Shape Control of a Carbon Fiber-Reinforced Polymer Reflector and Placement Optimization of the Actuators

Abstract

In this study, a method for the active shape control for carbon fiber-reinforced polymer (CFRC) reflectors using piezoelectric lead zirconate titanate (PZT) actuators is proposed. According to this method, a finite element model considering higher transverse shear deformation with independent voltage degrees of freedom is formulated by the Hamilton principle. An optimal shape controller that minimizes the discrete root mean square (RMS) error of a reflecting surface is applied. Then, the optimal arrangements of the PZT actuators are determined by numerical optimization methods, which are developed by modifying the classical Genetic Algorithm, with both single and multi-objective optimizations being studied. In the single optimization formulation, the number of actuators is considered as a constraint, and the RMS error of the reflector is regarded as the optimizing target. A hybrid method that combines the gradient projection method with an adaptive Genetic Algorithm is proposed to solve this problem. In the multi-objective optimization, the residual RMS errors and power consumption of the actuators are considered as the optimization targets. Pareto optimal solutions are obtained by an improved multi-objective Genetic Algorithm. Numerical examples are provided to show the effectiveness of the proposed methods.

1. Introduction

High-precision large reflectors are garnering widespread research attention. Many studies on improving reflector surface accuracies have been conducted. Pearson et al. [1] designed and analyzed a large and high-precision inflatable membrane reflector. Lang et al. [2] developed a design concept for high-precision shell reflectors with a structural optimization and a shape adjustment method. A pseudo-geometric method for the generation of the surface mesh geometry of deployable reflectors was proposed by Shi et al. [3]. Yuan et al. developed a technique for placing nodes off the working surface of a mesh reflector [4] and applied an additive structure, named as a self-standing-truss with hardpoints [5,6], to improve the surface accuracy of the reflector. Soykasap et al. [7] designed and analyzed a high-precision offset-stiffened spring-back reflector with carbon fiber-reinforced composite (CFRC) materials.

Active shape control is an effective approach to improve reflector surface accuracies. Smart materials and smart structures such as PVDF (poly vinylidene fluoride), PZT (piezoelectric lead zirconate titanate), MFC (macro fiber composite), and SMA (shape memory alloy material) actuators/sensors are widely used in active shape and vibration controls. Tabata et al. [8] investigated the feasibility of an active shape correction method for a deployable space antenna reflector. Fang [9] presented a high-precision adaptive control architecture with PVDF actuators for large membrane and thin shell reflectors, in which both an analytical model and an experimental system were developed. Wang [10] introduced an active shape adjustment method that used active cables to obtain a desired shape. Stein et al. [11] developed an active shape control technology for the reflecting surfaces using large spatially distributed actuator arrays. Andoh [12] presented an approach to shape control problems with a constrained number of discrete actuators. Lu et al. [13] used SMA actuators to improve the surface accuracy of a circular membrane structure. In their study, an analytical model and a solution method were developed, and the finite element method was then used to validate analytical model under typical thermal loads. Wu et al. [13] proposed a novel method to reduce the global and local surface error of a CFRC reflector using PZT and MFC actuators. Desmidt et al. [14] investigated the feasibility of using an active gore/seam cable-based control system to reduce the global root mean square errors arising from thermal loads. CFRC materials are widely used for high-precision reflectors owing to their high strength, high modulus, and low Coefficient Thermal Expansion (CTE). For improving the control efficiency, the arrangement of the actuators is essential to actively control the problems of space structures, especially for those with a weight budget and constrained energy supply. Many studies have investigated the geometry sizes, locations, and applied voltages of actuators for beam and plate structures. The previously studied methods for actuator arrangement optimization include gradient-based algorithms, group elimination methods, and random search algorithms.

Suleman and Gonicalves [15] optimized the sizes, thicknesses, and locations of the actuators for a rectangle cantilever composite beam to achieve maximum nodal displacements, minimum masses and minimum voltages for shape control task. Adali et al. [16] used a one-dimensional optimization algorithm to determine the locations of actuators for a laminated beam structure. Aldraihem et al. [17] optimized the lengths and locations of the actuators to maximize the weighted controllability. Barboni et al. [18] optimized the lengths and locations of the actuators for a simply supported beam. Bruant et al. [19] used a gradient method to optimize the lengths and locations of the actuators in a three-beam structure to achieve the minimum mechanical energy and maximum observability. These methods are based on the gradients of objective functions with respect to optimized variables. However, for discrete optimizing problems or complicated non-convex problems, the gradient-based methods are not suitable because the gradients of objective functions may be difficult to obtain and the solutions derived are the local optimum.

The random search algorithm is another type of optimization method for shape control problems. Dong and Huang [20] optimized a truss topology based on a multipoint approximate function and a Genetic Algorithm (GA). An optimized actuator was positioned for an adaptive truss using a two-level multipoint approximation method [21]. Liu and Lin [22] used a Simulated Annealing (SA) algorithm to track the shape of a smart plate and optimized the distribution of the voltage channels as well as the control voltages. Honda et al. [23] investigated multidisciplinary design optimization with the GA for vibration controls of smart laminated composite structures. Random search algorithms do not require gradients of the objective function. Therefore, variables are allowed to be discrete and the global optimum can be obtained. However, the accuracies of solutions from a random search algorithm may be lower than that from a gradient-based method due to its uncertain searching trajectory. For large adaptive space mirrors and reflectors, researchers have investigated the arrangement of the actuators. Delorenzo [24] used a linear quadratic Gaussian controller, combining with an efficient weight-selection technique to optimize the configurations of actuators and sensors. Kuo and Bruno [25] used a modified Simulated Annealing Algorithm to optimize the locations of the actuators of a honeycomb composite panel. Sheng and Kapania [26] used the GA to optimize the locations of the actuators for a primary mirror to achieve the minimum surface error caused by thermal distortion. Hill et al. [27] used the elimination method to determine the optimal grouping of the PVDF actuators for a membrane reflector. Wang et al. [28] studied the optimal design of actuator locations in a honeycomb mirror by using the Simulated Annealing Algorithm.

We have previously proposed a novel analysis method to reduce the global and local surface error of a CFRC reflector analytically with PZT (Piezo Direct, Burlingame, CA, USA) and MFC (MFC Manufacturing, Memphis, TN, USA) actuators employed [14]. For further research, in this article, the optimal placement of PZT actuators and the shape control experiment with PZT actuators are studied. Firstly, a finite element model with independent degrees of freedom of applied voltages is set up, and based on the finite element formulation, the optimal arrangement of the actuators is determined by a numerical method using the proposed modified GA. Both single and multi-objective optimizations are studied in optimizing the placement of the actuators. For the single-objective optimization, a hybrid method that combines the GA with the gradient projection method is proposed. The proposed method has higher convergence accuracy than the classical GA due to combining the random and deterministic methods. Numerical examples are provided to verify the effectiveness of this method. For the multi-objective optimization, the Pareto optimal solution of the shape control problem is derived. Unlike traditional approaches to converting a multi-objective optimization problem to a single-objective problems by enforcing the weighted sum of the objectives, here, a modified multi-objective Genetic Algorithm is proposed to derive the Pareto optimal solution.

2. Materials and Methods

A geometric model of a CFRC rib reflector (SGL Carbon, Wiesbaden, Germany) embedded with piezoelectric ceramic transducer (PZT) actuators is shown in Figure 1. The inscribed circle diameter of the hexagonal CFRC reflector is 1.0 m; the focal length of the reflector is 2.1 m. The height, length, and cut out height of ribs are 60 mm, 188.6 mm, and 20 mm, respectively, which were optimized in a previous study [29]. The fiber orientations of the CFRC are [0°/45°/−45°/90°]_s.

2.1. Displacements and Strain Assumption

A higher-order shear deformation theory was developed by Reddy [30]. It exhibits a higher accuracy in describing the displacement field in laminate elements, and it is employed to model the laminates. The displacements are assumed as follows:

$$u(x,y,z)=u(x,y,0)+z{\phi }_y(x,y,0)-\frac{4}{3h^2}{z}^3\left({\phi }_y+\frac{\partial w_0}{\partial x}\right),$$

(1)

$$v(x,y,z)=v(x,y,0)+z{\phi }_x(x,y,0)-\frac{4}{3h^2}{z}^3\left({\phi }_x+\frac{\partial w_0}{\partial y}\right),$$

(2)

$$w(x,y,z)=w(x,y,0)=w_0\left(x,y\right),$$

(3)

where h is thickness of the laminates; u, v, w are displacements in the x, y, and z directions; and ${\phi }_x$ and ${\phi }_y$ are rotation angles in the Y–Z and X–Z planes, respectively. The plane z = 0 is a neutral plane of the laminates along the thickness direction. Geometry equations are denoted in Equation (4). ${\epsilon }_{xx}$, ${\epsilon }_{yy}$, and ${\gamma }_{xy}$ are the plane strains of the plate; ${\gamma }_{yz}$ and ${\gamma }_{xz}$ are the shear strains along the thickness direction.

$$\left[\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\gamma }_{xy}\\ {\gamma }_{yz}\\ {\gamma }_{xz}\end{array}\right]=\left[\begin{array}{ccc}\frac{\partial }{\partial x}& 0& 0\\ 0& \frac{\partial }{\partial y}& 0\\ \frac{\partial }{\partial y}& \frac{\partial }{\partial x}& 0\\ 0& \frac{\partial }{\partial z}& \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}& 0& \frac{\partial }{\partial x}\end{array}\right]\left[\begin{array}{c}u\\ v\\ w\end{array}\right]=\nabla \mathit{d},$$

(4)

$\mathit{\epsilon }={\left[\begin{array}{ccccc}{\epsilon }_{xx}& {\epsilon }_{yy}& {\gamma }_{xy}& {\gamma }_{yz}& {\gamma }_{xz}\end{array}\right]}^T$ is a plate strain vector, $\mathit{d}={\left[uvw\right]}^T$ is a plate displacement vector, and $\nabla$ is a differential operator for the plate. For a three-dimensional (3-D) piezoelectric beam element, which can be used to model PZT actuators, the displacement assumption can be written as

$$u^{\prime }=u_0^{\prime }\left(x,0,0\right)+y{\phi }_z^{\prime }\left(x,0,0\right)+z{\phi }_y^{\prime }\left(x,0,0\right),$$

(5)

$$v^{\prime }=v_0^{\prime }\left(x,0,0\right),$$

(6)

$$w^{\prime }=w_0^{\prime }\left(x,0,0\right),$$

(7)

$u^{\prime }$,$v^{\prime }$, and $w^{\prime }$ denote displacements along the x-, y-, and z-directions, respectively. ${\phi }_y^{\prime }$ and ${\phi }_z^{\prime }$ are rotation angles in the X–Z and X–Y planes, respectively. Geometry equations of a 3-D beam are written as

$$\left[\begin{array}{c}{\epsilon }_x^{\prime }\\ {\gamma }_{xy}^{\prime }\\ {\gamma }_{xz}^{\prime }\\ {\kappa }_x^{\prime }\end{array}\right]=\left[\begin{array}{cccc}\frac{\partial }{\partial x}& 0& 0& 0\\ \frac{\partial }{\partial y}& \frac{\partial }{\partial x}& 0& 0\\ \frac{\partial }{\partial z}& 0& \frac{\partial }{\partial x}& 0\\ 0& 0& 0& \frac{\partial }{\partial x}\end{array}\right]\left[\begin{array}{c}{u}^{\prime }\\ v^{\prime }\\ w^{\prime }\\ {\phi }_x^{\prime }\end{array}\right]={\nabla }^{\prime }{\mathit{d}}^{\prime },$$

(8)

where ${\epsilon }_x^{\prime }$ denotes the axial strain; ${\gamma }_{xy}^{\prime }$ and ${\gamma }_{xz}^{\prime }$ denote the shear strains; ${\phi }_x^{\prime }$ is a twist angle of the beam in the X–Y plane; ${\kappa }_x^{\prime }$ is a twist angle per length along the x-direction; ${\mathit{\epsilon }}^{\prime }={\left[\begin{array}{cccc}{\epsilon }_x^{\prime }& {\gamma }_{xy}^{\prime }& {\gamma }_{xy}^{\prime }& {\kappa }_x^{\prime }\end{array}\right]}^T$ and ${\mathit{d}}^{\prime }={\left[\begin{array}{cccc}{u}^{\prime }& v^{\prime }& w^{\prime }& {\phi }_x^{\prime }\end{array}\right]}^T$ are a strain vector and a displacement vector of the beam, respectively; ${\nabla }^{\prime }$ is a differential operator for the beam.

2.2. Constitutive Equations

The constitutive equations of the orthogonal anisotropy laminates can be simplified as Equation (9).

$$\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\tau }_{yz}\\ {\tau }_{zx}\\ {\tau }_{xy}\end{array}\right]=\left[\begin{array}{ccccc}{\bar{Q}}_{11}& {\bar{Q}}_{12}& 0& 0& {\bar{Q}}_{16}\\ {\bar{Q}}_{12}& {\bar{Q}}_{22}& 0& 0& {\bar{Q}}_{26}\\ 0& 0& {\bar{Q}}_{44}& {\bar{Q}}_{45}& 0\\ 0& 0& {\bar{Q}}_{45}& {\bar{Q}}_{55}& 0\\ {\bar{Q}}_{16}& {\bar{Q}}_{26}& 0& 0& {\bar{Q}}_{66}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ 2{\gamma }_{yz}\\ 2{\gamma }_{zx}\\ 2{\gamma }_{xy}\end{array}\right]=\bar{\mathit{Q}}\mathit{\epsilon },$$

(9)

$\sigma ={\left[\begin{array}{ccccc}{\sigma }_{xx}& {\sigma }_{yy}& {\tau }_{yz}& {\tau }_{zx}& {\tau }_{xy}\end{array}\right]}^{{\rm T}}$ is a stress vector; $\bar{\mathit{Q}}$ is the simplified elastic matrix in the primary material coordinate; $\mathit{\epsilon }={\left[\begin{array}{ccccc}{\epsilon }_{xx}& {\epsilon }_{yy}& {\gamma }_{yz}& {\gamma }_{zx}& {\gamma }_{xy}\end{array}\right]}^T$ is a strain vector. A generalized constitutive equation of a 3-D piezoelectric beam in a local primary coordinate can be written as

$$\left[\begin{array}{c}{\sigma }_{xx}^{\prime }\\ {\tau }_{xy}^{\prime }\\ {\tau }_{xz}^{\prime }\\ T_{yz}^{\prime }\end{array}\right]=\left[\begin{array}{cccc}{\bar{Q}}_{11}^{\prime }& 0& 0& 0\\ 0& {\bar{Q}}_{66}^{\prime }& {\bar{Q}}_{65}^{\prime }& 0\\ 0& {\bar{Q}}_{65}^{\prime }& {\bar{Q}}_{55}^{\prime }& 0\\ 0& 0& 0& {\bar{Q}}_{\rho }\end{array}\right]\left[\begin{array}{c}{\epsilon }_{xx}^{\prime }\\ 2{\gamma }_{xy}^{\prime }\\ 2{\gamma }_{xz}^{\prime }\\ {\kappa }_x^{\prime }\end{array}\right]+\left[\begin{array}{ccc}{\bar{e}}_{11}^{\prime }& 0& {\bar{e}}_{31}^{\prime }\\ 0& 0& {\bar{e}}_{36}^{\prime }\\ {\bar{e}}_{15}^{\prime }& {\bar{e}}_{25}^{\prime }& 0\\ 0& 0& 0\end{array}\right]\left[\begin{array}{c}{E}_x^{\prime }\\ E_y^{\prime }\\ E_z^{\prime }\end{array}\right]={\bar{\mathit{Q}}}^{\prime }{\mathit{\epsilon }}^{\prime }+{\bar{\mathit{e}}}^{\prime }{\mathit{E}}^{\prime },$$

(10)

$$\left[\begin{array}{c}{D}_x^{\prime }\\ D_y^{\prime }\\ D_z^{\prime }\end{array}\right]=\left[\begin{array}{cccc}{\bar{e}}_{11}^{\prime }& 0& {\bar{e}}_{15}^{\prime }& 0\\ 0& 0& {\bar{e}}_{25}^{\prime }& 0\\ {\bar{e}}_{31}^{\prime }& {\bar{e}}_{36}^{\prime }& 0& 0\end{array}\right]\left[\begin{array}{c}{\epsilon }_{xx}^{\prime }\\ 2{\gamma }_{xy}^{\prime }\\ 2{\gamma }_{xz}^{\prime }\\ {\kappa }_x^{\prime }\end{array}\right]+\left[\begin{array}{ccc}{\bar{\chi }}_{11}^{\prime }& 0& 0\\ 0& {\bar{\chi }}_{22}^{\prime }& 0\\ 0& 0& {\bar{\chi }}_{33}^{\prime }\end{array}\right]\left[\begin{array}{c}{E}_x^{\prime }\\ E_y^{\prime }\\ E_z^{\prime }\end{array}\right]={\bar{\mathit{e}}}^{\prime }{\mathit{\epsilon }}^{\prime }+{\bar{\mathit{\chi }}}^{\prime }{\mathit{E}}^{\prime }.$$

(11)

Equation (10) is the mechanical equation and Equation (11) is the electric equation. ${\bar{\mathit{Q}}}^{\prime }$ is the simplified elastic matrix of the beam; ${\bar{\mathit{e}}}^{\prime }$ and ${\bar{\mathit{\chi }}}^{\prime }$ are the simplified piezoelectric constant matrix and dielectric constant matrix, respectively. The actuation mode of the PZT actuator used in this study is the d11 type; therefore, in matrices ${\mathit{e}}^{\prime }$ and ${\bar{\mathit{\chi }}}^{\prime }$, only ${\bar{e}}_{31}\ne 0$ and ${\bar{\chi }}_{11}\ne 0$.

2.3. Interpolation Formulation

For the CFRC laminates, Hermite interpolation functions can be used for deducing the element displacements. Four nodes of plate elements are used to model the laminates. Each node of a plate element contains seven degrees of freedom. For a plate element, a displacement vector can be denoted as

$${{\left[\begin{array}{ccc}u& v& w\end{array}\right]}_e}^{{\rm T}}=\mathit{N}{{\mathit{d}}_e}^{{\rm T}},$$

(12)

A nodal displacements vector can be written as

$${\mathit{d}}_e={\left[\begin{array}{cccc}{\mathit{d}}_1^0& {\mathit{d}}_2^0& {\mathit{d}}_3^0& {\mathit{d}}_4^0\end{array}\right]}^T,$$

(13)

where each node has seven degrees of freedom, shown as

$${\mathit{d}}_i^0=\left[u_{0\mathrm{i}}{v}_{0\mathrm{i}}{w}_{0\mathrm{i}}{\phi }_{\mathrm{xi}}{\phi }_{\mathrm{yi}}\right.{\left(\frac{\partial w}{\partial x}\right)}_i{\left.{\left(\frac{\partial w}{\partial y}\right)}_i\right]}^T.$$

(14)

The shape function matrix $\mathit{N}$ is a 3 × 28 matrix. ${\mathit{d}}_e$ is a nodal displacement vector with 28 elements. $\mathit{N}$ is defined using Hermite interpolation functions [31]. Substituting Equation (12) into Equation (4), a strain vector of the element can be expressed as

$$\mathit{\epsilon }=\nabla \mathit{N}{{\mathit{d}}_e}^{{\rm T}}=\mathit{B}{{\mathit{d}}_e}^{{\rm T}},$$

(15)

where $\mathit{B}$ is the strain matrix of the plate element.

For a piezoelectric beam element, a displacement interpolation formulation can be expressed as

$${\left[\begin{array}{cccc}{\mathit{u}}^{\prime }& {\mathit{v}}^{\prime }& {\mathit{w}}^{\prime }& {\mathit{\phi }}_{\mathit{x}}^{\prime }\end{array}\right]}_{\mathit{e}}^{\mathit{T}}={\mathit{N}}^{\prime }{{\mathit{d}}_{\mathit{e}}^{\prime }}^{\mathit{{\rm T}}}.$$

(16)

A strain vector can be further denoted as

$${\mathit{\epsilon }}^{\prime }={\nabla }^{\prime }{\left[\begin{array}{cccc}{\mathit{u}}^{\prime }& {\mathit{v}}^{\prime }& {\mathit{w}}^{\prime }& {\mathit{\phi }}_{\mathit{x}}^{\prime }\end{array}\right]}_{\mathit{e}}^{\mathit{T}}={\nabla }^{\prime }{\mathit{N}}^{\prime }{{\mathit{d}}_{\mathit{e}}^{\prime }}^{\mathit{{\rm T}}}={\mathit{B}}^{\prime }{{\mathit{d}}_{\mathit{e}}^{\prime }}^{\mathit{{\rm T}}},$$

(17)

where ${\mathit{d}}_{\mathit{e}}^{\prime }=\left[\begin{array}{cc}{d^{\prime }}_1^0& {d^{\prime }}_2^0\end{array}\right]$ and ${{\mathit{d}}^{\prime }}_i^0={\left[\begin{array}{cccccc}{u}_{0i}^{\prime }& v_{0i}^{\prime }& w_{0i}^{\prime }& {\phi }_{xi}^{\prime }& {\phi }_{yi}^{\prime }& {\phi }_{zi}^{\prime }\end{array}\right]}^T$ are an element displacement vector and a generalized nodal degree of freedom vector, respectively.

2.4. Variation Formulation

The generalized Hamilton principle is employed to deduce the equilibrium equations of the system. The Hamilton variation equation can be expressed as

$$\mathit{\delta }{\int }_{{\mathit{t}}_1}^{{\mathit{t}}_2}\left(\mathit{K}-\mathit{P}+\mathit{W}\right)\mathit{d}\mathit{t}=0,$$

(18)

where K is kinetic energy of the whole system, P is the generalized potential energy, and W is the external work. For the static analysis, the kinetic energy K is zero. In piezoelectric structures, the generalized potential energy P can be written as

$$\mathit{P}={\iiint }_{\mathit{V}}\left(\frac{1}{2}{\mathit{\epsilon }}^{\mathit{T}}\bar{\mathit{Q}}\mathit{\epsilon }+{\mathit{E}}^{\mathit{T}}{\bar{\mathit{e}}}^{\mathit{{\rm T}}}\mathit{\epsilon }+\frac{1}{2}{\mathit{E}}^{\mathit{T}}\mathit{\chi }\mathit{E}\right)\mathit{d}\mathit{V}=\frac{1}{2}{\iiint }_{\mathit{V}}{\left[\begin{array}{c}\mathit{\epsilon }\\ \mathit{E}\end{array}\right]}^{\mathit{{\rm T}}}\left[\begin{array}{cc}\bar{\mathit{Q}}& \bar{\mathit{e}}\\ {\bar{\mathit{e}}}^{\mathit{{\rm T}}}& \mathit{\chi }\end{array}\right]\left[\begin{array}{c}\mathit{\epsilon }\\ \mathit{E}\end{array}\right]\mathit{d}\mathit{V},$$

(19)

where $\bar{\mathit{Q}}$, $\bar{\mathit{e}}$, and $\mathit{\chi }$ are constant matrices of the material’s properties. $\mathit{\epsilon }$ and $\mathit{E}$ are a generalized strain vector and an electric field vector, respectively. For a piezoelectric beam element, its electric field can be written as

$${\mathit{E}}^{\prime }=\left[\begin{array}{c}{\mathit{E}}_{\mathit{x}}^{\prime }\\ {\mathit{E}}_{\mathit{y}}^{\prime }\\ {\mathit{E}}_{\mathit{z}}^{\prime }\end{array}\right]=-\left[\begin{array}{c}\frac{{\mathit{U}}_{\mathit{e}}^{\prime }}{\mathit{l}}\\ 0\\ 0\end{array}\right]=\mathit{L}{\mathit{U}}_{\mathit{e}}^{\prime }.$$

(20)

$U_e^{\prime }$ is the electric potential applied to a piezoelectric beam element. Using Equations (15), (19) and (20), the potential energy in a piezoelectric beam element can be further expressed as

$${\mathit{P}}_{\mathit{B}}={\frac{1}{2}\left[\begin{array}{c}{\mathit{d}}_{\mathit{e}}^{\prime }\\ {\mathit{U}}_{\mathit{e}}^{\prime }\end{array}\right]}^{\mathit{{\rm T}}}{\left[\begin{array}{cc}{\mathit{K}}_m^{\prime }& {\mathit{K}}_{\mathit{me}}^{\prime }\\ {{\mathit{K}}_{\mathit{me}}^{\prime }}^{\mathit{T}}& {\mathit{K}}_{\mathit{ee}}^{\prime }\end{array}\right]}_{\mathit{e}}\left[\begin{array}{c}{\mathit{d}}_{\mathit{e}}^{\prime }\\ {\mathit{U}}_{\mathit{e}}^{\prime }\end{array}\right],$$

(21)

$${\mathit{K}}_{\mathit{m}}^{\prime }={\iiint }_{\mathit{V}}{\mathit{B}}^{\prime \mathit{T}}{\bar{\mathit{Q}}}^{\prime }{\mathit{B}}^{\prime }\mathit{d}\mathit{V},$$

(22)

$${\mathit{K}}_{\mathit{me}}^{\prime }={\iiint }_{\mathit{V}}{\mathit{B}}^{\prime \mathit{T}}{\overline{\mathit{e}}}^{\prime }\mathit{L}\mathit{d}\mathit{V},$$

(23)

$${\mathit{K}}_{\mathit{ee}}^{\prime }={\iiint }_{\mathit{V}}{\mathit{L}}^{\mathit{T}}{\mathit{\chi }}^{\prime }\mathit{L}\mathit{d}\mathit{V}.$$

(24)

In Equation (21), ${\mathit{K}}_m^{\prime }$, ${\mathit{K}}_{\mathrm{me}}^{\prime }$, and ${\mathit{K}}_{\mathrm{ee}}^{\prime }$ are the mechanical stiffness matrix, the mechanical–electric matrix, and the electric matrix for beam and plate elements, respectively. For the CFRC laminates, the piezoelectric constant matrix ${\bar{\mathit{e}}}^{\prime }$ and the dielectric constant matrix ${\mathit{\chi }}^{\prime }$ are zero matrices. Their potential energy can be expressed as

$${\mathit{P}}_{\mathit{L}}=\frac{1}{2}{{\mathit{d}}_{\mathit{e}}}^{\mathit{T}}\left({\iiint }_{\mathit{V}}{\mathit{B}}^{\mathit{{\rm T}}}\bar{\mathit{Q}}\mathit{B}\mathit{d}\mathit{V}\right){\mathit{d}}_{\mathit{e}}=\frac{1}{2}{{\mathit{d}}_{\mathit{e}}}^{\mathit{T}}{\mathit{K}}_{\mathit{m}}{\mathit{d}}_{\mathit{e}},$$

(25)

In Equation (25), ${\mathit{K}}_m$ is the element stiffness matrix of the CFRC laminates. The work of the external forces can be written in a generalized form as

$$W={\left[\begin{array}{c}u\\ v\\ w\end{array}\right]}^{{\rm T}}{\mathit{F}}_n+{\iint }_S{\left[\begin{array}{c}u\\ v\\ w\end{array}\right]}^{{\rm T}}{\mathit{F}}_{s}dS+{{\int }_S\left[\begin{array}{c}{U}_x^{\prime }\\ U_y^{\prime }\\ U_z^{\prime }\end{array}\right]}^{{\rm T}}\mathit{Q}dS,$$

(26)

where ${\mathit{F}}_n$ and ${\mathit{F}}_s$ are the nodal force and surface force vectors; ${\mathit{U}}^{\prime }={\left\{\begin{array}{ccc}{U}_x^{\prime }& U_y^{\prime }& U_z^{\prime }\end{array}\right\}}^T$ and $\mathit{Q}$ are electric potential and electric charge applied to the piezoelectric actuators, respectively. Substituting Equations (21), (25) and (26) into Equation (18), the system variation formulation can be obtained as

$$\sum _b\left[\begin{array}{c}\begin{array}{c}\frac{1}{2}\delta {\left[\begin{array}{c}{\mathit{d}}_e^{\prime }\\ {\mathit{U}}_e^{\prime }\end{array}\right]}^{\mathrm{T}}{\left[\begin{array}{cc}{\mathit{K}}_m^{\prime }& {\mathit{K}}_{\mathrm{me}}^{\prime }\\ {{\mathit{K}}_{\mathrm{me}}^{\prime }}^{\mathbf{T}}& {\mathit{K}}_{\mathrm{ee}}^{\prime }\end{array}\right]}_e\left[\begin{array}{c}{\mathit{d}}_e^{\prime }\\ {\mathit{U}}_e^{\prime }\end{array}\right]\\ \end{array}\\ -\left(\delta {{\mathit{d}}_e^{\prime }}^{{\rm T}}{{\mathit{F}}_e^n}^{\prime }+\delta {{\mathit{U}}_e^{\prime }}^{{\rm T}}{\mathit{Q}}_e^{\prime }\right)\end{array}\right]+\sum _p\left[\begin{array}{c}\frac{1}{2}\delta {{\mathit{d}}_e}^{{\rm T}}{\mathit{K}}_m{\mathit{d}}_e\\ -\left(\delta {{\mathit{d}}_e}^{{\rm T}}{\mathit{F}}_e^n+\delta {{\mathit{d}}_e}^{{\rm T}}{\mathit{N}}^{{\rm T}}{\mathit{F}}_e^s\right)\end{array}\right]=0.$$

(27)

Equation (27) is a variation formulation of the element nodal displacements for the whole system. In Equation (27), b and p stand for piezoelectric beam elements and CFRC plate elements. ${\mathit{F}}_e^n$ and ${\mathit{F}}_e^s$ are vectors of the element nodal and surface forces, respectively; ${\mathit{Q}}_e^{\prime }$ denotes the electric charges in a beam element. Each element coordinate shall be transformed into a global coordinate system by multiplying it by a Jacobi matrix. Subsequently, a global stiffness matrix can be obtained by assembling the element stiffness matrices in the global coordinate system. Owing to the arbitrariness of the variation components, Equation (27) can be converted into algebraic equations in the global coordinate as

$${\left[\begin{array}{cc}{\mathit{K}}_m& {\mathit{K}}_{\mathrm{me}}\\ {\mathit{K}}_{me}^{\mathrm{T}}& {\mathit{K}}_{\mathrm{ee}}\end{array}\right]}_{global}\left[\begin{array}{c}\mathit{d}\\ \mathit{U}\end{array}\right]=\left[\begin{array}{c}\mathit{F}\\ \mathit{Q}\end{array}\right].$$

(28)

In Equation (28), $\mathit{d}$ is a vector of the global nodal displacements; $\mathit{U}$ is a voltage vector of all the piezoelectric elements; $\mathit{F}$ is a vector of the nodal external forces; and $\mathit{Q}$ is a vector of the external electric charges. By solving Equation (28), the nodal displacements and the actuator voltages can then be obtained.

3. Results

The discrete nodal root mean square (RMS) error is used to evaluate the surface accuracy of a reflector. As shown in Figure 2, a geometric schematic of a reflector is given.

The origin of the coordinate system is located at the vertex of the paraboloid. The y-axis passes the focal point of the paraboloid. A discrete root mean square (RMS) error of the reflector is defined as

$$f_{\mathrm{RMS}}=\sqrt{\frac{{\sum }_{i=1}^k{\left({{\delta }_i}^{Desired}-{\delta }_i^{Deformed}\right)}^2}{k}},$$

(29)

where ${\delta }_i^{Deformed}$ and ${{\delta }_i}^{Desired}$ are the deformed and desired z-coordinate values of the i-th node; and k is the number of nodes on the surface of the reflector. The deformed z-coordinate of the i-th node can be expressed as

$${\delta }_i^{Deformed}={\delta }_i^{Initial}+\nabla {\delta }_i,$$

(30)

where $\nabla {\delta }_i$ is the displacement of the i-th node along the z-direction. Let ${\mathit{\delta }}^D$ stand for a nodal position vector of the desired reflector surface, ${\mathit{\delta }}^I$ stands for a nodal position vector of the initial reflector surface, and $\nabla \mathit{\delta }$ stands for a nodal displacement vector. Assuming the total number of reflector nodal displacements is p, the number of nodal displacements on the reflector surface is q, and the number of voltage freedom degrees is r, an equivalent objective function can be defined as

$$f={\left[\nabla \mathit{\delta }-\left({\mathit{\delta }}^D-{\mathit{\delta }}^I\right)\right]}^T\left[\nabla \mathit{\delta }-\left({\mathit{\delta }}^D-{\mathit{\delta }}^I\right)\right]=\nabla {\mathit{\delta }}^T\nabla \mathit{\delta }-2\nabla {\mathit{\delta }}^T\left({\mathit{\delta }}^D-{\mathit{\delta }}^I\right)+{\left({\mathit{\delta }}^D-{\mathit{\delta }}^I\right)}^T\left({\mathit{\delta }}^D-{\mathit{\delta }}^I\right)$$

(31)

where the reflector nodal displacements vector $\nabla \mathit{\delta }$ is a q-dimensional vector and is determined only by the voltage vector $\mathit{U}$. According to Equation (28), the total nodal displacement vector can be denoted as

$$\mathit{d}={{\mathit{K}}_m}^{-1}\left(\mathit{F}-{\mathit{K}}_{\mathrm{me}}\mathit{U}\right),$$

(32)

where ${\mathit{K}}_m$ is a p-by-p nonsingular matrix, $\mathit{F}$ is a p-dimensional forces vector, and $\mathit{U}$ is an r-dimensional voltages vector; ${\mathit{K}}_{\mathit{me}}$ is a p-by-r mechanical-electric matrix. The nodal displacements vector of the reflector surface $\nabla \mathit{\delta }$ can be written as

$$\nabla \mathit{\delta }={\mathit{K}}_{\mathit{s}}\left(\mathit{F}-{\mathit{K}}_{\mathit{me}}\mathit{U}\right),$$

(33)

where ${\mathit{K}}_s$ is a q-by-r submatrix of ${{\mathit{K}}_m}^{-1}$ that consists of the rows corresponding to the nodal displacements on the reflector surface in matrix ${{\mathit{K}}_m}^{-1}$. After substituting Equation (33) into Equation (31), the objective function f can be expressed as a function in terms of the voltage vector $\mathit{U}$:

$$f\left(\mathit{U}\right)=\left({\mathit{F}}^T-{\mathit{U}}^T{{\mathit{K}}_{\mathit{me}}}^T\right)\left({{\mathit{K}}_s}^T{\mathit{K}}_s\right)\left(\mathit{F}-{\mathit{K}}_{\mathit{me}}\mathit{U}\right)={\mathit{K}}_s\left(\mathit{F}-{\mathit{K}}_{\mathit{me}}\mathit{U}\right)\left({\mathit{\delta }}^D-{\mathit{\delta }}^I\right)+{\left({\mathit{\delta }}^D-{\mathit{\delta }}^I\right)}^T\left({\mathit{\delta }}^D-{\mathit{\delta }}^I\right).$$

(34)

Equation (34) can be further simplified as

$$\left(\mathit{U}\right)={\mathit{U}}^T\mathit{A}\mathit{U}-{\mathit{b}}^T\mathit{U}+c,$$

(35)

where the parameter matrices are

$$\mathit{A}={{\mathit{K}}_{\mathrm{me}}}^T\left({{\mathit{K}}_s}^T{\mathit{K}}_s\right){\mathit{K}}_{\mathrm{me}},$$

(36)

$${\mathit{b}}^T=2{\left({\mathit{\delta }}^D-{\mathit{\delta }}^I\right)}^T{\mathit{K}}_s{\mathit{K}}_{\mathrm{me}}+2{\mathit{F}}^T\left({{\mathit{K}}_s}^T{\mathit{K}}_s\right){\mathit{K}}_{\mathrm{me}},$$

(37)

$$\mathit{c}=2{\left({\mathit{\delta }}^{\mathit{D}}-{\mathit{\delta }}^{\mathit{I}}\right)}^{\mathit{T}}{\mathit{K}}_s\mathit{F}+{\mathit{F}}^{\mathit{T}}\left({{\mathit{K}}_s}^{\mathit{T}}{\mathit{K}}_s\right)\mathit{F},$$

(38)

$\mathit{A}$ is an r-by-r positive definite matrix; $\mathit{b}$ is an r-dimensional vector; and c is a constant. So, an optimal shape controller can be written as a matrix form

$$\min f\left(\mathit{U}\right)={\mathit{U}}^T\mathit{A}\mathit{U}-{\mathit{b}}^T\mathit{U}+cs.t.{\mathit{U}}_{min}\le \mathit{U}\le {\mathit{U}}_{max},$$

(39)

Equation (39) is in the form of standard quadratic programming in terms of the voltage vector $\mathit{U}$ that contains the upper and lower bounds ${\mathit{U}}_{min}$ and ${\mathit{U}}_{max}$. A fast iteration method called the gradient projection method can be used to directly solve the optimization problem in a standard quadratic form. Therefore, the voltages of the actuators and the residual RMS errors can be calculated.

4. Optimal Placement of PZT Actuators

The arrangement of PZT actuators has a large impact on the control efficiency of an active reflector. A large number of actuators will increase the weight and reduce the reliability of the active reflector. So, the optimal distribution of PZT actuators must be investigated. The arrangement optimization of PZT actuators is a discrete variable optimization problem. The gradients of an objective function with respect to the discrete variables are difficult to obtain. So, gradient-based methods are not suitable for the arrangement optimization of actuators. The random search algorithm is another type of heuristic method, in which the gradient of the objective function is often not required. The Genetic Algorithm (GA) is a very effective and robust heuristic method for searching the optimal solution of non-convex problems without deriving gradients. It mimics the evolutionary behavior of living creatures. It is employed to optimize the placement of actuators and the corresponding voltages applied.

4.1. Basic Strategies of Genetic Algorithm for Actuator Placement Optimization

The Genetic Algorithm uses binary numbers to discretize and encode the optimizing variables. To model an actuator placement optimization problem, a certain number of candidate positions are prepared for the selection of the PZT actuators. Here, the total number of candidate positions is 72. A representation of the candidate positions and its physical meaning are shown in Figure 3, where the bold lines and thin lines represent one actuator placed and not placed at this candidate position, respectively. Thus, a 72-bit binary number λ is used to encode the actuator positions, “0” and “1” indicate no actuator and one actuator at a candidate position, respectively. The voltage applied to each actuator is a continuous variable in the optimization process.

The four basic steps of the Genetic Algorithm are selection, crossover, mutation and reorganization. For this problem, some modifications have been made to improve the iteration efficiency. The Roulette and Elite selection methods are employed for the selection, where Elite selection entails making the best individual in the current generation not participate in the crossover and mutation but directly reserve to the next generation. Then, adaptive crossover and mutation operators [32] are employed to act on the individuals selected. The probability of crossover and mutation can be written as

$$P_c=\left\{\begin{array}{c}{\mathit{P}}_{\mathit{c}1}-\frac{\left({\mathit{P}}_{\mathit{c}1}-{\mathit{P}}_{\mathit{c}2}\right)\left({\mathit{f}}^{\prime }-{\mathit{f}}_{\mathit{a}\mathit{v}\mathit{e}}\right)}{{\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{f}}_{\mathit{a}\mathit{v}\mathit{e}}}\\ {\mathit{P}}_{\mathit{c}1}\end{array}\right.\left\{\begin{array}{c}{f}^{\prime }>f_{ave}\\ f^{\prime }\le f_{ave}\end{array}\right.,$$

(40)

$$P_m=\left\{\begin{array}{c}{P}_{c1}-\frac{\left(P_{m1}-P_{m2}\right)\left(f^{\prime }-f_{ave}\right)}{f_{max}-f_{ave}}\\ P_{m1}\end{array}\right.\left\{\begin{array}{c}f>f_{ave}\\ f\le f_{ave}\end{array}\right.,$$

(41)

where P_c1, P_c2 are the minimum and maximum probabilities of crossover; P_m1 and P_m2 are the minimum and maximum probabilities of mutation, respectively; f is the fitness of an individual; $f^{\prime }$ is the larger one of fitness values in the two cross individuals; $f_{max}$ is the maximum value of fitness in each generation; $f_{ave}$ is the average value of fitness in each generation; and P_c and P_m are the final crossover and mutation probability, respectively.

To ensure the number of actuators is less than the allowed number in the final solutions, an adaptive penalty function is proposed, to be added to the original objective function f to form a new objective function. Therefore, a modified objective function can be derived from Equation (31) and an exterior penalty function. The modified objective function can be written as Equation (42).

$$P_c=\left\{\begin{array}{c}f\left(\mathit{\lambda },\mathit{U}\right)\\ f\left(\mathit{\lambda },\mathit{U}\right)+f^{\mathrm{penalty}}\left(g\right){\left[\mathit{max}\left({‖\mathit{\lambda }‖}_1-N,0\right)\right]}^2\end{array}\right.\begin{array}{c}{‖\mathit{\lambda }‖}_1\le N\\ {‖\mathit{\lambda }‖}_1>N\end{array}.$$

(42)

The dynamic penalty factor can be expressed as Equation (43).

$$f^{\mathrm{penalty}}\left(g\right)=R\left[1+\epsilon -\mathit{sin}\left(\frac{\pi g}{2G}\right)\right].$$

(43)

$\mathit{U}$ is the voltage vector. N is the maximum number of actuators allowed for shape control. R is a scaling parameter that can be chosen according to the original objective function, G is the total number of generations, and g is the number of the current generation. Equation (43) indicates that when the number of actuators does not exceed the allowed value, the modified function value equals the original function value. However, when the number of actuators exceeds the allowed value, the more the number exceeds it, the larger the modified objective function value. $\epsilon$ is another adjustable parameter that causes the value of the penalty function to be greater than zero when the number of allowed actuators is exceeded. The smaller the value of the objective function, the greater the probability that the individual will be selected for crossover and mutation. The iteration will stop when the maximum number of generations is reached or the objective function value does not change significantly.

For the optimization of the actuator placement and the corresponding voltages, both single-objective and multi-objective optimization simulation have been studied. For single-objective optimization, two load cases are studied. In the first case, a displacement load consisting of Zernike polynomials is studied. In the second case, in-orbit thermal loads of a special case are applied to a reflector. For multi-objective optimization, both the RMS error and the energy consumption of actuators are considered as optimizing objectives. They will be described in detail in Section 4.2 and Section 4.3.

4.2. Implementation of Genetic Algorithm for Single-Objective Optimization

• Case 1: Displacement load consists of Zernike polynomials

Zernike polynomials are often used to represent surface errors in optical engineering. They are a series of orthogonal polynomials consisting of power functions and harmonic functions. Lower-order members of the Zernike polynomials represent typical optical wave-front aberrations, such as the power, astigmatism, coma, and spherical aberration [33]. They can be used as a set of orthogonal basis error modes to form the surface errors of reflectors or optic mirrors. The second~eighth normalized Zernike polynomials in a unit hexagonal area are employed to form a reflector surface error. Their expressions are given in Appendix A. The displacement load is written as Equation (44):

$$S\left(\rho ,\theta \right)=\frac{1}{80}\sum _{i=2}^8{Z}_i\left(\rho ,\theta \right).$$

(44)

It is applied to nodes on the reflector surface. An equivalent RMS error of the reflector is considered as an optimizing function to reduce. The mathematical formulation of the optimization can be written as

$$\left\{\begin{array}{c}\min f\left(\mathit{\lambda },\mathit{U}\right)={‖{\mathit{\delta }}^I+\nabla \mathit{\delta }\left(\mathit{\lambda },\mathit{U}\right)-{\mathit{\delta }}^D‖}_2\\ s.t.{\left[\begin{array}{cc}{\mathit{K}}_m& {\mathit{K}}_{me}\\ {\mathit{K}}_{me}^T& {\mathit{K}}_{ee}\end{array}\right]}_{global}\left[\begin{array}{c}\mathit{d}\\ \mathit{U}\end{array}\right]=\left[\begin{array}{c}\mathit{F}\\ \mathit{Q}\end{array}\right]\\ U_{min}\le U_i\le U_{max}\\ {‖\mathit{\lambda }‖}_1\le N\end{array}\right..$$

(45)

In this optimization formulation, $\mathit{\lambda }$ and $\mathit{U}$ are actuator binary placements vector and the corresponding voltages vector, ${\mathit{\delta }}^I$ and ${\mathit{\delta }}^D$ are initial and desired surface nodal position vectors, and $\nabla \mathit{\delta }\left(\mathit{\lambda },\mathit{U}\right)$ is the nodal displacement vector of the reflector surface. U_max and $U_{min}$ are the maximum and minimum allowed voltages. N is the maximum number of allowed actuators. The modified Genetic Algorithm mentioned above is used to place the actuators. An equivalent RMS error is obtained by solving the quadratic programming problem (39). Then, the fitness value is calculated by Equation (42). An iteration flow chart for the single-objective optimization is shown in Figure 4.

• Numerical example

A numerical simulation is provided. The properties of the CFRC laminates and PZT actuators are listed in

Table 1. Materials’ properties.

Materials	Property		Value
Unidirectional carbon fiber layer	Elastic modulus	E1 (GPa)	145
		E2 (GPa)	11
	Poisson’s ratio	v12	0.25
	Shear modulus	G12 (GPa)	5.28
		G13 (GPa)	5.28
		G23 (GPa)	4.23
PZT actuators’ equivalent parameters	Elastic modulus	E1 (GPa)	26.53
	Shear modulus	τzx (GPa)	2.15
	Maximum voltages (V)		150
	Minimum voltages (V)		−100
	Equivalent length (mm)		80
	Poisson’s ratio		0.3
	Piezoelectric constant (pm/V)		360

.

The geometric properties of the CFRC reflector under investigation are shown in

Table 2. Structural parameters of the CFRC reflector.

Structural Parameter	Value
Inscribed circle diameter of the outermost grille	1.0 m
Focal length	2.1 m
Width of the grid	0.1886 m
Height of the grid	0.06 m
Excavation width of grid	0.1 m
Depth of the grid	0.02 m

.

The iteration processes for different numbers of actuators are shown in Figure 5. The abscissa represents the number of iterations, and vertical axis represents the minimum residual RMS error of each generation.

The algorithm quickly converges and the minimum RMS error becomes stable within 200 generations, as shown in Figure 5. The initial RMS error of the reflector is 30.91 µm. The residual RMS errors of 8, 10, 20, 30, 40 and 60 actuators are 11.86, 10.61, 7.52, 6.96, 6.68 and 6.46 µm, respectively. The initial and residual displacement contours for different numbers of actuators are shown in Figure 6. The corresponding placement of the PZT actuators and their voltages are shown in Figure 7. As shown, the residual RMS error decreases as the number of actuators increases. However, the decreasing rate of the RMS error gradually decreases with the increasing of the actuators. This implies that the residual RMS error is stable when the number of actuators reaches a critical value.

Simulations of increasing numbers of actuators are studied. The relationship between the residual RMS error and the allowed number of actuators is shown in Figure 8.

As shown, when the number of actuators is less than 30, the residual RMS error decreases rapidly as the number of actuators increases. After the number of actuators reaches 50, the residual RMS error does not decrease much even if the number of actuators is further increased. This means that the number of actuators has reached saturation for the shape-controlling task.

2.

In-orbit thermal load

Large space antennas orbiting the Earth always suffer from severe thermal effects, including an extremely cold black background, strong solar radiations, and complex heat fluxes of the Earth’s infrared radiation and albedo. Complex thermal effects yield complicated thermal deformation modes that correspond to deteriorated antenna performance owing to the reduction in the reflector surface shape accuracy. However, the optimal placement of actuators based on in-orbit thermal distortions has not been addressed previously. In this case, the placement optimization of actuators is investigated based on the surface errors induced by in-orbit thermal deformation.

To improve the control efficiency under in-orbit thermal effects, a typical day needs to be selected to develop and demonstrate this method. Assuming a reflector is operated in a geostationary orbit, typical dates are the spring equinox, autumn equinox, summer solstice, and winter solstice. During the spring equinox, a cycling reflector is under the most severe thermal effects, especially when flying into and out of the umbra area of Earth. Significant changes in solar radiation lead to large temperature gradients across the reflector [34]. Thus, the corresponding thermal distortion is more severe than any other days. A schematic of the in-orbit thermal condition during the spring equinox is shown in Figure 9.

To maintain the high surface accuracy of the reflector, the arrangement of a limited number of actuators must be optimized to minimize the RMS error during a cycling period. The change in the temperature distribution is assumed to be much slower than the responses of the adjustment process induced by the actuators. Thus, an optimizing objective function that is an integration of the discrete RMS errors in a time span is proposed. It can be expressed as

$$F\left(\mathit{\lambda },\mathit{U}\left(t\right)\right)={\int }_{t_s}^{t_e}{‖\left[{\mathit{\delta }}^I-{\mathit{\delta }}^D+\nabla {\mathit{\delta }}_v\left(\mathit{\lambda },\mathit{U}\left(t\right)\right)+\nabla {\mathit{\delta }}_t\left(\mathit{T}\left(t\right)\right)\right]‖}_2,$$

(46)

where $\mathit{U}\left(t\right)$ denotes the voltage vector that varies with time, $\mathit{T}\left(t\right)$ is the nodal temperature vector, and ts and te are the starting and ending times of the circle, respectively. $\nabla {\mathit{\delta }}_v$ and $\nabla {\mathit{\delta }}_t$ are the nodal displacement vectors induced by voltages applied and temperature distribution of the reflector at time point t. By dividing the integration interval with respect to time t into n equal sections with a midpoint and using the Simpson integration formula in each section, the objective function can be expressed as

$$F\left(\mathit{\lambda },\mathit{U}\left(t\right)\right)={\int }_{t_s}^{t_e}g\left(t\right)dt=\frac{\mathsf{\Delta }t}{6}{\sum }_{i=1}^n\left[g\left(t_i\right)+4g\left(\frac{t_i+t_{i+1}}{2}\right)+g\left(t_{i+1}\right)\right].$$

(47)

The equivalent RMS error at time t_i can be expressed as Equation (48).

$$g\left(t_i\right)={‖{\mathit{\delta }}^I-{\mathit{\delta }}^D+\nabla {\mathit{\delta }}_v\left(\mathit{\lambda },\mathit{U}\left(t_i\right)\right)+\nabla {\mathit{\delta }}_t\left(\mathit{T}\left(t_i\right)\right)‖}_2.$$

(48)

$\mathsf{\Delta }t=t_{i+1}-t_i$ is a small-time span. Minimizing this objective function can effectively improve the control efficiency of the orbital process. An optimization formulation can be written as

$$\{\begin{array}{c}\min F(\mathit{\lambda },\mathit{U})=\frac{\Delta t}{6}{\displaystyle \sum _{i=1}^n\left[g(t_i)+4g(\frac{t_i+t_{i+1}}{2})+g(t_{i+1})\right]}\\ s.t.g(t_i)={\Vert {\mathit{\delta }}^I-{\mathit{\delta }}^D+\nabla {\mathit{\delta }}_{\nu }(\mathit{\lambda },\mathit{U}(t_i))+\nabla {\mathit{\delta }}_t(\mathit{T}(t_i))\Vert }_2\\ {\mathit{K}}_{mm}\nabla {\mathit{\delta }}_{\nu }(\mathit{\lambda },\mathit{U}(t_i))+{\mathit{K}}_{mp}\mathit{U}(t_i)=\mathit{F}\\ \max \left|\mathit{U}(t_i)\right|\le U_{max}\end{array}.$$

(49)

NX TMG (originally integrated in I-DEAS) has been used to construct an in-orbit thermal model for the reflector’s time-dependent temperature distributions calculation on the day of the spring equinox. The distribution of the reflector nodal temperature is calculated at each time point. So, the corresponding thermal deformations at each time point $\nabla {\mathit{\delta }}_t\left(\mathit{T}\left(t_i\right)\right)$ can thus be obtained. Then, they can be substituted to Equation (48) for calculating the voltages and equivalent RMS error $g\left(t_i\right)$.

An iterative process for the above optimization problem can be described as follows. The GA is applied to determine the positions of the actuators. Then, $\mathit{U}\left(t_i\right)$ and $g\left(t_i\right)$ at each time point are derived by solving the constrained quadratic programming problem (39). Subsequently, all $g\left(t_i\right)$ shall be substituted into Equation (47) to calculate the objective function $F\left(\mathit{\lambda },\mathit{U}\right)$. The penalty mechanism Equation (42) will also be used to calculate the fitness of the GA. The iteration of the GA will stop if the fitness value does not change significantly.

• Numerical example

The material thermal properties used in this simulation are shown in

Table 3. Thermal properties.

Property	Value
Materials	Single-fiber layer prepreg		PZT actuator (average)
Thermal Conductivity (W/m K)	18.6		37.2
Specific Heat (J/kg·K)	850		460
CTE (/°C)	−1.01 × 10−6	3.81 × 10−6	8.00 × 10−6
Density (kg/m3)	1510		5060

.

Assume that the initial position of the reflecting surface coincides with the ideal position, which means ${\mathit{\delta }}^I={\mathit{\delta }}^D$. The surface RMS errors induced by the in-orbit thermal deformation during the day without any control are shown in Figure 10. The shaded area in Figure 10 represents the integral of the RMS error over time, which is used as the objective function for subsequent optimization. The small shaded area represents the integration from the RMS error y₁ corresponding to time t₁ to the RMS error y₂ corresponding to time t₂.

The area of the shadow represents the objective function value that shall be minimized. When the iteration stops, the optimized placement of the actuators is obtained. The iteration curve of the optimization corresponding to N = 35 is shown in Figure 11.

The iteration converges quickly, and the value of the objective function does not change when the number of generations is larger than 550. An actuator placement denoted by the binary vector $\mathit{\lambda }$ can be derived from the optimization. The optimization simulations are also studied when N equals 18, 48 and 54, respectively. Placement of the PZT actuators and the corresponding values of the objective functions are shown in Figure 12. Thick blue lines indicate where actuators are distributed, while thin lines indicate where there are no actuators. As shown in the figure, the placement of the actuators is symmetrical about a certain oblique axis with different quantity limitations of actuators. This is because the Sun is in the equatorial plane during the spring equinox. The external thermal loads on the reflector are symmetrical about a certain oblique axis during the circle.

The optimized values of the objective function are 291.67, 253.03, 243.84, and 241.95 µm·h corresponding to the conditions of 18, 35, 48, and 54 actuators, respectively. The value of the objective function decreases with the increasing of the actuators. The curve of the objective function value with respect to the number of actuators is shown in Figure 13a.

As shown in the figure, the decreasing rate of the objective value reduces with the increasing of the actuators. The curves of the residual RMS error corresponding to different numbers of actuators are shown in Figure 13b. It can be seen that the time and areas formed by the RMS errors become smaller as the number of actuators increases. However, the reduction becomes insignificant when the number of actuators is larger than 48. This phenomenon indicates that the actuators employed should not exceed a certain number.

4.3. Implementation of Genetic Algorithm for Multi-Objective Optimization

Besides the surface accuracy and weight, for adaptive space reflectors, the energy consumption of the actuators is also an important aspect. Therefore, the residual RMS errors of the reflector and energy consumption shall be considered as two optimizing objectives. The optimal configuration of the actuators and the corresponding voltages can be obtained by solving a multi-objective optimization problem.

A weighted summation of the objective functions method was commonly used in previous studies [35,36] for solving multi-objective optimization problems. This method is easy to implement, but the solution obtained is unique and may not meet engineering requirements and the weight coefficients are not easy to determine for some nonlinear programing problems. So, a novel multi-objective optimization approach for reflector shape control is investigated. The maximum allowed voltage and the maximum number of actuators are considered as constraints. Mathematical formulation of a multi-objective optimization can be written as

$$\{\begin{array}{c}\min f_1(\mathit{\lambda },\mathit{U})=\Vert {\mathit{\delta }}^I+\left\{\nabla \mathit{\delta }(\mathit{\lambda },\mathit{U})\right\}-\left\{{\mathit{\delta }}^D\right\}{\Vert }_2\\ \min f_2(\mathit{U})=\frac{{\Vert \left\{\mathit{U}\right\}\Vert }_2^2}{10000}\\ s.t.{\mathit{K}}_{mm}\nabla \mathit{\delta }+{\mathit{K}}_{mp}\mathit{U}=\mathit{F}\\ \max \left|\mathit{U}\right|\le U_{\max }\\ {\Vert \mathit{\lambda }\Vert }_1\le N\end{array},$$

(50)

where $f_1\left(\mathit{\lambda },\mathit{U}\right)$ and $f_2\left(\mathit{U}\right)$ are the equivalent residual RMS error and consuming energy of the actuators. The resistance of all the actuators is set to 10,000 ohms. A modified multi-objective Genetic Algorithm (MOGA), based on non-dominated individuals, is developed to derive the Pareto frontier. The Pareto frontier is defined as a set of all the non-dominated solutions in a feasible domain [37].

First, non-dominated individuals in each generation are chosen according to the values of the objective functions. Subsequently, their priority levels are set to be 1. Among the non-dominated solutions in each generation, the individuals whose number of actuators do not exceed N will be stored in a collection for further comparison. For each dominated individual, the number of individuals that dominate it is denoted by K. Subsequently, a priority function is proposed herein using the value of K and a penalty function, expressed as:

$$f_0^e\left(X_i\right)=\left\{\begin{array}{cc}\frac{1}{1+f^{penalty}\left(g\right)\cdot \mathit{max}\left({‖\left\{\lambda \right\}‖}_1-N,0\right)}& X_i\in P_j\\ \frac{1}{K_i^j+1+f^{penalty}\left(g\right)\cdot \mathit{max}\left({‖\left\{\lambda \right\}‖}_1-N,0\right)}& X_i\notin P_j\end{array}\right.,$$

(51)

where $K_i^j$ denotes a priority level of the i-th individual in the j-th generation, P_j denotes a non-dominated set of the j-th generation and $f^{penalty}\left(g\right)$ is a penalty factor as shown in Equation (43). N is the maximum allowed number of actuators. To ensure the diversity of the population and that more non-dominated solutions can be derived, a sharing function is used to obtain the fitness value of each individual. It can be denoted as

$$g\left(X_i,X_j\right)\left|i\ne j\right.=\left\{\begin{array}{cc}0& \sigma \left(X_i,X_j\right)>{\sigma }_{sharing}\\ 1-\frac{\sigma \left(X_i,X_j\right)}{{\sigma }_{sharing}}& \sigma \left(X_i,X_j\right)\le {\sigma }_{sharing}\end{array}\right.,$$

(52)

X_i and X_j are different individuals of each generation; $\sigma \left(X_i,X_j\right)$ is the Euclidean distance between X_i and X_j, and ${\sigma }_{sharing}$ is a critical value of the Euclidean distance. A fitness function of each individual can be expressed as

$$f^{fit}\left(X_i\right)=\frac{m\times f_0^e\left(X_i\right)}{{\sum }_{j=1,j\ne i}^{m}g\left(X_i,X_j\right)},$$

(53)

Here, m is the number of individuals in one generation. The adaptive crossover and mutation operators Equations (40) and (41) are also used in this algorithm. A flowchart of the algorithm is shown in Figure 14. A non-dominated solution filter is designed to reserve a certain number of non-dominated solutions. The reserved non-dominated solutions are chosen to yield 30–40% of the population. Based on the distance between the solutions, a filter is designed to reserve a certain number of solutions. It can be expressed as

$$\left\{\begin{array}{l}\mathit{min}\left(\sqrt{{\left(f_1\left(X_i\right)-f_1\left(X_j\right)\right)}^2+{\left(f_2\left(X_i\right)-f_2\left(X_j\right)\right)}^2}\right)\left|j=\mathrm{1,2},\cdots k\&j\ne i\right.>\epsilon ,reserve{X}_i\\ else,delete{X}_i\end{array}\right.$$

(54)

Here, $\epsilon$ is an adjustable value of the minimum distance between $f_2\left(X_i\right)$ and $f_2\left(X_j\right)$. The filter can remove crowded individuals and scatter reserved solutions in the solution domain. An optimization flow chart is shown in Figure 14.

• Numerical examples

The loads in case 1 are also used in this case. The results of the method for the weighted summation of objectives are obtained to compare with those of the modified algorithm. A normalized weighted-sum function of the objectives can be denoted as

$$f\left(X\right)={\alpha }_1\overline{f_1\left(X\right)}+{\alpha }_2\overline{f_2\left(X\right)},$$

(55)

Here, ${\alpha }_1$ and ${\alpha }_2$ are coefficients of weight; $\overline{f_1\left(X\right)}$ is the normalized RMS error and $\overline{f_2\left(X\right)}$ is the normalized energy consumption, as defined by Equation (42). Setting ${\alpha }_1={\alpha }_2=0.5$ implies that the weights of the two objective functions are equal. The allowed number of actuators is first set to be 72. The Pareto front of the non-dominated solutions of the two methods can be obtained.

After 1000 iterations, the Pareto fronts of the non-dominated solutions derived by the two methods are shown in Figure 15a–c, which show 40 non-dominated solutions reserved through the filter using the modified method. As shown in (a), more solutions are biased toward the objective of energy consumption and are not uniformly distributed. From (b) and (c), the number of non-dominated solutions of the modified method is more than that of the weighted-sum objective method and the solutions are distributed more widely.

Table 4. Single optimal solution and 15 non-dominated solutions by the modified GA.

Value of Objectives	Single RMS Error Optimization (µm)
f1	4.4
f2	45.4
Value of Objectives	Residual RMS Error (µm) and Energy Consumption
f1	4.9	5.2	5.4	5.8	5.4	6.2	7.0	7.9
f2	9.1	7.8	6.9	6.0	7.3	5.5	4.7	3.8
Value of Objectives	Residual RMS Error (µm) and Energy Consumption
f1	8.5	10.6	12.7	15.0	17.8	21.7	28.0
f2	3.4	2.7	2.1	1.7	1.3	0.88	0.42

lists the solutions of a single objective optimization and 15 non-dominated solutions using the modified GA.

As shown in Table 4, the energy consumption reduces as the residual RMS error increases. This implies that the two objectives are contradictory to each other. When the residual RMS error is 5.2 µm, the corresponding energy consumption is 7.8 in the Pareto set. Compared with the result of the single RMS error optimization, the residual RMS error is 4.4 µm, but the energy consumption is 45.4. The residual RMS error of the multi-objective optimization increased by 15.4% but the energy consumption reduced by 82.8%. The actuator voltages of these two solutions are shown in Figure 16.

As shown in Figure 16, the amplitude of the voltage distribution of the solution in the multi-objective optimal control (the red line) is much smaller than that of the single-objective optimal control (the black line). The maximum voltage in the multi-objective control is 79.8 V. However, nine actuators reached the maximum voltage amplitude of 150 V in the single-objective optimal control. The voltage range of the multi-objective optimal control is smaller than that of the single-objective optimal control. The allowed number of actuators significantly affects the multi-objective optimal control. Fewer actuators imply less alternative displacement patterns in the control process. When the number of allowed actuators is 30, 50, and 72, respectively, the corresponding Pareto frontiers are shown in Figure 17.

As shown in Figure 17, the values of the Pareto frontier are smaller when more actuators are allowed. The value of the Pareto frontier with 50 actuators is much smaller than that with 30 actuators. However, the difference between that of 50 actuators and 72 actuators is small. When the number of actuators is greater than a certain number, the Pareto fronts do not vary significantly.

4.4. Discussion

In this section, the placement of the actuators has been optimized to improve the control efficiency of the active reflector. For the single-objective optimization, the load consists of orthogonal Zernike polynomials and in-orbit thermal loads have been applied to the reflector, respectively. The optimal arrangement is derived to minimize the residual RMS error by the proposed method based on the GA. The control efficiency is improved by increasing the number of actuators, but it will reach to an approximate saturation condition when reaching a certain number of actuators. For the multi-objective optimization, the RMS error and the energy consumption are considered as the optimizing objectives. The Pareto frontiers and corresponding solutions are derived by the proposed method. The proposed method can contain more and widely distributed non-dominated solutions for engineering purposes compared with the traditional method. The obtained non-dominated solutions can significantly reduce the energy consumed by slightly increasing residual the RMS error.

5. Conclusions

In this paper, a novel shape control method for carbon fiber-reinforced composite (CFRC) reflectors with piezoelectric lead zirconate titanate (PZT) actuators is proposed. First, a finite element model considering higher transverse shear deformation with independent voltage degrees of freedom is formulated. Based on the finite element model, an optimal shape controller is developed for minimizing the root mean square (RMS) error. Then, the placement of the actuators has been optimized by using a single-objective and a multi-objective optimization method. For the single-objective optimization, an optimization method that combines the Genetic Algorithm with the gradient project method is proposed to minimize the residual RMS error. Two numerical examples are given to verify the feasibility of the proposed method. For the multi-objective optimization, both the residual RMS error and the energy consumption are considered as optimizing objectives. The proposed method can contain more widely distributed non-dominated solutions compared with the method of the weighted sum of the objective functions. The obtained non-dominated solutions can significantly reduce the energy consumption by slightly increasing the RMS error. Numerical examples are provided to verify the feasibility of the proposed optimal control method.