Actuator Fault Decoupled Residual Generation on Lateral Moving Aircraft

Implementation of time-scheduled maintenance is not suitable if it is applied for systems with many varieties of heavy workload and harsh environment since on that condition components degrade earlier that those under normal condition. Therefore it has been shifted to condition-based maintenance (CBM). One important aspect, among others, toward implementation of CBM method is fault isolation. The problem investigated in this paper is related todecouple residual generation for actuator fault isolation of an aircraft on lateral movement. The proposed solution for that problem is to implement combination of transformation matrix and special filter. Transformation matrix is used to convert feature locations of actuator faults to signature vectors. Moreover, the signature vectors will be processed further by special filter to generate decoupled residuals. It is assumed that the actuator is the only fault when the aircraft is on lateral movement. The result showed that special filter and transformation matrix can be designed so that the residual of aileron actuator fault is decoupled from the residual of rudder actuator fault.

Another previously proposed method in this area is methods based on fuzzy logic. The main advantage of this scheme is it does not require detailed mathematical equations. As the mathematical equation commonly required to model the system condition is rather complex and requires specific modeling expertise and efforts. Previous scheme implements fuzzy logic for residual evaluation to detect failure in robotic systems [7]. Another example of fuzzy logic implementation in this field is its implementation on Takagi-Sugeno system model [8]. The disadvantage of fuzzy logic in fault detection and localization is it requires ad-ho rules for a specific system condition. The develpment of this ad-hoc rules requires significant domain knowledge which is not transferrable between problem domains.
To address the shortcoming of fuzzy logic implementation for fault detection and localization requires the system mathematical model to increase the applicability of this method to larger range of cases.Several works related to fault detection or fault lozalization/ isolation based on mathematical model can be seen in [9][10]. Meanwhile, to compensate the disadvantage of artificial neural network in fault detection and localization, i.e. its inability to handle fault signal which is not contained in its prior knowledge, the concept of vector with specific direction can beused. This scheme can be implemented by a combination of special filter and transformation matrix for multi-sensor fault detection in web winding system [11]. In this paper, those methods will be modified for the purpose of actuator fault localization, more specifically to generate residual signal from the aileron actuator and rudder actuator which are decoupled in laterally moving aircraft. The benefit of vector concept is it allows us to handle new form of fault signals which is previously unseen, as it only requires the direction of signature vector. The magnitude of signature vector is allowed to vary according to the form of failure, which increases the range of fault signal that it can handles. In other words, the direction of signature vector only corresponds to failure location. Which imply that it is possible to decouple the faultsignal residuals by designing the direction of signature vectors.
The main contribution of this paper is in the use of transformation matrix to restrict the fault signals into separate linear subspaces. The combination of transformation matrix and the specifc filter is designed to generate a dynamics of the filter error signal with separate dynamics for each failure modes. The filter output contains information regarding the actuator fault signal, system state signal, and system input signal. This signal is then further transformed by the transformation matrix to extract the actuator fault signal and convert it into a signature vector. The effectiveness of this method will be shown through design of a decoupled actuator fault detection and localization on laterally moving aircraft.

Proposed Method 2.1. Mathematical Model of Plant with Actuator Failure
A plant experienceing an actuator failure can be modeled as where ( ) is the state vector, ( ) is the output vector, and ( ) is the input vector. The actuator failure effects on the dynamics are modeled by an actuator location matrix which is composed of columns of affected by the failures. The faultsignal is denoted by ( ).

Analysis of Decoupled Residual Generation for Actuator Fault Isolation
In this section, we discuss a modification of method proposed in [11], to handle actuator failure. To generate a residual vector, a special filter and transformation matrix is required. The filter is used to generate filter error signal which contain the actuator fault signal among other signals. The transformation matrix is to convert the actuator failure location signature into a signature vector with fixed direction. The error signal is defined as, where ( ) is the error signal and ( ) is the filter state. Therefore, the filter dynamic model can be written as, with is the filter process matrix, is a transformation matrix relating the value of system output and the filter states, and is a transformation matrix relating the plant input and the filter states. The filter states is designed to capture the possible failure modes and related to the system states by the following equations, One of the function of this specific filter is to generate the dynamics for the error signal between the filter output and the measured system states after transformations, ( ) ( ). This filter is designed such that in its steady state condition, the estimate error is zero if there is no failure. In the condition of failure, the special filter is designed to to generate decoupled residuals. The dynamic of filter error signal is given as, which can be rewritten as, In case of two actuator failures (denoted by and ), the error dynamics is equal to, These equations imply that the filter response depends on the plant input, the plant state, and the fault signals. To ensure that the response only depends on the actuator fault signals, then the following conditions have to be fulfilled, Under these conditions, the error dynamics become, Taking a special case where is a diagonal matrix, then it implies that, where is an identity matrix with two diagonal components. Each of the identity matrix columns is taken as the signature vectors.

Transformation Matrix Calculation
Our main contribution in this work is the introduction of transformation matrix which decouples the error signal. In this section we will discuss the details of calculation for this transformation matrix. From equation 6, it can be observed that matrix is composed of rows and columns, while matrix is composed of rows and columns. Matrix is assumed to directly measure state variables, such that [ ]. Matrix can be denoted as [ ] with is a matrix, while matrix is a ( ) matrix. As can be designed as a diagonal matrix, then eq. 8.1 can be separated into two parts [12], which to simplify the notation can be written as, The vector is the -th rowof matrix . From the discussion, we can observe that is an element of the left nullspace of matrix . Therefore, the bases for left nullspace of needs to be determined to calculate the value of . As in general is not a square full-rank matrix, the left null-space will have to be calculated by utilizing the SVD (Singular Value Decomposition) algorithm [13]. It implies that there exists such that the row vector can be written as, where is the -th basis of the left nullspace of the matrix, is the dimension of the left nullspace. This relation allows us to express equation 9 as, with is the -th column of identity matrix. The relation shows that each can be calculated independently, which leads to easier computation procedure.

Mathematical Model of Laterally Moving Aircraft
The derivation of mathematical model for laterally moving aircraft has been previously studied in [14]. The model is derived for a very large four-engined, passenger jet aircraft CHARLIE operating at 6100m height moving laterally with 0.8 mach airspeed can be modeled as a linear system, with, [ ] (20)

Fault Detection Filter and Transformation Matrix Design
It can be observed from matrix that the state space dimension of the model ( ) is equal to four, while the number of output signal can be observed from matrix ( ) to be equal to two. The input to the system is represented by the two columns of , with the first column corresponds to the aileron while the second column corresponds to the rudder. As it implies that there might be two sources of disturbance (fault signal), then the fault detection filter dimension ( ) is designed to be two. To ensure the stable convergence of the error between the estimated filter output and transformed state space, the eigenvalues of the fault detection filter matrix must have negative real value, which is designed to be at and . Following the previous discussion in section 3.A, the algorithm for filter design can be performed according to the pseudocode given in algorithm 1.
Using the model of lateral moving aircraft and the designed eigenvalue as the input to the algorithm genereates the following transformation matrix and fault detection filter matrix , while the filter input matrix G is equal to,

Research Method
To show the performance of the designed fault detection filter and transformation matrix, in this section we will perform comparison between conventional fault detection method based on comparison between the measured state and the estimated state represented in Figure 1 and the proposed fault detection filter with transformation matrix represented in Figure 2.
In both of the simulated schemes, the aircraft dynamics is represented by the upper blocks, while the lower block represents the fault detection schemes. In the conventional scheme, as the comparison is done in the subspace of system output, there is no guarantee of decoupling between fault signals. While in our proposed method, the comparison is done between the estimated states and the fault detection filter on the subspace selected by transformation matrix. Therefore, we can design the transformation matrix to give error signal ( ) which deliberately separates the fault signal as can be seen in equation 9 and equation 10.

Results and Analysis
The numerical simulation results are shown in Figure 3. The simulation is performed using backward Euler as the discritization method with a time step of 0.001 seconds where the numerical value of the model is given in eq.18-23. The system is given a constant vector of [ ] as its input, and the state value of all zeros as its initial condition. The fault signal is given in Figure 3.b in which the fault signal is composed of a step signal for the aileron and the oscillating signal in the rudder. Due to this fault signal, the output which should be composed of decaying sinusoids converging to the given input signals are continuously oscilliating as it can be observed in Figure 3.a.
The main aim of fault detection method is to detect the existence of this fault signal. The performance of both conventional fault detection method and the proposed fault detection method is given in Figure 3.c and  Figure 3.d, both the step fault signal and the oscillatingfault signal can be reconstructed faithfully as shown by the continuous line and the dashed line respectively. It can be seen that the reconstructed step signal has a fast exponential change rather than an idealstep change. This corresponds to the dynamics of the reconstructed aileron fault signal given by the chosen eigenvalue . On the other hand, careful observation suggests that the reconstructed oscillating fault signal has a slightly shifted phase compared to the actual oscillating fault signal in the rudder which is caused by the chosen eigenvalue . In this simulation, it is true that for more accurate fault signal reconstruction, the eigenvalues of matrix can be chosen to have larger negative values. However, larger negative eigenvalues will result in the fault detection filter becoming more sensitive to noise, hence in practice, the eigenvalues of matrix needs to be carefully chosen.

Conclusion
In this work, a method to design fault detection filter and transformation matrix is proposed. The introduction of transformation matrix improves the conventional fault detection 1893 filter by allowing it to completely decouple the fault signal. The effectiveness of this method is shown by an example of fault detection filter and transformation matrix design to decouple the fault signal on a lateral moving aircraft model. The simulation results showed that the design is able to decouple the aileron residual fault signal and rudder residual fault signal which leads to actuator fault isolation. Some interesting future works would be to combine this novel capability of fault signal decoupling with the control system capability to adapt to fault signal to design a reconfigurable control system.