1. Introduction
Space gravitational wave detectors are intended to detect the space-time ripples (i.e., gravitational wave signal) generated by some events in the universe (such as the merger of a binary black hole [
1] and an asymmetrical stellar collapse and explosion [
2]) from the low-frequency band. Since Einstein predicted the existence of gravitational waves in 1916 [
3,
4], the idea of measuring and observing gravitational waves by experimental means has been lingering in the minds of many scientists [
5]. It is a challenging task because the signal is very weak. Space gravitational wave detectors such as LISA [
6], DECIGO [
7], Taiji [
8], Tianqin [
9], etc., use inter-satellite distributed interferometers for scientific measurement tasks. The laser interference link is used to measure the tiny position change between free floating test masses. The laser is sent from one test mass to a remote test mass. When the gravitational wave signal passes through the interferometer’s optical path, it will stretch the space and change the original optical path length. The gravitational wave signal is obtained by measuring the change in the length of the interferometer’s laser path. The accuracy of measuring optical path length changes must be in nanoradians [
10]. We must eliminate variations in the length of the laser link that are affected by non-gravitational perturbations to ensure that these changes are only affected by gravitational waves.
The relatively mature and typical solution of the space gravitational wave detector is to use six sets of telescopes and optical platforms in three sets of spacecrafts (S/C) to launch lasers in deep space, between hundreds of thousands of kilometers and millions of kilometers away from Earth [
11,
12,
13,
14]. To reach picometer accuracy in space gravitational wave interferometry, it is necessary to shield any non-gravitational interference noise and perform a series of operations to control the spacecraft and its payload in parallel [
3]. In addition, the drag-free control of the test mass must be realized in the direction of the sensitive axis. For scientific measurement, the links, which act as a deep-space, large-scale, long-baseline inter-satellite interferometer, will provide length measurements with picometer accuracy. With high-precision attitude pointing and alignment, the optical path will have sufficient transmission integrity to ensure that the measurement task can be completed. There are two control schemes for the continuous maintenance and tracking of the inter-satellite optical path. One is realized by the spacecraft attitude control in cooperation with the telescope, and the other is realized by using the laser launcher to employ the Fast Steering Mirror (FSM) self-tuning control. In the engineering implementation, the FSM scheme will cause the optical path change and affect the ranging accuracy, so we opted to use the first scheme.
For the line-of-sight (LOS) tracking control in science mode, it is necessary to maintain a high-precision relative attitude for the spacecraft. Each spacecraft uses its two laser links to obtain the relative attitude of the other two distant spacecrafts. Due to the requirement of ultra-high-precision measurement, the spacecraft no longer controls its attitude through the star sensor signal; instead, this is controlled through differential wavefront sensing (DWS) [
15,
16], and the attitude error precision calculated by DWS measurement can reach the nanometer level.
The long-term, accurate capture of the optical path guarantees that the measurement task can be carried out effectively. The LOS tracking of the three groups of S/C is very critical in maintaining a stable laser link, which depends on the attitude adjustment of the S/C and telescope [
17]. In science mode, the spacecraft and telescope controls couple with the test mass control, and the free-falling test mass is captured by the drag-free control and electrostatic suspension control [
17,
18,
19]. Aiming at resolving the problem of coupling in each control loop of the drag-free control system, Fichter and Gath [
20,
21] decoupled the S/C control, telescope control, and test mass control for the drag-free control system in science mode. The proposed methods can decouple all control loops and design the controller through loop-shaping. The simulation and performance verification of each loop are carried out. Loop-shaping design is a method of balancing the sensitivity function and the complementary sensitivity function of the control loop in the full-frequency domain through the weighting function. It is an indirect shaping method in a specific frequency band. Based on the above considerations, we directly designed the controller in the finite frequency domain, which is easy to implement, has low order, and can make the concerned performance the best in the finite frequency domain.
For the ultra-fine pointing control of the optical path, Bauer [
15] simulated the precision pointing performance of LISA based on a model with 19 degrees of freedom. Basile [
22] used the PID controller based on quaternion feedback to simulate fine pointing attitude control of LISA. Zhang [
23] used sliding mode control technology combined with neural network optimization to control the pointing of the telescope with high precision. The above references mainly focused on attitude maneuver based on the satellite’s own attitude information.
In order not to generate interference that disrupts scientific measurements, a suitable controller needs to be designed, and the controller must achieve noise rejection and attitude tracking accuracy performance within the measurement bandwidth (MBW) [
22,
24]. Aiming at finding solutions for specific frequency noise suppression and the tracking control problem, Wu and Zocco [
25,
26] used loop shaping to trade off the control loop sensitivity function and the complementary sensitivity function through the weighting function in the full-frequency domain, and they also performed indirect specific frequency band shaping for noise suppression. Ren and Sun [
27,
28] used the finite frequency controller directly focused on the frequency domain of interest and optimized the system performance in a specific frequency band. According to the different requirements of different frequency bands, Pan, Mi, and Lian [
29,
30,
31] designed corresponding controllers and combined the ideological advantages of frequency division control to improve the performance of the control system.
The existing references lack the simulation research on LOS changes caused by the joint action of spacecraft and telescope attitude control loops when the detector performs the measurement task. The attitude pointing stability of the line-of-sight direction of the space gravitational wave detector’s telescope needs to reach a very safe level, such as nrad/Hz. At the same time, because the telescope’s line-of-sight direction is determined by the coupling of spacecraft attitude and telescope attitude, and in order to avoid the unnecessary and frequent attitude maneuvering of the spacecraft as much as possible, which will affect the high-precision line-of-sight alignment, it is necessary to further optimize and study the joint control of the LOS attitude control loop.
Since S/C attitude control and telescope attitude control work simultaneously, the tracking process needs to consider the performance changes that their interactions bring to LOS pointing. In order to maintain the stability of the spacecraft platform as much as possible and avoid its unstable movement or frequent movements that have a great adverse effect on LOS tracking, in this paper, we establish a linear solution model for telescope LOS error and the corresponding spacecraft and telescope attitude error based on DWS high-precision attitude measurement. Based on the attitude measurement model, we designed a frequency division controller to improve the attitude and pointing performance of LOS tracking. We chose the telescope attitude control loop frequency division as it is the faster response part, mainly relative to the high-frequency band within the MBW. The S/C attitude frequency division is mainly in the low-frequency band within MBW. A frequency division point search method based on the minimization of the cost function of the spacecraft and the telescope is used to reduce the frequent maneuvering of the spacecraft. In turn, the attitude angle power spectral density (PSD) of the entire control system is significantly reduced.
The subsequent subsections are arranged as follows.
Section 2 introduces the background requirements of the LOS tracking control of space gravitational wave detectors;
Section 3 focuses on the design of the frequency division control;
Section 4 carries out the simulation;
Section 5 presents our conclusions and a future scope.
3. Frequency Division Control
The LOS tracking control mechanism is shown in
Figure 2. It is a coupled model of S/C attitude control and telescope attitude control. Since S/C attitude control and telescope attitude control work simultaneously, the tracking process needs to consider the performance changes that their interactions bring to LOS pointing.
In the frequency domain, consider the LOS tracking error , the S/C attitude control, and the telescope attitude control error . Then, define the control demand function , , to balance the low and high-frequency attitude deviation distribution and the LOS deviation performance in each control loop. is the weight coefficient used to adjust the influence of the corresponding frequency components of attitude deviation and LOS tracking deviation in the system. Among them, is the frequency division transfer function matrix. In order to maintain the stability of the spacecraft platform as much as possible and to avoid a large adverse effect on LOS tracking due to its unstable motion or frequent motion, we chose the telescope attitude control loop frequency division as it is the faster response part, mainly relative to the high-frequency band within the MBW. The S/C attitude frequency division is mainly in the low-frequency band within the MBW.
The control torque of the S/C frequency division control slows down in the high-frequency band, and the response of the telescope control torque slows down in the low-frequency band. By changing the frequency division, we can achieve a kind of coordination and trade-off in the frequency division control of the telescope and the S/C. Among them are the following equations:
A frequency division point search method based on minimizing the cost function of the spacecraft and the telescope is used to perform the frequency division operation of the controller to improve the pointing stability of the system. Based on the idea of Pan [
29], our frequency division calculation algorithm is as follows:
where
is the cost function for S/C,
is the cost function for the telescopes,
is the frequency division, and
is the end frequency of the division interval. The cost functions for the frequency division control of the S/C and the telescope are determined by the following equation:
with
with
where
and
are constants, which are related to delay and deviation.
represents the changes in expectations of
. We expect that the S/C attitude control loop and the telescope attitude control loop will work together to reduce
to 0 as much as possible; we also consider the sensor and actuator errors in the calculation, and the lower limit value of
is set to 1 nrad. In addition,
The goal is to minimize the cost function. The design of the spacecraft cost function is focused on the trade-off of the consequences of the delay, and the telescope cost function is focused on the trade-off of the consequences of the bias of attitude pointing. When the high-frequency component of the LOS pointing performance evaluation function is greater than the given deviation requirement value, the accumulation of large deviations for a long time will inevitably affect the output performance of the LOS pointing and will not even meet the maximum pointing deviation requirement. At this time, if the frequency division can be appropriately increased, under the premise that the delay characteristics of the spacecraft have little effect on the LOS attitude and pointing deviation, the spacecraft attitude controller can reduce a part of the attitude deviation. Under the condition that the high-frequency component of the LOS pointing performance evaluation function is not large, and that the passive deviation requirement value is allowed, the frequency division will be appropriately reduced to give full play to the fast response characteristics of the telescope attitude and improve the performance of the LOS tracking control. When we fix the measurement parameters of the cost function, and when the division frequency is lower, the cost function value of the telescope attitude control is higher, and the cost function value of the spacecraft attitude control is lower. On the contrary, the higher the frequency division, the lower the cost function value of the telescope attitude control, and the higher the cost function value of the spacecraft attitude control.
We set the basic controller part as a finite frequency controller [
27], which satisfied the following: the output torque of the thruster is limited
, the output is limited
,
, the closed-loop system is asymptotically stable under the disturbance controller, which satisfied the following equation:
The system measurement equation and the output equation are as follows:
In addition,
where,
,
,
,
,
,
.
The dynamic output feedback controller is constructed as follows:
The closed loop system is written as follows:
where
We then designed , , to meet the following requirements:
- (1)
The closed-loop system is asymptotically stable in the absence of disturbances.
- (2)
Under the action of disturbance, the performance should be met using the following equation:
- (3)
The output torque limit is .
- (4)
The output limit is .
For the control system, given a positive scalar
,
,
, if there is a symmetric matrix
,
,
,
,
,
,
,
, a general matrix
,
,
,
,
,
,
,
M,
,
satisfies the following inequality constraints:
Our design requirements can be met if:
If there is a feasible solution to the inequality, the controller coefficient matrix can be solved by the equation below:
Theorem 1 (Lemma 1 (S-procedure) [
34]).
For a vector , the Hermitian matrix is , . The necessary and sufficient conditions for the establishment of the formula are: , . Proof of Condition 1. The closed-loop system is asymptotically stable in the absence of perturbations.
Suppose that the invertible matrix U and its corresponding inverse matrix V are partitioned as follows:
Let
,
,
, we have
Construct the following Lyapunov–Krasovskii functional
, then
since the closed-loop system satisfies in the absence of the perturbation:
Condition 1 is confirmed. □
Proof of Condition 2. The closed-loop system satisfies the performance index under the disturbance .
First, construct the Lyapunov–Krasovskii functional
and
because the closed loop is under disturbance; hence,
where
Under zero initial conditions, we perform the Fourier transform; noting that
and integrating the inequalities, we obtain the following equation:
Using Parseval’s theorem, the time domain is converted into the frequency domain as follows:
It is clear that when
is established, then we have
, and the closed-loop system can meet the limited frequency domain performance. Note that
, where
.
Therefore, according to Lemma 1,
, and the necessary and sufficient conditions are
, where
Let
, and
can be written as follows:
Therefore, we obtain . Condition 2 is confirmed. □
Proof of Condition 3. The closed-loop system satisfies the output torque limit .
When
,
. The integral of the formula from 0 to
∞ is
. Let
; note that when
, we obtain
, and let
. Then,
Condition 3 is confirmed. □
Proof of Condition 4. The closed-loop system satisfies the output limit
. Let
, then
Condition 4 is confirmed. □
4. Simulation
The simulation parameters are shown in
Table 2. Without loss of generality, set the initial DWS deviation as
rad, under the line-of-sight pointing constrain limit of
rad. The noise shape function of the sensors, actuator, and sun pressure are shown in
Figure 4. The attitude changes of the LOS tracking due to disturbance are shown in
Figure 5. It can be seen that under the disturbance, the LOS pointing error drifts with the noise, and the change in its PSD curve exceeds the LOS pointing requirements with respect to time, which is not tolerated.
Using loop shaping, finite frequency, and frequency division controller, the PSDs of the output torque of the S/C and the telescope control loop are shown in
Figure 6 and
Figure 7. Among them, the frequency division control adopts the frequency division point search method, which minimizes the cost function of the spacecraft and the telescope and performs an adaptive frequency division operation. The frequency division point changes, as shown in
Figure 8. The frequency division control, finite frequency control, and the loop-shaping
control pointing error of LOS tracking under disturbance as well as its corresponding attitude pointing PSD are shown in
Figure 9,
Figure 10,
Figure 11,
Figure 12 and
Figure 13.
Finite frequency control and frequency division control significantly improve the performance of the output torque compared to the loop-shaping
control, as shown in
Figure 6 and
Figure 7. For the LOS tracking scheme, when the S/C and telescope attitude control loops are jointly controlled, the stability of the output torque of the S/C attitude control loop using frequency division control is better than the other two control methods within the whole frequency domain. However, for the attitude control of the telescope, the output torque stability performance of the frequency division control is worse than that of the finite frequency control at the beginning of the low frequency due to the influence of the reduced sensitivity of the low-frequency signal. At the same time, we can see the performance of frequency division control in the high-frequency band. In the high-frequency band within the MBW, the output torque has significantly improved stability performance. In the high-frequency band outside the MBW, the output torque performance is inferior to the finite frequency control. Since we focused on measuring the performance within the MBW, this does not affect the overall superiority of the frequency division control over the other two control methods.
Adaptive frequency division was used for the frequency division controllers, as shown in
Figure 8. It can be seen that the optimal frequency division point slides from a higher frequency to a lower frequency and finally stabilizes at a constant value. According to the algorithmic implications of minimizing the cost function of the spacecraft and the telescope, it also shows that the control torque output of the spacecraft tends to speed up when the deviation is large, and that the control torque output of the spacecraft slows down when the deviation is small. Therefore, the frequency division control mechanism reduces the frequent maneuvering of the spacecraft to a certain extent.
Frequency division control enables faster and smoother tracking within the detector’s MBW, as shown in
Figure 9,
Figure 10 and
Figure 11. We compared the performance of the algorithm in terms of the convergence rate of the LOS tracking error and the overshoot. The finite frequency controller converges faster and with a lighter overshoot than the loop-shaping
controller. However, the optimal frequency division control, with the fastest convergence and the lightest overshoot, can achieve accurate LOS tracking and a more stable convergence within 2 s.
Frequency division control can achieve more stable LOS pointing stability within MBW, as shown in
Figure 12 and
Figure 13. For the LOS pointing stability of the telescope T1, the frequency division control has the best performance. The finite frequency control is superior to the loop-shaping
control in the high-frequency band and has little difference compared with the loop-shaping control in the low-frequency band. For telescope T2, due to the large overshoot, the line-of-sight pointing stability of the traditional loop-shaping
controller is not ideal. The best sequence for noise suppression is frequency division control, finite frequency control, and loop-shaping
control.