A Novel Approach to Repetitive Control via Sampled-data H ∞ Filters
Masaaki Nagahara1, Masaki Ogura2 , and Yutaka Yamamoto3

Abstract— A new paradigm of digital signal processing has
recently been proposed in view of sampled-data control theory.
This filter, called sampled-data H ∞ filter, approximates alinear phase characteristic in continuous-time domain, that is,a pure delay. By this property, we propose a new repetitive
controller including the sampled-data H ∞ optimal filter. Moreover, to achieve the stability, we introduce the Internal Model
Control (IMC). Our method solves the problem of stability,
implementability, and inter sample ripples in digital repetitive
control. A numerical example shows the effectiveness of our
method.

I. INTRODUCTION
Repetitive control is a control scheme to be designed for
tracking periodic reference signals or for rejecting periodic
disturbances [1]. This method was first introduced in thecontrol of magnetic power supply for proton synchrotron [2].
Since then, a number of theoretical and industrial studies
have been made on repetitive control, see surveys [3], [4]and references in there.
In view of the internal model principle [5], [6], repetitive
control system must include the periodic signal generator
Qc (s) =e−Ls
1 − e−Lsin the feedback loop. This enables us to track arbitrary
periodic references of period L with zero steady-state error.
There however arise two problems; stability and implementability.
The stability problem is due to the fact that the repetitive
control system is a neutral delay-differential system. By thisnature, the control system cannot be exponentially stable
if the transfer function of the plant is strictly proper [1],[6]. To remedy this, modified repetitive control [7], [1] wasproposed, in which the repetitive controller Qc (s) is replacedbyFc (s)e−Ls
Qmod(1)c (s) =
1 − Fc (s)e−Ls
where Fc (s) is a lowpass filter with a cutoff frequency ωc .
This repetitive control enables us to exponentially stabilize
strictly proper plants, and to well track reference signals up
to the cutoff frequency ωc .
The modified repetitive control is a good scheme forgeneral plants, there however remains the second problem:
implementability. This is due to the infinite dimensionality
The authors are with Graduate School of Informatics, Kyoto University,
Kyoto 606-8501, Japan1nagahara@ieee.org
ogura@acs.i.kyoto-u.ac.jp
3
yy@i.kyoto-u.ac.jp
2of the delay e−Ls , which is difficult to implement in an
analog device. To this problem, controller discretization is aneffective solution. In this approach, the delay e−Ls is conventionally replaced by a sampled-data system Hh z −m Sh ,where Hh and Sh are respectively the zero-order hold andthe ideal sampler with sampling period h, and m is a positive
integer satisfying L = mh. This leads to digital repetitive
control [8] with repetitive controller
Qd (z) =z −m.
1 − z −mThis is the signal generator for arbitrary discrete-time periodic signals of an integer period m, by which the control
system can achieve perfect tracking1 for the periodic signals
on the sampling instants.
However, it has another problem: intersample ripples. A
remedy for this is to adopt a generalized hold [9] or a multirate control [10]. On the other hand, the intersample behavior
in digital repetitive control can be taken into account bymodern sampled-data control theory [11], [12]. There havebeen several works on sampled-data repetitive control, see[13], [14], [15], [16], [17].
Motivated by these works, we propose a novel repetitive
control which solves the three problems mentioned above;• stability,• implementability,• and intersample ripples.
For the stability problem, we introduce the internal model
control (IMC) [18], instead of the modified repetitive control.
The principle of IMC is in incorporating the model in thecontroller to handle the intrinsic difficulty in the plant. In
our case, the plant may be strictly proper and non-minimum
phase (i.e., difficult to control), and hence we place the model
into our repetitive controller.
The main idea of the present article is that by using the
IMC structure, sampled-data H ∞ filters can be applied toeffectively solve the three problem listed above simultaneously. This filter was first proposed in [19], and since thena number of works have appeared on various digital signal
processing problems, such as sampling rate converters [20],[21], optimal wavelet expansion [22], fractional delay filters[23], JPEG noise reduction [24] and ∆Σ converters [25]. See
also a survey paper [26].
The sampled-data H ∞ filter is a digital one, which isdesigned to approximate the time delay e−Ls with respect tothe H ∞ norm of the sampled-data error system. It is noted[26] that the conventional Shannon reconstruction causes
1 Perfect tracking means that the steady-state tracking error is zero.

r
−

K

Pc

y

r
− +

Kc

Pc

y

Pc
Fig. 1.
Control System: K can be a continuous-time or a sampleddata repetitive controller, Pc is a stable plant which can have unmodeled
dynamics.
Fig. 2. Internal Model Control: Pc is a stable continuous-time plant and
Pc is a model of Pc . Kc is a continuous-time stable controller.

high-frequency noise due to the Gibbs phenomenon. Thisis because the conventional filter is designed to perfectly
reconstruct the original signal within the Nyquist frequency.
This is the same phenomenon as one in digital repetitive
control. That is, the intersample ripples in digital repetitive
control can be interpreted as the Gibbs phenomenon causedby perfectly tracking to the reference signal on the sampling
instants. In view of this, the sampled-data H ∞ filter can also
be very effective in digital repetitive control.
The organization of this article is as follows. In Section II,
we formulate the problem of repetitive controller design. We
adopt the IMC structure for repetitive control in continuoustime domain in Section III. Based on the discussion there,
we introduce in Section IV the sampled-data H ∞ filter in
repetitive control. Section V illustrates a design example toshow effectiveness of our method. Section VI concludes thisarticle.
A. Notations
R: the real numbers.
R+ : the positive real numbers.
C: the complex numbers.
C+ : the complex numbers s which satisfies Re[s] > 0.
jR
: the imaginary numbers.
H ∞ : the bounded analytic functions in C+ .
RH ∞ : the bounded rational functions in C+ .
RL∞ : the bounded rational functions on jR.
L2: the square integrable functions on R+ .
B(L2 ) : the linear bounded operators in L2 .
Sh
: the ideal sampler with period h.
Hh : the zero-order hold with period h.
II. R EPETITIVE C ONTROL P ROBLEM
We here formulate repetitive control problem. First of all,let us see the control system shown in Fig. 1. In this figure,Pc is a plant to be controlled. Throughout this paper, we
assume that the plant Pc is stable, or already stabilized
by another controller. The plant can be perturbed due touncertainty in modeling. The controller K we design is acontinuous-time or a sampled-data system. In both cases,
the controller should be achieve••(robust) stability,
and tracking reference r of period L.

As mentioned above, if the relative degree of Pc is not
zero, perfect tracking and stability never go together. Thus,
we relax the strict requirement of tracking. Our problem isformulated as follows.

Problem 1: Find a stabilizing controller K such that thesensitivity function S = (I + Pc K)−1 is sufficiently small at
frequencies
ωn :=

2πn
,
L

n = 0, 1, 2, . . . , N

where N is a given positive integer.
This is the main idea of modified repetitive control. We
however do not adopt the structure of (1), but introduce theinternal model control.
III. R EPETITIVE C ONTROL BY I NTERNAL M ODELC ONTROL
A. Sensitivity optimization
Internal model control (IMC) is a powerful method tocontrol a plant which is difficult to control due to delays,
nonlinearities, etc. The IMC structure is shown in Fig. 2. In
this figure, Pc is a continuous-time plant and Pc ∈ RH ∞ isa model of Pc .
At first, assume that Pc = Pc (i.e., there is no model
error). Then, by the theory of Youla parameterization [27],[28], the feedback system is internally stable if and only if
the controller Kc is in H ∞ . If Kc ∈ H ∞ , the sensitivity
function is given by
Sc (s) = 1 − Pc (s)Kc (s).
To solve our problem (Problem 1), we employ a weighting
function Wc ∈ RH ∞ which is a lowpass filter2 with cutoff
frequency ωc > ωN = 2πN L−1 for given positive integer
N . By this weighting function, we optimize the following
objective function.
Jc (Kc ) = k(e−Ls − Pc Kc )Wc k∞ ,Kc ∈ H ∞ .

(2)Roughly speaking, if this is sufficiently small, then we have
Pc (jωn )Kc (jωn ) ≈ ejωn L = 1,

n = 1, 2, . . . , N

and hence the sensitivity function Sc (s) will be sufficiently
small at {jωn }n=1,2,...,N .
The optimal controller Kc minimizes the H ∞ norm of
the error system shown in Fig. 3. This is a standard oneblock H ∞ control problem [27], [28]. In fact, we have thefollowing lemma.
2 We choose a lowpass W because we want to better track signals at lowcfrequencies ω1 , . . . , ωN .

e−Ls
w

z −m

Hh

+ e
−

Wc
Kc

Fig. 3.

Sh+

Pc

Fig. 4.

Error system for approximation of e−Ls .

r
− +

Fc

Digital Repetitive Controller.

Kd

Sh

inf

Kc ∈H ∞Pcy

Pc

Lemma 1: Assume Pc has no zeros on jR and let
γopt :=Hh

Jc (Kc ).

Then for any ε > 0, there exists Kε ∈ H ∞ such that
Jc (Kε ) < γopt + ε.

Fig. 5.

Sampled-data repetitive control system with IMC structure.

IV. IMC R EPETITIVE C ONTROL BY SAMPLED - DATA H ∞Moreover, the sub optimal filter Kε is decomposed as

FILTERSA. Sensitivity optimization
Kε = K1 + K2∞∞

where K1 ∈ RH and K2 ∈ H .
The proof is given in the appendix.
B. Robust stability conditions
We here consider uncertainty in the plant Pc . Let∆c := {∆c ∈ H ∞ : k∆c k∞ ≤ 1}.
Then we have the following lemma.
Lemma 2: Assume Kc ∈ H ∞ .
1) Assume the plant has multiplicative perturbations, thatis,
Pc = Pc (1 + ∆c Wm ),∆c ∈ ∆cwhere Wm ∈ H ∞ is a weighting function. Then thecontrol system shown in Fig. 2 is internally stable forall ∆c ∈ ∆c if
kPc Kc Wm k∞ < 1.
2) Assume the plant has additive perturbations, that is,
Pc = Pc + ∆c Wa ,∆c ∈ ∆cwhere Wa ∈ H ∞ is a weighting function. Then thecontrol system shown in Fig. 2 is internally stable forall ∆c ∈ ∆c if
kKc Wa k∞ < 1.
The proof is straightforward by using the small gain theorem[27].
The controller given in this section is the (sub) optimal
solution for the repetitive control with an IMC structure
shown in Fig. 2. It however includes the infinite-dimensional
K2 ∈ H ∞ and is difficult to implement. In the next section,
we solve this problem by using sampled-data H ∞ filters.

The basic digital repetitive controller is shown in Fig.
4. This is an approximation of the continuous-repetitive
controller Qc (s) = e−mhs (1 − e−mhs )−1 assuming that m
is a positive integer satisfying L = mh. By the equation
Sh Hh = I (identity) [11], the digital repetitive controller isgiven by Hh Qd Sh where Qd (z) = z −m (1 − z −m )−1 . This
Qd is the signal generator for arbitrary discrete-time periodic
signals of period m. By the discrete-time version of internal
model principle, the repetitive control system with Qd canperfectly track arbitrary m-periodic references. As mentioned
in the introduction, this system can lead to intersample
ripples. We here propose an alternative to Qd via IMC andsampled-data H ∞ filters. The proposed repetitive control
system is shown in Fig. 5 (cf. Fig. 2). In this figure, Fc
is a strictly proper RH ∞ function. This Fc is usually called
anti-aliasing filter. This however is not necessarily required
to have cutoff frequency within the Nyquist frequency πh−1 .
This is introduced to make the feedback system be L2 bounded. See [11]. The sampled-data H ∞ optimal filter isan approximation of the continuous-time delay e−Ls taking
account of intersample errors by using sampled-data H ∞control theory. This fact can be used in our IMC repetitive
control.
Assume that Pc = Pc , and the sampling frequency
2πh−1 is non-pathological [11]. Then, by using the Youlaparameterization, the feedback system is stable if and only
if the digital controller Kd is stable. If Kd is stable, thesensitivity function is given by
Ssd = I − Pc Hh Kd Sh Fc .
To solve our problem, we use the same weighting function
Wc as one in the previous section. By this weighting function, we optimize the following objective function.
Jsd (Kd ) = k(e−mhs − Pc Hh Kd Sh Fc )Wc k∞2(3)

where the norm is the L -induced norm of the sampleddata error system which is equivalent to the H ∞ norm of

Fc

e−mhs
w

Sh

Kd

+

+ e
−

Wc
Fc

Sh

Kd

Hh

Hh

Pd

Pc

Fig. 6. Error system for approximation of e−mhs where m is a positive
integer.

Fig. 7. Digital implementation of sampled-data repetitive controller: Pd is
the step-invariant discretization of Pc .

Fc+the lifted system [11]. Throughout this section, the norm
k·k∞ denotes the L2 -induced norm. The corresponding error
system is shown in Fig. 6 (cf. Fig. 3). The optimal filter,
called sampled-data H ∞ filter, can be obtained numerically
via lifting method [29] or fast-discretization [19], [30].

Sh

↑M

Kmr

Hh/M

Pc
Fig. 8.

Multirate repetitive controller with upsampler ↑ M .

B. Robust stability conditions
We here consider uncertainty in the plant Pc . Let∆c := {∆c ∈ B(L2 ) : k∆c k∞ ≤ 1},
where B(L2 ) is the set of the linear bounded operators in
L2 . We have the following lemma [11].
Lemma 3: Assume Kd is stable.
1) Assume the plant has multiplicative perturbations, thatis,
Pc = Pc (1 + ∆c Wm ), ∆c ∈ ∆c ,where Wm ∈ RH ∞ is a weighting function. Then thecontrol system shown in Fig. 5 is internally stable forall ∆c ∈ ∆c if
kPc Hh Kd Sh Fc Wm k∞ < 1.
2) Assume the plant has additive perturbations, that is,
Pc = Pc + ∆c Wa ,It is pointed out in [10] that in digital repetitive control
systems, the period of the hold device can shorter than thatof the sampler. Considering this advantage, we can adopt a
multirate controller shown in Fig. 8. instead of the controller
in Fig. 5 In this system, ↑ M is the upsampler defined by↑ M : {x[k]}∞k=0 7→ {x[0], 0, . . . , 0, x[1], 0, . . .}.
| {z }
M−1

The controller Kmr is designed to optimize the following
objective function
Jmr (Kmr ) = k{e−mhs − Hh/M Kmr (↑ M )Sh Fc }Wc k∞ (4)By using discrete-time lifting [11], this can be solved numerically, see [21]. Digital implementation as shown in Fig.
7 is also obtained by discrete-time lifting.

∆c ∈ ∆c ,∞where Wa ∈ RH is a weighting function. Then thecontrol system shown in Fig. 5 is internally stable forall ∆c ∈ ∆c if
kHh Kd Sh Fc Wa k∞ < 1.
C. Implementation
Our sampled-data controller in Fig. 5 includes both adiscrete-time controller Kd and a continuous-time plantmodel Pc . To implement this in a digital device, we equivalently transform the sampled-data system into a discretetime system. To do this, we introduce the step-invariant
transformation [11],Pd = Sh Fc Pc Hh .
This system is a linear time-invariant discrete-time system,by which our controller is transformed intoKd
K = HhSh Fc
1 − Pd Kd
This controller is illustrated in Fig. 7.

D. Multirate control

V. D ESIGN E XAMPLE
In this section, we show a design example. The nominal
plant Pc here is
Pc (s) =s−1.
(s + 2)(s + 3)(s + 4)(s + 5)

The period L of reference signals is L = 10. The sampling
period is h = 1. The weighting function is a lowpass filter
with cutoff frequency ωc = π/20 = 0.157, that is,π
Wc (s) =.
20s + πThe lowpass filter Fc is chosen as
1, T = 1/100.
Ts + 1The parameter Tε to design continuous-controller Kc (seethe appendix) is the same as T . With these parameters, we
design three controllers; the continuous-time H ∞ optimal
Kc minimizing (2), the sampled-data H ∞ optimal Kd minimizing (3), and the multirate controller Kmr minimizing (4).
Fig. 9 shows these controllers. In this figure, the inverse of
Pc is also plotted. The continuous-time controller mimics
Fc (s) =Bode Plot

100

0.6
0.4
0.2
0

50
Magnitude

Magnitude (dB)

−0.2
−0.4
−0.6
0

−0.8
−1
−1.2

−50 −1
10

10

0

10

1

10

−1.4
0

2

20

40

60

80

100

120

140

160

180

200

Time (sec)Frequency (rad/sec)

Fig. 9.
Controllers; continuous-time H ∞ sub-optimal Kc (dash), theinverse of Pc (dash-dots), sampled-data H ∞ filter Kd (dots), and multirate
sampled-data H ∞ filter Kmr (solid).

Fig. 11. Tracking Error; with no perturbation (solid) and perturbed (dash).

VI. C ONCLUSION
In this article, we have proposed a novel repetitive control
by and sampled-data H ∞ filters. The controller has the IMC
structure and the controller optimizes the sensitivity function.
A numerical example shows the effectiveness of our method.
It is an easy extension to robust controller design.

1.5

1Magnitude

0.5

R EFERENCES

0

−0.5

−1

−1.5
0

20

40

60

80

100

120

140

160

180

200

Time (sec)Fig. 10. Output of sampled-data H ∞ optimal repetitive control system;
delayed reference (dash) and output y under plant perturbation.

the gain of Pc−1 very well. The two vertical lines show the
Nyquist frequencies π (for Kd ) and 4π (for Kmr ). By using
the sampled-data H ∞ optimal Kd , we simulate the repetitive
control for a triangle input of period L = 10. We assume theplant has a multiplicative perturbation
Pc (s) = Pc (s) (1 + ∆c (s)) ,∆c (s) =s − 0.5.
s+1

Under this perturbation, the output is shown in Fig. 10. In
the first few steps, the output shows over-shoots, then it
converges the input (note that the input shown is delayed by
one step). To see the error precisely, we show the error in Fig.
11. We also show the error when there is no perturbation.
Since we did not consider the robust performance, theerror with perturbation is much larger than that without
perturbation.

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A PPENDIXP ROOF OF L EMMA 1
Let (m, g) be an inner-outer factorization of Pc Wc ∈RH ∞ , that is,
Pc (s)Wc (s) = m(s)g(s),where m is inner and g is outer. Then by the method ofabsorption of the outer factor [31], the optimal value γopt is
reduced toγopt = inf ∞ ke−Ls m∼ Wc − Qk∞ ,
Q∈H

(5)

where m∼ (s) = m(−s). Let {A, B, C} be a minimal
realization of m∼ (s)Wc (s). Then we havee−Ls m∼ (s)Wc (s) = Φ(s) + Ψ(s)where
Φ(s) := C(e−Ls I − e−LA )(sI − A)−1 B,Ψ(s) := Ce−LA (sI − A)−1 B.

Since Φ is an FIR (finite-impulse-response) system [32] (i.e.,Φ ∈ H ∞ ) and Ψ is a rational function, we have
γopt =

inf

Q1 ∈RH ∞kΨ − Q1 k∞ .

(6)

By Nehari theorem [27], this value γopt is equal to the Hankel
norm of Ψ, and let Q1,opt ∈ RH ∞ be the optimal Q in (6)and letQopt (s) := Q1,opt (s) + Φ(s).
This Qopt is in H ∞ and is the optimal Q of (5). Let ν be
the relative degree of the outer function g(s). Since Wc isstrictly proper and e−Ls is inner, for any ε > 0, there exists
Tε > 0 such that
Jc (Kε ) = ke−Ls Wc − mgKε k < γopt + ε,
whereQopt (s)= K1 (s) + K2 (s),g(s)(Tε s + 1)ν
Q1,opt (s)K1 (s) =∈ RH ∞ ,g(s)(Tε s + 1)ν
Φ(s)∈ H ∞.
K2 (s) =g(s)(Tε s + 1)ν
Kε (s) :=