Emmanuel Toinet

A finitely presented subgroup of the automorphism group of a right-angled Artin group

Let G Γ be a right-angled Artin group. We use geometric methods to compute a presentation of the subgroup H of Aut(G Γ ) consisting of the automorphisms that send each generator to a conjugate of itself. This generalizes a result of McCool on basis-conjugating automorphisms of free groups.

Introduction

A right-angled Artin group is a finitely-generated group subject to the relations that some of the generators commute. It can be described by a simplicial graph Γ with one vertex for each generator and one edge for each pair of commuting generators. At the two extremes are the free group F n of rank n (Γ is discrete), and the free abelian group n (Γ is complete). Rightangled Artin groups are sometimes called graph groups, or free partially commutative groups. In recent years, right-angled Artin groups have received considerable attention due to the fact that they contain many interesting subgroups, and also because of their actions on CAT(0) cube complexes. The automorphism group of an arbitrary right-angled Artin group is less known (see [D], [CV], [Mi], for example). For a general survey on right-angled Artin groups see [C].

Automorphisms of right-angled Artin groups were first studied by Servatius in [S]. Drawing on Nielsen automorphisms for free groups, Servatius defined four classes of automorphisms -consisting of inversions, partial conjugations, transvections, and symmetries (see Section 2) -, and conjectured that they generate Aut(G Γ ). Servatius proved his conjecture for some classes of right-angled Artin groups -for example, when Γ is a tree. Thereafter Laurence proved the conjecture for arbitrary right-angled Artin groups in [L]. More recently, Day gave a finite presentation for the automorphism group of a general right-angled Artin group (see [D]), which generalizes the presentation that McCool gave for the automorphism group of a free group (see [M1]).

Let Γ = (V, E) be a finite simplicial graph, and let G Γ be the right-angled Artin group associated to Γ. Recall that G Γ has the presentation:

G Γ = V | vw = wv, ∀ {v,w} ∈ E . Definition 1.1. We say that an automorphism ϕ of G Γ is vertex-conjugating if ϕ(v) is conjugate to v for all v ∈ V.
Vertex-conjugating automorphisms were first introduced by Laurence in [L], where they are called conjugating. They also appear in the recent work of Duncan and Remeslennikov (see [DR]). One of the steps in Laurence's proof of Servatius' conjecture was to show that the set of vertex-conjugating automorphisms coincides with the subgroup H of Aut(G Γ ) generated by the partial conjugations (see Section 2 for the definition of a partial conjugation). Let S denote the set of all partial conjugations of G Γ . In Section 3, we define a finite set R of relations satisfied by the elements of S. Our main result is the following:

Theorem 1.2. The group H has the presentation S | R .

In order to prove Theorem 1.2, we shall construct a finite, connected 2-complex with fundamental group H = S | R . Our proof is similar to that of [M3]. In [M3], McCool gave a finite presentation for the subgroup H of Aut(F n ) consisting of basis-conjugating automorphisms. Note that we cannot hope for a generalization of the presentation given in the theorem of [M3] (see Remark 3.2 below). Our proof will use the presentation Day gave for the automorphism group of a right-angled Artin group, that will be described in the next section.

I am grateful to Luis Paris, my Ph.D. thesis advisor, for his comments on earlier versions of this work.

Preliminaries

Let Γ = (V, E) be a finite simplicial graph, and let G Γ be the right-angled Artin group associated to Γ. Let v be a vertex of Γ. The link of v, denoted by lk(v), is the subset of V consisting of all vertices that are adjacent to v.

The star of v, denoted by st(v), is lk(v) ∪ {v}. We set L = V ∪ V -1 . Let x ∈ L. The vertex of x, denoted by v(x), is the unique element of V ∩ {x, x -1 }. We set lk L (x) = lk(v(x)) ∪ lk(v(x)) -1 , and st L (x) = st(v(x)) ∪ st(v(x)) -1 .
Let w be a word in V ∪ V -1 . The support of w, denoted by supp(w), is the subset of V of all vertices v such that v or v -1 is a letter of w. A word w in V ∪ V -1 is said to be reduced if it contains no subwords of the form vW v -1 or v -1 W v with supp(W ) ⊂ star(v). For a word w in V ∪ V -1 , we denote by |w| the length of w. The length of an element g of G Γ is defined to be the minimal length of any word representing g. Note that the length of g is equal to the length of any reduced word representing g. We say that an element g of G Γ is cyclically reduced if it can not be written vhv -1 or v -1 hv with v ∈ V, and |g| = |h| + 2. By [S], Proposition 2, every element of G Γ is conjugate to a unique (up to cyclic permutation) cyclically reduced element. The length of a conjugacy class is defined to be the minimal length of any of its representative elements. Observe that the length of a conjugacy class is equal to the length of a cyclically reduced element representing it. For an n-tuple of conjugacy classes W , we define the length of W , denoted by |W |, as the sum of the lengths of its elements (n ≥ 1).

Let v, w be vertices of Γ. We use the notation v ≥ w to mean lk(w) ⊂ st(v). We use the notation v ∼ w to mean v ≥ w and w ≥ v.

The Laurence-Servatius generators for Aut(G Γ ) are defined as follows:

Inversions: Let v ∈ V.
The automorphism ι v that sends v to v -1 and fixes all other vertices is called an inversion.

Partial conjugations: Let x ∈ L, and let Y be a non-empty union of connected components of Γ \ st(v(x)). The automorphism c x,Y that sends each vertex y in Y to x -1 yx and fixes all vertices not in Y is called a partial conjugation.

Transvections: Let v, w ∈ V be such that v ≥ w. The automorphism τ v,w that sends w to vw and fixes all other vertices is called a transvection.

Symmetries: Let ϕ be an automorphism of the graph Γ. The automorphism φ given by φ(v) = ϕ(v) for all v ∈ V is called a symmetry.

Our aim is to compute a presentation of the subgroup H of Aut(G Γ ) generated by the partial conjugations. Our proof will use the fact that partial conjugations are long-range Whitehead automorphisms.

Following [D], we call Whitehead automorphism every automorphism α of one of the following two types:

Type 1: α restricted to V ∪ V -1 is a permutation of V ∪ V -1 .
Type 2: There is an element a ∈ L, called the multiplier of α, such that α(a) = a, and for each x ∈ V, the element α(x) lies in {x, xa, a -1 x, a -1 xa}.

One can show that the set of type 1 Whitehead automorphisms is the subgroup of Aut(G Γ ) generated by inversions and symmetries.

Following [D], we say that a Whitehead automorphism α is long-range if α is of type 1 or if α is of type 2 and α fixes the vertices of lk(v(a)) (where a is the multiplier of α).

We denote by W the set of Whitehead automorphisms, by W 1 the set of Whitehead automorphisms of type 1, and by W 2 the set of Whitehead automorphisms of type 2. We also denote by W ℓ the set of long-range Whitehead automorphisms.

We use the following notation for type 2 Whitehead automorphisms. Let A be a subset of L, and let a ∈ L, such that a ∈ A and a -1 / ∈ A. Provided that it exists, (A, a) denotes the automorphism given by: (A, a)(a) = a, and, for all x ∈ V \ {v(a)}, The following remark will be of importance in our proof: In [D], Day proved that Aut(G Γ ) is generated by the Whitehead automorphisms, subject to the relations: (R7)

(A, a)(x) =        x if x / ∈ A and x -1 / ∈ A xa if x ∈ A and x -1 / ∈ A a -1 x if x / ∈ A and x -1 ∈ A a -1 xa if x ∈ A and x -1 ∈ A If A is a subset of L, we set A -1 = {a -1 | a ∈ A}. If A
(R1) (A, a) -1 = (A -a + a -1 , a -1 ), for (A, a) ∈ W 2 . ( R2
The entire multiplication table of W 1 -which forms a finite subgroup of Aut(G Γ ).

(R8) (A, a) = (L -a -1 , a)(L -A, a -1 ), for (A, a) ∈ W 2 . (R9) (A, a)(L -b -1 , b)(A, a) -1 = (L -b -1 , b), for (A, a) ∈ W 2 , b ∈ L such that b / ∈ A, b -1 / ∈ A. (R10) (A, a)(L -b -1 , b)(A, a) -1 = (L -a -1 , a)(L -b -1 , b), for (A, a) ∈ W 2 , b ∈ L such that b ∈ A, b -1 / ∈ A, and b = a.
Note that the relation (R8) is a direct consequence of the relations (R1) and (R2).

In order to prove Theorem 1.2, we need to introduce the following technical definitions.

Let α, β ∈ W, and let W be an n-tuple of conjugacy classes (n ≥ 1). Following [D], we say that βα is a peak with respect to W if:

|α.W | ≥ |W |, |α.W | ≥ |βα.W |,
and at least one of these inequalities is strict.

Let α 1 ,...,α k ∈ W (k ≥ 1). We say that α i is a peak of the factorization

α k • • • α 1 with respect to W if 1 ≤ i < k and α i+1 α i is a peak with respect to α i-1 • • • α i .W . We say that the factorization α k • • • α 1 is peak-reduced with respect to W if it has no peaks with respect to W . The height of a peak α i is |α i • • • α 1 .W |. for β ∈ W 1 . It remains to show that π 1 (K, W ) = H = S | R .
Let T be a maximal tree in C. We compute a presentation of π 1 (K, W ) using T . There will be a generator (V 1 , V 2 ; α) for each edge (V 1 , V 2 ; α) of K.

Since C is simply connected, we have

(6) (α.W, βα.W ; β) = 1 (in π 1 (K, W )),
for all α, β ∈ W 1 .

Let P be the set of combinatorial paths in the 1-skeleton K (1) of K. We define a map ϕ : P → Aut(G Γ ) as follows. For an edge e = (V 1 , V 2 ; α), we set ϕ(e) = α, and for a path

p = e ε k k • • • e ε 1 1 , we set ϕ(p) = ϕ(e k ) ε k • • • ϕ(e 1 ) ε 1 . Clearly, if p 1 and p 2 are loops at W such that p 1 ∼ p 2 , then ϕ(p 1 ) = ϕ(p 2 ).
Hence, ϕ induces a map ϕ : π 1 (K, W ) → Aut(G Γ ). It is easily seen that ϕ is a homomorphism. Then we see from ( 6) that ϕ maps π 1 (K, W ) to H. It follows immediately from the construction of K that ϕ : π 1 (K, W ) → H is surjective. Thus, it suffices to show that ϕ is injective. Let p be a loop at

W such that ϕ(p) = 1. We have to show that p ∼ 1. Write p = e ε k k • • • e ε 1 1
, where k ≥ 1 and ε i ∈ {-1,1} for all i ∈ {1,...,k}. Using the 2-cells arising from the relations (1), and the fact that W -1 1 = W 1 , we can restrict our attention to the case where p = e k • • • e 1 . Set α i = ϕ(e i ) for all i ∈ {1,...,k}. Note that α i ∈ S ∪ W 1 ⊂ W ℓ for all i ∈ {1,...,k}.

Let Z be a tuple containing each conjugacy class of length 2 of G Γ , each appearing once.

We shall prove that

p ∼ e ′ l • • • e ′ 1 , such that, if we set α ′ i = ϕ(e ′ i ) for all i ∈ {1,...,l}, then α ′ i ∈ W 1 or α ′ i ∈ W 2 ∩ Inn(G Γ )
for each i ∈ {1,...,l}. First, we examine the case where α k • • • α 1 is peak-reduced with respect to Z. Consider the following sequence of integers:

|Z|, |α 1 .Z|, |α 2 α 1 .Z|,..., |α k-1 • • • α 1 .Z|, |α k • • • α 1 .Z| = |Z|.
By Lemma 5.2 in [D], |Z| is a minimal element of {|α.Z| | α ∈ W ℓ }, and therefore is the minimum of the above sequence. On the other hand, since α k • • • α 1 is peak-reduced with respect to Z, there does not exist i ∈ {1,...,k-1} such that we have

|α i-1 • • • α 1 .Z| ≤ |α i • • • α 1 .Z|, |α i+1 • • • α 1 .Z| ≤ |α i • • • α 1 .Z|,
and at least one of these inequalities is strict. Therefore, the above sequence is a constant sequence, and we have

|α i • • • α 1 .Z| = |Z|,
for all i ∈ {1,...,k}. We argue by induction on i ∈ {1,...,k} to prove that α i • • • α 1 .Z is a tuple containing each conjugacy class of length 2 of G Γ , each appearing once. The result holds for i = 0 by assumption. Suppose that i ≥ 1, and that the result holds for i -1. Observe that a type 1 Whitehead automorphism does not change the length of a conjugacy class. Thus, we can assume that α i is a type 2 Whitehead automorphism. Since

|α i α i-1 • • • α 1 .Z| = |α i-1 • • • α 1 .
Z|, α i is trivial, or an inner automorphism by [D], Lemma 5.2. Thus, the result holds for i. In this case, p has already the desired form.

We define:

h p = max{|α i • • • α 1 .Z| | i ∈ {0,.
..,k}}, and:

N p = |{i | i ∈ {0,...,k} and |α i • • • α 1 .Z| = h p }|.
We argue by induction on h p . The base of induction is |Z| -the smallest possible value for h p by [D], Lemma 5.2. If h p = |Z|, then the factorization α k • • • α 1 is peak-reduced and we are done. Thus, we can assume that h p > |Z| and that the result has been proved for all loops p ′ with h p ′ < h p . Let i ∈ {1,...,k} be such that α i is a peak of height h p . An examination of the proof of Lemma 3.18 in [D] shows that e i+1 e i ∼ f j • • • f 1 such that, if we set β κ = ϕ(f κ ) for all κ ∈ {1,...,j}, then:

(7)

|β κ • • • β 1 α i-1 • • • α 1 .Z| < |α i α i-1 • • • α 1 .Z| ,
for all κ ∈ {1,...,j -1}. Therefore, we get p

∼ e k • • • e i+2 f j • • • f 1 e i-1 • • • e 1 = p ′ , and a new factorization α k • • • α i+2 β j • • • β 1 α i-1 • • • α 1 .
We argue by induction on N p . If N p = 1, then (7) implies that h p ′ < h p and we can apply the induction hypothesis on h p . If N p ≥ 2, then (7) implies that h p ′ = h p and N p ′ < N p , and we can apply the induction hypothesis on N p . This completes the induction. Now, using the 2-cells arising from the relations (5), we obtain p ∼

h s • • • h 1 g r • • • g 1 ,
where, if we set γ i = ϕ(g i ) for all i ∈ {1,...,r} and δ j = ϕ(h j ) for all j ∈ {1,...,s}, then δ i ∈ W 1 for all i ∈ {1,...,s} and γ j ∈ W 2 ∩ Inn(G Γ ) for all j ∈ {1,...,r}. Using (6), we obtain p ∼ g r • • • g 1 . Set Z = ∩ v∈V st(v). It follows from Servatius' Centralizer Theorem (see [S]) that the center Z(G Γ ) of G Γ is the special subgroup of G Γ generated by Z. Let Γ ′ be the full subgraph of Γ spanned by V \ Z. We have G Γ ′ ≃ Inn(G Γ ), where the isomorphism is given by v → ω v (see [D], Lemma 5.3, for example). Write

γ i = (L -lk L (c i ) -c -1 i , c i ), where c i ∈ V \ Z ∪ (V \ Z) -1 (i ∈ {1,...,r}). Since γ r • • • γ 1 = 1 (in Inn(G Γ )), we have c r • • • c 1 = 1 (in G Γ ′ ). Therefore c r • • • c 1 is a product of conjugates of defining relators of G Γ .
Using the 2-cells corresponding to the relations (1) and (3)(b), we deduce that p ∼ 1. We conclude that ϕ is injective, and thus H = π 1 (K, W ). Now, using the 2-cells arising from the relations (5) (with α = β), we obtain: 5) that these relations will result from the relations (1)-( 4) above. It is easily seen that the relations (1)-( 4) above are equivalent to those of R. We have shown that H has the presentation S | R . Remark 3.2. We cannot hope for a generalization of the presentation given in the theorem of [M3], since, in a general right-angled Artin group, the existence of one-term partial conjugations depends on the existence of domination relations between the vertices of Γ. (A one-term partial conjugation is a partial conjugation of the form c x,{y} with x ≥ y.)

  and B are subsets of L, and a is an element of L, we use the notations A -B for A \ B, A+B for A⊔B (if A∩B = ∅), A-a for A\{a} and A+a for A⊔{a} (if a / ∈ A).

  Remark 2.1. Let x ∈ L, and let Y be a non-empty union of connected components of Γ \ st(v(x)). Set A = Y ∪ Y -1 ∪ {x}, and a = x. Then the Whitehead automorphism (A, a) is nothing but the partial conjugation c x,Y . In particular, the Whitehead automorphism (L-lk L (a)-a -1 , a) is the inner automorphism ω a induced by a. Note that there is not a unique way to write a partial conjugation as a type 2 Whitehead automorphism. More specifically, if B ⊂ lk(v(a)), then the Whitehead automorphisms (A, a) and (A + B + B -1 , a) represent the same element of S.

  ) (A, a)(B, a) = (A ∪ B, a), for (A, a), (B, a) ∈ W 2 with A ∩ B = {a}.(R3) (B, b)(A, a)(B, b) -1 = (A, a), for (A, a), (B, b) ∈ W 2 such that a / ∈ B, a -1 / ∈ B, b / ∈ A, b -1 / ∈ A, and at least one of (a) A ∩ B = ∅ or (b) b ∈ lk L (a) holds. (R4) (B, b)(A, a)(B, b) -1 = (A, a)(Bb + a, a), for (A, a), (B, b) ∈ W 2 such that a / ∈ B, a -1 / ∈ B, b / ∈ A, b -1 ∈ A, and at least one of (a) A ∩ B = ∅ or (b) b ∈ lk L (a) holds. (R5) (Aa + a -1 , b)(A, a) = (Ab + b -1 , a)σ a,b , for (A, a) ∈ W 2 , b ∈ L such that b ∈ A, b -1 / ∈ A, b = a,and v(b) ∼ v(a). Here σ a,b denotes the type 1 Whitehead automorphism that sends a to b -1 and b to a, and fixes the other generators. (R6) σ(A, a)σ -1 = (σ(A), σ(a)), for (A, a) ∈ W 2 , and σ ∈ W 1 .

  (α.W, α.W ; (α(A), α(a))) = (α.W, W ; α -1 )(W, W ; (A, a))(W, α.W ; α), and then, using (6), (α.W, α.W ; (α(A), α(a))) = (W, W ; (A, a)), for all α ∈ W 1 , and (A, a) ∈ S. It then follows that H is generated by the (W, W ; (A, a)), for (A, a) ∈ S. We identify (W, W ; (A, a)) with (A, a) for all (A, a) ∈ S. Any relation in H = π 1 (K, W ) will come from the 2-cells of K. Then we see from (

Proof of the main theorem

In this section, we prove the following:

Theorem 3.1. The group H has a presentation with generators c x,Y , for x ∈ L and Y a non-empty union of connected components of Γ \ st(v(x)), and relations:

Proof : Our proof is based on arguments developed by McCool in [M2] and [M3] (similar arguments were used in [D]). Recall that S denote the set of partial conjugations. Let R denote the set of relations given in the statement of Theorem 3.1. We shall construct a finite, connected 2-complex K with fundamental group H = S | R .

We identify a partial conjugation with any of its representatives in W 2 (see Remark 2.1 above). Note that for every (A, a) ∈ W 2 , we have (A, a) ∈ S if and only if (Aa) -1 = Aa.

The vertices of K will be the set of n-tuples α.W , where α ranges over the set W 1 of type 1 Whitehead automorphisms. There will be a directed edge (α.W, βα.W ; β) labelled β joining α.W to βα.W for all α, β ∈ W 1 . In addition, there will be a loop (α.W, α.W ; (A, a)) labelled (A, a) at α.W for all α ∈ W 1 , and (A, a) ∈ S. This specifies the 1-skeleton K (1) of K.

We shall define the 2-cells of K. These 2-cells will derive from the relations (R1)-(R10) of [D]. Let K 1 be the 2-complex obtained by attaching 2-cells corresponding to the relations (R7) to K (1) . Note that, if C is the 2complex obtained from K 1 by deleting the loops (α.W, α.W ; (A, a)) for α ∈ W 1 , and (A, a) ∈ S, then C is just the Cayley complex of W 1 , and therefore is simply connected.

We now explore the relations (R1)-(R5) and (R8)-(R10) of [D] to determine which of these will give rise to relations on the elements of S.

The relation (R1) will give rise to the following:

(1) (A, a) -1 = (Aa + a -1 , a -1 ), for (A, a) ∈ S.

The relation (R2) will give rise to:

( From (R5), no relations arise (by the same argument as above).

From (R8), we obtain a relation which is a direct consequence of ( 1) and ( 2).

The relation (R9) will give rise to the following:

for (A, a) ∈ S, and b ∈ L such that b / ∈ A, and b -1 / ∈ A. From (R10), no relations arise (by the same argument as above).

We rewrite the relations (1)-(4) in the form:

where σ 1 ,...,σ k ∈ S, and ε 1 ,...,ε k ∈ {-1,1}. Let K 2 be the 2-complex obtained from K 1 by attaching 2-cells corresponding to the relations (1)-( 4). Note that the boundary of each of these 2-cells has the form:

Finally, the relations (R6) will give rise to the following:

(5) α(A, a)α -1 = (α(A), α(a)),

for (A, a) ∈ S, and α ∈ W 1 . Then K is obtained from K 2 by attaching 2cells corresponding to the relations (5). Observe that the boundary of each of these 2-cells has the form:

(β.W, β.W ; (α(A), α(a))) -1 (β.W, α -1 β.W ; α) -1 (α -1 β.W, α -1 β.W ; (A, a)) (α -1 β.W, β.W ; α),


Khalid El Majdoub

Fouad Giri

Hamid Ouadi

Luc Dugard

Zara Fatima

Chaoui

F Z Chaoui

Vehicle longitudinal motion modeling for nonlinear control

Keywords: vehicle longitudinal control, longitudinal slip, tire Kiencke"s model, speed control, Lyapunov stability

INTRODUCTION

Vehicle longitudinal motion control aims at ensuring passenger safety and comfort. It is an important aspect in dynamic collaborative driving i.e. when multiple vehicles should coordinate to share road efficiently while maintaining safety. In this respect, several works have been devoted to what is commonly referred to adaptive cruise control that consists in maintaining a specified headway between vehicles [START_REF] Ioannou | Autonomous Intelligent Cruise Control[END_REF][START_REF] Moon | Design, tuning, and evaluation of a full-range adaptive cruise control system with collision avoidance[END_REF]. Different control techniques have been used in these works including linear and adaptive control [START_REF] You | New adaptive approaches to real-time estimation of vehicle sideslip angle[END_REF], genetic fuzzy control [START_REF] Poursamad | Design of genetic-fuzzy control strategy for parallel hybrid electric vehicles[END_REF], sliding mode control [START_REF] Liang | Vehicle longitudinal brake control using variable parameter sliding control[END_REF]Nouveliere and Mammar, 2007), and scheduling gain control involving PIDs [START_REF] Ren | Motion control for a two-wheeled vehicle using a selftuning PID controller[END_REF]. However, most previous works on longitudinal control were based on simple models neglecting important nonlinear aspects of the vehicle such as rolling resistance, aerodynamics effects and road load. In some studies, the controller performances were not formally analyzed [START_REF] Ren | Motion control for a two-wheeled vehicle using a selftuning PID controller[END_REF]. In [START_REF] Yamakawa | A method of torque control for independent wheel drive vehicles on rough terrain[END_REF], longitudinal vehicle control has been studied focusing on torque management for independent wheel drive. It is worth noticing that in all previous studies on longitudinal vehicle control, the control design has been based on simple models not accounting for tire-road interaction.

In the present study, the problem of longitudinal vehicle control is revisited, for front wheel propelled vehicles, focusing on speed regulation. The aim is to design a controller that is able to tightly regulate the chassis and wheel velocities, in both acceleration and deceleration driving modes, despite changing and uncertain driving conditions. This problem has not been dealt with previously. A further originality of the present paper is that the control design relies upon a more complete model that accounts for most vehicle nonlinear dynamics including tire-road interaction. That is, the study includes two major contributions. First, a suitable control model is developed for the vehicle longitudinal behavior. In this respect, recall that a convenient model is one that is sufficiently accurate but remains simple enough to be utilizable in control design. To meet the accuracy requirement, the model must account not only for aerodynamic phenomena but also, and especially, for tire-road friction. Modeling the tire/road contact is a quite complex issue involving multiple aspects relevant to tire characteristics (e.g. structure, pressure) and to environmental factors (e.g. road load, temperature).

Several tire models have been proposed in the literature e.g. Guo"s model [START_REF] Guo | A non-steady and non-linear tire model under large lateral slip condition[END_REF], Pacejka"s model [START_REF] Pacejka | Magic Formula Tyre Model with Transient Properties[END_REF]), Dugoff"s model [START_REF] Dugoff | An analysis of tire traction properties and their influence on vehicle dynamic performance[END_REF]), Gim"s model [START_REF] Gim | An analytical model of pneumatic tyres for vehicle dynamic simulations part1: Pure slips[END_REF]), Kiencke"s model (Kiencke and Nielsen, 2004). In the present work, Kiencke"s model is retained because it proves to be a good compromise between accuracy and simplicity. The overall vehicle modeling is carried out according to the bicycle model principle. In addition to tire equations, the model includes chassis dynamics equations (obtained from fundamental dynamics and aerodynamics laws) and incorporates relevant practical prior knowledge e.g. the tire longitudinal slip is physically limited. The overall vehicle model turns out to be a combination of two nonlinear state-space representations describing, respectively, the acceleration and deceleration longitudinal driving modes. Its high complexity makes it hardly utilizable for control design but, due to its high accuracy, it proves to be quite suitable for simulator building. This model development is one major achievement of the present study. The second contribution is the design of a nonlinear controller that ensures global stabilization and longitudinal speed regulation during acceleration/deceleration driving modes. This is carried out using the Lyapunov design technique [START_REF] Khalil | Nonlinear Systems[END_REF], based on a simpler (but still accurate) version of the above simulation-oriented model.

It is formally proved that the developed controller actually achieves the stability and regulation objectives it was designed to. Furthermore, it is observed through numerical simulations that the controller is quite robust with respect to uncertainties on environmental characteristics.

The paper is organized as follows: Section II is devoted to modeling the acceleration/deceleration vehicle longitudinal behavior; the obtained model is used in Section III to design a controller and to analyze the resulting closed-loop system; the controller performances are illustrated in Section IV by numerical simulations. A conclusion and reference list end the paper.

MODELLING OF CHASSIS LONGITUDINAL MOTION

Except for aerodynamic forces, all external efforts acting on a vehicle are generated at the wheel-road contact. The understanding and modeling of the forces and torques developed at wheel-road contact is essential for studying properly the vehicle dynamics. These are discussed in the forthcoming subsections. In this respect, recall that the vehicle motion is composed of two types of displacements: translations along the z y x , , axes and rotations around these same axes (Fig. 1).

Kiencke's Tire Modeling

The tire is a main component of the wheel-road contact as it ensures three important functions (Kiencke and Nielsen, 2004): (i) bearing the vertical load and absorbing road deformations; (ii) producing longitudinal acceleration efforts and contributing to vehicle braking; (iii) producing the required transversal efforts that help the vehicle turning.

The efforts generated at the wheel-road contact include longitudinal (acceleration/deceleration) forces, lateral guiding forces and self alignment torque. The effect of these efforts on the vehicle behaviour is determined by the tire-road adhesion. For small load variations, the longitudinal coefficient of friction is characterized by the following ratio:

v tx F F (1)
where tx F denotes the longitudinal effort and v F the vertical load. The ratio is called longitudinal adhesion or friction coefficient. The value of this coefficient depends on the tire slip resulting from the deformation of the tire in contact with the road (Kiencke and Nielsen, 2004). The longitudinal slip is characterized by the coefficient defined as follows:

. in acceleration mode, i.e. . in deceleration mode, i.e.

W V V V , one has: W eff V W V r V V V 1 1 (2)
W V V V
, one has:

1 1 V W eff V W V r V V (3)
where eff r denotes the effective wheel radius, W designates the wheel angular velocity,

w V  is the speed of the tire-road contact, v V 
is the linear velocity of the wheel centre (Fig. 2). A similar deformation occurs when the wheel presents a slip angle i.e. the resulting lateral slip produces a lateral force ty F . Modeling the efforts at the wheel-road contact has been given a great deal of interest over the last years. In this respect, several tire models have been developed with quite different properties, e.g. [START_REF] Guo | A non-steady and non-linear tire model under large lateral slip condition[END_REF][START_REF] Pacejka | Magic Formula Tyre Model with Transient Properties[END_REF]Dugoff and Segel, 1990;[START_REF] Gim | An analytical model of pneumatic tyres for vehicle dynamic simulations part1: Pure slips[END_REF]Kiencke and Nielsen, 2004). For control design use, the most suitable tire model is one that presents the best accuracy/simplicity compromise. From this viewpoint, Kiencke"s model turns out to be a quite satisfactory choice (Kiencke and Nielsen, 2004). Indeed, this model is sufficiently accurate as it accounts for the main features such as the vertical load v F , slip angle , slip coefficient . On the other hand, it has already proved to be useful in designing simple estimators for state variables like slip angle and lateral efforts [START_REF] You | New adaptive approaches to real-time estimation of vehicle sideslip angle[END_REF]. In the present paper, this model will prove to be useful in control design.

Kiencke's Model

This was developed in (Kiencke and Nielsen, 2004) using the Burckhardt"s extended model to compute the friction coefficient . Accordingly, the latter is a function of the combined longitudinal/lateral slip coefficient and the forces acting on the tire. Fig. 3 gives a schematic representation of Kiencke"s wheel model where:

) 1 )( exp( ) exp( 1 2 5 4 3 2 1 v G F C V C C C C (4) 2 2 y (5) y z t F ty F tx F W V V V y z V V tx F ty F W Fig. 2. Forces applied on the wheel reff Mtz W V V V 1 and ) tan( ) 1 ( y (Acceleration) (6) 1 V W V V and ) tan( y (Deceleration) (7)
The other parameters and variables contained in equations ( 4) to ( 7) have the following meanings:

3 2 1 , , C C C
: parameters depending on the road state. C : coefficient depending on the maximal driving speed.

5 C : coefficient depending on the wheel allowed maximal load. : global (longitudinal/lateral) slip.

y : lateral slip.

G V : speed of the vehicle centre of gravity.

V F : vertical load. In Kiencke"s model, the longitudinal and lateral efforts are respectively described by ( 8)-( 9 (i) The tire behaviour is independent of the vehicle maximal speed. That is, the term in 4 C is neglected in equation ( 4).

(ii) The tire model depends linearly on the maximal load. Then, the term in 5 C is neglected in (4). (iii) The vehicle moves along a straight line. That is, the slip angle is null.

Using these remarks, equations (4) to (9) simplify to:

Rolling Resistance and Aerodynamic Resistance

It is obvious that tire deformation causes mechanical losses. Radial deformations are caused by the vertical load. Fig. 4 shows that the vertical force distribution (along the deformation area) is not uniform. The resulting force moves from the central point I to a point located at a distance max , rr d (Fig 4). This is called vertical load maximal trail and is defined as follows:

eff rr rr r d max , ( 14
)
When the wheel starts moving, a torque is generated provided that 0 max , rr d

. To bring back the vertical load strength to the central point I , a compensating torque rr M must be applied (by the driving motor) on the wheel. This is called rolling resistance torque and is given by:

eff rr v rr v rr r F d F M max , (15)

Aerodynamic Resistance

Aerodynamic resistance has naturally an impact on energy consumption onboard. Fluid mechanics laws are resorted to explain air flow around a moving vehicle. Accordingly, vehicle forms are continuously reinvented to improve aerodynamic performances (Power and Nicastri, 2000).

Aerodynamic efforts come in direct interaction with the vehicle, producing various forces (drag, lift and lateral) which in turn generate torques (yaw, roll and pitch). The aerodynamic drag force is significant when the vehicle moves at grand speed. The aerodynamic lift effort increases the rolling resistance (because the surface of tire-road contact grows up) improving tire steerability. On the other hand, the aerodynamic lift force decreases the vehicle adhesion which reduces its stability and security [START_REF] Milliken | Race car vehicle dynamics[END_REF]. In presence of front-wind, the aerodynamic resistance is represented by two forces: the aerodynamic drag force aex F and the aerodynamic lift force aez F . These are defined as follows: : air density depending on atmosphere pressure and ambient temperature.

2 2 ) ( 2 1 ) ( 2 1 a V z aez a V x aex V V S C F V V S C F ( 16
C x : aerodynamic drag coefficient.

C z : aerodynamic lift coefficient.

S : frontal projection area vehicle.

V V : vehicle speed.

V a : wind speed (positive in presence of front-wind, negative in presence of rear-wind).

Modeling of a Wheel Submitted to Driving Couple

Fig. 5 shows a one-wheel vehicle with mass v M . The wheel is driven by a couple m M . Let J denotes the inertia resulting from the wheel, the transmission shaft and the driving motor. Invoking the dynamic fundamental principle, one gets the following equation:

rr eff t W m M r F dt d J M (18)
where equation ( 15) has been used to account for the rolling resistance. Using the relation

W eff W r V , one gets: rr eff v eff t m eff w r F r F M J r V  (19)
Equation ( 19) together with (1), ( 2), ( 3) and ( 10) describe the one-wheel vehicle behavior. This is further illustrated by the schematic representation of Fig. 6.

Model of Two-Wheel Vehicle with One Driving Wheel

Longitudinal and transversal behaviors can be assumed to be decoupled when the steering angle is small. Then, we make use of the vehicle symmetry to perform a projection (of all forces) on the longitudinal axis reducing thus the four-wheel model into a (two-wheel). Fig. 7 illustrates the forces involved in a bicycle model. The involved notations are described in Table I. Now, the focus will be made on the bicycle model considering that only one wheel is submitted to a motor torque m M . For vehicle stability purpose, the front wheel is driving. The distribution of the vehicle load over the tires is computed applying the two dynamic fundamental laws. Doing so, one gets the following equations:

v v aex v tf V M F g M F  ) sin ( (20) 0 ) cos ( aez v vr vf F g M F F (21) 0 ) ( h F F l F l F tr tf r vr f vf (22)
Solving equations ( 21) and ( 22) for F vf and F vr yields:

g M F g M F g V g M F v aex v aez v v vf ) 1 ( ) sin ( cos ) 1 (  (23) g M F g M F g V g M F v aex v aez v v vr ) sin ( cos  (24)
where

r f l l h and r f f l l l
. From equation ( 20) it is easily seen that the vehicle speed undergoes the following equation:

) sin ( 1 aex v tf v v F g M F M V  (25)
F v V v V W F t M m Body of mass M v W t F v F

State-Space Representation of Vehicle Longitudinal Behavior

The equations obtained so far are now combined together to build-up a state-space representation of the vehicle longitudinal acceleration/deceleration behavior. The vehicle longitudinal dynamics are characterized by two state variables, i.e. vehicle (chassis) speed v V and front-wheel speed w V . As the slip coefficient depends on the current driving mode (acceleration or deceleration), the vehicle is characterized by two state-space representations. Each representation describes the vehicle in the corresponding operation mode. 10), ( 11), ( 13) and ( 15)-( 18), one obtains the following state-space representation:

State-Space Representation in Deceleration Mode

v w V V Combining (3), (
2 7 6 5 1 4 3 2 1 ) ( ) exp( a v v w v w v w m w V V V V V V V V M V  ) exp( ) ( ) exp( ) ( 2 10 9 2 8 v w a v v w a v v w V V V V V V V V V V (26a) 2 15 14 13 1 4 3 2 2 12 11 ) ( ) exp( ) ( a v v w v w v w a v v V V V V V V V V V V V  ) exp( ) ( ) exp( ) ( 2 18 17 2 16 v w a v v w a v v w V V V V V V V V V V (26b)
where the various parameters are defined in Table II. A more compact representation of (26a-b) is obtained introducing the notations:

m M u , w V x 1 , v V x 2 and: 2 2 7 2 1 6 5 1 2 1 4 2 1 3 2 2 1 1 ) ( ) exp( ) , ( a V x x x x x x x x x f ) exp( ) ( ) exp( ) ( 2 1 2 2 10 2 1 9 2 2 2 1 8 x x V x x x V x x x a a (27a) 2 2 15 2 1 14 13 1 2 1 4 2 1 3 2 2 2 12 11 2 1 2 ) ( ) exp( ) ( ) , ( a a V x x x x x x x V x x x f ) exp( ) ( ) exp( ) ( 2 1 2 2 18 2 1 17 2 2 2 1 16 x x V x x x V x x x a a (27b)
With the above notations, the state-space representation (26a-b) is given the following usual compact form:

) , ( ) , ( 2 1 2 2 2 1 1 1 1 x x f x x x f u x   (28)
C ' : 2 C 1 : J r eff 1 ' : J r eff 2 : ) ( 1 3 1 C C k v 2 ' : ) ( 1 3 1 C C k v 3 : 3 C k v 3 ' : 3 C k v 4 : ) exp( 2 1 C C k v 4 ' : ) exp( 2 1 C C k v 5 : J r C C k g M eff v rr v 2 3 1 )) ( ( cos ) 1 ( 5 ' : J r C C k g M eff v rr v 2 3 1 )) ( ( cos ) 1 ( 6 : J r C k g M eff v v 2 3 cos ) 1 ( 6 ' : J r C k g M eff v v 2 3 cos ) 1 ( 7 : J r C C k SC eff v rr z 2 3 1 )) ( ( ) 1 ( 2 1 7 ' : J r C C k SC eff v rr z 2 3 1 )) ( ( ) 1 ( 2 1 8 : J r C k SC ) 1 ( 2 1 2 eff 3 v z 8 ' : J r C k SC eff v z 2 3 ) 1 ( 2 1 9 : J r C C k g M eff v v 2 2 1 ) exp( cos ) 1 ( 9 ' : J r C C k g M eff v v 2 2 1 ) exp( cos ) 1 ( 10 : J r C C k SC eff v z 2 2 1 ) exp( ) 1 ( 2 1 10 ' : J r ) C exp( C k SC ) 1 ( 2 1 2 eff 2 1 v z 11 ) sin( g 11 ' :
) sin( g : 10), ( 11), ( 13) and ( 15)-( 18), one gets the following state-space representation:

x v SC M 2 1 ' 12 : x v SC M 2 1 : ) ( cos ) 1 ( 3 1 C C k g v 13 ' : ) ( cos ) 1 ( 3 1 C C k g v 14 : 3 cos ) 1 ( C k g v 14 ' : 3 cos ) 1 ( C k g v 15 : ) C C ( k SC ) 1 ( M 2 1 3 1 v z v 15 ' : ) C C ( k SC ) 1 ( M 2 1 3 1 v z v 16 : 3 ) 1 ( 2 1 C k SC M v z v 16 ' : 3 ) 1 ( 2 1 C k SC M v z v : ) exp( cos ) 1 ( 2 1 C C k g v 17 ' : ) exp( cos ) 1 ( 2 1 C C k g v : ) exp( ) 1 ( 2 1 2 1 C C k SC M v z v 18 ' : ) exp( ) 1 ( 2 1 2 1 C C k SC M v z v 2.7.2. State-Space Representation in Acceleration Mode w v V V Using (2), (
2 7 ' 6 ' 5 ' 1 ' 4 ' 3 ' 2 ' 1 ' ) ( ) exp( a v w v w v w v m w V V V V V V V V M V  ) exp( ) ( ) exp( ) ( ' 2 10 ' ' 9 ' 2 8 ' w v a v w v a v w v V V V V V V V V V V (29a) 2 15 ' 14 ' 13 ' 1 ' 4 ' 3 ' 2 ' 2 12 ' 11 ' ) ( ) exp( ) ( a v w v w v w v a v v V V V V V V V V V V V  ) exp( ) ( ) exp( ) ( ' 2 18 ' ' 17 ' 2 16 ' w v a v w v a v w v V V V V V V V V V V (29b)
where the various parameters are defined in Table II. Let us introduce the notations:

2 2 1 2 8 ' 2 2 7 ' 1 2 6 ' 5 ' 1 1 2 ' 4 ' 1 2 3 ' 2 ' 2 1 ' 1 ) ( ) ( ) exp( ) , ( a a V x x x V x x x x x x x x x f ) exp( ) ( ) exp( 1 2 ' 2 2 10 ' 1 2 ' 9 ' x x V x x x a (30a) 2 2 15 ' 1 2 14 ' 13 ' 1 1 2 ' 4 ' 1 2 3 ' 2 ' 2 2 12 ' 11 ' 2 1 ' 2 ) ( ) exp( ) ( ) , ( a a V x x x x x x x V x x x f ) exp( ) ( ) exp( ) ( 1 2 ' 2 2 18 ' 1 2 ' 17 ' 2 2 1 2 16 ' x x V x x x V x x x a a (30b)
where 1 , x u and 2

x are as in (27a-b). With these notations, the state-space representation (30a-b) is given the following more compact form: ,  (

) , ( ) , ( 2 1 ' 2 2 2 1 ' 1 ' 1 1 x x f x x x f u x   (
2 1 2 2 2 1 1 2 1 * 1 1 x x g x x x g u x x x   (32) with: ) , ( ) , ( 1 ) , ( ) , ( ) , ( 2 1 ' 1 2 1 2 1 1 2 1 2 1 1 x x f x x x x f x x x x g (33a) ) , ( ) , ( 1 ) , ( ) , ( ) , ( 2 1 ' 2 2 1 2 1 2 2 1 2 1 2 x x f x x x x f x x x x g (33b) ' 1 2 1 1 2 1 2 1 * 1 ) , ( 1 ) , ( ) , ( x x x x x x (33c) 2 ) ( 1 ) , ( 2 1 2 1 x x sign x x (33d)
In addition to vehicle aerodynamics, this model does account for tire-road contact effect. Furthermore, it will prove to be useful in control design (Section III).

Control-Oriented Model Ignoring Tire Dynamics

To better appreciate the benefit of accounting of tire-road contact, a comparison will be performed in Section IV between the controller obtained from ( 32)-( 33) and the one obtained from a simplified model neglecting tire-road contact. Specifically, the simplified model is obtained letting 0 (tire sliding negligence) which immediately implies that

w v V V (i.e. 2 1 x x
). Ignoring also the rolling resistance rr , one gets invoking the dynamic fundamentals principles for translation and rotation:

eff tf m eff w r F M J r V  ( 34
) ) sin ( 1 aex v tf v v F g M F M V  (35)
where aex F is defined by ( 16). It is readily seen from ( 34)-( 35) that the vehicle speed undergoes the following equation:

2 2 2 2 ) ( 2 1 sin a v x v v eff v eff m v eff eff v V V SC M g M r J M r M M r J r V  (36)
where the different parameters are defined in Table V. Introduce the following notations:

v eff eff M r J r 2 (37a) 2 2 2 2 2 ) ( 2 1 sin ) ( a x v v eff v eff V x SC M g M r J M r x f (37b)
Then, equation ( 36) can be given by the following more compact form where

m M u and v V x 2 : ) ( 2 2 x f u x  (38)
The first-order equation ( 38) is a simplified version of the second-order model ( 32)-( 33). As already mentioned, both models will be based upon in control design and the obtained controllers will be compared later in Section IV.

Simulation Model

Obvious physical considerations show that in real-life longitudinal motions, the vehicle and wheel speeds are always quite close to each other. On the other hand, there is no guarantee that the model ( 32)-( 33) ensures always that 2 1 x x . In the next lines, the model will be slightly modified so that , with:

1 2 ' 4 ' 1 2 3 ' 2 ' 2 1 ' 2 1 4 2 1 3 2 2 1 exp ) , ( exp ) , ( x x x x x x x x x x x x (39)
Indeed, it is readily seen that the functions and ' are continuous and:

0 exp ) , ( 0 exp ) , ( ' 4 ' 3 ' 2 ' 2 1 ' 4 3 2 2 1 x x x x if 2 1 x x
The continuity of and ' then guarantees the existence of 0 h and 1 0 l , such that:

0 ) , ( inf 0 ) , ( inf 2 1 ' 1 1 2 1 1 1 2 1 2 1 x x x x h l x x x x (40)
As a matter of fact, the size of h and l depends on the parameters As it accounts for the prior knowledge (47), the new model ( 52) turns out to be more accurate than the models (32)-( 33) and ( 38). However, model ( 52) is too complex to be used for control design, due e.g.

to the nonsmooth nature of the function ) (z r defined by ( 51). Therefore, that model can (and presently will) only be used to build up simulators of the vehicle longitudinal motion. Model ( 32)-( 33) is a quite satisfactory compromise between model (38) (simpler but less accurate) and model ( 52) (more accurate but more complex).

Example 1. To illustrate previous and forthcoming results, the example of a Citroën-2CV car, with the characteristics of Table VI, will be considered throughout in the rest of the paper. Using Table II and

VEHICLE LONGITUDINAL REGULATORS DESIGN

Reference Trajectory Generation

We seek a speed controller for the vehicle moving in longitudinal acceleration/deceleration modes.

The regulator design is based on the vehicle model ( 32) and the control objective is to enforce the vehicle chassis and wheel speeds,

w V x 1 and v V x 2
, to track their reference trajectories, denoted * w V and * v V , respectively. The latter are required to be time-differentiable. This requirement that can always be complied with by pre-filtering given (non-differentiable) speed setpoints:

d w r w d v r v V s T V V s T V 1 1 1 1 * * (53a) with ) 0 ( * w V 0 ) 0 ( * v V
, where the filter time constant V is let to be of the form:

d v d w V V ) 1 ( * (53b)
where * is a constant (representing the sliding) that is uniquely obtained from (32) letting there

d v V x 2 , d v V x ) 1 ( * 1 and setting 0 2 x  . Doing so, one gets 0 , ) 1 ( * 2 d v d v V V g
, due to (32).

Then (33b) yields:

0 , ) 1 ( , ) 1 ( 1 , ) 1 ( , ) 1 ( * ' 2 * * 2 * d v d v d v d v d v d v d v d v V V f V V V V f V V (53c)
This is an algebraic equation that must be solved to get the adequate value of * for any fixed

d v V . Recall that 1 , 1 , ) 1 ( * d v d v V V .
Then, it is readily checked that the desired speeds

d v d v d w d v V V V V ) 1 ( , , * will be in the validity domain v D if * belongs to the interval h h l 1 , .
Example 2. turns out to be % 10 %, 7 .

km/h) d v V V a = 0, θ = 0° V a = 10, θ =0° V a =

Speed Control Law Design

As the control objective is to enforce the vehicle speeds ( w V , v V ) to track their reference trajectories

( * w V , * v V
), let us introduce the following tracking errors:

* 2 2 * 1 1 v def w def V x z V x z (54)
From (32) it readily follows that the errors undergo the following equations:

* 2 1 2 2 * 2 1 1 2 1 * 1 1 , , ) , ( v w V x x g z V x x g u x x z     (55)
Let us consider the following positive definite Lyapunov function candidate:

2 2 1 1 2 1 z c z c ) z , z ( V ( 56
)
where 2 1 , c c are any positive design parameters. Derive )) ( ), ( (

2 1 t z t z V
with respect to time yields:

2 2 2 1 1 1 ) ( ) ( z z sign c z z sign c V    * 2 1 2 2 2 * 2 1 1 2 1 * 1 1 1 , ) ( , ) , ( ) ( v w V x x g z sign c V x x g u x x z sign c   * 2 1 2 2 1 2 * 2 1 1 1 2 1 * 1 1 1 ) , ( ) ( ) ( ) , ( ) , ( ) ( v w V x x g z sign z sign c V x x g c u x x c z sign   (57)
where equations ( 44) have been used in the second equality. Equation ( 46) suggests the following control law:

* 2 1 2 2 1 2 1 * 2 1 1 1 2 1 * 1 1 ) , ( ) ( ) ( ) ( ) , ( ) , ( 1 v w V x x g z sign z sign c z sign V c V x x g c x x c u   (58) with 0 c
is a new design parameter. Indeed, substituting the right side of (58) to u in (57) yields

V c V 
which in turn implies:

t c e ) 0 ( V ) t ( V (59)
Equation ( 48) holds for any 0 t provided that (58) has been applied over the interval ) , 0 [ t . On the other hand, substituting (58) in (32) yields the closed-loop system representation in the ) , ( 2

1 x x coordinates: ) , ( ) ) , ( ) ( )( ( 2 1 2 2 1 * 2 1 2 2 2 1 * 1 1 x x g x c V c V x x g z sign c z sign V c x v w     (60a)
Note that the derivatives * w V  and * v V  in (58) can be obtained using (53a). Specifically:

r v d v v r w d w w T V V V T V V V * * * *   (60b)
Theorem 1 (Main result). Consider the control system, illustrated by Fig. 10, consisting of the statespace model (32) in closed-loop with the regulator defined by the control law (58). Let the regulator parameters be chosen as follows:

* 2 * 1 * ) 0 ( 1 / ) 0 ( 1 ) 1 ( 1 v w l V c V V c V (61a) h v w V c V V c V 1 ) 0 ( 1 / ) 0 ( 1 1 * 2 * 1 * (61b)
for some 1 0 , arbitrarily chosen. Then, one has the following properties:

1) The domain v D defined by ( 41) is an invariant set of the closed-loop system (60a-b). 2) Whatever the initial condition

) 0 ( ) ( ) 0 ( ) 0 ( ) ( ) 0 ( c V V t x c V V c V V t x c V V v v w w , for all ) 0 [ t This in turn gives: 2 * 1 * 2 1 2 * 1 * ) 0 ( ) 0 ( ) ( ) ( ) 0 ( ) 0 ( c V V c V V t x t x c V V c V V v w v w or, equivalently: * 2 * 1 * * 2 1 * 2 * 1 * * ) 0 ( 1 ) 0 ( 1 ) ( ) ( ) 0 ( 1 ) 0 ( 1 v w v w v w v w V c V V c V V V t x t x V c V V c V V V , for all ) , 0 [ t (64)
Using (53b), it follows from (64) that, for all ), it follows that the control signal ) (t u is also bounded proving Part 2a. Part 2b is a direct consequence of (59). This completes the proof of Theorem 3.1 Remarks 1. 1) Theorem 3.1 ensures that the closed-loop system is asymptotically stable (in the ) , ( 21 z z -coordinates) with an attraction region containing the whole validity domain v D . 2) A crucial step in the regulator development was to find out a suitable Lyapunov function. In particular, quadratic like functions turned out to be helpless, due to the particular structure of the controlled system.

) , 0 [ t : * 2 * 1 * 2 1 * 2 * 1 * ) 0 ( 1 ) 0 ( 1 ) 1 ( ) ( ) ( ) 0 ( 1 ) 0 ( 1 ) 1 ( v w v w V c V V c V t x t x V c V V c V which,
3) Inequalities in (61a-b) can easily be fulfilled letting 4) Theorem 1 shows that the regulator (58) performs well when applied to the control design model (32). The question is whether such a good performance is preserved when the regulator is applied to the more accurate model ( 52), see (Figs. 12a). This question is investigated by simulation in Section IV.

Control Law Design Neglecting Tire Dynamics

Presently, the simplified control design model ( 38 From ( 41) it is follows that this error undergoes the following equation:

* 2 ) ( v V x f u z   (66)
Consider the Lyapunov function candidate:

2 ) ( 2 z z V (67)
Using ( 66)-(67), one gets the time-derivative of ) (z

V

with respect to time:

* 2 ) ( v V x f u z V   (68)
This suggests the following control law:

z c V x f u v 2 ) ( 1 * 2  (69)
with 0 c is a design parameter. Indeed, substituting (67) in (68) gives cV V  which shows that V is exponentially vanishing. Hence, the control objective (i.e. 0 z ) is ensured if the control law ( 69) is applied to the simplified model ( 38). The question is whether such a good performance is preserved when the control law ( 69) is applied to the more realistic model (52) (Fig. 12b). This question is investigated in Section IV.

Further Discussion on Practical Implementation

First, notice that the control action generated by a vehicle longitudinal controller (like (58)) constitutes the torque reference signal for the engine torque regulator (not discussed in this study). Accordingly, the longitudinal speed regulator and the engine torque regulator constitute together a more global cascade controller (Fig. 11). In cascade control jargon, the torque regulator is referred to "inner" or "slave" while the speed regulator is called "outer" or "master". For the global cascade controller to perform well the torque regulator must be designed so that the inner loop is much more rapid than the outer loop. The previous remarks are independent of the engine technological nature (thermal, electric or hybrid).

Recall also that the analytical design of the (outer) longitudinal speed regulator (58) (developed in Section III) relies on the longitudinal vehicle model ( 32)-( 33). The numerical values of the involved parameters depend on the particular vehicle under study (for instance, Table VI shows those of a Citroën-2CV). They should normally be provided by the vehicle manufacturer; otherwise they must be determined using model identification methods.

The controller design parameters ) , , , ( The practical application of the longitudinal speed regulator necessitates the numerical implementation of the control law (58) and speed measurements (

w V x 1 and v V x 2
). As the controlled system have relatively slow dynamics (due to its mechanical nature) a sampling frequency of 100 Hz would be convenient for data acquisition. Following the usual practice, the conditioning of data acquisition is made better if the signals are properly processed before sampling. Online pre-processing operations include signal filtering, amplification, modulation, demodulation etc. Given the simplicity of involved online algebraic operations (in control law and signal processing), the small number of online input/output measurements and the relatively low data-acquisition frequency, the required computation resources are relatively modest. A low-cost DSP and any one of today"s microprocessors or programmable logic controllers (PLC) would be sufficient for real-time implementation of the control algorithm. The major numerical implementation tasks are described by Table V. As mentioned previously, there is no simple way to find the best choice for the design parameters i.e. . Following the usual practice, suitable numerical values are selected using the heuristic "try-an-error" search method. Accordingly, the next values have been obtained:

. the model reference time constant in (53a) is set to s T r 1 , . the design parameters . the best choice of the design parameter c in the controller (69) turned out to be 2 c .

Control Performances in Easy Driving Conditions

Presently, the control performances of both controllers are illustrated in ideal driving conditions characterized by dry and flat road and weak front wind. Specifically, the driving conditions are defined

C C C .
Furthermore, it is supposed that all above values are perfectly known to the designer and used in the controller design. Figs. 13a-e show the control system responses obtained with both controllers in these easy driving conditions. For both, the speed responses, v V and w V , converge to their respective references (with settling time less than 1.75s) and present no overshoot in the acceleration mode (Figs.

13a-b). But, as expected, the performances of the sophisticated controller (58) are clearly better than

CONCLUSION

The problem of vehicle longitudinal control is addressed, both in acceleration and deceleration modes, based on the two new models defined by ( 32) and ( 52). The originality of these models lies in the fact that they explicitly accounts for the longitudinal slip (resulting from tire deformation) using Kiencke"s

Fig. 1 .

 1 Fig. 1. Degrees of freedom of a vehicle

  4

  Fig. 3. Kiencke"s wheel model

  Fig. 4. Rolling resistance

Fig. 5 .

 5 Fig. 5. One-wheel vehicle model Fig. 6. Schematic representation of a one-wheel vehicle submitted to a driving couple

Fig. 7 .

 7 Fig. 7. The forces acting on a bicycle type vehicle

  property of it. Doing so, one also will discard any risk of singularity in (27a-b) or (30a-b) by ensuring that 0

Fig. 8 .Figs.

 8 Fig. 8. Synoptic scheme of the simulation vehicle model (52)

  speed setpoints. The above references cannot both be freely chosen because they are linked by (32). As a matter of fact, the desired reference of vehicle speed d v V is first chosen by the user. Then, the wheel reference d w

Fig. 10 .

 10 Fig. 10. Theoretical control system analyzed in Theorem 1. It include the vehicle longitudinal model (32) in closed-loop with the regulator (58)

  ) is based upon. The control objective is to enforce

  suitable numerical values before online running of the control algorithm. As shown by simulation (Section IV), suitable values can simply be selected following the usual "try-an-error" search method.

Fig 13c .FigFig. 17 .

 13c17 Fig 13c.Control signal u . Solid: control signal with regulator (58). Dashed: control signal with regulator(69)

TABLE I .

 I NOTATIONS OF VEHICLE LONGITUDINAL MODEL.

TABLE II .

 II DEFINITION OF THE MODEL PARAMETERS

	Vehicle parameters in deceleration	Vehicle parameters in acceleration
	:	2

31) 2.8 Control-and Simulation-Oriented Models for Two-Wheel Vehicle With Single Driving Wheel

	2.8.1 Control-Oriented Models
	2.8.1.1 Control-Oriented Model Accounting For Tire Dynamics
	Combining the mode-dependent state space representations (28) and (31), one gets a single model
	representing the vehicle in all operation modes i.e.
			)	(	,	)
	(	,	)

Table VI

 VI

	, one gets the numerical values of the parameters in the functions	and ' defined by (39),
	see Table III.
	Fig. 9a that	(	, 1 x x	2	)	0	if	0.93	1 x	1	. On the other hand, Fig. 9b shows that	'	(	, 1 x 2 x	)	0
									2 x

It is readily seen, by simple checking, that conditions (40a) are fulfilled. Furthermore, it is seen from

  Table IV shows the values of the parameter

* , obtained by solving (53c), for different values of d v V in both acceleration and deceleration modes. It is readily observed that * belongs to the

TABLE IV .

 IV VALUE OF PARAMETER * (%) IN DIFFERENT OPERATING CONDITIONS (ALL SPEEDS ARE IN

TABLE VI .

 VI NUMERICAL CHARACTERISTICS OF THE CITROËN-2CV CAR(FROM CHAIBET, 2006)

	AD : Asphalt Dry, CW : Cobblestone Wet	AD	CW
	C 1	: Primary tire parameter	1.2801	0.5
	C 2	: Primary tire parameter	23.99	30
	C 3	: Primary tire parameter	0.52	0.2
	M v	: Chassis mass	560 kg
	J	: Wheel inertia	1000 kg m 2
	r eff	: Effective wheel radius	0.28 m
	rr	: Rolling resistance coefficient	0.025
		: Related height of the center of gravity	0.2
		: Related position of center of gravity	0.43
	K v	: Load correction factor	0.55
		: Density of air	1.202 kg/m 3
	C x	: Aerodynamic drag coefficient	0.5
	C z	: Aerodynamic lift coefficient	0.259
	S	: Frontal area vehicle	0.8 m 2

2 2 2 2

 (46)where (.) sat denotes the saturation function defined by:

It is readily seen from ( 46) that the auxiliary variable It is easily checked using (47) that (.) r is the unit rectangular function defined by:

Putting the second equation in (42) together with ( 48)-( 49), one gets the new model:

TABLE V. CONTROLLER IMPLEMENTATION ALGORITHM

Step #0# Choice of design parameter and sampling time

Step #1# Acquisition of parameters and states variables )

Step #4# Apply the control value to the engine

and Go to step #1#

SIMULATION

The performances of the sophisticated controller (58), obtained from model (32), will be compared using numerical simulations, to those of the simpler controller (69), obtained from the simpler and less accurate model (38). Both controllers are applied to the most accurate (simulation-oriented) vehicle longitudinal model (52), that not only account for tire dynamics but also for the validity domain ( 41) so that the model operation is singularity-free. That is, the simulation study will be performed, with

Matlab-Simulink, according to the experimental setups illustrated by Figs 12a and 12b, respectively.

The characteristics of the vehicle are those of a Citroën-2CV of Table VI.

Engine torque regulator

Chassis & wheel speed measurements those of (69). Indeed, the former is much more speedier, develops a smaller control effort (Fig. 13c-d) and ensures a weaker sliding (Fig. 13e).

Control Performances in Hard Driving Conditions

Presently, the driving conditions are much harder compared to the previous subsection. That is, the road is given the profile of Fig. 14 showing an ascendant stage followed immediately by a descent stage. Analytically, the varying road slop is defined as follows: . However, the deviation is much larger with the simpler controller (69) than with the sophisticated one (58). This fact is better illustrated by Fig. 16 that show the behavior of sliding is better with controller (58). Fig. 17 shows the wheel torque developed by both controllers. Clearly, the regulator (69) develops a huge torque (on the wheel) during the ascendant stage. In the descendant stage, the developed torque decays drastically taking small values. As a matter of fact, such behavior is practically unacceptable because it is harmful for (person and vehicle) safety and is costly from an energetic viewpoint. Clearly, the regulator (58) is better. The supremacy of (58) over ( 69) is now quantified considering the average torque and frictional work done by the front tires. These are respectively defined as follows:

where u is the torque applied to the wheels, tf F is the component of the frictional force projected onto the tangential direction of the contact surface, w V is the speed of contact tire-road and is the entire simulation time interval. Fig. 18a shows the variation of the average torque and frictional work with the maximum slop max . The comparison between the two controllers is dealt with considering two road states i.e.:

-Asphalt Dry ) 0.52 23.99, , 2801 . 1 ( ) , , (

The figure indicates that, in all driving conditions, the average torque of the vehicle being driven by the regulator ( 58) is much smaller than with the regulator (69). This confirms that energy consumption is lower with (58) and safety is better. From Fig. 18b it is seen that tire friction activity is much weaker when the vehicle is driven with the regulator (58), especially when the vehicle is going through a bump by cobblestone wet road condition. Then, a high tire friction activity is involved with the controller (69) which means that the tires are then subject high pressure. 52) also accounts for real-life considerations that physically limit the longitudinal slip value. Consequently, (52) turns out to be quite suitable for simulating the vehicle longitudinal behavior. The control design is based on the slightly simpler model (32) which ignores the slip limitation but still accounts for the tire longitudinal slip. It is shown that a speed controller can actually be obtained from that model using a Lyapunov type design technique. It is formally shown that the obtained controller, defined by (58), does meet its performances i.e. stability and perfect speed reference. It is also checked by simulation that the controller ( 58) is much better than the simpler one defined by ( 69) and ignoring longitudinal slip. The supremacy of ( 58) is especially appreciated in hard driving conditions i.e. crossing rampant and wet roads. In all simulated experiments, the vehicle longitudinal motion is represented by the most accurate and highly nonlinear model ( 52).


Abderrahim Elfadili
email: elfadili_abderrahim@yahoo.fr

Fouad Giri

Abdelmounime El Magri

Luc Dugard

Fatima Zara Chaoui

A El Fadili

Adaptive Nonlinear Control of Induction Motors through AC/DC/AC Converters

Keywords: Induction machines, speed and flux regulation, AC/DC/AC converters, PFC, Lyapunov methods, backstepping design technique

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

INTRODUCTION

It is widely recognized that the induction motor has become a main actuator for industrial purposes. Indeed, as compared to the DC machine, it provides a better power/mass ratio, simpler maintenance (as it includes no mechanical commutators) and a relatively lower cost. However, the problem of controlling induction motors is more complex because these are multivariable and highly nonlinear systems and some of their parameters are time-varying. From the technological viewpoint, a considerable progress has been made in power electronics over the last two decades. Reliable power converters have become available which makes technically possible flexible speed variation of electrical drives including induction machines. Accordingly, the speed variation of these machines is carried out by acting on the supply network frequency. Similarly, before the recent progress in modern power electronics, there was no effective and flexible way to vary the frequency of a supply network. In this respect, recall that the power networks may be either DC or AC but mono-phase (this is for instance the case in the electric traction domain). Therefore, three-phase DC/AC inverters turn out to be the only possible interface (between railway networks and 3-phase AC motors) due to their high capability to ensure flexible voltage and frequency variation.

The control problem at hand is to design a speed controller for the complex system including an AC/DC rectifier, a DC/AC inverter and an induction motor. The controller is expected to ensure a wide range speed regulation, despite mechanical parameters' variations. Indeed, the rotor inertia moment, the viscous friction coefficient and the load torque are presently allowed to be unknown and step-like varying.

The rectifier is (of?) a pulse width modulation (PWM ) type that features power regeneration, controlled DC-link voltage, small filter, 4-quadrant operation (bidirectional power transmission).

Nowadays, the use of PWM rectifiers has considerably spread in industry and concerns a wide range spectrum of applications, [START_REF] Xiong Jian | Simplified Control Circuit of Three Phase PWM Rectifier[END_REF], [START_REF] Marques | A PWM Rectifier Control System with DC Current Control Based on the Space Vector Modulation and Stabization[END_REF]. The point is that the association 'rectifier-inverter-motor' acts as a highly nonlinear load vis-à-vis to the main power AC network (which is supposed to provide almost perfect sinusoidal voltage to all other connected loads). Therefore, undesirable higher harmonics are generated (due to the load nonlinearity) and pollute the supply network. This harmonic pollution reduces the rectifier efficiency, induces voltage distortion in the AC supply network and induces electromagnetic compatibility problems [START_REF] Xiong Jian | Simplified Control Circuit of Three Phase PWM Rectifier[END_REF].

In the light of the above considerations, it turns out that the control objective must not only be motor speed regulation but also the current harmonics rejection through power factor correction (PFC), [START_REF] Singh | Improved Power Quality AC-DC Converter for Electric Multiple Units in Electric Traction[END_REF].

In most previous works on induction machine speed control, the control problem was simplified by ignoring the dynamics of the AC/DC/AC converters. Accordingly, the machine is supposed to be directly controlled by stator voltages. The simplified control problem has been dealt with using several control strategies ranging from simple techniques, e.g. field-oriented control [START_REF] Ortega | On Speed Control of Induction Motors[END_REF], to more sophisticated nonlinear approaches, e.g. passivity control [START_REF] Duarte-Mermoud | Induction motor control based on adaptive passivity[END_REF], direct torque control [START_REF] Jasinski | Direct Control for AC/DC/AC Converter-Fed Induction Motor With Active Filtering Function[END_REF], [START_REF] Barambones | Vector Control for Induction Motor Drives Based on Adaptive Variable Structure Control Algorithm[END_REF], or sliding mode control [START_REF] Traoré | Sensorless Induction Motor: High Order Sliding Mode Controller and Adaptive Interconnected Observer[END_REF]. A control strategy that ignores the presence of the AC/DC rectifier suffers at least from two main drawbacks. First, the controller design relies on the assumption that the DC voltage (provided by the AC/DC rectifier) is perfectly regulated; the point is that perfect regulation of the rectifier output voltage cannot be ensured when ignoring the rectifier load which is nothing but the 'inverter-motor' set. The second drawback is concerned by the entire negligence of the PFC requirement. That is, from a control viewpoint, it is not judicious to consider separately the invertermotor association, on one hand, and the power rectifier, on the other hand.

In the present work, we will develop a new control strategy that simultaneously accounts for all system components i.e. the AC/DC rectifier and the association 'DC/AC inverter-motor'. Our control strategy is featured by its multi-loops nature. First, a current loop is designed so that the coupling between the power supply network and the AC/DC rectifier operates with a unitary power factor.

Then, a second loop is designed to regulate the output voltage of the AC/DC rectifier so that the DClink between the rectifier and the inverter keeps on a constant voltage despite changes in the motor operation conditions. Finally, a bi-variable loop is designed to make the motor velocity track its varying reference value and regulate the rotor flux norm to its nominal value. All control loops are designed using the Lyapunov and backstepping techniques [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF]. Interestingly, the load torque, rotor inertia and friction coefficient are presently allowed to be unknown parameters subject to step-like changes. This parametric uncertainty is coped with, by providing the controller with a parameter adaptation capability. It will be formally proved that the proposed multi-loop nonlinear adaptive controller actually stabilizes (globally and asymptotically) the controlled system and meets its tracking objectives with a good accuracy. Specifically, the motor speed and rotor flux norm will be shown to perfectly track their references, whatever the initial conditions. The rectifier input current and output voltage are shown to match well their reference values. More precisely, the steady-state tracking errors for these two variables are both harmonic signals with amplitudes depending, among others, on the supply network frequency. These theoretical results are obtained by making a judicious use of adequate control theory tools, e.g. averaging theory and Lyapunov stability [START_REF] Khalil | Nonlinear systems[END_REF].

The paper is organized as follows: the system under study (i.e. the AC/DC/AC converter and induction motor association) is modeled and given a state space representation in Section 2; the controller design and the closed-loop system analysis are presented in Section 3; the controller performances are illustrated through numerical simulations in Section 4. For convenience, the main notations used throughout the paper are described in Table II given at the end of the paper.

MODELING 'AC/DC/AC CONVERTER-INDUCTION MOTOR' ASSOCIATION

The controlled system scheme is represented by Fig 1 . It includes an AC/DC boost rectifier, on one hand, and an 'inverter-induction motor' combination, on the other hand. The inverter is a DC/AC converter operating, like the AC/DC rectifier, according to the known pulse width modulation (PWM) principle.

AC/DC Rectifier Modeling

The power supply network is connected to an H-bridge converter consisting of four IGBT's with antiparallel diodes for bidirectional power flow mode. The converter should be controlled so that two main tasks are accomplished: (i) providing a constant DC link voltage; (ii) ensuring an almost unitary power factor connection with the power network. Applying Kirchhoff's laws, this subsystem is described by the following set of differential equations: }. Specifically:

       ON is S and OFF is S if OFF is S and ON is S if s 1 1 (1c)
The above (instantaneous) model describes accurately the physical inverter. Then, it is based upon to build up converter simulators. However, it is not suitable for control design due to the switched nature of the control input s . As a matter of fact, most existing nonlinear control approaches apply to systems with continuous control inputs. Therefore, the control design for the above converter will be performed using the following average version of (1a-b) [START_REF] Sira | Control Design Techniques in Power Electronics Devices[END_REF]:

2 1 1 1 1 1 x u L L v dt dx e   (2a) s i C x u C dt dx 2 1 2 1 1 1 2   (2b)
where:

e i x  1 , dc v x  2 , 1 u = s (2c)
are the average values over the cutting periods of e i , dc v and s , respectively.

Inverter-Motor Modeling

The induction model is based on the motor equations in the rotating  -and- axes and reads as:

J T i i J m J f dt d L r s r s        ) (       (3a)         s s r r s v m i bp ba dt di 1      (3b)         s s r r s v m i bp ba dt di 1      (3c)        r s sr r r p i aM a dt d      (3d)        r s sr r r p i aM a dt d      (3e)
where

 s i ,  s i ,   r ,   r ,  ,
and, L T , are the stator currents, rotor fluxes, angular speed, and load torque, respectively. Wherever they come in, the subscripts s and r refer to the stator and rotor, respectively. That is, p designates the number of pole-pairs, J the inertia of the motor-load set, and f is the friction coefficient. The remaining parameters are defined as follows:

r r L R a  , r s sr L L M b   , 2 2 2 / ) ( r s r sr s r L L R M R L     , ) / ( 1 2 r s sr L L M    , r sr L M p m  , s L m  1 1  . In (3a-e),   s s v v ,
are the stator voltage in the  -coordinates (Park's transformation of the three phase stator voltages). The inverter is featured by the fact that the stator α-and β-voltages can be controlled independently. To this end, these voltages are expressed in function of the corresponding control action (see e.g. [START_REF] Ryan | Modeling of Sinewave Inverters: A Geometric Approach[END_REF]): . The latter are defined by (1c) replacing there ( S , ' S ) by

( i S , ' i S ) ( 3 , 2 , 1  i ).
Now, let us introduce the state variables:

  3 x ,  s i x  4 ,  s i x  5 ,   r x  6 ,   r x  7 , (4b)
where the bar refers to signal averaging over cutting periods (just as in (2c)). Using the power conservation principle, the power absorbed by the DC/AC inverter is given by the usual expression

s ai i x P 2 
. On the other hand, the power released by the inverter is given by ) ( , it follows that:

  5 3 4 2 x u x u i s   (4c)
Then, substituting (4a-c) in (3a-e) yields the following state-space representation of the association 'inverter-motor': The state space equations thus obtained are put together to get a state-space model of the whole system including the AC/DC/AC converters and the induction motor. For convenience, the whole system's model is rewritten here for future reference:

J T x x x x J m x J f dt dx L      ) ( 4 7 6 5 3 3 (5a)
2 1 1 1 1 1 x u L L v dt dx e   (6a) ) ( 2 1 2 1 5 3 4 2 1 1 2 x u x u C x u C dt dx    (6b) J T x x x x J m x J f dt dx L      ) ( 4 7 6 5 3 3 (6c) 2 2 1 4 7 3 6 4 x u m x x bpx bax dt dx      (6d)

CONTROLLER DESIGN

Control Objectives

There are two operational control objectives:

AC/DC rectifier control design

Controlling rectifier input current to meet PFC

The PFC objective means that the input current rectifier should be sinusoidal and in phase (or opposite phase) with the AC supply voltage. Therefore, one seeks a regulator that enforces the current 1

x to tack a reference signal * 1 x of the form:

e v k x  * 1 (7)
At this point, k is any real parameter that is allowed to be time-varying. This parameter is positive when the induction machine operates in motor mode and negative in the generator mode. Introduce the current tracking error:

* 1 1 1 x x z   (8)
In view of (6a), the above error undergoes the following equation:

* 1 1 2 1 1 1 / / x L x u L v z e      (9)
To get a stabilizing control law for this first-order system, consider the quadratic Lyapunov function V  is a negative definite function of 1 z if the control input is chosen as follows:

  2 * 1 1 1 1 1 1 / ) / ( x x L v z c L u e     (10) with 0 1  c
is a design parameter. The properties of such a control law are summarized in the following proposition, the proof of which is straightforward. and its first time derivative are available, then one has the following properties:

Proposition 1. Consider the system, next called current (or inner) loop, composed of the current equation (6a) and the control law (10) where

1) The current loop undergoes the equation

1 1 1 z c z    with * 1 1 1 x x z  
. As 1 c is positive, this equation is globally exponentially stable, i.e. 1 z vanishes exponentially, whatever the initial conditions.

2) If in addition k converges (to a finite value), then the PFC requirement is asymptotically fulfilled in average i.e. the (average) input current 1

x tends (exponentially fast) to its reference e v k as

  t 

DC link voltage regulation

The aim is now to design a tuning law for the ratio k in (7) so that the rectifier output voltage x . This is the subject of the following proposition.

Proposition 2. Consider the power rectifier described by (6a-b) together with the control law [START_REF] Traoré | Sensorless Induction Motor: High Order Sliding Mode Controller and Adaptive Interconnected Observer[END_REF].

Under the same assumptions as in Proposition 1, one has the following properties:

1) The output voltage 2 x varies, in response to the tuning ratio k , according to the equation:

) ( 2 1 ) ( 2 1 5 3 4 2 1 2 2 2 x u x u C v z v k x C dt dx e e     (11)
2) The squared voltage (

2 2 x y 
) varies, in response to the tuning ratio k , according to the equation: or, equivalently:

) , ( / / 1 2 t x C v z C v k dt dy e e     (12) with C x u x u x t x / ) ( ) , (
2 1 1 1 x x u v x e  (14)
Also, from ( 7)-( 8), one immediately gets that

1 1 z v k x e   which together with (14) yields 2 1 2 1 1 ) ( x v z v k x u e e  
. This establishes [START_REF] Jasinski | Direct Control for AC/DC/AC Converter-Fed Induction Motor With Active Filtering Function[END_REF] due to (6b). Deriving x y  with respect to time and using [START_REF] Jasinski | Direct Control for AC/DC/AC Converter-Fed Induction Motor With Active Filtering Function[END_REF] yields relation [START_REF] Xiong Jian | Simplified Control Circuit of Three Phase PWM Rectifier[END_REF] and completes the proof of Proposition 2 

The ratio k stands up as a control signal in the first-order system defined by [START_REF] Xiong Jian | Simplified Control Circuit of Three Phase PWM Rectifier[END_REF]. As previously mentioned, the reference signal

e e y t x C t z E C t k E C k E z        ) , ( / ) cos( 2 / ) 2 cos( / 1 2 2 2    (15)
where one used the fact that

) cos( . . 2 t E v e e   and )) 2 cos( 1 ( 2 2 t E v e e   
. To get a stabilizing control law for the system (15), consider the following quadratic Lyapunov function:

2 2 2 5 . 0 z V  (16)
It is easily checked that the time-derivative 2 V  can be made negative definite in the state 2 z by letting:

  ) , ( ) cos( 2 ) 2 cos( 2 2 1 2 2 t x z c C t z E t E k E k e e         ref y C   (17a) where 0 2  c
is a design parameter. The point is that such an equation involves a periodic singularity due to the mutual neutralization of the first two terms on the left side of (17a). To get off this singularity and, besides, to avoid an excessive chattering in the solution, the two terms in cos(.) on the left side of (17a) are ignored, leading to the following approximate simpler solution:

  2 2 2 / ) , ( E t x z c C k     2 / E y C ref   (17b)
Bearing in mind the fact that the first derivative of the control ratio k must be available (Proposition 1), the following filtered version of the above solution is proposed:

  2 2 2 / ) , ( E t x z c C d k d k       2 / E y C d ref   (18)
At this point, the regulator parameters  

2

, c d are any positive real constants. The proof of the forthcoming proposition 3 (?) will make it clear how these should be chosen for the control objectives to be achieved. For now, let us summarize the main findings (?) in the following proposition.

Proposition 3. Consider the inner control loop consisting of the AC/DC rectifier described by (6a-b)

together with the control laws [START_REF] Traoré | Sensorless Induction Motor: High Order Sliding Mode Controller and Adaptive Interconnected Observer[END_REF] and (18). Using Proposition 1 (Part 1), it turns out that the inner loop undergoes, in the

) , , ( 2 1 k z z
-coordinates, the following equation, where

* 1 1 1 x x z   and ref y y z   2 :                                                                        0 ) cos( 2 ) 2 cos( 0 ) , ( 1
0 0 0 0 0 0 1 2 2 2 1 2 2 2 1 2 1 t z E C t k C E y t x E C d k z z d c E C d C E c k z z e e ref        (19) 

Motor speed and rotor flux norm regulation

The problem of controlling the rotor speed and flux norm is now addressed for the induction machine described by (5a-e). The speed reference ref  is any bounded and derivable function of time and its two first derivatives are available and bounded. These properties can always be achieved filtering the reference through second-order linear filters. The flux reference ref  is fixed to its nominal value. The controller design will now be performed in two steps using the tuning-functions backstepping adaptive technique [START_REF] Krstic | Nonlinear and Adaptive Control Design[END_REF]. First, introduce the tracking errors:

3 3 x z ref    (20) ) ( 2 7 2 6 2 ref 4 x x z     (21)
Step 1. It follows from (5a) and (5d-e) that the errors 3 z and 4 z undergo the differential equations:

J x f J T J x x x x m z L ref / / / ) ( 3 4 7 5 6 3         (22) ) ( 2 2 5 7 4 6 ref ref 4 x x x x aM z sr        ) ( 2 4 2 ref z a    (23)
In ( 22) and ( 23), the quantities ) (

) ( ) ( 3 3 3 1 z f T z c J ref L ref def          (24a)
On the other hand, the fact J , L T and f are unknown suggests the certainty equivalence from of equations (24a).

)  , respectively. Nevertheless, we retain the expressions of 1  and 1  as first stabilizing functions and introduce the new errors:

) ( ) ( ˆ3 3 3 1 z f T z c J ref L ref def          (24b) ) ( 2 2 4 2 ref ref ref 4 4 1 z a z c def          (25)
) ( 4 7 5 6 1 5 x x x x m z     (28) ) ( 2 5 7 4 6 1 6 x x x x aM z sr     (29)
Then, using the notations (24) to (29), the dynamics of the errors 3 z and 4 z , initially described by ( 22)-( 23), can be rewritten as follows:

J z J x f J T J z c J z c z L ref / / / / ) ( ~5 3 3 3 3 3 3           (30) 6 4 4 4 z z c z     (31)
where

J J J   , L L L T T T   and f f f   (32)
Step 2. The second design step consists in choosing the actual control signals, 2 u and 3 u , so that all errors ( 3 z , 4 z , 5 z , 6 z ) converge to zero. To this end, we should make (?) how these errors depend on the actual control signals ( 2u , 3 u ). We start focusing on 5 z ; it follows from (28) that:

) ( 4 7 4 7 5 6 5 6 1 5 x x x x x x x x m z             (33)
Assume that the parameters J , L T and f are constant or slowly time-varying and using (5a-e), (32) and (24), one gets from (33):

) ( / ) ( 3 6 2 7 2 1 5 3 2 5 u x u x x mm J z f J c z                               3 3 3 3 3 3 3 ) ( ) ( ) ( x f T z c J x J f J T z c J J f J c L ref L ref      (34) with   ) ( ) ( ) )( ( ) (
x x mbpx x x x x mpx x x x x a m f J f J c z c ref ref                    (35)
Similarly, it follows from (29) that 6 z undergoes the following differential equation:

) ( 2 5 7 5 7 4 6 4 6 1 6 x x x x x x x x aM z sr             (36)
Using (5a-e) and (25), it follows from (36):

) ( 2
                   ) ( 2 ) ( ) ( 2 ) )( ( 2
         (38)
To analyze the error system, composed of equations (30-31), ( 34) and (37-38), let us consider the following augmented Lyapunov function candidate: ) is:

J f J T J J z z z z V L /
J T T J f f J J J z z z z z z z z V L L / / / 6 6 5 5 4 4 3 3 4                (40)
Using (30), ( 34) and (37) in (40) and adding

z c z c z c z c V       J z f / 2 5  J z z / 5 3    2 5 3 3 3 3 5 3 3 3 ) )(( ( ) ( ~z c z c f J c z z z c J J J ref ref               ) ( ~3 5 3 f J c z z T J T L L                 2 5 3 3 5 3 3 ) ( ~z f J c x z x z f J f    ) ( 2 3 7 2 6 1 2 2 4 6 6 6 u x u x m x aM z z c z sr                             3 3 3 3 6 2 7 2 1 5 3 5 2 5 ) ( ) ( ) ( x f T z c J u x u x x m z c c z L ref      (41)
This suggests the following parameter adaptation laws : 32) and (42), the expressions of J ˆ, L T ˆand f ˆ can be calculated with the following equations:

L T L T     ~ , J J     ~ and f f     ~ (42) erehw ) )( ( ) ( 3 3 3 5 3 3 3 2 5 3 ref ref J z c f J c z z c z z c             (43a) 5 3 3 ) ( z f J c z L T     (43b) 2 5 3 3 5 3 3 ) ( z f J c x z x z f      (43c) from (
J J    ˆ , L T L T    ˆ , f f    ˆ (44)
Substituting the parameter adaptation laws (42) to

1 z z J z J f z c z c z c z c V           ) ( 2 3 7 2 6 1 2 6 6 4 2 6 u x u x m x aM z c z z sr           3 3 3 3 6 2 7 2 1 5 3 5 2 5 ) ( ) ( ) ( x z c u x u x x m z c c z f T ref J L                (45)
where 5 c and 6 c are new positive real design parameters. Equation (45) suggests that the control signals 3 2 , u u must be chosen so that the two quantities between curly brackets (on the right side of (45) are set to zero. Letting these quantities equal to zero and solving the resulting second-order linear equation system with respect to ( 3 2 , u u ), gives the following control law:

                                 2 6 6 4
        (48a)
By using the following inequality: As the right side of (49) is a negative definite function of the state vector ( 3 z , 4 z , 5 z , 6 z ), the closedloop system is globally asymptotically stable [START_REF] Khalil | Nonlinear systems[END_REF]. The result thus established is more precisely formulated in the following proposition: Proposition 4 (Speed regulation). Consider the closed-loop system composed of the induction machine, described by model [START_REF] Ortega | On Speed Control of Induction Motors[END_REF], and the nonlinear controller defined by the control law (46). Then, one has the following properties:

1) The closed-loop error system undergoes, in the ( 3 z , 4 z , 5 z , 6 z ) coordinates, the following equations:

         (51e)     5 3 3 3 5 3 3 3 3 5 3 5 ) ( ) ( ) )( ( ) , , , , ( ~z x f J c J f J T f J c z c z c f J c J J J f J T J J z z L ref def L             (51f) 2)
The above linear system is globally asymptotically stable with respect to the Lyapunov function 4 V .

Consequently, the errors ( 3 z , 4 z , 5 z , 6 z ) vanish exponentially fast, whatever the initial conditions  Remark 1. Note that the exponential nature of stability guarantees stability robustness with respect to modelling and measurements errors [START_REF] Khalil | Nonlinear systems[END_REF].

PFC Achievement

In the following proposition, it is shown that, for a specific class of reference signals, including periodic signals, the control objectives are achieved (in the mean) with an accuracy that depends, among others, on the network frequency e  . The following notations are needed to formulate results:

  T k z z Z 2 1 1  ;   T z z z z Z 6 5 4 3 2  ,   T L J f J T J J / / /   (52a) C E a / 2 0  ; 2 2 1 / E C d c a  ; C E a / 2 2  ; 2 3 / E d a  ; e   / 1  (52b) 7 7 2 ) 3 , 4 ( ) 4 , 3 ( 1              IR A A A (52c)               d a a c A 1 0 1 1 0 0 0 0 0 ;                      6 5 4 3 2 0 1 0 0 ) / ( 0 0 1 0 0 0 / 1 0 c J f c c J c A (52d) ) 4 , 3 ( O , ) 3 , 4 ( O null matrices in 4 3 R and 3 4 R , respectively.     7 1 2 0 0 0 0 0 0 ) cos( ) 2 cos( 0 ) , ( IR t z a t k a t Z f T e e      (52e)   7 3 0 0 0 0 / 1 0 IR a C g T    (52f)   7 2 1 0 0 0 0 / 1 0 IR E C d h T   (52g) T Z Z Z h ] 0 ) , ( 0 ) , ( 0 0 0 [ ) , ( 2 5 2 3 2 2       (52h)
Proposition 5. Consider the system including the AC/DC/AC power converters and the induction motor, connected in tandem as shown in Fig. 1. For control design purpose, the system is represented by its average model (6a-e). The system is set in closed-loop with the adaptive controller defined by the control laws [START_REF] Traoré | Sensorless Induction Motor: High Order Sliding Mode Controller and Adaptive Interconnected Observer[END_REF]18,46) and the parameter adaptive law (44), where all design parameters (i.e. 1) The resulting closed-loop system undergoes the state-space equation:

) , ( ) , ( ) , ( 2 2 1 2       Z h y h t Z g t Z f Z A Z ref    (53)

Simulation Protocol

The nonlinear adaptive controller, developed in Section 3, including the control laws [START_REF] Traoré | Sensorless Induction Motor: High Order Sliding Mode Controller and Adaptive Interconnected Observer[END_REF]18,46) and the parameter adaptive law (42-44), is now evaluated by simulation. The simulated system is given the following characteristics, corresponding to a real-life experimental set-up available in the GIPSA Lab (Grenoble, France):

. Supply network: Fig. 4 shows the changing inertia moment J and viscous friction coefficient f . The variations are 100% of the corresponding nominal values, approaching the physically allowed limits. The nonlinear controller developed in Section 3 will be applied.

) cos( . 2 ) ( t E t v e e

Control performances in absence of model uncertainties

The controller performances are illustrated by Figs 3 and 5 to 8. Fig. 6 shows that the DC-link voltage It is seen that the current amplitude changes whenever the speed reference or load torque vary (Fig. 2). However, the current frequency is insensitive to these changes. Specifically, the current remains (almost) always in phase or in opposed phase with the supply net voltage complying with the PFC requirement. This is further demonstrated by Fig. 5 which shows that the ratio k takes a constant value after transient periods following the changes in speed reference and load torque. The ratio k is negative whenever the network gets back some energy. This confirms Proposition 5 (part 2) and Remark 2c. Figs 2 and8 show that the motor speed and the rotor flux norm match well their respective references (no ripples) despite the variation of mechanical parameters (load torque L T , moment of inertia J and viscous friction coefficient f ). This confirms the result of Proposition 4 and Remark 2a. For both controlled variables ) , (

, the tracking quality is quite satisfactory at high speed as well as at low speed. The response time, after each change in speed reference and load torque, is less than s 0.1 . In this respect, let us note that the global convergence nature of the tracking results in Propositions 4 and 5 guarantee that all transients remain of finite duration, whatever the variable initial conditions and the unknown parameter jumps. Nevertheless, it is recommended to use any available prior knowledge to improve the choice of initial conditions in control laws and in parameter adaptive laws.

Control performances in presence of errors on model nominal parameters

The robustness of the proposed control to modeling errors is now illustrated by considering errors on the nominal values of some parameters. The parameters

CONCLUSION

The problem of controlling associations including AC/DC rectifier, DC/AC inverter and induction motor has been addressed. The system dynamics have been described by the averaged seventh order nonlinear state-space model (6a-g). Based on such a model, the multi-loops nonlinear adaptive controller defined by [START_REF] Traoré | Sensorless Induction Motor: High Order Sliding Mode Controller and Adaptive Interconnected Observer[END_REF]18,42,43,44,46) have been confirmed by a simulation study which further showed additional features of the proposed controller, e.g. its weak sensitivity to electrical model parameters, its supremacy over conventional regulators. To the authors' knowledge, it is the first time that a so complete formal control design and analysis framework is developed for induction motors and associated power converters.

APPENDIX : Proof of proposition 5

Proof. Part 1. Substituting the right sides of (46) to 2 u and 3 u in (13) leads to:

C t Z t x / ) , ( ) , ( 2     (55)
where we have used (52a-g). Substituting the right side of (55) to ), it follows that the system (53) is periodically time-varying. Therefore, the averaging theory turns out to be a suitable framework to analyze its stability (see e.g. [START_REF] Khalil | Nonlinear systems[END_REF]).

                                  0 ) cos( 2 ) 2 cos( 0 / 1 0 ) , ( 1 2 2 2 1 1 1 t z E C t k C E E C d y C t Z Z A Z e e ref
    N y t y ref ref / 2 *   ,     N t ref ref / 2 *     . ( 57b
)
Also, it is easily seen that

) ( / ) ( / ) ( ) ( t Z dt t dZ d dW W           with e   / 1 
. Then, it follows from (53) and using (57b), that the state vector W undergoes the following state equation:

  ) , ( / 2 ) , , ( ) , , ( 2 2
* 1 2 1 1       W h N y h W g W f W A W ref            (58a) with ) , ( ) , , ( 1    W f W f  , ) , ( ) , , ( 2 2 1      W W  (58b)
It readily follows from (52e) that:

    T w a k a W f 0 0 0 0 0 ) cos( ) 2 cos( 0 ) , , ( 1 2 0 1       (58c)
where the following notations are adopted in coherence with (52a):

  T T T W W W 2 1  ,   T w w w W 2 1 1  ,   T w w w w W 6 5 4 3 2  (59)
According to the system averaging theory, one gets stability results regarding the system of interest (58a-b) by analyzing the averaged system defined by:

) , ( ) ( ) ( 2 2 2 1 1      W h W g W f W A W      where 7 IR W  (60a) with        N d W f N 2 1 W f       2 0 1 0 1 , , lim , (60b)
     N d W W         2 0 2 1 0 2 1 , , N 2 1 lim ) ( (60c)      N d W h W h       2 0 2 2 0 2 2 ) , , , ( N 2 1 lim ) , ( (60d)
Note that the last term on the right side of (58a) has not been accounted for in (60a) because its average value is null, due to the periodicity (with period  2 ) of * ref y . From (58c), one has:

    T W f 0 0 0 0 0 0 0 1  (61)
In view of (61) the average system (60a) simplifies to:

) , ( ) ( 2 2 2 1     W h W g W A W     (62)
where the following notations are used in coherence with (59) and (60a):

  T T T W W W 2 1  ,   T w w w W 2 1 1  ,   T w w w w W 6 5 4 3 2  (63)
In view of (52f), the vector g assumes the following partition:

  T T T g g 4 1 0  with   T a C g 3 1 / 1 0   ,   T 0 0 0 0 0 4  (64)
This together with (52d) implies that (60a) can be decomposed in the following two state equations: A is in turn Hurwitz. Its characteristic polynomial is:

) ( 2 1 1 1 1 1 W g W A W       (65a) ) , ( 2
2 2 2 2    W h W A W   (
  2 1 2 1 2 1 3 1 ) ( ) ( det c c d c c d d c A I            (68)
Applying for instance the well known Routh's algebraic criteria, it follows that all zeros of the polynomial (68) have negative real parts if the coefficients ( A is Hurwitz, implying that the autonomous part of the linear system (65a) is globally exponentially stable. Then, one gets from (66) that the solution of the nonautonomous system (65a) satisfies: . Again, the exponential feature of the convergence is due to the linearity of (65a). Combining (66) and (69), it follows that the state vector

Fig. 1 .

 1 Fig.1. AC/DC/AC drive circuit with three-level inverter

  network voltage (with known constants e E  , ) and s is the switch position function taking values in the discrete set { 1 , 1 

L

  are the self-inductances. sr M denotes the mutual inductance between the stator and rotor windings.

  α-and β-axes (Park's transformation) of the three phase duty ratio system

  (i) Speed regulation: the machine speed  must track, as closely as possible, a given reference signal ref  , despite the parametric uncertainties (concerning the load torque L T , rotor inertia J and friction coefficient f ). (ii) PFC requirement: the rectifier input current e i must be sinusoidal and in phase or opposed phase with the AC supply voltage e v .As there are three control inputs at hand, namely 1 u , 2 u and 3 u , two more control objectives are added:(iii) Controlling the continuous voltage dc v making it track a given reference signal

.

  It can be easily checked that the time-derivative 1

  a given reference value dcref v . As mentioned above, dcref v is generally (but not mandatory) set to the nominal value of the stator voltage amplitude. The first step in designing such a tuning law is to establish the relation between the ratio k (control input) and the output voltage 2

1 .

 1 The power absorbed by the AC/DC rectifier is given by the well known expression On the other hand, the power released by the rectifier (toward the load including the capacity and the inverter) is given by

( 2 )

 2 of the squared DC-link voltage dc v x  is chosen to be constant (i.e. 0  ref y ), it is given the nominal value of stator voltage amplitude. Then, it follows from (12) that the tracking error

  ref

  virtual control signals. If these were the actual control signals, the error system (22)-(23) could be globally asymptotically stabilized



  positive design parameters and J ˆ, L T ˆand f ˆ are the estimates of J , L T and f respectively. Indeed, considering the Lyapunov function: This would prove the global asymptotic stability of the system (22)-(25). As the quantities are not the actual control signals, they cannot be let equal to 1  and 1

  time-derivative along the trajectory of the state vector ( 3 z , 4 z , 5 z , 6 z

  vanishes in practice because of the machine remnant flux. Substituting the control law (46) to ( 3 2 , u u ) on the right side of (45) yields:

  nominal value. Then, one has the following properties:

Remarks 2 . 5 (.

 25 positive integer N), and suppose them to be time derivable (up to second order for ref ) with bounded derivatives. Then, there exists a positive real *  such that, a) The motor speed and the rotor flux norm both converge to their respective references as a result of Proposition 4.b) Using Proposition 1 and the fact that the tuning parameter and its time derivative are available, we get that the error to zero. The importance of Proposition 5 (part 2) lies (partly) in the fact that the (time varying) parameter k does converge to a fixed value (up to a harmonic error that depends on e  ). This demonstrates that the PFC requirement is actually fulfilled with an accuracy that depends on e  . The larger e  , the more accurate the PFC quality. It will be seen in the next simulation study that the usual value Part 2) also demonstrates that the tracking objective is achieved (in the mean) for the DCThat is, these signals must vary slower than the network voltage.d) The fact that the tracking error the existence of output ripples. Proposition 5 (Part 2) ensures that the effect of ripples is not significant if e  is sufficiently large. It will be observed through simulations that the value

10 .

 10 phase 220V/50Hz . AC/DC/AC converters: Induction machine: it is a 7.5KW motor whose characteristics are summarized in TableI. The indicated values of design parameters using a 'try-and-error' search method and proved to be suitable.TABLE I. NUMERICAL VALUES OF CONSIDERED MOTOR CHARACTERISTICS is described by Figs. 2 to 4 which show that the reference signals and machineload are profiled so that the machine is enforced to operate, successively, both at high and low speeds and in motor and generator modes.The DC-link voltage reference is set to the constant value flux norm is set to its nominal value (1Wb ).

  and quickly settles down after each change in the speed reference or the load torque. As expected by Proposition 5 and commented on in Remark 2c, the DC link voltage dc v is subject to small amplitude ripples oscillating at the supply network frequency e  . The resulting input current e i is illustrated by Fig 7.

MFig. 2 . 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .

 23456789 Fig. 2. Applied load torque Fig. 3. Rotor speed Ω (rd/s) . Upper: reference and measured speed; lower: speed error control.

  has been designed and analyzed using tools from the Lyapunov stability and averaging theory. It was formally established that the proposed controller achieves the objectives it has been designed for, i.e. (i) an almost unitary power factor; (ii) a tight regulation of DC-link voltage (v dc ); (iii) a satisfactory rotor speed reference tracking over a wide range of mechanical parameter variations; (iv) a tight regulation of the rotor flux norm. These results

Part 2 .

 2 notations (52b-g) have been used. Putting together (56) and (51) in a global state space equation one gets (53) and proves Part 1. As dcref v , and ref  as well as their derivatives are constant or periodic (with period e N  

  positive which is actually the case. Hence, the matrix 1

2 

 2 exponentially stable equilibrium of the average system (62). Now, invoking the averaging theory (e.g. Theorem 10.4 in[START_REF] Khalil | Nonlinear systems[END_REF]), we conclude that there exists a positive real constants *  such that, applies to the differential equation (56) using the relation  , it depends continuously on  and: establishes Part 2-a of Proposition 5.To prove Part 2-b, let us obtain more insight on the equilibrium * W . In coherence with (52a), this is decomposed as follows: is readily checked using (52d) and (64) that: from (52b), (71b), (72), (74) and (75) that: with (76a-b), establishes Part 2-b and completes the proof of Proposition 5  TableII. Main notations : inverse of supply net frequency i.e.