# Hierarchical Model Updating Method for Vector Electric-Propulsion Satellites
## 1 Introduction  
In contrast with traditional chemical propulsion, electric propulsion has the characteristics of low thrust force and high specific impulse [1], which are conducive to reducing fuel consumption [2] and prolonging the life of satellites. Therefore, it meets the requirements for geostationary Earth orbit (GEO) satellites, low Earth orbit (LEO) satellites, small satellites, and deep space probes to work on orbit in the long run [3]. It has also become an important development direction for new high-performance and long-life spacecraft. In many countries, the manufacturing industry has shown the development trend of digitalization and intelligence [4]. As a technology that can realize the interaction and integration of the physical world and the digital world [5], the digital twin has attracted widespread attention. In 2010, the National Aeronautics and Space Administration (NASA) proposed to apply digital twins to flight training, accompanying flight, health monitoring, and parameter optimization [6]. By establishing a digital twin model of a satellite to perform optimization design and feedback control of real satellites, it is advantageous to reduce the cost and risk of decision-making. Since the simulation model is different from the real physical entity and is affected by uncertainties, it is unable to fully reflect the characteristics of the real system and reduces the prediction accuracy. There are two categories of uncertainty sources in a digital twin, which are aleatory uncertainty and epistemic uncertainty [7]. Aleatory uncertainty refers to the intrinsic randomness of a system, which is inevitable and irreducible. For example, a thruster may have uncertainties such as the manufacturing tolerance of parts, installation error, thermal deformation of structural parts, and nonaxisymmetric installationof cathodes [8], and these errors could cause the thruster to deviate from the nominal states. Epistemic uncertainty refers to uncertainty caused by limited data, lack of knowledge, or model simplifications and assumptions. It can be decreased when more information or data is available [9]. For example, many studies simplified the thruster to an ideal force model while ignoring the details of the mechanical structures when performing thrust calibration. The more simplifications are made, the more difficult it is to establish an accurate and reliable model. Therefore, it is essential to identify, describe and evaluate the uncertainties quantitatively, known as uncertainty quantification (UQ). The study of UQ includes the expression of uncertainty, the propagation of uncertainty, the identification of model parameters, and so on [10]. In this research, it refers in particular to the model updating method for the vector thruster satellite, with using the measurement data to estimate the errors and correct the simulation model so that the error between the model and the real satellite is minimized and the prediction level of the model could be improved. According to the sources of uncertainty, model updating includes formula updating and parameter updating. Formula updating refers to the addition and modification of certain items in a mathematical formula. For instance, decoupling the coupled subsystems for separate calculations will result in some loss of accuracy. Parameter updating involves the use of measurement data to adjust model parameters without modifying the formula structure. This paper focuses on the latter. Model parameter updating is essentially an optimization problem that can be solved using optimization algorithms, including traditional optimization methods and modern optimization methods. The former includes the least-squares method and filtering algorithm, while the latter includes particle swarm optimization (PSO) and genetic algorithms (GAs). All of them are widely used for parameter estimation. The modern optimization methods are time-consuming, whereas the traditional optimization method has a fast convergence speed, so the least-squares method is adopted in this paper.