The system dynamics model of the enhanced DFP spacecraft can refer to the previously published paper [
16]. A new disturbance attenuation and pointing control system is proposed, focusing on the integrated control strategy and coupling analysis, decoupling and feedback design, and amplitude-frequency response analysis and control bandwidth design.
3.1. Integrated Control Strategy and Coupling Analysis
The integrated application of the active vibration isolation control technology, the drag-free control technology and the spacecraft attitude control technology, the full-band disturbance attenuation and the high-precision pointing accuracy can be realized through the integrated control of the PM, SM and TM. The low-frequency disturbances are attenuated by the DFC loop and the high-frequency disturbances are attenuated by the AVIC loop. The DFC loop and the AVIC loop work together to achieve the payload’s full-frequency disturbance attenuation. Additionally, the PM is controlled to point to the target by utilizing the APC loop and the collision between the PM and the SM can be avoided by utilizing the ATC loop. The APC loop and the ATC loop work together to achieve the payload’s high-precision pointing requirement.
The control scheme of the enhanced DFP spacecraft is illustrated in
Figure 2, in which the subscripts S, P, and T represent the SM, the PM, and the TM, respectively, and are replaced by the X below;
,
,
,
,
, and
are the forces generated by the Earth, the thrusters, the flexible cables, the electromagnetic actuators, the other internal disturbances, and the other external disturbances, respectively;
,
,
,
,
, and
are the torques generated by the Earth, the flywheels, the flexible cables, the electromagnetic actuators, the other internal disturbances, and the other external disturbances, respectively;
and
are the control forces and torques, respectively;
is the attitude angle of the PM with respect to the geocentric inertial coordinate system;
is the attitude angle of the SM with respect to the PM;
is the relative displacement of the PM with respect to the SM;
is the relative displacement of the SM with respect to the TM; and
,
,
and
are the desired values of the
,
,
and
, respectively.
The four control loops of the enhanced DFP are coupled with each other, as shown in
Figure 3, which is mainly reflected in:
(1) Sensor measurement coupling: The 2D-PSDs deployed between the PM and SM simultaneously measure the relative displacement and relative attitude of the PM with respect to the SM. The measurement of the relative displacement and the measurement of relative attitude are used for the AVIC and the ATC loop, respectively. Therefore, there is sensor measurement coupling between the AVIC and ATC loops.
(2) Actuator output coupling: The electromagnetic actuators deployed between the PM and the SM simultaneously output the control forces and control torques acting on the PM. The control forces and the control torques are used for the AVIC and the APC loop, respectively. Therefore, there is actuator output coupling between the AVIC and APC loops.
(3) Action and reaction coupling: The electromagnetic actuators output the control force acting on the PM while producing reaction interference for the SM, so there is action and reaction coupling between the AVIC and DFC loops. At the same time, the electromagnetic actuators output the control torques acting on the PM while producing reaction interference for the SM, so there is action and reaction coupling between the APC and ATC loops. Similarly, the flexible cables deployed between the PM and the SM also cause action and reaction coupling among the control loops.
(4) Coupling of translation and rotation: The reaction forces caused by the electromagnetic actuators and the flexible cables act on the SM, and torques are generated due to the deviation from the centroid of the SM. As a result, there is a coupling of the translation and rotation between the AVIC and ATC loops.
3.2. Decoupling and Feedback Design
After decomposing the acceleration terms into the control action of the actuators, disturbance and coupling stiffness, the dynamics model is expressed as
where
is the acceleration caused by the thrusters;
is the angular acceleration caused by the flywheels;
and
are the acceleration and the angular acceleration caused by the electromagnetic actuators, respectively;
and
are the acceleration and the angular acceleration caused by other disturbances, respectively;
and
are the mass ratio and the inertia ratio of the PM to the SM, respectively; and
,
, and
are the coupling stiffnesses of the ATC loop, the AVIC loop and the DFC loop, respectively. With the assumption that the cross-coupling stiffnesses are all zeros,
,
, and
are diagonal matrices.
To facilitate feedback design and control bandwidth design, the decoupling is carried out based on pre compensation. The pre compensation terms are derived mathematically based on the dynamic model, without considering the uncertainties of system parameters. If the
,
,
,
,
and
are estimated or observable, feedforward compensation is carried out to eliminate the influence of various disturbances. The output of the actuators can be designed as
where
,
,
and
are the feedback control part of the actuators.
Based on the integrated control strategy of the enhanced DFP spacecraft, the control system is decoupled by pre-compensating, and the output of the actuators can be further designed as
where
,
,
and
are the feedback terms for controlling coordinate errors.
According to Equations (1) and (3), the decoupled control model of the enhanced DFP spacecraft is expressed as
After the decoupling design, the APC loop, the ATC loop, the AVIC loop, and the DFC loop all achieve control input decoupling and are decomposed into a single-input single-output control system. The feedback control of the enhanced DFP spacecraft is designed as follows. The controller of the APC feeds back the attitude pointing coordinate
, and the electromagnetic actuators output the control torques. The controller of the ATC feeds back the attitude tracking coordinate
, and the flywheels output the control torques. The controller of the AVIC feeds back the active vibration isolation coordinate
, and the electromagnetic actuators output the control forces. The controller of the DFC feeds back the drag-free coordinate
, and the thrusters output the control forces. Therefore, the feedback terms for controlling the coordinate errors are designed as
According to Equations (3) and (5), the ideal outputs of the actuators are expressed as
3.3. Amplitude-Frequency Response Analysis and Control Bandwidth Design
The new disturbance attenuation and pointing control system of the enhanced DFP spacecraft is converted into 12 single-input single-output control loops through the decoupling and feedback design. The amplitude-frequency response is analyzed and the control bandwidth is designed based on the Proportional-Integral-Differentive (PID) control algorithm. The general form of the PID controller can be written as
In the above expression, , and are the PID controller parameters; and are the feedback coordinate and its first-order derivative, respectively; and are the desired value of the feedback coordinate and its first-order derivative, respectively; and is the control output.
According to Equations (4) and (7), the sensitivity function and the complementary sensitivity function of the closed-loop control loop can be expressed as
where
and
are the stiffness and damping of coupling disturbances, respectively, which are caused by factors including the flexible cables and relative measurement.
The sensitivity function
and the complementary sensitivity function
indicate the ability of the closed-loop feedback control system to attenuate indirect disturbances and direct disturbances, respectively. The indirect disturbances caused by flexible cables and the sensor noises are related to the feedback coordinates. The direct disturbances directly influence the controlled object, including external disturbances such as atmospheric resistance and solar radiation pressure, internal disturbances such as motor rotation and solar panel vibration, and actuator output noises. To analyze the amplitude-frequency response of the disturbance attenuation performance of the closed-loop feedback control system, the sensitivity function
and the complementary sensitivity function
are rewritten as a combination of several minimum phase typical elements. The sensitivity function
and the complementary sensitivity function
can be expressed as
where
is the reciprocal of the time constant of the first-order inertia element;
and
are the undamped natural oscillation frequency and damping ratio of the second-order oscillation element, respectively; and
and
are the undamped natural oscillation frequency and damping ratio of the second-order differential element, respectively. The amplitude-frequency response of the minimum phase typical elements is clear.
Figure 4 displays an example of the amplitude-frequency response of the closed-loop feedback control system. The indirect disturbances in the high-frequency band can be attenuated by the sensitivity function
, while the direct disturbances in the low-frequency band can be attenuated by the complementary sensitivity function
.
The control bandwidth for each closed-loop control loop of the enhanced DFP spacecraft depends on the minimum phase typical element parameters. The APC loop attenuates the direct disturbances to improve the payload’s pointing accuracy, so the critical frequency of the complementary sensitivity function should be high, which means that the large control bandwidth is suitable for the APC loop. Furthermore, the ATC loop attenuates the direct disturbances to avoid a collision between the PM and the SM. Therefore, the critical frequency of the complementary sensitivity function should be high to generate large control bandwidth of the ATC loop. The AVIC loop is used to isolate the PM from the indirect disturbances in the high-frequency band, so the critical frequency of the sensitivity function should be low, which means that the small control bandwidth is suitable for the AVIC loop. The DFC loop is used to isolate the SM from the direct disturbances in the low-frequency band, so the critical frequency of the complementary sensitivity function should be high, which means that the large control bandwidth is suitable for the DFC loop.
The disturbance attenuation index of the disturbance attenuation and pointing control system exceeds −20 dB in the all-frequency band. In accordance with the design requirements of the amplitude-frequency response of each closed-loop control loop, the appropriate values of the minimum phase typical element parameters are determined and then the PID controller parameters are calculated. Comparing Equation (8) and Equation (9), the PID controller parameters can be expressed as
The design results are presented in
Table 2.