Brief paper Dual-modestructuredigitalrepetitivecontrolchemori/Temp/Leila/Repetitive...convergence rate than conventional RCs. DMRC requires the same data memory size as that of conventional

Automatica 43 (2007) 546–554www.elsevier.com/locate/automatica

Brief paper

Dual-mode structure digital repetitive control�

Keliang Zhoua,∗, Danwei Wangb, Bin Zhangb, Yigang Wangb, J.A. Ferreirac, S.W.H. de Haanc

aDepartment of Electrical Engineering, Southeast University, Nanjing 210096, ChinabSchool of EEE, Nanyang Technological University, 639798, SingaporecEPP, Delft University of Technology, 2628 CD, Delft, The Netherlands

Received 11 May 2005; received in revised form 22 August 2006; accepted 28 September 2006Available online 23 January 2007

Abstract

A flexible repetitive control (RC) scheme named “dual-mode structure repetitive control” (DMRC) is presented in this article. A robuststability criterion for DMRC systems is derived in terms of two parameters: odd-harmonic RC gain and even-harmonic RC gain. Severaluseful corollaries for the stability are addressed to reveal the compatibility of DMRC. The general framework of DMRC offers the flexibilityin the development of various RC controllers. Without additional complexity and loss of tracking accuracy, DMRC can achieve faster errorconvergence rate than conventional RCs. DMRC requires the same data memory size as that of conventional RC one. An application exampleof DMRC controlled PWM inverter illustrates the validity of our proposed DMRC scheme. Comparisons of DMRC, conventional RC andodd-harmonic RC highlight the advantages of the presented DMRC approach.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Internal model principle; Repetitive control; Odd-harmonic; Even-harmonic; Sample-data control

1. Introduction

Repetitive control (RC) (Hara, Yamamoto, Omata, &Nakano, 1988; Inoue, Nakano, Kubo, Matsumoto, & Baba,1981; Tomizuka, Tsao, & Chew, 1988), which is based onInternal Model Principle (Francis & Wonham, 1976), is de-veloped to track/eliminate periodic signals with a knownperiod by including their generator in a stable closed-loopsystem. Applications of RC have been widely reported indifferent fields, which include hard disk drives (Chew &Tomizuka, 1990), robotic manipulators (Cosner, Anwar, &Tomizuka, 1990), PWM inverters (Zhou & Wang, 2001;Zhou, Wang, & Low, 2000), PWM rectifiers (Zhou & Wang,2003), active power filters (Griñó, Costa-Castelló, & Fossas,2003), satellites (Broberg & Molyet, 1992), steel castings(Manayathara, Tsao, Bentsman, & Ross, 1996), and so on.

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Sam Geunder the direction of Editor M. Krstic.

∗ Corresponding author. Tel.: +86 25 83792260; fax: +86 25 83791696.E-mail addresses: [email protected] (K. Zhou), [email protected]

(D. Wang), [email protected] (S.W.H. de Haan).

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2006.09.018

In conventional RC systems, any reference signal with afundamental period N can be exactly tracked by including aperiodic signal generator 1/(zN −1) in the closed-loop system.Such a periodic signal generator needs at least N memory cells.Conventional RC controllers can eliminate both even and oddharmonics that are below Nyquist frequency by introducinginfinite gain at these harmonic frequencies.

For some systems, such as CVCF PWM converters, anodd-harmonic RC controller (Costa-Castelló, Grinó, & Fossas,2004; Griñó & Costa-Castelló, 2005) is proposed to reduceonly the odd-harmonic errors, which are dominant in theerrors. The even-harmonic periodic errors are neglected ortreated as non-repetitive errors. Compared with a conventionalRC controller, an odd-harmonic RC controller occupies lessdata memory and offers faster convergence rate of the trackingerror with identical RC gain (Zhou et al., 2006). However, theeven-harmonic errors will reside or might be amplified in anodd-harmonic RC system. The amplified even harmonic errorsmay lead to some undesired negative impacts on the system,e.g. dc voltage residues may cause the saturation of magneticcomponents, such as transformers and inductors.

A dual-mode structure RC (DMRC) is proposed in this paper.It can be used to improve the performance (error convergence

K. Zhou et al. / Automatica 43 (2007) 546–554 547

rate and tracking accuracy) of RC controller. More important,without additional complexity, the dual-mode structure is flexi-ble for housing various RC controllers, such as conventional RCcontroller, odd-harmonic RC controller, and so on. The robuststability condition and error convergence condition of DMRCsystems will be derived as an extension of conventional/odd-harmonic RC control system (Griñó & Costa-Castelló, 2005;Hara et al., 1988; Tomizuka et al., 1988). An application exam-ple of DMRC controlled PWM inverter is provided to demon-strate the effectiveness and advantages of DMRC.

2. Dual-mode structure digital repetitive control

2.1. Dual-mode periodic signal generator

As shown in Fig. 1, a conventional digital periodic signalgenerator Gg(z) (Tomizuka et al., 1988) with period N can bewritten as

Gg(z) = z−N

1 − z−N= 1

zN − 1, (1)

where N = T/Ts ∈ N with T and Ts being the fundamentalperiod of the signals and the sampling time. It is clear that, ifN is even, the Gg(z) in (1) has its poles at z = ej2m�/N , m =0, 1, . . . , N−1, wherein z=ej(2k+1)2�/N , k=0, 1, . . . , (N/2)−1 relating to odd-harmonic frequencies and z = ej4k�/N , k =0, 1, . . . , (N/2) − 1 relating to even-harmonic frequencies. Inother words, transfer function Gg(z) is used to generate funda-mental frequency and its harmonics.

Eq. (1) can be rewritten as follows:

Gg(z) = 1

2

(1

zN/2 − 1− 1

zN/2 + 1

)

= 1

2(Geg(z) + Gog(z)), (2)

where Geg(z) = 1/(zN/2 − 1) is an even-harmonic signal gen-erator, Gog(z)=−1/(zN/2 +1) is an odd-harmonic signal gen-erator. Notice that, the conventional repetitive signal generatorGg(z) contains two modes of signal generator: an odd-harmonicGog(z) and an even-harmonic Geg(z). Hereinafter, such a struc-ture is called “dual-mode structure”.

Furthermore, as shown in Fig. 2, a general dual-mode struc-ture prototype RC controller Gdrp(z) is proposed as follows:

Gdrp(z) = (keGeg(z) + koGog(z))Gf(z), (3)

z-Nw0 (z) w (z)

z-N/2

z-N/2

Gog (z)

Geg (z)

Gg (z)

w0 (z) w (z)1/2

Fig. 1. Conventional periodic signal generator.

z-N/2

z-N/2ke

ko +

+

Gog (z)

Geg (z)

e (z) udrp (z)Gf (z)

Fig. 2. Dual-mode structure prototype RC controller.

where even-harmonic RC gain ke �0 and odd-harmonic RCgain ko �0; Gf(z) is a compensation filter. Obviously, if ko =ke, Eq. (3) represents a conventional prototype RC controller(Tomizuka et al., 1988) with RC gain kr = 2ko = 2ke; if ke = 0,Eq. (3) is a prototype odd-harmonic RC controller (Griñó &Costa-Castelló, 2005) with RC gain kr = ko; and if ko = 0, Eq.(3) becomes a prototype even-harmonic RC controller with RCgain kr = ke. A RC controller, which is based on such a dual-mode periodic signal generator structure, is called “dual-moderepetitive controller” (DMRC).

Remark 1. Fig. 2 clearly indicates that, without additionalcomplexity, DMRC provides a general structure for various RCcontrollers, such as conventional RC controller, odd-harmonicRC controller and so on. The data memory size occupied byDMRC controller is the same as that of conventional RC one.The tracking error convergence rate of DMRC system can betuned by adjusting the RC gains of ko and ke. For example,if the odd-harmonic components dominate the errors, the RCgains could be chosen as ko > ke to improve the total errorconvergence rate, while eliminating both odd-harmonic andeven-harmonic errors; and vice versa. What is more, dual-modestructure can not only be applied to the digital RC controller,but also be easily employed to continuous-time RC controllerby replacing conventional kr/(e

L − 1) (Hara et al., 1988) withke/(e

L/2 − 1) − ko/(eL/2 + 1), where L is the period of the

continuous-time periodic signal.

2.2. Dual-mode structure repetitive control system

Fig. 3 shows a typical closed-loop control system with aplug-in DMRC controller, where R(z) is the reference input;Y (z) is the output, E(z) = R(z) − Y (z) is the tracking error;D(z) is the disturbance; Gp(z) is the transfer function of theplant, Gc(z) is the conventional feedback controller; Gdr(z) isa modified DMRC controller; ko is the odd-harmonic RC gain;ke is the even-harmonic RC gain; Udr(z) is the output of the RCcontroller; Gf(z) is a filter to stabilize the overall closed-loopsystem; Gogm(z) and Gegm(z) are modified periodic signal gen-erators for odd-harmonic signals and even-harmonic signals, re-spectively; Q1(z) and Q2(z) are low-pass filters to enhance thesystem robustness, with |Qi(ej�)|�1 (i =1, 2)(|Qi(ej�)| → 1at low frequencies and |Qi(ej�)| → 0 at high frequencies), e.g.Qi(z) = �i1z + �i0 + �i1z

−1 with 2�i1 + �i0 = 1, �i0 �0 and�i1 �0 (i = 1, 2) (Tomizuka et al., 1988).

548 K. Zhou et al. / Automatica 43 (2007) 546–554

--

ko

Udr (z)

+

Gogm (z)

Gc (z) Gp (z)U (z)

D (z)

Gf (z)

Q1 (z)

Q2 (z)+

R (z) + E (z) Y (z)

-

+

++

+

z-N/2

z-N/2

+

+ke

Gegm (z)

Dual-mode Repetitive Controller Gdr (z)

Fig. 3. Dual-mode digital repetitive control system.

The conventional controller Gc(z) is chosen so that the trans-fer function

H(z) = Gc(z)Gp(z)

1 + Gc(z)Gp(z)(4)

is asymptotically stable. Therefore, there exists an inverse func-tion Gfn(z) of H(z) (Tomizuka, 1987) such that

Gfn(z)H(z) = 1. (5)

Since the periodic signal generator introduce a N -step delayz−N , Gfn(z) can be implemented in the RC controller if Gfn(z)

is a non-causal filter (Note: N is much greater than the relativedegree of H(z)). And in practice, due to model uncertaintiesand load variations, it is impossible to obtain the exact transferfunction H(z). That is, the practical inverse function Gf(z) ofH(z) can be written as

Gf(z) = Gfn(z)(1 + �(z)), (6)

where �(z) denotes the uncertainties which are assumed to bebounded by |�(ej�)|�ε with ε being a positive constant, and�(z) is stable.

From Eqs. (5) and (6), we have

Gf(z)H(z) = Gfn(z)H(z)(1 + �(z)) = 1 + �(z). (7)

Let Gf(z)H(z)=NGH (�)ej�GH with z= ej�, from (5), (6) and(7), we have

NGH (�) = |Gfn(ej�)H(ej�)||1 + �(ej�)|�1 + ε. (8)

The plug-in DMRC Gdr(z) can be expressed as follows:

Gdr(z) = (koGogm(z) + keGegm(z))Gf(z)

=(

ko−z−N/2Q1(z)

1 + z−N/2Q1(z)+ ke

z−N/2Q2(z)

1 − z−N/2Q2(z)

)Gf(z).

(9)

Theorem 1. For the closed-loop DMRC system shown in Fig.3 with constraints (4)–(9), if the RC gains ke and ko satisfy thefollowing inequalities:

ke �0, ko �0, (10)

and

0 < ke + ko <2

1 + ε, (11)

then the closed-loop DMRC system is asymptotically stable.

Proof. See Appendix A. �

Remark 2. Theorem 1 offers a stability criterion for the closed-loop DMRC system. If the odd-harmonics and even-harmonicsare treated as two different periodic signals, DMRC system be-comes one special case of multiple-periods RC systems (Chang,Suh, & Oh, 1998; Yamada, Riadh, & Funahashi, 2000).

DMRC offers a general framework to develop various RCcontrollers, e.g. conventional RC controller, odd-harmonic RCcontroller, etc. In the following paragraphs, corollaries willshow that the stability criterions for conventional RC systems(Cosner et al., 1990) and odd-harmonic RC systems (Griñó &Costa-Castelló, 2005) are compatible to our Theorem 1.

Corollary 1. If the DMRC system in Fig. 3 with RC gainsko = ke = kr/2 and filters Q1(z) = Q2(z) = Q(z) fulfils thefollowing condition:

|Q2(z)(1 − krGf(z)H(z))| < 1, (12)

then it is asymptotically stable, and the RC gain kr satisfies

0 < kr <2

1 + ε. (13)

Proof. If filters Q1(z)=Q2(z)=Q(z) and RC gains ko =ke =kr/2, the DMRC system in Fig. 3 becomes a conventional RCsystem with RC gain kr and filter Q2(z).

From Fig. 3 and Eq. (A.1), the transfer function G(z) fromR(z) to Y (z) and the transfer function Gd(z) from D(z) to Y (z)

for such a DMRC system can be derived as

Y (z) = G(z)R(z) + Gd(z)D(z)

= H(z)(1 − z−NQ2(z)(1 − krGf(z)))

1 − z−NQ2(z)(1 − krGf(z)H(z))R(z)

+ (1 + Gc(z)Gp(z))−1(1 − z−NQ2(z))

1 − z−NQ2(z)(1 − krGf(z)H(z))D(z). (14)

Obviously, if |Q2(z)(1 − krGfH(z))| < 1, then all the polespi (i = 0, 1, . . . , N) of G(z) and Gd(z) in (14) are inside theunit circle |z| = 1. Since H(z) is asymptotically stable, theclosed-loop system transfer functions G(z) and Gd(z)in (14)are asymptotically stable. Corresponding criterion can be foundin conventional RC systems (Tomizuka et al., 1988).

The formula in (12) can be expressed as

N2Q(�)|1 − krNGH (�)ej�GH | < 1. (15)

Or equivalently,

|1 − krNGH (�) cos �GH − jkrNGH (�) sin �GH | < 1

N2Q(�)

.

(16)

K. Zhou et al. / Automatica 43 (2007) 546–554 549

Using norm definition and taking square on both sides (Wang& Ye, 2005), we have

0 < kr <1 − N4

Q(�)

N4Q(�)(krN

2GH (�))

+ 2 cos �GH

NGH (�)(17)

Since (1 − N4Q(�))/(krN

2GH (�)N4

Q(�))�0, to ensure thesystem stability conservatively, we can choose

0 < kr <2 cos �GH

NGH (�)� 2

NGH (�).

Similarly, the stability range of kr can be obtained as

0 < kr <2

1 + ε. �

Corollary 2. If DMRC in Fig. 3 with RC gains ko > 0 andke = 0 fulfils the following condition:

|Q1(z)(1 − koGf(z)H(z))| < 1 (18)

then the DMRC system in Fig. 3 is asymptotically stable, andthe RC gain ko should satisfy

0 < ko <2

1 + ε. (19)

Proof. If ke = 0, the DMRC system in Fig. 3 becomes an odd-harmonic RC system (Griñó & Costa-Castelló, 2005) with RCgain ko and filter Q1(z). Proof of Corollary 2 is similar to thatof Corollary 1. �

Theorem 2. If the closed-loop DMRC system with Qi(z) = 1(i = 1, 2) in Fig. 3 is asymptotically stable, then the error e(k)

in Fig. 3 converges asymptotically to zero when its spectralcontent corresponds to the frequencies of the roots of zN = 1.

Proof. The error transfer function T (z) is given by

T (z) = E(z)

R(z) − D(z)

= 1

1 + Gc(z)Gp(z)

1

1 + H(z)Gdr(z)

= 1

1 + Gc(z)Gp(z)

Gx(z)

Gx(z) + (1 + �(z))Gy(z), (20)

where

Gx(z) = 1 − z−NQ1(z)Q2(z) + z−N/2(Q1(z) − Q2(z)),

Gy(z)=(ko+ke)z−NQ1(z)Q2(z)+(keQ2(z)−koQ1(z))z

−N/2.

Since H(z) and G(z) are asymptotically stable, it is clearthat if Qi(z) = 1 (i = 1, 2), then

|T (ej�)| = 0, ∀z = ej� such that zN = 1. (21)

Therefore the error e(k) in Fig. 3 converges asymptoticallyto zero. �

Remark 3. The proofs of Corollaries 1 and 2 indicate that,Theorem 1 offers a more conservative stability condition forconventional RC systems than Theorem 2 in Cosner et al.(1990); Theorem 1 offers a more conservative stability condi-tion for odd-harmonic RC systems than Proposition 4 in Griñó& Costa-Castelló (2005).

Remark 4. Theorem 2 indicates repetitive errors, which in-clude even-harmonic errors and odd-harmonic errors, can becompletely eliminated by our proposed DMRC controller withQi(z) = 1 (i = 1, 2), even under modeling uncertainty.

Remark 5. The proof of Theorem 1 indicates the introductionof low-pass filter Qi(z) (i = 1, 2) with |Qi(ej�)|�1 will makeit easier to ensure all poles |pi | < 1 (i = 0, 1, . . . , N) of G(z)

and Gd(z) in Eq. (A.1), and then enhance the robustness of theDMRC system. On the other hand, as shown in the proof of The-orem 2, the introduction of Qi(z) (i=1, 2) with |Qi(ej�)| → 1(i = 1, 2) at low frequencies and |Qi(ej�)| → 0 (i = 1, 2) athigh frequencies, will cause the variation of the zeros of T (z)

(20), and yields the imperfect cancellation of periodic errors,especially at the high frequency band. The tracking accuracywill be reduced. Therefore, the introduction of low-pass filterQi(z) (i = 1, 2) brings a trade-off between tracking accuracyand system robustness in the DMRC system.

3. Application example

Consider a single-phase pulse-width modulation (PWM) in-verter system as shown in Fig. 4, where vc is the output voltage;io is the load current; vin is the PWM control input; dc voltagevdc = 130 V; Ln = 2 mH, Cn = 10 �F, Rn = 30 � are nominalcomponent values of the inductor L (with actual value 2.2 mH),the capacitor C (with actual value 12 �F) and the resistive loadR (with actual value 30 �), respectively; the reference input

nL io

S1 S2

S3 S4

Rn vc

+

-

vdc

vin Cn

PWM generator

FeedbackController

Repetitive Controller

y = vc

yref = vcref

vc

+-

eur

u

ic

ic

Fig. 4. Repetitive controlled single-phase PWM inverter.

550 K. Zhou et al. / Automatica 43 (2007) 546–554

1

0.8

0.6

0.4

0.2

00 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Frequency (Hz)

NG

H (

ω)

NGH

(ω)

Fig. 5. Magnitude NGH (�) of Gf (z)H(z).

yref(=vcref) is (fs=) 50 Hz, 100V (peak) sinusoidal voltage;sampling frequency f = 1/T = 10 kHz;N = f/fs = 200.

In discrete-time domain, the state-space equation (Zhou,Wang, & Low, 2000) for the inverter system with nominalparameters Ln, Cn, and Rn is given by

{vc(k + 1) = 0.816vc(k) + 8.611 × 10−5v̇c(k) + 0.176vin(k),

v̇c(k + 1) = −3260vc(k) + 0.571v̇c(k) + 3029vin(k),

y(k) = vc(k),

with a feedback controller Gc(z) as follows:

vin(k) = −0.0214vc(k) − 2.38 × 10−6v̇c(k) + 0.0286vcref(k),

then transfer function from yref to y for the closed-loop feedbacksystem with nominal parameters Ln, Cn and Rn can be obtainedas

Hn(z) = 0.9286z + 0.6964

z2 + 0.7321z − 0.0759

and with true parameters L, C and R, the transfer function fromyref to y is

H(z) = 0.7035z + 0.5436

z2 + 1.644z + 0.1066.

The plug-in DMRC controller is given by Eq. (9), whereQ1(z) = Q2(z) = (z + 2 + z−1)/4 is sufficiently enough tomake a good trade-off between high tracking accuracy andsystem robustness (Tomizuka et al., 1988; Zhou & Wang, 2001);Gf(z) = 1/Hn(z).

Thus, we have

Gf(z)H(z) = 1 + �(z)

= z2 + 0.7321z − 0.0759

0.9286z + 0.6964

0.7035z + 0.5436

z2 + 1.644z + 0.1066

with its magnitude characteristics given in Fig. 5.

From (8) and Fig. 5, we have ε ≈ 1.12−1=0.12. Accordingto Theorem 1, we have ko + ke < 2/(1 + ε) ≈ 1.79. In thefollowing simulations, we choose ko + ke = 0.4 < 1.79.

Fig. 6(a) shows the steady-state response of only feedbackcontrolled inverter. It can be seen that a 4.7V (amplitude),50 Hz (fundamental frequency, odd-harmonics) componentwith a −0.5V dc (even-harmonics) offset dominates the track-ing error. Figs. 6(b)–(d) show the steady-state responses ofvarious plug-in RC controlled inverters. Fig. 6(b) and (d)indicate that both conventional RC controller and DMRCcontroller (ke > 0 and ko > 0) can significantly reduce thefeedback controlled tracking error (both odd-harmonics andeven-harmonics) to be within negligible ±0.01 V. However,as shown in Fig. 6(c), a 0.25V (amplitude), 100 Hz (2nd har-monic frequency) component with a −0.59V dc offset stillresides in the odd-harmonic RC controlled tracking error.It clearly manifests that odd-harmonic RC controller can-not reduce the even-harmonic errors or might even magnifythem.

Fig. 7 shows the transient tracking error responses with vari-ous RC controllers (ko +ke =0.4) being plugged into the feed-back controlled inverter at time t=0.06 s. In terms of error con-vergence rate, as shown in Fig. 7, odd-harmonic RC (ko = 0.4and ke = 0) is the fastest, DMRC (ko = 0.32 and ke = 0.08)is the second fastest, DMRC (ko = 0.28 and ke = 0.12) is thethird fastest, and conventional RC (kr = 0.4, or ko = ke = 0.2)is the slowest. Since the odd-harmonic component significantlydominates the tracking error (as shown in Fig. 6(a)), the largerodd-harmonic gain ko is, the faster the error convergence rateis. In terms of tracking accuracy, as shown in Fig. 7, DMRC(ko =0.32 and ke =0.08), DMRC (ko =0.28 and ke =0.12) andconventional RC have comparable high accuracy, while thereare obvious even-harmonic errors residing in the odd-harmonicRC system. That is to say, odd-harmonic RC is not immune toeven-harmonic errors (Zhou et al., 2006). These even-harmonicerrors in the odd-harmonic RC system may lead to some unde-sired negative impacts, e.g. dc voltage residues may cause thesaturation of magnetic components.

Remark 6. DMRC is the extension of conventional RC andodd-harmonic RC. The design of Qi(z) for DMRC can be thesame as that of conventional/odd-harmonic ones. In our case,for a CVCF PWM inverter at sample rate 10 kHz, the majorityof its total harmonic distortion is below 1 kHz. To enhance thesystem robustness without significant loss of tracking accuracy,the bandwidth of Qi(z) (i = 1, 2) should be greater than 1 kHzand less than 5 kHz (Nyquist frequency). Q1(z) and Q2(z) canbe identical or different filters as long as they well meet theabove demand. Here Q1(z) = Q2(z) = (z + 2 + z−1)/4 areappropriate filters. Moreover, in practice, since the tracking er-ror components will change significantly with different plantsand different loads, and the errors also contain many frequen-cies, it is almost impossible to determine the optimal gain ratioof ko/ke theoretically. Following the basic tuning method de-scribed in Remark 1, a good choice of the gains ko and ke canbe found by experiments.

K. Zhou et al. / Automatica 43 (2007) 546–554 551

100

80

60

40

20

0

-20

-40

-60

-80

-100

100

80

60

40

20

0

-20

-40

-60

-80

-100

100

80

60

40

20

0

-20

-40

-60

-80

-100

100

80

60

40

20

0

-20

-40

-60

-80

-100

0.04 0.045 0.05 0.055 0.06

Time (sec)

Time (sec)

0.04 0.045 0.05 0.055 0.06

Time (sec)

6

4

2

0

-2

-4

-6

0.98 0.985 0.99 0.995 1

Time (sec)

0.98 0.985 0.99 0.995 1

Time (sec)

0.98 0.985 0.99 0.995 1

Time (sec)

0.98 0.985 0.99 0.995 1

Time (sec)

0.98 0.985 0.99 0.995 1

Time (sec)

0.98 0.985 0.99 0.995 1

0.02

0.015

0.01

0.005

0

-0.005

-0.01

-0.015

-0.02

0.02

0.015

0.01

0.005

0

-0.005

-0.01

-0.015

-0.02

Ou

tpu

t V

aria

ble

sO

utp

ut

Va

ria

ble

sO

utp

ut

Va

ria

ble

sO

utp

ut

Va

ria

ble

s

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9

Tra

ckin

g E

rro

rs (

V)

Tra

ckin

g E

rro

rs (

V)

Tra

ckin

g E

rro

rs (

V)

Tra

ckin

g E

rro

rs (

V)

d

c

b

aV

cref

ro

e = VCref

- Vc

e = VCref

- Vc

e = VCref

- Vc

Vc

Vc

Vcref

ro

ro

e = VCref

- Vc

Vcref

Vcref

Vc

Vcr

o

Fig. 6. Steady-state responses: (a) Feedback controller control; (b) conventional RC control (kr = 0.4); (c) odd-harmonic RC control (ko = 0.4 and ke = 0);(d) DMRC control (ko = 0.28 and ke = 0.12).

552 K. Zhou et al. / Automatica 43 (2007) 546–554

4

3

2

1

0

-1

-2

-3

-4

-5

Tra

nsie

nt T

rackin

g E

rror

(V)

0.1 0.2 0.30.15 0.25

Time (sec)

Traditional RC kr = 0.4

DMRC ko = 0.28, k

e = 0.12

DMRC ko = 0.32, k

e = 0.08

Odd - harmonic RC ko = 0.4, k

e = 0

Fig. 7. Various RC controlled transient tracking errors.

4. Conclusions

This paper proposed a dual-mode structure for RC controller.Without additional complexity, dual-mode structure offers theflexibility in the development of various RC controllers, such asconventional RC, odd-harmonic RC and so on. A robust stabil-ity criterion for DMRC systems is derived. Corollaries for thestability are addressed to reveal the compatibility of DMRC.The error convergence condition of DMRC systems is stated.Without loss of tracking accuracy, DMRC can be widely usedto improve the transient response of RC controlled systems,such as PWM converters, hard disks and so on. An applica-tion example of DMRC controlled inverter is given to show thepromising advantages of the DMRC controller: without loss oftracking accuracy, DMRC can offer faster error convergencerate than conventional RC; DMRC yields higher tracking ac-curacy than odd-harmonic RC.

Appendix A. Proof of Theorem 1

From Eqs. (4)–(9) and Fig. 3, the transfer function G(z)

from R(z) to Y (z) and the transfer function Gd(z) from D(z)

to Y (z) can be derived as follows:

Y (z) = G(z)R(z) + Gd(z)D(z)

= H(z)(1 + Gdr(z))

1 + H(z)Gdr(z)R(z) + (1 + Gc(z)Gp(z))

−1

1 + H(z)Gdr(z)D(z)

= H(z)(1 + Gf(z)(koGogm(z) + keGegm(z)))

1 + (1 + �(z))(koGogm(z) + keGegm(z))R(z)

+ (1 + Gc(z)Gp(z))−1

1 + (1 + �(z))(koGogm(z) + keGegm(z))

× D(z), (A.1)

where

Gogm(z) = −z−N/2Q1(z)

1 + z−N/2Q1(z),

Gegm(z) = z−N/2Q2(z)

1 − z−N/2Q2(z),

Qi(z) = |Qi(ej�)|ej�Qi(�) = NQi(�)ej�Qi(�) (i = 1, 2),

NQi(�) = |Qi(ej�)|�1 (i = 1, 2).

Let z = |z|ej� = aej� with |z| = a, then we have

Re[Gogm(z)] = Re

[ −z−N/2Q1(z)

1 + z−N/2Q1(z)

]

= Re

[−

(aN/2

NQ1(�)ej(N�/2−�Q1(�)) + 1

)−1]

(A.2)

and

Re[Gegm(z)] = Re

[z−N/2Q2(z)

1 − z−N/2Q2(z)

]

= Re

[(aN/2

NQ2(�)ej(N�/2−�Q2(�)) − 1

)−1]

.

(A.3)

Let bi = aN/2/NQi(�) and �i = N�/2 − �Qi(�) (i = 1, 2).If |z| = a�1, we have bi �aN/2 �a�1 (i = 1, 2). Eq. (A.2)can be rewritten as

Re[Gogm(z)] = Re

[ −1

b1ej�1 + 1

]

= Re

[ −(b1 cos �1 + 1) + jb1 sin �1

(b1 cos �1 + 1)2 + (b1 sin �1)2

]

= −b1 cos �1 − 1

b21 + 1 + 2b1 cos �1

.

If −(b1 cos �1 + 1)�0, we have

Re[Gogm(z)] = −(b1 cos �1 + 1)

b21 + 1 + 2b1 cos �1

�0;

If −(b1 cos �1 + 1) < 0, we have

b21 + 1 + 2b1 cos �1

−(b1 cos �1 + 1)= b2

1 − 1

−(b1 cos �1 + 1)− 2

� a2 − 1

−(b1 cos �1 + 1)− 2

� − 2 ∀|z| = a�1,

K. Zhou et al. / Automatica 43 (2007) 546–554 553

and then

Re[Gogm(z)] = −(b1 cos �1 + 1)

b21 + 1 + 2b1 cos �1

� − 1

2|z|�1. (A.4)

Eq. (A.3) can be rewritten as

Re[Gegm(z)] = Re

[1

b2ej�2 − 1

]

= Re

[(b2 cos �2 − 1) − jb2 sin �2

(b2 cos �2 − 1)2 + (b2 sin �2)2

]

= b2 cos �2 − 1

b22 + 1 − 2b2 cos �2

If b2 cos �2 − 1�0, we have

Re[Gegm(z)] = b2 cos �2 − 1

b22 + 1 − 2b2 cos �2

�0;

If b2 cos �2 − 1 < 0, we have

b22 + 1 − 2b2 cos �2

b2 cos �2 − 1= b2

2 − 1

b2 cos �2 − 1− 2

� a2 − 1

b2 cos �2 − 1− 2

� − 2 ∀|z| = a�1,

and then

Re[Gegm(z)] = b2 cos �2 − 1

b22 + 1 − 2b2 cos �2

� − 1

2, ∀|z|�1. (A.5)

From Eqs. (4)–(8), (A.4), (A.5) and ko �0, ke �0, 0 < ko +ke < 2/(1 + ε), we have

min|z|�1Re[koGogm(z) + keGegm(z)]

= min|z|�1(koRe[Gogm(z)] + keRe[Gegm(z)])

� min|z|�1

((−1

2

)ko + ke

(−1

2

))> − 1

2

2

1 + ε

� −∣∣∣∣ 1

1 + �(ej�)

∣∣∣∣ , ∀|z|�1. (A.6)

This implies that

1

1 + �(z)+ (koGogm(z) + keGegm(z)) �= 0, ∀|z|�1. (A.7)

Thus, all the poles pi (i = 0, 1, . . . , N) of the transfer func-tions G(z) and Gd(z) in (A.1) are inside the unit circle |z| = 1,i.e. |pi | < 1. Finally, the asymptotical stability of H(z) impliesthe asymptotical stability of the closed-loop DMRC system inFig. 3.

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Zhou, K., Low, K. S., Wang, D., Luo, F., Zhang, B., & Wang, Y. (2006).Zero-phase odd-harmonic repetitive controller for a single-phase PWMinverter. IEEE Transactions on Power Electronics, 21(1), 193–201.

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Keliang Zhou received his B.E. degree fromthe Huazhong University of Science and Tech-nology, Wuhan, China, in 1992, the M.Eng. de-gree from Wuhan University of Transportation,Wuhan, China, in 1995, and the Ph.D. degreefrom Nanyang Technological University, Singa-pore, in 2002. Currently he is a professor withDepartment of Electrical Engineering, SoutheastUniversity, Nanjing, China. His research inter-ests mainly involve power electronics and elec-tric machines drives, advanced control and itsapplications and renewable energy generation.

Dr. Zhou has authored or co-authored more than 30 published technicalarticles in the relevant areas.

554 K. Zhou et al. / Automatica 43 (2007) 546–554

Danwei Wang received his Ph.D. and MSEdegrees from the University of Michigan, AnnArbor in 1989 and 1985, respectively. He re-ceived his B.E. degree from the South ChinaUniversity of Technology, China in 1982. Since1989, he has been with the School of EEE,Nanyang Technological University, Singapore.Currently, he is an associate professor of theSchool of EEE, head of the division of Controland Instrumentation and director of the Centrefor Intelligent Machines, NTU. He has servedas general chairman, technical chairman and

various positions in international conferences, such as ICARCVs, IEEE RAMand ACCV. He is an associate editor of Conference Editorial Board, IEEEControl Systems Society, an associate editor of International Journal of Hu-manoid Robotics, and deputy chairman of IEEE Singapore Robotics and Au-tomation Chapter. He was a recipient of Alexander von Humboldt fellowship,Germany. His research interests include robotics, control theory and applica-tions. He has published many technical articles in the areas of iterative learn-ing control, repetitive control, robust control and adaptive control systems,as well as manipulator/mobile robot dynamics, path planning and control.

Bin Zhang received his BE and MSE degreesfrom Nanjing University of Science and Tech-nology, China, in 1993 and 1999, respectively.He is currently working toward the Ph.D. degreeat Nanyang Technological University, Singa-pore. His current research interests are ILC/RC,intelligent control, digital signal processing,and their applications to robot manipulators,power electronics and vibration suppression.

Yigang Wang received the B.E. and MSEdegrees from Harbin Institute of Technology,Harbin, China, in 2001 and 2003, respectively.His current research interests are repetitive andlearning control, robust, adaptive and multiratefiltering and control, with applications to PWMinverters, mechatronic system.

J.A. Ferreira received the BSc Eng., MSc Eng.,and Ph.D. degrees from Rand Afrikaans Uni-versity, Johannesburg, South Africa, in 1981,1983, and 1988, respectively, all in electricalengineering. In 1981, he was with the Instituteof Power Electronics and Electric Drives, Tech-nical University of Aachen, and worked in in-dustry at ESD (Pty.) Ltd. from 1982 to 1985.From 1986 to 1997, he was with the Faculty ofEngineering, Rand Afrikaans University, wherehe held the Carl and Emily Fuchs Chair ofpower electronics in later years. Since 1998, he

has been a Professor with Delft University of Technology, Delft, The Nether-lands. Dr. Ferreira was the Chairman of the South African Section of theIEEE from 1993 to 1994. He is the Founding Chairman of the IEEE Joint In-dustry Applications Society/Power Electronics Society (IAS/PELS) Beneluxchapter. He served as the IEEE Transactions on Industry Applications Re-view Chairman for the IEEE Industry Applications Society Power ElectronicDevices and Components Committee and is an Associate Editor of the IEEETransactions on Power electronics. He was a member of the IEEE PowerElectronics Specialists Conference Adcom and is currently the Treasurer ofthe IEEE PELS. He served as the Chairman of the CIGRE SC14 NationalCommittee of the Netherlands and was a Member of the Executive Commit-tee of the European Power Electronics Society.

Sjoerd de Haan is associate professor in powerelectronics within the Electrical Power Process-ing group of the Delft University of Technol-ogy since 1995. He obtained his MSc degreein Physics at the Delft University of technol-ogy in 1973. He started his carrier at TPD-TNO(applied physics research centre) and was sub-sequently employed by both Delft University ofTechnology and Eindhoven University of tech-nology. From 1993 until 1995 he was seniorresearcher with ECN (The Netherlands EnergyResearch Foundation) in Petten, where he was

in charge of research on electrical conversion for renewable energy sys-tems. His current research interests concerns electrical systems for dispersedgeneration, pulsed power and compact converters. De Haan was author andco-author of over a hundred papers.