J. Cent. South Univ. Technol. (2009) 16: 0789-0793

DOI: 10.1007/s11771-009-0131-2

Similarity measure application to fault detection of flight system

KIM J H, LEE S H, WANG Hong-mei(王洪梅)

(School of Mechatronics, Changwon National University, 9 Sarim-dong, Changwon, Gyeongnam, 641-773, Korea)

Abstract: Fault detection technique is introduced with similarity measure. The characteristics of conventional similarity measure based on fuzzy number are discussed. With the help of distance measure, similarity measure is constructed explicitly. The designed distance-based similarity measure is applicable to general fuzzy membership functions including non-convex fuzzy membership function, whereas fuzzy number-based similarity measure has limitation to calculate the similarity of general fuzzy membership functions. The applicability of the proposed similarity measure to general fuzzy membership structures is proven by identifying the definition. To decide fault detection of flight system, the experimental data (pitching moment coefficients and lift coefficients) are transformed into fuzzy membership functions. Distance-based similarity measure is applied to the obtained fuzzy membership functions, and similarity computation and analysis are obtained with the fault and normal operation coefficients.

Key words: similarity measure; fuzzy number, distance; non-convex membership function

1 Introduction

Data classification study has been applied in decision making [1-2], fault detection [3], and pattern classification [4-5]. Classification or clustering algorithm was developed on the basis of distance-based measures between different data groups. Generally, each data within the same group has higher degree of similarity as compared to data in different groups. Hence, the similarity measure is appropriate for classifying data with similar characteristics or for selecting different types of data from the entire data. Until now similarity measure is mainly based on fuzzy number and distance measures [6-11]. The explicit similarity measure presents the similar value between data groups or data to data set. Generally, the similar value means the degree of similarity. Most studies emphasize on the design of similarity measure on the basis of fuzzy number and center of gravity (COG) [6-7]. With the conventional similarity measure, it is easy to calculate the degree of similarity. However, fuzzy number is restricted to triangular and trapezoidal membership functions of a fuzzy set. This means that the similarity measure cannot be applied to general fuzzy membership functions. Whereas, the distance-based similarity measure is applicable to general fuzzy membership functions, including non-convex fuzzy membership functions. However, this similarity measure requires more time and consideration. Here, the similarity measure is considered from the viewpoint of certainty and uncertainty; and the measures of certainty and uncertainty are calculated by using the distance measure.

Accurate fault detection is essential in the case of aircraft stuck during normal aircraft operation. The flight control system is generally equipped with a lot of features in order to increase the safety of the aircraft, particularly in cases where the aircraft is damaged due to malfunctioning of the control surface system. If the extent of the damage is determined after the occurrence of the failure, the fault tolerant control system is capable of overcoming various faults in real time. Therefore, the pilot or the flight control system accomplishes the mission or returns to the safety region. By the flight test results, the deciding coefficients are obtained. These parameters are used to decide the fault or normal operations by transforming these coefficients into fuzzy membership function. The proposed distance-based similarity measure is applied to computing the degree of similarity with respect to fault or normal data set.

2 Flight systems and flight test

The estimation procedures are introduced with the help of well known airplane dynamics. Discussion on the control surface struck condition and the flight test procedure are also included.

2.1 Flight systems

Consider the well known aircraft dynamics along a combination of the longitudinal and lateral directions combining aircraft dynamics as follows [12-14]:

                          (1)

                          (2)

where  the state vector is x=[α  u  q  θ  β  p  r  In the state vector, p, q and r are the angular velocities; α is the angle of attack; β denotes the sideslip angle; u=δ_e is the elevator deflection;  and θ represent roll and pitch angles, respectively.

The system matrix A and input matrix B that contain the model parameters are found in Refs.[12-14]. In Eq.(2), it is generally considered that C=I and D=0. The estimated parameters are assumed to be constant during flight test maneuvering [15].

By applying Fourier transform to Eqs.(1) and (2), we have [12-14]

                     (3)

                        (4)

where jω denotes differential operator.

The least squares cost function is

?                    (5)

[Re(X?X)]^-1Re(X?Y)                      (6)

where  Y is the vector to be measured; X is a matrix of independent variable that denotes the given flight point,  and  is a parameter vector; ε is a complex error vector; and the discrete Fourier transform for the ith sampling time is obtained as X_i(ω)= X_i-1(ω)+ here  is constant for a given frequency and constant sampling interval, and ?t denotes sampling time. Furthermore, the estimated parameter covariance matrix is

[Re(X?X)]^-1

where  the equation error variance  can be estimated

from the residual [(Y-X)?(Y-X)], p is

the number of parameters to be estimated, and m is the number of frequency points of interest.

Frequency spacing of 0.02 Hz is found to be adequate, which leads to 50 frequencies being evenly distributed in the interval of 0.02-1.00 Hz for each transformed time-domain signal.

2.2 Control surface stuck condition and test procedure

Control surface failure is related to the failures of elevator, rudder, and aileron. To fix the elevator, the one-piece elevator is split into two. To fix the rudder, one more rudder is added [14]. Intentional control surface failures are considered, such as the right elevator failure, the left rudder failure, the left aileron failure and a combination of these failures. The flight test is performed by excluding the uncontrollability and untrimmability of the aircraft under post-failure conditions and the flight test is scheduled. Table 1 shows the flight test conditions for control surface stuck.

Table 1 Control surface stuck conditions

In the first flight test, the control input for the real-time parameter estimation is applied by using the knob and switching method [14]. The flight data are acquired for exciting the aircraft dynamics by using the above mentioned method [14-15]. However, it is found that the applied time interval is slightly inaccurate. In order to realize a constant control input and time interval, in the second stage of the flight test, the control input device is modified to use a RF modem and an R/C transmitter. Depending on the flight condition of the control surface stuck, the flight test results help engineers to adapt to the control surface stuck. After the ground test, the pilot maintains the trimmed level flight under the post-failure condition, and the engineer applies the control input by using a new device.

Considering the primary control of surface stuck and combination stuck, the aircraft must show controllability and trimmability under post-failure conditions. Fig.1 shows the photo of aircraft DURUMI-Ⅱ in take-off for the flight test. The flight test procedure involves the recovery of the aircraft from the fault state to the normal state [15]. Table 2 shows that the aircraft DURUMI-Ⅱ in take-off is successfully tested under fault conditions. Furthermore, it shows the trim value of the available primary control surface under post-failure conditions. Trim value is satisfied by 5.752 4 when it is in a normal mode.

Fig.1 Photo of DURUMI-Ⅱ in take-off

3 Similar measure and application to fault detection

For calculating similarity of data, similarity measure is introduced by two different approaches, i.e. fuzzy

Table 2 Elevator trim values in longitudinal mode

number and distance measure.

3.1 Similarity measure with fuzzy number and distance measure

The trapezoidal membership function  is denoted by fuzzy number (a, b, c, d, ω) [7]. A new method was presented to calculate the COG point of a generalized fuzzy number, and a new COG calculation method was derived based on the concept of the medium curve [12]. These COG points play an important role in the calculation of the fuzzy number based similarity measure. In Refs.[7-8], the degrees of similarities were derived by using the membership functions of the fuzzy number and COG, and a similarity measure to overcome the drawbacks of the existing similarity measures was proposed as follows:

                        (7)

where   and  denote the COG points of fuzzy numbers  and  respectively;  and  are expressed by a₄-a₁ and b₄-b₁, respectively.  is satisfied by 1 if +＞0, and 0 if +=0. However, similarity measure (Eq.(7)) can be used only for triangular or trapezoidal fuzzy membership functions.

To design similarity measure for general fuzzy membership function including non-convex fuzzy membership function data, other point of view is required. Here, with the help of axiomatic definition of similarity measure proposed by LIU [10], the similarity between the fuzzy sets is considered for two Gaussian-type membership functions.

The area of the shaded region can be considered as a component of the similarity measure. The two items 1-d((∪), [1]_X) and 1-d((∪), [0]_X) will hereafter be denoted as  and . In these two items, [1]_X and [0]_X are denoted by 1 and 0 for the whole universe of discourse, respectively.  denotes the common area of a membership function; whereas  represents the set information between the two membership functions. Hence, a proper similarity measure is obtained by combining the two values. In Theorem 1, the distance measure between fuzzy membership functions  and  is based on the well known Hamming distance measure.

Theorem 1  For any sets , ∈F(X),

s(, )=2-d((∩), [1]_X)-d((∪), [0]_X)  (8)

is the similarity measure, where d satisfies the Hamming distance.

Proofs can be followed easily. From the definition of LIU [10], (S1) refers to the commutativity of sets  and , which is clear from (Eq.(8)) itself. For (S2), d((∩), [1]_X) and d((∪), [0]_X) are equal to 1. Hence, s(,)=0 is easily obtained.

For arbitrary different sets  and , the inequality in (S3) is proved as follows:

s(, )≤2-d((∩), [1]_X)-d((∪), [0]_X)=

s(,)

Finally,  if

s(, )=2-d(, [1]_X)-d(, [0]_X)≥

2-d(, [1]_X)-d(, [0]_X)=s(,)

Similarly, s(, )≥s(, ) is also satisfied.

3.2 Similarity measure for fault detection

    The scatter diagram of coefficients   and  is shown in Fig.2. In the longitudinal mode, in order to obtain the failure status of the elevator, the following data are required: the pitching moment coefficient due to the changes of the elevator deflection  the pitching moment coefficient due to the change of the attack angle , and the lift coefficient due to the change of the attack angle . For constructing the fuzzy membership function, values of   and are divided into eight groups, and the number of data sets is normalized. In Fig.2, it can be observed that the normal and fault values of do not interchange, whereas those of  and  inter- change. By means of scatter diagrams, normal and fault fuzzy membership functions are illustrated in Fig.3. From Fig.2, the control surface stuck is determined by monitoring the value of  which facilitates the discrimination between normal and fault conditions. Data

Fig.2 Scatter diagrams of coefficients (a), (b) and (c)

points are described for elevator struck case, for example, ep10 means elevator 10? fixed. For elevator and ruder left (right) stuck case, e0r110 denotes elevator 0? and rudder left 10? fixed cases [15]. The similarity measure using is as follows:

(F_N, F_F)=2-d((F_N∩F_F), [1]_X)-d((F_N∪F_F), [0]_X) (9)

where  F_N and F_F denote the normal and fault fuzzy membership functions from Fig.3. Furthermore, other similarity measures using and  are defined as

Fig.3 Membership functions of coefficients (a), (b) and (c) with scatter diagram

 and , respectively. Six data points in Fig.2(c) are selected to calculate and analyze the characteristics. Points from a to f of in Fig.2(c) are as follows:

a=4.997 4, b=5.678 0, c=5.754 9, d=5.599 1,

e=5.5623 8, f=5.446 2.

The locations of the six points are in the mixed area under normal and fault conditions. It is necessary to calculate the similarity measure corresponding to the membership values. Hence, the membership values of points a and d are illustrated in Fig.3.

From these similarity measures, the mixed similarity measures are combined as follows:

                    (10)

where  w₁, w₂, and w₃ are the weighting factors for the similarity measure. The value of is classified into normal and fault conditions by  and for point p. Similarly, other similarities  and  are also classified from points a to f (see Table 3).

Table 3 Similarity measures for three coefficients

Table 3 implies that the fault decisions are accurate by similarity computation and comparison. If the membership functions of the parameters have different importance for their values, proper tuning of w₁, w₂ and w₃ will be performed. In Fig.3, the case of an elevator with 50% failure is illustrated, the decreased effectiveness of the elevator is also noticed, and the occurrence of a fault cannot be identified from the scatter diagram, especially for Fig.2(a), even though the data group is close. If an aircraft has a 20%-30% control surface loss as damage from severe circumstance such as  battle, it can be assumed that this can lead to a decrease of by up to 20%-30%. Then, the scatter data shown in Fig.2(a) are closed and mixed, and fault detection becomes more difficult. In such a case, the similarity measure helps to detect the fault.

4 Conclusions

(1) The characteristics of similarity measure are discussed by the derivation from the fuzzy number and the distance measure. The results from fuzzy number are easy to compute; however, they are strictly limited to the trapezoidal or triangular membership functions. The distance-based similarity measure is applicable to general membership functions.

(2) A similarity measure is proposed for the application of longitudinal mode unmanned aircraft vehicle fault detection. The pitching moment coefficients due to the change of elevator deflection, the pitching moment coefficient due to the change of the attack angle, and the lift coefficient due to the change of attack angle are transformed into fuzzy membership functions for measuring similarity. By computing the similarity measure, it is verified that the proposed similarity measure is also useful for the analysis of non convex fuzzy membership functions.

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(Edited by CHEN Wei-ping)

Foundation item: Project supported by the Second Stage of Brain Korea and Korea Research Foundation

Received date: 2008-11-25; Accepted date: 2009-03-18

Corresponding author: LEE S H, PhD; Tel: +82-55-213-3884; E-mail: leehyuk@changwon.ac.kr