# ADAPTIVE FUZZY TRACKING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS WITH UNKNOWN DISTRIBUTED TIME-VARYING DELAYS AND UNKNOWN CONTROL DIRECTIONS

Document Type : Research Paper

Authors

Department of Applied Mathematics, Xidian University, Xi'an 710071, P.R.China

Abstract

In this paper, an adaptive fuzzy control scheme is proposed for a class of perturbed strict-feedback nonlinear systems with unknown discrete and distributed time-varying delays, and the proposed design method does not require a priori knowledge of the signs of the control gains.
Based on the backstepping technique, the adaptive fuzzy controller is constructed. The main contributions of the paper are that (i) by constructing appropriate Lyapunov functionals and using the Nussbaum functions, the adaptive tracking control problem is solved for the strict-feedback unknown nonlinear systems with the unknown discrete and distributed time-varying delays and the unknown control directions (ii) the number of adaptive parameters is independent of the number of rules of fuzzy logic systems and system state variables, which reduces the computation burden greatly. It is proven that the proposed controller guarantees
that all the signals in the closed-loop system are bounded and the system output converges to a small neighborhood of the desired reference signal. Finally, an example is used to show the effectiveness of the
proposed approach.

Keywords

#### References

\bibitem{BoMsCh:Dfacmntdsuanucd}
A. Boulkroune, M. Msaad and H. Chekireb, {\it Design of a fuzzy adaptive controller
for MIMO nonlinear time-delay systems with unknown actuator nonlinearities and
unknown control direction}, Information Sciences, {\bf 180}\textbf{(24)} (2010), 5041-5059.
\bibitem{BoMsFa:Afcmnstvdssin}
A. Boulkroune, M. Msaad and M. Farza, {\it Adaptive fuzzy controller for
multivariable nonlinear state time-varying delay systems subject to input
nonlinearities}, Fuzzy Sets and Systems, {\bf 164} \textbf{(1)} (2011), 45-65.
\bibitem{BoMsFa:Odobfacnsucgn} A. Boulkroune, M. Msaad and M. Farza, {\it On the design of observer-based fuzzy
adaptive controller for nonlinear systems with unknown control gain sign}, Fuzzy Sets
and Systems, {\bf 201}\textbf{(16)} (2012), 71-85.
\bibitem{ChLiLiShLi:Dafcnstvd}
B. Chen, X. P. Liu, K. F. Liu, P. Shi and C. Lin,
{\it Direct adaptive fuzzy control for nonlinear systems with time-varying delays},
Information Sciences, {\bf 180}\textbf{(5)} (2010), 776-792.
B. Chen, S. C. Tong and X. P. Liu, {\it Fuzzy approximate disturbance decoupling of MIMO nonlinear systems by backstepping
approach}, Fuzzy Sets and Systems, {\bf 158}\textbf{(10)} (2007), 1097-1125.

\bibitem{ChJi:Atptvnpsumnn}
W. S. Chen and L. C. Jiao, {\it Adaptive tracking for periodically time-varying and nonlinearly parameterized systems using multilayer neural networks}, IEEE Trans. Neural Networks, {\bf 21}\textbf{(2)} (2010), 345-351.
\bibitem{ChJiLiLiLi:Anbofcsnsfstvd}
W. S. Chen, L. C. Jiao, J. Li and R. H. Li, {\it Adaptive NN backstepping output-feedback control for stochastic nonlinear strict-feedback systems with time-varying delays}, IEEE Trans. Systems, Man and Cybernetics-Part B: Cybernetics, {\bf 40}\textbf{(3)} (2010), 939-950.
\bibitem{ChJiLiLi:Abfcnpspd} W. S. Chen, L. C. Jiao, R. H. Li and J. Li, {\it Adaptive backstepping fuzzy control for nonlinearly parameterized systems with periodic disturbances}, IEEE Trans. Fuzzy Systems, {\bf 18}\textbf{(4)} (2010), 674-685.
\bibitem{ChZh:Gsabfcofsuhfgs}W. S. Chen and Z. Q. Zhang,
{\it Globally stable adaptive backstepping fuzzy control for output-feedback systems with unknown high-frequency gain sign},
Fuzzy Sets and Systems, {\bf 161}\textbf{(6)} (2010), 821-836.
\bibitem{DuShYa:Annccltsns}
H. B. Du, H. H. Shao and P. J. Yao, {\it
Adaptive neural network control for a class of
low-triangular-structured nonlinear systems}, IEEE Trans. Neural
Networks, {\bf 17}\textbf{(2)} (2006), 509-514.
\bibitem{FeCaRe:Dfcsgs}G. Feng, S. G. Cao, N. W. Rees and etc., {\it Design of fuzzy control systems with guaranteed stability}, Fuzzy Sets and Systems,
{\bf 85}\textbf{(1)} (1997), 1-10.
S. Filali, S. Bououden, M. L. Fas and A. Djebebla,{\it Robust adaptive fuzzy model predictive control and its application to an industrial surge tank problem}, ICIC Express Letters, {\bf 1}\textbf{(2)} (2007), 197-202.
\bibitem{DwLiNi:Anccnptds}
D. W. C. Ho, J. M. Li and Y. G. Niu,
{\it Adaptive neural control for a class of nonlinearly parametric time delay systems}, IEEE Transactions on Neural
Networks, {\bf 16}\textbf{(3)} (2005), 625 - 635.
\bibitem{HoGeReLe:Raccusfns}
F. Hong, S. S. Ge, B. Ren and T. H. Lee, {\it Robust adaptive control for a class of uncertain strict-feedback nonlinear systems}, Int. J. Robust Nonlinear Control, {\bf 19}\textbf{(7)} (2009), 746-767.
\bibitem{HuWaGu:Afofcdntdsucd}
C. C. Hua, Q. G. Wang and X. P. Guan, {\it Adaptive fuzzy output-feedback controller
design for nonlinear time-delay systems
with unknown control direction}, IEEE Trans. Systems Man Cybernet.-Part B, {\bf 39}\textbf{(2)} (2009), 363-374.
\bibitem{LiChZhFa:Resusnnddtvd}H. Y. Li, B. Chen, Q. Zhou and S. L. Fang, {\it Robust exponential
stability for uncertain stochastic neural networks with discrete and
distributed time-varying delays},
Physics Letters A, {\bf 372}\textbf{(19)} (2008), 3385-3394.
\bibitem{LiQi:Acnpsanff} W. Lin and C. J. Qian, {\it Adaptive control of nonlinearly
parameterized systems: a nonsmooth feedback framework}, IEEE
Trans. Automatic Control, {\bf 47}\textbf{(5)} (2002), 757-774.
\bibitem{LiChWeTo:Anoftccudtns}Y. J. Liu, C. L. P. Chen,
G. X. Wen and S. C. Tong, {\it Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems}, IEEE Trans. Neural Networks, {\bf 22}\textbf{(7)} (2011), 1162-1167.
\bibitem{LiToLi:Obaftccunms}
Y. J. Liu, S. C. Tong and T. S. Li, {\it Observer-based adaptive fuzzy tracking control for a
class of uncertain nonlinear MIMO systems}, Fuzzy Sets and Systems, {\bf 164}\textbf{(1)} (2011), 25-44.
\bibitem{LiToWaLiCh:Anofcdroocunss}
Y. J. Liu, S. C. Tong, D. Wang, T. S. Li and C. L. Chen,
{\it Adaptive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO systems}, IEEE Trans. Neural Networks, {\bf 22}\textbf{(8)} (2011), 1328-1334.
\bibitem{LiToWa: Afotccuns}Y. J. Liu, S. C. Tong and W. Wang, {\it Adaptive fuzzy output
tracking control for a class of uncertain nonlinear systems},
Fuzzy Sets and Systems, {\bf 160}\textbf{(19)} (2009), 2727 - 2754.
\bibitem{LiWa:Afcunns}Y. J. Liu and W. Wang, {\it Adaptive fuzzy control for a class of uncertain nonaffine nonlinear systems}, Information Sciences,
{\bf 177}\textbf{(18)} (2007), 3901-3917.
\bibitem{LiWaToLi£ºRatcnsbbfap}
Y. J. Liu, W. Wang, S. C. Tong and Y. S. Liu, {\it Robust adaptive tracking control for nonlinear systems based on bounds of fuzzy approximation parameters}, IEEE Trans. Systems, Man, and Cybernetics, Part A: Systems and Humans, {\bf 40}\textbf{(1)} (2010), 170-184.

\bibitem{Ry:Auascns}
E. P. Ryan, {\it A universal adaptive stabilizer for a class of nonlinear systems}, Syst. Control Lett., {\bf 16}\textbf{(3)} (1991), 209-218.
\bibitem{ToLiLi:Faobcmns} S. C. Tong, C. Y. Li and Y. M. Li, {\it Fuzzy adaptive observer backstepping control for MIMO nonlinear systems}, Fuzzy Sets and Systems,
{\bf 160}\textbf{(19)} (2009), 2755-2775.
\bibitem{ToLiLiLi:Obafbccsnsfs}
S. C. Tong, Y. Li, Y. M. Li and Y. J. Liu, {\it Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems}, IEEE Trans. Systems, Man, and Cybernetics, Part B: Cybernetics, {\bf 41}\textbf{(6)} (2011), 1693 - 1704.
\bibitem{ToLi:Obfacsfns}
S. C. Tong and Y. M. Li, {\it Observer-based fuzzy adaptive control for strict-feedback nonlinear systems}, Fuzzy Sets and Systems,
{\bf 160}\textbf{(12)} (2009), 1749-1764.

\bibitem{Wa:Afscdsa}L. X. Wang,
{\it Adaptive fuzzy systems and control design and stability
analysis}, NJ: Prentice-Hall, Englewood Cliffs, 1994.
\bibitem{WaCh:Aftcntdswvcc}
M. Wang and B. Chen, {\it Adaptive fuzzy tracking control of
nonlinear time-delay systems with unknown virtual control
coefficients}, Information Sciences, {\bf 178}\textbf{(22)} (2008), 4326-4340.
\bibitem{WaCh:Aftccpsfntds} M. Wang and
B. Chen, {\it Adaptive fuzzy tracking control for a class of perturbed
strict-feedback nonlinear time-delay systems}, Fuzzy Sets and
Systems, {\bf 159}\textbf{(8)} (2008), 949-967.
\bibitem{XiFrSh:Rcddsacc} L. Xie, E. Fridman and U. Shaked, {\it Robust $H_{\infty}$ control of distributed delay systems with application to the combustion control}, IEEE Trans. Automat. Control, {\bf 46}\textbf{(12)} (2001), 1930-1935.
\bibitem{XuCh:Rofcudds}S. Xu and T. Chen, {\it Robust $H_{\infty}$ output feedback control for uncertain distributed delay systems}, Eur. J. Control,
{\bf 9}\textbf{(6)} (2003), 566-574.
\bibitem{XuCh:Alafdusdd} S. Xu and T. Chen, {\it An LMI approach to the $H_{\infty}$ filter design for uncertain systems with distributed delays}, IEEE Trans. Circuits Syst. II, {\bf 51}\textbf{(4)} (2004), 195-201.
\bibitem{XuLaChZo:Addarfudds} S. Xu, J. Lam, T. Chen and Y. Zou, {\it A delay-dependent approach to robust $H_{\infty}$ filtering for uncertain distributed delay systems}, IEEE Trans. Signal Process., {\bf 53}\textbf{(10)} (2005), 3764-3772.
\bibitem{YaFeRe:Acbsgarafcsfns} Y. S. Yang,
G. Feng and J. S. Ren, {\it A combined backstepping and small-gain
approach to robust adaptive fuzzy control for strict-feedback
nonlinear systems}, IEEE Trans. Systems Man Cybernet.-Part A,
{\bf 34}\textbf{(3)} (2004), 406-420.
\bibitem{Yu:Alarfustvdd} X. G. Yu, {\it An LMI approach to robust $H{\infty}$ filtering for uncertain systems with time-varying distributed delays}, Journal of the Franklin Institute, {\bf345}\textbf{(8)} (2008), 877-890.
\bibitem{ZhGe:Ancmnstvdsudzgs}T. P. Zhang and S. S. Ge, {\it Adaptive neural control of MIMO nonlinear state time-varying
delay systems with unknown dead-zones and gain signs}, Automatica, {\bf43}\textbf{(6)} (2007), 1021-1033.
\bibitem{ZhYi:Afcamnsudz}T. P. Zhang and Y. Yi, {\it Adaptive fuzzy control for a class of MIMO nonlinear systems with
unknown dead-zones}, Acta Automatica Sinica, {\bf33}\textbf{(1)} (2007), 96-99.