Open Accessℋ 2 control of discrete‐time Markov jump linear systems with uncertain transition probability matrix: improved linear matrix inequality relaxations and multi‐simplex modelling
Author(s) -
Morais Cecília F.,
Braga Márcio F.,
Oliveira Ricardo C.L.F.,
Peres Pedro L.D.
Publication year - 2013
Publication title -
iet control theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.059
H-Index - 108
eISSN - 1751-8652
pISSN - 1751-8644
DOI - 10.1049/iet-cta.2012.1015
Subject(s) - mathematics , linear matrix inequality , markov chain , simplex , bounded function , state transition matrix , scalar (mathematics) , stochastic matrix , matrix (chemical analysis) , discrete time and continuous time , linear system , control theory (sociology) , symmetric matrix , mathematical optimization , eigenvalues and eigenvectors , computer science , mathematical analysis , control (management) , composite material , statistics , physics , materials science , quantum mechanics , artificial intelligence , geometry
This study is concerned with the problem of ℋ 2 state‐feedback control design for discrete‐time Markov jump linear systems (MJLS), assuming that the transition probability matrix is not precisely known, but belongs to a polytopic domain, or contains unknown or bounded elements. As a first contribution, the uncertainties of the transition probability matrix are modelled in terms of the Cartesian product of simplexes, called multi‐simplex. Thanks to this representation, the problem of robust mean square stability analysis with an ℋ 2 norm bound can be solved through convergent linear matrix inequality (LMI) relaxations constructed in terms of polynomial solutions. The proposed conditions yield a better trade‐off between precision and computational effort when compared with other methods. As a second contribution, new conditions in terms of LMIs with a scalar parameter lying in the interval (− 1, 1) are proposed for ℋ 2 state‐feedback control with complete, partial or no observation of the Markov chain. Owing to the presence of the scalar parameter, less conservative results when compared with other conditions available in the literature can be obtained, at the price of increasing the associated computational effort. Numerical examples illustrate the advantages of the proposed methodology.