# A New Design Method of  High-Order Modified Repetitive Control Systems for Reference Inputs with Uncertain Period-Time

Authors: Zhongxiang Chen; Kou Yamada; Tatsuya Sakanushi
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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. RH∞ denotes the set of stable proper real rational functions; H∞ denotes the set of stable causal functions; Rn denotes the n-dimensional Euclidean space; Rn×m is the set of all n×m real matrices; I is the identity matrix; the null matrix or null vector of appropriate dimension is denoted by 0; the superscript “T” stands for the transpose of a matrix; σ-{·} denotes the largest singular value of {·}; the symmetric terms in a symmetric matrix are denoted by *; for example, [AB*C]=[ABBTC].


## 2. Problem Formulation

Figure1 depicts the proposed high-order modified repetitive control system, where G(s)∈Rp×m(s) is a strictly proper plant with an assumption that p≤m. C1(s)∈Rm×p(s) is a forward compensator to enhance the robust stability of this closed-loop system and improve the transient response, C2(s)∈Rm×p(s) is typically called the learning filter and is used to compensate the high-order modified repetitive controller Cr(s),  y(t)∈Rp is the control output, and r(t)∈Rp is a periodic reference input. Uncertainty of period-time is modeled as relative uncertainty on L, bounded by Δ:
(1)Lδ≔L(1+δ),|δ|≤Δ<∞.
In Figure 1, the structure of the high-order modified repetitive controller Cr(s) consists of multiple periodic signal generators and can be expressed as
(2)Cr(s)=H(s)(I-H(s))-1=(I-H(s))-1H(s),(3)H(s)=∑i=1N‍Wiqi(s)e-sLi,(4)Li=i×L,
where Wi(i=1,…,N) are the weighting factors to modify the dynamic response in between the harmonic frequencies, and q(s)∈Rp×p(s) is a low-pass filter assumed as a diagonal matrix function.Figure 1
High-order modified repetitive control system forN=3.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 harmonicsl=0,…,ℏ of the reference input are assumed to lie in the passband of q(s). This can be guaranteed by the proper design of q(s), provided that all harmonics l=0,…,ℏ lie well below ωco, the cutoff frequency of q(s).Assumption 2.
In its passband,q(s) is assumed to equal its DC gain q(jω)=I for all ω<ωco. These two assumptions imply that
(5)Cr(s)=C^r(s):=(I-H^(s))-1H^(s),(6)H^(s)=∑i=1N‍Wie-sLi,
for all s=jωl(l=0,…,ℏ), where Li(i=1,…,N) are defined in (4) and ωl(l=0,…,ℏ) are the frequency components of the periodic reference input described as
(7)ωl≔2lπL,l=0,…,ℏ.
In this case, the N weighting factors Wi(i=1,…,N) can be determined by using the methods proposed in [5, 7, 10].
In addition, suppose that the strictly proper plant,G(s), is controllable and observable and its state-space description is
(8)x˙(t)=Apx(t)+Bpu(t),y(t)=Cpx(t),
where Ap∈Rn×n, Bp∈Rn×m, and Cp∈Rp×n are system matrices, and x(t)∈Rn, u(t)∈Rm, and y(t)∈Rp 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 plantG(s), a constant N, and a low-pass filter q(s), find the admissible compensators C1(s) and C2(s) such that the controller C(s) written as
(9)C(s)=C1(s)+C2(s)Cr(s)
robustly stabilizes the closed-loop system with high control precision for the periodic reference input with uncertainty in period-time, where Cr(s) 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 Figure1 is internally stable if and only if the controllers can be written as(10a)C(s)={(Y(s)-Q(s)N~(s))-1(X(s)+Q(s)D~(s)):|Y(∞)-Q(∞)N~(∞)|≠0(Y(s)-Q(s)N~(s))-1}(10b)={(X~(s)+D(s)Q(s))(Y~(s)-N(s)Q(s))-1:|Y~(∞)-N(∞)Q(∞)|≠0(Y~(s)-N(s)Q(s))-1},(11)Q(s)={Q0n(s)+(Y(s)Q~(s)-Q0n(s))H(s)}×{Q0d(s)+(N~(s)Q~(s)-Q0d(s))H(s)}-1∈H∞,
where Q0n(s), Q0d(s), Q~(s)∈RH∞, and rank  Q0d(s)=p, N(s), D(s), N~(s), and D~(s) are coprime factors ofG(s) on RH∞, and X(s), Y(s), X~(s), Y~(s)∈RH∞ satisfy
(12)[Y(s)X(s)-N~(s)D~(s)][D(s)-X~(s)N(s)Y~(s)]=[D(s)-X~(s)N(s)Y~(s)][Y(s)X(s)-N~(s)D~(s)]=[I00I].Proof (Necessity).
To prove the necessity, suppose that there exist admissible compensatorsC1(s) and C2(s), such that the controller C(s) 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 C(s) can be written as (10a) and (10b). Substitute C(s) in (9) into (10a) and the Q(s) in (11) can be represented as
(13)Q(s)={(Y(s)C1(s)-X(s))+(Y(s)C2(s)-Y(s)C1(s)+X(s))H(s)}×{(D~(s)+N~(s)C1(s))+(N~(s)C2(s)-N~(s)C1(s)-D~(s))H(s)}-1.
Let C1n(s)∈RH∞ and C1d(s)∈RH∞ be coprime factors of C1(s) on RH∞ satisfying
(14)C1(s)=C1n(s)C1d-1(s).
Under the assumptions in Section 2, the low-pass filter q(s) is a diagonal matrix function; this means that H(s) is also a diagonal matrix function, and substituting (14) into (13) yields
(15)Q(s)={(Y(s)C1n(s)-X(s)C1d(s))+(Y(s)C2(s)C1d(s)-Y(s)C1n(s)X(s)C1d(s)+X(s)C1d(s))H(s)}×{(D~(s)C1d(s)+N~(s)C1n(s))+(N~(s)C2(s)C1d(s)-N~(s)C1n(s)-D~(s)C1d(s))H(s)}-1.
Moreover, let C2n(s)∈RH∞ and C2d(s)∈RH∞ be coprime factors of C2(s)C1d(s) satisfying
(16)C2(s)C1d(s)=C2n(s)C2d-1(s).
Combining (15) and (16), (11) can be obtained, where
(17)Q0n(s)={Y(s)C1n(s)-X(s)C1d(s)}C2d(s),Q0d(s)={D~(s)C1d(s)+N~(s)C1n(s)}C2d(s),Q~(s)=C2n(s).
Note that Q0n(s), Q0d(s) and Q~(s), are RH∞ functions. As a consequence, the necessity can be proven if Q(s) is a H∞ matrix function. According to (2), (9), and (13), Q(s) can be rewritten as
(18)Q(s)={Y(s)C(s)-X(s)}{I+G(s)C(s)}-1D~-1(s).
According to (12), the equation
(19)D~(s)Y~(s)+N~(s)X~(s)=I
holds and D~-1(s) can be expressed as
(20)D~-1(s)=Y~(s)+G(s)X~(s).
Subsequently,
(21)Q(s)={Y(s)C(s)-X(s)}{I+G(s)C(s)}-1×{Y~(s)+G(s)X~(s)}=Y(s)C(s){I+G(s)C(s)}-1Y~(s)+Y(s)C(s){I+G(s)C(s)}-1G(s)X~(s)-X(s){I+G(s)C(s)}-1Y~(s)-X(s){I+G(s)C(s)}-1G(s)X~(s).
Because C(s) stabilizes the closed system, Q(s)∈H∞ can be proven according to internal stability [15]. Therefore, the necessity is obtained.
Sufficiency. Suppose that there is Q(s)∈H∞ written by (11) such that (10b) stabilizes the closed-loop system in Figure 1. Substituting (10b) into (9) yields
(22)C1(s)={X~(s)Q0d(s)+D(s)Q0n(s)}×{Y~(s)Q0d(s)-N(s)Q0n(s)}-1,(23)C2(s)=Q~(s){Y~(s)Q0d(s)-N(s)Q0n(s)}-1,
respectively. Therefore, the sufficiency is obtained.
This completes the proof.From Theorem1, Q~(s)∈RH∞, Q0d(s)∈RH∞, and Q0n(s)∈RH∞ are the free parameters that need to be determined. In the following, the design methods for Q~(s), Q0d(s), and Q0n(s) are based on BMIs and the control characteristic.


### 3.2. Design of Free ParametersQ0d(s) and Q~(s)

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 parametersQ0d(s) and Q~(s) such that Q(s) belongs to H∞ on the basis of state-space description. According to (2) and (11), Q(s) can be expressed as
(24)Q(s)=[Q0d(s)Y(s)Q~(s)]Qf(s),(25)Qf(s)=[ICr(s)]{[Q0d(s)-I00Q~(s)]I+[IN~(s)]×[Q0d(s)-I00Q~(s)][ICr(s)]}-1.
Because Q0n(s)∈RH∞, Y(s)∈RH∞, and Q~(s)∈RH∞, Q(s)∈H∞ means that the closed-loop system Qf(s) is stable. The problem of selecting the free parameters Q0d(s)∈RH∞ and Q~∈RH∞, while ensuring Q(s)∈H∞, is converted to design Q0d(s)∈RH∞ and Q~(s)∈RH∞, thus stabilizing the closed-loop system Qf(s). A general design method for Q0d(s) and Q~(s) was established, based on the state-space description and assuming a low-pass filter q(s), as
(26)q(s)=[AlBlCl0].
Then, from Figure 1, the state-space description of Cr(s) can be obtained as
(27)x˙r(t)=Ar0xr(t)+Ar1xr(t-L)+Bre(t),z(t)=Crxr(t-L),
where xr(t) is the state of Cr(s),
(28)Ar0:=[Al⋱Al],Ar1:=[BlW1Cl⋯BlWN-1ClBlWNClBlCl0⋱⋮BlCl0],Br=[BlT0⋯0]T,Cr=[W1Cl⋯WNCl].
Assume that the state-space descriptions of Q0d(s), Q~(s), and N~(s) are
(29)Q0d(s)=[A0dB0dC0dD0d],Q~(s)=[A~qB~qC~qD~q],N~(s)=[A~nB~nC~nD~n],
where D0d has full column rank. Draw the configuration of Qf(s) in Figure 2 and the state-space description of the closed-loop system Qf(s) with r(t)≡0 is given by
(30)x˙(t)=A0x(t)+A1x(t-L),
where
(31)x(t):=[xrT(t)xnT(t)xqT(t)x0T(t)]T,A0:=[Ar0-BrD0d†C~n0-BrD0d†C0d0A~nB~nC~q000A~q00-B0dD0d†C~n0A0d-B0dD0d†C0d],A1:=[Ar1000B~nD~qCr1000B~qCr10000000],D0d† is the left pseudoinverse of D0d. The left pseudoinverse of Q0d(s) is written as
(32)Q0d†(s)=[A0d-B0dD0d†C0d-B0dD0d†D0d†C0dD0d†]=[A-0dB-0dC-0dD-0d].
This means that the normal rank of Q0d(s) is p and Q0d(s) could also be represented as
(33)Q0d(s)=[A-0d-B-0dD0dC-0d-B-0dD0dD0dC-0dD0d]=[A0dB0dC0dD0d].Figure 2
Configuration ofQf(s).ReplacingA0d-B0dD0d†C0d, -B0dD0d†, D0d†C0d, and D0d†, with A-0d, B-0d, C-0d, and D-0d in (32), respectively, the system matrix A0 is written as
(34)A0=[Ar0-BrD-0dC~n0-BrC-0d0A~nB~nC~q000A~q00B-0dC~n0A-0d].
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 P>0 and Q>0 exist such that
(35)Π:=[PA0+A0TP+QPA1*-Q]<0,
then the time-delay system (30) is stable.Proof.
Choose the candidate Lyapunov functional to be(36)V(x(t),t)=xT(t)Px(t)+∫t-Lt‍xT(θ)Qx(θ)dθ,
where P=PT>0 and Q=QT>0. Calculating the derivative of V(t) for the time-delay system (30) yields
(37)V˙(x(t),t)=2xT(t)Px˙(t)+xT(t)Qx(t)-xT(t-L)Qx(t-L)=2xT(t)P(A0x(t)+A1x(t-L))≔ξT(t)Πξ(t),
where ξ(t):=[xT(t)xT(t-L)]T. Clearly, the stability condition for the time-delay system (30) is
(38)Π<0.
This completes the proof.Theorem1 requires that Q~(s) and Q0d(s) are RH∞ matrix functions, which means that Q~(s) and Q0d(s) should be stable. According to Lyapunov functional theory, the BMI condition (35) contains the stability condition of Q~(s). Hence, it is only necessary to provide the stability condition of Q0d(s), and the following theorem will provide the stability condition in the form of BMIs for Q0d(s).Theorem 3.
For system (30), if symmetric positive-definite matrices P>0, Q>0, and X>0 exist such that (35) and
(39)Ξ:=[XA-0d+A-0dTXXB-0dC-0dT*-I0**-D-0dD-0dT]<0
hold, then the time-delay system (30) is stable and Q0d(s)∈RH∞.Proof.
From Lyapunov functional theory,Q0d(s) is a RH∞ matrix function if there exists a symmetric positive-definite matrix X=XT>0 such that
(40)XA0d+A0dTX<0,
which is equivalent to
(41)XA-0d-XB-0dD0dC-0d-(B-0dD0dC-0d)TX+A-0dTX<0.
According to the results in [16], a sufficient condition guaranteeing (41) is that there exists a positive number λ>0 such that
(42)λXA-0d+λA0dTX+λ2XB-0dB-0dTX+C-0dTD0dTD0dC-0d<0.
Replacing λX and D0dTD0d with X and (D-0dD-0dT)-1 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 forQ~(s) and Q0d†(s) are required. Without loss of generality, Q~(s) and Q0d†(s) should be represented as
(43)Q~(s)=[A~q+ΔA~qB~q+ΔB~qC~q+ΔC~qD~q+ΔD~q],Q0d†(s)=[A-0d+ΔA-0dB-0d+ΔB-0dC-0d+ΔC-0dD-0d+ΔD-0d],
where A~q, B~q, C~q, D~q, A-0d, B-0d, C-0d, and D-0d are the given initial matrices, and ΔA~q, ΔB~q, ΔC~q, ΔD~q, ΔA-0d, ΔB-0d, ΔC-0d, and ΔD-0d 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 functionsQ~(s) and Q0d†(s), if symmetric positive-definite matrices P>0, Q>0, and X>0 and appropriately dimensioned matrices ΔA~q, ΔB~q, ΔC~q, ΔD~q, ΔA-0d, ΔB-0d, ΔC-0d, and ΔD-0d exist such that
(44)[P(A0+ΔA0)+(A0T+ΔA0T)P+QP(A1+ΔA1)*-Q]<0,[ΛX(B-0d+ΔB-0d)C-0dT+ΔC-0dT*-I0**-(D-0d+ΔD-0d)(D-0d+ΔD-0d)T]<0
hold, where Λ, ΔA0, and A1 are defined as
(45)Λ:=X(A-0d+ΔA-0d)+(A-0dT+ΔA-0dT)X,(46)ΔA0:=[0-BrΔD-0dC~n0-BrΔC-0d00B~nΔC~q000ΔA~q00ΔB-0dC~n0ΔA-0d],(47)ΔA1:=[0000B~nΔD~qCr1000ΔB~qCr10000000],
then the time-delay system (30) is stable and Q0d(s) is RH∞.Proof.
Replacing matricesA~q, B~q, C~q, D~q, A-0d, B-0d, C-0d, and D-0d with matrices A~q+ΔA~q, B~q+ΔB~q, C~q+ΔC~q, D~q+ΔD~q, A-0d+ΔA-0d, B-0d+ΔB-0d, C-0d+ΔC-0d, and D-0d+ΔD-0d, 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 ParameterQ0n(s)

As mentioned, the objective of this paper is to develop a new design method that results in a closed-loop system in Figure1 that is robustly stable and has high control precision for reference inputs with uncertain period-time. Hence, the free parameter Q0n(s) should be designed after the control characteristic. The output sensitivity matrix function is defined as the transfer matrix function from the reference input r(t) to the tracking error e(t):
(48)S(s):=(I+G(s)C(s))-1.
Having good tracking performance of the closed-loop system requires that
(49)σ-{S(s)}=σ-{(I+G(s)C(s))-1}
be as small as possible [15]. From (12), (10b), and (11), the sensitivity matrix function is written as
(50)S(s)={Y~(s)Q0d(s)-N(s)Q0n(s)}×{Q0d(s)+(N~(s)Q~(s)-Q0d(s))H(s)}-1×(I-H(s))D~(s),
and the largest singular value of the sensitivity matrix function is
(51)σ-{S(s)}≤σ-{Y~(s)Q0d(s)-N(s)Q0n(s)}×σ-{(Q0d(s)+(N~(s)Q~(s)-Q0d(s))H(s))-1}×σ-{(I-H(s))}σ-{D~(s)}.
Because of Q(s)∈H∞, D~(s)∈RH∞, and q(s) satisfying Assumptions1 and 2, when the reference signal r(t) is assumed to be a periodic signal with a known and fixed period-time L, the output y(t) follows the reference input r(t) with a small steady-state error because of
(52)σ-{I-H(jwl)}≃0,(l=0,…,ℏ).
However, for the reference input with unknown period-time, the frequency components ω-k(k=0,…,h) of the reference input are not equal to ωl(l=1,…,ℏ), which yields
(53)σ-{S(jω-k)}≄0,(k=0,…,h),
where ω-k(k=0,…,h) are defined as
(54)ω-k:=2kπL(1+δ),|δ|≤Δ<∞
for k=0,…,h. To obtain a small steady-state error, Q0n(s)∈RH∞ will be selected such that
(55)σ-{Y~(jω-k)Q0d(jω-k)-N(jω-k)Q0n(jω-k)}≃0(i=1,…,h).Hereby, the design method forQ0n(s) is detailed depending upon the sensitivity function. According to [19], N(s) is strictly proper when G(s) is a strictly proper system. N(s) is factorized as
(56)N(s)=Ni(s)No(s),
where Ni(s) is a square inner matrix function with Ni(0)=I and No(s) is an outer matrix function [15, 20]. Because No(s) is an outer matrix, there exists a stable m×p matrix No+(s) such that
(57)No(s)No+(s)=I.
For this reason, Q0n(s) and q-(s) are chosen by
(58)Q0n(s)=No+(s)q-(s)Y~(s)Q0d(s),(59)q-(s)=diag{1(1+τ1s)α1,…,1(1+τps)αp}∈RH∞,
where αi(i=1,…,p) is a positive integer to make Q0n(s) proper; then,
(60)Y~(s)Q0d(s)-N(s)Q0n(s)=(I-Ni(s)q-(s))Y~(s)Q0d(s).
Therefore, when q-(s) is chosen to satisfy
(61)σ-{I-Ni(jω-k)q-(jω-k)}≃0,(k=0,…,h),
the output y(t) follows the reference input r(t) with a small steady-state error that is caused by
(62)σ-{S(jω-k)}≃0
for k=0,…,h. The designs for Ni(s), No(s), and q-(s) 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 constantN, with low-pass filter q(s), and calculate the N weighting factors Wi(i=1,…,N) according to [5, 7, 10].Step 2.
According to [19], obtain the coprime factors N(s), D(s), N~(s), and D~(s) and parameters X(s), Y(s), X~(s), and Y~(s).Step 3.
Settle the initial matrix functionsQ~(s)∈RH∞ and Q0d†(s)∈RH∞, and obtain the state-space description of the closed-loop system (30).Step 4.
Solve the feasible problems (44) to obtain ΔA~q, ΔB~q, ΔC~q, ΔD~q, ΔA-0d, ΔB-0d, ΔC-0d, and ΔD-0d.Step 5.
Replace matricesA~q, B~q, C~q, D~q, A-0d, B-0d, C-0d, and D-0d with matrices A~q+ΔA~q, B~q+ΔB~q, C~q+ΔC~q, D~q+ΔD~q, A-0d+ΔA-0d, B-0d+ΔB-0d, C-0d+ΔC-0d, and D-0d+ΔD-0d, respectively. In addition, calculate the matrices A0d, B0d, C0d, and D0d using (33).Step 6.
To obtain good tracking performance, the low-pass filterq-(s) is settled to be the form of (59) with q-(0)=I and to satisfy (61).Step 7.
Using the above parameters, the high-order modified repetitive controllerC(s) in (9) can be obtained for system (8), where C1(s) and C2(s) 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.


## 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 Figure1 is internally stable if and only if the controllers can be written as(10a)C(s)={(Y(s)-Q(s)N~(s))-1(X(s)+Q(s)D~(s)):|Y(∞)-Q(∞)N~(∞)|≠0(Y(s)-Q(s)N~(s))-1}(10b)={(X~(s)+D(s)Q(s))(Y~(s)-N(s)Q(s))-1:|Y~(∞)-N(∞)Q(∞)|≠0(Y~(s)-N(s)Q(s))-1},(11)Q(s)={Q0n(s)+(Y(s)Q~(s)-Q0n(s))H(s)}×{Q0d(s)+(N~(s)Q~(s)-Q0d(s))H(s)}-1∈H∞,
where Q0n(s), Q0d(s), Q~(s)∈RH∞, and rank  Q0d(s)=p, N(s), D(s), N~(s), and D~(s) are coprime factors ofG(s) on RH∞, and X(s), Y(s), X~(s), Y~(s)∈RH∞ satisfy
(12)[Y(s)X(s)-N~(s)D~(s)][D(s)-X~(s)N(s)Y~(s)]=[D(s)-X~(s)N(s)Y~(s)][Y(s)X(s)-N~(s)D~(s)]=[I00I].Proof (Necessity).
To prove the necessity, suppose that there exist admissible compensatorsC1(s) and C2(s), such that the controller C(s) 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 C(s) can be written as (10a) and (10b). Substitute C(s) in (9) into (10a) and the Q(s) in (11) can be represented as
(13)Q(s)={(Y(s)C1(s)-X(s))+(Y(s)C2(s)-Y(s)C1(s)+X(s))H(s)}×{(D~(s)+N~(s)C1(s))+(N~(s)C2(s)-N~(s)C1(s)-D~(s))H(s)}-1.
Let C1n(s)∈RH∞ and C1d(s)∈RH∞ be coprime factors of C1(s) on RH∞ satisfying
(14)C1(s)=C1n(s)C1d-1(s).
Under the assumptions in Section 2, the low-pass filter q(s) is a diagonal matrix function; this means that H(s) is also a diagonal matrix function, and substituting (14) into (13) yields
(15)Q(s)={(Y(s)C1n(s)-X(s)C1d(s))+(Y(s)C2(s)C1d(s)-Y(s)C1n(s)X(s)C1d(s)+X(s)C1d(s))H(s)}×{(D~(s)C1d(s)+N~(s)C1n(s))+(N~(s)C2(s)C1d(s)-N~(s)C1n(s)-D~(s)C1d(s))H(s)}-1.
Moreover, let C2n(s)∈RH∞ and C2d(s)∈RH∞ be coprime factors of C2(s)C1d(s) satisfying
(16)C2(s)C1d(s)=C2n(s)C2d-1(s).
Combining (15) and (16), (11) can be obtained, where
(17)Q0n(s)={Y(s)C1n(s)-X(s)C1d(s)}C2d(s),Q0d(s)={D~(s)C1d(s)+N~(s)C1n(s)}C2d(s),Q~(s)=C2n(s).
Note that Q0n(s), Q0d(s) and Q~(s), are RH∞ functions. As a consequence, the necessity can be proven if Q(s) is a H∞ matrix function. According to (2), (9), and (13), Q(s) can be rewritten as
(18)Q(s)={Y(s)C(s)-X(s)}{I+G(s)C(s)}-1D~-1(s).
According to (12), the equation
(19)D~(s)Y~(s)+N~(s)X~(s)=I
holds and D~-1(s) can be expressed as
(20)D~-1(s)=Y~(s)+G(s)X~(s).
Subsequently,
(21)Q(s)={Y(s)C(s)-X(s)}{I+G(s)C(s)}-1×{Y~(s)+G(s)X~(s)}=Y(s)C(s){I+G(s)C(s)}-1Y~(s)+Y(s)C(s){I+G(s)C(s)}-1G(s)X~(s)-X(s){I+G(s)C(s)}-1Y~(s)-X(s){I+G(s)C(s)}-1G(s)X~(s).
Because C(s) stabilizes the closed system, Q(s)∈H∞ can be proven according to internal stability [15]. Therefore, the necessity is obtained.
Sufficiency. Suppose that there is Q(s)∈H∞ written by (11) such that (10b) stabilizes the closed-loop system in Figure 1. Substituting (10b) into (9) yields
(22)C1(s)={X~(s)Q0d(s)+D(s)Q0n(s)}×{Y~(s)Q0d(s)-N(s)Q0n(s)}-1,(23)C2(s)=Q~(s){Y~(s)Q0d(s)-N(s)Q0n(s)}-1,
respectively. Therefore, the sufficiency is obtained.
This completes the proof.From Theorem1, Q~(s)∈RH∞, Q0d(s)∈RH∞, and Q0n(s)∈RH∞ are the free parameters that need to be determined. In the following, the design methods for Q~(s), Q0d(s), and Q0n(s) are based on BMIs and the control characteristic.


## 3.2. Design of Free ParametersQ0d(s) and Q~(s)

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 parametersQ0d(s) and Q~(s) such that Q(s) belongs to H∞ on the basis of state-space description. According to (2) and (11), Q(s) can be expressed as
(24)Q(s)=[Q0d(s)Y(s)Q~(s)]Qf(s),(25)Qf(s)=[ICr(s)]{[Q0d(s)-I00Q~(s)]I+[IN~(s)]×[Q0d(s)-I00Q~(s)][ICr(s)]}-1.
Because Q0n(s)∈RH∞, Y(s)∈RH∞, and Q~(s)∈RH∞, Q(s)∈H∞ means that the closed-loop system Qf(s) is stable. The problem of selecting the free parameters Q0d(s)∈RH∞ and Q~∈RH∞, while ensuring Q(s)∈H∞, is converted to design Q0d(s)∈RH∞ and Q~(s)∈RH∞, thus stabilizing the closed-loop system Qf(s). A general design method for Q0d(s) and Q~(s) was established, based on the state-space description and assuming a low-pass filter q(s), as
(26)q(s)=[AlBlCl0].
Then, from Figure 1, the state-space description of Cr(s) can be obtained as
(27)x˙r(t)=Ar0xr(t)+Ar1xr(t-L)+Bre(t),z(t)=Crxr(t-L),
where xr(t) is the state of Cr(s),
(28)Ar0:=[Al⋱Al],Ar1:=[BlW1Cl⋯BlWN-1ClBlWNClBlCl0⋱⋮BlCl0],Br=[BlT0⋯0]T,Cr=[W1Cl⋯WNCl].
Assume that the state-space descriptions of Q0d(s), Q~(s), and N~(s) are
(29)Q0d(s)=[A0dB0dC0dD0d],Q~(s)=[A~qB~qC~qD~q],N~(s)=[A~nB~nC~nD~n],
where D0d has full column rank. Draw the configuration of Qf(s) in Figure 2 and the state-space description of the closed-loop system Qf(s) with r(t)≡0 is given by
(30)x˙(t)=A0x(t)+A1x(t-L),
where
(31)x(t):=[xrT(t)xnT(t)xqT(t)x0T(t)]T,A0:=[Ar0-BrD0d†C~n0-BrD0d†C0d0A~nB~nC~q000A~q00-B0dD0d†C~n0A0d-B0dD0d†C0d],A1:=[Ar1000B~nD~qCr1000B~qCr10000000],D0d† is the left pseudoinverse of D0d. The left pseudoinverse of Q0d(s) is written as
(32)Q0d†(s)=[A0d-B0dD0d†C0d-B0dD0d†D0d†C0dD0d†]=[A-0dB-0dC-0dD-0d].
This means that the normal rank of Q0d(s) is p and Q0d(s) could also be represented as
(33)Q0d(s)=[A-0d-B-0dD0dC-0d-B-0dD0dD0dC-0dD0d]=[A0dB0dC0dD0d].Figure 2
Configuration ofQf(s).ReplacingA0d-B0dD0d†C0d, -B0dD0d†, D0d†C0d, and D0d†, with A-0d, B-0d, C-0d, and D-0d in (32), respectively, the system matrix A0 is written as
(34)A0=[Ar0-BrD-0dC~n0-BrC-0d0A~nB~nC~q000A~q00B-0dC~n0A-0d].
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 P>0 and Q>0 exist such that
(35)Π:=[PA0+A0TP+QPA1*-Q]<0,
then the time-delay system (30) is stable.Proof.
Choose the candidate Lyapunov functional to be(36)V(x(t),t)=xT(t)Px(t)+∫t-Lt‍xT(θ)Qx(θ)dθ,
where P=PT>0 and Q=QT>0. Calculating the derivative of V(t) for the time-delay system (30) yields
(37)V˙(x(t),t)=2xT(t)Px˙(t)+xT(t)Qx(t)-xT(t-L)Qx(t-L)=2xT(t)P(A0x(t)+A1x(t-L))≔ξT(t)Πξ(t),
where ξ(t):=[xT(t)xT(t-L)]T. Clearly, the stability condition for the time-delay system (30) is
(38)Π<0.
This completes the proof.Theorem1 requires that Q~(s) and Q0d(s) are RH∞ matrix functions, which means that Q~(s) and Q0d(s) should be stable. According to Lyapunov functional theory, the BMI condition (35) contains the stability condition of Q~(s). Hence, it is only necessary to provide the stability condition of Q0d(s), and the following theorem will provide the stability condition in the form of BMIs for Q0d(s).Theorem 3.
For system (30), if symmetric positive-definite matrices P>0, Q>0, and X>0 exist such that (35) and
(39)Ξ:=[XA-0d+A-0dTXXB-0dC-0dT*-I0**-D-0dD-0dT]<0
hold, then the time-delay system (30) is stable and Q0d(s)∈RH∞.Proof.
From Lyapunov functional theory,Q0d(s) is a RH∞ matrix function if there exists a symmetric positive-definite matrix X=XT>0 such that
(40)XA0d+A0dTX<0,
which is equivalent to
(41)XA-0d-XB-0dD0dC-0d-(B-0dD0dC-0d)TX+A-0dTX<0.
According to the results in [16], a sufficient condition guaranteeing (41) is that there exists a positive number λ>0 such that
(42)λXA-0d+λA0dTX+λ2XB-0dB-0dTX+C-0dTD0dTD0dC-0d<0.
Replacing λX and D0dTD0d with X and (D-0dD-0dT)-1 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 forQ~(s) and Q0d†(s) are required. Without loss of generality, Q~(s) and Q0d†(s) should be represented as
(43)Q~(s)=[A~q+ΔA~qB~q+ΔB~qC~q+ΔC~qD~q+ΔD~q],Q0d†(s)=[A-0d+ΔA-0dB-0d+ΔB-0dC-0d+ΔC-0dD-0d+ΔD-0d],
where A~q, B~q, C~q, D~q, A-0d, B-0d, C-0d, and D-0d are the given initial matrices, and ΔA~q, ΔB~q, ΔC~q, ΔD~q, ΔA-0d, ΔB-0d, ΔC-0d, and ΔD-0d 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 functionsQ~(s) and Q0d†(s), if symmetric positive-definite matrices P>0, Q>0, and X>0 and appropriately dimensioned matrices ΔA~q, ΔB~q, ΔC~q, ΔD~q, ΔA-0d, ΔB-0d, ΔC-0d, and ΔD-0d exist such that
(44)[P(A0+ΔA0)+(A0T+ΔA0T)P+QP(A1+ΔA1)*-Q]<0,[ΛX(B-0d+ΔB-0d)C-0dT+ΔC-0dT*-I0**-(D-0d+ΔD-0d)(D-0d+ΔD-0d)T]<0
hold, where Λ, ΔA0, and A1 are defined as
(45)Λ:=X(A-0d+ΔA-0d)+(A-0dT+ΔA-0dT)X,(46)ΔA0:=[0-BrΔD-0dC~n0-BrΔC-0d00B~nΔC~q000ΔA~q00ΔB-0dC~n0ΔA-0d],(47)ΔA1:=[0000B~nΔD~qCr1000ΔB~qCr10000000],
then the time-delay system (30) is stable and Q0d(s) is RH∞.Proof.
Replacing matricesA~q, B~q, C~q, D~q, A-0d, B-0d, C-0d, and D-0d with matrices A~q+ΔA~q, B~q+ΔB~q, C~q+ΔC~q, D~q+ΔD~q, A-0d+ΔA-0d, B-0d+ΔB-0d, C-0d+ΔC-0d, and D-0d+ΔD-0d, 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 ParameterQ0n(s)

As mentioned, the objective of this paper is to develop a new design method that results in a closed-loop system in Figure1 that is robustly stable and has high control precision for reference inputs with uncertain period-time. Hence, the free parameter Q0n(s) should be designed after the control characteristic. The output sensitivity matrix function is defined as the transfer matrix function from the reference input r(t) to the tracking error e(t):
(48)S(s):=(I+G(s)C(s))-1.
Having good tracking performance of the closed-loop system requires that
(49)σ-{S(s)}=σ-{(I+G(s)C(s))-1}
be as small as possible [15]. From (12), (10b), and (11), the sensitivity matrix function is written as
(50)S(s)={Y~(s)Q0d(s)-N(s)Q0n(s)}×{Q0d(s)+(N~(s)Q~(s)-Q0d(s))H(s)}-1×(I-H(s))D~(s),
and the largest singular value of the sensitivity matrix function is
(51)σ-{S(s)}≤σ-{Y~(s)Q0d(s)-N(s)Q0n(s)}×σ-{(Q0d(s)+(N~(s)Q~(s)-Q0d(s))H(s))-1}×σ-{(I-H(s))}σ-{D~(s)}.
Because of Q(s)∈H∞, D~(s)∈RH∞, and q(s) satisfying Assumptions1 and 2, when the reference signal r(t) is assumed to be a periodic signal with a known and fixed period-time L, the output y(t) follows the reference input r(t) with a small steady-state error because of
(52)σ-{I-H(jwl)}≃0,(l=0,…,ℏ).
However, for the reference input with unknown period-time, the frequency components ω-k(k=0,…,h) of the reference input are not equal to ωl(l=1,…,ℏ), which yields
(53)σ-{S(jω-k)}≄0,(k=0,…,h),
where ω-k(k=0,…,h) are defined as
(54)ω-k:=2kπL(1+δ),|δ|≤Δ<∞
for k=0,…,h. To obtain a small steady-state error, Q0n(s)∈RH∞ will be selected such that
(55)σ-{Y~(jω-k)Q0d(jω-k)-N(jω-k)Q0n(jω-k)}≃0(i=1,…,h).Hereby, the design method forQ0n(s) is detailed depending upon the sensitivity function. According to [19], N(s) is strictly proper when G(s) is a strictly proper system. N(s) is factorized as
(56)N(s)=Ni(s)No(s),
where Ni(s) is a square inner matrix function with Ni(0)=I and No(s) is an outer matrix function [15, 20]. Because No(s) is an outer matrix, there exists a stable m×p matrix No+(s) such that
(57)No(s)No+(s)=I.
For this reason, Q0n(s) and q-(s) are chosen by
(58)Q0n(s)=No+(s)q-(s)Y~(s)Q0d(s),(59)q-(s)=diag{1(1+τ1s)α1,…,1(1+τps)αp}∈RH∞,
where αi(i=1,…,p) is a positive integer to make Q0n(s) proper; then,
(60)Y~(s)Q0d(s)-N(s)Q0n(s)=(I-Ni(s)q-(s))Y~(s)Q0d(s).
Therefore, when q-(s) is chosen to satisfy
(61)σ-{I-Ni(jω-k)q-(jω-k)}≃0,(k=0,…,h),
the output y(t) follows the reference input r(t) with a small steady-state error that is caused by
(62)σ-{S(jω-k)}≃0
for k=0,…,h. The designs for Ni(s), No(s), and q-(s) 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 constantN, with low-pass filter q(s), and calculate the N weighting factors Wi(i=1,…,N) according to [5, 7, 10].Step 2.
According to [19], obtain the coprime factors N(s), D(s), N~(s), and D~(s) and parameters X(s), Y(s), X~(s), and Y~(s).Step 3.
Settle the initial matrix functionsQ~(s)∈RH∞ and Q0d†(s)∈RH∞, and obtain the state-space description of the closed-loop system (30).Step 4.
Solve the feasible problems (44) to obtain ΔA~q, ΔB~q, ΔC~q, ΔD~q, ΔA-0d, ΔB-0d, ΔC-0d, and ΔD-0d.Step 5.
Replace matricesA~q, B~q, C~q, D~q, A-0d, B-0d, C-0d, and D-0d with matrices A~q+ΔA~q, B~q+ΔB~q, C~q+ΔC~q, D~q+ΔD~q, A-0d+ΔA-0d, B-0d+ΔB-0d, C-0d+ΔC-0d, and D-0d+ΔD-0d, respectively. In addition, calculate the matrices A0d, B0d, C0d, and D0d using (33).Step 6.
To obtain good tracking performance, the low-pass filterq-(s) is settled to be the form of (59) with q-(0)=I and to satisfy (61).Step 7.
Using the above parameters, the high-order modified repetitive controllerC(s) in (9) can be obtained for system (8), where C1(s) and C2(s) 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.


## 3.3.1.  Procedure

Step 1.
Select the constantN, with low-pass filter q(s), and calculate the N weighting factors Wi(i=1,…,N) according to [5, 7, 10].Step 2.
According to [19], obtain the coprime factors N(s), D(s), N~(s), and D~(s) and parameters X(s), Y(s), X~(s), and Y~(s).Step 3.
Settle the initial matrix functionsQ~(s)∈RH∞ and Q0d†(s)∈RH∞, and obtain the state-space description of the closed-loop system (30).Step 4.
Solve the feasible problems (44) to obtain ΔA~q, ΔB~q, ΔC~q, ΔD~q, ΔA-0d, ΔB-0d, ΔC-0d, and ΔD-0d.Step 5.
Replace matricesA~q, B~q, C~q, D~q, A-0d, B-0d, C-0d, and D-0d with matrices A~q+ΔA~q, B~q+ΔB~q, C~q+ΔC~q, D~q+ΔD~q, A-0d+ΔA-0d, B-0d+ΔB-0d, C-0d+ΔC-0d, and D-0d+ΔD-0d, respectively. In addition, calculate the matrices A0d, B0d, C0d, and D0d using (33).Step 6.
To obtain good tracking performance, the low-pass filterq-(s) is settled to be the form of (59) with q-(0)=I and to satisfy (61).Step 7.
Using the above parameters, the high-order modified repetitive controllerC(s) in (9) can be obtained for system (8), where C1(s) and C2(s) 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 plantG(s) with the following parameters:
(63)Ap=[132-5],Bp=[10],Cp=[10].
For this unstable control plant, it is easy to verify that the pairs (Ap,Bp) and (Ap,Cp) are controllable and observable, respectively. According to [19], choose the F and L as follows:
(64)F=[-3-4],L=[-1-4]T.
Then, the parameters N(s), D(s), N~(s), D~(s), X(s), Y(s), X~(s), and Y~(s) are given by
(65)[-D(s)X~(s)N(s)Y~(s)]=[-s2+4s-11s2+7s+1219s+43s2+7s+12s+5s2+7s+12s2+8s+13s2+7s+12],(66)[Y(s)X(s)N~(s)-D~(s)]=[s2+8s+13s2+5s+619s+43s2+5s+6s+5s2+5s+6-s2+4s-11s2+5s+6].
In this paper, to show the generality of this design method and obtain the weighting factors conveniently, N=5 was chosen. As is well known, the low-pass filter q(s) has a bandwidth restriction for a nonminimum phase plant. Here, we chose the robust low-pass filter q(s) to be
(67)q(s)=10.01s+1.
Considering improvements to the control precision for intermediate frequencies and that the low-pass filter q-(s) has no restriction on bandwidth, set q-(s) as
(68)q-(s)=10.001s+1.
To improve the system sensitivity at intermediate frequencies, we chose the high-order repetitive controller as noise robust [10]. The corresponding weighting factors Wi(i=1,…,5) could then be obtained directly from [10]:
(69)W=[515415315215115].
The initial matrix functions Q~(s) and Q0d-1(s) were set as
(70)Q~(s)=s2+5s+6s2+15s+50,Q0d-1(s)=s2+5s+16s2+5s+6.Then, the feasible problem is solved using the PENBMI solver to obtainQ~(s) and Q0d(s) as
(71)Q~(s)=4.5331s2+122.5857s+424.0851s2+90.9312s+1561.1741,Q0d(s)=0.1656s2+6.4867s+63.0101s2+32.3705s+219.2992.
Because N(s) is the minimum phase, the inner-outer factorization of N(s) can be set as
(72)N(s)=Ni(s)No(s),
where Ni(s)=I and No(s)=N(s). Then, Q0n(s) is
(73)Q0n(s)=N-1(s)q-(s)Y~(s)Q0d(s).According to the Nyquist stability criterion,Q0d(s)∈RH∞, Q~(s)∈RH∞, and N~(s)∈RH∞; then, Q(s)∈H∞ means that the Nyquist plot of det{Q0d(s)+(N~(s)Q~(s)-Q0d(s))H(s)} does not encircle the origin (0,0). Obviously, Figure 3 illustrates that Q(s)∈H∞. This demonstrates that the design method for the free parameters Q~(s) and Q0d(s) is much more effective than that in [12, 13].Figure 3
Nyquist plot ofdet{Q0d(s)+(N~(s)Q~(s)-Q0d(s))H(s)}.Using the above-mentioned parameters, the stabilizing controllerC(s), 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:
(74)r(t)=2sin(2πL+ΔLt),L=1[sec].
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 q-(s).Figure 4
Sensitivity function of the high-order repetitive control system.The simulation results using our method are shown in Figure5. 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 0.05%, and this system entered the steady state in the 11th period. When the period-time had an uncertainty level Δ of 5%, the largest steady-state peak-to-peak relative error was 0.15%. 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 Δ=11%, the largest steady-state peak-to-peak relative error of this control system increases to 1.6%. This tracking error is small and can be further reduced by modifying the cutoff frequency of the low-pass filter q-(s) 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.Figure 5
Tracking errors for the references with uncertain period-time.


## 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.



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*Source: 374328-2013-02-26.xml*