QIAN WU1, XIAOFENG LI1,*, YI ZHOU1, JUN YUAN1, YONG WANG1
1, Beijing Institute of Control Engineering, Beijing,100094, China.
Xiaofeng_Li_bice@163.com
First author: QIAN WU, wuqian945@163.com
Second author and corresponding author: XIAOFENG LI, Xiaofeng_Li_bice@163.com
The third author: YI ZHOU, yizhou_bice@163.com
Fourth author: JUN YUAN, Yuanjun502@gmail.com
Fifth author: YONG WANG, yongwang_bice502@163.com
Acknowledgement:
This research was supported by the National Natural Science Foundation of China (NSFC)project – Research on On-Orbit Adaptive Evolution Theories and Methods for Space Vehicle Control Software.
Abstract
This research presents a systematic framework to assess satellite attitude control systems in tracking mode. It aims at putting right the severe inadequacies found in assessment methods. The method developed here not only makes use of grey targets but also incorporates a modified Principal Component Analysis, thus relieving to some extent data insufficiency and uncertainty in satellite telemetry data as well as making environmental space a natural habitat for such operations. A framework for assessment, including six parameters—attitude expectancy, relative accuracy, stability, response time, angular velocity, and energy cost—was established. A modified Principal Component Analysis such as this relieves sources of error brought on by a blur, but it also provides for more balanced weighting. The grey number transformation quantifies measurement uncertainties and environment-induced perturbations inherent in operating in space environments. Test performed using real satellite telemetry data from 44 operational sequences indicates an improved discriminatory ability, yielding a maximum-minimum difference of 0.4362, compared to 0.2965 realized by existing TOPSIS technique-based assessment. The methodology achieves compatibility degree of 0.9238 while maintaining noise tolerance of 22.3%, which significantly outperforms existing standard techniques by a notable margin. An assessment of computing efficiency rationalizes application feasibility, exhibiting linear scaling behavior and reasonable processing overhead. The system successfully identifies optimum performing sequences defined by off-target distance ranges from 0.2502 to 0.3355, simultaneously adequately discriminating poorly performing systems. The outcome provides essential knowledge relevant to satellite mission planning, operational improvement, and development of automatic health monitoring facilities.
Keywords: satellite attitude control, performance evaluation, grey-target decision, principal component analysis, tracking mode, uncertainty quantification
- Introduction
Determination and the attitude control systems are an indispensable part of modern space projects; spacecraft need to be able to accurately locate themselves after all. But in any new situation they enter, these abilities must continue working reliably. With the ever-growing complexities of the current satellite mission tasks, combined with increased performance specifications and the corresponding operational requirements, it is important to establish a stringent performance evaluation process to achieve improved system practicality (Zhang et al., 2024).
There is also a critical demand for stringent evaluation protocols surrounding satellite systems. They have to work in an adaptive mode within diverse operational domains that are shot through with uncertainties and disturbances; hence, an effective evaluation framework becomes essential. Traditional satellite attitude control system effectiveness estimation techniques typically focus exclusively on single or isolated examples of component or operational functionality in a restricted mission environment. Therefore, these approaches are generally unable to make inclusive recognitions of the combined effects on global system behavior in this set of diverse mission scenarios.
They are also inadequate to perceive the uncertainty and incompleteness within attended telemetry data during its mission operating timescale extensions beyond a few minutes or seconds. Over such long timescales, changes in environmental parameters and noise in measurement terms typically make the precision of measurement data plummet (He et al., 2021). In addition, the complexity of quantifying general system behavior is further complicated by the existence of multiple representative operational modes, which need a simultaneous assessment in different performance terms to be evaluated by their corresponding weight terms.
The attitude determination methods have progressed from traditional sensor-based ones to multi-sensor integration systems, involving multiple measurement inputs for increased accuracy and reliability. Furthermore, the use of cheap missions in constellations of non-gyro-based satellites has been projected as possible replacements for expensive mission implementation that could provide a single array of magnetometer data for mission design (Carletta et al., 2020). The process of fusion requires designing performance assessment methods specific to particular domains, which can best utilize special characteristics and shortcomings typical to particular types of sensor modalities, maintaining data integrity and quality at all times.
Unscented Kalman filters demonstrate outstanding performance in handling nonlinear dynamics over a wide range of magnitudes due to their modern design (Garcia et al., 2019). In situations that are more challenging, like faulty sensors or unmodeled external disturbances, sophisticated filtering methods could drive significant improvements over current state-of-the-art baseline techniques due to their increased guaranteed robustness based on a novel architecture, while still achieving significant results in less problematic scenarios (Chu et al., 2025). In studies dealing with methodologies developed to measure advanced filters, it becomes imperative to develop all-inclusive platforms that can measure accuracy in addition to supporting fault tolerance and computational effectiveness in a variety of operating scenarios. In dealing with sensor fusion, it becomes imperative to develop general assessment protocols to account for the functional properties of all input sensors in addition to operating effectiveness of the entire system. Formation flying maneuvers represent extremely demanding scenarios in which attitude control systems have to be tightly synchronized while supporting the distinct mandates of each vehicle. The application of distributed control schemes requires sophisticated assessment techniques to quantify single-vehicle performance in tandem with aggregate group dynamics (Fan & Huang, 2021). The extension of reinforcement learning to the case of formations has shown promise in handling sophisticated constraint environments while maintaining group integrity (Cai et al., 2024).
The assessment techniques used by satellite attitude control systems have to meet especially stringent specifications not adequately met by available methodologies. The inherent nondeterminism typical of space environments, including transitory noise found in measurement and systems degradation, makes it difficult to define adequate performance metrics and sound assessment standards. Traditional measurement procedures often imply deterministic models, which can potentially ignore real operating environments’ stochastic nature, thus not delivering performance evaluations truly representative of the actual ability of the system. The performance of attitude control systems is defined in many dimensions, thus posing a grand challenge. This is because of the many intricate relationships between the numerous performance parameters, including accuracy, stability, response time, as well as the consumption of energy, which render in-depth analysis as well as measurement extremely burdensome. Traditional measurement approaches, in most cases, favor specific performance parameters, which most of the time lack the capability of assessing the performance of the system in all aspects comprehensively.
Existing space missions call for the creation of evaluation strategies that can efficiently handle incomplete information, variable operating conditions, as well as provide reliable performance measures in the presence of uncertainty. There is a pressing need for the creation of thorough evaluation frameworks capable of surmounting these challenges, alongside computational effectiveness and user friendliness, in consideration of the recent developments in satellite attitude control systems, in particular. Such evaluation frameworks must combine numerous control strategies as well as sensing technologies, in combination with mission-specific parameters characteristic of new space initiatives, while at the same time providing essential performance measures in the design of the system as well as decision-making procedures. - Theoretical Foundation
2.1 Gyroless Satellite Attitude Dynamics Model
The basic principles in the field of satellite attitude determination and control systems include basic mathematical concepts, measurement techniques, and signal processing theories, which have a crucial effect on the operational performance and efficiency of these systems. An in-depth understanding of these base theories is of immense importance to the development of efficient evaluation methods that maximize the performance and efficiency of modern spacecraft systems.
The determination of satellite attitude in gyroscopic sensor-free environments poses a variety of challenges and requires a complete understanding of the rotational dynamics pertinent to gyroscopic sensor-free spacecraft. Euler’s motion equations enable the description of basic attitude dynamics and reveal the connection between applied torque and resulting angular motion. In situations where gyroscopes are not available, the system will have to rely on other sensing mechanisms to determine angular velocity in addition to its classical attitude parameters.
The attitude kinematics can be formulated by using quaternions, to allow describing spacecraft orientations easier without the complications caused by singularities. Thus, a gyroless system can be defined by quaternion evolution, which is defined by the following differential equation:
Where represents the unit quaternion and denotes the skew-symmetric matrix formed from the angular velocity vector . In gyroless applications, the angular velocity must be estimated through indirect measurements and filtering techniques.
SVD-Laplace Particle Filter is one of the potential methods of attitude estimation that reduces the need for gyroscopic data based on the use of magnetometer measurements along with spacecraft dynamic models (Dahia et al., 2023). This method efficiently controls the nonlinear dynamics involved in attitude estimation while providing fast computation for onboard applications. Also, the particle filter design is able to efficiently manage the non-Gaussian properties of the noise as well as the uncertainty of magnetometer-based measurement sensors.
Usage of the Modified Unscented Kalman Filter has been identified as improving effectiveness in non-gyroscope satellite configurations, specifically in the case of failed sensors and deteriorating measurement precision (Pourtakdoust et al., 2022). These methodologies maintain estimation precision by skillfully dealing with the failure of sensors as well as the innate space environment’s noise, which is commonly detrimental to the magnetometer-derived data obtained.
2.2 Measurement System Principles
Modern satellite attitude determination systems combine various types of sensors in various operational conditions in order to optimize their measurement functionalities. An example of this is attitude determination systems using magnetometers, in which the Earth’s magnetic field is adopted as the reference vector. This provides three-dimensional measurements of the magnetic field, which can in turn be interpreted in order to determine the spacecraft’s attitude relative to the magnetic field’s direction.
Magnetometer calibration is one of the essential components of the effectiveness of measurement systems, particularly in applications that call for high accuracy. TRIAD+ filtering configurations provide essential paradigms for the simultaneous evaluation of attitude as well as magnetometer calibration (Soken & Sakai, 2020). These approaches are best suited in the compensation for systematic errors as well as the control of bias drifts, which can seriously impact measurement precision for longer periods of system operation.
Star sensors embedded in gyroscopic arrangements feature sophisticated measurement platforms combining extremely high-precision stellar data with continuous inertial information. The assessment techniques used by satellite attitude control systems have to meet especially stringent specifications not adequately met by available methodologies. The inherent nondeterminism typical of space environments, including transitory noise found in measurement and systems degradation, makes it difficult to define adequate performance metrics and sound assessment standards. Traditional measurement procedures often imply deterministic models, which can potentially ignore real operating environments’ stochastic nature, thus not delivering performance evaluations truly representative of the actual ability of the system. These approaches address systematic errors as well as bias drifts that can significantly impact measurement precision over extended operational periods，as shown in Table 1.
Table 1. Measurement System Characteristics and Performance Parameters
Sensor Type Measurement Principle Accuracy Range Update Rate Primary Limitations
Magnetometer Earth’s magnetic field vector 0.1° – 1.0° 1-50 Hz Magnetic disturbances, field model errors
Star Tracker Stellar position references 0.001° – 0.01° 0.1-10 Hz Sun/Earth interference, processing delays
Sun Sensor Solar illumination direction 0.01° – 0.5° 1-100 Hz Eclipse periods, reflected light
Gyroscope Angular velocity measurement 0.001°/s – 0.1°/s 100-1000 Hz Drift, temperature sensitivity
Solar Panels Photovoltaic current variations 1° – 5° 1-10 Hz Limited to sun-pointing applications
Additionally, the research of attitude determination using solar panel arrays in lieu of hardware-based solar sensors presents an effective method to derive orientation within mission applications predicated on aligning with the sun. The Unscented Kalman Filter applied as a sensor of solar radiation based on photovoltaic array systems has already demonstrated good results in a wide variety of mission scenarios (Baroni, 2020). It uses attitude estimation by means of solar radiation-induced direction changes on photovoltaic arrays with a software-only approach; there is no additional hardware-based solar sensor support.
Adaptive fusion sensors make it possible to integrate heterogeneous measurement sources efficiently, thereby ensuring adaptability where sensors have failures or reduced performance. Advanced fusion methods with Adaptive Network-Based Fuzzy Inference Systems ensure high accuracy in many applications, including satellite attitude estimation (Fakoor et al., 2021). Those methods adjust the fusion weights systematically based on measurements’ quality and the existing conditions of the measurement processes.
2.3 Signal Processing and Estimation Theory
Modern filtering techniques provide the theoretical foundations for present attitude estimation tools, providing mathematical constructs to bring about optimal state estimation in the presence of uncertainty. The theoretical foundation provided for the Unscented Kalman Filter avoids the pitfall of linear approximations by utilizing the formulation of the Unscented transform as distinct, deterministic sampling schemes, hence maintaining the higher-order statistical moments inherent in nonlinear functions.
The Unscented Transform generates sigma points according to specific rules that keep the mean as well as the covariance of the starting distribution but efficiently retain nonlinear phenomena. For an n-dimensional state vector, the computation of the sigma points is given by:
Where and represents a scaling parameter that controls the spread of sigma points around the mean state estimate.
Complementary filtering has been generally accepted as an effective methodology within the area of sensor fusion, mainly because it has low computational cost within certain conditions. Adaptive generalized complementary filters have improved ability to handle time-varying situations simultaneously and also overcome intrinsic uncertainties inherent within sensor data, enabling real-time operation (Wu et al., 2025). These filters combine efficiently the pros of high-pass and low-pass complementary filtering methods, thus efficiently optimizing data fusion processing from a set of heterogeneous sensors.
Particle filtering methods show remarkable effectiveness in dealing with non-Gaussian probability distributions and the complex nonlinear behavior typical of complex systems. Specifically, the SVD-Laplace Particle Filter considerably reduces the amount of computations involved in standard particle filtering methods but ensures a very good level of estimation precision remains intact (Dahia et al., 2023). The use of singular value decomposition aims to enhance the efficiency of particles simultaneously dealing with issues of computations.
Multi-sensor fusion theory offers a mathematical model that enables effective measurement data integration originating from different sources. Multi-sensor fusion-based methods find application in the field of optoelectronics (Kong et al., 2022). Such methodologies efficiently address the intricacies involved in the combination of sensors with varied attributes, update rates, and degrees of accuracy in the collective estimation schemes.
Optimal filtering techniques comprehensively cover measurement inaccurances along with cases of faulty sensors via the implementation of adjustable processing techniques. New applications of the robust Unscented Kalman Filter maintain estimation accuracy in adverse conditions (Chu et al., 2025). These involve incorporating mechanisms for accommodation of outliers, so that faulty measurements do not substantially impair estimation performance.
Pivot-based fusion of sensors represents new strategies specifically developed for low-resource platforms, e.g., mini-satellites. They optimize computation without reducing estimation quality (Pham et al., 2021). Pivot-based method identifies, in real time, the dominant sensors based on quality assessments as well as the availabilities, allowing for graceful performance degradation in the case of partial failure of the sensors.
The conceptual foundations provided in this chapter set the mathematical foundations essential for understanding gyroless satellite attitude determination systems, as well as the pertinent issues related to estimation and measurement. They constitute the foundation for formulating methodologies for performance evaluation as well as optimization strategies, capable of efficiently investigating as well as improving system performance in numerous operational scenarios as well as mission specifications.
- Methodology and Experimental Design
3.1 Performance Evaluation Methodology
3.1.1 Performance Indicator System Construction
This chapter outlines the comprehensive method of the evaluation of satellite attitude control system performance in tracking mode, combining grey-target decision theory with upgraded principal component analysis to address ambiguity as well as incompleteness of information in telemetry data. The research design framework provides systematic validation planning for the proposed evaluation method.
Performance evaluation indicator system for satellite attitude control systems in tracking mode includes six fundamental dimensions that comprehensively outline the system’s effectiveness. According to the mission’s demands as well as the operation’s qualities, the system of indicators contains various factors, such as attitude pointing precision, relative attitude pointing precision, pointing steadiness, attitude control response time, angular velocity during maneuvering, as well as the consumed energy in attitude control (Zhang et al., 2024).
Attitude pointing accuracy represents the deviation between expected and actual pointing directions after satellite attitude maneuvers. The pointing error can be expressed as:
Where represents the total rotation matrix including errors, denotes the ideal rotation matrix, and represents the initial unit vector. The attitude pointing accuracy is determined as .
Pointing stability characterizes the rate of attitude angle changes during stabilized operation periods. The three-axis attitude stability is calculated as:
Where represents instantaneous angular velocity within steady-state time, and denotes the time duration after maneuvering completion. According to mission requirements and operational characteristics, the indicator system includes attitude pointing accuracy, relative attitude pointing accuracy, pointing stability, attitude control response time, maneuvering angular velocity, and attitude control energy consumption, as detailed in Table 2.
Table 2. Performance Indicator System for Satellite Attitude Control System Evaluation
Performance Indicator Definition Calculation Method Unit Indicator Type
Attitude pointing accuracy Maximum pointing deviation degrees Cost-based
Relative pointing accuracy Normalized pointing error ratio Cost-based
Pointing stability Angular velocity variation deg/s Cost-based
Response time Attitude adjustment duration seconds Cost-based
Angular velocity Maneuvering rate deg/s Benefit-based
Energy consumption Control effort joules Cost-based
3.1.2 Grey Number Transformation
Performance indicators originate from distinct sources and exhibit uncertainty characteristics due to dynamic environmental factors, sensor noise, and measurement limitations. Grey number transformation addresses these uncertainties by uniformly characterizing indicator values using generalized standard grey numbers.
For any generalized grey number where , the conversion to generalized standard grey number follows:
Where represents the white part, represents the grey part, and denotes the unit grey factor.
In attitude tracking mode, the further each indicator deviates from expected values, the greater the uncertainty impact. All standardized indicator values can be transformed according to:
Where represents the statistical mean of indicator values. The probability distribution of the unit grey coefficient is generally assumed as when sufficient information is unavailable.
3.1.3 Standardization Process
Satellite data often exhibit inconsistencies in units and measurement scales, necessitating standardization before evaluation. Performance indicators are categorized into benefit-based and cost-based indicators based on their optimization direction. For benefit-based indicators, larger values are preferable, while cost-based indicators favor smaller values.
The standardization process follows different formulations for each indicator type. For benefit-based indicators:
This transformation uniformly converts all indicators into the [0,1] interval, where 1 represents optimal performance and 0 indicates failure to meet target requirements.
3.2 Grey-Target Decision Model
3.2.1 Improved Principal Component Analysis
Traditional principal component analysis can be affected by indicator correlation, leading to inaccurate weight determination. The improved method addresses correlation amplification effects by modifying the calculation approach for principal component weights.
The weight of indicator in traditional principal component analysis is expressed as:
Where represents eigenvalues of the indicator covariance matrix and denotes corresponding eigenvector elements.
3.2.2 Weight Determination Method
The improved principal component weight calculation method incorporates a correction factor to reduce correlation effects:
Where the correction factor is defined as:
This modification reduces the influence of high correlation coefficients on weight calculations, providing more balanced weight distributions across performance indicators.
3.2.3 Target Distance Calculation
The grey-target decision model evaluates performance by calculating distances between indicator sequences and optimal target centers. The target center is determined by comparing indicator performance across different operational processes:
where represents the weight of the -th indicator. Smaller target distances indicate superior performance levels.
3.3 Experimental Design and Validation Framework
3.3.1 Simulation Environment Configuration
The experimental validation utilizes telemetry data from Earth observation satellites operating in sun-pointing mode during attitude tracking operations. The simulation environment incorporates realistic operational conditions including orbital dynamics, environmental disturbances, and sensor characteristics representative of actual mission scenarios.
Figure 1. Performance Evaluation Methodology Flowchart
Figure 1 illustrates the complete methodology flowchart, demonstrating the systematic approach from data collection through performance evaluation. The flowchart emphasizes the parallel processing of weight determination and target center identification, which subsequently converge for final performance assessment.
3.3.2 Test Scenarios and Comparison Methods
The experimental validation encompasses multiple operational scenarios to comprehensively assess methodology effectiveness across diverse conditions. Primary test scenarios include normal sun-pointing operations, eclipse transition periods, large-angle maneuvering sequences, and degraded sensor operation modes. Each scenario provides unique challenges that test different aspects of the evaluation methodology.
Comparison methods include traditional performance evaluation approaches such as the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) and Fuzzy Comprehensive Evaluation (FCE). These established methods provide baseline references for assessing the improved grey-target decision methodology. The TOPSIS method evaluates system effectiveness by calculating relative proximity between evaluation metrics and superior solutions, while FCE utilizes fuzzy mathematical theory for comprehensive assessment under multiple influencing factors.
The experimental design incorporates 44 complete operational datasets extracted from sun-pointing modes under attitude tracking patterns, spanning operational periods from August 2021 to October 2021. This dataset provides sufficient diversity to validate methodology robustness across varying operational conditions and environmental disturbances.
3.3.3 Performance Metrics and Evaluation Criteria
Methodology validation employs multiple performance metrics to assess evaluation accuracy and reliability. Compatibility degree measurements determine how closely evaluation results align with alternative methods, with higher compatibility indicating greater representativeness and accuracy. The compatibility degree for method is calculated as:
Where represents the rank correlation coefficient between methods and .
Deviation analysis quantifies the extent to which assessed values differ from results obtained through alternative methods. Smaller deviation values indicate greater reliability and consistency with established evaluation approaches. The deviation is computed as:
Where and represent evaluated performance values for the -th process using methods and respectively.
The measure checks how successful tests and their measuring scale are under varied environments at picking out strong-performing units from weak ones alike. It shows that when a large value is obtained through one method, there becomes an increased ability to distinguish between high-performing units and low-performing ones. The measure evaluates how sensitive the test method is to results of performance and the ability of this to produce strong correlation output. Strong results can prove that the grey target decision-making approach used in this paper has better performance in the evaluation of various parameters.
In addition, it can be both computationally efficient and practical for engineering to use any higher power application. Nosetabs this way, that way airflow guiders is subjected to what extent System. Finally, an experimental setup is designed with a wide range of theoretical validity and engineering practical admissibility, thus providing data certification on the performance of the method under many different operating conditions or mission requirements.
- Results Analysis
4.1 Attitude Determination Accuracy Analysis
This chapter provides a thorough assessment of results obtained from validating performance assessment methodology, focusing on gray-target decision support effectiveness based on improved principal component analysis. The research covers investigations into accuracy of attitude determination, assessments of computing efficiency, and investigations into robustness within different operating modes, based on real satellite telemetry data within sun-point tracking scenarios.
The analysis of attitude determination precision shows significant improvements in evaluation accuracy and the ability to discriminate over current methods. Performance measures extracted from the suggested approaches reflect increased effectiveness under a variety of operational conditions and suggest an increased ability to discriminate between efficient and inefficient attitude control systems. Results of performance evaluation tests for various operating conditions reveal distinct accuracy characteristics of attitude control configurations. The grey-target decision model successfully identifies optimal performance processes with off-target distances ranging from 0.2502 to 0.3355 for top-performing systems, while poorly performing processes exhibit distances between 0.5498 and 0.6864. This discrimination range demonstrates the methodology’s effectiveness in capturing performance variations across diverse operational conditions.
The S14 process achieved the best overall performance with an off-target distance of 0.2502, exhibiting excellent performance across all six evaluation indicators. Attitude pointing accuracy reached 0.643, relative accuracy achieved 0.967, and energy consumption attained optimal levels at 1.000. Statistical analysis reveals that attitude pointing accuracy and attitude control response time demonstrate the highest sensitivity coefficients of 0.3753 and 0.3291 respectively, indicating their dominant influence on overall system performance assessment.
Figure 2. Performance Correlation Analysis
Figure 2 presents comprehensive performance correlation analysis across key evaluation dimensions. Panel A demonstrates the strong negative correlation between pointing accuracy and off-target distance (R² = 0.72), confirming the methodology’s ability to capture accuracy variations effectively. Panel B reveals the relationship between response time and overall performance assessment, showing moderate correlation (R² = 0.58) with increased scatter for mid-range processes. Panel C illustrates energy consumption impact on performance evaluation, exhibiting strong correlation (R² = 0.81) particularly for extreme performance cases. Panel D validates ranking consistency between the proposed method and TOPSIS approach, with correlation coefficient of 0.89 indicating strong agreement while maintaining enhanced discrimination capability.
Detailed statistical analysis reveals significant improvements in error distribution characteristics. The proposed methodology achieves 68% of measurements within acceptable error ranges, compared to 52% for traditional approaches. High-accuracy measurements increased from 23% to 31%, demonstrating enhanced precision in performance assessment. The coefficient of variation for attitude pointing accuracy decreased from 0.34 in traditional methods to 0.26 in the proposed approach, indicating more consistent evaluation results across different operational processes.
4.2 Computational Efficiency Analysis
Computational efficiency assessment demonstrates that the proposed methodology maintains practical applicability while providing enhanced evaluation capabilities. Processing time analysis reveals acceptable computational overhead relative to the significant improvements in evaluation accuracy and discrimination capability.
Execution time measurements across different dataset sizes show approximately linear scaling characteristics for the proposed methodology. Average processing time for the complete 44-process dataset requires 2.8 seconds on standard computational hardware, compared to 1.2 seconds for traditional TOPSIS and 1.8 seconds for fuzzy comprehensive evaluation methods. The improved principal component analysis contributes approximately 0.8 seconds additional processing time, representing acceptable overhead considering enhanced evaluation quality.
Figure 3. Computational Efficiency Comparison
Figure 3 shows the comparative computational efficiency of several evaluation methods. Our proposed method has linear scalability with respect to dataset size, resulting in manageable processing time even for large datasets. Even though it requires roughly 2.3 times the processing time of TOPSIS, the increased discrimination power and efficient treatment of uncertainty justify this computational investment for critical performance evaluation applications.
Memory utilization analysis indicates peak usage of 152 MB during grey number transformation processes, remaining well within practical limits for onboard implementation scenarios. CPU utilization patterns show efficient resource management with peak usage of 76% during covariance matrix eigenvalue computations. Parallel processing potential analysis reveals that grey number transformation operations can be effectively distributed, providing scalability options for large-scale evaluation scenarios.
4.3 Robustness Analysis
Robustness testing confirms the methodology’s stability under various perturbation conditions and parameter variations. The evaluation framework maintains consistent performance across diverse operational scenarios and demonstrates graceful degradation under adverse conditions.
Sensitivity analysis across key methodology parameters reveals stable performance characteristics within practical operational ranges. Grey coefficient distribution parameter variations (±25% from nominal values) produce maximum evaluation result changes of 4.1%, demonstrating acceptable robustness for real-world applications. Weight determination stability testing shows that covariance matrix perturbations up to 18% maintain evaluation ranking consistency above 88%.
Figure 4. Robustness Performance Analysis
Figure 4 demonstrates robustness performance across different noise levels for various evaluation methods. The proposed methodology maintains ranking correlation above 0.84 even with 20% noise contamination, significantly outperforming traditional approaches. At 15% noise level, the proposed method retains 88% ranking consistency compared to 76% for TOPSIS and 71% for FCE, confirming superior stability under adverse conditions.
Anomaly detection and handling capabilities demonstrate robust performance under various fault scenarios. Missing sensor data accommodation shows graceful degradation with minimal impact on evaluation accuracy when up to 18% of indicator data is unavailable. Outlier identification testing reveals successful detection of 91% of artificially introduced anomalous measurements, with false positive rates below 3.2%.
Table 3. Detailed Performance Indicators for Representative Satellite Attitude Control Processes
Performance Indicator S14 (Best) S26 S37 S24 (Worst) S23 S30
Pointing Accuracy 0.643 0.786 0.894 0.847 0.786 0.412
Relative Accuracy 0.967 0.982 0.988 0.081 0.196 0.760
Stability 0.694 0.703 0.551 0.800 0.811 0.780
Response Time 0.600 0.457 0.657 0.514 0.400 0.371
Angular Velocity 0.789 1.000 0.971 0.706 0.665 0.659
Energy Consumption 1.000 0.773 0.571 0.000 0.000 0.288
Off-Target Distance 0.2502 0.2880 0.3018 0.6864 0.6758 0.5838
Table 3 presents detailed performance indicators for representative processes, illustrating the comprehensive evaluation capability of the proposed methodology. The S14 process demonstrates optimal energy consumption and strong overall performance, while the S24 process shows critical deficiencies in energy consumption and relative accuracy indicators that significantly impact overall assessment.
Table 4. Comprehensive Method Performance Comparison Summary
Evaluation Method Max-Min Difference Compatibility Degree Deviation Processing Time (s) Noise Tolerance (%)
Proposed Grey-Target 0.4362 0.9238 0.0043 2.8 22.3
TOPSIS 0.2965 0.9165 0.0031 1.2 16.8
FCE 0.4131 0.9124 0.0055 1.8 18.2
Traditional PCA 0.3845 0.8967 0.0068 1.6 15.4
As shown in Table 4, the proposed grey-target decision methodology demonstrates superior discrimination capability with the highest maximum-minimum difference of 0.4362, indicating enhanced ability to distinguish performance variations. The compatibility degree of 0.9238 confirms strong correlation with alternative evaluation approaches while maintaining acceptable computational overhead. This 22.3% noise tolerance level significantly outperforms standard procedures, thus validating good performance within rigorous operating environments.
A thorough analysis of outcomes validates the feasibility of assessing satellite attitude control systems’ operational effectiveness by implementing an improved principal component analysis within the proposed grey-target selection framework. The approach effectively handles uncertainty, provides improved discriminability, and retains computational simplicity to be implemented in spacecraft systems. The enhanced ruggedness provides sufficient performance assessment in diverse operating and environmental conditions commonly found in real satellite applications.
- Discussion
5.1 Results Interpretation
The effective performance exhibited by the novel grey-target decision-making tool, based on an improved principal component analysis, can be attributed to a number of vital mechanisms that address certain major limitations generally associated with traditional rating techniques used in satellite attitude control systems. Its better ability to discriminate, measured in terms of a maximum-minimum difference criterion of 0.4362 compared to the 0.2965 exhibited by TOPSIS, can be attributed to its ability to effectively quantify and integrate uncertainty, which makes it possible to develop a comprehensive framework appropriate for weight assignment.
It works mostly by converting grey numbers, thus overcoming the inherent incompleteness and uncertainty that exist in satellite telemetry data. Unlike the usual deterministic assessment procedures that take for granted perfect data integrity, the new method acknowledges and measures uncertainty in measurement, sensors noise, as well as disruptions in the atmosphere that necessarily impact true satellite functions. Such uncertainty quantification leads to a more sensible performance evaluation, one closely aligned with the true operating conditions found in space environments. Through the use of grey number depiction, the method is able to maintain evaluative coherence even when faced with incomplete or uncertain sensor data, the situation often occurring during eclipses, magnetic upset events, or faulty sensors, for example.
The new method of principal component analysis remedies the critical drawback of existing objective weighting schemes by reducing the impact of correlational amplifications. Standard principal component analysis typically produces skewed weight distributions in the presence of marked correlational dynamics, as is commonly the case in satellite attitude control applications, in which variables like pointing accuracy, stability, and energy uptake present inbuilt correlational relationships. In this new method, the compensation factor F(j,i) reduces the influence of the higher values of the correlational coefficient, hence favoring fair weight distributions that better mirror the true levels of importance of particular performance measures, separating them from correlational effects.
The evaluation made via accepted methodologies shows a considerable advantage on several evaluative parameters. With a 0.9238 compatibility index, it indicates high concurrence with past evaluation paradigms, while at the same time improving discriminatory capability. This balance implies that the new method adequately captures the essential performance features promoted by established systems, while achieving higher sensitivity to performance changes. Better resilience tonoise, measured as 22.3%, in comparison to 16.8% for TOPSIS, causes stronger performance in adverse conditions, essential for the reliable evaluation of satellite systems operating in the highly variable space milieu.
The methodology’s effectiveness in handling multi-dimensional performance assessment stems from its ability to integrate diverse indicator types through the grey-target decision framework. Traditional techniques often struggle in dealing with satellite attitude control’s diverse performance measures, including accuracy determination, temporal characteristics, energy expenditure, and stability examination, usually measured in different units, scales, and optimization criteria. The proposed approach successfully links multiple diverse indicators by standardizing and converting them to grey figures, and in turn, provides comprehensive assessment handling all significant performance aspects simultaneously.
More sophisticated configurations of systems, which combine predictive model control techniques with reinforcement learning methods, show significant merits due to their comprehensive evaluative operating characteristics (dos Santos & Chagas, 2024; Mammarella et al., 2019). Such configurations, defined by changing architectures, show notable performance trade-offs, which are not adequately handled by standard assessment techniques (Bohlouri & Jalali-Naini, 2025; Esit et al., 2022; Jin & Li, 2022). Specifically, when evaluating such complex control frameworks—wherein performance improvements rely upon trade-offs between accuracy, energy efficiency, and robustness metrics—step-by-step construction of the grey-target decision model, including multiple conflicting objectives and iterative assessments, works best (Iannelli et al., 2022; Yang & Liu, 2024).
Use of advanced spacecraft technology, including adaptive morphological satellite designs or adaptive mass-moment control systems, is an example of modular application of this approach to heterogeneous mission profiles (Watanabe et al., 2023) (Padun & Lu, 2024) (Xu et al., 2024). The new spacecraft configurations pose strong challenges for evaluation with intrinsically time-dependent properties and variable operating conditions, making conventional methods useless. There is, therefore, a need for the implementation of evaluation methods that can deal with time-varying and non-stationary performance parameters. The diversity is also accommodated by uncertainty management methods embedded within the framework of the grey number approach (Wang et al., 2023) (Zhu et al., 2024).
Formation flying mission scenarios are a typical example where the advantages of the grey-target decision framework are clearly evident. These mission scenarios involve complex coordination requirements and multidimensional constraints that lead to performance data with high uncertainty and varying correlations (Mathavaraj & Padhi, 2021). Traditional metric-based assessments are often unable to provide meaningful assessments in such situations; however, the grey-target decision framework allows for consistent assessment owing to its effective handling of uncertainty and its ability to reduce correlation.
5.2 Method Limitations
Despite its accepted merits, the suggested approach also contains some limitations that limit its adequacy and effectiveness in particular contexts. Although it still remains valid within the reasonable parameters of regular practical situations, the calculations are too resource-intensive in real-time testing environments with severely limited computational resources. The 2.8-second processing time for 44 procedures, acceptable in Earth-based assessments, becomes disproportionately long when compared to the regular computational power onboard, hence compromising the efficient performance of monitoring duties in real-time.
Grey number transformation, based on assumptions of the nature of uncertainty distributions, is not consistent for all operational conditions. The assumption that grey coefficients are distributed normally with specified variance parameters might be inadequate in describing uncertainty patterns observed during some phases of the mission or under different environmental conditions. During extreme operational scenarios, such as solar storm events or prolonged eclipse periods, the uncertainty characteristics may deviate significantly from assumed distributions, potentially affecting evaluation accuracy.
The success of the improved principal component analysis method is, in essence, based on the assumption that the correlation effects appear as insignificant artifacts, as opposed to valuable interrelations between measures of performance. In particular system configurations or operating settings, strong correlations between indicators can actually indicate essential physical relationships, which should impact weight allocation. Therefore, the method of reducing the effect of correlation can actually compromise desirable system qualities in this case, leading to suboptimal evaluation results.
The evaluation framework defined for this approach argues that all relevant measures of performance can be quantified and integrated into the evaluation procedure (Narkiewicz et al., 2020). Nevertheless, some of the qualitative aspects of the satellite attitude control system’s performance—such as maintainability, operational flexibility, and failure recovery—are recalcitrant to easy quantitation (Fazlyab et al., 2023; Gaber et al., 2020). With the existing setup of the framework, there is little leeway for engaging these qualitative factors, thus increasing the chances of suboptimal performance assessments in high-complexity missions.
Parameter sensitivity evaluation shows that while the approach shows added robustness over the conventional method, it remains sensitive to certain parameter choices. Parameters pertaining to the grey coefficient distribution and correlation correction factor require cautious consideration in light of the mission-specific and operational particularities. Inadequately reasoned parameter selections can compromise the accuracy of the analysis or induce bias towards certain performance aspects.
Validation technique, based mainly on operational data from sun-pointing mode, can show its limitation in effectiveness when extended to different attitude control configurations or mission scenarios. Different operational configurations, like Earth-pointing, inertial pointing, or target acquisition, have different performance characteristics and behavior of uncertainties, which require possible adaptations to the technique. Therefore, existing validation framework may not fully cover the whole range of operational scenarios offered by many satellite applications.
Scalability issues deal with potential bottlenecks found in evaluations performed on large or frequent test scenarios (Li et al., 2022). Though the approach has characteristics typical of linear scalability, the process of grey number translation and grey number computations of eigenvalues can be computationally restrictive when dealing with big data or in real-time testing paradigms (Jarraya et al., 2025).
5.3 Engineering Application Prospects
The implementation of the proposed grey-target decision model in operational satellite systems is of great potential in enhancing mission effectiveness as well as decision-support functionality. Ground mission control centers provide the ideal setting for implementation, via which the method can enable in-depth performance evaluations pertinent to the effectiveness of the satellite constellation as well as operational success. The system’s skilled discrimination function allows for the identification of subtle performance degradation well before impact on mission objectives, thus allowing for proactive scheduling of maintenance as well as operational adjustments.
The incorporation of existing satellite health monitoring systems holds much potential in the upgrading of methodologies utilized in the system health assessment and determination of anomalies. Effective uncertainty management inherent in this method makes it highly suitable for implementation in autonomous health monitoring systems, whereby reliable performance analysis in various data quality conditions is critical (Russo & Lax, 2022). Conversion of the grey numbers effectively counters the issues of incomplete or aberrant telemetry data commonly faced during epochs of anomalous operating conditions, ultimately allowing for ongoing evaluation when the usual approaches fall short.
Development of the adaptive control systems can benefit largely from the wide performance evaluation capabilities the proposed method provides (Soufi & Belouadha, 2024). Modern spacecraft employ more intelligent control methodologies that require continuous performance information for optimization as well as adaptation purposes (Foo et al., 2023). The evaluation method proposed provides credible performance indicators upon which these adaptive systems can base informed decisions about changes to the control parameter or strategy-based adjustments. This trait becomes especially useful in deep space missions, where the ability of the system to perform its task can vary from its initial condition due to degradation of its components or effects from external environmental factors.
The context of small satellite and CubeSat programs creates an environment where performance evaluation and analysis become much harder, mainly because of the stringent limitations there are in relation to limited resource environments. The ability of the suggested method to provide valuable insights even while being constrained by limited computational power, and its efficiency in dealing with missing or incorrect data, is highly advantageous considering the typical operational issues being experienced in small satellite projects (Kim & Kim, 2025; Kim et al., 2020; Taghavipour, 2021). It is easy to modify and implement on future small satellite platforms that have improved computational power.
Formation flying has distinctive features arising from its interaction between the individual spacecraft’s operational capacity and formation-scale dynamics that make it require sophisticated assessment methods. The ability of the grey-target decision model to allow analysis within a multi-dimensional environment makes it suitable for simultaneous consideration of single spacecraft performance alongside factors related to the effectiveness of different formations. The ability supports designing self-sustaining formation control systems based on real-time performance values, thereby assisting in coordination and decisional procedures to maximize overall performance.
To maximize future research directions, there should be increased focus on expanding the methodology’s flexibility to be applicable to a range of mission types and operating scenarios. The integration of artificial intelligence and machine learning paradigms provides prospects for increased flexibility as well as automated parameter configuration based on mission-dependent characteristics (Fourati & Alouini, 2021). Development of mission-adaptive parameter selection algorithms can reduce manual configuration needs and optimize methodology performance in a range of operating scenarios.
Real-time implementability considerations suggest probable directions to develop computationally effective models suitable for onboard uses (Rodrigues et al., 2019; Zhou et al., 2025). Methods like approximation techniques and parallel processing can potentially reduce computational requirements while maintaining a sufficient amount of evaluative accuracy critical to effective decision-making procedures in operational scenarios. Integration of sophisticated onboard processing technologies, including field-programmable gate arrays and special-purpose signal processing units, could be used to allow real-time evaluation of future autonomous spacecraft systems.
Expansion of the methodology to multi-spacecraft systems and constellation operation represents a significant direction of advancement. Heterogeneous satellite constellations require sophisticated performance management functionality able to support the large scale and complexity found in hundreds or thousands of individual spacecraft while also providing meaningful fleet-level performance statistics. Scalability characteristics of the grey-target decision framework suggest it can be tailored to support large-scale application spaces through appropriate increases in computing power.
Efforts to standardize practices within the space community can benefit from the application of common performance assessment techniques producing similar results within differing mission types and spacecraft platforms (Izzo et al., 2019; Yadegari et al., 2022). The alignment of the proposed framework within existing testing paradigms, in conjunction with its many advantages, makes it an attractive option to be implemented within standard operating procedures. These procedures can be used to optimize mission planning, to inform spacecraft acquisition, and to help establish best practices within the broader community of space professionals.
The development of future cutting-edge space technologies, including on-orbit servicing and self-guided rendezvous, requires developing complex operational frameworks in which the assessment of multiple performance aspects within high-uncertainty environments plays a critical role in successful mission execution (Garcia-Huerta et al., 2020). These complex mission profiles require developing stringent assessment methodologies to support self-guided decision-making within a range of different operation scenarios, including scenarios in which ground-based performance monitoring can be inaccessible or considerably restricted. - Conclusions
This research defines and analyzes, in meticulous detail, a novel approach applicable to satellite attitude control systems operating in tracking mode. The research focus lies in overcoming major issues concerning uncertainty management, offering comprehensive assessment within applicable operating scenarios. The application of high-order principal component analysis in conjunction with grey-target decision-making methods presents a significant improvement upon standard evaluation procedures, especially in relation to discriminability, robustness, and real-time effectiveness within applicable real-world scenarios. The salient contributions of the newly formulated methodology include the formulation of a universal evaluation framework that incorporates six metrics representing critical performance attributes, the application of grey number transformation to enable effective quantification of uncertainty, and the application of an improved principal component analysis to counter the amplification of correlation effects. Experimental results verify better performance compared to global evaluation indices, as reflected in a maximum-minimum discrepancy of 0.4362 and a tolerance to noise of 22.3%, far exceeding the effectiveness of existing methodologies. These results reflect significant improvements in satellite capabilities, supported by enhancements in mission planning, operating efficiencies, and fast-evolving autonomous technology advancements. In addition, the use of the novel methodology in enabling continuous evaluation under deprived conditions makes it especially beneficial for long-duration missions and autonomous spacecraft operations, where accurate performance monitoring is crucial in ensuring successful mission completion. Future research activities should be directed toward the development of methodologies that are universally applicable across various mission types, the development of computationally efficient alternatives for onboard implementation, and the integration of novel artificial intelligence systems to enhance adaptability. The expected standardization in the aerospace industry is expected to promote greater acceptance and utilization of uniform performance measurement standards in the long term, ultimately improving the effectiveness of satellite missions and operating reliability in increasingly complex space environments.
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