Abstract— An algorithm of dual identification and control is proposed. The uncertainties affecting the model are assumed to be norm-bounded. A new limited complexity polytope updating approach is presented and used for control. The robust performance obtained in case of the exact polytope or limited complexity polytope is proved to be convergent to the nominal performance for a given model parameters and uncertainties bounds. I. INTRODUCTION N this paper we present an algorithm of dual identification and robust control. Such algorithm was intensively studied in past years and many papers were published in the subject (see for example [1]-[6]). This extensive study is due to the improvements obtained by considering the connections between identification and control. Indeed, various dual algorithms were presented with different structures of controlled model or uncertainties affecting this model, various control objectives or different identification algorithms. In this paper we suppose that the structured and unstructured uncertainties affecting the model are norm bounded leading to to the Unknown But Bounded Error approaches (UBBE) which updates set-membership such as polytopes, orthotopes, parallelotopes, ellipsoids. In particular, we present a new limited complexity polytope updating procedure. Our main objective is to prove that the robust performance obtained by such dual control algorithm converges, in a certain sense, to the nominal robust performance obtained by a given model and known uncertainties bounds. The expression of this nominal performance is presented in section 2. In section 3, we present the set-membership updating, especially a limited number of faces polytope. In section 4,the algorithm of identification and robust control is detailed and the main results are developed. Simulations results are presented in section 5 and concluding remarks in section 6. Notation. Manuscript received October 16, 2006. S. Maraoui is Assistant at the Institut Supérieure de Sciences Appliquées et de Technologies Gabes Tunisia. He is also a Phd student in the Unit of Research (UR) ATSI at the ENIM (email: saber.maraoui@issatgb.rnu.tn ). H. Messaoud is Associate Professor at the Ecole National d'Ingénieur de Monastir (ENIM). He is also responsible of the UR – ATSI (email: hassani.messaoud@enim.rnu.tn ). G. Garcia is Professor at the Institut National de Sciences Appliquées de Toulouse – France. He is member of the LAAS-CNRS, 7 avenue du Colonel Roche, 31077 Toulouse Cédex 4, France, in the group MAC (email: garcia@laas.fr ). x is a sequence of real numbers ( ) { } 0 ∞ = = k x x k , ( ) max ∞ = k x x k , ( ) lim sup →∞ = ss k x x k is a semi-norm, ( ) ( ) { } , , = … t s x x s x t , () () { } max ,..., ∞ = t s x x s x t , α β ∠ is the angle between α and β. II. ROBUST PERFORMANCE A. Plant model The controlled LTI process is described by the following model. ( ) ( ) ( ) ( ) ( ) -1 -1 -1 q yk =qBq uk+ ν k A (1) Where y, u and ν are the output, the control input and the model uncertainty respectively. q -1 is the backward operator. A(q -1 ) and B(q -1 ) are polynomials in q -1 defined by: A(q -1 ) =1 + a 1 q -1 + ... + a n q -n (2) B(q -1 ) = b 0 + b 1 q -1 + … + b m q -m (3) The model uncertainty ν(k) will be supposed of the form. ( ) ( ) ( ) ( ) ν = + + k y u w k w k ek (4) Where w y and w u represent finite memory uncertainty representing the neglected dynamics in the model added to the output and the input respectively ([7]). e(k) is an external perturbation. All these uncertainties are supposed bounded as: 1 1 − −τ ∞ ∞ − −τ ∞ ∞ ∞ ⎧ ≤δ ⎪ ⎪ ≤δ ⎨ ⎪ ≤δ ⎪ ⎩ k y y k k u u k e w y w u e (5) Where τ is the length of the memory of perturbation. The plant model can be rewritten as: ( ) ( ) ( ) =ϕ θ+ν T y k k k (6) ϕ(k) is the regressor vector and θ is the parameters vector given respectively by: ( ) ( ) ( ) ( ) ( ) 1, , , 1, , ϕ = − − − − − − ⎡ ⎤ ⎣ ⎦ … … T k y k y k n uk uk m (7) [ ] 1 0 , , , , , T n m a a b b θ = … … (8) Consider a Tow Degree of Freedom (TDF) controller of the form: Robust Adaptive Control using Limited Complexity Polytopes Saber Maraoui, Hassani Messaoud and Germain Garcia I Soumis à ECC 20007, Kos – Greece, July 2 -5