Robust Repetitive Control by Sampled-data H ∞ Filters
Masaaki Nagahara
Graduate School of Informatics
Kyoto University
Kyoto, 606-8501, JAPAN

Yutaka Yamamoto
Graduate School of Informatics
Kyoto University
Kyoto, 606-8501, JAPANnagahara@ieee.org

yy@i.kyoto-u.ac.jp

Abstract— In this article, we study digital repetitive control
to track continuous-time periodic references. Sampled-data
control gives a very natural platform for studying digital
repetitive control, and in particular, sampled-data H ∞ filtering
is particularly suitable for this. The link is provided by introducing internal model control. This leads to a close connection
between robust repetitive control and sampled-data H ∞ noise
reduction, or regularization. With this connection, it becomes
possible to employ a variety of filtering techniques, such as
multirate filtering, for robust digital repetitive control. Design
examples show the effectiveness of our method in robustness indigital repetitive control.

dimensionality of the delay e−Ls , which is difficult to implement with an analog device. For this problem, controller
discretization is an effective solution. In this approach, thedelay e−Ls is conventionally replaced by a sampled-datasystem Hh z −m Sh , where Hh and Sh are, respectively, thezero-order hold and the ideal sampler with sampling periodh, and m is a positive integer satisfying L = mh. This leads
to digital repetitive control [8] with repetitive controller

I. INTRODUCTION
Repetitive control is a control scheme which is designed
for tracking periodic reference signals or for rejecting periodic disturbances [1]. This method was first introduced in
the control 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] andreferences therein.
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(1)
Qmodc (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 provides a reasonable
scheme for general plants; however, there remains the second problem: implementability. This is due to the infinite

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.
While this appears to be a reasonable compromise, it
inevitably induces intersample ripples, which clearly limits
the tracking capability. Although a partial solution may beobtained by introducing a generalized hold [9] or a multirate
control [10], one wants to have an overall performance
estimate and control of such intersample ripples. This can beeffectively handled by modern sampled-data control theory[11], [12], and there are indeed several works on sampleddata repetitive control: [13], [14], [15], [16], [17].
While motivated by these works, we here propose anothernovel repetitive control scheme viewed from the viewpoint ofdigital filter design. This approach makes use of the internal
model control (IMC) [18], instead of the modified repetitive
control, and gives an effective solution to the following
problems discussed above:
stability,
implementability,• and intersample ripples.
••

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 objective of the present article is to show that by using
the IMC structure, sampled-data H ∞ filters can effectively
solve the three problems simultaneously listed above. Thisfilter was first proposed in [19], and since then a number
1 Perfect tracking means that the steady-state tracking error is zero.

r
−

K

Pc

y

Fig. 1. Digital repetitive control system: K is a sampled-data repetitive
controller, Pc is a stable plant which can have unmodeled dynamics.

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
to the H ∞ norm of the sampled-data error system. It isnoted [26] that the conventional Shannon reconstruction
causes high-frequency errors due to the Gibbs phenomenon.
This is because the conventional filter is designed to perfectly reconstruct the original signal within the Nyquist
frequency. Without confining ourselves to this idea of perfectreconstruction (however below the Nyquist frequency), it
is possible to obtain a better overall performance by using
sampled-data H ∞ filters.
We have proposed this approach in [27], but the robustness
was not discussed there. In this article, we also show thatrobust repetitive control can be described by sampled-data
H ∞ noise reduction or regularization. Regularization is usedin machine learning [28] or inverse problems [29], [30].
Using this technique, we can well control a trade-off between
fitting of the given input data and robustness against model
uncertainties. In regularization, robustness can be achievedby reducing a norm of the solution. In the same vein,
we apply regularization to achieve robustness in repetitive
control.
The organization of this article is as follows. In Section
II, we formulate the problem of repetitive controller design.
We introduce IMC-based digital repetitive control in Section
III, where we discuss the relation between repetitive control
and sampled-data H ∞ filtering. We also consider robust
stability there. Section IV illustrates design examples to show
effectiveness of our method. Section V concludes this article.
A. Notation
The subscripts ‘c’ and ’d’ indicate, respectively,“continuous-time” and “discrete-time.” We use boldface letters (e.g., K) for sampled-data systems (i.e., systems including both continuous- and discrete-time signals); in particular,
Sh and Hh are the ideal sampler with period h and thezero-order hold with period h, respectively. L2 denotes thesquare integrable functions on R+ (the strictly positive real
numbers), and B(L2 ) the linear bounded operators in L2 .
II. D IGITAL R EPETITIVE C ONTROL P ROBLEM
Here we formulate a repetitive control problem. Consider
the control system shown in Fig. 1. In this figure, Pc is aplant to be controlled. Throughout this article, we assume

Sh+

Fig. 2.

z −m

HhConventional digital repetitive controller.

that the plant Pc is stable, or already stabilized by anothercontroller. The plant can be perturbed due to uncertainty inmodeling. The controller K we design is a sampled-datasystem. In this setting, the control system should achieve• (robust) stability,• and tracking reference r of period L.
As mentioned above, if the relative degree of Pc is nonzero,
perfect tracking and stability are incompatible. Thus, wemust relax the strict requirement of perfect tracking. Our
problem is formulated as follows.
Problem 1: Find a (robustly) stabilizing controller K
such that the sensitivity function S = (I + Pc K)−1 issufficiently small at frequencies
2πn, n = 0, 1, 2, . . . , N
Lwhere 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 aninternal model control structure.
ωn :=III. R EPETITIVE C ONTROL WITH IMC S TRUCTURE BYS AMPLED - DATA H ∞ F ILTERS
A. Sensitivity Optimization
The conventional digital repetitive controller is shown in
Fig. 2. This is an approximation of the continuous-time repetitive controller Qc (s) = e−mhs /(1 − e−mhs ) assuming thatm is a positive integer satisfying L = mh. By the equation
Sh Hh = I (identity) [11], the digital repetitive controller
is given by Hh Qd Sh where Qd (z) = z −m /(1 − z −m ).
This Qd is the signal generator for arbitrary discrete-time
periodic signals of period m. By the discrete-time versionof the internal model principle, the repetitive control systemwith Qd can perfectly track arbitrary m-periodic references.
Note however that this can lead to intersample ripples, since
this accomplishes tracking only at sampled points. Here we
propose an alternative to Qd via the internal model control(IMC) scheme and sampled-data H ∞ filters. The proposed
repetitive control system is shown in Fig. 3. In this figure, Pc
is a stable continuous-time plant and Pc denotes a model of
Pc . The system Fc is assumed to be stable and strictly proper.
This Fc is usually called an anti-aliasing filter. This however
is not necessarily required to have a cutoff frequency lower
than the Nyquist frequency π/h. This is introduced to makethe feedback system to be L2 -bounded (see [11]).
The sampled-data H ∞ optimal filter is an approximationof 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
for the moment that Pc = Pc , and the sampling frequency

r
− +

Fc

Kd

Sh

Hh

Pc

y

e−mhs
w1

Wc1

+
−

v

Pc
+
Fig. 3.

Sampled-data repetitive control system with IMC structure.

Fc

Sh

Kd

Hh

e

Pc

n

Wc2
w2

e−mhs
w

+ e
−

Wc1
Fc

Fig. 4.
integer.

Sh

Kd

Hh

Pc

2π/h 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
S = 1 − P c H h Kd S h F c .
To solve our problem, we use a weighting function Wc1
which is stable and lowpass. By using this weighting function, we optimize the following objective function(2)

where the norm is the L2 -induced norm of the sampled-data
error system which is equivalent to the H ∞ norm of thelifted system [11]. Roughly speaking, if this is sufficiently
small, then we have Pc Hh Kd Sh Fc ≈ e−mhs in low frequencies determined by the cut-off frequency of Wc1 . Thenthe sensitivity function S can be small at {jωn }n=1,2,...,N .
The corresponding error system is shown in Fig. 4. The
optimal filter, called sampled-data H ∞ filter, can be obtained
numerically by the fast-sampling method [19], [31] or H ∞discretization [32].
B. Robust Stability Conditions
We consider uncertainty in the plant Pc . Let D := {∆ ∈B(L2 ) : k∆k∞ ≤ 1}, where B(L2 ) is the set of the linear
bounded operators in L2 . We have the following lemma [11].
Lemma 1: Assume Kd is stable.
1) The control system shown in Fig. 3 for any Pc of theform (multiplicative perturbations) Pc = Pc (1+∆Wc2 )where ∆ ∈ D and Wc2 is a stable weighting function,is internally stable if
kPc Hh Kd Sh Fc Wc2 k∞ < 1.

Error system for robust repetitive controller.

C. Robust Control

Error system for approximation of e−mhs where m is a positive

J1 (Kd ) = k(e−mhs − Pc Hh Kd Sh Fc )Wc1 k∞

Fig. 5.

(3)2) The control system shown in Fig. 3 for any Pc of theform (additive perturbations) Pc = Pc + ∆Wc2 where∆ ∈ D, and Wc2 is a stable weighting function, isinternally stable if kHh Kd Sh Fc Wc2 k∞ < 1.

Here we formulate a robust controller design problem inour repetitive control framework. For simplicity, we consider
only the case of multiplicative perturbations, although analmost parallel discussion applies to the case of additive
perturbations.
To relate the robust stability condition (3) with our objective function J1 in (2), we define the following two-blockobjective function
 −mhs(e− Pc Hh Kd Sh Fc )Wc1
J2 (Kd ) =(4)
Pc Hh Kd Sh Fc Wc2∞
where Wc1 and Wc2 are stable weighting functions. It should
be noted that the weighting functions Wc1 and Wc2 must be
chosen considering the trade-off between performance androbustness. As mentioned above, our goal is to track periodic
signals in low frequencies, and hence the weighting function
Wc1 is chosen to be a lowpass filter. On the other hand, if
Wc2 is also a lowpass filter, this will fundamentally limit the
tracking performance since the filter cannot have enough gainin low frequencies. Usually the noise n have more energy inhigh frequencies, so we choose a highpass Wc2 , in particular,
to have the property Wc2 (0) = 0. We also assume that thenorm kWc2 k∞ be much smaller than kWc1 k∞ since a large
Wc2 can seriously degrade the tracking performance. See theexample in IV-B below.
The corresponding block diagram for designing a robust
controller is shown in Fig. 5. This diagram provides an interpretation of the design as noise reduction. Continuous-timenoise n is added before A/D (Analog-to-Digital) conversion(i.e., Sh Fc ), and the filter Kd attempts to reduce the noise n
and reconstruct the input signal v. This interpretation givesa recipe for designing the analog device Fc (anti-aliasing
filter). That is, if Wc2 (s) is a highpass filter with cutoff
frequency ωc2 , then Fc (s) can be chosen as a lowpass filter
with the same (or a lower) cutoff frequency. This however
can decrease the tracking performance in high frequencies.
Another interpretation of the robust controller design isregularization [28], [29], [30]. The design in the previous
subsection to minimize (2) is a kind of inverse problem; weseek a sampled-data (delayed) inverse system Hh Kd Sh of
the uncertain continuous-time system “Fc Pc ” whose inverse
cannot be stable and causal. This type of inverse problem

Fc

Sh

Kd

+

Hh

Pd
Fig. 6. Digital implementation of sampled-data repetitive controller: Pd is
the step-invariant discretization of Pc .

+

Fc

Sh

↑M

Kd

Hh/M

Pc

This is a multirate filtering problem addressed in [21]. The
optimal filter can be numerically obtained by using thediscrete-time lifting [11], see [21]. Digital implementation
as shown in Fig. 6 is also obtained by discrete-time lifting.
Robust controller as proposed in III-C can also be designedby the following objective function −mhs(e− Pc Hh/M Kd (↑M )Sh Fc )Wc1.
J4 (Kd ) =Pc Hh/M Kd (↑M )Sh Fc Wc2∞(6)
The optimal Kd can be numerically obtained by fastsampling method [31] or H ∞ discretization [32], withdiscrete-time lifting.
IV. D ESIGN E XAMPLES

Fig. 7.

Multirate repetitive controller with upsampler ↑ M .

is called ill-posed, for which regularization is a powerfultool. Regularization limits the norm of an inverse system toobtain a robust inverse against uncertainty of a target systemor data. We can see that the objective function (4) limits
the H ∞ norm of Pc Hh Kd Sh Fc with weighting function(regularization parameter) Wc2 . Then the gain of the resulting
filter Kd is reduced in high frequencies determined by Wc2 .
See the design example in IV-B.
The optimal controller which minimizes (4) can be computed numerically by fast-sampling method [31] or H ∞discretization [32].
D. Implementation
Our sampled-data controller in Fig. 3 includes both adiscrete-time controller Kd and a continuous-time plantmodel Pc . To implement this with a digital device, weequivalently convert the sampled-data system into a discretetime system. To this end, 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 − P d Kd
This controller is illustrated in Fig. 6.
E. Multirate Filtering and Control
It is pointed out in [10] that in digital repetitive control
systems, the period of the hold device can be shorter than thatof the sampler. Considering this advantage, we can adopt a
multirate controller shown in Fig. 7 instead of the controller
in Fig. 3 In this system, ↑ M is the upsampler [33] definedby↑ M : {x[k]}∞k=0 7→ {x[0], 0, . . . , 0, x[1], 0, . . .}.
| {z }M −1

Assuming that Pc = Pc , the controller Kd is designed to
optimize the following objective function
J3 (Kd ) = k{e−mhs − Pc Hh/M Kd (↑ M )Sh Fc }Wc1 k∞ . (5)In this section, we show design examples. The nominal
plant Pc here is
Pc (s) =s−1
2s3 + 3s2 + 4s + 5.

The period L of reference signals is 20. The sampling period
h is 1, and the delay steps in designing the H ∞ filter is m =L/h = 20. The first example shows efficiency of multirate
structure of digital repetitive control. The second example
discusses robustness of repetitive controllers.
A. Multirate Control
Here we show the effectiveness of multirate filters discussed in III-E. We assume that there is no perturbation in
the plant, that is, Pc = Pc .
The weighting function is a lowpass filter with cutoff
frequency ωc = π/20, that is,
Wc1 (s) =1,
Tc s + 1

Tc =

20
.
π

The lowpass filter Fc is chosen as
1
1
, T =
.
Ts + 1
100
With these parameters, we design two controllers; thesampled-data H ∞ optimal Kd1,opt minimizing J1 in (2) andthe multirate controller Kd3,opt minimizing J3 in (5). The
Bode gain plots of these controllers are shown in Fig. 8. In
this figure, the inverse of Pc is also plotted. The controllers
well simulate the shape of |Pc (jω)−1 | in the low frequencies,
while they are attenuated near the Nyquist frequencies (π and
4π). The shapes of the filters in the high frequencies result
from the mini-max optimization with the trade-off between
mimicking Pc−1 and reducing imaging [33] due to the zeroorder hold Hh or Hh/M (and the upsampler ↑ M in the caseof M = 4).
Then, by using these filters, we simulate the repetitive
control for a rectangular reference of period L = 20. Fig. 9shows the time response of the single-rate repetitive control
system. This figure shows that there remain ripples aroundthe edges of rectangle. The time response of the multirate
repetitive control system is shown in Fig. 10. We can clearly
see that the multirate structure substantially reduces the
ripples in Fig. 9, and shows the effectiveness of the multirate
Fc (s) =

20
50
10

40
30

0
Gain (dB)Gain (dB)

20
10
0

-10

-20

-10
-30

-20
-30

-40 -1
10-40 -110010Frequency (rad/sec)

10

1

Fig. 8. Controllers; single-rate sampled-data H ∞ filter Kd1,opt (dash),multirate sampled-data H ∞ filter Kd3,opt (solid), and the inverse of Pc(dots).

1

Amplitude

0.4
0.2
0
-0.2
-0.4
-0.6

Fig. 11. Controllers; nominal controller Kd3,opt (solid) and robust controller
Kd4,opt (dash).

Tc s
20, Tc =.
(7)
Tc s + 1π
The number 0.05 is a trade-off parameter between performance and robustness, see the discussion in III-C. The otherparameters are the same as ones in the previous subsection.
Fig. 11 shows the Bode gain plots of the designed filters.
While the robust controller Kd4,opt is quite close to thenominal controller Kd3,opt in low frequencies, it has much
lower gain in high frequencies. This attenuation is due to theeffect of taking account of high-frequency uncertainty (7) in
our robust design. This can also be interpreted as a result of
regularization discussed in III-C. The difference between thepeaks of two gains in high frequencies is about 10 dB.
Using these filters, we give simulation results of therepetitive control for a triangular reference signal of period
L = 20. The plant is assumed to have a multiplicative
perturbation defined bys−1
Pc (s) = Pc (s) (1 + ∆(s)) , ∆(s) =.
s+1
Fig. 12 shows the tracking error of the repetitive control
system with the nominal controller Kd2,opt . Due to the effectof the plant perturbation, the error shows persistent responseand decreases very slowly. On the other hand, by using
the robust controller Kd4,opt , the error decays much fasterand exhibits a dramatic improvement in performance. These
results show the present robust controller design is effective
against uncertainty in the plant.
Wc2 (s) = 0.05 ·

-0.8
-1
10

20

30

40

50
60
Time (sec)

70

80

90

100

Fig. 9. Time response of single-rate sampled-data H ∞ optimal repetitive
control system; delayed reference (dots) and output y (solid).

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0

1

B. Robust Control
Here we show an example of robust controller design
discussed in III-C and III-E.
We design two controllers; multirate nominal controller
Kd3,opt minimizing (5) with M = 4, and multirate robust
controller Kd4,opt minimizing (6) with the weighting function

0.6

0

10

controller when a faster hold device is available in therepetitive control system, although the sampler acts slowly.
That is, multirate filtering is efficient for digital repetitive
control.

0.8

Amplitude

010Frequency (rad/sec)

10

20

30

40

50
60
Time (sec)

70

80

90

100

Fig. 10. Output of multirate (M = 4) sampled-data H ∞ optimal repetitive
control system; delayed reference (dots) and output y (solid).

1
0.8
0.6

Amplitude

0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1

0

100

200

Fig. 12.

300

400

500
600
Time (sec)

700

800

900

1000

Tracking error by nominal controller.

1
0.8
0.6
0.4

Error

0.2
0
-0.2
-0.4
-0.6
-0.8
-1

0

100

200

Fig. 13.

300

400

500
600
Time (sec)

700

800

900

1000

Tracking error by robust controller.

V. C ONCLUSION
In this article, we have proposed a novel repetitive control
by sampled-data H ∞ filters. The controller has an IMC
structure and is designed by solving a one-block sampleddata H ∞ control problem, which is equivalent to a sampleddata H ∞ filtering problem. We have also proposed a robust controller design, which is described by a two-block
sampled-data H ∞ control problem. We have discussed a
relation between the robust controller design and noise
reduction or regularization. Design examples show the effectiveness of our method.
R EFERENCES[1] S. Hara, Y. Yamamoto, T. Omata, and M. Nakano, “Repetitive control
system: a new type servo system for periodic exogenous signals,” IEEE
Trans. Automat. Contr., vol. 33, pp. 659–668, 1988.
[2] T. Inoue, M. Nakano, T. Kubo, S. Matsumoto, and H. Baba, “Highaccuracy control of a proton synchrotoron magnet power supply,” inProc. of IFAC 8th World Congress, 1981, pp. 216–221.
[3] Y. Yamamoto, “Learning control and related problems in infinitedimensional systems,” in Essays on Control: Perspectives in the
Theory and its Applications, H. L. Trentelman and J. C. Willems,
Eds. Birkhauser, 1993.
[4] L. Cuiyan, Z. Dongchun, and Z. Xianyi, “A survey of repetitive
control,” in Proc. of IEEE/RSJ Intl. Conf. on IROS, 2004, pp. 1160–
1166.
[5] B. A. Francis and W. M. Wonham, “The internal model principle forlinear multivariable regulators,,” J. Appl. Math. Optimiz., vol. 2, pp.
170–194, 1975.

[6] Y. Yamamoto and S. Hara, “Relationships between internal andexternal stability for infinite-dimensional systems with applications
to servo problem,” IEEE Trans. Automat. Contr., vol. 33, no. 11, pp.
1044–1052, 1988.
[7] S. Hara, T. Omata, and M. Nakano, “Synthesis of repetitive control
systems and its application,” in Proc. of 24th IEEE CDC, 1985, pp.
1387–1392.
[8] K.-K. Chew and M. Tomizuka, “Digital control of repetitive errors
in disk drive systems,” IEEE Control Syst. Mag., vol. 10, no. 1, pp.
16–20, 1990.
[9] S. Hara, M. Tetsuka, and R. Kondo, “Ripple attenuation in digital
repetitive control systems,” in Proc. of 29th IEEE CDC, 1990, pp.
1679–1684.
[10] H. Fujimoto, F. Kawakami, and S. Kondo, “Multirate repetitive control
and applications,” in Proc. of ACC, 2003, pp. 2875–2880.
[11] T. Chen and B. Francis, Optimal Sampled-Data Control Systems.
Springer, 1995.
[12] Y. Yamamoto, “Digital control,” in Wiley Encyclopedia of Electricaland Electronics Engineering, J. G. Webster, Ed. John Wiley & Sons,
1999.
[13] A. Langari and B. A. Francis, “Sampled-data repetitive control systems,” in Proc. of ACC, 1998, pp. 3234–3235.
[14] A. Langari, Sampled-Data Repetitive Control Systems. PhD dissertation, 1997.
[15] H. Ishii and Y. Yamamoto, “Periodic compensation for sampled-data
H ∞ repetitive control,” in Proc. of 37th IEEE CDC, 1998, pp. 331–
336.
[16] G. O. C. Hean, M. Nagahara, and Y. Yamamoto, “Repetitive control
via sampled-data H ∞ control,” in Proc. of IFAC Workshop on Periodic
Control Systems, 2004, pp. 561–565.
[17] H. Fujioka, Y. Kobayashi, and H. Hayakawa, “A design of sampleddata H∞ repetitive control,” Trans. of the ISCIE, vol. 19, no. 11, pp.
413–417, 2006 (in Japanese).
[18] M. Morari and E. Zafiriou, Robust Process Control. Prentice Hall,
1989.
[19] P. P. Khargonekar and Y. Yamamoto, “Delayed signal reconstruction
using sampled-data control,” in Proc. of 35th IEEE CDC, 1996, pp.
1259–1263.
[20] H. Ishii, Y. Yamamoto, and B. A. Francis, “Sample-rate conversion
via sampled-data H ∞ control,” in Proc. of 38th IEEE CDC, 1999,
pp. 3440–3445.
[21] M. Nagahara and Y. Yamamoto, “A new design for sample-rate
converters,” in Proc. of 39th IEEE CDC, 2000, pp. 4296–4301.
[22] K. Kashima, Y. Yamamoto, and M. Nagahara, “Optimal wavelet
expansion via sampled-data control theory,” IEEE Signal Processing
Lett., vol. 11, no. 2, pp. 79–82, 2004.
[23] M. Nagahara and Y. Yamamoto, “Optimal design of fractional delay
filters,” in Proc. of 42th IEEE CDC, 2003, pp. 6539–6544.
[24] H. Kakemizu, M. Nagahara, A. Kobayashi, and Y. Yamamoto, “Noise
reduction of JPEG images by sampled-data H ∞ optimal ε filters,” inProc. of SICE Annual Conf., 2005, pp. 1080–1085.
[25] M. Nagahara, T. Wada, and Y. Yamamoto, “Design of oversamplingdelta-sigma DA converters via H ∞ optimization,” in Proc. of IEEEICASSP, vol. III, 2006, pp. 612–615.
[26] Y. Yamamoto, “A new approach to signal processing via sampled-data
control theory,” Australian J. Electrical and Electronics Engineering,
vol. 2, no. 2, pp. 141–149, 2005.
[27] M. Nagahara, M. Ogura, and Y. Yamamoto, “A novel approach torepetitive control via sampled-data H ∞ filters,” in Proc. of ASCC,
2009 (submitted).
[28] C. M. Bishop, Pattern Recognition and Machine Learning. Springer,
2006.
[29] A. Neumaier, “Solving ill-conditioned and singular linear systems: A
tutorial on regularization,” SIAM Review, vol. 40, pp. 636–666, 1998.
[30] G. C. Goodwin, “Inverse problems with constraints,” in Proc. of IFAC
15th World Congress, 2002, pp. 35–48.
[31] Y. Yamamoto, A. G. Madievski, and B. D. O. Anderson, “Approximation of frequency response for sampled-data control systems,”
Automatica, vol. 35, pp. 729–734, 1999.
[32] L. Mirkin and G. Tadmor, “Yet another H ∞ discretization,” IEEE
Trans. Automat. Contr., vol. 48, no. 5, pp. 891–894, 2003.
[33] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Prentice Hall,
1993.