Abbas Dideban
email: adideban@semnan.ac.ir

M Zareiee
email: mzareiee@semnan.ac.ir

Hassane Alla
email: hassane.alla@inpg.fr

Controller synthesis with very simplified linear constraints in PN model

Keywords: Discrete Event Systems (DES), Petri Nets, Supervisory control, Controller synthesis, Forbidden states

HAL is

INTRODUCTION

A unifying framework for the control of discrete event systems is provided by Ramadge andWonham (1987, 1989), and so far their supervisory control theory is the most general and comprehensive theory presented for logical DES. In [START_REF] Ramadge | Supervisory control of a class of discrete event processes[END_REF][START_REF] Ramadge | The control of discrete event systems.Dynamics of discrete event systems[END_REF], it is shown that we can model the systems by using of formal languages and finite automata. Methods exist for designing controllers based on automata system models, however these methods often involve exhaustive searches or simulations of the system behaviour, making them impractical for systems with large numbers of states and transitions causing event [START_REF] Moody | Petri net supervisorfor DES with uncontrollable and unobservable transition[END_REF]. For solving these problems, we can model systems with Petri nets (PNs). Petri nets are a very appropriate and useful tool for the study of discrete event systems because of their modelling power and their mathematical properties. In the last decade, the research in the field of controller synthesis of discrete event systems has become one of the most active domains [START_REF] Achour | Supervisory controller of Petri nets under partial Observation[END_REF][START_REF] Giua | Control of safe ordinary Petri Nets using unfolding[END_REF]. In [START_REF] Giua | Generalized Mutual Exclusion Constraints on Nets with Uncontrollable Transitions[END_REF] it is shown that it is possible to represent the forbidden states with linear constraints. With the idea presented by [START_REF] Yamalidou | Feedback control of petri nets based on place invariants[END_REF] we can design a controller by adding a control place to the PN mode. Each control place corresponds to a constraint. However, when the number of forbidden states is large, then the number of constraints has the same size. In Giua (1992b) it is shown that we can simplify some constraints. But there is no systematic method for simplifying these constraints in general case. This simplification is done by using the PNs structural properties. In [START_REF] Dideban | From forbidden state to linear constraints for the optimal supervisory control[END_REF] a systematic method for reducing the size and the number of constraints was introduced. It was applicable for safe and conservative PNs. In [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF], an efficient method for constructing a controller based on safe PN is presented. A set of linear constraints allow forbidding the reachability of specific states. The number of these so-called forbidden states, and consequently the number of constraints are large and lead to a large number of control places. A systematic method for reducing the size and the number of constraints for safe PN was introduced there. With applying the idea in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF], it is possible that sometimes the final number of constraints may remian large. In this paper our objective is to develop the idea in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF] and to present a systematic method for safe PN which allows to do more simplification than in the previous approaches. In this paper, we introduce the concept of Partial Place invariant and then, we present the simplification method. We apply this method to a concrete example to highlights its advantages. We use constraints which are equivalent to forbidden states. These constraints can be calculated in two different ways. They can be given directly as specifications or they can be deduced thanks to the Kumar approach [START_REF] Kumar | Supervisory control of deterministic Petri nets with regular specification languages[END_REF]. By applying the idea in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF] and the new approach, it is possible to make a powerful simplification. The advantage of the new idea is to reduce more the number of forbidden states. This paper is oraganized as follows: in section 2, some important definitions are introduced. In section 3, the idea of passage from forbidden states to linear constraints will be introduced. The idea in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF] and a practical example are presented in section 4. The basic original idea of simplification and the calculation of control places are presented in section 5. Finally, the conclusion is given in the last section.

PRELIMINARY PRESENTATION

In this paper, it is supposed that the reader is familiar with the PNs basis (David & Alla, 2005) and the theory of supervisory control [START_REF] Ramadge | Supervisory control of a class of discrete event processes[END_REF][START_REF] Ramadge | The control of discrete event systems.Dynamics of discrete event systems[END_REF]. We briefly introduce the idea that was presented in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF]. For more detail, the reader can refer to this reference. . A PN is represented by a quadruplet R={P, T, W, M 0 } where P is the set of places, T is the set of transition, W the incidence matrix and M 0 is the initial marking. Here we show the number of tokens in a place P i with m i . The PN is assumed to be safe; then the marking of each place is Boolean.

M R denotes the set of PN reachable markings. In M R , two subsets can be distinguished; the set of authorized states M A and the set of forbidden states M F . The set of forbidden states correspond to two groups: 1) The set of reachable states (M F ′ ) which either don't respect the specifications or are deadlock states.

2) The set of states for which the occurrence of uncontrollable events leads to states in M F ′ .

The set of authorized states are the reachable states without the set of forbidden states:

M A =M R \M F
Among the forbidden states, an important subset is constituted by the border forbidden states denoted as M B [START_REF] Kumar | Supervisory control of deterministic Petri nets with regular specification languages[END_REF].

Definition 1. let M B be the set of border forbidden states:

} , | { i j A j c F i B M M M and M → ∈ ∃ ∑ ∈ ∃ ∈ = σ σ M M M
Where ∑ c is the set of controllable transitions.

In this paper we use the concept of support instead of marking as presented in definition 2. Definition 2. the function Support(X) of a vector

N X } 1 , 0 { ∈ is: Support(X) = The set of marked places in X.
Definition 3. let R be a PN , P the set of its places , M(M 0 ) the set of reachable markings from M 0 and k a constant. a place invariant is obtained if there is a sub-set of places P ′ ={P 1 , P 2 ,…, P r } included in P and a weighing vector (q 1 , q 2 ,…, q r ) for which all the weights q i are positive integers such that:

) ( , ... 0 2 2 1 1 M m k m q m q m q i r r M ∈ ∀ = + + +
Then we say P ′ is a place invariant, (David & Alla 2005).

But if we remove some places from the set P ′ , what happens? It follows the definition of partial invariant: Definition 4. let P ′ ={P 1 ,P 2 ,…,P r } be a place invariant in PN model R, P i1 ={P 1 ,P 2 ,…,P L }for which{1,2,..L}⊂ {1,2,..r}, is a partial place invariant and it satisfy the following equation:

q 1 m 1 +q 2 m 2 +…+q L m L ≤k , ∀m∈M(M 0 ) 3. FORBIDDEN STATES TO LINEAR CONSTRAINTS Let [ ] ) ,..., , ( 2 1 N T i i m m m M M =
Be a forbidden state in set M B and Support(M i )={P i1 P i2 P i3 …P in } the set of marked places of M i . From a forbidden state, a linear constraint can be constructed [START_REF] Giua | Generalized Mutual Exclusion Constraints on Nets with Uncontrollable Transitions[END_REF].

The linear constraint deduced from the forbidden state M i is given below. The state M i does not verify this relation. Therefore, by applying this relation, M i will be forbidden.

∑ = - ≤ n k ik n m 1 1
Where n=Card [support (M i )], is the number of marked places of M i , and m ik is the marking of place P ik of state M i . Let M (M T =[m 1 ,m 2 ,…,m N ]) be a general marking and M i be a forbidden state. The constraint (forbidden state M i ) is denoted by c i and can be rewritten as follows

1 ] ) support( [ Card . i - ≤ M M M T i For example if: 2 3 )] ( rt Card[suppo ] 1 , 0 , 0 , 0 , 1 , 1 , 0 [ 7 3 2 ≤ + + ⇒ = ⇒ = m m m M M i T i
Verifying the relation in above is equivalent to forbidding state M i when the PN model is conservative. However, in a safe PN not necessarily conservative, this equivalence is not always true. In follow we recall the concept of over state that was introduced in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF]. Definition 5. let M 2 =P 21 P 22 …P 2m be an accessible state, M 1 =P 11 P 12 …P 1n will be an over-state of M 2 if:

2 1 M M ≤ .
That means all of the places in M 1 are in M 2 . For example M 1 =P 1 P 3 is an over-state of M 2 =P 1 P 3 P 6 P 9 . The name over-state is used because the constraint corresponding to an over-state holds constraint. For example, the constraint m 4 +m 6 ≤1 that is correspond to the over-state M 1 =P 4 P 6 holds both following constraints: m 1 +m 4 +m 6 +m 9 ≤3 , m 2 +m 4 +m 6 +m 9 ≤3 These two constraints forbid states M 6 =P 1 P 4 P 6 P 9 and M 7 =P 2 P 4 P 6 P 9 . P 4 P 6 is an over-state of both states P 1 P 4 P 6 P 9 and P 2 P 4 P 6 P 9 which could be verified by M 1 ≤M 6 and M 1 ≤M 7 . Thus by using only the constraint m 4 +m 6 ≤1, both states M 6 and M 7 will be forbidden. Remark 1. With each over-state b i , we associate a constraint c i :

b i =(P i1 P i2 P i3 …P in ) → c i =(P i1 P i2 P I3 …P in ,n-1). That means: m i1 +m i1 +…+m in ≤n-1. The concept of over-state could be used for the reduction the number of constraints deduced from states or over-states [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF]. In the next section we introduce these ideas and the problem to solve.

RECUCTION THE NUMBER OF CONSTRAINTS

Now we explain the idea in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF]. Suppose that in a system we have the set of authorized states M A and the set of border forbidden states M B . In the first step of reduction, we must construct the sets B 1 (All the over-states of border forbidden states) and A 1 (All the over-states of authorized states). Then we remove from B 1 , the states or over-states that exist in A 1 and call this set B 2 . In the next step we remove the states which their over-states are in B 2 from it and we call it B 3 . Sometimes it is possible, to simplify the reduced set B 3 , by application of the idea that is presented in [START_REF] Dideban | Synthesis of discrete controllers by simplification of constraints and conditions[END_REF]. But sometimes it is not the simplest set. The rules to choose the final over-states are similar to the rules of the Quine-McCluskey method to simplify the logical expressions (Morris Mano 2001).

Introduction of a practical example

Consider a system with three machines and two robots. The closed loop PN model of this system is presented in fig. 1. The start command of each machine is accomplished by controllable transitions c 1 , c 2 , c 3 and the stop command is accomplished by uncontrollable transitions f 1 , f 2 , f 3 . Two Robots are used to free three machines. All of machines can be turned on or off independently. The end of process for each machine when at least one of the robots are not free, is forbidden. From constructing the reachability graph, we can calculate the set of border forbidden states. As it is obvious in fig. 2, the existence of uncontrollable events leads to the forbidden states. For example when the system is in state (P 1 P 4 P 5 P 8 P 10 ), it is possible to fire the uncontrollable event f 1 , while it is not authorized by the specifications. This state is called as a forbidden state. And the set of border forbidden states can be determined from these forbidden states.

In this example the set of border forbidden states is: After this step we remove from B 1 , the states that are common in B 1 and A 1 , and the residual set is named B 2 .

Remark 2. we don't write the set of over-states of authorized states A 1 since with having M A we can remove the common states from B 1 . By the method presented in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF], the controller can be synthesized by final selection of simplified over-states.

Controller synthesis

Final selection is similar to the McCluskey method for simplifying logical terms. In this method we construct a table (table1

{ : 3 → × B M R B is as: ⎩ ⎨ ⎧ - ≤ = . 0 ) ( 1 ) , ( not if M of state over is b M b b M R i j i j j i
The covering of a marking is an integer number:

∑ = ) , ( ) ( j i i b M R M Cv
Cv(M i ) ≥ 1 means that forbidden state M i is covered by at least one over-state.

Table 1: Relation between over-states and border forbidden states

P2P4P6P7P9 P2P4P6P8P9 P2P4P6P7P10 P2P4P6P8P10 P2P4P5P8P9 P2P4P5P7P10 P2P3P6P8P9 P2P3P6P7P10 P1P4P6P8P9 P1P4P6P7P10 P2P3P5P8P10 P1P4P5P8P10 P1P3P6P8P10 P 2 P 4 P 6 1 1 1 1 0 0 0 0 0 0 0 0 0 P 2 P 4 P 8 0 1 0 1 1 0 0 0 0 0 0 0 0 P 2 P 6 P 8 0 1 0 1 0 0 1 0 0 0 0 0 0 P 4 P 6 P 8 0 1 0 1 0 0 0 0 1 0 0 0 0 P2P4P10 0 0 1 1 0 1 0 0 0 0 0 0 0 P2P6P10 0 0 1 1 0 0 0 1 0 0 0 0 0 P 4 P 6 P 10 0 0 1 1 0 0 1 0 0 1 0 0 0 P 2 P 8 P 10 0 0 0 1 0 0 0 0 0 0 1 0 0 P 4 P 8 P 10 0 0 0 1 0 0 0 0 0 0 0 1 0 P 6 P 8 P 10 0 0 0 1 0 0 0 0 0 0 0 0 1 C v (Mj) 1 4 4 1 0 1 1 2 1 1 1 1 1 1
The final constraints must cover all the border forbidden states. This means that with applying the constraints equivalent to the final sets, all the border forbidden states are forbidden. These constraints are as follow:

B

Before using the method in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF], the number of constraints and equivalently the number of border forbidden states for our example was 13, but with using this method, the number of constraints is 9. It is clear that the number of simplified constraints is large, and for designing a controller by the method presented by [START_REF] Yamalidou | Feedback control of petri nets based on place invariants[END_REF], the number of control places is also large. However, is it possible to simplify these constraints more? The answer to this question is given by a method based on the concept of partial place invariant that was defined in section 2.

INTRODUCTION OF THE NEW IDEA FOR SIMPLIFICATION

We consider the partial invariant for safe and conservative PNs. From the invariant relations we can construct partial invariants. For example:

m 1 +m 2 +…+m n = k ⇒ m 1 +m 2 +…+m n-1 ≤ k
In safe PN in general case, it is not possible to construct this equation directly. In the following, we present some properties based on partial invariants that in many cases allow us to simplify the set of constraints representing the forbidden states. Property 1.; suppose that m i1 , m i2 are the number of tokens in the places P i1 , P i2 respectively. If P i1 P i2 is not in the set of over-states of authorized states we have:

m i1 +m i2 ≤1
Proof: Suppose that the constraint is not true. Then we have: m i1 + m i2 > 1 So, for safe PNs we have: m i1 +m i2 =2 →m i1 = m i2 =1 This means P i1 P i2 is in the set of over-states of authorized states that it is not true. Then: m i1 +m i2 ≤1 Property 2 (Extension of Property 1). suppose that the constraint m i1 +…+m in ≤1 is true. If all of the over-states {P i1 P i(n+1) ,…,P in P i(n+1) } are not in the set of over-states of authorized states, we have: m i1 +…+m in +m i(n+1) ≤1

Proof: The proof of this property is similar to the proof of property1. Suppose that this relation is not true, so we can write: m i1 +…+m in +m i(n+1) >1 hence we have:

m i1 +…+m in +m i(n+1) = 2 m i(n+1) = 1 m i(n+1) ≤ 1 (for safe PN) m i1 +…+ m in = 1 m i1 +…+ m in ≤ 1 m i1 +…+ m in = 1 ⇒ ∃ m ik = 1( k ∈[1,n])
Then P ik P i( n+1) is an over-state of authorized states that is not true, then:

m i1 + …+ m in + m i(n+1) ≤1 Remark 3. A constraint m 1 + m 2 +…+ m k ≤ k can be presented by (P 1 P 2 … P k , k).
Property3. Let M = {(P 1 P k … P j , k),...,(P r P k … P j , k)} be the subset of constraints verified the authorized states. If the authorized states verify the partial invariant m 1 + m 2 +... + m r ≤ 1 the r constraints are equivalent to one constraint as bellow:

(

m 1 + m 2 +…+ m r ) + m k +…+ m j ≤ k Proof: Necessary condition: ∀ q ∈ {1,…, r} m q + m i +…+ m j ≤ k ⇒ m i +…+ m j ≤ k
And also from the constraint, we have: m 1 + m 2 +... + m r ≤ 1 By adding two constraints, we have:

(

m 1 + m 2 +…+ m r ) + m k +…+ m j ≤ k +1
We show that the value k+1 is not any time accessible. Suppose that k+1 is accessible. We have:

m 1 +m 2 +…+m r =1 , m i +…+m j = k
From the r constraints and equation m i +…+m j =k , we have:

m 1 =m 2 =…=m r =0 Then: (m 1 +m 2 +…+m r )=0
That is not true.

Sufficient condition:

(

m 1 +…+ m r ) + m i +…+ m j ≤ k ∀ q ∈ {1,…, r} m q ≥ 0 ⇒ ∀ q ∈ {1,…,r} m q + m i +…+ m j ≤ k
So it is obvious that with using these properties the 9 constraints have reduced to one constraint that is good reason for utilization of these properties.

In this example, there is not the final selection, but generally by the method as the same was presented in last section, we can select the final result.

Calculation of control places

To calculate the control places corresponding for each linear constraint, we will use the method developed in Yamalidou et al. (2006). Now we briefly present this method that is based on concept of P-invariant. Suppose that we have the set of constraints as L.m p ≤ b that m p is the marking vector of system, L is a n×n matrix, b is a n c ×1 vector, n c is the number of constraints and n is the number of places. In this method for each constraint, we add a place to the PN model of the system. In Yamalidou et al. (2006), it is proved that this places guarantees that these constraints are respected. Suppose W p be the incidence matrix of the system. For each constraint we add a row to W p and we show this row as W c , corresponds to the incidence matrix of the controller. and W c is calculated as follows: As it is obvious from fig. 3, with adding P c1 , forbidden states can not be reached..

W c = -L.W p

CONCLUSION

In this paper, we have presented a systematic method to reduce the number of linear constraints corresponding to the forbidden states for a safe PN. This is realized by using nonreachable states and by building the constraints using a systematic method. The important concept of over-state has been used; it corresponds to a set of markings which keep the same property (forbidden or authorized). From the forbidden states, the set of over-states is calculated. The utilization of non-reachable markings allows great simplification of the constraints. In this paper, we have extended the idea presented in [START_REF] Dideban | Reduction of Constraints for Controller Synthesis based on Safe Petri Nets[END_REF]. We have used the partial invariant for more simplifications. Finally, we have shown that sometimes it is possible to simplify without invariant or partial invariant. By this method we can arrive to very simplified constraints deduced from the forbidden states.

Fig. 1 .

 1 Fig. 1. closed loop PN model Fig.2, gives part of reachability graph of the PN model presented in fig 1. The construction of the reachability graph is stopped when a forbidden state is reached. The state after these forbidden states is presented by .

  ) where the first row represents the set of border forbidden states M B , and the first column is the set of simplified over-states B 2 . The relation between border forbidden states and the simplified over-states can be expressed by the following definition: Definition 6. let B 3 ={b 1 ,b 2 ,…,b m } be the set of simplified over-states and M B ={M 1 ,M 2 ,…,M N } be the set of border forbidden states. The relation } 1 , 0

W

  c is added to W p as follow:If the initial marking of system is m p0 , the initial marking for the added places is calculated as follow:m s0 =b-L.m p0Now we calculate the control places for our example. The simplified constraint was (P 2 P 4 P 6 P 8 P 10 , 2), Then we have:L=[0 1 0 1 0 1 0 1 0 1] ⇒W c =[-1 0 0 -1 0 0 -1 0 0 1 1], M s0 =2The PN model of the final controller after adding the control place is indicated in fig 3. The control place and corresponding arcs are shown with gray color.

Fig. 3 .

 3 Fig. 3. Controlled PN model

= (P 2 P 4 P 6 P 8 P 10 , 2)

Now, the sets of over-states of M A and M B are constructed. The set of over-states of M B is shown with B 1 and M A with A 1 .

Hence the set of over-states of M B is:

Generally, by application of these properties, it is possible to simplify the constraints. However, for our example, it is not possible to do more simplification using these properties. Suppose that the sets A ' 1 and B ' 1 be the sets of authorized and forbidden states, respectively: property 1 and3 for states {P 1 P 4 P 5 , P 2 P 4 P 5 } we have: m 1 + m 2 + m 4 + m 5 ≤ 2. For states {P 2 P 5 P 7 , P 4 P 5 P 7 }, the constraint m 2 + m 4 ≤ 1 is not verified, but is it necessary? We show that it is not necessary. When the borne of constraints is more than 1, for having

The following propertie generalizes this idea.

Property 4 suppose that we have the set of constraints:

Consider n overstates as follow:

If none of these states are not in the set of over-states of authorized states, then we can reduce these constraints to one constraint as follow: P i1 P i2 P 1 P 2 …P n that means:

Proof: the proof of this property is clear. Suppose that the inequality m i1 +m i2 +m 1 +m 2 +…+m n ≤ n is not true then m i1 +m i2 +m 1 +m 2 +…+m n =n+1,, that means (n+1) places is marked in the set of P = (P i1 , P i2 , P 1 , P 2 , …, P n ) then at least one of the condition in the property is violated. So our supposition is not true. Property 5. Suppose that we have the constraints C 1 = (P i1 P i2 …P im P 1 P 2 …P n , n) and C 2 = {(P i(m+1) P 1 P 2 …P n , n), that verify the authorized states. If all of the over-states content n+1 places from the sets P = (P i(m+1) , P i1 P i2 …P im P 1 P 2 …P n ,) are not exist in the sets of over-states of authorized states, the two constraints C 1 and C 2 can be replaced by one constraint as follow :

Proof: the proof of this property is similar to the proof of property 4. Suppose that the inequality m i1 +m i2 +…+m im +m i(m+1) +m 1 +m 2 + … +m n ≤ n is not true then m i1 +m i2 +…+m im +m i(m+1) +m 1 +m 2 +…+m n = n+1, that means (n+1) places is marked in the set of P = (P i1 , P i2 , …, P im ,P i(m+1), P 1 , P 2 , …, P n ) then at least one of the condition in the property is violated. So our supposition is not true.

More simplification in our example by new idea

B

We can select the subset C 1 that contains two common places and one different place. C 1 = {(P 2 P 4 P 6 , 2), (P 2 P 4 P 8 , 2).

The over-state P 6 P 8 is in the set of over-states of authorized states. So it is impossible to use the property 2, but the states P 2 P 6 P 8 , P 4 P 6 P 8 , are not in the set of over-states of authorized states, then it is possible to use the property 4.

C 1 = {(P 2 P 4 P 6 , 2), (P 2 P 4 P 8 , 2) ⇒ C ' 1 = {(P 2 P 4 P 6 P 8 , 2)

For applying the property 5 on the sets of C ' 1 and C 2 = {(P 2 P 4 P 10 , 2)}, it is clear that the over-states P 2 P 8 P 10 , P 4 P 8 P 10 , P 2 P 6 P 10 , P 4 P 6 P 10 are not in the set of over-states of authorized states, then, we arrive to the simplified constraint: The simplified sets C ' 2 and C ' 3 and C ' 4 are the same. Then the final set is: C '


Florent Loiselay
email: florentloiselay@vedecom.fr

Leyla Arioua Habarek

Joao Oliveira

Ali Alhoussein

SiC & GaN comparison for high switching frequency, high efficiency 7 kW Boost Converter

come     SiC & GaN comparison for high switching frequency, high efficiency 7 kW Boost Converter

Introduction

Electric vehicles require both isolated and non-isolated power converters with high power density which can be obtained with high switching frequency thanks to the use of wide band gap (WBG). With the GaN technology, for non-isolated converter, more than 4 kW power transfer is rarely achieved for one switching cell. For example, in order to reach 7,5 kW, 3 interleaved 2,5 kW cells switching at 100 kHz were needed [START_REF] Chupryn | An Interleaved DC/DC Converter for Automotive Applications with GaN Power Semiconductors[END_REF], or also a 10 kW inverter can be obtained with 3 inverter leg of 3,3 kW power each [START_REF] Li | Design consideration of high power GaN inverter[END_REF]. Many publications deal with SiC based boost converters, but mostly for 600 and 800 V applications (using 1200 V SiC mosfets) [START_REF] Calderon-Lopez | High power density DC-DC converter with SiC MOSFETs for electric vehicles[END_REF][START_REF] Sewergin | Modular bidirectional full-SiC DC-DC converter for automotive applications[END_REF]. With converter interleaving, the 7 kW and 11 kW boost converter presented in this paper can be used as a building brick to design converters in the 30 to 50 kW range. Some publications compare the 650 V SiC and GaN components performance at lower power and current range [START_REF]A Performance Comparison of GaN E-HEMTs Versus SiC MOSFETs in Power Switching Applications[END_REF][START_REF] Li | SiC/GaN power semiconductor devices: a theoretical comparison and experimental evaluation under different switching conditions[END_REF][START_REF] Li | Sic/GaN power semiconductor devices theoretical comparison and experimental evaluation[END_REF][START_REF] Chowdhury | Comparison of 600V Si, SiC and GaN power devices[END_REF]. For a 900W application, 400 to 200 V synchronous buck application switching at 200 kHz, better results were obtained with the 30 A 650 V GaN component from GaN System than with the SiC 900V 35 A components from Cree (650 V components from Cree were not available yet at the time) [START_REF]A Performance Comparison of GaN E-HEMTs Versus SiC MOSFETs in Power Switching Applications[END_REF]. Double pulse testing also concluded that this 30 A 650 V GaN components from GaN System has lower switching loss than SiC components with similar rating. [START_REF] Li | SiC/GaN power semiconductor devices: a theoretical comparison and experimental evaluation under different switching conditions[END_REF][START_REF] Li | Sic/GaN power semiconductor devices theoretical comparison and experimental evaluation[END_REF]. Comparison of Si, SiC and GaN in simulation at 5 A also concluded that the GaN-HEMT have the best on-state and switching performance [START_REF] Chowdhury | Comparison of 600V Si, SiC and GaN power devices[END_REF].

GaN and SiC 650V components available

Many manufactures are involved with GaN components with rating under 30A. For our 7 kW applications, focus was given to components with higher rating as indicated in Table 1. For a boost converter, the switching loss in the lower hard commutating transistor is directly related with the switching energy Etotal generally indicated in the datasheet (sum of the switching energy at turn ON and at turn OFF). A low switching energy is a good indicator on the possibility to increase the switching frequency and to increase the power density. The values in Table 1 are directly derived from the data and curves of the corresponding datasheets. The third to last column indicates the switching energy at a specified current and the second to last column the switching energy at 25 A, in both cases at 400V. Only for GS66516B, the switching energy at 25 A is not given in datasheets but obtained from simulation [START_REF] Oliveira | Switching Behavior and Comparison of Wide Bandgap Devices for Automotive Applications[END_REF]. Simulations were not performed for the GS66508B components, the only information available is that it has a very low switching energy at 15 A. Regarding the GaN components, it can be noted that the GaN system GS66516B and the Transphorm components TP65H035G4WS have very similar resistance at 125°C as well as very similar switching energy at 25A. The GaN component from VisIC V22TC65S1A also has a similar resistance but a lower switching energy. From the data, a factor of merit (FOM) [START_REF] Oliveira | Switching Behavior and Comparison of Wide Bandgap Devices for Automotive Applications[END_REF][START_REF] Meiβner | Switching Behavior and Comparison of 600V SMD Wide Bandgap Power Devices[END_REF] is also computed and indicated in the last column multiplying the switching energy with the resistance at 125°C. The lower the FOM, the better is the switch for hard switching application (boost, buck, inverter, PFC…).

Table 1 Electrical properties of 650V SiC and

GaN components with rating of 30A or higher

Design of the WBG based converter

A 400V boost converter based on GS66516B 25 mohm components from GaN System was designed at VEDECOM. It has been tested in two versions with a small and a big heatsink as can be seen in Fig. 1 (b andc). The 2 GS66516B switches are placed on an Insulated Metal Substrate (IMS) as shown in Fig. 1 Volumes of those converters are summarized in Table 2.

Optimisation with the GaN System converter

The GaN System prototype in the small heatsink version, is tested up to 3,9 kW at 200 kHz. Nonetheless, the temperatures were already being measured over 100°C at 3,9 kW and 200kHz. It was thus decided to do a second generation prototype incorporating some improvements, the gate resistance for turn ON is reduced from 10 ohms to 8 ohms in order to reduce the switching losses at turn ON, the IMS reference and parameters are changed in order to improve the thermal behavior.

The second generation prototype was first tested till up to 100kHz 7,34kW, with the bigger heatsink. It was then also tested at the same operation point with the smaller heatsink, the impact of the heatsink reduction size on the GaN components temperature was inferior to 5°C and the impact on efficiency was neglectable.

Efficiency results

Efficiency of the SiC and GaN converters have been measured with 305V at input and 400V at output using a high accuracy measurement power analyzer. Results at different switching frequencies and at different output powers are shown in Fig. 2 for the GaN converter based on GS66516B, Fig. 3 for the SiC converter based on C3M0060065K, Fig. 4 for the SiC converter based on C3M0060025K and Fig. 5 for the GaN converter based on the V22TC65S1A.

Fig. 2

GaN converter based on GS66516B For the two SiC converters, the pronounced drop in efficiency at 100 kHz 8.2 kW is due to the excessive heating of the inductor which was designed for a 7.5 kW application. Indeed at 100 kHz the current ripple is stronger, at 8.2 kW, this leads to a gradual entry into the saturation zone and additional heating. As shown in Fig. 4 and Fig. 5, the addition of a small fan on the inductor (0,9 W consumption, 37 cm 3 volume), allows to improve the efficiency at 8,2 kW 100kHz.

Fig. 5 GaN converter based on V22TC65S1A

For all the tests performed, the temperature of the V22TC65S1A GaN component were stabilized at temperature lower than 70°C.

Efficiency results comparison

In order to compare the efficiencies obtained with SiC & GaN, the results from the previous sections are superimposed on Fig. 6, Fig. 7 and Fig. 8.

Fig. 6

GaN & SiC converter efficiency comparison at 100 kHz At 100 kHz switching frequency, the converter with GS66516B was tested up to 7,34 kW and the converter with the other reference up to after 8 kW. From 4 to 8 kW, the best efficiency was obtained with the SiC 25 mohm components. Regarding the GaN, the GaN System and the VisIC converter had almost identical efficiencies.

Fig. 7

GaN & SiC converter efficiency comparison at 200 kHz Fig. 8 GaN VisIC & SiC Cree converters efficiency comparison at 300 kHz and SiC Cree 60 mohm 500 kHz operation At 200 kHz the Gan System converter was tested up to 4.3 kW, the GaN VisIC and the SiC Cree converters up to 8.2 kW. At 300kHz, only the SiC converter and the GaN VisIC converter could be tested. Only the SiC converter with the 60 mohm reference was operated at 500 kHz. For the 100, 200 and 300 kHz switching frequencies, at higher power from 6 to 8 kW, the best efficiency was always obtained with the SiC converter using the 25 mohm C3M0025065K components from Cree. Regarding the GaN converters, similar efficiencies were obtained with the GaN System and the VisIC references, but with VisIC, ilt was possible to go higher in power and in switching frequency thanks to a much better thermal impedance junction to ambiant and probably also thanks to slightly lower switching loss.

Inductor size reduction with higher switching frequency

For both SiC converters, the 3*47 uH inductors in series were replaced with a 47 uH inductor, cooled with a small fan (0,9 W consumption, 37 cm 3 volume). Both SiC converters were tested successfully at 8,2 kW 300kHz with the new inductor. In both cases, the replacement of the inductor had no significant impact on the overall efficiency at 8,2 kW. At 8,2 kW the increase of the switching frequency to 300 kHz thus allowed to reduce the size of the inductor from 129 to 80 cm 3 .

Scope measurements

The experimental oscilloscope measurements obtained during the tests are indicated in Fig. 9, 10, 11 and 13. The measurements show higher ringing and overshoots with the SiC which can partially be explained by the TO-247 package and by a board layout with higher power loop inductance. For both GaN converters, the power and drive loop inductance were fully optimised. The ringing and overshoot on the converter with VisIC components were smaller than with the SiC but bigger than with GaN System.

Method for switching loss estimation of both transistors

In the datasheet, as indicated in section 2 and in Table 1, the switching energy are given for a given current and a given temperature. This corresponds to the hard switching loss of the lower transistor in the boost application. Nonetheless, in the boost, other losses are related with the frequency, losses related with the ripple, losses in the complementary switch (dead time loss, Eqrr loss), losses due to dynamic Ron and losses due to the raise of temperature when the switching frequency is increased. To cover all loses in both transistors which are proportional to the switching frequency, we propose here to define a new parameter as indicated in Eq. ( 1):

𝐸 𝑏𝑜𝑡ℎ 𝑐𝑜𝑛𝑣𝑒𝑟𝑡𝑒𝑟 𝑠𝑤𝑖𝑡𝑐ℎ𝑒𝑠 = Ploss(f1) -Ploss(f2) 𝑓1 -𝑓2 (1)
In order not to be impacted by the varying inductor loss, f1 and f2 must be taken high enough, so that the ripple current is low enough so that the impact on the inductor loss can be neglected. In the case of our converter 305V to 400V with 141 uH inductor, for a frequency of 200 kHz or higher, the peak to peak ripple current in the inductor is 2,5 A or smaller. Applying Eq. ( 1) to the results of section 4.1, the Fig. 12, can be obtained:

11 kW converter

In this section, the power rating of the converter is extended to 11 kW. An inductor with current rating higher than 30 A must be used. To keep minimizing the volume and weight of the inductor, a high switching frequency of 200 kHz is selected. The new inductor is obtained by paralleling two 47uH inductor with cooling by the small fan. From the results shown on Fig. 6, 7 and 8 the converter with the 25 mohm SiC component has the best efficiency at high power for all switching frequency. It was thus selected for the power extension. Efficiency at different output power is shown in Fig. 13 and scope measurements in Fig. 14. After careful review of GaN and SiC components available in the market, the best components were selected. Those components were then tested in dedicated boost converters and excellent performances were achieved with both GaN and SiC components with similar high efficiency and power superior to 8 kW. The possibility to increase the switching frequency were the highest with the Cree SiC components and with the VisIC GaN components thanks to low switching loss as well as reduced overall thermal impedance. With the SiC converter, using the increase of the switching frequency, a reduction of the size, weight and cost of the inductor at 300 kHz was accomplished. This reduction come as a trade off with a slightly decreased efficiency compared to 100 kHz operation. Those results can be useful for both converter designers as well as system engineers comparing the respective advantage of an 8 kW converter switching at 100 or 300 kHz. With all components selected, SiC from Cree, GaN from GaN System and GaN from VisIC, converters with high power density from 10 to 13 kW/Liter were realized. The ringing and overshoot on the converter with GaN VisIC components were smaller than with the SiC but bigger than with GaN System. Our results have thus also shown a trade off between EMC behaviour on one side and efficiency, power capability and power density on the other side. In a simplistic approach, the smaller was the component package, the best was the EMC behaviour, the bigger was the component package, the better was the overall thermal impedance and thus the power capability. A method to measure and visualize the overall switching loss of the converter was also shown. Finally, a 11 kW converter version of the SiC converter was also tested. In this paper, experimental results were given to compare GaN and SiC performance for hard switching application at 400V and power range superior to 4kW. For lower power and lower voltage many other GaN manufacturers are involved. The results of this paper can also only be partially extrapolated for soft switching comparison of SiC and GaN. For a final selection in an automotive application, robustness to short circuit, reliability, cost, and availability will also need to be considered.

Reference

Fig. 1

 1 Fig. 1 Picture of (a) IMS (b) GaN converter (small heatsink) (c) GaN converter (big heatsink)

Fig. 3 Fig. 4

 34 Fig. 3SiC converter based on C3M0060065K

Fig. 9

 9 Fig. 9 Measurement of converter with GS66516B GaN components, 305 to 400V, 7,34 kW, 100 kHz, 141 uH and time zoom on the commutation

Fig. 10

 10 Fig. 10 Measurement of converter with V22TC65S1A GaN components, 305 to 400V, 7,32 kW, 200 kHz, 141uH and time zoom at 5,9 kW on the commutation

Fig. 11

 11 Fig. 11 Measurement of converter with C3M0060065K SiC components, 305 to 400V, 8,2 kW 300 kHz 47 uH inductor with fan and time zoom on the commutation

Fig. 12

 12 Fig. 12 Total switching energy of the converter at converter operating temperature

Fig. 13

 13 Fig. 13 Efficiency of SiC converter based on C3M0025065K with different inductors

Fig. 14

 14 Fig. 14 Measurement of converter with C3M0025065K SiC component, 305 to 400V, 10,8 kW 200 kHz 23,5 uH inductor with fan and time zoom on the commutation

Table 2

 2 Volume of the converters

4 Converters tests results


From rags to draperies: depictions of London paupers in the Victorian visual arts

Résumé Du chiffon au drapé : la représentation des pauvres londoniens dans les arts visuels victoriens Si les romans victoriens décrivent dans les rues de Londres des gueux en haillons, traités comme des matières informes, animalisées, objectifiées par le regard empli d'effroi d'un narrateur ou d'un personnage, le sujet est plus délicat à traiter dans les arts visuels. Des peintres tels que Hubert von Herkomer, Frank Holl et Luke Fildes tirent renommée et succès d'une représentation stylisée et empathique de la misère, qui s'inspire parfois de leurs illustrations de presse, moins contraintes par les codes de la peinture académique. Aussi les peintures, photographies et illustrations de miséreux si affligés qu'ils en semblent morts, effondrés contre les murs de villes hostiles à leur présence, se tiennent constamment entre esthétisation et documentation, entre hommage au genre pictural et obédience au réalisme social. F.G. Watts travaille la matière des chiffons et loques qui couvrent les pauvresses à la façon d'un marbre, Frank Holl emprunte la structuration de ses tableaux à celle des représentations de vierges à l'enfant de la Renaissance italienne, les photographes John Thomson et Adolphe Smith, documentant les rues de Londres, relayés par la peinture de genre de Augustus Edwin Mulready, concentrent le regard sur des mères isolées ou des enfants mendiants, Gustave Doré dispose les mendiants en frise : la délicatesse de ces dispositifs sert autant la dénonciation de l'abjecte misère que l'embellissement, la revalorisation des pauvres. Le gueux solitaire est magnifié, notamment par le biais du vêtement, quand la masse de mendiants hagards rassemblés, à l'exemple de ceux de Dudley Hardy et de Charles Paul Renouard, apparaît dangereuse. Un regard moderne porté sur l'altérité s'élabore ici, non sans hommage appuyé aux maîtres anciens.

In "Night Walks", a chronicle of his nocturnal perambulations in the streets of London, Charles Dickens records how his persona, the insomniac Uncommercial Traveller, nearly trips over a vagrant lying down on the ground: Suddenly, a thing that in a moment more I should have trodden upon without seeing, rose up at my feet with a cry of loneliness and houselessness, struck out of it by the bell, the like of which I never heard. We then stood face to face looking at one another, frightened by one another. The creature was like a beetle-browed hair-lipped youth of twenty, and it had a loose bundle of rags on, which it held together with one of its hands. It shivered from head to foot, and its teeth chattered, and as it stared at me -persecutor, devil, ghost, whatever it thought me -it made with its whining mouth as if it were snapping at me, like a worried dog. Intending to give this ugly object money, I put out my hand to stay it -for it recoiled as it whined and snapped -and laid my hand upon its shoulder. Instantly, it twisted out of its garment, like the young man in the New Testament, and left me standing alone with its rags in my hands1 .

The gaze the narrator and the pauper exchange emphasizes their reciprocal fright. The narrator's subsequent humane attempt to touch and assist the miserable man is counterbalanced by this reification and animalization: he is described as "a thing", an "ugly object" who utters harrowing noises. Notable is his "loose bundle of rags", which recalls the tattered clothes street labourers scavenged to make a living. The narrator is left with a ragged piece of clothing in his hands, which he compares to the garment Christ's young disciple leaves in the guards' hands as he flees naked from the garden of Gethsemane (Mark 14:51-52). The experience of urban misery is recounted by way of a comparison with a Biblical episode. The wretch identified by his abject rags is thus ennobled by this allusion.

Describing ragged and slouching paupers such as the one Dickens portrays here in his literary sketches raised other difficulties when it came to the visual arts 2 . Urban poverty was increasingly represented on canvas during the century and social reportage was emerging as a new epistemological field. Yet, it was still considered an unfit subject for art. Painters especially were constrained by artistic conventions, but also by ideological considerations 3 . The Victorian tendency to idealize or sentimentalize the depictions of modern subjects in painting is to be found in many pictorial representations of misery. The critical reception was generally averse to realistic depictions of poverty. Exhibiting them on the walls of galleries or hanging them in middle-class homes was therefore challenging [START_REF] Julian | Hard Times: Social Realism in Victorian Art[END_REF] . Still, committed patrons did acquire the "social realist" pictures that Hubert von Herkomer, Frank Holl and Luke Fildes produced in the 1870s. Fildes' row of paupers wearing grimy rags in Applicants for Admission to a Casual Ward of 1874 5 (figure 3) created a sensation when exhibited at the Royal Academy. But the reception of their representations of social issues was not always positive and Fildes, Holl and Herkomer later turned to the more lucrative genre of portraiture. They had all been successful press illustrators for The Graphic, but it was easier to depict physical and psychological destitution in press illustrations than in painting, which was constrained by codes and conventions [START_REF]On the cross-influences between the arts and the illustrated press, see Caroline ARSCOTT[END_REF] . An ever-increasing production of textual reportages in book or periodical form evoked the topic of the vast multitudes of poor labourers and mendicants who roamed the streets of London and these reports were often illustrated with engravings and photographs [START_REF]See for example Henry Mayhew's London Labour and the London poor (1851-1861[END_REF] . These too are nonetheless characterized by a tension between documentation and aestheticization. Indeed, the depiction of misery in the visual arts particularly problematizes the issue of the gaze of the artist as well as that of the beholder. The artists who wished to raise awareness about social ills had to conciliate this intention with social expectations and artistic conventions as well as with their own aesthetic principles.

This chapter focuses on a particular motif in the visual arts: the representation, in a corpus of paintings, illustrations and photographs, of paupers who have collapsed on the ground or are slouching or sitting against walls or on bridges. These figures are sometimes so destitute or apathetic that they seem to be dead. Their averted or closed eyes deprive them of the gaze they might have returned to the spectator. Their bodily postures and their tattered garments are clear visual signs of their utter social degradation. The artists, however, were motivated by artistic considerations and had to keep within the bounds of the visually acceptable and so the figures are often handled artistically. They are generally based on previous iconographical models and their rags are depicted like fluid folds that sometimes evoke draperies. Exactly as Dickens ennobles the pauper's rags by means of a comparison, the garments are transfigured by the treatment of the figures' clothes. In so doing, the indigents are somewhat divested of their working-class identity and removed from their socio-economic context.

The "human refuse": G.F. Watts's draped pauper

Around 1849-1850, the painter F.G. Watts painted four social subjects which stand apart from his neoclassical and symbolist production. He was in search of a new language that would allow him to address high-minded topical subjects. The Song of the Shirt depicts a desperate seamstress. Like numerous contemporary paintings on the theme, it was drawn from a poem published by Thomas Hood (1799-1845) in 1843. Irish Famine is a frontal presentation of a starving Irish family. Found Drowned represents a woman washed ashore under a bridge, presumably a suicide. Under a Dry Arch shows a forlorn female pauper. Watts had read Carlyle's denunciations of social inequality in Past and Present (1843) and he was eager to broach social issues 8 . Yet he never adopted the visual style of realism. Incidentally, Watts did not exhibit these canvases before 1881. He lent them to annual exhibitions of the Whitechapel Art Gallery as part of the collective venture led by thinkers and artists to educate the working class and improve their lives 9 . During his career, Watts often denounced the scourges of his society, such as child prostitution, women's exploitation or animal cruelty. His mode to do so was often that of allegory or symbolism. He thus modelled the human figure on classical and Michaelangelesque sculptural prototypes (mostly the nude or classically draped figures) as these, to him, were immediately legible and could best express his idea. The human figures in his four social subjects thus personify ideas and emblematise the universal condition of man.

Under a Dry Arch 10 (figure 1) depicts a poor woman who has found shelter under a bridge, perhaps to die there. The broad handling of paint and the drab colours convey her dire situation. She is alone in a blurry and austere setting and there are no clues as to her situation. Watts' painting is a discomforting image of the ravages of the economic and social context and their impact on women. Yet his pictorial treatment of the woman's clothes aestheticizes his subject. Her garment is an artistic flowing mass of folds that hardly suggests realistic rags. Watts thus addresses this social subject in his usual vein -that is, with a classicizing language and in an allegorical mode.

In his work on avant-garde photography, George Didi-Huberman compares the Parisian serpillières and chiffons to survivals of Antiquity: their shapes somehow retain something of the classical drapery, and the piece of clothing acquires an independent life just as it falls on the pavement 11 . The classical motif thus becomes a base object. Yet the visual reminiscence of the antique allows for a displacement of the gaze: what seems at first an informal heap of folds, a miserable and impure piece of refuse, is elevated to a beautiful and noble motif. The chiffon, or rag, is a leitmotif in photography and this points to its centrality in the imaginary of modernity.

Watts's image too allows for a displacement of the gaze. In a comment on the painting, he associated his figure with debris: "my old woman under the archway I hope, arouses pity for human refuse [START_REF] Watts | The Pointless Meaningfulness of Watts's Work[END_REF] ". Her posture, gaunt face and crouching position equate her to a heap of rags and drapery, and this creates a "visual paradox [START_REF]Le paradoxe visuel proposé par ce tas de voiles ou de haillons[END_REF] ". The garment is indeed a reminiscence of the Parthenon marbles that Watts copied in the British Museum as show the intricate, diagonal and rectilinear folds of the orange-tinged cloak on the upper part of her body and the fluid, flowing folds of the green gown which covers her legs. The antique formula ennobles the figure of the pauper and realism here meets allegory. Even at the moment of her probable death -the rippling garment is like life flowing away from her -, the pathetic woman remains artistic. The woman's ugliness reflects the physical effects of poverty on her body and face. Watts nonetheless aestheticizes this realist subject thanks to visual echoes drawn from religious iconography. Her bodily posture, the attitude of her head, the play on the light and her pallor call to mind paintings of the Virgin swooning at the sight of her son crucified on the Cross 14 . This fits the tendency in much Victorian art to represent working-class figures as allegories of redemption. Here, though, salvation is not to be found in the Church: she turns her back on St Paul's Cathedral, which looms out of the blurred darkness of the cityscape. This institution is reduced to a distant shadow because it disregards the fate of the poor and this, for Mark Bills, exemplifies Watts's interest in "Dickensian social realism [START_REF] Mark | Dickens and the Painting of Modern Life[END_REF] ". The grey-brown hues evoke the filth and pollution of a foul urban world which corrupts its inhabitants, in particular women. The bridge is a motif that symbolises the contrast between the frailty of humanity and the immutability of solid buildings; it also functions as a liminal threshold between life and death and imprisons the figure in a tomb-like space.

Watts refrained from exhibiting his "social subjects" until quite late. He may have found it problematic to produce an image of suffering which, to follow T. Ghosh's analysis, could be "inextricably enmeshed with liberal guilt and pleasurable consumption [START_REF] Tanushree | Gifting Pain: the Pleasures of Liberal Guilt in London, a Pilgrimage and Street Life in London[END_REF] ". Misery elicited an uneasy tension between the ethical and the aesthetic and the potentiality that the work should occasion the scopophilic gaze may have left the artist in a quandary. On the other hand, the aestheticization of the subject and its inscription within iconographical models enabled Watts to tackle the issue of destitution in the medium of painting.

A mass of crumpled folds: Frank Holl's impoverished mother and child

Frank Holl was particularly interested in the reality of the labouring and destitute classes, whom he represented in a style that reflected his academic training and his emulation of the Old Masters. He felt sincere empathy for the harsh conditions of his fellow men but, like many other artists, he looked to previous iconographical models when drawing the human figure. Many contemporary paintings of female paupers and their children are visual reminders of religious iconography. Besides, mothers are often represented as maternal and devoted, which corresponds to middle-class constructions of woman in a culture that sanctified the family 17 . In a drawing published in The Graphic in 1888 18 , Holl represents a mother carrying a child who has settled on a doorstep. The bundle against her bosom almost hides her face and seems to contain another baby. The shape of a cross on the door suggests that this is a church. According to his daughter, Holl was inspired by a scene he had actually witnessed [START_REF] Mark | Emerging from the Shadows[END_REF] . The group is organised in an academic triangular composition. The mother is like a secular version of the Madonna with Jesus and the Infant John the Baptist placed on a crumpled and sketchy layer of cloth. What predominates is the mass of clothing. The painter used thin hatchings and thick strokes to render the materiality of the gown and the movement of the folds. White dabs of gouache suggest the uncomfortable presence of snow. Parts of the figures' bodies are lit and stand out against the dark heap of fabric. The mother's face is half-shown and merely suggested by swift lines. Her neatly combed hair, on the other hand, is delineated with thin, flowing lines. Although the woman is de-individuated, her attitude and the treatment of her hands express pathos and the preservation of maternal instinct. The visual reminders of religious iconography and the handling of the garment ennoble the pauper woman. Holl's treatment of working-class figures was always elevated and the way he depicted the folds of their rags attests to the legacy of the antique.

Holl made a version of this work in a painting [START_REF]Masculinities in Victorian Painting[END_REF] but a comparison of the two shows that the medium of illustration enabled him to tackle this dire subject in a freer and bolder way. The mother is sitting erect: she has not collapsed on the floor. She turns to the child comfortingly and her eyes are open. This individualizes her and the image is less discomforting than the drawing. A basket laden with flowers points to her occupation and the urban setting contains the comforting presence of a church in the background. The dispositif, thus, orientates the spectator's gaze: in a context in which paintings were exhibited in venues mostly frequented by the middle class, this sentimentalized depiction of a poor mother and her child who are situated socially could elicit the beholders' empathy without upsetting their sensibilities or offending their opinions [START_REF] Casteras | Seeing the Unseen. Pictorial Problematics and Victorian Images of Class, Poverty, and Urban Life[END_REF] . The drawing, on the contrary, leaves the beholder with no clues as to the situation and social class of the woman, who is thus universalized. Holl concentrates on the pathetic plight of a desperate mother and children who have reached utter degradation by falling to the ground like a heap of clothes. Both works point to different strategies that are conditioned by the medium used, but in both cases the artistic treatment makes these paupers a legitimate subject in art.

The swathed abject "crawler"

Representing wretched paupers in an artistic mode was also a concern of early photographers, even when they aimed at documenting their society. In 1877 John Thomson and Adolphe Smith published a photo-reportage entitled Street Life in London which concentrates on the lives of workers and beggars. In particular, they describe "the Crawlers of Saint Giles's", old women who had become so wretched that they had to crawl in order to beg. These "wrecks of humanity" are plunged in "a lethargic condition which can scarcely be preferable to death itself": they are "constantly dozing, and yet are never asleep22 ". Their physical attitudes reflect that state: "Some of them are unable to lie down for days. They sit on the hard stones step of the workhouse, their heads reclining on the door23 ". Their immobility is only interrupted when they crawl like animals, occasionally exchanging gazes with the passer-by when, "with one eye open", they "look up languidly".

The chapter is illustrated with a photograph of a "crawler" evoked in the text -an old woman characterized by "abject misery". She lives on the street with her young son who is "covered in rags" but does not appear in the image. He has placed his own son in her care so that she may "redeem herself" -a highly religious term -and earn her meagre subsistence of tea and bread. The photograph shows her with the baby, who is only protected by "the dirty and worn shawl which covers the poor woman's shoulder". Yet although she exemplifies the category of the crawlers, she is not represented as an utmost figure of abjection. The treatment of her clothes, on the contrary, is quite artistic and she is ennobled by the visual allusion to the Madonna and child. The setting is reduced to a minimum. The reality of urban filth is reflected in the work on the grain of the photograph, with traces, grooves or stains on the wall and a rather blurred though geometrical background. The figure is placed close to the spectator and at the centre of the photograph. The "crawler" seems like a variation on the picturesque old beggar. The position of her head evokes the crawlers' characteristic lethargy mentioned in the text, but it also recalls the iconography of martyrology. Most of all, her striped cloak is a graceful swathe that espouses her arms and envelops the infant. As for her gown, it is rendered with clearly delineated folds and creases that differ from the rags evoked in the text. The treatment of the garment is therefore quite artistic.

In their preface, Thomson and Smith stated that their objective was to reflect a reality. Ghosh, however, points to the ambivalence of such publications, which were devised as works of social investigation but were also advertised as commodified artistic books24 . This to Ghosh generates "an interplay of […] disparate spectatorial motivations25 ". Such images indeed conflate curiosity for the Other, a more or less authentic desire for social justice and the impulse to aestheticize the abject so as to make it fit for artistic production and consumption.

The picturesque attire of the child-beggar

Contemporary photography of misery is thus often poised between the documentary and the aesthetic. This was especially the case with images of street "urchins". Poor Jo, Homeless 26 by Oscar Rejlander (1813-1875) is a case in point. Originally entitled Night in Town and conceived as a pendant to another beggar boy, Day in Town. Thus renamed, the work alludes to the streetsweeper of Dickens's novel Bleak House (1852-1853) It thus moves the representation away from reality and inscribes it within the fictional. Poor Jo, Homeless represents a ragamuffin wearing rags and sitting on a doorstep. His body is curled up and his face buried in his crossed arms. His posture evokes prostration and lethargy, but it is also considerably closer to the artistic poses of académies than to real-life situations. The artist had a young model pose in a minimalized and stylized décor, his body covered with artistically arranged rags. His shirt is depicted with graceful folds as it sophisticatedly falls off his shoulder. The figure fits into the tradition of the picturesque child beggars initiated by Bartolomé Esteban Murillo (1617-1682 [START_REF] Tijana | studies the genealogy of beggars from Spanish to Dutch or Flemish and then to English painting in "The Indecorous Appeal of Beggar Boys: Murillo, de Lairesse and Gainsborough[END_REF] ). All in all, the photograph stands as an aestheticized image of infantile destitution.

Such beggar-child imagery was a favourite subject with the genre painter Augustus Edwin Mulready (1844-1905). Like Thomas Kennington (figure 5), Mulready produced innumerable sentimentalized depictions of children begging, working, resting or sleeping on the streets of London. In particular, he relished in images of children who are sleeping but seem almost dead, perhaps. Representing children enabled the artist to make his paintings of poverty more acceptable, and possibly more saleable. The artist probably intended to make the public aware of the plight of such children. However, his images do not contain signs that he aimed at denouncing an iniquitous social system. On the contrary, the sentimentalized treatment alleviates the topic of poverty.

This element is prominent in his painting A Recess on a London Bridge 28 (figure 6). Mulready chooses the symbolic motif of the bridge on which a ragged child is sleeping. The treatment of his clothes enhances the artistic dimension of the image: the shirt's folds are carefully delineated, with marked brushstrokes that indicate the fluidity of the garments. The setting is reduced to the geometrical structure of the parapet and bench. The subject and the theatrical treatment of light and shade emulate Murillo's beggar children and his dramatic baroque tenebrism 29 . The boy's face is illuminated by dramatic effects of light while his body is partly in the shade. The starry, inky blue sky, which occupies a third of the picture, also evokes a stage backdrop. The painting does not depict an individual story of destitution; rather, it is related to the artistic and the fictional.

The work is therefore ambivalent. It depicts a social reality -the death of street infantswhile flaunting the artistic means to do so. The stylish ribbon that adorns the urchin is a traditional iconographic motif that recalls the unrealistic drapes Murillo placed on his beggars [START_REF]See for example Young Boys playing dice[END_REF] . But its blood-red colour also connotes death. It. A shadow seems to gradually invade the boy's body, as if suggesting his inexorable demise. His head is slightly tilted backwards so that the beholder may focus on his delicate beauty. Mulready departs from the figure of the picturesque Southern beggar: his model corresponds to current stereotypes of English beauty: the doll-like face, unspoiled skin, delicate closed eyes and regular fair hair emphasize the impression of purity. Moreover, he turns his face to the sky, perhaps an allusion to the theme of the comfort found in the afterlife. Death -and the bridge is highly symbolic here -is perhaps more promising than life on earth. However, the article "a" in the title indicates that this is just any bridge. The topographic, economic and social context is thus not made explicit. Representing a beggar child enabled the artist to appeal to sentiment yet it certainly prevented him from indicting the system. This canvas stands rather as an allegory of misery.

Mulready's repeated representations of exhausted young beggars on bridges, which are variations on his prolific production of children working or sleeping on the streets, attest to the contemporary commodification of images of misery. The sentimentalizing trend of many Victorian images of infantile poverty points to the artistic and ideological norms that restricted British artists in a way that differed from practices in France. One comparison between Mulready's and Kennington's maudlin portrayals of children with the starker and more realistic depictions of urban urchins produced by Fernand Pelez (1848-191331 ) shows their different approaches: in presenting artistic beggars, the artists ran the risk of anesthetizing the desire to awaken consciences.

Gustave Doré's friezes of tattered paupers

In 1872, Gustave Doré and William Blanchard Jerrold (1826-1884), a writer and journalist, published London, a Pilgrimage32 , an illustrated travelogue that provides numerous scenes of urban destitution. The book reflects the contemporary interest in social investigation. It was also designed as a gift-book and it sold well, despite its lukewarm critical reception. Images of poverty thus acquire the status of "leisure-time commodities33 ". Doré's depictions of the city are imaginative visions rather than documentary pictures. They recall his illustrations for tales, fairy tales or Dante's works. London holds several illustrations that represent ragged sitting or slouching vagrants: they are either resting or sleeping on or under bridges34 . Most are organised in frieze-like compositions. The reality of their dire poverty is aestheticized, among other means by the treatment of the figures' ragged clothes and battered hats. The visual language is based on pre-existing iconographical compositions and motifs.

The first of these engravings, Resting on the Bridge (p. XII), shows six figures on a bench organised around two groups of three paupers, each arranged in a pyramidal structure -one with a woman who holds a baby on her lap and rests her head against a child's, and the other with a girl holding two infants. Doré plays on the contrasting effects produced by the lines and the masses: the flowing folds of the paupers' clothes and cloak relieve the geometrical structure and straight lines of the pavement, parapet and sky. The figures are focused upon by the vertical format and constricted space. Despite the vacant plane in the foreground, the beholder is directly confronted to the figures. However, more than a depiction of London's misery, the illustration evokes the iconography of the Holy family resting during the Flight into Egypt, with the veiled woman functioning as a profane Madonna. Doré handles the rags as rippling draperies heightened with white strokes, which attests to his emulation of the Old Masters.

Such artistic treatment of beggars resting or sleeping on bridges is found in many other illustrations. Asleep under the Stars (figure 7), which illustrates a chapter entitled "London Charity" (ch. XXI, p. 179), depicts a row of beggars of all ages who are similarly placed in a frieze-like composition. The choice of the symmetrical stacks and central recess -enhanced by the horizontal format -allows Doré to play again on the contrast between the geometrical structure of the bridge and the disorderly figures of ragged sleepers. The text below the illustration reads: "The aged, the orphan, the halt, the blind 35 , of London, would fill an ordinary city. When the struggle for life is so severe as it is in England in the happiest times, the wounded and disabled and invalided must be in considerable numbers" (p. 179). There is some variety in the figures: older and younger men and women alternate with children. The seated figures are more or less slumping on the bench; their poses are contrived and yet artistic. What stands out again are the masses of their rags. The fluid and elaborate garments depicted in dark tones contrast with the light grid against which they are placed -an effect which is amplified by the undulating line of the heads. The upper half of the picture shows a murky starry sky that conjures up the impression of cosmic calm. The scene is placed at some distance from the viewer thanks to the inclusion of a portion of the cobbled street in the foreground. All this gives the impression that the illustration does not describe real homeless people but models posing on a stage. This theatrical dimension is present in another engraving entitled The Penny Gaff Frequenters (p. 165). Except for the vertical format, this illustration adopts the same composition, with a bridge under a night sky. This time, Saint Paul's cathedral is visible in the background; there are also fewer figures: an elderly couple and a child wearing hats are slumped on a bench. This looks like another example of homeless people sleeping rough in the urban environment. However, the image appears in a chapter devoted to "play" and to the theatre. The text actually blurs the distinction between reality and the theatre. Among the popular forms of entertainment described, such as penny gaffs, that is, cheap theatrical productions, is a play entitled the "Starving Poor of Whitechapel" at the "Garrick Theatre", attended by an empathetic audience: indeed, it is explained that the public "understood only too well what hard life in Whitechapel meant; and had spent nights with the stars, upon the stones of London" (p. 164). What becomes unclear, consequently, is whether the illustration refers to the actors performing on the stage or to the real scenes that inspired the spectacle. The conflation between the social realm and the world of the theatre further underlines the artistic dimension of Doré's illustrations.

The chapter on "play" contains another night-scene on a bridge that highlights the artistic and non-realist dimension of Doré's depictions of paupers. A Cold Resting Place (figure 8) (p. 177) is a new variation on the figure of the young girl with infants on her laps. An elaborate cloak envelops her, and her rags are treated like artistic drapery. The image is a step away from the documentary: the figure appears as a universal emblem of misère -a modern-day, juvenile and destitute adaptation of the traditional iconographical subjects of Roman Charity or of the Madonna and child. The privileged motif of the sleeping paupers in drape-like tatters thus fits a pictorial strategy that allows Doré to tackle a topical reality while achieving artistic effects above all.

Ragtag multitudes: the disruptive collective

Such aestheticized and allegorized sleeping or slumping beggars garbed in artistic rags allowed for a safe mode of contemplation. The artistic treatment could somewhat annul their political and social charge. A few visual works, though, represent flocks of destitute paupers lying down on the ground. The effect here is more unsettling. Sans Asile (1889) by Dudley Hardy (1867-1922) depicts an almost indistinct mass of rough sleepers lying against each other in Trafalgar Square [START_REF] Edwin | The painting's precise references have not been found. There is an engraving in the Art Journal[END_REF] . They seem like heaps of bodies in shabby clothes and are all crammed and tangled. Only a few figures stand out. The scene is inspired by a real event that took place in 1887 and which William Booth, who founded the Salvation Army, describes in In Darkest England and the Way Out (1890): "The existence of these unfortunates was somewhat rudely forced upon the attention of Society in 1887, when Trafalgar Square became the camping ground of the Homeless Outcasts of London37 ". Booth then describes similar night-scenes in order to raise awareness about the situation of the mass of Londoners who were deprived of decent housing and clothing. He focuses on the postures of the vagrants lying in huddled groups as well as on their ungainly rags, which provide scant comfort against the elements. His descriptions of such social situations are often recorded in dramatic scenes that emphasize the visual.

Hardy chose to place the figures of homeless people on the right-hand side of his canvas, under one of the monumental lions. The statue is cropped -perhaps an ironical comment on the grandeur of Britain. The solid sculpture and the blurred architecture of London in the background contrast with the randomly bodies of the vagrants, who adopt different positions: some are sleeping while others are awake and stare in opposite directions. A few figures are standing, but most are reclining or sitting, propped one against the other or against the pedestal. A man moves away from the group. Unlike Doré's rows of beggars, there is no orderly alignment. The various positions and gazes produce a sense of confusion. The accidental bodily poses of the sleeping figures make them look like dead people. A bandaged ex-soldier conjures up the image of a battlefield strewn with corpses. The treatment of the garments makes it even more difficult to distinguish the bodies and this is emphasized by the random groupings of entire or fragmented figures whose rags are amalgamated. Yet the fact that some of the figures are erect and awake, with tense or desperate expressions on their faces, gives an unusually social and political potential to the painting. The figures are packed, many of them reduced to heaps of clothing, and yet some of them are individualized. The composition is bold: the space between the group and the spectator on the left creates a sense of distance but the figures on the right extend towards the picture plane. The urban flâneur thus seems to have stumbled by chance upon this spectacle of social destitution and is directly confronted to it. This scene of utter bodily degradation turns into a blatant exposition of contemporary economic hardships.

The French artist Charles Paul Renouard (1845-1924) founded his effects on a similar mode when he illustrated another real event for The Graphic. "Our Homeless London Poor, St James's Park at Mid-day"38 (figure 9) shows a group of homeless women and men lying down in a corner of the park. The title and the recognizable buildings on the horizon clearly place the scene at the heart of London. The cropped composition shows the influence of photography. The figures are scattered in various positions on the field and the beholder is given both a panoramic view and a close perspective. The bodies are layered from top to bottom on the image and the effect created is that the foreground slips down from the picture plane towards the viewer. Another field on the left, separated from the central one by a lane, is strewn with bodies. This reinforces the sense of a multitude of passive, de-individualized men and women who literally invade the viewer's space. A few of the figures are sitting and chatting, but most of them are asleep. They wear distinctive garments but most look like inert bundles. The artist plays on crosshatchings and on contrasting light and dark tones to suggest the heavy material, the fluid folds and the deep creases. Very few faces are discernible. Their immobility connotes idleness and unemployment, as the title indicates (they sleep in the middle of the day). Most adopt the relaxed attitudes of rest and apathy, but the random placing of the bodies makes them look like corpses. A woman on the right is entirely muffled in a white cloak that looks like a pall. Above her, a dog sniffs at its immobile master.

One figure stands out in this general arrangement. At the exact centre of the image, a man quietly rests on his elbow, looking at the beholders as if to interpellate them. As M.E. Allen notes, he "stares ahead with an easy assurance, as if daring the viewer to displace him [START_REF] Also | The Art of G.F. Watts[END_REF] ". The figure is a variation on the iconographical subject of the leisured idler basking in the sun who, here, embodies the issue of working-class unemployment. Allen notes that political activists as well as socialist, religious and charity groups staged meetings and marches to mobilize the crowds assembled in the Park. The government authorities then tried to expel both the homeless and the militants. The issue of the politicization of the suffering poor was a much-debated topic, and the quotation from Shakespeare's Midsummer Night's Dream (Act III, sc. 2) -"Sleep, that sometimes shuts up sorrow's eye" -rings like an ironical comment on the central figure's gaze. Renouard provides a trenchant commentary on a social ill. The garments, hats and shoes are carefully delineated but the numerous figures are nonetheless treated as random heaps of clothing and rags. The figures are deprived of their gazes, which de-individuates them, except for the central figure, who engages in a direct ocular confrontation with the beholder and brings in a disruptive dimension.

All in all, the images studied here point to an ambivalent form of scopophilia -contemplating the artistic image of a destitute human being whose poverty makes him an "Other" to the wealthier observer walking around the gallery -or perhaps also walking around London. Such images came in the context of the advent of the spectatorial and the spectacular. Still, as this chapter has tentatively shown, artistic depictions of misery were constrained by artistic conventions which are reflected by the artistic treatment of bodies and garments. Many artists tried to conciliate the wish to reach effective visual effects and the desire to focus on the human dimension. While this tendency somewhat lessened the issue of misery, it nevertheless contributed to open the public's eye to human suffering. Significantly, a number of avant-garde artists, mostly Van Gogh (1853-1890) and Picasso (1881-1973), were fascinated with and inspired by some of these images of poverty. Such works, therefore, fit into the genealogy of modernity.

Anne-Florence GILLARD-ESTRADA Université de Rouen-Normandie ERIAC

8

  See David A. STEWART, "Reality, Artifice, and the Politics of Evolution: Watts and Carlyle in the Earnest Age", Victorian Poetry, Autumn-Winter, 1995, vol. 33, n° 3/4, Word and Image, p. 476-498; Lindsay ERRINGTON, Social and Religious Themes in English Art 1840-1860, New York, Garland, 1984. 9 In the early 1880s Watts became involved in the teaching work which social reformers and committed artists conducted in the destitute area of Whitechapel. See Mrs. Russell Barrington, F.G. Watts. Reminiscences, London, George Allen, 1905, p. 90-92; Mrs BARNETT, Canon Barnett, His Life, Work and Friends, London, John Murray, 1918, vol. I, p. 161 and II, p. 67-68. 10 G.F. WATTS, Under a Dry Arch, c. 1848-1850, oil on canvas, 137 x 101.5 cm, Watts Gallery, Crompton. 11 Georges DIDI-HUBERMAN, Nimpha moderna. Essai sur le drapé tombé, Paris, Gallimard, 2002, coll. « Art et Artistes », p. 13.

Charles DICKENS, "Night Walks", in Charles DICKENS, The Uncommercial Traveller, Daniel TYLER (éd.), Oxford, OUP,

2015, p. 126-134, here p. 131-132 (first published in Dickens' journal All the Year Round in July 1861).

Even a realist painting like Fildes' Applicants for Admission to a Casual Ward (figure4) shows a woman and children or huddled kinfolk that correspond to coded representations of the family.

John THOMSON and Adolphe SMITH, Street Life in London, London, Sampson Low, Searle & Rivington, 1877, p. 81-82.

Ibid, p. 82. All subsequent quotations are from this page.

Tanushree GHOSH, "Gifting Pain: the Pleasures of Liberal Guilt in London, a pilgrimage and Street Life in London", art. cit.

Ibid., p 92.

Oscar REJLANDER, Poor Jo, Homeless, c. 1860, photograph, carbon print, 20.3 x 15.7 cm, J. Paul Getty Museum. URL: https://victorianweb.org/photos/rejlander/6.html (last accessed 18 February 2022).

See Fernand Pelez, 1848-1913: la parade des humbles, Isabelle COLLET (dir.), Paris Musées, 2009.

William Jerrold BLANCHARD and Gustave DORÉ, London, a Pilgrimage, London, Grant & Co., 1872. See online: https://gallica.bnf.fr/ark:/12148/bpt6k10470488.image (last accessed 18 February 2022). All subsequent references are to this edition.

Tanushree GHOSH, "Gifting Pain: the Pleasures of Liberal Guilt in London, a Pilgrimage and Street Life in London", art. cit., p. 92.

Another of Doré's favoured modes of composition is the positioning of beggars against a wall or a door: they appear as heaps of rags arranged in a more or less pyramidal disposition.

This is an echo of Luke 14:21: "The poor, the maimed, the halt, and the blind" ("halt" meaning "lame, handicapped").

William BOOTH, In Darkest England and the Way Out, New York, London, Funk & Wagnalls, 1890, Chapter 3, "The Homeless", p. 25.

Charles Paul Renouard, "Our Homeless London Poor, St James's Park at Mid-day", The Graphic,

Michelle Elizabeth ALLEN, Cleansing the City: Sanitary Geographies in Victorian London, Athens, Ohio UP, p. 135.