# A Predefined-Time Control for the Laser Acquisition in Space Gravitational Wave Detection Mission
Furthermore, the characteristics of scanning methods such as spiral scanning, raster scanning, Lissajous scanning and rose scanning have been compared and analyzed [18]. [PERSON] [19] conducted a detailed study for the acquisition phase of the GRACE Follow-on mission. [PERSON] [20] introduced the inter-satellite laser link experiment of the SLLEX mission and gave a detailed description of the acquisition scenario. [PERSON] [21] designed a control law for the acquisition process of the LISA mission and conducted a simulation to verify the pointing performance. The pointing error was less than 8 nrad/Hz\({}^{1/2}\) in the frequency band 1-100 mHz. [PERSON] [22] proposed an approach for the acquisition control of LISA formation to build the six laser links one by one, and the simulation results indicated that the LISA acquisition requirements can be met in a timely manner. [PERSON] [23; 24] designed a control system for the acquisition phase of the LISA constellation, built the six laser links one by one with the spiral scanning guidance low and implemented a Kalman filter to reject the disturbance. The simulations demonstrated that an acquisition strategy with the proposed control method is feasible. [PERSON] [25] proposed an acquisition scheme based on the differential power sensing method for the gravity measurement mission: two satellites scanned uncertainty cones simultaneously and independently. The laser beams were power-modulated at different frequencies instead of the commonly used single-way scanning method. The numerical simulation and experiments achieved a success rate of more than 99% and the acquisition time was significantly reduced. [PERSON] [26] proposed a high-speed, high-precision, fully automatic acquisition scheme for the Taiji Program. They developed an improved differential power sensing combined with the down-sampling algorithm and the match filter algorithm and achieved 1\(\upmu\)rad resolution and 220 s scanning time. Most of the current studies about the acquisition have focused more on the field of laser communication and less on the field of the GW detection. In addition, the studies shown above indicate that current schemes for the acquisition phase of the GW detection build the laser link one by one for six links which means more time costs. To reduce the acquisition time, we proposed a new configuration of satellite for the acquisition phase. In recent years, many space applications require stringent time response constraints. To satisfy the requirements for the time response constraints, the finite-time stability [27; 28; 29; 30] was proposed. Nevertheless, the finite-time stability often depends on the initial conditions of the system; that is to say, the finite time varies with the initial conditions. To solve this problem, the fixed time stability [31; 32; 33] was proposed where the time can be guaranteed to be a constant under the change of initial conditions. However, it is hard to obtain the direct relationship between the tuning parameters and the fixed stabilization time. Furthermore, a new method called prescribed-time stability [34; 35] control was proposed where the control gains are functions of the desired convergence time, whereas the convergence time is a conservative estimation of the upper bound of the stabilization time so that it is often lager than the true convergence time. To overcome these problems above, [PERSON] [36] developed the predefined-time stability control where the settling times can be considered as an explicit parameter and the predefined time is the minimum upper bound of the fixed stabilization time. As a new control strategy and powerful control method for the applications with stringent time response constraints, the predefined-time stability has been successfully applied in many fields and has achieved good results. [PERSON] [37] proposed a predefined-time robust stabilization for the robotic manipulators, [PERSON] [38] proposed a predefined-time attitude control for the rigid spacecraft and [PERSON] [39] proposed