American Institute of Mathematical Sciences

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October  2010, 6(4): 861-880. doi: 10.3934/jimo.2010.6.861

Adaptive control of nonlinear systems using fuzzy systems

Tayel Dabbous 1,

1. 

Dep. of Elec. Eng., Higher Technological Institute, Ramadan 10th City, Egypt

Received  March 2007 Revised  June 2010 Published  September 2010

In this paper we consider the adaptive control problem for a class of systems governed by nonlinear differential equations. Using Takagi-Sugeno approach, we have proposed a fuzzy model, which is linear in nature, the behavior of which is close to that of the unknown (nonlinear) plant. Based on this fuzzy model, we have proposed certain control structure with the help of which the plant output is capable of tracking certain desired trajectory. Using a suitable objective function and variation arguments, we have developed a set of necessary conditions with the help of which the parameters of the proposed fuzzy model and controller can be determined. Based on these necessary conditions, a numerical scheme is presented for computing the unknowns. Further, the question of continuous dependence of the proposed estimator and controller on system parameters (robustness) has been studied. Finally, the proposed adaptive control scheme has been applied to two different examples to illustrate the effectiveness of the proposed adaptive control scheme.
Keywords: system identification, robustness., fuzzy systems, optimal control, .
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Tayel Dabbous. Adaptive control of nonlinear systems using fuzzy systems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 861-880. doi: 10.3934/jimo.2010.6.861
References:
[1]

N. Ahmed, "Elements of Finite Dimensional Systems and Control Theory," Pitman Monographs and Surveys in Pure and Applied Mathematics, 37,, Longman Scientific and Technical, (1988).   Google Scholar

[2]

M. Alata and K. Demirli, Adaptive control of a class of nonlinear systems with first order Parameterized Sugeno fuzzy approximator,, IEEE Trans. on Syst., (2001), 410.   Google Scholar

[3]

G. Cybenko, Approximation by superposition of sigmoidal functions,, Math. Contr. Signals Syst., 2 (1989), 303.  doi: 10.1007/BF02551274.  Google Scholar

[4]

T. Dabbous, Filtering of linear partially observed stochastic systems: The fuzzy logic approach,, Dynamic and Contr., 11 (2001), 315.  doi: 10.1023/A:1020898203511.  Google Scholar

[5]

T. Dabbous, Fuzzy optimal control for bilinear stochastic systems with fuzzy parameters,, Dynamic and Contr., 11 (2001), 243.  doi: 10.1023/A:1015224002970.  Google Scholar

[6]

T. Dabbous and M. Bayoumi, Optimal control for partially observed nonlinear deterministic systems with fuzzy parameters,, Dynamic and Contr., 11 (2001), 353.  doi: 10.1023/A:1020867121258.  Google Scholar

[7]

A. Fradkov, Speed-gradient scheme and its applications in adaptive control,, Aut. Remote Control, 40 (1979), 1333.   Google Scholar

[8]

F. Girosi and T. Poggio, Network and best approximation property,, Biol. Cybern., 63 (1990), 169.  doi: 10.1007/BF00195855.  Google Scholar

[9]

T. Johansen and B. Foss, Constructing NARMAX models using ARMAX models,, Int. J. Contr., 58 (1993), 1125.  doi: 10.1080/00207179308923046.  Google Scholar

[10]

T. Johansen and P. Ioannu, Robust adaptive control of minimum phase nonlinear systems,, Adaptive Control Signal Processing, 10 (1996), 61.  doi: 10.1002/(SICI)1099-1115(199601)10:1<61::AID-ACS387>3.0.CO;2-H.  Google Scholar

[11]

L. Karsenti, Adaptive tracking strategy for a class of nonlinear systems,, IEEE Trans. on Aut. Contr., 43 (1998), 1272.   Google Scholar

[12]

M. Krstic and L. Kanellakopoulos, "Nonlinear Adaptive Control Design,", Wiely, (1995).   Google Scholar

[13]

Panteley E. Loria, A. and H. Nijmeijer, Control of chaotic duffin equation with uncertainty in all parameters,, IEEE Trans. on Circuit Sys., 45 (1998), 1252.  doi: 10.1109/81.736558.  Google Scholar

[14]

R. Marino and P. Tomel, Global adaptive feedback control of nonlinear systems: Part II: Nonliear parameterization,, IEEE Trans. Aut. Control, 38 (1993), 17.  doi: 10.1109/9.186309.  Google Scholar

[15]

R. Marino and P. Tomel, "Nonlinear Adaptive Control Design: Geometric, Adaptive and Robust,", Prentic-Hall, (1995).   Google Scholar

[16]

A. Morris and A Montague, Artificial neural networks: Studies in process modeling and control,, Trans. ICHEME, (1994), 3.   Google Scholar

[17]

J. Park and T. Sandberg, Universal approximation using radial function networks,, Neural Comp., 3 (1991), 246.  doi: 10.1162/neco.1991.3.2.246.  Google Scholar

[18]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its application to modeling and control,, IEEE Trans. on Sys., (1985), 116.   Google Scholar

[19]

L. Wang, "Adaptive Fuzzy Systems and Control,", Prentice-Hall, (1994).   Google Scholar

[20]

L. Wang and J. Mendel, Back propagation fuzzy system as nonlinear system identifier,, Proc. IEEE Int. Conf. on Fuzzy Systems, (1992), 1409.   Google Scholar

[21]

A. Yesidirek and F. Lewis, Feedback linearization using neural networks,, Automatica, 31 (1995), 1659.  doi: 10.1016/0005-1098(95)00078-B.  Google Scholar

[22]

T. Zhang and C. Hang, Adaptive control of first order systems with nonlinear parameterization,, IEEE Trans. Aut. Control, 45 (2000), 1512.  doi: 10.1109/9.871761.  Google Scholar

show all references

References:
[1]

N. Ahmed, "Elements of Finite Dimensional Systems and Control Theory," Pitman Monographs and Surveys in Pure and Applied Mathematics, 37,, Longman Scientific and Technical, (1988).   Google Scholar

[2]

M. Alata and K. Demirli, Adaptive control of a class of nonlinear systems with first order Parameterized Sugeno fuzzy approximator,, IEEE Trans. on Syst., (2001), 410.   Google Scholar

[3]

G. Cybenko, Approximation by superposition of sigmoidal functions,, Math. Contr. Signals Syst., 2 (1989), 303.  doi: 10.1007/BF02551274.  Google Scholar

[4]

T. Dabbous, Filtering of linear partially observed stochastic systems: The fuzzy logic approach,, Dynamic and Contr., 11 (2001), 315.  doi: 10.1023/A:1020898203511.  Google Scholar

[5]

T. Dabbous, Fuzzy optimal control for bilinear stochastic systems with fuzzy parameters,, Dynamic and Contr., 11 (2001), 243.  doi: 10.1023/A:1015224002970.  Google Scholar

[6]

T. Dabbous and M. Bayoumi, Optimal control for partially observed nonlinear deterministic systems with fuzzy parameters,, Dynamic and Contr., 11 (2001), 353.  doi: 10.1023/A:1020867121258.  Google Scholar

[7]

A. Fradkov, Speed-gradient scheme and its applications in adaptive control,, Aut. Remote Control, 40 (1979), 1333.   Google Scholar

[8]

F. Girosi and T. Poggio, Network and best approximation property,, Biol. Cybern., 63 (1990), 169.  doi: 10.1007/BF00195855.  Google Scholar

[9]

T. Johansen and B. Foss, Constructing NARMAX models using ARMAX models,, Int. J. Contr., 58 (1993), 1125.  doi: 10.1080/00207179308923046.  Google Scholar

[10]

T. Johansen and P. Ioannu, Robust adaptive control of minimum phase nonlinear systems,, Adaptive Control Signal Processing, 10 (1996), 61.  doi: 10.1002/(SICI)1099-1115(199601)10:1<61::AID-ACS387>3.0.CO;2-H.  Google Scholar

[11]

L. Karsenti, Adaptive tracking strategy for a class of nonlinear systems,, IEEE Trans. on Aut. Contr., 43 (1998), 1272.   Google Scholar

[12]

M. Krstic and L. Kanellakopoulos, "Nonlinear Adaptive Control Design,", Wiely, (1995).   Google Scholar

[13]

Panteley E. Loria, A. and H. Nijmeijer, Control of chaotic duffin equation with uncertainty in all parameters,, IEEE Trans. on Circuit Sys., 45 (1998), 1252.  doi: 10.1109/81.736558.  Google Scholar

[14]

R. Marino and P. Tomel, Global adaptive feedback control of nonlinear systems: Part II: Nonliear parameterization,, IEEE Trans. Aut. Control, 38 (1993), 17.  doi: 10.1109/9.186309.  Google Scholar

[15]

R. Marino and P. Tomel, "Nonlinear Adaptive Control Design: Geometric, Adaptive and Robust,", Prentic-Hall, (1995).   Google Scholar

[16]

A. Morris and A Montague, Artificial neural networks: Studies in process modeling and control,, Trans. ICHEME, (1994), 3.   Google Scholar

[17]

J. Park and T. Sandberg, Universal approximation using radial function networks,, Neural Comp., 3 (1991), 246.  doi: 10.1162/neco.1991.3.2.246.  Google Scholar

[18]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its application to modeling and control,, IEEE Trans. on Sys., (1985), 116.   Google Scholar

[19]

L. Wang, "Adaptive Fuzzy Systems and Control,", Prentice-Hall, (1994).   Google Scholar

[20]

L. Wang and J. Mendel, Back propagation fuzzy system as nonlinear system identifier,, Proc. IEEE Int. Conf. on Fuzzy Systems, (1992), 1409.   Google Scholar

[21]

A. Yesidirek and F. Lewis, Feedback linearization using neural networks,, Automatica, 31 (1995), 1659.  doi: 10.1016/0005-1098(95)00078-B.  Google Scholar

[22]

T. Zhang and C. Hang, Adaptive control of first order systems with nonlinear parameterization,, IEEE Trans. Aut. Control, 45 (2000), 1512.  doi: 10.1109/9.871761.  Google Scholar

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