Fault Isolation and Identification of a Four-Single-Gimbal Control Moment Gyro On-board a 3-axis Stabilized Satellite

Control moment gyros are known for their applications in attitude stabilization. These actuators are susceptible to malfunction, which results in faults and failures. Therefore, diagnosing the faults can improve the reliability of completing a mission while reducing maintenance costs. Thus, a model-based fault diagnosis method is proposed here. The intended algorithm is an enhanced version of previous work by the author. The enhancement employs a condensed approach to alleviate the delay caused by the filter’s confidence in its estimations. A case-study on a closed-loop controlled satellite is provided along with an extensive Monte Carlo simulation to evaluate the proposed method’s performance. The results show that the enhanced method can achieve superior performance while requiring less computational resources by eliminating extra grid search loops.


INTRODUCTION
A control moment gyros (CMG) consists of a flywheel gimballed about its axes. The torques generated by the rotation of the flywheels' rotation axes provide a means to control the satellite's orientation in space.
Many researchers have examined fault isolation and identification (FII) in the past few years. Parameter estimation methods (Jiang et al., 2008;Li et al., 2011;Rahimi et al., 2015Rahimi et al., , 2020Rahimi, Dev Kumar, et al., 2019;Ye et al., 2015), can achieve fault detection and isolation at the same time as these methods estimate the system parameters directly which offers time, source, and size of the fault altogether.
To the author's knowledge, the FII framework for CMGs as satellite actuators has not been explored extensively in the published literature. Rahimi et al. (2020) have proposed a hierarchical model-based fault detection, isolation, and identification (FDII) approach that employs a binary search to update the parameter estimation filter's posterior covariance matrix elements. The proposed method in (Rahimi et al., 2020) provides an enhancement on a previously developed detection and isolation method in (Rahimi et al., 2017) by reducing the loops in the search for posterior covariance matrix elements from all elements to only the diagonal elements and further reducing it by moving from a multiple-value grid search to a binary grid search. The enhanced method in (Rahimi et al., 2020) provides speed and accuracy improvements in the estimator compared to (Rahimi et al., 2017). However, the author believes that further improvement can be made in the estimation process. Therefore, this paper proposes an enhancement on the method provided in (Rahimi et al., 2020) by further simplifying the grid search process and using fewer loops to reduce the estimator's computational requirements without sacrificing the performance. Furthermore, the performance of the proposed enhancement is evaluated on a satellite stabilized by a four-single-gimbal-CMG (4SGCMG) cluster in comparison to the previous methods (Rahimi et al., 2020) and (Rahimi et al., 2017) to provide evidence on the superior performance of the novel approach.
The remaining of this manuscript contains the problem definition in Section 2. The proposed enhancement in Section 3. The case-study in Section 4. The simulation setup, along with the results and discussions in Section 5, and finally, the conclusions in Section 6.

PROBLEM DEFINITION
Considering the nonlinear system: ∈ ℝ denote state, control input, system parameters, and measurements, respectively.
∈ ℝ , ∈ ℝ , ∈ ℝ denote the additive process noise for states, additive process noise for parameters, and additive measurement noise, respectively.
(•) and (• ) denote a nonlinear process model and a nonlinear measurement model, respectively. The objective is to develop an estimation method to isolate the location and identify the severity of the faults in the system with less computation time and more accuracy than (Rahimi et al., 2020) and (Rahimi et al., 2017).
This requires that the system component faults can affect the physical system parameters (Sobhani-Tehrani et al., 2014), under which the Eq. (1) will have: (2) where 0 ∈ ℝ denotes the nominal parameter values and ∈ ℝ denotes the fault parameters containing fault elements. Employing Eq.
(2), one can treat the nonlinear FII as a parameter estimation problem.

METHODOLOGY
The proposed FII in this study focuses on the second stage of a hierarchical approach by Rahimi et al. (2020) to propose a novel adaptation mechanism for the FII. Hence, fault detection is briefly explained, while the adaptation mechanism is further detailed.

Fault Detection
To detect the faults in the system, a residual is generated using (Rahimi et al., 2017) =̂−̂ (3) where ̂ and ̂ are the estimated outputs for the system and model, respectively, with scalar defined as = (4) The threshold 0 (Rahimi et al., 2017) triggers a fault when > 0 . The next step in the process is updating the system model (Ω ) to match the current state of the system referred to as FII, detailed in the next section.

Filter Adaptation Algorithm
The posterior estimates covariance matrix elements represent the confidence in the estimated values (Bisht & Singh, 2014). The issue arises as the filter converges and subsequently becomes slow to adapt to sudden changes. To address this, elements must be adapted to avoid latency following a fault detection. To achieve this, Rahimi et al. (2017) proposed the covariance adaptation algorithm (CAUKF) using the unscented Kalman filter (UKF) algorithm (2017). Rahimi et al. improved the method using a binary grid covariance adaptation algorithm (GAUKF) (2020), where the idea is to only consider the binary choice of "increase" or "no change" for the diagonal elements of the . In this study an improved version of the adaptation is proposed as simplified binary grid covariance adaptation algorithm (SGAUKF), as outlined in Table 1. For the proposed approach in this study, in step 2 of Table 1, the choices (combinations) are only formed for the even or odd diagonal elements. This would reduce the order of options by half resulting in a loop with 2 /2 instead of 2 compared to the CAUKF. In Table 1, is used as a provisional variable for finding diagonal elements of the .
is the binary combination to add to the previous step's posterior covariance −1 , is the candid superimposed posteriori covariance, and is the residual for estimates of the next time step using as posterior covariance. The rationale for only adjusting the odd or even elements in the diagonal for the posterior matrix is the fact that since the parameters that are being estimated are interrelated through the system dynamics, once the filter is adjusted to less confidence in half of its elements, it will try to compensate for the other elements as well. Hence, not having to check for all binary options reduces the computation time without sacrificing the performance. In some cases, it provides better performance, as will be seen in the results section.

Fault Isolation and Identification
The FII can be accomplished using Eq. (2). The number of possibilities corresponds to the sum of all -combinations of the faulty units as where, at time step , denotes the number of faulty units, and is the total number of available units.
A dual state/parameter estimation (Rahimi et al., 2020) is employed to estimate the system states and fault parameters in the system and achieves the FII in one step. At this stage, ̂ and ̂ are introduced as parameter estimates for the system and model, respectively, using Ω . A normalized residual is calculated using Eq. (4) as = , where is the weight matrix for normalization.

CASE STUDY: FAULT DIAGNOSIS FOR CMGS
The proposed FII is evaluated using an ACS of a three-axis stabilized satellite with a 4SGCMG cluster. The simulation setup is shown in Figure 2, with details as follows:

Controller
A sliding mode controller is used to attain the desired attitude of ∈ ℝ 4×1 and ∈ ℝ 3×1 (Kumar et al., 2018). The quaternion and angular velocity errors are obtained as   (Kim et al., 2015).

Fault Scenario
The total number of possibilities for faulty scenarios using Eq. (5) adds up to 16. To easily refer to each scenario, Table  2 assigns each possibility a number where FW abbreviates faulty wheels. For example, in scenario 0, no wheel is faulty, and in scenario 15, all wheels are faulty.
Having the building components of the FII detailed so far, the proposed fault diagnosis for this case study can be formulated as follows:

RESULTS AND DISCUSSIONS
For performance evaluations, numerical simulations were conducted in MATLAB with an integration step size of = 0.01 for a total simulation time of 200 sec. The simulation was run on an Intel® computer with i7-4790 3.60 GHz CPU and 16.00 GB RAM. A list of other simulation parameters is provided in Table 3 where the Euler angles to quaternion conversions follow the 3-2-1 sequence. The initial and desired conditions for roll, pitch, and yaw are (−90°, 10°, 10°) and (0°, 0°, 0°), respectively.
= 10 −5 and = 10 −4 are used for the UKF. Additive noise is considered for the measurements in Eq. (21) and (22) when discussing the signal to noise ratio (SNR) in the following sections. The fault scenario considered is an outof-phase abrupt, as detailed in Table 4. The system response and fault detection results are not provided as this study focuses on isolation. Hence, only the results for FII in Table  4 scenario with Monte Carlo (MC) simulations are provided. In the following sections, simulation results were obtained for three SNRs: 45dB, 55dB, and 75dB; however, only the 45dB (worst-case scenario) results are reported unless explicitly mentioned otherwise.  Table 4. Case-study out-of-phase abrupt fault

Fault Isolation and Identification
where the (•) outputs summation of the matrix diagonal elements and is the total CM diagonal elements. The results shown in Table 7 compare the accuracy of different methods by Rahimi et al. as discussed earlier, using the definition provided in Eq. (24). Table 7 shows the inferior CAUKF accuracy in a closedloop controlled system and the superior performance of the SGAUKF compared to both the CAUKF and GAUKF. The inferior accuracy of the CAUKF in such a system setting is owed to its inability to adapt as quickly to unknown system changes as compared to the other two methods GAUKF and SGAUKF. Table 8 provides the MC simulations' average execution time for each method. The results show that the CAUKF has the highest execution time since it must loop through extensive grids. GAUKF, however, demands less time than CAUKF since it is only doing a binary grid search; nevertheless, SGAUKF has further decreased the computation time by cutting the loops required for the posterior covariance from 2 to 2 /2 while also improving the accuracy. Therefore, SGAUKF improves accuracy and reduces computation time compared to others. The reduction in execution time is not extreme due to the processing power allocation required by other parts of the algorithm that still take time and are more dominant.   Table 9. RSME comparison where ̂ and are the estimated and true parameter values, respectively. is the number of parameters, and and denote the start and end time for the RMSE calculation in the simulation, respectively.

The
and are introduced to form the period before a fault occurs (pre-fault), during fault (faulty), and after the fault recovery (post-fault) and calculate RSMEs for these periods and list them in Table 9. Table 9 values are calculated for each parameter and averaged over the range of pre-fault, faulty, and post-fault periods. It can be seen from the results in Table 9 that the SGAUKF has a superior performance in comparison to other methods. It is evident that during the fault, RSMEs are larger, and the difference is directly proportional to the noise level. Table 10 summarizes the computation time for the FII, where most allocation is in the isolation module given the iterative composition. The computation time increases with the noise level as more iterations are required to complete the adaptation in the posterior covariance matrix.

CONCLUSIONS
A modified adaption method (SGAUKF) was proposed in this paper to adjust the elements of the UKF posterior matrix so that abrupt changes in system parameters can be isolated promptly. for GAUKF and 75% for CAUKF. Despite insignificant accuracy improvement, computation time was decreased from 8.74 to 6.36 sec, i.e., ~27%, compared to GAUKF, proving the SGAUKF as a promising option for FII of nonlinear systems with comparable accuracy to GAUKF and less computation time simpler structure requirements.

ACKNOWLEDGMENT
The author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and the University of Windsor. For the last ten years, he has been involved in various industrial research, technology development, and systems engineering projects/contracts related to the control & diagnostics of satellites, UAVs, and commercial aircraft subsystems. His research work is primarily focused on model-based and data-driven fault detection, diagnostics, and prognosis; machine learning and intelligent systems; linear & nonlinear controller/observer design; and avionics, sensors, and measurement technologies. More recently, he has started working on smart factories and intelligent industrial automation in industry 4.0 in collaboration with industrial partners using computer vision, machine learning and artificial intelligence. He is a member of IEEE and AIAA institutions in addition to the PHM society.