Enhanced fractional multi-period repetitive controller with fractional time-delay for high-precision servo systems

Abstract

Repetitive control (RC) is widely used for achieving high-precision tracking and effective rejection of periodic disturbances in servo and vibration-prone systems. Conventional single-period and multi-period repetitive controllers, however, rely on integer delay models that cannot exactly represent the true fundamental periods of multi-frequency periodic signals, leading to frequency mismatch and residual steady-state errors. To address this limitation, this paper proposes a Fractional Multi-Period Repetitive Controller (FMPRC) that preserves non-integer delay lengths within the discrete-time multi-period internal model, thereby maintaining accurate frequency alignment between the controller and actual periodic components. The total delay is decomposed into integer and fractional components, where the fractional delay is realized using either infinite impulse response (IIR) or finite impulse response (FIR) filters. Unlike conventional delay rounding approaches, the proposed formulation embeds fractional-delay modeling directly into the internal model while preserving unity gain, causality, and closed-loop stability under the plug-in RC framework. Stability conditions and practical implementation aspects of the FMPRC are analyzed, and both IIR- and FIR-based realizations are systematically compared. Simulation studies on high-precision servo systems subject to multi-frequency and time-varying periodic disturbances demonstrate that the proposed FMPRC significantly improves steady-state tracking accuracy and periodic disturbance rejection compared to conventional single-period and multi-period repetitive controllers. These results demonstrate the effectiveness of the proposed FMPRC as multiple-frequency vibration suppression strategy for high-precision motion and servo control systems.

Keywords

servo systems repetitive controller fractional time-delay multi-period signal high-precision tracking disturbance rejection

1. Introduction

Tracking and rejecting periodic signals are common challenges in many control applications. Repetitive controller (RC), based on the internal model principle introduced by (Francis and Wonham, 1975), is a well-known control strategy used to achieve high-precision tracking of periodic references and effective rejection of periodic disturbances. RC is a learning-based controller, similar to iterative learning control (Chen et al., 2024; Huang and Li, 2014; Li et al., 2024; Wu et al., 2021; Xie et al., 2023), where the current control input is derived from the previous period or cycle. As the iterations increase, the performance of the system in performing repetitive tasks improves. Recently, RC has been implemented in many control applications such as wearable walking exoskeletons (Huang et al., 2024), bearingless induction motor (Ye et al., 2024), line-of-sight stabilization (Feng et al., 2024), piezo-actuated nanoscanners (Li et al., 2022), islanded microgrid (Ma et al., 2019), and high-speed raster scanning (Li et al., 2021).

In modern digital control systems, RC is predominantly implemented in discrete-time. As a result, substantial research efforts have focused on discrete-time internal model structures and their practical realizations. In discrete-time, the internal model is implemented using delay operators that reproduce the signal period, enabling asymptotic periodic tracking when accurate period matching is achieved. The discrete-time RC model is expressed as $\frac{β (z) z^{- N_{r}}}{1 - β (z) z^{- N_{r}}}$ (Chew and Tomizuka, 1990). This discrete-time model has a finite dimension, with $N_{r}$ evenly spaced poles located on the unit circle. At higher frequencies, the poles are pushed inward due to the presence of the low-pass filter $β (z)$ . This model is capable of tracking or compensating periodic signals only at frequencies $f_{r} = \frac{j}{T_{r}}$ , for $j = 1, 2, \dots, \frac{N}{2},$ meaning it can only handle the basis frequency and its harmonics up to the Nyquist frequency of the repetitive signal (Kurniawan et al., 2014b). In addition, the discrete-time internal model is limited to handling a single base frequency $f_{r}$ . Therefore, if any frequency component other than the harmonics or a slight variation in the base frequency appears in the reference or disturbance signal, the model fails to track or compensate for that component, leading to a significant degradation in performance. As a result, this model is referred to as a single-period repetitive controller (SPRC), as it is only capable of handling one base frequency.

When the actual base frequency of the reference or disturbance deviates slightly from its nominal value ${\bar{f}}_{r}$ , such as $f_{r} = {\bar{f}}_{r} + δ_{f}$ , the tracking performance of SPRC deteriorates. This is due to the fact that the effectiveness of the RC depends on the controller’s delay being exactly matched to the actual period of the reference or disturbance. To overcome this issue, a design called higher-order SPRC (HO-SPRC) was developed in (Jamil et al., 2020; Kurniawan et al., 2023; Pipeleers et al., 2008; Steinbuch, 2002; Steinbuch et al., 2007), which modifies the internal model structure to improve robustness against variations in the period. HO-SPRC incorporates multiple time-delays (e.g., $z^{- N_{r}}, z^{- {2 N}_{r}}, \dots, z^{- {h N}_{r}}$ ), where $h$ represents the order of the SPRC. The use of multiple time-delays increases the internal model gains surrounding the base frequency and its harmonics, enhancing the performance of reference tracking and disturbance rejection at frequencies near the nominal ones. However, this approach is only effective for small variations in frequency, where $δ_{f}$ is relatively small. In cases of significant frequency shifts, or when the reference or disturbance consists of multiple uncorrelated frequency components (i.e., not harmonics of each other), both SPRC and HO-SPRC are inadequate in addressing these issues.

To resolve this issue, the multi-period repetitive controller (MPRC) (Yamada et al., 2000; Owens, L. M. and and Banks, 2004; Longman RW, Yeol JW, 2005; Yan et al., 2019; Kurniawan et al., 2020) was developed to track or compensate for such repetitive signals. These types of signals are commonly encountered in applications such as power electronics, electro-mechanical systems with multiple rotating machines, and industrial printing (Blanken et al., 2020; Pérez-Arancibia et al., 2010; Rashed et al., 2013;). In the MPRC, multiple base periods $T_{1}, T_{2}, \dots, T_{m}$ are utilized to create corresponding time delays (e.g., $z^{- N_{1}}, z^{- N_{2}}, \dots, z^{- N_{m}}$ ) to handle each frequency component present in both the reference and disturbance signals. However, most MPRC designs assume that the delay lengths $N_{1}, N_{2}, \dots, N_{m}$ are integer values, which are straightforward to implement in digital controllers. In practice, however, the delay lengths $N_{1}, N_{2}, \dots, N_{m}$ may be fractional, due to the controller’s sampling period specification. Typically, when implementing discrete-time MPRC with fractional delays, the fractional delay lengths are rounded to the nearest integer values. Unfortunately, this approach introduces a mismatch between the controller’s periods and the actual periods of the reference or disturbance, leading to a degradation in the tracking accuracy of the MPRC system. This limitation becomes particularly critical when the sampling period results in non-integer delay lengths, causing persistent frequency mismatch in the internal model. Therefore, a different approach that preserves fractional delay components is required to maintain accurate frequency alignment and fully exploit the internal model principle.

This paper aims to enhance the performance of conventional MPRC by preserving the fractional delay lengths in the internal model, resulting in what we call the fractional multi-period repetitive controller (FMPRC). The proposed approach improves the tracking accuracy of closed-loop systems by explicitly incorporating fractional delay lengths for accurate frequency component matching in the reference and disturbance signals. Each fractional delay is decomposed into integer and fractional parts, with the fractional component approximated using either IIR or FIR filters. In this study, the FMPRC with an IIR filter is referred to as IIR-FMPRC, while the FMPRC with a FIR filter is denoted as FIR-FMPRC. A stability analysis of the closed-loop system using these controllers is provided, along with simulation results and comparative studies to validate the tracking improvement achieved by the proposed design. To highlight the originality of this work, the main contributions are summarized as follows:

• A FMPRC is proposed by systematically extending the conventional MPRC framework to preserve non-integer delay lengths within the internal model, thereby enhancing frequency alignment between the controller and the actual periodic signals and improving steady-state tracking accuracy.

• A unified fractional-delay modeling framework is developed within the plug-in discrete-time MPRC structure, leading to two realizations: IIR-FMPRC and FIR-FMPRC based on Thiran and Lagrange approximations, respectively.

• The proposed approach embeds fractional-delay modeling directly into the multi-period internal model while preserving unity gain and causality of the delay operators.

• A stability analysis and practical design guideline are provided to ensure implementability and fair comparison with conventional SPRC and MPRC schemes.

Compared with recent RC extensions that enhance robustness against frequency variations, model uncertainties, and external disturbances, such as notch-based internal models (Kurniawan et al., 2025), decentralized structures for multivariable systems (Kurniawan et al., 2025), composite strategies combining adaptive disturbance rejection control (ADRC) with higher-order repetitive control (HORC) (Zhang et al., 2025), Padé-based preview RC formulations in continuous-time settings (Lan and Zhao, 2024), as well as fractional-order ADRC-based RC designs where fractional dynamics are introduced in the observer rather than in delay modeling (Lan et al., 2025), the proposed FMPRC addresses a fundamentally different issue: the period mismatch caused by integer delay rounding in discrete-time multi-period RC. Rather than modifying the feedback structure or introducing auxiliary compensation schemes, this work directly preserves non-integer discrete-time delay lengths within the internal model itself, ensuring accurate frequency alignment for all periodic components appearing in both the reference and disturbance signals.

The remainder of this paper is organized as follows: Section 2 provides the preliminaries, introducing the basis concepts of plug-in RC and the design of discrete-time single-period RC (SPRC) and multi-period RC (MPRC). Section 3 presents the proposed method, including the synthesis, stability conditions, and design implementation. Section 4 discusses the simulation results and comparisons for three distinct cases. Finally, Section 5 offers concluding remarks.

2. Preliminaries: Design of SPRC and MPRC

Consider the plug-in RC-based closed-loop control system shown in Figure 1, where $r (z), y (z), v (z), e (z)$ correspond to reference, output, disturbance, and error signals, respectively, $u_{R} (z)$ is an RC output and $u (z)$ is a control input, $P (z)$ is a plant model, and $C_{c} (z)$ is a conventional feedback controller.

Figure 1.

A closed-loop system structure incorporating a plug-in RC.

A single-period RC (SPRC) and multiple-period RC (MPRC) as plug-in controllers in Figure 1 can be expressed as follows

C_{s} (z) = \frac{U_{R} (z)}{e (z)} = [\frac{β (z) z^{- D}}{1 - β (z) z^{- D}}] L_{f} (z),

(1)

C_{M} (z) = \frac{U_{R} (z)}{e (z)} = \underset{i n t e r n a l m o d e l}{\underset{⏟}{[\frac{M (z)}{1 - M (z)}]}} L_{f} (z),

(2)

where

M (z)

in (2) is equal to

M (z) = 1 - (\prod_{i = 1}^{m} [1 - β_{i} (z) z^{- D_{i}}])

(3)

Here, $z^{- D}$ represents a one-period time-delay for the single basis frequency, while $z^{- D_{i}}$ in (3) denotes the one-period time-delay for the basis frequency $f_{i}$ . The delay length $D_{i}$ is given by $D_{i} = \frac{T_{i}}{T_{s}}$ , where $T_{i} = \frac{1}{f_{i}}$ is the period of the $i$ -th basis frequency, and $T_{s}$ is the sampling time. Additionally, $β (z)$ is a low-pass filter, and $L_{f} (z)$ is a learning function. The schematic representations of single-period and multi-periods RCs are shown in Figure 2.

Figure 2.

Schematic representations of (a) single-period RC, and (b) multi-period RC.

To further understand the realization of the block diagram $M (z)$ , two-period RC can be considered, derived as follows

M (z) = 1 - (1 - β_{1} (z) z^{- D_{1}}) (1 - β_{2} (z) z^{- D_{2}}),

(4)

= β_{1} (z) z^{- D_{1}} + β_{2} (z) z^{- D_{2}} - {β_{1} (z) β}_{2} (z) z^{- D_{1} - D_{2}} .

It can be noticed that $M (z)$ is essentially a chain of low-pass filters and multiple time-delays, which can be realized according to the block diagram shown in Figure 3.

Figure 3.

Schematic representation of $M (z)$ .

The low-pass filter $β (z)$ in (1) and (2) is included to enhance the robustness of the RC against high-frequency components, but this comes at the cost of reduced tracking accuracy for high-frequency harmonics. Typically, this filter is designed as a moving average filter, providing unity gain and zero phase shift, with the following form

β (z) = β_{0} + \sum_{i = 1}^{b} (β_{i} z^{i} + β_{i} z^{- i}),

(5)

where

β_{0} + 2 \sum_{i = 1}^{b} β_{i} = 1

. This filter offers unity gain at frequencies below its bandwidth and contributes a zero phase shift across all frequencies. In addition to

β (z)

, the learning function

L_{f} (z)

is also part of the RC transfer function and is crucial for stabilizing the closed-loop system. The design of

L_{f} (z)

can take various forms, such as the inverse model (Chew and Tomizuka, 1990; Tomizuka et al., 1989), the lead phase model (Mitrevska et al., 2018; Zhang et al., 2008), the IIR filter (Kurniawan, Cao and Man, 2014), and others, as long as it serves to stabilize the closed-loop system.

Throughout this work, two main assumptions are considered:

Assumption 1

The continuous-time open-loop plant model $P (s)$ is assumed to be known. Using the sampling time $T_{s}$ , the corresponding discrete-time model $P (z)$ shown in Figure 1 is obtained and used for learning function synthesis and stability analysis.

Assumption 2

The non-harmonic basis frequencies $f_{i}, i = 1, \dots, m$ , composing both the reference and disturbance signals are assumed to be known. This assumption allows accurate construction of the delay terms $D_{i} = \frac{T_{i}}{T_{s}}$ and the internal model.

3. Fractional multi period RC (FMPRC)

In this work, the proposed FMPRC is introduced by modifying the internal model of MPRC, as shown in equation (2), with a particular focus on modifying the delay terms $z^{- D_{i}}$ in (3). In conventional MPRC, the delay terms $z^{- D_{i}}$ are restricted to integer values. However, in practical applications, the actual periodic components may not correspond to integer multiples of the sampling period. Rounding the delay to the nearest integer introduces frequency mismatch in the internal model, which may violate the internal model principle and lead to residual steady-state errors. By decomposing the delay into integer and fractional parts and approximating the fractional delay, the proposed FMPRC achieves more accurate frequency alignment between the internal model and the true periodic components. This improved alignment enhances steady-state tracking accuracy and disturbance rejection performance. To address this, the delay terms $z^{- D_{i}}$ are now modified to

z^{- D_{i}} = z^{- D_{I_{i}}} z^{- d_{F_{i}}},

(6)

where

z^{- D_{I_{i}}}

and

z^{- d_{F i}}

, respectively, represent the integer and fractional delay terms of the i-th based frequency. Here,

D_{I_{i}} < D_{i}

are positive integer numbers and

d_{F_{i}} > 0

are positive fractional numbers. Since the fractional terms

z^{- d_{F_{i}}}

are not practical for direct implementation, they will be approximated using two types of filters: an IIR filter and an FIR filter.

3.1. IIR filter-based approximation

We begin by approximating $z^{- d_{F_{i}}}$ using a stable and causal IIR filter $Ι_{i}^{P} (z)$ , where the subscript i denotes the i-th basis frequency and the superscript $P$ indicates the filter order. Fractional delay approximations based on IIR filters have been previously investigated in the context of periodic tracking control and harmonic compensation for single-period or odd-harmonics RC structures (Chen et al., 2022; Kurniawan et al., 2022). In this work, however, such fractional delay modeling is embedded into the multi-period repetitive control (MPRC) framework to accurately represent multiple basis frequencies. Following the Thiran formulation (Laakso et al., 1996; Thiran, 1971), the coefficients of $Ι_{i}^{P} (z)$ are determined as

z^{- d_{F_{i}}} \approx Ι_{i}^{P} (z) = \frac{a_{i, P} z^{P} + a_{i, P - 1} z^{P - 1} + \dots + a_{i, 0}}{a_{i, 0} z^{P} + a_{i, 1} z^{P - 1} + \dots + a_{i, P}}

(7)

Here

a_{i, 0}, a_{i, 1}, \dots, a_{i, P}

represent the parameters of the filter to be designed. These parameters are then computed according to the following

a_{i, j} = {(- 1)}^{j} \frac{P!}{j! (P - j)!} \prod_{k = 0}^{P} \frac{d_{F_{i}} - P + k}{d_{F_{i}} - P + j + k},

(8)

\forall i = 1, 2, \dots, m, j = 1, 2, \dots, P .

The IIR filter $Ι_{i}^{P} (z)$ , determined according to (7)-(8) is both causal and stable filter, which implies that its poles lie inside the unit circle. This ensures that $Ι_{i}^{P} (z)$ can be realized as a standalone block, offering an overall stable internal model. Furthermore, $Ι_{i}^{P} (z)$ provides unity gain across all frequencies, as expected from the magnitude characteristics of $z^{- d_{F_{i}}}$ . Later, the fractional MPRC using an IIR filter-based approximation is referred to IIR-FMPRC, and the multi-periods term in equation (3) is modified to

M_{I} (z) = 1 - (\prod_{i = 1}^{m} [1 - β_{i} (z) Ι_{i}^{P} (z) z^{- D_{I_{i}}}]) .

(9)

Then, the schematic implementation of

M_{I} (z)

for the IIR-based approximation is obtained by modifying Figure 3 and is illustrated in Figure 4(a). Finally, the proposed IIR-FMPRC is formulated by

C_{M_{I}} (z) = \frac{u_{R} (z)}{e (z)} = \underset{i n t e r n a l m o d e l}{\underset{⏟}{[\frac{M_{I} (z)}{1 - M_{I} (z)}]}} L_{I} (z) .

(10)

Figure 4.

Schematic implementation of the proposed fractional multi-period repetitive controller: (a) IIR-FMPRC based on the IIR filter approximation, and (b) FIR-FMPRC based on the FIR filter approximation.

3.2. FIR filter-based approximation

Next, we aim to approximate $z^{- d_{F_{i}}}$ using an FIR filter $F_{i}^{C} (z)$ , where the subscript $i$ denotes the $i$ -th basis frequency, and the superscript $C$ represents the order of the FIR filter. In the FIR-based approximation approach, the fractional delay terms $z^{- d_{F_{i}}}$ are approximated using FIR filters. The coefficients of these filters are derived through Lagrange interpolation. Specifically, the fractional delay $z^{- d_{F_{i}}}$ are computed as follows

z^{- d_{F_{i}}} \approx F_{i}^{C} (z) = \sum_{j = 0}^{C} g_{i, j} z^{- j} .

(11)

Here

g_{i, j}

correspond to the FIR’s coefficients which are calculated as follows

g_{i, j} = \prod_{k = 0, k \neq j}^{C} \frac{d_{F_{i}} - k}{j - k}, j = 0, 1, \dots, C .

(12)

The constant $C$ in (11) and (12) represents the order of the Lagrange interpolation and is typically chosen such that $C \approx 2 d_{F_{i}}$ (Zhao and Ye, 2017). Similar to the IIR filter $Ι_{i}^{P} (z)$ , the FIR filter $F_{i}^{C} (z)$ is expected to exhibit similar behavior. Therefore, $F_{i}^{C} (z)$ , determined according to (11)–(12), is also a causal and bounded filter, implying that its zeros lie inside the unit circle. This ensures that $F_{i}^{C} (z)$ can be realized as a standalone block, contributing to an overall stable internal model. Furthermore, $F_{i}^{C} (z)$ provides unity gain across all frequencies, as it is shown that $\sum_{j = 0}^{C} g_{i, j} = 1$ , which aligns with the expected magnitude characteristics of $z^{- d_{F_{i}}}$ . Subsequently, the fractional MPRC, which employs an approximation based on an FIR filter, is also referred to as the FIR-FMPRC. In this case, the multi-period term in equation (3) is modified to

M_{F} (z) = 1 - (\prod_{i = 1}^{m} [1 - β_{i} (z) F_{i}^{C} (z) z^{- D_{I_{i}}}]) .

(13)

Similar to the IIR-based case, the schematic implementation of $M_{F} (z)$ for the FIR-based approximation is derived by modifying Figure 3 and is shown in Figure 4(b). Then, the proposed FIR-FMPRC is expressed as

C_{M_{F}} (z) = \frac{u_{R} (z)}{e (z)} = \underset{i n t e r n a l m o d e l}{\underset{⏟}{[\frac{M_{F} (z)}{1 - M_{F} (z)}]}} L_{f} (z) .

(14)

3.3. Stability analysis and design implementation

Next, the stability of the closed-loop system shown in Figure 1 is analyzed, assuming that the plug-in FMPRC expressed in (10) is used. Since the internal model in (10) has a similar structure to that in (1), with the only difference being the replacement of the term $β (z) z^{- D}$ by $M_{I} (z)$ , the stability conditions of the FMPRC can be derived from those of the SPRC. The stability conditions of the SPRC are well-documented in the literature (Chew and Tomizuka, 1990; Cosner et al., 1990; Tomizuka et al., 1989). Therefore, the stability conditions of the FMPRC are as follows:

1. The closed-loop plant model $P_{c} (z)$ must be a stable transfer function. It is given by

P_{c} (z) = \frac{C_{c} (z) P (z)}{1 + C_{c} (z) P (z)} .

(15)

2.The system follows the small gain theorem for the closed-loop system incorporating the plug-in RC, as shown in Figure 1. Specifically, the condition is

{‖ [1 - L_{f} (z) P_{c} (z)] M_{I} (z) ‖}_{\infty} < 1 .

(16)

The condition in (16) can be further written as

{‖ [1 - L_{f} (z) P_{c} (z)] M_{I} (z) ‖}_{\infty} < 1, | 1 - L_{f} (z) P_{c} (z) | | M_{I} (z) | < 1, \forall z = e^{j ω}, 0 < ω < \frac{π}{T_{s}} | 1 - L_{f} (z) P_{c} (z) | < \frac{1}{| M_{I} (z) |} .

(17)

From (15), the closed-loop plant model is defined as the transfer function from $r (z)$ to $y (z)$ without including the repetitive controller (RC). This distinction is important because the open-loop plant P(s) may be unstable or marginally stable. Therefore, a baseline controller $C_{c} (z)$ , such as P, PI, PID, lead, or lag control, can be employed to stabilize the plant and ensure the stability of $P_{c} (z)$ . The learning function $L_{f} (z)$ in (10) is responsible for ensuring asymptotic convergence of the tracking error $e (k)$ . To satisfy condition (17), $L_{f} (z)$ is selected as $L_{f} (z) = k_{L} {P_{c}}^{- 1} (z)$ , where ${P_{c}}^{- 1} (z)$ , denotes the inverse model of $P_{c} (z)$ . When $P_{c} (z)$ is stable and minimum-phase, $L_{f} (z)$ can be directly implemented. For stable non-minimum-phase plants, a zero-phase error tracking controller is adopted following in (Tomizuka et al., 1989). The gain $k_{L}$ is tuned to balance convergence speed and closed-loop stability. Since $L_{f} (z)$ approximates the inverse of $P_{c} (z)$ , its improperness may arise when the relative degree of $P_{c} (z)$ is greater than 0. To ensure implementability, a delay term $z^{- r_{d}}$ is introduced, as shown in Figure 5, where $r_{d}$ represents the relative degree of $P_{c} (z)$ .

Figure 5.

Design implementation of FMPRC.

The above stability conditions are derived under the assumption of an accurate plant model. In practical implementations, however, modeling errors and unmodeled dynamics may introduce deviations between the nominal closed-loop plant $P_{c} (z)$ and the actual system. The effect of such perturbations is discussed in the following remark.

3.3.1. Remark 1

Consider plant uncertainties modeled as a bounded perturbation $Δ_{P} (z)$ affecting the closed-loop plant $P_{c} (z)$ . In this case, the small-gain-based stability condition becomes: ${‖ [1 - L_{f} (z) P_{c} (z) (1 + Δ_{P} (z))] M_{I} (z) ‖}_{\infty} < 1$ . If this inequality holds, asymptotic stability of the closed-loop system is guaranteed. It should be noted that the learning function $L_{f} (z)$ is designed as an approximate inverse of the nominal model $P_{c} (z)$ . When the actual plant deviates from its nominal model due to perturbations, $L_{f} (z)$ no longer perfectly compensates the plant dynamics. As a result, the magnitude of the term ${[1 - L}_{f} (z) P_{c} (z) (1 + Δ_{P} (z))]$ tends to increase, making the left-hand side of the above inequality larger. While stability can still be preserved if the condition remains satisfied, the convergence rate may become slower and the transient becomes longer. It is also important to emphasize that the fractional-delay modeling adopted in the proposed FMPRC does not alter the fundamental small-gain stability structure, since the IIR- and FIR-based delay filters are designed to be causal, stable, and unity-gain approximations.

Finally, to facilitate practical implementation, the design procedure of the proposed FMPRC is organized into a step-by-step guideline, as summarized in Table 1.

Table 1.

Practical design guidelines for the proposed FMPRC.

Step no.	Description
1	Obtain the discrete-time open-loop plant model $P (z)$ from the continuous-time model $P (s)$ using sampling time. The open-loop model $P (z)$ may be unstable or marginally stable.
2	Design a baseline feedback controller $C_{c} (z)$ and obtain the closed-loop plant $P_{c} (z)$ according to (15), such that $P_{c} (z)$ is stable. For simplicity, a proportional controller can be used to preserve the plant order while ensuring all poles lie inside the unit circle.
3	Identify the non-harmonic basis frequencies ( $f_{1}, \dots, f_{m}$ ) composing the reference and disturbance signals, and compute the corresponding periods ( $T_{1}, \dots, T_{m}$ ).
4	Compute the discrete-time one-period delays $z^{- D_{i}}$ for each basis frequency, where $D_{i} = T_{i} / T_{s}$ .
5	Decompose each delay $D_{i}$ into an integer part $D_{I i}$ and a fractional part $d_{F i}$ .
6	Select the low-pass filter $β (z)$ according to (5), with bandwidth $f_{β}$ chosen above the dominant frequency components of the reference and disturbance signals. Increasing $β_{0} (0 < β_{0} < 1)$ in (5) increases the bandwidth, and vice versa.
7	Approximate the fractional delay $z^{- d_{F i}}$ using either an IIR filter based on (7)–(8) or an FIR filter based on (11)–(12). Filter orders 1 or 2 are typically sufficient in resembling the magnitude and phase characteristics of $z^{- d_{F i}}$ , while higher orders increase the complexity of the multi-period internal model.
8	Design the learning function $L_{f} (z)$ as the inverse model of $P_{c} (z)$ for minimum-phase systems. For non-minimum-phase systems, a zero-phase tracking controller can be adopted.
9	Select the learning gain $k_{L}$ , typically starting from unity ( $k_{L} = 1$ ). Increasing the learning gain accelerates convergence but increases control effort, while smaller values yield reduced control effort with slower convergence.
10	Construct the FMPRC using (10) for the IIR-based realization or (14) for the FIR-based realization.

4. Results and discussion

Consider a servomotor (Kurniawan, Cao and Man, 2014) given by a continuous-time model as follows

P (s) = \frac{θ (s)}{V (s)} = \frac{1.74}{0.0268 s^{2} + s},

(18)

where

θ (s)

is the angular position (rad), and

V (s)

is the input voltage (volt). Please note that an open-loop plant (18) is marginally stable, as one of its poles is located at

s = 0

. A discrete-time model approximated by ZOH approach sampled at

T_{s} = 0.007 s

is obtained as

P (z) = \frac{0.0015 z + 0.0013}{z^{2} - 1.77 z + 0.7701},

(19)

The discrete-time model is also marginally stable, with one of its poles located at $z = 1$ . Then, a proportional conventional controller $C_{c} (z) = 10$ is added to obtain the stabilized plant model as

P_{c} (z) = \frac{0.0146 z + 0.0134}{z^{2} - 1.756 z + 0.7835} .

(20)

The plant model in (20) is now a stable, minimum-phase system with one zero and two complex poles, all located inside the unit circle. Specifically, the zero is at $z = - 0.9178$ , and the poles are at $z = 0.878 \pm i 0.112$ . The closed-loop model in (20) also indicates that $P_{c} (z)$ has a relative degree of 1.

To provide information regarding the simulation environment, the computational platform and implementation details are briefly described below. Simulations were performed on an HP ZBook Firefly G8 equipped with an Intel® Core™ i7-1165G7 processor, 8 GB RAM, running Windows 11, using MATLAB/Simulink R2022b. For future experimental validation, a dual-servo setup is planned, where one motor generates the reference motion and the other introduces independently adjustable periodic disturbances, including non-harmonic frequencies.

4.1. Case 1: Tracking periodic reference $r (k)$ and suppressing periodic disturbance $v$ ( k )

In Case 1, we consider the simultaneous tracking of the periodic reference $r (k)$ and suppression of the periodic disturbance $v (k)$ . The reference $r (k)$ is modeled in (21) and illustrated in Figure 6(a), while the disturbance $v (k)$ is modeled in (22) and shown in Figure 6(b). From (21) and (22), it is evident that $r (k)$ and $v (k)$ have different basis frequencies: the basis frequency of $r (k)$ is 1 Hz, and the basis frequency of $v (k)$ is 1.75 Hz. This results in two distinct basis periods: $T_{1} = 1 s$ and $T_{2} = 0.571 s$ . Given the sampling time $T = 0.007 s$ , the corresponding fractional delays are $D_{1} = 142.86$ and $D_{2} = 81.63$

r (k) = {\begin{array}{l} \frac{π}{3}, \\ \frac{2 π}{2} \sin (2 π K T), \\ - \frac{π}{3} \end{array} \begin{array}{l} \frac{2 π}{3} \sin (2 π K T) > \frac{π}{3}, \\ | \frac{2 π}{3} \sin (2 π K T) | \leq \frac{π}{3} \\ \frac{2 π}{3} \sin (2 π K T) < \frac{π}{3} . \end{array}},

(21)

v (k) = 0.2 \sin (3.5 π K T) + 0.1 \sin (7 π k T),

(22)

Figure 6.

(a) Periodic reference $r (k)$ and (b) periodic disturbance $v (k)$ .

We begin by implementing the IIR-FMPRC design strategy presented in Section 3. Table 2 summarizes the IIR filter representations of the fractional time-delays

z^{- D_{1}}

and

z^{- D_{2}}

for two different filter orders,

P = 1

and

P = 2

. From Table 2, it can be observed that the typical approach involves rounding the fractional delays

D_{1}

and

D_{2}

to the nearest integers, resulting in integer delays of

D_{1} = 143

and

D_{2} = 82

, which are easier to implement. However, this method introduces steady-state errors due to the period mismatch, and these errors will appear in the simulation output. Additionally, Table 2 illustrates that the IIR filters, approximated using the Thiran formula (7)–(8), have a causal form for both

P = 1

and

P = 2

. Furthermore, all of the filters exhibit unity gain (0 dB), with their poles located inside the unit circle. Specifically, the poles of

I_{1}^{1} (z)

I_{2}^{1} (z)

I_{1}^{2} (z)

, and

I_{2}^{2} (z)

are at

- 0.0753, {- 0.1643, 0.0664}, - 0.227,

and

{0.3275, 0.0461}

, respectively. Therefore, the IIR filters used to approximate the fractional delay components

z^{- d_{F_{1}}}

and

z^{- d_{F_{2}}}

are stable, causal, and maintain unity gain across all frequencies.

Table 2.

IIR- and FIR-based filter representations for the fractional time-delays $z^{- D_{1}}$ and $z^{- D_{2}}$ with different filter orders.

Filter	$D_{i, {i = 1, 2}}$	Round $(D_{i})$	$D_{I_{i}}$	$d_{F_{i}}$	$I_{i}^{P} (z)$
IIR	$D_{1} = 142.86$	143	142	0.86	$I_{1}^{1} (z) = \frac{0.0753 z + 1}{z + 0.0753}$
	$D_{1} = 142.86$	143	141	1.86	$I_{1}^{2} (z) = \frac{- 0.0109 z^{2} + 0.0979 z + 1}{z^{2} + 0.0979 z - 0.0109}$
	$D_{2} = 81.63$	82	81	0.63	$I_{2}^{1} (z) = \frac{0.227 z + 1}{z + 0.227}$
	$D_{2} = 81.63$	82	80	1.63	$I_{2}^{2} (z) = \frac{- 0.0151 z^{2} + 0.2814 z + 1}{z^{2} + 0.2814 z - 0.0151}$
FIR	$D_{1} = 142.86$	143	142	0.86	$F_{1}^{1} (z) = 0.14 + 0.86 z^{- 1}$
	$D_{1} = 142.86$	143	141	1.86	$F_{1}^{2} (z) = - 0.0602 + 0.2604 z^{- 1} + 0.7998 z^{- 2}$
	$D_{2} = 81.63$	82	81	0.63	$F_{2}^{1} (z) = 0.37 + 0.63 z^{- 1}$
	$D_{2} = 81.63$	82	80	1.63	$F_{2}^{2} (z) = - 0.1165 + 0.6031 z^{- 1} + 0.5143 z^{- 2}$

Next, we proceed with the implementation of the FIR-FMPRC design strategy. Table 2 also provides a summary of the FIR filter representations for the fractional time-delays $z^{- D_{1}}$ and $z^{- D_{2}}$ at two different filter orders, $C = 1$ and $C = 2$ . From Table 2, it can be seen that the filters remain causal and maintain a unity gain (0 dB), as indicated by the total coefficients of the filter being equal to 1. Consequently, the FIR filters used to approximate the fractional delay $z^{- d_{F_{1}}}$ and $z^{- d_{F_{2}}}$ are both stable and implementable. Let us choose the low-pass filters $β_{1} (z) = β_{2} (z) = 0.25 z^{- 1} + 0.5 + 0.25 z$ , designed according to (5). The learning function $L_{f} (z)$ is designed as the inverse model of $P_{c} (z)$ as shown in (20). The learning function gain is set to $k_{L} = 1$ . It can be observed that (20) has a relative degree of one, making $L_{f} (s)$ an improper transfer function, as the degree of the numerator is one greater than that of the denominator. Therefore, the design implementation shown in Figure 5 can be applied. As a result, the proposed IIR-FMPRC in (10) and FIR-FMPRC in (14) can now be synthesized and implemented.

All comparative analyses are conducted within the same plug-in RC framework as shown in Figure 1 to ensure structural consistency. The basis of comparison focuses on the type of internal model employed. The SPRC serves as the fundamental benchmark, while the conventional MPRC represents the multi-period case without fractional-delay enhancement. The proposed IIR-FMPRC and FIR-FMPRC extend this structure by incorporating fractional-delay modeling. Controllers with adaptive frequency estimation are not included in the direct comparison, as such methods introduce additional transient effects that would obscure the specific contribution of the fractional internal model. Therefore, to ensure a fair comparison, the proposed IIR-FMPRC and FIR-FMPRC are evaluated alongside the SPRC and MPRC, as defined in equations (1) and (2), respectively. The initial comparison is conducted based on the magnitude characteristics of their respective internal models. The internal models of SPRC, MPRC, IIR-FMPRC and FIR-FMPRC are presented as follows

S P R C = \frac{β_{1} (z) z^{- 143}}{1 - β_{1} (z) z^{- 143}},

(23)

M P R C = \frac{M (z)}{1 - M (z)}, \to M (z)

(24)

M (z) = β_{1} (z) z^{- 143} + β_{1} (z) z^{- 82} - β_{1}^{2} (z) z^{- 225},

I I R - F M P R C = \frac{M_{I} (z)}{1 - M_{I} (z)}, \to M_{I} (z)

(25)

M_{I} (z) = β_{1} (z) I_{1}^{1} (z) z^{- 142} + β_{1} (z) I_{2}^{1} (z) z^{- 81} - β_{1}^{2} (z) I_{1}^{1} (z) I_{2}^{1} (z) z^{- 223},

F I R - F M P R C = \frac{M_{F} (z)}{1 - M_{F} (z)}, \to M_{F} (z)

(26)

M_{F} (z) = β_{1} (z) F_{1}^{1} (z) z^{- 142} + β_{1} {(z) F}_{2}^{1} (z) z^{- 81} - β_{1}^{2} (z) F_{1}^{1} (z) F_{2}^{1} (z) z^{- 223} .

Here $I_{1}^{1} (z)$ , $I_{2}^{1} (z)$ , $F_{1}^{1} (z)$ , and $F_{2}^{1} (z)$ are all are listed in Table 2. The magnitude-frequency characteristics of the SPRC and MPRC internal models are illustrated in Figure 7, whereas those of the IIR-FMPRC and FIR-FMPRC internal models are depicted in Figure 8. As shown in Figure 7(a), the SPRC targets only a single base frequency and its harmonics, with peaks appearing at 0.999 Hz, 1.998 Hz, 2.997 Hz, and so on. These frequencies exhibit a slight deviation from the desired peaks at 1 Hz, 2 Hz, 3 Hz, etc. This minor mismatch is attributed to the rounding of the delay length $D_{1}$ to the nearest integer. Unlike SPRC, which targets a single basis frequency and its harmonics, MPRC, as shown in Figure 7(b), is designed to target two basis frequencies and their respective harmonics. However, similar to SPRC, MPRC also exhibits slight mismatches. The first basis frequency and its harmonics align closely with those observed in SPRC. Additionally, slight deviations are present in the second basis frequency, where peaks appear at approximately 1.74 Hz, 3.484 Hz, and so on, instead of the desired 1.75 Hz, 3.5 Hz, etc. These mismatches in both basis frequencies result from rounding the delay lengths $D_{1}$ and $D_{2}$ to the nearest integers.

Figure 7.

Magnitude-frequency characteristics: (a) SPRC internal model, (b) MPRC internal model.

Figure 8.

Magnitude-frequency characteristics: (a) IIR-FMPRC internal model, (b) FIR-FMPRC internal model.

In contrast to SPRC and MPRC, the IIR-FMPRC and FIR-FMPRC, as illustrated in Figures 8(a) and (b), accurately align their frequency peaks with the desired base frequencies. The first base frequency exhibits peaks at 1 Hz, 2 Hz, and so on, while the second base frequency shows peaks at 1.75 Hz, 3.5 Hz, etc. This precision indicates that employing both IIR and FIR filters to approximate fractional delays successfully maintains the desired frequency peaks, thereby enhancing the tracking performance. Notably, there is a difference between IIR-FMPRC and FIR-FMPRC in terms of peak magnitudes, with IIR-FMPRC displaying slightly higher peak values compared to FIR-FMPRC. This phenomenon will later affect the tracking performance, especially during the steady state.

Figure 9 illustrates the tracking error responses of the system using SPRC, MPRC, IIR-FMPRC, and FIR-FMPRC. The tracking error responses for all four controllers, covering both the overall and transient phases, are shown in Figures 9(a) and (b), respectively. Figure 9(b) indicates that SPRC achieves a faster transient response and a lower transient peak compared to MPRC, IIR-FMPRC, and FIR-FMPRC. This behavior is expected, as SPRC involves only a single delay term, whereas IIR-FMPRC and FIR-FMPRC employ dual delay terms, resulting in longer effective time delays for the control input.

Figure 9.

Comparison of tracking errors $e (k)$ for SPRC, MPRC, IIR-FMPRC, and FIR-FMPRC under Case 1: (a) overall time response, and (b) transient behavior.

Figure 10 presents the steady-state tracking performance of the considered controllers. As shown in Figure 10(a), SPRC exhibits a larger steady-state tracking error compared to MPRC, IIR-FMPRC, and FIR-FMPRC. The enlarged view in Figure 10(b), corresponding to the interval from t = 9 to 9.5 s, further highlights that both IIR-FMPRC and FIR-FMPRC provide improved tracking accuracy relative to MPRC. In addition, IIR-FMPRC achieves a slightly smaller steady-state error than FIR-FMPRC. In addition to the tracking errors shown in Figures 9 and 10, the following performance metrics are also evaluated

\max - e = \max | e (k) |,

(27)

\max - ess = \max | e_{s s} (k) |,

(28)

rms - ess = \sqrt{\frac{1}{τ_{e n d} - τ_{s s}}} \sum_{k = τ_{s s}}^{τ_{e n d}} e^{2} (k) .

(29)

where

e_{s s} (k)

is steady-state error defined as

e (k)

for

k = τ_{s s}, τ_{s s} + 1, \dots, τ_{e n d}, τ_{s s} = \frac{5}{0.007} = 714,

and

τ_{e n d} = r o u n d (\frac{15}{0.007}) = 2143 .

Figure 10.

Steady-state tracking performance for Case 1: (a) tracking errors (b) enlarged view over $t = 9 - 9.5$ s.

Complementing the steady-state error metrics, two global performance indices are also considered to provide a more comprehensive evaluation of the tracking behavior. Specifically, the Integral of Squared Error (ISU) and the Integral of Time-weighted Absolute Error (ITAE) are defined as

ISU = \sum_{k = 1}^{τ_{e n d}} e^{2} (k) T_{s},

(30)

ITAE = \sum_{k = 1}^{τ_{e n d}} k T_{s} | e (k) | T_{s},

(31)

where

T_{s}

is the sampling period. The ISU metric captures the overall accumulated tracking error energy, while ITAE penalizes long-lasting errors and emphasizes convergence behavior over time. These indices complement the steady-state metrics and enable a clearer assessment of both transient and long-term tracking performance.

Table 3 summarizes the performance metrics for Case 1, including max-e, max-ess, rms-ess, ISU, and ITAE. Although SPRC achieves the smallest max-e, consistent with its slightly faster initial transient response observed in Figure 9, it exhibits larger steady-state errors as reflected by its higher max-ess and rms-ess values. In contrast, both IIR-FMPRC and FIR-FMPRC provide improved steady-state accuracy, with IIR-FMPRC attaining the lowest rms-ess among all controllers. From a global perspective, the ISU and ITAE results further clarify the performance differences. While ISU reflects the overall accumulated tracking error energy, ITAE emphasizes convergence speed over time. Despite SPRC showing a faster initial response, its larger ITAE value indicates slower overall convergence. The proposed FMPRC schemes achieve significantly lower ITAE values, demonstrating faster error decay and improved long-term convergence while maintaining reduced steady-state errors. Overall, these results confirm that FMPRC enhances both convergence characteristics and steady-state tracking performance.

Table 3.

Comparison of max-e, ISU, ITAE, max-ess, and rms-ess for Cases 1-3.

Case	Method	Filter order	max-e (rad)	ISU (rad)	ITAE (rad)	max-ess (rad)	rms-ess (rad)
1	SPRC	—	0.7139	0.1524	2.4337	0.0577	0.0235
	MPRC	—	0.8561	0.3204	1.4533	0.0211	0.0097
	IIR-FMPRC	1	0.8690	0.3231	0.8963	0.0240	0.0039
	IIR-FMPRC	2	0.8707	0.3233	0.8947	0.0240	0.0039
	FIR-FMPRC	1	0.8688	0.3229	0.9379	0.0267	0.0045
	FIR-FMPRC	2	0.8690	0.3231	0.8998	0.0250	0.0040
2	SPRC	—	0.7340	0.1549	2.0425	0.1012	0.0399
	MPRC	—	0.8789	0.3250	1.5558	0.0476	0.0177
	IIR-FMPRC	1	0.8926	0.3281	1.1072	0.0607	0.0170
	IIR-FMPRC	2	0.8942	0.3283	1.1060	0.0608	0.0170
	FIR-FMPRC	1	0.8925	0.3280	1.1432	0.0609	0.0171
	FIR-FMPRC	2	0.8927	0.3281	1.1101	0.0605	0.0170
3	SPRC	—	0.7718	0.2373	2.1039	0.0442	0.0178
	MPRC	—	1.0534	0.5050	2.3557	0.0254	0.0124
	IIR-FMPRC	1	1.0655	0.5072	1.6673	0.0262	0.0049
	IIR-FMPRC	2	1.0663	0.5076	1.6686	0.0262	0.0049
	FIR-FMPRC	1	1.0672	0.5072	1.7166	0.0295	0.0056
	FIR-FMPRC	2	1.0668	0.5073	1.6735	0.0274	0.0051

In addition to performance evaluation, the computational burden of the proposed FMPRC is briefly discussed. With two basis frequencies and delay lengths D₁ = 142.86 and D₂ = 81.63, the implementation uses integer delay buffers of 142 and 81 samples, with fractional parts realized using low-order IIR or FIR filters. The per-cycle floating-point operation (FLOP) count is derived from the control law update equation in the z-domain:

U_{R} (z) = β (z) z^{- D} U_{R} (z) + β (z) z^{- D} L_{f} (z) e (z)

. For SPRC, this yields approximately 16 FLOP per cycle. For MPRC with

m = 2

, the internal model

M (z) = β_{1} (z) z^{- D_{1}} + β_{2} (z) z^{- D_{2}} - {β_{1} (z) β}_{2} (z) z^{- D_{1} - D_{2}}

introduces two additional filter branches and one cross-term, increasing the count to approximately 33 FLOP per cycle. The proposed FMPRC appends a fractional-delay filter per basis frequency on top of this base cost, namely a Thiran IIR all-pass filter for IIR-FMPRC or a Lagrange FIR filter for FIR-FMPRC, with the resulting per-cycle counts for all controllers summarized in Table 4. All variants remain O(1) since the filter order is a fixed design constant. Assuming a conservative hardware baseline of 10 MFLOPS, the heaviest case (IIR-FMPRC, P = 2) requires only ∼5.7 μs, representing 0.08% of the 7 ms sampling budget, confirming real-time feasibility.

Table 4.

Computational complexity of SPRC, MPRC, and the proposed FMPRC variants per control cycle (T_s = 7 ms; hardware baseline ≥10 MFLOPS).

Controller	FLOP/cycle	Complexity	Ratio vs SPRC	Est. exec. time	Budget used	Real-time feasible?
SPRC	∼16	O(1)	1×	∼1.6 μs	0.02%	Yes
MPRC	∼33	O(1)	2.1×	∼3.3 μs	0.05%	Yes
IIR-FMPRC (P = 1)	∼47	O(1)	2.9×	∼4.7 μs	0.07%	Yes
IIR-FMPRC (P = 2)	∼57	O(1)	3.6×	∼5.7 μs	0.08%	Yes
FIR-FMPRC (C = 1)	∼41	O(1)	2.6×	∼4.1 μs	0.06%	Yes
FIR-FMPRC (C = 2)	∼45	O(1)	2.8×	∼4.5 μs	0.06%	Yes

4.2. Case 2: Tracking periodic reference $r (k)$ and suppressing time-varying periodic disturbance $v$ ( k )

In Case 2, we evaluate the tracking performance under a time-varying disturbance while maintaining the same reference signal as in Case 1. The disturbance $v (k)$ is modeled in (30) and depicted in Figure 11. As shown in (30), the system initially operates with a base frequency of 1 Hz for the first 7 seconds, after which the frequency shifts to 1.75 Hz. Notably, the second frequency is not a harmonic of the first, and vice versa. Given the presence of two distinct base frequencies (1 Hz and 1.75 Hz), the same controller design as in Case 1 remains applicable

v (k) = {\begin{cases} 0.2 \sin (2 π k T) + 0.1 \sin (4 π k T), \\ 0.2 \sin (3.5 π k T) + 0.1 \sin (7 π k T), \end{cases} \begin{array}{l} k T \leq 7 \\ k T > 7 \end{array},

(32)

Figure 11.

Time-varying periodic disturbance $v (k)$ .

Figure 12 illustrates the tracking errors of the four controllers for Case 2. Similar to Case 1, Figure 12(b) demonstrates that SPRC achieves a faster initial transient response and a lower transient peak compared to MPRC, IIR-FMPRC, and FIR-FMPRC. However, as shown in Figure 13(a), the tracking performance of SPRC deteriorates significantly when the disturbance frequency changes at $t = 7$ s, indicating its inability to handle frequency variations. Figure 13 further presents the steady-state tracking performance for Case 2. As observed in Figure 13(a), both IIR-FMPRC and FIR-FMPRC achieve improved tracking accuracy compared to MPRC. The enlarged view in Figure 13(b) confirms that the proposed IIR-FMPRC and FIR-FMPRC deliver superior steady-state tracking performance compared to both SPRC and MPRC under time-varying periodic disturbances.

Figure 12.

Tracking error responses $e (k)$ of SPRC, MPRC, IIR-FMPRC, and FIR-FMPRC for Case 2: (a) overall time response, and (b) transient response.

Figure 13.

Steady-state tracking performance for Case 2: (a) tracking errors $e (k)$ in steady-state, and (b) enlarged view over $t = 9 - 10$ s.

The performance metrics for Case 2 are summarized in Table 3. Similar to Case 1, SPRC achieves the smallest max-e, reflecting its faster initial transient response. However, it exhibits larger steady-state errors, as indicated by higher max-ess and rms-ess values compared to the other controllers. In contrast, both IIR-FMPRC and FIR-FMPRC provide improved steady-state accuracy, with IIR-FMPRC achieving the smallest rms-ess among all methods. From a global performance perspective, the ISU and ITAE results further clarify these differences. While SPRC shows competitive transient behavior, its larger ITAE indicates slower overall convergence. The proposed FMPRC schemes achieve lower ITAE values and maintain reduced steady-state errors, confirming their superior convergence and steady-state tracking performance under time-varying disturbance conditions.

4.3. Case 3: Tracking periodic reference $r (k)$ and suppressing periodic disturbance $v (k)$ under plant parameter variations

To further assess robustness against plant uncertainty, an additional simulation is conducted by varying all parameters of the nominal plant model in (18) by −30%. This variation represents modeling errors and practical implementation deviations. The periodic reference and disturbance signals remain the same as in Case 1 to ensure a consistent and fair comparison. All controller structures and parameters are kept identical to those used in the previous cases, allowing a direct evaluation of how SPRC, MPRC, IIR-FMPRC, and FIR-FMPRC tolerate plant perturbations without redesign or retuning. The tracking error responses for Case 3 are shown in Figure 14, including (a) the transient time response and (b) the steady-state response. Compared with Case 1, as illustrated previously in Figure 9(b), the transient responses of all methods become slower under plant parameter variations. In terms of steady-state behavior, Figure 14(b) shows a pattern similar to that observed in Case 1 in Figure 10(a), with tracking errors remaining within the range of approximately −0.05 to 0.05 rad. These results indicate that although convergence is slower under plant perturbations, the steady-state tracking accuracy remains comparable to the nominal case.

Figure 14.

Tracking error responses $e (k)$ of SPRC, MPRC, IIR-FMPRC, and FIR-FMPRC for Case 3: (a) transient behavior, and (b) steady-state performance.

The performance metrics for Case 3 are also summarized in Table 3. Compared to Case 1, plant parameter variations of −30% lead to a general increase in error indices for all controllers, reflecting the impact of model mismatch. Similar to Case 1, SPRC achieves the smallest max-e due to its faster initial transient response. However, its steady-state errors remain larger compared to the FMPRC-based methods. Both IIR-FMPRC and FIR-FMPRC maintain improved steady-state accuracy under plant perturbation, with lower rms-ess values than SPRC and MPRC. From a global perspective, the ISU and ITAE values in Case 3 are generally larger than those in Case 1, indicating slower overall convergence due to plant parameter variations. This behavior is expected since the learning function is designed based on the nominal model, and plant perturbations reduce compensation accuracy. Nevertheless, the stability condition remains satisfied, and steady-state tracking performance is still preserved. The FMPRC schemes continue to exhibit lower steady-state errors compared to SPRC and MPRC, demonstrating that the proposed fractional-delay modeling maintains steady-state tracking performance despite plant deviations from nominal parameters.

While the proposed FMPRC demonstrates improved steady-state tracking performance, several practical limitations should be noted. The computational and implementation complexity increases with higher-order IIR/FIR filters and with a larger number of non-harmonic basis frequencies included in the internal model, since each additional basis frequency introduces extra delay elements and fractional-delay approximation blocks, thereby increasing memory usage and design complexity. Moreover, the framework assumes that the basis frequencies $f_{i}, i = 1, \dots, m$ , are known a priori; variations in these frequencies may lead to delay mismatch and degraded tracking accuracy. The stability and asymptotic convergence guarantees also depend on the accuracy of the plant model $P (z)$ , and significant model variations may violate the convergence performance and stability conditions. These aspects indicate that future work should enhance robustness both with respect to frequency variations and plant model uncertainties.

5. Conclusion

This paper has presented a fractional multi-period repetitive controller (FMPRC) that enhances conventional MPRC by incorporating fractional time-delay into the internal model. By decomposing the delay into integer and fractional components and realizing the fractional delay using IIR- or FIR-based filters, the proposed FMPRC achieves accurate alignment with multiple basis frequencies, thereby reducing frequency mismatch and residual steady-state errors. Simulation results under multi-frequency, time-varying periodic disturbances, and plant perturbations demonstrate that FMPRC consistently outperforms SPRC and conventional MPRC in terms of tracking accuracy and disturbance rejection, particularly for steady-state performance. Robustness analyses under plant parameter variations further confirm that the proposed framework preserves closed-loop stability and steady-state tracking accuracy, although convergence speed may slightly degrade due to model mismatch. It should be noted that the current framework assumes the basis frequencies to be known a priori; variations or drift in these frequencies may lead to delay mismatch and degraded tracking accuracy. As a direction for future work, experimental validation using a planned dual-servo platform will be conducted, where one motor generates the reference motion and the other introduces independently adjustable periodic disturbances, including non-harmonic components. In addition, the proposed framework can be extended to scenarios involving unknown or time-varying disturbance frequencies by integrating adaptive or data-driven frequency estimation mechanisms.

Footnotes

Acknowledgments

The authors used AI-based language assistance tools for minor editing and clarity enhancement. All technical content and scientific interpretations remain the sole responsibility of the authors.

ORCID iD

Edi Kurniawan

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Research and Innovation Agency (BRIN), Indonesia, through the Research Assistant (RA) program No. 166/HK/II/2024.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Enhanced fractional multi-period repetitive controller with fractional time-delay for high-precision servo systems

Abstract

Keywords

1. Introduction

2. Preliminaries: Design of SPRC and MPRC

3. Fractional multi period RC (FMPRC)

3.1. IIR filter-based approximation

3.2. FIR filter-based approximation

3.3. Stability analysis and design implementation

3.3.1. Remark 1

4. Results and discussion

4.1. Case 1: Tracking periodic reference r ( k ) and suppressing periodic disturbance v ( k )

4.2. Case 2: Tracking periodic reference r ( k ) and suppressing time-varying periodic disturbance v ( k )

4.3. Case 3: Tracking periodic reference r ( k ) and suppressing periodic disturbance v ( k ) under plant parameter variations

5. Conclusion

Footnotes

Acknowledgments

ORCID iD

Funding

Declaration of conflicting interests

References

4.1. Case 1: Tracking periodic reference $r (k)$ and suppressing periodic disturbance $v$ ( k )

4.2. Case 2: Tracking periodic reference $r (k)$ and suppressing time-varying periodic disturbance $v$ ( k )

4.3. Case 3: Tracking periodic reference $r (k)$ and suppressing periodic disturbance $v (k)$ under plant parameter variations