Journal of Robotics, Networking and Artificial Life
Vol. 5(3); December (2018), pp. 149–152
DOI: 10.2991/jrnal.2018.5.3.1; ISSN 2405-9021 print; 2352-6386 online
https://www.atlantis-press.com/journals/jrnal

Research Article

Sticking Fault Detecting Method for CARIMA Model
Toyoaki Tanikawa1, Henmi Tomohiro2,*
Advanced Course in Industrial and Systems Engineering, National Institute of Technology, Kagawa College, Takamatsu, Kagawa 761-8058, Japan
Department of Electrical and Computer Engineering, National Institute of Technology, Kagawa College, Takamatsu, Kagawa 761-8058, Japan

12

ARTICLE INFOABSTRACT

Article HistoryThis paper proposes a sticking fault detecting method for controlled auto-regressive integrated moving average model (CARIMA)
which detect the sticking fault of control input and feedback signal. It consists of model estimation using recursive least square
method with the forgetting factor and fault detection. In the fault detection, an evaluation function is introduced, and it generates
a fault signal from the input and output data. Numerical simulations are performed, and it is shown that this method can detect
the sticking fault.

Received 10 August 2018
Accepted 19 October 2018
Keywords
Fault detection
sticking fault
CARIMA model© 2018 The Authors. Published by Atlantis Press SARL.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

2.1. Problem Statement

If a system with a fault continues to be operated, it can cause aserious accident or a considerable damage. Thus, it is important
to detect faults and compensate them, and many fault detection
methods have been proposed [1,2]. The advantage of fault detection is that the safety is improved, you can cope with the fault
more promptly, and sometimes the system can be controlled
compensating the fault. There are two kinds in fault detection,
signal-based detection and model-based one. The signal-based
detection is for example a method using spectral analysis, statistical signal analysis or pattern recognition, while the model-­based detection uses an observer or a parameter estimation [3].
In model-­based detection, a general method detecting additive
faults is proposed by Isermann [4].

First, we assume the control object is single input and single output
system and expressed in Eq. (1) as following CARIMA model:

In detecting a fault where control input or feedback signal is fixed,a general detection method is not established. In this paper, we
propose a model-based detection method for the sticking faultof control input and feedback signal on a system expressed as acontrolled auto-regressive integrated moving average (CARIMA)
model. Then, we discuss its effectiveness performing a numericalsimulation.

2. STICKING FAULT DETECTING METHOD
In this section, we discuss how to detect sticking fault of control
input and feedback signal on the general linear systems.
* Corresponding author. Emails: toyoakipd0@gmail.com; henmi@t.kagawa-nct.ac.jp;
www.kagawa-nct.ac.jp

x (k)
∆(1)
A[z −1 ] = 1 + a1z −1 + a2 z −2 + …+ an z − n 	
−1−1−2−m
B[z ] = b0 + b1z + b2 z + …+ bm z
C[z −1 ] = 1 + c1z −1 + c2 z −2 +  + cl z − l

A[z −1 ] ym (k) = z − km B[z −1 ]um (k) + C[z −1 ]where ym(k) represents a system output, um(k) is a control input, x (k)is a white Gaussian noise with 0 mean and km is a time delay. z−1 is abackshift operator where z−1ym(k) = ym(k − 1) and Δ is a difference
operator defined as Δ = 1 − z−1. System parameters, a, b and c in
Eq. (1) changes gradually or do not change at all. We suppose thereference value of the control system changes depending on time.
Let us define the sticking fault. When the sticking fault of control
input occurs, the input to the system um(k) becomes a white noiseand expressed in Eq. (2):(normal)
u(k)um (k) = +u(k)(fault)xu
 f(2)

where u(k) is a control input calculated by the controller, uf is a constant value and xu(k) is a white Gaussian noise with 0 mean. Whenthe sticking fault of feedback signal occurs, the feedback signal tothe controller y(k) becomes a white noise and expressed in Eq. (3):(normal) y (k )y(k) =  m+()(xfault)yky f(3)

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150

where yf is a constant value and xy(k) is a white Gaussian noise with0 mean.
Figure 1 shows a block diagram of the fault detection for theabove-mentioned faults. This method consists of two parts, modelestimation and fault analysis.

2.3. Fault Detection

2.2. Model Estimation
Least square method with the forgetting factor [5] is used for themodel estimation. In this method, the cost function is given inEq. (4):N

The weight of a past input–output data in cost function IN(q )becomes smaller as the time goes by due to the forgetting factor.
Thus, the estimated parameter p(k) is calculated using resent data
mainly. Thus, this parameter estimation method is also effective forsystems where its parameters change.

Figure 2 shows the outline of the fault detection. To detect sticking
fault, it uses the estimated system parameters and the input–output
data of past nd steps called detecting period. The fault state is evaluated by the following Eq. (8):2I N (p ) = ∑l N − k  y(k ) − y ( k|p ) 	(4)k =1
where p is a system parameter vector, l is a forgetting factor wherey ( k|p ) is an estimated output and N is the number of the0 < l ≤ 1, 
input–output data. The estimated parameter p which minimizes
the cost function on CARIMA model is obtained by the following
recursive algorithm [Eq. (5)]P (k - 1)y (k)¨(k )l + y T (k ) P (k - 1)y (k )P (k - 1)y (k)y T ( k ) P (k - 1) ù1é
P (k ) = ê P (k - 1) ú 	(5)lël + y T (k ) P (k - 1)y (k ) úû¨(k ) = Dy (k ) -y T (k)q(k - 1)
h (k) = Dy (k ) -y T (k)q(k)p(k) = q(k - 1) +¨(k) is a priori error and h(k) is a posterior error. The parameter
vector q(k ) and the data vector y (k) are defined in Eq. (6) as follows:q(k) = [c1 (k),  , c l (k)]T
a1 (k), , an (k), b0 (k),, bm (k), 	(6)y (k) = [− ∆y(k − 1),, − ∆y(k − n), ∆u(k − km ), ,∆u (k − km − m), h (k − 1), , h (t − l )]T
where â, b and ĉ represent the estimated parameter of a, b and c in
Eq. (1) respectively. The initial value of the covariance matrix P(k)is given in Eq. (7):V (k) =å[z -1 ]y(i) - z - km D B
 [z -1 ]u(i)}2{D A

ki = k - nd +1å [z -1 ]y(i)}2 + b{D Ai = k - n +1k(8)d[z -1 ]where b is an infinitesimal value to avoid dividing by 0. A−1
and B[z ] are the estimated system parameter polynomials atstep k − nd − nm which are gained from q(k - nd - nm ) calculated in Eq. (5), where nm is the order of the system defined by
nm = max{m, n} + 1 . It is assumed that q(k - nd - nm ) is a goodestimation of the actual system at step k because we are focusing on
systems whose parameters do not change abruptly.
We assume a situation that a fault has occurred. The estimatedparameters become useless values after the fault. However, at amoment like Figure 2, estimated parameter q(k - nd - nm ) is stillvalid and only the input–output data after the fault are used for theevaluation value V(k). Then V(k) becomes bigger than 1 as being
mentioned in Section 3, and a fault signal is generated after satisfying the following inequality [Eq. (9)]:V(k) > Vth	(9)where Vth is a threshold value set as Vth > 1. After detecting the fault,
it stops updating the estimated parameters in Eq. (5).

3. STICKING FAULT EVALUATION VALUE

We discuss the size of the evaluation value V(k) on the normal stateP(0) = g I	(7)and the fault one. First, let us define the error polynomials of thewhere I is an identity matrix. g is usually set as a large value like
103 or 104.

system parameters, EA[z−1] and EB[z−1] as Eq. (10): [z -1 ] - A[z -1 ]E A[z -1 ] = A [z -1 ] - B[z -1 ] 	(10)E [z -1 ] = B
BFigure 1 | Block diagram of sticking fault detection

Figure 2 | Outline of fault detection

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Using Eqs. (1–3) and (10) and ignoring the infinitesimal value b,evaluation function in Eq. (8) on the normal state can be deformedas Eq. (11):V (k) =å {DE y(i) - z DE u(i) + Cx (i)} 	(11)å {z DBu(i) + DE y(i) + Cx (i)}
- km

Ai2

B

- km2Ai

where the polynomial expression is abbreviated like B instead ofB[z−1], and ∑ i means ∑ ik= k − nd +1. Assuming that the terms of error
polynomials and that of white noise x(i) are relatively small compared to the term of u(i) in Eq. (11), V (k)  1 is gained.
On the other hand, after the sticking fault of the input has occurred,
the evaluation value at a moment described in Figure 2 is deformed
using Eq. (2) of the fault state and Eqs. (1), (3) and (10), andexpressed as Eq. (12):å {- z D B (u(i) - x (i)) + DE y(i) + Cx (i)} 	(12)V (k) =å {DE y(i) + z DBx (i) + Cx (i)}2- kmui

2- kmAiAu

while the evaluation value on a sticking fault of the feedback signal
is deformed using Eq. (3) of the fault state, and expressed as Eq. (13):∑ {− zV (k) =i− km}

 (i) + ∆ A
x (i)∆ Buy∑ {∆ Ax (i)}2i2(13)yBecause the terms of error polynomials and white noises are smaller
than that of u(i) as assumed before, V (k)  1 is obtained in bothEqs. (12) and (13). Therefore, the evaluation value becomes greater
after the sticking fault occurs, and the fault can be detected by Eq. (9).
However, the threshold Vth has to be set based on an input–output
data on an actual system or a simulation results because the evaluation
value V(k) is dependent on the estimation error and disturbances.
Systems usually have limitations of the control input. When designing a controller considering the limitation, higher and lower constant limits are generally used. However, Eqs. (11–13) state thatΔu(i) has to have a value, and Δu(i) can be 0 when the input is limited by the constant value. Thus, the limitation has to be dependent
on time, and is given by Eq. (14):where km = 1. We set the disturbance characteristics polynomials
C[z−1] as C[z−1] = 1. The system parameter polynomials do notchange throughout the simulation. x(k), xu(k) and xy(k) are set as azero mean white Gaussian noise with the variance of 0.012, 0.12 and0.0052 respectively.
In the model estimation, we assume that the order of the systemparameter polynomials l, m and n and the time delay km are known.
Then, the estimated parameter vector p(k) is given in Eq. (16):p(k ) = [a1 (k), a 2 (k ), b0 (k ), b1 (k )]T 	(16)We set g to 103 and l to 0.98. In the fault detection, nd = 20, Vth =1.5 and b = 10−10 are used.
Generalized predictive control [6] which uses CARIMA model
to predict the output is used as a controller. When designing it,Eq. (14) is used for the internal model, and N1 = 1, N2 = 5, NU = 5,l(j) = 0.01, and time constant of the reference trajectory t = 0.3 s aretaken. For the input limitation, A = 1, w = 0.1p, uh = 4 and ul = −4
are used.
Figure 3 shows a simulation result where the sticking fault of control input occurs at a step 400. r(k) represents reference value andthe dashed line on the graph V(k) is the threshold value Vth. The
evaluate value V(k) is normally smaller than 1, and it becomes
greater after the fault occurs. It surpasses Vth at a step 406, and thenthe fault signal is generated. V(k) takes relatively great value on thebeginning due to the error of the parameter estimation.
Figure 4 shows a result of the sticking fault of feedback signal. Like
Figure 3, V(k) becomes greater after the fault, and the fault signal isgenerated at a step 404.

5. CONCLUSION
We proposed the sticking fault detecting method in which the evaluation function is introduced. It evaluates the input–output data onìumax (k), uc (k ) > umax (k)ïu (k) = íuc (k), umin (k) £ uc (k) £ umax (k)ïîumin (k), uc (k) < umin (k)	(14)umax (k) = Asin (w k) + uhumin (k ) = Asin (w k ) + ul
where uc(k) is the control input calculated by the controller, andA, w, uh and ul are setting parameters.

4. SIMULATION EXAMPLE
To verify the effectiveness of this fault detecting method, we apply
it to a system expressed as Eq. (15):A[z −1 ] = 1 − 1.307 z −1 + 0.3768z −2
B[z −1 ] = 0.05158 + 0.03729z −1(15)151

Figure 3 | Detection result on control input sticking fault

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ACKNOWLEDGMENTS
This work was supported by JSPS KANKENHI (Grant Number15K21591).

REFERENCES

Figure 4 | Detection result on feedback sticking fault

a detecting period, using the past estimated parameter calculatedby the recursive least square method. Performing a numerical simulation, it is confirmed that the sticking fault of control input andfeedback signal can be detected.

[1] M. Takahashi, Self-repairing control utilizing instability byfailures, (Performance Assessment and Design of ControlSystems), Syst. Control Inform. 58 (2014), 111–116 (in Japanese)(Special issue).
[2] A. Inoue, M. Deng, Y. Shinichi, S. Okazaki, Fault detection to aprocess control experimental system by using an adaptive observer,
Trans. Soc. Inst. Control Eng. 43 (2007), 621–623 (in Japanese).
[3] T. Hukuda, Diagnosis on Machinery (part 1) - the outline andan evaluation of its diagnostic reliability, J. Jpn. Soc. Safety Eng.
36 (1997), 247–252 (in Japanese).
[4] R. Isermann, Model-based Fault-detection and Diagnosis –Status and Applications, IFAC Automatic Control in Aerospace,
Saint-Petersburg, Russia, 2004, pp. 49–60.
[5] L. Ljung, T. Söderström, Theory and Practice of Recursive
Identification, MIT Press, Cambridge, MA, 1983.
[6] D.W. Clarke, C. Mohtadi, P.S. Tuffs, Generalized predictive
control—part I. The basic algorithm, Automatica 23 (1987),
137–148.

Authors Introduction
Dr. Tomohiro Henmi

Mr. Toyoaki Tanikawa

He received his associate degree ofengineering from National Instituteof Technology, Kagawa College in2017. He is currently doing bachelorof engineering at National Instituteof Technology, Kagawa College.

He is an Assistant Professor at theDepartment of Electro-MechanicalEngineering, National Instituteof Technology, Kagawa College,Japan. He received his PhD fromOkayama University, Japan in 2005.
His research interests include robotics and control of underactuatednon-linear systems.