This paper presents a robust second-order feedback type iterative learning control (ILC) for a class of uncertain fractional-order singular systems. Sufficient conditions for the robust convergence of the proposed type of learning control algorithm, with respect to the bounded external disturbance and uncertainity, have been established and specified for time domain. Finally, simulation examples are presented to illustrate the performance of the proposed fractional order ILC.
AhnHSMooreKChenY (2007) Iterative learning control robustness and monotonic convergence for interval systems, London: Springer-Verlag London Limited.
2.
AnliEOzkolI (2010) Classical and fractional-order analysis of the free and forced double pendulum. Engineering2: 935–949.
3.
ArimotoSKawamuraSMiyazakiF (1984) Bettering operation of robots by learning. Journal of Robotic Systems2: 123–140.
4.
BagleyRLTorvikPJ (1983) Fractional calculus – a different approach to the analysis of viscoelastically damped structures. AIAA J21: 741–748.
5.
BaleanuDDiethelmKScalasE (2012) Fractional calculus models and numerical methods, Series on Complexity, Nonlinearity and Chaos, Boston: World Scientific.
6.
BaleanuDDefterliOAgrawalOP (2009) A central difference numerical scheme for fractional optimal control problems. Journal of Vibration and Control15: 583–597.
7.
BienZHuhK (1989) High-order iterative learning control algorithm. IEE Proc. Part. -D136: 105–112.
BristowdATharayilMAlleyneAG (2006) A survey of iterative learning control: A learning-based method for high-performance tracking control. IEEE Control Syst. Mag26: 96–114.
10.
BuzurovicI (2007) Dynamic model of medical robot represented as descriptor system. International journal of information and systems sciences2: 316–333.
11.
CajićMLazarevićPM (2014) Fractional order spring/spring-pot/actuator element in a multibodysystem: Application of an expansion formula. Mechanics Research Communications62: 44–56.
12.
ChenYGongZWenC (1998) Analysis of a high-order iterative learning control algorithm for uncertain nonlinear systems with state delays. Automatica34: 345–353.
13.
Chen YQ and Moore KL (2001) On Dα type iterative learning control. In: Proceedings of the 40th IEEE Conference on Decision and Control. Orlando, FL USA, 2001, pp. 4451–4456.
14.
DaiL (1989) Singular Control Systems,118 Lecture Notes in Control and Information Sciences, New York: Springer-Verlag.
15.
DiethelmKFordNJ (2002) Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications265: 229–248.
16.
FangCHChangFR (1993) Analysis of stability robustness for generalized state-space systems with structured perturbations. Systems and Control Letters21: 109–114.
17.
HilferR (2000) Applications of Fractional Calculus in Physics, River Edge, NJ: World Scientific.
18.
JangTChoiCAhnH (1995) Iterative learning control in feedback systems. Automatica31: 243–248.
19.
IbrirS (2009) LMI approach to regularization and stabilization of linear singular systems: the discrete-time case. World Academy of Science, Engineering and Technology52: 276–279.
20.
KaczorekT (2011) Singular fractional linear systems and electrical circuits. Int. J. Appl. Math. Comput. Sci21: 379–384.
21.
KaczorekT (2013) Singular fractional continuous-time and discrete-time linear systems. Acta mechanica et automatica7: 26–33.
22.
KilbasAASrivastavaHMTrujilloJJ (2006) Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier.
23.
Kosmatopoulos EN and Christodoulou MA (1992) Stability analysis of differential-algebraic neural networks. In: Proceeding 12th World Conference of IFAC. Sydney, Australia 1992, 6: 1–5.
24.
LanYHZhouY (2013) Dα-type iterative learning control for fractional-order linear time-delay systems. Asian Journal of Control15: 669–677.
25.
Lan YH and Jun LV (2012) P-type Iterative Learning Control of Fractional Order Nonlinear Time-delay Systems. 24th Chinese Control and Decision Conference (CCDC), 2012, pp.1027–1031.
26.
Lazarević M (2003a) Higher order pd–type iterative learning control for lti system. In: Proceedings of the International Carpathian Control Conference (ICCC2003),Tatranska Lomnica, Slovak Republic, 26–29 May, pp. 512–515.
27.
Lazarević MP (2003b) PDα-type iterative learning control for fractional LTI system. In: Proceedings of the International Carpathian Control Conference (ICCC2003), Tatranska Lomnica,Slovak Republic, 26–29 May, pp. 869–872.
28.
Lazarević MP (2004) PDα-type iterative learning control for fractional LTI system. In: Proceedings of the 16th International Congress of Chemical and Process Engineering, CHISA 2004, Praha, Czech Republic 22–26 August.
29.
Lazarević MP (2010) Iterative Learning Feedback Control for Nonlinear Fractional Order System-PDα Type. In: Proceedings of the 4th IFAC Workshop Fractional Differentiation and its Applications, FDA2010. Badajoz, Spain, 18–20 October, Ar.FDA–148.
30.
Lazarević MP (2014) Advanced Topics on Applications of Fractional Calculus on Control Problems, System Stability And Modeling. In Mladenov V and Mastorakis N (eds) Advanced Topics on Applications of Fractional Calculus on Control Problems, System Stability And Modeling, pp. 43–104.
31.
LeeHSBienZ (1996) Study on robustness of iterative learning control with non-zero initial error. Int. J. of Control64: 345–359.
32.
Li Y, Chen YQ and Ahn HS (2009) Fractional order iterative learning control. In: Proceedings of the 2009 ICROS-SICE Int. Joint Conference, Fukuoka, Japan, August, pp. 3106–3110.
33.
LiYChenYQAhnHS (2011a) Fractional-order iterative learning control for fractional-order linear systems. Asian Journal of Control13: 54–63.
34.
Li Y, Chen YQ and Ahn HS (2011b) On the PDα−Type Iterative Learning Control for the Fractional-Order nonlinear Systems. In: Proceedings of the 2011 American Control Conference, San Francisco,CA,USA, June 29–July, 2011, pp. 779–782.
35.
LiYChenYQAhnHS (2013) A survey on fractional-order iterative learning control. Journal of Optimization Theory and Applications156: 127–140.
36.
Liu S, Wang JR and Wei W (2014) Iterative learning control based on a noninstantaneous impulsive fractional-order system. Journal of Vibration and Control. Epub ahead of print. DOI: 10.1177/1077546314545638.
37.
MachadoJT (1997) Analysis and design of fractional-order digital control systems. J Syst. Anal. Model Simulat27: 107–122.
Mc-ClamrochNH (1986) Singular systems of differential equations as dynamic models for constrained robot systems, Proc. of the 1986 IEEE International Conf. on Robotics and Automations3: 21–28.
40.
MillerKSRossB (1993) An introduction to the fractional calculus and fractional differential equations, New York: Wiley.
41.
MüllerPC (1993) Stability of Linear Mechanical Systems with Holonomic Constraints. Applied Mechanics Review46: 160–164.
42.
MüllerPC (2005) Modelling and control of mechatronic systems by the descriptor approach. Journal of theoretical and applied mechanics43: 593–607.
43.
MooreKL (1993) Iterative learning control for deterministic systems Advances in Industrial Control, London: Springer-Verlag.
44.
MonjeCAChenYVinagreBM (2010) Fractional-order Systems and Controls: Fundamentals and Applications, London: Springer-Verlag.
45.
OldhamKBSpanierJ (1974) The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, New York: Academic Press.
46.
Ortiguiera M and Coito F (2008) The initial conditions of Riemann-Liouville and Caputo derivatives. In: Proceedings of the 6th EUROMACH Nonlinear Dynamics Conference.
Piao FX, Zhang QL and Wang ZF (2006) Iterative learning control for uncertain singular system. Proceedings of the 2006 IEEE International Conference on Information Acquisition, 20–23 August, pp. 1469–1474.
49.
PiaoFXZhangQLWangZF (2007) Iterative Learning Control for a Class of Singular Systems. Acta automatica sinica33: 658–659.
RosenbrockHH (1974) Structural properties of linear dynamical systems. Int. J. Control20: 191–202.
52.
SabatierJFargesCTrigeassouJC (2014) Fractional systems state space description: some wrong ideas and proposed solutions. Journal of Vibration and Control20: 1076–1084.
53.
SamkoSGKilbasAAMarichevIO (1993) Fractional Integrals and derivatives: Theory and Applications, Switzerland: Gordon & Breach.
54.
Senping T, Gang H and Lun Z (2007) Iterative learning control for the singular systems with delay. In: Proceedings of the 26th Chinese Control Conference July 26–31, 2007, Zhangjiajie, Hunan, China, pp. 535–538.
55.
SkruchPMitkowskiW (2013) Fractional-order models of the ultracapacitors. In: MitkowskiWKacprzykJBaranowskJ (eds) Advances in the Theory and Applications of Non-integer Order Systems, Switzerland: Springer, pp. 281–294.
56.
Song X, Liu L and Wang Z (2012) Stabilization of singular fractional-order systems·a linear matrix inequality approach. In: Proceedings of the IEEE International Conference on Automation and Logistics,15–17 August Zhengzhou, China, pp. 19–23.
57.
UchiyamaM (1978) Formulation of high-speed motion pattern of a mechanical arm by trial (in Japanese). Trans. SICE (Soc. Instrum. Control Eng.)14: 706–712.
58.
YaoYZhuangJChang-YinS (2013) Suffcient and necessary condition of admissibility for fractional-order singular system. Acta automatica sinica39: 2160–2164.
59.
Vardulakis AI, Karampetakis NP, Antoniou E, et al. (2003) A descriptor package for Mathematica. In: Proceedings of the 11th Mediterranean Conference on Control and Automation, 18–20, June 2003, Rhodes, Greece.
60.
XieSLXieZDLiuYQ (1999) Learning control algorithm for state tracking of singular systems with delay. Systems Engineering and Electronics21: 10–16.
61.
Xu JX, Wang XW and Heng LT (1995) Analysis of continuous iterative learning control systems using current cycle feedback. In: Proceedings of the American Control Conference. Seattle, Washington, USA, pp. 4221–4225.