risk dominance is the binary relation on the set of fuzzy variables denoted by n,τ and defined as follows: For two ξ 1 and ξ 2 be two fuzzy variables,

ξ 1 n,τ ξ 2 if E[ξ 1 ] ≥ E[ξ 2 ] FLPM n,τ [ξ 1 ] ≤ FLPM n,τ [ξ 2 ] . (21)
The strict dominance denoted by n,τ is defined with at least one strict inequality in [START_REF] Seidl | Poverty measurement: a survey, Welfare and Efficiency in Public Economics[END_REF].

The following result characterizes the new dominance relation n,τ in the three following cases: (1) the two fuzzy variables have disjoint supports and τ is less than the minimum of the lower bounds of the two supports, (2) the two fuzzy variables are symmetric and τ is between the lower bounds of the two supports and (3) one of the two fuzzy variables is a crisp number and the other one is a fuzzy variable with τ as its upper bound. Notice that the three results of this theorem can be interpreted as follows:

1. The first case means that, in absence of risk, the "best" fuzzy variable is the one with greater expected return.

2. According to the second case, when two distributions have equal means, it is more suitable to choose the less risky one.

3. The third case reveals that: if two distributions have the same expected return value which is below to the target, in the most case , the "best" distribution is the one which make "certain" to get this value.

We now state our result.

Theorem 1. Let ξ 1 and ξ 2 be two fuzzy variables with distribution functions Φ 1 and Φ 2 . Assume that (21) holds. Then:

1. If Φ 1 (τ -) = Φ 2 (τ -) = 0, then ξ 1 n,τ ξ 2 if and only if E[ξ 1 ] > E[ξ 2 ]. 2. If    E[ξ 1 ] = E[ξ 2 ] Φ 1 (τ -) = 0 Φ 2 (τ -) > 0 , then ξ 1 n,τ ξ 2 . 3. If E[ξ 1 ] = E[ξ 2 ] = τ -r (with r > 0)
, Φ 1 is a degenerate distribution that assigns credibility 1 to τ -r with r > 0, and Φ 2 is a non-degenerate distribution with Φ 2 (τ ) = 1, then:

ξ 1 n,τ ξ 2 if and only if n > 1.
To establish this proof, we recall the Jensens' Inequality for fuzzy variable introduced earlier by Liu [START_REF] Liu | Uncertainty Theory: An introduction to its axiomatics foundations[END_REF] (Theorem 1.59, page 68): "Let ξ be a fuzzy variable and f : R → R a strictly convex function. If E [ξ] and

E[f (ξ)] are finite, then f (E[ξ]) < E[f (ξ)]."
We now establish the proof of the Theorem.

Proof : 1) Let us assume that Φ 1 (τ -) = Φ 2 (τ -) = 0. By relation (19), we have FLPM n,τ [ξ 1 ] = FLPM n,τ [ξ 2 ] = 0. (⇒) Assume on the contrary that ξ 1 α,τ ξ 2 and E[ξ 1 ] ≤ E[ξ 2 ]
. This inequality and the equality imply that there is not any strict inequality between the means or the fuzzy lower partial moments of the fuzzy variables ξ 1 and ξ 2 . This contradicts ξ 1 n,τ ξ 2 . Therefore, we have:

E[ξ 1 ] > E[ξ 2 ]. (⇐) Assume that E[ξ 1 ] > E[ξ 2 ]. Thus, the equality FLPM n,τ [ξ 1 ] = FLPM n,τ [ξ 2 ] = 0 and the definition of n,τ imply ξ 1 n,τ ξ 2 . 2) Assume that E[ξ 1 ] = E[ξ 2 ], Φ 1 (τ -) = 0, Φ 2 (τ -) > 0.
That means FLPM n,τ [ξ 1 ] = 0 and FLPM n,τ [ξ 2 ] > 0, according to relation [START_REF] Ravallion | Poverty comparisons[END_REF].

3) Let us assume that Φ 1 is a degenerate distribution that assigns credibility 1 to τ -r with r > 0, and Φ 2 is a non-degenerate distribution that has Φ 2 (τ ) = 1 and

E[ξ 1 ] = E[ξ 2 ] = τ -r.
Let us set f (y) = (τ -y) n for y ≤ τ, and r > 0. According to the fact that Φ 1 is a degenerate distribution function that assigns credibility 1 to τ -r, we have

τ -∞ (τ -y) n dΦ 1 (y) = r n and f (E[ξ 1 ]) = r n .
f is strictly convex as n > 1 . By the Inequality of Jensens and the fact that

E[ξ 1 ] = E[ξ 2 ], we have: E[f (ξ 2 )] = τ -∞ (τ -y) n dΦ 2 (y) > f (E[ξ 1 ]) = r n . Finally, we have τ -∞ (τ -y) n dΦ 2 (y) > τ -∞ (τ -y) n dΦ 1 (y). Thus ξ 1 n,τ ξ 2 .
We can prove the converse case in the same way.

The following example compares two trapezoidal fuzzy variables by means of the mean-risk dominance for the target value τ = 1 2 and the order n = 2. Example 1. Let ξ 1 = (-1, -1 2 , 3 2 , 2) and ξ 2 = (-2, 0, 1, 3) be two trapezoidal fuzzy variables.

We have

E[ξ 1 ] = E[ξ 2 ] = 1 2
. By taking τ = 1 2 and n = 2. We have: FLPM n,τ [ξ 1 ] = 19 24 ≤ FLPM n,τ [ξ 2 ] = 31 24 . It follows that ξ 1 2, 1 2 ξ 2 .

Let us justify that n,τ is not a complete relation on the set of fuzzy variables.

Remark 2. Let ξ 1 = (1, 4, 5) and ξ 2 = (2, 3, 4) be two fuzzy variables, n = 2 and τ = 4. We have

E[ξ 1 ] = 7 2 , E[ξ 2 ] = 3, FLPM 2,4 [ξ 1 ] = 3 2 and FLPM 2,4 [ξ 2 ] = 4 3 . Thus, E[ξ 1 ] > E[ξ 2 ] and FLPM 2,4 [ξ 1 ] > FLPM 2,4 [ξ 2
]. Hence ξ 1 2,4 ξ 2 and ξ 2 2,4 ξ 1 . Thereby, n,τ is not a complete relation.

Let us end this Subsection by establishing some properties satisfied by the mean-risk dominance relation on the set of fuzzy variables. More precisely, the two first properties justify that n,τ is a pre-order on the set of fuzzy variables. The third property establishes indifference part of n,τ and the fourth property establishes sufficient condition on supports of two trapezoidal fuzzy variables in order to compare them. On the other hand, we have:

FLPM n,τ [ξ] = τ -∞ (τ -u) n-1 Φ 1 (u)du and FLPM α,τ [η] = τ -∞ (τ -u) n-1 Φ 2 (u)du.
By the fact that a > a , b > b , c > c and d > d , we have Φ 1 (r) ≤ Φ 2 (r), ∀r ∈ R, according to Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF]. That leads to FLPM n,τ [ξ] ≤ FLPM n,τ [η]. As E[ξ] ≥ E [η] and FLPM n,τ [ξ] ≤ FLPM n,τ [η], we conclude that ξ n,τ η.

The mean risk-dominance can generate a mean-risk model of the form maximize

E[ξ] -λFLPM n,τ [ξ] ( 22
)
in the sense of a trade-off analysis where λ > 0, is a trade-off coefficient. Following the work of Ogryczak and Ruszczynski [START_REF] Ogryczak | On consistency of stochastic dominance and mean-semideviation models[END_REF], in the following paragraph, we introduce the two notions of consistency and λ-consistency of a mean-risk model with the first or the second order dominance of fuzzy variables, introduced and studied by Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF], defined by:

• ξ 1 1 ξ 2 if ∀r ∈ R, Φ 1 (r) ≤ Φ 2 (r). • ξ 1 2 ξ 2 if ∀t ∈ R, t -∞ [Φ 2 (r) -Φ 1 (r)
]dr ≥ 0. We then establish some results on these two notions.

Consistency of mean-risk models with respect to first and second order dominances

Let us define those notions.

Definition 3. Let k, n ∈ {1, 2}, λ ∈ R * + , τ ∈ R, ξ 1
and ξ 2 be two fuzzy variables.

1. A mean-risk model is said to be consistent with a relation dominance of order k if:

ξ 1 k ξ 2 ⇒ ξ 1 n,τ ξ 2 .
2. A mean-risk model is said to be λ-consistent with a relation dominance of order k if:

ξ 1 k ξ 2 ⇒ E[ξ 1 ] ≥ E[ξ 2 ] E[ξ 1 ] -λFLPM n,τ [ξ 1 ] ≥ E[ξ 2 ] -λFLPM n,τ [ξ 2 ]
The following result establishes consistency of the mean-risk model with each order dominance. Proposition 5.

1. A mean-risk model is consistent with the first order dominance.

2.

A mean-risk model is consistent with the second order dominance if the downside risk is the expected loss.

Proof : Let k ∈ {1; 2}, n ∈ N * , τ ∈ R, ξ 1 and ξ 2 be two fuzzy variables with credibility distributions Φ 1 and Φ 2 such that ξ 1 k ξ 2 . Let us prove that ξ 1 n,τ ξ 2 . 1) For k = 1. a) Assume that ξ 1 1 ξ 2 and we prove that

ξ 1 α,τ ξ 2 . ξ 1 1 ξ 2 ⇒ ∀r ∈ R, Φ 1 (r) ≤ Φ 2 (r), that is, ∀r ∈ R, Cr{ξ 1 ≤ r} ≤ Cr{ξ 2 ≤ r} (23) and ∀r ∈ R, Cr{ξ 1 ≥ r} ≥ Cr{ξ 2 ≥ r} (24)
According to the definition of 1 . On the other hand, we have: 23) and ( 24), we conclude that

E[ξ i ] = +∞ 0 Cr{ξ i ≥ r} dr - 0 -∞ Cr{ξ i ≤ r} dr ∀i ∈ {1; 2} According to (
E[ξ 1 ] ≥ E[ξ 2 ].
In the same manner, we have:

FLPM n,τ [ξ i ] = α τ -∞ (τ -x) n-1 Cr{ξ i ≤ x}dx ∀i ∈ {1; 2} These last relations lead to FLPM n,τ [ξ 1 ] < FLPM n,τ [ξ 2 ]. Finally, we obtain ξ 1 n,τ ξ 2 . b) Since ∀r ∈ R, Φ 1 (r) ≤ Φ 2 (r) then ∀t ∈ R, t -∞ [Φ 2 (r) -Φ 1 (r)] dr ≥ 0. We easily obtain the proof. 2) For k = 2.
Let us assume that ξ 1 2 ξ 2 . The following equality

E[ξ i ] = +∞ 0 (1 -Φ i (r)) dr - 0 -∞ Φ i (r) dr, ∀i ∈ {1, 2}, leads to: E[ξ 1 ] -E[ξ 2 ] = +∞ -∞ [Φ 2 (r) -Φ 1 (r)] dr.
By using the characterization of 2 and by the fact that ξ 1 2 ξ 2 , we obtain

+∞ -∞ [Φ 2 (r) -Φ 1 (r)] dr ≥ 0, that means E[ξ 1 ] ≥ E[ξ 2 ].
On the other hand, by using relation ( 8), we get:

FLPM 1,τ [ξ 1 ] -FLPM 1,τ [ξ 2 ] = τ -∞ (τ -u) 1 (Φ 1 -Φ 2 )(u)
. By using the definition of 2 and by the fact that ξ 1 2 ξ 2 , we obtain

τ -∞ (Φ 2 -Φ 1 )(u) du ≥ 0, that is, FLPM 1,τ [ξ 1 ] ≤ FLPM 1,τ [ξ 2 ]. Finally, by the fact that E[ξ 1 ] ≥ E[ξ 2 ] and FLPM 1,τ [ξ 1 ] ≤ FLPM 1,τ [ξ 2 ], we conclude that ξ 1 1,τ ξ 2 .
The following result establishes the 1-consistency of the mean-risk model with each order dominance.

Corollary 2.

1. A mean-risk model is 1-consistent with the first order dominance.

A mean-risk model is 1-consistent with the second order dominance if

the downside risk is the expected loss.

Proof : The proof is easily deduced from Proposition 5 by taking λ = 1.

As a consequence of the previous results, the next result determines optimal solutions of mean-risk models as non dominated elements of order dominances. In this view, the determination of such optimal solutions becomes simple since Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF] characterized such non dominated fuzzy variables. Proposition 6.

1. Optimal solutions of model ( 22) are non dominated fuzzy variables with respect to first order dominance 1 .

2. Optimal solutions of model [START_REF] Sen | Poverty: an ordinal approach to measurement[END_REF] are non dominated fuzzy variables with respect to second order dominance 2 if the downside risk is the expected loss.

Proof : 1) Let be n ∈ N * , τ ∈ R. Let us consider an optimal solution ξ 0 of model [START_REF] Sen | Poverty: an ordinal approach to measurement[END_REF] which is not efficient with respect to first dominance 1 , that is, there exist a fuzzy variable η 0 such that η 0 1 ξ 0 . According to Proposition 5, we have:

η 0 1 ξ 0 ⇒ η 0 n,τ ξ 0 (25)
By ( 21), we have:

(25)⇒ E[η 0 ] ≥ E[ξ 0 ] FLPM n,τ [η 0 ] ≤ FLPM n,τ [ξ 0 ]
with at least one strict inequality, which leads to

E[η 0 ]-λFLPM n,τ [η 0 ] > E[ξ 0 ]-λFLPM n,τ [ξ 0 ], ∀λ > 0.
Therefore, this last inequality contradicts the fact that ξ 0 is an optimal solution of model [START_REF] Sen | Poverty: an ordinal approach to measurement[END_REF].

2) The proof is the same for optimal solutions with respect to second order dominance 2 and n = 1.

By considering model ( 22) as a general form of models taking into consideration investors preferences through targets values1 , Proposition 6 justifies that optimal portfolios obtained through optimization models defined in Sadefo et al. [START_REF] Sadefo | Moments and semimoments for fuzzy portfolio selection[END_REF] belong to the sets of best portfolios respectively with first and order dominance, defined in Tassak et al. [START_REF] Tassak | Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns[END_REF].

In the last Section of this paper, we present an application of a FLPM as a tool for poverty measurement in fuzzy context. The obtained poverty index is analyzed.

Application of FLPM for poverty measurement

In many situations, an individual's income can be given without any precision or can not completely be known, it is reasonable to describe that income by a fuzzy variable ξ. For example, in many african countries, people carry out one or several economic's activities to improve their welfare. In other words, besides their professional permanent job, most individuals practise other ones to face to increasing real life difficulties or expensive level of life. We assume that permanent jobs generate incomes which are currently or officially known and can vary during a time period and we describe such type of income by an equipossible fuzzy variable ξ = (a, b) (on left of Figure 1). Meanwhile, secondary activities namely are not generally taken into account From formula [START_REF] Hagenaars | A class of poverty indices[END_REF], we have the following FLPMs:

• For ξ = (a, b, c), becomes F LP M n,z [ξ] =        0 if z ≤ a 1 2 (z -a) n if a ≤ z ≤ b 1 2 (z -a) n + 1 2(n+1)(c-b) (z -b) n+1 if b ≤ z ≤ c 1 2 (z -a) n + 1 2(n+1)(c-b) [(z -b) n+1 -(z -c) n+1 ] if z ≥ c . (26)
• For ξ = (a, b),

F LP M n,z [ξ] =    0 if z ≤ a 1 2 (z -a) n if a ≤ z ≤ b 1 2 [(z -a) n + (z -b) n ] if z ≥ b . (27)
In the following, we will use FLPM to evaluate poverty in such population where active population have income described by previous trapezoidal fuzzy variable. More specifically, the expected loss (expected gap income) with respect to a threshold z is given by F LP M 1,z [ξ] = E[max(z -ξ, 0)]. Thereby, we introduce the following notions of poverty on individuals. Definition 4. Let i and j be two individuals with incomes respectively described by fuzzy numbers ξ i = (a i , b i , c i ) and ξ j = (a j , b j , c j ) and z a nonnegative real number.

1. i is said to be poor with respect to the poverty line

z if E[ξ i ] < z. 2. i is said to be almost more poor than j if E[ξ i ] ≤ E[ξ j ].
3. i is said to be more poor than j if a i ≤ a j , b i ≤ b j and c i ≤ c j .

We now introduce our poverty index based on the expected loss. Thus, we assume that an individual is symbolized by a positive integer.

Definition 5. Let P be a population of N individuals, z ∈ (0, ∞), Q z = {i ∈ P, E[ξ i ] < z} the set of poor individuals with respect to the poverty line z, q z = |Q z | the number of poor individuals with respect to the poverty line z. The poverty index I with respect to the poverty line z is given by:

I(z) = 1 N qz k=1 F LP M k 1,z z . Remark 3. 1.
If the individual i is more poor than the individual j, then i is almost more poor than j.

The generic term of the index

F LP M i 1,z z
is the expected income gap ratio of individual i with respect to the poverty line z and F LP M i 1,z is his expected gap income.

Let us establish the variation of the index with respect to the poverty line.

Lemma 1. The expected income gap ratio is nondecreasing with respect to the poverty line z.

Proof: According to (26), we have:

( F LP M 1,z [ξ] z ) =        0 if z ≤ a a 2z 2 if a ≤ z ≤ b a 2z 2 + 1 4(c-b)z 2 (z -b)(z + b) if b ≤ z ≤ c a 2z 2 + 1 4z 2 (c + b) if z ≥ c .
We conclude that F LP M 1,z [ξ] is nondecreasing with respect to z by the fact that its derivative is a nonnegative function.

In the following, we examine if the new index satisfies some intuitive properties among which two meaningful and well-known axioms of a poverty index proposed by Sen [START_REF] Sen | Poverty: an ordinal approach to measurement[END_REF] and defined by:

• Poverty Monotonicity (PM):
For all x, y ∈ [0, ∞) n and z ∈ (0, ∞), if Q x,z = Q y,z and x = y except for x i > y i with i ∈ P, then I(x, z) < I(y, z).

• Transfer Sensitivity (TM): For all x, y ∈ [0, ∞) n and z ∈ (0, ∞), if y is obtained from x by a progressive transfer among the poor, then I(y, z) < I(x, z).

We start with two first properties.

Proposition 7.

1. The poverty index is nondecreasing with respect to the poverty line z.

The poverty index satisfies PM.

Proof: 1) Let z 1 and z 2 be two poverty lines such that z 1 ≤ z 2 , ξ a fuzzy variable that describes an unknown income with its distribution function Φ. As q z 1 ≤ q z 2 , we have:

I(z 2 ) -I(z 1 ) = 1 N ( qz 1 k=1 F LP M k 1,z 2 z 2 - F LP M k 1,z 1 z 1 ) + 1 N qz 2 k=qz 1 F LP M k 1,z 2 z 2
.

According to Lemma 1, we have:

F LP M k 1,z 2 z 2 ≥ F LP M k 1,z 1 z 1
, therefore I(z 2 ) ≥ I(z 1 ).

2) Let us consider an individual j whose income is described by ξ j = (a j , b j , c j ), z a poverty line and θ a nonnegative real number that generates the new income ξ j = (a j + θ, b j + θ, c j + θ). Without loss of generality, let us assume that i is the only one in the population P who has an improved income. Let us prove that the new poverty index I (z) is less than I(z). We distinguish two cases: (i) If E[ξ j ] ≥ z, then j is not poor any longer with respect to z and we have: ). We only need to prove that F LP M j 1,z ≤ F LP M j 1,z . Without loss of generality, let us assume that a j ≤ z ≤ b j . We have two cases: (ii1) If

I (z) = 1 N qz-1 k=1 F LP M k
a j + θ ≤ z ≤ b j + θ, then F LP M j 1,z = 1 2 (z -a j -θ) ≤ 1 2 (z -a j ). Thus F LP M j 1,z ≤ F LP M j 1,z . (ii2) If z < a j + θ, then F LP M j 1,z = 0 and F LP M j 1,z ≤ F LP M j
1,z . In all cases, we get: I (z) ≤ I(z).

Most poverty index on random variables that exist in the literature violate the Transfer Sensitivity axiom. It is easy to justify that our new index also violates it. Thereby, we characterize the set of poverty lines on which this axiom is satisfied. For that, we propose a new version of TS axiom, namely the Restricted Transfer Sensitivity (RTS) axiom, that is satisfied by the index. Let us define the Transfer Sensitivity (RTS): ∃z 0 > 0, for all x, y ∈ [0, ∞) n and z ∈ (0, z 0 ], if y is obtained from x by a progressive transfer among the poor, then I(y, z) < I(x, z). Thus we have the following result. Proof: Let be N individuals (N ∈ N) whose incomes are respectively described by semi-trapezoidal fuzzy variables ξ k = (a k , b k , c k ) and z a real number such that z ∈ (0, min 1≤k≤N b k ]. Without loss of generality, let us assume that i and j are the only two individuals whose incomes are transferred such that i is more poor than j before the transfer (a j ≥ a i , b j ≥ b i ) and the two individuals remain poor after the transfer with respect to the poverty line z. Let us consider a nonnegative real number θ such that θ ≤ min(a j -a i , b j -b i )(that means, j remains less poor than i after the transfer) and the new incomes after the transfer are described by ξ i = (a i + θ, b i + θ) and ξ j = (a j -θ, b j -θ). The new poverty index is:

I (z) = 1 N ( k∈Qz\{i,j} F LP M k 1,z z + F LP M i 1,z z + F LP M j 1,z z
). We have to prove that

I (z) ≤ I(z). It suffices to prove that F LP M i 1,z + F LP M j 1,z ≤ F LP M i 1,z + F LP M j 1,z . Furthermore, without loss of generality, we assume that z ∈ [a j , b i ] = [a j ∨ a i , b j ∧ b i ]. Let us set: T = F LP M i 1,z + F LP M j 1,z -F LP M i 1,z -F LP M j 1,z . We have: T = 1 2 (2θ(z -a j ) + θ 2 -2θ(z -a i ) + θ 2 ) = θ(a i -a j + θ). As θ ≤ min(a j -a i , b j -b i ), we have T ≤ 0.
Finally, we obtain: I (z) ≤ I(z).

Corollary 3. The poverty index satisfies the Transfer Sensitivity axiom between two individuals such that one is poor than another if their incomes are described by equipossible fuzzy variables.

Interpretation 1. The previous result seems more natural to transfer incomes between two poor individuals in order to reduce poverty when their incomes are completely known or declared in a time period.

Let us end this Section by examining what can be the critical poverty line.

The concept of poverty is a relative and subjective's concept, in fact: -a person who gets a car is rich if he belongs to a population where most of individuals move on foot or by bicycle.

-a person who gets a car is not necessary rich if he belongs to a population where most of individuals get a car. Thus, it seems natural that people do not agree on the poverty's status of an individual. So, it is more suitable and interesting to classify individuals with respect to a critical poverty level rather than looking for a target value which can help to check whether an individual is really poor or rich. As it seems more natural, the poverty line that induces a minimum poverty level is the one that could guarantee welfare to a large number of individuals in the population. Therefore, the critical poverty line is obtained by minimizing the poverty index I(z) and we propose the following model:    minimize I(z) w.r.t z ∈ (0, +∞) [START_REF] Zheng | Poverty orderings[END_REF] Following that idea, individuals whose incomes are less than the critical poverty line can really be considered as poor individuals in the population.

In the following, we propose a numerical example where model ( 28) is applied to determine the critical poverty line with ten given fuzzy incomes.

Example 2. Let us consider for a time period of four years the following incomes2 from ten employees in Cameroon provided by the ministry of Finance and described by semi-trapezoidal fuzzy variables By using Matlab with "fmincon" command at the initial point z 0 = 306, the implementation of model [START_REF] Zheng | Poverty orderings[END_REF] gives the following results: the critical poverty line is obtained at z * = 296.8 and the corresponding poverty index is I(z * ) = 0.16.

On the other hand, one can evaluate different expected incomes in order to identify those which are below the critical poverty line. We observe that except employees 4 and 5, whose expected incomes are respectively 339.7 and 373.5 (beyond the critical poverty line), the other employees can be considered as poor in this group. It seems that "secondary activities" influence weakly on an employee's status: it seems evident because those activities don't generate regular or certain incomes.

Concluding remarks

In this paper, we introduce Fuzzy Lower Partial Moment (FLPM) of a fuzzy variable which evaluates downside risk of a fuzzy risk and generalizes semimoment of a fuzzy variable. It preserves, on fuzzy variables, some well-known properties of Lower Partial Moment for ramdom variables. Moreover, we supplemented the two (first and second) order dominances on fuzzy variables by defining and studying a third dominance relation for binary comparisons of fuzzy variables through their benefits and risks by means of the FLPM. We establish conditions under which the set of best portfolios of mean-risk model, an optimization program to determine optimal portfolios by means of the new dominance, belong to the set of best portfolios with respect to the first or second order dominance. Those obtained properties stipulate consitency and λ-consistency (λ is the trade off coefficient between expected return and FLPM of a given portfolio defining the mean-risk model) of the new dominance with the two known dominances. By considering individuals' incomes as fuzzy variables, we define a new poverty index based on FLPM and studied its properties. More precisely, we established that the new index satisfies the poverty monotonicity and violates Transfered sensitivity. We then deduce the set of individuals' incomes under which the index satisfies the given property. We propose a model to generate a critical poverty line useful to identify poor and non poor individuals. We implement a proposed model for getting the critical poverty line in a numerical example based on a typical sample of ten employees in Cameroon.

Proposition 4 .

 4 Let n ∈ N * and τ ∈ R, ξ, η, ζ three given fuzzy variables. The fuzzy mean-risk dominance n,τ satisfies the following properties:

4 and

 4 E[η] = a +b +c +d 4 , we have E[ξ] ≥ E[θ].

Figure 1 :

 1 Figure 1: Two fuzzy variables describing fuzzy incomes

1 N

 1 we get I (z) ≤ I(z). (ii) If E[ξ j ] < z, then j remains poor with respect to z and we have I (z) =

Proposition 8 .

 8 Restricted Transfer Sensitivity axiom of the new index Let us consider N individuals (N ∈ N) whose incomes are respectively de-scribed by semi-trapezoidal fuzzy variables ξ k = (a k , b k , c k ), k ∈ {1, ..., N } and z a poverty line. The poverty index I(z) satisfies the RTS axiom between two given individuals such that one is poor than another if z 0 = min 1≤k≤N b k ].

  Let ξ be a fuzzy variable, τ ∈ R and n ∈ N * . The Fuzzy Lower Partial Moment (FLPM) of ξ with order n and target value τ is defined as

	3 Fuzzy Lower Partial Moments and Mean-
	risk dominance of fuzzy variables
	3.1 Definitions, specific cases and examples
	Definition 1.

In model[START_REF] Sen | Poverty: an ordinal approach to measurement[END_REF], the trade-off coefficient λ can illustrate in a particular case, an investor's preference level, of mean to risk or inversely.

Those incomes are expressed in thousand of francs CFA

The source of different "official incomes" is: "MINFI-Direction de la solde"and "unknown incomes" are obtained through further investigations.