Active Control of the Space-borne Antenna Reflector Considering Thermal Load

In order to improve the shape accuracy of the space-borne antenna reflector under the interference of uncertain load, the active control method of the reflector shape was studied. First, a finite element model of the thermal and mechanical coupling of the space-borne antenna reflector was established. Taking the uncertain load into account, the radial basis function (RBF) neural network was used to estimate the error, and the adaptive controller was designed according to the estimation error. Based on the Lyapunov stability theory, the stability of the controller was analyzed. Finally, the adaptive control of the space-borne antenna reflector was simulated numerically. The simulation results show that the method of RBF neural network adaptive control is effective by reducing the shape error of the reflector in a short time when considering the uncertain load.


Introduction
With the rapid development of communications, space science, and earth observation, the demand for large-aperture space-borne antenna reflectors is increasing. According to antenna theory, it is known that large-aperture antenna can transmit a large amount of high-resolution data [1,2], and high shape accuracy (small RMS error) can achieve high operating frequency band and wide frequency band. Many experiments have proved that thermal deformation is the main factor for the shape deviation of the reflector. In this case, it is necessary to actively control the shape of the reflector [3].
Active shape control of the reflector is divided into static shape control and dynamic shape control. The static shape control of the reflector assumes that the external load changes slowly. Song X S et al. [4] established an influence coefficient matrix according to the magnitude of the influence of each actuator on the deformation of the grid reflector. Furthermore, the least square method is used to solve the voltage value of each actuator, and the effectiveness of this method is verified by experiments. Wu et al. [5] used piezoelectric ceramic transducer (PZT) and large fiber composite (MFC) actuators to reduce the overall and local (high-order) surface errors of the reflector respectively. The antenna may also be subjected to thermal shocks when satellite in orbit. This rapid external load may cause vibrations on the reflector. In addition, static shape control only considers the final control force without considering the loading process. Unreasonable loading process will also lead to the vibration of the structure. Based on the above factors, the static shape control of the reflector has certain limitations.
In order to eliminate the vibration of the reflector structure, scholars have adopted a dynamic shape control method to control the reflector. Xun et al. [6] took a large cable net antenna as the research object, and used a fast model predictive control method to estimate the input voltage curve of the actuator. Luo et al. [7] proposed a hybrid control algorithm based on FLC, and designed a controller based on this algorithm to enhance the attenuation of free vibration of large truss structures.
= [ ] is the strain vector of the element, = [ ] is the displacement vector of the element. ∇ is the derivative operator. But the strain here should be the total strain, which is the sum of the force and thermal expansion.
{ } = { } + { } (2) { } is the elastic strain and { } is the thermal strain. ∆T represent the temperature increment, the anisotropic linear expansion coefficient is 1 , 2 , 3 , and the thermal strain is supposed to be The relationship between stress and elastic strain is obtained as So the relationship between stress and total strain is Where [ ] is the elastic coefficient matrix. Using the same element and the same shape function to interpolate the temperature increment and displacement inside the element with the nodal temperature rise and nodal displacement, the expression of the temperature increment and displacement are given as The element elastic strain energy is given as } Substituting the above formula into the strain energy of the element, the following equation can be obtained as Where [ ] is the element stiffness matrix.
Its extreme condition is given as ∂U ∂{ } = 0 The following equation can be obtained as [ ]{ } = { } (6) Where K is the total stiffness matrix of the structure, and the displacement of each node of the reflector thermal deformation can be obtained by solving equation (6).

Controller design
Considering the damping of the system, the system dynamics equation is given as is the total mass matrix of the structure, is the total damping matrix, is the total stiffness matrix, ( ) is the force of the actuator, and is the external force of the structure (thermal shock or external random interference load, etc. ), ̈ is the nodal acceleration, ̇i s the nodal velocity, is the nodal displacement.
Transform the modal coordinates of Eq. (7), the following equation is obtained as In fact, the lower-order modes play a greater role, so the mode truncation method is used to reduce is the first k-order mode shape matrix, is the remaining mode shape matrix, is the first k-order mode coordinates, and is the remaining-order mode coordinates. After truncating the mode, = , multiply both sides of the Eq. (7) by , = = ( 1 2 , 2 2 , ⋯ 2 ) Eq. (7) can be simplified to Due to various uncertainties, the mass matrix, damping matrix and stiffness matrix will always be different from the theoretical values. In practical applications, the expression of ( ) is generally unknown. Therefore, in order to control the space-borne antenna reflector system, it is necessary to estimate the function of ( ).
RBF neural network can approximate any nonlinear function with a compact set and arbitrary precision. This paper uses RBF neural network to approximate ( ), and then realizes the control of the space-borne antenna reflector [8,9].
Assuming that the function obtained by the RBF approximation is ̂( ), the adaptive control law is given as

Stability analysis
For the closed-loop system of the above formula, the following Lyapunov function is designed for stability analysis. due to ℎ̃= ( ℎ̃) Organizing the above formula, the following equation is given as Design the following weight adaptive algorithm ̃̇= ℎ + 1 ‖ ‖̂ ̂̇= ℎ + 1 ‖ ‖̂(21) It can be proved that the above adaptive algorithm can ensure the stability of the system by designing appropriate control parameters.
Substituting equation (21) According to ̇≪ 0, the following inequalities can be derived as It can be obtained from equation (22) that increasing the eigenvalue of Q and reducing the eigenvalue of P can ensure that ̇≪ 0 can also reduce the convergence value of X and improve the convergence effect of the system.

Control simulation
Establish the finite element model of the reflector in the finite element analysis software, and derive the mass matrix and stiffness matrix of the reflector model, combining the RBF neural network and the above-mentioned adaptive weight algorithm to realize adaptive control of the space-borne antenna reflector model.

Simulation parameters
The damping parameters in the reflector model are set as: damping coefficient 1 =2× 10 −3 , 2 =10 −4 , the system parameters and the parameters in the controller are shown in Table 1 and Table 2. 0.01 The evaluation standard of the control effect is the root mean square value of the surface error of the reflector, and the root mean square value (RMS) is defined as is the displacement of the i-th node, is the ideal displacement of the i-th node, and k is the number of nodes.

RBF neural network
The neural network topology diagram is shown in Figure 2. It can be seen from the figure that the RBF neural network is composed of three layers: input layer, middle layer and output layer.

Figure 2. RBF neural network topology
The first layer is the input layer, which represents the node displacement and velocity error of the reflecting surface. The second layer is the middle layer. The activation function of each middle layer node uses the radial basis function. The third layer is the output layer, which represents the estimated value of ( ). The output layer result is related to the radial basis function and weight coefficient. The equation of the neural network model based on the radial basis function is given as input layer middle layer output layer ( ) This paper uses the standard Gaussian function as the activation function, which can be expressed as is the base function center of the j-th central layer unit, and represents the variance. There are three parameters to be learned by the RBF neural network: , , and the weight between the intermediate layer and the output layer.
The structure of the RBF neural network is 38-50-1, the input of the neural network is = [ ,̇], the parameters of the Gaussian function are set to = (−0.1: 0.1: 50), = 1 , the initial weight coefficients are set to 0.1. Figure 3 is the RMS error curve of the RBF neural network adaptive control reflector. It can be seen from the figure that the initial profile error root mean square value is about 700 μm. When the control is stable, the root mean square value reaches approximately 0.8 μm. The simulation results can be concluded that the RBF neural network adaptive control can reduce the shape error of the reflector to a certain extent.    Fig. 5(a), and at the end of the control, the displacement error of every free node is reduced to below 2.5 μm, which is represented in Fig. 5(d). Fig. 5 visually shows that the adaptive control of RBF neural network method is valid to solve the reflector shape control problem.

Conclusion
This paper takes the space-borne antenna reflector as the research object, and adopts RBF neural network adaptive controller to control the surface error after thermal deformation. Taking into account the interference of uncertainty load, the active control of the reflector system is realized. The follow conclusion can be obtained.
(1) The thermal-mechanical coupling finite element model of the space-borne antenna reflector is theoretically established. Based on the established finite element model, an adaptive control law of surface accuracy considering the uncertainty is proposed.
(2) When there is a large load disturbance from the outside, the RBF neural network adaptive control has relatively small fluctuations and good robustness.