This study is concerned with the problem of H2 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 H2 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 H2 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. © The Institution of Engineering and Technology 2013.71216651674Ji, Y., Chizeck, H.J., Controllability, stabilizability and continuoustime markovian jump linear-quadratic control (1990) IEEE Trans. Autom. Control, 35 (7), pp. 777-788Ji, Y., Chizeck, H.J., Jump linear quadratic gaussian control: Steadystate and testable conditions (1990) Control Theory Adv. Technol, 6, pp. 289-319Feng, X., Loparo, K.A., Ji, Y., Chizeck, H.J., Stochastic stability properties of jump linear systems (1992) IEEE Trans. Autom. Control, 37 (1), pp. 38-53Costa, O.L.V., Fragoso, M.D., Stability results for discrete-time linear systems with markovian jumping parameters (1993) J. Math. Anal. Appl, 179, pp. 154-178Costa, O.L.V., Fragoso, M.D., Discrete-time lq-optimal control problems for infinite markov jump parameter systems (1995) IEEE Trans. Autom. Control, 40 (12), pp. 2076-2088Costa, O.L.V., Do Val, J.B.R., Geromel, J.C., A convex programming approach to h2-control of discrete-time markovian jump linear systems (1997) Int. J. Control, 66 (4), pp. 557-579Costa, O.L.V., Assumpção Filho, E.O., Boukas, E.K., Marques, R.P., Constrained quadratic state feedback control of discrete-time markovian jump linear systems (1999) Automatica, 35 (4), pp. 617-626Boukas, E.K., (2005) Stochastic Switching Systems: Analysis and Design, , Birkhäuser, Berlin, GermanyCosta, O.L.V., Fragoso, M.D., Marques, R.P., (2005) Discrete-Time Markovian Jump Linear Systems, , Springer-Verlag, New York, NY, USAVargas, A.N., Furloni, W., Do Val, J.B.R., Control of observed markov jump linear systems with constraints on state and input: A necessary optimality condition (2007) Proc. Third IFAC Symp. System, Structure and Control (SSSC07), pp. 250-255. , Foz do Iguassu, PR, Brazil, OctoberYang, T., Networked control system: A brief survey (2006) IET Control Theory Appl, 153 (4), pp. 403-412Zhang, L., Gao, H., Kaynak, O., Network-induced constraints in networked control systems -A survey (2013) IEEE Trans. Ind. Inf, 9 (1), pp. 403-416Xiong, J., Lam, J., Gao, H., Ho, D.W.C., On robust stabilization of markovian jump systems with uncertain switching probabilities (2005) Automatica, 41 (5), pp. 897-903De Souza, C.E., Robust stability and stabilization of uncertain discretetime markovian jump linear systems (2006) IEEE Trans. Autom. Control, 51 (5), pp. 836-841Karan, M., Shi, P., Kaya, C.Y., Transition probability bounds for the stochastic stability robustness of continuous-and discretetime markovian jump linear systems (2006) Automatica, 42, pp. 2159-2168Oliveira, R.C.L.F., Vargas, A.N., Do Val, J.B.R., Peres, P.L.D., Robust stability, h2 analysis and stabilisation of discrete-time markov jump linear systems with uncertain probability matrix (2009) Int. J. Control, 82 (3), pp. 470-481Boukas, E.K., Guaranteed cost for stochastic systems with unknown transition jump rate (2009) Proc. 2009 American Control Conf., pp. 4422-4427. , St. Louis, MO, USA, JuneXiong, J., Lam, J., Robust h2 control of markovian jump systems with uncertain switching probabilities (2009) Int. J. Syst. Sci, 40 (3), pp. 255-265Luan, X., Liu, F., Shi, P., Finite-time filtering for non-linear stochastic systems with partially known transition jump rates (2010) IET Control Theory Appl, 4 (5), pp. 735-745Ma, S., Boukas, E.K., Chinniah, Y., Stability and stabilization of discrete-time singular markov jump systems with time-varying delay (2010) Int. J. Robust Nonlinear Control, 20 (5), pp. 531-543Ma, S., Zhang, C., Zhu, S., Robust stability for discrete-time uncertain singular markov jump systems with actuator saturation (2011) IET Control Theory Appl, 5 (2), pp. 255-262Shen, M., Yang, G.-H., New analysis and synthesis conditions for continuous markov jump linear systems with partly known transition probabilities (2012) IET Control Theory Appl, 6 (14), pp. 2318-2325Zhang, L., Boukas, E.K., Stability and stabilization of markovian jump linear systems with partly unknown transition probabilities (2009) Automatica, 45 (2), pp. 463-468Zhang, L., Boukas, E.K., H∞ control of a class of extended markov jump linear systems (2009) IET Control Theory Appl, 3 (7), pp. 834-842Zuo, Z., Liu, Y., Wang, Y., Li, H., Finite-time stochastic stability and stabilisation of linear discrete-time markovian jump systems with partly unknown transition probabilities (2012) IET Control Theory Appl, 6 (10), pp. 1522-1526Zhang, Y., He, Y., Wu, M., Zhang, J., State estimation for markovian jump systems with time-varying delay and partial information on transition probabilities (2012) IET Control Theory Appl, 6 (16), pp. 2549-2555Oliveira, R.C.L.F., Bliman, P.-A., Peres, P.L.D., Robust lmis with parameters in multi-simplex: Existence of solutions and applications (2008) Proc. 47th IEEE Conf. Decision and Control, pp. 2226-2231. , Cancun, Mexico, DecemberDo Val, J.B.R., Geromel, J.C., Gonçalves, A.P., The h2-control for jump linear systems: Cluster observations of the markov state (2002) Automatica, 38 (2), pp. 343-349Braga, M.F., Morais, C.F., Oliveira, R.C.L.F., Peres, P.L.D., Robust stability and stabilization of discrete-time markov jump linear systems with partly unknown transition probability matrix (2013) Proc. 2013 American Control Conf., pp. 6800-6805. , Washington DC USA JuneLöfberg, J., Yalmip: A toolbox for modeling and optimization in matlab (2004) Proc. 2004 IEEE Int. Symp. Computer Aided Control Systems Design, pp. 284-289. , http://control.ee.ethz.ch/~joloef/yalmip.php, Taipei, Taiwan, SeptemberSturm, J.F., Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones (1999) Optim. Methods Softw, 11 (1-4), pp. 625-653. , http://sedumi.ie.lehigh.eduBoukas, E.K., H∞ control of discrete-time markov jump systems with bounded transition probabilities (2009) Optim. Control Appl. Methods, 30 (5), pp. 477-494Zhang, L., Boukas, E.K., Mode-dependent filtering for discretetime markovian jump linear systems with partly unknown transition probabilities (2009) Automatica, 45 (6), pp. 1462-1467Bliman, P.-A., An existence result for polynomial solutions of parameter-dependent lmis (2004) Syst. Control Lett, 51 (3-4), pp. 165-169Oliveira, R.C.L.F., Peres, P.L.D., Parameter-dependent lmis in robust analysis: Characterization of homogeneous polynomially parameterdependent solutions via lmi relaxations (2007) IEEE Trans. Autom. Control, 52 (7), pp. 1334-1340Scherer, C.W., Higher-order relaxations for robust lmi problems with verifications for exactness (2003) Proc. 42nd IEEE Conf. Decision and Control, pp. 4652-4657. , Maui, HI, USA, DecemberScherer, C.W., Relaxations for robust linear matrix inequality problems with verifications for exactness (2005) SIAM J. Matrix Anal. Appl, 27 (2), pp. 365-395Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., Linear matrix inequalities in system and control theory (1994) SIAM Studies in Applied Mathematics, p. 39. , Philadelphia, PAGahinet, P., Apkarian, P., A linear matrix inequality approach to h∞ control (1994) Int. J. Robust Nonlinear Control, 4 (4), pp. 421-448Iwasaki, T., Skelton, R.E., All controllers for the general h∞ control problem: Lmi existence conditions and state-space formulas (1994) Automatica, 30 (8), pp. 1307-1317Pipeleers, G., Demeulenaere, B., Swevers, J., Vandenberghe, L., Extended lmi characterizations for stability and performance of linear systems (2009) Syst. Control Lett, 58 (7), pp. 510-518Blair, W.P., Sworder, D.D., Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria (1975) Int. J. Control, 21 (5), pp. 833-841