Roderick Edwards

Aude Maignan

Complex Self-reproducing Systems

Keywords: self-reproducing systems, self-organizing systems, cellular automata, substitution systems, L-Systems

Cellular automata and L-Systems are well-known formal models to describe the behaviour of biological processes. They are discrete dynamical systems, each of which can have complex and varied behaviour. Here, we study a class of substitutive systems incorporating properties of both cellular automata and L-systems, that exhibits self-reproducing behaviour. A one-dimensional array of cells is considered, each cell has a set of modes or states which are determined by a number from Z/nZ * (n prime). The behaviour of a cell depends on the states of its neighbours and obeys to an additive rule. It has also a cell-division mode, which allows the line of cells to grow. The behaviour of such a model can be complex, but, using algebraic techniques, we prove that it can describe a reproducing system.

Introduction

Biological processes [START_REF] Barnes | Introduction to Modelling for Biosciences[END_REF][START_REF] Murray | Mathematical Biology: II. Spatial Models and Biomedical Applications[END_REF] are often modelled by a continuous formalism in which the behaviour of the system is described by differential equations. The smooth variation of each variable is described as a function of others [START_REF] Junck | Ten Equations that Changed Biology: Mathematics in Problem-Solving Biology Curricula[END_REF][START_REF] Turchin | Evolution in population dynamics[END_REF][START_REF] Yoshida | Evolutionary tradeoff between defence against grazing and competitive ability in a simple unicellular alga, Chlorella vulgaris[END_REF]. Another type of modelling uses a discrete formalism called multiagent formalism. Cellular Automata (CA) and L-Systems are parts of multi-agent systems which are homogeneous (all cells are identical). CA are defined by a lattice of sites, an alphabet of symbols and an evolution rule. Each cell evolves in discrete time steps according to some deterministic rules that depend only on local neighbours. In the configuration space, CA trajectories are likely to merge with time, and after many time steps, trajectories starting from almost all initial states become concentrated onto attractors. In these cases, completely disordered starting states evolve to more structured states by a process of self-organisation. In other cases, complex behaviour can appear. In his well-known papers [START_REF] Wolfram | Computation theory of cellular automata[END_REF][START_REF] Wolfram | Universality and complexity in cellular automata[END_REF], Stephen Wolfram proposed a classification of cellular automaton rules into four classes.

1. Class I: Evolution leads to a homogeneous state. 2. Class II: Evolution leads to a set of separated simple stable or periodic structures. 3. Class III: Evolution leads to a chaotic pattern. [START_REF] Junck | Ten Equations that Changed Biology: Mathematics in Problem-Solving Biology Curricula[END_REF]. Class IV: Evolution leads to complex localised structures, sometimes long-lived. This classification is based on observations of typical behaviours. K. Sutner [START_REF] Sutner | Classification of cellular automata, Encyclopedia of Complexity and System Science[END_REF], but also F. Bagnoli and al. [START_REF] Bagnoli | Damage Spreading and Lyapunov Exponents in Cellular Automata[END_REF] and J.M. Baetens and al. [START_REF] Baetens | Phenomenological study of irregular cellular automata based on Lyapunov exponents and Jacobians[END_REF] give a formalisation of those classes. For instance, the elementary rule 30 of Wolfram exhibits many different behaviours of class I,II,III and IV. On the other hand, an additive CA with fixed and periodic boundary conditions over a finite field Z n are more often of class I and II. This problem of determining dynamical possibilities for additive CA has been studied by Wolfram [START_REF] Martin | Algebraic Properties of Cellular Automata[END_REF], who has obtained some powerful theoretical results thanks to algebraic tools.

Wolfram has also studied substitution systems [START_REF] Wolfram | A New Kind of Science[END_REF], discrete dynamical systems such that, at every step, each site is replaced by a fixed block of new sites. Substitution systems are are clearly designed so that the number of sites can change. The theory of such systems was first developed by Aristid Lindenmayer and Przemyslaw Prusinkiewicz for modelling plant growth [START_REF] Lindenmayer | Mathematical models for cellular interaction in development (Parts I and II)[END_REF][START_REF] Prusinkiewicz | The algorithmic beauty of plants[END_REF], using a framework called L-systems. An L-system is defined as a t-tuple G = (V, ω, P ) where V is an alphabet, ω is the initial state of the system and P is a set of production rules. One of the simplest and best-known examples is the modelling of the anabaena algae. The alphabet is reduced to two states V = {A, B} and only two rules are required:

A → AB B → A
At each step, each letter A is transformed to the two letter sequence AB and each letter B is transformed to A. If the initial state is A, for example, we obtain the sequence ABAABABAABAABABA AABABA after 6 steps. The family of possible transformation rules is huge and the study of the behaviours of L-systems is done on a case-by-case basis.

The aim of this paper is to combine properties of L-systems and additive CA in order to create a simple model which describes a self-reproducing "organism". The algebraic properties of this new model allow one to describe theoretically its behaviour, particularly its reproducing behaviour.

The model is described in Section 2. In Section 3 we present our analysis of its behaviour.

Model description

We consider a one dimensional structure in which the evolution of a particular site depends on its own value and those of its nearest neighbours. Sites are arranged around a circle (so as to give periodic boundary conditions). This structure is named a sequence. The number of sites of a sequence depends on time and is denoted by N (t). The values of the sites of a sequence denoted a also depend on time and are denoted by a

(t) 0 , . . . , a (t)
N (t)-1 . The possible site values are elements of a finite set Z * n = {1, 2, ..., n -1} (n prime). The value of a site of the neighbour-dependent substitution system we consider is computed by an operator Φ (α,β) , which depends on two parameters (α, β) ∈ Z 2 n and is defined by a simple additive rule of the form (taking the site index i modulo N (t)): a

(t+1) i = αa (t) i-1 + βa (t) i + αa (t) i+1 [mod n] if αa (t) i-1 + βa (t) i + αa (t)
i+1 [mod n] = 0, -and an expansive rule such that a (t+1) i is divided into two sites of value 1 if αa

(t) i-1 + βa (t) i + αa (t)
i+1 [mod n] = 0 (i.e., 0 → 11).

We call such a system a Della Dora system, because it was first proposed to us by Jean Della Dora. The family of Della Dora systems is parameterised by α, β and n.

Subsequences are open in the sense that, by themselves, they carry no information about what is to either side. Thus we allow the possibility that a subsequence may be a full sequence, with periodic repetitions on either side, though it may, of course, be surrounded by something else.

Because of the expansive rule, the index of a site can change over time. But at a given step, a sequence may be represented by a characteristic polynomial

A (t) (x) = N (t)-1 i=0 a (t)
i x i , and the additive rule can be represented by multiplication of the characteristic polynomial by a fixed Di-polynomial in x, namely T (x) = αx+β +αx -1 , according to A (t+1) (x) = T (x)A (t) (x) mod(x N (t) -1). This approach has been developed by Wolfram [13] for a pure additive CA. The new expansive rule modifies (but does not destroy) the algebraic properties of reference [START_REF] Martin | Algebraic Properties of Cellular Automata[END_REF]. For instance, let us consider the commutative ring Z 3 . The sequences built on Z * 3 are composed of sites of value 1 or 2. Six different dynamics can be defined by the six Di-polynomials

x + x -1 , 2x + 2x -1 , x + 1 + x -1 , x + 2 + x -1 , 2x + 1 + 2x -1 , and 2x + 2 + 2x -1 . We present 3 examples which correspond to 3 different behaviours. Example 1. First, consider the Di-polynomial T (x) = x + 2 + x -1 on Z * 3 .
The sites evolve such that, at each time step, the value of a site a

(t+1) i -takes the value a (t) i-1 + 2a (t) i + a (t) i+1 [mod 3] if this value is not equal to 0;
else, splits into two sites, both of which take the value 1. This is a specific L-system and can be defined using this following transformation rules, where the site in brackets is modified according to the rule defined by the arrow. 1 1 1 → 1 1 1 2 → 2 1 2 1 → 11 1 2 2 → 1 2 1 1 → 2 2 1 2 → 11 2 2 1 → 1 2 2 2 → 2 The evolution of this system can be represented graphically at successive time steps by successive lines. Sites with value one are represented by a light colour; sites with value two are dark.

Figure 1 shows the system's behaviour when the initial configuration is (1112). For every initial configuration, N (t) is bounded and the system converges to a fixed point or a cycle.

This example thus belongs to Wolfram's class II and describes a synchronisation phenomenon. This synchronisation is a strong case of a self-organizing process where a global coordination arises out of an initially disordered system.

The behaviour of the second example is more complex.

Example 2. Consider the system with Di-polynomial T (x) = 2x + 1 + 2x -1 on Z * 3 and the 0 → 11 rule. This is defined in detail by the following L-system :

1 1 1 → 2 1 1 2 → 1 1 2 1 → 11 1 2 2 → 2 2 1 1 → 1 2 1 2 → 11 2 2 1 → 2 2 2 2 → 1
Figure 2 shows the system's behaviour when the initial configuration is (1112111). The behaviour is complicated and there is no obvious nested structure, so it belongs to Wolfram's class III.

Between the above examples, there exists a class of dynamics where the behaviour is both rich and predictable. Example 3. Consider the system with Di-polynomial T (x) = 2x + 2 + 2x -1 on Z * 3 , again with the rule 0 → 11. The transformation rule is described in the following table:

1 2 1 → 2 1 2 2 → 1 1 1 1 → 11 1 1 2 → 2 2 2 1 → 1 2 2 2 → 11 2 1 1 → 2 2 1 2 → 1
The behaviour is complicated but nested structure can be identified. Moreover nested structures give birth to new nested structures. This belongs to Wolfram's class IV .

More generally, the discrete phase portrait of such systems is difficult to analyse because the dimension of the phase space evolves over time. Nevertheless, algebraic tools bring powerful results. In Example 3 above, the open subsequences 11...11 = 1 k with k ≥ 4 always produce a subsequence of 1s of at least the same length after each time step (when k is exactly 4, the subsequence of 1s remains the same length: the open subsequence 21 4 2 produces another sequence containing 21 4 2). This property is at the root of a very interesting type of behaviour: self-replication. We have, in fact, that

Proposition 1. a (t) i-1 = a (t) i = a (t) i+1 = 1 implies that a (t+1) i is transformed into two sites of value 1 if and only if 1 is a root of its Di-polynomial. So in Z n (n > 2), Φ (α,β) gives behaviours of Wolfram class IV if (x-1)|x(αx+β +αx -1 ). Equivalently, Φ (α,β) gives behaviours of Wolfram class IV if 2α + β = 0 [mod n].
We call systems with this property, 2α + β = 0 [mod n], Della Dora systems of type 1.

Every configuration in a Della Dora system of type 1 has a unique successor at every time step. However, a configuration may have several distinct predecessors. αx 2 + βx + α is not prime and Φ (α,β) is not injective so these dynamics are irreversible.

The following section describes the properties of Della Dora systems of type 1. We will see how the behaviours of these systems can be described by a finite number of subsequences. This corresponds to a self-organizing phenomenon. We will also see that over time, specific subsequences give birth to a copy of themselves. This corresponds to a self-reproducing phenomenon.

3 Analysis of Della Dora systems of type 1

First definitions

Cycles in phase space can be determined by a polynomial approach. A P -periodic cycle of an additive CA is a list of P sequences, each of length N , (a (t) , a (t+1) , ..., a (t+P -1) ) , such that their corresponding characteristic polynomials, A (t+i) , (i ∈ {0, 1, ...P -1}) obey

A (t+i+1) (x) = T (x)A (t+i) (x) mod(x N -1)
and

A (t+i) (x) = T (x) P A (t+i) (x) mod(x N -1) .
The cycles of the dynamical system Φ (α,β) are the cycles of the additive dynamics defined by the Di-polynomial T (α,β) , such that every sequence occurring in the cycle has no null sites (a 0 site would lead to a splitting 0 → 11 and thus could not be part of a cycle).

For Example 3, it is possible to give a formal proof showing that there are only two types of cycles, one of periodicity P = 1 and another of periodicity P = 2. The cycles of periodicity 2 can be denoted by ((2211) k , (1122) k ) where k is an integer and (2211) k and (1122) k are the successive states of the cycle (i.e., Φ (2,2) ((2211

) k ) = (1122) k and Φ (2,2) ((1122) k ) = (2211) k ).
Here, (2211) k denotes a sequence of length 4k such that (2211) k = 22112211...2211. With the same notation, the cycles of periodicity 1 can be denoted by ((21) k ). All these cycles are reached only by themselves. In general, it is possible also to have transient approaches to cycles. An example in the system with α = 2, β = 1 and n = 5 (a Della Dora system of type 1) is the sequence (141) for which Φ (2,1) (141) = (131) which is then part of the cycle ((131), (424)).

Over time, an open subsequence 11...11 = 1 k (k ≥ 4) produces a subsequence of 1s with length that grows (exponentially) without bound when k > 4, and with length that may remain fixed or may grow when k = 4, with no further variation in behaviour. If we consider these as sites in a 'base state,' then information resides in regions in which some 'not-1' sites occur. The characteristic sequences of a sequence a are maximal open subsequences of sites of Z n which do not contain 1 k with k ≥ 4 and neither begin nor end with a 1. In effect, the occurrence of expanding subsequences of 1s separate the characteristic sequences, and the subsequence 1111 can be considered a separator: if b and c are two characteristic sequences of a sequence a, changing the values of any sites of b will never affect values of sites of c. Characteristic sequences b and c evolve independently. They can have periodic behaviour, they can vanish, and they can also grow. If 1111 appears at some time step from the interior of a characteristic sequence, it creates new independent subsequences. This is, in that case, a reproduction process.

A sequence containing the subsequence 1111 we call a broken sequence; otherwise it is unbroken. Broken sequences consist of one or more characteristic sequences separated by separators of the form 1 k with k ≥ 4. Unbroken sequences cannot be considered characteristic sequences, since they do not have 1111 to either side.

Dynamics of characteristic sequences

The dynamics of a characteristic sequence (which is open) is sightly different from the dynamics of a (closed) sequence. Let Φ o denote this new dynamic applied to an open sequence s = s 0 ...s N -1 surrounded by multiple sites with value 1. Then, Φ o (s) = α + β + αs 0 , α + βs 0 + αs 1 , . . . , αs i-1 + βs i + αs i+1 , . . . , αs N -2 + βs N -1 + α, αs N -1 + β + α , where, as before, we apply the expansive rule 0 → 11 where applicable. The new computation of the extremal sites is a consequence of the fact that s is bordered by sites of value 1. The independent evolution of characteristic sequences can now be expressed by Φ(b1 * c1 * ) = Φ o (b)1 * Φ o (c)1 * where (b1 * c1 * ) is a closed sequence constituted by two characteristic sequences, and * is a non specified integer greater than or equal to 4. It is convenient to introduce a more compressed notation:

Φ o (b; c) = Φ o (b); Φ o (c).
This notation can easily be extended to more than two characteristic sequences.

For Example 3, using this new notation, Φ 2 o (2) = Φ o (222) = 2; 2. So, when the characteristic sequence (2) appears, after two steps it splits into two characteristic sequences [START_REF] Bagnoli | Damage Spreading and Lyapunov Exponents in Cellular Automata[END_REF]. The periodicity of reproduction of the characteristic sequence (2) is therefore 2. Figure 4 represents the Petri net of this simple example of behaviour. More complex examples can be represented in this way. For instance, let us consider again the dynamics Φ (2,1) over Z * 5 . Figure 5 shows the Petri net associated with the characteristic sequence 233213. The same characteristic sequences can appear several times in a sequence. Let S a denotes the set of distinct characteristic sequences that are contained in the sequence a.

For Example 3, it is easy to show that 1 k (k ∈ N and k ≥ 4) are the unique separators. We can consider 1111 a weak separator, i.e., a separator that persists but whose length does not increase. Two characteristic sequences that are separated by a weak separator correspond to independent entities which stay linked geographically. For example, Φ (2,2) (21 4 2) = (121 4 21) and the sequence of 1's does not grow longer. On the other hand, if the separator is not weak, the characteristic sequences move apart.

Behaviour analysis

For all Della Dora systems of type 1 we have studied, with various choices of α, β and n, we have proven that the behaviour of any sequence that does not belong to a cycle, and is not on a transient leading to a cycle, can be described by a finite number of characteristic sequences.

More precisely, let a be the initial state. Even if the sequence continually grows, there exists a finite set of characteristic sequences S depending on a, such that for all integers l, S Φ l (a) ⊂ S. In order to prove this strong property, new terms have to be defined.

Broken sequences grow without bound: lim t→∞ N (t) = ∞. This is true even if there is only one separator and it is weak, because, even though the length of the separating sequence may remain fixed, it feeds new sites into the characteristic sequence at either side. Unbroken sequences may break and then grow without bound, or may not, such as is the case when they fall into a cycle, possibly after a transient. Of course, if a sequence grows without bound, it cannot be part of a cycle or a transient leading to a cycle, and if a sequence does not grow without bound, it must reach a cycle because the accessible state space is then finite. This still leaves open the possibility, a priori, of an unbroken sequence that never breaks but grows without bound. In order to deal with both unbroken sequences and characteristic sequences, and in particular to investigate whether or not they eventually split, we consider the internal dynamics of an arbitrary subsequence.

Let Φ r denote the partial evaluation of an open sequence s such that: Φ r (s) = αs 0 + βs 1 + αs 2 , . . . , αs i-1 + βs i + αs i+1 , . . . , αs N -3 + βs N -2 + αs N -1

where the expansive rule 0 → 11 is once again applied. Note that Φ r (s) does not transform the end sites of the open sequence s, because that would require information on sites outside of s.

We say that a subsequence s intrinsically separates if there exists a number of steps k such that Φ k r (s) contains a separator. When a subsequence separates intrinsically, the value of the first site is never influenced by the value of the site which is just after the subsequence.

Unbroken sequences that eventually cycle clearly do not separate (break). Characteristic sequences can also cycle in the sense that, using our condensed notation, Φ P o (s) = s and these also clearly do not separate. An example, in the system with α = 2, β = 1 and n = 5, is s = 3 for which

Φ 2 o (3) = Φ o (424) = 3.
Characteristic sequences can also be on transients leading to cycles, such as s = 4 in the above example, for which Φ o (4) = 3. If the characteristic sequence s cycles, there exists an integer i s ( 0 ≤ i s ≤ 3) such that (s1 is ) k also cycles, but γ(s1 is ) k (γ ∈ Z n ) does not necessary cycle any more. For instance (424) 10 cycles but 4(424) 10 does not cycle and does not separate intrinsically. The cyclic part of Φ t o (4(424) 10 ) decreases when t increases and Φ 10 o (4(424) 10 ) = 31124333114; 3; (424) 8 . In other examples, the cyclic part can also totally disappear. We call a disturbed cycle a characteristic sequence of the form * (s1 is ) k * where s is a cycle and * represents any subsequence of Z n that does not contain a separator, and is small enough such that * (s1 is ) k * does not separate intrinsically. * can also be the empty set, for instance, s = 11211211324434434423 is such that Φ 5

(2,1) (s) = s but Φ 5 o (s) = 3; 212111221112113244344343112 and s is already a disturbed cycle. Our objective, then, is to show that every subsequence above a certain size either separates intrinsically or is a transient to a disturbed cycle, or eventually cycles, whether considered as an unbroken sequence or as a characteristic sequence. In the case of intrinsically separable subsequences, even if, over the k steps required to guarantee separation, some of the resulting characteristic sequences became longer, there is still a finite size to which they can grow in k steps. In the case of disturbed cycles, we have observed that there exists a finite small integer λ such that for all integers l > λ, the number of attached cycles s of Φ l o ( * (s1 is ) k * ) decreases or remains the same and there exists an integer l such that Φ l o ( * (s1 is ) k * ) is composed of intrinsically separable subsequences and cycles. Thus, over time, a disturbed cycle disappears and generates a set of characteristic sequences. The size of those characteristic sequences is bounded and the set is finite.

This will provide a proof that the number of characteristic sequences generated by any initial sequence is bounded. Consequently, for any given initial broken sequence, the entire future behaviour can be described by a finite number of characteristic sequences.

We have developed an R program whose arguments are n of the finite set Z * n , α and β of the dynamics and a length N for subsequences. The program computes the list of all the subsequences of length N that do not separate intrinsically using the separator 1111.

For Example 3, running our R program shows that every subsequence of length at least 18 either intrinsically separates or is a disturbed cycle or eventually cycles.

Thus, the behaviour of every broken sequence can be defined by a finite set of characteristic sequences.

More generally, we conjecture that for any Della Dora system of type 1, there exists a length N such that every characteristic sequence or unbroken sequence of length N intrinsically separates or is a transient to a disturbed cycle or a cycle or eventually cycles. This will imply that the entire future behaviour of every sequence can be described by a finite number of unbroken sequences and a finite number of characteristic sequences and the separators. Finally the evolution of any sequence would be representable by a finite Petri net.

Conclusion

Cellular automata are capable of very complex behaviour. They model a large range of biological phenomena. For instance, Wolfram has used them to describe natural systems from snowflakes to mollusc shells. The class of Della-Dora systems are less simple in construction than cellular automata and can also have very complex behaviour. The idea is to add the possibility of modelling the important biological process of mitosis. In this extended abstract, we have focussed on the study of a subclass of such systems, which we call Della-Dora systems of type 1. These are irreversible systems that give birth to very interesting self-organizing and self-reproducing behaviours. In future work, the behaviour of Della-Dora system of type 1 will be explored as a simulation of the type of behaviour seen in algae or filaments in a Petri dish. A formalisation for the classification of Della-Dora systems of type 1 will also be explored. For Cellular Automata, quantitative measures have been proposed to identify the Wolfram classes. In our case, the number of sites evolves and these measures must be adapted.

Fig. 1 .

 1 Fig. 1. Example of synchronisation: T (x) = x + 2 + x -1 over Z * 3 with substitution rule 0 → 11.

Fig. 2 .

 2 Fig. 2. Example of chaotic behaviour : T (x) = 2x + 1 + 2x -1 over Z * 3 , with substitution rule 0 → 11.

Fig. 3 .

 3 Fig. 3. Example of self replication: T (x) = 2x + 2 + 2x -1 over Z * 3 , with substitution rule 0 → 11.

Fig. 4 .

 4 Fig. 4. Petri net of the characteristic sequence (2) at time t, t + 1 and t + 2 under the dynamics of the system with Di-polynomial T = 2x 2 + 2x + 2 over Z * 3 .

Fig. 5 .

 5 Fig. 5. Petri net for the characteristic sequence (233213) under the dynamics of the system with Di-polynomial T = 2x 2 + x + 2 over Z * 5 .


Kolli Abdelfatah

Olivier Béthoux
email: olivier.bethoux@lgep.supelec.fr

Member, IEEE Alexandre De Bernardinis
email: alexandre.de-bernardinis@ifsttar.fr

Member, IEEE Eric Labouré

A Kolli

G Coquery

Space Vector PWM Control Synthesis for a H-Bridge Drive in Electric Vehicles

Keywords: drives, Inverters, Space vector pulse width modulation (SVPWM), Permanent magnet machines, Semiconductor device reliability I

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

-by using multilevel inverters topologies used in a high power traction drive [START_REF] Dixon | Asymmetrical Multilevel Inverter for Traction Drives Using Only One DC Supply[END_REF].

-by redistributing the control efforts in a four-wheel independently driven electric vehicles [START_REF] Wang | Fault-Tolerant Control with Active Fault Diagnosis for Four-Wheel Independently Driven Electric Ground Vehicles[END_REF]… Furthermore, faults may also happen on sensors and can be taken into account by active fault-tolerant control systems. For example in [START_REF] Diallo | A fault tolerant control architecture for Induction Motor Drives in Automotive Applications[END_REF]- [START_REF] Benbouzid | Advanced Fault-Tolerant Control of Induction-Motor Drives for EV/HEV Traction Applications: From Conventional to Modern and Intelligent Control Techniques[END_REF], authors consider a high-performance induction-motor drive for an EV or a hybrid one (HEV). The proposed systems detect a sensor loss or a sensor recovery and dynamically change its strategies to sustain the best control performances.

Besides, faults may also occur in the electrical machine and can be considered using fault-tolerant designs, as example for a redundant IPM (Interior Permanent Magnet) motors structure [START_REF] Bianchi | Design Of a Fault-Tolerant IPM Motor for Electric Power Steering[END_REF].

In this paper, a specific fault-tolerant drive topology patented by Valeo Company [START_REF] Sousa | Combined Electric Device For Powering and Charging[END_REF]- [START_REF] De-Sousa | Method and electric combined device for powering and charging with compensation means[END_REF] is studied.

It allows both traction and charger operating modes with the same power electronics devices [START_REF] Sousa | A Combined Multiphase Electric Drive and Fast Battery Charger for Electric Vehicles Topology and Electric Propulsion Efficency Analysis[END_REF]- [START_REF] Lacroix | Design of a Battery-Charger Controller for Electric Vehicle Based on RST Controller[END_REF]. Furthermore, the traction operating mode is fault-tolerant as it can operate with only two phases of the electrical machine.

The considered topology uses six half bridges instead of a three classical one, leading to a much higher complexity of the control strategies but offering new voltage configurations. So, these degrees of freedom are used to design new control methods and will be compared with classical ones in this paper. The next issue is to investigate whether the innovative control methods can be reconfigured in a switch failure case. Both presentations are based on space vector pulse width modulation technique (SVPWM), which allows the proper analysis of this discrete control problem and to synthesize suitable control strategies.

The paper is organized as follows: Section II deals with power electronic architecture tolerant to semiconductor failure in EV drive. Section III discusses modeling and control drive requirements.

Section IV focuses on SVPWM control synthesis for normal operating mode. Comprehensive analysis of the failure is conducted in section V. Finally, the section VI draws conclusions and perspectives.

II.POWER ELECTRONIC ARCHITECTURE TOLERANT TO SEMICONDUCTOR FAILURE IN EV DRIVE

Semiconductors are an important source of failure in traction systems. Investigations have highlighted the failure rate per element in a traction system [START_REF] Schwab | Reliability Evaluation of a Permanent Magnet Synchronous Motor Drive for an Automotive Application[END_REF]. Authors assess that the semiconductor failures and their driver auxiliary circuit are the most recurrent in electric automotive applications. Using reliability prediction methods, authors in [START_REF] Schwab | Reliability Evaluation of a Permanent Magnet Synchronous Motor Drive for an Automotive Application[END_REF] point out that there is a reasonably equal repartition between faults occurring in the semiconductors and their auxiliary circuits. They estimate that about 40% of failures can be attributed to semiconductors faults and 39 to their auxiliary circuits. Driver circuits should be reliable enough, to prevent additional sources of failure [START_REF] Dulau | A New Gate Driver Integrated Circuit for IGBT Devices with Advanced Protections[END_REF].

Automotive is a constraining environment: start-stop cycles, power and thermal cycles, confined areas with thermal constraints [START_REF] Bouarroudj-Berkani | Fatigue des composants électroniques de puissance-Physique de défaillance[END_REF]. In this severe environment, power semiconductors are the most sensitive elements in terms of lifetime and reliability. They are subject to electro-thermal and also mechanical stresses, accelerating their failure mechanisms [START_REF] Bouarroudj | Comparison of Stress Distributions and Failure Modes During Thermal Cycling and Power Cycling on high Power IGBT Modules[END_REF]. These failures are mainly induced by power and thermal cycling of the semiconductor device [START_REF] Coquery | High Temperature Reliability on Automotive Power Modules Verified by Power Cycling Tests up to 150°C[END_REF]. For example, an interesting investigation conducted on a Renault Kangoo electric vehicle has been carried out in [START_REF] Coquery | Power module lifetime estimation from chip temperature direct measurement in an automotive traction inverter[END_REF]. The authors have demonstrated that direct silicon temperature measurement of an operating traction inverter is feasible.

As a result, authors establish an accurate thermal model of the IGBT traction inverter which can be applied in real-time EV application. Several papers have also been published on the topic of power modules accelerated ageing tests (thermal cycling and power cycling) in order to estimate the level of semiconductors expectancy lifetime [START_REF] Bouarroudj | Comparison of Stress Distributions and Failure Modes During Thermal Cycling and Power Cycling on high Power IGBT Modules[END_REF]- [START_REF] Coquery | High Temperature Reliability on Automotive Power Modules Verified by Power Cycling Tests up to 150°C[END_REF] and to study the failure mechanisms and effects [START_REF] Coquery | High Temperature Reliability on Automotive Power Modules Verified by Power Cycling Tests up to 150°C[END_REF]- [START_REF] Ciappa | Selected Failure Mechanisms of Modern Power Modules[END_REF]- [START_REF] Vallon | Converter Topology for Reliability Test Bench Dedicated to PWM Inverters[END_REF]. It has been reported that repetitive failures have two different sources: an external one which can be, for example, linked to faulty control signal [START_REF] Richardeau | Failures-tolerance and remedial strategies of a PWM multicell inverter[END_REF] and an internal one, linked to physical degradation of the power semiconductor. Examples of these internal failures are: bond wire lift off, gate leakage failure and damages on semiconductors chip and solder [START_REF] Coquery | High Temperature Reliability on Automotive Power Modules Verified by Power Cycling Tests up to 150°C[END_REF], SiAl contact ageing [START_REF] Vallon | Converter Topology for Reliability Test Bench Dedicated to PWM Inverters[END_REF], electro-migration effect [START_REF] Ciappa | Selected Failure Mechanisms of Modern Power Modules[END_REF], latch-up [START_REF] Lefebvre | Experimental Behavior of Single-Chip IGBT and COOLMOS Devices Under Repetitive Short-Circuit Conditions[END_REF]. These internal failures can modify the operating state of the semiconductor and induce abnormal behavior like OC or SC states. These power switch malfunctions decrease the number of discrete voltages that the voltage inverter can apply to the machine. Consequently this new limitation prevents a suitable magnetic field rotation in the rated torque and speed frame. Moreover the rated control algorithm imposes voltages that are not necessarily applied to the motor. In conclusion these abnormal states affect the power converter function, which impacts the motor operating behavior and consequently the traction powertrain, compromising the vehicle driver safety. Indeed, a switch failure in standard 3-leg inverter influences drastically the motor behavior and prevents continuity of service [START_REF] Richardeau | Fault-Tolerant Inverter for on-Board Aircraft EHA[END_REF].

Current technology provides "smart" drivers that can provide early detection of a fault occurring in the power switch and return a feedback fault signal. For instance cross conduction is usually handled by a soft turn-off once a fast desaturation detector raises the alarm. This information should be exploited by the control strategy interface in the objective of diagnostic, protection (fault confining) and reconfiguration to an adequate strategy.

In case of failure affecting a machine phase, a particular remedial strategy has been reported in [START_REF] Bianchi | Innovative Remedial Strategies for Inverter Faults in IPM Synchronous Motor Drives[END_REF].

After fault isolation which inhibits one motor phase, authors propose to connect the motor star point to an additional fourth leg. This choice is performed in order to allow the flow of the neutral current, while the other two motor phases remain supplied. Fig. 1 presents this fault tolerant architecture.

However, the proposed solution requires the use of contactors for fault isolation; this is a drawback in electric vehicle applications, in particular regarding cost and reliability of such solutions. Welchko et al. listed many interesting fault-tolerant AC motor drive topologies that unfortunately face the same drawback [START_REF] Welchko | Fault tolerant three-phase AC motor drive topologies: a comparison of features, cost, and limitations[END_REF]. This paper investigates another solution, which offers a greater ability for reconfiguration. The power structure is composed of 3 H-bridges, each of them supplying a separate PMSM motor phase. The proposed architecture does not require the fault-isolation contactors. As shown in Fig. 2, the 3H-Bridge converter uses a machine without neutral point. The advantage of this architecture is to allow the application of the full DC link voltage to each PMSM stator winding [START_REF] Sousa | Combined Electric Device For Powering and Charging[END_REF]- [START_REF] Sousa | A Combined Multiphase Electric Drive and Fast Battery Charger for Electric Vehicles Topology and Electric Propulsion Efficency Analysis[END_REF]- [START_REF] Mecrow | Fault-Tolerant Permanent Magnet Machine Drives[END_REF]. However, with this specific architecture, the zero-sequence current is not structurally rejected, unlike the classical solution with a star neutral point topology [START_REF] Sousa | Method of Evaluating the Zero-Sequence Inductance Ratio for Electrical Machines[END_REF]. The control of the converter should fulfill two requirements: optimal working in normal mode, and in degraded mode.

Obviously the proposed 3 H-bridge architecture based on 12 switches involves a higher number of semiconductor devices, but offers many redundancy possibilities in degraded mode using only two bridges. Two points may be noted:

-A competing configuration using the same number of power switches [START_REF] Sousa | A Combined Multiphase Electric Drive and Fast Battery Charger for Electric Vehicles Topology and Electric Propulsion Efficency Analysis[END_REF] is reported in [START_REF] Reddy | Decoupled Space-Vector PWM Strategies for a Four-Level Asymmetrical Open-End Winding Induction Motor Drive with Waveform Symmetries[END_REF]- [START_REF] Corzine | Performance characteristics of a cascaded two-level converter[END_REF].

This paper proposes a four-level voltage source inverter topology (dual-inverter) but uses two isolated and unequal dc-link voltages. Other authors suggest segmenting the power source with two converters feeding double-star synchronous machines. This architecture is fault-tolerant and ensures minimum loss control under fault phase conditions [START_REF] Shamsi-Nejad | Fault Tolerant and Minimum Loss Control of Double-Star Synchronous Machines Under Open Phase Conditions[END_REF]- [START_REF] Ertugrul | Fault tolerant motor drive system with redundancy for critical applications[END_REF].

-A major advantage of our structure is, besides the operation in traction mode, the ability to operate in battery recharge mode with the same hardware, namely the power converter and the electric machine. This is achieved through the PM machine windings, which has a phase middle point for direct connection to the recharging network (Fig. 2). In this mode, the motor phases used as a filter inductances are subdivided in two half windings. The middle point is used to inject currents [START_REF] Sousa | A Combined Multiphase Electric Drive and Fast Battery Charger for Electric Vehicles Topology and Electric Propulsion Efficency Analysis[END_REF]- [START_REF] Lacroix | Design of a Battery-Charger Controller for Electric Vehicle Based on RST Controller[END_REF].

C Middle-point stator winding

Grid connection

V dc Separate phase PMSM V a V b V c T 1 D 1 T 3 D 3 T 4 D 4 T 2 D 2

III.MODELING AND CONTROL DRIVE REQUIREMENTS

The set of the PMSM and its 3-H bridge inverter is first described in a control model approach (III-A and III-B). Secondly, this model highlights the key issues that the control algorithm must satisfy (III-C). Next, the listed requirements allow the discussion of the innovative PWM control method and to classify classic and new strategies (III-D to III-F). Finally the chosen method is implemented in a MATLAB/Simulink software platform (III-G).

A. Drive Control Modeling In Concordia reference frame

The studied motor is a non-saturated non-salient pole permanent magnet synchronous machine and no damping effect is considered. The three-stator winding is constructed to give a sinusoidal air-gap flux distribution; similarly, specially shaped rotor magnets provide sinusoidal flux within the air gap. The Concordia transform (CT) has proved to simplify motor drive control [START_REF] Önel | Induction Motor Bearing Failure Detection and Diagnosis: Park and Concordia Transform Approaches Comparative Study[END_REF]. Using the generalized power invariant CT enables to diagonalize the stator inductance matrix.

               +             +       ⋅ =       + + ⋅ = β α β α β α β α e e i i dt d L 0 0 L i i R V V e i dt d L i R V c c 0 0 0 0 0 (1)
Because the motor phases are not electrically connected one to the other, the sum of the three phase currents is not constrained. Thus, for the proposed motor drive architecture, zero-sequence current is not forced to be null and (1) reveals two fictitious machines associated to 2 orthogonal eigenspaces related to the 0-line and the α-β frame [START_REF] Scuiller | Comparison of two 5-phase Permanent Magnet Machine Winding Configurations. Application on Naval Propulsion Specifications[END_REF]- [START_REF] Martin | Space Vector Control of 5-phase PMSM Supplied by 5 H-bridge VSIs[END_REF]- [START_REF] Kestelyn | Multi-phase System Supplied by SVM VSI : A New Fast Algorithm to Compute Duty Cycles[END_REF].

1. First, the Zero-sequence fictitious machine is associated to the leakage inductance L 0 and back EMF e 0 component induced by the rotor rotation. Assuming perfectly balanced back electromotive forces in the three windings, e 0 is null. Therefore, this fictitious machine does not produce any electromagnetic torque. However, its current i 0 creates unnecessary ohmic losses.

2. Second, the main fictitious machine is associated to the cyclic inductance L c and the back EMF e αβ components. Thus, the instantaneous electromagnetic torque is given by: ( )

β β α α i e i e 1 T em ⋅ + ⋅ Ω = (2)
Where Ω is the motor speed. The main motor drive objectives are to produce a desired torque while minimizing the Joule losses [36]. These goals are achieved by controlling the inverter power switches.

The next subsection explains the dedicated 3H-Bridge converter potentials.

B. Space Vector PWM Control Approach

As presented in Fig. 2, the stator winding terminals are directly connected to one H-bridge. Each Hbridge has four discrete states but can only produce three discrete voltage levels, namely +V dc , 0 and -V dc . Together the 3-phase H-Bridge inverter provides 27 switching states. To match the PMSM multimachine representation, the output voltage vectors are described in the Concordia reference frame (Fig. 3). The stator voltages are divided into four main families [START_REF] Martin | Contribution to the Voltage Supply of Multi-Phase Synchronous Permanent Magnet Machines: Normal and Fault Operating Modes[END_REF] according to magnitude and zero-sequence component values. These families are summarized in the following Table I:

TABLE I STATOR VOLTAGES FAMILY DECOMPOSITION FAMILIES VOLTAGE VECTORS MAGNITUDE V 0 VALUE Family I 15, 17 , 18, 23, 24, 26 3 2 2
⋅ dc V 3 1 ⋅ ± dc V Family II 6, 8 , 12, 16, 20, 22 2
⋅ dc V 0 Family III 2, 3 , 4, 7, 10, 19 3 2 dc
V 3 1 ⋅ ± dc V
Family IV 5, 9, 11, 13, 21, 25

3 2 dc V 3 1 2 ⋅ ± dc V Family V 1 0 0
Family VI 14, 27 0 To control both fictitious machines, the PMSM controller requires a particular space vector

[V 0 * V α * V β * ]
T . The 3-H bridge inverter can only provide an average value of this voltage setpoint, using pulse-width modulation (PWM) technique. PWM also produces voltage harmonics that can be controlled by monitoring switching frequency. Consequently, the PWM generation is a 4-dimensional problem: Two dimensions are devoted to the main machine, one dimension is linked to the zerosequence machine and the last one permits to meet the time constraint. Therefore 4 different discrete voltage vectors {V 1 , V 2 , V 3 , V 4 } among the 27 inverter possibilities are necessary for solving this Space Vector PWM problem [38]- [START_REF] Zhao | Space Vector PWM Control of Dual Three-phase Induction Machine using Vector Space Decomposition[END_REF]. Noting T sw the desired switching period, the 4 associated durations {∆t 1 , ∆t 2 , ∆t 3 , ∆t 4 } are computed using (3):

                ∆ ∆ ∆ ∆                 =                 4 3 2 1 4 3 2 1 4 3 2 1 04 03 02 01 * * * 0 1 1 1 1 1 2 / 1 t t t t V V V V V V V V V V V V T V V V sw β β β β α α α α β α (3)
The set of four voltage vectors is chosen to guarantee that each ∆t k is positive. In 3-H bridge case there are numerous ways of choosing such a set. The main issue of this section is to select the most relevant one. The next sub-section clarifies the PMSM SVPWM specifications.

C. Control Drive Requirements

The SVPWM has to control the 3-H bridge inverter in order to maximize the drive performances and minimize the global power losses.

The first objective is to optimize the electromechanical functioning in the torque -speed frame [START_REF] Zeraoulia | Electric Motor Drive Selection Issues for HEV Propulsion Systems: A Comparative Study[END_REF].

At high speed, the ohmic losses can be neglected (L c ω >> R) and the base speed Ω b can be expressed as follows [START_REF] Stulrajter | Permanent Magnet Synchronous Motor Control Theory[END_REF]- [START_REF] Miller | Propulsion Systems for Hybrid Vehicles[END_REF].

( )

2 max _ 2 2 max _ ) ( 2 3 αβ αβ I p L K V c b ⋅ ⋅ + = Ω Ω (4)
Where K Ω is motor constant, p is pole pairs, I αβ_max is rms current value at maximum torque operating and V αβ_max is the maximum inverter output voltage supplied to the PMSM. In a vehicular application, the DC link voltage is limited by the maximal battery voltage. Consequently, to extend the PMSM speed range, the SVPWM has to provide the largest possible magnitude [START_REF] Miller | Propulsion Systems for Hybrid Vehicles[END_REF].

The second objective is the power efficiency optimization. This has several aspects: 1.Minimize switching losses and ensure their equal-distribution between the 3H-bridges. The switching losses are an important part of the semiconductor power losses [START_REF] Xiong | New Physical Insights on Power MOSFET Switching Losses[END_REF]. Reducing these losses is important to improve the inverter reliability because thermal stress is a key factor of the failure mechanism [START_REF] Coquery | Power module lifetime estimation from chip temperature direct measurement in an automotive traction inverter[END_REF]. For the same reason, ensuring fair switching repartition is crucial.

2.Ability to feed the zero-sequence machine with the smallest voltage (V 0 ). In ideal PMSM, the back EMF e 0 component is null and consequently the average V 0 voltage has to be zero. But, even in this case, the set of four voltages may generate a zero-sequence voltage inducing zero-sequence current ripple (∆i 0 ). This current undulation has to be minimized to prevent pointless losses. This last point is crucial because the zero-sequence inductance value (L 0 ) is generally very small due to good phase magnetic coupling.

The third objective is the robustness of SVPWM control regarding:

1.The temporal accuracy of application times of the four chosen vectors. The control implementation is based on microcontroller and FPGA logic circuit using a digital timer. Timer precision as well as driver circuit and power switch delays introduce uncertainties in the real application. Moreover, internal dead time is mandatory to prevent leg short-circuit but it makes the actual PMSM voltage dependent on the current sign. This is another reason for the uncertainty on the voltage value actually applied.

2.The PMSM parameters. Any winding unbalance produces a non-null zero-sequence back EMF (e 0 ). In this real world context, the SVPWM sequence has to be able to generate a small non zerosequence (V 0 ) with little harmonics content in order to counteract the PMSM defect.

In the following subsections, we use these specifications to analyze existing SVPWM techniques and provide innovative SVPWM strategies. Next subsection describes the two standard SVPWM methods.

D. Standard SVPWM Control Methods

Previous studies on SVPWM modulation techniques have already suggested two different strategies for a set of PMSM and 3-H bridge inverter.

The first method (Method I) discussed in [START_REF] Kestelyn | Multi-phase System Supplied by SVM VSI : A New Fast Algorithm to Compute Duty Cycles[END_REF] employs the 1 st and 6 th vector families. Fig. 4 -A shows the 3D representation of the selected voltages in the Concordia reference frame. Note that each selected vector develops a nonzero component in the zero-sequence subspace.

The second standard SVPWM strategy (Method II) utilizes the single vector of family V, the two vectors of family VI and the six vectors of family II [START_REF] Martin | Space Vector Control of 5-phase PMSM Supplied by 5 H-bridge VSIs[END_REF]. Fig. 4 -B depicts the 3D representation of the discrete voltages defined by this method. To date, 2 vector families have been neglected in the PWM generation. It is therefore interesting to explore other SVPWM strategies. The following sets out the proposed space vector control methods.

α β 0 V 23 V 24 V 26 V 17 V 18 V 15 V 27 V 14 α-β plane α β 0 V 27 V 14 V 22 V 20 V 8 V 16 V 12 V 6 V 1 α-β plane
Both of the standard methods are restricted in the use of possibilities offered by the 3-H bridge inverter. Families III and IV are unused and each standard method is based on a unique main family (namely family I and family II for method I and method II respectively). It is interesting to explore other voltage combinations to solve (2) and to check their ability to respect the listed drive requirements. The following subsection sets out new space vector control methods.

E.Innovative SVPWM Control Methods

The diagram depicted in Fig. 5 shows the vector families used by the three proposed control technique.

In Method III, family VI is voluntarily eliminated in order to reduce the instantaneous voltage of the zero-sequence fictitious machine. However, the maximum achievable voltage may be reduced since family III has small magnitude vectors. To tackle this problem, we propose Method IV which uses the largest voltages vectors (families I and II). To enable equation (3) resolution, Method IV uses also family VI; these two vectors give the ability to reach the zero-sequence demand. Method V selects the smallest vectors (family III); the aim is to examine whether it leads to a relevant solution for small voltage requirement (generally corresponding to low motor speed). Note that family IV is rejected due to its high contribution in 0-axis.

available voltages

Method III

Family I, Family II, Family III and Family V

Method IV

Family I, Family II and Family VI

Method V

Family II, Family III and Family VI Fig. 5. Vectors set for the innovative proposed SVPWM strategies.

F. Comparative Study

Standard and suggested voltage vectors set have to be compared using the same criteria. Drive requirements impose the 3 mentioned constraints: speed range, power efficiency and sensitivity regarding parameter and control uncertainties. given by:

                  - ∆ - + ∆ - + - ∆ - ∆ - - ∆ - + ∆ - + ∆ ∆         =         β β β β α α α α β α
V V t ) V V ( t ) V V ( ) 2 T )( V ( t t 2 T V V t ) V V ( t ) V V ( ) 2 T )( V ( t t V V V V V V V V T 1 V V (5)
Varying the two remaining times ∆t 1 and ∆t 2 arbitrarily from 0 to T sw /2 allows the exploration of all the possibilities. Referring to (4), each base speed corresponds to the highest voltage magnitude that the corresponding strategy can achieve on a complete electrical turn. Table III summarizes the computed results. Method III and Method IV have the best performance regarding this criterion. From table III, we can notice that the first four methods allow roughly the same speed range whereas, as expected, method V prevents high speeds. But minimizing the torque ripple conducts to reduce the α-β current ripple (refer to (2)) and consequently α-β voltage ripple (refer to (1)). The scalar projection of methods vectors onto αβ plane defines sectors. As depicted in Fig. 6, there are six sectors for the standard methods and twelve for the proposed ones. From this representation, it is clear that the reduction of the voltage differences and therefore of the voltage ripple involves using adjacent vectors in a defined sector. 2) Power efficiency optimization: The reference vector is located in a unique sector associated to a single set of four voltages including, for some methods vector, V 27 and V 14 (Fig. 6). To minimize switching losses, it is crucial to construct a sequence guaranteeing an equitable distribution of the number of switches per leg, to obtain a limited number of switches and to avoid excessive switching while sector changing. Taking the first sector as an example, Fig. 7 illustrates the turn-on and turn-off numbers. The 3H-bridges are represented by three rectangles. Points inside them depict the number of switches achieved from the beginning of the half period: red points depict current switches while black ones symbolize previous switches. For instance, a single Hbridge needs two switches, 1 turn-on and 1 turn-off, to switch from the normalized voltage +1 to -1 or vice versa.

V 23 V 26 V 17 V 24 V 15 V 18 α β V 27 V 14 Sector I (A) V 20 V 8 V 16 V 22 V 6 V 12 α β V 1 Sector I (B) V 27 V 14 V 23 V 26 V 17 V 24 V 15 V 18 α β V 1 V 19 V 2 V 7 V 10 V 3 V 4 V 20 V 8 V 16 V 22 V 6 V 12 Sector I (C) V 23 V 26 V 17 V 24 V 15 V 18 α β V 20 V 8 V 16 V 22 V 6 V 12 Sector I (D) V 14 V 27 V 20 V 8 V 16 V 22 V 6 V 12 α β V 19 V 2 V 7 V 10 V 3 V 4 Sector I (E) V 27 V 14
Finally every method gives the same number of switches (namely six) and ensures their equal distribution (2 per bridge) except Method II. This latter requires 7 switches which are necessarily asymmetrically distributed. For each method, the selected sequence also involves a temporal evolution of the zero-sequence voltage component. This point is a crucial issue since it drives current ripple and related ohmic losses. For this purpose, ( 5) is used to determine any voltage with a null mean zero-sequence component. However, the instantaneous value of zero sequence voltage is not necessarily null, as shown in Fig. 8. The resulting current ripple is filtered by the zero-sequence impedance of the motor. As the switching period T sw has a low value compared to electric motor time constant (L 0 / R), a fair approximation of this ripple ∆i 0 (Fig. 8) is given by the δi 0 (t) peak to peak evaluation where δi 0 (t) is computed, on each switching period as:

[+1 +1 +1] V 27 [+1 -1 +1] V 24 [+1 -1 -1] V 23 [-1 -1 -1] V 14 H-Bridge 1 T sw /2 H-Bridge 2 H-Bridge 3 (A) [0 0 0] V 1 [+1 -1 0] V 22 [+1 0 -1] V 20 [-1 -1 -1] V 14 H-Bridge 1 T sw /2 H-Bridge 2 H-Bridge 3 (B) [0 0 0] V 1 [+1 0 0] V 19 [+1 -1 0] V 22 [+1 -1 -1] V 23 [0 0 0] V 1 H-Bridge 1 T sw /2 H-Bridge 2 H-Bridge 3 (C) [-1 -1 -1] V 14 [+1 -1 -1] V 23 [+1 -1 0] V 22 [+1 +1
t 0 0 0 0 1 i (t) V ( ).d L ∆ = τ τ ∫ (6)
The corresponding results are illustrated in Fig. 9. Clearly, the current ripple is highly dependent on the method used and on the voltage magnitude (and therefore on the motor speed) as seen in Fig. 9-A to Fig. 9-E. In low speed conditions, methods using family VI to achieve a null α-β voltage generate a important ∆i 0 . That is the reason why, with respect to this criterion, methods I, II, IV and V have poor performance at low speed. However, method III is particularly interesting. It makes use of null voltage vector V 1 to reduce the desired voltage magnitude;

hence this method produces a very low ∆i 0 at low speed.

Similarly, this ripple remains low at high speed because the two discrete voltage with non-null 0-component are applied consecutively and have the smallest 0-component (families I and III).

Lastly, for the current ripple, it is not interesting to combine method V at low speed with any other method at high speed (Fig. 9-E).

T sw /2 t V 0 j (t) For the first case, ( 5) is solved using the same incremental step (namely T sw /50) for ∆t 1 and ∆t 2 .

T sw ∆t 1j ∆t 2j ∆t 3j ∆t 4j ∆t 4j ∆t 3j ∆t 2j ∆t 1j 0 ∆t 1i ∆t 2i ∆t 3i ∆t 4i ∆t 4i ∆t 3i ∆t 2i ∆t 1i i 0 j (t) ∆ ∆ ∆ ∆i 0 j ∆ ∆ ∆ ∆i 0 i

G. Space Vector PWM Implementation

In electric automobile applications, PM synchronous motors are driven using field-oriented control [START_REF] Hussein | Electric and Hybrid Vehicles; Design and Fundamentals[END_REF]. This technique is applied in synchronous rotating frame (0dq) in order to get DC quantities which can be regulated without steady-state error by simple PI correctors [START_REF] Alemayehu | Vector control of direct drive six phase permanent magnet synchronous generators[END_REF]. Fig. 11 shows the generic self-control diagram that is implemented in this study. The control loop purpose is to force the i d and i 0 currents to zero and to meet the torque demand using only i q current. This can be calculated using the following expression [START_REF] Semail | Vectorial Formalism for analysis and design of polyphase synchronous machines[END_REF]: Duty cycle determination

* em * q T K 1 i ⋅ = Ω (7)
2 / 1 V V V 1 1 1 1 v v v v v v v v v v v v T t t t t * * * 0 1 4 h 3 2 1 4 h 3 2 1 04 03 02 01 4 3 2 1             ×             × =             ∆ ∆ ∆ ∆ - β α β β β α α α V 0 V α V β

Strategy

H. Results and Discussions

As shown in Fig. 11, the combination of PMSM and 3 H-bridge inverter associated with its dedicated control scheme have been tested in simulation using Matlab/Simulink software tools. The DC link voltage is kept constant along the operation. The PMSM model does not include a mechanical model.

Only steady operation at a fixed speed (specifically 955 rpm) and with a constant reference torque (namely 60 N.m) are examined. Fig. 13 shows the results obtained with two control methods (i.e. I and III). Furthermore it should be noted that all methods have the same complexity since they are based on solving equation (3). For Control design, the combination of a PMSM and a 3-H bridge inverter offer important degrees of freedom. The SVPWM synthesis reveals a significant number of strategies. This exhaustive study was based on a comprehensive inventory of the EV drive requirements. The quantitative analysis allows the identification of a new and more attractive option. Simulation results confirm the interest of this choice.

The drive architecture was analyzed in nominal mode. The second crucial issue is to explore its abilities to operate in degraded mode. The following section gives an overview of the degraded mode possibilities and proposes fault-tolerant control where feasible.

IV. EXHAUSTIVE ANALYSIS OF FAILURE MODE

The studied system may encounter fault conditions. As mentioned in section II, power switches have the largest failures occurrence and their occurrence has very significant implications. Switch breakdown may result in switch open circuit (OC) or switch short-circuit (SC). In practice, the SC switch failure is most common compared to the OC failure (85% for SC and 15% for OC) [START_REF] Louis | Control of Non-conventional Synchronous Motors[END_REF]. For that reason, section V: 1) explains drive remaining possibilities once a single power switch becomes inoperative,

2)

suggests fault tolerant mode control algorithm.

For simplicity, a semiconductor breakdown (precisely (T 2 , D 2 )) of the third bridge (i.e. phase C) is considered.

Remaining Discrete Space Vectors: Fig. 15 represents switch states during a short -circuit failure on a single power switch (T 2 , D 2 ). In this case, the common gate driver leg forces power switch (T 3 , D 3 ) to

be immediately in open state.

When such a fault occurs, the affected H-bridge can only produce two different levels instead of three, namely 0 and +V dc (example in Fig. 15). As a result, the inverter is only able to produce eighteen discrete space vectors, that is to say half the normal case. These cases are listed in Table VI while Fig. 16 illustrates them in Concordia reference frame. Exhaustive Research of a Remedial Strategy: In faulty drive, the first objective is to obtain a rotating voltage in the α-β frame by using combinations of the remaining discrete voltages. The second objective is to realize a voltage with a null mean zero-sequence component during a switching period (<V 0 > = 0). The zero sequence current i 0 is essentially limited by the zero sequence inductance whose value is low. Hence, as the three voltage components (V α , V β , V 0 ) still need to be controlled, the faulty PWM generation remains a 4-dimensional problem. Any achievable voltage is determined by using [START_REF] Dixon | Asymmetrical Multilevel Inverter for Traction Drives Using Only One DC Supply[END_REF] while varying ∆t 1 and ∆t 2 arbitrarily from 0 to T sw /2. In this equation, {V1, V2, V3, V4} is a set of four selected voltages out of the remaining eighteen ones. Consequently, the number of possible combinations is 3060 (C 18 4 = 3060). A given voltage can be achieved by different voltage sets. On the contrary, some voltages may never be realized. Obviously, solving (5) is constrained by positive times ∆t 3 and ∆t 4 .

TABLE VI STATOR VOLTAGES DECOMPOSITION

VECTORS V a V b V c V 0 V α V β V 1 0 0 0 0 0 0 V 3 0 0 +1 3 3 - 6 1 - 2 2 V 4 0 -1 0 - 3 3 6 1 - 2 2 V 6 0 -1 +1 0 0 -2 V 7 0 +1 0 3 3 - 6 1 2 2 V 9 0 +1 +1 ) 3 3 ( 2 ⋅ - 6 1 2 ⋅ 0 V 10 -1 0 0 - 3 3 - 6 1 2 ⋅ 0 V 12 -1 0 +1 0 - 2 3 - 2 2 V 13 -1 -1 0 - ) 3 3 ( 2 ⋅ - 6 1 - 2 2 V 15 -1 -1 +1 - 3 3 - 6 1 2 ⋅ -2 V 16 -1 +1 0 0 - 2 3 2 2 V 18 -1 +1 +1 3 3 - 6 1 4 ⋅ 0 V 19 +1 0 0 3 3 6 1 2 ⋅ 0 V 21 +1 0 +1 ) 3 3 ( 2 ⋅ 6 1 - 2 2 V 22 +1 -1 0 0 2 3 - 2 2 V 24 +1 -1 +1 3 3 6 1 2 ⋅ -2 V 25 +1 +1 0 ) 3 3 ( 2 ⋅ 6 1 2 2 V 27 +1 +1 +1 3
Fig. 17-A shows the SV reachable area defined in case of a single power switch SC. It is a half hexagon; the missing part is due to inability to meet the V 0 = 0 specification. Consequently, it shows that a rotating voltage cannot be obtained using the remaining discrete vectors.

As given in Table VI, the high zero-sequence component of vector V 25 (2•(√3)/3) prevents null V 0 in the critical region. Among the 18 remaining vectors, the only ones which permit to reduce this high voltage are V 4 , V 10 and V 13 . V 13 is the opposite of V 25 (see Fig. To summarize, a satisfactory behavior in degraded operating mode can't be obtained without a reconfiguration of the power converter structure. Indeed, considering a phase disconnection of the faulty H-bridge, it could lead to a more suitable degraded operating mode. This topic will be studied in a future work and presented in a forthcoming paper.

V. CONCLUSION

This paper has considered H-Bridge power converter for a three-phase PM synchronous motor in an electric vehicle (EV).

Firstly, using a methodical approach, innovative space vector PWM (SVPWM) control methods has been proposed. Furthermore, using the EV drive requirements, these suggested methods has been compared to standard SVPWM ones. Among these, a specific one has combined all the advantages: minimization of switching losses, balancing of switching rate between the three H-bridges, insensitivity to duty-cycle, maximization of the drive performance, and reduction of the zero-sequence current ripple. This proposed method has been successfully implemented in simulation with a selfcontrolled PM motor model using Matlab-Simulink® environment.

Secondly, power switch failure mode has been fully investigated. Exhaustive analysis of the most common breakdown demonstrates that short-circuit (SC) is critical state regarding the margins left in the control strategy. Future works will be focused on the SVPWM control methods in normal and degraded operating modes.

Finally, it has been demonstrated that no satisfactory post-failure reconfiguration can be applied if the disconnection of the PMSM phase from the failing H-bridge is not considered. The work presented in this paper has been a first step in defining fault tolerant control for an EV power train.

Fig. 1 .

 1 Fig.1. Fault-tolerant scheme of 4-leg inverter drive[START_REF] Bianchi | Innovative Remedial Strategies for Inverter Faults in IPM Synchronous Motor Drives[END_REF].

Fig. 2 .

 2 Fig. 2. Studied architecture: PMSM drive fed by 3 H-Bridge inverters.

Fig. 3 .

 3 Fig. 3. 3H-Bridge inverter switching states and space vectors.

Fig. 4 .

 4 Fig. 4. Selected vectors of standard SVPWM modulation methods. (A) Method I, (B) Method II.

Fig. 6 .

 6 Fig. 6. Scalar projection of methods vectors onto α-β plane. Associated Sectors. (A) Method I,

Fig. 7 .

 7 Fig. 7. Switching distribution in the 3H-bridges. (A) Method I, (B) Method II, (C) Method III,

Fig. 8 .Fig. 9 .

 89 Fig. 8. Zero-sequence voltage and current waveforms in generic case.

Fig. 10 Fig. 10 .

 1010 Fig.10shows the achievable voltage vectors for each method, represented by points in each

Fig. 11 .

 11 Fig. 11. Synoptic of proposed control architecture. The control loop generates sinusoidal reference voltages V 0αβ * in the stationary 0αβ frame. The 3-H

Fig. 12 .

 12 Fig. 12. Algorithm diagram of SVPWM modulation technique.

Fig. 13 -Fig. 13 .Fig. 14 .

 131314 Fig.13-A and Fig.13-Bshow instantaneous electromagnetic torque obtained for a SVPWM switching

Fig. 15 .

 15 Fig. 15. Short-circuit (SC) switch failure case.

Fig. 16 .

 16 Fig. 16. Discrete SV in case of a single power switch in S-C.

17

 17

  -A) and the zero-sequence component of the two vectors V 4 and V 10 are half of V 25 one. The diagram depicted in Fig. 17-B illustrates a typical SVPWM combination containing V 4 and V 25 . It shows that for a given duty cycle (σ) of V 25 ,double this value (2σ) must be applied to V 4 in order to generate a null V 0 . Consequently, the scalar projection of the resulting reference voltage is always located in the healthy region.

Fig. 17 .

 17 Fig. 17. Achievable reference voltages in case of single power switch in short-circuit states.

  Table V summarizes the whole comparative analysis. Advantages and drawbacks of each modulation technique are evaluated against each performance criterion. In conclusion, the innovative SVPWM Method III seems to be the best candidate for the electric drive control. It meets all the abovementioned control drive specifications. To complete this comparison, the following subsection presents the SVPWM control implementation of this proposed method and a competing method, namely Method I.

			TABLE IV
		NUMBER OF ACHIEVABLE REFERENCE VECTORS
	V 0 constraint	Method I	Method II	Method III	Method IV	Method V
	V 0 = 0	331	1305	1326	1002	332
	|V 0 | ≤ (V dc /100).	453	2885	4142	1190	582

ACKNOWLEDGEMENT

Authors would like to thank the French ministry of Industry, in the context of the project SOFRACI "Structure d'Onduleur Fort Rendement A fonction Charge Intégrée", 2009 to 2012, for its financial support to this research work.

VI.REFERENCES