http://iet.metastore.ingenta.com
1887

Linear matrix inequality approach to local stability analysis of discrete-time Takagi–Sugeno fuzzy systems

Linear matrix inequality approach to local stability analysis of discrete-time Takagi–Sugeno fuzzy systems

For access to this article, please select a purchase option:

Buy article PDF
$19.95
(plus tax if applicable)
Buy Knowledge Pack
10 articles for $120.00
(plus taxes if applicable)

IET members benefit from discounts to all IET publications and free access to E&T Magazine. If you are an IET member, log in to your account and the discounts will automatically be applied.

Learn more about IET membership 

Recommend Title Publication to library

You must fill out fields marked with: *

Librarian details
Name:*
Email:*
Your details
Name:*
Email:*
Department:*
Why are you recommending this title?
Select reason:
 
 
 
 
 
IET Control Theory & Applications — Recommend this title to your library

Thank you

Your recommendation has been sent to your librarian.

This study deals with the problem of local stability analysis and the computation of invariant subsets of the domain of attraction (DA) for discrete-time Takagi–Sugeno fuzzy systems. Based on the fuzzy Lyapunov functions, new sufficient conditions and an iterative scheme are proposed in order to prove the local stability and to estimate the DA. The mean value theorem and polytopic type bounds on the gradient of the membership functions are used to consider the relation between the membership functions at samples k and k + 1. Each step of the iterative procedure consists of linear matrix inequalities (LMIs) or single-parameter minimisation problems subject to LMI constraints, which are solvable via convex optimisations. Finally, examples compare the proposed conditions with existing tests.

References

    1. 1)
      • 1. Tanaka, K., Wang, H.O.: ‘Fuzzy control systems design and analysis’ (John Wiley and Sons, New York, USA, 2001).
    2. 2)
      • 2. Khalil, H.K.: ‘Nonlinear systems’ (Prentice-Hall, Upper Saddle River, NJ, 2001, 3rd edn.).
    3. 3)
      • 3. Boyd, S., Ghaoui, , El, L., Féron, E., Balakrishnan, V.: ‘Linear matrix inequalities in systems and control theory’ (SIAMPhiladelphia, USA, 1994).
    4. 4)
      • 4. Kim, E., Lee, H.: ‘New approaches to relaxed quadratic stability condition of fuzzy control systems’, IEEE Trans. Fuzzy Syst., 2000, 8, (5), pp. 523534 (doi: 10.1109/91.873576).
    5. 5)
      • 5. Tanaka, K., Ikeda, T., Wang, H.O.: ‘Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs’, IEEE Trans. Fuzzy Syst., 1998, 6, (2), pp. 250265 (doi: 10.1109/91.669023).
    6. 6)
      • 6. Tuan, H.D., Apkarian, P., Narikiyo, T., Yamamoto, Y.: ‘Parameterized linear matrix inequality techniques in fuzzy control system design’, IEEE Trans. Fuzzy Syst., 2001, 9, (2), pp. 324332 (doi: 10.1109/91.919253).
    7. 7)
      • 7. Vadivel, P., Sakthivel, R., Mathiyalagan, K., Thangaraj, P.: ‘Robust stabilisation of non-linear uncertain Takagi–Sugeno fuzzy systems by H control’, IET Control Theory Appl., 2012, 6, (16), pp. 25562566 (doi: 10.1049/iet-cta.2012.0626).
    8. 8)
      • 8. Chadli, M., Guerra, T.M.: ‘LMI solution for robust static output feedback control of discrete Takagi–Sugeno fuzzy models’, IEEE Trans. Fuzzy Syst., 2012, 20, (6), pp. 11601165 (doi: 10.1109/TFUZZ.2012.2196048).
    9. 9)
      • 9. Ko, J.W., Park, P.G.: ‘Further enhancement of stability and stabilisability margin for Takagi–Sugeno fuzzy systems’, IET Control Theory Appl., 2012, 6, (2), pp. 313318 (doi: 10.1049/iet-cta.2011.0009).
    10. 10)
      • 10. Ko, J.W.: ‘State-feedback H switching control for Takagi–Sugeno fuzzy systems based on partitioning the range of fuzzy weights’, IET Control Theory Appl., 2012, 6, (15), pp. 24602466 (doi: 10.1049/iet-cta.2012.0067).
    11. 11)
      • 11. Ding, B.C., Sun, H.X., Yang, P.: ‘Further studies on LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno's form’, Automatica, 2006, 42, (3), pp. 503508 (doi: 10.1016/j.automatica.2005.11.005).
    12. 12)
      • 12. Ding, B.C.: ‘Homogeneous polynomially nonquadratic stabilization of discrete-time Takagi–Sugeno systems via nonparallel distributed compensation law’, IEEE Trans. Fuzzy Syst., 2010, 18, (5), pp. 9941000 (doi: 10.1109/TFUZZ.2010.2053210).
    13. 13)
      • 13. Guerra, T.M., Vermeiren, L.: ‘LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno's form’, Automatica, 2004, 40, (5), pp. 823829 (doi: 10.1016/j.automatica.2003.12.014).
    14. 14)
      • 14. Guerra, T.M., Kruszewski, A., Bernal, M.: ‘Control law proposition for the stabilization of discrete Takagi–Sugeno models’, IEEE Trans. Fuzzy Syst., 2009, 17, (3), pp. 724731 (doi: 10.1109/TFUZZ.2008.928602).
    15. 15)
      • 15. Kruszewski, A., Wang, R., Guerra, T.M.: ‘Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time TS fuzzy models: a new approach’, IEEE Trans. Autom. Control, 2008, 53, (2), pp. 606611 (doi: 10.1109/TAC.2007.914278).
    16. 16)
      • 16. Lee, D.H., Park, J.B., Joo, Y.H.: ‘Improvement on nonquadratic stabilization of discrete-time Takagi–Sugeno fuzzy systems: multiple-parameterization approach’, IEEE Trans. Fuzzy Syst., 2010, 18, (2), pp. 425429.
    17. 17)
      • 17. Lee, D.H., Park, J.B., Joo, Y.H.: ‘Approaches to extended non-quadratic stability and stabilization conditions for discrete-time Takagi–Sugeno fuzzy systems’, Automatica, 2011, 47, (3), pp. 534538 (doi: 10.1016/j.automatica.2010.10.029).
    18. 18)
      • 18. Lee, D.H., Park, J.B., Joo, Y.H.: ‘A fuzzy Lyapunov function approach to estimating the domain of attraction for continuous-time Takagi–Sugeno fuzzy systems’, Inf. Sci., 2011, 185, (1), pp. 230248 (doi: 10.1016/j.ins.2011.06.008).
    19. 19)
      • 19. Lee, D.H., Park, J.B., Joo, Y.H.: ‘Further improvement of periodic control approach for relaxed stabilization condition of discrete-time Takagi–Sugeno fuzzy systems’, Fuzzy Sets Syst., 2011, 174, (1), pp. 5065 (doi: 10.1016/j.fss.2011.02.013).
    20. 20)
      • 20. Lee, D.H., Park, J.B., Joo, Y.H.: ‘A new fuzzy Lyapunov function for relaxed stability condition of continuous-time Takagi–Sugeno fuzzy systems’, IEEE Trans. Fuzzy Syst., 2011, 19, (4), pp. 785791 (doi: 10.1109/TFUZZ.2011.2142315).
    21. 21)
      • 21. Xie, X., Ma, H., Zhao, Y., Ding, D.-W., Wang, Y.: ‘Control synthesis of discrete-time T–S fuzzy systems based on a novel non-PDC control scheme’, IEEE Trans. Fuzzy Syst., 2013, 21, (1), pp. 147157 (doi: 10.1109/TFUZZ.2012.2210049).
    22. 22)
      • 22. Zhang, H., Xie, X.: ‘Relaxed stability conditions for continuous-time T–S fuzzy-control systems via augmented multi-indexed matrix approach’, IEEE Trans. Fuzzy Syst., 2011, 19, (3), pp. 478492 (doi: 10.1109/TFUZZ.2011.2114887).
    23. 23)
      • 23. Chen, Y.-J., Ohtake, H., Tanaka, K., Wang, W.-J., Wang, H.O.: ‘Relaxed stabilisation criterion for discrete T–S fuzzy systems by minimum-type piecewise non-quadratic Lyapunov function’, IET Control Theory Appl., 2012, 6, (12), pp. 19181925 (doi: 10.1049/iet-cta.2010.0697).
    24. 24)
      • 24. Chen, Y.-J., Ohtake, H., Tanaka, K., Wang, W.-J., Wang, H.O.: ‘Relaxed stabilization criterion for T–S fuzzy systems by minimum-type piecewise-Lyapunov-function-based switching fuzzy controller’, IEEE Trans. Fuzzy Syst., 2012, 20, (6), pp. 11661173 (doi: 10.1109/TFUZZ.2012.2196049).
    25. 25)
      • 25. Lam, H.K.: ‘LMI-based stability analysis for fuzzy-model-based control systems using artificial T–S fuzzy model’, IEEE Trans. Fuzzy Syst., 2011, 19, (3), pp. 505513 (doi: 10.1109/TFUZZ.2011.2116027).
    26. 26)
      • 26. Tan, W., Packard, A.: ‘Stability Region analysis using polynomial and composite polynomial Lyapunov functions and Sum-of-Squares programming’, IEEE Trans. Autom. Control, 2008, 53, (2), pp. 565571 (doi: 10.1109/TAC.2007.914221).
    27. 27)
      • 27. Topcu, U., Packard, A.: ‘Local stability analysis for uncertain nonlinear systems’, IEEE Trans. Autom. Control, 2009, 54, (5), pp. 10421047 (doi: 10.1109/TAC.2009.2017157).
    28. 28)
      • 28. Bernal, M., Guerra, T.M.: ‘Generalized nonquadratic stability of continuous-time Takagi–Sugeno models’, IEEE Trans. Fuzzy Syst., 2010, 18, (4), pp. 815822 (doi: 10.1109/TFUZZ.2010.2049113).
    29. 29)
      • 29. Bernal, M., Sala, A., Jaadari, A., Guerra, T.M.: ‘Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions’, Fuzzy Sets Syst., 2011, 185, (1), pp. 514 (doi: 10.1016/j.fss.2011.07.008).
    30. 30)
      • 30. Lee, D.H.: ‘Domain of attraction analysis for continuous-time Takagi–Sugeno fuzzy systems: an LMI approach’. Proc. 51st IEEE Conf. Decision and Control, Hawaii, USA, December 2012, pp. 61876192.
    31. 31)
      • 31. Pan, J., Fei, S., Guerra, T.M., Jaadari, A.: ‘Non–quadratic local stabilisation for continuous-time Takagi–Sugeno fuzzy models: a descriptor system method’, IET Control Theory Appl., 2012, 6, (12), pp. 19091917 (doi: 10.1049/iet-cta.2011.0380).
    32. 32)
      • 32. Pan, J., Guerra, T.M.: ‘Nonquadratic stabilization of continuous T–S fuzzy models: LMI solution for a local approach’, IEEE Trans. Fuzzy Syst., 2012, 20, (3), pp. 594602 (doi: 10.1109/TFUZZ.2011.2179660).
    33. 33)
      • 33. Lee, D.H.: ‘Local stability and stabilization of discrete-time Takagi–Sugeno fuzzy systems using bounded variation rates of the membership functions’. Proc. IEEE Symp. Series on Computational Intelligence, Singapore, April 2013(to be published).
    34. 34)
      • 34. Gahinet, P., Nemirovski, A., Laub, A.J., Chilali, M.: ‘LMI control toolbox’ (Natick, MathWorks, 1995).
    35. 35)
      • 35. Löfberg, J.: ‘YALMIP: a toolbox for modeling and optimization in MATLAB’. Proc. IEEE Int. Symp. Computer Aided Control Systems Design.Taipei, Taiwan, September 2004, pp. 284289.
    36. 36)
      • 36. Strum, J.F.: ‘Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones’, Optimization Methods and Software, 1999, 11–12, pp. 625653http://sedumi.mcmaster.ca (doi: 10.1080/10556789908805766).
    37. 37)
      • 37. Buck, R.C.: ‘Advanced calculus’ (McGraw-Hill, 1978, 3rd edn.).
    38. 38)
      • 38. Chakraborty, A., Seiler, P., Balas, G.J: ‘Nonlinear region of attraction analysis for flight control verification and validation’, Control Eng. Pract., 2011, 19, pp. 335345 (doi: 10.1016/j.conengprac.2010.12.001).
    39. 39)
      • 39. Jarvis-Wloszek, Z.: ‘Lyapunov based analysis and controller synthesis for polynomial systems using sum-of-squares optimization’. PhD dissertation, University of California, Berkeley, jagger.me.berkeley.edu/~zachary/.
    40. 40)
      • 40. Tan, W., Packard, A.: ‘Searching for control Lyapunov functions using Sum of Squares Programming’. Proc. 42nd Annual Allerton Conf. Communication, Control and Computing, September–October 2004, pp. 210219.
    41. 41)
      • 41. Mozelli, L.A., Palhares, R.M., Souza, F.O., Mendes, E.M.: ‘Reducing conservativeness in recent stability conditions of TS fuzzy systems’, Automatica, 2009, 45, (6), pp. 15801583 (doi: 10.1016/j.automatica.2009.02.023).
    42. 42)
      • 42. Rockafellar, R.: ‘Convex analysis’ (Princeton University, 1970).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2013.0033
Loading

Related content

content/journals/10.1049/iet-cta.2013.0033
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading
Errata
An Erratum has been published for this content:
Errata
This is a required field
Please enter a valid email address