## High-order $$\mathcal{D}^{\alpha}$$-type iterative learning control for fractional-order nonlinear time-delay systems.(English)Zbl 1263.93099

Summary: This paper presents a high-order $$\mathcal{D}^{\alpha}$$-type Iterative Learning Control (ILC) scheme for a class of fractional-order nonlinear time-delay systems. First, a discrete system for $$\mathcal{D}^{\alpha}$$-type ILC is established by analyzing the control and learning processes, and the ILC design problem is then converted to a stabilization problem for this discrete system. Next, by introducing a suitable norm and using a generalized Gronwall-Bellman Lemma, a sufficient condition for the robust convergence with respect to the bounded external disturbance of the control input and the tracking errors is obtained. Finally, the validity of the method is verified by a numerical example.

### MSC:

 93C15 Control/observation systems governed by ordinary differential equations 68T05 Learning and adaptive systems in artificial intelligence 34A08 Fractional ordinary differential equations
Full Text:

### References:

 [1] Podlubny, I.: Fractional Differential Equations, pp. 12–18. Academic Press, New York (1999) · Zbl 0924.34008 [2] Hilfer, R.: Application of Fractional Calculus in Physics, pp. 22–24. World Scientific, Singapore (2000) · Zbl 0998.26002 [3] Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D.: Fractional-Order Systems and Controls: Fundamentals and Applications, pp. 431–438. Springer, New York (2010) · Zbl 1211.93002 [4] Li, Y., Chen, Y.Q., Ahn, H.S.: Fractional-order iterative learning control for fractional-order linear systems. Asian J. Control 13(1), 54–63 (2011) · Zbl 1248.93085 [5] Luo, Y., Chen, Y.Q.: Fractional order [proportional derivative] controller for a class of fractional order systems. Automatica 45, 2446–2450 (2009) · Zbl 1183.93053 [6] Krishna, B.T.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 91, 386–426 (2011) · Zbl 1203.94035 [7] Lan, Y.H., Huang, H.X., Zhou, Y.: Observer-based robust control of 0<1 fractional-order uncertain systems: a linear matrix inequality approach. IET Control Theory Appl. 6, 229–234 (2012) [8] Xu, J.X., Tan, Y.: Linear and Nonlinear Iterative Learning Control, pp. 47–51. Springer, Berlin (2003) · Zbl 1021.93002 [9] Ahn, H.S., Moore, K.L., Chen, Y.Q.: Iterative Learning Control: Robustness and Monotonic Convergence for Interval Systems, pp. 38–48. Springer, London (2007) · Zbl 1162.93025 [10] Hoelzle, D.J., Alleyne, A.G., Johnson, A.: Basis task approach to iterative learning control with applications to micro/robotic deposition. IEEE Trans. Control Syst. Technol. 19(5), 1138–1148 (2011) [11] Ruan, X., Bien, Z.Z., Wang, Q.: Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue norm. IET Control Theory Appl. 6, 707–714 (2012) [12] Chen, Y.Q., Moore, K.L.: On D {$$\alpha$$} -type iterative learning control. In: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, USA, pp. 32–37 (2001) [13] Lazarevic, M.P.: PD {$$\alpha$$} -type iterative learning control for fractional LTI system. In: Proceedings of the 16th International Congress of Chemical and Process Engineering, Prague, Czech Republic, pp. 271–275 (2004) [14] Li, Y., Ahn, H.-S., Chen, Y.Q.: Iterative learning control of a class of fractional order nonlinear systems. In: 2010 IEEE International Symposium on Intelligent Control, Yokohama, Japan, pp. 779–783 (2010) [15] Sun, M., Wang, D., Wang, Y.: Varying-order iterative learning control against perturbed initial conditions. J. Franklin Inst. 347, 1526–1549 (2010) · Zbl 1202.93053 [16] Luo, Y., Chen, Y.Q., Pi, Y.G.: Dynamic high order periodic adaptive learning compensator for cogging effect in permanent magnet synchronous motor servo system. IET Control Theory Appl. 5, 669–680 (2011) [17] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002) · Zbl 1014.34003 [18] Deng, W.H.: Smoothness and stability of the solutions for nonlinear fractional differential equations. Nonlinear Anal. 72, 1768–1777 (2010) · Zbl 1182.26009
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.