In this study, the authors focus on the stability analysis for Markovian jump linear systems with partly known transition rates in the continuous-time domain and partly known transition probabilities in the discrete-time domain. By using the properties of the transition rates and transition probabilities, two new sufficient conditions are derived for the stochastic stability of the continuous-time and discrete-time Markovian jump linear systems, respectively. The main advantage of the proposed stability conditions is that the total number of linear matrix inequalities (LMIs) in the proposed stability conditions is much less than that in some existing results. Based on the presented stability conditions, two state feedback controllers are designed for the considered systems in terms of LMIs. In addition, two kinds of stability criteria are developed for the stochastic stability of the considered systems with incomplete transition descriptions by the existence of the unique positive definite solution of the coupled Lyapunov matrix equations. Finally, two numerical examples and one practical example are provided to verify the correctness of the theoretical results.

References

1. 1)
  - 9. Zhang, L., Lam, J.: ‘Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions’, IEEE Trans. Autom. Control, 2010, 55, (7), pp. 1695–1701.
2. 2)
  - 15. Ding, L., Gao, H., Xia, K., et al: ‘Adaptive sliding mode control of mobile manipulators with Markovian switching joints’, J. Appl. Math., 2012, 16, pp. 3800–3844.
3. 3)
  - 4. Costa, O.L.V., Fragoso, M.D., Marques, R.P.: ‘Discrete-time Markovian jump linear systems’ Springer-Verlag, London, UK, 2005).
4. 4)
  - 16. Zheng, B.C., Yang, G.H.: ‘Sliding mode control for Markov jump linear uncertain systems with partly unknown transition rates’, Int. J. Syst. Sci., 2014, 45, (10), pp. 1999–2011.
5. 5)
  - 2. Costa, O.L.V., Val, J.B., Geromel, J.C.: ‘Coutinuous-time state-feedback H2 control of Markovian jump linear systems via convex analysis’, Automatica, 1999, 35, pp. 259–268.
6. 6)
  - 10. Zhang, Y., He, Y., Wu, M., et al: ‘Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices’', Automatica, 2011, 47, (1), pp. 79–84.
7. 7)
  - 14. Costa, O.L.V., Fragoso, M.D., Todorov, M.G.: ‘A detector-based approach for the control of Markov jump linear systems with partial information’, IEEE Trans. Autom. Control, 2015, 60, (5), pp. 1219–1234.
8. 8)
  - 1. Boukas, E.K..: ‘Stochastic switching systems: analysis and design’ (Springer Science & Business Media, Boston, USA, 2007).
9. 9)
  - 8. Zhang, L., Boukas, E.K.: ‘Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities’, Automatica, 2009, 45, (2), pp. 463–468.
10. 10)
  - 3. Ji, Y., Howard, J.C.: ‘Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control’, IEEE Trans. Autom. Control, 1990, 35, (7), pp. 777–788.
11. 11)
  - 12. Morais, C.F., Braga, M.F., Oliveira, R.C.L.F., et al: ‘H∞ and H2 control design for polytopic continuous-time Markov jump linear systems with uncertain transition rates’, Int. J. Robust Nonlinear Control, 2016, 26, (3), pp. 599–612.
12. 12)
  - 7. Zhang, L., Boukas, E.K., Lam, J.: ‘Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities’, IEEE Trans. Autom. Control, 2008, 53, (10), pp. 2458–2464.
13. 13)
  - 5. Feng, X., Loparo, K., Ji, Y., et al: ‘Stochastic stability properties of jump linear systems’, IEEE Trans. Autom. Control, 1992, 37, (1), pp. 38–53.
14. 14)
  - 6. Shi, P., Boukas, E.K., Shi, Y.: ‘On stochastic stabilization of discrete-time Markovian jump systems with delay in state’, Stoch. Anal. Appl., 2003, 21, (4), pp. 935–951.
15. 15)
  - 13. Luan, X., Zhao, S., Shi, P., et al: ‘H∞ filtering for discrete-time Markov jump systems with unknown transition probabilities’', Int. J. Adapt. Control Signal Process., 2014, 28, (2), pp. 138–148.
16. 16)
  - 11. Shen, M., Yang, G.H.: ‘New analysis and synthesis conditions for continuous Markov jump linear systems with partly known transition probabilities’, IET Control Theory Applic., 2012, 6, (14), pp. 2318–2325.

Stochastic stability analysis of Markovian jump linear systems with incomplete transition descriptions

References

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