Advertisement

Soft Computing

, Volume 23, Issue 23, pp 12277–12293 | Cite as

Variable-structure backstepping controller for multivariable nonlinear systems with actuator nonlinearities based on adaptive fuzzy system

  • Mohammed Haddad
  • Farouk Zouari
  • Abdesselem BoulkrouneEmail author
  • Sarah Hamel
Foundations
  • 31 Downloads

Abstract

In this paper, a novel robust adaptive fuzzy control is presented for a quite general class of multivariable nonlinear systems with actuators’ nonlinearities (saturation with dead zone) and uncertain dynamics. The backstepping concept in combination with the variable-structure control framework and Lyapunov approach is used to design this adaptive fuzzy control. The fuzzy systems are incorporated in the controller for approximating online the unknown system dynamics. In the controller design and stability analysis, the control gain matrices, which are not necessarily symmetric and definite, are decomposed via the so-called SDU matrix decomposition lemma into a product of three main useful matrices, namely a symmetric definite-positive matrix, a diagonal constant matrix with + 1 or − 1 in its main diagonal and a unity upper triangular matrix. It is shown that the proposed adaptive fuzzy control is able to ensure the uniform ultimate boundedness of all solutions of the closed-loop system, as well as the convergence of the underlying tracking errors. Finally, in a numerical simulation framework, the effectiveness of the presented controller is illustrated on two practical examples.

Keywords

Fuzzy control Adaptive backstepping control Variable-structure control MIMO nonlinear systems Actuator nonlinearities Robot manipulator Helicopter system 

Notes

Compliance with ethical standards

Conflict of interest

The author declares that they have no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

References

  1. Boulkroune A, M’Saad M (2011) A fuzzy adaptive variable-structure control scheme for uncertain chaotic MIMO systems with sector nonlinearities and dead-zones. Expert Syst Appl 38(12):14744–14750CrossRefGoogle Scholar
  2. Boulkroune A, Tadjine M, M’Saad M, Farza M (2008a) Unified approach for design of indirect adaptive output-feedback fuzzy controller. Intell Syst Technol Appl 5(1–2):83–103zbMATHGoogle Scholar
  3. Boulkroune A, Tadjine M, M’Saad M, Farza M (2008b) How to design a fuzzy adaptive control based on observers for uncertain affine nonlinear systems. Fuzzy Sets Syst 159(8):926–948zbMATHCrossRefGoogle Scholar
  4. Boulkroune A, Tadjine M, M’Saad M, Farza M (2009) Adaptive fuzzy controller for non-affine systems with zero dynamics. Int J Syst Sci 40(4):367–382MathSciNetzbMATHCrossRefGoogle Scholar
  5. Boulkroune A, M’Saad M, Chekireb H (2010a) Design of a fuzzy adaptive controller for MIMO nonlinear time-delay systems with unknown actuator nonlinearities and unknown control direction. Inf Sci 180(24):5041–5059MathSciNetzbMATHCrossRefGoogle Scholar
  6. Boulkroune A, Tadjine M, M’Saad M, Farza M (2010b) Fuzzy adaptive controller for MIMO nonlinear systems with known and unknown control direction. Fuzzy Sets Syst 161(3):797–820MathSciNetzbMATHCrossRefGoogle Scholar
  7. Boulkroune A, M’Saad M, Farza M (2011) Adaptive fuzzy controller for multivariable nonlinear state time-varying delay systems subject to input nonlinearities. Fuzzy Sets Syst 164(1):45–65MathSciNetzbMATHCrossRefGoogle Scholar
  8. Boulkroune A, M’Saad M, Farza M (2012) Fuzzy approximation-based indirect adaptive controller for multi-input multi-output non-affine systems with unknown control direction. IET Control Theory Appl 6(17):2619–2629MathSciNetCrossRefGoogle Scholar
  9. Boulkroune A, Msaad M, Farza M (2014) State and output feedback fuzzy variable structure controllers for multivariable nonlinear systems subject to input nonlinearities. Int J Adv Manuf Technol 71:539–556CrossRefGoogle Scholar
  10. Boulkroune A, M’saad M, Farza M (2017) Adaptive fuzzy system-based variable-structure controller for multivariable nonaffine nonlinear uncertain systems subject to actuator nonlinearities. Neural Comput Appl 28(11):3371–3384CrossRefGoogle Scholar
  11. Bounar N, Boulkroune A, Boudjema F (2014) Adaptive fuzzy control of doubly-fed induction machine. J Control Eng Appl Inform 16(2):98–110Google Scholar
  12. Chen M, Tao G (2016) Adaptive fault-tolerant control of uncertain nonlinear large-scale systems with unknown dead zone. IEEE Trans Cybern 46(8):1851–1862CrossRefGoogle Scholar
  13. Chen M, Ge SS, How BVE (2010) Robust adaptive neural network control for a class of uncertain MIMO nonlinear systems with input nonlinearities. IEEE Trans Neural Netw 21(5):796–812CrossRefGoogle Scholar
  14. Farrell J, Polycarpou M, Sharma M (2004) On-line approximation based control of uncertain nonlinear systems with magnitude, rate and bandwidth constraints on the states and actuators. In: Proceedings of the American control conference. IEEE, Boston, Ma, USA, vol 3, pp 2557–2562Google Scholar
  15. Farrell J, Sharma M, Polycarpou M (2005) Backstepping-based flight control with adaptive function approximation. J Guid Control Dyn 28(6):1089–1102CrossRefGoogle Scholar
  16. Gambler A (2004) Multivariable adaptive state-space control: a survey. In: Proceedings of the IEEE 5th Asian control conference, vol 1, pp 185–191Google Scholar
  17. Haddad M, Boulkroune A (2016) Adaptive fuzzy system-based variable-structure controller for uncertain MIMO nonlinear systems subject to actuator nonlinearities. In: Proceedings of the 8th international conference on modelling, identification and control (ICMIC). IEEE, pp 1002–1019Google Scholar
  18. Hsu L, Costa RR, Lizarralde F (2007) Lyapunov/passivity-based adaptive control of relative degree two MIMO systems with an application to visual servoing. IEEE Trans Autom Control 52(2):364–371MathSciNetzbMATHCrossRefGoogle Scholar
  19. Hu QL, Ma GF, Xie LH (2008) Robust and adaptive variable structure output feedback control of uncertain systems with input nonlinearity. Automatica 44(2):552–559MathSciNetzbMATHCrossRefGoogle Scholar
  20. Hu C, Yao B, Wang Q (2013) Performance-oriented adaptive robust control of a class of nonlinear systems preceded by unknown dead zone with comparative experimental results. IEEE/ASME Trans Mechatron 18(1):178–189CrossRefGoogle Scholar
  21. Humusoft (2002) CE 150 helicopter model, user’s manual. Humusoft, PragueGoogle Scholar
  22. Kulkarni A (2011) Wavelet based control for a class of delayed nonlinear systems with input constraints. Expert Syst Appl 38(3):1993–1998CrossRefGoogle Scholar
  23. Labiod S, Guerra TM (2007) Direct adaptive fuzzy control of a class of nonlinear systems with input saturation. IFAC Proc 40(21):169–174zbMATHCrossRefGoogle Scholar
  24. Lai G, Liu Z, Zhang Y, Chen CP (2016) Adaptive fuzzy tracking control of nonlinear systems with asymmetric actuator backlash based on a new smooth inverse. IEEE Trans Cybern 46(6):1250–1262CrossRefGoogle Scholar
  25. Li Y, Tong S (2014) Adaptive fuzzy output-feedback control of pure-feedback uncertain nonlinear systems with unknown dead zone. IEEE Trans Fuzzy Syst 22(5):1341–1347CrossRefGoogle Scholar
  26. Li Y, Tong S, Li T (2013) Direct adaptive fuzzy backstepping control of uncertain nonlinear systems in the presence of input saturation. Neural Comput Appl 23(5):1207–1216CrossRefGoogle Scholar
  27. Li Y, Tong S, Li T (2014) Adaptive fuzzy output-feedback control for output constrained nonlinear systems in the presence of input saturation. Fuzzy Sets Syst 248:138–155MathSciNetzbMATHCrossRefGoogle Scholar
  28. Li Y, Tong S, Li T (2015) Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control direction and unknown dead-zones. IEEE Trans Fuzzy Syst 23(4):1228–1241CrossRefGoogle Scholar
  29. Li Y, Tong S, Li T (2016) Hybrid fuzzy adaptive output feedback control design for uncertain MIMO nonlinear systems with time-varying delays and input saturation. IEEE Trans Fuzzy Syst 24(4):841–853CrossRefGoogle Scholar
  30. Li H, Wang L, Du H, Boulkroune A (2017a) Adaptive fuzzy backstepping tracking control for strict-feedback systems with input delay. IEEE Trans Fuzzy Syst 25(3):642–652CrossRefGoogle Scholar
  31. Li H, Bai L, Wang L, Zhou Q, Wang H (2017b) Adaptive neural control of uncertain nonstrict-feedback stochastic nonlinear systems with output constraint and unknown dead zone. IEEE Trans Syst Man Cybern Syst 47(8):2048–2059CrossRefGoogle Scholar
  32. Liu Z, Wang F, Zhang Y, Chen X, Chen CLP (2015) Adaptive tracking control for a class of nonlinear systems with a fuzzy dead-zone input. IEEE Trans Fuzzy Syst 23(1):193–204CrossRefGoogle Scholar
  33. Monopoli RV (1975) Adaptive control for systems with hard saturation. In: Proceedings of the IEEE conference on decision and control, Houston, Texas, USA, pp 841–843Google Scholar
  34. Park JH, Park GT (2001) Robust adaptive controller using universal approximators for nonlinear systems under input constraint. In: Proceedings of the international symposium on industrial electronics (ISIE). IEEE, Pusan, South Korea, vol 3, pp 1881–1886Google Scholar
  35. Polycarpou M, Farrell J, Sharma M (2003) On-line approximation control of uncertain nonlinear systems: issues with control input saturation. In: Proceedings of the American control conference, Denver, Colorado, USA, vol 1, pp 543–548Google Scholar
  36. Selmic RR, Lewis FL (2000) Deadzone compensation in motion control systems using neural networks. IEEE Trans Autom Control 45(5):602–613MathSciNetzbMATHCrossRefGoogle Scholar
  37. Strang G (1980) Linear algebra and its applications, 2nd edn. Academic Press, New WorkzbMATHGoogle Scholar
  38. Tao G (2014) Multivariable adaptive control: a survey. Automatica 50(11):2737–2764MathSciNetzbMATHCrossRefGoogle Scholar
  39. Tao G, Kokotovic PV (1994) Adaptive sliding control of plants with unknown dead-zone. IEEE Trans Autom Control 39(1):59–68zbMATHCrossRefGoogle Scholar
  40. Tong S, Li Y (2013a) Adaptive fuzzy output feedback control of MIMO nonlinear systems with unknown dead-zone input. IEEE Trans Fuzzy Syst 21(1):134–146CrossRefGoogle Scholar
  41. Tong S, Li Y (2013b) Adaptive fuzzy decentralized output feedback control for nonlinear large-scale systems with unknown dead-zone inputs. IEEE Trans Fuzzy Syst 21(5):913–925CrossRefGoogle Scholar
  42. Tong S, Liu C, Li Y (2010) Fuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynamical uncertainties. IEEE Trans Fuzzy Syst 18(5):845–861CrossRefGoogle Scholar
  43. Tong SC, Li YM, Feng G, Li TS (2011) Observer-based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems. IEEE Trans Syst Man Cybern Part B 41(4):1124–1135CrossRefGoogle Scholar
  44. Tong S, Li Y, Shi P (2012) Observer-based adaptive fuzzy backstepping output feedback control of uncertain MIMO pure-feedback nonlinear systems. IEEE Trans Fuzzy Syst 20(4):771–785CrossRefGoogle Scholar
  45. Tong S, Wang T, Li Y, Chen B (2013) A combined backstepping and stochastic small-gain approach to robust adaptive fuzzy output feedback control. IEEE Trans Fuzzy Syst 21(2):314–327CrossRefGoogle Scholar
  46. Wang LX (1994) Adaptive fuzzy systems and control: design and stability analysis. Prentice-Hall, Englewood CliffsGoogle Scholar
  47. Wang D, Huang J (2002) Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form. Automatica 38(8):1365–1372MathSciNetzbMATHCrossRefGoogle Scholar
  48. Xu B, Shi ZK, Yang CG, Sun FC (2014) Composite neural dynamic surface control of a class of uncertain nonlinear systems in strict-feedback form. IEEE Trans Cybern 44(12):2626–2634CrossRefGoogle Scholar
  49. Zhang T, Ge SS, Hang CC (2000) Adaptive neural network control for strict-feedback nonlinear systems using backstepping design. Automatica 36(12):1835–1846MathSciNetzbMATHCrossRefGoogle Scholar
  50. Zhang TP, Wen H, Zhu Q (2010) Adaptive fuzzy control of nonlinear systems in pure feedback form based on input-to state stability. IEEE Trans Fuzzy Syst 18(1):80–93CrossRefGoogle Scholar
  51. Zhong YS (2005) Globally stable adaptive system design for minimum phase SISO plants with input saturation. Automatica 41(9):674–692MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LAJUniversity of JijelJijelAlgeria
  2. 2.Université de Tunis El ManarTunisTunisia

Personalised recommendations