# Envelope-Based Variable-Gain Control Strategy for Vibration Suppression of Solar Array Using Reaction Wheel Actuator

Authors: Song Wu; Guoan Tang; Bifa Chen
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

The orbital operation of spacecraft can excite the long-drawn and low-frequency vibration of the solar array, which is prone to affecting the task execution of the system. To address this issue, an envelope-based variable-gain control strategy is proposed to suppress vibration of the solar array using the reaction wheel (RW) actuator. The RW actuator is individually mounted on the solar array to provide reaction torque through the speed change of its rotor. The governing equation of motion of the solar array actuated by a RW actuator is deduced with the state space representation. The control relation between the measured bending moment and the rotational speed of the RW actuator with the constant-gain coefficient is firstly developed and demonstrated in numerical simulation. Changing the gain coefficient to be inversely proportional to the envelope function of vibration, a variable-gain control strategy is proposed to improve the damping effect of the RW actuator. Simulation results show that the vibration suppression performance of the RW actuator is improved compared to the constant-gain control. As the actual on-orbit natural frequency of the solar array is not always exactly known, the robustness of the control system is analyzed for the deviation between the estimated and the actual natural frequency values. The proposed variable-gain control is also experimentally verified using a simplified elastic plate model. Experimental results indicate that the vibration attenuation time is decreased to 29.1% and 50.22% compared to the uncontrolled and the constant-gain controlled states, respectively.


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## Body

## 1. Introduction

With the trend of larger expansion in dimensions, the deployable appendages of the spacecraft, such as the solar array, the antenna, and the manipulator, possess remarkable flexibility in a fully deployed state [1, 2]. The ultralow natural frequency of the large-scale flexible appendage fn is the main reason for slow vibration attenuation. The vibration can result from various on-orbit excitations, e.g., the spacecraft’s attitude adjusting, the payload pointing, and the spacecraft docking [3–5]. Besides, due to the absence of air resistance and weak friction between mechanisms under low gravity, the inherent modal damping ratio ζn of the appendage is quite small. According to the exponential law of vibration attenuation, for vibration amplitude decaying by ς times, the required time is estimated by −lnς/2πfnζn. If the empirical values of fn and ζn are taken as 0.05 Hz and 0.005, respectively, and the required ς=5%, then the required time is 1907.1 s. In such cases, it takes a long time to sufficiently attenuate the excited vibration, which is unfavorable for the spacecraft attitude control and payload accurate pointing. In addition, the longtime vibration may accumulate mechanical damage to the flexible appendages.Various methods have been developed to suppress the vibration of the spacecraft’s flexible appendages in the last decades. These methods can mainly be divided into the feedforward control such as the input shaping technique, the passive vibration control (PVC), and the active vibration control (AVC). The input shaping technique was applied to reduce the vibration of the appendages during spacecraft orbital maneuvering, solar array sun pointing, or manipulator transporting [6, 7]. For example, Na et al. [4] used the built-in actuator at the array base to control the vibration of a solar array. However, the input shaping technique does not work when the structural vibration direction is different from the actuating direction of the built-in actuator. Moreover, it is not suitable for accelerating the residual vibration attenuation of the structure.Adopting the passive or active control method satisfied the needs of accelerating vibration attenuation [8–11]. By applying damping materials or passive dampers to the structures, the passive control methods were implemented and demonstrated to be highly stable and efficient [12–14]. However, for the large-scale flexible appendages, the partially covered damping materials or passive dampers are not able to efficiently improve the low-order modal damping of the whole system. The active control methods perform well by combining a network of sensor-controller-actuator and provide a significant vibration suppression effect [15–17]. The actuator selection for different application situations is one of the issues in AVC. Piezoelectric material is a type of widely used actuator [18–20]. It is also employed as the sensor in active control [21–24]. However, the suppression effect is not obvious when the piezoelectric actuator is applied to large-scale flexible space structures. To solve this issue, Casella et al. [25] combined both the piezoelectric material and the on–off air jet thrusters as the actuators, but the compensation of the air jet working medium of the thrusters is an obstacle in space application. Sun et al. [26] proposed a novel vibration suppression scheme for solar arrays by using the cable-driven parallel robot actuators, which achieved a remarkable vibration suppression effect. Whereas for the deployable space structures, the involved cables cause issues in deploying process design. Meanwhile, its robustness is also an issue in practical application. Hu et al. [27] used a stiffness-variable joint mechanism for two hinged flexible plates. Both simulation and experimental results demonstrated the significant damping effect induced by the innovative joint mechanism. However, the actuator is only suitable for the sort of flexible structures embedded with joint mechanisms.The reaction wheel (RW) is usually implemented in spacecraft attitude control, which has the benefits of light weight and high efficiency with consumption of renewable electricity [28–32]. An active control method to suppress the vibration of the solar array through the RW actuator was studied in [33]. Numerical simulations of a solar array showed that, by using a RW actuator with approximately 2% of the structural mass, the vibration attenuation time was reduced to one-third of that from the uncontrolled state. However, for aerospace applications, the requirements of the RW actuator, such as the volume and weight, are very strict. Therefore, it is desirable to maximize the actuator’s vibration suppression capability. The sliding mode control [34, 35], the fuzzy logic [36], and the artificial algorithm [37] were investigated and integrated into the vibration control strategy design for the space flexible appendages. Bonding the piezoelectric actuator to the flexible panel, Liu et al. [38] developed a hybrid control scheme for the attitude stabilization and vibration suppression of the flexible spacecraft by considering model parameter uncertainty, measurement error, actuator faults, and other factors. It is worth pointing out that finding the optimal gains for the controller is very crucial and challenging for achieving the best performance of vibration suppression in the control system, which is still an open research problem. The variable gain control methods have been developed by other researchers in various engineering applications previously [39–43]. Combining the variable gain control into the design of a vibration suppression strategy has the potential in improving the damping performance of the actuators. Due to the fact that the gain adjustment rules require complex procedures and the calculations of the relevant parameters usually depend on experience [44–46], it is challenging to construct a feasible variable-gain control strategy for AVC in actual implementation.In this paper, we focus on developing a variable gain control strategy for the RW actuator to maximize its capability in suppressing vibration of the solar array. The main contribution of this paper is the variable-gain control strategy through the envelope function of vibration to improve the damping performance of the RW actuator for vibration suppression of the solar array. The numerical models of a full-scale solar array with variable-gain control and constant-gain control are developed to evaluate and validate the proposed algorithm. An elastic plate model is used in the laboratory for feasibility investigation. The design, implementation, and verification of the proposed strategy are presented to highlight the significant vibration suppression performance of the RW actuator. The control relation between the measured bending moment and the rotational speed of the RW actuator with the constant gain is firstly applied to the solar array. It is figured out from the simulation results that a larger amplitude of the rotational speed of the RW actuator implies a larger output reaction torque in the control, which leads to a better damping effect. In our work, adjusting the control gain to be linear to the inverse of the vibration envelope is the strategy to increase the output reaction torque of the RW actuator. The proposed variable-gain control strategy is proved to be able to improve the damping performance of the RW actuator compared to the constant-gain control.The rest of the paper is organized as follows. In Section2, dynamic modelling for vibration suppression of the solar array by utilizing the RW actuator is deduced, and the damping effect of the RW actuator is demonstrated through a full-scale flexible solar array with numerical simulation. The design, numerical verification, and robustness analysis of the envelope-based variable-gain control strategy are discussed in Section 3. Experimental results are presented in Section 4 to illustrate the effectiveness and feasibility of the presented method. Conclusions are drawn in Section 5.


## 2. Dynamic Modelling and Effect of Vibration Suppression for Solar Array

### 2.1. Vibration Suppression for Solar Array Utilizing RW Actuator

For vibrating structures dominated by the first-order mode, the physical quantities describing the structural motion state can be approximately expressed by the modal coordinateξ1 and its corresponding modal shape function φ, in which ξ1 is a temporal function and φ is a spatial function. For example, as shown in Figure 1, the solar array has the geometric characteristic of a large length-to-width ratio. The root of the solar array connecting to the spacecraft body is fixed based on the assumption that the mass and moment of inertia of the spacecraft body are much larger than that of the solar array [47]. For the sake of illustration, a Cartesian coordinate system is established on the solar array. The X-axis is parallel to the normal direction of the panel surface. The Y-axis is parallel to the extension direction. The Z-axis follows the right-hand rule. In this paper, the first-order out-of-plane bending mode of the solar array along the X-axis is assumed to be dominant.Figure 1
Structural diagram of the solar array with a RW actuator.To suppress the first-order out-of-plane bending vibration, a RW actuator is mounted at the midspan of the free side of the solar array, as shown in Figure1. The reaction torque induced by the speed change of the RW actuator is applied as the active force. The angular displacement around the Z‐axis at the RW mounted position is designated as θ. The root bending moment Mb is defined as the moment around the Z‐axis at the connected position between the root of the flexible solar array and the spacecraft body. Both Mb and θ can be expressed as linear functions of the modal coordinate ξ1.Denoting the linear proportionality coefficients asφM and φθ, Mb and θ can be written as
(1)Mb=φMξ1,θ=φθξ1,respectively. Such that
(2)Mb=φMφθθ.The control law for the rotational speed of RWΩ is designed to be proportional to the root bending moment Mb:
(3)Ω=ηMb=ηφMξ1.The rotating axis ofΩ is parallel to the Z‐axis. The control gain η can be given by
(4)η=ΩmaxMb,max=ΩmaxθmaxφθφM.Here,Ωmax denotes the permitted maximum rotational speed of the RW actuator; the peak values of θ,Mb during vibration are θmax,Mb,max, respectively. Specifying η by Equation (4) results that the RW actuator rotates at Ωmax when the structural vibration reaches the peak with no overload.Denoting the rotational acceleration of the RW asΩ̇, according to the moment of momentum theorem, the induced reaction torque applied to the solar array is Ta=−IΩ̇. Considering Equation (3), the reaction torque Ta can be written as
(5)Ta=−IηṀb,in which I is the rotating moment of inertia of the RW about its rotational axis. Controlling the rotational speed Ω by the designed algorithm, the induced torque Ta can be used to suppress the vibration of the solar array.Performing a first-order time derivative on both sides of Equation (2) and substituting the result into Equation (5) by considering Equation (4) gives
(6)Ta=−IΩmaxθmaxθ̇.The minus sign “−” guarantees the direction of the reaction torque Ta always opposite to the direction of the angular velocity θ̇. As a result, the RW actuator can always do negative work to the solar array. That is, Ta plays a role of consuming the vibration mechanical energy of the solar array, so it can be regarded as artificial damping.The motion equation of the flexible solar array with a RW actuator can be expressed in the first-order modal coordinate as(7)ξ¨1+2ω1ζ1ξ̇1+ω12ξ1=φθTa.Here,ω1 and ζ1 are the first-order natural frequency (rad/s) and modal damping ratio, respectively. The detailed derivation of Equation (7) can be consulted from [33]. Rewriting Equation (7) in a state-space representation:
(8)ẋ=Ax+Bu,where
(9)x=ξ̇1ξ1T,u=Ta,A=−2ω1ζ1ω12−10,B=φθ0T.They can be referred to as the state vector (x), the input vector (u), the state matrix (A), and the input matrix (B), respectively.On the other hand, Equation (1) can be equivalently written as
(10)y=Cx,where
(11)y=Mb,C=0φM.Here,y and C are the output vector and the output matrix, respectively.Equation groups (8) and (10) are the typical state-space form of a dynamic system [48]. The abovementioned modal parameters can be obtained by numerical approach, e.g., finite element method.


### 2.2. Numerical Simulations and Analysis

The solar array with a deployed area of6.2m×25m utilized in the Tianhe core module of the Chinese Space Station is adopted to validate the effectiveness of the vibration control method. The corresponding finite element model with a full scale is constructed using MSC/PATRAN, and it is shown in Figure 2.Figure 2
The finite element mesh of the flexible solar array and its first-order bending mode.The flexible panels are meshed by shell elements. The driving mechanisms, the deployable structures, and the storing tubes and boxes are modelled using beam elements with measured physical parameters. Interested readers can consult Reference [49] for more detailed information. A torsional spring element with a modulus of 106N⋅m/rad is employed to connect the root of the solar array and the spacecraft body. The end of the spring element with connection to the spacecraft body is fixed. The finite element mesh (blue grid) and its first-order out-of-plane bending vibration mode are shown in Figure 2. The modal analysis is performed with MSC/NASTRAN. The total mass of the solar array model is 594.2kg. The RW actuator selected in the simulation weighed 1kg, which accounted for about 0.17% of the model in this study case. The moment of inertia of the current RW is 0.005kg⋅m2. The maximum rotational speed is 6000RPM, i.e., Ωmax=628.3rad/s.The modal parameters of the full-size flexible solar array model utilized in the numerical simulation are listed as follows. The first-order natural frequency of the model isω1=0.2746rad/s. The modal parameters mentioned in Equation (1) are φM=29.15N⋅m and φθ=0.003rad, respectively. Considering the testing experiences for similar structures, the modal damping ratio of the model is taken as ζ1=0.005 (very small as suggested by the aerospace agency).To make the simulation more realistic, a band-pass filter is adopted for the sensor signal to eliminate the DC (direct current) drift and high-frequency noise. The band-pass filter is a 2nd-order Butterworth filter with a high cut-off frequency of 0.06118 Hz and a low cut-off frequency of 0.02622 Hz. Namely, the band width of the filter is determined through taking the first-order bending natural frequency of the solar array as the central frequency and floating up and down by 40%. The constant-gain control law expressed in Equation (3) is firstly employed to verify the vibration suppression effect. The simulation process is executed with a fixed-step size of 0.01 s. The peak value of the root bending moment is approximately 150N⋅m once disturbed. Thus, the initial conditions of the model are given by
(12)ξ10=0,ξ̇10=1.5s−1.As shown in Figure3, the controller realized by Equation (3) is involved to obtain the speed signal for the RW actuator. The control gain is 3.76N⋅m⋅s/rad given by Equation (4). If the gain coefficient is 0, the reaction torque Ta equals to 0 during the whole simulation process, which can be regarded as the state without control.Figure 3
Diagram of the feedback control with constant gain.The simulation results of the bending momentMb with and without control are plotted as the solid and the dashed lines in Figure 4, respectively. Compared to the dashed line (without control), the attenuation of the solid line (with control) is accelerated. The comparison demonstrates that the reaction torque Ta can be used to suppress the vibration of the solar array through designing the rotational speed Ω according to the control law given by Equation (3).Figure 4
Response curve of the root bending moment with constant-gain control.The rotational speedΩ is shown in Figure 5. It is much larger and close to its permitted maximum value Ωmax in the early stage. However, according to Equations (3) and (5), the peak values of Ω and the reaction torque Ta decrease with the attenuation of vibration. For example, the rotational speed of the RW decays to 267.4rad/s at 384.3 s, which is less than half of Ωmax. In this paper, the amplitude of the rotational speed is defined as a measure of the capability of the RW actuator utilized for vibration suppression. A larger amplitude of the rotational speed, which is closer to Ωmax, is preferred, for which the capability of the RW actuator is more sufficiently utilized to provide a better control effect. Hence, it indicates that the capability of the RW actuator has the potential to be improved for providing a better control effect.Figure 5
Rotational speed of the RW actuator with constant-gain control.


## 2.1. Vibration Suppression for Solar Array Utilizing RW Actuator

For vibrating structures dominated by the first-order mode, the physical quantities describing the structural motion state can be approximately expressed by the modal coordinateξ1 and its corresponding modal shape function φ, in which ξ1 is a temporal function and φ is a spatial function. For example, as shown in Figure 1, the solar array has the geometric characteristic of a large length-to-width ratio. The root of the solar array connecting to the spacecraft body is fixed based on the assumption that the mass and moment of inertia of the spacecraft body are much larger than that of the solar array [47]. For the sake of illustration, a Cartesian coordinate system is established on the solar array. The X-axis is parallel to the normal direction of the panel surface. The Y-axis is parallel to the extension direction. The Z-axis follows the right-hand rule. In this paper, the first-order out-of-plane bending mode of the solar array along the X-axis is assumed to be dominant.Figure 1
Structural diagram of the solar array with a RW actuator.To suppress the first-order out-of-plane bending vibration, a RW actuator is mounted at the midspan of the free side of the solar array, as shown in Figure1. The reaction torque induced by the speed change of the RW actuator is applied as the active force. The angular displacement around the Z‐axis at the RW mounted position is designated as θ. The root bending moment Mb is defined as the moment around the Z‐axis at the connected position between the root of the flexible solar array and the spacecraft body. Both Mb and θ can be expressed as linear functions of the modal coordinate ξ1.Denoting the linear proportionality coefficients asφM and φθ, Mb and θ can be written as
(1)Mb=φMξ1,θ=φθξ1,respectively. Such that
(2)Mb=φMφθθ.The control law for the rotational speed of RWΩ is designed to be proportional to the root bending moment Mb:
(3)Ω=ηMb=ηφMξ1.The rotating axis ofΩ is parallel to the Z‐axis. The control gain η can be given by
(4)η=ΩmaxMb,max=ΩmaxθmaxφθφM.Here,Ωmax denotes the permitted maximum rotational speed of the RW actuator; the peak values of θ,Mb during vibration are θmax,Mb,max, respectively. Specifying η by Equation (4) results that the RW actuator rotates at Ωmax when the structural vibration reaches the peak with no overload.Denoting the rotational acceleration of the RW asΩ̇, according to the moment of momentum theorem, the induced reaction torque applied to the solar array is Ta=−IΩ̇. Considering Equation (3), the reaction torque Ta can be written as
(5)Ta=−IηṀb,in which I is the rotating moment of inertia of the RW about its rotational axis. Controlling the rotational speed Ω by the designed algorithm, the induced torque Ta can be used to suppress the vibration of the solar array.Performing a first-order time derivative on both sides of Equation (2) and substituting the result into Equation (5) by considering Equation (4) gives
(6)Ta=−IΩmaxθmaxθ̇.The minus sign “−” guarantees the direction of the reaction torque Ta always opposite to the direction of the angular velocity θ̇. As a result, the RW actuator can always do negative work to the solar array. That is, Ta plays a role of consuming the vibration mechanical energy of the solar array, so it can be regarded as artificial damping.The motion equation of the flexible solar array with a RW actuator can be expressed in the first-order modal coordinate as(7)ξ¨1+2ω1ζ1ξ̇1+ω12ξ1=φθTa.Here,ω1 and ζ1 are the first-order natural frequency (rad/s) and modal damping ratio, respectively. The detailed derivation of Equation (7) can be consulted from [33]. Rewriting Equation (7) in a state-space representation:
(8)ẋ=Ax+Bu,where
(9)x=ξ̇1ξ1T,u=Ta,A=−2ω1ζ1ω12−10,B=φθ0T.They can be referred to as the state vector (x), the input vector (u), the state matrix (A), and the input matrix (B), respectively.On the other hand, Equation (1) can be equivalently written as
(10)y=Cx,where
(11)y=Mb,C=0φM.Here,y and C are the output vector and the output matrix, respectively.Equation groups (8) and (10) are the typical state-space form of a dynamic system [48]. The abovementioned modal parameters can be obtained by numerical approach, e.g., finite element method.


## 2.2. Numerical Simulations and Analysis

The solar array with a deployed area of6.2m×25m utilized in the Tianhe core module of the Chinese Space Station is adopted to validate the effectiveness of the vibration control method. The corresponding finite element model with a full scale is constructed using MSC/PATRAN, and it is shown in Figure 2.Figure 2
The finite element mesh of the flexible solar array and its first-order bending mode.The flexible panels are meshed by shell elements. The driving mechanisms, the deployable structures, and the storing tubes and boxes are modelled using beam elements with measured physical parameters. Interested readers can consult Reference [49] for more detailed information. A torsional spring element with a modulus of 106N⋅m/rad is employed to connect the root of the solar array and the spacecraft body. The end of the spring element with connection to the spacecraft body is fixed. The finite element mesh (blue grid) and its first-order out-of-plane bending vibration mode are shown in Figure 2. The modal analysis is performed with MSC/NASTRAN. The total mass of the solar array model is 594.2kg. The RW actuator selected in the simulation weighed 1kg, which accounted for about 0.17% of the model in this study case. The moment of inertia of the current RW is 0.005kg⋅m2. The maximum rotational speed is 6000RPM, i.e., Ωmax=628.3rad/s.The modal parameters of the full-size flexible solar array model utilized in the numerical simulation are listed as follows. The first-order natural frequency of the model isω1=0.2746rad/s. The modal parameters mentioned in Equation (1) are φM=29.15N⋅m and φθ=0.003rad, respectively. Considering the testing experiences for similar structures, the modal damping ratio of the model is taken as ζ1=0.005 (very small as suggested by the aerospace agency).To make the simulation more realistic, a band-pass filter is adopted for the sensor signal to eliminate the DC (direct current) drift and high-frequency noise. The band-pass filter is a 2nd-order Butterworth filter with a high cut-off frequency of 0.06118 Hz and a low cut-off frequency of 0.02622 Hz. Namely, the band width of the filter is determined through taking the first-order bending natural frequency of the solar array as the central frequency and floating up and down by 40%. The constant-gain control law expressed in Equation (3) is firstly employed to verify the vibration suppression effect. The simulation process is executed with a fixed-step size of 0.01 s. The peak value of the root bending moment is approximately 150N⋅m once disturbed. Thus, the initial conditions of the model are given by
(12)ξ10=0,ξ̇10=1.5s−1.As shown in Figure3, the controller realized by Equation (3) is involved to obtain the speed signal for the RW actuator. The control gain is 3.76N⋅m⋅s/rad given by Equation (4). If the gain coefficient is 0, the reaction torque Ta equals to 0 during the whole simulation process, which can be regarded as the state without control.Figure 3
Diagram of the feedback control with constant gain.The simulation results of the bending momentMb with and without control are plotted as the solid and the dashed lines in Figure 4, respectively. Compared to the dashed line (without control), the attenuation of the solid line (with control) is accelerated. The comparison demonstrates that the reaction torque Ta can be used to suppress the vibration of the solar array through designing the rotational speed Ω according to the control law given by Equation (3).Figure 4
Response curve of the root bending moment with constant-gain control.The rotational speedΩ is shown in Figure 5. It is much larger and close to its permitted maximum value Ωmax in the early stage. However, according to Equations (3) and (5), the peak values of Ω and the reaction torque Ta decrease with the attenuation of vibration. For example, the rotational speed of the RW decays to 267.4rad/s at 384.3 s, which is less than half of Ωmax. In this paper, the amplitude of the rotational speed is defined as a measure of the capability of the RW actuator utilized for vibration suppression. A larger amplitude of the rotational speed, which is closer to Ωmax, is preferred, for which the capability of the RW actuator is more sufficiently utilized to provide a better control effect. Hence, it indicates that the capability of the RW actuator has the potential to be improved for providing a better control effect.Figure 5
Rotational speed of the RW actuator with constant-gain control.


## 3. Design, Simulation, and Analysis of Envelope-Based Variable-Gain Control

To overcome the issue that the amplitude of the rotational speed decreases with the attenuation of vibration, it is proposed to involve a variable-gain control algorithm based on the envelope function of vibration in the feedback control law. The controller design, as well as the numerical verification and the robustness analysis of the control system, is going to be discussed in this section.


### 3.1. Controller Design

With the action of the damping force (i.e., the inherent structural damping and the reaction torque), vibration amplitude of the solar array decreases over time with a given initial condition. The envelope of the root bending momentMb gradually converges to 0 from Mb,max. Hence, determining the gain coefficient η according to Equation (4) guarantees Ω not exceed its permitted limit in the beginning of the control. Whereas the structure vibration amplitude decreases, amplifying the gain coefficient also makes the RW operate in the permitted operating range. For example, the relationship of Mb and Ω can be redesigned as
(13)Ω=γηMb.Here,γ is the design factor to amplify the gain coefficient appropriately.The control effect of the RW decreases with the vibration attenuation by adopting the constant-gain control. Inspired by that, the gain amplifying coefficientγ can be designed to be linear to the inverse of the envelope function of vibration. The response of the bending moment Mbt in the time domain presents the characteristics of periodic vibration and relatively slow attenuation. The slow attenuation part is the so-called envelope function in mathematics, and it can be denoted as EMbt. During the free vibration attenuation, the ideal envelope function of the root bending moment EMbt is a monotonically decreasing function with a maximum value Mb,max. The reciprocal of EMbt is a monotonically increasing function. Therefore, the amplifying coefficient γ is accordingly designed as
(14)γ=Mb,maxEMbt.Considering Equation (4), Equation (14) can be described as
(15)γ=ΩmaxηEMbt=ΩmaxEηMbt.To avoid the exception that the denominator (i.e.,EMbt) is zero in the numerical calculations, the maximum value of γ should be limited. A simple way is to apply a hardbound γmax to the computation of γ. Substituting Equation (15) into Equation (13), Mb and Ω are linked via
(16)Ω=minΩmaxEηMbt,γmax×ηMbt.In the digital signal process, the envelope function can be computed by various methods, such as Hilbert transform [50, 51] or squaring and low-pass filtering [52, 53]. In this research, the envelope of the measured response signal is solved by using the squaring and low-pass filtering method. Designate the envelope function EMbt=M¯. If the low-pass filter is taken as a 2nd-order Butterworth low-pass filter with a cutoff frequency Ωprad/s, the squaring of M¯ in the Laplace domain satisfies [54]
(17)LN=GsL2Mb2=2Ωp2s2+2Ωps+Ωp2LMb2,in which L⋅ denotes the Laplace transformation, Gs is the transfer function of the filter, and s is the Laplace variable,
(18)N=M¯2.Rewriting Equation (17) in the time domain:
(19)N¨+2ΩpṄ+Ωp2N=2Ωp2Mb2.Performing the first- and second-order time derivatives on both sides of Equation (18), respectively, we have
(20)Ṅ=2M¯M¯̇,N¨=2M¯M¯¨+2M¯̇2.Substituting Equations (18) and (20) into Equation (19), the relation of M¯ and Mb can be constructed:
(21)2M¯M¯¨+2M¯̇2+2Ωp2M¯M¯̇+Ωp2M¯2=2Ωp2Mb2.Unit Equations (13) and (14) to eliminate γ:
(22)Mb=ΩM¯ηΩmax.Substituting Equation (22) into Equation (21) and moving the item on the right-hand side of the equation to the left-hand side lead to
(23)M¯¨+1M¯M¯̇2+2ΩpM¯̇+Ωp22M¯−Ωp2Ω2M¯η2Ωmax2=0.Then, substituting the first equation of Equation (1) into Equation (22),
(24)ΩM¯ηΩmax=φMξ1.Recalling Equation (24), one can get
(25)Ω=ηΩmaxφMξ1M¯,Ω̇=ηΩmaxφMξ̇1M¯−ξ1M¯̇M¯2.Simultaneous formulas (5), (7), (23), and (25), the equation group of the closed-loop system is constituted by ξ1 and M¯:
(26)ξ¨1+2ω1ζ1ξ̇1+ω12ξ1=aξ1M¯̇M¯2−ξ̇1M¯,M¯¨+2ΩpM¯̇+Ωp22M¯=bM¯ξ12−1M¯M¯̇2,where a=IΩmax2φθ2/θmax,b=Ωp2φM2, and a,b>0.Designatingy1y2y3y4T=ξ̇1ξ1M¯̇M¯T, the equation group (26) can be written equivalently as follows:
(27)ẏ1=−2ω1ζ1y1−ω12y2+ay2y3y42−ay1y4,ẏ3=−2Ωpy3−Ωp22y4+by4y22−1y4y32.Perform a Taylor expansion of functiony4 at time t0, and denote y4=y40+Δy4y40>0, where Δy4 is the corresponding small increment. Then,
(28)ẏ4=Δẏ4,1y4=1y40+Δy4=1y4011+Δy4/y40≈1y401−Δy4y40,1y42=1y4021−Δy4y402≈1y4021−2Δy4y40.Substituting Equation (28) into Equation (27), and adding two identical equations ẏ2=y1 and Δẏ4=y3, one can obtain
(29)ẏ1=−2ω1ζ1y1−ω12y2+ay402y2y3−ay40y1−2ay403y2y3Δy4−ay402y1Δy4,ẏ2=y1,ẏ3=−2Ωpẏ3−Ωp22y40+by40y22−1y40y32+1y402y32Δy4−Ωp22Δy4−by402y22Δy4,Δẏ4=y3.Considering the assumption of small perturbation and weak damping,y1,y2,y3, and Δy4 are small quantities. Thus, neglecting the higher-order terms, the corresponding state-space representation of Equation (29) can be built up:
(30)ẏs=Asys+Bsus.Here,(31)As=−2ω1ζ1+ay40−ω120100−2Ωp−Ωp2210,ys=y1y2y3Δy4,Bs=0010,us=−Ωp22y40=constant,in which 0 is a 2×2 zero matrix.The characteristic equation of the closed-loop system (30) is
(32)λ2+2ω1ζ1+ay40λ+ω12λ2+2Ωpλ+Ωp22=0.The solutions of Equation (32) are
(33)λ1,2=−2ω1ζ1+a/y40±2ω1ζ1+a/y402−4ω122,λ3,4=−22Ωp.It can be seen from Equation (33) that the real parts of λii=1,2,3,4 are negative, and it is independent of y40. Or, namely, the system is stable with a local Taylor expansion of y4 at any time such that the stability in the whole process can be guaranteed. Thus, the closed-loop system is stable in small perturbation.


### 3.2. Numerical Verification

In this section, to make the different control strategies comparable, the numerical model of the flexible solar array, as shown in Figure2, is used to examine the effect of the proposed variable-gain control strategy.The feedback control diagram with envelope-based variable-gain control is updated based on Figure3, and it is shown in Figure 6. The gain amplifying coefficient γ, which is constructed according to Equation (15), is involved in the controller to improve the reaction torque provided by the RW actuator. The bound γmax is taken as 10 in this study. There is a delay in calculating the envelope function of the response signal by using the squaring and low-pass filtering method. The correct envelope value of the signal is output at about the second natural period of vibration after the simulation starts. Therefore, the rotational speed Ω should be limited in case exceeding its permitted operating range. To this end, a saturation module is utilized to limit the absolute value of its output within Ωmax, which is also applied to the negative values of Ω.Figure 6
Diagram of the feedback control with variable gain.The transient responses are revisited with the same initial condition, as shown in Equation (12). Figure 7 reveals two response curves of Mb, in which the black bold line and the black dashed line are the responses with envelope-based variable-gain and constant-gain control, respectively. Compared to the response with constant-gain control, the vibration attenuation speed with envelope-based variable-gain control is increased, which verifies the effectiveness of the proposed approach. Taking the threshold as 5% of Mb,max, the time required for vibration amplitude damping to the threshold is called vibration attenuation time. The vibration attenuation times in different cases are listed in Table 1. As shown, the result with constant-gain control is 1367.6 s, which is reduced to 657.5 s when adopting envelope-based variable-gain control. That is, the vibration attenuation time with variable-gain control is decreased to 48.1% from the state with constant-gain control and to 30.1% from the state without control.Figure 7
Response curve of the root bending moment with variable-gain control.Table 1
The vibration attenuation times of the simulation model.
Without controlConstant-gain controlVariable-gain controlVibration attenuation time (s)2181.71367.6657.5The rotational speedΩ with envelope-based variable-gain control is shown in Figure 8. As seen, in the first 600 s, the RW actuator rotates at a peak value near Ωmax, which implies the output reaction torque is improved. Therefore, involving the variable-gain control algorithm enables us to maximize the capability of the RW actuator and improve its damping effect.Figure 8
Rotational speed of the RW with variable-gain control.


### 3.3. Robustness Analysis

The envelope-based variable-gain control strategy is designed and verified for the configuration of the solar array characterized by nominal parameters. However, once the spacecraft is orbiting, some of its structural characteristics can result different from the ones measured during ground tests or derived from a numerical model. Among all the possible uncertainties, the ones that have a major impact on the performance of the vibration suppression of the solar array are associated with the natural frequency. In the following, these uncertainties reflect on the estimated errors of the first-order bending natural frequency, which result in the effect of the capability of the proposed control strategy to reduce the vibrations. Takingω1=0.2746 rad/s as the benchmark, it is assumed that the predetermined frequency for the control system design exists relative errors of ±5%, ±10%, ±15%, and ±20%, respectively. Adopting the method mentioned above, the high cut-off and the low cut-off frequencies of the band-pass filter are redesigned under different frequency error settings. The corresponding transient responses of the solar array are revisited with the same model and initial conditions.Compared to the vibration attenuation timet0=2181.7s without control, the vibration attenuation times with control and their relative ratios to t0 (i.e., relative attenuation time) under different estimated error settings are listed in Table 2. From Table 2, the vibration suppression still works effectively when the predetermined frequency deviates to a certain extent from the exact frequency (0.2746rad/s). It reveals that the control system is robust when there are errors in the predetermined frequency. Unfortunately, if the error is too large, for example, an error of -20%, the relative attenuation time is only decreased to 52.16%. Therefore, in order to achieve a better vibration suppression effect, it is advisable to provide the natural frequency of the solar array as accurately as possible.Table 2
Vibration suppression effect under different relative error conditions.
Relative errorVibration attenuation time (s)Relative attenuation time (%)10657.530.1425%670.330.723-5%713.432.70410%671.130.765-10%779.435.72615%672.730.847-15%901.841.34820%696.031.909-20%1137.952.16


## 3.1. Controller Design

With the action of the damping force (i.e., the inherent structural damping and the reaction torque), vibration amplitude of the solar array decreases over time with a given initial condition. The envelope of the root bending momentMb gradually converges to 0 from Mb,max. Hence, determining the gain coefficient η according to Equation (4) guarantees Ω not exceed its permitted limit in the beginning of the control. Whereas the structure vibration amplitude decreases, amplifying the gain coefficient also makes the RW operate in the permitted operating range. For example, the relationship of Mb and Ω can be redesigned as
(13)Ω=γηMb.Here,γ is the design factor to amplify the gain coefficient appropriately.The control effect of the RW decreases with the vibration attenuation by adopting the constant-gain control. Inspired by that, the gain amplifying coefficientγ can be designed to be linear to the inverse of the envelope function of vibration. The response of the bending moment Mbt in the time domain presents the characteristics of periodic vibration and relatively slow attenuation. The slow attenuation part is the so-called envelope function in mathematics, and it can be denoted as EMbt. During the free vibration attenuation, the ideal envelope function of the root bending moment EMbt is a monotonically decreasing function with a maximum value Mb,max. The reciprocal of EMbt is a monotonically increasing function. Therefore, the amplifying coefficient γ is accordingly designed as
(14)γ=Mb,maxEMbt.Considering Equation (4), Equation (14) can be described as
(15)γ=ΩmaxηEMbt=ΩmaxEηMbt.To avoid the exception that the denominator (i.e.,EMbt) is zero in the numerical calculations, the maximum value of γ should be limited. A simple way is to apply a hardbound γmax to the computation of γ. Substituting Equation (15) into Equation (13), Mb and Ω are linked via
(16)Ω=minΩmaxEηMbt,γmax×ηMbt.In the digital signal process, the envelope function can be computed by various methods, such as Hilbert transform [50, 51] or squaring and low-pass filtering [52, 53]. In this research, the envelope of the measured response signal is solved by using the squaring and low-pass filtering method. Designate the envelope function EMbt=M¯. If the low-pass filter is taken as a 2nd-order Butterworth low-pass filter with a cutoff frequency Ωprad/s, the squaring of M¯ in the Laplace domain satisfies [54]
(17)LN=GsL2Mb2=2Ωp2s2+2Ωps+Ωp2LMb2,in which L⋅ denotes the Laplace transformation, Gs is the transfer function of the filter, and s is the Laplace variable,
(18)N=M¯2.Rewriting Equation (17) in the time domain:
(19)N¨+2ΩpṄ+Ωp2N=2Ωp2Mb2.Performing the first- and second-order time derivatives on both sides of Equation (18), respectively, we have
(20)Ṅ=2M¯M¯̇,N¨=2M¯M¯¨+2M¯̇2.Substituting Equations (18) and (20) into Equation (19), the relation of M¯ and Mb can be constructed:
(21)2M¯M¯¨+2M¯̇2+2Ωp2M¯M¯̇+Ωp2M¯2=2Ωp2Mb2.Unit Equations (13) and (14) to eliminate γ:
(22)Mb=ΩM¯ηΩmax.Substituting Equation (22) into Equation (21) and moving the item on the right-hand side of the equation to the left-hand side lead to
(23)M¯¨+1M¯M¯̇2+2ΩpM¯̇+Ωp22M¯−Ωp2Ω2M¯η2Ωmax2=0.Then, substituting the first equation of Equation (1) into Equation (22),
(24)ΩM¯ηΩmax=φMξ1.Recalling Equation (24), one can get
(25)Ω=ηΩmaxφMξ1M¯,Ω̇=ηΩmaxφMξ̇1M¯−ξ1M¯̇M¯2.Simultaneous formulas (5), (7), (23), and (25), the equation group of the closed-loop system is constituted by ξ1 and M¯:
(26)ξ¨1+2ω1ζ1ξ̇1+ω12ξ1=aξ1M¯̇M¯2−ξ̇1M¯,M¯¨+2ΩpM¯̇+Ωp22M¯=bM¯ξ12−1M¯M¯̇2,where a=IΩmax2φθ2/θmax,b=Ωp2φM2, and a,b>0.Designatingy1y2y3y4T=ξ̇1ξ1M¯̇M¯T, the equation group (26) can be written equivalently as follows:
(27)ẏ1=−2ω1ζ1y1−ω12y2+ay2y3y42−ay1y4,ẏ3=−2Ωpy3−Ωp22y4+by4y22−1y4y32.Perform a Taylor expansion of functiony4 at time t0, and denote y4=y40+Δy4y40>0, where Δy4 is the corresponding small increment. Then,
(28)ẏ4=Δẏ4,1y4=1y40+Δy4=1y4011+Δy4/y40≈1y401−Δy4y40,1y42=1y4021−Δy4y402≈1y4021−2Δy4y40.Substituting Equation (28) into Equation (27), and adding two identical equations ẏ2=y1 and Δẏ4=y3, one can obtain
(29)ẏ1=−2ω1ζ1y1−ω12y2+ay402y2y3−ay40y1−2ay403y2y3Δy4−ay402y1Δy4,ẏ2=y1,ẏ3=−2Ωpẏ3−Ωp22y40+by40y22−1y40y32+1y402y32Δy4−Ωp22Δy4−by402y22Δy4,Δẏ4=y3.Considering the assumption of small perturbation and weak damping,y1,y2,y3, and Δy4 are small quantities. Thus, neglecting the higher-order terms, the corresponding state-space representation of Equation (29) can be built up:
(30)ẏs=Asys+Bsus.Here,(31)As=−2ω1ζ1+ay40−ω120100−2Ωp−Ωp2210,ys=y1y2y3Δy4,Bs=0010,us=−Ωp22y40=constant,in which 0 is a 2×2 zero matrix.The characteristic equation of the closed-loop system (30) is
(32)λ2+2ω1ζ1+ay40λ+ω12λ2+2Ωpλ+Ωp22=0.The solutions of Equation (32) are
(33)λ1,2=−2ω1ζ1+a/y40±2ω1ζ1+a/y402−4ω122,λ3,4=−22Ωp.It can be seen from Equation (33) that the real parts of λii=1,2,3,4 are negative, and it is independent of y40. Or, namely, the system is stable with a local Taylor expansion of y4 at any time such that the stability in the whole process can be guaranteed. Thus, the closed-loop system is stable in small perturbation.


## 3.2. Numerical Verification

In this section, to make the different control strategies comparable, the numerical model of the flexible solar array, as shown in Figure2, is used to examine the effect of the proposed variable-gain control strategy.The feedback control diagram with envelope-based variable-gain control is updated based on Figure3, and it is shown in Figure 6. The gain amplifying coefficient γ, which is constructed according to Equation (15), is involved in the controller to improve the reaction torque provided by the RW actuator. The bound γmax is taken as 10 in this study. There is a delay in calculating the envelope function of the response signal by using the squaring and low-pass filtering method. The correct envelope value of the signal is output at about the second natural period of vibration after the simulation starts. Therefore, the rotational speed Ω should be limited in case exceeding its permitted operating range. To this end, a saturation module is utilized to limit the absolute value of its output within Ωmax, which is also applied to the negative values of Ω.Figure 6
Diagram of the feedback control with variable gain.The transient responses are revisited with the same initial condition, as shown in Equation (12). Figure 7 reveals two response curves of Mb, in which the black bold line and the black dashed line are the responses with envelope-based variable-gain and constant-gain control, respectively. Compared to the response with constant-gain control, the vibration attenuation speed with envelope-based variable-gain control is increased, which verifies the effectiveness of the proposed approach. Taking the threshold as 5% of Mb,max, the time required for vibration amplitude damping to the threshold is called vibration attenuation time. The vibration attenuation times in different cases are listed in Table 1. As shown, the result with constant-gain control is 1367.6 s, which is reduced to 657.5 s when adopting envelope-based variable-gain control. That is, the vibration attenuation time with variable-gain control is decreased to 48.1% from the state with constant-gain control and to 30.1% from the state without control.Figure 7
Response curve of the root bending moment with variable-gain control.Table 1
The vibration attenuation times of the simulation model.
Without controlConstant-gain controlVariable-gain controlVibration attenuation time (s)2181.71367.6657.5The rotational speedΩ with envelope-based variable-gain control is shown in Figure 8. As seen, in the first 600 s, the RW actuator rotates at a peak value near Ωmax, which implies the output reaction torque is improved. Therefore, involving the variable-gain control algorithm enables us to maximize the capability of the RW actuator and improve its damping effect.Figure 8
Rotational speed of the RW with variable-gain control.


## 3.3. Robustness Analysis

The envelope-based variable-gain control strategy is designed and verified for the configuration of the solar array characterized by nominal parameters. However, once the spacecraft is orbiting, some of its structural characteristics can result different from the ones measured during ground tests or derived from a numerical model. Among all the possible uncertainties, the ones that have a major impact on the performance of the vibration suppression of the solar array are associated with the natural frequency. In the following, these uncertainties reflect on the estimated errors of the first-order bending natural frequency, which result in the effect of the capability of the proposed control strategy to reduce the vibrations. Takingω1=0.2746 rad/s as the benchmark, it is assumed that the predetermined frequency for the control system design exists relative errors of ±5%, ±10%, ±15%, and ±20%, respectively. Adopting the method mentioned above, the high cut-off and the low cut-off frequencies of the band-pass filter are redesigned under different frequency error settings. The corresponding transient responses of the solar array are revisited with the same model and initial conditions.Compared to the vibration attenuation timet0=2181.7s without control, the vibration attenuation times with control and their relative ratios to t0 (i.e., relative attenuation time) under different estimated error settings are listed in Table 2. From Table 2, the vibration suppression still works effectively when the predetermined frequency deviates to a certain extent from the exact frequency (0.2746rad/s). It reveals that the control system is robust when there are errors in the predetermined frequency. Unfortunately, if the error is too large, for example, an error of -20%, the relative attenuation time is only decreased to 52.16%. Therefore, in order to achieve a better vibration suppression effect, it is advisable to provide the natural frequency of the solar array as accurately as possible.Table 2
Vibration suppression effect under different relative error conditions.
Relative errorVibration attenuation time (s)Relative attenuation time (%)10657.530.1425%670.330.723-5%713.432.70410%671.130.765-10%779.435.72615%672.730.847-15%901.841.34820%696.031.909-20%1137.952.16


## 4. Experimental Verification

### 4.1. Experimental Setup for Vibration Suppression

In this section, a real example is used for feasibility investigation. The test article is an elastic plate with two additional weights. Figure9 details the experimental setup. The antishock sponges with a thickness of 0.5 mm are pasted on both surfaces of the plate. The two additional weights and antishock sponges are utilized to adjust the natural frequency and damping of the model, respectively. To reduce the influence of gravity, the plate is laterally placed to ensure that the bending vibration is perpendicular to the gravity direction. One side of the elastic plate is fixed on a sliding table to simulate the disturbance from the spacecraft body. The other side is equipped with a brushed-DC motor (BDCM), which serves as a RW actuator to provide reaction torque.Figure 9
Photograph of the experimental setup for vibration suppression.In this case, the total mass of the DC motor is 5.8 g. Parameters of the experimental apparatus are listed in Table3. By experimental test and analysis, the natural frequency corresponding to the first-order out-of-plane bending mode of the model is 0.805 Hz.Table 3
Parameters of the experimental apparatus.
Length, height, and thickness of elastic plate370mm×150mm×0.3mmMass of elastic plate126 gMass of additional weight 167 gMass of additional weight 2115 gDistance from fixed side to additional weight 1217.5 mmDistance from fixed side to additional weight 2362.5 mmMass of DC motor5.8 gInstead of measuring the bending moment as done in the numerical simulation, in the experiment, the strain signal of the vibrating structure is measured to acquire the input of the control system. To acquire the signal with a larger SNR (signal-to-noise ratio), the measuring point is arranged at a location with higher stresses. Denote the longitudinal strain of the measured position asεl, then the modal coordinate can be solved by
(34)ξ1t=εlφε,where φε is the first-order modal coefficient corresponding to εl. Meanwhile, Equation (34) indicates that the measured strain signal is proportional to ξ1. Therefore, according to Equations (3) and (34), the rotational speed and the measured strain signal can be linked via
(35)Ω=ηφMφεεl.Equation (35) states that similar to measuring the root bending moment Mb, the measured strain signal is linear to the speed signal of the RW actuator in the experiment. However, it should be noted that Equation (35) holds, if and only if the linear relationships exist between the modal coordinate and Mb,εl, as stated in Equations (1) and (34).The strain gauges, with a resistance of350Ω, are bonded on the elastic plate near the fixed side to acquire the local strain signal εl. The input voltage of the full-bridge strain circuit is 1.5 V. The output voltage of the strain bridge circuit is amplified 3000 times linearly by a strain amplifier and sampled by an analog-to-digital (A/D) module. Let Vɛ be the strain amplifier’s output voltage, which is also called strain voltage. Through the A/D module, the corresponding digital value of Vɛ is acquired and regarded as the input of the control system.When the elastic plate is disturbed,Vɛ changes with the vibration of the elastic plate. According to the predesigned algorithm, the driving signal voltage Vd commanding the speed of the DC motor is solved and output to the motor driver through a D/A (digital-to-analog) module. In this paper, the motor driver is ESCON-50/5 produced by MAXON Corporation. The speed signal voltage range is 0∼2.4V, corresponding to the rotational speed 0~14800 RPM linearly. The direction of the rotational speed is commanded by a logic high or low signal.The slip table administrated by a stepper motor is adopted to excite the vibration of the elastic plate model. The excitation is performed as follows. The slip table undergoes two periods of sinusoidal motion at0.8Hz along the normal direction of the plate surface. The speed function ve of the excitation table is given in mm/s by
(36)ve=15sin1.6πt,0≤t≤2.5s,0,t>2.5s.The vibration response is measured by strain gauges during the entire time period (from the beginning of the excitation to the end of the experiment). In order to verify the suppression effectiveness, the vibration attenuation with and without control is measured.


### 4.2. Experimental Performance of Vibration Suppression Subjected to Different Control Strategies

As aforementioned, the rotational speedΩ and the strain εl are linear to the electric voltage Vd and Vε, respectively. Recalling Equation (35), the driving signal voltage Vd is proportional to the strain voltage Vε:
(37)Vd=κVε,where κ is the proportional gain coefficient in the experiment. It is required that the motor rotates within the maximum ratings and near the maximum speed when Vε reaches its peak value. Therefore, the coefficient κ is estimated by Vdmax/Vεmax, and it is taken as −3.5 in this case, in which Vdmax is the permitted maximum value of the driving voltage, which equals to 2.4 V in this case.The response curve of the strain voltageVε with constant-gain control is shown as the solid line in Figure 10. Compared to the response without control (the dashed line), the vibration attenuation process is accelerated. The results verify that the vibration attenuation of the elastic plate can be accelerated by applying the reaction torque based on the control relation (37).Figure 10
Measured response curve of the strain voltage with constant-gain control.The envelope-based variable-gain control strategy is also validated based on the elastic plate model. Recalling Equations (15) and (16), the gain amplifying coefficient γ in the experiment is accordingly designed to be
(38)γ=minVdmaxEκVεt,γmax.The calculation module forγ is added to the control program according to Equation (38). The envelope-based variable-gain control strategy is employed for vibration suppression of the plate model with the same experimental settings. The response curve of the strain voltage Vε is plotted as the solid line in Figure 11. Compared to the measured result with constant-gain control (the dashed line), the attenuation speed with the proposed variable-gain control is increased.Figure 11
Measured response curve of the strain voltage with variable-gain control.The vibration attenuation time is used as a metric to demonstrate the vibration suppression efficiency. The vibration amplitude thresholdVε∗=0.05EmaxVεt. Based on the measured responses, the vibration attenuation times without control, with constant-gain control, and with envelope-based variable-gain control are listed in Table 4, respectively. Compared to 153.4 s and 89 s, the vibration attenuation time with envelope-based variable-gain control is reduced to 29.1% of that from the uncontrolled state and to 50.22% of that from the constant-gain controlled state.Table 4
The vibration attenuation time of the experiment model.
Without controlConstant-gain controlVariable-gain controlVibration attenuation time (s)153.48944.7


## 4.1. Experimental Setup for Vibration Suppression

In this section, a real example is used for feasibility investigation. The test article is an elastic plate with two additional weights. Figure9 details the experimental setup. The antishock sponges with a thickness of 0.5 mm are pasted on both surfaces of the plate. The two additional weights and antishock sponges are utilized to adjust the natural frequency and damping of the model, respectively. To reduce the influence of gravity, the plate is laterally placed to ensure that the bending vibration is perpendicular to the gravity direction. One side of the elastic plate is fixed on a sliding table to simulate the disturbance from the spacecraft body. The other side is equipped with a brushed-DC motor (BDCM), which serves as a RW actuator to provide reaction torque.Figure 9
Photograph of the experimental setup for vibration suppression.In this case, the total mass of the DC motor is 5.8 g. Parameters of the experimental apparatus are listed in Table3. By experimental test and analysis, the natural frequency corresponding to the first-order out-of-plane bending mode of the model is 0.805 Hz.Table 3
Parameters of the experimental apparatus.
Length, height, and thickness of elastic plate370mm×150mm×0.3mmMass of elastic plate126 gMass of additional weight 167 gMass of additional weight 2115 gDistance from fixed side to additional weight 1217.5 mmDistance from fixed side to additional weight 2362.5 mmMass of DC motor5.8 gInstead of measuring the bending moment as done in the numerical simulation, in the experiment, the strain signal of the vibrating structure is measured to acquire the input of the control system. To acquire the signal with a larger SNR (signal-to-noise ratio), the measuring point is arranged at a location with higher stresses. Denote the longitudinal strain of the measured position asεl, then the modal coordinate can be solved by
(34)ξ1t=εlφε,where φε is the first-order modal coefficient corresponding to εl. Meanwhile, Equation (34) indicates that the measured strain signal is proportional to ξ1. Therefore, according to Equations (3) and (34), the rotational speed and the measured strain signal can be linked via
(35)Ω=ηφMφεεl.Equation (35) states that similar to measuring the root bending moment Mb, the measured strain signal is linear to the speed signal of the RW actuator in the experiment. However, it should be noted that Equation (35) holds, if and only if the linear relationships exist between the modal coordinate and Mb,εl, as stated in Equations (1) and (34).The strain gauges, with a resistance of350Ω, are bonded on the elastic plate near the fixed side to acquire the local strain signal εl. The input voltage of the full-bridge strain circuit is 1.5 V. The output voltage of the strain bridge circuit is amplified 3000 times linearly by a strain amplifier and sampled by an analog-to-digital (A/D) module. Let Vɛ be the strain amplifier’s output voltage, which is also called strain voltage. Through the A/D module, the corresponding digital value of Vɛ is acquired and regarded as the input of the control system.When the elastic plate is disturbed,Vɛ changes with the vibration of the elastic plate. According to the predesigned algorithm, the driving signal voltage Vd commanding the speed of the DC motor is solved and output to the motor driver through a D/A (digital-to-analog) module. In this paper, the motor driver is ESCON-50/5 produced by MAXON Corporation. The speed signal voltage range is 0∼2.4V, corresponding to the rotational speed 0~14800 RPM linearly. The direction of the rotational speed is commanded by a logic high or low signal.The slip table administrated by a stepper motor is adopted to excite the vibration of the elastic plate model. The excitation is performed as follows. The slip table undergoes two periods of sinusoidal motion at0.8Hz along the normal direction of the plate surface. The speed function ve of the excitation table is given in mm/s by
(36)ve=15sin1.6πt,0≤t≤2.5s,0,t>2.5s.The vibration response is measured by strain gauges during the entire time period (from the beginning of the excitation to the end of the experiment). In order to verify the suppression effectiveness, the vibration attenuation with and without control is measured.


## 4.2. Experimental Performance of Vibration Suppression Subjected to Different Control Strategies

As aforementioned, the rotational speedΩ and the strain εl are linear to the electric voltage Vd and Vε, respectively. Recalling Equation (35), the driving signal voltage Vd is proportional to the strain voltage Vε:
(37)Vd=κVε,where κ is the proportional gain coefficient in the experiment. It is required that the motor rotates within the maximum ratings and near the maximum speed when Vε reaches its peak value. Therefore, the coefficient κ is estimated by Vdmax/Vεmax, and it is taken as −3.5 in this case, in which Vdmax is the permitted maximum value of the driving voltage, which equals to 2.4 V in this case.The response curve of the strain voltageVε with constant-gain control is shown as the solid line in Figure 10. Compared to the response without control (the dashed line), the vibration attenuation process is accelerated. The results verify that the vibration attenuation of the elastic plate can be accelerated by applying the reaction torque based on the control relation (37).Figure 10
Measured response curve of the strain voltage with constant-gain control.The envelope-based variable-gain control strategy is also validated based on the elastic plate model. Recalling Equations (15) and (16), the gain amplifying coefficient γ in the experiment is accordingly designed to be
(38)γ=minVdmaxEκVεt,γmax.The calculation module forγ is added to the control program according to Equation (38). The envelope-based variable-gain control strategy is employed for vibration suppression of the plate model with the same experimental settings. The response curve of the strain voltage Vε is plotted as the solid line in Figure 11. Compared to the measured result with constant-gain control (the dashed line), the attenuation speed with the proposed variable-gain control is increased.Figure 11
Measured response curve of the strain voltage with variable-gain control.The vibration attenuation time is used as a metric to demonstrate the vibration suppression efficiency. The vibration amplitude thresholdVε∗=0.05EmaxVεt. Based on the measured responses, the vibration attenuation times without control, with constant-gain control, and with envelope-based variable-gain control are listed in Table 4, respectively. Compared to 153.4 s and 89 s, the vibration attenuation time with envelope-based variable-gain control is reduced to 29.1% of that from the uncontrolled state and to 50.22% of that from the constant-gain controlled state.Table 4
The vibration attenuation time of the experiment model.
Without controlConstant-gain controlVariable-gain controlVibration attenuation time (s)153.48944.7


## 5. Conclusion

In this paper, an envelope-based variable-gain control strategy for the reaction wheel (RW) actuator is proposed to suppress the bending vibration of the solar array. Both numerical simulations and experimental studies are provided to demonstrate the rationality and validity of the proposed variable-gain control in improving the vibration suppression performance of the RW actuator compared to the constant-gain control. Taking the finite element model of a full-scale solar array as the object, the damping effects of the RW actuator subjected to different control strategies are compared by using the vibration attenuation time as a metric. The vibration attenuation time without control and with constant-gain control are 2181.7 s and 1367.6 s, respectively, and it is reduced to 657.5 s when the envelope-based variable-gain control is adopted. The numerical simulations considering deviations between the estimated and the exact natural frequency values are performed to analyze the robustness of the control system. It is concluded that the proposed technique is robust within a deviation range of ±20%. To obtain a better control effect, a more accurate estimation value of the natural frequency is required. Experiments on an elastic plate model further prove that the proposed variable-gain control method is a valid and feasible technology. Compared to 153.4 s and 89 s, the vibration attenuation time with envelope-based variable-gain control is reduced to 29.1% of that from the uncontrolled state and to 50.22% of that from the constant-gain controlled state.



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*Source: 6055819-2022-05-20.xml*