Finite Element Dynamic Modeling of Smart Structures and Adaptive Backstepping Control

Abstract

Smart structures with topological configurations that integrate perception and actuation have complex geometric features. The simplification of these features can lead to deviations in dynamic characteristics, making it difficult to establish an accurate dynamic model. Uncertainties, such as material nonlinearity, hysteresis in elastic deformation, and external disturbances, affect the trajectory tracking accuracy of the smart structure’s actuation function. This paper proposes a modeling method that combines finite element unit bodies and orthogonal characteristic mode reduction to construct an accurate dynamic model of the smart structure and design an adaptive backstepping controller. Nonlinear dynamic equations are derived through a finite element analysis of the structure, and the orthogonal characteristic mode reduction method is employed to reduce computational complexity while ensuring model accuracy. An adaptive backstepping controller is designed to mitigate model uncertainties and achieve precise trajectory tracking control. Simulation and experimental results demonstrate that the proposed method can effectively handle the nonlinearity and modeling errors of smart structures, achieving high-precision trajectory tracking and verifying the accuracy of the dynamic model as well as the robustness of the controller.

1. Introduction

Smart structures refer to advanced engineering structures that integrate sensors, actuators, controllers, and information processing systems, endowing them with self-perception, self-diagnosis, self-adaptation, and even autonomous decision-making capabilities [1]. In real-time, such structures can sense changes in the external environment (load, temperature, vibration, damage, etc.) and respond through their built-in intelligent system, optimizing performance and enhancing safety [2,3]. In modern engineering, smart structures have attracted extensive attention due to their ability to sense changes in the external environment and respond automatically. They have enormous application potential in many fields, such as aerospace [4], mechanical engineering [5], and civil engineering [6].

Topological optimization methods can seek the optimal material distribution within a given design space and achieve an integrated structure and function design for smart structures through mathematical modeling and algorithm innovation. Sena et al. [7] designed a smart structure with shunt piezoelectric ceramics by establishing an equilibrium equation between the maximum harvested power and the minimum structural damage and using a multi-objective optimization method. De Almeida et al. [8] proposed a numerical framework that simultaneously applies density-based multi-material conductive flexible mechanism topology optimization and a composite multi-layer geometric projection method for optimizing the size, position, and orientation of embedded piezoelectric stack actuators to achieve an integrated design of actuators, sensors, and load-bearing structures. Senatore et al. [9] proposed an adaptive structural integrated topology optimization scheme based on the ground structure method. By linearizing the formula into a mixed-integer linear problem that can be solved to global optimality, the simultaneous optimization of structural layout and actuator position is achieved. However, due to the complex geometric features, such as hollowing, fractals, and multi-scale coupling, smart structures with topological configurations encounter numerous difficulties in dynamic modeling. The topological optimization generates irregular pore structures such as dendritic supports and honeycomb cells by removing redundant materials and removing traditional regular structures’ geometric symmetry and periodicity, which leads to a strong localization of vibration modes, making it difficult for the reduced-order models based on modal truncation to accurately describe the broadband response [10,11].

The dynamic behaviors in multi-scale designs (such as macro-structure–micro-material integration) vary significantly at different scales. Traditional single-scale modeling methods cannot describe the cross-scale energy transfer mechanism, and existing methods (such as homogenization theory) have to ignore topological details, resulting in relatively large errors when calculating the coupling stiffness matrix [12]. Some smart structures achieve adaptive deformation through topological optimization (such as origami-inspired reconfigurable structures), and their configurations change dynamically during service with changes in load or control input (such as hinge-opening and -closing, cell shape switching, etc.). At this time, dynamic modeling needs to consider the nonlinear update of the stiffness matrix caused by the time-varying topology, leading to increased computational costs and challenges in meeting the modeling accuracy requirements [13,14].

The rapid response of smart structures usually requires precise trajectory control. However, it is affected by unmodeled dynamic characteristics, external disturbances, the hysteresis of structural materials, and other uncertain factors. $\mu$-synthesis technology combines linear theory with the H-infinity control framework. A structured uncertainty model can analyze the system’s characteristics under various uncertain conditions. It optimizes performance indicators through numerical algorithms to design a controller that can ensure system stability and robustness [15]. Moutsopoulou et al. [16] used the $\mu$-synthesis and reduced-order H-infinity feedback optimal output controller combination method to construct a smart structure controller that includes the nominal model and uncertainties, solving the oscillation suppression problem. Zhang et al. [17] considered the system model uncertainty caused by parameter perturbation and modeling error and proposed a dynamic controller design method for piezoelectric smart beams using $\mu$-synthesis technology.

During external disturbances, sliding mode control forces the system state to slide along a predetermined trajectory by designing a sliding surface, and has strong robustness against external disturbances and system parameter perturbations. The influence of disturbances can be controlled within the range designed by the sliding surface without affecting the system’s stability. This can quickly suppress the deviation caused by disturbances and improve the system’s response speed [18,19]. Alizadeh et al. [20] proposed a three-degrees-of-freedom arm robot super-twisting fast non-singular terminal sliding mode control strategy to address the chattering and slow convergence speed problems in sliding mode control. Under model mismatches and external disturbances, the trajectory tracking accuracy, robustness, and convergence time were improved. By combining sliding mode control with adaptive algorithms, observers, etc., some successful control strategies for enhancing intelligent structures’ dynamic performance have been proposed [21,22,23]. However, due to the complexity and uncertainty of the topological configuration, it is difficult to establish an accurate dynamic model, and the above control methods face problems such as low control accuracy and large computational load.

Recent studies have made notable progress in advanced control strategies for structural systems. For instance, an optimal nonlinear fractional-order controller designed for passive/active base isolation buildings equipped with friction-tuned mass dampers was shown to be effective in handling nonlinearities and improving seismic response mitigation [24]. This controller leverages fractional calculus to enhance its flexibility in adjusting dynamic characteristics. However, its application is primarily focused on base isolation systems with specific damper configurations, which may not directly adapt to the time-varying topological features of smart structures. Another study introduced a new seismic control framework combining optimal $P{I}^{\lambda }{D}^{\mu }$ controller series with a fuzzy PD controller, which integrates the advantages of fractional-order control and fuzzy logic to handle system uncertainties [25]. While this framework exhibits strong robustness in seismic scenarios, its reliance on predefined control structures may limit its performance when dealing with the complex and varying dynamics of topologically optimized smart structures.

Additionally, a new control approach for the seismic control of buildings equipped with active mass dampers, namely the optimal fractional-order brain emotional learning-based intelligent controller, has demonstrated superior adaptive capabilities by mimicking human emotional learning mechanisms [26]. This method shows promise in handling uncertain dynamic behaviors. However, its application in the context of smart structures with integrated perception and actuation capabilities, especially those with topological configurations, remains underexplored.

Adaptive backstepping control does not require an accurate system model compared to traditional control methods. It can estimate system parameters and external disturbances in real-time through an adaptive mechanism, demonstrating superiority in addressing the control problems in smart structures with topological configurations [27]. We used topological optimization methods to obtain the topological configuration of smart structures and analyzed their sensing and actuation characteristics [28]. This paper proposes to combine the dynamic modeling based on finite element bodies with the orthogonal eigenmode order reduction method to establish an accurate dynamic model of smart structures, design an adaptive backstepping controller, and conduct simulation and experimental verification studies. The main contributions of this paper are as follows:

• By combining the orthogonal eigenmode reduction method with the finite element method, a precise dynamic model of a smart structure with a topological configuration was constructed;
• Considering the uncertainties of the smart structure dynamic model, such as nonlinearity and modeling deviations, an adaptive backstepping controller was designed to achieve precise trajectory tracking control of the smart structure’s actuation function.

The rest of this paper is structured as follows: In Section 2, a dynamic model of the smart structure is presented. The proposed control method is derived and the stability of the control system is proved in Section 3. The corresponding simulations and experiments are presented in Section 4. Finally, conclusions are drawn in Section 5.

2. Smart Structural Dynamics Modeling Based on the Finite Element Method

Topological optimization structures have complex shapes. Ignoring minor features or simplifying boundary conditions often means models fail to accurately reflect actual dynamic characteristics. The core concept of the finite element method is to discretize complex mechanical structures into a finite number of element bodies, conduct a mechanical analysis of each element body separately, and finally integrate the calculation results of each element body to obtain the comprehensive mechanical analysis results for the overall structure.

The finite element method discretizes the continuous body into finite element bodies. Assume that the node displacement of each element body is $u^e$ within the element body. Through node displacement interpolation, the displacement field can be obtained

$$\mathit{u}=\mathit{N}{\mathit{u}}^e$$

(1)

where $\mathit{N}$ is a shape function matrix.

The strain $\mathit{\epsilon }$ and the stress $\mathit{\sigma }$ of the element body can be calculated through the displacement field, and we obtain

$$\mathit{\epsilon }=\mathit{B}{\mathit{u}}^e,\mathit{\sigma }=\mathit{D}\mathit{\epsilon }$$

(2)

where B is the train-displacement matrix, and D is the material constitutive matrix.

The kinetic energy $T^e$ and potential energy $U^e$ of the element body can be obtained, respectively, as follows:

$$T^e={\displaystyle \frac{1}{2}}{\int }_{{\mathsf{\Omega }}^e}\rho {\dot{\mathit{u}}}^T\dot{\mathit{u}}d\mathsf{\Omega },U^e={\displaystyle \frac{1}{2}}{\int }_{{\mathsf{\Omega }}^e}{\mathit{\epsilon }}^T\mathit{\sigma }d\mathsf{\Omega }$$

(3)

Using the Lagrange equation, the dynamic equation of the element body can be expressed as

$${\mathit{M}}^e{\ddot{\mathit{u}}}^e+{\mathit{K}}^e{\mathit{u}}^e={\mathit{F}}^e$$

(4)

where ${\mathit{M}}^e$ is the mass matrix of the element body, and ${\mathit{M}}^e={\int }_{{\mathsf{\Omega }}^e}\rho {\mathit{N}}^T\mathit{N}d\mathsf{\Omega }$; ${\mathit{K}}^e$ is the stiffness matrix of the element body, and ${\mathit{K}}^e={\int }_{{\mathsf{\Omega }}^2}{\mathit{B}}^T\mathit{DB}d\mathsf{\Omega }$; $F^e$ is the load vector of the element body, and ${\mathit{F}}^e={\int }_{{\mathsf{\Omega }}^e}{\mathit{N}}^T\mathit{f}d\mathsf{\Omega }+{\int }_{{\mathsf{\Gamma }}^e}{\mathit{N}}^T\mathit{t}d\mathsf{\Gamma }$.

By assembling the matrices of all the element bodies into a global matrix and introducing the damping matrix $\mathit{D}(\dot{\mathit{q}},\mathit{q})$, the nonlinear dynamic model of the smart structure can be derived as

$$M\left(\mathit{q}\right)\ddot{\mathit{q}}+D(\dot{\mathit{q}},\mathit{q})\dot{\mathit{q}}+K\left(\mathit{q}\right)\mathit{q}=H\left(\mathit{q}\right)\mathit{u}$$

(5)

where $\mathit{q}\in {\mathbb{R}}^{n_q\times 1}$ is the position of the grid node, $n_q$ is the number of grid nodes, which is taken as 8935 in this paper. $M\left(\mathit{q}\right)\in {\mathbb{R}}^{n_q\times n_q}$ is the always-invertible mass matrix, $D(\dot{\mathit{q}},\mathit{q})\in {\mathbb{R}}^{n_q\times n_q}$ is the system damping matrix, and $K\left(\mathit{q}\right)\in {\mathbb{R}}^{n_q\times n_q}$ is the stiffness matrix, $H\left(\mathit{q}\right)\in {\mathbb{R}}^{n_u\times n_q}$ is the direction matrix of the executive force of the smart structure, $n_u$ is the number of driving force inputs, and $\mathit{u}\in {\mathbb{R}}^{n_u\times 1}$ is the matrix of the actuator driving force.

Assume that the system damping $D(\dot{\mathit{q}},\mathit{q})$ is proportional damping which is proportional to the mass matrix and the stiffness matrix, and can be expressed as

$$D(\dot{\mathit{q}},\mathit{q})={\alpha }_{D}M\left(\mathit{q}\right)+{\beta }_{D}K\left(\mathit{q}\right)$$

(6)

where ${\alpha }_D$ and ${\beta }_D$ are the damping coefficients.

Due to the contradictory problems regarding the number of units, model accuracy, and computational complexity, in order to reduce the computational complexity while ensuring model accuracy, this paper adopts the orthogonal intrinsic model order reduction method. The core task lies in analyzing the given data set to find a set of optimal orthogonal basis functions so that these functions can represent the original data with minimal error. These basis functions are obtained through feature decomposition of the data covariance matrix and correspond to the eigenvectors of the covariance matrix. The eigenvalues represent the degree of the contribution of each basis function to the variance of the original data; that is, the energy of the original data contained in that basis function. Suppose $S_q$ is the set of system states $\mathit{q}$:

$$S_q=\left(\mathit{q}\left(t_1\right),\mathit{q}\left(t_2\right),\dots ,\mathit{q}\left(t_s\right)\right)\in {\mathbb{R}}^{n_q\times s}$$

(7)

where $s\in \mathbb{N}$ is the set quantity of system states $\mathit{q}$ collected. By performing singular value decomposition on (7), we obtain

$$S_q=\mathcal{V}\mathsf{\Sigma }{U}^{\top }={\mathcal{V}}_r{\mathsf{\Sigma }}_r{U}_r^{\top }+\Delta$$

(8)

where $\Delta$ is the order reduction error of the model, and $\Delta ={\mathcal{V}}_{\overline{r}}{\mathsf{\Sigma }}_{\overline{r}}{U}_{\overline{r}}^{\top }$. State q can be decomposed into a lower-order state $q_r\in {\mathbb{R}}^r(r\ll n)$ and a higher-order state $q_{\overline{r}}\in {\mathbb{R}}^{n-r}$, which is expressed as

$$\mathit{q}=\left[\begin{array}{cc}{\mathit{V}}_r\hfill & {\mathit{V}}_{\overline{r}}\hfill \end{array}\right]\left[\begin{array}{c}{\mathit{q}}_r\hfill \\ {\mathit{q}}_{\overline{r}}\hfill \end{array}\right]$$

(9)

Based on the above analysis, the smart structural dynamics model after order reduction can be simplified as follows:

$$M_r\left({\mathit{q}}_r\right){\ddot{\mathit{q}}}_r+D_r\left({\dot{\mathit{q}}}_r,{\mathit{q}}_r\right){\dot{\mathit{q}}}_r+K_r\left({\mathit{q}}_r\right){\mathit{q}}_r=H_r\left({\mathit{q}}_r\right)\mathit{u}$$

(10)

where

$$\left\{\begin{array}{c}{M}_r\left({\mathit{q}}_r\right)={\mathit{V}}_r^{\top }M\left(\mathit{q}\right){\mathit{V}}_r\hfill \\ D_r\left({\dot{\mathit{q}}}_r,{\mathit{q}}_r\right)={\mathit{V}}_r^{\top }D(\dot{\mathit{q}},\mathit{q}){\mathit{V}}_r\hfill \\ K_r\left({\mathit{q}}_r\right)={\mathit{V}}_r^{\top }K\left(\mathit{q}\right){\mathit{V}}_r\hfill \\ H_r\left({\mathit{q}}_r\right)={\mathit{V}}_r^{\top }H\left(\mathit{q}\right)\hfill \end{array}\right.$$

(11)

Let $\Delta M_r\left({\mathit{q}}_r\right)$, $\Delta D_r\left({\mathit{q}}_r,{\dot{\mathit{q}}}_r\right)$, and $\Delta K_r\left({\mathit{q}}_r\right)$ be represented as the uncertain terms in the system dynamics model, respectively; then, (5) can be rewritten as follows:

$$M_{r0}\left({\mathit{q}}_r\right){\ddot{\mathit{q}}}_r+D_{r0}\left({\mathit{q}}_r,{\dot{\mathit{q}}}_r\right){\dot{\mathit{q}}}_r+K_{r0}\left({\mathit{q}}_r\right){\mathit{q}}_r=H_r\left({\mathit{q}}_r\right)\mathit{u}+{\mathit{f}}_r$$

(12)

where ${\mathit{f}}_r=-\Delta M_r\left({\mathit{q}}_r\right){\ddot{\mathit{q}}}_r-\Delta D_r\left({\mathit{q}}_r,{\dot{\mathit{q}}}_r\right){\dot{\mathit{q}}}_r-\Delta K_r\left({\mathit{q}}_r\right){\mathit{q}}_r$, and satisfied with $\left|{\mathit{f}}_r\right|\le {\overline{\mathit{f}}}_r$, ${\overline{\mathit{f}}}_r$ is the upper boundary of uncertainty.

Assuming that ${\mathit{\eta }}_{\mathbf{1}}={\mathit{q}}_r$ and ${\mathit{\eta }}_{\mathbf{2}}={\dot{\mathit{q}}}_r$, the state equation of the system dynamics model can be expressed as follows:

$$\left\{\begin{array}{c}{\dot{\mathit{\eta }}}_1={\mathit{\eta }}_2\hfill \\ {\dot{\mathit{\eta }}}_2=M_{r0}^{-1}\left(\mathit{\tau }+{\mathit{f}}_r-D_{r0}{\mathit{\eta }}_2-K_{r0}{\mathit{\eta }}_1\right)\hfill \end{array}\right.$$

(13)

where $\mathit{\tau }=H_r\mathit{u}$.

3. Adaptive Backstepping Control and Stability Analysis

The position error and velocity error of the smart structure can be given by the following:

$${\mathit{e}}_1={\mathit{\eta }}_{d1}-{\mathit{\eta }}_1,{\mathit{e}}_2=\mathit{\xi }-{\mathit{\eta }}_2$$

(14)

where ${\mathit{e}}_1\in {\mathbb{R}}^{3\times 1}$ is the position error, and ${\mathit{e}}_2\in {\mathbb{R}}^{3\times 1}$ is the velocity error, ${\mathit{\eta }}_{d1}\in {\mathbb{R}}^{3\times 1}$ and ${\mathit{\eta }}_1\in {\mathbb{R}}^{3\times 1}$ are the desired trajectory and the actual trajectory, respectively, $\mathit{\xi }\in {\mathbb{R}}^{3\times 1}$ is the virtual control. Differentiating the position error $e_1$ with respect to time, we obtain

$${\dot{\mathit{e}}}_1={\dot{\mathit{\eta }}}_{d1}-{\dot{\mathit{\eta }}}_1={\dot{\mathit{\eta }}}_{d1}-{\mathit{\eta }}_2={\dot{\mathit{\eta }}}_{d1}+{\mathit{e}}_2-\mathit{\xi }$$

(15)

The Lyapunov function of the first step can be constructed as follows:

$$V_1={\displaystyle \frac{1}{2}}{\mathit{e}}_1^{\mathrm{T}}{\mathit{e}}_1$$

(16)

Using Derivative (16) with respect to time, we obtain

$${\dot{V}}_1={\mathit{e}}_1^{\mathrm{T}}{\dot{\mathit{e}}}_1={\mathit{e}}_1^{\mathrm{T}}\left({\dot{\mathit{\eta }}}_{d1}+{\mathit{e}}_2-\mathit{\xi }\right)$$

(17)

let $\mathit{\xi }={\dot{\mathit{\eta }}}_{d1}+k_1{\mathit{e}}_1$, (17) can be rewritten as

$${\dot{V}}_1=-{\mathit{e}}_1^{\mathrm{T}}{k}_1{\mathit{e}}_1+{\mathit{e}}_1^{\mathrm{T}}{\mathit{e}}_2$$

(18)

where $k_1\in {\mathbb{R}}^{3\times 3}$ is a positive definite matrix.

Deriving the velocity error ${\mathit{e}}_2$ with respect to time, and substitute into (17), we obtain

$${\dot{\mathit{e}}}_2=\dot{\mathit{\xi }}-M_{r0}^{-1}\left(\mathit{\tau }+{\mathit{f}}_r-D_{r0}{\mathit{\eta }}_2-K_{r0}{\mathit{\eta }}_1\right)$$

(19)

where $\dot{\mathit{\xi }}={\ddot{\mathit{\eta }}}_{d1}+k_1{\dot{\mathit{e}}}_1$.

The Lyapunov function of the second step can be constructed as follows:

$$V_2=V_1+{\displaystyle \frac{1}{2}}{\mathit{e}}_2^{\mathrm{T}}{\mathit{e}}_2$$

(20)

Deriving (20) with respect to time, we obtain

$${\dot{V}}_2={\dot{V}}_1+{\mathit{e}}_2^{\mathrm{T}}{\dot{\mathit{e}}}_2={\dot{V}}_1+e_2^{\mathrm{T}}\left(\dot{\mathit{\xi }}-M_{r0}^{-1}\left(\mathit{\tau }+{\mathit{f}}_r-D_{r0}{\mathit{\eta }}_2-K_{r0}{\mathit{\eta }}_1\right)\right)$$

(21)

Let the control law of the smart structure be

$$\mathit{\tau }=M_{r0}\left(\dot{\mathit{\xi }}+k_2{\mathit{e}}_2+{\mathit{e}}_1\right)+D_{r0}{\mathit{\eta }}_2+K_{r0}{\mathit{\eta }}_1-{\mathit{f}}_r$$

(22)

where $k_2\in {\mathbb{R}}^{3\times 3}$ is a positive definite matrix.

Substituting (22) into (21), we obtain

$${\dot{V}}_2=-{\mathit{e}}_1^{\mathrm{T}}{k}_1{\mathit{e}}_1-{\mathit{e}}_2^{\mathrm{T}}{k}_2{\mathit{e}}_2$$

(23)

As $k_1$ and $k_2$ are positive definite matrices, ${\dot{V}}_2\le 0$. Since ${\mathit{f}}_r$ is bounded and uncertain, and the deviated dynamic parameters are not easy to obtain, this will affect the control accuracy. Therefore, we consider the uncertainty of ${\mathit{f}}_r$ and propose an adaptive controller based on the uncertainty of the smart structure, which is given by the following:

$$\mathit{\tau }=M_{r0}\left(\dot{\mathit{\xi }}+k_2{\mathit{e}}_2+{\mathit{e}}_1\right)+D_{r0}{\mathit{\eta }}_2+K_{r0}{\mathit{\eta }}_1-{\widehat{\mathit{f}}}_r$$

(24)

where ${\widehat{\mathit{f}}}_r$ is the estimated value of ${\mathit{f}}_r$.

The Lyapunov function of the third step can be constructed as follows:

$$V_3=V_1+V_2+{\displaystyle \frac{1}{2k_3}}{\left(M_{r0}^{-1}{\tilde{\mathit{f}}}_r\right)}^{\mathrm{T}}\left(M_{r0}^{-1}{\tilde{\mathit{f}}}_r\right)$$

(25)

where $k_3\in {\mathbb{R}}^{3\times 3}$ is a positive definite matrix, ${\tilde{\mathit{f}}}_r={\widehat{\mathit{f}}}_r-{\mathit{f}}_r$ is the estimated error of the uncertainty in the kinetic model.

Deriving (25) with respect to time, we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_3& ={\dot{V}}_1+{\dot{V}}_2+{\displaystyle \frac{1}{k_3}}{\left(M_{r0}^{-1}{\tilde{\mathit{f}}}_r\right)}^{\mathrm{T}}\left(M_{r0}^{-1}{\dot{\widehat{\mathit{f}}}}_r\right)\hfill \\ \hfill & ={\dot{V}}_1+{\mathit{e}}_2^{\mathrm{T}}\left(M_{r0}^{-1}\left(\mathit{\tau }+{\mathit{f}}_r-D_{r0}{\mathit{\eta }}_2-K_{r0}{\mathit{\eta }}_1\right)-\dot{\mathit{\xi }}\right)+{\displaystyle \frac{1}{k_3}}{\left(M_{r0}^{-1}{\tilde{\mathit{f}}}_r\right)}^{\mathrm{T}}\left(M_{r0}^{-1}{\dot{\widehat{\mathit{f}}}}_r\right)\hfill \end{array}$$

(26)

Substituting (24) into (26) and simplifying it yields

$${\dot{V}}_3=-{\mathit{e}}_1^{\mathrm{T}}{k}_1{\mathit{e}}_1-{\mathit{e}}_2^{\mathrm{T}}{k}_2{\mathit{e}}_2+{\left(M_{r0}^{-1}{\tilde{\mathit{f}}}_r\right)}^{\mathrm{T}}\left({\displaystyle \frac{1}{k_3}}{M}_{r0}^{-1}{\dot{\widehat{\mathit{f}}}}_r-{\mathit{e}}_2\right)$$

(27)

When the adaptive law is satisfied with $\dot{\widehat{{\mathit{f}}_r}}=M_{r0}{k}_3{\mathit{e}}_2$, then,

$${\dot{V}}_3=-{\mathit{e}}_1^{\mathrm{T}}{k}_1{\mathit{e}}_1-{\mathit{e}}_2^{\mathrm{T}}{k}_2{\mathit{e}}_2\le 0$$

(28)

4. Numerical Validation

4.1. Simulations

Based on the motion characteristics of the three-degrees-of-freedom parallel mechanism, the mapping relationship between the elastic body and the three-degrees-of-freedom parallel mechanism was established. Using the strain energy of the motion degrees of freedom as the stiffness measure, Zhu et al. [28] proposed a difference strain energy method combined with the Jacobian matrix to obtain smart structures with three rotational degrees of freedom and three translational degrees of freedom, respectively, as shown in Figure 1.

To verify the effectiveness of the modeling method that combines the finite element method and orthogonal characteristic mode reduction, as well as its adaptive backstepping controller, we conducted a simulation analysis using MATLAB (R2024b). Specifically, a random disturbance signal was added: $\left(2.5\sin \right(t),5\sin (t),5\cos (t\left)\right)\pm \mathrm{rand}(3,1)$. The controller parameters were set as shown in

Table 1. Parameters of adaptive backstepping controller.

Controller	Parameters
ABSC	$k_1=\mathrm{diag}(100,100,100),k_2=\mathrm{diag}(20,20,20),k_3=\mathrm{diag}(30,30,30)$

.

The simulation analysis was conducted using both the traditional PID controller and the ABSC controller proposed in this paper. The trajectory tracking simulation results of the smart structure under uncertain disturbances are shown in Figure 2.

The trajectory tracking error graphs are presented in Figure 3. Figure 3a indicates that the average absolute errors of the traditional PID controller in the Z-axis, Y-axis, and X-axis directions are $0.400$, $0.139$, and $0.840$, respectively. Figure 3b shows that the average absolute errors of the ABSC controller in the Z-axis, Y-axis, and X-axis directions are $0.00021$, $0.00020$, and $0.00036$, respectively. The simulation analysis results demonstrate that the ABSC controller proposed in this paper can compensate for the dynamic uncertainties of the intelligent structure, thereby improving the trajectory tracking accuracy.

4.2. Experiments

The smart structure performance test experimental platform consists of a smart structure, a DC servo motor, a dynamic strain gauge, a laser interferometer, and a computer. The DC servo motor uses the PMM60-L model, which has a rated torque of 0.637 N·m, and is controlled in torque mode. The dynamic strain gauge utilizes the ST-3C model, which features 10 independent data acquisition channels and a sampling frequency of 1 Hz, and is connected to the computer via the RS-232 interface. Its data comes from the strain signals collected by the 120-5AA type strain gauge. The laser interferometer adopts the XL-80 model of Renishaw (Shanghai) Trading Co, Ltd, Shanghai, China. The composition of the smart structure performance test platform is shown in Figure 4.

The trajectory tracking performance of the smart structure was compared using the traditional PID controller, adaptive control [29], and the control method proposed in this paper. The trajectory tracking performance and tracking errors of the X-axis, Y-axis, and Z-axis are shown in Figure 5. When the traditional PID controller was used, the average tracking errors of the smart structure in the x, y, and z directions were 0.113, 0.166, and 0.233, respectively. When the adaptive controller was used, the average tracking errors of the smart structure in the x, y, and z directions were 0.085, 0.078, and 0.092, respectively. When the ABSC controller proposed in this paper was used, the average tracking errors of the smart structure in the x, y, and z directions were 0.037, 0.035, and 0.044, respectively.

5. Conclusions

The paper proposes a dynamic modeling method combining finite element unit bodies and orthogonal characteristic mode reduction for smart structures, and designs an adaptive backstepping controller. Through finite element analysis, the nonlinear dynamic equations of the structure are derived, and the orthogonal characteristic mode reduction method is used to reduce computational complexity while ensuring model accuracy. The adaptive backstepping controller is designed to address model uncertainties and achieve precise trajectory tracking control. The simulation and experimental results show that the method can effectively handle the nonlinearity and modeling errors of smart structures, realize high-precision trajectory tracking, and verify the accuracy of the dynamic model and the robustness of the controller.