##
**Design of noise and period-time robust high-order repetitive control, with application to optical storage.**
*(English)*
Zbl 1138.93347

Summary: Repetitive control is useful if periodic disturbances or setpoints act on a control system. Perfect (asymptotic) disturbance rejection is achieved if the period time is exactly known. The improved disturbance rejection at the periodic frequency and its harmonics is achieved at the expense of a degraded system sensitivity at intermediate frequencies. A convex optimization problem is defined for the design of high-order repetitive controllers, where a trade-off can be made between robustness for changes in the period time and for reduction of the error spectrum in-between the harmonic frequencies. The high-order repetitive control algorithms are successfully applied in experiments with the tracking control of a CD-player system.

### MSC:

93B51 | Design techniques (robust design, computer-aided design, etc.) |

93B35 | Sensitivity (robustness) |

68P20 | Information storage and retrieval of data |

93C95 | Application models in control theory |

PDFBibTeX
XMLCite

\textit{M. Steinbuch} et al., Automatica 43, No. 12, 2086--2095 (2007; Zbl 1138.93347)

Full Text:
DOI

### References:

[1] | Ben-Tal, A., & Nemirovski, A. (2001). Lectures on modern convex optimization: Analysis, algorithms and engineering applications, MPS-SIAM series on optimization.; Ben-Tal, A., & Nemirovski, A. (2001). Lectures on modern convex optimization: Analysis, algorithms and engineering applications, MPS-SIAM series on optimization. · Zbl 0986.90032 |

[2] | Bodson, M., Sacks, A., & Khosla, P. (1994). Harmonic generation in adaptive feedforward cancellation schemes. IEEE Transactions on Automatic Control (September) 1939-1944.; Bodson, M., Sacks, A., & Khosla, P. (1994). Harmonic generation in adaptive feedforward cancellation schemes. IEEE Transactions on Automatic Control (September) 1939-1944. · Zbl 0816.93048 |

[3] | Cao, Z., & Ledwich, G. (2001). Tracking variable periodic signals with fixed sampling rate. In Proceedings of the 40th IEEE conference on decision and control (Vol. 5, pp. 4885-4890).; Cao, Z., & Ledwich, G. (2001). Tracking variable periodic signals with fixed sampling rate. In Proceedings of the 40th IEEE conference on decision and control (Vol. 5, pp. 4885-4890). |

[4] | Chang, W. S.; Such, I. H.; Kim, T. W., Analysis and design of two types of digital repetitive control systems, Automatica, 31, 5, 741-746 (1995) · Zbl 0825.93447 |

[5] | Chang, W. S., Suh, I. H., & Oh, J. H. (1998). Synthesis and analysis of digital multiple repetitive control systems. In Proceedings of the American control conference (pp. 2687-2691).; Chang, W. S., Suh, I. H., & Oh, J. H. (1998). Synthesis and analysis of digital multiple repetitive control systems. In Proceedings of the American control conference (pp. 2687-2691). |

[6] | Chen, Y. Q., Ding, M., Xiu, L., Ooi, K., & Tan, L. (2003). Optimally designed parsimonious repetitive learning compensator for hard disc drives having high track density. US Patent 6,574,067 (pp. 1-13).; Chen, Y. Q., Ding, M., Xiu, L., Ooi, K., & Tan, L. (2003). Optimally designed parsimonious repetitive learning compensator for hard disc drives having high track density. US Patent 6,574,067 (pp. 1-13). |

[7] | Chen, Y. Q., & Moore, K. L. (2002). Harnessing the nonrepetitiveness in iterative learning control. In Proceedings of the 41st IEEE conference on decision and control (pp. 3350-3355).; Chen, Y. Q., & Moore, K. L. (2002). Harnessing the nonrepetitiveness in iterative learning control. In Proceedings of the 41st IEEE conference on decision and control (pp. 3350-3355). |

[8] | Chew, K. K., & Tomizuka, M. (1990). Digital control of repetitive errors in disk drive systems. In Proceedings of the American control conference (pp. 540-548).; Chew, K. K., & Tomizuka, M. (1990). Digital control of repetitive errors in disk drive systems. In Proceedings of the American control conference (pp. 540-548). |

[9] | Choi, G. H., Oh, J. H., & Choi, G. S. (1999). Repetitive tracking control of a coarse-fine actuator. In Proceedings of the IEEE/ASME international conference on advanced intelligent mechatronics (pp. 335-340).; Choi, G. H., Oh, J. H., & Choi, G. S. (1999). Repetitive tracking control of a coarse-fine actuator. In Proceedings of the IEEE/ASME international conference on advanced intelligent mechatronics (pp. 335-340). |

[10] | Choi, G. S.; Lim, Y. A.; Choi, G. H., Tracking position control of piezoelectric actuators for periodic reference inputs, Mechatronics, 12, 5, 669-684 (2002) |

[11] | Dötch, H. G. M., Smakman, H. T., Van den Hof, P. M. J., & Steinbuch, M. (1995). Adaptive repetitive control of a compact disc mechanism. In Proceedings of the 1995 IEEE conference on decision and control (pp. 1720-1725), New Orleans, December.; Dötch, H. G. M., Smakman, H. T., Van den Hof, P. M. J., & Steinbuch, M. (1995). Adaptive repetitive control of a compact disc mechanism. In Proceedings of the 1995 IEEE conference on decision and control (pp. 1720-1725), New Orleans, December. |

[12] | Francis, B. A.; Wonham, W. M., The internal model principle for linear multivariable regulators, Applied Mathematics and Optics, 2, 170-194 (1975) · Zbl 0351.93015 |

[13] | Fung, R. F.; Huang, J. S.; Chien, C. G.; Wang, Y. C., Design and application of a continuous repetitive controller for rotating mechanisms, International Journal of Mechanical Sciences, 42, 9, 1805-1819 (2000) · Zbl 0968.70008 |

[14] | Gotou, M., Ueta, E., Nakamura, A., & Matsuo, K. (1991). Development of multirate sampling repetitive learning servo system and its application to a compact camcorder. In IEEE/RSJ international workshop on intelligent robots and systems IROS ’91 (pp. 647-654), November 1991.; Gotou, M., Ueta, E., Nakamura, A., & Matsuo, K. (1991). Development of multirate sampling repetitive learning servo system and its application to a compact camcorder. In IEEE/RSJ international workshop on intelligent robots and systems IROS ’91 (pp. 647-654), November 1991. |

[15] | Guo, L., Reducing the manufacturing costs associated with hard disk drives—a new disturbance rejection control scheme, IEEE/ASME Transactions on Mechatronics, 2, 2, 77-85 (1997) |

[16] | Hara, S.; Yamamoto, Y.; Omata, T.; Nakano, M., Repetitive control system—a new type servo system, IEEE Transactions on Automatic Control, AC-33, 659-668 (1988) · Zbl 0662.93027 |

[17] | Hillerström, G., Adaptive suppression of vibrations—a repetitive control approach, IEEE Transactions on Control Systems Technology, 4, 1, 72-77 (1996) |

[18] | Inoue (1990). Practical repetitive control system design. In Proceedings of the 29th conference on decision and control (pp. 1673-1678), December.; Inoue (1990). Practical repetitive control system design. In Proceedings of the 29th conference on decision and control (pp. 1673-1678), December. |

[19] | Kim, B. S.; Li, J.; Tsao, T. C., Two-parameter robust repetitive control with application to a novel dual-stage actuator for noncircular machining, IEEE/ASME Transactions on Mechatronics, 9, 4, 644-652 (2004) |

[20] | Kim, D. H.; Tsao, T. C., Robust performance control of electrohydraulic actuators for electronic cam motion generation, IEEE Transactions on Control Systems Technology, 8, 2, 220-227 (2000) |

[21] | Köroğlu, H., & Morgül, Ö. (1999). Discrete-time LQ optimal repetitive control. In Proceedings of the American control conference (Vol. 5, pp. 3287-3291).; Köroğlu, H., & Morgül, Ö. (1999). Discrete-time LQ optimal repetitive control. In Proceedings of the American control conference (Vol. 5, pp. 3287-3291). |

[22] | Köroğlu, H.; Morgül, Ö., Time-varying repetitive control for better transient response and stochastic behaviour, Electronics Letters, 37, 17, 1101-1102 (2001) |

[23] | Lee, R. C.H.; Smith, M. C., Robustness and trade-offs in repetitive control, Automatica, 34, 7, 889-896 (1998) · Zbl 0961.93019 |

[24] | Li, J., & Tsao, T. C. (1999). Rejection of repeatable and non-repeatable disturbances for disk drive actuators. In Proceedings of the American control conference (pp. 3615-3619).; Li, J., & Tsao, T. C. (1999). Rejection of repeatable and non-repeatable disturbances for disk drive actuators. In Proceedings of the American control conference (pp. 3615-3619). |

[25] | Luo, Z., & Mahawan, B. (1998). Repetitive control of tracking systems with time-varying periodic references. In Proceedings of the 37th IEEE conference on decision and control (Vol. 2, pp. 1247-1249).; Luo, Z., & Mahawan, B. (1998). Repetitive control of tracking systems with time-varying periodic references. In Proceedings of the 37th IEEE conference on decision and control (Vol. 2, pp. 1247-1249). · Zbl 1006.93538 |

[26] | Manayathara, Th. J.; Tsao, T.-C.; Bentsman, J.; Ross, D., Rejection of unknown periodic load disturbances in continuous steel casting process using learning repetitive control approach, IEEE Transactions on Control Systems Technology, 4, 3, 259-265 (1996) |

[27] | Messner, W., & Bodson, M. (1994). Design of adaptive feedforward controllers using internal model equivalence. In Proceedings of the American control conference (pp. 1619-1623).; Messner, W., & Bodson, M. (1994). Design of adaptive feedforward controllers using internal model equivalence. In Proceedings of the American control conference (pp. 1619-1623). |

[28] | Moon, J.; Lee, M.; Chung, M., Repetitive control for the track-following servo system of an optical disk drive, IEEE Transactions on Control Systems Technology, 6, 5, 663-670 (1998) |

[29] | Moore, K. L., & Chen, Y. Q. (2002). On monotonic convergence of high-order iterative learning update laws. In Proceedings of the IFAC World congress, Barcelona, Spain.; Moore, K. L., & Chen, Y. Q. (2002). On monotonic convergence of high-order iterative learning update laws. In Proceedings of the IFAC World congress, Barcelona, Spain. |

[30] | Rogers, E.; Owens, D. H., Stability analysis for linear repetitive processes (1992), Springer: Springer Berlin · Zbl 0772.93072 |

[31] | de Roover, D.; Bosgra, O. H.; Steinbuch, M., Internal model based design of repetitive and iterative learning controllers for linear multivariable systems, International Journal of Control, 73, 10, 914-929 (2000) · Zbl 1006.93602 |

[32] | Schootstra, G., & Steinbuch, M. (1998). Control system for a process that exhibits periodic disturbances. US Patent nr. 5,7842,272, July, 21.; Schootstra, G., & Steinbuch, M. (1998). Control system for a process that exhibits periodic disturbances. US Patent nr. 5,7842,272, July, 21. |

[33] | Singh, T.; Vadali, S. R., Robust time delay control, ASME Journal of Dynamic Systems, Measurement and Control, 115, 2(A), 303-306 (1993) · Zbl 0775.93040 |

[34] | Stan, S. G., Optimization of the CD-rom system towards higher data throughputs (1999), PhD thesis: PhD thesis Technische Universiteit Eindhoven |

[35] | Steinbuch, M., Repetitive control for systems with uncertain period-time, Automatica, 38, 12, 2103-2109 (2002) · Zbl 1018.93023 |

[36] | Steinbuch, M., van den Eerenbeemt, J. , Weiland, S., & Singh, T. (2004). On noise and period-time sensitivity in high order repetitive control. In IEEE 43rd IEEE conference on decision and control (pp. 1295-1300), IEEE, Bahamas, United States.; Steinbuch, M., van den Eerenbeemt, J. , Weiland, S., & Singh, T. (2004). On noise and period-time sensitivity in high order repetitive control. In IEEE 43rd IEEE conference on decision and control (pp. 1295-1300), IEEE, Bahamas, United States. |

[37] | Steinbuch, M.; van Groos, P. J.M.; Schootstra, G.; Wortelboer, P. M.; Bosgra, O. H., \( \mu \)-synthesis of a compact disc player, International Journal of Robust and Nonlinear Control, 8, 169-189 (1998) · Zbl 0900.93241 |

[38] | Steinbuch, M., & Norg, M. L. (1998). Advanced motion control: An industrial perspective. European Journal of Control 278-293.; Steinbuch, M., & Norg, M. L. (1998). Advanced motion control: An industrial perspective. European Journal of Control 278-293. · Zbl 0925.93706 |

[39] | Steinbuch, M., & Schootstra, G. (1998). Filter, repetitive control system and learning control system both provided with such filter. US Patent nr. 5,740,090, April 14.; Steinbuch, M., & Schootstra, G. (1998). Filter, repetitive control system and learning control system both provided with such filter. US Patent nr. 5,740,090, April 14. |

[40] | Tomizuka, M., Zero phase error tracking algorithm for digital control, Journal of Dynamic Systems, Measurement and Control, 109, 65-68 (1987) · Zbl 0621.93017 |

[41] | Tomizuka, M., Tsao, T. C., & Chew, K. K. (1988). Discrete-time domain analysis and synthesis of repetitive controllers. In Proceedings of the American control conference (pp. 860-866), June.; Tomizuka, M., Tsao, T. C., & Chew, K. K. (1988). Discrete-time domain analysis and synthesis of repetitive controllers. In Proceedings of the American control conference (pp. 860-866), June. |

[42] | Tsao, T. C., & Nemani, M. (1992). Asymptotic rejection of periodic disturbances with uncertain period. In Proceedings of the American control conference (pp. 2696-2699).; Tsao, T. C., & Nemani, M. (1992). Asymptotic rejection of periodic disturbances with uncertain period. In Proceedings of the American control conference (pp. 2696-2699). |

[43] | Xu, L., & Yao, B. (2001). Adaptive robust repetitive control of a class of nonlinear systems in normal form with applications to motion control of linear motors. In Proceedings of the IEEE/ASME international conference on advanced intelligent mechatronics (Vol. 1, pp. 527-532).; Xu, L., & Yao, B. (2001). Adaptive robust repetitive control of a class of nonlinear systems in normal form with applications to motion control of linear motors. In Proceedings of the IEEE/ASME international conference on advanced intelligent mechatronics (Vol. 1, pp. 527-532). |

[44] | Yamada, M., Riadh, Z., & Funahashi, Y. (2000). Design of robust repetitive control systems for multiple periods. Proceedings of the 39th IEEE conference on decision and control (Vol. 4, pp. 3739-3744).; Yamada, M., Riadh, Z., & Funahashi, Y. (2000). Design of robust repetitive control systems for multiple periods. Proceedings of the 39th IEEE conference on decision and control (Vol. 4, pp. 3739-3744). |

[45] | Yamada, M.; Riadh, Z.; Funahashi, Y., Design of discrete-time repetitive control system for pole placement and application, IEEE/ASME Transactions on Mechatronics, 4, 2, 110-118 (1999) |

[46] | Yau, W., & Tsai, W. (1999). Repetitive control design for linear servo systems. In Proceedings of the American control conference (Vol. 5, pp. 3728-3732).; Yau, W., & Tsai, W. (1999). Repetitive control design for linear servo systems. In Proceedings of the American control conference (Vol. 5, pp. 3728-3732). |

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.