, CunptrRu 2. STnSTLISATIoN pAR REToun p'ÉtRt

D. and L. C. , il existe un champ de vecteurs Z de classe C*, ffi ) 1, tel que: (rrvl*1t'2v(x):-0, vr e IR") et (r, pp.1-4

Z. Soit and . Le-champ-de-vecteurs-défini-par,

, Ce champ de vecteurs est de classe C^, ffi ) 1 et vérifie Z(r,0):0. De plus

, 1t 2v(x) > 0, vz e IR") et (t'rvç*1-0 <+ L2v(r): 0), p.18

, En effet, comme Y(r,0) :0, alors d'après

. Nous-avons-donc, LyV (r) L 2V (x) : LyV (x)L sV (r) qui entraîne, que: LyV(x)L7V(r) 2 0

. Nous-avons-Également, L2V(r) -0 <-L2V(x) :0, qui implique, en utilisant (2.17), que: LyV(x) -0 <+, pp.2-2

. Maintenant, 0 et qrrc 2 est de classe C-, nous pouvons développer Z(r,u) sous la forme: Z(x,u): u(zo(*) * uzr(ae, p.20

, où Zo et ZL sont des champs de vecteurs de classe C*-1

, Nous définissons à présent deux suites de fonctions (F7,)r2o et (Gt)t>o , de classe C', à partir desquelles sera formulé le premier résultat de cette section

. Soient-(-f, G&)Ëx, définies par: (i) r'(") : V(r), Vc ? IR, ii) F1,(o) : g <+ Ly, p.0

D. 'un-point-de-vue-géométrique, il s'agit de la commande qui permettrait de maintenir l'état du système sur la surface de discontinuité o:0, s'il n'était pas soumis aux perturbations ïetg

, En utilisant (7.7), nous avons

. Si, CB) est non singulière, alors la condition (7.8) entraine que: uc: -Krt(t) -K2r(t -h)

C. La-partie-discontinue-de-la-loi-de-commande,

, (t)l * W2lx (t -lr) l + W3 ) sgn(a

, où les gains l[, W2 et I4l3 sont strictement positifs et sont choisis assez grands pour garantir I'attractivité de la surface de commutation

, Théorème 7

. Soit-le-système, 1) dont les perturbations uérifient les conditions de bornitude (7.2) Alors il apparaît un mode de glissement, si Ia loi d,e command, ) uérifie: Ë'tll9ll qWu knll9ll 1wz et, pp.0-1

B. Annexe and . Lemme,

. Lemme,

, Si g : lR + lR est une fonction continue qui s'annul e en zéro alors elle n'est pas nécessairement décomposable sous la forme: g(y): alk@)

?. Avec-p, . In, and . Co,

, Remarque Sans perte"de généralité, nous pouvons supposer que p : 1 et que g est impair

, En effet, yi peut toujours se mettre sous la forme: al:yf',f'|.:y#yuz+!

. Ainsi-si-l-'écriture, 1) est justifiée, alors g peut aussi se mettre sous la forme: s@): y"È$g@)

\. , ):a' k(a)

, Preuve du lemme D'après la remarque ci-dessus, il suffit de montrer que g n'est pas toujours décomposable sous la forme

, Soit (go)o6ry url? suite de fonctions définie par: ço@) : a# et (ae, p.7

, La fonction généralement utilisée est

, La surface o : 0 est alors attractive sur tout le domaine de fonctionnement )/ si : oà <0

, Stabilité du mode de glissement La surface de glissement étant atteinte

, équation du mode de glissement que nous ne pouvons pas déterminer par les méthodes classiques d'analyse des équations différentielles, en raison de la présence d'un second membre discontinu. La méthode de la commande équivalente (c/. [UTK78, UTK92]) répond à ce problème. Elle consiste à admettre qu'en mode de glissement tout se passe comme si le système était piloté par une commande dite "équivalente, Elle est définie commeétant la commande qui rend la surface o invariante dans le temps, c'est à dire telle que, p.0

, La résolution de cette équation permet d'obtenir z"o qui, reinjectée dans l'équation d'état du système, donne l'équation du mode de glissement: x:f"s:f(t

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