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Constrained model predictive control synthesis for uncertain discrete-time Markovian jump linear systems

Constrained model predictive control synthesis for uncertain discrete-time Markovian jump linear systems

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This study is concerned with model predictive control (MPC) for discrete-time Markovian jump linear systems subject to polytopic uncertainties both in system matrices and in transition probabilities between modes. The multi-step mode-dependent state-feedback control law is utilised to minimise an upper bound on the expected worst-case infinite horizon cost function. MPC designs for three cases: unconstrained case, constrained case and constrained case with low online computational burden (LOCB) are developed, respectively. All of them are proved to guarantee mean-square stability. In the constrained case, the minimisation of the expected worst-case infinite horizon cost function and constraints handling are dealt with in a separate way. The corresponding algorithm is proved to guarantee both the mean-square stability and the satisfaction of the hard mode-dependent constraints on inputs and states. To reduce the computational complexity, an algorithm with LOCB is developed by making use of the affine property of the solution to linear matrix inequalities. Finally, a numerical example is given to illustrate the proposed results.


    1. 1)
      • 12. Park, B., Kwon, W.H.: ‘Robust one-step receding horizon control of discrete-time Markovian jump uncertain systems’, Automatica, 2002, 38, (7), pp. 12291235 (doi: 10.1016/S0005-1098(02)00017-1).
    2. 2)
      • 17. Wen, J., Liu, F.: ‘Robust receding horizon dual-mode control for Markov jump systems’, Electr. Mach. Control, 2009, 13, (6), pp. 919925(in Chinese).
    3. 3)
      • 21. Wen, J., Liu, F., Nguang, S.: ‘Feedback predictive control for constrained fuzzy systems with Markovian jumps’, Asian J. Control, 2012, 14, (3), pp. 795806 (doi: 10.1002/asjc.344).
    4. 4)
      • 1. Costa, 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’, Automatica, 1999, 35, (4), pp. 617626 (doi: 10.1016/S0005-1098(98)00202-7).
    5. 5)
      • 3. Feng, X., Loparo, K.A., Ji, Y., Chizeck, H.J.: ‘Stochastic stability properties of jump linear systems’, IEEE Trans. Autom. Control, 1992, 37, (1), pp. 3852 (doi: 10.1109/9.109637).
    6. 6)
      • 8. Huang, H., Li, D., Xi, Y.: ‘Design and input-to-state practically stable analysis of the mixed H2/H feedback robust model predictive control’, IET Control Theory Appl., 2012, 6, (4), pp. 498505 (doi: 10.1049/iet-cta.2011.0187).
    7. 7)
      • 35. Cao, X.: ‘Stochastic learning and optimization a sensitivity-based approach’ (Springer Science + Business Media, LLC, 2007).
    8. 8)
      • 30. Wan, Z., Kothare, M.V.: ‘An efficient off-line formulation of robust model predictive control using linear matrix inequalities’, Automatica, 2003, 39, (5), pp. 837846 (doi: 10.1016/S0005-1098(02)00174-7).
    9. 9)
      • 26. Li, D., Xi, Y.: ‘Constrained feedback robust model predictive control for polytopic uncertain systems with time delays’, Int. J. Syst. Sci., 2011, 42, (10), pp. 16511660 (doi: 10.1080/00207720903576530).
    10. 10)
      • 27. De Oliveira, M.C., Bernussou, J., Geromel, J.C.: ‘A new discrete-time robust stability condition’, Syst. Control Lett., 1999, 37, (4), pp. 261265 (doi: 10.1016/S0167-6911(99)00035-3).
    11. 11)
      • 19. Wen, J., Liu, F.: ‘Receding horizon control for constrained Markovian jump linear systems with bounded disturbance’, J. Dyn. Syst. Meas. Control, 2011, 133, (1), pp. 011005-110 (doi: 10.1115/1.4002709).
    12. 12)
      • 23. Blackmore, L., Bektassov, A., Ono, M., Williams, B.C.: ‘Robust, optimal predictive control of jump Markov linear systems using particles’. Proc. 10th Int. workshop Hybrid Systems: Computation and Control, Pisa, Italy, April 2007, pp. 104117.
    13. 13)
      • 16. Liu, F., Cai, Y.: ‘Constrained predictive control of Markov jump linear systems based on terminal invariant sets’, Acta Autom. Sin., 2008, 34, (4), pp. 496499(in Chinese) (doi: 10.3724/SP.J.1004.2008.00496).
    14. 14)
      • 28. Daafouz, J., Bernussou, J.: ‘Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties’, Syst. Control Lett., 2001, 43, (5), pp. 355359 (doi: 10.1016/S0167-6911(01)00118-9).
    15. 15)
      • 15. Patrinos, P., Sopasakis, P., Sarimveis, H.: ‘Stochastic model predictive control for constrained networked control systems with random time delay’. Proc. 18th IFAC World Congress, Milano, Italy, June 2011, pp. 1262612631.
    16. 16)
      • 34. Wan, Z., Kothare, M.V.: ‘Efficient robust constrained model predictive control with a time varying terminal constraint set’, Syst. Control Lett., 2003, 48, (5), pp. 375383 (doi: 10.1016/S0167-6911(02)00291-8).
    17. 17)
      • 18. Wen, J., Liu, F.: ‘Feedback predictive control for uncertain jump systems’, Control Decis., 2010, 25, (6), pp. 916920(in Chinese).
    18. 18)
      • 13. Vargas, A.N., Do Val, J.B.R., Costa, E.F.: ‘Receding horizon control of Markov jump linear systems subject to noise and unobserved state chain’. Proc. 43rd IEEE Conf. Decision and Control, Atlantis, Paradise Island, Bahamas, December 2004, pp. 43814386.
    19. 19)
      • 9. Park, B., Lee, J., Kwon, W.H.: ‘Receding horizon control for linear discrete systems with jump parameters’. Proc. 36th IEEE Conf. Decision and Control, San Diego, California, USA, December 1997, pp. 39563957.
    20. 20)
      • 29. Daafouz, J., Riedinger, P., Iung, C.: ‘Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach’, IEEE Trans. Autom. Control, 2002, 47, (11), pp. 18831887 (doi: 10.1109/TAC.2002.804474).
    21. 21)
      • 24. Li, D., Xi, Y.: ‘Design of robust model predictive control based on multi-step control sets’, Acta Autom. Sin., 2009, 35, (4), pp. 433437.
    22. 22)
      • 22. Kothare, M.V., Balakrishnan, V., Morari, M.: ‘Robust constrained model predictive control using Linear Matrix Inequalities’, Automatica, 1996, 32, (10), pp. 13611379 (doi: 10.1016/0005-1098(96)00063-5).
    23. 23)
      • 20. Wen, J., Liu, F., Nguang, S.: ‘Sampled-data predictive control for uncertain jump systems with partly unknown jump rates and time-varying delay’, J. Frankl. Inst., 2012, 349, (1), pp. 305322 (doi: 10.1016/j.jfranklin.2011.11.004).
    24. 24)
      • 2. Costa, O.L.V., Fragoso, M.D., Marques, R.P.: ‘Discrete-time Markov jump linear systems’ (Springer, 2005).
    25. 25)
      • 14. Vargas, A.N., Furloni, W., Do Val, J.B.R.: ‘Constrained model predictive control of jump linear systems with noise and non-observed Markov state’. Proc. American Control Conf., Minneapolis, Minnesota, USA, June 2006, pp. 929934.
    26. 26)
      • 4. Boukas, E.K., Benzaouia, A.: ‘Stability of discrete-time linear systems with Markovian jumping parameters and constrained control’, IEEE Trans. Autom. Control, 2002, 47, (3), pp. 516521 (doi: 10.1109/9.989152).
    27. 27)
      • 7. Genuit, B.A.G., Lu, L., Heemels, W.P.M.H.: ‘Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach’, IET Control Theory Appl., 2012, 6, (8), pp. 10151028 (doi: 10.1049/iet-cta.2010.0709).
    28. 28)
      • 36. Löfberg, J.: ‘YALMIP: a toolbox for modeling and optimization in MATLAB’. Proc. IEEE Int. Symp. Computer Aided Control Systems Design, Taipei, Taiwan, China, September 2004, pp. 284289.
    29. 29)
      • 11. Park, B., Kwon, W.H., Lee, J.: ‘Robust receding horizon control of discrete-time Markovian jump uncertain systems’, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 2001, 84, (9), pp. 22722279.
    30. 30)
      • 5. Fang, Y., Loparo, K.A.: ‘Stochastic stability of jump linear systems’, IEEE Trans. Autom. Control, 2002, 47, (7), pp. 12041208 (doi: 10.1109/TAC.2002.800674).
    31. 31)
      • 31. Kozin, F.: ‘A survey of stability of stochastic systems’, Automatica, 1969, 5, (1), pp. 95112 (doi: 10.1016/0005-1098(69)90060-0).
    32. 32)
      • 33. Bernardini, D., Bemporad, A.: ‘Scenario-based model predictive control of stochastic constrained linear systems’. Proc. Joint 48th IEEE Conf. Decision and Control and 28th Chinese Control Conf., Shanghai, China, December 2009, pp. 63336338.
    33. 33)
      • 25. Li, D., Xi, Y.: ‘Constrained robust feedback model predictive control for uncertain systems with polytopic description’, Int. J. Control, 2009, 82, (7), pp. 12671274 (doi: 10.1080/00207170802530883).
    34. 34)
      • 10. Do Val, J.B.R., Başar, T.: ‘Receding horizon control of Markov jump linear systems’. Proc. American Control Conf., Albuquerque, New Mexico, USA, June 1997, pp. 31953199.
    35. 35)
      • 37. Sturm, J.F.: ‘Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones’, Optim. Methods Softw., 1999, 11, (1–4), pp. 625653 (doi: 10.1080/10556789908805766).
    36. 36)
      • 6. Zou, Y., Chen, T., Li, S.: ‘Network-based predictive control of multirate systems’, IET Control Theory Appl., 2010, 4, (7), pp. 11451156 (doi: 10.1049/iet-cta.2008.0577).
    37. 37)
      • 32. Kushner, H.: ‘Introduction to stochastic control’ (Holt, Rinehart and Winston, Inc., 1971).

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