Sticking Fault Detecting Method for CARIMA Model

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 modelbased detection uses an observer or a parameter estimation [3]. In modelbased detection, a general method detecting additive faults is proposed by Isermann [4].


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 modelbased 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.

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.

Problem Statement
First, we assume the control object is single input and single output system and expressed in Eq. (1) as following CARIMA model: where y m (k) represents a system output, u m (k) is a control input, x (k) is a white Gaussian noise with 0 mean and k m is a time delay. z −1 is a backshift operator where z −1 y 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): where u(k) is a control input calculated by the controller, u f is a constant value and x 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): where y f is a constant value and x 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.

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): where p is a system parameter vector, l is a forgetting factor where 0 1 < ≤ l , y k |p ( ) is an estimated output and N is the number of the input-output data. The estimated parameter p which minimizes the cost function on CARIMA model is obtained by the following recursive algorithm [Eq. (5)] ( ) 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: 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): where I is an identity matrix. g is usually set as a large value like 10 3 or 10 4 .
The weight of a past input-output data in cost function I N (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 for systems where its parameters change. -is a good estimation of the actual system at step k because we are focusing on systems whose parameters do not change abruptly.

Fault Detection
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 q ( ) k n n d m --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)]: 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).

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

CONCLUSION
We proposed the sticking fault detecting method in which the evaluation function is introduced. It evaluates the input-output data on Using Eqs. (1-3) and (10) and ignoring the infinitesimal value b, evaluation function in Eq. (8) on the normal state can be deformed as Eq. (11): where the polynomial expression is abbreviated like B instead of B[z −1 ], and ∑ i means ∑ = − + i k n k d 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), and expressed as Eq. (12): 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): 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.
where u c (k) is the control input calculated by the controller, and A, w, u h and u l are setting parameters.

SIMULATION EXAMPLE
To verify the effectiveness of this fault detecting method, we apply it to a system expressed as Eq. (15): .