INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, e-ISSN 1841-9844, 14(3), 359-374, June 2019.

Effect of Soft Errors in Iterative Learning Control
and Compensation using Cross-layer Approach

G.-M. Jeong, K. Lee, S.-I. Choi, S.-H. Ji, N. Dutt

Gu-Min Jeong
School of Electrical Engineering
Kookmin University, Seoul, Korea
gm1004@kookmin.ac.kr

Kyoungwoo Lee*
Department of Computer Science
Yonsei University, Seoul, Korea
*Corresponding author: kyoungwoo.lee@gmail.com

Sang-Il Choi
Department Applied Computer Engineering
Dankook University, Korea
choisi@dankook.ac.kr

Sang-Hoon Ji
Advanced Robotics Group
KITECH, Ansan, Korea
robot91@kitehc.re.kr

Nikil Dutt
Department of Computer Science
UC Irvine, CA, USA
dutt@ics.uci.edu

Abstract: In this paper, we study the effects of radiation-induced soft errors in
iterative learning control(ILC) and present the compensation techniques to make the
ILC systems robust against soft errors. Soft errors are transient faults, which occur
temporarily in memories where the energetic particles strike the sensitive region in
the transistors mainly under abnormal conditions such as high radiation, high tem-
perature, and high pressure. These soft errors can cause bit value changes without
any notification to the controller, affect the stability of the system, and result in
catastrophic consequences. First, we investigate and analyze the effects of soft er-
rors in the ILC systems. Our analytical study shows that when a single soft error
occurs in the output data from the ILC, the performance of the learning control is
significantly degraded. Second, we propose novel learning methods by incorporating
the existing techniques across the system abstraction levels in the ILC to compensate
for soft-error-induced incorrect output. The occurrence of soft errors is estimated by
using a monotonic convergence of the erroneous outputs in a cross-layer manner, and
our proposed methods can significantly reduce these negative impacts on the system
performance. Under the assumption of soft error occurrence, our analytic study has
proved the convergence of the proposed methods in the ILC systems and our simula-
tion results show the effectiveness of the proposed methods to efficiently reduce the
impacts of soft-error-induced outputs in the ILC systems.
Keywords: Soft error, iterative learning control, compensation.

1 Introduction

With the advances in the highly integrated semiconductor and microelectronic manufactur-
ing techniques, the variability of electronic devices has drastically increased in real time. On the

Copyright ©2019 CC BY-NC



360 G.-M. Jeong, K. Lee, S.-I. Choi, S.-H. Ji, N. Dutt

other hand, soft errors or transient faults induced by various sources (e.g., heat and cosmic ra-
diation) have become one of the critical threats to the system stability [4]. Soft errors, different
from hard errors, which are permanent faults in the system and cannot be corrected without
replacement, are difficult to detect and can be as critical as hard errors to the system reliability
in certain application areas [4,10,14,15,17,19,22]. The probability of soft error occurrence can
be extremely low because these errors generally occur under abnormal situations. However, a
single soft error can result in a bit value change, that is, from 0 to 1, or vice versa, in the memory
without notifications to the application if there is no error detection or correction mechanism.
Thus, depending on the bit location of the soft error, it can be critical to the stability of the
controller.

Until now, there have been several researches for fault tolerant techniques to reduce the neg-
ative impacts of the soft errors on embedded systems. In particular, cross-layer-based approaches
have been considered as promising techniques to analyze the trade-off among the performance,
power, and reliability and have been presented to maximize the reliability with minimal overheads
of the power and performance in various embedded systems [3,8,12,13,21].

Iterative learning control(ILC) systems have been widely deployed and used for various
conventional and emerging control applications [1,2,5,7,9,16,20]. A single error or an incorrect
output can cause a catastrophic consequence because these control systems can be used for
mission- and safety-critical systems. However, no research has been conducted to protect the ILC
systems from soft errors, to the best of our knowledge, while the soft error rate is exponentially
increasing with technology scaling. To this end, we analyze the effects of soft errors in the ILC
and propose compensation methods to increase the reliability of the ILC systems against soft
errors, in particular, techniques in a cross-layer manner. First, we discuss the impacts of a single
soft error on the output in the ILC systems and show that the soft error significantly affects the
performance of the learning control. For an intensive analysis, we have selected an example with
4-byte floating point number [6] and have shown that a soft error in the ILC systems can lead to
a large number of errors in the value and incorrect outputs eventually as threats to the system
stability and reliability. Indeed, a soft error can significantly reduce the speed of the learning
curve in a linear discrete-time system. Therefore, it is required to design and develop mitigation
techniques to protect the ILC systems from soft errors. Second, we propose novel compensation
methods against soft errors for the ILC systems. We have made a few assumptions about the
soft errors in the ILC systems and investigated the new detection and compensation methods. In
particular, from a monotonic convergence of the output error, we have estimated the occurrence
of the critical soft errors that significantly affect the system output and result in catastrophic
consequences. We propose four compensation methods: (i) exact rollback method, (ii) recovery
with the previous iteration value, (iii) recovery with the adjacent value, and (iv) recovery with
the desired value. In addition, the methodology and effectiveness of these four compensation
techniques have been evaluated on the basis of their advantages and disadvantages. Based on
the assumptions that we have made for the soft errors in this study, the convergence of each
method has been presented. Our extensive simulation results show that the negative impacts of
the soft errors can be successfully canceled with the proposed compensation methods.

The remainder of this paper is organized as follows. In Section 2, related works are briefly
summarized for the soft-error related problems and the ILC for discrete-time systems. In Section
3 and 4, the effect of the soft error in the ILC is discussed and a new learning method to
compensate for the effect of the soft error is introduced, respectively. An illustrative example is
described in Section 5 and the conclusion follows in Section 6.



Effect of Soft Errors in Iterative Learning Control
and Compensation using Cross-layer Approach 361

2 Related works

In this section, the preliminary results for the researches on the soft-errors and the ILC are
briefly discussed.

2.1 Soft errors

Several decades of technology scaling have brought us to a point where transistors have
become extremely susceptible to even small fluctuations in the voltage levels, slight noise in
the power supply, signal interference, and cosmic particle strikes [4]. Any of these effects can
temporarily toggle the logic value of a transistor, and therefore, it is called a transient fault
or soft error. These soft errors are not permanent and nondestructive, that is, resetting of the
device restores the normal behavior. An energetic particle such as an alpha particle, a neutron,
or a free proton, can form a diffusion region of a CMOS transistor and produce a charge that can
result in toggling of the logic value of the gates or flip-flops. This phenomenon of change in the
logic state is called a transient fault or a soft error. Soft errors can result in erroneous program
states, incorrect outputs, and, eventually, system crashes.

Soft errors have already been proved to cause significant fiscal damages [22]. For example,
SUN blamed the soft errors for the crash of their million-dollar line SUN flagship servers [14].
More recently, Hewlett Packard acknowledged that a large installed base of a 1024-CPU server
system in the Los Alamos National Laboratory has been frequently crashing owing to soft er-
rors [15]. In another incident, the soft errors brought a billion-dollar automotive factory to halt
every month [10], [17]. Further, highly integrated reliability-sensitive embedded devices such
as mobile health-care systems and anti-lock braking systems (ABSs) in the automotive engine
control units (ECUs) are significantly threatened by the exponentially increasing soft error rates
with technology scaling. Thus, it is necessary to combat the soft errors in embedded systems in
both emerging and traditional computing environments.

Recently, several selective protection techniques have been proposed in order to combat soft
errors in a cost-efficient manner by protecting the failure critical data only [11, 13, 18, 21]. For
instance, partially protected caches(PPC) [11] utilizes the knowledge of the content and device
hardware capabilities to selectively store the critical data in a more reliable hardware (e.g.,
protected cache) and non-critical data in a less reliable one (e.g., non-protected cache). In [3], a
novel memory cell design method considering robustness to soft-errors has been proposed for a
deep learning accelerator.

Cross-layer-based techniques have been presented to maximize protection at a minimal cost
by coupling and cooperating different error control schemes across the system abstraction layers
from the hardware layer to the application layer in mobile embedded systems [12]. In [8], a
cross-layer approach has been introduced to robust face recognition against soft errors. However,
these techniques have been proposed to mitigate the impacts of the soft errors on domain specific
embedded systems or general computer systems, and not control systems, in particular.

2.2 ILC for linear discrete time system

Some of the preliminary results for the ILC are briefly discussed, considering the monotonic
convergence of the output errors [5]- [9].

We consider a linear time invariant(LTI) system described by

x(i + 1) = Ax(i) + Bu(i)

y(i) = Cx(i) (1)



362 G.-M. Jeong, K. Lee, S.-I. Choi, S.-H. Ji, N. Dutt

where, u ∈ R1, x = [x1, · · · ,xn]T ∈ Rn, and y ∈ R1 are the input, the state, and the output of the
system, respectively. A, B and C are matrices of appropriate dimensions. Let ud(i), xd(i), and
yd(i) represent the input, the state, and the output corresponding to the desired trajectory. Also
let the desired output yd(i), i ∈ [σ,N + σ − 1], be given and u[i,j] := [u(i), · · · ,u(j)]T ,y[i,j] :=
[y(i), · · · ,y(j)]T . Here σ denotes the relative degree, which means that CAσ−1B 6= 0 and
CAiB = 0, i < σ − 1.

The transfer function of the system is represented as

G(z) =
β1z

n−1 + · · · + βn
zn + α1zn−1 + · · · + αn

. (2)

Here, we assume that the relative degree, σ, is known as a priori (i.e., β1 = · · · = βσ−1 = 0).
Based on the relative degree, we can obtain the input-to-output relation as

y(i + σ) = CAσx(i) + CAσ−1Bu(i). (3)

In addition, the following input-to-output mapping is used for ILC:

y[σ,N+σ−1] = Hx(0) + Ju[0,N−1], (4)

H =
[

(H0)
T , · · · , (HN−1)T

]T
,

Jd0 =



J0 0 · · · 0
J1 J0 · · · 0
...

...
...

...
JN−1 JN−2 · · · J0


 ,

where Hl = CAσ and Jl = CAσ+l−1B. The time interval for the output of interest is [σ,N +σ−1]
for minimum phase systems. Also, at every iteration, we set xk(0) = xd(0).

Considering (4), an input update law is given as

uk+1
[0,N−1] = u

k
[0,N−1] + Se

k
[σ,N+σ−1], (5)

where ek
[l,m]

= yd
[l,m]
−yk

[l,m]
and S ∈ RN×N is the learning gain matrix.

The monotonic convergence of the above ILC method is shown in the following lemma.

Lemma 1. For the uncertain system (1), if the condition

||I −JS|| ≤ ρ < 1, (6)

holds for all k, the output error ek
[σ,N+σ−1] monotonically converges to 0 as k →∞. That is,

||ek+1
[σ,N+σ−1]|| ≤ ρ||e

k
[σ,N+σ−1]||, lim

k→∞
||ek[σ,N+σ−1]|| = 0. (7)

Proof: From (4) and (5), we can obtain

ek+1
[σ,N+σ−1] = Ju

d
[0,N−1] −Ju

k+1
[0,N−1]

= Jud[0,N−1] −J(u
k
[0,N−1] + Se

k
[σ,N+σ−1])

= [I −JS]ek[σ,N+σ−1].

Taking the norms on the both sides and from (6), we can obtain (7). 2



Effect of Soft Errors in Iterative Learning Control
and Compensation using Cross-layer Approach 363

3 Effect of soft error in iterative learning control

In this section, we analyze the effect of soft errors on the ILC. It is shown that the learning
performance is significantly affected by the soft errors in a linear discrete time system.

3.1 Illustrative example of ILC

Figure 1 shows the basic structure of the ILC [1]. As shown in Figure 1 and equation (5),
the input of the next iteration is determined from the output error and input of the current
iteration. Since a large amount of memory should be used for the ILC, it can be easily affected
by the soft errors in the memory.

Figure 1: Basic ILC configuration for discrete time systems (modified from the figure in [1])

For the analysis of the effect of soft errors, let us consider an example of a linear discrete-time
single-input and single-output(SISO) systems.

Let us consider the following linear discrete-time system:

x(i + 1) =


 0 0 −0.11 0 0

0 1 0


x(i) +


 10

0


u(i)

y(i) =
[

1 0.5 0.0625
]
x(i). (8)

This system has two minimum-phase zeros (z = −0.25) and the relative degree of the system
is 1. The desired trajectory is given as follows:

yd(i) =

{
0, i = 0
0.2 sin(0.1π(i− 1)), 1 ≤ i ≤ 21.

(9)

Here, we set N = 21, u(21) = 0 and S = αI21×21 = 0.1I21×21. Since the system (8) and the
learning gain α satisfy the conditions ||J0|| >

∑N−1
i=1 ||Ji|| with 1 > 0.7361 and |1−αJ0| < 1 with

0.9 < 1, which are sufficient conditions for the convergence in [16], the convergence condition
||I − 0.1J|| < 1 is satisfied.

The input update law is expressed as

uk+1
[0,20]

= uk[0,20] + 0.1e
k
[1,21]. (10)

Figure 2 shows the outputs and inputs when there is no soft error. As shown in Figure 2(c),
the root mean square(RMS) error for the output error approaches 0 as k →∞.



364 G.-M. Jeong, K. Lee, S.-I. Choi, S.-H. Ji, N. Dutt

-0.3

-0.2

-0.1

 0

 0.1

 0.2

 0.3

 0  5  10  15  20

In
p

u
ts

i

yy1
yy2
yy3

(a) Inputs for different values of k

-0.5

-0.25

 0

 0.25

 0.5

 0  5  10  15  20

O
u

tp
u

ts

i

yyd
yy1
yy2
yy3

(b) Desired output and the outputs for different val-
ues of k

 0

 0.4

 0.8

 1.2

 1.6

 0  5  10  15  20  25  30

T
h

e
 R

M
S

 e
r
r
o

r

k

(c) RMS error for different values of k

Figure 2: Outputs and inputs when there is no soft error

3.2 Effect of soft errors to ILC

Let us assume that a soft error occurs in the memory. If we assume that there is single
bit error in the 4-byte floating point variable, the effect of the soft error is different, depending
on the bit location among the 32 bits where a single soft error occurs. Table 1 summarized the
structure of the 4-byte floating point variable [6].

Table 1: Binary format for floating point variable with single precision (32bits) [6]

Parts Sign Exponent Fraction
Number of bits 1 8 23

Table 2 presents an example of the effect of the single soft error on the 4-byte floating point
variable with a value of 0.2. As summarized in Tables 1 and 2, the soft errors in the sign bit,
exponent part, and the upper part of the fraction lead to a large number of errors introduced
and, eventually, can be critical to the control system. In real embedded systems, 1-bit error can
be overcome by ECC(Error Correction Code). However when there are 3 or more bit errors,
ECC cannot check those errors. Also the error values can be larger than that of 1-bit error.
Assuming 1-bit error in MSB can show the effect of soft-errors in ILC. In order to show the effect
of soft-errors, we assume just 1-bit error for the simulation without loss of generality.

By considering above mentioned aspects, let us discuss the effects of the soft error on the



Effect of Soft Errors in Iterative Learning Control
and Compensation using Cross-layer Approach 365

Table 2: Effect of soft error on the 4-byte floating point variable with a value of 0.2

Part Error bit Approximate vaule (≈)
Sign 1(MSB) -0.200000

Exponent

2 6.805647 × 1035
3 0.000000
4 0.000000
5 0.000003
6 0.000781
7 0.012500
8 0.800000
9 0.400000

Fraction

10 0.137500
11 0.231250
12 0.215625
13 0.192188
14 0.196094
15 0.201953
16 0.200977
17 0.199512
18 0.199776
19 0.200122
20 0.200061
21 0.199969
22 0.199985
23 0.200008
24 0.200004
25 0.199998
26 0.199999

27-32 0.200000

ILC of the previous example for system (8). For the previous example, let us assume that there
are soft errors in the memory. To analyze the effect of soft errors, we also assume the followings:

• 4-byte floating point variables are used for the storage and processing of all the data.

• 1-bit soft error occurs in yk(4) in the most significant bit(MSB) of the 4-byte memory.

• A soft error occurs at every 4-th iteration corresponding to the iteration number k.

With this setting as an example, since the MSB of the 4-byte floating number variable is
the sign bit, the sign of yk(4) changes at every 4-th iteration. The learning is performed under
these assumptions for the soft errors.

Figure 3 shows the effects of a soft error for the previous example. As shown in Figure 3, a
soft error can significantly degrade the performance of the learning control. In other words, the
soft errors reduce the learning speed and affect the convergence of the output error in the ILC.



366 G.-M. Jeong, K. Lee, S.-I. Choi, S.-H. Ji, N. Dutt

-0.3

-0.2

-0.1

 0

 0.1

 0.2

 0.3

 0  5  10  15  20

In
p

u
ts

i

yy1
yy2
yy3

(a) Inputs for different values of k

-0.5

-0.25

 0

 0.25

 0.5

 0  5  10  15  20

O
u

tp
u

ts

i

yyd
yy1
yy2
yy3

(b) Desired output and the outputs for dif-
ferent values of k

 0

 0.4

 0.8

 1.2

 1.6

 0  5  10  15  20  25  30

T
h

e
 R

M
S

 e
r
r
o

r

k

(c) RMS error for different values of k

Figure 3: Outputs and inputs when there are soft errors induced

4 Cross-layer approach to compensate for the soft error

In this section, we propose new detection and compensation methods for the soft errors in
the ILC by obtaining the monotonic convergence with the cross-layer approach, as described in
Figure 4. As summarized in Table 2, depending on the location of the soft error, its effect on
the control system varies. The errors in the upper part of the 4-byte data format are critical to
the control systems, while the other errors are negligible.

Similarly, with regard to the monotonic convergence of the output error, it is not required
to detect all the soft errors. In this study, we detect the soft error that prohibits the monotonic
convergence, and further, we compensate for the soft error with other relevant data available.

Figure 4: Compensation of soft error in a cross-layer manner

For further analysis, some assumptions are made by considering the soft error imposed on
the system as follows:



Effect of Soft Errors in Iterative Learning Control
and Compensation using Cross-layer Approach 367

• (A1) A soft error occurs in the system output, that is, yk.

• (A2) At iteration k, only one soft error occurs. Hence soft-error occurs once at iteration
k.

• (A3) There is no contiguous soft error corresponding to iteration k. In other words, when
there is a soft error at iteration k, there is no soft error at k + 1 and there may be a soft
error at k + 2.

With regard to (A1), soft errors can occur in the system, restored memories of the ILC,
or the learning controller. Different from other variables, the system output yk can have soft
errors within all the components such as the system, restored memory, and learning controller.
By considering these aspects, we make an assumption (A1).

Using (A2), we can detect the location of the soft error in the 4-byte floating point variable.
Further, using (A3), we can derive the monotonic convergence with the output error of the
previous iteration.

It should be noted that depending on the compensation method used, we can neglect or
alleviate the related assumptions.

4.1 Detection of critical soft errors using cross-layer approach

If condition (6) is satisfied, ||ek+1
[σ,N+σ−1]|| ≤ ρ||e

k
[σ,N+σ−1]|| should be guaranteed for some

ρ < 1.
However, as shown in Figure 3, and Tables 1 and 2, depending on the location of the soft

error, the monotonic convergence cannot be guaranteed.
Among all the soft errors, we only detect those errors affecting the monotonic convergence.

Therefore, when ||ek+1
[σ,N+σ−1]|| ≥ ||e

k
[σ,N+σ−1]||, we can conclude that a critical soft error has

occurred. After detecting a soft error, we can find the location and compensate for the soft
error.

Considering these facts, we propose a soft error detection algorithm to find the location of
error j and iteration k for the ILC as follows :

Detection of critical soft errors in a cross-layered manner

• Step 1: For the learning curve, validate whether ||ek||∞ ≥ ||ek−1||∞.

• Step 2: If the error ||ek||∞ is larger than or equal to ||ek−1||∞, it is determined that a
critical soft error occurs at iteration k.

• Step 3: Determine j, which makes ||ek(j)|| = ||ek||∞, that is, the location of the largest
value at iteration k.

The following lemma shows that we can detect the location of the soft error with the as-
sumptions (A1)-(A3).

Lemma 2. (Determining time j and iteration k, where soft error occurs. )
Let us assume that a soft error satisfying (A1)-(A3) occurs in the process of the ILC in the

uncertain system (1). Using the above algorithm, the location of the soft error, that is, time j
and iteration k can be determined.



368 G.-M. Jeong, K. Lee, S.-I. Choi, S.-H. Ji, N. Dutt

Proof: If there is no soft error, (7) holds for ek and ek−1. However, if there is a critical soft
error, we can obtain ||ek|| ≥ ||ek−1||. Thus, we can conclude that a critical soft error occurs at
iteration k. 2

When there is a critical soft error at iteration k, let us assume that the soft error occurs in
j, that is, ek(j) contains the soft error.

Let us denote ẽk(j) as the real output error when there is no soft error, and ẽk by replacing
ek(j) with ẽk(j) in ek. From the definition of the ∞-norm, we can obtain the following equations:

||ek||∞
= max{||ek(σ)||, · · · , ||ek(j)||, · · · , ||ek(N + σ − 1)||} (11)
||ẽk||∞

= max{||ek(σ)||, · · · , ||ẽk(j)||, · · · , ||ek(N + σ − 1)||} (12)

Further, according to the relationship among ek, ẽk, and ek−1, the following equations hold:

||ek||∞ ≥ ||ek−1||∞, ||ẽk||∞ < ||ek−1||∞. (13)

Therefore, to satisfy (13), the following equation should be guaranteed by using (11) and (12).

||ek||∞ = ||ek(j)|| (14)

Therefore, if we select the maximum value among the output errors at iteration k, we can deter-
mine j.

4.2 Compensation of soft errors

After detecting a soft error in yk(j), we can compensate for the soft error in several ways.
For instance, as shown in Figure 5, we can use the stored data for yk(j), the value of the previous
iteration yk−1(j), the adjacent value yk(j − 1), or the desired output yd(j).

Figure 5: Recovery of the soft error induced yk(j) with related data

In general, for the resilient design of a soft error, the exact rollback method has been widely
used. After periodically copying the value of yk to the buffers, we can restore the value when
an error is detected. Although we need additional buffer spaces for the rollback, it is a simple



Effect of Soft Errors in Iterative Learning Control
and Compensation using Cross-layer Approach 369

and effective technique to reduce the memory errors. If advanced error-detection methods can
be supported, it can be used without assumptions (A2) and (A3).

However, with regard to (A1), this method cannot be used when the soft error occurs within
the system, that is, before copying the data to the buffer. For the soft errors in the memory or
learning controller, the rollback method can be used irrespective of the variables.

The value of the previous iteration yk−1(j) can be used to compensate for yk(j). Since the
learning is performed according to the iteration number k, it is easy to prove the monotonic
convergence. In addition, it can be used when there is a soft error within the system or before
copying the data to the buffer.

Further, we can use the adjacent value yk(j−1). For real-time control, this approach can be
used for the compensation of soft errors. In this example, we do not need additional buffers. If
more advanced error-detection methods can be supported, it can be used, relaxing assumptions
(A2) and (A3).

Otherwise, the desired value yd(j) can be applied for the compensation. Since the error
ek(j) becomes zero, monotonic convergence can be derived.

Based on these facts, we can summarize the compensation method as follows:

Compensation method 0 - exact rollback

• Step 1: Restore the measured output value yk to an additional buffer at iteration k.

• Step 2: When a soft error is detected at iteration k and time j, recover it with yk(j) from
the buffer.

Compensation method 1- using the value of the previous iteration yk−1(j)

• Step 1: Restore the output value yk−1 to the memory at iteration k − 1.

• Step 2: When a soft error is detected at iteration k and time j, replace yk(j) with yk−1(j)
.

Compensation method 2- using the adjacent value yk(j − 1)

• Step 1: When a soft error is detected at iteration k and time j, replace yk(j) with yk(j−1)
.

Compensation method 3- using the desired value yd(j)

• Step 1: When a soft error is detected at iteration k and time j, replace yk(j) with yd(j) .

With the presented compensation methods, monotonic convergence is proved in the theorems
below. For compensation methods 1 and 3, convergence is given based on (A1)-(A3), whereas,
for the compensation method 2, we need another condition for the magnitude of the compensated
error.

Theorem 3. (Convergence of the ILC when using compensation method 1)
Let us assume that a soft error satisfying (A1)-(A3) occurs at the process of the ILC in the
uncertain system (1). Let us apply the input update law (5), the critical soft error detection
method, and compensation method 1 to the uncertain system (1). If the condition (6) holds for
all k, the output error ||ek

[σ,N+σ−1]||∞ converges to 0 as k →∞.



370 G.-M. Jeong, K. Lee, S.-I. Choi, S.-H. Ji, N. Dutt

Proof: Let us consider that a soft error occurs in yk(j). That is, ||ek||∞ ≥ ||ek−1||∞ and
||ek||∞ = ||ek(j)||.

From Lemma 2, we can detect j and k. Further, we compensate for the data using yk−1(j)
instead of yk(j). Let us denote the compensating value yk−1(j) as ŷk(j) and the compensated k-th
iteration data as êk, respectively. From these values, we obtain êk = [ek(σ), · · · , êk(j), · · · ,ek(N+
σ − 1)], where êk(j) = yd(j) − ŷk(j) = yd(j) −yk−1(j) = ek−1(j).

since ||êk(j)|| = ||ek−1(j)||, from (13) and (14), we can obtain

||êk||∞ ≤ ||ek−1||∞. (15)

Further, since there is no soft error at iteration k + 1 from (A3), the following equation holds:

||ek+1||∞ < ρ||êk||∞,ρ < 1. (16)

From (15) and (16), the output error approaches 0 as k →∞.
2

Theorem 4. (Convergence of the ILC when using compensation method 2)
Let us assume that a soft error satisfying (A1)-(A3) occurs at the process of ILC in the uncer-
tain system (1). Let us apply the input update law (5), the critical soft error detection method
and compensation method 2 to the uncertain system (1). If the compensated error satisfies
||yd(j)−yk(j−1)|| ≤ ||ek−1||∞ and condition (6) holds for all k, the output error ||ek[σ,N+σ−1]||∞
converges to 0 as k →∞.

Proof: We can detect j and k with the detection method. Let us compensate for the data
using yk(j − 1) instead of yk(j). Let us denote the compensating value yk(j − 1) as ŷk(j)
and the compensated k-th iteration error as êk, respectively. From these values, we obtain
êk = [ek(σ), · · · , êk(j), · · · ,ek(N + σ − 1)], where êk(j) = yd(j) − ŷk(j) = yd(j) −yk(j − 1).

If ||êk(j)|| = ||yd(j) −yk(j − 1)|| ≤ ||ek−1||∞, from (13) and (14), we can obtain

||êk||∞ ≤ ||ek−1||∞. (17)

Further, since there is no soft error at iteration k + 1 from (A3), the following equation holds:

||ek+1||∞ < ρ||êk||∞,ρ < 1. (18)

From (17) and (18), the output error approaches 0 as k →∞.
2

Theorem 5. (Convergence of the ILC when using compensation method 3)
Let us assume that a soft error satisfying (A1)-(A3) occurs in the process of the ILC in the
uncertain system (1). Let us apply the input update law (5), the critical soft error detection
method and compensation method 3 to the uncertain system (1). If the condition (6) holds for
all k, the output error ||ek

[σ,N+σ−1]||∞ converges to 0 as k →∞.

Proof: We can detect j and k with the detection method. Let us compensate for the data
using yd(j) instead of yk(j). Then, for the compensated data at iteration k, we obtain êk =
[ek(σ), · · · , êk(j) = 0, · · · ,ek(N + σ − 1)].

Hence, from (13) and êk(j) = 0, we can obtain

||êk||∞ ≤ ||ẽk||∞ < ||ek−1||∞. (19)



Effect of Soft Errors in Iterative Learning Control
and Compensation using Cross-layer Approach 371

Further, since there is no soft error at iteration k + 1 from (A3), the following equation holds:

||ek+1||∞ < ρ||êk||∞,ρ < 1. (20)

From (19) and (20), the output error approaches 0 as k →∞.
2

-0.3

-0.2

-0.1

 0

 0.1

 0.2

 0.3

 0  5  10  15  20

In
p

u
ts

i

yy1
yy2
yy3

(a) Inputs for different values of k

-0.5

-0.25

 0

 0.25

 0.5

 0  5  10  15  20

O
u

tp
u

ts
i

yyd
yy1
yy2
yy3

(b) Desired output and the outputs for dif-
ferent values of k

 0

 0.4

 0.8

 1.2

 1.6

 0  5  10  15  20  25  30

T
h

e
 R

M
S

 e
r
r
o

r

k

(c) RMS error for different values of k

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

 0

 0.002

 0  5  10  15  20  25  30

D
if

f.
 b

e
t.

 F
ig

. 
2

(c
) 

a
n

d
 F

ig
. 

6
(c

)

k

(d) Differences between the output errors
when there is no soft error and when it is
compensated using method 1

Figure 6: Outputs and inputs when soft error compensation method 1 is used

5 Simulation results

Let us apply the proposed methods to the case in Section 3.2. The soft error condition
in the previous example satisfies assumptions (A1)-(A3) and we can apply the detection and
compensation methods to this example.

Figure 6 shows the results of compensation method 1. As shown in Figure 6, we can observe
that the output error converges to 0 as k →∞ and the compensation method works well although
there is a slight performance degradation as compared with the case without soft errors. Figure
6(d) shows the differences between the output errors when there is no soft error and when the
soft error is compensated using compensation method 1.

Figure 7 shows ||ek||∞ trajectory when there is no soft error and when the induced soft
error is compensated using the compensation methods. For compensation methods 1 and 3,
we can observe that the error approaches 0 as k → ∞, whereas for compensation methods 2,
||yd(j)−yk(j−1)|| ≤ ||ek−1||∞ cannot always be satisfied, and the monotonic convergence cannot
be guaranteed.



372 G.-M. Jeong, K. Lee, S.-I. Choi, S.-H. Ji, N. Dutt

In this example, when using compensation method 3, as compared to compensation method
1, êk(j) becomes 0, and it reduces the speed of convergence. As shown in Figure 7, compen-
sation method 1 is the best method in this example, among compensation methods 1, 2, and
3, considering the monotonic convergence in the ILC. Further, depending on the location and
characteristics of the soft errors, we can combine the presented four compensation methods for
achieving a better performance.

 0

 0.1

 0.2

 0.3

 0.4

 0.5

 0  5  10  15  20  25  30

y
y
1

k

Method 1
Method 2
Method 3

Figure 7: ||ek||∞ according to k for the different compensation methods

6 Conclusion

In this paper, we have studied the effects of soft errors in the ILC and proposed novel
compensation methods for the soft errors in the ILC. First, we have analyzed the effects of soft
errors in the ILC. It has been shown that a soft error can affect the speed of the learning curve
and make the controller lose the learning history. Second, based on the detection algorithm for
the critical soft error that affects the monotonic convergence, we have proposed compensation
methods for the ILC. The convergence of the proposed method has been proved under the as-
sumptions for soft errors. From the experimental results, we have demonstrated the effectiveness
of our proposed methods.

Furthermore, it is expected that several directions of researches could be developed. We
could make more complicated assumptions such as multiple soft errors and extend the results
to the various control applications. As an example in the ILC, depending on the location and
characteristics of the soft errors, we could combine the four presented compensation methods for
achieving better performance. These remain as future work.

Funding

This research was supported by the Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1A09083894),
and also supported by the National Research Foundation of Korea (NRF) funded by the Korean
Government (MSIP) (NRF-2016R1A5A1012966).

Author contributions

The authors contributed equally to this work.



Effect of Soft Errors in Iterative Learning Control
and Compensation using Cross-layer Approach 373

Conflict of interest

The authors declare no conflict of interest.

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