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.

This paper proposes an improved approach to ℋ ₂ and ℋ _∞ robust state feedback control design for discrete-time polytopic time-invariant linear systems based on Linear Matrix Inequalities (LMIs) with a scalar parameter. The synthesis conditions, that depend on a real parameter lying in the interval (-1,1), become LMIs for fixed values of the scalar, reducing to standard conditions in the literature when the scalar is equal to zero. At the price of line searches combined with LMIs, less conservative results for robust state feedback control are obtained. The closed-loop stability and the ℋ ₂ and ℋ _∞ guaranteed costs are certified by means of an affine parameter-dependent Lyapunov function. The validity and the efficiency of the method are illustrated by means of examples and exhaustive numerical comparisons.

This paper deals with the problem of designing reduced-order robust dynamic output feedback controllers for discrete-time Markov jump linear systems (MJLS) with polytopic state space matrices and uncertain transition probabilities. Starting from a full order, mode-dependent and polynomially parameter-dependent dynamic output feedback controller, sufficient linear matrix inequality based conditions are provided for the existence of a robust reduced-order dynamic output feedback stabilising controller with complete, partial or none mode dependency assuring an upper bound to the ${\mathcal{H}}_{\infty }$ or the ${\mathcal{H}}_2$ norm of the closed-loop system. The main advantage of the proposed method when compared to the existing approaches is the fact that the dynamic controllers are exclusively expressed in terms of the decision variables of the problem. In other words, the matrices that define the controller realisation do not depend explicitly on the state space matrices associated with the modes of the MJLS. As a consequence, the method is specially suitable to handle order reduction or cluster availability constraints in the context of ${\mathcal{H}}_{\infty }$ or ${\mathcal{H}}_2$ dynamic output feedback control of discrete-time MJLS. Additionally, as illustrated by means of numerical examples, the proposed approach can provide less conservative results than other conditions in the literature.