An improved linear matrix inequality (LMI) approach is proposed to deal with the problems of stability, mode-dependent and mode-independent stabilization of discrete-time Markov jump linear systems (MJLS) with partly unknown transition probability matrix. As a first contribution, the uncertain parameters of the transition probability matrix are modeled in terms of the Cartesian product of simplexes, called multi-simplex. Then, convergent LMI relaxations with improved trade-off between precision and computational effort are proposed for the stability analysis of this class of MJLS. Finally, new design conditions based on LMIs with a scalar parameter are proposed for state feedback control, in both mode independent and mode dependent scenarios, providing less conservative results when compared to other conditions available in the literature, as illustrated by numerical examples. © 2013 AACC American Automatic Control Council. 67846789Boeing,Eaton,Halliburton,Honeywell,MathWorksJi, Y., Chizeck, H.J., Controllability, stabilizability and continuoustime Markovian jump linear-quadratic control (1990) IEEE Trans. Autom. Control, 35 (7), pp. 777-788. , JulyJi, Y., Chizeck, H.J., Jump linear quadratic Gaussian control: Steady-state solution and testable conditions (1990) Control Theory Adv. Techn, 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. 38-53. , JanuaryCosta, O.L.V., Fragoso, M.D., Stability results for discrete-time linear systems with Markovian jumping parameters (1993) J. Math. Anal. Appl, 179, pp. 154-178. , OctoberCosta, 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. 2076-2088. , DecemberCosta, 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, pp. 557-579. , MarchBoukas, E.K., (2005) Stochastic Switching Systems: Analysis and Design, , Berlin, Germany: BirkhäuserCosta, O.L.V., Fragoso, M.D., Marques, R.P., (2005) Discrete-Time Markovian Jump Linear Systems, , New York, NY, USA: Springer-VerlagXiong, 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. 897-903. , MayDe Souza, C.E., Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems (2006) IEEE Trans. Autom. Control, 51 (5), pp. 836-841. , MayKaran, M., Shi, P., Kaya, C.Y., Transition probability bounds for the stochastic stability robustness of continuous-and discrete-time Markovian jump linear systems (2006) Automatica, 42, pp. 2159-2168. , DecemberOliveira, 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. 470-481. , MarchMa, S.P., Boukas, E.K., Robust quadratic control of discrete-time singular Markov jump systmes with bounded transition probabilities (2009) Proc. 2009 Amer. Control Conf., pp. 4044-4049. , St. Louis, MO, USA, JuneBoukas, E.K., Guaranteed cost for stochastic systems with unknown transition jump rate (2009) Proc. 2009 Amer. Control Conf., pp. 4422-4427. , St. Louis, MO, USA, JuneMa, 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. 531-543. , MarchZhang, L., Boukas, E.K., Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities (2009) Automatica, 45 (2), pp. 463-468. , FebruaryZhang, L., Boukas, E.K., Mode-dependent filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities (2009) Automat-ica, 45 (6), pp. 1462-1467. , JuneOliveira, 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 Control, pp. 2226-2231. , Cancun, Mexico, DecemberBliman, P.-A., An existence result for polynomial solutions of parameter-dependent LMIs (2004) Syst. Control Letts, 51 (3-4), pp. 165-169. , MarchBoyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., (1994) Linear Matrix Inequalities in System and Control Theory, , Philadelphia, PA: SIAM Studies in Applied MathematicsGahinet, P., Apkarian, P., A linear matrix inequality approach to Hcontrol (1994) Int. J. Robust Nonlinear Control, 4 (4), pp. 421-448. , July-AugustIwasaki, T., Skelton, R.E., All controllers for the general Hcontrol problem: LMI existence conditions and state-space formulas (1994) Automatica, 30 (8), pp. 1307-1317. , AugustScherer, C.W., Higher-order relaxations for robust LMI problems with verifications for exactness (2003) Proc. 42nd IEEE Conf. Decision 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. 365-395. , JuneOliveira, R.C.L.F., Peres, P.L.D., Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations (2007) IEEE Trans. Autom. Control, 52 (7), pp. 1334-1340. , JulyLöfberg, J., YALMIP: A toolbox for modeling and optimization in MATLAB (2004) Proc. 2004 IEEE Int. Symp. on Comput. Aided Control Syst. Des., 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. Method Softw, 11 (1-4), pp. 625-653. , http://sedumi.ie.lehigh.edu/