, Application à l'estimation d'état des systèmes à commutation 97
, , p.99
,
,
,
,
, Conditions de rang pour la résolution d'un système d'équations112
Conditions de rang pour la résolution d'un système d'équations ,
Alors l'équation matricielle (A.5) est équivalente avec: Axi =bi pour tout i = 1, ... , p et ces équations sont solvables si et seulement si rang ,
, ===}: Les conditions (A.7) signifient que chaque bi est une combinaison linéaire des colonnes de A et donc par conséquent la condition (A.6) est satisfaite, Nous démontrons que la condition (A.7) est équivalente à la condition (A.6)
,
, , vol.116, p.118
, Nous notons avec x(k,k 0 ,x(k 0 )) la solution à l'instant k du système autonome discret
, Les définitions précédentes restent valables pour les systèmes à temps discret, mais il convient d'apporter certaines modifications concernant les résultats généraux sur la stabilité exponentielle
La trajectoire d'équilibre Xe 0 est exque : llx(k,x(ko),ko)ll:::; allx(ko)IICk-ko Y k, vol.1 ,
, Méthode directe de Lyapunov La méthode directe de Lyapunov permet d'étudier la stabilité d'un système autour d'une trajectoire d'équilibre sans avoir à résoudre l'équation d'état. L'existence d'une fonction V(x,t), appelée fonction de Lyapunov, le signe de cette fonction ainsi que celui de sa dérivée donnent une information sur la stabilité du système, Si Br =Rn alors l'origine est globalement exponentiellement stable. ? B.1.2
, Rn x R+-+ R+ une fonction continue. V est dite propre définie positive si: 1. Y tER+, Y xE Rn
V(x,t) = 0 ==> x= 0 3. Y tER+ ,
, Une fonction V(x,t) de classe C 1 est une fonction de Lyapunov locale (respectivement globale) au sens large pour un système différentiel si elle est propre définie positive et s'il existe un voisinage de l'origine mo tel que: y xE mo (respectivement xE Rn) V(x,t) = âV~:,t) + ( 8 v~:,t)) T f
, Si V(x,t) < 0 alors V est appelée fonction de Lyapunov au sens strict
, Si un système différentiel admet une fonction de Lyapunov locale au sens large (respectivement au sens strict) alors la trajectoire d'équilibre Xe _ 0 est localement stable
, Si la fonction de Lyapunov est globale alors on parle de stabilité globale (respective
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