全捷联光学导引头视场角及输入约束下制导控制一体化设计

全捷联光学导引头视场角及输入约束下制导控制一体化设计 Integrated guidance and control design considering angle of view constraint of strap-down optical seeker and input constraint 1 first-author 邵茂森（1994-），男，四川巴中人，硕士研究生，2018年于哈尔滨工程大学获得学士学位，主要从事制导与控制方面的研究。E-mail：shaomaosen@ mail.nwpu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424794&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424792&type= 22.01333618 32.00404358 邵茂森 （1994-），男，四川巴中人，硕士研究生，2018年于哈尔滨工程大学获得学士学位，主要从事制导与控制方面的研究。E-mail： shaomaosen@ mail.nwpu.edu.cn shaomaosen@ mail.nwpu.edu.cn 2 corresp E-mail： fengqingyundan@163.com E-mail： fengqingyundan@163.com 徐　骋（1980-），男， 江苏江阴人，博士，研究方向包括飞行器导航制导与控制技术、协同控制技术、故障诊断与容错控制技术等。E-mail：fengqingyundan@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424796&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424798&type= 22.01333618 32.00401306 徐　骋 （1980-），男， 江苏江阴人，博士，研究方向包括飞行器导航制导与控制技术、协同控制技术、故障诊断与容错控制技术等。E-mail： fengqingyundan@163.com fengqingyundan@163.com 1 1 西北工业大学，陕西 西安 710072 西北工业大学，陕西 西安 710072 Northwestern Polytechnical University， Xi'an 710072， China Northwestern Polytechnical University， Xi'an 710072， China 复杂系统控制和智能协调技术重点实验室，北京 100074 复杂系统控制和智能协调技术重点实验室，北京 100074 Science and Technology on Complex System Control and Intelligence Agent Cooperation Laboratory， Beijing 100074， China Science and Technology on Complex System Control and Intelligence Agent Cooperation Laboratory， Beijing 100074， China 以全捷联光学制导飞行器打击运动目标为背景，本文将捷联解耦原理和状态约束控制方法相结合，提出了一种同时考虑导引头视场角约束和执行机构输入约束的新型制导控制一体化设计方法。首先，建立了具有严格反馈形式的全捷联制导控制一体化模型，针对模型的不确定性设计了一种对未知上界平方估计的自适应律，基于积分型障碍Lyapunov函数结合动态面控制解决了导引头视场角约束的问题，同时采用饱和函数光滑逼近以及Nussbaum增益函数解决了执行机构输入约束的问题。最后，通过Lyapunov理论证明了所设计制导控制一体化方法的一致有界稳定。典型场景的数值仿真结果表明，该方法可以同时满足捷联光学导引头8°，8.5°，9°的视场角约束以及舵执行机构10°的输入约束。 This paper proposes a new type of guidance control for optically guided vehicles that considers the seeker’s field of view constraint and the actuator input constraint simultaneously. Specifically， to allow for a strapdown optically guided vehicle to strike a moving target against a background， we integrate the strapdown decoupling principle with the state constraint control method. This integrated design method follows three key steps： firstly， full strapdown guidance and control with strict feedback is modelled； secondly， an adaptive law for estimating the unknown upper bound square is designed for model uncertainty； thirdly， based on an integral obstacle Lyapunov function combined with dynamic surface control， the seeker’s field of view angle constraint problem is solved. Moreover， the actuator input constraint problem is solved using a smooth approximation to the saturation function in combination with the Nussbaum gain function. Finally， application of the Lyapunov theory proves that the design of the integrated guidance and control method is uniformly bound and stable. Numerical simulation results of typical scenarios show that this method can simultaneously meet the 8°， 8.5°， and 9° field of view constraints of strapdown optical seekers and the 10° input constraints of rudder actuators. 捷联光学导引头 制导控制一体化 状态约束 输入约束 strap-down optical seeker integrated design of guidance and control line-of-sight constraint input constraint 1 引　言 1 传统意义上，飞行器制导控制系统的设计是以制导回路与控制回路的时标分离和动力系数的小扰动线性化为基本前提。然而，随着飞行器系统性能要求的提高，其制导控制系统需要满足多任务、多工作模式、大包络机动飞行等特点。这要求提高制导回路带宽，使得制导回路与控制回路之间的时标分离特性不再突出，因此，导引与控制一体化设计成为一种新的设计理念 ［ 1 1 ］ 。 现代制导武器的发展方向是小型化、数字化、低成本化，而传统飞行器的平台式导引头框架结构具有体积大、抗过载冲击能力差的缺点，因此直接刚性地与弹体固连的全捷联式导引头引起研究人员的广泛关注。此类导引头因具有结构简单、成本低、可靠性高 ［ 2 2 ］ 等优点，而成为近年来的研究热点，其中捷联光学制导更是重要的发展方向 ［ 3 3 ］ 。然而，此类导引头也给制导控制系统设计带来了新的挑战。一方面，在对目标进行精确打击时，相比传统导引头体制，与飞行器本体固连的光学导引头在较大的姿态调节下目标更容易脱离视场，而现有设计方法没有从理论上完全解决该问题，这成为此类飞行器应用于机动目标打击的瓶颈 ［ 4 4 - 6 6 4-6 ］ 。另一方面，捷联光学导引头直接获取的是相对于飞行器本体的体视线角信息，其兼具弹-目相对运动与姿态运动信息 ［ 7 7 ］ 。因此，为了确保该视线角在制导控制过程中始终能够满足导引头视场角约束，制导控制一体化也是一种设计方案。 实际上，在飞行器对目标实施精确打击的过程中，执行机构还存在输入约束的问题。舵机作为执行机构对于控制信号的输入会存在一个最大的阈值，当输入超过此阈值时其输出会产生饱和现象，这使得实际的控制结果与理想的控制效果会存在较大误差，故在制导控制一体化设计时必须考虑输入约束的问题。 针对制导控制一体化设计的视场角约束和输入约束问题，目前已有一些研究成果。文献［ 8 8 ］提出了在不违反视场角约束的情况下如何选择导航增益以实现所有可能的冲击角。文献［ 9 9 ］通过使用加权最优控制理论变化的系数来解决视场角约束问题。以上两种方法的共同假设是：导弹的攻角很小，忽略了姿态对视场角的影响，这在实际飞行中是不可能的。文献［ 10 10 ］基于时变滑模和积分型障碍Lyapunov函数设计了新型制导方法，同时满足视场角约束和攻击角约束，假设自动驾驶仪是理想工作状态。除此之外，一些先进的控制方法也应用于视场角约束，如滑模控制 ［ 11 11 ］ 、反步法控制 ［ 12 12 ］ 、积分型障碍Lyapunov函数 ［ 13 13 ］ 和依附黎卡提方程的状态约束 ［ 14 14 ］ 。这些方法在约束视场角时都基于攻角为零的假设，然而实际攻击过程中，对目标的跟踪主要通过姿态调整来实现，因此零攻角的假设显然不合理。文献［ 15 15 ］中加入了弹体系来考虑攻角问题，提出了二维场景下的一体化设计方法，文献［ 16 16 ］将该方法拓展到三维空间。但上述方法都只是考虑视场角约束的制导控制一体化设计，并没有考虑输入约束问题。 现有的制导控制一体化设计都没有同时考虑体视场角约束和输入约束，因此本文首先基于捷联解耦光学导引头的体视线解耦模型和经典的制导控制一体化模型，在考虑体视线角约束和输入约束的情况下，建立了一体化设计数学模型。然后，基于Lyapunov函数和动态面控制方法，设计了自适应律估计不确定性干扰，提出了具有视场角约束和输入约束的制导控制一体化方法。该方法有效地解决了视场角约束和输入约束的问题。 2 数学建模 1 将目标视为质点，全捷联光学导引头导弹与目标在纵向平面中的相对运动关系如 图1 图1 所示。其中， <math id="M1"><mi>X</mi><mi>O</mi><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424802&type= 8.38199997 2.87866688 为惯性坐标系； <math id="M2"><mi>M</mi><mo>,</mo><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424806&type= 7.78933382 3.21733332 分别代表导弹和目标； <math id="M3"><msub><mrow><mi>A</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424810&type= 11.17599964 4.14866638 分别为导弹与目标的法向加速度，其方向分别与导弹和目标的速度 <math id="M4"><msub><mrow><mi>V</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424815&type= 11.43000031 4.14866638 方向垂直； <math id="M5"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424818&type= 2.70933342 2.79399991 为弹目相对距离； <math id="M6"><msub><mrow><mi>q</mi></mrow><mrow><mi>L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424822&type= 3.30200005 4.57200003 为惯性系视线角； <math id="M7"><msub><mrow><mi>θ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>T</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424827&type= 8.97466660 4.14866638 分别为导弹与目标的航迹角； <math id="M8"><msub><mrow><mi>x</mi></mrow><mrow><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424831&type= 3.21733332 3.72533321 为弹体纵轴方向，由此得到 <math id="M9"><mi>ϑ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424835&type= 2.03200006 2.96333337 为弹体俯仰角， <math id="M10"><msub><mrow><mi>q</mi></mrow><mrow><mi>B</mi><mi>L</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424839&type= 4.57200003 4.57200003 为相对弹体系的体视线高低角。 2.1　全捷联制导控制一体化模型 2 根据考虑视场角约束的全捷联制导控制一体化设计方法 ［ 17 17 ］ ，二维模型为： <math 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mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>z</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424866&type= 11.76866722 3.72533321 ，系统输入 <math id="M16"><mi>u</mi><mo>=</mo><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424873&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424871&type= 8.63599968 2.79399991 ； <math id="M17"><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>b</mi><mi>l</mi></mrow></msub><mo>=</mo><mo>-</mo><mover accent="true"><mi>ϑ</mi><mo>˙</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424877&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424875&type= 15.91733265 4.57200003 ， <math id="M18"><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424879&type= 4.23333359 3.72533321 ， <math id="M19"><msub><mrow><mi>Δ</mi></mrow><mrow><mi>α</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424887&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424883&type= 3.64066648 3.72533321 和 <math id="M20"><msub><mrow><mi>Δ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424891&type= 3.97933316 3.72533321 为有界不确定项；动力系数定义如下： <math display="block" id="M21"><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mover accent="true"><mi>R</mi><mo>˙</mo></mover></mrow><mrow><mi>R</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424901&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424897&type= 16.42533302 8.38199997 ， <math display="block" id="M22"><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mi>q</mi><mi>S</mi><msubsup><mrow><mi>C</mi></mrow><mrow><mi>y</mi></mrow><mrow><mi>α</mi></mrow></msubsup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mo stretchy="false">(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>L</mi></mrow></msub><mo>-</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>m</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424905&type= 42.75666428 7.87400007 ， <math display="block" id="M23"><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">33</mn></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mi>q</mi><mi>S</mi><msubsup><mrow><mi>C</mi></mrow><mrow><mi>y</mi></mrow><mrow><mi>α</mi></mrow></msubsup></mrow><mrow><mi>m</mi><msub><mrow><mi>V</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424914&type= 21.67466736 8.72066593 ， <math display="block" id="M24"><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">43</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>q</mi><mi>S</mi><mi>L</mi><msubsup><mrow><mi>m</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi>α</mi></mrow></msubsup></mrow><mrow><msub><mrow><mi>J</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424920&type= 20.91266632 8.72066593 ， <math display="block" id="M25"><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">44</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>q</mi><mi>S</mi><msup><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mi>m</mi></mrow><mrow><mi>z</mi></mrow><mrow><mover accent="true"><mi>ω</mi><mo>¯</mo></mover></mrow></msubsup></mrow><mrow><msub><mrow><mi>J</mi></mrow><mrow><mi>z</mi></mrow></msub><msub><mrow><mi>V</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424929&type= 22.60600090 8.72066593 ， <math display="block" id="M26"><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>q</mi><mi>S</mi><mi>L</mi><msubsup><mrow><mi>m</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi>δ</mi></mrow></msubsup></mrow><mrow><msub><mrow><mi>J</mi></mrow><mrow><mi>z</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424937&type= 19.98133469 8.72066593 . 2.2　输入约束设计 2 在导弹飞行过程中控制信号输入与舵机的实际输出并不完全相符。因为存在一个最大的阈值 <math id="M27"><msub><mrow><mi>u</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=17424948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424943&type= 5.41866684 3.72533321 ，故舵机的实际输出是一个饱和函数 <math id="M28"><mi>s</mi><mi>a</mi><mi>t</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424956&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424952&type= 9.82133293 3.55599999 ，如 图2 图2 所示，其中实线为饱和函数曲线。 输出的饱和曲线存在一个尖锐的角，这给制导控制一体化设计和稳定性分析都带来了极大挑战。本文引入一个平滑函数来代替饱和函数，消除了 <math id="M29"><mi>s</mi><mi>a</mi><mi>t</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424972&type= 9.82133293 3.55599999 产生的尖锐角问题，平滑函数如 图2 图2 中虚线所示，具体形式如下： <math display="block" id="M30"><mi>h</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>×</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mtext> </mtext><mi>h</mi><mfenced separators="|"><mrow><mfrac><mrow><mi>u</mi></mrow><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></mfrac></mrow></mfenced><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mfrac><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi><mo>/</mo><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></msup><mo>-</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>u</mi><mo>/</mo><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></msup></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mi>u</mi><mo>/</mo><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>u</mi><mo>/</mo><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424979&type= 72.30532837 9.56733322 . 因此饱和函数可以展开为： <math display="block" id="M31"><mi>s</mi><mi>a</mi><mi>t</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424988&type= 34.12067032 3.55599999 ， 其中 <math id="M32"><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425002&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17424997&type= 7.78933382 3.55599999 是平滑函数相对于饱和函数的误差，并且满足： <math display="block" id="M33"><mfenced open="|" close="|" separators="|"><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≤</mo><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">h</mi><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0.2384</mn><msub><mrow><mi>u</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=17425010&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425005&type= 63.50000000 4.31799984 . 建立新的二维模型为： <math display="block" id="M34"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>B</mi><mi>L</mi></mrow></msub></mrow><mrow><mi>R</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>q</mi><mi>b</mi><mi>l</mi></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>q</mi><mi>l</mi></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">33</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>α</mi></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">43</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">44</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi>h</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>ω</mi><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo> </mo><mo> </mo><mo> </mo><mo>=</mo><mo>-</mo><mi>c</mi><mi>u</mi><mo>+</mo><mi>w</mi></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425014&type= 55.20266724 29.37933159 . 其中： <math id="M35"><mi>c</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425025&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425022&type= 8.12800026 2.79399991 ， <math id="M36"><mi>w</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425032&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425028&type= 2.96333337 2.79399991 是待设计的辅助控制信号。 3 捷联制导控制一体化设计 1 3.1　捷联制导控制一体化设计 2 在搭建的二维一体化模型上进行全捷联制导控制一体化设计，设计结果应满足： （1）整个系统都是稳定有界的； （2）体视线角始终在视场角约束的范围之内， <math id="M37"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mi>Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425531&type= 13.71599960 4.82600021 ， <math id="M38"><mi>Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425043&type= 2.87866688 3.21733332 为最大视线角； （3）执行机构的输入不会到达饱和状态， <math id="M39"><mfenced open="|" close="|" separators="|"><mrow><mi>u</mi></mrow></mfenced><mo><</mo><msub><mrow><mi>u</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=17425055&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425051&type= 14.98600006 4.06400013 ， <math id="M40"><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>=</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425059&type= 14.13933277 3.72533321 ， <math id="M41"><msub><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425068&type= 3.72533321 3.72533321 为最大舵偏角。 基于动态面控制，利用反步法和积分障碍Lyapunov函数（iBLF）相结合设计捷联制导控制一体化设计如下： <math display="block" id="M42"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>c</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>R</mi><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>R</mi></mrow></mfrac></mrow></mfenced><mo>-</mo><mi>R</mi><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>d</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn><mi>d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>c</mi></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>d</mi></mrow></msub><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">2</mn><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>c</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn><mi>d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>d</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn><mi>d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>c</mi></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>d</mi></mrow></msub><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">3</mn><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn><mi>c</mi></mrow></msub><mo>=</mo><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">33</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn><mi>d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn><mi>d</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn><mi>d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn><mi>c</mi></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn><mi>d</mi></mrow></msub><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">4</mn><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn><mi>d</mi></mrow></msub></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">5</mn><mi>c</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">43</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">44</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn><mi>d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">41</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">)</mo><mo> </mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">5</mn><mi>d</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn><mi>d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">5</mn><mi>c</mi></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">5</mn><mi>d</mi></mrow></msub><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">5</mn><mi>c</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">0</mn></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">5</mn><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo>=</mo><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><mi>c</mi><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>u</mi><mo>+</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><mi>N</mi><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>χ</mi><mo>,</mo><mover accent="true"><mi>χ</mi><mo>˙</mo></mover><mo>=</mo><mi>γ</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mover accent="true"><mi>w</mi><mo>¯</mo></mover></mtd></mtr><mtr><mtd><mi>w</mi><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">)</mo><mover accent="true"><mi>w</mi><mo>¯</mo></mover><mo> </mo></mtd></mtr><mtr><mtd><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo>=</mo><mo>-</mo><mi>c</mi><mi>u</mi><mo>+</mo><mi>w</mi></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425075&type= 69.25733185 108.88134003 . 式中： <math id="M43"><mo> </mo><mo> </mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</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=17425088&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425084&type= 16.67933464 4.14866638 ； <math id="M44"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</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=17425096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425092&type= 16.08666801 4.14866638 ； <math id="M45"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425100&type= 16.08666801 4.14866638 ； <math id="M46"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">41</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn><mo> </mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425108&type= 16.34066582 4.14866638 。 3.2　自适应律设计 2 在导弹稳定飞行的过程中，其气动参数、角速率、舵偏角和攻角都是有界的，即存在： <math display="block" id="M47"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>b</mi><mi>l</mi></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>l</mi></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>α</mi></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>,</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>ω</mi><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425116&type= 67.05599976 4.91066647 . 令 <math id="M48"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425122&type= 56.21866989 4.40266657 ，为干扰值，那么全捷联制导控制一体化方法中对干扰的估计为 <math id="M49"><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>,</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425130&type= 21.67466736 4.99533367 ，可由以下设计的自适应律得到： <math display="block" id="M50"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">32</mn></mrow></msub><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">41</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">42</mn></mrow></msub><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425140&type= 48.09066391 32.17333603 ， 其中 <math id="M51"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">32</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">41</mn></mrow></msub><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">42</mn></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425148&type= 50.63066483 4.14866638 。 4 稳定性分析 1 4.1　相关引理 2 引理1 ［ 18 18 ］ ：定义积分障碍Lyapunov函数： <math display="block" id="M52"><mi>V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>z</mi></mrow></msubsup><mrow><mfrac><mrow><mi>σ</mi><msubsup><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mo stretchy="false">(</mo><mi>σ</mi><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425157&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425154&type= 48.34466553 8.89000034 ， 其中： <math id="M53"><mi>z</mi><mo>=</mo><mi>x</mi><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425165&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425161&type= 16.67933464 3.72533321 为跟踪误差， <math id="M54"><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo><</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425169&type= 12.27666664 4.06400013 为受约束的状态， <math id="M55"><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425204&type= 3.30200005 3.72533321 为连续可微的状态跟踪指令，满足 <math id="M56"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfenced><mo><</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425186&type= 13.63133430 4.91066647 。 如果状态满足 <math id="M57"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfenced><mo><</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425186&type= 13.63133430 4.91066647 ，则有： <math display="block" id="M58"><mfrac><mrow><msup><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>≤</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mfrac><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425192&type= 30.31066895 8.46666718 . Lyapunov函数 <math id="M59"><mi>V</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425202&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425199&type= 13.37733269 4.48733330 对 <math id="M60"><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425204&type= 3.30200005 3.72533321 的偏导数为： <math display="block" id="M61"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>V</mi></mrow><mrow><mo>∂</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mfrac><mo>=</mo><mi>z</mi><mfenced separators="|"><mrow><mfrac><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced></mtd></mtr><mtr><mtd><mi>ρ</mi><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn><mi>z</mi></mrow></mfrac><mi mathvariant="normal">l</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>-</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425210&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425208&type= 62.39933777 20.57399940 . 引理2 ［ 19 19 ］ ：对于实数 <math id="M62"><mi>m</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>p</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425212&type= 23.45266724 3.80999994 ，若它们满足等式 <math id="M63"><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mi>p</mi></mrow></mrow><mo>+</mo><mrow><mrow><mn mathvariant="normal">1</mn></mrow><mo>/</mo><mrow><mi>q</mi></mrow></mrow><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425216&type= 24.38400078 4.14866638 ，则有： <math display="block" id="M64"><mi>m</mi><mi>n</mi><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><msup><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mi>q</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425220&type= 28.78666496 8.38199997 . 引理3 ［ 20 20 ］ ：对于 <math id="M65"><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425224&type= 9.56733322 3.72533321 ，令 <math id="M66"><mi mathvariant="normal">χ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><mi mathvariant="bold">R</mi><mo>:</mo><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>≤</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425227&type= 32.93532944 4.06400013 <math id="M67"><mi>t</mi><mo>≥</mo><mn mathvariant="normal">0</mn><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425231&type= 9.73666668 3.64066648 与 <math id="M68"><mi mathvariant="bold">N</mi><mo>=</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>×</mo><mi mathvariant="bold">χ</mi><mo>⊂</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425236&type= 29.46400070 3.89466691 均为开集合，考虑系统： <math display="block" id="M69"><mover accent="true"><mi>η</mi><mo>˙</mo></mover><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425239&type= 16.67933464 3.72533321 ， 式中： <math id="M70"><mi mathvariant="bold-italic">η</mi><mo>=</mo><msup><mrow><mo stretchy="false">[</mo><mi mathvariant="bold-italic">ω</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo>⊂</mo><mi mathvariant="bold">Ν</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425244&type= 26.24666595 3.97933316 ，函数 <math id="M71"><mi>h</mi><mo>:</mo><msub><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>×</mo><mi mathvariant="bold">N</mi><mo>→</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425247&type= 27.68600082 4.14866638 关于 <math id="M72"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425252&type= 1.10066664 2.79399991 分段连续且关于 <math id="M73"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425257&type= 2.03200006 2.79399991 满足局部一致Lipschitz条件。假设存在函数 <math id="M74"><mi mathvariant="bold-italic">U</mi><mo>=</mo><msup><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>→</mo><msub><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425261&type= 21.42066574 3.97933316 及 <math id="M75"><mi mathvariant="bold-italic">V</mi><mo>=</mo><mi mathvariant="bold">χ</mi><mo>→</mo><msub><mrow><mi mathvariant="bold">R</mi></mrow><mrow><mo>+</mo></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425265&type= 19.81200027 4.57200003 在各自定义域正定且连续可导，并使得 <math id="M76"><mi mathvariant="bold-italic">V</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425272&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425270&type= 11.76866722 2.96333337 ，当 <math id="M77"><mfenced open="|" close="|" separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>→</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425273&type= 12.44600010 4.06400013 时存在： <math display="block" id="M78"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced><mo stretchy="false">)</mo><mo>≤</mo><mi>U</mi><mo stretchy="false">(</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced><mo stretchy="false">)</mo><mo>≤</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mfenced open="‖" close="‖" separators="|"><mrow><mi mathvariant="bold-italic">ω</mi></mrow></mfenced><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425278&type= 51.47733307 4.06400013 ， 式中 <math id="M79"><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425283&type= 8.21266651 4.57200003 均为 <math id="M80"><msub><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">∞</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425289&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425287&type= 4.82600021 3.72533321 类函数。令 <math id="M81"><mi>W</mi><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo>+</mo><mi>U</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425296&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425292&type= 31.32666588 3.72533321 且 <math id="M82"><mi>x</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>∈</mo><mi mathvariant="bold">χ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425298&type= 13.20800018 3.80999994 ，如果不等式满足： <math display="block" id="M83"><mover accent="true"><mi>W</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>η</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>W</mi></mrow><mrow><mo>∂</mo><mi>η</mi></mrow></mfrac><mi>h</mi><mo>≤</mo><mo>-</mo><mi>L</mi><mi>W</mi><mo>+</mo><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425306&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425303&type= 45.97399902 8.38199997 ， 式中 <math id="M84"><mi>L</mi><mo>,</mo><mi>D</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425310&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425308&type= 13.71599960 3.21733332 ，则 <math id="M85"><mi>x</mi><mo>∈</mo><mi mathvariant="bold">χ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425312&type= 7.95866632 3.64066648 且 <math id="M86"><mi mathvariant="bold-italic">ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425318&type= 2.62466669 2.79399991 有界。 引理4 ［ 21 21 ］ ： <math id="M87"><mi>V</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425345&type= 7.78933382 3.55599999 和 <math id="M88"><mi>χ</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425326&type= 6.43466663 3.72533321 是光滑函数，并且在 <math id="M89"><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425357&type= 8.29733276 3.55599999 上， <math id="M90"><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425334&type= 14.64733315 3.55599999 。 <math id="M91"><mi>N</mi><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425339&type= 8.89000034 3.72533321 是一个Nussbaum增益函数，如果以下不等式成立， <math id="M92"><mi>V</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425345&type= 7.78933382 3.55599999 和 <math id="M93"><mi>χ</mi><mo stretchy="false">(</mo><mo>·</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425351&type= 6.43466663 3.72533321 在 <math id="M94"><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425357&type= 8.29733276 3.55599999 一定是有界的。 <math display="block" id="M95"><mtable><mtr><mtd><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>V</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>K</mi><mi>t</mi></mrow></msup><mo>+</mo><mfrac><mrow><mi>T</mi></mrow><mrow><mi>K</mi></mrow></mfrac><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>K</mi><mi>t</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>K</mi><mi>t</mi></mrow></msup></mrow><mrow><mi>γ</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mi>N</mi><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mover accent="true"><mi>χ</mi><mo>˙</mo></mover><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><mi>K</mi><mi>τ</mi></mrow></msup></mrow><mi mathvariant="normal">d</mi><mi>τ</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425363&type= 56.55733109 17.94933319 ， 其中： <math id="M96"><mi>K</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425369&type= 9.65200043 2.79399991 ， <math id="M97"><mi>T</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425375&type= 9.56733322 2.79399991 ， <math id="M98"><mi>γ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425381&type= 8.63599968 3.64066648 ， <math id="M99"><mi>ψ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425387&type= 2.11666679 3.64066648 是一个正的变量。 4.2　稳定性分析 2 定义推导过程中的边界层误差变量为： <math display="block" id="M100"><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mi>d</mi></mrow></msub><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mi>c</mi></mrow></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">2,3</mn><mo>,</mo><mn mathvariant="normal">4,5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425393&type= 40.63999939 4.57200003 . 将 式（11） 式（11） 代入 式（14） 式（14） ，由设计原则可得： <math display="block" id="M101"><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mi>i</mi><mi>c</mi></mrow></msub><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>,</mo><mo> </mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mi>i</mi><mi>c</mi></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mo> </mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">2,3</mn><mo>,</mo><mn mathvariant="normal">4,5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425399&type= 63.83866882 8.46666718 . 根据引理2可得： <math display="block" id="M102"><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mo>-</mo><mfrac><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><mo>-</mo><mfrac><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425405&type= 63.66933060 8.72066593 . 定义自适应律得估计偏差如下： <math display="block" id="M103"><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3,4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425411&type= 41.40200043 4.99533367 . 可知估计误差导数为： <math display="block" id="M104"><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mo>˙</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover accent="true"><mrow><mover accent="true"><mi>D</mi><mo>˙</mo></mover></mrow><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3,4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425419&type= 44.70399857 5.75733376 . 根据 式（11） 式（11） 和 式（14） 式（14） 可得： <math display="block" id="M105"><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>c</mi></mrow></msub></mrow><mrow><mi>R</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>b</mi><mi>l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425425&type= 38.18466568 7.61999989 . 同理可得： <math display="block" id="M106"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>c</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>l</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">33</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">43</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">44</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">5</mn><mi>c</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mtext>         </mtext><msub><mrow><mi>Δ</mi></mrow><mrow><mi>ω</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn><mi>d</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi>u</mi></mrow></mfenced><mover accent="true"><mi>u</mi><mo>˙</mo></mover><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn><mi>d</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425433&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425431&type= 66.04000092 27.68600082 . 将 式（11） 式（11） 代入 式（20） 式（20） 可得： <math display="block" id="M107"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>R</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>R</mi></mrow></mfrac></mrow></mfenced><mo>-</mo></mtd></mtr><mtr><mtd><mtext>          </mtext><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>b</mi><mi>l</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>l</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>α</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">41</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>ω</mi><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi>u</mi></mrow></mfenced><mo stretchy="false">(</mo><mo>-</mo><mi>c</mi><mi>u</mi><mo>+</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn><mi>d</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425435&type= 63.58466721 40.38600159 . 设计积分型障碍Lyapunov函数如下： <math display="block" id="M108"><mi>V</mi><mo>=</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425438&type= 48.25999832 3.80999994 ， 其中各个变量定义为： <math display="block" id="M109"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>σ</mi></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>σ</mi><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425442&type= 56.98066711 46.31266785 . 根据引理1对 式（23） 式（23） 求导可得： <math display="block" id="M110"><mfenced open="{" close="" separators="|"><mrow><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><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover accent="true"><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mover accent="true"><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><msub><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mover accent="true"><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><msub><mrow><mover accent="true"><mi>s</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425447&type= 51.98533249 36.06800079 . 将 式（21） 式（21） 代入 式（24） 式（24） 可得： <math display="block" id="M111"><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><mo>-</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>R</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>R</mi></mrow></mfrac><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mfrac><mrow><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>R</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>b</mi><mi>l</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfrac><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>-</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425450&type= 146.89666748 21.92866707 <math display="block" id="M112"><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>l</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mfrac><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfrac><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425454&type= 111.42134094 8.72066593 ， <math display="block" id="M113"><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>α</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mfrac><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfrac><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">32</mn></mrow></msub><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425458&type= 102.02333832 8.72066593 ， <math display="block" id="M114"><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>ω</mi><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><mfrac><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mrow></mfrac><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">42</mn></mrow></msub><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">41</mn></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425462&type= 110.23600006 8.72066593 ， <math display="block" id="M115"><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>-</mo><mi>c</mi><mi>u</mi><mo>+</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn><mi>d</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo stretchy="false">(</mo><mo>-</mo><mi>c</mi><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>u</mi><mo>+</mo><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>ω</mi><mo>+</mo><mover accent="true"><mi>ω</mi><mo>¯</mo></mover><mo>-</mo><mover accent="true"><mi>ω</mi><mo>¯</mo></mover><mo>-</mo><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn><mi>d</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mtext>       </mtext><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo stretchy="false">(</mo><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>+</mo><mover accent="true"><mi>h</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>ω</mi><mo>-</mo><mover accent="true"><mi>ω</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>γ</mi></mrow></mfrac><mo stretchy="false">(</mo><mfrac><mrow><mo>∂</mo><mi>h</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><msup><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><mi>χ</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mover accent="true"><mi>χ</mi><mo>˙</mo></mover></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425468&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425466&type= 116.92467499 14.56266594 . 由于 <math id="M116"><mi>R</mi><mo>></mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425471&type= 30.05666542 4.40266657 ，因此由 式（12） 式（12） 可得： <math display="block" id="M117"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>b</mi><mi>l</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>,</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>q</mi><mi>l</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>Δ</mi></mrow><mrow><mi>α</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mo>,</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>ω</mi><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425476&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425474&type= 44.95800018 11.51466751 . 根据引理2可知： <math display="block" id="M118"><mtable><mtr><mtd><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>≥</mo><mn mathvariant="normal">2</mn><mroot><mrow><mfrac><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mrow><mrow/></mroot><mo>≥</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>≥</mo><mn mathvariant="normal">2</mn><mroot><mrow><mfrac><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mrow><mrow/></mroot><mo>≥</mo><msub><mrow><mi>d</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>D</mi><mo>̂</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>-</mo><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mi>i</mi><mi>c</mi></mrow></msub><mo>≤</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><msubsup><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425479&type= 67.14067078 39.87799835 <math display="block" id="M119"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>R</mi></mrow></mfrac><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>R</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo stretchy="false">)</mo><mfrac><mrow><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>R</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>R</mi></mrow></mfrac><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>R</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo stretchy="false">)</mo><mfrac><mrow><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>R</mi></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi>s</mi></mrow><mrow><mi>l</mi></mrow></msub><msub><mrow><mi>s</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>s</mi></mrow><mrow><mi>l</mi></mrow></msub><msub><mrow><mi>y</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo stretchy="false">(</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>y</mi></mrow><mrow><mi>l</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425483&type= 66.29399872 41.74066544 . 式中： i =2，3，4，5， j =1，2，3，4， l =2，3，4。 根据 式（7） 式（7） 可得： <math display="block" id="M120"><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>≥</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>σ</mi></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mi mathvariant="normal">d</mi><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425486&type= 39.53932953 8.46666718 . 将 式（27） 式（27） 代入 式（25） 式（25） 可得： <math display="block" id="M121"><mtable columnalign="left"><mtr><mtd><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mo>=</mo><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>V</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>≤</mo></mtd></mtr><mtr><mtd><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>σ</mi></mrow><mrow><msup><mrow><mi>Q</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mi>d</mi><mi>σ</mi><mo>-</mo><mfenced separators="|"><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>R</mi></mrow></mfrac></mrow></mfenced><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><msubsup><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>-</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>R</mi></mrow></mfrac></mrow></mfenced><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfrac><mo>-</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo> </mo><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfrac><mo>-</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mrow></mfrac><mo>-</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><msubsup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">32</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">42</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">32</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">42</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">41</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>γ</mi></mrow></mfrac><mo stretchy="false">(</mo><mfrac><mrow><mo>∂</mo><mi>h</mi></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><msup><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><mi>χ</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><msup><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425490&type= 147.40466309 47.49800110 如果设计参数满足以下条件： <math display="block" id="M122"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≥</mo><mi>K</mi><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>R</mi></mrow></mfrac><mo>≥</mo><mfrac><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>≥</mo><mfrac><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>≥</mo><mfrac><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>-</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn><mi>R</mi></mrow></mfrac><mo>≥</mo><mfrac><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfrac><mo>-</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>a</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">23</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>≥</mo><mfrac><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfrac><mo>-</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>≥</mo><mfrac><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>τ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mrow></mfrac><mo>-</mo><msubsup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>-</mo><mfrac><mrow><msub><mrow><mover accent="true"><mi>b</mi><mo>¯</mo></mover></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>≥</mo><mfrac><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>≥</mo><mi>K</mi><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo>≥</mo><mi>K</mi><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">32</mn></mrow></msub><mo>≥</mo><mi>K</mi><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">42</mn></mrow></msub><mo>≥</mo><mi>K</mi><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425494&type= 70.95066071 50.03799820 <math display="block" id="M123"><mtable columnalign="left"><mtr><mtd><msup><mrow><mi>Δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">32</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><mfrac><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">42</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">31</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">41</mn></mrow></msub></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425501&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425498&type= 66.29399872 17.69533348 . 则可得到： <math display="block" id="M124"><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mo>≤</mo><mo>-</mo><mi>K</mi><mi>V</mi><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>γ</mi></mrow></mfrac><mo stretchy="false">(</mo><mfrac><mrow><mo>∂</mo><mi>h</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><msup><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><mi>χ</mi><mo>-</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mover accent="true"><mi>χ</mi><mo>˙</mo></mover><mo>+</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425505&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425504&type= 69.76532745 8.55133343 . 对 式（31） 式（31） 积分可得： <math display="block" id="M125"><mtable columnalign="left"><mtr><mtd><mover accent="true"><mi>V</mi><mo>˙</mo></mover><mo>≤</mo><mi>V</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>K</mi><mi>t</mi></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><mi>Δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>K</mi></mrow></mfrac><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>K</mi><mi>t</mi></mrow></msup><mo stretchy="false">)</mo><mo>+</mo></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfrac><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>K</mi><mi>t</mi></mrow></msup></mrow><mrow><mi>γ</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mo>∂</mo><mi>h</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>u</mi></mrow></mfrac><msup><mrow><mi>χ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mtext> </mtext><mi>χ</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mover accent="true"><mi>χ</mi><mo>˙</mo></mover><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>K</mi><mi>τ</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425508&type= 61.89133835 18.71133423 故可得到以下结论： （1）根据引理4和 式（32） 式（32） 可得出 <math id="M126"><mi>V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425572&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425569&type= 3.04800010 2.87866688 在 <math id="M127"><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425516&type= 8.46666718 4.48733330 是有界的，即 <math id="M128"><mi>V</mi><mo>≤</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425524&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425521&type= 26.16200066 4.40266657 ； （2）当时 <math id="M129"><mi>t</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425528&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425526&type= 7.95866632 2.79399991 ， <math id="M130"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo><</mo><mi>Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425531&type= 13.71599960 4.82600021 ，即满足体视线角约束。假设 <math id="M131"><mi>t</mi><mo>=</mo><msub><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425536&type= 8.38199997 3.72533321 ， <math id="M132"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>≥</mo><mi>Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425541&type= 13.63133430 4.82600021 ，则根据引理3，当 <math id="M133"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>→</mo><mi>Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425550&type= 13.80066681 4.82600021 时， <math id="M134"><msub><mrow><mi>V</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>→</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425557&type= 12.95400047 3.80999994 ，则 <math id="M135"><mi>V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425572&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425569&type= 3.04800010 2.87866688 变为无界，与假设相违背，故结论成立； （3）因为Lyapunov函数 <math id="M136"><mi>V</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425572&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425569&type= 3.04800010 2.87866688 有界，故所有闭环信号，包括误差 <math id="M137"><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425575&type= 2.03200006 3.72533321 和边界层误差 <math id="M138"><msub><mrow><mi>y</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425583&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425580&type= 2.62466669 4.57200003 都是有界的。 5 仿　真 1 为了证明所提出的考虑视场角和输入约束的制导控制一体化方法的有效性，本节进行了捷联光学导引头导弹对运动目标打击的数学仿真，打击目标加速度特性设置为 <math id="M139"><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">20</mn><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425588&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425586&type= 25.23066711 4.48733330 。 表1 表1 给出了模型参数， 表2 表2 给出了该方法的控制参数， 表3 表3 给出了初始仿真参数。 气动参数 数 据 弹体参数 数 据 <math id="M140"><msubsup><mrow><mi>C</mi></mrow><mrow><mi>y</mi></mrow><mrow><mi>α</mi></mrow></msubsup></math> 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425625&type= 3.72533321 3.38666677 -1.57 <math id="M147"><msub><mrow><mi>J</mi></mrow><mrow><mi>z</mi></mrow></msub><mo>/</mo><mo stretchy="false">(</mo><mi mathvariant="normal">k</mi><mi mathvariant="normal">g</mi><mo>·</mo><msup><mrow><mi mathvariant="normal">m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425640&type= 14.47799969 4.31799984 0.85 <math id="M148"><msubsup><mrow><mi>m</mi></mrow><mrow><mi>z</mi></mrow><mrow><mover accent="true"><mi>ω</mi><mo>¯</mo></mover></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425650&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425646&type= 4.06400013 3.38666677 -0.05 参数 数据 参数 数据 参数 数据 <math id="M149"><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425659&type= 2.53999996 3.21733332 0.6 <math id="M150"><msub><mrow><mi>k</mi></mrow><mrow><mn>11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425668&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425665&type= 3.47133350 3.21733332 0.2 <math id="M151"><msub><mrow><mi>k</mi></mrow><mrow><mn>12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425681&type= 3.47133350 3.21733332 0.1 <math id="M152"><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425685&type= 2.53999996 3.21733332 0.5 <math id="M153"><msub><mrow><mi>k</mi></mrow><mrow><mn>21</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425689&type= 3.47133350 3.21733332 0.2 <math id="M154"><msub><mrow><mi>k</mi></mrow><mrow><mn>22</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425693&type= 3.47133350 3.21733332 0.1 <math id="M155"><msub><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425699&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425697&type= 2.53999996 3.21733332 5 <math id="M156"><msub><mrow><mi>k</mi></mrow><mrow><mn>31</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425701&type= 3.47133350 3.21733332 0.2 <math id="M157"><msub><mrow><mi>k</mi></mrow><mrow><mn>32</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425705&type= 3.47133350 3.21733332 0.1 <math id="M158"><msub><mrow><mi>k</mi></mrow><mrow><mn>4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425711&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425709&type= 2.53999996 3.21733332 5 <math id="M159"><msub><mrow><mi>k</mi></mrow><mrow><mn>41</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425713&type= 3.47133350 3.21733332 0.2 <math id="M160"><msub><mrow><mi>k</mi></mrow><mrow><mn>42</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425717&type= 3.47133350 3.21733332 0.1 <math id="M161"><msub><mrow><mi>k</mi></mrow><mrow><mn>5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425722&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425719&type= 2.53999996 3.21733332 60 <math id="M162"><msub><mrow><mi>τ</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425726&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425724&type= 2.45533323 3.21733332 0.04 <math id="M163"><msub><mrow><mi>τ</mi></mrow><mrow><mn>3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425728&type= 2.45533323 3.21733332 0.08 <math id="M164"><msub><mrow><mi>τ</mi></mrow><mrow><mn>4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425732&type= 2.45533323 3.21733332 0.08 <math id="M165"><msub><mrow><mi>τ</mi></mrow><mrow><mn>5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425736&type= 2.45533323 3.21733332 0.1 <math id="M166"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425743&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425741&type= 1.69333339 3.13266683 25 <math id="M167"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425745&type= 1.26999998 2.37066650 1 参 数 数 据 <math id="M168"><msub><mrow><mi>x</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425751&type= 12.86933327 3.97933316 0 <math id="M169"><msub><mrow><mi>y</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425759&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425756&type= 12.69999981 4.14866638 2 000 <math id="M170"><msub><mrow><mi>V</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mo>·</mo><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425761&type= 21.59000015 4.14866638 500 <math id="M171"><msub><mrow><mi>θ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425765&type= 14.05466747 3.97933316 -40 <math id="M172"><msub><mrow><mi>ω</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>·</mo><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425769&type= 22.43666649 4.14866638 -2 <math id="M173"><msub><mrow><mi>ϑ</mi></mrow><mrow><mi>M</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425775&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425772&type= 14.22399998 3.97933316 -50 <math id="M174"><msub><mrow><mi>x</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425778&type= 12.44600010 3.97933316 2 000 <math id="M175"><msub><mrow><mi>y</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425782&type= 12.27666664 4.14866638 0 <math id="M176"><msub><mrow><mi>V</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mi mathvariant="normal">m</mi><mo>·</mo><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425785&type= 21.16666603 4.14866638 200 <math id="M177"><msub><mrow><mi>θ</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425790&type= 13.63133430 3.97933316 5 分别要求8°，8.5°，9°的体视线角约束条件和10°的舵偏角输入约束条件，仿真结果如 图3 图3 所示。 图3 图3 为3种视场角约束条件下导弹飞行过程中视场角的变化情况； 图4 图4 为3种视场角约束下的舵偏角。 由 图3 图3 ~ 图4 图4 可以看出，在不同场景下，所设计的制导控制一体化方法满足体视场角约束（8°，8.5°，9°），且满足输入约束（ <math id="M178"><mfenced open="|" close="|" separators="|"><mrow><mi>u</mi></mrow></mfenced><mo><</mo><msub><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>,</mo><msub><mrow><mi>u</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">10</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425807&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=17425806&type= 31.32666588 4.06400013 ）。 图5 图5 ~ 图6 图6 分别为在8°的体视场角约束条件和10°的舵偏角输入约束条件下的弹目运动轨迹和虚拟控制量跟踪误差。 1,2;3,4; 从 图5 图5 可以看出，设计的一体化方法能保证导弹成功拦截目标。从 图6 图6 可以看出，所设计的一体化方法中的虚拟控制量跟踪误差收敛至零。 为了证明本文设计的输入约束的有效性，与未考虑体视线角约束和未考虑输入约束的一体化方法（iBLF-IGC）对比。令本文设计的导引律为CIBLF-IGC，在视场角约束为8°，输入约束为10°以及相同的导弹参数及仿真打击条件下进行对比仿真，仿真结果如 图7 图7 所示。 由 图7 图7 可以得出，在加入体视线角约束后，体视线角被限制在约束界限（8°）之内，而iBLF-IGC方法却不能满足视线角的约束要求。由 图8 图8 可以得出，在加入输入约束后，执行机构的输入被限制在约束界限（10°）之内，满足设计要求，验证了所设计一体化方法的有效性。 6 结　论 1 本文针对捷联光学导引头导弹飞行过程中存在的视场角约束以及导弹执行机构存在的输入约束问题，结合积分型障碍Lyapunov函数、动态面法和Nussbaum函数，提出了一种具有体视线角约束和输入约束的制导控制一体化设计方法。以积分型障碍Lyapunov函数和动态面法来处理视场角约束问题和建模存在的不确定性问题。利用光滑函数和Nussbaum函数，在一体化设计方法中就对输入问题进行处理，而不是仅依靠饱和函数强行处理输入饱和问题，这种方法对于解决输入约束问题都有着参考价值。在理论上证明了系统的有界稳定后，又进行数学仿真，结果表明设计的具有视场角约束和输入约束的一体化方法，成功地使视场角约束在8°，8.5°，9°内，输入约束在10°内，满足了视线角约束和输入约束的要求。 参考文献 孙宝彩 ， 范军芳 . 全捷联小型弹药制导系统设计 ［J］. 红外与激光工程 ， 2014 ， 43 （ 6 ）： 1960 - 1965 . 全捷联小型弹药制导系统设计 SUN B C ， FAN J F . 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