Fractional order polytopic systems: robust stability and stabilisation
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RESEARCH
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Fractional order polytopic systems: robust stability and stabilisation Christophe Farges, Jocelyn Sabatier* and Mathieu Moze * Correspondence: Jocelyn. [email protected] University of Bordeaux, IMS Laboratory (CRONE Team), CNRS UMR 5218, 351 Cours de la Libération, 33405 Talence, France
Abstract This article addresses the problem of robust pseudo state feedback stabilisation of commensurate fractional order polytopic systems (FOS). In the proposed approach, Linear Matrix Inequalities (LMI) formalism is used to check if the pseudo-state matrix eigenvalues belong to the FOS stability domain whatever the value of the uncertain parameters. The article focuses particularly on the case of a fractional order ν such that 0 < ν < 1, as the stability region is non-convex and associated LMI condition is not as straightforward to obtain as in the case 1 < ν < 2. In relation to the quadratic stabilisation problem previously addressed by the authors and that involves a single matrix to prove stability of the closed loop system, additional variables are then introduced to decouple system matrices in the closed loop system stability condition. This decoupling allows using parameter-dependent stability matrices and leads to less conservative results as attested by a numerical example. Keywords: Fractional order systems, inear Matrix Inequalities, Robust control, State feedback, Polytopic systems
Introduction As for linear time invariant integer order systems, it is now well known that stability of a linear fractional order system depends on the location of the system poles in the complex plane. However, pole location analysis remains a difficult task in the general case. For commensurate fractional order systems, powerful criteria have been proposed. The most well known is Matignon’s stability theorem [1]. It permits to check the system stability through the location in the complex plane of the dynamic matrix eigenvalues of the system pseudo-state space representation. Matignon’s theorem is in fact the starting point of several results in the field [2,3]. This is the case of the Linear Matrix Inequalities (LMI) stability conditions recently proposed by the authors [4]. These conditions are used to synthesise a stabilising pseudo-state feedback whatever the system fractional order ν in the set ]0,2[. Although much progress has been made in the field of fractional system stability, linear time invariant fractional systems robust stability remains an open problem. Among the existing results and only for interval fractional systems, the stability issue was discussed in [5-7]. As commented in [5,8], the result is rather conservative. To reduce the conservatism, in [8], a new robust stability checking method was proposed for interval uncertain systems, where Lyapunov inequality is used for finding the maximum
© 2011 Farges et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which perm
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