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Article(id=1241381046915232116, tenantId=1146029695717560320, journalId=1238841944844054536, issueId=1241381046445470068, articleNumber=null, orderNo=null, doi=10.12347/j.ycyk.20231006001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1696521600000, receivedDateStr=2023-10-06, revisedDate=1703692800000, revisedDateStr=2023-12-28, acceptedDate=null, acceptedDateStr=null, onlineDate=1773897905329, onlineDateStr=2026-03-19, pubDate=1705248000000, pubDateStr=2024-01-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773897905329, onlineIssueDateStr=2026-03-19, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773897905329, creator=13701087609, updateTime=1773897905329, updator=13701087609, issue=Issue{id=1241381046445470068, tenantId=1146029695717560320, journalId=1238841944844054536, year='2024', volume='45', issue='1', pageStart='1', pageEnd='132', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773897905217, creator=13701087609, updateTime=1773903111898, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241402884936495824, tenantId=1146029695717560320, journalId=1238841944844054536, issueId=1241381046445470068, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241402884936495825, tenantId=1146029695717560320, journalId=1238841944844054536, issueId=1241381046445470068, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=38, endPage=46, ext={EN=ArticleExt(id=1241381047158501750, articleId=1241381046915232116, tenantId=1146029695717560320, journalId=1238841944844054536, language=EN, title=Quadrotor Attitude Control Method Based on Cascade LADRC, columnId=1239133503154081962, journalTitle=Journal of Telemetry, Tracking and Command, columnName=TT & C Communication and Navigation, runingTitle=null, highlight=null, articleAbstract=

For the quadrotor UAV attitude adjustment process, the existence of parameter uncertainty and external environmental interference often cause certain difficulties to the attitude control. This paper studies and proposes a quadrotor attitude control method based on cascade linear active disturbance rejection. First, the dynamic attitude model of the quadrotor UAV was established, and a cascade PID dualloop control architecture was proposed to decompose the attitude control task into two internal and external loops. The Levant differentiator is used to extract control parameters to enhance tracking capabilities. In addition, the original linear active disturbance rejection controller is optimized to better eliminate the impact of external random disturbances on the system. The MATLAB Simulink environment was used to simulate the proposed control method. The results show that compared with traditional method, this method can better suppress the influence of the system caused by perturbations and enhance the tracking ability of the system to the desired signals, which significantly improves the precision and stability of the attitude adjustment of the quadcopter UAV, and improves the precision and robustness of the attitude control of the quadcopter UAV.

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针对四旋翼无人机姿态调整过程中，由于参数不确定性及外界环境干扰，往往给姿态控制带来一定困难。本文研究提出一种基于串级线性自抗扰的四旋翼姿态控制方法。首先建立四旋翼无人机的动力学姿态模型，提出采用串级PID双环路控制架构，将姿态控制任务分解为内外两个环路。采用Levant微分器提取控制参量，以强化跟踪能力。此外，对原线性自抗扰控制器进行优化，旨在更好地消除外界随机扰动对系统的影响。利用MATLAB Simulink环境对提出的控制方法进行模拟，结果表明相比传统方法，该方法能更好地抑制系统受扰动引起的影响，增强系统对期望信号的跟踪能力，从而明显提高了四旋翼无人机姿态调整的精度与稳定性，以及四旋翼无人机姿态控制的精度和鲁棒性。

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常哲　1997年生，硕士研究生。

陈鹏云　1987年生，副教授，硕士生导师。

李佳成　1996年生，硕士研究生。

石上瑶　1971年生，博士，正高级工程师。

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常哲　1997年生，硕士研究生。

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常哲　1997年生，硕士研究生。

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陈鹏云　1987年生，副教授，硕士生导师。

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陈鹏云　1987年生，副教授，硕士生导师。

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李佳成　1996年生，硕士研究生。

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李佳成　1996年生，硕士研究生。

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石上瑶　1971年生，博士，正高级工程师。

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石上瑶　1971年生，博士，正高级工程师。

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外环输入：期望角度ϕdθdψd.
使用公式(3)计算虚拟控制量U1，U2，U3，U4.
使用公式(4)解耦姿态角ϕ，θ，ψ.
使用公式(8)计算误差e.
使用公式(11)计算观测信号，，
使用公式(16)和(17)求解期望信号的微分dϕd，dθd，dψd.
内环输入：以上中间控制信号.
使用公式(14)计算稳定信号d.
内环输出：修正信号uϕ，uθ，uψ.
外环输出：最终信号ϕ，θ，ψ.
), ArticleFig(id=1241396563042422994, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1241381046915232116, language=CN, label=算法I：, caption=

串级ILADRC姿态控制

, figureFileSmall=null, figureFileBig=null, tableContent=
外环输入：期望角度ϕdθdψd.
使用公式(3)计算虚拟控制量U1，U2，U3，U4.
使用公式(4)解耦姿态角ϕ，θ，ψ.
使用公式(8)计算误差e.
使用公式(11)计算观测信号，，
使用公式(16)和(17)求解期望信号的微分dϕd，dθd，dψd.
内环输入：以上中间控制信号.
使用公式(14)计算稳定信号d.
内环输出：修正信号uϕ，uθ，uψ.
外环输出：最终信号ϕ，θ，ψ.
), ArticleFig(id=1241396563122114772, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1241381046915232116, language=EN, label=Table 1, caption=

Parameter initialization

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物理量数值物理量数值
无人机质量m=0.65kgObxb轴的转动惯量Ix=7.5×10-3
重力加速度g=9.8m/s2Obyb轴的转动惯量Iy=7.5×10-3
臂长l=0.32 mObyb轴的转动惯量Iz=1.3×10-2
升力系数b=3.1×10-5空气阻力系数K4=K5=0.1
扭力系数7.5×10-7空气阻力系数K6=0.15
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参数初始化

, figureFileSmall=null, figureFileBig=null, tableContent=
物理量数值物理量数值
无人机质量m=0.65kgObxb轴的转动惯量Ix=7.5×10-3
重力加速度g=9.8m/s2Obyb轴的转动惯量Iy=7.5×10-3
臂长l=0.32 mObyb轴的转动惯量Iz=1.3×10-2
升力系数b=3.1×10-5空气阻力系数K4=K5=0.1
扭力系数7.5×10-7空气阻力系数K6=0.15
), ArticleFig(id=1241396563281498331, tenantId=1146029695717560320, journalId=1238841944844054536, articleId=1241381046915232116, language=EN, label=Table 2, caption=

Cumulative error under different disturbance

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干扰模式算法QψQϕQθ∑Qi
无ADRC0.356 20.356 20.651 51.363 9
无ILADRC0.033 10.033 10.283 60.349 8
正
弦		ADRC0.467 10.467 10.761 51.695 7
正弦ILADRC0.253 40.253 40.490 40.997 2
白噪声ADRC0.614 40.614 40.9012.129 8
白噪声ILADRC0.241 60.241 60.485 50.968 7
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不同干扰下的累计误差

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干扰模式算法QψQϕQθ∑Qi
无ADRC0.356 20.356 20.651 51.363 9
无ILADRC0.033 10.033 10.283 60.349 8
正
弦		ADRC0.467 10.467 10.761 51.695 7
正弦ILADRC0.253 40.253 40.490 40.997 2
白噪声ADRC0.614 40.614 40.9012.129 8
白噪声ILADRC0.241 60.241 60.485 50.968 7
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基于串级线性自抗扰的四旋翼姿态控制方法
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常哲 1 , 陈鹏云 1 , 李佳成 1 , 石上瑶 2
遥测遥控 | 测控通信与导航 2024,45(1): 38-46
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遥测遥控 | 测控通信与导航 2024, 45(1): 38-46
基于串级线性自抗扰的四旋翼姿态控制方法
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常哲1, 陈鹏云1, 李佳成1, 石上瑶2
作者信息
  • 1中北大学航空宇航学院　太原　030051
  • 2中北大学软件学院　太原　030051

    • 常哲　1997年生，硕士研究生。

    陈鹏云　1987年生，副教授，硕士生导师。

    李佳成　1996年生，硕士研究生。

    石上瑶　1971年生，博士，正高级工程师。

Quadrotor Attitude Control Method Based on Cascade LADRC
Zhe CHANG1, Pengyun CHEN1, Jiacheng LI1, Shangyao SHI2
Affiliations
  • 1.School of Aerospace Engineering, North University of China, Taiyuan 030051, China
  • 2.School of Software, North University of China, Taiyuan 030051, China
出版时间: 2024-01-15 doi: 10.12347/j.ycyk.20231006001
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摘要
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针对四旋翼无人机姿态调整过程中，由于参数不确定性及外界环境干扰，往往给姿态控制带来一定困难。本文研究提出一种基于串级线性自抗扰的四旋翼姿态控制方法。首先建立四旋翼无人机的动力学姿态模型，提出采用串级PID双环路控制架构，将姿态控制任务分解为内外两个环路。采用Levant微分器提取控制参量，以强化跟踪能力。此外，对原线性自抗扰控制器进行优化，旨在更好地消除外界随机扰动对系统的影响。利用MATLAB Simulink环境对提出的控制方法进行模拟，结果表明相比传统方法，该方法能更好地抑制系统受扰动引起的影响，增强系统对期望信号的跟踪能力，从而明显提高了四旋翼无人机姿态调整的精度与稳定性，以及四旋翼无人机姿态控制的精度和鲁棒性。

四旋翼无人机  /  线性自抗扰控制  /  串级PID控制  /  Levant微分器

For the quadrotor UAV attitude adjustment process, the existence of parameter uncertainty and external environmental interference often cause certain difficulties to the attitude control. This paper studies and proposes a quadrotor attitude control method based on cascade linear active disturbance rejection. First, the dynamic attitude model of the quadrotor UAV was established, and a cascade PID dualloop control architecture was proposed to decompose the attitude control task into two internal and external loops. The Levant differentiator is used to extract control parameters to enhance tracking capabilities. In addition, the original linear active disturbance rejection controller is optimized to better eliminate the impact of external random disturbances on the system. The MATLAB Simulink environment was used to simulate the proposed control method. The results show that compared with traditional method, this method can better suppress the influence of the system caused by perturbations and enhance the tracking ability of the system to the desired signals, which significantly improves the precision and stability of the attitude adjustment of the quadcopter UAV, and improves the precision and robustness of the attitude control of the quadcopter UAV.

Quad-rotor UAV  /  Linear active disturbance rejection control  /  Cascade PID control  /  Levant differentiator
常哲, 陈鹏云, 李佳成, 石上瑶. 基于串级线性自抗扰的四旋翼姿态控制方法. 遥测遥控, 2024 , 45 (1) : 38 -46 . DOI: 10.12347/j.ycyk.20231006001
Zhe CHANG, Pengyun CHEN, Jiacheng LI, Shangyao SHI. Quadrotor Attitude Control Method Based on Cascade LADRC[J]. Journal of Telemetry, Tracking and Command, 2024 , 45 (1) : 38 -46 . DOI: 10.12347/j.ycyk.20231006001
正文
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0　引言
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作为一种新兴的运载工具，四旋翼无人机可以低速、低空飞行，能够实现垂直起降、定点悬停等动作，非常适合环境较为狭小的场合，多用于军事和航空运动领域[1]。由于其轻便灵巧，国内外高校和各大科研机构青睐之并加大对其研究力度。目前四旋翼无人飞行器已成为控制领域研究的热点[2]。
但是，四旋翼无人机在复杂环境下飞行时受外界扰动的影响较大，无法保证系统的稳定性和精确性。为了提高四旋翼无人机在复杂环境下的操作性能，相关学者投入大量精力，提出了很多经典控制算法。目前，对于四旋翼无人机的控制方法主要分为线性控制方法和非线性控制方法两种。线性控制技术包括机动性模型控制，这种方法最简单，是利用四旋翼无人机的线性等效机动性模型设计PID[3]或LQR[4]等线性控制器以实现位置/姿态控制。常用的非线性控制方法包括滑模控制[5]、反馈线性化[6]、反步控制[7]，这些方法能够解决强耦合的非线性问题。但以上方法在系统受到外界干扰时并不能完全保证系统的性能。现阶段很多方法都需要依赖四旋翼无人机精确的系统模型，但四旋翼无人机系统本身由于存在模型不确定性和未知的外界扰动，很难建立精确的数学模型。因此，需要建立起既不依赖于无人机系统精确数学模型，又能有效抑制未知扰动的飞行控制方法投入使用。
自抗扰控制技术[8](Active Disturbance Rejection Control, ADRC)利用现代控制理论分析方法在经典控制理论的基础上构建出一种不精准依赖被控对象的控制方法。通用线性ADRC控制器结构(Linear Active Disturbance Rejection Control, LADRC)[9]的提出使ADRC参数概念更直观。随着研究的不断深入，ADRC在四旋翼无人机控制中被广泛使用。文献[10]利用浸入与不变控制的目标系统映射方法改进线性控制律，文献[11]构建了一种新型非线性光滑函数以解决观测器观测精度不足的问题，文献[12]利用试验测量及数值仿真的方法获取模型参数后，采用设计控制分配系数的方法以解决控制冗余的问题，文献[13-15]分别根据串级控制的思想采用内外环控制方法以提高系统的响应速度和抗干扰性。文献[16]采用不同阶数的线性自抗扰控制，共同作用在四旋翼飞行器的速度控制上，减小了姿态角的收敛速度和跟踪误差。文献[17]、文献[18]分别以基于正弦函数和构建一种新型光滑非线性函数来改进观测器以有效减小抖振。
以上方法均提高了系统稳定性并增强了系统的误差跟随能力。因此，本文设计了一种基于串级线性自抗扰的四旋翼姿态控制方法。首先，改进扩张状态观测器，对误差跟踪精度进行提高；其次，在内环加入Levant微分器对控制系统的抗干扰能力和鲁棒性进行优化；最后，在内外环双回路采用串级PID控制以降低整个控制系统对于被控对象的依赖性。使用Matlab Simulink进行仿真，结果表明：所提方法具有较高精度、较好稳定性，可以用于四旋翼无人机的姿态控制。
1　四旋翼动力学建模
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四旋翼无人机有4个控制输入但具有6个自由度，为了描述四旋翼无人机的运动状态，一般使用Newton-Euler公式来建立无人机运动学和动力学模型。构建大地坐标系和机体坐标系后可以建立四旋翼运动学方程。大地坐标系Oexeyeze的坐标原点Oe为地面上选定的一点，Oexe轴指向无人机机头方向,Oeze轴垂直于地面指向地心的反方向，xeOeye为水平面，符合右手螺旋定则；机体坐标系Obxbybzb为固连坐标系，坐标原点Ob为无人机重心位置，Obxb轴选取为与四旋翼飞行器垂直十字架的一个支架重合，Obyb轴与另一个支架重合，Obzb轴垂直于xbObyb面指向飞行器上方，符合右手螺旋定则。坐标系统建立如图1所示。
四旋翼的位置可以用[xyz]T来描述，滚转、俯仰和偏航的角度可以分别用[ϕ，θ，ψ]T表示。大地坐标系与机体坐标系的转换矩阵为Q，通常使用欧拉角法得到的大地坐标系到机体坐标系的旋转矩阵如下：
四旋翼无人机是非常典型的强耦合欠驱动MIMO模型，平移与旋转方程为：
式中，Fi是电机产生的推力，m是负载，Ki是阻力系数，p、q、r是机体角速度，KT是升力系数，KM是扭力系数，l是每个旋翼末端到重心的距离即飞行器半径长度，ωi是电机转速，Ii是围绕每个轴体的转动惯量。
使用虚拟控制输入Ui控制四旋翼的运动方式，表示如下：
在小角度飞行的情况下可近似认为，由于只关注姿态控制的情况，为了合理简化模型，这里选择忽略运动模型里的耦合项，根据拉格朗日方程，十字形四旋翼无人机的动力学方程可以表示为[19]
式中，u1=U1/m，ui=Ui/Ij，i=2，3，4，j=x，y，z均为模型控制输入。
2　控制器设计
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2.1　改进自抗扰控制器
ADRC的理论基础是以简单的积分型系统为标准模型，任何偏差或扰动都看作为总的扰动项。利用微分器提取被控对象的微分信号，同时安排过渡过程，采用非线性扩张观测器对系统中的各种扰动进行实时估计和补偿，使原本充满不确定性的系统在行为上等同于标准模型，连续消除模型误差和扰动的影响；控制环路中利用PID三项反馈原理提取误差信号，并通过非线性组合实现控制量的调节，使系统始终保持在一个稳定状态，降低了系统复杂性，有效提升了控制设计的可操作性和可理解性，实现了系统模型的简化和规格化。
为了跟踪匹配不连续型的输入信号并提取系统状态的近似微分信息，利用二阶最速开关系统构造出一种离散形式的非线性微分跟踪器为
式中，h为采样周期，v(k)为第k时刻的输入信号，δ为决定跟踪快慢的参数，fst(*)函数是最速控制综合函数，表示为
对应的二阶扩张观测器设计为
控制律设计为
其中，fal函数表示为
LADRC简化了跟踪微分器(Tracking Differentiator, TD)配置过程，将扩张状态观测器(Extended State Observer, ESO)和非线性状态误差反馈(Nonlinear State Error Feedback, NLSEF)进行线性化，将各参数与控制器和观测器的频率联系到一起，从而通过带宽确定线性状态误差反馈(Linear State Error Feedback, LSEF)和线性扩张状态观测器(Linear Extended State Observer, LESO)参数。与非线性自抗扰控制架构下的扩张状态观测器相比，线性扩张状态观测器基于线性模型来预测扩张状态量，这比非线性观测器模型设计要简单直接得多。此外，由于扩张状态观测器本身就具备对内外各种扰动进行估计和补偿的功能，所以在控制策略设计上，原本需要考虑非线性状态误差的反馈控制律，可以进一步极简化为常见的PD组合控制方式。采用线性扩张观测设计模型和经典PD控制策略，大大减轻了控制器设计的难度。虽然模型预测能力不如非线性观测器，但在通用的控制任务中效果足够理想，这也是它被普遍采用的原因。线性自抗扰控制(LADRC)结构如图2所示。
将线性扩张状态观测器进行误差扰动改进，对于n阶系统进行扩张得到扩张方程为
对应的线性扩张状态观测器表示为
从而得到
四旋翼无人机作为本文的被控对象，是典型的二阶系统，所以对于二阶系统进行状态扩张得到
根据系统方程和扩张状态方程可以得到
控制器结构如图3所示。
2.2　串级PID控制
串级控制不仅控制精度更高，还可通过内环控制调节内环的惯性，从而提升整体系统的响应速度。此外，在串级结构中，若辅助回路的参数变化或受到外界干扰，由于可以由辅助回路自己进行控制，所以对主回路的影响会大为降低，这有利于显著提升整体系统的抗干扰能力。
传统PID控制模型为：
其传递函数为：Gc=kP+kI/s+kDs。式中，kP、kI和kD为三个可调参数。串级PID控制[20]是一种广泛使用的多环控制策略，包含两个或两个以上的PID控制环。外环输入是系统要跟踪的命令信号，输出是内环PID控制器的设定值；内环输入是系统的当前状态信号，控制器根据误差来调节系统状态，使之跟踪外环设定值，输出系统的控制量。同时，内外环PID控制器互相独立，外环响应相对较慢，用于调节整体动态性能；内环响应相对较快，可以提升系统的稳定性。将内外环路分离，整体设计更简洁方便，通过内外环分工合作，内环处理细微动态响应，外环监测整体动态趋势。此结构能够更好地协同各环节，从而实现控制精度和响应速度的优化，并且提高系统的抗干扰性能。将串级PID用于姿态双回路的控制，如图4所示。
2.3　串级线性自抗扰控制
本文所设计的姿态控制器基于二阶LADRC，相对于LADRC加入Levant微分器[21]，由Levant微分器、ILESO和LSEF三部分组成，利用观测器对所有扰动进行实时估计和补偿。为使得信号的测量误差和输入噪声具有鲁棒性，基于滑模技术对输入信号进行初始处理，如式(16)所示：
其中，
且有C>0，是v(t)导数的Lipschitz常数上界。
姿态控制器如图5所示，结构简单，算法清晰明了。使用改进观测器建立动态模型，基于系统模型和测量数据估计系统的状态，不断更新扰动的估计值，再通过线性反馈策略补偿系统的状态估计误差进而抵消扰动，通过以上的关键元素相互配合，使得改进自抗扰控制器能够有效抑制系统的扰动，提高系统的稳定性和鲁棒性。同时，双重回路控制能够进一步抑制控制信号出现偏差，相较于单级控制方法对于期望信号的跟踪能力更加突出。
改进控制结构总体图如图6所示。
姿态控制模块算法描述如下：
3　仿真与分析
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3.1　仿真条件
为了验证所提出方法的有效性，基于正余弦初始设定开展姿态角的控制仿真试验。将本文设计控制器记为ILADRC并与ADRC进行实验对比。试验条件为：Simulink(MATLAB R2021b)，仿真时间设定为10 s。对四旋翼无人机参数进行初始化见表1。
3.2　结果分析
为更好验证所设计控制方法的有效性，设置偏航角ψ和滚转角ϕ为正弦输入，俯仰角θ为余弦输入，并分别加入正弦扰动和白噪声对系统进行干扰。期望信号的振幅均为1，频率均为1，数学表达如式(18)所示。
由于滚转通道与偏航通道的预期期望相同且可以实现完全解耦，所以二者输出相同，使用同一张图进行展示。无扰动时的仿真结果如图7所示。
在第4 s至第7 s加入正弦扰动，振幅为0.1，频率为100 r/s，仿真结果如图8所示。
在第4 s至第7 s加入白噪声扰动，噪声功率为0.001，仿真结果如图9所示。
由图7~图9可以看出，当系统加入扰动的时候，ILADRC相较于ADRC对源信号的追踪能力更强，以图9为例，当白噪声信号起作用时，ILADRC的输出信号始终在期望值附近，要比ADRC的输出信号更加稳定，能够有效抑制扰动对系统产生的影响。
对于姿态角的跟踪误差效果如图10~图12所示：
为描述抗干扰性能，采用式(18)进行定量计算：
式中，Qi表示在i方向上的跟踪误差累积。不同干扰下的计算结果见表2。
由上表可以看出，无干扰、正弦干扰和白噪声干扰情况下，ILADRC跟踪误差累计分别约为ADRC的26%、59%和45%。同时从图10~图12可以看出，ILADRC的跟踪误差要明显小于ADRC，不仅能快速跟踪期望信号，而且跟踪精度高；在期望非零的初始时刻，ILADRC的跟踪效果好于ADRC，具有更快的收敛速度；在4~7 s时加入干扰，对于不同的干扰模式，ILADRC都可以快速稳定并收敛于平均期望，在抗干扰方面展现出更加优异的性能。
4　结束语
收起
针对四旋翼无人机姿态调整过程中存在的不确定性和外部扰动，本文提出了一种基于线性自抗扰的四旋翼姿态控制方法，利用Matlab Simulink进行仿真，可以得到以下结论：
① 改进线性自抗扰观测器能够更精准地估计扰动，并且在有噪声干扰的时候能够实现对扰动的快速抑制。
② 改进串级控制系统使用内外环路双控制，相较于单级控制的传统自抗扰控制器能够利用逐级控制，更好地实现对期望信号的快速跟踪并减小累计误差。
③ 仿真结果表明：本文所设计的控制方法较传统自抗扰控制能够有效协同实现信号的精确跟踪控制，提高系统的稳定性。
基金
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  • 国家自然科学基金(51909245; 62003314)
  • 山西省重点研发计划(202202020101001)
  • 2022年中北大学科技立项资助项目(20221876)
文献
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2024年第45卷第1期
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doi: 10.12347/j.ycyk.20231006001
  • 接收时间：2023-10-06
  • 首发时间：2026-03-19
  • 出版时间：2024-01-15
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  • 收稿日期：2023-10-06
  • 修回日期：2023-12-28
基金
国家自然科学基金(51909245; 62003314)
山西省重点研发计划(202202020101001)
2022年中北大学科技立项资助项目(20221876)
作者信息
    1中北大学航空宇航学院　太原　030051
    2中北大学软件学院　太原　030051
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