#### Abstract

This paper considers the design of high-order modified repetitive control systems for periodic reference inputs with uncertain period-time. The objective of this work was to develop a new design method so that the closed-loop high-order modified repetitive control system is robustly stable with high control precision for periodic reference inputs with uncertain period-time. The parametrization of all stabilizing controllers containing three free parameters is proposed based on the Youla-Kucera parameterization. Moreover, to obtain the free parameters, the constraint conditions were converted into stability conditions in the form of Bilinear Matrix Inequalities that can be solved using the available software. In addition, the high control precision is guaranteed by designing the free parameters after the control characteristic of this control system. The validity and effectiveness of the proposed design method were verified by numerical examples.

#### 1. Introduction

In practical applications, many control systems, such as industrial robots, computer disk drives, and rotating machine tools, have to deal with periodic reference or disturbance signals. For this class of systems, Hara et al. [1] proposed a repetitive controller that applies the internal model principle [2], which states that if a reference or disturbance signal can be regarded as the output of an autonomous system, including this system in a stable feedback loop guarantees asymptotically tracking or rejecting performance. This control technique has been proven to be an efficient control scheme for handling periodic signals and has been applied extensively.

Generally, most repetitive controller designs in the literature suffer from two major drawbacks. One is the requirement of exact knowledge of the period-time of reference or disturbance signals [3]. This means that in practical applications, either the period-time is required to be a constant or an accurate measurement of the periodicity is necessary. The other drawback is due to the Bode sensitivity integral [4]: the perfect reduction at harmonic frequencies is counteracted by amplification of noise at intermediate frequencies.

To address these problems, the so-called high-order repetitive control has been established [5–13]. Inoue [6] first proposed the high-order repetitive controller to improve the control precision and cancel out the phase lag. Based on that work [6], Chang et al. [7] provided an optimization method to determine the weighting factors by minimizing the infinity norm of the relative error transfer function for the high-order repetitive controller. For small variations in the period-time, in [8], an approach is presented to design a high-order repetitive controller such that robustness for period changes is obtained. Inspired by the results in [7], Steinbuch et al. [9, 10] proposed a unified framework able to reproduce the results of both [7] and [8]. In contrast with the conventional design approaches that are focused on the perfect nominal periodic performance of the closed-loop system, Pipeleers et al. [5, 11] presented a systematic, semidefinite programming-based scheme to compute high-order repetitive controllers, which yields an optimal trade-off between nonperiodic performance and robust periodic performance. For multiple-period systems that have a similar structure to the high-order repetitive control system, solutions are given by Yamada et al. [12] and Yamada et al. [13] using the Youla-Kucera parametrization (YKP) without solving the problem of designing the free parameters.

In this paper, we consider the problem of designing a high-order modified repetitive control system to track or reject external signals with uncertain period-time.

A new structure of high-order modified repetitive controller was established inspired by the conventional high-order repetitive controller. Two useful methodologies, the YKP and the Bilinear Matrix Inequalities (BMIs) approach, were adopted to design compensators in the high-order modified repetitive control system. The objective was to develop a new and simple design method that not only provides good control performance while preserving the system’s stability, but can also be efficiently implemented with available software. The approach taken in this paper was as follows. First, the parametrization of all stabilizing controllers was derived on the basis of the YKP. Next, the constraint conditions of the free parameters were transformed into stability conditions in the form of BMIs. In addition, according to the control characteristic, a low-pass filter was introduced to specify the intermediate frequencies. Finally, numerical examples for a single-input/single-output (SISO) tracking system were specified to illustrate the validity and effectiveness of the proposed design method. The results presented in this paper show that the high-order modified repetitive control system can provide high control precision for periodic signals with a certain vibration in period-time. Moreover, the control precision at the intermediate frequencies was satisfied by modifying the cutoff frequency of low-pass filters in compensators.

*Notations*. denotes the set of stable proper real rational functions; denotes the set of stable causal functions; denotes the -dimensional Euclidean space; is the set of all real matrices; is the identity matrix; the null matrix or null vector of appropriate dimension is denoted by ; the superscript “” stands for the transpose of a matrix; denotes the largest singular value of ; the symmetric terms in a symmetric matrix are denoted by ; for example, .

#### 2. Problem Formulation

Figure 1 depicts the proposed high-order modified repetitive control system, where is a strictly proper plant with an assumption that . is a forward compensator to enhance the robust stability of this closed-loop system and improve the transient response, is typically called the learning filter and is used to compensate the high-order modified repetitive controller , is the control output, and is a periodic reference input. Uncertainty of period-time is modeled as relative uncertainty on , bounded by : In Figure 1, the structure of the high-order modified repetitive controller consists of multiple periodic signal generators and can be expressed as where are the weighting factors to modify the dynamic response in between the harmonic frequencies, and is a low-pass filter assumed as a diagonal matrix function.

Like the current high-order repetitive control design approaches in [5, 7, 10], this work relied on the following two nonrestrictive assumptions.

*Assumption 1. *All harmonics of the reference input are assumed to lie in the passband of . This can be guaranteed by the proper design of , provided that all harmonics lie well below , the cutoff frequency of .

*Assumption 2. *In its passband, is assumed to equal its DC gain for all . These two assumptions imply that
for all , where are defined in (4) and are the frequency components of the periodic reference input described as
In this case, the weighting factors can be determined by using the methods proposed in [5, 7, 10].

In addition, suppose that the strictly proper plant, , is controllable and observable and its state-space description is
where , , and are system matrices, and , , and denote state, control input, and control output, respectively.

The design problem considered in this paper can be stated as follows: for a given strictly proper plant , a constant , and a low-pass filter , find the admissible compensators and such that the controller written as
robustly stabilizes the closed-loop system with high control precision for the periodic reference input with uncertainty in period-time, where is defined in (2).

#### 3. High-Order Modified Repetitive Control System Design

##### 3.1. The Parametrization of All Stabilizing Controllers

In this section, we derive the parametrization of all stabilizing controllers, with the structure of the high-order modified repetitive controller for periodic reference inputs with uncertain period-time. The necessary and sufficient conditions for the parametrization of all stabilizing controllers by means of the YKP are proposed by the following main theorem.

Theorem 1. *The control system in Figure 1 is internally stable if and only if the controllers can be written as**
where , , , and rank , , , , and are coprime factors of on , and , , , satisfy
*

*Proof (Necessity). *To prove the necessity, suppose that there exist admissible compensators and , such that the controller with a high-order repetitive control structure written as (9) stabilizes the closed-loop system. By means of the YKP [14], the parametrization of all stabilizing controllers can be written as (10a) and (10b). Substitute in (9) into (10a) and the in (11) can be represented as
Let and be coprime factors of on satisfying
Under the assumptions in Section 2, the low-pass filter is a diagonal matrix function; this means that is also a diagonal matrix function, and substituting (14) into (13) yields
Moreover, let and be coprime factors of satisfying
Combining (15) and (16), (11) can be obtained, where
Note that , and , are functions. As a consequence, the necessity can be proven if is a matrix function. According to (2), (9), and (13), can be rewritten as
According to (12), the equation
holds and can be expressed as
Subsequently,
Because stabilizes the closed system, can be proven according to internal stability [15]. Therefore, the necessity is obtained.*Sufficiency*. Suppose that there is written by (11) such that (10b) stabilizes the closed-loop system in Figure 1. Substituting (10b) into (9) yields
respectively. Therefore, the sufficiency is obtained.

This completes the proof.

From Theorem 1, , , and are the free parameters that need to be determined. In the following, the design methods for , , and are based on BMIs and the control characteristic.

##### 3.2. Design of Free Parameters and

In [12, 13], the designs of the free parameters use the Nyquist stability criterion by manual examination, which is very difficult and inefficient. To resolve this problem, it was essential to establish an efficient and easy design method for the free parameters. In this subsection, an efficient design method for the free parameters using BMIs that can be solved with existing software is presented.

The first step is to design the free parameters and such that belongs to on the basis of state-space description. According to (2) and (11), can be expressed as Because , , and , means that the closed-loop system is stable. The problem of selecting the free parameters and , while ensuring , is converted to design and , thus stabilizing the closed-loop system . A general design method for and was established, based on the state-space description and assuming a low-pass filter , as Then, from Figure 1, the state-space description of can be obtained as where is the state of , Assume that the state-space descriptions of , , and are where has full column rank. Draw the configuration of in Figure 2 and the state-space description of the closed-loop system with is given by where is the left pseudoinverse of . The left pseudoinverse of is written as This means that the normal rank of is and could also be represented as

Replacing , , , and , with , , , and in (32), respectively, the system matrix is written as By employing the Lyapunov functional method to system (30), the following result can be derived.

Theorem 2. *For system (30), if symmetric positive-definite matrices and exist such that
**
then the time-delay system (30) is stable. *

*Proof. *Choose the candidate Lyapunov functional to be
where and . Calculating the derivative of for the time-delay system (30) yields
where . Clearly, the stability condition for the time-delay system (30) is
This completes the proof.

Theorem 1 requires that and are matrix functions, which means that and should be stable. According to Lyapunov functional theory, the BMI condition (35) contains the stability condition of . Hence, it is only necessary to provide the stability condition of , and the following theorem will provide the stability condition in the form of BMIs for .

Theorem 3. *For system (30), if symmetric positive-definite matrices , , and exist such that (35) and
**
hold, then the time-delay system (30) is stable and . *

* Proof. *From Lyapunov functional theory, is a matrix function if there exists a symmetric positive-definite matrix such that
which is equivalent to
According to the results in [16], a sufficient condition guaranteeing (41) is that there exists a positive number such that
Replacing and with and and applying the Schur complement lemma show that (42) is equivalent to (39).

This completes the proof.

To avoid zero solutions, the initial transfer matrices for and are required. Without loss of generality, and should be represented as where , , , , , , , and are the given initial matrices, and , , , , , , , and are the unknown matrices that need to be determined. In the following theorem, modified stability conditions are proposed in the form of BMIs.

Theorem 4. *For given initial matrix functions and , if symmetric positive-definite matrices , , and and appropriately dimensioned matrices , , , , , , , and exist such that
**
hold, where , , and are defined as
**
then the time-delay system (30) is stable and is . *

* Proof. *Replacing matrices , , , , , , , and with matrices , , , , , , , and , respectively, in (35) and (39), the constraints (44) can be obtained.

This completes this proof.

Constraints (44) are the BMIs that can be solved by BMI solvers, such as PENBMI [17] and bmibnb in YALMIP [18].

##### 3.3. Design of Free Parameter

As mentioned, the objective of this paper is to develop a new design method that results in a closed-loop system in Figure 1 that is robustly stable and has high control precision for reference inputs with uncertain period-time. Hence, the free parameter should be designed after the control characteristic. The output sensitivity matrix function is defined as the transfer matrix function from the reference input to the tracking error :
Having good tracking performance of the closed-loop system requires that
be as small as possible [15]. From (12), (10b), and (11), the sensitivity matrix function is written as
and the largest singular value of the sensitivity matrix function is
Because of , , and satisfying *Assumptions 1 and 2*, when the reference signal is assumed to be a periodic signal with a known and fixed period-time , the output follows the reference input with a small steady-state error because of
However, for the reference input with unknown period-time, the frequency components of the reference input are not equal to , which yields
where are defined as
for . To obtain a small steady-state error, will be selected such that

Hereby, the design method for is detailed depending upon the sensitivity function. According to [19], is strictly proper when is a strictly proper system. is factorized as where is a square inner matrix function with and is an outer matrix function [15, 20]. Because is an outer matrix, there exists a stable matrix such that For this reason, and are chosen by where is a positive integer to make proper; then, Therefore, when is chosen to satisfy the output follows the reference input with a small steady-state error that is caused by for . The designs for , , and are described in detail in [21–24].

Hence, by applying this design method to the free parameters, both stability and good control performance can be guaranteed for the high-order modified repetitive control system. Consequently, the following procedures to design the stabilizing high-order modified repetitive control system are provided.

###### 3.3.1. Procedure

*Step 1. *Select the constant , with low-pass filter , and calculate the weighting factors according to [5, 7, 10].

*Step 2. *According to [19], obtain the coprime factors , , , and and parameters , , , and .

*Step 3. *Settle the initial matrix functions and , and obtain the state-space description of the closed-loop system (30).

*Step 4. *Solve the feasible problems (44) to obtain , , , , , , , and .

*Step 5. *Replace matrices , , , , , , , and with matrices , , , , , , , and , respectively. In addition, calculate the matrices , , , and using (33).

*Step 6. *To obtain good tracking performance, the low-pass filter is settled to be the form of (59) with and to satisfy (61).

*Step 7. *Using the above parameters, the high-order modified repetitive controller in (9) can be obtained for system (8), where and are written as (22) and (23), respectively.

The design procedure proposed in this section is applicable for both SISO linear systems and multiple-input/multiple-output (MIMO) linear systems by simply modifying the dimensions of some matrices.

#### 4. Numerical Examples

In this section, numerical examples are generated to illustrate the validity and effectiveness of the proposed approach. Consider a SISO strictly proper plant with the following parameters: For this unstable control plant, it is easy to verify that the pairs and are controllable and observable, respectively. According to [19], choose the and as follows: Then, the parameters , , , , , , , and are given by In this paper, to show the generality of this design method and obtain the weighting factors conveniently, was chosen. As is well known, the low-pass filter has a bandwidth restriction for a nonminimum phase plant. Here, we chose the robust low-pass filter to be Considering improvements to the control precision for intermediate frequencies and that the low-pass filter has no restriction on bandwidth, set as To improve the system sensitivity at intermediate frequencies, we chose the high-order repetitive controller as noise robust [10]. The corresponding weighting factors could then be obtained directly from [10]: The initial matrix functions and were set as

Then, the feasible problem is solved using the PENBMI solver to obtain and as Because is the minimum phase, the inner-outer factorization of can be set as where and . Then, is

According to the Nyquist stability criterion, , , and ; then, means that the Nyquist plot of does not encircle the origin . Obviously, Figure 3 illustrates that . This demonstrates that the design method for the free parameters and is much more effective than that in [12, 13].

Using the above-mentioned parameters, the stabilizing controller , which contains a high-order modified repetitive control structure in (9), was obtained. To demonstrate its effectiveness, sensitivity function plots were made and simulations were conducted for the reference inputs: The magnitude of the resulting sensitivity function is plotted in Figure 4, for a system with and without repetitive control. From this figure, it can be concluded that the high-order controller provided a better control performance when the period-time had a limited uncertainty. When the reference had intermediate frequency components, the control performance became worse than that without the repetitive controller. However, the control performance at the intermediate frequencies was improved by modifying the cutoff frequency of the low-pass filter .

The simulation results using our method are shown in Figure 5. Clearly, the high-order modified repetitive control system was stable. For the period-time without variation, the largest steady-state peak-to-peak relative error was as small as , and this system entered the steady state in the 11th period. When the period-time had an uncertainty level of , the largest steady-state peak-to-peak relative error was . This result shows that for a certain vibration in period-time, the high-order modified repetitive controller continued to provide high control precision. As seen in Figure 4, when , the largest steady-state peak-to-peak relative error of this control system increases to . This tracking error is small and can be further reduced by modifying the cutoff frequency of the low-pass filter because it has no bandwidth restriction. The results shown in Figure 5 demonstrate that the design method presented here provided not only good stability, but also satisfactory tracking performance for the reference inputs with uncertain period-time. Compared with conventional design methods, the design method proposed in this paper widens the range of the uncertainty in period-time and has a wider range of practical applications, which makes it very practical.

#### 5. Conclusion

This paper presents a new design method for high-order modified repetitive control systems for periodic reference inputs with uncertain period-time. A high-order modified repetitive control system was established and the parametrization of all stabilizing controllers was derived based on the YKP. Moreover, the constraint conditions of free parameters were converted into stability conditions in the form of BMIs, which can be solved using existing software. In addition, the control precision of this closed-loop control system was guaranteed by analyzing its control characteristic. From the simulation results, we found that the high-order repetitive controller provided high control precision with a certain vibration in period-time and that the control precision at intermediate frequencies could be satisfied by modifying the cutoff frequency of low-pass filters in compensators.

However, the design of this high-order modified repetitive control system was subject to a control plant without uncertainty and disturbance, because the low-pass filter that was used to specify the intermediate frequencies had bandwidth restrictions to address the problem of robust stabilization. For this reason, the control precision will be degraded and the uncertain range in period-time will be narrowed. This issue requires further attention, particularly in cases where there exist uncertainties in the plant. In such cases, the optimal performance needs to be considered, as do the largest cutoff frequencies and widening of the uncertain range in the period-time.

#### Acknowledgments

The first author would like to express his gratitude to the China Scholarship Council (CSC) for its support and scholarship. The authors gratefully acknowledge the reviewers for their helpful comments and suggestions, which have improved the presentation.