Variable parameter linear tracking differentiator and its application in altazimuth large area telescope

变参数线性跟踪微分器及其在地平式大视场望远镜中的应用 Variable parameter linear tracking differentiator and its application in altazimuth large area telescope corresp first-author E-mail： yxxair@163.com E-mail： yxxair@163.com 杨晓霞（1984-），女，山东德州人，博士，副研究员，2005年于山东大学获得学士学位，2010年于中国科学院数学与系统科学研究院获得博士学位，主要从事大型光电望远镜先进控制方法的研究。E-mail：yxxair@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920218&type= 22.01333237 32.00399780 杨晓霞 （1984-），女，山东德州人，博士，副研究员，2005年于山东大学获得学士学位，2010年于中国科学院数学与系统科学研究院获得博士学位，主要从事大型光电望远镜先进控制方法的研究。E-mail： yxxair@163.com yxxair@163.com 邓永停（1987-），男，山东潍坊人，博士，研究员，2015年于中科院长春光机所获得博士学位，主要从事望远镜精密跟踪控制术的研究及电子学设计方面的研究。E-mail： dyt0612@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920230&type= 22.01332855 32.00399780 邓永停 （1987-），男，山东潍坊人，博士，研究员，2015年于中科院长春光机所获得博士学位，主要从事望远镜精密跟踪控制术的研究及电子学设计方面的研究。E-mail： dyt0612@163.com dyt0612@163.com 中国科学院 长春光学精密机械与物理研究所，吉林 长春 130033 中国科学院 长春光学精密机械与物理研究所，吉林 长春 130033 Changchun Insititute of Optics， Fine Mechanics and Physics， Chinese Academy of Sciences， Changchun 130033， China Changchun Insititute of Optics， Fine Mechanics and Physics， Chinese Academy of Sciences， Changchun 130033， China 为了对具有速度和加速度限制的物理信号安排合理的过渡过程，提出了一种带有速度和加速度饱和限制的变参数线性跟踪微分器。将跟踪微分器的快速因子设计为跟踪误差减函数的形式，即使在初始跟踪误差非常大的时候也不会产生非常大的加速度，同时结合速度和加速度饱和的限制，使用同一组参数就可以对大范围的阶跃信号给出与物理系统反馈能力相匹配的过渡过程信号。仿真计算表明，所设计的跟踪微分器的跟踪效果与经典的非线性跟踪微分器几乎相同，计算复杂度却明显降低。将所设计的跟踪微分器应用于地平式大视场望远镜的快速调转定位中，与工程中常用的位置分段方式进行对比，响应时间缩短了大约15%~37%，有效提高了望远镜的搜索效率。变参数跟踪微分器不但能够起到过渡作用，还能最大限度地激发物理系统的反馈能力。该方法参数设置简单，易于工程中的调试和实现。 To develop a reasonable transition process for physical signals with speed and acceleration limits， we propose a variable parameter linear tracking differentiator with speed and acceleration saturations. The parameter of the tracking differentiator is designed as a decreasing function of the tracking error， which avoids large acceleration shock in the initial stage when the tracking error is relatively large. Using the same set of parameters， the designed tracking differentiator can give reasonable transition signals that match the feedback capabilities of physical systems for a wide range of step signals. Simulation results show that the proposed tracking differentiator can produce nearly the same tracking effect as that of the classical nonlinear tracking differentiator and with reduced computational complexity. We applied the designed tracking differentiator to the rapid positioning of an altazimuth large area telescope. Compared with the commonly used position segmentation method， the response time was reduced by 15%–37%， thereby improving the search efficiency of the telescope. The proposed variable parameter tracking differentiator can not only play a transitional role but can also maximize the feedback capabilities of physical systems. The parameter setting of the proposed method is simple as well as easy to debug and implement in engineering practice. 伺服控制 跟踪微分器 变参数 过渡过程安排 地平式大视场望远镜 servo control tracking differentiator variable parameter transition process arrangement altazimuth large area telescope 1 引 言 1 自抗扰控制方法（Active Disturbance Rejection Control， ADRC ［ 1 1 - 2 2 1-2 ］ ）是一种基于过程误差来抑制或消除误差的控制方法，在工业和工程领域得到了广泛的应用 ［ 3 3 - 8 8 3-8 ］ 。跟踪微分器（Tracking Differentiator， TD ［ 9 9 ］ ）是自抗扰控制方法的一个主要组成部分，主要用于对不连续或带随机噪声的信号提取连续信号及微分信号。大量的理论研究工作针对TD的收敛性 ［ 10 10 - 11 11 10-11 ］ 、微分信号的提取能力以及滤波能力 ［ 12 12 - 14 14 12-14 ］ 展开，很多TD的改进形式 ［ 15 15 - 16 16 15-16 ］ 被提出并应用在不同领域中 ［ 17 17 - 18 18 17-18 ］ 。事实上，TD的一个主要作用是用于对控制系统中的参考信号进行过渡过程安排，过渡过程信号作为参考信号和反馈信号之间的一个桥梁，当参考信号发生大的跳变时，仍然能够保证反馈信号跟踪一个连续渐变的过渡过程信号，从而使得控制误差不会出现大的跳变，进而保证控制系统的稳定性和鲁棒性。 TD又分为线性TD（Linear TD， LTD）和非线性TD（Nonlinear TD， NLTD）。在安排过渡过程时，LTD的优点在于算法复杂度低，简单易实现，缺点在于跟踪不同幅值的阶跃信号时，需要调节快速因子参数，否则容易造成LTD输出信号的速度和加速度过大或者过小，起不到过渡作用。NLTD的优点在于通过使用最速控制综合函数，使得在一定加速度的限制下能够最快地跟踪参考输入，但是其算法复杂度高、计算量大，并且在跟踪不同幅值的阶跃信号时也存在输出速度过大或过小的问题。 在使用跟踪微分器安排过渡过程时，应该使得输出的过渡过程信号的速度、加速度与物理系统的反馈能力相匹配，否则，过渡过程信号变化太快起不到过渡作用，变化太慢会影响系统的最大跟踪能力。本文研究了一种带速度和加速度饱和限制的变参数线性跟踪微分器，它既能够在参考信号和反馈信号之间起过渡桥梁的作用，又能够充分发挥受控物理对象的最大速度和最大加速度能力。 2 TD性能分析 1 经典的安排过渡过程的工具是LTD和NLTD。 2.1　线性跟踪微分器 2 LTD的离散形式如下： <math display="block" id="M1"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><mo stretchy="false">(</mo><mo>-</mo><msubsup><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920240&type= 90.59333038 10.07533360 ， 其中： <math id="M2"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920248&type= 3.47133350 4.57200003 为设定值， h 是离散化的采样步长参数， r 1 为快速因子，是系统唯一的可调参数，它决定系统的收敛速度。当设定值 <math id="M3"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≡</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920254&type= 16.76399994 4.82600021 为常值时，定义 x 1 第一次到达0.99 p 0 的时间为过渡时间 T 0 ，定义 V maxLTD 和 a maxLTD 分别是LTD输出的最大速度和最大加速度，也就是 <math id="M4"><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920260&type= 31.91933441 3.97933316 ， <math id="M5"><msub><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920266&type= 31.41133499 3.97933316 ， 表1 表1 给出了LTD的最大速度和最大加速度与参数和设定值之间的关系。 2 r 1 3 p 0 =1 3 p 0 =10 T 0 V maxLTD a maxLTD T 0 V maxLTD a maxLTD 5 1.326 1.84 25 1.32 18.44 250 10 0.663 3.69 100 0.66 36.97 1 000 15 0.441 5.56 225 0.44 55.60 2 250 20 0.331 7.43 400 0.33 74.32 4 000 25 0.264 9.31 625 0.26 93.13 6 250 30 0.220 11.20 900 0.22 112.00 9 000 35 0.188 13.10 1 225 0.18 131.07 12 250 40 0.165 15.01 1 600 0.16 150.16 16 000 45 0.146 16.94 2 025 0.14 169.40 20 250 50 0.131 18.86 2 500 0.13 188.67 25 000 55 0.119 20.81 3 025 0.11 208.13 30 250 60 0.109 22.76 3 600 0.10 227.68 36 000 65 0.101 24.73 4 225 0.10 247.33 42 250 70 0.093 26.70 4 900 0.09 267.05 49 000 75 0.087 28.69 5 625 0.08 286.92 56 250 80 0.082 30.69 6 400 0.08 306.92 64 000 85 0.077 32.69 7 225 0.07 326.92 72 250 90 0.072 34.69 8 100 0.07 346.97 81 000 95 0.069 36.75 9 025 0.06 367.52 90 250 100 0.065 38.74 10 000 0.06 387.42 100 000 使用MATLAB的数据拟合工具，可以得到拟合表达式： <math display="block" id="M6"><mtable><mtr><mtd><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">6.671</mn></mrow><mrow><msup><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">1.004</mn></mrow></msup></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0.335</mn><mtext> </mtext><mn mathvariant="normal">3</mn><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">1.031</mn></mrow></msup><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></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=37920284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920280&type= 65.02400208 14.90133286 图1 图1 给出了当 p 0 =1时离散计算结果与拟合函数曲线之间的关系。可以看出，过渡过程时间 T 0 与参数 r 1 成反比，并且与最终的稳态响应值没有关系，这说明使用LTD安排过渡过程时，只要参数 r 1 确定，跟踪稳态值1和跟踪稳态值100或者10 000的时间是相同的，这就导致LTD给出的 V maxLTD 和 a maxLTD 均与稳态值 p 0 相关， p 0 越大， V maxLTD 和 a maxLTD 越大，而如果 V maxLTD 和 a maxLTD 远远超过物理系统的反馈速度和加速度能力，那么物理系统的反馈信号就无法跟踪LTD所给出的过渡过程信号，此时LTD所给出的信号就起不到过渡作用。 1;2;3; 另一方面，通过简单地在LTD的输出信号上加入速度和加速度饱和环节，是不能够解决这个问题的，因为通过仿真计算可以看出，此时LTD的输出会发生振荡。设 p 0 =100， r 1 =15， h =0.001，根据 式（2） 式（2） 可以推算出LTD给出的最大速度和最大加速度如下： <math display="block" id="M7"><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">547</mn><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">22500</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920320&type= 47.41333389 4.14866638 . 设物理系统反馈的最大速度 <math id="M8"><msub><mrow><mi>V</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=37920329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920326&type= 6.60400009 3.80999994 和最大加速度 <math id="M9"><msub><mrow><mi>a</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=37920335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920332&type= 5.50333309 3.72533321 如下： <math display="block" id="M10"><msub><mrow><mi>V</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">21</mn><mo>,</mo><msub><mrow><mi>a</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">12</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920341&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920338&type= 30.98800087 4.14866638 . 图2 图2 给出了加入 式（4） 式（4） 的速度和加速度上界后LTD的输出曲线。 2.2　非线性跟踪微分器 2 离散最速跟踪微分器利用fhan函数实现，它的加速度通过参数 r 限定，具体形式如下： <math display="block" id="M11"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mtext>                          </mtext><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920355&type= 57.99666977 15.40933418 ， 其中： h 是采样周期， r 是快速因子， h 0 是滤波因子，可以大于采样周期 h ，fhan是非线性函数，具体形式如下： <math display="block" id="M12"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>d</mi><mo>=</mo><mi>r</mi><msup><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>h</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mn mathvariant="normal">8</mn><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>d</mi><mo stretchy="false">)</mo><mo>/</mo><mn mathvariant="normal">2</mn></mtd></mtr><mtr><mtd><mi>a</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="normal">f</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="normal">f</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">f</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo>=</mo><mo>-</mo><mi>r</mi><mfenced separators="|"><mrow><mi>a</mi><mo>/</mo><mi>d</mi></mrow></mfenced><mi mathvariant="normal">f</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>              </mtext><mo>-</mo><mi>r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="normal">f</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920361&type= 68.07199860 43.85733414 ， 其中： <math id="M13"><mi mathvariant="normal">f</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920369&type= 65.78600311 3.97933316 ， <math id="M14"><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920375&type= 8.46666718 3.97933316 是符号函数。经过计算可以得到 T 0 ， V maxNLTD 以及 a maxNLTD 与参数 r 和 p 0 的关系如下： <math display="block" id="M15"><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2</mn><mroot><mrow><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>r</mi></mrow></mfrac></mrow><mrow/></mroot><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mroot><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>r</mi></mrow><mrow/></mroot><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920380&type= 66.12466431 9.22866726 ， 其中 V maxNLTD 和 a maxNLTD 分别是NLTD的最大速度和最大加速度。可以看出，NLTD的加速度是具有上限的，但是最大速度跟稳态值 p 0 有关，没有上限，因此带有速度饱和 V max 的NLTD设计如下： <math display="block" id="M16"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>,</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920387&type= 88.56133270 16.42533302 . 与LTD不同，给NLTD加入速度饱和限制之后不会引起信号的波动。 图3 图3 给出了带有速度饱和的NLTD输出曲线，其中 p 0 =100， V max =21， r =12， h =0.001， h 0 =0.01。 带有速度饱和的NLTD可以给出与物理系统反馈能力相匹配的过渡过程信号，在实现过渡作用的同时能够充分发挥系统的最大反馈能力。但是，由于非线性函数fhan的计算比较复杂，不利于在工程中的实时实现。本文给出一种变参数LTD，合理地将参数 r 设计为跟踪误差的函数，克服了LTD的不足，并且比带有速度饱和的NLTD更简单易实现。 3 变参数线性跟踪微分器设计 1 连续的VLTD设计如下： <math display="block" id="M17"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</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 stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920403&type= 73.40599823 10.07533360 ， 其中时变参数 r 1 （ t ）设计如下： <math display="block" id="M18"><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi>σ</mi><mroot><mrow><mi>r</mi></mrow><mrow/></mroot></mrow><mrow><mroot><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow/></mroot></mrow></mfrac><mo>,</mo><mi mathvariant="normal">当</mi><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mi>σ</mi><mroot><mrow><mi>r</mi></mrow><mrow/></mroot><mo>,</mo><mtext>    </mtext><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mtext>   </mtext><mi mathvariant="normal">当</mi><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920411&type= 78.73999786 19.98133469 ， 其中： r 可以设计为反馈物理信号的最大加速度， <math id="M19"><mi>σ</mi><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920419&type= 8.46666718 3.30200005 是唯一的可调参数。将 式（10） 式（10） 代入 式（9） 式（9） 可以得到VLTD的一般形式如下： <math display="block" id="M20"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</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 stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>r</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><mi>σ</mi><mroot><mrow><mi>r</mi></mrow><mrow/></mroot><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mroot><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow/></mroot><mtext>   </mtext><mi mathvariant="normal">当</mi><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≥</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mo>-</mo><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>r</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><mi>σ</mi><mroot><mrow><mi>r</mi></mrow><mrow/></mroot><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mtext>                                               </mtext><mi mathvariant="normal">当</mi><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo><</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920426&type= 142.15533447 19.47333336 . 可以看出，VLTD与LTD的区别在于将快速因子设置为一个时变函数，它设计为跟踪误差的减函数，以此保证初期跟踪误差非常大时，过渡过程的加速度也不会非常大。当跟踪误差小于1时，VLTD就变为LTD。 3.1　VLTD的收敛性分析 2 对VLTD的收敛性进行分析，得到如下定理。 定理1，设 p 0 是常值，对于任意的初值 x 1 （0）和 x 2 （0）=0，VLTD都能在有限时间内收敛至线性区间内，其中线性区间定义为跟踪误差的绝对值小于1的区间。 证明：设 <math id="M21"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920439&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920435&type= 50.88466644 4.82600021 ， <math id="M22"><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><mi>σ</mi><mroot><mrow><mi>r</mi></mrow><mrow/></mroot><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920446&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920441&type= 17.18733406 4.31799984 那么VLTD可以简化为： <math display="block" id="M23"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mroot><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced></mrow><mrow/></mroot><mo> </mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mo stretchy="false">|</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≥</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mtext>      </mtext><mo> </mo><mtext>   </mtext><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mo stretchy="false">|</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo><</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920448&type= 97.62065887 18.28800011 . 因此只需证明对于任意的 <math id="M24"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920456&type= 8.55133343 4.48733330 和 <math id="M25"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920510&type= 14.98600006 4.48733330 ，存在时间点 <math id="M26"><msub><mrow><mi>t</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=37920783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920781&type= 8.80533314 3.72533321 使得 <math id="M27"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920482&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920479&type= 17.69533348 4.57200003 。 情形1：如果 <math id="M28"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920486&type= 17.44133186 4.57200003 ，那么当 t > 0时系统的变化规律如下： <math display="block" id="M29"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920493&type= 47.92133331 9.65200043 . 上述微分方程的解如下： <math display="block" id="M30"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mi>t</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mi>t</mi></mrow></mfenced><mo>/</mo><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mi>t</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920501&type= 62.99200058 4.57200003 . 由于 <math id="M31"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920510&type= 14.98600006 4.48733330 ，所以有： <math display="block" id="M32"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></mfenced><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mi>t</mi></mrow></msup><mo><</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920521&type= 69.25733185 4.57200003 . 对于情形1，当系统的初值处于线性区间时，系统始终处于线性区间内。 情形2：如果 <math id="M33"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920535&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920531&type= 15.07066536 4.48733330 ，那么系统的变化规律如下： <math display="block" id="M34"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow/></mroot></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920539&type= 54.10200119 10.92199993 . 整理第二个方程得到： <math display="block" id="M35"><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow/></mroot></mrow></mfrac><mo>=</mo><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msup><mrow><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow/></mroot><mo stretchy="false">)</mo></mrow><mrow><mi>'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920547&type= 60.95999908 9.65200043 . 在时间区间 <math id="M36"><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920694&type= 8.97466660 3.55599999 上对等式两边进行积分，得到： <math display="block" id="M37"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><mfenced separators="|"><mrow><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msup><mrow><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow/></mroot><mo stretchy="false">)</mo></mrow><mrow><mi>'</mi></mrow></msup></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920566&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920562&type= 57.15000153 7.87400007 . 进一步整理得： <math display="block" id="M38"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>τ</mi><mo>-</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><mrow/></mroot><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920574&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920570&type= 60.70600128 4.82600021 . 将 式（19） 式（19） 在时间区间 <math id="M39"><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=37920582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920578&type= 8.55133343 3.55599999 上进行积分，得到： <math display="block" id="M40"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot><mi>t</mi><mo>-</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><mrow/></mroot><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920587&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920585&type= 87.46067047 8.21266651 如果存在时间点 <math id="M41"><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>∈</mo><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=37920729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920725&type= 14.39333248 4.48733330 ，使得 <math id="M42"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920600&type= 15.40933418 4.48733330 ，那么结论成立。而如果对于任意的 <math id="M43"><mi>τ</mi><mo>∈</mo><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=37920611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920608&type= 13.88533401 3.55599999 都有 <math id="M44"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920615&type= 14.98600006 4.48733330 ，那么： <math display="block" id="M45"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><mrow/></mroot><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo>≥</mo><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920624&type= 25.31533241 7.87400007 . 整理 式（21） 式（21） 得： <math display="block" id="M46"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>≤</mo><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mtext>  </mtext><mi>t</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920632&type= 74.16799927 6.85799980 . 显然，不等式的右边是关于 t 的二次函数，当 <math id="M47"><mi>t</mi><mo>></mo><mn mathvariant="normal">4</mn><mfenced separators="|"><mrow><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>/</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920638&type= 34.88266373 6.85799980 时， 式（22） 式（22） 的右边相对于 t 是一个单调递减函数，并且： <math display="block" id="M48"><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mtext>  </mtext><mi>t</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920649&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920645&type= 81.44934082 6.85799980 . 因此，一定存在 <math id="M49"><msub><mrow><mi>t</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=37920783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920781&type= 8.80533314 3.72533321 ，使得： <math display="block" id="M50"><mo>-</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mroot><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920658&type= 74.76066589 4.82600021 . 从而有： <math display="block" id="M51"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">)</mo><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920666&type= 15.40933418 4.48733330 . 结论成立。 情形3：如果 <math id="M52"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>≤</mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920675&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920672&type= 18.88066673 4.48733330 ，那么系统的变化规律如下： <math display="block" id="M53"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>/</mo><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow/></mroot></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920677&type= 53.93266296 10.92199993 . 整理第二个方程可得： <math display="block" id="M54"><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>/</mo><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow/></mroot><mo>=</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msup><mrow><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow/></mroot><mo stretchy="false">)</mo></mrow><mrow><mi>'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920688&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920684&type= 68.74933624 4.48733330 . 将 式（27） 式（27） 的两端在时间区间 <math id="M55"><mo stretchy="false">[</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920694&type= 8.97466660 3.55599999 上积分得： <math display="block" id="M56"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msub><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>τ</mi></mrow></msubsup><mrow><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msup><mrow><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow/></mroot><mo stretchy="false">)</mo></mrow><mrow><mi>'</mi></mrow></msup><mi mathvariant="normal">d</mi><mi>t</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920704&type= 53.67866135 7.87400007 . 进一步整理得： <math display="block" id="M57"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mi>τ</mi><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><mrow/></mroot><mo>-</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920712&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920709&type= 67.56400299 4.82600021 . 将上述方程的两边在时间区间 <math id="M58"><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=37920716&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920714&type= 8.55133343 3.55599999 上积分得： <math display="block" id="M59"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><mn mathvariant="normal">2</mn><mo>-</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot><mtext>  </mtext><mi>t</mi><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><mrow/></mroot><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920722&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920719&type= 98.38265991 7.87400007 如果存在 <math id="M60"><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>∈</mo><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=37920729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920725&type= 14.39333248 4.48733330 ，使得 <math id="M61"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920734&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920731&type= 18.96533203 4.48733330 ，那么结论成立。如果对于任意的 <math id="M62"><mo>∀</mo><mi>τ</mi><mo>∈</mo><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=37920740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920737&type= 16.08666801 3.55599999 都有 <math id="M63"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>≤</mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920746&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920743&type= 18.71133423 4.48733330 ，那么有： <math display="block" id="M64"><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mi>t</mi></mrow></msubsup><mrow><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><mrow/></mroot><mi mathvariant="normal">d</mi><mi>τ</mi></mrow><mo>≥</mo><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920748&type= 28.70199966 7.87400007 . 因此有： <math display="block" id="M65"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>></mo><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot></mrow></mfenced><mtext>  </mtext><mi>t</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920756&type= 73.82933807 6.85799980 . 显然，不等式右边是一个关于 t 的二次函数，当 <math id="M66"><mi>t</mi><mo>></mo><mn mathvariant="normal">4</mn><mfenced separators="|"><mrow><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfenced><mo>/</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920767&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920762&type= 38.26933289 6.85799980 时，不等式右边是一个关于 t 的单调递增函数，并且有： <math display="block" id="M67"><mtable><mtr><mtd><munder><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">m</mi></mrow><mrow><mi>t</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot></mrow></mfenced><mtext>  </mtext><mi>t</mi><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920777&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920771&type= 59.52066422 13.12333298 所以存在 <math id="M68"><msub><mrow><mi>t</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=37920783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920781&type= 8.80533314 3.72533321 ，使得： <math display="block" id="M69"><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>/</mo><mn mathvariant="normal">2</mn><mo>+</mo><mn mathvariant="normal">4</mn><msub><mrow><mi>σ</mi></mrow><mrow><mi>r</mi></mrow></msub><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mroot><mrow><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo></mrow><mrow/></mroot></mrow></mfenced><mtext>  </mtext><mi>t</mi><mo>+</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo stretchy="false">)</mo><mo>></mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920786&type= 71.37400055 6.85799980 . 因此有： <math display="block" id="M70"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">)</mo><mo>></mo><mo>-</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920794&type= 18.96533203 4.48733330 . 定理结论成立。 上述结论说明，对于任意的初值 x 1 （0），VLTD必定能在有限时间内收敛至线性区间内。 3.2　VLTD的速度和加速度上限分析 2 设 <math id="M71"><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><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=37920806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920802&type= 6.01133299 3.72533321 为VLTD到达速度上限的时间点，那么有 <math id="M72"><msub><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920823&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920810&type= 19.55800056 4.48733330 ，也就是： <math display="block" id="M73"><mtable columnalign="left"><mtr><mtd><mo>-</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo><mo>-</mo></mtd></mtr><mtr><mtd><mn mathvariant="normal">2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn mathvariant="normal">0</mn><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920829&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920825&type= 44.87333298 10.41399956 进一步整理得： <math display="block" id="M74"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mfenced open="|" close="|" separators="|"><mrow><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920832&type= 64.17733002 4.82600021 . 如果 <math id="M75"><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920840&type= 30.31066895 4.82600021 ，那么有： <math display="block" id="M76"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo><</mo><mi>σ</mi><mroot><mrow><mi>r</mi></mrow><mrow/></mroot><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920852&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920847&type= 31.83466530 5.07999992 . 如果 <math id="M77"><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920893&type= 30.31066895 4.82600021 ，那么有： <math display="block" id="M78"><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>σ</mi><mroot><mrow><mi>r</mi></mrow><mrow/></mroot><mo>/</mo><mroot><mrow><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920863&type= 53.42466354 6.09600019 . 也就是： <math display="block" id="M79"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mi>σ</mi><mroot><mrow><mi>r</mi><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced></mrow><mrow/></mroot><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920871&type= 55.45666885 6.09600019 . 根据VLTD的特点， <math id="M80"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>≥</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920879&type= 34.45933533 4.82600021 ，因此VLTD的最大速度满足： <math display="block" id="M81"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo></mrow></mfenced><mo>≥</mo><mi>σ</mi><mroot><mrow><mi>r</mi><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>/</mo><mn mathvariant="normal">8</mn></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920884&type= 34.37466812 5.33400011 . 不等 式（40） 式（40） 说明当 <math id="M82"><mtext> </mtext><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>v</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>≥</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920893&type= 30.31066895 4.82600021 时，VLTD的速度输出最大值不存在上限。 另一方面，可以推导出VLTD在初始时刻达到最大加速度 <math id="M83"><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920899&type= 4.99533367 3.04800010 ，并且减速阶段的加速度明显比加速阶段的加速度小。为了避免VLTD减速阶段的加速度过小，一般选取参数 <math id="M84"><mi>σ</mi><mo>></mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920906&type= 8.46666718 2.79399991 ，这样就会导致加速阶段的加速度会超过物理系统的最大反馈加速度，因此，在VLTD的加速度输出上加入加速度饱和上限即可。 通过上面的分析可知，有必要对VLTD的速度和加速度输出加入饱和限制，记 x 3 为VLTD的加速度输出， <math id="M85"><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">V</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920917&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920913&type= 12.27666664 3.80999994 为VLTD的速度上限，那么带有速度和加速度饱和限制的离散VLTD形式设计如下： <math display="block" id="M86"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo>_</mo><mi mathvariant="normal">V</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>r</mi><mo>-</mo><mn mathvariant="normal">2</mn><mi>σ</mi><mroot><mrow><mi>r</mi></mrow><mrow/></mroot><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>/</mo><mroot><mrow><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mrow/></mroot><mo> </mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mtext> </mtext><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>≥</mo><mn mathvariant="normal">1</mn></mtd></mtr><mtr><mtd><mo>-</mo><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>r</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>-</mo><mn mathvariant="normal">2</mn><mi>σ</mi><mroot><mrow><mi>r</mi></mrow><mrow/></mroot><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo> </mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mtext> </mtext><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>-</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo><</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920922&type= 147.91265869 35.47533035 . 3.3　仿真设计 2 设物理系统的最大速度和最大加速度分别为 V max =20和 a max =12，VLTD的参数取值如下：采样步长 h =0.001， <math id="M87"><mi>σ</mi><mo>=</mo><mn mathvariant="normal">2.3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921188&type= 12.27666664 2.79399991 ， r =12，速度上限 <math id="M88"><msub><mrow><mi>V</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">V</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">D</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">20</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920941&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920937&type= 20.82799911 3.80999994 ，加速度上限为 r ，稳态响应值 p 0 =100，50，10，1， 图4 图4 分别给出了加入速度和加速度限制前后的VLTD输出对比曲线。可以看出，加入速度和加速度饱和限制之后，VLTD的收敛性没有受到影响，而且在 p 0 较大，也就是VLTD本身输出的最大速度超过 V maxVLTD 时，速度饱和才起作用；而在 p 0 较小时，也就是VLTD本身输出的最大速度不超过 V maxVLTD 时，速度饱和不起作用。一旦 <math id="M89"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920952&type= 1.86266661 2.79399991 确定之后，无论 p 0 取值如何，加速度饱和都会起作用，因为只要 <math id="M90"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37920952&type= 1.86266661 2.79399991 > 1，那么初始段的加速度一定会大于 r 。 1,2;3,4; 将带速度和加速度饱和的VLTD（41）与带速度饱和的NLTD（7）进行比较，其中VLTD的参数取值与上面仿真相同，NLTD的参数取值为 r =12， h 0 =0.01，稳态值 p 0 取为100， 图5 图5 给出了两者的对比曲线。可以看出，VLTD可以达到与NLTD几乎相同的效果，并且从加速度曲线上可以看出，在接近稳态值时，VLTD的加速度曲线更平滑，这有利于反馈信号的平稳快速地跟踪。另外，VLTD的计算复杂度远远小于NLTD，在使用时能够节省计算内存空间，有利于算法的实时性。 4 VLTD在地平式大视场望远镜中的应用 1 4.1　地平式大视场望远镜搜索路径分析 2 大视场望远镜主要针对地球同步轨道（GEO）、同步转移轨道（GTO）及中高轨（MEO）的目标进行搜索、捕获、跟踪和测量。不同轨道的目标特性不同，相应的工作模式有所不同。由于极轴式望远镜受结构设计的限制，望远镜口径不宜过大，因此将米级大视场望远镜设计为地平式结构。由于地平式望远镜方位轴和俯仰轴的耦合作用，无论何种工作模式都会牵扯到方位轴的频繁变步长调转，即使是对固定赤纬的目标进行搜索。 本文以GEO目标为例进行说明，GEO目标的星下点赤纬范围为±15°，设观测站址的经纬度为 <math id="M91"><mo stretchy="false">(</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921024&type= 10.83733368 4.82600021 ， A 是地平式望远镜的方位角， E 是地平式望远镜的俯仰角。以望远镜俯仰角大于10°作为目标可观测的判别条件，可以计算出观测站能有效观测±15°赤纬带的GEO目标的星下点经度为 <math id="M92"><msub><mrow><mi>σ</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub><mo>±</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921029&type= 10.41399956 3.80999994 的目标，其中 <math id="M93"><msub><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921035&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921033&type= 2.87866688 3.72533321 由站址纬度 <math id="M94"><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921036&type= 3.21733332 4.57200003 决定。以两个不同纬度的站址为例，设 <math id="M95"><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">s</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">24</mn><mo>°</mo><mi mathvariant="normal">N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921040&type= 16.59466553 4.74133301 ， <math id="M96"><msub><mrow><mi>ϕ</mi></mrow><mrow><mi mathvariant="normal">s</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">46</mn><mo>°</mo><mi mathvariant="normal">N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921044&type= 16.59466553 4.74133301 ，望远镜视场为2.5°×2.5°，搜索路径采取“Z”字型模式，以星下点纬度不变，经度以2.5°作为阶跃步长，计算在不同纬度观测站观测±15°赤纬带目标时方位角和俯仰角的变化情况，并且分别计算了每次调转时方位角和俯仰角分别调转的步长，得到的曲线如 图6 图6 和 图7 图7 所示。 对比 图6 图6 和 图7 图7 可以看出，观测站点纬度越低，观测GEO目标带的俯仰角越大，能有效观测的目标带经度范围就越大，调转单个视场时，方位角需要调转的步长就越大。对于地平式望远镜来说，即使固定步长扫描同一赤纬的空间目标，方位轴仍是进行变步长的位置调转。另外，由于地平式望远镜两个轴的耦合运动，即使扫描同一赤纬的空间目标，望远镜的俯仰角并不是固定不变的，而是需要进行变步长的位置调转，上述两种情况中俯仰轴分别进行了0~2°不等的位置调转。在跟踪模式下，当切换被跟踪目标时，也需要调转方位轴和俯仰轴位置，而且调转的步长与所切换的两个目标的相对位置有关。 4.2　传统的位置环分段控制模式 2 对于地平式大视场望远镜来说，无论是进行搜索模式还是跟踪模式，都会频繁涉及到望远镜两个轴的变步长位置阶跃，而且阶跃步长大小不一。传统的望远镜主轴控制，一般对位置控制环采用分段控制的方式，当位置误差较大时，位置环输出常值；当位置误差较小时，则进行闭环解算。位置环输出的具体形式如下： <math display="block" id="M103"><msup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub><mo> </mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mfenced><mo>></mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>N</mi></mrow></msub></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>i</mi></mrow></msub><mo> </mo><mtext>  </mtext><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mtext> </mtext><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo stretchy="false">)</mo><mo> </mo><mo> </mo><mo> </mo><mi mathvariant="normal">i</mi><mi mathvariant="normal">f</mi><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></mfenced><mo>≤</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921094&type= 58.58933258 16.42533302 ， 其中： <math id="M104"><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921098&type= 24.38400078 3.21733332 ， <math id="M105"><msup><mrow><mi>ω</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921101&type= 3.47133350 3.04800010 是位置控制环的输出， <math id="M106"><msub><mrow><mi>e</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>=</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub><mo>-</mo><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921104&type= 18.20333290 3.72533321 是位置误差， <math id="M107"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921110&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921108&type= 4.06400013 3.72533321 是位置参考值， <math id="M108"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921112&type= 1.86266661 2.79399991 是位置反馈值， <math id="M109"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo><</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>N</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921118&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921116&type= 27.77066803 3.72533321 是预设的阈值， <math id="M110"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921120&type= 33.52799988 3.72533321 是位置控制环输出的常值， <math id="M111"><msub><mrow><mi>C</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921123&type= 3.72533321 3.72533321 是位置闭环控制器。该方法的缺点在于分段的过程比较依赖经验值，例如 <math id="M112"><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921126&type= 2.53999996 3.72533321 ， <math id="M113"><msub><mrow><mi>ω</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921128&type= 3.30200005 3.72533321 以及分段数 N 的选择都需要通过多次试验，才能使所设计的分段参数与物理系统的实际响应相匹配，否则如果切入闭环的速度过大或者过小，都会影响系统的响应品质。另外，一旦物理系统的最大速度或者最大加速度发生改变（例如系统的转动惯量发生改变、或者驱动器的驱动电流、驱动电压发生改变等），均需要对分段进行重新设计，否则会出现因分段与系统响应能力不匹配所造成的响应过慢或者超调等现象。 4.3　VLTD在地平式大视场望远镜快速定位中的应用 2 将带有速度和加速度饱和限制的VLTD应用于望远镜轴系的快速调转定位中，并且与传统的位置分段控制方式进行对比实验。实验对象是某1 m级地平式大视场望远镜的方位轴，轴系采用交流永磁同步电机驱动，驱动器和控制器硬件平台采用DSP+FPGA，驱动电源采用48 V的开关电源，驱动电流的最大值为10 A，位置反馈信号采用29位绝对式编码器测量，速度反馈信号通过位置反馈信号差分并滤波后得到，控制方式采用电流+速度+位置的三闭环方式，控制框图如 图8 图8 和 图9 图9 所示。其中， 图8 图8 的位置环采用传统的位置分段控制方式， 图9 图9 的位置环采用VLTD进行位置参考信号的过渡。除了位置误差的处理方式不同，两者的三环控制器设计完全相同，电流控制器的设计采用传统的PI控制器，速度环控制器采用ADRC控制方式，具体设计方法见文献［ 19 19 ］，此处不再赘述，位置环控制器采用最简单的比例控制器。 在进行位置环分段以及VLTD的设计之前，均需要对系统的最大反馈能力进行摸底测试，得到系统的最大反馈速度为20.5 （°）/s，最大加速度约为19.6 （°）/s 2 。在不存在机械限位的前提下，根据就近调转原则，方位轴的最大调转步长为180°。为了验证系统对不同步长的位置响应能力，用两种实验方法分别进行2°~180°之间不同阶跃步长的位置阶跃实验，其中位置环分段参数取值为： N =8， <math id="M114"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.01</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921137&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921136&type= 16.42533302 3.89466691 ， <math id="M115"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.017</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921138&type= 18.28800011 3.89466691 ， <math id="M116"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.28</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921141&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921140&type= 16.42533302 3.89466691 ， <math id="M117"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1.1</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921144&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921142&type= 14.56266594 3.89466691 ， <math id="M118"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">2.2</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921145&type= 14.56266594 3.89466691 ， <math id="M119"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">11</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921149&type= 12.61533356 3.89466691 ， <math id="M120"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">22</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921153&type= 12.61533356 3.89466691 ， <math id="M121"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">33</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921155&type= 12.61533356 3.89466691 ； <math id="M122"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.1</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921157&type= 22.77533340 4.48733330 ， <math id="M123"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.2</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921159&type= 22.77533340 4.48733330 ， <math id="M124"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.5</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921161&type= 22.77533340 4.48733330 ， <math id="M125"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1.9</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921163&type= 22.77533340 4.48733330 ， <math id="M126"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">3.8</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921165&type= 22.77533340 4.48733330 ， <math id="M127"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">5.3</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921167&type= 22.77533340 4.48733330 ， <math id="M128"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">11.8</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921169&type= 24.63800049 4.48733330 ， <math id="M129"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">16.7</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921171&type= 24.63800049 4.48733330 ， <math id="M130"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">9</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">20</mn><mtext> </mtext><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo><mo>/</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921173&type= 20.82799911 4.48733330 。 图10 图10 ~ 图13 图13 分别给出了几个典型的实验结果，其中 p 0 =2°，10°，30°和180°。可以看出，使用位置分段的方式均能够达到无超调的响应效果，但是由于分段的设置，在减速的过程中不能够充分发挥系统的最大减速能力，所以减速段的时间明显变长。 使用VLTD过渡的实验中，参数取值为 h =0.001， <math id="M131"><mi>σ</mi><mo>=</mo><mn mathvariant="normal">2.3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=37921188&type= 12.27666664 2.79399991 ， r =19， V max_VLTD =20。为了验证VLTD的输出速度作为速度前馈的作用，分别完成了带有速度前馈和不带有速度前馈的实验，几组典型的实验结果如 图14 图14 ~ 图17 图17 所示。从实验结果可以看出，使用带速度和加速度限制的VLTD安排的过渡过程能够按照预设的速度和加速度引导望远镜轴系运动，在加速和减速段都能够充分发挥出系统的反馈能力。 表2 表2 列出了实验中位置响应进入5″误差带的时间 t 1 和1″误差带的时间 t 2 ，它定量地描述了不同实验方法的效果。可以看出，使用VLTD安排过渡过程的方式可以使得动态响应时间明显变短，并且利用VLTD的第二个输出作为速度前馈能够进一步缩短响应时间，与位置分段方法相比，响应时间缩短了大约15%~37%，有效地提高了望远镜的搜索效率，对于大视场望远镜具有重要的意义。 2 设定值/（°） 2 时间 2 位置分 段方法 2 VLTD方法 不带速度前馈 带速度前馈 2 2 t 1 1.394 1.301 1.085 t 2 1.526 1.487 1.252 2 10 t 1 2.946 2.258 2.025 t 2 3.074 2.436 2.181 2 30 t 1 4.574 3.147 2.847 t 2 4.708 3.308 2.948 2 180 t 1 12.171 10.652 10.351 t 2 12.301 10.802 10.452 s 对比实验中只给出了一种分段取法，虽然通过增加分段数 N 以及优化分段策略可以进一步提高该方法的响应速度，但是优化过程需要进行大量的实验，而且分段数 N 越多，程序的计算复杂度越高，也不利于程序的二次移植。而本文给出的VLTD设计方法，参数设定方法简单，只需测量物理系统的最大反馈速度和加速度，就可以进行参数的设定，且程序可以进行快速移植。 5 结 论 1 本文给出了一种为物理信号安排过渡过程的工具——变参数线性跟踪微分器，它是在分析已有线性跟踪微分器安排过渡过程的不足基础上提出的，所给出的过渡过程能够很好地与物理系统的反馈能力相匹配，在起到过渡作用的同时，又能够充分发挥系统的最大反馈能力。从理论上对VLTD的收敛性进行了证明，并且给出了仿真计算。将VLTD应用于地平式大视场望远镜的快速调转控制中，替代传统的位置环基于位置误差的分段控制方式，实验结果表明，在各种不同大小步长的位置阶跃响应中，响应时间可以缩短大约15%~37%，大大提高了望远镜的搜索效率。所设计的安排过渡过程方法的参数简单易设置，利于在工程中的调试和实现。 本文中的分析以及实验均是针对常值信号进行的，VLTD也可应用于对时变信号的过渡过程安排，在后续工作中会对VLTD跟踪时变信号的性质进行分析。 参考文献 韩京清 . 自抗扰控制器及其应用 ［J］. 控制与决策 ， 1998 ， 13 （ 1 ）： 19 - 23 . doi: 10.3321/j.issn:1001-0920.1998.01.005 http://dx.doi.org/10.3321/j.issn:1001-0920.1998.01.005 自抗扰控制器及其应用 HAN J Q . Auto-disturbances-rejection controller and its applications ［J］. Control and Decision ， 1998 ， 13 （ 1 ）： 19 - 23 . （in Chinese） . doi: 10.3321/j.issn:1001-0920.1998.01.005 http://dx.doi.org/10.3321/j.issn:1001-0920.1998.01.005 Auto-disturbances-rejection controller and its applications HAN J Q . From PID to active disturbance rejection control ［J］. IEEE Transactions on Industrial Electronics ， 2009 ， 56 （ 3 ）： 900 - 906 . doi: 10.1109/tie.2008.2011621 http://dx.doi.org/10.1109/tie.2008.2011621 From PID to active disturbance rejection control ZHANG D Y ， YAO X L ， WU Q H ， et al . ADRC based control for a class of input time delay systems ［J］. Journal of Systems Engineering and Electronics ， 2017 ， 28 （ 6 ）： 1210 - 1220 . ADRC based control for a class of input time delay systems 赵希梅 ， 吴岑 . 基于滑模自抗扰的PMLSM电流偏差解耦控制 ［J］. 光学 精密工程 ， 2022 ， 30 （ 4 ）： 431 - 441 . doi: 10.37188/OPE.20223004.0431 http://dx.doi.org/10.37188/OPE.20223004.0431 基于滑模自抗扰的PMLSM电流偏差解耦控制 ZHAO X M ， WU C . Current deviation decoupling control based on sliding mode active disturbance rejection for PMLSM ［J］. Opt. Precision Eng. ， 2022 ， 30 （ 4 ）： 431 - 441 . （in Chinese） . doi: 10.37188/OPE.20223004.0431 http://dx.doi.org/10.37188/OPE.20223004.0431 Current deviation decoupling control based on sliding mode active disturbance rejection for PMLSM YUAN Y ， YU Y ， WANG Z D ， et al . A sampled-data approach to nonlinear ESO-based active disturbance rejection control for pneumatic muscle actuator systems with actuator saturations ［J］. IEEE Transactions on Industrial Electronics ， 2019 ， 66 （ 6 ）： 4608 - 4617 . doi: 10.1109/TIE.2018.2864711 http://dx.doi.org/10.1109/TIE.2018.2864711 A sampled-data approach to nonlinear ESO-based active disturbance rejection control for pneumatic muscle actuator systems with actuator saturations 黄浦 ， 杨秀丽 ， 修吉宏 ， 等 . 音圈致动快速反射镜的降阶自抗扰控制 ［J］. 光学 精密工程 ， 2020 ， 28 （ 6 ）： 1365 - 1374 . 音圈致动快速反射镜的降阶自抗扰控制 HUANG P ， YANG X L ， XIU J H ， et al . Reduced-order active disturbance rejection control of fast steering mirror driven by VCA ［J］. Opt. Precision Eng. ， 2020 ， 28 （ 6 ）： 1365 - 1374 . （in Chinese） Reduced-order active disturbance rejection control of fast steering mirror driven by VCA SHI D W ， XUE J ， ZHAO L X ， et al . Event-triggered active disturbance rejection control of DC torque motors ［J］. IEEE/ASME Transactions on Mechatronics ， 2017 ， 22 （ 5 ）： 2277 - 2287 . Event-triggered active disturbance rejection control of DC torque motors LI X ， ZHOU W L ， LUO J ， et al . A new mechanical resonance suppression method for large optical telescope by using nonlinear active disturbance rejection control ［J］. IEEE Access ， 7 ： 94400 - 94414 . doi: 10.1109/ACCESS.2019.2928050 http://dx.doi.org/10.1109/ACCESS.2019.2928050 A new mechanical resonance suppression method for large optical telescope by using nonlinear active disturbance rejection control 韩京清 ， 王伟 . 非线性跟踪-微分器 ［J］. 系统科学与数学 ， 1994 ， 14 （ 2 ）： 177 - 183 . 非线性跟踪-微分器 HAN J G ， WANG W . Nonlinear tracking-differentiator ［J］. Journal of Systems Science and Mathematical Sciences ， 1994 ， 14 （ 2 ）： 177 - 183 . （in Chinese） Nonlinear tracking-differentiator ZHANG H H ， XIAO G X ， YU X H ， et al . On convergence performance of discrete-time optimal control based tracking differentiator ［J］. IEEE Transactions on Industrial Electronics ， 2021 ， 68 （ 4 ）： 3359 - 3369 . On convergence performance of discrete-time optimal control based tracking differentiator GUO B Z ， ZHAO Z L . Weak convergence of nonlinear high-gain tracking differentiator ［J］. IEEE Transactions on Automatic Control ， 2013 ， 58 （ 4 ）： 1074 - 1080 . doi: 10.1109/TAC.2012.2218153 http://dx.doi.org/10.1109/TAC.2012.2218153 Weak convergence of nonlinear high-gain tracking differentiator YU J D ， JIN S H . Sliding mode tracking differentiator with adaptive gains for filtering and derivative estimation of noisy signals ［J］. IEEE Access ， 2021 ， 9 ： 86017 - 86024 . Sliding mode tracking differentiator with adaptive gains for filtering and derivative estimation of noisy signals GHANES M ， BARBOT J P ， FRIDMAN L ， et al . A second order sliding mode differentiator with a variable exponent ［C］. 2017 American Control Conference （ACC）. May 24 - 26 ， 2017 . Seattle， WA， USA. IEEE ， 2017 ： 3300 - 3305 . A second order sliding mode differentiator with a variable exponent XIE Y D ， ZHANG H H ， SHE L H ， et al . Design and implementation of an efficient tracking differentiator ［J］. IEEE Access ， 7 ： 101941 - 101949 . Design and implementation of an efficient tracking differentiator BU X W ， WU X Y ， CHEN Y X ， et al . Design of a class of new nonlinear disturbance observers based on tracking differentiators for uncertain dynamic systems ［J］. International Journal of Control， Automation and Systems ， 2015 ， 13 （ 3 ）： 595 - 602 . doi: 10.1007/s12555-014-0173-6 http://dx.doi.org/10.1007/s12555-014-0173-6 Design of a class of new nonlinear disturbance observers based on tracking differentiators for uncertain dynamic systems SHEN W ， LIU X Y ， SU X Y . High-precision position tracking control of electro-hydraulic servo systems based on an improved structure and desired compensation ［J］. International Journal of Control， Automation and Systems ， 2021 ， 19 （ 11 ）： 3622 - 3630 . doi: 10.1007/s12555-020-0705-1 http://dx.doi.org/10.1007/s12555-020-0705-1 High-precision position tracking control of electro-hydraulic servo systems based on an improved structure and desired compensation ZHANG H H ， XIE Y D ， XIAO G X ， et al . A simple discrete-time tracking differentiator and its application to speed and position detection system for a maglev train ［J］. IEEE Transactions on Control Systems Technology ， 2019 ， 27 （ 4 ）： 1728 - 1734 . A simple discrete-time tracking differentiator and its application to speed and position detection system for a maglev train YANG Z B ， JI J L ， SUN X D ， et al . Active disturbance rejection control for bearingless induction motor based on hyperbolic tangent tracking differentiator ［J］. IEEE Journal of Emerging and Selected Topics in Power Electronics ， 2020 ， 8 （ 3 ）： 2623 - 2633 . Active disturbance rejection control for bearingless induction motor based on hyperbolic tangent tracking differentiator YANG X X ， DENG Y T ， ZHANG B ， et al . Extended state observer-based control with an adjustable parameter for a large ground-based telescope ［J］. Research in Astronomy and Astrophysics ， 2021 ， 21 （ 12 ）： 316 . doi: 10.1088/1674-4527/21/12/316 http://dx.doi.org/10.1088/1674-4527/21/12/316 Extended state observer-based control with an adjustable parameter for a large ground-based telescope 10.37188/OPE.20223023.3045 TP273 A 1004-924X（2022）23-3045-13 杨晓霞,邓永停,赵金宇等.变参数线性跟踪微分器及其在地平式大视场望远镜中的应用[J].光学精密工程,2022,30(23):3045-3057. 杨晓霞,邓永停,赵金宇等.变参数线性跟踪微分器及其在地平式大视场望远镜中的应用[J].光学精密工程,2022,30(23):3045-3057. YANG Xiaoxia,DENG Yongting,ZHAOJinyu ,et al.Variable parameter linear tracking differentiator and its application in altazimuth large area telescope[J].Optics and Precision Engineering,2022,30(23):3045-3057. YANG Xiaoxia,DENG Yongting,ZHAOJinyu ,et al.Variable parameter linear tracking differentiator and its application in altazimuth large area telescope[J].Optics and Precision Engineering,2022,30(23):3045-3057. 国家自然科学基金资助项目（No.12122304，No.11973041）；中国科学院青年创新促进会会员项目（No.2019218） 2022-09-30 2022-11-26 2022-12-10 2022-12-20 光学精密工程 23 30