Malek Ghanes
email: ghanes@ensea.fr

Gang Zheng
email: zheng@ensea.fr

On Sensorless Induction Motor Drives: Sliding Mode Observer and Output Feedback Controller

In this paper, a sensorless output feedback controller is designed in order to drive the Induction Motor (IM) without the use of flux and speed sensors. Firstly, a new sliding mode observer that uses only the measured stator currents is synthesized to estimate the speed, the flux and the load torque. Secondly, a current-based field oriented sliding mode control is developed so as to steer the estimated speed and flux magnitude to the desired references. Stability analysis based on Lyapunov theory is also presented in order to guarantee the closed loop stability of the proposed observer-control system. Two experimental results for a 1.5-kW IM are presented and analyzed by taking into account the unobservability phenomena of the Sensorless Induction Motor (SIM).

I. INTRODUCTION

Vector controlled IM drives are wide spread electromechanical conversion systems with efficient energy [START_REF] Boglietti | Energy-efficient motors[END_REF] for highdynamic performance applications, where motion control or high precision speed control is needed [START_REF] Tsuji | A Wide-Range Velocity Measurement Method for Motion Control[END_REF]. Since rotational transducers and their associated digital or analogue circuits give extra costs and are often complex and rather fragile, reducing the robustness of the total system, there is an increasing interest in industry in control schemes without rotational sensors-the so-called SIM control. We refer the reader to [START_REF] Holtz | Sensorless control of induction machines-with or without signal injection[END_REF] for a tutorial account on the topic.

Given that high-performance controllers are readily available when speed and flux are known, it seems reasonable to estimate speed and flux, in the spirit of observer and control theories, to replace in the control scheme the actual speed and flux by its estimation. Several theoretical and practical solutions have been proposed in the literature. For instance, in [START_REF] Kubota | Speed sensorless filed-oriented control of induction motor with rotor resistance adaptation[END_REF] an algorithm for simultaneous estimation of motor speed and rotor resistance is proposed. A semi-global exponential rotor velocity and flux tracking algorithm is proved in [START_REF] Feemster | Sensorless rotor velocity tracking control for induction motors[END_REF]. In [START_REF] Marino | A nonlinear tracking control for sensorless induction motors[END_REF], a second-order control algorithm is designed including a rotor speed observer. An estimator of rotor speed and an indirect field-oriented control with a sliding mode are proposed in [START_REF] Barambones | A sensorless variable structure control of induction motor drives[END_REF]. In [START_REF] Pea | Sensorless Control of Doubly-Fed Induction Generators Using a Rotor-Current-Based MRAS Observer[END_REF], based on the model reference adaptive system (MRAS), a sensorless method estimating the rotor position and speed from the machine rotor currents is presented. Reduced and full order observers for flux and speed estimation of sensorless induction-motor drives are analyzed in [START_REF] Harnefors | Complete Stability of Reduced-Order and Full-Order Observers for Sensorless IM Drives[END_REF]. In [START_REF] Comanescu | Decoupled Current Control of Sensorless Induction-Motor Drives by Integral Sliding Mode[END_REF] the problems of current decoupling control and controller tuning associated with sensorless vector-controlled induction motor drives are studied. Parallel identification schemes for both speed and stator resistance of sensorless induction motor drives are proposed in [START_REF] Zaky | Wide-Speed-Range Estimation With Online Parameter Identification Schemes of Sensorless Induction Motor Drives[END_REF] for a wide range of speed estimation. Nevertheless for most of these contributions, rotor flux information and/or knowledge of load torque (and/or acknowledge load torque considered constant) are needed for controller implementation. On the other hand, removing the speed sensors affects the IM observability property at very low speed. Important contributions in this direction have been reported in [START_REF] Ghanes | Observability Study and Observer-Based Interconnected Form for Sensorless Induction Motor[END_REF], where under some operating conditions (low speed) the IM is not observable. Furthermore, strategies based on IM spatial saliency methods with fundamental excitation and high frequency signal injection [START_REF] Holtz | Sensorless control of induction machines-with or without signal injection[END_REF], extended Kalman filter techniques and adaptive system approaches [START_REF] Montanari | Speed sensorless control of induction motor based on indirect Field-orientation[END_REF] have been studied. The sensorless control of IM allowing operation at very low speed can also be found in [START_REF] Edelbaher | Low-Speed Sensorless Control of Induction Machine[END_REF], [START_REF] Feemster | Sensorless rotor velocity tracking control for induction motors[END_REF], [START_REF] Ghanes | Observability Study and Observer-Based Interconnected Form for Sensorless Induction Motor[END_REF], [START_REF] Ghanes | Cascade and highgain observers comparison for sensorless closed-loop induction motor control[END_REF], [START_REF] Leppanen | Speed-Sensorless Induction Machine Control for Zero Speed and Frequency[END_REF], [START_REF] Staines | Sensorless Control of Induction Machines at Zero and Low Frequency Using Zero Sequence Currents[END_REF].

The first main contribution of this paper is to design a new sliding mode observer for the SIM in presence of unknown load torque. And experimental results show the performances of the observer thanks to a dedicated Sensorless Observer Benchmark to test observers in open loop. Secondly, a sensorless control scheme is designed, which is based on a combination of field oriented control (FOC) methodology and robust sliding mode technique. Sufficient conditions are given to guarantee the stability of the whole closed-loop system. Furthermore, a Sensorless Control Benchmark is designed in order to test and to evaluate the performance of sensorless controllers. Theoretical and experimental results show the feasibility of the proposed method.

II. OBSERVER DESIGN

A. IM Model

In the rotating (d-q) reference frame, the IM dynamic model (1) reads ( [START_REF] Chiasson | Modeling and High-Performance Control of Electric Machines[END_REF])

                     ωr = mφ rd i sq -cω r -1 J T l φ rd = -aφ rd + aM sr i sd ρ = pω r + a Msr φ rd i sq isd = -γi sd + abφ rd + pω r i sq + a Msr φ rd i 2 sq +m 1 V sd isq = -γi sq -bpω r φ rd -pω r i sd -a Msr φ rd i sd i sq +m 1 V sq (1)
where respectively i sd -i sq , V sd -V sq , φ rd -ρ, ω r -T l denote the stator currents and stator voltages, the norm of flux and angle of flux, the speed and load torque, where the subscripts s and r refer to the stator and rotor. The parameters a, b, c, γ, Υ, m and m 1 are defined by a

= (R r /L r ), b = (M sr /ΥL s L r ), c = (f v /J), γ = L 2 r Rs+M 2 sr Rr ΥLsL 2 r , Υ = 1 -(M 2 sr /L s L r ) , m = (pM sr /JL r ), m 1 = (1/ΥL s ),
where R s and R r are the resistances, L s and L r are the self-inductances, M sr is the mutual inductance between the stator and rotor windings, p is the number of pole-pairs, J is the inertia of the system (motor and load) and f v is the viscous damping coefficient. The control inputs are the stator voltages. Only stator current and stator voltage are measurable. Furthermore, an operating domain D is defined as follows:

Definition 1: Operation Domain D: Φ rd max , I sd max , I sq max , ω r max and T l max are respectively the actual maximum values for the flux, currents, speed and load torque such that

|φ rd | ≤ Φ rd max , |i sd | ≤ I sd max , |i sq | ≤ I sq max , |ω r | ≤ ω max r , |T l | ≤ T l max .

B. Quick Review on the Observability Phenomena of SIM

In [START_REF] Ghanes | Observability Study and Observer-Based Interconnected Form for Sensorless Induction Motor[END_REF], we have demonstrated that IM observability cannot be established in the particular case when the fluxes φ rα , φ rβ and the speed ω r are constant even if the higher derivatives of outputs are used. This operating case coincides with the following physically interpretations:

1) when the fluxes are constant ( φrα = φrβ = 0), or equivalently the excitation voltage is zero (ω s = 0), it implies that: pω r + Rr Te Rr + f v (Fig. 1) (see [START_REF] Ghanes | Observability Study and Observer-Based Interconnected Form for Sensorless Induction Motor[END_REF] for more details).

-6 ω r T l @ @ @ @ @ @ @ @ -M Fig. 1. Unobservability curve in the map (T l , ωr) .

Obviously, the observability is lost gradually when we approach this unobservability curve.

C. Sliding Mode Observer design

Several observers such as the sliding mode observer have been developed to estimate rotor speed. Sliding mode observer [START_REF] Barbot | Sliding Mode Observers" chapter 4, book[END_REF] appears as an important technique, since it offers many good properties, such as good performance against unmodeled dynamics, insensitivity to parameter variations, external disturbance rejection and fast dynamic response [START_REF] Utkin | Sliding mode control design principles and applications to electric drives[END_REF].

Since the measurements of the motor are given in the classical fixed stator frame (a,b,c) while the observer is written in the frame of the rotating rotor field (d-q), it is thus necessary to carry out a change of reference from the measures. Initial measurements are transformed from the three-phase reference frame to a diphasic reference frame by using the following Concordia equations:

   V α = 2 3 (V a -1 2 V b -1 2 V c ); V β = 2 3 (V b -V c ) i α = 2 3 (i a -1 2 i b -1 2 i c ); i β = 2 3 (i b -i c )
where V a , V b , V c and i a , i b , i c are the three-phase voltages of IM and three-phase currents (supposed balanced).

The next step consists in passing in the turning reference frame by the Park transformation. This transformation requires the calculation of rotor field frame angle with respect to the fixed reference frame. This calculation is carried out starting from the equations of (2) to (4), just as the calculation of the new measurements to the frame (d-q). dρ dt

= pω r + α r M sr φrd i sq (2)
V sd = cos(ρ)V α + sin(ρ)V β (3)
V sq = -cos(ρ)V α + sin(ρ)V β (4)
Let us consider system (1) with the outputs y = i sd i sq .

It is easy to check that φ rd and ω r of (1) is not observable. Hence classical observer design techniques cannot be applied. However, we notice that (1) has a stable zero dynamics [START_REF] Isidori | Nonlinear control systems[END_REF] about φ rd , with the output y 1 = i sd and a > 0 in (1), hence by simply using an estimator, it is possible to estimate or detect φ rd . With the estimation of φ rd , then ω r of (1) becomes observable. Hence we can use a sliding mode observer to recover ω r . Remark 1: Since system (1) stays in D defined in Definition 1, thus it is Bounded Input Bounded State (BIBS).

For system (1), we propose the following sliding mode observer:

         φrd = -a φrd + aM sr i sd isd = -γi sd + E 1 ab φrd + E 1 a Msr φrd i 2 sq +m 1 V sd + E 1 λ 1 sgn (i sd -îsd ) ωr = E 1 m φrd i sq -E 2 cω r + E 2 λ 2 sgn (ω r -ωr ) (5)
where sgn( * ) :

   1 if * > 0 -1 if * < 0 ∈ [-1, 1] . if * = 0 (6)
with the auxiliary state ωr = E2λ1sgn(i sd -î sd ) pisq where

E 1 = 0, if | φrd -φ rd | > ǫ 1, if | φrd -φ rd | ≤ ǫ; E 2 = 0, if îsd = i sd 1, if îsd = i sd ;
where ǫ is a small positive real and the estimation of load torque T l given by the following equation

Tl = -E 3 Jλ 2 sgn (ω r -ωr ) (7)
where E 3 = 0, if ωr = ωr 1, if ωr = ωr . Theorem 1: For system (1), the proposed observer (5) can exponentially converge to φ rd and estimate ω r and T l .

Proof: Assuming that E 1 = 0 (if E 1 = 1 we directly move to the next step), the observation error dynamics of φ rd , noted as ε φ rd = φ rd -φrd , is defined as: εφ rd = -aε φ rd with a > 0, and it implies exponential convergence of φrd to φ rd .

Given a sufficient small real ǫ > 0, we can always find t 1 , such that when t > t 1 , we have ε φ rd ≤ ǫ, hence we have E 1 = 1 after t > t 1 . Then we consider the convergence of îsd to i sd . For this, we note ε i sd = i sd -îsd . Since when t > t 1 we have E 1 = 1 and ε φ rd ≤ ǫ where ǫ is a sufficient small real, then

εi sd = ab - aM sr i 2 sq φ rd φrd ε φ rd + pω r i sq -λ 1 sgn (i sd -îsd )
Let us consider the following Lyapunov function:

V εi sd = 1 2 ε 2 i sd . Since system (1) stays in D, hence if λ 1 = max{pω r i sq + ab - aM sr i 2 sq φ rd φrd ε φ rd } + ζ i sd (8)
with ζ i sd > 0 and ε φ rd ≤ ǫ, we have

Vε i sd = pωrisq + (ab - aMsri 2 sq φ rd φrd )ε φ rd -λ1sgn (εi sd ) εi sd ≤ -ζi sd |εi sd | = - √ 2ζi sd V 1/2 ε φ rd
which implies the convergence of îsd to i sd in a finite time, noted as t 2 . Hence after t > t 2 > t 1 , we have E 2 = 1 and εi sd = ε i sd = 0, which gives pω r i sq + ab - Consequently we have

ωr = ωr + Ė(t) = mφ rd i sq -cω r - 1 J T l + Ė(t)
Analogously, by taking the Lyapunov function:

V εω r = 1 2 ε 2 ωr
with ε ωr = ωr -ωr , and if

λ 2 = max{- 1 J T l + mi sq ε φ rd + cE(t) + Ė(t)} + ζ ωr (9)
with ζ ωr > 0, we have

Vεω r = - 1 J T l + mi sq ε φ rd + cE(t) + Ė(t) ε ωr -λ 2 sgn (ω r -ωr ) ε ωr ≤ -ζ ωr |ε ωr | = - √ 2ζ ωr V 1/2 εω r
which signifies the convergence of ωr to ωr in a finite time, noted as t 3 . Hence after t > t 3 > t 2 , we have E 3 = 1 and εωr = ε ωr = 0, which gives Tl = T l -Jmi sq ε φ rd -JcE(t) -J Ė(t). Since ωr and φrd converge exponentially to ω r and φ rd respectively, which implies ε φ rd , E(t) and Ė(t) vanish exponentially. Consequently Tl converges exponentially to T l .

III. FOC VIA SLIDING MODE TECHNIQUES

In this section, a controller is designed by combining the FOC method ( [START_REF] Blaschke | The principle of field orientation applied to the new transvector closed-loop control system for rotating field machines[END_REF]) with Sliding Mode Control method (SMC, [START_REF] Utkin | Sliding mode control design principles and applications to electric drives[END_REF], [START_REF] Sabanovic | Application of sliding modes to induction motor control[END_REF]). The design procedure is based on the well-known assumption of current-fed IM (see [START_REF] Khalil | Sensorless Speed Control of Induction Motor[END_REF], [START_REF] Chiasson | Modeling and High-Performance Control of Electric Machines[END_REF]).

A. Field Oriented Control. Consider the IM dynamic model given by (1) in the (d-q) reference frame. In this frame the electromagnetic torque T em = pMsr Lr φ rd i sq is proportional to the product of φ rd and i sq . Thus by holding constant the magnitude of the rotor flux, a linear relation between i sq and T em is obtained. In order to cancel the nonlinear dynamics of i sd and i sq , the system is forced into current-command mode by using high gain feedback (see [START_REF] Khalil | Sensorless Speed Control of Induction Motor[END_REF], [START_REF] Chiasson | Modeling and High-Performance Control of Electric Machines[END_REF]). More precisely, the following PI current controllers

V sd = Ki vd t 0 (i * sd -i sd )dt + Kp vd (i * sd -i sd ) V sq = Ki vq t 0 (i * sq -i sq )dt + Kp vq (i * sq -i sq ) (10)
are used to force i sd and i sq to track their respective references i * sd and i * sq and produce fast responses when large feedback gains are used. Hence, assuming that i * sd and i * sq as the new inputs, it follows that

ωr = mφ rd i * sq -cω r -T l J φ rd = -aφ rd + aM sr i * sd (11)
In order to solve the flux and speed trajectory tracking problem, the following assumption is introduced. where κ = aM sr and ∆a is the uncertainty term of parameter a. In order to design a flux sliding mode controller, we define the flux tracking error e φ rd = φ rd -φ * rd where φ * rd is the flux reference. Then the associated error dynamics is

ėφ rd = -ae φ rd + κi * sd -aφ * rd -φ * rd + ∆aφ rd ( 13
)
¿From SMC theory, let us define the φ rd flux sliding manifold as follows

σ φ rd = e φ rd -(k φ rd -a) t 0 e φ rd (τ )dτ.
Then the associated Lyapunov function is selected as V σ φ rd = 1 2 σ 2 φ rd , where its time derivative is given by

Vσ φ rd = σ φ rd [σ φ rd 1 + σ φ rd 2 i * sd + ∆aφ rd ] (14)
with σ φ rd 1 = -k φ rd e φ rd -aφ * rd -φ * rd and σ φ rd 2 = κ. Therefore, the sliding mode controller can be designed as follows i * sd,eq = -

l φ rd σ φ rd + σ φ rd 1 + η φ rd sgn(σ φ rd ) σ φ rd 2 (15)
with the sign function is defined by [START_REF] Comanescu | Decoupled Current Control of Sensorless Induction-Motor Drives by Integral Sliding Mode[END_REF]. Consequently, [START_REF] Isidori | Nonlinear control systems[END_REF] becomes

Vσ φ rd = -l φ rd V σ φ rd +σ φ rd [-η φ rd sgn(σ φ rd )+∆aφ rd ]. (16)
Choosing l φ rd > 0 and η φ rd > max{ ∆aφ rd } (defined hereafter), it follows that Vσ φ rd ≤ 0. As V σ φ rd is contracting. ¿From Assumption (1-a-b), then max{φ rd } := K max φ rd can not be greater than max{φ rd (0), φ * rd } + |∆e φ rd (0)|. Consequently, ∆aφ rd is bounded and can be set as

η φ rd = ∆a max K max φ rd + b φ rd (17)
with b φ rd a small positive constant. Hence [START_REF] Kubota | Speed sensorless filed-oriented control of induction motor with rotor resistance adaptation[END_REF] becomes

Vσ φ rd = -l φ rd V σ φ rd -b φ rd |σ φ rd | < - √ 2b φ rd V 1/2
which implies that we have σ φ rd = 0 in a finite time. Therefore [START_REF] Leppanen | Speed-Sensorless Induction Machine Control for Zero Speed and Frequency[END_REF] becomes

ėφ rd = (k φ rd -a)e φ rd . (18)
Hence, the flux tracking error e ωr exponentially converges to 0 for (k φ rd -a) < 0.

Choosing [START_REF] Khalil | Sensorless Speed Control of Induction Motor[END_REF] to force φ rd to track its reference φ * rd ensures that the flux is properly established in the motor. Hence, after the IM is fluxed (φ rd = φ * rd = constant), the electromagnetic torque (T em ) can be rewritten as T em = K T i * sq , where K T is the motor torque constant defined by K T = pMsr Lr φ rd . As a consequence, the linear relationship between the input i * sq and the speed dynamics ωr is obtained. Then, the speed control is obtained through the input i * sq via a speed controller described below.

Speed controller design. Consider the mechanical equation of (11) including uncertainties as follows

ωr = -cω r + hi * sq + d ωr (19)
where h = mφ rd and d ωr = -∆cω r -T l J is the term uncertainty. Defining the speed tracking error e ωr = ω r -ω * r , it follows

ėωr = -ce ωr + hi * sq -cω * r -ω * r + d ωr (20)
Define now the sliding manifold as follows

σ ωr = e ωr -(k ωr -c) t 0 e ωr (τ )dτ (21)
and the candidate Lyapunov function associated to the sliding manifold ( 21) is defined as V σω r = 1 2 σ 2 ωr . By computing its time derivative, we obtain

Vσω r = σ ωr σ ωr = σ ωr [σ ωr 1 + σ ωr2 i * sq + d ωr ]
where σ ωr 1 = -ke ωr -cω * r -ω * r and σ ωr2 = h.

Then, the speed controller reads i * sq,eq = -l ωr σ ωr + σ ωr 1 + η ωr sgn(σ ωr ) σ ωr2 [START_REF] Tsuji | A Wide-Range Velocity Measurement Method for Motion Control[END_REF] with the sign function is defined by [START_REF] Comanescu | Decoupled Current Control of Sensorless Induction-Motor Drives by Integral Sliding Mode[END_REF]. Then [START_REF] Tsuji | A Wide-Range Velocity Measurement Method for Motion Control[END_REF] becomes

Vσω r = -l ωr V σω r + σ ωr [-η ωr sgn(σ ωr ) + d ωr ].
By choosing l ωr > 0 and η ωr > max{ d ωr } (defined hereafter). From Assumption (1-a-b) then max{ω r } := K max ωr can not be greater than max{|ω r (0)|, |ω * r |} + |∆e ωr (0)|. Consequently, as d ωr = ∆cω r + T l J then d ωr is bounded. Finally, from Assumption (1-c) η ωr is set as

η ωr = ∆cK max ωr + ̺ J + b ωr (23)
with b ωr a small positive constant. Following the same procedure used for flux, we have Vσω r < -√ 2b ωr V 1/2 , which means we obtain σ ωr = 0 in a finite time. Therefore [START_REF] Sabanovic | Application of sliding modes to induction motor control[END_REF] implies ėωr = (k ωr -c)e ωr (24) which makes that the speed tracking error e ωr exponentially converges to 0 for (k ωr -c) < 0. Proposition 1: Consider IM model ( 11) and suppose that Assumption 1 is satisfied. Then under the action of speed controller [START_REF] Tsuji | A Wide-Range Velocity Measurement Method for Motion Control[END_REF] and flux controller [START_REF] Khalil | Sensorless Speed Control of Induction Motor[END_REF], the rotor speed and the flux track their desired trajectories exponentially.

Proof:

Using V c = 1 2 σ 2 φ rd + 1 2 σ 2 ωr as a Lyapunov function candidate, then the time derivative gives Vc = -l φ rd V σ φ rd + [-η φ rd sgn(σ φ rd ) + ∆aφ rd ] -l ωr V σω r + [-η ωr sgn(σ ωr ) + d ωr ] < - √ 2bV 1/2
where b = min{b φ rd , b ωr }. It implies that after a finite time we have ėφ rd = (k φ rd -a)e φ rd and ėωr = (k ωr -c)e ωr , hence we have exponential convergence of the rotor speed and flux.

IV. STABILITY ANALYSIS OF THE CLOSED-LOOP SYSTEM

In order to implement controllers ( 15) and ( 22), the speed/flux measures are replaced by their estimates resulting in the new controllers with the speed ωr and the flux φrd are given by observer [START_REF] Chiasson | Modeling and High-Performance Control of Electric Machines[END_REF].

i * sd φrd = -l φrd σ φrd + k φrd e φrd + aφ * rd + φ * rd -η φrd sgn(σ φrd ) κ (25)
The IM observer must be fluxed to ensure estimated speed tracking. In order to avoid the singularity in (26), the flux (5) is initialized with initial conditions different from zero. In practice, electrical engineers overcome this singularity by starting to track firstly the flux φ rd to its reference φ * rd = constant. The same trick is adopted for the estimated flux φrd by adding an offset ε = 0.05W b such as

i * sq (ωr) = -lωrσ ωr + kω r e ωr + cω * r + ω * r -ηωrsgn(σ ωr ) max{ φrd , ε}m (27)
Theorem 2: Consider system (1) where ( 5) is an associated sliding mode observer which is initialized in D, suppose that Assumption (1-a-b) holds. Under the action of controllers defined in ( 25) and ( 27), the tracking errors of speed and flux converge asymptotically to zero.

Proof: By choosing the whole Lyapunov function as

V = V 1 + V 2 where V 1 = 1 2 σ 2 φrd + 1 2 ε 2 φ rd and V 2 = 1 2 σ 2 ωr + 1 2 ε 2
ωr , then we have V = V1 + V2 where

V1 = σ φrd σ φrd + ε φ rd εφ rd = σ φ rd ė φrd -(k φ rd -a)e φrd + ε φ rd φrd -φrd ≤ σ φrd -ae φrd + κi * sd φrd -aφ * rd -φ * rd + ∆a φrd -(k φ rd -a)σ φrd e φrd -aε 2 φ rd = σ φrd -l φrd σ φrd + ∆aφ rd -η φrd sgn(σ φrd ) -aε 2 φ rd ≤ -l φrd σ 2 φrd -b φ rd σ φrd -aε 2 φ rd < -µ1V1
with µ 1 = min{2l φrd , 2a} and Before evaluating the performances of the proposed observer-controller in closed loop, it is necessary to test and validate the performances of the proposed observer in open loop. For that a dedicated benchmark (Fig. 7) is defined to test observers on and near the unobservability curve 1 .

V < -µ 1 V 1 -µ 2 V
1 see section II-B for more details Zone 1. The initial conditions of speed and stator pulsation are such that the IM is observable (from 0s to 2s). Zone 2. The stator voltages is forced to zero corresponding to constant fluxes while the rotor velocity remains constant, making the state unobservable between 4 and 5 seconds and between 6 and 7 seconds. Zone 3. Between 5 and 6 seconds, the rotor moves with a constant acceleration, allowing to check the observer convergence when the state is slightly observable. Zone 4. The IM is driven outside the unobservability conditions. Practically, to apply this benchmark, the main difficulty lies in the simultaneous control of speed and stator pulsation so that the slip pulsation ω g = ω s -pω does not exceed a limiting value ω g = R r M sr i q /L r φ d , which corresponds to the highest admissible stator current. The reference slip pulsation is given in Fig. 3.c. In order to respect the above condition, it is necessary to drive the speed of the motor by a connected synchronous motor which is controlled to follow the speed trajectory. Simultaneously, the frequency of the voltages applied to the IM stator follows the stator pulsation reference shown in Fig. 3. Moreover robustness tests are defined by realistic variation of stator resistance and stator inductance. This benchmark is applied on an experimental setup. The frequency of the voltages applied to the stator of IM is controlled by classical U/f control which is independent of motor measurements and estimated state. At the same time, the speed of the IM is controlled by the connected synchronous motor using speed measurement. The sliding mode observer uses only the measurement of stator voltages and stator currents. The block scheme used in experimental set-up to test the sliding mode observer in open loop is presented in Fig. 2. The block 'sliding mode observer' uses only the stator current measurements in the reference rotating frame (d -q) to estimate the speed, flux, load torque and the angle of flux (after using the transformation of Concordia and Park). The block 'U/f Controller' provides classical U/f control which is independent of motor measurements and estimated state. This block controls the frequency of the voltages applied to the stator (defined in the 'Observer Benchmark') by its inverter. At the same time, the track of the reference speed trajectory (defined in the 'Observer Benchmark') is imposed to the IM by the connected also synchronous motor via its inverter.

The parameters 2 λ 1 and λ 2 , of observer ( 5) are chosen as λ 1 = 1000, λ 2 = 800.

Remark 2: Due to experimental conditions (temperature, ...), the identified parameters are not exactly the real parameters of IM. The control experimental conditions are nearly different compared to the identification conditions and moreover the identification methodology has a certain uncertainty in its results. Thus this case is already a first robustness test.

Speed estimation (Fig. 3)

On Fig. 3 the speed responses for identified parameters (nominal case) and case with stator resistance variation are shown. For robustness test case (+50%of R s ), the speed is affected a little compared to nominal case when the observer is under observable conditions and near unobservable conditions. Load torque and flux estimation (Fig. 4 and6)

Fig. 4 and 6 display the current and flux responses for identified parameters (nominal case) and case with stator resistance variation. The +50% variation on R s affects clearly the flux response (Fig. 6) under observable and unobservable conditions according to the nominal case.

On the other hand, the tracking in load torque (Fig. 4) is not affected with this positive resistance variation when the IM is under observable conditions but it is sensible a little on the unobservable conditions. By comparing the experimental results obtained with the interconnected high gain observer given by [START_REF] Ghanes | Observability Study and Observer-Based Interconnected Form for Sensorless Induction Motor[END_REF], it can be remarked that the proposed sliding mode observer is stable near unobservable curve while the interconnected high gain becomes unstable. The main reason for this difference between the behavior of the two observers near unobservable curve lies in their estimation error gains. The proposed sliding mode observer uses gains which are preliminary fixed. The interconnected high gain observer uses gains which are computed 2 guidelines parameters tuning are given in Appendix at each iteration of the observer by solving Lyapunov equation.

B. Part II: Observer-controller experimental results

Now the proposed FOC-SMC using sliding mode observer in closed loop, is tested on an experimental set-up. A Sensorless Control Benchmark defines the adequate reference trajectories to evaluate the performances of the sensorless control algorithms under the following operating conditions (Fig. 7).

• Area 1. Low speed with nominal load (from 1s to 3s).

• Area 2. High speed with nominal load (from 4s to 6s).

• Area 3. Very low speed (zero frequency) with nominal load (the IM is unobservable3 from 7s to 9s). The block scheme chart used in experimental set-up to test the law control with observer in closed loop is presented in Fig. 8. The block "Sliding Mode Observer" uses only the stator current measurements in the reference rotating frame (d -q) to estimate the currents, speed, flux amplitude and flux angle. The block "Sliding Mode and Field Oriented Control" contains the proposed controller. This block uses the estimates of speed, flux amplitude and flux angle given by the block "Sliding Mode Observer" and the current measurements after using the transformation of Park and Concordia. Then, it gives the inputs control in the reference fixed frame (a,b,c) after using the inverse transformations of Park and Concordia. These control inputs drive the inverter to impose the speed and flux reference trajectories (defined by the "Control Benchmark"). The track of the reference load torque trajectory (also defined in the "Control Benchmark") is imposed by the connected synchronous motor.

1) Case with identified parameters: Fig. 9 and 10 show the experimental results in case where identified parameters are used to design the flux and speed sliding mode control with sliding mode observer (observer-controller scheme). We can remark that both systems "Control+Observer" give good performances i) in term of trajectory tracking: the motor speed (Fig. 9.b) tracks correctly its reference (Fig. 9.a) even under unobservable conditions (between 7 and 9 sec), nevertheless it appears a small static error when the motor is under unobservable conditions, ii) in term of perturbation rejection: the load torque is very well rejected under low speed and high speed (Fig. 9: 1.5 s, Fig. 9: 2.5 s and Fig. 9: 5 s). For estimated flux (Fig. 10), the same conclusion is given and moreover it exists a small peaking at the beginning (Fig. 10.b) which is due to initial conditions. Control efforts V sd and V sq are shown in figure [START_REF] Harnefors | Complete Stability of Reduced-Order and Full-Order Observers for Sensorless IM Drives[END_REF]. In figure 11 the measured and reference currents of i sd and i sq are displayed. 2) Robustness case: To check the robustness of the designed Control-Observer with respect to motor parameters variation, a stator resistance variation of +50% is considered. The results that we have obtained are depicted in Figures 13 and14. It can be noticed that for motor speed (Fig. 13), the increase of the rotor resistance affects the performances of both "Control+Observer" at high and low speed conditions compared with the previous case. It appears a small static error at the time when the load torque is applied (Fig. 13: 1.5 s and Fig. 13: 5 s) and when the load torque is removed (Fig. 13: 2.5 s). When the motor is under unobservable conditions (between 7 and 9 sec), the static error is improved. For estimated flux (Fig. 14), the results are nearly similar compared with the previous case.

VI. CONCLUSION

In this paper, a step by step sliding mode observer has firstly been designed to estimate the speed, flux and the load torque of SIM even at very low speed conditions. Based on Lyapunov theory, the convergence of the proposed observer is proved. Experimental results show that the proposed observer is stable near and under unobservable conditions. The robustness of the proposed observer is verified according to +50% stator resistance variation where it was found that the flux and speed estimations are sensible to this variation.

Secondly, a field oriented control combined with sliding mode controller is designed to steer the estimated speed and flux magnitude to their desired trajectories and stability analysis based on Lyapunov theory is given. Furthermore, sufficient conditions to guarantee the convergence of the whole control-observer system are presented.

The proposed controller using the designed observer in closed loop has been tested and validated by an experimental set-up using the reference trajectories of a realistic sensorless control benchmark. The robustness of the proposed controller using a sliding mode observer was experimentally verified with respect to significant test (resistance variation).

pφ 2 dp 2 φ 2 d; 2 )

 222 = ω s = 0 or T em = -Kω r where T em is the electromagnetic torque and K = Rr if the speed motor is constant, thus T em = (f v ω r +T l ) = -Kω r . This last equation defines the unobservability curve in the map (T l , ω r ) with M = p 2 φ 2 d

aMsri 2 sq φ rd φrd ε φ rd -λ 1 2 sq φ rd φrd ε φ rd

 212 sgn (i sd -îsd ) = 0. Finally we have ωr = ω r + ab -aMsri pisq = ω r + E(t) where E(t) = ab -aMsri 2 sq φ rd φrd ε φ rd pisq . Since system (1) stays in D defined in Definition 1, it can be seen that E(t) and Ė(t) are bounded and they converge to zero exponentially. Hence we have the exponential convergence of ωr to ω r .

Assumption 1 :

 1 a-The state initial conditions of the IM are in the physical domain D. b-The desired trajectories (φ * rd and ω * r ) are in the physical domain D. c-The actual load torque is assumed to be bounded by a maximal fixed value ̺. This maximal value is chosen in accordance to the realistic torque characteristics of the chosen drive: |T l | < ̺. B. Sliding Mode Control. Flux controller design. From (11), consider the following IM flux dynamic equation with uncertainties φrd = -aφ rd + ∆aφ rd + κi * sd (12)

e

  and i * sq (ωr) = -lωrσ ωr + kω r e ωr + cω * r + ω * rηωrsgn(σ ωr ) φrd = φrd -φ * rd e ωr = ωr -ω * r σ φrd = e φrd -(k φ rd -a) t 0 e φrd dτ σ ωr = e ωr -(k ωr -c) t 0 e ωr dτ.

1 /2 2 with µ 2

 122 V2 = σ ωr σωr + εωr ε ωr = σ ωr ( ėωr -(k ωr -c)e ωr ) + ε ωr ωr -ωr = σ ωr -ce ωr + hi * sq (ω r ) -cω * r -ω * r + d ωr -(k ωr -c)σ ωr e ωr -ζ ωr |ε ωr | = σ ωr [-l ωr σ ωr + d ωr -η ωr sgn(σ ωr )] -ζ ωr |ε ωr | ≤ -l ωr σ 2 ωr -b ωr |σ ωr | -ζ ωr |ε ωr | < -µ 2 V = min{ √ 2b ωr , √ 2ζ ωr } since |x| + |y| ≥ |x| + |y|. Hence we have

  convergence of speed and flux to zero.V. EXPERIMENTAL RESULTSHere, the tests have been performed with the following IM values:

Fig. 2 .

 2 Fig. 2. Pie chart of the sliding mode observer.

Fig. 3 .Fig. 4 .

 34 Fig. 3. wr and ŵr

  Fig. 6. φrd

Fig. 7 .

 7 Fig. 7. Control benchmarkThe parameters4 of controllers[START_REF] Ghanes | Cascade and highgain observers comparison for sensorless closed-loop induction motor control[END_REF] and (25)-(27) are chosen as follows: Kp vd = 2, Kp vq = 2, K Ivd = 0.05, K Ivq = 0.05, η φ rd = 10, k φ rd = -80, l φ rd = 4, η ωr = 5, k ωr = -40, l ωr = 2. Those of the observer are done in section (V-A).

Fig. 8 .

 8 Fig. 8. Pie chart of the observer-controller

Fig. 9 .

 9 Fig. 9. ω * r , ωr, ωr.

Fig. 10

 10 Fig. 10. φ * rd , φrd .

Fig. 12 Remark 3 :

 123 Fig. 11. V sd and Vsq

Fig. 13

 13 Fig. 13. ω * r , ωr, ωr.

Fig. 14

 14 Fig. 14. φ * rd , φrd .

see section II-B for more details

guidelines parameters tuning are given in Appendix

ACKNOWLEDGMENTS

Authors would like to thanks A. Glumineau and R. Boisliveau for the use of the experimental set-up located at IRCCyN (see [24]).

Appendix: Parameters tuning of the observer-controller scheme

-The gains λ 1 and λ 2 of the observer (5) are chosen to satisfy equations ( 8) and ( 9) respectively.

-For the controller given by [START_REF] Ghanes | Cascade and highgain observers comparison for sensorless closed-loop induction motor control[END_REF], the parameters Kp vd , Kpvq, K Ivd , K Ivq are determined as follows : Considering the dynamic equations of i sd and isq given by (1) without nonlinearities and coupling terms

Writing the transfer function which lies the stator currents of (28) with their references given by [START_REF] Ghanes | Cascade and highgain observers comparison for sensorless closed-loop induction motor control[END_REF] as a second order system in closed loop, it follows

By imposing ζ = 1 to avoid peaking and a currents bands-widths F BD at least less than a middle of Fe = 1/Te where Te = 200µs is the sampling

the parameters Kp vd , Kpvq, K Ivd , K Ivq can be established:

where K Ivd = Kp vd T i vd and K Ivq = Kpvq T ivq .

-For the controller given by ( 25)-( 27), l φ rd > 0, lω r > 0 and the parameters k φ rd , kω r , η φ rd and ηω r are chosen to satisfy respectively ( 18), ( 24), ( 17) and ( 23).

Malek GHANES


A I Solomon
email: a.i.solomon@open.ac.uk

G E H Duchamp
email: gduchamp2@free.fr

P Blasiak
email: pawel.blasiak@ifj.edu.pl

A Horzela

K A Penson
email: penson@lptl.jussieu.fr

From Quantum Mechanics to Quantum Field Theory: The Hopf route

We show that the combinatorial numbers known as Bell numbers are generic in quantum physics. This is because they arise in the procedure known as Normal ordering of bosons, a procedure which is involved in the evaluation of quantum functions such as the canonical partition function of quantum statistical physics, inter alia. In fact, we shall show that an evaluation of the non-interacting partition function for a single boson system is identical to integrating the exponential generating function of the Bell numbers, which is a device for encapsulating a combinatorial sequence in a single function.

We then introduce a remarkable equality, the Dobinski relation, and use it to indicate why renormalisation is necessary in even the simplest of perturbation expansions for a partition function.

Finally we introduce a global algebraic description of this simple model, giving a Hopf algebra, which provides a starting point for extensions to more complex physical systems.

Introduction

The motivation for this work is as follows: Recent analysis of perturbative quantum field theory (pQFT) has shown that at the heart of this theory there lies a Hopf algebra description. Since pQFT is a subtle and as yet not completely solved area, it is of pedagogical value to examine simpler systems to illustrate how these algebraic constructs may arise. In this note we examine the simple theory given by the quantum partition function Z of a non-relativistic boson system, a completely solvable model (at least in the free boson case). On the basis of a combinatorial methodology, we show that certain combinatorial numbers, known as Bell numbers, play a central role in such an analysis. The graphical approach to the description of these combinatorial numbers leads to a sort of non-relativistic analogue of Feynman diagrams. En route, we also remark how that even in this simple case, uncritical expansion methods may lead to divergences, which can be cured by regularization; this imitates the renormalization methods of pQFT. This theory does not depend on space or time; so it is a non-field theory. However, the description of the Quantum Partition Function that we employ uses the coherent state parameter z as integration variable; and this plays the role of integration over space and time in the relativistic case.

The core of the note then deals with how an elementary Hopf algebra arises naturally from this approach. Although the resulting Hopf algebra, which we call BELL (since it arises directly from a graphical description of the combinatorial Bell numbers), is rather too uncomplicated, (technically it is commutative and co-commutative), it serves as a starting point for an extension to more complicated algebraic systems. Indeed, in later work we show that extensions and deformations of this algebra can emulate that of pQFT, thus supplying an alternate, algebraic, route from the physical path which takes us from the non-relativistic arena to that of relativistic quantum field theory.

Bell and Stirling Numbers; Exponential Generating Function

The Bell number B(n) simply gives the number of partitions of {1, 2, . . . , n}. Thus for the set {1, 2, 3}, we have {1, 2, 3}, {1, 2}{3} {1, 3}{2} {3, 2}{1} {1}{2}{3} so that B(3) = 5. Equivalently, B(n) gives the number of ways in which n distinct objects may be distributed among n identical containers, some of which may remain empty. The first few values of B(n) are

B(n) = 1, 1, 2, 5, 15, 52, 203, . . . , n = 0, 1, 2, . . . .

A compact way of defining a combinatorial sequence is by means of a generating function.

For the Bell numbers, the exponential generating function is given by

∞ n=0 B(n)
x n n! = exp(e x -1).

(

) 1
We shall give a proof of Eq.( 1) after we introduce a graphical description for the Bell numbers in Section 3.2. A slight generalization of the exponential generating function for the Bell numbers is given by that defining the Bell polynomials B n (y):

exp (y(e x -1)) = ∞ n=0 B n (y) x n n! (2)
With this notation B(n) ≡ B n (1).

Stirling numbers (of the second kind) S(n, k) are related to the Bell numbers. These are defined as the number of ways of putting n different objects into k identical containers, leaving none empty. From this definition we have B(n) = n k=1 S(n, k). Bell and Stirling numbers are fundamental in quantum theory. This is because they arise naturally in the normal ordering problem of bosonic operators. By the normal ordering operation N f (a, a † ) we mean reorder the boson operators in f (a, a † ) so that all annihilation operators are on the right. For canonical boson creation and annihilation operators {a † , a} satisfying [a, a † ] = 1, the Stirling numbers of the second kind S(n, k) intervene through [2,3]

N (a † a) n = n k=1 S(n, k)(a † ) k a k . (3)
The corresponding Bell numbers B(n) = n k=1 S(n, k) are simply the expectation values

B(n) = z|(a † a) n |z z=1 (4)
taken in the coherent state defined by

a|z = z|z (5)
for z = 1.

Graphs

Bell number diagrams

We now give a graphical representation of the Bell numbers, based on work of Brody, Bender and Meister [START_REF] Bender | Quantum field theory of partitions[END_REF][START_REF] Bender | Combinatorics and field theory[END_REF], which we have extended in [6]. Consider labelled lines which emanate from a white dot, the origin, and finish on a black dot, the vertex. We shall allow only one line from each white dot but impose no limit on the number of lines ending on a black dot. Clearly this simulates the definition of S(n, k) and B(n), with the white dots playing the role of the distinguishable objects, whence the lines are labelled, and the black dots that of the indistinguishable containers. The identification of the graphs for 1, 2 and 3 lines is given in Figure 1 below. We have concentrated

S(1,1)=1 S(3,3)=1 S(3,2)=3 S(3,1)=1 n=1: n=3: S(2,2)=1 S(2,1)=1 n=2: B(1)=1 B(2)=2 B(3)=5 V 1 V 1 3 V 1 2 V 1 V 2 V 2 V 3 Figure 1. Graphs for B(n), n = 1, 2, 3.
on the Bell number sequence and its associated graphs since, as we shall show, there is a sense in which this sequence of graphs is generic in the evaluation of the Quantum Partition Function. By this we mean the following: The exponential generating function for the Bell polynomials is identical to the (integrand of) the Canonical Quantum Partition Function of a non-interacting single-mode boson model. Then, when interactions are present these may be incorporated by the use of suitable strengths associated with the vertices of the graphs. That is, we can represent the perturbation expansion of an interacting model by the same sequence of graphs as in Figure 1, with suitable vertex multipliers (denoted by the V terms in the same figure).

The Connected Graph theorem

We introduce this important theorem [START_REF] Ford | Proc.Nat.Acad[END_REF] in order to obtain the exponential generating function of the Bell numbers. The theorem states:

Theorem 1 Connected Graph Theorem: If C(x) = ∞ n=1 c(n)x n /n! is

the exponential generating function of a set of CONNECTED labelled graphs then A(x) = exp C(x) is the exponential generating function of ALL the graphs.

In Figure 2 we see that there is only one connected graph for each n, so

C(x) = x/1! + x 2 /2! + x 3 /3! + . . . = e x -1 whence A(x) = exp(e x -1)
gives the exponential generating function for the Bell numbers.

n = 1 n = 2 n = 3

Quantum Partition Function

We now show that the Canonical Partition Function of Quantum Statistical mechanics is simply related to the exponential generating function of the Bell polynomials.

Free boson gas and Bell polynomials

The canonical partition function Z associated with a quantum mechanical hamiltonian H is given by

Z = Tr exp(-βH) ( 6
)
where β is the usual inverse temperature ( 1/k Boltzmann T ). We start with the elementary case of the hamiltonian for the single-mode free boson gas H = ǫa † a (ignoring an additive constant), ǫ > 0. The usual computation of the partition function, exploiting the completeness property

∞ n=0 |n n| = I, is immediate: Z = Tr exp(-βǫa † a) = ∞ n=0 n|e -βǫn |n = ∞ n=0 e -βǫn = 1 -e -βǫ -1 . (7)
However, we may use any complete set to perform the trace. We choose coherent states as defined in Eq.( 5) above, which are explicitly given by

|z = e -|z| 2 /2 n (z n /n!)a † n |0 . (8)
For these states the completeness or resolution of unity property is

1 π d 2 z|z z| = I ≡ dµ(z)|z z|. (9)
The appropriate trace calculation is

Z = 1 π d 2 z z| exp(-βǫa † a)|z = (10) = 1 π d 2 z z| : exp(a † a(e -βǫ -1)) : |z , (11)
where we have used the following well-known relation [START_REF] Klauder | Fundamentals of Quantum Optics Benjamin[END_REF][START_REF] Louisell | Quantum Statistical Properties of Radiation[END_REF] for the forgetful normal ordering operator : f (a, a † ) : which means normally order the creation and annihilation operators in f forgetting1 the commutation relation [a, a † ] = 1:

N exp(xa † a) =: exp(a † a(e x -1)) : .

We therefore obtain, integrating over the angle variable θ and the radial variable r = |z|,

Z = 1 π 2π 0 dθ ∞ 0 r dr exp r 2 (e -βǫ -1) , (13)
which gives us Z = (1e -βǫ ) -1 as before. We rewrite the above equation to show the connection with our previously-defined combinatorial numbers. Writing y = r 2 and x = -βǫ, Eq.( 13) becomes

Z = ∞ 0 dy exp(y(e x -1)) . ( 14
)
This is an integral over the classical exponential generating function for the Bell polynomials as given in Eq.( 2). This leads to the combinatorial form for the partition function

Z = ∞ 0 dy ∞ n=0 B n (y) x n n! . ( 15
)

General partition functions

We now apply this graphical approach to the general partition function in second quantized form, with the usual definition for the partition function Eq.( 6). In general the hamiltonian is given by H = ǫw(a, a † ), where ǫ is the energy scale, and w is a string (= sum of products of positive powers) of boson creation and annihilation operators. The partition function integrand F for which we seek to give a graphical expansion, is

Z(x) = F (x, z) dµ(z) , (16)
where

F (x, z) = z| exp(xw)|z = (x = -βǫ) = ∞ n=0 z|w n |z x n n! = ∞ n=0 W n (z) x n n! = exp ∞ n=1 V n (z) x n n! , (17)
with obvious definitions of W n and V n . The sequences {W n } and {V n } may each be recursively obtained from the other [START_REF] Pourahmadi | [END_REF]. This relates the sequence of multipliers {V n } of Figure 1 to the hamiltonian of Eq.(6). The lower limit 1 in the V n summation is a consequence of the normalization of the coherent state |z . Although Eq.( 15) is remarkably simple in form, it is often by no means a straightforward matter to evaluate the analogous integral for other than the free boson system considered here. Further, as we show below, we may not interchange the integration and the summation, as each individual y integral diverges. At this point the analogue of this model to perturbative quantum field theory becomes more transparent. For example, to calculate the partition Function Z(x), we must integrate over the Partition Function Integrand F (x, z). Here, the integration over the coherent state parameter z plays the role of the space-time integration of Quantum Field Theory. As we have just remarked, with reference to Eq.( 15) for the non-interacting case, and as in pQFT (perturbative quantum field theory), term-by-term integration, corresponding to fallaciously interchanging the summation and integration actions, results in infinities at each term, as we now verify for the non-interacting case. Thus even here an elementary form of renormalization is necessary, if we resort to term-wise integration.

Considerations such as these have led many authors, as noted in reference [START_REF] Kreimer | Knots and Feynman Diagrams[END_REF], to consider a global, algebraic approach to pQFT; and this we shall do in Section 6.

But first we show how divergences arise.

The Dobiński Relation and Divergence

Divergence

The celebrated Dobiński relation for the Bell numbers is given by (see, for example, [START_REF] Wilf | Generatingfunctionology[END_REF]):

B(n) = 1 e ∞ k=0 k n k! , n = 1, 2, . . . ( 18
)
and for the Bell polynomials by

B n (y) = e -y ∞ k=0 k n k! y k . ( 19
)
These relations have been extended in [START_REF] Penson | Hierarchical Dobiński-type relations via substitution and the moment problem[END_REF]. Using the Dobiński relation Eq.( 19) we can rewrite the expression for the partition function for the free single-mode boson Eq.( 15) as

Z = ∞ 0 dy ∞ n=0 B n (y) x n n! (20) = ∞ 0 ∞ n=0 x n n! {e -y ∞ k=0 k n k! y k }dy (21)
We may attempt to write this as a perturbation expansion (in x ≡ -βǫ) by interchanging the integration and summation, thus:

Z = ∞ n=0 x n n! ∞ k=0 k n k! ∞ 0 {e -y y k }dy (22) = ∞ n=0 x n n! ∞ k=0 k n k! Γ(k + 1) (23) = ∞ n=0 x n n! ∞ k=0 k n . ( 24
)
Therefore the resulting "perturbation expansion" is of the form

Z = ∞ 0 b n x n b n = 1 n! ∞ k=0 k n = ∞
with each term in the expansion being infinite. Thus even in this simple case of a free boson expansion, a perturbation expansion obtained from the illegal interchange of integration and summation is divergent in every term.

Regularization

In this section we illustrate how the method of regularization may be applied to alleviate the problem of divergences. Writing the partition function Eq.( 14) as

Z = ∞ 0 e -αy dy (α = 1 -e -βǫ ) (25)
immediate evaluation gives Z = 1/α as in Eq.( 7). However, if we expand

Z = ∞ 0 ∞ n=0 (-αy) n /n! dy (26)
and exchange the order of integration, we obtain

(FALSE) Z = ∞ n=0 (-α) n /n! ∞ 0 y n dy (27)
each term of which diverges. The regularization method is to write

Z = lim M →∞ M 0 ∞ n=0 (-αy) n /n! dy (28)
and, since the integration is over a finite range, we may interchange integration and summation, to obtain

Z = lim M →∞ ∞ n=0 (-α) n /n! M 0 y n dy (29) = lim M →∞ ∞ n=0 {(-α) n /n!}{M (n+1) /(n + 1)} (30) = lim M →∞ ∞ n=0 {(-αM ) (n+1) /(n + 1)!}(-1/α) (31) = lim M →∞ {(e -M α -1)(-1/α)} (32) = 1/α (33)
as before. Note that y bounded by a finite M implies the eminently reasonable assumption of a maximum energy for the coherent state as defined in Eq.( 5).

Hopf Algebra structure

So far we have discussed the analytic and combinatorial structure of our simple model. Modern approaches to perturbative quantum field theory introduce a global algebraic structure. We describe the analogue of this Hopf Algebra structure in the context of our model, which we refer to as BELL below. We first introduce a basic Hopf Algebra, which we call POLY, generated by a single parameter x. This is essentially a standard one-variable polynomial algebra on which we impose the Hopf operation of co-product, co-unit and antipode, as a useful pedagogical device for describing their properties. The Hopf structure associated with our model described above is simply a multi-variable extension of POLY. It may be shown that the foregoing structure POLY satisfies the axioms of a commutative, co-commutative Hopf algebra.

BELL

We now briefly describe the Hopf algebra BELL which is appropriate for the diagram structure introduced in this note, defined by the diagrams of Figure 1.

BELL as an extension of POLY

It may be seen that BELL is a multivariable version of POLY. To show this, we code the diagrams by letters. We use an infinite alphabet Y = {y 1 , y 2 , y 3 • • •} = {y k } k≥1 and code each connected diagram with one black spot and k white spots with the letter y k . An unconnected diagram will be coded by the product of its letters. In this way, the diagrams of Figure 1 are coded as follows:

• first line : y 1 • second line : y 2 1 and y 2 • third line : y 3 1 and y 1 y 2 and y 3 . Thus one sees that each diagram of weight n with k connected components is coded bijectively by a monomial of weight n (the weight of a monomial y i 1 y i 2 • • • y ir is just the sum of the indices r j=1 i j ) and k letters. The algebra BELL is coded by commutative polynomials in the infinite alphabet Y ; that is, the coding is an isomorphism BELL → R[Y ]. As an aside, one may note that the basis elements of BELL are sometimes referred to as forests. As above, it may be shown that the foregoing structure BELL satisfies the axioms of a commutative, co-commutative Hopf algebra.

Discussion

In this note we have shown the value of the combinatorial method, which leads inter alia through the graphical approach, to a Hopf algebra description of a simple non-relativistic free boson system. We named the resulting (commutative, co-commutative) Hopf algebra BELL. To extend this simple description to more realistic models, it is convenient to introduce a symmetrized form of BELL; i.e. where the black and white spots appear symmetrically [START_REF] Duchamp | Feynman graphs and related Hopf algebras[END_REF]. Thus we have introduced the Hopf algebra DIAG where a typical series of graphs is given in Figure 3 below. Note that since the black and white spots play symmetrical roles, in the figure we permit Figure 3. Some Graphs of DIAG ourselves to exchange their order from that used in, for example, Figure 1. We may also apply "weights" L m to the black spots as well. Although the resulting Hopf algebra DIAG is still commutative/co-commutative, it may in turn be extended to a non-commutative Hopf algebra (LDIAG) which forms the basis of an extension of this algebraic approach to a description sufficiently general to emulate those algebras which describe pQFT. This extension is described in detail in a subsequent artcle in this conference [START_REF] Minh | An interface between physics and number theoryThese proceedings[END_REF].

Figure 2 .

 2 Figure 2. Connected Graphs for B(n), n = 1, 2, 3.

6. 1 .

 1 The Hopf Algebra POLY POLY consists of polynomials in x (say, over the real field R, for example).The standard algebra structure of addition and associative multiplication is obtained in the usual way, by polynomial addition and multiplication. The additional Hopf operations are: (i) The coproduct ∆ : POLY-→POLY × POLY is defined by ∆(e) = e × e (unit e) ∆(x) = x × e + e × x (generator x) ∆(AB) = ∆(A)∆(B) otherwise so that ∆ is an algebra homomorphism. (ii) The co-unit ǫ satisfies ǫ(e) = 1 otherwise ǫ(A) = 0. (iii) The antipode S : POLY-→POLY satisfies S(e) = e; on the generator x, S(x) = -x. It is an anti-homomorphism, i.e. S(AB) = S(B)S(A).

( i )

 i Each distinct diagram is an individual basis element of BELL; thus the dimension is infinite. (Visualise each diagram in a "box".) The sum of two diagrams is simply the two boxes containing the diagrams. Scalar multiples are formal; for example, they may be provided by the V coefficients. (ii) The identity element e is the empty diagram (an empty box). (iii) Multiplication is the juxtaposition of two diagrams within the same "box". BELL is generated by the connected diagrams; this is a consequence of the Connected Graph Theorem (as discussed above in 3.2). Since we have not here specified an order for the juxtaposition, multiplication is commutative. (iv) The coproduct ∆ : BELL-→BELL × BELL is defined by ∆(e) = e × e (unit e) ∆(y) = y × e + e × y (generator y) ∆(AB) = ∆(A)∆(B) otherwise so that ∆ is an algebra homomorphism. (v) The co-unit ǫ satisfies ǫ(e) = 1 otherwise ǫ(A) = 0. (vi) The antipode S : BELL-→BELL satisfies S(e) = e; on a generator x, S(y) = -y. It is an anti-homomorphism, i.e. S(AB) = S(B)S(A).

This procedure generally alters the value of the operator to which it is applied.

Acknowlegements

The authors wish to acknowledge support from the Agence Nationale de la Recherche (Paris, France) under Program No. ANR-08-BLAN-0243-2 and from PAN/CNRS Project PICS