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In this work, we propose a solution to the performance problem described by 𝐻 (cid:2998) adaptive observer Here, a linear polytopic varying systems is considered and also submitted to actuator faults. A new convex optimization theorem in terms of linear matrix (LMI) is given. In fact the solution guaranties 𝐻 (cid:2998) performance. To resolve the BMI problem, we introduce a new variable in order separate the Lyapunov function and the observer gain. The corresponding proofs of convergence are developed based on a polyquadratic approach. The efficiency and the performances of the proposed adaptive observer are illustrated on controlling a VTOL air craft.