Comparison of Piezoelectric Stack-Based Passive and Active Vibration Suppression Systems for Satellite Solar Panels

Abstract

This study proposes a piezoelectric device for vibration damping in satellite solar panels. The design features a structural arrangement with piezoelectric stacks configured in a V-shape and hinged to the main yoke structure. The satellite structure is modeled using an Euler–Bernoulli beam finite element framework, incorporating the electro-mechanical coupling of active elements through equivalent nodal piezoelectric loads. Various shunt circuits are designed to mitigate vibrations, with a parametric study conducted to optimize the key circuit parameters. Additionally, a filtered PID active suppression system is developed and tuned using a meta-heuristic algorithm to determine optimal controller gains. Numerical simulations are performed to evaluate and compare the effectiveness of the proposed vibration suppression systems, demonstrating the efficiency of the smart structure configuration and providing performance analysis.

1. Introduction

Satellites frequently incorporate large, flexible components such as antennas and solar panels, which are essential to fulfill the several demands of their mission, whether for communication, Earth observation, or scientific exploration. Modeling the structural dynamics of the flexible spacecraft is essential to understanding the behavior of these flexible multi-body systems in the harsh environment of space. Such spacecraft typically consist of structural elements like beams, plates, trusses, and other simple hinged structures, which collectively exhibit intricate and coupled dynamic characteristics [1]. The equivalent beam idealization of solar panels and other spacecraft appendages was widely utilized in the literature to facilitate preliminary investigations into control-structure interactions and the design of vibration suppression systems, as demonstrated in studies such as [2,3,4]. When the spacecraft undergoes rapid and extensive movements, such as orbital transfers or attitude adjustments, these motions often induce significant vibrations in the flexible components. This interplay underscores the critical coupling between the rigid-body dynamics of the spacecraft and the structural vibrations of its flexible appendages, which may be challenging in dynamic modeling and control [5].

Common techniques employed to mitigate control-structure interaction phenomena include restricting force and torque commands, limiting the size of appendages, and implementing maneuver control systems that filter out the critical frequencies of the appendages [6]. However, to mitigate undesirable interactions between control systems and structural dynamics without limiting the satellite design space, there is increasing interest in both passive and active vibration damping techniques. A representative and successful example of flexible appendage vibration control is given by the passive dampers installed on the Hubble space telescope [7] to dampen the oscillations of the solar panels. Due to the successful application of piezo-ceramics in the active vibration suppression of structures [8,9], recent research has explored the use of piezoelectric materials to minimize vibrations in satellites. These materials are a proven technology for smart structures [10,11], offering high control authority in a compact form, making them particularly well suited for space applications [12]. In fact, these materials have several advantageous properties, including broad operational bandwidth, high reliability, compact size, and ease of installation, making them an appealing choice for vibration mitigation applications in space systems [13]. For instance, in [14], a support truss structure with embedded piezostacks was proposed to damp vibrations on a satellite during launch operations. This study presented a direct comparison between the damping capabilities of the smart structure in the passive damping condition and those in the active control condition, where a fuzzy controller was specifically designed. In [15], distributed P-shaped piezoelectric stack sensors/actuators have been employed for the vibration suppression of a flexible satellite antenna by means of a Proportional-Integral-Derivative (PID) controller.

In the framework of satellite solar panel vibration damping, it can be found that in [16], piezoelectric patches attached to the surface of the solar panels were utilized to develop an active control system based on a PID controller; a similar approach was exploited in [17]. Similarly, Refs. [18,19] proposed a Positive Position Feedback (PPF) strategy to generate control signals for the piezoelectric patches and compared the results with the ones obtained implementing a Linear Quadratic Gaussian (LQG) architecture in [20]. PPF was experimentally implemented in [21], within a synergetic control loop with a sliding mode controller, to damp out flexible panel oscillations during attitude control of the “Platform Integrating Navigation and Orbital Control Capabilities Hosting Intelligence Onboard” (PINOCCHIO) test rig. Non-linear control systems have been explored in [22], where a hybrid adaptive sliding mode/Lyapunov controller was designed to suppress the flexible panel oscillations, while in [23], a modified sliding mode controller was proposed. Furthermore, piezostacks have been employed for solar panel vibration damping in [24,25], where an Offset Piezoelectric Stack Actuator (OPSA), positioned at a vertical distance from the primary structure using a rigid support, was introduced. The piezostacks have been coupled with piezoelectric patches providing feedback signals for a direct velocity feedback controller. Piezoelectric stacks have also been used in [26,27], where the active elements were embedded along the edges of the solar panel and worked as an Active Tension Device (ATD). It was demonstrated that, through the tensile force produced, the vibration frequencies of the structure could be effectively increased, thereby enabling faster vibration attenuation. This study aims to present a V-stack piezoelectric system installed on the conventional yoke supporting structure of solar panels to mitigate their vibrations.

Within the framework of passive structural damping systems, shunted circuits have been utilized to dissipate the vibrational energy of primary structures, offering an appealing solution due to their minimal hardware and software requirements for implementing vibration suppression [28]. Studies such as [28,29] have investigated distributed piezoelectric patches connected to both purely resistive and resonant circuits to suppress satellite solar panel vibrations, with the resonant circuits showing superior performance. To the best of the authors’ knowledge, piezostack devices combined with shunt circuits have not yet been explored for vibration suppression in large flexible appendages of spacecraft. This approach holds significant potential to enhance spacecraft controllability with minimal impact on size and weight. Accordingly, this work also aims to provide a systematic analysis of the capabilities of shunted circuits when used with the proposed piezoelectric stacks configuration and to evaluate which damping enhancement technique, shunted circuits or active control, offers superior performance.

This paper is organized as follows: in Section 2, the numerical model of the smart structure is detailed, presenting the equivalent beam realization of the solar panels and the inclusion of the piezoelectric elements in the finite element model. In Section 3, the control systems architectures are presented, and in Section 4, the results of the numerical analyses are discussed.

2. Numerical Model

2.1. Satellite Structure

In this work, the classical triangular satellite yoke configuration is considered, and, taking into account the symmetries of the spacecraft structure, only one yoke/panel assembly is studied. The solar panel is hinged to the yoke, which is joined to the satellite body by means of a fixed constraint. The piezoelectric stacks, each measuring 8 cm in length and with a square cross-section, are attached to the main yoke structure at an angle of 45 degrees relative to the longitudinal axis. This configuration forms a V-shaped arrangement on both sides of the yoke, enabling the piezostacks to respond to the bending displacements of the solar panel through two support beams affixed to the yoke’s vertices. Both the yoke and support beams present a square hollow cross-section and are made of the same material of the solar panel, i.e., aluminum. The geometrical and stiffness characteristics of the yoke and solar panel have been taken from [5,30], respectively, while the piezoelectric stacks data are taken from [31,32]; all data are resumed in

Table 1. Geometrical and material parameters of the satellite.

Parameters	Values
Satellite body
Mass of the body $m_b$ [kg]	150
Moments of inertia $J_{XX},J_{YY},J_{ZZ}$ [kg · m2]	100, 100, 100
Yoke
Yoke length, $l_Y$ [m]	2
Cross-section width, $W_Y$ [m]	$50\times {10}^{-3}$
Cross-section thickness, $t_Y$ [m]	$1\times {10}^{-3}$
Solar panel
Length of the panel, a [m]	2.0
Width of the panel, b [m]	2.0
Distance between the hinges $b_0$ [m]	1.6
Honeycomb core thickness, $2h_c$ [m]	0.0197
Sandwich surface sheets thickness, $h_s$ [m]	$0.15\times {10}^{-3}$
Honeycomb cell length, $l_c$ [m]	$6.35\times {10}^{-3}$
Honeycomb cell wall thickness, ${\delta }_c$ [m]	$0.0254\times {10}^{-3}$
Elastic modulus, $E_0$ [Pa]	$6.89\times {10}^{10}$
Density, ${\rho }_0$ [kg  m−3]	$2.8\times {10}^3$
Poisson’s ratio, $\nu$	0.33
Hinge stiffness, $k(\mathrm{N}·\mathrm{m}/\mathrm{rad})$	500
Piezostack
Piezostack elastic modulus, $E_p$  [Pa]	$4.5\times {10}^{10}$
Piezostack density, ${\rho }_p$ [kg  m−3]	7700
Piezostack cross-section, $A_p$ [mm2]	$10\times 10$
Piezoelectric charge coefficient, $d_{33}\left[{\displaystyle \frac{m}{V}}\right]$	$425\times {10}^{-12}$
Dielectric dissipation factor, $tan\left(\delta \right)$	$17\times {10}^{-3}$
Maximum and minimum Voltage	$V_{max}=250$ V, $V_{min}=-150$ V

. The spacecraft structural scheme is shown in Figure 1, where the piezoelectric active elements are highlighted in red.

The solar panel structural model is defined as a sandwich panel with a honeycomb core and upper and lower thin surface sheets. Since local structural phenomena are not relevant to the aim of this work, an equivalent beam idealization of the solar panel is introduced; this approach also enables the realization of a low-order numerical model, which is well suited for the control system design purposes of this study. Therefore, the sandwich structure is first reduced to an isotropic equivalent rectangular plate, following the homogenization procedure proposed in [33], which is in turn considered an equivalent beam. This simplification is motivated by the fact that this work aims to study the performance of vibration suppression systems in terms of control authority and reliability rather than the structural aspect in terms of strain and stresses acting on the panel. Therefore, the equivalent beam model of the sandwich panel is exploited to realize a low-order model of the smart structure [34,35]. In detail, The Euler–Bernoulli beam theory forms the basis of the numerical model of the structure, which is implemented in a Finite Element (FE) framework and defined in a three-dimensional space. Each beam finite element is modeled with two nodes, providing six degrees of freedom per node, three for translations and three for rotations, which can be collected in the generalised displacement vector $\mathbf{q}$$={\left[u,v,w,\varphi ,\theta ,\psi \right]}^T$. Cubic Hermite shape functions have been utilized to interpolate bending displacements, while linear shape functions have been applied for axial and torsional displacements.

2.2. Piezoelectric Stacks Model

To develop the structural numerical model, the Finite Element Modeling (FEM) of piezoelectric stacks has first been addressed. Piezoelectric stacks consist of n piezoelectric layers stacked atop one another, with internal electrodes embedded between them. When an electric voltage is applied, the stack predominantly undergoes axial deformation, either elongating or contracting along its primary axis, while deformations in other directions are negligible [36]. Conversely, if an axial displacement is imposed on the piezostack, it generates an electric charge Q, which can flow as a current i representing a measure of structural deformation rate. Given the operational principles of piezostacks, their behavior is governed by the constitutive equations of the piezoelectric effect, expressed in the strain-charge form as outlined in the ANSI/IEEE Standard. These equations can be simplified and reduced to apply exclusively along the stack’s primary axis (direction 3), which aligns with the piezoelectric poling direction. The resulting governing equations read as [31]

$$\left\{\begin{array}{c}{\epsilon }_{33}=S_{33}{\sigma }_{33}+d_{33}E\hfill \\ D_{33}=d_{33}{\sigma }_{33}+{ϵ}_{33}E\hfill \end{array}\right.$$

(1)

where ${\epsilon }_{33}$ represents the strain, $S_{33}$ denotes the material compliance, ${\sigma }_{33}$ is the stress, $d_{33}$ stands for the piezoelectric charge coefficient, ${ϵ}_{33}$ is the dielectric constant, and E corresponds to the electric field. The total axial deformation of the stack can be expressed by applying the first equation in Equation (1) to each individual piezoelectric layer, with thickness t, obtaining the following

$${{\epsilon }_{33}}^T={\displaystyle \frac{\sum {{\epsilon }_{33}}^{i}t}{L}}={\displaystyle \frac{nt}{L}}\left(S_{33}{\sigma }_{33}-d_{33}{\displaystyle \frac{V}{t}}\right)$$

(2)

with V and L denoting the applied voltage and piezostack length. Therefore, the stress can be derived from Equation (2) and assuming that $nt=L$, it reads as

$${\sigma }_{33}={\displaystyle \frac{{{\epsilon }_{33}}^T}{S_{33}}}-{\displaystyle \frac{d_{33}}{S_{33}t}}V$$

(3)

The piezoelectric-induced forces must be included into the finite element framework; therefore, in order to compute the equivalent nodal force vector due to the piezostack deformation, the virtual work principle is applied as [37]

$$\partial L^{*}=\int {\epsilon }^T\sigma d\mathsf{\Omega }=0$$

(4)

the infinitesimal integration volume element being $d\mathsf{\Omega }$.

Within the FE framework used in this work, the piezostack deformation can be related to the nodal displacement vector $\delta$ through the strain-displacement matrix $\mathbf{B}$, which is obtained by taking the spatial derivative of the shape function matrix with respect to the beam axis coordinate [38], according to the relation

$$\epsilon =\mathbf{B}\delta$$

(5)

Therefore, Equation (4) can be expressed as

$$\partial L^{*}={\int \int }_{\mathsf{\Omega }}{\mathbf{B}}^T{\displaystyle \frac{1}{S_{33}\mathrm{L}}}\mathbf{B}dAdL\delta -{\int \int }_{\mathsf{\Omega }}{\mathbf{B}}^T{\displaystyle \frac{d_{33}}{S_{33}t}}VdAdL=0$$

(6)

and the equivalent nodal force vector, in the axial direction only of the beam FE local frame, is computed from the second term of Equation (6) as

$${\tilde{\mathbf{F}}}_V={\left[\begin{array}{cccccc}-{\displaystyle \frac{d_{33}{E}_{p}A}{t}}\hfill & 0\hfill & 0\hfill & {\displaystyle \frac{d_{33}{E}_{p}A}{t}}\hfill & 0\hfill & 0\hfill \end{array}\right]}^{\mathrm{T}}V$$

(7)

with $E_p$ and A denoting the stack Young’s modulus and its cross-section area, respectively.

Once the equivalent nodal force vector is defined, the electric current across the stack must be computed. By applying the second equation in Equation (1) to each stack layer, the total electric displacement is expressed as

$$\sum D_3^i=\sum d_{33}^i{\sigma }_{33}^i+\sum {ϵ}_{33}^{i}E$$

(8)

and by substitution of the stress in Equation (3), it is obtained that

$$n{D}_3={\displaystyle \frac{L}{t}}{d}_{33}{\displaystyle \frac{{{\epsilon }_{33}}^T}{S_{33}}}+{\displaystyle \frac{n{d}_{33}^2}{S_{33}t}}V-{\displaystyle \frac{n{ϵ}_{33}}{t}}V$$

(9)

Therefore, integrating Equation (9) over the piezostack cross-sectional area A and recalling that the electric charge is computed as $Q=nA{D}_3$, and by substitution of Equations (5) in (9), the piezostack charge reads as [39]

$$\begin{array}{cc}\hfill Q& ={\displaystyle \frac{E_{p}A{d}_{33}}{t}}\left[\begin{array}{c}-11\end{array}\right]\delta +\left({\displaystyle \frac{E_{p}An{d}_{33}^2}{t}}-{\displaystyle \frac{nA{ϵ}_{33}}{t}}\right)V\hfill \\ \hfill & ={\mathbf{K}}_{\delta V}\delta +K_{VV}V\hfill \end{array}$$

(10)

and the electric current is eventually obtained by time derivation $i=\dot{Q}$.

2.3. FEM Model

Defining the piezostack finite element characteristics, the whole structure was modeled in a Euler–Bernoulli beam FE framework. Therefore, computing the structural consistent mass and stiffness matrices and applying the clamped boundary conditions at the yoke-body connection, the satellite FE dynamic model is obtained as

$$\left[\begin{array}{cc}{\mathbf{M}}_{\mathbf{11}}\hfill & {\mathbf{M}}_{\mathbf{12}}\hfill \\ {\mathbf{M}}_{\mathbf{12}}^T\hfill & {\mathbf{M}}_{\mathbf{22}}\hfill \end{array}\right]\left[\begin{array}{c}{\ddot{\delta }}_1\hfill \\ {\ddot{\delta }}_2\hfill \end{array}\right]+\left[\begin{array}{cc}{\mathbf{K}}_{\mathbf{11}}\hfill & {\mathbf{K}}_{\mathbf{12}}\hfill \\ {\mathbf{K}}_{\mathbf{12}}^T\hfill & {\mathbf{K}}_{\mathbf{22}}\hfill \end{array}\right]\left[\begin{array}{c}{\delta }_1\hfill \\ {\delta }_2\hfill \end{array}\right]=\left[\begin{array}{c}{\mathbf{F}}_V\hfill \\ \mathbf{0}\hfill \end{array}\right]V$$

(11)

where ${\delta }_1$ and ${\delta }_2$ are the unknown and known displacement vectors [40], respectively, and ${\mathbf{F}}_V$ is the global piezoelectric force vector. It is worth noting that structural damping has not been included in Equation (11) in order to consider the worst design scenario for vibration-damping systems. Therefore, considering that non-homogeneous boundary conditions are applied to the FE model to account for the satellite’s body movements as disturbances to the flexible structure, the governing equations are expressed as [41]

$${\mathbf{M}}_{\mathbf{11}}{\ddot{\delta }}_1+{\mathbf{K}}_{\mathbf{11}}{\delta }_1={\mathbf{F}}_{V}V-{\mathbf{M}}_{\mathbf{12}}{\ddot{\delta }}_2-{\mathbf{K}}_{\mathbf{12}}{\delta }_2$$

(12)

where the right-hand term is the effective force vector including the contribution of non-homogeneous boundary conditions [42].

After developing the numerical model of the structure, a modal reduction technique was applied. This approach ensures that the numerical model remains sufficiently accurate while significantly reducing the number of degrees of freedom. The reduced model is computationally efficient and well suited for designing effective vibration suppression systems, enabling the optimization of control strategies while maintaining the necessary fidelity for dynamic analysis. In particular, the eigenvalues $\omega$ and eigenvectors $\mathsf{\Phi }$ of the homogeneous system, derived from Equation (11), have been computed. Subsequently, only the j lowest eigen-modes were retained to compute the following reduced matrices [43]

$$\begin{array}{cc}\hfill & {\mathbf{M}}_{11}^{\mathbf{j}}={\mathsf{\Phi }}_j^T{\mathbf{M}}_{\mathbf{11}}{\mathsf{\Phi }}_j,{\mathbf{K}}_{11}^{\mathbf{j}}={\mathsf{\Phi }}_j^T{\mathbf{K}}_{\mathbf{11}}{\mathsf{\Phi }}_j,{\mathbf{F}}_V^{\mathbf{j}}={\mathsf{\Phi }}_j^T{\mathbf{F}}_V,\hfill \\ \hfill & {\mathbf{M}}_{12}^{\mathbf{j}}={\mathsf{\Phi }}_j^T{\mathbf{M}}_{\mathbf{12}},{\mathbf{K}}_{12}^{\mathbf{j}}={\mathsf{\Phi }}_j^T{\mathbf{K}}_{\mathbf{12}},{\delta }_1^j={\mathsf{\Phi }}_j^T{\delta }_1.\hfill \end{array}$$

(13)

Therefore, the reduced system governing equations read as

$${\mathbf{M}}_{11}^{\mathbf{j}}{\ddot{\delta }}_j+{\mathbf{K}}_{11}^{\mathbf{j}}{\delta }_1={\mathbf{F}}_V^{\mathbf{j}}V-{\mathbf{M}}_{12}^{\mathbf{j}}{\ddot{\delta }}_2-{\mathbf{K}}_{12}^{\mathbf{j}}{\delta }_2$$

(14)

and is complemented by the electric current equations for each piezostacks, which can be expressed in matrix form as

$$\mathbf{i}={\mathbf{K}}_{\delta V}{\mathfrak{J}}_p{\mathit{\varphi }}_j{\dot{\delta }}_1^j+{\mathbf{K}}_{VV}\dot{\mathbf{V}}$$

(15)

where ${\mathfrak{J}}_p$ is a support row vector consisting of zeros except for entries corresponding to the displacements of the piezoelectric stack finite element nodes. Additionally, another support vector, ${\mathfrak{J}}_t$, is introduced to extract the vertical displacement at the solar panel tip, expressed as $w_t={\mathfrak{J}}_t{\mathit{\varphi }}_j{\delta }_1^j$, which is regarded as an output of interest for the comparative analysis of the vibration suppression systems.

The block diagram of the resolving system, combining Equations (14) and (15), is reported in Figure 2.

3. Control Systems

In this section, the two control systems considered for damping the solar panel vibrations of the satellite are detailed. First, the passive control systems based on shunt circuits are presented. Then, the active control system based on a filtered PID controller is introduced and the tuning procedure of the controller parameters is discussed.

3.1. Shunt Circuits

Shunt circuits offer an effective approach for preserving structural integrity and mitigating oscillations by suppressing unwanted vibrations arising on critical components while requiring only minimal electrical components and a simple hardware setup. Leveraging both series and parallel configurations, they provide a reliable means of dissipating vibration energy, promoting smoother operation, and enhancing structural longevity [44]. The vibration damping control framework utilizing shunt circuits implemented in this study is depicted schematically in Figure 3, where it can be noted that the electrical current through the piezoelectric stack, representing the output of the resolving system, is fed back into the circuit. Subsequently, the voltage generated across the circuit serves as the input, enabling the piezoelectric stacks to perform the stabilizing control action.

In this work, three circuits, whose electrical schemes are reported in Figure 4, are implemented:

• RC: a dissipative only circuit with a resistance R in parallel to the piezo capacity C;
• ‖RLC: a resonant circuit with resistance, impedance L, and capacity C in parallel;
• RL‖C: a resonant circuit with resistance and impedance in series both in parallel to the capacity.

The transfer functions $G_{Vi}\left(s\right)$ of the studied circuits, which relate the input current through the piezoelectric stack to the output applied voltage, are defined as follows [45,46]

$$\begin{array}{cc}\hfill {G_{Vi}}^{RC}\left(s\right)& ={\displaystyle \frac{R}{1+sRC}}\hfill \\ \hfill {G_{Vi}}^{RL\parallel C}\left(s\right)& ={\displaystyle \frac{s}{C\left(s^2+\frac{1}{RC}s+\frac{1}{LC}\right)}}\hfill \\ \hfill {G_{Vi}}^{\parallel RLC}\left(s\right)& ={\displaystyle \frac{s+\frac{R}{L}}{C\left(s^2+\frac{R}{L}s+\frac{1}{LC}\right)}}\hfill \end{array}$$

(16)

3.2. Active Control

The active control architecture is based on a pair of filtered PID (fPID) controllers. These controllers measure the error between the desired and actual voltage across a piezostack, which is used as a sensor, and send an input voltage to another piezostack, acting as an actuator, to suppress the oscillations of the main structure. Specifically, the filtered PID controllers are designed to minimize the error function $\epsilon \left(t\right)=V_{\mathrm{ref}}\left(t\right)-V_p\left(t\right)$, where $V_{\mathrm{ref}}\left(t\right)=0$ V is the reference voltage. The term $V_p\left(t\right)=R_p{i}_s$ represents the voltage output of an operational amplifier with resistance $R_p$. This amplifier receives as input the electric current $i_s$ from the sensing piezostack, which can be considered short-circuited (i.e., the voltage across the piezo is zero) [36]. It is common for operational amplifiers configured as current amplifiers to exhibit low input resistance [47]. This is crucial to accurately sense the input current without introducing significant voltage drops or loading effects. Consequently, in this work, a resistance value of $R_p=10$ $\mathsf{\Omega }$ was chosen, ensuring minimal impact on the input signal. In this work, the pairs of sensing-actuator piezostacks are positioned on both sides of the yoke, with one piezostack operating in sensing mode and the others in actuation mode.

The input voltage to the piezostacks actuator is computed according to the following expression [37]

$$V_a\left(t\right)=K_p\left[\epsilon \left(t\right)+{\displaystyle \frac{1}{{\tau }_i}}{\int }_0^t\epsilon \left(\tau \right)d\tau +{\epsilon }_D\left(t\right)\right]$$

(17)

where the parameters $K_p$ and ${\tau }_i$ represent the proportional gain and integral time constant of the fPID controller, while the filtered derivative term ${\epsilon }_D\left(t\right)$ is obtained by solving the following first-order differential equation

$${\tau }_{DF}{\dot{e}}_D+e_D={\tau }_D\dot{e}$$

(18)

being ${\tau }_D$ and ${\tau }_{DF}$, the derivative and filter time constants, respectively.

The block diagram of the active control system architecture is shown in Figure 5.

The controller parameter tuning is addressed by means of a Population Decline Swarm Optimizer ($P_{D}SO$), which is a meta-heuristic optimization algorithm. In this work, the objective function of the optimization algorithm is the root mean square (rms) of the normalized tip deflection error [48]

$$f_{obj}=rms\left(1-{\displaystyle \frac{w_t\left(t\right)}{w_0}}\right)$$

(19)

where $w_t\left(t\right)$ and $w_0$ are the actual solar panel tip displacement and its steady state value, respectively. The input given to the smart structure for the optimization simulations is a step vertical displacement to the satellite body with amplitude $w_b=0.1$ m. The simulation time was chosen as $T_s=20$ s, which is sufficiently long to capture the entire transient response, while remaining short enough to avoid excessive computational cost.

The particles $P^i$ of the optimization algorithm at each iteration $\lambda$ are characterized by four coordinates, i.e., the controller parameters, such that $P_{\lambda }^i={\left[\begin{array}{cccc}{K}_p^i& {\tau }_i^i& {\tau }_D^i& {\tau }_{DF}^i\end{array}\right]}_{\lambda }$ and the optimization problem reads as

$$\begin{array}{c}min{f}_{obj}\left(P_{\lambda }^i\right)\\ \mathrm{s}.\mathrm{t}.\\ K_{pmin}\le K_p\le K_{pmax}\\ {\tau }_{\mathrm{imin}}\le {\tau }_i\le {\tau }_{\mathrm{imax}}\\ {\tau }_{Dmin}\le {\tau }_D\le {\tau }_{Dmax}\\ {\tau }_{DFmin}\le {\tau }_{DF}\le {\tau }_{DFmax}\end{array}$$

(20)

where the boundaries of the research space were determined through a trial-and-error procedure. For more details on the optimization algorithm, the interested reader is referred to [37].

4. Results

In this section, the results of the numerical analyses are presented. First, the modal analysis of the smart structure is performed to define the reduced-order model used for the control system design. Next, the shunt circuits are designed through a parametric study, while the fPID controller is tuned using a meta-heuristic optimization algorithm. Finally, the performance of the tuned control systems is compared.

4.1. Modal Analysis

The modal analysis of the structure was first performed in order to select the number of modes to be retained in the reduced order model. To validate the reliability of the developed FEM model, the smart structure was also modeled using the commercial software ANSYS, and the results from the modal analysis were compared. The ANSYS model was created using BEAM189 finite elements, which implement Euler–Bernoulli beam theory, and SHELL181 finite elements, which are based on Mindlin–Reissner shell theory. A fixed constraint was applied at the yoke vertex connecting the smart structure to the satellite body, while revolute joints were used to simulate the torsional stiffness of the hinges linking the yoke to the solar panels. Additionally, multi-point constraints were employed to enforce the coupling of the remaining displacement components at the hinge’s location. Lanczos’ method [49] was employed to extract the natural modes of the structure. The computed natural frequencies for the first six modes are presented in

Table 2. Natural frequencies of the smart structure.

Mode	ANSYS [Hz]	Present [Hz]	% Error
1—OOP bending	1.49	1.47	1.35%
2—IP bending	3.7	3.6	2.7%
3—OOP bending	8.67	8.52	1.7%
4—torsion	9.14	9.1	0.4%
5—OOP bending	81.54	80.14	1.7%
6—IP bending	109.1	107.6	1.4%

, where it is shown that the present model accurately predicts the natural frequencies of the smart structure, with a maximum error of 2.7% compared to the ANSYS model. To determine the number of natural modes to include in the reduced-order model, the mass participation factor for displacements along the Z-axis was computed for each natural mode, and it was found that the first five modes account for 95.2% of the total mass participation. Therefore, the first $j=5$ modes were selected for the reduced-order model.

4.2. Tuning of Shunt Circuits

The tuning of the shunted circuits electrical components is first presented in this section. Given the electric capacity of the piezostacks [32] $C=7.7$$\mathsf{\mu }$F, the inductance L and resistance R of the shunt circuits have been tuned carrying out a parametric study which aims to find the combination that provides for the minimum value of the objective function in Equation (19) when the satellite body, and subsequently the yoke, is subjected to a step input of the vertical translation with amplitude $w_b=0.1$ m. The results obtained from the parametric analysis are reported in Figure 6 where it is shown that the best performance under satellite body imposed displacement are given by the resonant RLC circuits.

More in detail, for the parallel ‖RLC configuration, it is found that increasing the impedance value lowers the objective function for all the resistance range studied, but the point of minimum moves to higher resistance values. Moreover, Figure 6b shows that for impedance values higher than 200 H, the objective function does not exhibit any improvements. On the other hand, for the series RL‖C configuration, it is found that increasing the L value over 10 H leads to worse results. The RC circuit results, instead, have shown that, even if it provides a certain damping contribution to the main structure, it is less efficient with respect to the resonant circuits. The tuning results are summarized in

Table 3. Shunt circuit optimal parameters.

	RC	‖RLC	RL‖C
$min\left(f_{obj}\right)$	0.281	0.119	0.129
$R_{\mathrm{opt}}$ [$\Omega$]	360	91	300
$L_{\mathrm{opt}}$ [H]	−	300	10

where it is found that the minimum objective function value is obtained for the ‖RLC circuit with resistance and impedance values of $R=91$$\Omega$ and $L=300$ H. It is here noted that, in order to implement high impedance while maintaining a small solenoid size, a synthetic impedance may be employed by means of an Antoniou synthetic inductor [50], shown in Figure 7, whose equivalent impedance is given by $L_{\mathrm{eq}}={\displaystyle \frac{R_1{R}_3{C}_4{R}_5}{R_2}}$ [51].

The numerical simulation results obtained for the tuned shunted circuits are reported in Figure 8 for comparison. The time histories of the solar panel tip vertical displacement, corresponding to the minimum points in Table 3, are also compared with the structural response of the open-loop case. It can be noted that the RC circuit provides an amount of damping to the structure but with a very high settling time. On the other hand, the time responses of the RL‖C circuit presents an overshoot of $0.23$ m, lower than the RC one, and completely suppressing the solar panel bending vibrations within 5 s. The best performance is given by the tuned ‖RLC circuit, being able to suppress the vibration in less than 5 s while presenting an overshoot of $0.2$ m.

4.3. Tuning of the fPID Controller

As mentioned in the previous section, a $P_{D}SO$ optimization algorithm was employed for the tuning of the fPID controller parameters. The boundaries of the research space for the optimization problem in Equation (20) were determined through an initial trial-and-error approach and were defined as follows: ${10}^{-3}\le K_P\le 10$, ${10}^{-3}\le {\tau }_i\le {10}^3$, ${10}^{-3}\le {\tau }_D\le 10$, and ${10}^{-3}\le {\tau }_{DF}\le 10$. Moreover, the algorithm parameters were set as follows: The cognitive and social acceleration constants, which influence particle velocity, were assigned values of $c_c=c_s=2.025$. The inertia factor, balancing global and local search strategies, was varied within the range $\left[{\mu }_{min},{\mu }_{max}\right]=\left[0.4,0.9\right]$. The maximum number of iterations was set to $\Lambda =20$, the population decline factor to $\chi =0.5$, and the number of consecutive iterations with a constant swarm size to $\Delta =5$. Three optimizations were performed, each considering a different maximum number of particles, $P_{max}={\left[\begin{array}{ccc}5\hfill & 10\hfill & 20\hfill \end{array}\right]}_{\lambda }$, to evaluate the convergence of the procedure. The algorithm was executed 10 times for each $P_{max}$, and the final fitness function values were analyzed to determine the minimum, maximum, median, and standard deviation ($\sigma$) of the optimization results. The computed convergence indices are presented in

Table 4. $P_{D}SO$ optimization results.

${\mathit{P}}_{\mathit{max}}$	${{\mathit{f}}_{\mathit{obj}}}^{\mathit{min}}$	${{\mathit{f}}_{\mathit{obj}}}^{\mathit{max}}$	$\mathit{Median}\left({\mathit{f}}_{\mathit{obj}}\right)$	${\mathit{\sigma }}_{{\mathit{f}}_{\mathit{obj}}}$	${\mathit{K}}_{\mathit{P}}$	${\mathit{\tau }}_{\mathit{i}}$	${\mathit{\tau }}_{\mathit{D}}$	${\mathit{\tau }}_{\mathit{DF}}$
5	$0.139$	$0.165$	$0.146$	$0.007$	$5.02$	$606.35$	$0.024$	$0.98$
10	$0.101$	$0.114$	$0.111$	$0.004$	$3.34$	$648.12$	$0.012$	$0.92$
20	$0.101$	$0.109$	$0.103$	$0.002$	$3.34$	$648.12$	$0.012$	$0.92$

, which shows that the minimum objective function value is ${f_{obj}}^{min}=0.101$. This minimum is already achieved with a maximum particle count of $P_{max}=10$. The coordinates corresponding to ${f_{obj}}^{min}$ are $\left[\begin{array}{cccc}3.34\hfill & 648.12\hfill & 0.012\hfill & 0.92\hfill \end{array}\right]$.

Figure 9 illustrates the optimization convergence trends. It is worth noting that small swarm sizes possibly cause convergence to local minima. However, the re-initialization mechanism introduced through demographic reduction [37] enables the exploration of different trajectories, ensuring good convergence even with a limited number of particles.

The displacement response of the solar panel tip, corresponding to the optimal point identified through the optimization procedure, is presented in Figure 10. The figure illustrates that the active control system effectively suppresses the bending vibrations of the solar panel within 4 s, achieving a maximum overshoot of $0.22$ m. This demonstrates the superior performance of the active control architecture compared to shunted circuit approaches, particularly in terms of settling time, while maintaining comparable levels of overshoot. These results highlight the advantages of the active control system when the smart structure is subjected to a step vertical displacement of the satellite.

4.4. Performance Comparison

In order to compare the performance of the designed control systems, the voltage required by both the ‖RLC circuit, which exhibits the best solar panel tip time response among the studied shunted circuits, and the fPID controller is analyzed. The voltage output through each piezostack actuator is shown in Figure 11. It is worth noting that, in the case of the ‖RLC circuit, the upper and lower piezostack voltages are perfectly anti-phase, indicating that the imposed displacement excites purely bending vibrations. By comparing the actuation voltages in Figure 11, it can be observed that the ‖RLC circuit has a lower maximum peak, equal to $186.4 V$, while the fPID controller requires a maximum voltage equal to the piezostack saturation value in the very first phase of the transient. Therefore, even though the fPID controller provides slightly better performance in terms of settling time, it requires a higher actuation voltage, reaching saturation, which could pose a problem in the presence of system uncertainties.

In order to study the system response to a possible operative condition of the satellite, an attitude maneuver, corresponding to a rotation of the satellite body around the global X axis, was considered, and the solar panel tip displacement was analyzed in terms of local vibration amplitude ${\overline{w}}_t$. The effect of the solar panel vibrations on the motion of the satellite body was also studied by computing the error $\overline{\theta }$ between the desired attitude angle and the actual one. The attitude motion is governed by a bang-bang driving torque, which is computed as

$$T^b={\displaystyle \frac{J_{XX}{\theta }_f}{{\tau }_p\left(t_f-{\tau }_p\right)}}$$

(21)

where $J_{XX}=184$ kg·m² is the satellite moment of inertia around the X axis, ${\tau }_p=10$ s is the width of the pulse, ${\theta }_f=55$ deg is the final attitude angle [25], and $t_f=20$ s is the total maneuver time width. The obtained results are presented in Figure 12. It can be observed that the attitude motion of the satellite body induces undesired vibrations on the solar panel, which present peaks of ${\overline{w}}_t^{OL}=0.31$ m. The solar panel vibrations also cause oscillations around the desired attitude angle of the satellite body, with a maximum amplitude of ${\overline{\theta }}^{OL}=0.16$ deg. Conversely, both closed-loop systems significantly reduce the solar panel maximum displacement to an absolute value of ${{\overline{w}}_t}^{\parallel RLC}=0.165\mathrm{m}$ for the shunted system and ${{\overline{w}}_t}^{fPID}=0.157\mathrm{m}$ for the fPID system, therefore showing a better performance of the latter. This behavior is also noticed in the satellite body vibrations for which the maximum amplitude is reduced to ${\overline{\theta }}^{\parallel RLC}=0.084\mathrm{deg}$ for the shunted system and ${\overline{\theta }}^{fPID}=0.08\mathrm{deg}$ for the fPID system. It can also be noted that the vibration settling time is 5 s for the shunted circuit and 3 s for the fPID-controlled system, which proves the greater effectiveness of the active control system. However, it is noteworthy that the actuation voltage required over time by the shunted circuit is lower than that of the fPID controller. Specifically, the shunted circuit reaches a maximum voltage of $V^{\parallel RLC}=187.5\mathrm{V}$, while the fPID controller requires a maximum voltage of $V_a^{fPID}=211\mathrm{V}$. The electric current required in the circuit was also computed to estimate the power consumption of the smart device. Each active element was modeled using an equivalent parallel RC circuit, where the capacitance $C_p$ corresponds to that of the piezoelectric element, and the resistance $R_p$ is determined based on its relationship with the dielectric loss factor $tan\left(\delta \right)={\displaystyle \frac{1}{\omega R_p{C}_p}}$, where $\omega$ represents the excitation frequency used to estimate $tan\left(\delta \right)$, as specified in the material datasheet [52]. The results obtained are reported in Figure 12g,h, where it is shown that the fPID control system also provide higher current with a maximum absolute peak of $i^{fPID}=0.7\mathrm{A}$ while a maximum value of $i^{\parallel RLC}=0.17\mathrm{A}$ is registered when using the shunted circuit. Last, the electric power is compared in Figure 12i,j, where it is shown that the shunted system asks for a maximum power peak of $P^{\parallel RLC}=25.6\mathrm{W}$, while the fPID controlled system presents a peak of $P^{\parallel fPID}=109.7\mathrm{W}$. Therefore, from the obtained results, it is shown that the fPID closed-loop system demonstrates slightly better vibration suppression performance but at the cost of requiring higher actuation voltage and current, which in turn increases electric power demand.

Since various factors can influence the system parameters, potentially leading to deviations from their nominal values, there is inherent uncertainty in estimating the performance of the vibration suppression system. Therefore, in order to study the effects of these uncertainties on the closed loop systems, a probabilistic model is introduced, and a range of variations around relevant structural and piezoelectric parameters is considered. Variation ranges for the satellite’s structural parameters have been defined based on previous studies in the literature [53], as reported in

Table 5. Parameter variation ranges of the probabilistic model.

Parameter	Variation Range
Solar panel length	$\left[24\right]$ m
Young’s modulus	$\pm 8\%$
Mass density	$\pm 4\%$
Piezoelectric stack capacity	$\pm 10\%$
Piezoelectric charge coefficient	$\pm 12\%$
Dielectric dissipation factor	$\pm 10\%$
Voltage saturation	$\pm 10\%$

. Moreover, in order to study the scalability of the vibration suppression systems, the length of the solar panel was also considered as a variable, having a range of $\left[24\right]$ m. Moreover, the variations in piezoelectric material parameters have been selected to account for the effects of temperature, which can range from −65 °C to +125 °C in Low Earth Orbit (LEO) [54]. Specifically, the variation range of $d_{33}$ was determined according to the material datasheet, while the variation of $tan\left(\delta \right)$ was defined based on the findings in [55]. To define a probability density distribution that maximizes uncertainty, the principle of maximum entropy, introduced by Weaver and Shannon [56], is applied. Given that the expected values of the uncertain parameters coincide with their nominal values [57], it can be shown that the probability distribution with the highest entropy is the uniform distribution [58].

To assess the impact of the considered uncertainties on the performance of the vibration suppression systems, 1000 Monte Carlo simulations were carried out, taking into account the satellite’s attitude motion and randomly selecting the uncertain parameter values within the defined ranges. The results obtained were analyzed in terms of the Integral of Square Error (ISE) of the solar panel tip displacement and maximum power requested during the transient response. The results, standardized by means of a z-score method [59], are shown in Figure 13, where the linear regression line is also reported in red. The results indicate that the parameter with the greatest influence on the performance index ISE($w_t$) for both closed-loop systems is the solar panel length. However, the maximum deviations of the performance index from the mean value are comparable. In contrast, a significant difference is observed in the maximum power demand. Specifically, for the shunted circuit system, the values of $Z[max(P\left)\right]$ lie within the range $[-1.031.72]$, indicating a low correlation, whereas for the fPID system, the range is $[-2.312.63]$, with an inverse correlation. Additionally, a strong direct correlation is found between $Z[max(P\left)\right]$ and the charge coefficient in the shunted circuit system, while in the fPID system, there is a slight inverse relationship.

5. Conclusions

This study investigated the effectiveness of a piezoelectric device for vibration damping in satellite solar panels, with a focus on comparing the performance of passive and active control systems. The results highlighted that, while the active control system (implemented through a filtered PID controller architecture) achieves superior performance in terms of transient response and settling time, it requires significantly higher actuation voltages, reaching the piezostacks saturation level. In contrast, the ‖RLC shunt circuit, optimized through a parametric study, demands lower actuation voltage, with peak voltage values well below saturation, making them more robust against potential system uncertainties while ensuring comparable transient response. These findings suggest a trade-off between transient performance and actuation voltage requirements for this kind of vibration suppression system, offering insights into the practical application of active and passive vibration control strategies in satellite structures.