A note on convergence property of iterative learning controller with respect to sup norm

Pergamon

Aumna~ica, Vol. 33, No. 8, pp. 1591-1593, 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain ooO5-1098/97 $17.00 + 0.W

@?J 1997

PII: sooos-lo98(97)ooo68-x

Technical Communique

A Note on Convergence Property of Iterative Learning Controller with Respect to Sup Norm* HAK-SUNG Key Words-Learning

LEEt and ZEUNGNAM

BIENS

control; convergence; norms.

Abstract-We

discuss some undesirable phenomena associated with the A-norm that is used in the proof of convergence of the iterative learning controller (ILC). Specifically, we first report that huge overshoot may be observed in the output trajectory error, even though the applied learning algorithm has been proved to be exponentially convergent with respect to the A-norm. This is because of the way in which the A-norm converges differs from that of the sup norm. We then investigate the relation between the A-norm and the sup norm. Finally, we present a simple method to remedy the undesirable feature of the conventional ILC. 0 1997 Elsevier Science Ltd.

A-norm. Therefore, for real-world operation of an ILC, it is necessary to investigate the behavior of the ILC in the sense of the sup norm, From the A-norm definition (2), it is easily observed that (3) for A >O. Thus the A-norm is equivalent to the sup norm (Naylor and Sell, 1971). Therefore the conditions of convergence can be obtained equivalently in the sup norm. However, it is not certain that the routes of convergence are equivalent. In fact, we can observe a huge overshoot in the sense of the sup norm, even though exponential convergence is guaranteed with respect to the A-norm. In this paper, we discuss this in more detail, and investigate the relation between the A-norm and the sup norm. Furthermore, we propose a simple learning algorithm that resolves this undesirable phenomenon. In the following, the subscripts k and d, as in uk(t) and uJt). are employed to denote the iteration number and to indicate the desired time function respectively.

1. Introduction The A-norm defined in Arimoto et al. (1984) has been adopted in many papers (Hauser, 1987; Bondi et al., 1988; Heinzinger et al., 1992; Hwang et a/., 1993) as the topological measure in the proof of the convergence property for a newly proposed iterative learning controller (ILC). More specifically, the learning control input uk(t) E C[O, T] is proved to be convergent to the desired control input ud(t) E C[O, T] by showing that the inequality II%%+1(.)llASP Il~u/J*)IIA

(1)

2. Main results Consider the following sequence of continuous functions:

holds, where 6~~ = ud(t) - uk(t) and p < 1. C[O, T] denotes the vector space of all continuous functions defined on [0, T]. Here, the formal definition (Arimoto et al., 1984) of the A-norm for a function f: [0, T] + R” is given by

j%(r), h(t), h(t), .

1

(4)

where fk(.) E C[O, T], and for each k = 0, 1,2,. . , it is assumed that

where Ilf(t)lL

d maxi,,,,

If;Wl.

Ilfk+l(‘)llA

In real operation of an ILC, including computer simulation and real implementation, it is natural to apply the learning algorithm until a certain measure of error is within the prespecified allowable bound E. Since the control objectives may depend on the type of error, however, the A-norm may not be a satisfactory measure of error in certain types of application. When the control and/or state variables are specified in terms of voltage, for example, the absolute magnitude (i.e. the sup norm) of the variables is of major concern in protecting hardware components from failure. In general control applications such as tracking, the sup norm is more suitable for error measure on performance than the

(5)

=&ll,i(‘)I/A

with pk < 1. The condition p* < 1 implies that llfk(.)llA monotonically decreases as k + m. Now let t: be the time when ecAr Ilfk(t)llm takes its maximum value, i.e.

Ilh(~)llA =ozETee-h’ IlhWlL

= e-“‘: IlM~)ll-,

P-5)

and let tf be the time when Ilfk(t)III takes its maximum value, i.e. a,“n7 IlM)llx

*Received 10 October 1995; received in final form 10 March 1997. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato. Corresponding author Zeungnam Bien. Tel. +824286934024; Fax +8242869 3410. t Information Technology Laboratory, LG Corporate Institute of Technology, 16 Woomyeon-Dong, Seocho-ku, Seoul, 137-140 Korea. $ Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, 373-l Kusong-dong, Yusong-ku, Taejeon, 305701 Korea.

= Ilh(mcv

(7)

By using tk, and t$, we can establish the following relation between the A-norm and the sup norm. Lemma 1. Let fk(.) E C[O, T], k = 0, 1, 2, .

IlM~h

5 e-*‘i IlM%

(8)

Ilfk(‘h

z

(9)

@”

t: s t:. 1591

If A > 0 then

Ilf*(t%

WY

1592

Technical Communique

Proof. One may easily show that (8) and (9) hold by observing that Ilf*(.)ll~ =“F:xre-“’

where A=[_;

_;],

B=[;],

C=[O

11.

(24

Il.Mr)llr = eWAGIlh(rk,)ll= Let the desired output trajectory be given as

5 em”‘: “~2~ Ilfd~)lh

= ,~nrTe-“’

Il.Mr)ll=~ IIfk(r)Il=~e-“r~

IM%

(12)

And (10) is obvious from (8) and (9). This completes proof.

the 0

By using Lemma 1, it is possible to obtain the behavior of the iterative learning controller with respect to the sup norm as follows. Theorem 1. Let {fk(r)}& be a sequence of continuous functions defined on [0, T], and let t: and t: be the times when e-“’ Ilh(t)]lr and Ilfk(t)llx respectively take their maximum values. If = Pk Il.h(~)llA

Ilfkfl(~h

with pk < 1, for k = 0, 1,2,.

(13)

, then

l~f~+,(t~“)ll~~~e*‘~+‘-‘X)

Ilf*(r%

(14)

llf*+,(r~+‘)lll~~e^‘~+‘-~~

Ilfk(r:)lix.

(1%

Proof: From (8) and (9) in Lemma 1, one finds that ll~+,(r!+‘)II~~e*~+’

Ilf*+,(.)llA

1 <-ehW~G) -Pk

yd(r) = 12t(l -t),

01)

=ieArk+’

Ilfk(.)ll*

Ilfk(&lll,

(16)

This proves (14). And (15) can be proved in a similar way. 0 From Theorem 1, we can obtain the lower and upper bounds of respect to maxacrrT ilf&)ll- with maxcsrsr IIXdr)lll. Define k-l

r&, P ri + C (r; - r;) - rl’, ,=I Ir-l r& 4 rZ+ C (r: - r;) - r(:, r=l

(17)

05 t 5 1.

(25)

Based on the result in Hwang et al. (1993), suppose that the following learning law is applied: &+1(r) = u/Jr) + I]%(r)

- R8Y/&)l,

(26)

where Syk(r) = y,,(r) - yk(t) denotes the error in the output trajectory at the kth iteration. If IlI,-rcBII,
(27)

then it is already known (Hwang et al., 1993) that the update law (26) makes the error between yk(t) and yd(t) approach zero as k + m. Now let us assume that I and R are chosen as l/1.3 and 40 respectively. Note that the large value of R does not affect the condition of convergence (27). Figure 1 shows that there is a huge overshoot in the sense of the sup norm, even though IIyd(t) - yk(t) II A monotonically decreases as shown in Fig. 2 with A = 25. Figure 2 indicates that, as far as the hnorm is concerned, II yd(t) - yk(t)llA =O for k = 25, whereas rnaxasrsr IIy&t) - yk(t)IJx is still huge for k = 25. One reason for this huge overshoot is that the amount of updated input is very large. In the learning law (26), the amount of updated input depends on I’, R and the error. Since the iterative learning control is an open-loop-type controller, it is possible to have a large error at the terminal time. This large error becomes a large amount of the updated input with the help of P and R. And, at the next iteration, this large input makes the error larger near the terminal time. But, although the error becomes larger and larger near the terminal time, its A norm still decreases, as shown in Fig. 2, because of the nature of the A norm defined in (2). Note that, in calculating the A norm with a large value of A, the errors near the terminal time are extremely less weighted than those near the start time. In general, for the proof of the convergence property of ILC, it is sufficient to prove the existence of a A that satisfies (l), but not much is known of the value of A. So, if we apply an ILC that is proved to converge in the sense of the A norm, with a possibly large value of A, we may have a huge tracking error, which is not allowable in practice or even for computer simulation. Since the huge tracking error comes from the open-loop nature of ILC, one way of resolving this undesirable phenomenon is to make the amount of updated input small near at the terminal time. This can be accomplished by a

(18)

PknPPoPl”.Pk-l,

(19)

and let Mk = ,,z~~ Ilf&)IL

= Il.h(r~)ll-L

for k =

0, L2,.

. (20)

Then 1 1 “‘PM&~,, 5 Mk 5 T e”6Mo. Te Pn

Pn

(21)

As (21) shows, there may exist a huge value of Ilfk(r)lJr depending on rfb,, rky,, and A, even though Ilfk(.)llh monotonically decreases.

IO'lod-

Example system:

1. Consider

the following

i(r) = Ax(r) + h(r), y(r) = Cx(r),

linear

time-invariant (22) (23)

Fig. 1. Trend of maximum values of IIyd(t) -yk(t)ll, R=40.

when

Technical Communique 0.01,

I

iI

o.ooo

.I

o.om

_. i

1

_

Theorem 2. Let (_&(t)}be a sequence of continuous functions defined on [0, T], and let tt be the time when e-*’ Ilfk(t)llm takes its maximum value. If

Ilh+1(~)ll* < IlM.)ll*

(30)

then “z;Tk Ilfk+&Nlm *

‘“=k

*

Ilh(t)ll-

= Ilhmll==.

(31)

Proof. If we set T = t: in Lemma 1, it is obvious that ,.,

maTk Ilh0)ll= *

5

Fig. 2. Trend

10

20

25 xofnemibn

30

35

40

45

of A-norm values of IIyd(t) -y&)11* h=25whenR=40.

54

r(t)cB Ii_< I, t E [o, 7-l.

Ilf*+~(~)ll~ < Ilh(~)llr

e-“*+I IIR+drl+‘)ll~~

= e-“‘: IlMt~)II=.

with

(33)

From (33)

simple modification of the learning gain I. To be specific consider again the learning law (26), where F is constant. We can replace F with a function of time, F(t), as long as r(t) satisfies the following convergence condition: 11 i -

(32)

Let t$+’ be the time when II_. fk+,(t)llx takes its maximum ~. value-on [O,ti]. Then

.I

0

= Ilhmr.

,,F2;k llh-+,(t)Ilr A

= Ilfk+l(t:+‘)l~=
Ilfk(t:)IIl.

(34)

= ,)Fz;k IlM)ll=I

(35)

Since rk,2 t:+‘, it follows from (34) that

(28) ,,ZEaxkIlf+d~)ll= I

One candidate for r(t) has the form

r(t) = e-?‘I,

(29)

where y > 0. From (26) and (29) with y > 0, one can easily find that the amount of updated input near at terminal time would be small. Therefore, even if a large error is produced near the terminal time, it does not become the large updated input. Example 2 shows that the huge overshoot can be avoided using such a simple modification as shown above. Exumpfe 2. Consider again the system in Example 1. Here we use (29) as the learning gain, with y =4. As shown in Fig. 3, the overshoot is lowered to such a degree that it is practically acceptable. Lemma 1 entails the following implication about the behavior of iterative learning with respect to the sup norm.

< Ilhm-

This completes the proof.

0

Theorem 2 implies that if t: > 0 then there exists a time interval where Ilf~(t)llx monotonically decreases. And, if then for k = 1,2,. t: = T ma*istsr IIMW monotonically decreases if and only if (Ifk(.) IIAmonotonically decreases. 3. Concluding remarks In this paper, the behavior of an interative learning law has been examined with respect to the sup norm. The possibility of huge overshoot has been shown, and a simple modification has been proposed to resolve the difficulty. Depending upon applications, the need may arise to analyze the behavior of ILC in terms of a different norm. For example, when the variables represent voltage and/or currents and the power is of critical concern, analysis with respect to L, may be appropriate.

2.5 :

:

References Arimoto, S., Kawamura, S. and Miyazaki, F. (1984) Bettering operation of robots by learning. J. Robotic Syst.

:

2-

1,123-140. g

‘-”

.* I -

1-

0.5

0

0

5

10

15

20

25 Xofrmlion

30

35

40

Fig. 3. Trend of maximum values of [[y,,(t) -yk(t)llr R=4Oandy=4.

45

50

when

Bondi, P., Casalina, G. and Gambardella, L. (1988) On the iterative learning control theory for robotic manipulators. IEEE J. Robotics Autom. 4, 14-21. Hauser, J. E. (1987) Learning control for a class of nonlinear systems. In Proc. 26th IEEE Cot@ on Decision and Control, Los Angeles, CA, pp. 859-860. Heinzinger, G., Fenwick, D., Parden, B. and Miyazaki, F. (1992) Stability of learning control with disturbances and uncertain initial conditions. IEEE Trans. Autom. Control AC-37,110-114. Hwang. D. H., Kim, B. K. and Bien, Z. (1993) Decentralized itera!rve learning control methods for large scale linear dynamic systems. hat. J. Syst. Sci. 242239-2254. Naylor, A. W. and Sell, G. R. (1971) Linear Operator Theory in Engineering and Science. Holt, Rinehart and Winston, New York.