×

Robust learning control for a class of nonlinear systems with periodic and aperiodic uncertainties. (English) Zbl 1051.93039

The authors consider the output tracking problem for a nonlinear system having a triangular form \[ \begin{cases} \dot x_1= x_2+ w_1(x_1, t)+ \phi^T_1(x_1)\gamma(t)\\ \dot x_2= x_3+ w_2(x_1,x_2, t)+ \phi^T_2(x_1, x_2)\gamma(t)\\ \vdots\\ \dot x_n= u+ w_n(x_1,\dots, x_n, t)+ \phi^T_n(x_1,\dots, x_n)\gamma(t)\\ y= x_1\end{cases} \] where the functions \(w_i\) represent bounded (with known bounds) unstructured uncertainties, the functions \(\phi_i\) are known, and the function \(\gamma\) is unknown but periodic. The signal to be tracked is generated by a linear stable system. The method proposed by the authors is a combination of robust learning control and the backstepping procedure. A control law (consisting of three terms) is constructed in order to learn and approximate the unknown periodic function (by the use of a repetitive learning mechanism) and to suppress the unstructured bounded uncertainties (by using a robust control technique). The proposed control law is applied to the control of a chaotic Van der Pol attractor.

MSC:

93B51 Design techniques (robust design, computer-aided design, etc.)
93C10 Nonlinear systems in control theory
93D15 Stabilization of systems by feedback
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arimoto, S. (1985). Mathematical theory of learning with application to robot control. In K.S. Narendra (Ed.), Adaptive and Learning Systems: Theory and Application; Arimoto, S. (1985). Mathematical theory of learning with application to robot control. In K.S. Narendra (Ed.), Adaptive and Learning Systems: Theory and Application
[2] Arimoto, S.; Kawamura, S.; Miyazaki, F., Bettering operation of robots by learning, Journal of Robotic Systems, 1, 123-140 (1984)
[3] Ghosh, J.; Paden, B., Nonlinear repetitive control, IEEE Transactions on Automatic Control, 45, 949-954 (2000) · Zbl 0970.93540
[4] Jiang, Z.-P., & Praly, L. (1991). Iterative designs of adaptive controllers for systems with nonlinear integrators. Proceedings of the 30th IEEE Conference on Decision and Control; Jiang, Z.-P., & Praly, L. (1991). Iterative designs of adaptive controllers for systems with nonlinear integrators. Proceedings of the 30th IEEE Conference on Decision and Control
[5] Kanellakopoulos, I.; Kokotović, P. V.; Morse, A. S., Systematic design of adaptive controllers feedback linearizable systems, IEEE Transactions on Automatic Control, 36, 1241-1253 (1991) · Zbl 0768.93044
[6] Kokotović, P.; Arcak, M., Constructive nonlinear controlA historical perspective, Automatica, 37, 637-662 (2001) · Zbl 1153.93301
[7] Krstić, M.; Kanellakopoulos, I.; Kokotović, P., Nonlinear and Adaptive Control Design (1995), Wiley: Wiley New York · Zbl 0763.93043
[8] Moore, K., Iterative Learning Control for Deterministic Systems (1993), Springer: Springer Berlin · Zbl 0773.93002
[9] Park, B. H.; Kuc, T.-Y.; Lee, J. S., Adaptive learning control of uncertain robotic systems, International Journal of Control, 65, 725-744 (1996) · Zbl 0864.93079
[10] van der Pol, B., Forced oscillations in a circuit with nonlinear resistance (receptance with reactive triode), London, Edinburgh and Dublin Philosophical Magazine, 3, 65-80 (1927)
[11] Xu, J.-X., Analysis of iterative learning control for a class of nonlinear discrete-time systems, Automatica, 33, 1905-1907 (1997) · Zbl 0885.93031
[12] Xu, J.-X.; Qu, Z., Robust iterative learning control for a class of nonlinear systems, Automatica, 34, 983-988 (1998) · Zbl 1040.93519
[13] Xu, J.-X.; Viswanathan, B.; Qu, Z., Robust learning control for robotic manipulators with an extension to a class of nonlinear system, International Journal of Control, 73, 858-870 (2000) · Zbl 1006.93557
[14] Yamamoto, Y.; Hara, S., Relationship between internal and external stability for infinite-dimensional systems with applications to a servo problem, IEEE Transactions on Automatic Control, 33, 1044-1052 (1988) · Zbl 0661.93062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.