An ISS-modular approach for adaptive neural control of pure-feedback systems. (English) Zbl 1137.93367

Wang, Cong; Hill, David J.; Ge, S. S.; Chen, Guanrong

Automatica 42, No. 5, 723-731 (2006).

Summary: Controlling non-affine non-linear systems is a challenging problem in control theory. In this paper, we consider adaptive neural control of a completely non-affine pure-feedback system using radial basis function neural networks. An ISS-modular approach is presented by combining adaptive neural design with the backstepping method, Input-to-State Stability (ISS) analysis and the small-gain theorem. The difficulty in controlling the non-affine pure-feedback system is overcome by achieving the so-called “ISS-modularity” of the controller-estimator. Specifically, a neural controller is designed to achieve ISS for the state error subsystem with respect to the neural weight estimation errors, and a neural weight estimator is designed to achieve ISS for the weight estimation subsystem with respect to the system state errors. The stability of the entire closed-loop system is guaranteed by the small-gain theorem. The ISS-modular approach provides an effective way for controlling non-affine non-linear systems. Simulation studies are included to demonstrate the effectiveness of the proposed approach.

MSC:

93C40	Adaptive control/observation systems
93C10	Nonlinear systems in control theory
93B52	Feedback control
93D25	Input-output approaches in control theory
92B20	Neural networks for/in biological studies, artificial life and related topics

Keywords:

adaptive neural control; pure-feedback systems; non-affine systems; input-to-state stability; small-gain theorem

References:

[1]	Apostol, T. M., Mathematical analysis (1963), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0126.28202
[2]	Choi, J. Y.; Farrell, J. A., Adaptive observer backstepping control using neural networks, IEEE Transactions on Neural Networks, 12, 5, 1103-1112 (2001)
[3]	Christofides, P. D.; Teel, A. R., Singular perturbations and input-to-state stability, IEEE Transactions on Automatic Control, 41, 11, 1645-1650 (1996) · Zbl 0864.93086
[4]	Dong, X.; Chen, G.; Chen, L., Adaptive control of the uncertain Duffing oscillator, International Journal of Bifurcation and Chaos, 7, 7, 1651-1658 (1997) · Zbl 0967.93512
[5]	Ferrara, A.; Giacomini, L., Control of a class of mechanical systems with uncertainties via a constructive adaptive/second order VSC approach, Transactions of ASME, Journal of Dynamic Systems, Measurement and Control, 122, 1, 33-39 (2000)
[6]	Ge, S. S.; Hang, C. C.; Lee, T. H.; Zhang, T., Stable adaptive neural network control (2001), Kluwer Academic: Kluwer Academic Norwell, USA · Zbl 1001.93002
[7]	Ge, S. S.; Wang, C., Direct adaptive NN control of a class of nonlinear systems, IEEE Transactions on Neural Networks, 13, 1, 214-221 (2002)
[8]	Ge, S. S.; Wang, C., Adaptive NN control of uncertain nonlinear pure-feedback systems, Automatica, 38, 671-682 (2002) · Zbl 0998.93025
[9]	Haykin, S., Neural networks: A comprehensive foundation (1999), Prentice-Hall: Prentice-Hall New Jersey · Zbl 0934.68076
[10]	Hill, D. J., A generalization of the small-gain theorem for nonlinear feedback systems, Automatica, 27, 1043-1045 (1991)
[11]	Hunt, L. R.; Meyer, G., Stable inversion for nonlinear systems, Automatica, 33, 1549-1554 (1997) · Zbl 0890.93046
[12]	Isidori, A., Nonlinear control systems II (1999), Springer: Springer London · Zbl 0924.93038
[13]	Jiang, Z. P.; Teel, A. R.; Praly, L., Small-gain theorem for ISS systems and applications, Mathematics of Control, Signals, and Systems, 7, 95-120 (1994) · Zbl 0836.93054
[14]	Kanellakopoulos, I.; Kokotovic, P. V.; Morse, A. S., Systematic design of adaptive controller for feedback linearizable systems, IEEE Transactions on Automatic Control, 36, 11, 1241-1253 (1991) · Zbl 0768.93044
[15]	Kokotovic, P.; Arcak, M., Constructive nonlinear control: A historical perspective, Automatica, 37, 637-662 (2001) · Zbl 1153.93301
[16]	Krstic, M.; Kanellakopoulos, I.; Kokotovic, P., Nonlinear and adaptive control design (1995), Wiley: Wiley New York · Zbl 0763.93043
[17]	Kurdila, A. J.; Narcowich, F. J.; Ward, J. D., Persistency of excitation in identification using radial basis function approximants, SIAM Journal of Control and Optimization, 33, 2, 625-642 (1995) · Zbl 0831.93015
[18]	Kwan, C.; Lewis, F. L., Robust backstepping control of nonlinear systems using neural networks, IEEE Transactions on Systems, Man and Cybernetics, Part A, 30, 753-766 (2000)
[19]	Lewis, F. L.; Jagannathan, S.; Yeildirek, A., Neural network control of robot manipulators and nonlinear systems (1999), Taylor & Francis: Taylor & Francis London
[21]	Mareels, I. M.Y.; Hill, D. J., Monotone stability of nonlinear feedback systems, Journal of Mathematical Systems, Estimation, and Control, 2, 275-291 (1992) · Zbl 0776.93039
[22]	Polycarpou, M. M.; Mears, M. J., Stable adaptive tracking of uncertain systems using nonlinearly parametrized on-line approximators, International Journal of Control, 70, 3, 363-384 (1998) · Zbl 0945.93563
[23]	Seto, D.; Annaswamy, A. M.; Baillieul, J., Adaptive control of nonlinear systems with a triangular structure, IEEE Transactions on Automatic Control, 39, 1411-1428 (1994) · Zbl 0806.93034
[24]	Sontag, E. D., Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34, 435-443 (1989) · Zbl 0682.93045
[25]	Sontag, E. D.; Wang, Y., New characterizations of input-to-state stability, IEEE Transactions on Automatic Control, 41, 1283-1294 (1996) · Zbl 0862.93051
[26]	Vidyasagar, M., Nonlinear systems analysis (1993), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0900.93132
[27]	Wang, D.; Huang, J., Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form, Automatica, 38, 1365-1372 (2002) · Zbl 0998.93026
[28]	Zhang, Y.; Peng, P. Y.; Jiang, Z. P., Stable neural controller design for unknown nonlinear systems using backstepping, IEEE Transactions on Neural Networks, 11, 1347-1359 (2000)