Distributed vibration control for large-scale flexible space structure improved by fuzzy logic system

2.1. Dynamic model

The structure of large scale flexible spacecraft is shown in Fig. 1, it includes three parts: spacecraft bus, large antenna and payload. It can be regarded as the fixed end of the rigid body, and the influence of rigid body motion is ignored. The support truss structure and reflector in the antenna are composed of a large number of carbon fiber rods and cables. The full-scale finite element model of flexible spacecraft structure is established by Patran, and the 50 order modal analysis is carried out. The results are shown in Table 1.

Fig. 1Structural sketch of large scale flexible spacecraft

The modal analysis results show that the large-scale flexible spacecraft has the characteristics of low and dense frequency. In the results of finite element analysis, there are two first-order bending modes in the $OYZ$ plane, and they do not correspond strictly to the $Y$-axis and $Z$-axis directions. The frequencies of the 4th to 50th order modes are between 0.347Hz and 0.73 Hz, which contains a large number of local modes of the reflector surface, as well as the combination of high-order bending and torsion. We assumed that the modal damping of each order ${\zeta }_i=$0.005.

2.2. Equivalent model of piezoelectric actuator

The main body of the piezoelectric actuator rod is a cylinder structure with a length of 250 mm, an inner diameter of 10 mm and an outer diameter of 20 mm, which is stacked by piezoelectric ceramic plates. The elastic modulus in the driving direction $E_{33}=$48.31 GPa, and the piezoelectric strain coefficient in the driving direction $d_{33}=$5.93×10^-10 m/V. The number of piezoelectric plates $n=$1250 and the voltage limit $V_{max}$ is 120 V. The output displacement of piezoelectric actuator under zero stress state is shown in Eq. (1):

If the displacement caused by the driving voltage of piezoelectric actuator is equivalent to that caused by tension at ends, the equivalent tensile force is shown in Eq. (2):

The equivalent relationship between the force at ends and the load voltage at ends is shown in Eq. (3):

where $k_m$ is the degradation rate of actuator performance caused by factors such as the bonding and electrodes, $k_m<$1, defined as 0.9; $k$ is the proportional coefficient of equal force and driving voltage, $k=F_{max}/V_{max}=$135.

2.3. Multi-subsystem control model

The whole governing equation of large-scale flexible space structure is:

where $M$ is the total mass matrix, $D$ is the total damping matrix, $K$ is the total stiffness matrix, $F$ is the control force and $∆$ is the external disturbance.

Assuming that the large-scale flexible space structure is divided into $n$ substructures, the governing equation of each substructure can be written in a similar form as the Eq. (5):

where ${\mathrm{\Delta }}_i$ is the force from the adjacent subsystems, which could be considered as an external disturbance of current subsystem.

3.1. Design and layout of actuators and sensors

As shown in Fig. 2, it is assumed that the $Y$-direction and $Z$-direction displacements at the end of each span on the triangular support truss of antenna can be measured by sensors 1-5. The piezoelectric actuator group is arranged in the middle of each span, as is shown in Fig. 2. Each piezoelectric actuator group is composed of three actuating rods, as shown in Fig. 3. The A and B bars are loaded with opposite voltage to suppress $Y$-axis vibration. The C bar is loaded with voltage to suppress $Z$-axis vibration. According to the $Y$-direction and $Z$-direction displacement information transmitted by sensor $i$, the actuator group $i$ executes control instructions and suppresses the vibration of each module. The disturbance excitation occurs at the end of the payload.

Fig. 2Actuators and sensors layout

Fig. 3Working diagram of actuators

3.2. Switch control and PD control

The control system of switch control method is simple. It doesn’t need accurate dynamic model. When the velocity is positive, the maximum voltage value is set at the ends of the piezoelectric actuator to produce the force, which opposite to the velocity direction. When the velocity is negative, the voltage at the ends of the piezoelectric actuator switches to reverse rapidly, and the speed of the system tends to zero gradually. Actually, the piezoelectric actuators can only load positive voltage. The control rate is shown in Eq. (6):

where $x_{i,r}$ is the displacement of node $r$ at the $i$th substructure, and ${\dot{x}}_{i,r}$ is the velocity of node $r$ at the $i$th substructure.

PD control also does not require an accurate dynamic model. The error between the original position and the measured position is $e\left(t\right)$. Taking $e\left(t\right)$ as the input, proportional component $k_p$ can control the deviation rapidly. Differential component $k_d$ reflects the change rate of deviation, which introduces an early correction signal to speed up the system action and reduce the adjustment time before the deviation is too large. The control rate is shown in Eq. (7):

Since the actuators and sensors are considered as the collocated pairs, the input matrix $B_i$ is equal to the transposition of the output matrix $C_i$. So, the Eq. (5) is rewritten by the input fore $F$ calculated by the switch control rate Eq. (6) or the PD control rate Eq. (7):

The original dynamical system $M_i{\ddot{X}}_i+D_i{\dot{X}}_i+K_i{X}_i$ is obviously asymptotic stable, as well as $M_i>$0, $D_i>$0, and $K_i\ge$ 0. Since the $k_{swtich}$ > 0, $k_p$> 0, $k_d$> 0, and $C^{T}C$≥ 0, the Eq. (8) and Eq. (9) are therefore asymptotic stable, as well as $D_i+k_{swtich}{C}^{T}C>0$, $D_i+k_p{C}^{T}C>0$ and $K_i+k_d{C}^{T}C\ge 0$. In the actual engineering application and the following simulation tests, the sensors and the actuators cannot be collocated strictly in the same position, and the control command is discrete. Therefore, the spillover is easy to occur if the parameters are too large. Then, we proposed the improved method by fuzzy logic system.

3.3. Control method based on fuzzy logic system

Through the fuzzy logic system, a coefficient $K$ is set on the parameters of the original control rate, and the selection of $K$ is determined by the fuzzy logic system, as shown in Fig. 4. When the displacement disturbance $e$ and the rate of change $ec$ are large, $K$ is taken as the large value. The control voltage calculated by the original control rate can maximize the actuating capacity of actuators. When the displacement disturbance e and the rate of change $ec$ are small, $K$ is taken as the small value to ensure the stability and accuracy of the control.

Fig. 4Schematic diagram of controller structure

The input of fuzzy inference system are the end ($Y$ direction or $Z$ direction) displacement $e$ and the change of rate $ec$ which reflect the overall disturbance. The output variable is the parameter $K$. The input value $e$ is transformed into the domain of discourse [–10, 10] and $ec$ is transformed into the domain of discourse [–10, 10] by the scale change factor $K_e$ and $K_{ec}$. The set of 5 language description variables corresponding to $e$ and $ec$ is {negative large: NB, negative medium: NM, zero equal: ZE, positive medium: PM, positive large: PB}. Output $K$ is in the domain of discourse [1, 6], and the corresponding set of 5 language description variables is {level 1, level 2, level 3, level 4, level 5}. The triangular membership function ‘trimf’ is used to construct the membership function, shown in Fig. 5 and Fig. 6. 5×5 fuzzy rules are designed, as shown in Table 2. The Mamdani fuzzy controller with the above input and output rules is designed in the Matlab fuzzy toolbox. We use the maximum-minimum fuzzy synthesis formula, and the defuzzification method of area center.

Fig. 5Input e/ec membership function

Fig. 6Output K membership function