Abstract
The main problem in control theory is controlling the output of a system to achieve the asymptotic tracking of desired signals and/or asymptotic rejection of disturbances. Among the existing approaches to asymptotic tracking and rejection, tracking via an internal model, which can handle the exogenous signal (The term “exogenous signal” is used to refer to both the desired signal and the disturbance when there is no need to distinguish them.) from a fixed family of functions of time, is one of the important approaches [1]. The basic concept of tracking via the internal model originated from the internal model principle (IMP) [2, 3]. The IMP states that if any exogenous signal can be regarded as the output of an autonomous system, then the inclusion of this signal model, i.e., the internal model, in a stable closed-loop system can ensure asymptotic tracking and asymptotic rejection of the signal. Given that the exogenous signals under consideration are often nonvanishing, the characteristic roots of these autonomous systems that generate these exogenous signals are neutrally stable. To produce asymptotic tracking and asymptotic rejection, if a given signal has a certain number of harmonics, then a corresponding number of neutrally stable internal models (one for each harmonic) should be incorporated into the closed-loop based on the IMP. Repetitive control (RC, or repetitive controller, also abbreviated as RC) is a specialized tracking method via an internal model for the asymptotic tracking and rejection of general T-periodic signals [4].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The term “exogenous signal” is used to refer to both the desired signal and the disturbance when there is no need to distinguish them.
- 2.
A system incorporating internal models is called an internal model system for simplicity.
- 3.
In this book, the term “modified” in [4] is replaced with the more descriptive term “filtered”.
References
Isidori, A., Marconi, L., & Serrani, A. (2003). Robust Autonomous Guidance: An Internal Model Approach. London: Springer.
Francis, B. A., & Wonham, W. M. (1976). The internal model principle of control theory. Automatica, 12(5), 457–465.
Wonham, W. M. (1976). Towards an abstract internal model principle. IEEE Transactions on Systems, Man, and Cybernetics-Part B, 6(11), 735–740.
Hara, S., Yamamoto, Y., Omata, T., & Nakano, M. (1988). Repetitive control system: a new type servo system for periodic exogenous signals. IEEE Transactions on Automatic Control, 33(7), 659–668.
Hillerström, G. (1994). On Repetitive Control. Ph.D. Thesis, Luleå University of Technology, Sweden.
Lee, R. C. H., & Smith, M. C. (1998). Robustness and trade-offs in repetitive control. Automatica, 34(7), 889–896.
Safonov, M., & Athans, M. (1981). A multiloop generalization of the circle criterion for stability margin analysis. IEEE Transaction on Automatic Control, 26(2), 415–422.
Khalil, H. K. (2002). Nonlinear Systems. Upper Saddle River, NJ: Prentice-Hall.
Rabaha, R., Sklyarb, G. M., & Rezounenkoc, A. V. (2005). Stability analysis of neutral type systems in Hilbert space. Journal of Differential Equations, 214(2), 391–428.
Quan, Q., Yang, D., & Cai, K.-Y. (2010). Linear matrix inequality approach for stability analysis of linear neutral systems in a critical case. IET Control Theory and Applications, 4(7), 1290–1297.
Hale, J. K., & Lunel, S. M. V. (1993). Introduction to functional differential equations. New York: Springer.
Messner, W., Horowitz, R., Kao, W.-W., & Boals, M. (1991). A new adaptive learning rule. IEEE Transactions on Automatic Control, 36(2), 188–197.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Quan, Q., Cai, KY. (2020). Robustness Analysis of Repetitive Control Systems. In: Filtered Repetitive Control with Nonlinear Systems. Springer, Singapore. https://doi.org/10.1007/978-981-15-1454-8_4
Download citation
DOI: https://doi.org/10.1007/978-981-15-1454-8_4
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-1453-1
Online ISBN: 978-981-15-1454-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)