Sticking Fault Detecting Method for CARIMA Model

Toyoaki Tanikawa; Henmi Tomohiro

doi:10.2991/jrnal.2018.5.3.1

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Volume 5, Issue 3, December 2018, Pages 149 - 152

Sticking Fault Detecting Method for CARIMA Model

Authors

Toyoaki Tanikawa¹, Henmi Tomohiro²^{, *}

¹Advanced Course in Industrial and Systems Engineering, National Institute of Technology, Kagawa College, Takamatsu, Kagawa 761-8058, Japan

²Department of Electrical and Computer Engineering, National Institute of Technology, Kagawa College, Takamatsu, Kagawa 761-8058, Japan

^*Corresponding author. Emails: toyoakipd0@gmail.com; henmi@t.kagawa-nct.ac.jp; www.kagawa-nct.ac.jp

Corresponding Author

Henmi Tomohiro

Received 10 August 2018, Accepted 19 October 2018, Available Online 1 December 2018.

DOI: 10.2991/jrnal.2018.5.3.1 How to use a DOI?
Keywords: Fault detection; sticking fault; CARIMA model
Abstract: This 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.
Copyright: © 2018 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. INTRODUCTION

If a system with a fault continues to be operated, it can cause a serious 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].

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 fault of control input and feedback signal on a system expressed as a controlled auto-regressive integrated moving average (CARIMA) model. Then, we discuss its effectiveness performing a numerical simulation.

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.

2.1. Problem Statement

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

(1)A[z−1]ym(k)=z−kmB[z−1]um(k)+C[z−1]ξ(k)ΔA[z−1]=1+a1z−1+a2z−2+…+anz−nB[z−1]=b0+b1z−1+b2z−2+…+bmz−mC[z−1]=1+c1z−1+c2z−2+⋯+clz−l

where y_m(k) represents a system output, u_m(k) is a control input, ξ(k) is a white Gaussian noise with 0 mean and k_m is a time delay. z⁻¹ is a backshift operator where z⁻¹y_m(k) = y_m(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 the reference 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 u_m(k) becomes a white noise and expressed in Eq. (2):

(2)um(k)={u(k) (normal)uf+ξu(k)(fault)

where u(k) is a control input calculated by the controller, u_f is a constant value and ξ_u(k) is a white Gaussian noise with 0 mean. When the sticking fault of feedback signal occurs, the feedback signal to the controller y(k) becomes a white noise and expressed in Eq. (3):

(3)y(k)={ym(k) (normal) yf+ξy(k)(fault)

where y_f is a constant value and ξ_y(k) is a white Gaussian noise with 0 mean.

Figure 1 shows a block diagram of the fault detection for the above-mentioned faults. This method consists of two parts, model estimation and fault analysis.

2.2. Model Estimation

Least square method with the forgetting factor [5] is used for the model estimation. In this method, the cost function is given in Eq. (4):

(4)IN(θ)=∑k=1NλN−k[y(k)−y^(k|θ)]2

where θ is a system parameter vector, λ is a forgetting factor where 0 < λ ≤ 1, ŷ(k|θ) is an estimated output and N is the number of the input–output data. The estimated parameter θ^ which minimizes the cost function on CARIMA model is obtained by the following recursive algorithm [Eq. (5)]

(5)θ^(k)=θ^(k−1)+P(k−1)ψ(k)λ+ψT(k)P(k−1)ψ(k)ϵ(k)P(k)=1λ[P(k−1)−P(k−1)ψ(k)ψT(k)P(k−1)λ+ψT(k)P(k−1)ψ(k)]ϵ(k)=Δy(k)−ψT(k)θ^(k−1)η(k)=Δy(k)−ψT(k)θ^(k)

∈(k) is a priori error and η(k) is a posterior error. The parameter vector θ^(k) and the data vector ψ (k) are defined in Eq. (6) as follows:

(6)θ^(k)=[a^1(k),⋯,a^n(k),b^0(k),⋯,b^m(k), c^1(k),⋯,c^l(k)]T ψ(k)=[−Δy(k−1),⋯,−Δy(k−n), Δu(k−km),⋯,Δu(k−km−m), η(k−1),⋯,η(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):

(7)P(0)=γI

where I is an identity matrix. γ is usually set as a large value like 10³ or 10⁴.

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

2.3. Fault Detection

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 n_d steps called detecting period. The fault state is evaluated by the following Eq. (8):

(8)V(k)=∑i=k−nd+1k{ΔA^[z−1]y(i)−z−kmΔB^[z−1]u(i)}2∑i=k−nd+1k{ΔA^[z−1]y(i)}2+β

where β is an infinitesimal value to avoid dividing by 0. Â[z⁻¹] and B^[z−1] are the estimated system parameter polynomials at step k − n_d − n_m which are gained from θ^(k−nd−nm) calculated in Eq. (5), where n_m is the order of the system defined by n_m = max{m, n} +1. It is assumed that θ^(k−nd−nm) is a good estimation 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 estimated parameters become useless values after the fault. However, at a moment like Figure 2, estimated parameter θ^(k−nd−nm) is still valid and only the input–output data after the fault are used for the evaluation 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)]:

(9)V(k)>Vth

where V_th is a threshold value set as V_th > 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 state and the fault one. First, let us define the error polynomials of the system parameters, E_A[z⁻¹] and E_B[z⁻¹] as Eq. (10):

(10)EA[z−1]=A^[z−1]−A[z−1] EB[z−1]=B^[z−1]−B[z−1]

Using Eqs. (1–3) and (10) and ignoring the infinitesimal value β, evaluation function in Eq. (8) on the normal state can be deformed as Eq. (11):

(11)V(k)=∑i{ΔEAy(i)−z−kmΔEBu(i)+Cξ(i)}2∑i{z−kmΔBu(i)+ΔEAy(i)+Cξ(i)}2

where the polynomial expression is abbreviated like B instead of B[z⁻¹], and ∑_i means ∑i=k−nd+1k. Assuming that the terms of error polynomials and that of white noise ξ(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), and expressed as Eq. (12):

(12)V(k)=∑i{−z−kmΔB^(u(i)−ξu(i))+ΔEAy(i)+Cξ(i)}2∑i{ΔEAy(i)+z−kmΔBξu(i)+Cξ(i)}2

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):

(13)V(k)=∑i{−z−kmΔB^u(i)+ΔA^ξy(i)}2∑i{ΔA^ξy(i)}2

Because the terms of error polynomials and white noises are smaller than that of u(i) as assumed before, V(k) ≳ 1 is obtained in both Eqs. (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 V_th 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 that Δu(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):

(14) u(k)={umax(k),uc(k)>umax(k)uc(k), umin(k)≤uc(k)≤umax(k)umin(k),uc(k)<umin(k) umax(k)=Asin(ωk)+uh umin(k)=Asin(ωk)+ul

where u_c(k) is the control input calculated by the controller, and A, ω, u_h and u_l 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):

(15)A[z−1]=1−1.307z−1+0.3768z−2B[z−1]=0.05158+0.03729z−1

where k_m = 1. We set the disturbance characteristics polynomials C[z⁻¹] as C[z⁻¹] = 1. The system parameter polynomials do not change throughout the simulation. ξ(k), ξ_u(k) and ξ_y(k) are set as a zero mean white Gaussian noise with the variance of 0.01², 0.1² and 0.005² respectively.

In the model estimation, we assume that the order of the system parameter polynomials l, m and n and the time delay k_m are known. Then, the estimated parameter vector θ^(k) is given in Eq. (16):

(16)θ^(k)=[a^1(k),a^2(k),b^0(k),b^1(k)]T

We set γ to 10³ and λ to 0.98. In the fault detection, n_d = 20, V_th = 1.5 and β = 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, λ(j) = 0.01, and time constant of the reference trajectory τ = 0.3 s are taken. For the input limitation, A = 1, ω = 0.1π, u_h = 4 and u_l = −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 and the dashed line on the graph V(k) is the threshold value V_th. The evaluate value V(k) is normally smaller than 1, and it becomes greater after the fault occurs. It surpasses V_th at a step 406, and then the fault signal is generated. V(k) takes relatively great value on the beginning 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 is generated 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 on a detecting period, using the past estimated parameter calculated by the recursive least square method. Performing a numerical simulation, it is confirmed that the sticking fault of control input and feedback signal can be detected.

ACKNOWLEDGMENTS

This work was supported by JSPS KANKENHI (Grant Number 15K21591).

Authors Introduction

Mr. Toyoaki Tanikawa

He received his associate degree of engineering from National Institute of Technology, Kagawa College in 2017. He is currently doing bachelor of engineering at National Institute of Technology, Kagawa College.

Dr. Tomohiro Henmi

He is an Assistant Professor at the Department of Electro-Mechanical Engineering, National Institute of Technology, Kagawa College, Japan. He received his PhD from Okayama University, Japan in 2005. His research interests include robotics and control of underactuated non-linear systems.

REFERENCES

[1]M Takahashi, Self-repairing control utilizing instability by failures, (Performance Assessment and Design of Control Systems), Syst. Control Inform., Vol. 58, 2014, pp. 111-116. (in Japanese) (Special issue).

[2]A Inoue, M Deng, Y Shinichi, and S Okazaki, Fault detection to a process control experimental system by using an adaptive observer, Trans. Soc. Inst. Control Eng., Vol. 43, 2007, pp. 621-623. (in Japanese).

[3]T Hukuda, Diagnosis on Machinery (part 1) - the outline and an evaluation of its diagnostic reliability, J. Jpn. Soc. Safety Eng., Vol. 36, 1997, pp. 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 and T Söderström, Theory and Practice of Recursive Identification, MIT Press, Cambridge, MA, 1983.

[6]DW Clarke, C Mohtadi, and PS Tuffs, Generalized predictive control—part I. The basic algorithm, Automatica, Vol. 23, 1987, pp. 137-148.

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Journal: Journal of Robotics, Networking and Artificial Life
Volume-Issue: 5 - 3
Pages: 149 - 152
Publication Date: 2018/12/01
ISSN (Online): 2352-6386
ISSN (Print): 2405-9021
DOI: 10.2991/jrnal.2018.5.3.1 How to use a DOI?
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

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ris enw bib

TY  - JOUR
AU  - Toyoaki Tanikawa
AU  - Henmi Tomohiro
PY  - 2018
DA  - 2018/12/01
TI  - Sticking Fault Detecting Method for CARIMA Model
JO  - Journal of Robotics, Networking and Artificial Life
SP  - 149
EP  - 152
VL  - 5
IS  - 3
SN  - 2352-6386
UR  - https://doi.org/10.2991/jrnal.2018.5.3.1
DO  - 10.2991/jrnal.2018.5.3.1
ID  - Tanikawa2018
ER  -

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