Predefined-time sliding mode control for rigid spacecraft

刚性航天器的预定义时间滑模控制 Predefined-time sliding mode control for rigid spacecraft 1 2 first-author 赛华阳（1996-），男，河南泌阳人，博士研究生，2018年于西北农林科技大学获得学士学位，主要从事超声医疗机器人、机器人滑模控制等方面的研究。E-mail： saihuayang18@mails.ucas.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630229&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630233&type= 22.01332855 32.00400543 赛华阳 （1996-），男，河南泌阳人，博士研究生，2018年于西北农林科技大学获得学士学位，主要从事超声医疗机器人、机器人滑模控制等方面的研究。E-mail： saihuayang18@mails.ucas.ac.cn saihuayang18@mails.ucas.ac.cn 1 3 corresp E-mail： xuzhenbang@ciomp.ac.cn E-mail： xuzhenbang@ciomp.ac.cn 徐振邦（1982-），男，内蒙古通辽人，研究员，2005年、2010年于中国科学技术大学分别获得学士、博士学位，主要从事空间智能机器人、空间遥感器设计、振动控制方面的研究。E-mail： xuzhenbang@ciomp.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630245&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630252&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630248&type= 22.01332855 32.00395966 徐振邦 （1982-），男，内蒙古通辽人，研究员，2005年、2010年于中国科学技术大学分别获得学士、博士学位，主要从事空间智能机器人、空间遥感器设计、振动控制方面的研究。E-mail： xuzhenbang@ciomp.ac.cn xuzhenbang@ciomp.ac.cn 1 1 1 中科院长春光学精密机械与物理研究所 中科院空间光学系统在轨制造与;集成重点实验室，吉林 长春 130033 中科院长春光学精密机械与物理研究所 中科院空间光学系统在轨制造与;集成重点实验室，吉林 长春 130033 CAS Key Laboratory of On-orbit Manufacturing and Integration for Space Optics System， Changchun Institute of Optics， Fine Mechanics and Physics， CAS， Changchun 130033， China CAS Key Laboratory of On-orbit Manufacturing and Integration for Space Optics System， Changchun Institute of Optics， Fine Mechanics and Physics， CAS， Changchun 130033， China 中国科学院大学，北京 100049 中国科学院大学，北京 100049 University of Chinese Academy of Sciences， Beijing 100049， China University of Chinese Academy of Sciences， Beijing 100049， China 中国科学院大学材料与光电研究中心，北京 100049 中国科学院大学材料与光电研究中心，北京 100049 Materials and Optoelectronics Research Center， University of Chinese Academy of Sciences， Beijing 100049， China Materials and Optoelectronics Research Center， University of Chinese Academy of Sciences， Beijing 100049， China 针对刚性航天器在姿态跟踪控制中存在的系统不确定及外界干扰等问题，提出了一种预定义时间滑模控制器（PTSMC）。首先，给出了以四元数为姿态参数的航天器姿态跟踪控制系统，利用误差四元数和误差角速度设计了预定义时间滑模面。然后，考虑了航天器系统的不确定性和外界干扰设计了一种非保守上界的PTSMC，并通过边界层技术降低了系统抖动。最后，通过设计Lyapunov函数，证明了所提出的控制器的预定义时间稳定性和系统收敛时间上界的非保守性。仿真结果表明，刚性航天器的姿态跟踪误差精度可达1.5×10 -6 rad，角速度跟踪误差精度可达2×10 -6 rad/s。与现有的预定义时间控制器相比，所提出的控制器的稳定时间上限是更加非保守的，与传统PD控制和非奇异终端滑模控制相比，所提出的控制器具有更高的跟踪精度和鲁棒性。通过3自由度气浮平台的姿态跟踪实验进一步说明了控制方案的有效性，其中角度跟踪误差小于0.1 rad，位置跟踪误差小于0.2 m。 To minimize system uncertainty and external disturbance in attitude tracking control for rigid spacecraft， a predefined-time sliding mode controller （PTSMC） is proposed. First， the spacecraft attitude tracking system is developed with quaternion parameterization， and the predefined time sliding surface is designed using an error quaternion and error angular velocity. Then， considering the uncertainties and external disturbances of the spacecraft system， a PTSMC with a non conservative upper bound is designed， and the noise of the system is reduced using boundary layer technology. Finally， by designing the Lyapunov function， the predefined-time stability of the proposed controller and the non conservative upper bound of the system convergence are demonstrated. The simulation results show that using the proposed approach， the attitude tracking accuracy of rigid spacecraft can reach 1.5×10 - 6 rad， and the angular velocity tracking accuracy can reach 2×10 - 6 rad/s. Compared with the existing predefined time control and non singular terminal sliding mode control， the upper bound of the stabilization time of the proposed control is more non conservative and has higher tracking accuracy and robustness. The effectiveness of the control scheme is further illustrated by the attitude tracking experiment of the 3 DOF airborne platform. The angle tracking error is less than 0.1 rad， and the position tracking error is less than 0.2 m. 航天器 姿态跟踪 预定义时间控制 滑模控制 spacecraft attitude tracking predefined-time control sliding mode control 1 引　言 1 近年来，航天器的姿态跟踪问题由于在空间应用中的重要作用而得到了广泛的研究 ［ 1 1 - 2 2 1-2 ］ 。航天器的姿态调整是一个复杂的控制问题，因为其存在相互作用的非线性运动学和动力学模型，以及空间环境中不可预测的外界扰动 ［ 3 3 - 4 4 3-4 ］ 。 滑模控制（Sliding Mode Control， SMC）是处理具有不确定性和外界扰动的非线性系统的最有效的方法之一 ［ 5 5 ］ 。由于其对系统不确定性和外界干扰的强鲁棒性，SMC已被广泛应用到航天器的姿态控制中。但传统的SMC算法只能获得渐近稳定的结果，这意味着航天器的姿态跟踪误差需要很长的时间才能收敛到平衡点，而许多任务要求航天器实现快速的姿态调整 ［ 6 6 ］ 。为了满足这一要求，有限时间控制的概念被提出，它可以提高系统误差的收敛速度，使其在有限的时间内收敛。终端滑模控制就是一种典型的有限时间控制方法，其中，一种新型的非奇异终端滑模控制（Non-singular Terminal Sliding Mode Control， NTSMC）已经被应用到航天器的姿态调整中 ［ 7 7 ］ 。但有限时间控制方案的收敛时间依赖于系统的初始状态，因此难以提前获得航天器准确的姿态稳定时间界限。与有限时间控制方案不同的是，固定时间控制可以保证系统的收敛时间独立于系统的初始值，即系统收敛时间的上界是一个不依赖于系统初始状态的常数 ［ 8 8 ］ 。目前，一些固定时间的SMC也已经被成功地应用于航天器的姿态跟踪控制中 ［ 1 1 ， 9 9 ］ 。 虽然与有限时间控制相比，固定时间在系统稳定时间的建立上有显著的优势，但固定时间控制器的控制参数与稳定时间之间的关系往往是复杂的。为了进一步解决这个问题，一种更先进的控制概念，即预定义时间控制被提出 ［ 10 10 ］ 。预定义时间控制为被控系统提供了一种先进的稳定性特征，即通过调整预定义时间参数可以很容易地确定系统的稳定性上界，从而为系统的行为提供高度的确定性。在文献［ 10 10 - 12 12 10-12 ］中，预定义时间控制被用来和SMC相结合以提高一阶控制系统的鲁棒性，但这些方法仅能保证系统在SMC中到达阶段的预定义时间稳定性。在文献［ 13 13 - 15 15 13-15 ］中，预定义时间滑模控制的相关理论被进一步提出，并被拓展到二阶控制系统。此外，在文献［ 3 3 ］中，预定义时间控制被应用于航天器的姿态跟踪控制中。但上述预定义时间控制方案的稳定时间上界往往过于保守，这意味着控制器会远提前于所设定的稳定时间完成航天器的姿态调整。对于航天器姿态控制而言，过快的姿态调节会导致系统能量的浪费，甚至使控制力矩超过系统所允许的范围。 在本文中，我们提出了一种应用于刚性航天器姿态跟踪控制的新型预定义时间滑模控制（Predefined-Time Sliding Mode Control， PTSMC）方案用于解决现有预定义时间控制方案的稳定时间上界过于保守的问题。首先，我们设计了一种新型滑模变量和预定义时间滑模面，然后提出了一种新型PTSMC方案，并采用边界层技术降低了系统抖振。接着，通过Lyapunov函数证明了所提出的控制器的预定义时间稳定性和系统收敛上界的非保守性。最后通过仿真实验，分别与现有的预定义时间控制方案，NTSMC方案和比例微分（Proportional-Derivative， PD）控制进行了对比，并利用3自由度气浮平台进行了实验验证，证明了所提出控制方案的有效性和优越性。 2 刚性航天器模型和性质 1 如 图1 图1 所示，在建立航天器的姿态控制系统时常建立3个参考坐标系，包括惯性坐标系 F I ，参考坐标系 F O 和航天器本体坐标系 F B 。在本文中， F I 被选择为以地球为中心的惯性框架， F B 的原点位于被控航天器的质心，并围绕 F I 以 ω 0 ∈ <math id="M1"><mi mathvariant="bold">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630256&type= 2.62466669 2.79399991 + 的轨道速率运行。 F O 与 F B 有相同的原点，其滚转轴 X O 指向飞行器的飞行方向，俯仰轴 Y O 垂直于飞行器的运行轨道，偏航轴 Z O 指向地心。 刚性航天器的姿态可以通过航天器本体坐标系 F B 相对于惯性坐标系 F I 的姿态四元数进行表示，即（ q v ， q 4 ）∈ <math id="M2"><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>×</mo><mi mathvariant="bold">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630278&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630275&type= 11.17599964 2.96333337 。刚性航天器的运动学方程可以表示为： <math display="block" id="M3"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">q</mi><mo>˙</mo></mover></mrow><mrow><mi>v</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630281&type= 34.29000092 18.20333290 , 其中： <math id="M4"><mi mathvariant="bold-italic">ω</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630290&type= 10.15999985 3.38666677 表示航天器相对于本体坐标系 F B 的角速度， I 3 表示单位矩阵， <math id="M5"><msup><mrow><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></mrow><mrow><mo>×</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630297&type= 5.58799982 3.80999994 表示为由向量元素构建的偏斜对称矩阵，即： <math display="block" id="M6"><msup><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mo>×</mo></mrow></msup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced><mtext>  </mtext><mo>∀</mo><mi>a</mi><mo>=</mo><mo stretchy="false">[</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">]</mo><mtext> </mtext><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630306&type= 66.88666534 13.54666710 . 对于航天器的姿态跟踪问题，期望姿态的四元数可以表示为 <math id="M7"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mtext>  </mtext><msub><mrow><mi>q</mi></mrow><mrow><mi>d</mi><mn mathvariant="normal">4</mn></mrow></msub><msup><mrow><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>×</mo><mi mathvariant="bold">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630317&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630314&type= 37.76133347 5.07999992 ，它满足 <math id="M8"><msubsup><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>d</mi><mi>v</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>q</mi></mrow><mrow><mi>d</mi><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630321&type= 24.38400078 4.82600021 。令 <math id="M9"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi mathvariant="bold-italic">d</mi></mrow></msub><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630328&type= 11.59933376 4.23333359 表示为期望的角速度，则期望的运动学方程可以表示为： <math display="block" id="M10"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">q</mi><mo>˙</mo></mover></mrow><mrow><mi>d</mi><mi>v</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn 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mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630336&type= 38.69266510 18.20333290 . 相似地，将 <math id="M11"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><msubsup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msup><mrow><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630345&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630343&type= 28.02466965 5.07999992 表示为航天器姿态误差的四元数，其中 <math id="M12"><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>∈</mo><mi mathvariant="bold">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630349&type= 9.14400005 4.06400013 表示航天器本体坐标系 <math id="M13"><msub><mrow><mi>F</mi></mrow><mrow><mi>B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630357&type= 3.97933316 3.72533321 相对于参考坐标系 <math id="M14"><msub><mrow><mi>F</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630363&type= 4.23333359 3.72533321 的相对姿态。 <math id="M15"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630369&type= 11.17599964 4.23333359 表示角速度误差，航天器的误差姿态运动学方程可以表示为： <math display="block" id="M16"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo>˙</mo></mover></mrow><mrow><mi>v</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>e</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630375&type= 34.29000092 18.20333290 . 航天器的角速度误差可以通过下式给出： <math display="block" id="M17"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mi mathvariant="bold-italic">ω</mi><mo>-</mo><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630384&type= 22.26733208 3.72533321 ， 其中， <math id="M18"><mi mathvariant="bold-italic">C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630410&type= 2.53999996 2.79399991 表示从 <math id="M19"><msub><mrow><mi>F</mi></mrow><mrow><mi>O</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630401&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630398&type= 4.23333359 3.72533321 到 <math id="M20"><msub><mrow><mi>F</mi></mrow><mrow><mi>B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630404&type= 3.97933316 3.72533321 的坐标变换矩阵，并且矩阵 <math id="M21"><mi mathvariant="bold-italic">C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630410&type= 2.53999996 2.79399991 可以通过下式进行计算： <math display="block" id="M22"><mi mathvariant="bold-italic">C</mi><mo>=</mo><mo stretchy="false">(</mo><msubsup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630416&type= 53.17066193 4.65666676 . 此外，航天器的姿态动力学方程可以表示为： <math display="block" id="M23"><mi mathvariant="bold-italic">J</mi><mover accent="true"><mi mathvariant="bold-italic">ω</mi><mo>˙</mo></mover><mo>+</mo><msup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mo>×</mo></mrow></msup><mi mathvariant="bold-italic">J</mi><mi mathvariant="bold-italic">ω</mi><mo>=</mo><mi mathvariant="bold-italic">u</mi><mo>+</mo><mi mathvariant="bold-italic">d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630420&type= 31.91933441 3.13266683 ， 其中： <math id="M24"><mi mathvariant="bold-italic">J</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>×</mo><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630426&type= 12.95400047 3.38666677 是航天器关于本体坐标系 <math id="M25"><msub><mrow><mi>F</mi></mrow><mrow><mi>B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630431&type= 3.97933316 3.72533321 的惯性矩阵， <math id="M26"><mi mathvariant="bold-italic">u</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630438&type= 9.65200043 3.38666677 和 <math id="M27"><mi mathvariant="bold-italic">d</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630442&type= 9.90600014 3.38666677 分别表示系统的控制力矩和未知但有界的扰动力矩。 结合式（ 3 3 ），（ 5 5 ）和（7），航天器姿态的误差运动学和动力学方程可以被写为： <math display="block" id="M28"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo>˙</mo></mover></mrow><mrow><mi>v</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ω</mi><mo>˙</mo></mover></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>×</mo></mrow></msup><mi mathvariant="bold-italic">J</mi><mi mathvariant="bold-italic">ω</mi><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mi mathvariant="bold-italic">J</mi><mi mathvariant="bold-italic">ω</mi><mo>-</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">u</mi><mo>-</mo><mi mathvariant="bold-italic">d</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630451&type= 65.95532990 17.86466599 . 考虑到航天器模型的不确定性，可以假设： <math display="block" id="M29"><mi mathvariant="bold-italic">J</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mi mathvariant="bold-italic">J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630457&type= 18.62666702 3.80999994 ， 其中， <math id="M30"><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630463&type= 2.96333337 3.72533321 和 <math id="M31"><mi mathvariant="normal">Δ</mi><mi mathvariant="bold-italic">J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630469&type= 4.23333359 2.79399991 分别表示已知的系统的名义惯性矩阵和惯性矩阵 <math id="M32"><mi mathvariant="bold-italic">J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630474&type= 1.77800000 2.79399991 的不确定部分。 式（8） 式（8） 中给出的动力学方程可以被写为： <math display="block" id="M33"><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ω</mi><mo>˙</mo></mover></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>×</mo></mrow></msup><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="bold-italic">ω</mi><mo>+</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="bold-italic">ω</mi><mo>-</mo><mi mathvariant="bold-italic">u</mi><mo>-</mo><mi mathvariant="bold-italic">ρ</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630479&type= 52.15466690 10.41399956 ， 其中，系统耦合不确定部分 <math id="M34"><mi mathvariant="bold-italic">ρ</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630485&type= 9.65200043 3.89466691 被定义为如下形式： <math display="block" id="M35"><mi mathvariant="bold-italic">ρ</mi><mo>=</mo><mi mathvariant="bold-italic">d</mi><mo>-</mo><mi mathvariant="normal">Δ</mi><mi mathvariant="bold-italic">J</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><mi mathvariant="bold-italic">J</mi><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>-</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>×</mo></mrow></msup><mi mathvariant="normal">Δ</mi><mi mathvariant="bold-italic">J</mi><mi mathvariant="bold-italic">ω</mi><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mi mathvariant="normal">Δ</mi><mi mathvariant="bold-italic">J</mi><mi mathvariant="bold-italic">ω</mi><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630488&type= 80.85666656 5.07999992 参考文献［ 1 1 ］，系统耦合不确定部分的上界可以通过下式确定： <math display="block" id="M36"><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ρ</mi></mrow></mfenced><mo>≤</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630492&type= 43.09533310 5.07999992 . 本文要解决的问题如下：通过设计控制输入 <math id="M37"><mi mathvariant="bold-italic">u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630496&type= 2.11666679 2.79399991 ，使包含系统不确定性和外界干扰的刚性航天器的姿态误差 <math id="M38"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630500&type= 2.79399991 3.72533321 和角速度跟踪误差 <math id="M39"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630504&type= 3.55599999 3.72533321 在给定的时间内收敛到零。 3 预定义时间稳定性的基本理论 1 首先介绍关于有限时间、固定时间和预定时间稳定性的一些的定义和引理以用于控制器预定义时间稳定性的证明。 考虑一个非线性系统： <math display="block" id="M40"><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>;</mo><mi mathvariant="bold-italic">ρ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630511&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630507&type= 36.32199860 4.91066647 ， 其中： <math id="M41"><mi>x</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630515&type= 9.56733322 3.38666677 表示系统状态，向量 <math id="M42"><mi mathvariant="bold-italic">ρ</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>b</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630525&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630523&type= 9.65200043 3.89466691 代表系统（13）的系统参数。函数 <math id="M43"><mi>f</mi><mo>:</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630529&type= 15.32466698 3.89466691 可以被认为是非线性和连续的，并且它的原点被假设为系统（13）的平衡点，即 <math id="M44"><mi>f</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>;</mo><mi mathvariant="bold-italic">ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630537&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630534&type= 18.20333290 3.97933316 。 定义1.全局有限时间稳定性 ［ 16 16 ］ ：如果系统（13）的原点是全局渐近稳定的，且系统（13）的任何解 <math id="M45"><mi mathvariant="bold-italic">x</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630544&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630540&type= 11.93799973 4.48733330 在某个有限时刻达到平衡点，即 <math id="M46"><mo>∀</mo><mi>t</mi><mo>≥</mo><mi>T</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630548&type= 24.63800049 4.48733330 ，则系统（13）的平衡点被认为是全局有限时间稳定的，其中 <math id="M47"><mi>T</mi><mo>:</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><msub><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>⋃</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630558&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630555&type= 26.50066566 4.74133301 被称为稳定时间函数。 定义2.固定时间稳定性 ［ 8 8 ］ ：如果系统（13）的原点是全局有限时间稳定的且稳定时间函数有界，即 <math id="M48"><mo>∃</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn><mo>:</mo><mo>∀</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>T</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630561&type= 52.57799911 4.65666676 ，那么系统（13）的平衡点是固定时间稳定的。 定义3.预定义时间稳定性 ［ 17 17 ］ ：对于系统（13）的系统参数 <math id="M49"><mi mathvariant="bold-italic">ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630569&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630566&type= 2.03200006 3.64066648 和常数 <math id="M50"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ρ</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630572&type= 24.29933357 4.82600021 ，如果系统（13）的平衡点是固定时间稳定的，且稳定时间函数 <math id="M51"><mi>T</mi><mo>:</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi mathvariant="bold">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630577&type= 15.57866764 3.21733332 满足以下条件： <math display="block" id="M52"><mi>T</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub><mtext>   </mtext><mo>∀</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630583&type= 33.61266708 4.65666676 ， 则系统（13）的平衡点被认为是预定义时间稳定的，且 <math id="M53"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630607&type= 3.55599999 3.72533321 叫做预定义时间。此外，如果稳定时间函数满足 <math id="M54"><munder><mrow><mi>s</mi><mi>u</mi><mi>p</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>∈</mo><mi mathvariant="bold">R</mi></mrow></munder><mi>T</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630595&type= 23.79133224 7.02733374 ，那么 <math id="M55"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630607&type= 3.55599999 3.72533321 被称作强预定义时间，否则， <math id="M56"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630607&type= 3.55599999 3.72533321 被称作弱预定义时间。 理论1. ［ 13 13 ］ 如果存在满足以下条件的连续正定径向无界函数 <math id="M57"><mi>V</mi><mo>:</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi mathvariant="bold">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630616&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630613&type= 16.00200081 3.21733332 ： <math display="block" id="M58"><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo><mo>≤</mo><mo>-</mo><mfrac><mrow><mi>γ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mi>V</mi><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>β</mi><mi>V</mi><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630622&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630619&type= 56.72666931 8.80533314 ， 其中： <math id="M59"><mi mathvariant="bold-italic">x</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>\</mo><mo stretchy="false">{</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630625&type= 18.03400040 3.80999994 ； <math id="M60"><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630633&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630630&type= 16.42533302 3.64066648 为满足 <math id="M61"><mi>k</mi><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630636&type= 10.15999985 3.64066648 ， <math id="M62"><mi>k</mi><mi>q</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630645&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630642&type= 10.15999985 3.64066648 的正常数。那么，系统（13）的平衡点是预定义时间稳定的，预定义时间为 <math id="M63"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630648&type= 3.55599999 3.72533321 。如果设参数向量 <math id="M64"><mi mathvariant="bold-italic">ρ</mi><mo>=</mo><msup><mrow><mo stretchy="false">[</mo><mi>α</mi><mtext> </mtext><mi>β</mi><mtext> </mtext><mi>p</mi><mtext> </mtext><mi>q</mi><mtext> </mtext><mi>k</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630657&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630654&type= 31.75000000 4.23333359 ，则预定义时间函数满足 <math id="M65"><msub><mrow><mi>T</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630659&type= 15.74800014 4.82600021 ，其中 <math id="M66"><mi>γ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630664&type= 7.61999989 3.97933316 可以通过下式计算得出： <math display="block" id="M67"><mi>γ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>q</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>α</mi></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630670&type= 45.88933182 10.58333302 ， 其中： <math id="M68"><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630676&type= 9.82133293 4.14866638 分别是满足 <math id="M69"><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>k</mi><mi>p</mi></mrow><mrow><mi>q</mi><mo>-</mo><mi>p</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630683&type= 19.64266586 8.80533314 和 <math id="M70"><msub><mrow><mi>m</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>k</mi><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi><mo>-</mo><mi>p</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630689&type= 19.64266586 8.80533314 的正常数； <math id="M71"><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630694&type= 6.77333355 3.55599999 是gamma函数，满足 <math id="M72"><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>t</mi></mrow></msup><msup><mrow><mi>t</mi></mrow><mrow><mi>z</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="normal">d</mi><mi>t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630699&type= 31.91933441 7.87400007 。 4 预定义时间滑模控制器设计与稳定性分析 1 4.1　预定义时间滑模控制器设计 2 首先，设计如下滑模变量： <math display="block" id="M73"><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>=</mo><mo>⌊</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><msup><mrow><mo>⌉</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mo>⌊</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo>⌉</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>β</mi><mo>⌊</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo>⌉</mo></mrow><mrow><mi>q</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630703&type= 55.03333664 8.80533314 ， 其中： <math id="M74"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630708&type= 3.55599999 3.72533321 和 <math id="M75"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630712&type= 2.79399991 3.72533321 分别是航天器的角速度跟踪误差和姿态跟踪误差； <math id="M76"><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630716&type= 12.95400047 3.64066648 分别是定义的正常数且满足 <math id="M77"><mn mathvariant="normal">0</mn><mo><</mo><mi>p</mi><mo><</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>q</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630720&type= 25.56933212 3.64066648 ； <math id="M78"><mi>γ</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>q</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630723&type= 13.63133430 4.57200003 分别是满足 <math id="M79"><mi>γ</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>m</mi><mi>q</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>q</mi><mo>-</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mfrac><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>α</mi></mrow><mrow><mi>β</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630727&type= 40.55533600 10.58333302 ， <math id="M80"><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>k</mi><mi>p</mi></mrow><mrow><mi>q</mi><mo>-</mo><mi>p</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630732&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630730&type= 15.15533447 8.80533314 ， <math id="M81"><msub><mrow><mi>m</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>k</mi><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi><mo>-</mo><mi>p</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630735&type= 19.64266586 8.80533314 关系的正常数； <math id="M82"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630741&type= 4.48733330 3.72533321 是定义的滑动阶段的预定义时间常数； <math id="M83"><mo>⌊</mo><mo>·</mo><mo>⌉</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630745&type= 4.82600021 3.47133350 表示 <math id="M84"><mo stretchy="false">|</mo><mo>·</mo><mo stretchy="false">|</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630750&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630748&type= 13.12333298 3.97933316 。 接着，定义如下的预定义时间滑模面： <math display="block" id="M85"><mi mathvariant="bold-italic">s</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>+</mo><mo>⌊</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo>⌉</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630752&type= 22.09799957 6.60400009 . 根据上述的滑模变量（17）和滑模面（18），所提出的PTSMC可以表示如下： <math display="block" id="M86"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">u</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mfrac><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo><mi mathvariant="bold-italic">η</mi><mo>-</mo><mi mathvariant="bold-italic">a</mi></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo></mrow></mfrac><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="bold-italic">η</mi><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>+</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>×</mo></mrow></msup><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="bold-italic">ω</mi><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="bold-italic">ω</mi></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">η</mi><mo>=</mo><mfrac><mrow><mi>γ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">s</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>β</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">s</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mi>p</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mi>β</mi><mi>q</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630757&type= 59.85933304 55.37200165 ， 其中： <math id="M87"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630762&type= 4.48733330 3.72533321 表示系统到达阶段的预定义时间常数； <math id="M88"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630768&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630766&type= 12.95400047 4.14866638 是定义的正常数并满足不等 式（12） 式（12） ，对于向量 <math id="M89"><mi mathvariant="bold-italic">x</mi><mo>∈</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630770&type= 9.73666668 3.38666677 ， <math id="M90"><mo stretchy="false">|</mo><mi mathvariant="bold-italic">x</mi><mo stretchy="false">|</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630774&type= 3.47133350 3.47133350 表示分别对向量的每个元素取绝对值。 图2 图2 给出了（17）~（19）中所提出的PTSMC的控制信号流程图，其中 <math id="M91"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630778&type= 3.04800010 3.72533321 是使系统在预定义时间内收敛的滑模控制器， <math id="M92"><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630781&type= 3.04800010 3.72533321 是用于抵抗系统不确定性和外界扰动的鲁棒控制器。 4.2　稳定性分析 2 对于所提出的PTSMC的稳定性分析可以分为到达阶段和滑动阶段进行分析。同时，为了简化证明过程，我们将 式（10） 式（10） 重写为以下形式： <math display="block" id="M93"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ω</mi><mo>˙</mo></mover></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">u</mi><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mi mathvariant="bold-italic">ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630792&type= 43.68799973 5.07999992 ， 其中， <math id="M94"><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>×</mo></mrow></msup><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="bold-italic">ω</mi><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="bold-italic">ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630796&type= 71.45866394 5.07999992 ， <math id="M95"><mi>b</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630800&type= 17.61066628 4.65666676 。 4 . 2 . 1　系统到达阶段的稳定性分析 3 所提出的滑模面（18）关于时间的导数为： <math display="block" id="M96"><mtable><mtr><mtd><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>˙</mo></mover><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ω</mi><mo>˙</mo></mover></mrow><mrow><mi>e</mi></mrow></msub><mo>+</mo><mfrac><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">ω</mi><mo>˙</mo></mover></mrow><mrow><mi>e</mi></mrow></msub></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mfrac><mrow><msup><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mi>p</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mi>β</mi><mi>q</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630802&type= 45.88933182 29.63333130 . 带入 式（20） 式（20） ，可以得到： <math display="block" id="M97"><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>˙</mo></mover><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">u</mi><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mi mathvariant="bold-italic">ρ</mi><mo>+</mo><mfrac><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mo>+</mo><mi>b</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">ω</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">u</mi><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mi mathvariant="bold-italic">ρ</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mfrac><mrow><msup><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mi>p</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mi>β</mi><mi>q</mi><mo stretchy="false">|</mo><msup><mrow><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">|</mo><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630805&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630804&type= 142.32467651 16.42533302 . 将 式（22） 式（22） 带入所提出的控制输入（19），有： <math display="block" id="M98"><mtable><mtr><mtd><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>˙</mo></mover><mo>=</mo><mfrac><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo><mi mathvariant="bold-italic">η</mi><mo>-</mo><mi mathvariant="bold-italic">a</mi></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo></mrow></mfrac><mo>-</mo><mi mathvariant="bold-italic">η</mi><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mi mathvariant="bold-italic">ρ</mi><mo>+</mo><mfrac><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo><mfenced separators="|"><mrow><mfrac><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo><mi mathvariant="bold-italic">η</mi><mo>-</mo><mi mathvariant="bold-italic">a</mi></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mo>+</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo></mrow></mfrac><mo>-</mo><mi mathvariant="bold-italic">η</mi><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mi mathvariant="bold-italic">ρ</mi></mrow></mfenced></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mfrac><mrow><msup><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mi>p</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mi>β</mi><mi>q</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630806&type= 136.90600586 40.72467041 . 对 式（23） 式（23） 进一步化简可以得到： <math display="block" id="M99"><mtable columnalign="left"><mtr><mtd><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>˙</mo></mover><mo>=</mo><mo>-</mo><mi mathvariant="bold-italic">η</mi><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mi mathvariant="bold-italic">ρ</mi><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mi mathvariant="bold-italic">ρ</mi><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630808&type= 52.32400131 17.61066628 . 根据 式（12） 式（12） 和（19），有 <math id="M100"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>≥</mo><mi mathvariant="bold-italic">ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630810&type= 47.32866669 4.74133301 ，因此 式（24） 式（24） 可以推出以下不等式： <math display="block" id="M101"><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>˙</mo></mover><mo>≤</mo><mo>-</mo><mi mathvariant="bold-italic">η</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>γ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">s</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>β</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">s</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630812&type= 58.16600037 8.80533314 . 考虑一个Lyapunov候选函数为 <math id="M102"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">|</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">|</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630814&type= 16.76399994 4.65666676 ，则它的一阶导数为： <math display="block" id="M103"><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mover accent="true"><mi mathvariant="bold-italic">s</mi><mo>˙</mo></mover><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo><mo>≤</mo><mo>-</mo><mfrac><mrow><mi>γ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">s</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>β</mi><mo stretchy="false">|</mo><mi mathvariant="bold-italic">s</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><mi>γ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>β</mi><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630816&type= 60.45199966 19.47333336 . 根据理论1，我们可以得到，系统在到达阶段是预定义时间稳定的，且预定义时间为 <math id="M104"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630835&type= 4.48733330 3.72533321 。 4 . 2 . 2　系统滑动阶段的稳定性分析 3 一旦系统的跟踪轨迹被约束到滑模面上，即当 <math id="M105"><mi>t</mi><mo>></mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630822&type= 10.49866676 3.72533321 时 <math id="M106"><mi mathvariant="bold-italic">s</mi><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630825&type= 8.12800026 2.79399991 ，系统进入滑动阶段，根据 式（17） 式（17） 和（18），可以得到： <math display="block" id="M107"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mi>γ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>β</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630828&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630827&type= 54.10200119 8.80533314 . 设滑动阶段的Lyapunov候选函数为 <math id="M108"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo stretchy="false">|</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630829&type= 19.55800056 4.65666676 ，于是可以得到： <math display="block" id="M109"><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mfrac><mrow><mi>γ</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>β</mi><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630834&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630832&type= 61.89133835 8.80533314 . 同样地，根据理论1，系统在预定义时间 <math id="M110"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630835&type= 4.48733330 3.72533321 内沿滑模面收敛到平衡点。 根据上述证明，我们可以得到，所提出的控制器可以在系统不确定和存在外界扰动的情况下实现刚性航天器的预定义时间收敛，其收敛时间为 <math id="M111"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630837&type= 22.26733208 3.80999994 。 备注1. 由于存在系统的耦合不确定性 <math id="M112"><mi mathvariant="bold-italic">ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630839&type= 2.03200006 3.64066648 ，即使系统的初始误差 <math id="M113"><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630842&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630841&type= 11.17599964 3.72533321 ，系统误差也会在预定义时间 <math id="M114"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630851&type= 3.55599999 3.72533321 之前收敛到原点，即 <math id="M115"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630851&type= 3.55599999 3.72533321 为弱预定义时间。同时，我们可以考虑一种特殊情况，当 <math id="M116"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ρ</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630847&type= 13.03866673 4.82600021 成立时， 式（25） 式（25） 和（26）中的等号成立，此时 <math id="M117"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630849&type= 3.55599999 3.72533321 为系统的最小收敛时间上界，根据定义3， <math id="M118"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630851&type= 3.55599999 3.72533321 为强预定义时间。相较于现有的预定义时间控制方法，如文献［ 5 5 - 9 9 5-9 ］中所提出的控制方案，本文中的Lyapunov函数的导数仅在 式（25） 式（25） 考虑耦合不确定时进行了缩放，而其他控制方案则在推导过程中进行了多次缩放，因此本文中所提出收敛时间上界是更不保守的。 备注2. 考虑到符号函数 <math id="M119"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630854&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630853&type= 10.66800022 3.97933316 会导致系统控制输入的抖振，为了改善符号函数导致的系统抖振，将所提出的控制器（19）中的 <math id="M120"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630855&type= 11.17599964 3.97933316 修改为 <math id="M121"><mfrac><mrow><mi mathvariant="bold-italic">s</mi></mrow><mrow><mo stretchy="false">|</mo><mi mathvariant="bold-italic">s</mi><mo stretchy="false">|</mo><mo>+</mo><mn mathvariant="normal">0.01</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630858&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630857&type= 15.91733265 8.63599968 。同时，将 式（19） 式（19） 中的函数 <math id="M122"><mi mathvariant="bold-italic">a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630859&type= 1.94733346 2.79399991 修改为如下形式： <math display="block" id="M123"><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msup><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>α</mi><mi>p</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>p</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>+</mo><mi>β</mi><mi>q</mi><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo stretchy="false">)</mo><mtext>  </mtext><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo><mo>></mo><mi>ε</mi></mtd></mtr><mtr><mtd><mi>ε</mi><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>×</mo><mn mathvariant="normal">1</mn></mrow></msub><mtext>   </mtext><mo stretchy="false">|</mo><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy="false">|</mo><mo>≤</mo><mi>ε</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630861&type= 62.48400116 15.57866764 ， 其中， <math id="M124"><mi>ε</mi><mo>∈</mo><mi mathvariant="bold">R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630863&type= 7.87400007 3.13266683 是一个很小的正常数。 5 实验验证与分析 1 在本章节中，为了证明所提出控制方案在系统收敛时间的保守性和跟踪误差精度等方面的优越性，分别与现有的预定义时间控制方案，NTSMC方案和传统PD控制方案进行仿真对比。所有的数值仿真实验均基于Simulink/MATLAB 2020a进行，基本采样时间设为10 - 4 s。 与文献［ 7 7 ］中给出的航天器系统的参数相同，假设名义惯性矩阵为 <math id="M125"><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630865&type= 2.96333337 3.72533321 =［20　1.2　0.9；1.2　17　1.4；0.9　1.4　15］ kg·m 2 ，惯性矩阵的不确定度设为 <math id="M126"><mi mathvariant="normal">Δ</mi><mi mathvariant="bold-italic">J</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630867&type= 4.23333359 2.79399991 =diag（2，2，3） kg·m 2 。期望的角速度设为 <math id="M127"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630869&type= 3.89466691 3.72533321 =0.05［sin（π t /100） sin（2π t /100） sin（3π t /100）］ rad/s，外界扰动设为 <math id="M128"><mi mathvariant="bold-italic">d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630872&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630871&type= 2.28600001 2.87866688 =［0.1sin（ t ） 0.2 sin（1.2 t ） 0.3sin（1.5 t ）］ N·m。系统的初始状态设为 <math id="M129"><msub><mrow><mi mathvariant="bold-italic">q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630874&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630873&type= 3.13266683 4.57200003 =［0.3　 - 0.2　 - 0.3　0.883 2］ T rad和 <math id="M130"><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630876&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630875&type= 3.72533321 3.72533321 =［0.06　 - 0.04　0.05］ T rad/s。对于所提出的PTSMC方案，其控制参数分别设为 <math id="M131"><mi>p</mi><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630877&type= 8.55133343 3.64066648 ， <math id="M132"><mi>q</mi><mo>=</mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630880&type= 8.55133343 3.64066648 ， <math id="M133"><mi>k</mi><mo>=</mo><mn mathvariant="normal">0.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630883&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630882&type= 12.19200039 2.79399991 ， <math id="M134"><mi>α</mi><mo>=</mo><mn mathvariant="normal">4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630884&type= 8.63599968 2.79399991 ， <math id="M135"><mi>β</mi><mo>=</mo><mn mathvariant="normal">0.25</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630886&type= 14.39333248 3.64066648 ， <math id="M136"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630890&type= 20.57399940 3.72533321 ， <math id="M137"><mi>ε</mi><mo>=</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630894&type= 12.86933327 3.04800010 ， <math id="M138"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1.8</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630898&type= 13.63133430 3.72533321 ， <math id="M139"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">17.2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630902&type= 15.49400043 3.72533321 ， <math id="M140"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630907&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630905&type= 9.73666668 3.72533321 。 5.1　与现有的预定义时间控制方案的对比仿真实验 2 考虑文献［ 14 14 ］中所提出的二阶预定义时间滑模控制器（Second Order Predefined-Time Sliding Mode Control， SOPTSMC）： <math display="block" id="M141"><mtable><mtr><mtd><mi>u</mi><mo>=</mo><mo>-</mo><msubsup><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">|</mo><mi>σ</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msup><mo stretchy="false">)</mo><mo>⌊</mo><mi>σ</mi><msup><mrow><mo>⌉</mo></mrow><mrow><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msup><mo>-</mo><mi>k</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>σ</mi><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo>⌊</mo><mo>⌊</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo>⌉</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">2</mn><msubsup><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><mo stretchy="false">)</mo><mo>⌊</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo>⌉</mo></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msup><msup><mrow><mo>⌉</mo></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630909&type= 123.86732483 13.29266548 ， 其中， <math id="M142"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>/</mo><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></msup><mi mathvariant="normal">Γ</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>/</mo><mn mathvariant="normal">2</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>/</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630913&type= 49.02199936 10.41399956 ， <math id="M143"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mfrac><mrow><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></msubsup><mi mathvariant="normal">Γ</mi><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>/</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630917&type= 48.76800156 10.32933331 ，控制参数分别满足 <math id="M144"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630923&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630921&type= 11.26066685 3.72533321 ， <math id="M145"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630927&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630925&type= 9.73666668 4.57200003 ， <math id="M146"><mn mathvariant="normal">1</mn><mo>≤</mo><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630932&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630930&type= 16.42533302 4.82600021 ， <math id="M147"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630934&type= 11.26066685 3.72533321 ， <math id="M148"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630940&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630938&type= 9.82133293 3.72533321 ， <math id="M149"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630942&type= 9.82133293 4.57200003 ， <math id="M150"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630946&type= 9.73666668 4.57200003 ， <math id="M151"><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630952&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630950&type= 9.73666668 4.57200003 ， <math id="M152"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630954&type= 13.12333298 4.57200003 。 为了公平比较，修改控制器（30）中的 <math id="M153"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630960&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630958&type= 1.69333339 2.79399991 为本文中所提出的 <math id="M154"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630961&type= 2.79399991 3.72533321 ， <math id="M155"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630965&type= 11.59933376 3.97933316 修改为本文中在备注2中所设定的 <math id="M156"><mfrac><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">|</mo><mi>σ</mi><mo stretchy="false">|</mo><mo>+</mo><mn mathvariant="normal">0.01</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630971&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630969&type= 16.25600052 8.63599968 。其他参数参考文献［ 14 14 ］中给出的相应参数值，即 <math id="M157"><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1.2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630973&type= 13.54666710 4.57200003 ， <math id="M158"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630977&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630975&type= 9.73666668 4.57200003 ， <math id="M159"><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630981&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630979&type= 13.54666710 4.57200003 ， <math id="M160"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630983&type= 9.73666668 4.57200003 ， <math id="M161"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630988&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630986&type= 14.56266594 3.97933316 ， <math id="M162"><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630991&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630990&type= 9.82133293 4.57200003 将其预定义参数设为 <math id="M163"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630992&type= 20.57399940 3.72533321 。 对比仿真结果如图 3 3 ~ 5 5 所示。 图3 图3 和 图4 图4 分别表示与SOPTSMC控制方案相比的姿态跟踪误差和角速度跟踪误差，从图中可以看出，给定相同的预定义收敛时间 <math id="M164"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">6</mn><mtext> </mtext><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630995&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630994&type= 31.24200058 3.89466691 ，所提出控制方案中的跟踪误差在4 s后收敛到零，而SOPTSMC方案的实际收敛时间小于3 s。一般而言，我们总是希望航天器系统跟踪误差的实际收敛时间更接近于所设定的预定义收敛时间，这将有利于系统收敛时间的合理设置，因为更快的收敛时间往往意味着更多的能量消耗。 图5 图5 显示了控制力矩的对比，从图中可以看出，所提出控制方案的初始力矩约为50 N，远小于SOPTSMC方案中的初始输出力矩。此外，所提出的控制方案的控制力矩更加平滑，抖振更小，这对于刚性航天器的姿态跟踪控制是更有利的。 5.2　与NTSMC方案和传统PD控制方案的对比仿真实验 2 参考文献［ 7 7 ］中的所提出的NTSMC方案 <math display="block" id="M165"><mtable columnalign="left"><mtr><mtd><mi>u</mi><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>×</mo></mrow></msup><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>ω</mi><mo>-</mo><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mi>C</mi><msub><mrow><mi>ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>-</mo><mi>C</mi><msub><mrow><mover accent="true"><mi>ω</mi><mo>˙</mo></mover></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>λ</mi><msub><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>L</mi><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mn 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stretchy="false">|</mo></mrow><mrow><mi>r</mi></mrow></msup><mi>s</mi><mi>i</mi><mi>g</mi><mi>n</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext>                  </mtext><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo><mo>≥</mo><mi>δ</mi></mtd></mtr><mtr><mtd><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo stretchy="false">|</mo><mi>x</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msup><mtext> </mtext><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mtext>   </mtext><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo><mo><</mo><mi>δ</mi></mtd></mtr></mtable></mrow></mfenced></mtd></mtr><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo 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stretchy="false">(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mo stretchy="false">|</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><mrow><mi>T</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631009&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631007&type= 67.30999756 67.39466858 ， 其中：控制参数满足 <math id="M166"><mn mathvariant="normal">0</mn><mo><</mo><mi>r</mi><mo><</mo><mn mathvariant="normal">1</mn><mo><</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>≤</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631011&type= 29.12533188 4.23333359 ， <math id="M167"><mi>a</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>r</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><msup><mrow><mi>δ</mi></mrow><mrow><mi>r</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631015&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631014&type= 23.96066666 8.80533314 ， <math id="M168"><mi>b</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><msup><mrow><mi>δ</mi></mrow><mrow><mi>r</mi><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631016&type= 24.55333328 8.80533314 ， <math id="M169"><mn mathvariant="normal">0</mn><mo><</mo><mi>δ</mi><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631018&type= 15.32466698 2.79399991 ， <math id="M170"><mn mathvariant="normal">0</mn><mo><</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631021&type= 16.51000023 3.72533321 。同样地，为了公平地进行比较，将控制器（31）中的 <math id="M171"><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631024&type= 11.93799973 3.97933316 修改为 <math id="M172"><mfrac><mrow><mi>S</mi></mrow><mrow><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo><mo>+</mo><mn mathvariant="normal">0.01</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631027&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631026&type= 16.67933464 8.63599968 ，其他控制参数仍采用参考文献［ 7 7 ］中给出的控制参数 <math id="M173"><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1,1</mn><mo>,</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631028&type= 27.85533524 4.82600021 ， <math id="M174"><msub><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mn mathvariant="normal">20,5</mn><mo>,</mo><mn mathvariant="normal">10</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631030&type= 31.75000000 4.82600021 ， <math id="M175"><mi>λ</mi><mo>=</mo><mn mathvariant="normal">1.3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631033&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631032&type= 12.27666664 2.79399991 ， <math id="M176"><mi>r</mi><mo>=</mo><mn mathvariant="normal">0.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631034&type= 12.02266598 2.79399991 ， <math id="M177"><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631036&type= 13.20800018 3.72533321 ， <math id="M178"><mi>δ</mi><mo>=</mo><mn mathvariant="normal">0.01</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631039&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631038&type= 14.39333248 2.79399991 ， <math id="M179"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631040&type= 13.71599960 3.72533321 ， <math id="M180"><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi mathvariant="normal">、</mi><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">、</mi><msub><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631043&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631042&type= 13.29266548 3.97933316 与本方案中给定的参数值相同。 参考文献［ 18 18 ］中给出的PD控制方案： <math display="block" id="M181"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">u</mi><mo>=</mo><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">k</mi></mrow><mrow><mi>p</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">k</mi></mrow><mrow><mi>d</mi></mrow></msub><msub><mrow><mi>e</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msubsup><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>e</mi></mrow><mrow><mo>×</mo></mrow></msubsup><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>-</mo></mtd></mtr><mtr><mtd><msubsup><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="bold-italic">C</mi><msub><mrow><mi mathvariant="bold-italic">ω</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>×</mo></mrow></msup><msub><mrow><mi mathvariant="bold-italic">J</mi></mrow><mrow><mn 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631046&type= 3.04800010 3.72533321 和 <math id="M183"><msub><mrow><mi mathvariant="bold-italic">k</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631049&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631048&type= 3.21733332 3.72533321 为给定的控制参数，分别定义 <math id="M184"><msub><mrow><mi mathvariant="bold-italic">k</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">6</mn><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">k</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">8</mn><msub><mrow><mi mathvariant="bold-italic">I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631051&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23631050&type= 26.75466537 4.14866638 。 图6 图6 和 图7 图7 分别显示了所提出的控制方案与PD控制和NTSMC方案的姿态跟踪误差和角速度误差，从图中可以看出，PD控制中的姿态跟踪误差渐进地收敛到零点。同时，由于PD控制对系统的耦合不确定性不具有鲁棒性，因此它有较差的误差跟踪精度。相较于NTSMC方案，所提出的控制方案中系统的跟踪误差精度要远高于NTSMC方案，其姿态跟踪误差精度高于1.5×10 - 6 rad，角速度跟踪误差精度高于2×10 - 6 rad/s。最重要的是，PD控制的跟踪误差是渐进收敛的，它的收敛时间趋近于无穷大，NTSMC是有限时间收敛的，但它的收敛时间取决于系统的初始状态。而本文中所提出的PTSMC方案的收敛时间可以通过控制器中的预定义参数直接进行调节，对于航天器的姿态控制而言，这有利于它在规定的时间内完成姿态调整。 总之，仿真结果表明，本文所提出的PTSMC方案相较于目前存在的预定义时间控制方案，其收敛时间的上界更加不保守。相较于现有的NTSMC和PD控制方案，有更高的跟踪精度和鲁棒性，并具有预定义时间收敛的特性。同时，在所提出的控制器中只需要采集航天器的实际姿态与角速度，这对于航天器的姿态控制是可实现的，可以被应用在航天器的实际姿态控制中。 5.3　基于三自由度气浮平台的实验验证 2 由于在地面难以开展空间状态的飞行器姿态跟踪实验，为了进一步验证所提出的控制方案在实际应用中的有效性，本章节利用本课题组研制的三自由度自由飞行机器人气浮式平台 ［ 19 19 ］ 模拟飞行器姿态控制，进行了多次姿态跟踪控制实验，实验装置如 图8 图8 所示。该装置的运动是一个刚体的平面运动，包括平面 x ， y 轴方向的移动，和一个绕自身的转动量 ϕ 。 为了测试文中所提出的控制方案，分别进行了四次关于直线轨迹的跟踪实验，同时，在直线轨迹的跟踪过程中伴随平台角度的跟踪。四次实验的轨迹跟踪结果如 图9 图9 所示，实验均在气浮平台所在的大理石平台上进行。可以看出，四次实验均获得了较高的位置跟踪精度。对于轨迹1，在跟踪轨迹末端位置误差变大，这是由于轨迹1中气浮平台运行时间较长，平台后期供气不足导致的。对于轨迹1和轨迹2，气浮平台的初始位置与命令轨迹均有较大误差，但气浮平台在文中所提出的控制方案的作用下，均能迅速地跟踪到指令轨迹。 进一步地，在四次跟踪实验中，气浮平台的关于 x ， y 轴和转动量 ϕ 随时间的指令变化量与实际变化量如 图10 图10 所示。在每次实验中，控制器的预定义时间被设为20 s，从图中可以看出，各自由度的姿态跟踪误差大约均在10 s内收敛，这说明了所提出的预定义时间滑模控制器的有效性。同时，从实验结果可以看出，气浮平台具有优异的姿态跟踪性能，其中，角度跟踪误差小于0.1 rad， x 和 y 轴方向的跟踪误差小于0.2 m。 1;2; 6 结　论 1 本文在保证性能的前提下，解决了航天器的预定义时间姿态跟踪问题。通过Lyapunov控制理论，我们严格证明了所提出的控制方案对于航天器的姿态跟踪的调整时间是可以通过控制参数被指定的。同时，相较于现有的预定义时间控制策略，文中所提出的控制方案的收敛时间是更不保守的，这有利于航天器姿态调整时间的合理设置。最后，通过仿真实验证明刚性航天器的姿态跟踪误差精度可达1.5×10 - 6 rad，角速度跟踪误差精度可达2×10 - 6 rad/s，同时也说明了本方案相较于现有控制方案的优越性，包括更不保守的收敛时间上界，良好的抗干扰性和较高的跟踪误差精度。通过3自由度气浮平台的姿态跟踪实验说明了控制方案的有效性，角度跟踪误差小于0.1 rad，位置跟踪误差小于0.2 m。 参考文献 XIA Y Q ， ZHU Z ， FU M Y ， et al . 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