During the last decades numerous papers dealing with the design of static robust output feedback control schemes to stabilize uncertain systems have been published (Benton and Smith, 1999; Crusius and Trofino, 1999; El Ghaoui and Balakrishnan, 1994; Geromel, De Souza and Skelton, 1998; Henrion, Tarboriech and Garcia, 1999; Kose and Jabbari, 1999; Li Yu and Jian Chu, 1999; Mehdi, Al Hamid and Perrin, 1996; Pogyeon, Young Soo Moon and Wook Hyun Kwon, 1999; Tuan, Apkarian, Hosoe and Tuy, 2000; Veselý, 2002). Various approaches have been used to study the two aspects of the robust stabilization problem, namely conditions under which the linear system described in state space can be stabilized via output feedback and the respective procedure to obtain a stabilizing or robustly stabilizing control law. The necessary and sufficient conditions to stabilize the linear continuous time invariant system via static output feedback can be found in (Kucera, and De Souza, 1995; Veselý, 2001). In the above and other papers, the authors basically conclude that despite the availability of many approaches and numerical algorithms the static output feedback problem is still open. Recently, it has been shown that an extremely wide array of robust controller design problems can be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality (BMI) constraint. The BMI has been introduced in (Goh, Safonov and Papavassilopoulos, 1995). In this paper, the BMI problem of robust controller design with output feedback is reduced to a LMI problem (Boyd et al, 1994). The theory of Linear Matrix Inequalities has been used to design robust output feedback controllers in (Benton and Smith, 1999; Crusius and Trofino, 1999; El Ghaoui and Balakrishnan, 1994; Henrion, Tarboriech and Garcia, 1999; Li Yu and Jian Chu, 1999; Tuan, Apkarian, Hosoe, and Tuy, 2000; Veselý, 2001). Most of the above works present iterative algorithms in which a set of LMI problems are repeated until certain convergence criteria are met. The VK iteration algorithm proposed in (El Ghaoui and Balakrishnan, 1994) is based on an alternative solution of two convex LMI optimization problems obtained by fixing the Lyapunov matrix or the gain controller matrix. This algorithm is guaranteed to converge, but not necessarily, to the global optimum of the problem depending on the starting conditions. The main criticism formulated by control engineers against modern robust analysis and design methods for linear systems concerns the lack of efficient easy to use and systematic numerical tools. This is especially true when analyzing robust stability as affected by highly structured uncertainty with BMI, for which no polynomial-time algorithm has been proposed so far (Henrion, Alzelier and Peaucelle, 2002). This paper is concerned with the class of uncertain linear systems that can be described as x (t)=(A0+ A1θ1+···+ Apθp) x (t)(1) where θ=[θ1... θp]∈ Rp is a vector of uncertain and possibly time varying real parameters. The system represented by (1) is a polytope of linear affine systems which can be described by a list of its vertices x (t)= Acix (t), i= 1, 2,..., N(2) where N= 2p. The system represented by (2) is quadratically stable if and only if there is a common Lyapunov matrix P> 0 such that