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Robust Control of Positive 2-Dimensional Systems with Bounded Realness Property

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Abstract

As presented in this paper, we explore the control of a discrete-time two–dimensional (2-D) system using the Lyapunov approach. The Giovane–Roesser model (G–R) for 2-D systems was introduced, and we presented the asymptotic stability analysis for this class of systems while maintaining the strictly bounded real (SBR) property. In the next step, we solve the stability problem in the presence of uncertainties in the system while preserving the SBR condition. We design state feedback and output feedback controllers to control 2-D discrete-time systems with preceding uncertainties, introducing algorithms to design such controllers. In order to ensure the validity of our findings, we present the simulation results as numerical and practical examples.

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Data Availibility Statement

This paper is a theoretical study in which all data needed to plot the figures of this paper are in the illustrative example section. This study did not use any data set.

References

  1. Agathoklis, P.: Lower bounds for the stability margin of discrete two-dimensional systems based on the two-dimensional Lyapunov equation. IEEE Trans. Circuits Syst. 35(6), 745–749 (1988)

    MathSciNet  Google Scholar 

  2. Agathoklis, P., Jury, E.I., Mansour, M.: The importance of bounded real and positive real functions in the stability analysis of 2-D system. Proc. IEEE Int. Symp. Circuits Syst. 1, 124–127 (1991)

    Google Scholar 

  3. Agathoklis, P., Jury, E.I., Mansour, M.: The importance of bounded real and positive real functions in the stability analysis of 2-d system. IEEE Int. Symp. Circuit Syst. 1, 124–127 (1991)

    Google Scholar 

  4. Anderson, B.D.O.: A system theory criterion for positive real matrices. SIAM J. Control 5(2), 171–182 (1967)

    MathSciNet  MATH  Google Scholar 

  5. Anderson, B.D.O., Moore, J.B.: Optimal Control: Linear Quadratic Methods. Courier Corporation, North Chelmsford (2007)

    Google Scholar 

  6. Anderson, B.D.O., Vongpanitlerd, S.: Network Analysis and Synthesis: A Modern Systems Theory Approach. Courier Corporation, North Chelmsford (2013)

    Google Scholar 

  7. Anderson, B.D.O., Agathoklis, P.J.E.I., Mansour, M.: Stablity and the matrix Lyapunov equation for discrete 2-dimensional systems. IEEE Trans. Circuits Syst. 33, 261–267 (1986)

    MATH  Google Scholar 

  8. Aravena, J.L., Shafiee, M., Porter, W.A.: State models and stability for 2-D filters. IEEE Trans. Circuits Syst. 37(12), 1509–1519 (1990)

    MATH  Google Scholar 

  9. Benzaouia, A., Mesquine, F., Benhayoun, M.: Stabilization of 2D continuous systems with multi-delays and saturated control. Stud. Syst. Decis. Control 124, 187–199 (2018)

    MATH  Google Scholar 

  10. Chen, C.W., Tsai, J.S.H., Shieh, L.S.: Modeling of variable coefficient Roesser model for systems described by second-order partial differential equation. Circuits Syst. Signal Process. 22(5), 423–463 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Cui, Y., Shen, J., Chen, Y.: Stability analysis for positive singular systems with distributed delays. Automatica 94, 170–177 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Dhawan, A., Kar, H.: Optimal guaranteed cost control of 2-D discrete uncertain systems: an lmi approach. Signal Process. 87(12), 3075–3085 (2007)

    MATH  Google Scholar 

  13. Du, C., Xie, L., Zhang, C.: H infinity control and robust stabilization of two-dimensional systems in Roesser models. Automatica 37(2), 205–211 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Du, C., Xie, L., Zhang, C.: H \(\infty\) control and robust stabilization of two-dimensional systems in Roesser models. Automatica 37(2), 205–211 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Ebihara, Y., Ito, Y., Hagiwara, T.: Exact stability analysis of 2-D systems using LMIs. IEEE Trans. Autom. Control 51(9), 1509–1513 (2006)

    MathSciNet  MATH  Google Scholar 

  16. Eising, R.: State-space realization and inversion of 2-D systems. IEEE Trans. Circuits Syst. 27(7), 612–619 (1980)

    MathSciNet  MATH  Google Scholar 

  17. Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications, vol. 50. John Wiley & Sons, New Jersey (2011)

    MATH  Google Scholar 

  18. Fornasini, E., Marchesini, G.: State-space realization theory of two-dimensional filters. IEEE Trans. Autom. Control 21(4), 484–492 (1976)

    MathSciNet  MATH  Google Scholar 

  19. Fornasini, E., Marchesini, G.: Stability analysis of 2-D systems. IEEE Trans. Circuit Syst. 27, 1210–1217 (1980)

    MATH  Google Scholar 

  20. Fornasini, E., Valcher, M.E., Pinto, R.: A polynomial matrix approach to the structural properties of 2D positive systems. Linear Algebra Appl. 473(2–3 SPEC. ISS.), 458–473 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Hinamoto, T.: The bounded-real lemma for two-dimensional systems. In: 1990 IEEE International Conference on Systems Engineering, pp. 464–467 (1990)

  22. Kaczorek, T.: The singular general model of 2D systems and its solution. IEEE Trans. Autom. Control 33(11), 1060–1061 (1988)

    MATH  Google Scholar 

  23. Kaczorek, T.: Positive realization in canonical form of the 2D Roesser type model. Proc. IEEE Conf. Decis. Control 1, 335–336 (1997)

    Google Scholar 

  24. Kaczorek, T , ed.: Two-dimensional linear systems. Lecture notes in control and information Sciences,vol. 68. Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 55–60 (1985)

  25. Kaczorek T.: Realization problem for positive 2D hybrid systems. In: 2007 International Workshop on Multidimensional (nD) Systems, nDS , pp. 131–135 (2007)

  26. Kaczorek, T.: Positive 1D and 2D Systems. Springer Science and Business Media, Berlin (2012)

    MATH  Google Scholar 

  27. Kurek, J.E.: Stability of positive 2-D system described by the Roesser model. IEEE Trans. Circuits Syst. I 49(4), 531–533 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Liu, D.: Lyapunov stability of two-dimensional digital filters with overflow nonlinearities. IEEE Trans. Circ. Syst. I. Fundam. Theory Appl. 45(5), 574–577 (1998)

    MathSciNet  MATH  Google Scholar 

  29. Liu, L.J., Zhao, X., Di, W.: Stability of switched positive linear time-delay systems. IET Control Theory Appl. 13(7), 912–919 (2019)

    MathSciNet  MATH  Google Scholar 

  30. Lodge, J., Fahmy, M.: Stability and overflow oscillations in 2-D state-space digital filters. IEEE Trans. Acoust. Speech Signal Proc. 29(6), 1161–1171 (1981)

    MathSciNet  Google Scholar 

  31. Lu, W.-S., Lee, E.: Stability analysis for two-dimensional systems. IEEE Trans. Circuits Syst. 30(7), 455–461 (1983)

    MathSciNet  MATH  Google Scholar 

  32. Marszalek, W.: Two-dimensional state-space discrete models for hyperbolic partial differential equations. Appl. Math. Model. 8(1), 11–14 (1984)

    MathSciNet  MATH  Google Scholar 

  33. Mastorakis, N. E.: New necessary stability conditions for 2-D systems. IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 47(7), 1103–1105 (2000)

    MathSciNet  MATH  Google Scholar 

  34. Mirmomeni, M., Lucas, C., Araabi, B.N., Shafiee, M.: Forecasting sunspot numbers with the aid of fuzzy descriptor models. Space Weather (2007). https://doi.org/10.1029/2006SW000289

    Article  Google Scholar 

  35. Porter, W., Aravena, J.: 1-D models for m-D processes. IEEE Trans. Circuits Syst. 31(8), 742–744 (1984)

    MATH  Google Scholar 

  36. Rami, M.A., Tadeo, F.: Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits Syst. II 54(2), 151–155 (2007)

    Google Scholar 

  37. Roesser, R.: A discrete state-space model for linear image processing. IEEE Trans. Autom. Control 20(1), 1–10 (1975)

    MathSciNet  MATH  Google Scholar 

  38. Sadabadi, M., Shafiee, M.: Ar order determination of 3-D Arma models based on minimum eigenvalue (MEV) criterion and instrumental variable method. J. Control 3(2), 25–32 (2009)

    Google Scholar 

  39. Sadabadi, M.S., Shafiee, M., Karrari, M.: Parameter estimation of two-dimensional linear differential systems via Fourier based modulation fun. IFAC Proc. Vol. (IFAC-PapersOnline) 17(1 PART 1), 14385–14390 (2008)

    Google Scholar 

  40. Sadabadi, M.S., Shafiee, M., Karrari, M.: Two-dimensional ARMA model order determination. ISA Trans. 48(3), 247–253 (2009)

    MATH  Google Scholar 

  41. Shafiee, M.: Lyapunov theory for nonstationary 2D systems. In: IEEE International Symposium on Circuits and Systems. IEEE. (1991)

  42. Shu, Z., Lam, J., Gao, H., Baozhu, D., Ligang, W.: Positive observers and dynamic output-feedback controllers for interval positive linear systems. IEEE Trans. Circuits Syst. I 55(10), 3209–3222 (2008)

    MathSciNet  Google Scholar 

  43. Tsai, J.S.H., Li, J.S., Shieh, L.-S.: Discretized quadratic optimal control for continuous-time two-dimensional systems. IEEE Trans. Circuits Syst. I 49(1), 116–125 (2002)

    MathSciNet  MATH  Google Scholar 

  44. Valcher, M.E.: On the internal stability and asymptotic behavior of 2-D positive systems. IEEE Trans. Circuits Syst. I 44(7), 602–613 (1997)

    MathSciNet  MATH  Google Scholar 

  45. Vidyasagar, M.: Nonlinear systems analysis. Class. Appl. Math. 42, 1–492 (2002)

    MathSciNet  MATH  Google Scholar 

  46. Xie, L., Fu, M., de Souza, C.E.: H/sub infinity / control and quadratic stabilization of systems with parameter uncertainty via output feedback. IEEE Trans. Autom. Control 37(8), 1253–1256 (1992)

    MATH  Google Scholar 

  47. Xu, S., Lam, J., Lin, Z., Galkowski, K.: Positive real control for uncertain two-dimensional systems. IEEE Trans. Circuits Syst. I 49(11), 1659–1665 (2002)

    MathSciNet  MATH  Google Scholar 

  48. Xu, S., Lam, J., Lin, Z., Galkowski, K., Paszke, W., Sulikowski, B., Rogers, E., Owens, D.H.: Positive real control of two-dimensional systems: Roesser models and linear repetitive processes. Int. J. Control 76(11), 1047–1058 (2003)

    MathSciNet  MATH  Google Scholar 

  49. Yashiki, S., Matsumoto, N., Takahashi, T., Endo, T.: Bounded real property of time-delay time-varying systems and its application to learning control. Proc. IEEE Int. Conf. Control Appl. 2, 1417–1422 (1994)

    Google Scholar 

  50. Yeganefar, N., Yeganefar, N., Ghamgui, M., Moulay, E.: Lyapunov theory for 2-D nonlinear Roesser models: application to asymptotic and exponential stability. IEEE Trans. Autom. Control 58(5), 1299–304 (2012)

    MathSciNet  MATH  Google Scholar 

  51. Yu, L., Kongzhi, L., Yu, Y.: Optimal guaranteed cost control for uncertain discrete-time linear systems. Control Theory Appl. 16(5), 1971–1977 (1999)

    MathSciNet  Google Scholar 

  52. Zak, S.H.: On state-space models for systems described by partial differential equations. In: The 23rd IEEE Conference on Decision and Control, pp. 571–576 (1984)

  53. Zamani, I., Shafiee, M.: Stability analysis of uncertain switched singular time-delay systems with discrete and distributed delays. Optim. Control Appl. Methods 36(1), 1–28 (2015)

    MathSciNet  MATH  Google Scholar 

  54. Zamani, M., Shafiee, M., Zamani I.: Stability analysis of singular 2-D positive systems. In: 2021 29th Iranian Conference on Electrical Engineering (ICEE), pp. 626–631 (2021)

  55. Zeheb, E., Shorten, R., Sajja, S.S.K.: Strict positive realness of descriptor systems in state space. Int. J. Control 83(9), 1799–1809 (2010)

    MathSciNet  MATH  Google Scholar 

  56. Zhang, G., Zou, W., Zhang, X., Xuefeng, H., Zhao, Y.: Singular value decomposition based sample diversity and adaptive weighted fusion for face recognition. Digit. Signal Process. 62, 150–156 (2017)

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran

    Mahmoud Zamani & Masoud Shafiee

  2. Electrical and Electronic Engineering Department, Shahed University, Tehran, Iran

    Iman Zamani

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  1. Mahmoud Zamani
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Zamani, M., Zamani, I. & Shafiee, M. Robust Control of Positive 2-Dimensional Systems with Bounded Realness Property. Differ Equ Dyn Syst (2022). https://doi.org/10.1007/s12591-022-00615-w

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