Estabilidade robusta de sistemas lineares através de desigualdades matriciais lineares

Abstract

Sufficient conditions for the analysis of stability of linear systems with polytopic uncertainties are presented in this paper. The robust stability is guaranteed by the existence of a parameter dependent Lypaunov function obtained from the feasibility test of a set of linear matrix inequalities (LMIs) formulated at the vertices of the uncertainty polytope. Three conditions are presented, and the results are also compared with the analysis based on quadratic stability (same Lyapunov function for the entire set of uncertainties), for continuous as well as discrete-time systems. The first condition exploits the use of some extra variables (matrices) in the LMIs, and the second one uses a larger number of LMIs. These two conditions have recently appeared in the literature and are less conservative than quadratic stability. The third condition, proposed in this paper, combines the two ideas, yielding better results, and contains the previous conditions as particular cases. Several examples are presented to illustrate the Artigo submetido em 13/03/03 1a. Revisão em 20/10/03 Aceito sob recomendação do Ed. Assoc. Prof. Liu Hsu numerical performance of the LMI conditions in terms of efficiency and computational complexity.

Robust D-stability via discrete controllers for continuous time uncertain systems using LMIs with a scalar parameter

“…Given that P(α) > 0, inequality (13) indicates that µ > 0. Note that adopting P (α) = P = X = X T and µ → 0, (11) reduce to the quadratic stabilizability conditions presented in (6).…”

“…However, it is important to note that although the use of a polynomial Lyapunov matrix yields the sufficient and necessary conditions for stability analysis 1 Marco A. C. Leandro is with The Electronics Engineering Division, Instituto Tecnológico de Aeronáutica, DCTA-ITA-IEE, São José dos Campos, SP, 12228-900, Brazil (email:marcoacleandro@gmail.com) 2 José R. Colombo Júnior is with The Electronics Engineering Division, Instituto Tecnológico de Aeronáutica, DCTA-ITA-IEE, São José dos Campos, SP, 12228-900, Brazil (email:colombojrj@gmail.com) 3 Karl Heinz Kienitz is with The Electronics Engineering Division, Instituto Tecnológico de Aeronáutica, DCTA-ITA-IEE, São José dos Campos, SP, 12228-900, Brazil (email:kienitz@ita.br) [9], [10], so far, there are no extensions to the use of these conditions for the synthesis of a robust gain. Given this, the synthesis of a robust gain for polytopic uncertain systems can be approached with QS [11] or using a Lyapunov matrix depending on the affine form on the parameter and LMIs relaxations [12], [13], [14], [6], [5], [15] in many cases involving the search for a scalar parameter [16].…”

Compensadores dinâmicos para sistemas discretos no tempo com parametros variantes e aplicação a um sistema fuzzy Takagi-Sugeno

“…Observação 4 A complexidade numérica para a solucão de problemas baseados em LMIs está relacionada ao número escalar de variáveis de decisão e ao número de linhas das LMIs consideradas (Leite et al, 2004). Assim, pode-se verificar que o algoritmo (28)é numericamente mais complexo que o (27).…”

“…Dentre as formas de representação de sistemas LPV, a modelagem na forma politópica tem sido largamente explorada na literatura, com contribuições interessantes de vários pesquisadores brasileiros com colaboradores, reportadas em revistas e congressos nacionais (p.ex. : Oliveira et al, 2002;Leite et al, 2004;Araújo et al, 2010) e internacionais (p.ex. : Trofino and Souza, 2001;Geromel and Colaneri, 2006;Montagner et al, 2005;Souza and Trofino, 2005;Oliveira and Peres, 2008;Mozelli and Palhares, 2011); veja também a bibliografia mencionada nestas referências.…”

Uncertain Discrete-Time Systems with Delayed State: Robust Stabilization with Performance Specification via LMI Formulations

“…As a consequence, their achieved results may be very conservative, specially when applied to uncertain time-invariant systems. See, for instance, the works of Leite & Peres (2003), de Oliveira et al (2002) and Leite et al (2004). Perhaps the main…”