An Optimization Method for Design Solutions to Active Reflective Surface Control Systems Based on Axiomatic Design and Multi-Criteria Decision Making

Abstract

The design of an Active Reflective Surface Control System (ARCS) is a complex engineering task involving multidimensional and multi-criteria constraints. This paper proposes a novel methodological approach for ARCS design and optimization by integrating Axiomatic Design (AD) and Multi-Criteria Decision Making (MCDM) techniques. Initially, a structured design plan is formulated within the axiomatic design framework. Subsequently, four MCDM methods—Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), Entropy Weight Method (EWM), Multi-Criteria Optimization and Compromise Solution (VIKOR), and the integrated TOPSIS–Grey Relational Analysis (GRA) approach—are used to evaluate and compare the alternative solutions. Additionally, fuzzy information axioms are used to calculate the total information content for each alternative to identify the optimal design. A case study is conducted, selecting the optimal actuator for a 5 m diameter scaled model of the Five-hundred-meter Aperture Spherical radio Telescope (FAST), followed by digital control experiments on the chosen actuator. Based on the optimal design scheme, an ARCS prototype is constructed, which accelerates project completion and substantially reduces trial-and-error costs.

1. Introduction

Among the radio telescopes worldwide that currently employ active reflector surface control systems, typical examples include the Green Bank Telescope (GBT) in the United States, the Shanghai Tianma (TM), and the Guizhou FAST. Additionally, the planned Xinjiang QTT (QiTai radio Telescope) with a 110 m aperture, slated for completion in 2028, will also utilize an active reflector surface control system. The active reflector surface control system is primarily comprised of three elements: sensors, controllers, and actuators. Its function is to monitor and adjust the shape and position of the reflector in real time. The performance of the reflector in forming a parabolic surface is of paramount importance to the overall functionality of the telescope. Consequently, the ARCS is indispensable for guaranteeing that the system operates correctly and enables precise observation. The conflicting requirements and component dependencies that arise during the design of active reflector surface control systems have long been a research focus for scholars both domestically and internationally. Wang [1] realized reflector deformation correction by designing an actuator submodule. Zhang [2] designed a reflector adjustment scheme to improve the stability and reliability of structural equipment. Wang [3] established the active control model of the reflector panel using the idea of discretization, which provides a great reference value for the active control of the main reflector of the FAST radio telescope. One research team studying complex networks and visualizations has realized the analysis of ring beam structures, digital models of cable networks [4], and digital twin models of reflector units [5]. Zhang [6] addresses the complex design challenges of multi-objective and multi-constraint active reflective surface control systems using motor selection as a case study. Through the application of a monotonic co-design methodology, the control problem is embedded within an integrated design framework that encompasses the motor, actuator, and reflective surface panels. This approach effectively transforms the co-design challenge into a multi-objective optimization problem, allowing for a more comprehensive solution to the control requirements in active reflective systems. The designs mentioned above primarily emphasize component and functional enhancements, with limited research on the overarching theoretical framework of the design. Consequently, it is essential to develop a comprehensive, scientific, and integrated system approach to coordinate the output of each component for the completion of the overall system design. The research proposed in this study is based on this consideration.

The AD is a design theory based on the mathematical principles proposed by Suh [7]. This method is used for designing large or complex systems [8,9]. The application of the AD principle allows for both selecting the optimal option from a set of criteria and identifying the most suitable alternative. For example, Zhou [10] used AD and design structure matrix (DSM) as the methodological guidance and theoretical framework to realize the modular design scheme of agricultural micro-tiller. Yang [11] used axiomatic design to evaluate product design alternatives based on the information content of the product design decision matrix (PDDM), and Liu [12] proposed a multicriteria decision-making method based on SF-AD to evaluate HMI alternatives. Utilizing axiomatic design, this study examines the functional requirements of the ARCS, identifies the key structural design parameters, and completes the overall scheme development. Subsequently, the principal structural design parameters identified during the overall design phase, such as surface configuration and support structure, are subjected to further elaboration. Thereafter, the design criteria and methods for each of these parameters are defined in turn. This process results in a comprehensive set of design methods and systems tailored to the ARCS. The system-level axiomatic design breaks down the complex original system into a hierarchical structure, constructs a design matrix, and derives a solution. This approach enhances the systematic and scientific aspects of the design process, incorporates hierarchical structuring, and provides theoretical and methodological support for the design of active reflector surface-control systems, ultimately leading to improved design quality.

The multi-criteria decision-making (MCDM) method can systematically consider and weigh these criteria. The MCDM is a decision analysis method that deals with multiple criteria and schemes. Saleh [13] used the MCDM to evaluate suppliers of medical equipment procurement. Lu [14] proposed a fuzzy group MCDM method for the comprehensive performance evaluation of energy storage systems in various application scenarios. Sánchez [15] proposed an MCDM method to classify dangerous NEOs. Zebra [16] utilized the TOPSIS method to rank future renewable energy options, assisting policymakers in selecting the optimal energy alternatives for rural electrification. Similarly, Alqahtani [17] developed an integrated model using two MCDM techniques, further enhancing decision-making frameworks in energy planning. In this study, the alternatives were evaluated according to the design specifications and scope using the MCDM method. At present, there is no research on the application of AD and MCDM in ARCS.

In view of this, this study applies AD and MCDM to the design of an ARCS, which can not only solve the problem of how to reasonably construct each component to ensure functional independence but also independently evaluate each scheme, eliminate the process of pairwise comparison and priority allocation of schemes, and improve decision-making efficiency. This combination ensures that the system not only meets the design axioms in theory but also has excellent comprehensive performance in practical applications.

2. Methods

2.1. Principles of Axiomatic Design

Axiomatic Design (AD) tackles the complex relationship between a system’s functions and its structure by developing mathematical models and design equations involving Customer Attributes (CAs), Functional Requirements (FRs), Design Parameters (DPs), and Process Variables (PVs) [18]. This methodology generates solutions by mapping and decomposing across various domains. The design process follows a zigzag iterative approach, with data mapping between the functional, structural, and process domains [19]. As shown in Figure 1, functional requirements (FRs) are first defined, followed by mapping the design solution to the structural domain, then iterating back to the functional domain for adjustments. This cycle repeats until all functional requirements are met.

Axiomatic design is governed by two key principles [7]: the independence axiom and the information axiom. The independence axiom asserts that functional requirements should not be interdependent. The information axiom posits that among the designs meeting the independence criterion, the one with the least information content is considered the best solution.

2.2. Multi-Criteria Decision Process

The alternatives were assessed according to the design specifications and scope, and those found to be unsuitable were eliminated using the established filtering criteria. Following this, the total information for the chosen alternatives was computed to improve the precision of the design scheme selection process. Both qualitative and quantitative data related to the alternative options were evaluated using TOPSIS-Gray correlation analysis in conjunction with the fuzzy information axiom.

The TOPSIS method indicates the extent of match between a scheme and the demand by analyzing the distance in their positions. On the other hand, the gray correlation method assesses the correlation of demand based on shape similarity [20,21]. The calculation steps are outlined below [22,23,24]:

Step 1. Normalize the raw data:

$$X_{ij}={\displaystyle \frac{x_{ij}-min\left(x_{ij}\right)}{max\left(x_{ij}\right)-min\left(x_{ij}\right)}}$$

(1)

Step 2. Solve the weighted normalization matrix:

$$Z={\left(z_{ij}\right)}_{m\times n}$$

(2)

In the formula, $z_j^{+}=\underset{i}{max}{z}_{ij}={\omega }_j,z_j^{-}=\underset{i}{min}{\mathcal{Z}}_{ij}=0,i=1,2,\dots m,j=1,2,\dots n$

Step 3. Evaluate the various characteristics of active reflective surface control systems. Calculate the positive and negative ideal solutions of the weighted normalized matrix Z:

$$Z^{+}=(z_1^{+},z_2^{+},\cdots z_n^{+})=\omega$$

(3)

$$Z^{-}=(z_1^{-},z_2^{-},\cdots z_n^{-})=0$$

(4)

In the formula, $z_j^{+}=\underset{i}{max}{z}_{ij}={\omega }_j,z_j^{-}=\underset{i}{min}{z}_{ij}=0,i=1,2,\dots m,j=1,2,\dots n$

Step 4. Calculate the Euclid distances $d_i^{+}$ and $d_i^{-}$ from the alternatives to the positive and negative ideal solutions:

$$d_i^{+}=∥z_i-A^{+}∥=\sqrt{\sum _{j=1}^n{\left(z_{ij}-z_j^{+}\right)}^2}$$

(5)

$$d_i^{-}=∥z_i-A^{-}∥=\sqrt{\sum _{j=1}^n{\left(z_{ij}-z_j^{-}\right)}^2}$$

(6)

Step 5. Calculate the gray relational analysis coefficient matrix of alternative proposals and the positive and negative ideal solutions:

$$R^{+}=\left(r_i{j}^{+}\right)(m\times n)$$

(7)

$$R^{-}=\left(r_i{j}^{-}\right)(m\times n)$$

(8)

$$r_{ij}^{+}={\displaystyle \frac{\underset{i}{min}\underset{j}{min}|z_j^{+}-z_{ij}|+\epsilon \underset{i}{max}\underset{j}{max}|z_j^{+}-z_{ij}|}{|z_j^{+}-z_{ij}|+\epsilon \underset{i}{max}\underset{j}{max}|z_j^{+}-z_{ij}|}}={\displaystyle \frac{\epsilon {\omega }_j}{{\omega }_j-z_{ij}+\epsilon {\omega }_j}}$$

(9)

$$r_{ij}^{-}={\displaystyle \frac{\underset{i}{min}\underset{j}{min}|z_j^{-}-z_{ij}|+\epsilon \underset{i}{max}\underset{j}{max}|z_j^{-}-z_{ij}|}{|z_j^{-}-z_{ij}|+\epsilon \underset{i}{max}\underset{j}{max}|z_j^{-}-z_{ij}|}}={\displaystyle \frac{\epsilon {\omega }_j}{z_{ij}+\epsilon {\omega }_j}}$$

(10)

In this formula, the resolution is usually taken as $\epsilon$ = 0.5.

Step 6. Calculate the gray relational analysis degree of each machinery and the positive and negative ideal solutions $r^{+},r^{-}:$

$$r_i^{+}={\displaystyle \frac{1}{n}}\sum _{j=1}^n{r}_{ij}^{+}$$

(11)

$$r_i^{-}={\displaystyle \frac{1}{n}}\sum _{j=1}^n{r}_{ij}^{-}$$

(12)

Step 7. Perform the dimensionless processing of Euclidean distances $d_i^{+},d_i^{-}$ and correlation degrees $r^{+},r^{-},$ respectively:

$$D_i^{+}={\displaystyle \frac{d_i^{+}}{\underset{i}{max}{d}_i^{+}}}$$

(13)

$$D_i^{-}={\displaystyle \frac{d_i^{-}}{\underset{i}{max}{d}_i^{-}}}$$

(14)

$$R_i^{+}={\displaystyle \frac{r_i^{+}}{\underset{i}{max}{r}_i^{+}}}$$

(15)

$$R_i^{-}={\displaystyle \frac{r_i^{-}}{\underset{i}{max}{r}_i^{-}}}$$

(16)

Step 8. Combine the Euclidean distances and gray relational degrees after non-dimensionalization:

$$T_i^{+}={\displaystyle \frac{D_i^{-}}{D_i^{-}+D_i^{+}}}$$

(17)

$$S_i^{+}={\displaystyle \frac{R_i^{-}}{R_i^{-}+R_i^{+}}}$$

(18)

Step 9. Calculate the relative closeness:

$$Y_i^{+}={\nu }_1{T}_i^{+}+{\nu }_2{S}_i^{+}$$

(19)

In the formula, ${\nu }_1+{\nu }_2=1$

Step 10. The solutions are ranked based on their corresponding values of $Y_i^{+}$. A higher $Y_i^{+}$ value indicates a more favorable perception of the solution, implying that it better meets the desired criteria. Conversely, a lower $Y_i^{+}$ value suggests that the solution is less satisfactory and does not adequately fulfill the requirements.

The fuzzy information axiom involves conducting a qualitative analysis [25,26,27]. First, a membership function is defined, with its type being specified by the intervals ‘0’, ‘0.5’, and ‘1’. The fuzzy evaluation index is expressed as a real number within the interval from 0 to 1, which indicates the degree of membership. When the membership degree changes under varying conditions, it can be modeled using a function—this is referred to as the membership function. Choosing the right membership function is essential for accurately capturing fuzzy concepts. For linear distributions, a triangular or trapezoidal function is generally applied. In contrast, for nonlinear distributions, a nonlinear membership function, such as one based on a normal or Cauchy distribution, is selected. After that, the design and system ranges are established using fuzzy numbers, and the probability of meeting the design goal corresponds to the area of the overlapping range (Figure 2). Finally, the information content is calculated using Equation (20), which quantifies the amount of information.

$$I_{\mathit{definiteness}}={log}_2{\displaystyle \frac{Fuzzyareaofsystemrange}{Commonrange}}$$

(20)

In summary, this paper proposes a design process for an active reflective surface control system based on axiomatic design and multi-criteria decision making, applying hierarchical design, fuzzy information axiomatics, and TOPSIS-gray correlation analysis (Figure 3).

3. Axiomatic Design of Systems

3.1. Structural Hierarchy Design Process

The feasibility of the proposed design method is evaluated through the application of the AD and MCDM approaches to the control module of the active reflecting surface in the 500 m aperture spherical radio telescope. This research was conducted on the digital twin experimental platform of the Joint Laboratory for Space Debris Monitoring and LEO Satellite Networking Facility of Collaboration. The active reflecting surface features a 5 m basic aperture and a 3 m spherical radius. The objective is to develop a digital control system that is capable of actively deforming the reflecting surface through precise actuator manipulation and real-time monitoring of the surface status.

The system should minimize weight, size, power consumption, and cost while achieving its functional goals, addressing the SWaP-C (Size, Weight, Power, and Cost) equation. The functional structure derived from stepwise decomposition is shown in Figure 4.

The functional requirements and system constraints are as follows:

FR0 = Realization of active reflective surface operation mechanism

FR1 = Reflection range

FR2 = Active deformation

FR3 = Monitoring surface shape

C0 = The size is 0.5 times the same power system.

C1 = The cost is not higher than the same power active reflector system.

C2 = Less power consumption

3.1.1. The First Layer of Analysis

The surface shape design determines the reflection range; active deformation is realized by control, and the sensing unit provides feedback to monitor the surface shape. The following DPs responded to the FRs listed above:

DP0 = Active reflector system

DP1 = Surface design

DP2 = Control surface shape

DP3 = Sensing unit

The following is the first-level design equation.

$$\left\{\begin{array}{c}\mathrm{FR}1\\ \mathrm{FR}2\\ \mathrm{FR}3\end{array}\right\}=\left[\begin{array}{ccc}X& O& O\\ X& X& O\\ O& X& X\end{array}\right]\left\{\begin{array}{c}\mathrm{DP}1\\ \mathrm{DP}2\\ \mathrm{DP}3\end{array}\right\}$$

(21)

3.1.2. The Second Layer of Analysis

The surface shape design must meet the requirements of reflection range and accuracy, adopt high-precision partitions, and reasonably connect each unit panel. The functional requirement FR1, as defined above, may be decomposed with DP1 in mind as:

FR1.1 = Reflection range

FR1.2 = Meet the accuracy requirements

FR1.3 = Connection unit panel

FR1.4 = Face shape partition

The surface area was affected by the minimum unit panel, division design, node connection mode, and numbering. The corresponding DPs can be expressed as follows:

DP1.1 = Triangle

DP1.2 = Unit panel division

DP1.3 = Node connection mode

DP1.4 = Numbering

The design matrix for the above set of FRs and DPs is as follows:

$$\left\{\begin{array}{c}\mathrm{FR}1.1\\ \mathrm{FR}1.2\\ \mathrm{FR}1.3\\ \mathrm{FR}1.4\end{array}\right\}=\left[\begin{array}{cccc}X& O& O& O\\ X& X& O& O\\ O& O& X& O\\ X& O& O& X\end{array}\right]\left\{\begin{array}{c}\mathrm{DP}1.1\\ \mathrm{DP}1.2\\ \mathrm{DP}1.3\\ \mathrm{DP}1.4\end{array}\right\}$$

(22)

DP2 realizes the radial motion of panel nodes along the spherical surface by deformation. The functional requirement FR1 as defined above may be decomposed with DP2 in mind as:

FR2.1 = Support panel

FR2.2 = Deformation capacity

FR2.3 = Drive unit

FR2.4 = Control module

It is essential that the support structure provides adequate support for the driving device, sensing unit, and panel assembly. The actuator, driven by the motor, enables deformation, while the main control chip oversees the FR2.4 functionality. The corresponding DPs may be stated as follows:

DP2.1 = Supporting surface

DP2.2 = Actuators

DP2.3 = Motor control

DP2.4 = Main control chip

The design matrix for the above set of FRs and DPs is as follows:

$$\left\{\begin{array}{c}\mathrm{FR}2.1\\ \mathrm{FR}2.2\\ \mathrm{FR}2.3\\ \mathrm{FR}2.4\end{array}\right\}=\left[\begin{array}{cccc}X& O& O& O\\ X& X& O& O\\ O& X& X& O\\ O& O& X& X\end{array}\right]\left\{\begin{array}{c}\mathrm{DP}2.1\\ \mathrm{DP}2.2\\ \mathrm{DP}2.3\\ \mathrm{DP}2.4\end{array}\right\}$$

(23)

The sensing unit needs to feedback sensing data, surface data, and circuit information. The functional requirement FR3, as defined above, may be decomposed with DP3 in mind as follows:

FR3.1 = Measurement and sensing

FR3.2 = Feedback surface data

FR3.3 = Feedback circuit information

FR3.1 is realized by the laser ranging sensor. FR3.2 surface shape data are calculated for motor and sensor data in the driving unit, and a circuit diagnosis module is set up to realize the C2 fast response. The corresponding DPs may be stated as follows: DP3.1 = Laser ranging

DP3.2 = Motor feedback data

DP3.3 = Circuit diagnosis module

The design matrix for the above set of FRs and DPs is as follows:

$$\left\{\begin{array}{c}\mathrm{FR}3.1\\ \mathrm{FR}3.2\\ \mathrm{FR}3.3\end{array}\right\}=\left[\begin{array}{ccc}X& O& O\\ O& X& O\\ O& O& X\end{array}\right]\left\{\begin{array}{c}\mathrm{DP}3.1\\ \mathrm{DP}3.2\\ \mathrm{DP}3.3\end{array}\right\}$$

(24)

3.1.3. The Third Layer of Analysis

DP1.3 needs to realize traction, connection, and fixation functions. The functional requirement FR1.3, as defined above, may be decomposed with DP1.3 in mind as follows:

FR1.3.1 = Traction

FR1.3.2 = Connection

FR1.3.3 = Fixation

The steel cable is affixed to the screw via the connection terminal and attached to the connecting piece using a bolt structure. Furthermore, the unit panel is directly secured to the connecting piece through the same bolt mechanism. The corresponding DPs may be stated as follows:

DP1.3.1 = Steel wire rope

DP1.3.2 = Node connection piece

DP1.3.3 = Bolt

The design matrix for the above set of FRs and DPs is as follows:

$$\left\{\begin{array}{c}\mathrm{FR}1.3.1\\ \mathrm{FR}1.3.2\\ \mathrm{FR}1.3.3\end{array}\right\}=\left[\begin{array}{ccc}X& O& O\\ O& X& O\\ X& X& X\end{array}\right]\left\{\begin{array}{c}\mathrm{DP}1.3.1\\ \mathrm{DP}1.3.2\\ \mathrm{DP}1.3.3\end{array}\right\}$$

(25)

DP2.1 needs to realize the function of supporting the sphere and fixing. The functional requirement FR2.1, as defined above, may be decomposed with DP2.1 in mind as follows:

FR2.1.1 = Carrying objects

FR2.1.2 = Fixation

The supporting frame forms a supporting space-bearing object, and the fixed base can prevent the bottom of the supporting spherical structure from tilting. The corresponding DPs may be stated as follows:

DP2.1.1 = Bearing frame

DP2.1.2 = Fixed seat

The design matrix for the above set of FRs and DPs is as follows:

$$\left\{\begin{array}{c}\mathrm{FR}2.1.1\\ \mathrm{FR}2.2.2\end{array}\right\}=\left[\begin{array}{cc}X& O\\ O& X\end{array}\right]\left\{\begin{array}{c}\mathrm{DP}2.1.1\\ \mathrm{DP}2.2.2\end{array}\right\}$$

(26)

In order to enhance the smoothness of reflector deformation and optimize motor output torque and precision, the DP2.2 actuator was subjected to further decomposition. The functional requirement FR2.2, as previously defined, can be decomposed with reference to DP2.2 as follows:

FR2.2.1 = Provide power

FR2.2.2 = Converted to radial motion

FR2.2.3 = Smooth deformation

FR2.2.4 = Improve torque and accuracy

Due to the constraints and some limitations of the complex network and visualization research institute design team, the actuator is simplified to meet the requirements through the direct coupling reducer, flexible coupling, and direct action screw. The corresponding DPs may be stated as follows:

DP2.2.1 = Motor

DP2.2.2 = Screw structure

DP2.2.3 = Flexible coupling

DP2.2.4 = Direct interconnection

The design matrix for the above set of FRs and DPs is as follows:

$$\left\{\begin{array}{c}\mathrm{FR}2.2.1\\ \mathrm{FR}2.2.2\\ \mathrm{FR}2.2.3\\ \mathrm{FR}2.2.4\end{array}\right\}=\left[\begin{array}{cccc}X& O& O& O\\ O& X& O& O\\ O& X& X& O\\ X& X& O& X\end{array}\right]\left\{\begin{array}{c}\mathrm{DP}2.2.1\\ \mathrm{DP}2.2.2\\ \mathrm{DP}2.2.3\\ \mathrm{DP}2.2.4\end{array}\right\}$$

(27)

The functional requirement FR2.3, as defined above, may be decomposed with DP2.3 in mind as follows:

FR2.3.1 = Node connection piece

FR2.3.2 = Position control

The motor control program enables both forward and reverse rotation, along with speed regulation. The position PID algorithm compares the actual measured position with the target position, thereby ensuring precise control and enabling the active reflector node to accurately reach its intended position. The corresponding DPs may be stated as follows:

DP2.3.1 = Motor control program

DP2.3.2 = Incremental PID algorithm

The design matrix for the above set of FRs and DPs is as follows:

$$\left\{\begin{array}{c}\mathrm{FR}2.3.1\\ \mathrm{FR}2.3.2\end{array}\right\}=\left[\begin{array}{cc}X& O\\ O& X\end{array}\right]\left\{\begin{array}{c}\mathrm{DP}2.3.1\\ \mathrm{DP}2.3.2\end{array}\right\}$$

(28)

The DP2.4 main control chip needs to realize the functions of regulated power supply, signal transmission, electrical isolation, optocoupler, and control module connecting to external devices. The functional requirement FR2.4, as defined above, may be decomposed with DP2.4 in mind as follows:

FR2.4.1 = Power supply

FR2.4.2 = Signal transmission

FR2.4.3 = Electric isolation

FR2.4.4 = Optical coupler

FR2.4.5 = Connect peripherals

Control and data acquisition involve the custom main control chip to achieve the functional requirements. The corresponding DPs may be stated as follows:

DP2.4.1 = Voltage regulator module

DP2.4.2 = TTL to RS-485

DP2.4.3 = PWM output

DP2.4.4 = Limit sensor interface

DP2.4.5 = PCB board design

The design matrix for the above set of FRs and DPs is as follows:

$$\left\{\begin{array}{c}\mathrm{FR}2.4.1\\ \mathrm{FR}2.4.2\\ \mathrm{FR}2.4.3\\ \mathrm{FR}2.4.4\\ \mathrm{FR}2.4.5\end{array}\right\}=\left[\begin{array}{cccccc}X& O& O& O& O& O\\ X& X& O& O& O& O\\ X& O& X& O& O& O\\ X& O& O& X& O& O\\ X& X& X& X& X& O\end{array}\right]\left\{\begin{array}{c}\mathrm{DP}2.4.1\\ \mathrm{DP}2.4.2\\ \mathrm{DP}2.4.3\\ \mathrm{DP}2.4.4\\ \mathrm{DP}2.4.5\end{array}\right\}$$

(29)

The functional structure is obtained via stepwise decomposition (Figure 5).

3.2. Entire Design Matrix

The design matrix layout was used by Suh [7] and Carnevalli [28]. FRs are displayed in rows, DP are displayed in columns, and ‘X’ indicates that there is a dependency between an FR and DP. We evaluate whether the DP in each cell will affect the FR, and if so, we mark the cell with ‘X’. If there is no effect, it is 0. For example, the triangular element panel affects the reflection accuracy, and the corresponding cell is marked as ‘X’.

Figure 6 depicts the interrelationship between the system’s comprehensive functional requirements and its design parameters. The dark cells along the diagonal indicate that the resulting structure assumes the form of a lower triangular matrix, thereby signifying a decoupled design.

3.3. Solution

According to the overall design matrix, the design scheme of the surface shape block, control, and sensing unit is obtained. As shown in Figure 6, the surface is composed of 500 aluminum triangular reflector units, which are divided into 10 circles and laid on the cable net. The surface shape design affects the control mode. The control structure is composed of a support surface, actuator, motor, transmission mechanism, flexible coupling, direct reducer, and supporting parts. The sensing unit is composed of laser ranging, motor feedback data, and circuit feedback information.

The DP1.1 triangular element panels are connected to each other through DP1.3.2 node connectors. The supporting surface of DP2.1 includes the supporting frame and the fixed base. The DP2.2 actuator is installed under the node of the panel to drive the node to move radially along the spherical surface so that the DP1.1 panel moves. The DP3.1 laser ranging sensor is installed on the support surface to measure the distance from the node connector to the sensor.

The two threads of the DP2.2.2 screw mechanism are connected with the DP1.3.2 node connecting piece and the DP2.2.3 flexible coupling, respectively. When the output shaft of the DP2.2.1 motor rotates, the motor power is transmitted from the DP2.2.4 direct coupling reducer to the flexible coupling. The screw mechanism converts the rotation motion of the motor output into the linear motion of the screw, thus forming the axial relative displacement of the node-connecting piece and the support plate.

The mapping relationship between FRs and DPs, derived from the aforementioned decompositions, forms a lower triangular matrix that adheres to the independence axiom. The third-layer DPs are the specific methods required for the design of the ARCS, and the amount of information is the smallest, which satisfies the information axiom. Therefore, the design scheme divided by axiomatic design is a reasonable design.

4. Multi-Criteria Decision Making

Through precise position adjustment, dynamic shape adjustment, error compensation, improving system stability, and optimizing signal quality, the actuator ensures the high performance and high reliability of the ARCS. Therefore, the choice of actuator is very important. This paper employs MCDM to identify the key performance indicators of the actuator, including motion freedom, reliability, and connection type. Furthermore, a solid model of the FAST system is constructed for control experiments.

4.1. Evaluation

According to the requirements of the FAST solid model for the performance and structure of the actuator, the actuator criterion range is determined as shown in

Table 1. Actuator criteria.

Criteria	Description
A. Filtering criteria
A1. Degrees of freedom	Freedom of motion
A2. Reliability	Start–stop performance and life
A3. Connection type	Installation location of the actuator
A4. Carrying capacity/kg	Maximum load capacity at run time
A5. Drive distance/cm	Maximum distance that the actuator can drive
B. Evaluation criteria
B1. Maximum speed	Maximum operating speed of the actuator
B2. Reproducibility	Ability of the actuator to return to the same position
B3. Cost	Purchase and installation costs

. In the pre-screening of 11 actuators, 5 actuators were eliminated because the degree of freedom and the driving distance did not meet the application requirements. The selected actuator options are shown in

Table 2. Actuator specifications and scope.

	A6	A7	A8	A9	A10	A11
Degrees of freedom	Two degrees of freedom	Two degrees of freedom	Two degrees of freedom	Two degrees of freedom	Two degrees of freedom	Two degrees of freedom
Drive type	Electricity	Electricity	Electricity	Electricity	Electricity	Electricity
Control type	Servo motor	Servo motor	Servo motor	Servo motor	Servo motor	Servo motor
Reliability	General	Good	Good	Better	Good	Better
Structural complexity	Simpler	General	Complication	Relatively simple	Relatively complicated	General
Bearing capacity	0–9	0–20	0–11	0–13	0–15	0–17
Drive distance	0–200	0–280	0–150	0–400	0–600	0–650
Speed	0–150	0–400	0–200	0–300	0–200	0–200
Reproducibility	0.02	0.045	0.1	0.05	0.03	0.04
Cost (×1000)	20.5	22.5	18.5	25	35.5	32

.

A TOPSIS-gray correlation analysis is used to deal with quantitative attributes. We solve the weighted normalization matrix as follows:

$$X=\left[\begin{array}{cccc}0& 0& 0& 0.12\\ 1& 1& 0.312& 0.24\\ 0.18& 0.2& 1& 0\\ 0.36& 0.6& 0.375& 0.382\\ 0.55& 0.2& 0.124& 1\\ 0.72& 0.2& 0.25& 0.79\end{array}\right]$$

Based on the design of the demand vector $w=(0.45,0.15,0.1,0.3)$, we solve the weighted normalization matrix:

$$z=\left[\begin{array}{cccc}0& 0& 0& 0.36\\ 0.45& 0.15& 0.031& 0.072\\ 0.081& 0.03& 0.1& 0\\ 0.162& 0.09& 0.038& 0.115\\ 0.248& 0.03& 0.012& 0.3\\ 0.324& 0.03& 0.025& 0.237\end{array}\right]$$

We calculate the positive and negative ideal solutions of the weighted normalized matrix Z:

$$Z^{+}=\left[0.450,0.150,0.100,0.360\right]$$

$$Z^{-}=\left[0,0,0,0\right]$$

We calculate the Euclidean distance from each machine to the positive and negative ideal solutions:

$${d_i}^{+}=\left(0.552,0.238,0.491,0.353,0.251,0.200\right)$$

$${d_i}^{-}=\left(0.036,0.481,0.132,0.221,0.391,0.403\right)$$

We calculate the gray relational analysis coefficient matrix of alternative proposals $R^{+}$, $R^{-}$:

$$R^{+}=\left[\begin{array}{cccc}0.333& 0.6& 0.692& 1\\ 1& 1& 0.765& 0.439\\ 0.379& 0.652& 1& 0.385\\ 0.439& 0.789& 0.784& 0.479\\ 0.527& 0.652& 0.719& 0.789\\ 0.641& 0.652& 0.750& 0.647\end{array}\right]$$

$$R^{-}=\left[\begin{array}{cccc}1& 1& 1& 0.385\\ 0.333& 0.6& 0.879& 0.758\\ 0.735& 0.882& 0.692& 1\\ 0.581& 0.714& 0.856& 0.662\\ 0.476& 0.882& 0.949& 0.429\\ 0.410& 0.882& 0.9& 0.487\end{array}\right]$$

We calculate the gray relational analysis degree of each machinery and the positive and negative ideal solutions $r^{+}$, $r^{-}$:

$$r^{+}=\left(0.656,0.801,0.604,0.623,0.672,0.672\right)$$

$$r^{-}=\left(0.846,0.642,0.827,0.703,0.684,0.67\right)$$

The dimensionless treatment is carried out to obtain the following: $D_i^{+}$, $D_i^{-}$, $R_i^{+}$, $R_i^{-}$:

$$D_i^{+}=\left(1.000,0.431,0.889,0.639,0.455,0.362\right)$$

$$D_i^{-}=\left(0.075,1.000,0.274,0.459,0.813,0.730\right)$$

$$R_i^{+}=\left(0.819,1.000,0.754,0.778,0.839,0.839\right)$$

$$R_i^{-}=\left(1.000,0.759,0.978,0.831,0.809,0.792\right)$$

The dimensionless Euclid distance and gray correlation degree are combined. When ${\nu }_1={\nu }_2=0.5$, $T_i^{+}$$S_i^{+}$ can be calculated:

$$T_i^{+}=\left(0.447,1.000,0.514,0.619,0.826,0.785\right)$$

$$S_i^{+}=\left(1.000,0.595,0.934,0.735,0.632,0.577\right)$$

Calculate the relative $Y_i^{+}$:

$$Y_i^{+}=\left(0.7235,0.798,0.724,0.677,0.729,0.681\right)$$

A comparative analysis of the six alternative solutions was conducted using four methods: TOPSIS, EWM, VIKOR, and an integrated TOPSIS–GRA approach. Each method evaluated the alternatives based on their proximity to an ideal solution, offering a comprehensive perspective on rankings and preferences across various assessment criteria. The results are summarized in the

Table 3. The relative scores and rankings generated by each method.

Alternative	TOPSlS Relative Closeness	TOPSIS Ranking	Entropy-Weighted Score	Entropy Ranking	VIKOR Qi	VIKOR Ranking	TOPSIS and Grey Relational Analysis Integration	Yi + Ranking
A6	0.581588	3	0.042529	4	1	6	0.7235	4
A7	0.653621	2	0.04286	3	0.029449	1	0.798	1
A8	0.053011	6	0.027392	6	0.70424	5	0.724	3
A9	0.715366	1	0.056303	1	0.215041	2	0.677	6
A10	0.160129	5	0.032012	5	0.428575	4	0.729	2
A11	0.399471	4	0.043794	2	0.428003	3	0.681	5

below, with each column representing the relative scores and rankings generated by each method.

As illustrated in

Table 4. Experimental results analysis.

TOPSIS Analysis	The TOPSIS method computes a relative closeness score for each alternative, ranking them based on proximity to the ideal solution. Alternative A9 exhibits the highest closeness score of 0.715366, securing the top position in the TOPSIS ranking. A7 follows closely with a score of 0.653621, ranked second, while A8 scores the lowest, with a relative closeness value of 0.053011, placing it in the sixth position. These results suggest that A9 and A7 are the most desirable alternatives when considering all criteria collectively under TOPSIS.
EWM	The EWM, which assigns weights based on the degree of information entropy, highlights A9 as the top-ranking alternative, with a weighted score of 0.056303, followed by A11 with a score of 0.043794. A8, again, ranks the lowest, with a score of 0.027392, reinforcing its relative inadequacy across the criteria when assessed independently. This outcome reflects the entropy weighting’s emphasis on criteria dispersion, where the most informative criteria receive higher weighting.
VIKOR	The VIKOR method, focused on balancing compromise solutions, assigns the lowest Qi score 0.029449 to A7, designating it as the most preferred alternative. A9 ranks second, with Qi = 0.215041, indicating a competitive compromise solution, while A6 records the highest Qi score of 1, positioning it as the least desirable under this approach. VIKOR’s results suggest that A7 provides the most balanced solution when weighing ideal and anti-ideal distances.
TOPSIS-GRA	The integrated TOPSIS-GRA approach further corroborates the results, with A7 achieving the highest composite score of 0.798, followed closely by A10 at 0.729. A9, which ranked first in the standalone TOPSIS method, shows a relative decline, obtaining the lowest score of 0.677 in this integration, ranking sixth. This integrated approach offers a nuanced view by combining the strengths of TOPSIS and GRA, allowing for a more robust assessment by accounting for both similarity to the ideal solution and relational closeness among alternatives.

, the outcomes of the various methodologies exhibit a notable degree of consistency, particularly for A7, which emerges as a highly promising approach and consistently ranks within the top two in the majority of the methodologies. A8 consistently ranks as one of the least-favorable alternatives across all the methods, underscoring its limited suitability given the evaluated criteria. A9, while top-ranked in the TOPSIS and entropy weighting analyses, does not achieve the same position in VIKOR and TOPSIS-GRA, indicating that its performance may vary based on the prioritization and weighting mechanisms inherent in each method.

The fuzzy information axiom is applied to quantify the information content of qualitative attributes. Taking reliability as an example, the design range area is calculated as (2 + 3) × 1/2 = 2.5, and its information content is computed using Equation (20). The same method is applied to calculate the information content for other attributes, leading to the total information content for each alternative.

Table 5. Amount of information on actuator options.

	A6	A7	A8	A9	A10	A11
$I_{Reliability}$	-	0.322	-	3.322	0.322	3.322
$I_{Structuralcomplexity}$	0.7	1.7	-	1.7	4.7	1.7
$I_{Bearingcapacity}$	3.32	4.39	3.58	3.81	4.0	4.17
$I_{Drivedistance}$	7.64	8.46	7.23	8.64	8.9	8.68
$I_{Speed}$	7.23	8.64	7.64	8.23	7.64	7.64
$I_{Reproducibility}$	4.64	3.46	2.32	3.32	4.07	3.64
$I_{Cost}$	0	0	0	0	0	0
Information content	-	26.96	-	29.022	29.632	29.152

presents the information content of all attributes, where the results indicate that A7 has the lowest total information content. Due to reliability and structural complexity falling outside the expected range, A6 and A8 have infinite information content. The ranking of total information content is A7 < A9 < A11 < A10. Actuators with lower information content demonstrate more stable performance in response speed and control accuracy, minimizing the likelihood of frequent deviations or anomalies due to design uncertainties.

4.2. Control Experiments

Based on the design requirements and demand analysis described earlier, this study employs LabVIEW to develop host software for controlling a single motor. The LabVIEW program design encompasses two key components: the front panel interface and the program block diagram. The front panel is required to display motor speed and motion trajectory, while also providing buttons for functionalities such as forward and reverse rotation, returning to the initial position after stopping, and mid-operation halt. Figure 7 illustrates the front panel of the motor position control system. Figure 8 shows both the physical model and the engineering diagram of the actuator. In one node, the host computer controls a single motor, which in turn controls four actuators running at 11 mm. We carry out a number of experiments to determine the accuracy of the control, and we use the record to obtain the position of the actuator, as shown in Figure 9.

In this experiment, the average error when the four actuators adjust the panel deformation to reach the specified positions is presented in

Table 6. The average error of the actuator reaching the specified position.

Program	A7	A9	A10	A11
Average error/mm	0.213	0.296	0.312	0.358

. A maximum error within 0.35 mm is considered to indicate good synchronization. The experimental results confirm that schemes A7, A9, and A10 meet this criterion effectively.

4.3. Results

This study applied MCDM methods, including TOPSIS, EWM, VIKOR and TOPSIS-GRA, to theoretically rank six design alternatives. The results indicated that both A7 and A9 demonstrated high overall performance across the different evaluation methods. However, based on the information content calculated through axiomatic design, A7 exhibited a lower information content than A9, suggesting that A7 provides greater efficiency in system decoupling and functional implementation. Thus, A7 was selected as the optimal solution.

To further validate the accuracy of the theoretical analysis, control experiments were conducted to assess the tracking performance of the selected alternative relative to its reference position. The experimental results showed that A7 achieved a high degree of alignment with the reference trajectory, demonstrating superior control accuracy and dynamic adaptability. These findings confirmed the effectiveness and reliability of combining axiomatic design with MCDM methods in optimizing the ARCS.

Building on the previously established actuator design process, the development of the reflective panel morphology, deformation mode, and the corresponding control and sensing systems culminated in the creation of a 1:100 scale physical model of the FAST. This model consists of 495 aluminum triangular panels, along with 225 servo motors and actuators, complemented by multiple sensors. To achieve a 12% reduction in the weight of the reflective surface frame, pre-stressed steel cables replaced specific truss structures, maintaining the overall structural stiffness and strength. The reflective surface panels were produced using molds. While smaller panels enhance manufacturing precision, they would require a greater number of molds, leading to increased costs in processing, transportation, installation, and manufacturing. The triangular design of the reflective panels successfully decreased the number of mold types from 35 to 14, resulting in improved cost efficiency while still fulfilling performance specifications. The construction process further confirmed the design’s feasibility and effectiveness.

5. Conclusions

(1) The design methodology proposed in this study is tailored for the ARCS design process and is intended to serve as a foundation for broader applications in other system designs. This approach aims to enhance design efficiency, reduce development cycles, and improve design reliability. During the preliminary design phase, axiomatic design should be used to establish the design scheme. In the decision-making phase, initial filtering criteria are recommended to screen and identify alternative solutions, followed by the use of the fuzzy information axiom and grey relational analysis to evaluate the remaining options, ultimately selecting the optimal solution. Axiomatic design introduces system-level metrics, such as reliability, structural complexity, and repeatability, ensuring a comprehensive evaluation that meets functional requirements while optimizing multi-dimensional performance. By leveraging the decoupling and dynamic feedback mechanisms of axiomatic design, this approach enables greater adaptability and stability in response to dynamic demands, significantly enhancing the accuracy of priority ranking. This methodology represents a novel attempt, as similar methods have not yet been reported in the literature. The software system developed to support axiomatic design has been granted a software copyright registration.

(2) Based on the design approach, the functional requirements of the active reflector system are evaluated, leading to the determination of its main structure and design parameters. Subsequently, a solid model of the FAST system is created, followed by the design and control testing of the active reflector system. This approach allows for active structural deformation, thereby enhancing the overall performance of the system. It also achieves the decoupling of the equilibrium state from the overall output of the SWaP-C equation, confirming both the method’s effectiveness and the scheme’s rationality. It presents a theoretical framework and methodological guidance that are vital for optimizing active reflector design and enhancing system performance. Typical applications include curved surface deformation systems in maritime vessels and concentrating subsystems in tower-based solar power generation. Therefore, the design method can be used not only to guide the design of the active reflector but also to guide the design of complex curved surface deformation systems such as the ship’s curved surface deformation system and the concentrating subsystem.

(3) The ARCS represents a broad research area, with equipment requirements that vary based on specific application scenarios. This study established a design process using a 5 m diameter FAST model as an example. However, in model testing, the control system may not fully capture the complex feedback demands of the actual FAST system. For instance, coordinating 225 actuators and servo motors in the full-scale system may introduce nonlinear dynamic coupling, thereby increasing the complexity of control. Additionally, this study does not provide an in-depth analysis of the robustness and real-time performance of the control algorithms. In practical applications, more efficient and precise control strategies may be necessary to meet the full system’s operational demands. Further research will be conducted in the future with a view to enhancing the applicability of the methodology.