An algebraic design for the simultaneous stabilization of two systems

In this report we present a uniied description for studying the problem of the simultaneous stabilization of two plants. Three approaches for the simultaneous sta-bilizability are deened. The rst one corresponds to the deenition commonly used in the literature. For the third one, we show that, like for the rst one, the design of a simultaneous compensator leads to a divisibility condition in the ring of RH 1. A simple formulation of the existence condition for simultaneous stabilization is proposed. Moreover , the equivalence between the existence conditions for the rst and third approaches is shown. Finally an explicit method is given to compute simultaneous compensators. Approche alg ebrique pour la stabilisation simultan ee de deux syst emes. R esum e : Dans ce rapport de recherche nous pr esentons le probl eme de la stabilisation simultan ee de deux syst emes. Nous d eenissons ce probl eme de trois mani eres. La premi ere correspond a la d eenition habituelle pr esent ee dans la litt erature. Nous montrons que pour la troisi eme m ethode ainsi que pour la premi ere, l' etude du compensateur simultan e conduit a une condition de divisibilit e dans RH 1 et a ce propos, nous proposons une nouvelle formulation de la condition d'existence. De plus nous montrons l' equivalence des conditions d'existence pour la premi ere et la troisi eme approche. Finalement une m ethode de synth ese explicite est donn ee aan de construire des correcteurs simultan es pour deux syst emes. Mots-cl e : Stabilisation simultan ee, stabilisation forte, interpolation de fonctions ra-tionnelles unimodulaires dans RH 1 , param etrage de Youla. An algebraic design for the simultaneous stabilization of two systems. 3 Notations. R p s] : The set of proper rational functions with real coeecients. Rs] : The set of rational functions with real coeecients. RH 1 : The set of proper and stable rational functions with real coeecients. H : The set of Hurwitz's polynomials. IRs] : The set of polynomials with real coeecients. R +e : Positive reals including 0 and 1. U : The set of units in RH 1 .