IFAC Workshop on Adaptation and Learning in Control andSignal Processing, and IFAC Workshop on Periodic Control
Systems, Yokohama, Japan, August 30 – September 1, 2004

REPETITIVE CONTROL VIA SAMPLED-DATA
H ∞ CONTROLGeorge Ooi Chong Hean ∗,1 Masaaki Nagahara ∗∗,2
Yutaka Yamamoto ∗∗,3∗∗∗
Texas Instruments
Department of Applied Analysis and Complex DynamicalSystems, Graduate School of Informatics, Kyoto
University, Kyoto 606-8501, JAPAN

Abstract: Repetitive control has been implemented in various industrial applications to allow systems to track or reject unknown periodic signals with a ﬁxedperiod. Repetitive controllers are usually implemented by digital computers, inwhich undesired ripples may occur between sampling instants. In this work, weapproach this problem via sampled-data H ∞ control. We show that our approachsigniﬁcantly attenuates the ripples without sacriﬁcing the tracking performance.
Keywords: Repetitive control, Sampled-data control, Ripple attenuation

1. INTRODUCTION

ripples between the sampling instants (Nakano etal., 1989; Hara et al., 1990).

Repetitive control (Nakano et al., 1989) is a control to be designed for tracking periodic reference signals and for rejecting periodic disturbances. It has been widely implemented in various industrial applications. For example, highaccuracy tracking in magnet power supply for
proton synchrotron (Inoue et al., 1981), controlof robot manipulators which carry out repetitive
tasks (Omata et al., 1987), and so forth.

There are several literatures on digital repetitive
control systems investigating the ripples phenomena (Franklin and Emami-Naeini, 1986; Urikuraand Nagata, 1987; Hara et al., 1990). These researches are, however, in the discrete-time. Totackle the ripple phenomenon, we have to consider the intersample behavior. Therefore, westrongly propose to introduce sampled-data control (Chen and Francis, 1995) into repetitive control. Sampled-data control can take the intersample behavior into account, and hence the ripples
in repetitive control systems can be naturallyconsidered. From this point of view, several articles have studied repetitive control via sampleddata control (Langari and Francis, 1998; Ishii andYamamoto, 1998). Langari and Francis propose
to design the controller by using the induced
power-norm (Langari and Francis, 1998), whileIshii and Yamamoto take the L2 induced norm
(i.e., H ∞ norm) and the optimal ﬁlter is periodically time-varying (Ishii and Yamamoto, 1998).
Based on these researches, we propose a design

From a practical point of view, it is easier to implement the repetitive controllers digitally. Digital
repetitive controllers can make the steady-state
tracking error vanish at the sampling instants,
due to the internal model principle (Nakano etal., 1989). However, the output of the plant isstill continuous-time, and it is well-known that
digital repetitive control may result in undesired
1joji@acs.i.kyoto-u.ac.jp
nagahara@i.kyoto-u.ac.jp
3 yy@i.kyoto-u.ac.jp; author to whom all correspondence should be addressed.
2561

er
−W (s)F (s)lim ed [k] = 0.

ewhk∈Z
k→∞

Q[z]

K[z]

HhP (s)Remark 2. The sampled signal of e(t), that is,e(kh), k = 0, 1, 2, . . . , will not vanish as k →∞. Let the reference signal r(t) be a sinusoidal
function ejω0 t , ω0 := 2π/T , then the steady state
response of e(t) is given as follows (Yamamoto andAraki, 1994):

yFig. 1. Sampled data repetitive control system
method to ﬁnd the H ∞ optimal linear timeinvariant controller via sampled-data H ∞ control. Although sampled-data systems are generally inﬁnite-dimensional, we introduce the fastsampling/fast-hold approximation (Yamamoto etal., 1999; Yamamoto et al., 2002) to make the system ﬁnite-dimensional. By a numerical example,
we show that our repetitive controller attenuates
the intersample ripples, and simultaneously maintains excellent tracking performance.

e(t) = ejω0 t −∞1 P (jωn )Hh (jωn )h n=−∞

 0 h]F (jω0 )ejωn t ,× Sd [ejω0 h ]K[jω(2)where := K[z]Q[z],
K[z]
−1
Sd [z] := 1 + K[z]G,d [z]
Gd [z] := (Sh P F Hh )[z],1 − e−sh,s2πnωn := ω0 +,h2. PROBLEM FORMULATION

Hh (s) :=Consider the repetitive control system shown in
Figure 1, where the signal r is the reference input
and y is the plant output. In this block diagram,P (s) is a continuous-time plant to be controlled,F (s) an anti-aliasing ﬁlter, and W (s) a frequency
weight of the error e to be estimated. We assume
that the plant P (s) is strictly proper, the ﬁlter
F (s) and the weighting function W (s) are strictly
proper and stable. We have two controllers in thediagram; Q[z] is the repetitive controller:

n = 0, ±1, ±2, . . . .

Substituting t = kh, k = 0, 1, 2, . . . into theequation (2), we obtaine(kh) = ejωkh∞F (jω0 ) n=−∞ P (jωn )Hh (jωn ) jωkh− ∞.
en=−∞ F (jωn )P (jωn )Hh (jωn )1(1)zM − 1and K[z] is a digital controller (assumed to beproper and linear time-invariant) to stabilize thefeedback system and to improve the performance.
While the controllers are discrete-time systems,
the plant is a continuous-time one, and hence wehave the sampler Sh and the zero-order hold Hhwhere h is the sampling time.

If

Q[z] =F (jω0 )∞∞n=−∞ P (jωn )Hh (jωn )= 1,Fn=−∞ (jωn )P (jωn )Hh (jωn )(3)then we have e(kh) = 0. For example, we havee(kh) = 0 if F (s) = 1, that is, we do not use any
anti-aliasing ﬁlter, or if the plant is fully bandlimited up to the Nyquist frequency ωN := π/h,that is,

Our objective here is toP (jω) = 0,

• make the tracking error e(t) vanish at thesampling instants t = kh (k ∈ Z) as k → ∞,• attenuate the intersample ripples between
the sampling instants.

|ω| ≥ ωN ,and the frequency of the input signal is ω0 < ωN .
However, in general, the assumption (3) will hold,and hence, e(kh) will not vanish as k → ∞.
In order to attain the latter objective, we design
the controller K[z] via sampled-data control. Our
design problem is as follows:

The former is achievable by the repetitive controller (1) because of the internal model principle (Nakano et al., 1989). Deﬁne the ﬁltered andsampled error ed [k] := (Sh F e)[k] = (F e)(kh),k = 0, 1, 2, . . . , then we have the following theorem:

Problem 3. Let T be the system from r to ew
in Figure 1. Find the controller K[z] (properand linear time-invariant) which stabilizes thefeedback system and minimizes the L2 inducednorm of T :Theorem 1. Assume that the controller K[z] internally stabilizes the feedback system shown in
Figure 1, and M = T /h is a positive integer. Then,
we haveT  := supw∈L2562

T w
.
w

ew

ed

r
G(s)

ef

Sh/Newr

T

Hh/NrduFig. 3. Fast-sampling/fast-hold approximation
HhShQ[z]K[z]ukedrdFig. 2. Sampled data system TGdN [z]This problem is a sampled-data H ∞ optimal control problem. In the next section, we discuss how
we should solve this problem.

3. SAMPLED-DATA H ∞ CONTROL FOR
RIPPLE ATTENUATION

Q[z]Fig. 4. Discretized system TN

Deﬁne the state-space realizations of P (s), F (s)and W (s) as


A B
AF BF
P (s) :=, F (s) :=
,C 0
CF 0


AW BW
W (s) :=.
CW 0

• discretize the continuous-time output by asampler with sampling period h/N ,The feedback system in Figure 1 can be arranged
as a generalized form as in Figure 2 with⎡⎤
Ac B1 B2
W (s) −W (s)P (s)G(s) ==: ⎣ C1 0 0 ⎦F (s) −F (s)P (s)C2 0 0whereK[z]where N is a positive integer (see Figure 3).
With large N , the discretized signals may be
a good approximation of the continuous signals,
and hence we can control intersample ripples.
Moreover, we have the following theorem.
Theorem 4. For the sampled-data system T inFigure 2, there exist discrete-time systems {TN :N = 1, 2, . . . } such that⎤A0 0Ac = ⎣ −BF C AF 0 ⎦ ,−BW C 0 AW⎤⎡⎡ ⎤0B
B1 = ⎣ BF ⎦ , B2 = ⎣ 0 ⎦ ,BW0

0 0 CW
C1 =, C2 = 0 CF 0 .
C 0 0⎡

lim T  = TN .

N →∞The proof is given in (Yamamoto et al., 1999).
The discretized system TN is obtained as follows(see Figure 4).

Assume that (Ac , B2 ) is stabilizable and (C2 , Ac )is detectable. Our objective here is to minimize
the H ∞ norm of T (the system from r to ew ) inorder to attenuate the intersample ripples.
To solve our sampled-data H ∞ problem, we employ the fast-sampling/fast-hold method (Yamamotoet al., 1999; Yamamoto et al., 2002). The fastsampling/fast-hold technique is a method for approximating the performance of sampled-data systems. The procedure is as follows:• discretize the continuous-time input by ahold with sampling period h/N ,TN = Fl (GdN , KQ),⎡⎤
Ad B1dN
Bd
GdN [z] = ⎣ C1dN D11dN D12dN ⎦C200

where563

ed

Referencerd

10.8

GdN [z]0.6

Q[z]Amplitude0.4

K[z]

0.2
0−0.2
−0.4

Fig. 5. Standard discrete-time system for controller K[z]−0.6−0.8

Ac h−10

Ad := e , h
Bd :=eAc t B2 dt,
0B1dN :=N −1
B1NAdN⎤⎡. . . B1N ,0.8

1

Let the plant P (s), the anti-aliasing ﬁlter F (s)and the weight W (s) be as
1,(s + 1)(3s + 1)
20
F (s) =,
0.01s + 1
1
W (s) =,
2s + 1
P (s) =N −1
C1 AdN⎢⎢
D11dN := ⎢⎣0.6

Fig. 6. Reference signal

N −2AdNB1N0
C1 B1N..
.

0.4
Time [sec]C1⎢ C1 AdN ⎥⎥⎢C1dN := ⎢⎥,..
⎦⎣.
⎡0.200..
.

⎤... 0... 0⎥⎥. . .. ⎥ ,. .⎦respectively. We take the periodic reference signalr as follows:⎧0 ≤ t < 1/4,
⎨ 4t,r(t) = −4t + 2, 1/4 ≤ t < 3/4,⎩
4t − 4, 3/4 ≤ t < 1N −2
N −3
B1N C1 AdN
B1N . . . 0
C1 AdN
⎤0⎥
⎢
CB
12N⎥⎢
⎥⎢..
⎥,⎢
.
D12dN := ⎢⎥−2⎥⎢ N⎦⎣
C1 AidN B2N⎡of period T = 1.0 [sec] (see Figure 6). The
sampling period is chosen to be h = 0.05 [sec].
Thus, there are M = 20 sampling instants ineach period of the reference. We employ the fastsampling/fast-hold method with the discretizing
rate N = 4.

i=0AdN := eAc h/N , h/N
B1N :=eAc t B1 dt,
0 h/NeAc t B2 dt.
B2N :=Figure 7 illustrates transient response of the output y(t) (0 ≤ t ≤ 10) of the system designed
via sampled-data H ∞ design and the conventional
one. It is obvious that the proposed design showsa better transient response compared to the conventional one. Figure 8 shows the steady stateresponse (20 ≤ t ≤ 22). We can see that the conventional control shows intersample ripples, while
our control attenuates the ripples better. To see
the diﬀerence precisely, we depict the steady state
error in Figure 9, which proves that the proposed
method eﬀectively attenuates the intersample ripples, and also shows a better tracking performancein high-frequency.

0In Figure 4, the controller consists of Q[z] andK[z]. Our problem is therefore a constrained controller design, which is diﬃcult to solve. We haveassumed that K[z] is proper, and hence we shift
Q[z] to the upper block as shown in Figure 5,
and we ﬁnd K[z] which is proper and linear timeinvariant. As a consequence, we can obtain the
optimal controller via a standard discrete-time
H ∞ control.
4. DESIGN EXAMPLES

5. CONCLUSION

In this section, we present a numerical example
to illustrate the eﬀectiveness of our method. We
compare our method with a conventional one,that is, a discrete-time LQR design (Nakano etal., 1989).

We have proposed a sampled-data H ∞ designfor attenuating intersample ripples in sampleddata repetitive control. We demonstrate the ef-564

REFERENCES

Reference and Output Signals
25

Chen, T. and B. Francis (1995). Optimal SampledData Control Systems. Springer.
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Hara, S., M. Tetsuka and R. Kondo (1990). Ripple
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pp. 3724–3729.

20

Amplitude

151050−5−1001234Time [sec]567Fig. 7. Transient response: reference (dot), proposed (solid) and conventional (dash)

Steady State Response
10.8
0.60.4
Amplitude0.2
0−0.2−0.4
−0.6
−0.8
−120

20.5

21Time [sec]21.5

22Fig. 8. Steady state response: reference (dot),proposed (solid) and conventional (dash)

Steady State Error
0.1
0.08
0.06

Amplitude

0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
20

20.5

21
Time [sec]21.5

22Fig. 9. Steady state error: proposed (solid) andconventional (dash)fectiveness of our method by comparing it withan existing method.

565