,1IpKJIa).'1;H Ha Hh.rrrapcKaTa aKa,n;eMIDI Ha Hay:KHTe Comptes rendus de l' Academie bulgare des Sciences Tome 62, No 8,2009 SCIENCES ET INGENIERIE Theorie des systemes RELAXED ROBUST STABILITY ANALYSIS Svetoslav Savoy, Ivan Popchev (Submitted on April 14, 2009) Abstract Robust stability of poly topic systems is analysed via affine Lyapunov func- tion (LF). It is shown that when the pairwise inequalities between the entries of the uncertain vector are taken into account less conservative and relaxed conditions are obtained. Key words: affine Lyapunov function, matrix polytope, LMI, robust sta- bility 1. Introduction. Stability analysis of linear systems subjected to struc- tured real parametric uncertainty belonging to a compact vector set has been recognized as a key issue in the analysis of control systems. Robust stability can- not be directly assessed using convex optimization. In order to reduce the gap between quadratic and robust stability, attempts for reducing the conservatism of LMI methods have been made for more than a decade. Aimed at going beyond parameter-independent LFs, LMI techniques were proposed to derive quadratic in the state candidates for Lyapunov functions, which are affine [5,6,10], quadratic [1] and recently polynomial [2-4,9], in the uncertain parameter. Robust stability is verified through convex optimization problems formulated in terms of param- eterized LMIs, which can be efficiently solved by polynomial-time algorithms. The objective of this research is to find computable, less conservative and relaxed robust stability conditions via affine LFs, in a case when the uncertain vector a: belongs to the unit simplex. It is actually motivated by several recently obtained results [5,6,10], aimed at solving the same problem, which exhibit some common shortcomings (sources of conservatism). The main contributions are: (i) This work is supported by the Bulgarian Academy of Sciences under grant No 010077/2007. 1001