access icon free Quasi-convex combination method and its application to the stability analysis of 2D discrete-time Roesser systems with time-varying delays

This study is concerned with the problem of stability analysis of two-dimensional (2D) discrete-time Roesser systems with time-varying delays. First, based on an augmented Lyapunov–Krosovskii functional, a less conservative stability criterion incorporating time-varying terms is established by utilising a general free-matrix-based inequality. Next, in order to eliminate the time-varying terms without introducing redundant constraints, a quasi-convex combination method is proposed. Then, compared with the criteria derived via the other inequalities, the conservatism analysis is given to prove the proposed criterion can lead to a better result theoretically. Finally, a numerical example is presented to illustrate the advantage of the presented method.

Inspec keywords: matrix algebra; Lyapunov methods; delays; discrete time systems; stability; time-varying systems; stability criteria

Other keywords: 2D discrete-time Roesser systems; quasi-convex combination method; augmented Lyapunov-Krosovskii functional; two-dimensional discrete-time Roesser systems; stability analysis; general free-matrix-based inequality; less conservative stability criterion; conservatism analysis; time-varying delays

Subjects: Distributed parameter control systems; Time-varying control systems; Discrete control systems; Linear algebra (numerical analysis); Stability in control theory

References

    1. 1)
      • 21. Du, C., Xie, L.: ‘HControl and filtering of two-dimensional systems’ (Springer, Berlin, 2002).
    2. 2)
      • 6. Duan, Z., Xiang, Z., Karimi, H.R.: ‘Delay-dependent exponential stabilization of positive 2D switched state-delayed systems in the Roesser model’, Inf. Sci., 2014, 272, pp. 173184.
    3. 3)
      • 3. Bracewell, R.N.: ‘Two-dimensional imaging’ (Prentice-Hall, Englewood Cliffs, NJ, 1995).
    4. 4)
      • 8. Hien, L.V., Trinh, H.: ‘A novel approach to exponential stability of continuous-time Roesser systems with directional time-varying delays’, J. Franklin Inst., 2017, 354, pp. 10231041.
    5. 5)
      • 36. Zhang, C.K., He, Y., Jiang, L., et al: ‘Delay-variation-dependent stability of delayed discrete-time systems’, IEEE Trans. Autom. Control, 2016, 61, pp. 26632669.
    6. 6)
      • 19. Hien, L.V., Trinh, H.: ‘Delay-dependent stability and stabilisation of two-dimensional positive Markov jump systems with delays’, IET Control Theory Appl., 2017, 11, pp. 16031610.
    7. 7)
      • 24. He, Y., Wu, M., Liu, G.P., et al: ‘Output feedback stabilization for a discrete-time system with a time-varying delay’, IEEE Trans. Autom. Control, 2008, 53, pp. 23722377.
    8. 8)
      • 34. Jiang, X., Han, Q.L., Yu, X.: ‘Stability criteria for linear discrete-time systems with interval-like time-varying delay’. Proc. American Control Conf., Portland, OR, USA, 2005, pp. 28172822.
    9. 9)
      • 31. Seuret, A., Gouaisbaut, F., Fridman, E.: ‘Stability of discrete-time systems with time-varying delays via a novel summation inequality’, IEEE Trans. Autom. Control, 2015, 60, pp. 27402745.
    10. 10)
      • 27. Huang, H., Feng, G.: ‘Improved approach to delay-dependent stability analysis of discrete-time systems with time-varying delay’, IET Control Theory Appl., 2010, 4, pp. 21522159.
    11. 11)
      • 45. Park, P.G., Ko, J.W., Jeong, C.: ‘Reciprocally convex approach to stability of systems with time-varying delays’, Automatica, 2011, 47, pp. 235238.
    12. 12)
      • 25. Feng, Z., Lam, J., Yang, G.H.: ‘Optimal partitioning method for stability analysis of continuous/discrete delay systems’, Int. J. Robust Nonlinear Control, 2015, 25, pp. 559574.
    13. 13)
      • 16. Shen, J., Wang, W.: ‘Stability and positive observer design for positive 2D discrete-time system with multiple delays’, Int. J. Syst. Sci., 2017, 48, pp. 11361145.
    14. 14)
      • 13. Ahn, C.K.: ‘l2l Elimination of overflow oscillations in 2-D digital filters described by Roesser model with external interference’, IEEE Trans. Circuits Syst. II Express Briefs, 2013, 60, pp. 361365.
    15. 15)
      • 38. Zhang, C.K., He, Y., Jiang, L., et al: ‘Summation inequalities to bounded real lemmas of discrete-time systems with time-varying delay’, IEEE Trans. Autom. Control, 2017, 62, pp. 25822588.
    16. 16)
      • 40. Ghous, I., Xiang, Z.: ‘H stabilization of 2-D discrete switched delayed systems represented by the Roesser model subject to actuator saturation’, Trans. Inst. Meas. Control, 2015, 37, pp. 12421253.
    17. 17)
      • 44. Kaczorek, T.: ‘Two-dimensional linear systems’ (Springer-Verlag, Berlin, 1985).
    18. 18)
      • 22. Xu, J., Nan, Y., Zhang, G., et al: ‘Delay-dependent H control for uncertain 2-D discrete systems with state delay in the Roesser model’, Circuit Syst. Signal Process., 2013, 32, pp. 10971112.
    19. 19)
      • 4. Roesser, R.P.: ‘A discrete state-space model for linear image processing’, IEEE Tran. Autom. Control, 1975, 20, pp. 110.
    20. 20)
      • 28. Meng, X., Lam, J., Du, B., et al: ‘A delay-partitioning approach to the stability analysis of discrete-time systems’, Automatica, 2010, 46, pp. 610614.
    21. 21)
      • 30. Nam, P.T., Trinh, H., Pathirana, P.N.: ‘Discrete inequalities based on multiple auxiliary functions and their applications to stability analysis of time-delay systems’, J. Franklin Inst., 2015, 352, pp. 58105831.
    22. 22)
      • 43. Hien, L.V., Trinh, H.: ‘Stability of two-dimensional Roesser systems with time-varying delays via novel 2D finite-sum inequalities’, IET Control Theory Appl., 2016, 10, pp. 16651674.
    23. 23)
      • 42. Huang, S., Xiang, Z.: ‘Delay-dependent robust H control for 2-D discrete nonlinear systems with state delays’, Multidimension. Syst. Signal Process., 2014, 25, pp. 775794.
    24. 24)
      • 37. Zhang, C.K., He, Y., Jiang, L., et al: ‘An improved summation inequality to discrete-time systems with time-varying delay’, Automatica, 2016, 74, pp. 1015.
    25. 25)
      • 12. Hien, L.V., Trinh, H.: ‘Stability analysis of two-dimensional Markovian jump state-delayed systems in the Roesser model with uncertain transition probabilities’, Inf. Sci., 2016, 367, pp. 403417.
    26. 26)
      • 9. Chen, S.F.: ‘Delay-dependent stability for 2D systems with time-varying delay subject to state saturation in the Roesser model’, Appl. Math. Comput., 2010, 216, pp. 26132622.
    27. 27)
      • 2. Xu, S., Lam, J., Lin, Z., et al: ‘Positive real control of two-dimensional systems: Roesser models and linear repetitive processes’, Int. J. Control, 2003, 76, pp. 10471058.
    28. 28)
      • 7. Rogers, E., Galkowski, K., Paszke, W., et al: ‘Multidimensional control systems: case studies in design and evaluation’, Multidimens. Syst. Signal Process., 2015, 26, pp. 895939.
    29. 29)
      • 32. Xie, X., Yue, D., Zhang, H., Xue, Y.: ‘Fault estimation observer design for discrete-time Takagi-Sugeno fuzzy systems based on homogenous polynomially parameter-dependent Lyapunov functions’, IEEE Trans. Cybern., 2017, 47, pp. 25042513.
    30. 30)
      • 33. Kwon, O.M., Park, M.J., Park, J.H., et al: ‘Stability and stabilization for discrete-time systems with time-varying delays via augmented Lyapunov-Krasovskii functional’, J. Franklin Inst., 2013, 350, pp. 521540.
    31. 31)
      • 20. Li, X., Lam, J., Cheung, K.C.: ‘Generalized H model reduction for two-dimensional discrete systems’, Multidimens. Syst. Signal Process., 2016, 27, pp. 359382.
    32. 32)
      • 26. Kwon, O.M., Park, M.J., Park, J.H., et al: ‘Improved robust stability criteria for uncertain discrete-time systems with interval time-varying delays via new zero equalities’, IET Control Theory Appl., 2012, 6, pp. 25672575.
    33. 33)
      • 14. Ahn, C.K., Shi, P., Wu, L.: ‘l-gain performance analysis for two-dimensional Roesser systems with persistent bounded disturbance and saturation nonlinearity’, Inf. Sci., 2016, 333, pp. 126139.
    34. 34)
      • 17. Xie, X.P., Zhang, Z.W., Hu, S.L.: ‘Control synthesis of Roesser type discrete-time 2-D T-S fuzzy systems via a multi-instant fuzzy state-feedback control scheme’, Neurocomputing, 2015, 151, pp. 13841391.
    35. 35)
      • 29. Nam, P.T., Pathirana, P.N., Trinh, H.: ‘Discrete Wirtinger-based inequality and its application’, J. Franklin Inst., 2015, 352, pp. 18931905.
    36. 36)
      • 18. Hien, L.V., Trinh, H.: ‘Observers design for 2-D positive time-delay Roesser systems’, IEEE Trans. Circuits Syst. II Express Briefs, 2017, DOI: 10.1109/TCSII.2017.2723425.
    37. 37)
      • 15. Wang, L., Wang, W., Gao, J., et al: ‘Stability and robust stabilization of 2-D continuous-discrete systems in Roesser model based on KYP lemma’, Multidimens. Syst. Signal Process., 2017, 28, pp. 251264.
    38. 38)
      • 35. Chen, J., Lu, J., Xu, S.: ‘Summation inequality and its application to stability analysis for time-delay systems’, IET Control Theory Appl., 2016, 10, pp. 391395.
    39. 39)
      • 11. Fornasini, E., Marchesini, G.: ‘Doubly-indexed dynamical systems: state-space models and structural properties’, Math. Syst. Theory, 1978, 12, pp. 5972.
    40. 40)
      • 1. Whalley, R.: ‘Two-dimensional digital filters’, Appl. Math. Modell., 1990, 14, pp. 304311.
    41. 41)
      • 5. Li, X., Gao, H.: ‘Robust finite frequency H filtering for uncertain 2-D Roesser systems’, Automatica, 2012, 48, pp. 11631170.
    42. 42)
      • 10. Ghous, I., Xiang, Z.: ‘Reliable H control of 2-D continuous nonlinear systems with time varying delays’, J. Franklin Inst., 2015, 352, pp. 57585778.
    43. 43)
      • 41. Peng, D., Hua, C.: ‘Improved approach to delay-dependent stability and stabilisation of two-dimensional discrete-time systems with interval time-varying delays’, IET Control Theory Appl., 2015, 9, pp. 18391845.
    44. 44)
      • 39. Paszke, W., Lam, J., Galkowski, K., et al: ‘Robust stability and stabilisation of 2D discrete state-delayed systems’, Syst. Control Lett., 2004, 51, pp. 277291.
    45. 45)
      • 23. Gao, H., Chen, T.: ‘New results on stability of discrete-time systems with time-varying state delay’, IEEE Trans. Autom. Control, 2007, 52, pp. 328334.
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2017.0653
Loading

Related content

content/journals/10.1049/iet-cta.2017.0653
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading