Mode switching control for drag-free satellite based on region of attraction

First, authors give the reference coordinate system and dynamics models. In space gravitational wave detection, the drag-free satellite has 2 cubic TMs with an angle of 60° between them (see Fig. 1A). Three reference coordinate systems are used (see Fig. 1B). The inertial reference coordinate system is the Earth mean equator J2000 coordinates. Orbit Coordinate System (OCS) originates at the satellite mass center O_r on the ideal orbit, with z_r-axis pointing from Earth's mass center to the satellite's mass center and the x_r-axis perpendicular to the z_r-axis in the orbit plane and coinciding with the velocity. Cavity Coordinate System (CCS) originates at the center of the satellite cavity, with x_i-axis directing toward the laser link and z_i-axis perpendicular to the x_i-axis in the orbital plane. As for dynamics and motion models, the motion model of the satellite relative to the ideal orbit is established in the inertial coordinates, considering the error vector r = [x, y, z]^T of the satellite relative to the ideal orbit, the control vector u_s = [u_sx, u_sy, u_sz]^T acting on the satellite, and the interference vector d_s = [d_sx, d_sy, d_sz]^T acting on the satellite. The motion model of the TM relative to the cavity is given in OCS, with the position vector ρ_i = [x_i, y_i, z_i]^T from the cavity center to the TM’s mass center, the vector c_ai = [c_aix, c_aiy, c_aiz]^T from the satellite’s mass center to the cavity center, the control vector u_i = [u_ix, u_iy, u_iz]^T acting on the TM, and the interference vector d_i = [d_ix, d_iy, d_iz]^T acting on the TM. Besides, the solar radiation pressure, the reaction force, the position measurement noise, and the reaction force noise are formulated.

Then, authors investigate the controller design, and analyze the stability in view of the Lyapunov function and the region of attraction. Unlike traditional satellite control systems, the drag-free satellite’s control system includes the inner loop and the outer loop (see Fig. 2). The inner loop is used to control the TM including the TM’s capture control and the TM’s high-precision control, while the outer loop is used to control the satellite. For the drag-free satellite, the PID control law u_s is designed and the thruster model P_m is obtained by polynomial fitting with the anode voltage and the mass-flow rate. For the TM’s capture controller, the sliding mode control is selected due to its good robustness. According to the state equation of the TM movement model relative to the cavity and the exponential approach law, the 3-axis control law u_i of the TM is obtained. After capture, the H_∞ mixed-sensitivity controller is designed to improve the control accuracy and reduce the noise effect. By linearizing the state equation of the TM movement model and decoupling the system with the inverse matrix of the plant, the mixed-sensitivity problem is expressed as an optimization problem and the system control law is obtained. The sliding mode controller (i.e., subsystem 1) and the H_∞ mixed-sensitivity controller (i.e., subsystem 2) can be respectively proved stable via their the Lyapunov function V_t1 and V_t2. Furthermore, letting V_t1(t_f) > V_t2(t_f) where t_f is the switching time, the switching system is global stable according to the piecewise Lyapunov function method. Although the system satisfies the global stability requirements, it cannot guarantee the local stability of the nonlinear system after switching. In order to avoid system instability, Subsystem 1 is required to switch after entering the ROA of Subsystem 2. The ROA is defined as Ω(ρ,t) = {X_h ∊ Rⁿ | 0 ≤ V_h(X_h,t) ≤ ρ(t)} where Rⁿ is the Euclidean space, V_h(X_h,t) is the Lyapunov function of the nonlinear Subsystem 2, and ρ(t) is the continuously differentiable function. The sum of squares (SOS) method is used to calculate the ROA of Subsystem 2. Through calculation, the ROA is V_t2 ≤ 3.129×10^-15, and V_t1 is made greater than 3.129×10^-15 at time t_f, which ensures the system is globally stable.

Finally, authors show the numerical results and draw the conclusion. To test the performance of the switching control method, the numerical simulation is carried out. In simulation, the initial states of the TM relative to the cavity are (200 μm, 5 μm/s, 200 μm, 5 μm/s, 200 μm, 5 μm/s). The initial relative states of the satellite are (0, 0, 0, 0, 0, 0). The position coordinates of the TM1 and TM2 are (0.3 m, 0, 0) and (−0.3 m, 0, 0) in OCS, respectively. The maximum root mean square voltage of the Subsystem 1 and Subsystem 2 are 96 and 7 V, respectively. Results show that the relative motion errors of the satellite on the 3 axes are controlled within 0.1 nm and the control force is always within the thrust range, and there is no saturation. To study the performance of the proposed switching rule, the control errors of TM1 and TM2 under the switching rule of the ROA and the non-ROA are compared and analyzed, as shown in Figs. 7 and 8. It can be seen that Subsystem 1 is switched to Subsystem 2 under the constraints of the switching rule, the accuracy of the controlled system is improved and remaining stable for a long time after switching, and the relative motion error of the TM1 and TM2 is controlled within 2 nm. By comparison, the system of the non-ROA switching rule frequently switches after the initial switching, and the control accuracy is low. In conclusion, the design of switching rules is important, and the research in this paper will provide a reference for the switching rule design of the drag-free satellite.