Presenting the mathematical model to optimize the reliability of the satellite attitude determination and control system

Presenting the mathematical model to optimize the reliability of the satellite attitude determination and control system Akbar Mansouri, Akbar Alem-Tabriz a Department of Industrial Engineering, Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran b Department of Industrial Management, Management and Accounting Faculty, Shahid Beheshti University, Tehran, Iran Abstract The key issue in this study has been the integration of redundancy allocation and optimization of failure rates. The fact that we only need to work with parallel allocation and increase the number of components in a subsystem in parallel to improve the reliability or availability of parallel-series systems is necessary, but not enough. It is worth mentioning that in this research, the improvement of failure rates of different components in the system has been studied. In the meantime, it is important to note that with careful study of the effects of each of these approaches and the costs imposed on the system, the design problem data will be formed. Considering that more effort to improve the reliability of components leads to less redundancy allocation and vice versa, the optimization problem is performed to determine the exact number of redundancies along with determining the exact amount of improvements in complete failure rates. In this research, the satellite attitude determination and control system, the structure of the studied system and its components is introduced, then the reliability in this system is modeled and optimized with a mathematical approach based on the combination of reliability allocation and redundancy allocation. Keyword: Reliability, Satellite, Attitude determination and control system, redundancy allocation problem, Genetic algorithm.


1-Introduction
Satellites are human-made devices that are deliberately sent into space to go around the earth or other planets. The importance of satellites for telecommunications and the study of terrestrial resources and research and military and espionage applications is growing. Part of the scientific and specialized research that is being done in space-based laboratories has never been able to take on a practical dimension on Earth. To control the safety of sensitive systems, such as those in nuclear power plants, chemical processes, and the astronaut system, it is necessary to have readymade safety systems to complete on-line regulatory systems. Ready-to-use systems can automatically operate to keep the system safe and to prevent catastrophic outcomes under various conditions, however, ready-to-operate safety systems may not be able to perform the expected performance due to hidden failures. Therefore, it is very important to investigate these problems during testing and maintenance, and diagnose and resolve maintenance testing in accordance with the manufacturer's recommendations, which are generally cautious recommendations. In sensitive safety applications, the periodic maintenance period is sometimes determined by the supervisory team, for example, strict rules apply to intermittent maintenance to keep the level of uncertainty low. In general, intermittent and frequent testing can increase the likelihood of damage being detected. However, it may cause the system to crash faster due to the increased unnecessary costs imposed on resources. Therefore, logical and efficient testing of maintenance strategy is of great importance in ready-made safety systems.
In this research, by identifying the structure of the satellite attitude determination and control system, as well as the components used in it, the reliability of this system is analyzed and optimized. For this purpose, by investigating the information related to the system under study, the assumptions of modeling and optimizing the system reliability is determined. In this regard, practical and systemic constraints is considered. Finally, report the obtained results.
In this paper, a joint reliability-redundancy optimization approach for satellite attitude determination and control system is presented. This model is developed on a multiple-mode system that each mode has a specific reliability block diagram. The various subsystems of the system under study have active, cold-ready or k-out-of-n strategy, which are identified in the reliability block diagram of each mode. In this study, the percentage of improvements made to the failure rates of the components are integrated, and the failure rate of the components used in the closed continuous interval can be changed. 4 researchers. Fyffe et al., [8] were the first to present the mathematical model of the general problem of redundancy allocation. The goal of their proposed model was to maximize system reliability by considering weight and cost constraints. They solved this problem with the help of dynamic planning. Nakagawa and Miyazaki [9] presented a non-linear planning problem with a solution to optimize reliability. In fact, by changing the example of Fyffe et al., They solved the problem by using the exact method of substituting constraints, and showed that in the case of multiple constraints, this approach is better than the dynamic planning method. However, in order to increase the reliability, one or a set of the following measures can be performed. These measures can be implemented based on the assumptions and requirements of each system.  Optimization of redundancy level.
 Optimizing the selection of the type of components used in the system.  Optimization of existing technical activities in order to regulate failure and repair rates.
In the following, some conducted research on the optimization of the redundancy allocation problem will be showed in Table 1. Table 1 about here

3-Satellite attitude determination and control system
In this research, the satellite attitude determination and control system will be investigated. High efficiency, reliability and health are the most important criteria in the engineering of space systems.
Reliability refers to the possibility of the system working properly over a period of time. The first and most important step in preventing a system from malfunctioning is to detect failure in that system. Because in this case, there will be a good time to prevent problems in the system. Satellites are examples of self-contained, important, and costly space systems that are sent on relatively longterm missions. One of the most important parts of satellite that is directly related to health is the subsystem of satellite attitude determination and control system. The task of this subsystem is to determine and control the situation in space and neutralize the disturbing environmental disturbances and torques on the satellite. Therefore, this subsystem is always known as a subject of study to provide breakdown identification because improving the subsystem reliability of satellite attitude determination and control system directly affects the reliability of the satellite in space. In order to design the subsystem of satellite attitude determination and control system, in most cases, three reaction wheels are used in line with the three main axes of the satellite. Each of 5 these actuator has an electric motor and a heavy disk. By applying current to the electric motor, torque is produced, which changes the speed of the angles of the motor axis. The opposite direction is produced. Changing the speed of the reaction wheel by applying the necessary control algorithm to the motors causes the motor to reach the required speed from zero speed. After producing the necessary torque, the motor shuts off again. This change in speed causes the required tower radiation to be generated to achieve the desired state of the satellite. In order to increase reliability, a reaction or spare reaction cycle is used under the systems based on the reaction wheel. According to Figure 1, this actuator is located next to the main reaction cycle and as soon as one of the wheels fails or irreparable damage occurs in them, the spare reaction wheel will replace the defective cycle in the system. In general, the modules of system studied in this research are shown in the figure 2.

4-Problem Statement
In this section, the satellite attitude determination and control system, the structure of the system under study and its components are introduced, then model and optimize the reliability in this system according to the mathematical approach based on the combination of reliability allocation and redundancy allocation. In the following, we will introduce the components, structure and functions of the satellite attitude determination and control system.

4-1-Functions of attitude determination and control system
The functions of the attitude determination and control system include the following:  Adjust the satellite in the desired direction despite external disturbance torques  Determine the attitude of satellites using sensors  The situation switching by actuators  Satellite orientation for the mission The sub-system for determining and controlling the situation to perform each of its tasks in the form of control modes and obtaining the required attitude in each mode, consists of two parts: determination and control. The attitude determination section includes sensors and attitude determination algorithms, and the attitude control section includes actuators and attitude control algorithms. 6 The attitude determination and control system consists of several parts, which include the following:  Table 2 introduces the parameters and Table 3 introduces the variables of optimizing the reliability of satellite attitude determination and control problem. Table 2 about here   Please insert Table 3 about here 7

4-5-Assumptions
 The function of the components used in the system is independent of each other  The components failure used in the system is independent of each other  The components of the system under study are binary  The components cannot be repaired and returned to the system after failure  The parameters related to the cost and weight of the components in the system are deterministic and definite

4-6-Mathematical modeling
Based on the structure of the attitude determination and control system under each of the functional modes and also based on the arrangement of components according to the reliability block diagrams presented in the section 3-4, the reliability of the system under each functional modes is calculated. As can be seen from the reliability block diagram provided, the structure of the attitude determination and control system in the various functional modes of the hybrid structure includes active, cold-standby and k-out-of-n. In this section, we first model the reliability level of the attitude determination and control system in each of the functional modes under the conditions that the components used in the system follow any desired life distribution. Then, by placing the probability density function and the reliability of the exponential distribution in equations 1 to 5, the reliability of the system is presented in the functional modes of de tumbling, Coarse pointing, fine pointing, sun pointing and safe.
Now, under the assumptions provided in Section 4-5, according to the information provided about the system for determining and controlling the attitude of the satellite base, as well as calculations for the reliability of the system under different functional modes, the mathematical optimization model allocates reliability and redundancy allocation for the system under study is presented below. In this model, the reliability of each functional mode based on the arrangement of components and system structure in the block of other names provided for each of these modes is calculated. Also, the failure rate of the components used in the fixed system is assumed. It is used in the display system, so in the set of equations presented to calculate the reliability of the system under different functional modes, the probability density function and the reliability function of the display distribution are placed.
The objective function of the above mathematical model (6) is to maximize the minimum reliability of the attitude determination and control system under different functional modes. Constraints (7 to 11) calculates system reliability in the functional modes of de tumbling, coarse pointing, fine pointing, sun pointing and safe respectively. Constraint (12) guarantees that the total costs incurred in the system includes the costs of redundancy allocation and failure rate improvement does not exceed the maximum allowable cost. Constraint (13) ensures that the weight of the components allocated to the system does not exceed the maximum allowable weight specified for it. Constraints (14) to (19) ensure that redundant components to the system do not    Parents are first selected to perform the crossover operator, then the children are generated using a uniform crossover operator. The operation of this operator is described in references [30] and [31]. In this operator, for each gene in the selected parent chromosome, a number between zero and one is randomly generated, then the child's chromosomes are quantified in linear composition from the parent chromosomes.
Mutations are also performed on each array of the chromosome matrix. In this operator, after selecting the desired parent, a random number between zero and one is generated for each gene in the parent chromosome, and the values of the parent chromosome genes are mutated at a certain rate of mutation. Now, if the random number generated is less than the desired mutation rate, the corresponding gene on the parent chromosome will be randomly mutated, but if the random number generated is larger than the mutation rate, the gene will not be mutated on the parent chromosome [31]. For example, Figure 9 shows how the mutation operator executes on a chromosome.

Please insert Figure 9 about here
Using a random search method (RS) to solve the presented model can be a lower bound for minimization problems and a lower bound for maximization problems compared to other solution methods. In fact, proving the intelligent performance of meta-heuristic algorithms can be demonstrated by comparing them to an RS. So that, these algorithms must always be more powerful than an RS. Therefore an RS is provided to validate the proposed algorithm. The proposed RS Pseudocode is shown in Figure 10 [32]. Figure 10 about here

6-Results and discussions
In this section, the proposed model is solved. For this purpose, first a numerical example is presented on the case study. The table 4 shows the parameters and the information needed to solve the problem for a numerical sample example. 16 Please insert Table 4 Table 5 to tune on the appropriate levels. Due to the choice of twolevel experimental factor design, each of the experiments is considered at two levels, high and low.
The method of advancing the response surface methodology is such that in addition to the upper and lower limits, the axial points using the middle limits as well as a number of central points (in this research 5 central points are added to the design) is also considered. For this algorithm, according to the three available parameters, factor 2 3 is considered. For this purpose, we performed the experiment in MINITAB 16 software for the algorithm and tune the best level for the test result.

Please insert Table 5 about here
According to the explanations provided, the nonlinear regression equation for the proposed genetic algorithm, which shows the relationship between the parameters of the algorithm and the value of the objective function, is obtained. Now it is enough to solve the model (41) to get the optimal parameters of the genetic algorithm.  Table 6. Table 6 about here After obtaining the tuned parameters of the genetic algorithm, the example presented in Table 4 is solved using the genetic algorithm developed in this study and its results are reported. As shown in Figure 11, the convergence diagram of the genetic algorithm is shown in consecutive iterations, and in Tables 7 and 8, the information about the variables of the mathematical optimization model provided under the available maximum budget is clearly presented. Figure 11 about here Please insert Table 7 about here   Please insert Table 8 about here

7-Sensitivity analysis and validation
In this section, in order to validate the genetic algorithm developed to solve the mathematical model, which was discussed in Section 4-6, we compare the results of this algorithm with the solutions obtained from the random search method obtained under different amounts of the maximum budget. . Given that we know that the solutions obtained from the random search method are always solutions far from global optimization, so these solutions are a good criterion for evaluating the performance of the genetic algorithm. As can be seen in Table 9, genetics in all cases has found better solutions than random search, which indicates the efficient operation of this algorithm to solve the current problem. Figure 12 also shows the effect of increasing the maximum budget available on the reliability of the system under study. It is worth mentioning that in this paper, the 2018 version of MATLAB software has been used to implement the random search method and genetic algorithm. Table 9 about here Please insert Figure 12 about here

8-Conclusions
In this research, the satellite attitude determination and control system, the structure of the studied system and its components is introduced, then the reliability in this system was modeled and optimized according to the mathematical approach based on the combination of reliability allocation and redundancy allocation. By studying this system, it was determined that in general, the satellite attitude determination and control system has five different functional modes. These modes include de tumbling, coarse pointing, fine pointing, sun pointing and safe. In the following, each of these functional modes is introduced and the reliability block diagram of the system is presented under each of these modes. In this system, the function of the components used in the system and the components failure are independent of each other. Failed components do not damage the system as a whole. Also component are binary. In the attitude determination and control system, components cannot be repaired and returned to the system after failure. In the mathematical optimization model presented in this study, the parameters related to the cost and weight of the components in the system are deterministic and definite, and the system has the maximum cost and maximum allowable weight for the components used in it. The various subsystems of the system under study have active, cold-ready or k-out-of-n strategy, which are identified in the reliability block diagram of each mode. Also, systems where the components are cold-standby, there is a possibility that the switch will fail. In this study, the failure rate of the components used in the system is constant, so the life distribution of the components used in the system is exponential. In this study, the percentage of improvements made to the failure rates of the components is integer, and the failure rate of the components used in the closed continuous interval can be changed. The mathematical optimization model presented in this study falls into the category of optimization problems related to mixed integer nonlinear programming (MINLP).
Solving these problems always involves a lot of mathematical complexity. Therefore, the development of exact solution methods for these problems is very difficult and in most cases impossible. On the other hand, the redundancy allocation problem in terms of computational time falls into the class of NP-HARD problems. Therefore, metaheuristic algorithms should be used to solve the proposed mathematical model. In this study, a genetic algorithm was used to solve the proposed model. Also, to valid the results obtained from the developed genetic algorithm, a random search algorithm was used, which according to the obtained outputs, the genetic algorithm