Advertisement

International Journal of Fuzzy Systems

, Volume 21, Issue 8, pp 2588–2599 | Cite as

Direct Adaptive Fuzzy Control of Nonlinear Descriptor Systems

  • Naeimeh Fakhr Shamloo
  • Ali Akbarzadeh KalatEmail author
  • Luigi Chisci
Article
  • 12 Downloads

Abstract

This paper deals with direct adaptive fuzzy control for uncertain affine nonlinear descriptor systems. Two cases are considered: in the first one, it is assumed that the control gain is known, while in the second one, it is an unknown-but-bounded symmetric positive definite matrix. To account for uncertainties in the system dynamics, a fuzzy system is employed to directly approximate the unknown ideal controller. The adjustable parameters of the fuzzy system are updated by either a Lyapunov-based adaptative law in the first case, or a gradient descent algorithm minimizing a suitable quadratic cost function in the second case. Furthermore, an auxiliary compensating signal is designed to guarantee that the tracking error asymptotically vanishes in both cases. Simulation results show how the proposed methods exhibit satisfactory performance thus demonstrating their effectiveness.

Keywords

Nonlinear descriptor systems Adaptive fuzzy control Lyapunov theory 

References

  1. 1.
    Xu, S., Lam, J.: Robust control and filtering of singular systems. Springer, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Tischendorf, C.: Coupled systems of differential algebraic and partial differential equations in circuit and device simulation-modeling and numerical analysis. Habilitationsschrift, Berlin (2003)Google Scholar
  3. 3.
    You, L.-S., Chen, B.-S.: Tracking control designs for both holonomic and non-holonomic constrained mechanical systems: a unified viewpoint. Int. J. Control 58(3), 587–612 (1993)zbMATHGoogle Scholar
  4. 4.
    Zhou, L., Ho, D., Zhai, G.: Stability analysis of switched linear singular systems. Automatica 49, 1481–1487 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Yip, E., Sincovec, R.: Solvability, controllability, and observability of continuous descriptor systems. IEEE Tran. Autom. Control 26(3), 702–707 (1981)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Yu, R., Wang, D.: Structural properties and poles assignability of LTI singular systems under output feedback. Automatica 39, 685–692 (2003)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Dai, L.: Singular control systems. Spring-Verlag, Berlin (1989)zbMATHGoogle Scholar
  8. 8.
    Lio, Y., Kao, Y., Gu, S., Karimi, H.: Soft variable structure control design for singular systems. J. Franklin Inst. 352, 1613–1626 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Wu, A.G., Duan, G.R., Zhou, B.: Solution to descriptor Sylvester matrix equations. IEEE Trans. Autom. Control 53(3), 811–815 (2008)zbMATHGoogle Scholar
  10. 10.
    Xu, S., Van Dooren, P., Stefan, R., Lam, J.: Robust stability and stabilization for descriptor systems with state delay and parameter uncertainty. IEEE Trans. Autom. Control 47(7), 1122–1128 (2002)zbMATHGoogle Scholar
  11. 11.
    Shu, Y., Zhu, Y.: Stability and optimal control for uncertain continuous-time singular systems. Eur. J. Control 34, 16–23 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Mukundan, R., Dayawansa, W.: Feedback control of singular system—proportional and derivative feedback of the state. Int. J. Syst. Sci. 14(6), 615–632 (1983)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ren, J., Zhang, Q.: Robust \({H_\infty }\) control for uncertain descriptor systems by proportional-derivative state feedback. Int. J. Control 83(1), 89–96 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Leduc, H., Peaucelle, D., Pittet, C.: Adaptive control LMI-based design for descriptor systems rational in the uncertainties. IFAC-PapersOnLine 49(13), 135–140 (2016)Google Scholar
  15. 15.
    Zhang, L., Huang, B.: Robust model predictive control of singular systems. IEEE Trans. Autom. Control 49(6), 1000–1006 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Han, Y., Kao, Y., Gao, C., Jiang, B.: Robust sliding mode control for uncertain discrete singular systems with time-varying delays. Int. J. Syst. Sci. 48(4), 818–827 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Li, Y., Li, K., Tong, S.: Finite-time adaptive fuzzy output feedback dynamic surface control for MIMO non-strict feedback systems. IEEE Trans. Fuzzy Syst. 27(1), 96–110 (2018)Google Scholar
  18. 18.
    Wang, N., Tong, S., Li, Y.: Observer-based adaptive fuzzy control of nonlinear non-strict feedback system with input delay. Int. J. Fuzzy Syst. 20(1), 236–245 (2018)MathSciNetGoogle Scholar
  19. 19.
    Ghavidel, H.F., Akbarzadeh Kalat, A.: Observer-based robust composite adaptive fuzzy control by uncertainty estimation for a class of nonlinear systems. Neurocomputing 230, 100–109 (2017)zbMATHGoogle Scholar
  20. 20.
    Li, Y., Tong, S.: Fuzzy adaptive back stepping decentralized control for switched nonlinear large-scale systems with switching jumps. Int. J. Fuzzy Syst. 17(1), 12–21 (2015)MathSciNetGoogle Scholar
  21. 21.
    Su, S., Chang, J., Chen, S.: The study on direct adaptive fuzzy controllers. Int. J. Fuzzy Syst. 8(3), 150–159 (2006)MathSciNetGoogle Scholar
  22. 22.
    Liu, Y., Tong, S., Wang, W., Li, Y.: Observer-based direct adaptive fuzzy control of uncertain nonlinear systems and its applications. Int. J Control Autom. Syst. 7(4), 681–690 (2009)Google Scholar
  23. 23.
    Wang, J., Rad, A.B., Chan, P.T.: Indirect adaptive fuzzy sliding mode control: part I: fuzzy switching. Fuzzy Sets Syst. 122(1), 21–30 (2001)zbMATHGoogle Scholar
  24. 24.
    Shi, W., Wang, D., Li, B.: Indirect adaptive fuzzy prescribed performance control of feedback linearisable MIMO nonlinear systems with unknown control direction. IET Control Theory Appl. 11(7), 953–961 (2017)MathSciNetGoogle Scholar
  25. 25.
    Li, Y., Tong, S.: Adaptive fuzzy output-feedback stabilization control for a class of switched nonstrict-feedback nonlinear systems. IEEE Trans. Cybern. 47, 1007–1016 (2017)Google Scholar
  26. 26.
    Zhao, X., Shi, P., Zheng, X.: Fuzzy adaptive control design and discretization for a class of nonlinear uncertain systems. IEEE Trans. Cybern. 46, 1476–1483 (2016)Google Scholar
  27. 27.
    Wang, H., Liu, P., Niu, B.: Robust fuzzy adaptive tracking control for nonaffine stochastic nonlinear switching systems. IEEE Trans. Cybern. 48, 2462–2471 (2018)Google Scholar
  28. 28.
    Li, Y.-X., Yang, G.-H.: Adaptive asymptotic tracking control of uncertain nonlinear systems with input quantization and actuator faults. Automatica 72, 177–185 (2016)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Peng, Z., Wang, D., Wang, J.: Predictor-based neural dynamic surface control for uncertain nonlinear systems in strict-feedback form. IEEE Trans. Neural Netw. Learn. Syst. 28(9), 2156–2167 (2017)MathSciNetGoogle Scholar
  30. 30.
    Peng, Z., Wang, J., Wang, D.: Distributed maneuvering of autonomous surface vehicles based on neurodynamic optimization and fuzzy approximation. IEEE Trans. Control Syst. Technol. 26(3), 1083–1090 (2018)MathSciNetGoogle Scholar
  31. 31.
    Liang, H., Zhang, Y., Huang, T., Ma, H.: Prescribed performance cooperative control for multiagent systems with input quantization. IEEE Trans. Cybern. Early Access (2019)Google Scholar
  32. 32.
    Kaheni, M., Hadad Zarif, M., Akbarzadeh Kalat, A., Fadali, M.: Soft variable structure control of linear systems via desired pole paths. Inf. Technol. Control 47, 447–457 (2018)Google Scholar
  33. 33.
    Kaheni, M., Hadad Zarif, M., Akbarzadeh Kalat, A., Fadali, M.: Radial pole paths SVSC for linear time invariant multi input systems with constrained inputs. Asian J Control Early Access (2018)Google Scholar
  34. 34.
    Liang, H., Zhang, Z., Ahn, C.: Event-triggered fault detection and isolation of discrete-time systems based on geometric technique. Express Briefs, Early Access, IEEE Trans. Circuits Syst. II (2019)Google Scholar
  35. 35.
    Newcomb, R.W., Dziurla, B.: Some circuits and systems applications of semistate theory. Circuits Syst. Signal Process. 8(3), 235–260 (1989)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Wang, L.X., Mendel, J.M.: Fuzzy basis functions, universal approximation, and orthogonal least squares learning. IEEE Trans. Neural Netw. 3(5), 807–814 (1992)Google Scholar
  37. 37.
    Lin, C., Lam, J., Wang, J., Yang, G.: Analysis on robust stability for interval descriptor systems. Syst. Control Lett. 42, 267–278 (2001)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Yoshiyuki, J., Terra, M.H.: On the Lyapunov theorem for descriptor systems. IEEE Trans. Autom. Control 47, 1926–1930 (2002)zbMATHGoogle Scholar
  39. 39.
    Takagi, T., Sugeno, M.: Fuzzy identification of systems and its aplications to modeling and control. IEEE Trans. Syst. Man Cybern. 15, 116–132 (1985)zbMATHGoogle Scholar
  40. 40.
    Wang, L.X.: Adaptive fuzzy systems and control: design and stability analysis. Prentice-Hall, Englewood Cliffs, New Jersey (1994)Google Scholar
  41. 41.
    Chang, Y.C.: Robust tracking control for nonlinear MIMO systems via fuzzy approaches. Automatica 36, 1535–1545 (2000)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Li, H.X., Tong, S.C.: A hybrid adaptive fuzzy control for a class of nonlinear MIMO systems. IEEE Trans. Fuzzy Syst. 11, 24–34 (2003)Google Scholar
  43. 43.
    Slotine, J.E., Li, W.: Applied nonlinear control. Prentice-Hall, Englewood Cliffs, NJ (1991)zbMATHGoogle Scholar
  44. 44.
    Nekoukar, V., Erfanian, A.: Adaptive fuzzy terminal sliding mode control for a class of MIMO uncertain nonlinear systems. Fuzzy Sets Syst. 179(1), 34–49 (2011)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Shi, W., Li, B.: Adaptive fuzzy control for feedback linearizable MIMO nonlinear systems with prescribed performance. 2017, Fuzzy Sets and Syst. (In Press) (2011)Google Scholar
  46. 46.
    Chang, Y.C.: Adaptive fuzzy based tracking control for nonlinear SISO systems via VSS and H-infinity approaches. IEEE Trans. Fuzzy Syst. 9, 278–292 (2001)Google Scholar
  47. 47.
    Guo, H.G., Zhang, B.J.: Observer-based variable universe adaptive fuzzy controller without additional dynamic order. Int. J. Autom. Comput. 11, 418–425 (2014)Google Scholar
  48. 48.
    Erzberger, H.: Analysis and design of model following systems by state-space techniques. In: Proc. Of JACC; 572–581 (1968)Google Scholar
  49. 49.
    Wang, C.-H., Liu, H.-L., Lin, T.-C.: Direct adaptive fuzzy-neural control with state observer and supervisory controller for unknown nonlinear dynamical systems. IEEE Trans. Fuzzy Syst. 10(1), 39–49 (2002)Google Scholar
  50. 50.
    Labiod, S., Guerra, T.M.: Direct and indirect adaptive fuzzy control for a class of MIMO nonlinear systems, advances in robot manipulators. Ernest Hall (Ed.) (2010)Google Scholar
  51. 51.
    Labiod, S., Guerra, T.M.: Adaptive fuzzy control of a class of SISO nonaffine nonlinear systems. Fuzzy Sets and Syst. 158, 1126–1137 (2007)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Peng, Z., Wang, J., Wang, J.: Constrained control of autonomous underwater vehicles based on command optimization and disturbance estimation. IEEE Trans. Ind. Electron. 66(5), 3627–3635 (2019)Google Scholar
  53. 53.
    Peng, Z., Wang, J., Han, Q.-L.: Path-following control of autonomous underwater vehicles subject to velocity and input constraints via neurodynamic optimization. IEEE Trans. Ind. Electron. Early Access (2018)Google Scholar

Copyright information

© Taiwan Fuzzy Systems Association 2019

Authors and Affiliations

  • Naeimeh Fakhr Shamloo
    • 1
  • Ali Akbarzadeh Kalat
    • 1
    Email author
  • Luigi Chisci
    • 2
  1. 1.Faculty of Electrical Engineering and RoboticShahrood University of TechnologyShahroodIran
  2. 2.Department of Information Engineering (DINFO)Florence UniversityFirenzeItaly

Personalised recommendations