Zhong Liu
email: liuzhong@sia.cn

Didier Theilliol
email: didier.theilliol@univ-lorraine.fr

Liying Yang
email: yangliying@sia.cn

Yuqing He
email: heyuqing@sia.cn

Jianda Han
email: jdhan@sia.cn

Mode Transition and Fault Tolerant Control under Rotor-tilt Axle Stuck Fault of Quad-TRUAV

Keywords: Tilt rotor unmanned aerial vehicle, transition procedure, nonlinear control, backstepping, stuck fault, fault tolerant control

Tilt rotor unmanned aerial vehicle (TRUAV) is a transformable aircraft with abilities of hovering and high-speed cruise. Its transition control is still a difficult point due to varying structure and dynamic behaviour. The main contribution of this paper is to put forward a transition control method based on backstepping for a quad-TRUAV, and fault tolerant control is researched to keep its cruise ability under rotor-tilt axle stuck fault. Explicit mode transition would be avoided by regarding rotor-tilt angle as a control input generated from decoupling with virtual control variables. Numerical results in transition procedure and facing rotor-tilt axle faults are provided to illustrate control effectiveness.

INTRODUCTION

To break limitations caused by typical structures of conventional unmanned aerial vehicles (UAVs), plentiful hybrid aircraft are researched for wide applications. These aircraft own abilities of hovering and high-speed cruise, so are always accompanied with better flexibility and more endurance than fixed-wing UAVs (FWUAVs) and rotorcraft UAVs (RUAVs), see [START_REF] Saeed | A review on the platform design, dynamic modeling and control of hybrid UAVs[END_REF]. This paper focuses on tilt rotor UAV (TRUAV) that relies on wings and rotors for generating lift together, and is one of hybrid aircraft, see Liu et al. (2017a).

With fixed rotor-tilt angle at π/2 or 0, the control of TRUAV is similar with conventional RUAV or FWUAV, and regarded in helicopter mode or airplane mode. However, the rotor-tilt angle is not invariable in the transition procedure for acceleration or deceleration. In this procedure, TRUAV is with varying structure and dynamics, which would lead to complex coupling of control effects from rotors and aerodynamic parts. To deal with this problem, gain scheduling (GS) methods are always applied especially in practice, see [START_REF] Muraoka | Transition flight of quad tilt wing VTOL UAV. 28th congress of the international council of the aeronautical sciences (ICAS)[END_REF], [START_REF] Zhao | Robust transition control of a Martian coaxial tiltrotor aerobot[END_REF], and [START_REF] Liu | Transition control of tilt rotor unmanned aerial vehicle based on multi-model adaptive method[END_REF]. In these methods, rotor-tilt angle plays the role of a flag value representing TRUAV structure, and a set of linear controllers in different angles can be designed for interpolation [START_REF] Muraoka | Transition flight of quad tilt wing VTOL UAV. 28th congress of the international council of the aeronautical sciences (ICAS)[END_REF] and [START_REF] Liu | Transition control of tilt rotor unmanned aerial vehicle based on multi-model adaptive method[END_REF]) or direct switch [START_REF] Zhao | Robust transition control of a Martian coaxial tiltrotor aerobot[END_REF]) with explicit mode transition. To connect the variation of this flag value with varying dynamics, a special flight envelope called tilt corridor has to be considered in the transition procedure, see [START_REF] Liu | Transition control of tilt rotor unmanned aerial vehicle based on multi-model adaptive method[END_REF]. With this idea, the stability of final flight status in the helicopter mode or airplane mode is focused, but the stability in the transition procedure has not be proved in theory under GS idea.

According to typical tilt corridor in Liu et al. (2017a), different rotor-tilt angles limit different maximum and minimum flight velocities of TRUAV. That means rotor tilt and velocity variation are affected close with each other. As for actual platform, rotor-tilt axle is usually driven by servo through link mechanisms or chains. If it is stuck at one fixed angle, the transition procedure would be broken off, which might be fatal. Fault tolerant control (FTC) methods for aerial vehicles and other systems have been researched for a long time, and many active and passive FTC strategies have been presented, see [START_REF] Qi | A review on fault diagnosis and fault tolerant control methods for single-rotor aerial vehicles[END_REF], [START_REF] Jiang | Fault-tolerant control systems: A comparative study between active and passive approaches[END_REF][START_REF] Zhang | Bibliographical review on reconfigurable fault-tolerant control systems[END_REF]. However, this special stuck fault for TRUAV has not been considered by any existing reference. The main contribution of this paper focuses on the design of a hierarchical transition control method for the quad-TRUAV, and a fault tolerant strategy for rotor-tilt axle stuck fault is researched further. In this method, the rotortilt angle is regarded as a control input with similar effect as angle of attack. To deal with coupled control inputs and to decouple original dynamics, virtual control variables are considered by backstepping, and decoupling module is applied for control inputs. After rotor-tilt axle stuck fault, a degraded dynamic model is considered, and the decoupling module is also redesigned.

Reminding parts of this paper are organized as follows. Section 2 introduces the longitudinal modeling of quad-TRUAV. Section 3 introduces details of transition controller, including nonlinear controller for virtual control variables and decoupling module for control inputs. Section 4 introduces actuator stuck fault and fault tolerant strategy based on degraded model. Section 5 presents some numerical results to show the control effectiveness. Section 6 ends the whole paper with conclusion.

QUAD-TRUAV MODELING

Quad-TRUAV is equipped with two pairs of tiltable rotors that tilt together as shown in Fig. 1. With this structure, the impact caused by the rotor downwash could be alleviated effectively, and would be ignored in TRUAV modeling. The definitions of wind-axis coordinate system (O w x w y w z w ), body-axis coordinate system (O b x b y b z b ), and north-east-down(NED) coordinate system (O e x e y e z e ) are also shown in Fig. 1 according to [START_REF] Cai | Unmanned rotorcraft system[END_REF][START_REF] Stengel | Flight dynamics[END_REF]. Based on these coordinate systems, some states, including flight velocity V , flight height h, angle of attack α, pitch angle θ, and pitch rate q, are defined.

In transition procedure, only longitudinal model is considered. Similar with FWUAVs, dynamics and kinematics equations of TRUAV in wind-axis coordinate system can be formulated as follows according to [START_REF] Stengel | Flight dynamics[END_REF]:

V = F x m , (1) ḣ
= V sin(θ -α), (2) α
= F z mV + q, (3) q
= M y I y , (4)
θ = q, (5) where m is mass, I y and M y are rotational inertia and resultant moment in y-direction of body-axis coordinate system, F x and F z are resultant forces in x-and z-direction of wind-axis coordinate system. Further considering these forces and moment, F * is the addition of three components from rotors (F * r ), aerodynamic parts (F * a ), and gravity (F * g ), where * represents x or z for different directions, and resultant moment is the addition of two components from rotors (M yr ) and aerodynamic parts (M ya ).

Rotors are driven by brushless direct current motors. Generated forces are proportional to the square of rotor speeds, see [START_REF] Pounds | Modelling and control of a large quadrotor robot[END_REF]. Further transform them into wind-axis coordinate system, following component forces and moment are obtained:

F xr = 2C t ρAR 2 cos(α + i n )(Ω 2 f + Ω 2 b ), (6) F zr = -2C t ρAR 2 sin(α + i n )(Ω 2 f + Ω 2 b ), (7) M yr = 2C t ρAR 2 x r sini n (Ω 2 f -Ω 2 b ), ( 8
) where i n represents rotor-tilt angle, C t is tension coefficient, ρ is air density, A is rotor area, R is rotor radius, Ω f and Ω b are forward and backward rotor speeds, and x r is longitudinal distance from center of gravity to rotors.

As for aerodynamic forces and moment, all aerodynamic parts can be transformed into the same size by scaling aerodynamic coefficients. According to [START_REF] Zhao | Robust transition control of a Martian coaxial tiltrotor aerobot[END_REF], following aerodynamic forces and moment are obtained: 11) where S and c are the equivalent wing area and mean chord length, C D , C L , and C M are aerodynamic coefficients, and subscripts D, L, and M mean drag, lift, and pitch moment. In these coefficients, some items are disregarded for simplifications of nonlinear modeling and controller design. δ e is deflection of elevator.

F xa = 1/2ρV 2 S • C D = 1/2ρV 2 S • C D0 , (9)
F za = 1/2ρV 2 S • C L = 1/2ρV 2 S • (C L0 + C Lα α), (10) M ya = 1/2ρV 2 Sc • C M = 1/2ρV 2 Sc • (C M0 + C Mδ δ e ), (
Considering gravity further, following component forces are formed:

F xg = -mgsin(θ -α), (12)
F zg = mgcos(θ -α),
(13) where g is gravitational acceleration.

Introducing forces and moments (6)∼( 13) into (1)∼( 5), mathematical model of quad-TRUAV can be formulated as a typical nonlinear system as follows:

ẋ(t) = f (x(t), u o (t)), (14) y(t) = h(x(t)) Ix(t), (15) where x(t) = [V h γ α q] T is state vector, γ = θ -α is defined as track angle, u o (t) = [i n Ω f Ω b δ e ]
T is control input vector, y(t) is output vector, and I is an unit matrix to assume all states are measurable.

Note that, regarding i n as a control input in this paper, explicit mode transition in conventional GS method could be avoided. And this value is no necessarily limited in [0, π/2] for necessary control effect. More details of TRUAV modeling and parameters could refer to [START_REF] Pounds | Modelling and control of a large quadrotor robot[END_REF] and [START_REF] Zhao | Robust transition control of a Martian coaxial tiltrotor aerobot[END_REF].

TRACKING CONTROL SYNTHESIS

Based on a dynamic decoupling approach, the structure of tracking controller for transition procedure would be formulated firstly in this section. According to this structure, details of nonlinear controller and decoupling design would be introduced mainly.

Decoupling-based Controller Formulation

With generalized representations in ( 14) and ( 15), every state in x(t) can be regarded as a degree-of-freedom (DoF). Because of under-actuated characteristics and varying structure, some DoFs of TRUAV are coupled and with complex coupling of control inputs. It is obviously shown in ( 6)∼( 8) and ( 11) with i n as one control input. To avoid considering this coupling in controller design directly, virtual control variables are defined as follows:

v 1 (t) F V -gsinγ, v 2 (t) = v 21 (t) v 22 (t) F α -ρS 2m C Lα V α M q ,
where intermediate variables 16)

F V = 2ρAR 2 m C t cos(α + i n )(Ω 2 f + Ω 2 b ), (
F α = - 2ρAR 2 mV C t sin(α + i n )(Ω 2 f + Ω 2 b ), (17)
M q = 2ρAR 2 x r I y C t sini n (Ω 2 f -Ω 2 b ) + ρS c 2I y • V 2 C Mδ δ e
(18) include all coupled control inputs.

With above definitions, nominal quad-TRUAV model can be reformed with a simple form as follows:

V = - ρS 2m C D0 V 2 + v 1 (t), ḣ = V sinγ, γ = ρS 2m C L0 V - gcosγ V -v 21 (t), α = - ρS 2m C L0 V + gcosγ V + v 21 (t) + q, q = ρSc 2I y C M0 V 2 + v 22 (t).
For further controller design, above equations are formulated generally follows:

ẋ1 (t) = f 1 (x 1 (t), v 1 (t)), (19) ẋ2 (t) = f 2 (x 1 (t), x 2 (t), v 2 (t)), (20) where x(t) is divided into 2 parts: x 1 (t) = V and x 2 (t) = [h γ α q] T , output vectors y 1 (t) = x 1 (t) and y 2 (t) = x 2 (t).
Note that, coupled items about states and control inputs are considered in virtual control variables. ( 19) and (20) decouple original quad-TRUAV model into several subsystems corresponding to x 1 (1) and x 2 (t), in which subsystems corresponding to [h γ] T and [α q] T are also relatively independent. In this way, nonlinear controllers could be designed for virtual control variables, and control coupling does not need to be considered in nonlinear controllers design. Then, with the forms of intermediate variables ( 16)∼(18), decoupling module for control inputs can be designed. The structure of transition controller is summarized in Fig. 2, where subscript re represents reference value. The purpose of this controller is to ensure stability and reference tracking in transition procedure lim

t→+∞ (C re y(t) -r(t)) = 0,
where r(t) is reference vector [V re h re α re ] T , and matrix

C re =
1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 directly because of the unit output matrix in (15). 19) and ( 20) with virtual control variables v 1 (t) and v 2 (t). So many nonlinear feedback control methods could be applied here for stability and reference tracking.

Considering ( 19) with unit output matrix in (15), it is a single-input-single-output system. Feedback linearization can be applied directly to formulate following closed-loop system [START_REF] Khalil | Nonlinear systems[END_REF]):

ẋ1 (t) = -k 1 (x 1 (t) -x 1re (t)) + ẋ1re (t), (21)
where k 1 is a positive constant, and x 1re (t) is the reference value. To track reference velocity V re in TRUAV transition procedure, following virtual control variable is obtained:

v 1 (t) = -k V (V -V re ) + Vre + ρS 2m C D0 V 2 , (22)
where k V is a positive parameter. Intermediate variable F V in v 1 (t) could also be obtained for next subsection.

Further considering (20), this subsystem can be reformulated as follows: ẋ21 (t) = f 21 (x 1 (t), ẋ22 (t)) + g 21 (x 22 (t)), (23) ẋ22 (t) = f 22 (x 1 (t), x 22 (t)) + g 22 (v 2 (t)), (24) where x 21 (t) = [h α] T and x 22 (t) = [γ q] T . Considering ẋ22 (t) in ( 23) due to α = -γ+q, subsystems corresponding to [h γ] T and [α q] T could be analyzed separately as shown in Fig. 2. ( 23) and ( 24) are with strict feedback form, and this form is caused by inherently under-actuated characteristics of UAV. In this way, a hierarchical controller from bacskstepping would be a good choice, see [START_REF] Khalil | Nonlinear systems[END_REF] and [START_REF] Loza | Output tracking of systems subjected to perturbations and a class of actuator faults based on HOSM observation and identification[END_REF]. Concretely speaking, for (23), assume there is a Lypunov function V 21 (x 21 (t)) satisfies stability conditions with x 22 (t) = x 22re (t), which means + (x 22 (t) -x 22re (t)) T ( ẋ22 (t) -ẋ22re (t)) < 0, the stability of x 2 (t) can be obtained eventually.

V 21 (x 21 (t)) > 0,
With above idea, for tracking reference flight height h re and reference angle of attack α re , following virtual control variables are obtained:

v 21 (t) =k γ (γ -γ re ) -γre + ρS 2m C L0 V - gcosγ V + (h -h re )V sinγ -sinγ re γ -γ re , (25)
v 22 (t) = -k q (q -q re ) + qre - ρSc 2I y C M0 V 2 -(α -α re ), ( 26
)
where

γ re = -sign(V (h -h re )) |h -h re | H π, (27)
q re = -k α (α -α re ) + αre -k γ (γ -γ re ) + γre -(h -h re )V sinγ -sinγ re γ -γ re , (28)
H, k γ , k α , and k q are all positive parameters, and sign(•) represents sign calculation. Intermediate variables F α and M q in v 2 (t) could also be obtained for decoupling design.

The detailed stability analysis can be seen in Appendix A.

Decoupling Design for Control Inputs

There is no doubt that coupled items in original model would not "decrease" or "increase". Now that original model is decoupled by virtual control variables, that means serious coupling is included in v 1 (t) and v 2 (t). In controller design based on decoupled dynamics, a decoupling module with inputs as virtual control variables and outputs as control inputs should be designed, see Nordfeldt et al. (2006). However, decoupling designs are absolutely different facing different controlled plants, and dependent on actuator dynamics that could be represented as following generalized form [START_REF] Ahmed | Flight control of a small helicopter in unknown wind conditions[END_REF]):

v 1 (t) v 2 (t) = m 1 (x 1 (t), x 2 (t), u o (t)) m 2 (x 1 (t), x 2 (t), u o (t)) . ( 29
)
With above equation, decoupling modules are inverse functions of m 1 (•) and m 2 (•) for unknown vector u o (t).

Note that, actuator fault has not been introduced here, so outputs of decoupling module are control inputs u o (t) exactly, as shown in (29). However, in faulty case, outputs should be rewritten as inputs of actuator module, as shown in Fig. 2. Decoupling design for quad-TRUAV is not more than a algebraic problem. Algebraic forms of intermediate variables ( 16)∼( 18) would be analyzed mainly for control inputs, because they include all coupled control inputs.

According to ( 16) and ( 17), α + i n = arctan(-V Fα F V ). In transition procedure, i n could compensate the control effect of angle of attack as follows:

i n = arctan(- V F α F V ) -α. ( 30
)
The reference value of α can be set according to target value of rotor tilt angle as follows:

α re =    α τ , i n > 0 1 T α s + 1 arctan(- V F α F V ), i n ≤ 0 , (31)
where α τ is a constant reference, and T α is a time constant of the low-pass filter, which is used to remove underlying algebraic loop in case that α re depends on α directly. For normal transition procedure, α τ can be set according to required acceleration. Obviously, with ( 30) and ( 31), quad-TRUAV can be stabilized at rotor-tilt angle 0, and i n takes the whole control effect in transition procedure until it reaches given target value 0. With application of low-pass filter in (31), i n would less than 0 momentarily to compensate control effect in the end of transition procedure because of the delayed response of α to arctan(-V Fα F V ). For other control inputs, Ω 2 f + Ω 2 b are included in ( 16) and ( 17). So

Ω 2 f + Ω 2 b = m F 2 V + (V F α ) 2 2ρAR 2 C t . ( 32
)
As shown in ( 18), rotors and elevator are all included in M q , and could take control effect concurrently in transition procedure. To deal with this redundancy, define rotor control weight η firstly. With the form of M q and η ∈ [0, 1],

Ω 2 f -Ω 2 b = ηM q 2ρAR 2 xr Iy C t sini n , ( 33
)
δ e = (1 -η)M q ρSc 2Iy • V 2 C Mδ . ( 34
)
η should represent the weight of control effect from rotors according to TRUAV structure or flight velocity. Similar weight value is also applied in GS method, but there is not a theoretical form of this value. For decoupling here,

η = 1 - V 2 V 2 c
, where V c is the regular cruise speed in airplane mode.

In general, control inputs i n , Ω f , Ω b , and δ e are obtained by ( 30) and ( 32)∼(34). Note that, if

Ω 2 f + Ω 2 b < Ω 2 f -Ω 2 b , that would cause Ω 2 b < 0, which is unpractical. To ensure Ω 2
f ≥ 0 and Ω 2 b ≥ 0, reasonable planning for V re and h re is necessary, and low-pass filter in (31) is also helpful.

FAULT TOLERANT DESIGN

Actuator Stuck Fault

According to [START_REF] Qi | Self-healing control against cctuator stuck failures under constraints: application to unmanned helicopters[END_REF], actuator stuck fault can be defined as follows:

u o (t) = Φ(t)u i (t) + (I -Φ(t))u F . Above equation corresponds to actuator module in Fig. 2, where u o (t) represents control inputs of controlled plant in ( 14) and actuator outputs, u i (t) is actuator inputs, and u F is a constant vector representing the magnitude of stuck failure. Φ(t) = diag(φ 1 (t), ..., φ i (t), ..., φ m (t)), φ i (t) = 1 means ith actuator is fault-free, and φ i (t) = 0 represents ith actuator is faulty and lock-in-place.

For following FTC strategy, assume fault information can be estimated accurately and timely by fault detection and diagnosis (FDD) module. Because there are lots of mature methods can be applied here, see [START_REF] Blanke | Diagnosis and fault-tolerant control[END_REF] and [START_REF] Qi | KF-based adaptive UKF algorithm and its application for rotorcraft UAV actuator failure estimation[END_REF], FDD for matrix Φ and vector u F of actuator fault is not considered further in this paper.

Degraded Model Approach

Different with TRUAV control at any rotor-tilt angle, the rotor-tilt axle could not respond to actuator input after stuck fault, and faulty model can be formulated as ( 19) and ( 20) with v 1 (t) and v 2 (t) replaced by

v 1 F (t) m 1 (x 1 (t), x 2 (t), u o (t)| in=in F ), and v 2 F (t) m 2 (x 1 (t), x 2 (t), u o (t)| in=in F ),
where subscript F means equation with fault, and i n F represents faulty rotor-tilt angle. In this case, a control input is lost. A degraded model can be used to ensure partial control performance, see [START_REF] Souanef | Fault tolerant L 1 adaptive control based on degraded models[END_REF].

To obtain this degraded model, dynamics of controlled plant should be analyzed: similar with fixed-wing aircraft, TRUAV would not be accompanied with diverging states with stable flight height and angle of attack [START_REF] Stengel | Flight dynamics[END_REF]), so angle of attack control and flight height control are much more important for safe flight. That means only ẋ2

(t) = f 2 (x 1 (t), x 2 (t), v 2 F (t)) (35
) needs to be kept as degraded model, and x 1 (t) is in a open loop as follows:

ẋ1 (t) = f 1 (x 1 (t), v 1 F (t)
) compared with closed-loop form in fault free case (21).

Assume i n F can be measured or estimated by FDD module after a small delay. With above idea, discarding tracking control of flight velocity, virtual control variable v 2 F (t) could still be obtained from ( 25)∼( 28) to track references of flight height and angle of attack, and re-decoupled control inputs are constructed as follows and (34):

α re = α re F , Ω 2 f + Ω 2 b = F α -2ρAR 2 mV C t sin(α + i n F )
,

Ω 2 f -Ω 2 b = ηM q 2ρAR 2 xr Iy C t sini n F ,
where α re F is a adjustable reference value for angle of attack after stuck fault.

With reconstructed decoupling, height tracking is still feasible, and angle of attack is stabilized at α re F . According to quad-TRUAV dynamics model, flight velocity would research a new stabilized value when t → +∞ as follows:

V ∞ lim t→+∞ V (t) = - 2mF α∞ ρSC D0 tan(α re F + i n F ) = 2mg ρS(C L0 + C Lα α re F + C D0 tan(α re F + i n F )) , (36)
where F α∞ lim t→+∞ F α is the stabilized value of F α .

Note that, according to (36), new stabilized velocity V ∞ is determined by given reference value of angle of attack α re F . This provides some control freedom of velocity facing rotor-tilt axle stuck fault, and V ∞ can be regarded as a redesigned reference value. V ∞ is always researchable with suitable α re F , and actuator saturation is not easy to meet due to degraded model approach. However, the rang of angle of attack should be paid attention in case of stall.

NUMERICAL RESULTS

For following numerical results, model parameters in Section 2 are listed as m = 2.71, I y = 0.0816, c = 0.2966,

Results of Fault-free Transition Procedure

Set the initial values of V and h as 0.001 and 5, and their final reference values as 23 and 6. Curves of states and control inputs are shown in Fig. 3 and Fig. 4. In the beginning of numerical simulation, quad-TRUAV is with 90 • rotor-tilt angle, and this value is reduced for acceleration after 2s. Then, α is stabilized at α τ (α τ = 0 here), and i n takes main control effect. When i n reaches 0, TRUAV should be set in airplane mode, and α re should be with varying value according to (31). Because of the delayed response of α, rotor-tilt angle researches negative value for compensation, and stabilized at 0 in the end that means transition from helicopter mode to airplane mode is realized. For actual servo actuators, negative rotor-tilt angle is realizable. The flight height is also kept in whole transition procedure. Above numerical results represent the effectiveness of transition controller in Section 3.

Curve of rotor control weight is shown in Fig. 5. With this weight value, to ensure Ω 2 f ≥ 0 and Ω 2 b ≥ 0, continuous step signals are applied for velocity and height references with fixed slops as 2m/s 2 and 1m/s, which could also slow down the descending rate of rotor-tilt angle.

Results under Rotor-tilt Axle Stuck Fault

Assume the stuck fault at 30 • as a sample, and this faulty value can be measured or estimated 0.2s later, which can be regarded as FDD delay. Set the initial values of V Further consider the value of α re F . If faster or lower flight velocity is needed, α re F can be set as a negative or positive value according to (36). As shown in Fig. 9, setting α re F as -10 • , 0, and 10 • with 70 • rotor-tilt axle stuck fault, redesigned velocity references 22.27m/s, 20.49m/s, and 18.1m/s can be obtained, and actual velocity would be stabilized at them in the end, respectively. Although rate of convergence is slow, quad-TRUAV is still with the ability of tracking velocity reference in a degree by α re F . However, the permissible range of angle of attack should be taken into consideration.

Above numerical results are all with 0.2s FDD delay. To test the effect of FDD delay for FTC, following performance criteria are defined:

e 1 =

Fig. 1 .

 1 Fig. 1. Quad-TRUAV and coordinate systems

Fig. 2 .

 2 Fig. 2. Transition controller structure 3.2 Nonlinear Controller Design As analyzed in above subsection, quad-TRUAV model has been transformed into generally nonlinear form (19) and (20) with virtual control variables v 1 (t) and v 2 (t). So many nonlinear feedback control methods could be applied here for stability and reference tracking.

Fig. 3 .

 3 Fig. 3. Curves of states in transition procedure

Fig. 5 .

 5 Fig. 5. Curve of controller weight

te 0 (

 0 Fig. 7. Curves of states with FTC compared with fault-free states

  and V21 (x 21 (t))| x22=x22re < 0 with nonzero state vector x 21 (t). Further define e 22 (t) = x 22 (t) -x 22re (t) and alternative Lyapnov function

	V 2 (x 21 (t), e 22 (t)) = V 21 (x 21 (t)) +	1 2	e T 22 (t)e 22 (t).
	Obviously,
	V(x 21 (t), e 22 (t)) = V21 (x 21 (t))| x22=x22re + ( V21 (x 21 (t))
	-V21 (x 21 (t))| x22=x22re ) + (x 22 (t)
	-x 22re (t)) T ( ẋ22 (t) -ẋ22re (t)).
	If v 2 (t) is constructed and ensures
	( V21 (x 21 (t)) -V21 (x 21 (t))| x22=x22re )

Table 1 .

 1 Criteria with different FDD delays With 45 • stuck fault, same initial values and reference values with above example, α re F = 0, and t e = 100, Q = diag(1, 10 6 , 10 5 ), R = diag(10, 0.01, 0.01, 10 5 ), criteria with different FDD delays are listed in Table1. According to this table, larger FDD delay leads to worse tracking performance and more energy consumption, and might result in instability in spite of FTC. That means proposed FTC strategy requires real-time performance of FDD methods.6. CONCLUSIONIn conclusion, this paper proposes a new idea of transition controller for quad-TRUAV. Different with GS methods, rotor-tilt angle is regarded as a control input, and virtual control variable including more than one control inputs is considered to decouple original dynamics. In this way, explicit mode transition is avoided. Hierarchical nonlinear controller based on backstepping and decoupling module are designed for virtual control variables and control inputs, respectively. Moreover, rotor-tilt axle stuck fault is considered further. By reconstructing decoupling module, FTC strategy could ensure the cruise ability by setting a fixed angle of attack reference, and keep flight height tracking. Comparing defined performance criteria, realtime FDD methods are required for above FTC strategy.

	FDD delay (s)	0.1	0.2	0.3	0.33	0.34
	Criterion e 1	13.87	15.98	18.56	31.97	+∞
	Criterion e 2	723224 723248 723271 723413 +∞

would be replaced by V ∞ . So e 1 represents tracking performance, and e 2 represents consumed energy.

This study was supported by the National Natural Science Foundation of China (Nos. U1608253 and U1508208).

Appendix A. STABILITY PROOF OF NONLINEAR CONTROLLER

As for V , with virtual control variable ( 22), the closed-loop system can be formed as follows:

When t → +∞, V would be asymptotically stable at V re .

As for h, with state γ and its reference value (27), Lyapunov function can be chosen as

h could track constant reference value when t → +∞, and parameter H can limit the maximum value of h re . As for α, state q can be set as

With Lyapunov function V α (α -α re ) = 1 2 (α -α re ) 2 , the stability of α can be proven.

Based on the idea of backstepping, further define e γ = γγ re and e q = q -q re , consider Lyapunov functions

With virtual control variables ( 25) and ( 26

That means h, γ and α, q are all stabilized according to Lyapunov theory. Further introduce (25) into above q re , and ( 28) is obtained.

In general, the stabilization of nominal quad-TRUAV model is ensured by backstepping, and a hierarchical nonlinear controller is formulated for reference tracking.


Jordi Gómez-Segura

Javier Campo
email: jcampo@unizar.es

Inhar Imaz

Klaus Wurst
email: klaus.wurst@uibk.ac.at

Jaume Veciana

Philippe Gerbier
email: gerbier@univ-montp2.fr

Daniel Ruiz-Molina

New insights into ligand inductive effects on the stability of Mn 12 singlemolecule magnets. Synthesis, magnetic and thermal characterization of complexes [Mn 12 O 12 (O 2 CR) 16 (H 2 O) 4 ]⋅xH 2 O. (R = C 2 H; x = 3 and R = C 2 Ph; x = 4) and their thermolysis reaction products

    New insights into ligand inductive effects on the stability of Mn12 single molecule magnets. Synthesis, magnetic and thermal characterization of complex [Mn12O12(O2CC̸ ≡CH)16(H2O)4].3H2O and the corresponding manganese(II) propiolate [Mn3(O2CC̸ ≡CH)6(H2O)4]2H2O derived by thermolysis reaction.

Introduction

The so-called single-molecule magnets (SMM) represent a new kind of nanoscopic molecular materials that offer potential access towards the ultimate high-density information storage devices and quantum computing applications. 1 Outstanding magnetic properties such as out-of-phase AC magnetic susceptibility signals and stepwise magnetization hysteresis loops are representative of an extensive number of large metal complexes that function as SMM. Different families of SMM have been so far reported based on manganese, 2 nickel, 3 cobalt, 4 iron 5 and mixtures of metals. 6 The well-known Mn 12 family possess one of the highest T B so far reported (ca. 4K). Structurally, the metal core [Mn 12 (µ 3 -O) 12 ] comprises a central [Mn IV 4 O 4 ] 8+ cubane held within a non-planar ring of eight Mn III ions by eight µ 3 -O 2 -ions. Peripheral ligation is provided by sixteen carboxylate groups that are amenable to variation and accomplished by three or four H 2 O molecules. 7 Small structural variations of the Mn 12 core induced by environmental and lattice network characteristics have been shown to significantly influence their magnetic properties and quantum tunnelling behaviour. For example, pure Jahn-Teller isomers of a given Mn 12 complex that crystallize in the same space group and differ only in the identity of one solvent molecule of crystallization at one position have been prepared and shown to exhibit different magnetization relaxation rates. 8 Recent X-ray crystallographic studies have also shown that the presence of non-axial isomers in the crystal lattice of Mn 12 clusters may provide a possible explanation for the intriguing tunnelling behaviour of singlemolecule magnets. 9 To gain more knowledge on how subtle structural modifications of the [Mn 12 (µ 3 -O) 12 ] structure may influence the stability and magnetic properties of Mn 12 clusters, we have initiated a new synthetic strategy based on the systematic investigation of a number of Mn 12 complexes differing on the nature of peripheral carboxylic ligands. In previous works we have already presented a detailed study of the series CX 3 CO 2 H (X = F, Cl and Br), which exhibited a systematic variation of their volume and Lewis base character. 10,11 These studies showed that the relative thermal stability of Mn 12 complexes arises from an interplay between steric and ligand inductive effects. In a further step, herein we present the synthesis, thermal study and magnetic characterization of two new Mn . The propiolic acid of complex 2 was chosen on the basis of its remarkable acidity and small molecular volume. 12 Such a combination is expected to allow a specific study of the acidity effect on the thermal stability of the dodecamangenese core in the absence of remarkable steric congestions. Consequently. this is the ultimate step on the quest to elucidate the interplay between acidity and volume on the relative stability of Mn 12 clusters. Moreover, complex 3 bearing phenylpropiolic carboxylate ligands, less acid but more bulky, has also been prepared for comparison purposes. Finally, the thermal stability of complexes 1 and 2 has been explored and the resulting information used to induce the thermal degradation of complex 1 into the new layered complex

[Mn 3 (O 2 CC≡CH) 6 (H 2 O) 4 ]⋅2H 2 O (3).

Experimental

All reagents and solvents were used as received and no further purification was carried out. Elemental analyses were performed by the Servei d'Anàlisi Química of the Universitat Autònoma de Barcelona. FT-IR spectra were collected on a Perkin Helmer spectrometer in the range of 400-4000 cm -1 as KBr discs. Complex 1 was synthesized as described in ref. 13

Syntheses of complexes

Crystallography

A rhombohedral X-ray quality crystal was selected and mounted on a Nonius Kappa CCD diffractometer equipped with monochromatic Mo Kα (λ= 0.71073Å) radiation. The structure of 3 was solved by direct methods, using SHELXTL 14 software and refined by full-matrix least-squares method with SHELXTL software. The hydrogen atoms at carbon atoms were calculated, using a riding model, hydrogen atoms of water molecules were found and refined regular with isotropic temperature factors. All non-hydrogen. The details of the crystallographic data and refinement are summarized in Table 1.

Measurements

DC magnetic measurements were collected on a Quantum Design MPMS superconducting SQUID magnetometer equipped with a 5.5 T magnet and operational range of temperature 1.8-300 K. Ac magnetic measurements were collected in a 1.0 Oe ac field at the frequencies 9-99 Hz with no applied dc field. TG/DTA measurements were simultaneously performed on a Setaram Labsys TG-TDA 12 thermogravimetric analyzer under an N 2 + O 2 flow (60 mL.min -1 ) at a heating rate of 10 °C.min -1

Results and Discussion

Synthesis and magnetic characterization of 1 and 2

Complexes [Mn 12 O 12 (O 2 CC≡CR) 16 (H 2 O) 4 ]⋅xH 2 O (1: R = H, x = 3; and 2: R = C 6 H 5 , x = 4) were obtained following a synthetic methodology recently described by us. 13 Such methodology is based on the use of complex [Mn 12 O 12 (O 2 C t Bu) 16 (H 2 O) 4 ] (4) as the starting material for the ligand substitution reaction with the corresponding carboxylic acid (eq. 1). The presence of tert-butyl groups significantly increases its solubility in organic solvents and favours the displacement of the substitution equilibrium to completion due to the combined effects of the acidity and the steric demand.

Complex 1 was obtained as a brown microcrystalline powder at the bottom of the diffusion tube, being only moderately soluble in polar organic solvents such as acetonitrile. On the contrary, complex 2 was collected as small black crystals with a good solubility in a large variety of polar to weakly polar solvents. Unfortunately, these crystals gave too poorly resolved diffraction patterns to allow any further refinement, in spite of several recurrent crystallization experiments. Then, complexes 1 and 2 were characterized by elemental analysis, spectroscopic techniques and magnetic measurements. Ac magnetic susceptibility data were studied for polycrystalline samples of complexes 1 and 2 in the 1.8-10 K range with a 1 Oe ac field oscillating in the frequency range of 1-1150 Hz. The external dc magnetic field was held at zero. The resulting plot of the out-of-phase susceptibility χ M ″ signal versus temperature at three different frequencies is shown in Figure 1. As can be seen there, complex 2 exhibits frequency-dependent out-of-phase ac signals in the temperature range of 4-7 K whereas complex 1 Fig. 1 AC magnetic measurements for complex 2 (a) and 3 (b) in a 1 Oe ac field oscillating at the frequencies (♦) 2.11 (■) 5.7 and (▲) 11.1 Hz, with no applied dc field. The out of phase susceptibility (χ '' ) corresponds to the curves near the temperature axis in each plot. The in-phase curves (χ') have an higher response.

exhibits frequency-dependent out-of-phase ac signals in the temperature range of 2-4 K. The magnetization relaxation rate (1/τ) determined from the position of the peak maximum follows the Arrhenius equation as expected for a thermally activated Orbach process with an energy barrier of 52.5 and 61.3 K for complexes 1 and 2, respectively. Both, the observation of two different magnetization relaxation mechanisms (fast and slow relaxing species) and the respective effective energy barriers are characteristics of several other Mn 12 complexes. Ref Magnetization hysteresis curves were also obtained for a polycrystalline sample of complexes 1 and 2 at three different temperatures (1.8, 2.0 and 2.5 K). The sample was oriented in eicosane such that the axial anisotropy axis of the molecule is parallel with the external field. The crystals were first added to eicosane and torqued in a 5 kOe field. The temperature was then raised above the melting point of eicosane (312-318 K), held there for 30 minutes and then reduced to room temperature. Because of the predisposition of the crystals to torque, the crystals were oriented such that the magnetoanisotropy axis (i.e., z-axis) of each molecule is parallel with respect to the external field. Then, samples are magnetically saturated in a +2.0 T field, and afterwards the field was swept down to -2.0 T, and cycled back +2.0 T. In the case of complex 1, as the field is decreased from +2.0 T to zero, there is a considerable step at zero field that can be related with the predominance of the fast relaxing species rather than to the presence of thermally induced tunnelling. On the contrary, the magnetization hysteresis curves of complex 2 clearly exhibit steps on switching the external magnetic field. The presence of such steps has been already clearly attributed to resonance quantum mechanical tunnelling of the magnetization direction. If all the molecules changed their direction of magnetization by thermal activation over the barrier, then the hysteresis loop would be a smooth function with no steps.

Thermal studies

Thermogravimetric (TGA) and differential thermal analyses (DTA) experiments carried out on polycrystalline samples of complexes 1 and 2 are shown in Fig. 2. In the case of complex 2, the DTA curve obtained by heating the sample at a constant rate of 10 °C.min -1 shows a single strong exothermic peak at 118 °C, accompanied by an abrupt weight loss in the corresponding thermogravimetric experiment. The resulting tiny brown material was associated with a decomposition product, as confirmed by IR spectroscopy. Indeed the IR spectrum of such brown material shows a pattern characteristic of MnCO 3 , indicating that the exothermal process may be described by the following decarbonylation reaction (Eq. 2). 12

M(

O 2 CC≡CH) X → M (X/2) CO 3 + CO 2 + CO + HC≡CH + C 2)
The TGA and DTA curves of complex 2 are also shown in Fig. 2 (bottom). As can be seen there, the thermal response of complex 2 differs considerably from that observed for complex 1. A first weight loss of 32 %, less abrupt than that of sample of 2 is charaterized by two exotherms at 155 °C and 192 °C, respectively. The second weight loss which begins at ca. 295 °C corresponds to a weak exothermic process and leads at 500 °C to a black residue, with an overall weight loss of 62.5 %. The first exothermic process may be attributed to a decarboxylation process, as previously described for complex 2. However, in this case, the resulting phenylacetylenic radicals may react together on a polymerization process contributing in more or less degree to the thermal processes. The second weight loss, which begins at 295 ºC, can be attributed to the decarboxylation reaction converting MnCO 3 into manganese oxides.

Complexes 1 and 2 have pKa values of 1.84 and 2.25, respectively, and a comparable steric hindrance contribution (ca. 37 Å 3 ). 16 Therefore, the larger thermal stability of complex 2 may be well explained by the greater value of its pKa, i.e., its decreased inductive effect, confirming the importance of the pKa acidity on the thermal stability of the dodecamangenese core. In fact, the replacement of a hydrogen atom by a phenyl ring attached to a triple bond has already been shown to affect dramatically the reactivity of the latter, especially in the case of thermally-induced polymerizations 12 . Moreover, the temperature of decomposition of complexes 1 and 2 lies within the range of those described for other [Mn 12 O 12 (O 2 CR) 16

Synthesis and structural characterization of complex 3

In a previous work we have already reported how the thermal treatment of some Mn 12 clusters in solution leads to the obtaining of low nuclearity clusters by a decarboxylation process. For this reason, acetonitrile solutions of complexes 1 and 2 were also refluxed for 24 h under stirring and ambient conditions. As expected, the replacement of the acetylenic hydrogen atoms by a phenyl rings affects drastically the thermal stability of the Mn 12 complexes not only in the solid state but also in solution. Indeed, whereas complex 2 remains unaltered during the thermal treatment, complex 1 experiences a thermally induced reaction, which on cooling, leads to the obtaining of a red-brown microcrystalline precipitate. Such precipitate was characterized as complex [Mn 3 (O 2 CC≡CH) 6 (H 2 O) 4 ] (3) by elemental analysis and spectroscopic techniques. This implies a fragmentation of the metal core [Mn 12 (µ 3 -O) 12 ] with a reduction of the mixed-valence Mn(III)Mn(IV) pattern into the Mn(II) oxidation state. This fact can be explained most likely via a catalytic decarboxylation mechanism in the presence of strong acids as described elsewhere. 15 Co-ordinated carboxylates possessing a weak Lewis base character exhibit an appreciable lability that favours their release in solution. As a consequence, the metal core of the complex becomes co-ordinatively unsaturated which, in turn, results in unstable cationic carboxylato-Mn III species more sensitive to decarboxylation process via homolytic fragmentation (Eq. 2). 15 Further aggregation of cationic carboxylato-Mn II species yields the Mn II trinuclear linear complex 4 (Eq. 3).

The recrystallization of 3 from boiling acetonitrile gave wellformed rhombohedral deep red crystals suitable for X-ray diffraction studies. The X-ray structural studies revealed that 3 crystallizes with two additionnal water molecules in an extended sheet-like structure with a monoclinic space group P2 1 /n; a = 9.2800(2) Å, b = 9.4132(2) Å, c = 14.9675(3) Å, β = 99.630(1)°, and Z= 2 (Table 1) 2 and 203 parameters for 1889 reflections with I>2σ(I). Crystallographic parameters are given on Table 1 As shown in Fig. 3a However some differences can be highlighted, specially with the structure of complex 5. In the Mn 3 core of 3 two of the six carboxylates exhibit a symmetrical syn-syn bridging mode, with bond distances of 2.129 and 2.193 Å that separates the Mn ions with a distance of 3.425 Å. The remaining four carboxylates exhibit a twofold coordination mode, within and between trimeric units. In the case of complex 5, four of the six carboxylates are coordinated in the syn-syn mode to Mn ions. The remaining carboxylates are connected in a twofold coordination mode to the Mn ion located between them within the same trimeric unit and with Mn ions of neighbouring trimeric units. On the other side, the differences between complexes 3, 5 and 6 come from the different steric constraints given by acetate and propiolate groups. For comparison purposes, a top view of the tiles in the plane showing the inner organic ligands is shown in Fig. 4. Comparison of the transverse dimensions measured for the tiles of the structures shows that the tile size adapts to the steric hindrance generated by the inner organic groups directed to the centre of the tile. Interestingly, comparison of the constitutive tiles of 3 and 6 in the plane, shed some light on the reasons why such layered structures are obtained using propiolic acid or acetic whereas linear polymeric 19 or discrete molecular complexes (Mn 3 , Mn 6 ) are characterized with other more bulky acids. 10,11 From the examination of Fig. 4, it is clear that the cavity inside the tile can only accommodate ligands for which the van der Waals' volume does not exceed the value of ca. 37 Å 3 (C 2 H fragment). [START_REF] Viertelhaus | formate ligands which lead to 3-D networks[END_REF] With a volume larger (43 Å 3 ) the use of trifluoroacetate ligands leads to discrete complexes. 21

Mn III (O 2 CC≡CH) 2 + (O 2 CC≡CH) -→ Mn II (O 2 CC≡CH) + (O 2 CC≡CH) -+ CO 2 + HC≡C ⋅ (2) 3 Mn II (O 2 CC≡CH) + (O 2 CC≡CH) -→ Mn 3 (O 2 CC≡CH) 6 ( 3
)

Magnetic properties of complex 3

Ac magnetic susceptibility data at different frequencies (0.1 < v < 1k Hz range) and two orientations of the crystal with respect to the axis of the magnetometer, i.e. the applied magnetic field direction, were obtained for a single crystal of 3. In one of the orientations, the reciprocal vector [10-1]*, which is perpendicular to the layers of polymeric Mn trimers, lies perpendicular whereas in the other one lies parallel to the axis of the magnetometer. The ac susceptibility curves for both orientations are shown in Fig. 5.

As can be seen in Fig. 5, the in-phase signal exhibits a peak at 2.2 K associated to the presence of long range magnetic order. The stronger signal of the peak, which does not exhibit any frequency dependence, was obtained when the reciprocal vector [10-1]* lies perpendicular to the axis of the magnetometer. This fact can be explained by the presence of stronger fluctuations or predominant ferromagnetic interactions within the polymeric layers, whereas antiferromagnetic interactions take place between different layers, i.e., when the reciprocal vector [10-1]* lies parallel to the axis of the magnetometer. Such assignation can be justified by considering the structural arrangement of complex 3. Indeed, according with the linear disposition of the Mn 2+ cations within the repetitive trimeric unit, it is expected that the Mn ion with double multiplicity couples antiferromagnetically with each Mn ion located on a fourfold multiplicity site, leading to a ferrimagnetic entity with a spin S=5/3 µ B , as described by the Hamiltonian,

) • • • ( 2 1 3 1 , 3 3 2 3 , 2 2 1 2 , 1 S S S S S S J J J H + + - =
where S i =5/2 and J 1,2 is negative and J 3,1 is equal to zero.

Then, the resulting net magnetic moment S=5/3 of each trimeric unit couple ferromagnetically between them along the 2-D layers, which in turn couple antiferromagnetically to each other leading to a 3-D long range magnetic order below 2.2 K. Variable-temeprature dc magnetization measurements of a single crystal oriented with the reciprocal vector [1 0 -1] * perpendicular to the external magnetic field, have also been measured as function of temperature at several different magnetic fields ranging from 20 Oe to 10 kOe. The resulting magnetization curves, normalized by the applied external field, are shown in Fig. 6. At low fields (H<150), the sample experiences a transition from a paramagnetic to an antiferromagnetic phase (antiferromagnetically coupled planes). However, an increase of the magnetic field over 0.1 kOe induces variations on the magnetization curves. In this case, the magnetic phase transition is associated with a transition from a paramagnetic to a ferrimagnetic phase (ferromagnetic interactions within the layer). Finally, an increase of the external magnetic field over 10 kOe induces a systematic decrease of the susceptibility moment (M/H) most likely due to saturation effects. Finally, magnetization versus magnetic field measurements was carried out at 1.8 K for two different orientations of the crystal (see Fig. 7). From there, two main conclusions can be extracted. First, at magnetic fields higher than 3 kOe (H > 3kOe) the magnetization curves for both orientations are similar, as can be observed in the upper insert of the Fig. 7. From the same figure, we may also extract the second important detail about the easy axis of magnetization, which must be contained within the polymeric layers. Another remarkable fact is that at fields H>30 kOe the crystal increases its magnetization showing that the ferrimagnetic triplets are passing from the fundamental state to an energy level with higher magnetic moment before reaching complete saturation, apparently for fields H>55 kOe (not measurable in our experimental set-up). Finally, in the low field region (H < 50 Oe) we can notice that both curves are similar whereas for 50 Oe<H < 3 kOe the curves split. The magnetic structure in the ordered state cannot be determined from this limited experimental set of macroscopic measurements. As previously mentioned, a simple model that justifies such results is that the Mn 3 trimers couple antiferromagnetically to give a net ferrimagnetic moment, which in turn interacts ferromagnetically with other trimers of the same layer. Finally the layers interact antiferromagnetically with other layers giving an antiferromagnetic structure at zero field. The direction of the magnetic moment lies inside the layer although a weak canting (one layer with its neighbours) cannot be excluded. Then, when a magnetic field is applied perpendicular to the [1 0 -1] * direction a metamagnetic phase transition occurs for fields higher than 50 Oe due to the very low anisotropy of the Mn trimers. If the magnetic field is applied along to the [1 0 -1] * vector then the system reaches gradually the saturated state. However, to fully verify this hypothesis, further neutron diffraction experiments to resolve the magnetic structure are currently underway.

Conclusions

Propiolic acid comprises an unsaturated organic chain attached to the carboxylate group CO 2 -. On the basis of the linear structure, steric hindrance between adjacent carboxylates in the coordination sphere is highly minimized. As a consequence, any disruption of the [Mn 12 (µ 3 -O) 12 ] core is prone to be originated mostly by marked inductive effects due to its dissociation constant (pK a =1.84), i.e. weak Lewis base character. As a consequence, complex 2 results in a lower thermal stability than that of the phenyl derived carboxylate (pK a =2.25), favouring oxidative decarboxylation process. In contrast to complex 2, the corresponding phenylpropiolate Mn 12 complex shows a wider thermal stability domain and side reactions involving phenyl radicals are unlikely to undergo explosive degradation. The magnetic measurements performed on the new complexes 2, 3, show a characteristic behaviour or Mn 12 Single Molecule Magnet. On the other hand, the model deduced from our magnetic results measurements obtained with the twodimensional 4 complex shows an interesting intralayer ferrimagnetic behaviour resulting from the antiferromagnetic coupling inside Mn 3 cores and ferromagnetic interactions between the trimeric units. The layers interact antiferromagnetically between them.

[Mn 3 (

 3 O 2 CC≡CH) 6 (H 2 O) 4 ]⋅2H 2 O (3). Complex 1 (0.100g, 4.8⋅10 -5 mol) were suspended in 5 ml of acetonitrile, heated to reflux for 5 min and left undisturbed at room temperature in an open Erlenmeyer flask. Complex 3 was collected as deep red rhombohedral crystals by slow evaporation of the solvent. The crystals were washed with a minimum amount of acetonitrile and dried on a frit. Elemental analysis. Calc. for Mn 3 C 18 H 18 O 18 : C, 31.44; H, 2.62. Found: C, 31.54; H, 2.66. IR (KBr, cm -1 ): 3411 (m), 3270 (s), 2115 (s), 1714 (w), 1583 (s), 1540 (s), 1403 (s), 771 (m), 624 (m).

  (H 2 O) 4 ] complexes, where R stands for C(CH 3 ) 3 (T d = 150 °C), CF 3 (T d = 130 °C) or CHCl 2 (T d = 110 °C).

Fig. 2 .

 2 Fig. 2. TGA (plain symbols) and DTA (open symbols) curves of compounds 1 (top), 2 (bottom) in the temperature range 25-400 °C (heating rate: 10 °C.min -1 ) under an air flow.

, complex 3

 3 consists of a repetitive unit composed of three MnO 6 octahedra with a linear conformation. The central Mn(II) site of each trinuclear unit is connected to both lateral Mn(II) ions through six carboxylate ligands, whereas remaining coordinative positions of each external Mn ions are occupied by two water molecule ligands. Then, the basic trimeric units are connected to each other within the same plane to yield a 2-D structure (Fig 2b) consisting in a paving of hexamanganese pseudorectangular tiles, in which the organic groups are pointing toward the interior of the cavity. A similar arrangement have been already found in two related extended complexes [Mn 3 (O 2 CCH 3 ) 4 (H 2 O) 4 ]•8H 2 O (5), 17 and [Mn 3 (O 2 CCH 3 ) 6 (H 2 O) 4 ]⋅2H 2 O (6).18 In all the cases, the carboxylates link the central and the external ion of the trimeric unit via an µ-oxo bridge. Simultaneously, they are also bridging two Mn(II) ions of different trimeric units, through the O-C-O pathway with an anti configuration. As a result, the 2D organisation and the connection between the trimeric blocks are the same in all the cases.

Fig. 4 .Fig. 3 .

 43 Fig. 4. (Top) View of the tiles in the plane showing the inner organic ligands: a) 5 b) 6, and c) 3 structure. (Bottom) Side view of the tile for complex 3, emphasizing the inner organic ligands by its representation with the space-filling model. Color keys, Pink: manganese atoms, red: oxygen atoms, grey: carbon atoms.

Fig. 5 .Fig. 6 .

 56 Fig.5. In-phase ac susceptibility curves vs. temperature at different frequencies (que frecuencias?) measured for two orientations of a single crystal of 3.

Fig. 7 .

 7 Fig. 7. Magnetization versus magnetic field isotherms at 1.8 K measured for two orientations of a single crystal of 4

[Mn 12 O 12 (O 2 CC≡CH) 16 (H 2 O) 4 ]⋅3H 2 O (1).

	To a solution of 4
	(0.100g, 4⋅10 -5 mol) in dichloromethane (5 ml) an excess (35
	equiv.) of HC≡CCO 2 H (0.100g, 1.4⋅10 -3 mol) was added under
	stirring for few minutes. Isolation was achieved by slow
	diffusion of hexanes (5 mL) into this solution. Complex 1 was
	recovered as a brownish precipitate at the bottom of the diffusion
	tube. The solid was washed with dichloromethane and dried on a
	frit. Elemental analysis. Calc. for Mn 12 C 48 H 30 O 51 : C, 27.40; H, 1.43. Found: C, 27.02; H, 1.47. IR (KBr, cm -1 ): 3439 (s), 3275
	(s), 2109 (s), 1584 (s), 1400 (s), 1375 (s), 775 (m), 623 (m), 602
	(m).

[Mn

12 O 12 (O 2 CC≡CC 6 H 5 ) 16 (H 2 O) 4 ]⋅4H 2 O (2).

  Same procedure as described above, but replacing HC≡CCO 2 H by C 6 H 5 C≡CCO 2 H. Complex 2 was collected as small black needleshaped crystals. The crystals were washed with pentane and dried in air. Elemental analysis. Calc. for Mn 12 C 144 H 96 O 52 : C, 52.11; H, 2.89. Found: C, 52.14; H, 2.88. IR (KBr, cm -1 )

	: 3435
	(s), 3056 (w), 2217 (s), 1569 (m), 1411 (s), 759 (m), 689 (m),
	617 (m).

Table 1 :

 1 Crystallographic data for 3

	Formula	Mn 3 C 18 H 18 O 18
	Formula weight	687.14
	T /°C	223(2)
	Crystal system	monoclinic
	Space group	P2 1 /n
	a/Å	9.2800(2)
	b/Å	9.4132(2)
	c/Å β/°	14.9675(3) 99.630(1)
	V/Å 3	1289.05(5)
	Z	2
	D c /g cm -3 µ(Mo-Kα)/mm -1	1.77 1.53
	R1	0.0193
	wR2	0.0486
		a	b	c

+ 16 RCO 2 H → [Mn 12 O 12 (O 2 CR) 16 (H 2 O) 4 ] + 16 t BuCO 2 H (1)

Acknowledgements

This work was supported by the Spanish government under MAT 2002-0043, the Generalitat de Catalunya under XI2002-9 and the EU under the project Nanomagiqc. P. G. is grateful to the Région Languedoc-Roussillon for its financial support through the Communeauté de Travail des Pyrénées network DeNaMol. D. R.-M. also wants to thank Prof. David Hendrickson for fruitful discussions on the chemistry of Mn 12 clusters.

(LINE INCLUDED FOR SPACING ONLY -DO NOT DELETE THIS TEXT)


Thierry Pairault
email: pairault@ehess.fr

Hillman

AFRICA AND THE NEW SILK ROADS

Keywords: China, Africa, New Silk Roads, BRI F2, F5, O53, O55

This contribution raises the question of the place the New Silk Roads strategy could reserve for Africa and the 54 economies that form it. To answer this question, I will begin by focusing on Africa's economic importance for China (first investment, second trade) and then examine the strategical part that Africa as a whole and African states are likely to play. This research will be conducted on the basis of existing documents, whether official reports (administrative or managerial), scientific studies or more journalistic information. As far as possible, it will favour first-hand Chinese documents. We resolutely aim for axiological neutrality in our approach.

[1]

The Joint Communiqué issued on 16 May 2017 at the end of the Leaders Roundtable of the Belt and Road Forum for International Cooperation unequivocally highlights the economic purpose of the New Silk Roads Strategy: "We reaffirm our shared commitment to build open economy, ensure free and inclusive trade, oppose all forms of protectionism including in the framework of the Belt and Road Initiative. We endeavor to promote a universal, rules-based, open, non-discriminatory and equitable multilateral trading system with WTO at its core" 1 .

One glance at the various maps showing the New Silk Roads 2 is enough to understand that their main objective is definitively to penetrate the markets of European countries and even more blatantly those of the most developed among them (see figure 1). In 2016, maritime transport between China and Europe is obviously prevailing, carrying 94 percent of trade by weight and 64 percent by value; in contrast, rail carried less than 1 percent by volume and just over 2 percent by value 3 . The new inland Silk Roads clearly do not concern Africa since they run through the Eurasian continent, but what is actually emerging from the new maritime road? The Lloyd's list 4 enumerating the hundred largest container terminals ranks the Tangier Med terminal both 50 th among terminals in the world and 1 st among terminals in Africa. Next, this list indicates that the Alexandria terminal ranks 88 th in the world and 2 nd in Africa. In both cases, they are North African terminals focusing primarily on Mediterranean and especially European traffic -and somewhat African for Tangier Med. The current insignificance with regard to the world trade of African ports is still underlined if we observe that their [2] throughput in standard 5 containers represents in 2016 only 4% of the world port throughput of containers. Figure 2 shows that container terminals traffic, both in terms of throughput and value, is notably low in Africa compared to Europe, not to mention Asia. [START_REF] Djibouti | DIGITAL SILK ROAD Djibouti's case differs considerably from the previous one[END_REF] Therefore, in such a context, what place could the New Silk Roads strategy reserve for Africa and the 54 economies that form it? To answer this question, I will begin by focusing on Africa's economic importance for China (first investment, second trade) and then examine the strategical part that Africa as a whole and African states are likely to play.

AFRICA: OF LITTLE IMPORTANCE FOR INVESTORS

If we turn to the Chinese investment statistics collected specifically by MofCOM to reveal the countries involved in the New Silk Roads strategy, it is then obvious that African countries are not included -with the exception of Egypt alone because of the Suez Canal and the ports of Alexandria and Port Said. It should also be noted that none of the major economies of the European Union are included among the countries belonging to the New Silk Roads strategy. It is normal since they are not the roads but the targets. In 2016, these favoured European destinations attracted investment flows more than six times higher than those destined for Africa and stocks three times higher. [START_REF] Africa | OF SOME IMPORTANCE FOR LEARNING OTHERNESS China's current economic performance causes some authors -such as Arvind Subramanian 54[END_REF] In 2017, according to MofCOM data, non-financial investments directly related to projects under the New Silk Roads strategy (i.e. excluding projects carried out for other reasons either political or economic) amounted to 14.4 billion dollars. By comparison, the amount of 5 Twenty-foot equivalent unit (TEU). China and the United States were initially selected because they constitute three economies of roughly equal importance in terms of GDP). The considerable repatriation of capital recorded the following year forces the analysis to be refined and the role of the Netherlands, Luxembourg and the United Kingdom to be considered. These last three countries have financial practices that lack transparency mainly for the first two, but also frequently for the second, which at the same time suffers the anticipated effects of Brexit. In other words, one can wonder what is the exact origin (European or not) of these investments in Africa and repatriation of capital from Africa especially since we could note that the Chinese companies had operated a transfer of their funds from Luxembourg -become less attractive -towards the Netherlands -always accommodating. 10 If we look at three other leading European countries, we will see that, over the 2013-2016 8 See 2017 nian wo dui "yidai yilu" yanxian guojia touzi hezuo qingkuang [China's investment co-operation in countries along the "One Belt and One Road" in 2017], http://fec.mofcom.gov.cn/article/fwydyl /tjsj/201801/20180102699450.shtml. 9 Eurostat online database is available at http://ec.europa.eu/eurostat. 10 See http://pairault.fr/sinaf/index.php/statistiques/956.

[4] period, Germany would invest half as much each year as China, while Italy would do slightly better than China. As for France, the statistics presented here would a priori justify the acrimonious criticism to which it is often subjected; however, looking at the data country by country, one will observe that capital repatriation concern almost exclusively three oil countries (Angola, Congo and Gabon) whose foreign capital movements may be random and cyclical.

Table 2 provides data on FDI stocks held in Africa. It shows several things. First, the amount of stocks by country of origin of investments is relatively stable over the four years 2013-2016, or is slightly increasing. Then, in 2016, the amounts of Chinese and US FDI stocks are respectively 14% and 20% of the amount of EU-28 FDI stock, or in other words, the amount of EU-28 FDI stock is five times that of the US and seven times that of China. Even if we subtract the amount of Dutch FDI stock due to the indeterminable origin of the funds invested, the amount of EU-28 FDI stock in Africa is more than four times that of China. Finally, the most salient observation is that, although China was able to build up very quickly in Africa a stock of FDI comparable to the French stock (the first of 40 billion dollars, the second of 46 billion dollars in 2016 according to Eurostat), this means unequivocally that the amount of foreign investments in Africa whatever their origin is extremely low. Thus, China's ability to finance African infrastructure investments first, and then its industrial investments, can be a key factor for the continent's development even if it creates financial dependence on China.

AFRICA: OF CONTAINED COMMERCIAL IMPORTANCE

Figure 3 compares the growth in the volume of trade (exports plus imports) between China and Africa, on the one hand, and the European Union on the other. The volume of trade with Africa, which started from a very low level, increased 59 fold between 1995 and the peak year (2014), while that with the European Union increased only 19 fold. Nevertheless, the European Union is far from being the fool in a Sino-African farce, as some would have us believe, either out of pessimism or out of revengeful spirit. During these twenty years, the volume of China's trade with the European Union has increased by 507 billion dollars,

[5] compared with only 146 billion dollars for that with Africa. Let us draw a trend curve for both sets of data. We apply a second rank polynomial trend line for each resulting in a determination coefficient of 94% (European Union) and 92% (Africa). Second derivatives of the regression equations reveal that China's trade with the European Union is growing at a rate almost one-third faster than that with Africa. In other words, it is difficult to expect that in the short or medium term such a trend reversal that Africa could become so economically important for China that the current situation would be turned upside down. Let us examine the situation further. In 2016, 95% of Chinese imports of African products are composed of primary commodities (in UNCTAD parlance) 11 , of which 68% are ores and metals12 (40%) and fuels 13 (60%). However, what would a priori best mark China's dependence on Africa would be the growth in the amount of imports of ores, metals and fuels from 0.3 billion dollars in 1995 to 1.2 billion in 2003 and then to 14.3 billion in 2016. But such dependence, if proven, would remain localized to a small number of countries. Table 3 shows that for both Ores and metals and Fuels, 80% of Chinese imports originate mainly from three African countries for [6] each of the commodities. Even so, only two countries might play a significant role: South Africa for Ores and metals and Angola for Fuels. South Africa only ranks fifth among Ores and metals suppliers far behind Australia (29%), Chile (10%) and Brazil (10%) and next to Peru (6%). Also Angola among the fuels suppliers comes behind Russia (11%) and Saudi Arabia (9%), but neighbouring Oman, Iraq, Australia, Iran and Indonesia. Obviously the Chinese government is seeking to diversify its sources of supply in order to limit as much as possible its potential dependence on any African or non-African country. Conversely, while China has a weak and fully assumed dependence on African products, Africa appears much more dependent on China, as shown in Table 4. This table includes the fifteen countries whose exports to China account for at least 10% of their total exports in 2016. Some countries are dependent on China because they export food products14 (Eritrea, Zimbabwe…), agricultural raw materials15 (Equatorial Guinea, Gambia…), ores and metals16 (Sierra Leone, Mauritania and seven others), fuels17 (Sudan, Congo and three others) or manufactured goods 18 (Ethiopia and Zimbabwe). Considering the SITC classification, Africa is mostly dependent on its exports of ores and metals to China (28% of its exports in this category), agricultural raw materials (22%), and to a lesser extent on its fuels exports (14%). Overall, however, Africa's dependence on fuels is most significant: its fuel exports amount to [7] 37% of its total exports (all categories added together) to the world with China accounting for 5% and fifteen other destinations for 32% (the Unites States and India being almost equal with China: 4% each).Africa's dependence on China is also evident through its imports of Chinese-manufactured goods (see Table 5). This table classifies manufactured goods according to their degree of manufactures and it draws a comparison between the United States, the European Union and China -all three having roughly the same economic weight in terms of GDP. As suppliers of manufactured goods, these three economic entities are of very different calibres; the United States weighs very little (6%); China weighs (24%) four times more than the United States and two thirds of the European Union (35%). But what is certainly most important when comparing China and the European Union is the distribution of manufactured goods according to their technical sophistication. In this respect, China is the main supplier for low-tech goods (the two lowest categories) while the European Union is the foremost supplier for high-tech goods (the two highest categories). All these evidences support the conclusion that Africa as a whole is a minor economic partner for China -it being understood that the reverse is not true. The Chinese strategy is primarily a commercial strategy: selling services and goods. Yet Africa, from this trade perspective, weighs only a very limited weight even with 1.2 billion inhabitants, or 16% of the world population, it is at best only a potential market which, as such, does not justify any priority today. However, if we move from economics to politics, the outlook could change.

AFRICA: OF POTENTIAL DEPENDENCE ON CHINA

[8]

AFRICA: OF STRATEGICAL IMPORTANCE

At a March 8, 2005 press conference in Beijing, Chinese Foreign Minister Wang Yi stated that the New Silk Roads strategy was not a "geopolitical tool" bushi diyuan zhengzhi de gongju. [START_REF] Yi | Yidai yilu" shi zhongguo xiang shijie tigong de gonggong chanpin[END_REF] If this were really the case, our contribution could end there and we could emphatically conclude that Africa is not concerned with the New Silk Roads strategy. However, we must take into account that China is at the centre of a great global geopolitical game, and as such Africa is concerned whether it like it or not.

From the Bandung Conference in 1955 to the present day, Africa's role and place in China's foreign policy has significantly changed. Six African countries participated in the conference (Libya, the Gold Coast -now Ghana -, Ethiopia, Liberia, Sudan and Somalia) as well as delegations from two soon to be independent countries (Tunisia and Morocco) not to mention Algeria at war. China's actions in Africa have been from the outset more than revolutionary -they have been designed for a pragmatic and long-term strategic perspective. Algeria was strategically important to China and supporting Algerian independence served Chinese strategy. Once Algeria gained its independence, it worked as a bridgehead for Chinese aid to liberation movements in Africa and for China's accession to the UN. So it's no wonder that Wen Jiabao, in a press conference in Davos on 14 September 2011, could state: "I can tell you that at a time when China was experiencing the greatest poverty and was trading and generously helping Africa, it never took a drop of oil or a gram of ore from Africa". [START_REF] Shaolei | Zhongguo meiyou cong Feizhou caiguo yidi you, laguo yi dun kuang[END_REF] Until the early 1980s, China sought to gain recognition for African countries that only really took the step once China had been admitted to the United Nations on 25 October 1971: twenty African countries had recognised China before its admission, but four years later there were forty. In other words, the support of African countries for China was less unanimous than the history rewritten by Beijing suggests ("it is our black brothers from Africa who brought us there", Mao Zedong is supposed to have said [START_REF]] was written by Wang Taiping, a career diplomat, and published by the very official Beijing Press as part of a program of the Ninth Five-Year Plan subsidizing the publication of "key works" (zhongdian tushu). The polemic seems to have arisen from an article by Xiong Xianghui published in a journal of the Chinese Communist Party ("Mao Zedong shuoguo zhei ju hua ma?[END_REF] Roads were aimed at Chinese small or medium-sized enterprises24 as opposed to large staterun enterprises managed directly by the central government. He advised them to give priority to Southeast Asia, because it is better known because China has direct land and maritime relations with this region, ancient and extensive economic relations, but also because many Chinese have resided there for generations. Eastern Europe is not to be neglected; however, it will interest companies more for its market than for its investment opportunities. On the other hand, African countries and their often unstable political situation were clearly not destination to favour. 25 [10]

On the other hand, Wei Jianguo -former vice-minister of Commerce and current vicepresident of a Chinese think tank (China Center for International Economic Exchanges) -is developing a much more geopolitical vision that is shaking up Ou Xiaoli's mercantilist approach. In an interview given in August 2017 to the 21st Century Business Herald [START_REF] Erde | yidai yilu' kaichuang zhong, fei hezuo de xin shida[END_REF] , he considers that the strategy of the New Silk Roads (which he sometimes describes as a "landmark initiative", an "international initiative") is the form taken by the third wave of Chinese reforms, which are still moving towards greater internationalisation of China after those launched in 1978 (initial opening) and 2001 (entry into the WTO). However, he notes that the world is currently dominated by Trump's United States unilateralism and the protectionism of the European Union -which he attacks quite violently in another text 27 . In this context, China would have no choice but to ally itself with Africa, starting by matching the initiative of the New Silk Roads and the Agenda 2063 of the African Union. In doing so, China and Africa would unite and become more powerful. And to proclaim that China and Africa are enjoying an "all-weather partnership" (quan tian hou huoban guanxi). More precisely and by replacing it in the hierarchy of the Chinese partnerships, it would only be a "Comprehensive strategic parnership" (quanmian zhanlüe hezuo huoban guanxi) as announced by Xi Jinping on 5 December 2015 at FOCAC in Johannesburg 28 .

About twenty African countries individually enjoy a partnership with China whose designation may vary with the time of signatures, the depth of the relations established... Whatever the case, these partnerships, even the so-called strategic ones, impose no deep involvement for any of the parties, their only virtue is to discard topics that anger Beijing like Taiwan or human rights [START_REF] Honghua | Partnership Strategy of China: Progress, Evaluation and Prospects[END_REF] . Since the FOCAC conference in Johannesburg in 2015, these partnerships have gained new importance as they have become a tool serving the "five major pillars" (wu da zhizhu) of the New Silk Roads strategy. As a result, some African countries have sought Beijing's endorsement as a participating country in this strategy. At the Sino-African Investment Forum held in Marrakech on 28 and 29 November 2017, the Moroccan Minister of Industry, Moulay Hafid Elalamy, proclaimed that "All partner countries of the Silk Roads [strategy] benefit from Chinese investments. If you are not a signatory, you are excluded". [START_REF] Le | Le Maroc et les nouvelles routes de la soie : la troisième voie[END_REF] Morocco is actually the most striking example of this approach.

[11] [12]

MOROCCO: A COUNTER ON THE MARITIME SILK ROAD

Morocco is "the first Arab country in north-west Africa to sign" -"Arab", but not "African". Are we to understand that, strategically speaking, Morocco is not looked upon as an African country by China and that its integration into the New Silk Roads must be analysed outside Africa? Geographically speaking at least, Morocco is the fifth (or seventh) African country officially accredited as participating in the construction of New Silk Roads behind Egypt, Ethiopia, Kenya and Djibouti (and perhaps also Madagascar and South Africa). [START_REF] Zhang Zhuomin | Moluoge: Yiqie zhengzai gaibian[END_REF] On July 11, 2018, China announced that two other Arab countries (Libya and Tunisia) had joined the club of countries participating in the construction of the New Silk Roads. 38 Beyond the somewhat disproportionate enthusiasm of the Moroccan press for this event, what is striking is that Morocco has implemented a rather proactive and successful free zone policy that distinguishes it from many African countries.

The Chinese government is encouraging the creation of special economic zones in Africa to serve as incubators for Chinese companies (in this case mainly privately owned SMEs) attempting to internationalise. The Chinese conception of these zones is to provide SMEs with in situ training while protecting them from bad surprises thanks to the sponsorship of the Chinese company organising their zone [START_REF] Thierry | China in Africa: Goods Supplier, Service Provider rather than Investor[END_REF] . In the Moroccan case, Chinese companies in the automotive sector (as in the aeronautics and other sectors) will not need the support of their national partners in a delocalised bubble. They have much better, the Western car manufacturers implanted in China of which they already are the subcontractors: so subcontractors therefore know what the manufacturer ask for who, in turn, know what its subcontractors can supply. [START_REF] Savage | Quelles ont été les coulisses des négociations du Maroc avec les grands groupes automobiles?[END_REF] Of course, Renault has been producing in the Tangier-Med zone since 2012; PSA Peugeot Citroën is also planning to start production at its Kenitra site in 2019. It is still the probable arrival of Volkswagen and the setting-up of an assembly line in the Kenitra area. But it could also be Toyota, Hyundai or even Fiat. As for Ford, although it does not intend to invest immediately, it is already using the services of subcontractors attracted by the dynamics of the Renault ecosystem in the Moroccan free zones to supply its Valencia plant in Spain. Renault also manages the production of the joint ventures founded by Nissan and Dongfeng and, since 2016, a joint venture under its name. 42 There is therefore a Renault ecosystem, stretching from China to Morocco, which can provide Chinese companies setting up in Morocco under its wing with all the services that MOFCOM requires from special economic zones under its supervision. 43 Peugeot is not lagging behind and is ordering the strengthening of its own ecosystem. On 22 February 2017, several Peugeot sourcing managers met with equipment and engine manufacturers from the Dongfeng group to organise their co-operation in Morocco. [START_REF] Dongfeng | Dongfeng ling bujian jituan ying lai PSA jituan keren[END_REF] One of the largest automotive suppliers, Faurecia, also has an ecosystem stretching from China to Morocco, where it can welcome its Chinese subcontractors.

In the case of Morocco, its inclusion in the group of countries under the New Silk Roads strategy is actually a means for the Chinese government to assert the role it grants Morocco: in the service of Chinese interests for better access to the European Union market (and, secondarily, in a more distant future, to the African market).

[14] from pirate attacks the merchant ships frequenting the Maritime Silk Road; this by itself does not justify the inclusion of Djibouti among the countries participating in the New Silk Roads. By contrast, the role assigned to Djibouti in the Digital Silk Road, most certainly does.

I retrace elsewhere the history of China Merchants in Djibouti [START_REF] Pairault | La China Merchants à Djibouti: de la route maritime à la route numérique de la soie[END_REF] . Here I shall confine myself to recalling the organisation chart of the joint-venture (East Aden Holding Co) in charge of the operation and maintenance of the Djibouti free zone (see Figure 4). On the one hand, Great Horn Investment, which represents the interests of the State of Djibouti, holds 40% of the shares in the operating company (East Aden). On the other hand, a consortium (International Djibouti Industrial Parks Operation) represents the interests of the five Chinese companies that hold 60% of the shares in the operating company. The major Chinese shareholder is actually the China Merchants Group which holds two thirds of the shares of the consortium (66.7%) through three entities of which China Merchants Ports is the main Chinese stakeholder in the project. The Port of Dalian Authority Group, 23% owned by China Merchants Ports, holds 25% of the consortium shares; as a result, China Merchants Group has 73% of the votes in the consortium. The last shareholder is IZP Network Technologies which, despite its modest shareholding (8.2%), should play a significant role. authorising certain non-financial institutions to act as intermediaries between debtors and creditors and to organise their payments by Internet and bank cards; it was selected in 2013 to join the seventeen experimental institutions in the field of electronic payments. 47 After the acquisition, IZP restructures Globebill whose development is so considerable that China Merchants decides, in November 2015, to invest 60 million dollars (15% of the shares). 48 IZP and the China Merchants Group have also set up the Silk Road E-Merchants which is to serve as a platform for a Global Port Alliance which would already encompass 29 ports and 55 terminals with a capacity of 80 million standard containers. This alliance is supposed to forge an ecosystem integrating the administration of port resources, organisation of a geographical information system, and management of big data, digital customs clearance and all operations facilitating the marketing of Chinese products. Under this scheme, payments would be managed by the International Business Settlement (IBS), another of IZP's subsidiaries, which would dedicate itself to the internationalisation of the yuan "while building efficient and direct financial clearing infrastructure aiming at building a financial highway for the globalisation of Chinese businesses". 49 This was only anticipating a decision by the Chinese government on 13 July 2018 that identifies 22 cities (mainly ports) as venues for cross-border e-commerce pilot zones in an effort to boost cross-border e-commerce and upgrade foreign trade. [START_REF]Guowuyuan changwu hui jueding xin she yi pi kua jing dianzi shangwu zonghe shiyan qu, chixu tuijin duiwai kaifang cujin waimao zhuanxing shengji[END_REF] As a logical extension of the previous structures, IZP created, in partnership with the Djiboutian State (20%) 51 , the first full-service bank established by Chinese companies abroad, Silk Road International Bank (SIB), which opened its doors on 17 January 2018. Here's what Luo Feng, president of IZP, says 52 :

[16]

countries along the New Silk Roads, whose power and financing capacities are limited, where enterprises find it difficult to finance themselves and where social development is hampered.

Establishing the relevant institutions in these regions to provide financial services will not only promote local trade, but also provide timely assistance so that financial services can make a breakthrough from scratch. For us, this means a phenomenal business opportunity.

We are implementing a Next Generation Settlement Network (NGSN) worldwide to build the first international settlement network using the NGSN standard that will globally integrate the regional clearing capabilities of each bank and their subsidiaries around the world, promoting a global development strategy.

We are building an international renminbi settlement system to provide banks with low-cost money transfer channels to increase their profit margins, organise payments in real time seven days a week, 24 hours a day, and improve the customer experience [...]

In the future, we will also implement a digital money system and the appropriate payment, settlement, supply chain financing, factoring and other tools. One can imagine the huge market this represents. As a financial hub, the new technologies and models adopted by the SIB should create a new financial architecture for the construction of the New Silk Roads in all its aspects.

This commitment should probably not be interpreted as an advocacy by geek entrepreneurs eager for advanced techniques but rather as a strain from entrepreneurs seeking to compensate for the inadequacies of financial tools at their disposal. Unlike Western countries whose financial presence around the world is old and well established, post-Mao China has not yet had time to build a brick-and-mortar financial network not the least along the Silk Roads. In other words, if these Chinese companies are attempting to technologically leapfrog their Western competitors 53 , it is more by an unavailability of the offline world than by an addiction to the online world. However, this concern to work out a solution may also be seen as a desire for domination -even if the choice of Africa could be more a matter of choosing a pilot site where standards and legal requirements would be laxer than in developed countries.

Figure 3 .

 3 Figure 3. -Trade with China (1995-2016)

Figure 4 .

 4 Figure 4. -Operating and maintaining Djibouti free trade zone

6

  See UNCTAD's online database (http://unctadstat.unctad.org). The table "Container port throughput, annual, 2010-2016" provides the total number of containers handled per port, expressed in twenty-foot equivalent (TEU), for individual countries. This figure covers the loading, unloading, repositioning and transhipment of containers. New Silk Roads" was 144.3 billion dollars, which is over tenfold more than China invested in these countries. This amount of new turnkey contracts represents 54.4% of all contracts of this type signed during the same period by Chinese contractors and corresponds to a series of 7,217 contracts. 8 China's strategy is primarily a commercial strategy: China sells services (build and finance infrastructure construction) to partner countries in order to carry on the New Silk Roads advancement and to dispatch Chinese products whose final destinations are the European countries. Africa, in this trade perspective, has only a limited importance. Should China's investment performance in Africa, as reported above, be regarded as unique? I have compiled the following two tables (see tables 1 and 2) by consulting the Eurostat website (the statistical office of the European Union).9 The extracted data is quite challenging to interpret. Let's look to start the first three years of Table1, from 2013 to 2015 (previous years being unavailable). A primary reading would lead to the conclusion that the European Union would have invested ten times more in Africa in 2015 than China, which itself would have invested ten times more than the United States (the European Union,

7 See 2016 nian zhongguo duiwai zhijie touzi tongji gongbao [2016 statistical Bulletin of China's Outward Foreign Direct Investment], 2017, p. 45 sqq, http://fec.mofcom.gov.cn/article/tjsj/tjgb/201711 /20171102669889.shtml.

Figure 2.-Regional container terminals traffic throughput and value (2016) Source: UNCTAD's online database (http://unctadstat.unctad.org)

[3] new turnkey contracts signed in countries labelled "

Table 1

 1

		. -Net FDI outflows in billions of dollars
	Economies	2013	2014	2015	2016
	European Union (28)	17,8	21,1	31,0	-14,2
	United Sates	1,1	1,4	0,3	-0,5
	China	3,4	3,2	3,0	2,4
	Netherlands	8,9	14,5	22,6	-6,8
	Luxembourg	2,1	1,7	-0,8	-2,6
	United Kingdom	3,9	3,4	n.d.	-7,7
	Germany	1,5	0,5	0,6	1,1
	France	-1,9	-0,2	2,3	-0,4
	Italy	2,8	3,7	3,9	2,4
	Source: Eurostat [bop_fdi6_flow]

Table 2

 2

		. -FDI stocks to Africa in billions of dollars
	Economies	2013	2014	2015	2016
	European Union (28)	248	282	310	289
	Netherlands	96	109	130	117
	United Sates	44	55	54	55
	United Kingdom	44	55	53	53
	France	46	49	52	46
	China	26	32	35	40
	Italy	14	16	20	22
	Luxembourg	20	22	24	19
	Germany	8	9	8	10
	Japan	n.d.	8	8	9
	Portugal	5	7	6	6
	Spain	5	5	5	5
	Source: Eurostat [bop_fdi6_pos]

Table 3 .

 3 -China's potential dependence onAfrica (2016)

	Commodities by	as a percentage of	as a percentage of	rank among the
	country of origin	China's imports of	Chinese imports of	countries from
		these commodities	these goods from	which imports
		from Africa	the World	originate
	Ores and Metals
	South Africa	50%	5%	5
	Zambia	14%	1%	15
	Congo (DRC)	14%	1%	16
	Fuels
	Angola	63%	8%	3
	Congo (Brazzaville)	10%	1%	20
	South Sudan	7%	1%	23

Source: author's calculations from UNCTAD's online database.

Table 4

 4

		. -Africa's potential dependence on China (2016)
	countries with at	Share of exports to China in the total of this export category
	least 10% of its total exports to China	All products	All food items	Agricultural raw materials	Ores and metals	Fuels	Manufactu-red goods
	Sierra Leone	71%	16%	91%	87%	0%	4%
	Congo (DRC)	50%	0%	22%	60%	37%	1%
	Sudan	50%	8%	7%	26%	78%	4%
	Angola	46%	2%	61%	0%	48%	0%
	Congo	44%	2%	64%	2%	59%	0%
	Eritrea	39%	56%	4%	59%	0%	7%
	Mauritania	35%	3%	1%	85%	0%	2%
	Central African Rep.	30%	0%	37%	0%	0%	1%
	Gambia	23%	0%	96%	44%	0%	0%
	Zambia	19%	8%	16%	24%	0%	2%
	Zimbabwe	18%	33%	6%	10%	0%	14%
	Equatorial Guinea	14%	0%	98%	0%	9%	0%
	Cameroon	13%	0%	30%	1%	15%	1%
	Niger	10%	26%	7%	0%	0%	7%
	Ethiopia	10%	13%	1%	51%	0%	16%
	Africa	10%	3%	22%	28%	14%	2%
	Source: author's calculations from UNCTAD's online database.

Table 5 .

 5 -Africa's imports of manufactured goods (2016) If we go up value chains, China's participation in the supply of high-tech manufactured goods to Africa is probably much smaller than suggested in Table3. Therefore, Africa's dependence on China is multiple. It is both a direct dependence (purchase of inexpensive goods "designed by China") and the mediatisation of a dependence on Western countries (acquisition of goods "made in China") since China is often the channel through which African countries can gain an affordable access to advanced Western techniques.

				by degree of manufacturing
	as a percentage of total imports in the category	Manufactured goods	Labour-intensive and resource-intensive manufactures	Low-skill and technology-intensive manufactures	Medium-skill and technology-intensive manufactures	High-skill and technology-intensive manufactures	GDP billion dollars
	United States	6%	1%	4%	7%	7%	18 730
	European Union (28)	35%	25%	26%	42%	34%	16 494
	China	24%	39%	27%	20%	19%	11 218
	Other economies	35%	35%	43%	31%	40%
	Source: author calculations from UNCTAD's online database.

This opposition between low-tech and high-tech is all the more significant as many high-tech goods "made in China" are manufactured under the control of Western companies -including European ones. Some Western goods made in China are no more than low-cost variants under dedicated brands. For example, as far as truck manufacturers are concerned, Shacman trucks are Man low-cost trucks made in China, just as Foton trucks are Daimler-Benz low-cost trucks or Dongfeng trucks are Volvo, Nissan and Renault low-cost trucks.

  ). At the time of the vote, of the 20 African countries that officially recognised China, only 17 were unconditionally supporting Beijing and voted to reject draught resolution A/L632 requiring a[9] two-thirds majority vote. With a four-vote majority against this resolution, China was guaranteed entry: 26 African countries approved resolution A/l630 thereby voting in favour of China 22 . It is thus an opportunistic reaction which is prolonged through a series of recognition by a wide range of African countries, the number of which doubles within three years. Once this key period had passed, the number of recognition continued to grow, but at a slower pace since over the next forty years(1976-2015), the number of recognition by African countries increased by only eleven, to which were added two others in 2016 and a new one in 2018 -leaving Taiwan with only one African ally (Swaziland, recently renamed eSwatini).Today, there are 193 UN member states, including 54 African countries, representing nearly one third (28%) of the membership. The day China is recognised by all 54, it will be able to boast the support of an entire continent and, whether or not it is "chequebook diplomacy", "debt diplomacy"…, its political weight will be more than confirmed. At the same time, China supports the Polisario Front and the independence of Western Sahara (Sahrawi Arab Democratic Republic) which is partially recognised by the UN. Also, Somaliland, which maintains good relations with Ethiopia, is reported to be watched longanimously by Beijing, which plans to help modernise Hargeisa airport 23 . The "conquest" of Africa is not quite over yet, but Chinese leaders are preparing for it.

Africa's strategic importance to China can be inferred from two articles, each with its own specific concern. A September 8, 2017 article in the 21st Century Business Herald reported Ou Xiaoli's (Inspector with the Western Provinces Development Department at the National Development and Reform Commission) comments that incentives to take the New Silk

  Morocco has recognised China since 1958. Unlike Sino-Algerian relations marked by the exaltation of a combat fraternity which, at least in political statements, would justify the current Chinese presence in Algeria, Sino-Moroccan relations have soon been marked by a somewhat coolness since the ban and then the dissolution of the Moroccan Communist Party (MCP) at the end of the 1950s.31 Today, the photo of Ali Yata -a Moroccan of Algerian descent who became Secretary General of the MCP 32 -shaking hands with Mao Zedong in 1959 in Beijing serves to build a new alliance between the two countries based no longer solely on politics, but on economics33 . Afterwards the relationship was improved by a series of visits by Chinese leaders to Morocco 34 accompanied by counter-visits, including that of the King Mohammed VI in 2002 and 2016. Discussions for the implementation of a strategic partnership would have formally started in 2012 initially with the idea of signing in 2014 which, in practice, has been delayed until 2016.35 On November 17 of the following year, "Morocco and China signed a Memorandum of Understanding to promote together the construction of the Silk Road Economic Belt and the 21 st -century Maritime Silk Road".36 This step must not give rise to an interpretation endorsing Wei Jiangguo's theses. Reading the official communiqué, which provides only a minimum service, several observations should be made at the outset. It is only a memorandum of understanding, i.e. a text which affirms a convergence of views and intentions, but which does not set any objectives involving a legal commitment by the parties. Consequently, it is a text with more political than economic significance, which is reinforced by the fact that it was signed with the [Chinese] Ministry of Foreign Affairs and not with the [Chinese] Ministry of Commerce. If Morocco's inclusion in the new Silk Roads' strategy is primarily political here, it is nonetheless significant of a certain evolution: the opening of this strategy to Africa. Finally, the communiqué states that "Ali Yata, l'authentique défenseurde la cause nationale", published in Al Bayane on 14 November 2013 and reprinted by the Maghress aggregator, https://www.maghress.com/fr/albayane/119095. Ali Yata rencontre Mao", in special issue of Diplomatica retracing "55 ans d'amitié et de coopération entre la Chine et le Maroc" », revue de la Fondation diplomatique, 30 September 2013, p. 7, https://issuu.com/diplomatica/docs/d59w. 34 Yang Shangkun in 1992, Jiang Zemin in 1999, Wu Bangguo in 2005, Hu Jintao in 2006, Li Changchun in 2008, Wen Jiabao in 2012, Yu Zhengsheng in 2014; see "Zhongguo tong Moluoge de guanxi", [China's relationship with Morocco], on the website of the Chinese Ministry of Foreign Affairs http://www.fmprc.gov.cn/chn/ /gxh/cgb/zcgmzysx/fz/1206_35/1206x1/t359843.htm. 35 See, special issue of Diplomatica, op. cit., p. 21; Rachid Rhattat, "La relation économique et commerciale sino-marocaine : De la coopération au partenariat stratégique", L'Année du Maghreb, IX, 2013, p. 242 ; "Zhonghua renmin gongheguo he Moluoge wangguo guanyu jianli liang guo zhanlüe huoban guanxi de lianhe shengming" [Joint Statement of the People's Republic of China and the Kingdom of Morocco on establishing a Strategic Partnership between the two countries], 11 May 2016,

31 Jamaâ Baida, "Le communisme au Maroc pendant la période coloniale

(1912-1956)

", Proceedings of the Conference 'European Totalitarianism in the Mirrors of Contemporary Arab Thought', Beirut, October 6-8, 2010, Hg. Manfred Sing (Orient Institute Studies, 1), §22, https://www.perspectivia.net/publikationen /orient-institut-studies/1-2012/baida_communisme. 32 33 "1959 : http://www.xinhuanet.com/politics/2016-05/11/c_1118849465.htm. 36 Wang Qingyun, "Wang Yi tong Moluoge waijiao yu guoji hezuo dachen Bulida juxing huitan" [Wang Yi Holds Talks with Moroccan Minister of Foreign Affairs and International Cooperation Bourita], Zhongguo ribao [China Daily], 17 November 2017, http://cn.chinadaily.com.cn/2017-11/17/content_34654540.htm.

  All of[13] them maintain close relations in China with all the companies in the Chinese automotive sector since they are associated with manufacturers: Ford with Jiangling; Volkswagen with Shanghai Automotive Industry Corporation (SAIC) and First Automobile Works (FAW), the first and eleventh Chinese manufacturers; Peugeot and Renault with Dongfeng, the third Chinese manufacturer.41 The relations between Peugeot and Dongfeng are generally well known not only because the first work with the second in China, but also because the second entered PSA's capital (14%) as part of a capital increase in 2014. Renault's relations with Dongfeng, however, are less well known. France's largest car manufacturer sold its shares in Dongfeng Trucks (also known as Dongfeng Commercial Vehicles in Chinese) to Volvo Trucks (which had already bought Renault Trucks in 2001 and Nissan Diesel in 2006).

UNCTAD SITC Classes 0 + 1 + 2 + 3 + 4 + 68 + 667+ 971.

UNCTAD SITC Classes 27 + 28 + 68.

UNCTAD SITC Classes 0 + 1 + 22 + 4.

UNCTAD SITC Classes 2 less 22, 27 and 28.

UNCTAD SITC Classes 27 + 28 + 68.

UNCTAD SITC Classes 3.

Paul Tavernier, « L'année des Nations Unies (décembre 1970 -décembre 1971). Questions juridiques », Annuaire français de Droit international, 1971, n°17, p. 553-556.

Voir Zhongguo jiang bangzhu suomali lan kuojian jichang [China will help Somaliland expand its airport], Jingwai touzi [China Invest], 3 Septembre 2014, https://chinainvestin.com/index.php/zh/invest-in-2/news/1257.

 24 In Chinese minying qiye or minqi; the concept does not identify with what is generally understood by "private enterprises". These are limited companies under private law whose capital is totally, partially or not at all

public.25 See Zhou Xiaoxiao, « Minqi 'yidai yilu' yanxian zenme touzi: Zhuanjia jianyi shouxuan dongnanya shi » [How to invest in countries along the New Silk Roads: one expert advises to focus on Southeast Asia], 21 shiji jingji baodao [21st Century Business Herald], 8 September 2017, http://epaper.21jingji.com/html/2017-09/08/content_70140.htm.