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 "title": "Repetitive control",
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 "excerpt": "Repetitive control (RC) is a feedback method that embeds a periodic-signal generator inside the control loop, so systems track periodic references or reject periodic disturbances with near-zero steady-state error, used in power converters and disk drives.",
 "snippet": "Repetitive control (RC) is a feedback method that embeds a periodic-signal generator inside the control loop, so systems track periodic references or reject periodic disturbances with near-zero steady-state error, used in power converters and disk drives.",
 "node": "technology.engineering.engineering.electrical.electronics",
 "markdown": "# Repetitive control\n\nRepetitive control (RC) is a feedback method that embeds a generator of periodic signals inside the control loop, so that a dynamic system tracks periodic references or rejects periodic disturbances with near-zero steady-state error.<sup>[1](https://www.ieee-jas.net/en/article/doi/10.1109/JAS.2025.125297)</sup> It handles both tracking and disturbance rejection with the same mechanism: any signal that repeats with a fixed period is driven to zero error in the steady state, provided its period is known and the closed loop remains stable.<sup>[1](https://www.ieee-jas.net/en/article/doi/10.1109/JAS.2025.125297)</sup>\n\n| Key fact | Value | Source |\n|---|---|---|\n| Continuous-time internal model | \\( 1/(1-e^{-sT}) \\), infinite gain at \\( 2 \\cdot \\pi \\cdot k / T \\) for \\( k = 0, \\pm 1, \\pm 2, \\ldots \\) | <sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup> |\n| Discrete-time internal model | \\( z^{-N}/(1-z^{-N}) \\), with \\( N \\) samples per period | <sup>[3](https://faculty.washington.edu/chx/teaching/loopshaping/Chen_Tomizuka_IEEE_TCST2012-0116_RC_final.pdf)</sup> |\n| Steady-state promise | Perfect tracking and rejection for periodic signals of fixed period | <sup>[1](https://www.ieee-jas.net/en/article/doi/10.1109/JAS.2025.125297)</sup> |\n| Fundamental limit | No conventional RC with the pure-delay internal model can exponentially stabilize a strictly proper plant | <sup>[4](https://doi.org/10.1109/9.1274)</sup> |\n| Standard fix | Replace the delay \\( e^{-Ls} \\) by \\( q(s)e^{-Ls} \\), a low-pass filter trading accuracy for stability | <sup>[4](https://doi.org/10.1109/9.1274)</sup> |\n| Frequency requirement | The disturbance period must be known; a 7.5 Hz disturbance against an 8 Hz design is not rejected | <sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup> |\n| Main application domains | Power converters and drives, disk drives, nanopositioning, print-belt systems | <sup>[5](https://doi.org/10.1109/ojia.2021.3137589)</sup> |\n\n## How it works\n\nRC rests on the internal model principle of Francis and Wonham (1975): to track or reject a signal without steady-state error, its generator must sit inside a stable feedback loop.<sup>[6](https://doi.org/10.1007/bf01447855)</sup> A periodic signal of period \\( T \\) is generated by the transfer function \\( 1/(1-e^{-sT}) \\), which has poles on the imaginary axis at \\( 2 \\cdot \\pi \\cdot k / T \\) and therefore infinite gain at every harmonic of the fundamental frequency.<sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup> Placed in the loop, this infinite gain assures zero-error tracking at each of these frequencies.<sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup>\n\nThe generator is realized as a positive feedback loop around a pure delay \\( e^{-sT} \\): the error of the previous period is replayed to correct the current one.<sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup> Stability is analyzed with the small-gain theorem and time-lag system theory; the closed loop is a neutral-type time-lag system (a delay system whose stability depends on delayed state derivatives), because the operation is continuous, with the state at the start of each period equal to the state at the end of the previous one.<sup>[4](https://doi.org/10.1109/9.1274)</sup> This structure has a hard consequence: because the unfiltered generator has poles of arbitrarily high frequency, no conventional repetitive controller with the pure-delay internal model can exponentially stabilize a strictly proper plant.<sup>[4](https://doi.org/10.1109/9.1274)</sup> Yamamoto and Hara (1986) showed that stabilization requires the plant to have relative degree zero, that is, full-rank direct feedthrough; if there is no direct feedthrough the loop can never be stabilized.<sup>[7](https://doi.org/10.9746/sicetr1965.22.830)</sup>\n\n## How it is done\n\nThe dead time \\( L \\) is set to the period of the reference or disturbance signal.<sup>[8](https://onlinelibrary.wiley.com/doi/10.1002/asjc.413)</sup> In digital implementation, the delay becomes \\( z^{-N} \\), a buffer of \\( N \\) samples per period; only harmonics below the [Nyquist frequency](https://www.edgepedia.org/nyquist-frequency) can be canceled.<sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup> A typical design chooses the sampling period well below the plant time constant.<sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup>\n\nThe digital controller decomposes into three parts: the internal model \\( z^{-N_o}/(1-z^{-N_o}) \\), which ensures zero steady-state error; a low-pass filter \\( Q(z) \\), which provides robustness; and a compensator \\( G_s(z) \\), which guarantees closed-loop stability.<sup>[9](https://www.sciencedirect.com/science/article/abs/pii/S0967066117300710)</sup> The cutoff of \\( Q \\) is the central design tradeoff: performance depends largely on it, and a null-phase FIR filter is commonly used so that the internal-model pole frequencies are not shifted.<sup>[8](https://onlinelibrary.wiley.com/doi/10.1002/asjc.413)</sup> The learning gain \\( k_r \\) sets the convergence rate, with closed-loop poles approximately solving \\( z^N = 1 - k_r \\).<sup>[10](https://www.mdpi.com/2079-9292/7/12/446)</sup> The compensator often inverts the plant dynamics using the zero-phase error tracking (ZPETC) filter, which cannot invert non-minimum-phase zeros, and the robustness-filter design itself remains largely ad hoc.<sup>[11](https://www.precisionmechatronicslab.com/wp-content/uploads/2021/02/J18g.pdf)</sup> RC is usually added as a plug-in augmentation of an existing nominal controller.<sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup>\n\n## Origin\n\nThe method grew out of Japanese research on power supplies, where periodic ripple disturbances dominate, and early work on linear single-input single-output plants demonstrated its practical usefulness.<sup>[4](https://doi.org/10.1109/9.1274)</sup> The theoretical foundations were laid in the mid-1980s: Hara, Omata, and Nakano published stability conditions and synthesis methods in 1986 in the Transactions of the Society of Instrument and Control Engineers,<sup>[12](https://doi.org/10.9746/sicetr1965.22.36)</sup> and Yamamoto and Hara analyzed the internal model principle and stabilizability of repetitive control systems the same year.<sup>[7](https://doi.org/10.9746/sicetr1965.22.830)</sup> S. Hara and colleagues then published the modified, low-pass-filtered formulation in a 1988 IEEE Transactions on Automatic Control paper.<sup>[4](https://doi.org/10.1109/9.1274)</sup> In the United States, Tomizuka published the zero phase error tracking algorithm for digital control in 1987,<sup>[13](https://doi.org/10.1115/1.3143822)</sup> and Tomizuka, Tsao and Chew analyzed and synthesized discrete-time repetitive controllers in 1989.<sup>[14](https://doi.org/10.1115/1.3153060)</sup> A closely related precursor is the betterment process of Arimoto, Kawamura and Miyazaki (1984).<sup>[15](https://doi.org/10.1002/rob.4620010203)</sup>\n\n## Variants\n\n**Modified or filtered RC** replaces \\( e^{-Ls} \\) by \\( q(s)e^{-Ls} \\), with \\( q(s) \\) proper, stable, and below unity in magnitude above a cutoff \\( \\omega_c \\); this converts the neutral closed loop into a retarded one, at the price of tracking accuracy, since the desired poles at \\( 2 \\cdot k \\cdot \\pi \\cdot j / L \\) are altered by \\( q(s) \\).<sup>[4](https://doi.org/10.1109/9.1274)</sup> **Plug-in RC** augments an existing nominal controller with the repetitive compensator.<sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup> A plug-in design procedure was published by Tsai and Yao (2002).<sup>[16](https://doi.org/10.1109/tcst.2002.1014674)</sup> **Odd-harmonic RC** uses a half-period delay with negative feedback, providing infinite gain only at odd harmonics.<sup>[17](https://pmc.ncbi.nlm.nih.gov/articles/PMC9698222/)</sup> The odd-harmonic generator for digital plug-in control was treated by Griñó and Costa-Castelló.<sup>[18](https://doi.org/10.1016/j.automatica.2004.08.006)</sup> **High-order RC** improves robustness to signal-frequency variation through optimal performance trade-offs, as analyzed by Pipeleers and colleagues (2008).<sup>[19](https://doi.org/10.1016/j.automatica.2008.02.028)</sup> **Fractional-delay RC** handles non-integer samples per period caused by reference-frequency variation; Lagrange-interpolation-based designs, known as Fractional Order Repetitive Control (FORC), are the most frequently used approach in power converters, and Taylor-series-expansion designs recast the fractional-delay filter as a Farrow structure of sub-filters that can be retuned online.<sup>[9](https://www.sciencedirect.com/science/article/abs/pii/S0967066117300710)</sup> **Multi-period RC** targets \\( L \\) distinct disturbance periods with an \\( L \\)-period internal model.<sup>[20](https://www.uscamsl.com/Publications/Journal/Automatica_2010.pdf)</sup> Chen and Tomizuka recast the repetitive controller as a repetitive disturbance observer (RDOB), whose central component extracts the repetitive signal rather than low-pass filtering it, with a time-varying learning parameter for faster transients.<sup>[3](https://faculty.washington.edu/chx/teaching/loopshaping/Chen_Tomizuka_IEEE_TCST2012-0116_RC_final.pdf)</sup> **Adaptive RC** adjusts to period variations.<sup>[21](https://doi.org/10.1109/87.481769)</sup> Gaussian process RC extends the idea beyond periodic internal models through kernel design.<sup>[22](https://doi.org/10.1016/j.automatica.2022.110273)</sup>\n\n## Applications\n\n[Power electronics](https://www.edgepedia.org/power-electronics) is a major domain: RC tracks periodic signals and rejects periodic disturbances in converters and drives, improving steady-state behavior and harmonic distortion within limited bandwidth, where passive filters would reduce efficiency and add weight and volume.<sup>[5](https://doi.org/10.1109/ojia.2021.3137589)</sup> In disk drives, RC is used for track following.<sup>[23](https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/iet-cta.2011.0754)</sup> Multi-period designs have canceled simultaneous 70, 120, and 407 Hz disturbances in hard disk drives.<sup>[20](https://www.uscamsl.com/Publications/Journal/Automatica_2010.pdf)</sup> Other documented uses include nanopositioning stages,<sup>[11](https://www.precisionmechatronicslab.com/wp-content/uploads/2021/02/J18g.pdf)</sup> mechanical ventilation,<sup>[24](https://doi.org/10.1109/tcst.2023.3250966)</sup> industrial print-belt systems,<sup>[25](https://doi.org/10.1016/j.mechatronics.2024.103187)</sup> robot arm control, and regulation in vehicles.<sup>[3](https://faculty.washington.edu/chx/teaching/loopshaping/Chen_Tomizuka_IEEE_TCST2012-0116_RC_final.pdf)</sup>\n\n## Limitations and alternatives\n\nThe neutral-type pole chain approaches the imaginary axis, and the \\( H_\\infty \\) norm of the sensitivity can become unbounded at high frequencies unless the augmented system is bi-proper, so unmodeled high-frequency dynamics are a genuine instability risk.<sup>[26](https://hal.science/hal-04788778/document)</sup> By Bode's integral theorem, the sensitivity has a comb-like magnitude with gain amplification at non-repetitive frequencies, a waterbed effect (suppressing error at some frequencies amplifies it at others) that is worse when large non-periodic disturbances are present, as in hard disk drives.<sup>[3](https://faculty.washington.edu/chx/teaching/loopshaping/Chen_Tomizuka_IEEE_TCST2012-0116_RC_final.pdf)</sup> The exact disturbance frequency must be known in advance, and RC is highly sensitive to frequency uncertainty; with a fixed sampling period, performance decays significantly when the disturbance period changes, and fixed-RC failed to track after period changes in servo-motor experiments.<sup>[27](https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/iet-cta.2013.1059)</sup> The internal model also places an infinite number of open-loop poles on the stability boundary, forcing sacrifice of high-frequency harmonic performance.<sup>[28](https://onlinelibrary.wiley.com/doi/10.1002/asjc.409)</sup>\n\nCompared with [iterative learning control](https://www.edgepedia.org/iterative-learning-control) (ILC), RC operates continuously with periods back to back and no state reset, while ILC resets initial conditions before each trial; RC has one-dimensional dynamics in the time domain, ILC two-dimensional ones, and both rely on the internal model principle with similar filter-design guidelines.<sup>[29](https://lirias.kuleuven.be/retrieve/294118)</sup> In a common formulation the two differ only in the internal model's location, at the system output for RC and at the input for ILC.<sup>[30](https://eprints.soton.ac.uk/272583/1/Freeman_JACSP_2012.pdf)</sup> Compared with resonant control, RC is simpler and needs fewer computational resources but more memory.<sup>[2](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)</sup>\n\n## References\n\n1. [Repetitive Control: Basic Concept, Fundamental Theory, and Practical Applications (IEEE/CAA Journal of Automatica Sinica, 2025 survey)](https://www.ieee-jas.net/en/article/doi/10.1109/JAS.2025.125297)\n2. [Reduction of repetitive errors in tracking of periodic signals: theory and application of repetitive control (educational design paper, UPC repository copy)](https://upcommons.upc.edu/bitstreams/5b48d130-f7fe-4aea-b5ee-413a3c930325/download)\n3. [New Repetitive Control With Improved Steady-State Performance and Accelerated Transient (Chen & Tomizuka, IEEE TCST)](https://faculty.washington.edu/chx/teaching/loopshaping/Chen_Tomizuka_IEEE_TCST2012-0116_RC_final.pdf)\n4. [S. Hara and colleagues (1988). Repetitive control system: a new type servo system for periodic exogenous signals. IEEE Transactions on Automatic Control.](https://doi.org/10.1109/9.1274)\n5. [Mi Tang and colleagues (2021). State of the Art of Repetitive Control in Power Electronics and Drive Applications. IEEE Open Journal of Industry Applications.](https://doi.org/10.1109/ojia.2021.3137589)\n6. [B. A. Francis, W. M. Wonham (1975). The internal model principle for linear multivariable regulators. Applied Mathematics & Optimization.](https://doi.org/10.1007/bf01447855)\n7. [Yutaka YAMAMOTO, Shinji HARA (1986). The Internal Model Principle and Stabilizability of Repetitive Control Systems. Transactions of the Society of Instrument and Control Engineers.](https://doi.org/10.9746/sicetr1965.22.830)\n8. [Repetitive controller design for optimal performance (Asian Journal of Control)](https://onlinelibrary.wiley.com/doi/10.1002/asjc.413)\n9. [Taylor series expansion based repetitive controllers for power converters, subject to fractional delays (Control Engineering Practice)](https://www.sciencedirect.com/science/article/abs/pii/S0967066117300710)\n10. [Comparison of Different Repetitive Control Architectures: Synthesis and Comparison. Application to VSI Converters (Electronics, MDPI)](https://www.mdpi.com/2079-9292/7/12/446)\n11. [Improving Robustness Filter Bandwidth in Repetitive Control by Considering Model Mismatch (Asian Journal of Control, 2018)](https://www.precisionmechatronicslab.com/wp-content/uploads/2021/02/J18g.pdf)\n12. [Shinji HARA, Tohru OMATA, Michio NAKANO (1986). Stability Condition and Synthesis Methods for Repetitive Control Systems. Transactions of the Society of Instrument and Control Engineers.](https://doi.org/10.9746/sicetr1965.22.36)\n13. [Masayoshi Tomizuka (1987). Zero Phase Error Tracking Algorithm for Digital Control. Journal of Dynamic Systems Measurement and Control.](https://doi.org/10.1115/1.3143822)\n14. [Masayoshi Tomizuka, Tsu-Chin Tsao, Kok-Kia Chew (1989). Analysis and Synthesis of Discrete-Time Repetitive Controllers. Journal of Dynamic Systems Measurement and Control.](https://doi.org/10.1115/1.3153060)\n15. [Suguru Arimoto, Sadao Kawamura, Fumio Miyazaki (1984). Bettering operation of Robots by learning. Journal of Robotic Systems.](https://doi.org/10.1002/rob.4620010203)\n16. [Mi-Ching Tsai, Wu-Sung Yao (2002). Design of a plug-in type repetitive controller for periodic inputs. IEEE Transactions on Control Systems Technology.](https://doi.org/10.1109/tcst.2002.1014674)\n17. [Design of Fractional Order Odd-Harmonics Repetitive Controller for Discrete-Time Linear Systems with Experimental Validations](https://pmc.ncbi.nlm.nih.gov/articles/PMC9698222/)\n18. [Robert Griñó, Ramon Costa-Castelló (2004). Digital repetitive plug-in controller for odd-harmonic periodic references and disturbances. Automatica.](https://doi.org/10.1016/j.automatica.2004.08.006)\n19. [Goele Pipeleers and colleagues (2008). Robust high-order repetitive control: Optimal performance trade-offs. Automatica.](https://doi.org/10.1016/j.automatica.2008.02.028)\n20. [A new method for synthesizing multiple-period adaptive-repetitive controllers and its application to the control of hard disk drives (Pérez-Arancibia, Tsao, Gibson, Automatica 2010)](https://www.uscamsl.com/Publications/Journal/Automatica_2010.pdf)\n21. [G. Hillerstrom (1996). Adaptive suppression of vibrations - a repetitive control approach. IEEE Transactions on Control Systems Technology.](https://doi.org/10.1109/87.481769)\n22. [Noud Mooren, Gert Witvoet, Tom Oomen (2022). Gaussian process repetitive control: Beyond periodic internal models through kernels. Automatica.](https://doi.org/10.1016/j.automatica.2022.110273)\n23. [Robust approach to repetitive controller design for uncertain feedback control systems (IET Control Theory & Applications)](https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/iet-cta.2011.0754)\n24. [Joey Reinders and colleagues (2023). Repetitive Control for Lur’e-Type Systems: Application to Mechanical Ventilation. IEEE Transactions on Control Systems Technology.](https://doi.org/10.1109/tcst.2023.3250966)\n25. [Multirate repetitive control for an industrial print-belt system (Mechatronics, 2024)](https://doi.org/10.1016/j.mechatronics.2024.103187)\n26. [Repetitive Control: Concept, Limitations, Potential (peer-reviewed review, time-delay systems viewpoint)](https://hal.science/hal-04788778/document)\n27. [Digital design of adaptive repetitive control of linear systems with time-varying periodic disturbances (IET Control Theory & Applications)](https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/iet-cta.2013.1059)\n28. [A new adaptive control for periodic tracking/disturbance rejection (Asian Journal of Control)](https://onlinelibrary.wiley.com/doi/10.1002/asjc.409)\n29. [A unified framework for analysis and design of iterative learning and repetitive control systems](https://lirias.kuleuven.be/retrieve/294118)\n30. [A common setting for the design of iterative learning and repetitive controllers with experimental verification (Int. J. Adaptive Control and Signal Processing, 2012)](https://eprints.soton.ac.uk/272583/1/Freeman_JACSP_2012.pdf)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Engineering and manufacturing › Electrical and electronics engineering*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 21, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n",
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 "speakable": "Repetitive control is a feedback method that embeds a periodic-signal generator inside the control loop, so systems track periodic references or reject periodic disturbances with near-zero steady-state error, used in power converters and disk drives."
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