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<div> <div id="article-table-of-contents" data-ismobile="false"> <div class="TableOfContents_tableOfContents__0jAuv"> <div class="TableOfContents_tableOfContents__title__CH9Lz"> 在这一页上</d我v> <a class="TableOfContents_tableOfContents__item__JJil1" href="#abstract">文摘</一个> <a class="TableOfContents_tableOfContents__item__JJil1" href="#introduction">介绍</一个> <a class="TableOfContents_tableOfContents__item__JJil1" href="#conclusions">结论</一个> <a class="TableOfContents_tableOfContents__item__JJil1" href="#data-availability">数据可用性</一个> <a class="TableOfContents_tableOfContents__item__JJil1" href="#conflicts-of-interest">的利益冲突</一个> <a class="TableOfContents_tableOfContents__item__JJil1" href="#acknowledgments">确认</一个> <a class="TableOfContents_tableOfContents__item__JJil1" href="#references">引用</一个> <a class="TableOfContents_tableOfContents__item__JJil1" href="#copyright">版权</一个> <a class="TableOfContents_tableOfContents__item__JJil1" href="#related-articles">相关文章</一个> </div> </div> </div> </div> </div> <div class="ant-col ThreeColumnLayout_middleColumn__sbd41"> <div class="ArticleMiddleSection_articleMiddleSection__gAv9z"> <div class="articleHeader"> <div class="articleHeader__specialIssue"> <span class="articleHeader__specialIssue_tag">特殊的问题</年代p一个n> <h2 class="articleHeader__specialIssue_title">分析方法模型</h2> <a href="//www.newsama.com/journals/jmath/si/821741/" class="articleHeader__specialIssue_url" target="_blank" rel="noreferrer">把这个特殊的问题</一个> </div> <strong>研究文章|开放获取</年代trong> <div class="articleHeader__meta"> <span>体积2022年|</年代p一个n> <span>文章的ID7185131|</年代p一个n> <span class="articleHeader__meta_doiLink"><a href="https://doi.org/10.1155/2022/7185131" aria-label="Doi-link" target="_blank" rel="noreferrer">https://doi.org/10.1155/2022/7185131</一个></span> </div> <div class="simpleShowMore"> <button class="simpleShowMore__button">显示引用</button> </div> <h1 class="articleHeader__title">影响反应曲线的选择代表点的确定的分数阶模型的准确性</h1> <div class="articleHeader__authors"> <span class="articleHeader__authors_author"><strong>胡安·j·古德</年代trong><a href="//www.newsama.com/cdn-cgi/l/email-protection" aria-label="Mail Option"><span role="img" class="anticon"> <svg version="1.0" xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="0 0 1401 1101"> <path fill="#6ba439" d="M132.5 2.1C101.4 6 71.1 20.5 48.3 42.4c-22.7 21.7-37.5 48.3-44.6 80l-2.2 10.1v837l2.2 10.1c6.7 30 19.3 53.9 39.8 75.3 23.3 24.3 56.4 40.9 90 45 12.8 1.6 1122.2 1.6 1135 0 38.9-4.8 75.9-25.6 100-56.4 15-19.3 24.2-39 29.8-63.9l2.2-10.1v-837l-2.2-10.1c-7.1-31.8-21.9-58.3-44.6-80.1-20.2-19.2-43.9-31.7-73.6-38.5l-9.6-2.3L705 1.4c-321.9-.1-568.5.2-572.5.7zm1123.4 99.5c1.7.4 3.1 1 3.1 1.4 0 .5-125.6 87.7-279 194L700.9 490.2 422 297C268.5 190.7 143 103.4 143 103c0-.5 1-1.1 2.3-1.3 3.5-.7 1107.1-.8 1110.6-.1zM401 404c164.4 113.8 299.4 207 300 207 .6 0 135.6-93.2 300.1-207 164.5-113.9 299.3-207 299.5-207 .2 0 .4 170.9.4 379.7 0 417.9.5 386-6.1 398.3-6.2 11.6-19 21.4-31.9 24.4-5.8 1.4-67.4 1.6-562 1.6s-556.2-.2-562-1.6c-12.9-3-25.7-12.8-31.9-24.4-6.6-12.3-6.1 19.6-6.1-398.3 0-208.8.2-379.7.5-379.7S236.6 290.1 401 404z"></path> </svg></span></a><a href="https://orcid.org/0000-0003-4210-2454" aria-label="Redirect to Doi" target="_blank" rel="noreferrer"><span role="img" class="anticon"> <svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 256 256" xml:space="preserve" width="1em" height="1em"> <path d="M256 128c0 70.7-57.3 128-128 128S0 198.7 0 128 57.3 0 128 0s128 57.3 128 128z" fill="#a6ce39"></path> <path d="M86.3 186.2H70.9V79.1h15.4v107.1zm22.6-107.1h41.6c39.6 0 57 28.3 57 53.6 0 27.5-21.5 53.6-56.8 53.6h-41.8V79.1zm15.4 93.3h24.5c34.9 0 42.9-26.5 42.9-39.7C191.7 111.2 178 93 148 93h-23.7v79.4zM88.7 56.8c0 5.5-4.5 10.1-10.1 10.1s-10.1-4.6-10.1-10.1c0-5.6 4.5-10.1 10.1-10.1s10.1 4.6 10.1 10.1z" fill="#fff"></path> </svg></span></a><sup>1</年代up></span> <span class="articleHeader__authors_author">和巴勃罗•加西亚Bringas<一个href="https://orcid.org/0000-0003-3594-9534" aria-label="Redirect to Doi" target="_blank" rel="noreferrer"><span role="img" class="anticon"> <svg xmlns="http://www.w3.org/2000/svg" viewbox="0 0 256 256" xml:space="preserve" width="1em" height="1em"> <path d="M256 128c0 70.7-57.3 128-128 128S0 198.7 0 128 57.3 0 128 0s128 57.3 128 128z" fill="#a6ce39"></path> <path d="M86.3 186.2H70.9V79.1h15.4v107.1zm22.6-107.1h41.6c39.6 0 57 28.3 57 53.6 0 27.5-21.5 53.6-56.8 53.6h-41.8V79.1zm15.4 93.3h24.5c34.9 0 42.9-26.5 42.9-39.7C191.7 111.2 178 93 148 93h-23.7v79.4zM88.7 56.8c0 5.5-4.5 10.1-10.1 10.1s-10.1-4.6-10.1-10.1c0-5.6 4.5-10.1 10.1-10.1s10.1 4.6 10.1 10.1z" fill="#fff"></path> </svg></span></a><sup>2</年代up></span> </div> <div class="simpleShowMore"> <button class="simpleShowMore__button">显示更多</button> </div> <div class="articleHeader__academicEditor"> <strong>学术编辑器:</年代trong> <span>穆罕默德Alomari</年代p一个n> </div> <div class="articleHeader__timeline"> <div class="articleHeader__timeline_item "> <strong>收到了</年代trong> <span>2022年2月15日</年代p一个n> </div> <div class="articleHeader__timeline_item "> <strong>接受</年代trong> <span>2022年4月29日(</年代p一个n> </div> <div class="articleHeader__timeline_item articleHeader__timeline_item_sticky"> <strong>发表</年代trong> <span>2022年6月3日</年代p一个n> </div> </div> </div> <div class="articleBody"> <div class="xml-content"> <h4 class="header" id="abstract">文摘</h4> <p>本文的一般程序确定一个分数阶+空载(FFOPDT)模型。这个过程是基于拟合三个过程反应曲线上任意点,处理信息在哪里获得从一个简单的开环测试。简化的一般识别过程也被认为是,只有点对称位于反应曲线选择。提出了对称的过程被应用到以下组代表点:(5 - 50 - 95%),(10 - 50 - 90%),(15 - 50 - 85%),(20 - 50 - 80%),(25 - 50 - 75%)和(30 - 50 - 70%)。相应的FFOPDT模型参数解析表达式对这些组对称点。为了显示这个过程的有效性和适用性为分数阶模型的识别和洞察的影响选择的一组对称点识别模型的准确性,提出了一些数值例子。这个识别过程给好的结果相比与其他整数和分数阶识别方法。最后,在这种背景下提出了结论和最后的讲话。</p> <span class="end-abs"></span> <h4 id="introduction">1。介绍</h4> <p>虽然有可能获得一个模型流程解析法的物理模型,它更频繁的动态信息过程是通过实验tests-empirical模式过程反应曲线,关键控制系统的增益和振荡的周期稳定的极限,或通过继电器反馈<一个href="#B1">1</一个>]。</p> <p>从控制器的角度来看,控制过程包括过程本身、终端控制元件和测量仪器。在控制研究和设计和优化控制器的反馈控制回路,有必要控制过程的动态行为信息,通常的形式降维数学模型(<一个href="#B2">2</一个>]。控制流程模型提供了控制器之间的动态信息和测量仪表的输出信号。这个模型应该是简单的,但与此同时,它应该提供可靠的信息控制在操作点过程的行为。这些信息通常包括增益、时间常数(s)和明显的空时的控制过程。</p> <p>尽管过程控制的相当大的进步在过去的几十年里,众所周知,比例积分微分(PID)控制器广泛成功应用在工业控制应用中,成为工业过程控制标准(后<一个href="#B3">3</一个>]。学术和工业社会的承认,是应用最广泛的控制流程模型的一阶控制算法,辅助阳极,和二阶+空载(FOPDT、DPPDT SOPDT);然而,在某些情况下,这些模型不能代表了过程动力学与所需的精度。这是已经建议,例如,(<一个href="#B4">4</一个>),表明有必要过程有更好的信息来获取最优PID调优规则lag-dominated流程。</p> <p>实践经验表明,控制器调优通常可以通过很少的工厂信息的角度设计技术标准。使用一个开环步测试实验流程模型识别是广泛覆盖技术文献;见,例如,(描述的识别过程<一个href="#B5">5</一个>- - - - - -<一个href="#B8">8</一个>]。</p> <p>一些报纸也报道了基于拟合多个代表点的识别算法过程中瞬态响应一个阶跃变化(<一个href="#B9">9</一个>- - - - - -<一个href="#B11">11</一个>]。技术文献,有几个两到三点为FOPDT识别方法,DPPDT和SOPDT模型基于信息从反应曲线。</p> <p>在两点程序的情况下,可以考虑以下方法和时间:(<一个href="#B12">12</一个>)(35 - 85%),(<一个href="#B13">13</一个>(28.3 - -63.2%),或(<一个href="#B14">14</一个>)(33 - 70%)。</p> <p>三分的方法,以下引用和对应的时间可以被认为是:[<一个href="#B15">15</一个>(2 - 70 - 90%)和(5 - 70 - 90%),(<一个href="#B16">16</一个>)由斯塔克(15 - 45 - 75%)和(<一个href="#B17">17</一个>(14 - 55 - 91%)。</p> <p>123 c识别方法(<一个href="#B11">11</一个>)使用时间设置(25 - 75%)FOPDT DPPDT和(25 - 50 - 75%)SOPDT模型,分别。这种方法最近延长(<一个href="#B18">18</一个>)的识别repeated-pole +空载(RPPDT)模型。</p> <p>在前面提到的引用,这些点的位置是不同的:在某些情况下,选择给出任何解释;在别人,重要的点与模型参数选择密切相关。在一些方法,例如,(<一个href="#B11">11</一个>),这些点不是任意固定而被选择为了优化确定了模型参数。很明显,确定模型的准确性取决于这些点的选择过程的反应曲线,将在稍后讨论。</p> <p>分数阶微积分在过去的几十年里,出现了大量的学术和工业努力专注于获得更精确的建模方法和识别现实世界的现象;见,例如,(<一个href="#B19">19</一个>- - - - - -<一个href="#B21">21</一个>]。有各种各样的分数阶模型技术基于技术文献的反应曲线,最常见的方法是一个基于非线性优化;见,例如,(<一个href="#B22">22</一个>- - - - - -<一个href="#B25">25</一个>]。在这些方法中,获得的分数阶模型参数一般是通过最小化过程反应之间的误差曲线和分数阶阶跃响应模型。这些技术要求更高的计算工作与现有的分析方法相比,其主要特点是简单的应用程序。然而,没有很多分数阶建模过程的分析技术基于过程的反应曲线。在[<一个href="#B26">26</一个>),提出了一些策略,以确定FFOPDT模型的参数利用阶跃响应数据。它结合了数值计算和图形估计。这个参考可以被视为一种分数阶的开创性工作情况。Integral-based估计方法提出了(<一个href="#B27">27</一个>,<一个href="#B28">28</一个>),健壮的对测量噪声的存在。这两个方法可以被看作是一个扩展的方法中存在的经典案例(<一个href="#B3">3</一个>]。</p> <p>即使分数模型已经证明技术上优越,工业采用分步方法需要更多的分析(<一个href="#B29">29日</一个>]。</p> <p>此外,最近的调查列出了分数阶PID控制器作为一个新兴的工具在过程控制领域,其成功的主要原因是他们提供的固有可靠性更高的自由度和调优操作参数(<一个href="#B30">30.</一个>- - - - - -<一个href="#B33">33</一个>]。在这种背景下,最近的一些应用程序的控制算法可以被认为是在<一个href="#B34">34</一个>- - - - - -<一个href="#B37">37</一个>]。在一些现有的方法设计分数阶PID控制器,一个简单的模型的过程是利用目标控制器的优化参数(<一个href="#B38">38</一个>- - - - - -<一个href="#B43">43</一个>]。</p> <p>根据上述,获得结构简单分数阶模型的过程具有十分重要的意义,是非常有用的在实际设计整数和分数阶控制系统。因为过程阶跃响应是简单的物理解释和识别算法integer-order模型基于拟合多个代表点过程的反应曲线很容易申请,这可以被认为是自然的扩展这些方法对于分数阶的情况。为此,FFOPDT模型的通用识别过程进行了本文基于拟合三分在瞬态过程阶跃输入响应。在相同的框架中,提出了一种通用识别过程的简化,考虑到选择的点对称的过程反应曲线。相应的FFOPDT模型参数解析表达式好几套对称点是为了显示获得的适用性提出过程,洞察组对称点的影响确定了模型的准确性。</p> <p>本文的组织结构如下:部分<一个href="#sec2">2</一个>致力于提供数学基础和背景。节<一个href="#sec3">3</一个>的一般程序识别FFOPDT模型从任何三分过程反应曲线。节<一个href="#sec4">4</一个>,一个简化的一般识别过程被认为是,点在哪里选择对称过程反应曲线。相应的FFOPDT模型参数解析表达式好几套对称点也获得了在这一节中。提供了一些例子<一个href="#sec5">5</一个>显示该程序的有效性和适用性获得识别方法确定FFOPDT相比与其他整数和分数阶模型参数识别方法。此外,一些数值模拟的结果也说明在本节中,为了得到洞察的影响选择组对称点的识别模型的准确性。提出了结论和最后的讲话<一个href="#sec6">6</一个>。</p> <h4 id="theoretical-background">2。理论背景</h4> <p>在本节中,一些基本概念和定义部分简要讨论了微积分。</p> <p>分数微积分是集成的泛化和分化noninteger基本算子<年代p一个nclass="nowrap"> <svg height="13.8999pt" id="M1" style="vertical-align:-3.94351pt" version="1.1" viewbox="-0.0498162 -9.95639 21.1022 13.8999" width="21.1022pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.0091,0,0,-0.0091,0,3.784)"> <path d="M490 97L476 124C442 96 405 70 398 70C392 70 390 78 396 114C419 243 448 379 463 432L457 436C446 436 431 439 418 442C393 447 368 451 343 451C281 451 204 418 155 381C74 320 24 206 24 107C24 23 59 -12 88 -12C118 -12 155 5 191 34C236 70 290 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-68 275 89 275 269V271Z" id="g190-42"></path> </g> </g> </svg></span></span>一个和<我>t</我>操作的范围和吗<我>α</我>∈<年代p一个nclass="nowrap"> <svg height="8.73791pt" id="M3" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -8.68809 9.84705 8.73791" width="9.84705pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <path d="M712 0L497 285C582 312 628 378 628 466C628 609 518 662 390 662H70V0H236V271H310L505 0H712ZM491 604C551 577 584 532 584 464C584 401 550 345 491 321V604ZM447 320C428 317 411 315 392 315H236V618H397C412 618 432 613 447 610V320ZM626 44H528L363 271C392 271 422 270 451 275L626 44ZM192 44H114V618H192V44Z" id="g197-29"></path> </g> </svg>。</年代p一个n></p> <p>有几种不同的定义部分运营商(<一个href="#B44">44</一个>]。部分集成的一个最常用的定义是Riemann-Liouville定义:<年代p一个nclass="equation_break" id="EEq2"><span class="left_break" id="L4"></span><span class="middle_break"><span class="center_break" id="M4"> <svg height="30.6442pt" 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transform="matrix(.0091,0,0,-0.0091,70.2,12.704)"> <path d="M556 236V289H56V236H556Z" id="g54-33"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,75.76,12.704)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,70.2,22.752)"> <use xlink:href="#g50-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,81.59,18.968)"> <use xlink:href="#g113-103"></use> </g> <g transform="matrix(.013,0,0,-0.013,92.12,19.151)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,96.64,18.968)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,101.07,19.151)"> <use xlink:href="#g113-42"></use> </g> </g> </svg> <svg height="30.6442pt" id="M4-2" version="1.1" viewbox="0 0 150.224 30.6442" width="150.224pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B4-2"> <g transform="matrix(.013,0,0,-0.013,0,18.968)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,21.95,10.46)"> <use xlink:href="#g113-50"></use> </g> <rect height="0.65243" width="25.2362" x="12.46" y="15.314"></rect> <g transform="matrix(.013,0,0,-0.013,12.46,28.193)"> <path d="M485 501C480 550 475 615 475 650H42V622C119 616 127 606 127 525V126C127 44 119 34 41 28V0H311V28C220 34 212 43 212 126V584C212 612 216 615 244 615H318C382 615 404 608 420 587C435 568 445 541 455 497L485 501Z" id="g113-132"></path> </g> <g transform="matrix(.013,0,0,-0.013,21.24,28.376)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,25.75,28.193)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,33.14,28.376)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,38.89,18.981)"> <use xlink:href="#g119-89"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,47.65,5.43)"> <use xlink:href="#g50-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,47.65,29.578)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,54.77,19.151)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,59.29,18.968)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,66.63,18.968)"> <path d="M535 230V280H52V230H535Z" id="g117-33"></path> </g> <g transform="matrix(.013,0,0,-0.013,77.16,18.968)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,83.45,19.151)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,88.03,12.704)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,93.32,12.704)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,98.88,12.704)"> <path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path> </g> <g transform="matrix(.013,0,0,-0.013,106.05,18.968)"> <use xlink:href="#g113-103"></use> </g> <g transform="matrix(.013,0,0,-0.013,116.58,19.151)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,121.09,18.968)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,127.38,19.151)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,134.09,18.968)"> <use xlink:href="#g190-101"></use> </g> <g transform="matrix(.013,0,0,-0.013,140.98,18.968)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,147.26,18.968)"> <use xlink:href="#g113-45"></use> </g> </g> </svg> <svg class="alt_equation" height="54.7941pt" id="I4" version="1.1" viewbox="0 0 163.9 54.7941" width="163.9pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 -7.65)" xlink:href="#B4-1"></use> <use transform="matrix(1 0 0 1 13.59 24.15)" xlink:href="#B4-2"></use> </svg></span></span><span class="right_break" id="R4"> <svg class="label" height="11.44pt" id="M4-3" version="1.1" viewbox="0 0 15.038 11.44" width="15.038pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B4-3"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <path d="M411 139C381 76 368 73 314 73H129L267 224C351 317 400 380 400 469C400 574 322 635 233 635C176 635 128 610 98 576L40 495L63 476C89 515 130 568 198 568C274 568 316 519 316 436C316 347 251 264 191 192C142 134 85 76 30 21V0H403C415 45 425 89 438 131L411 139Z" id="g190-243"></path> </g> <g transform="matrix(.013,0,0,-0.013,10.54,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg> <svg class="alt_label" height="54.7941pt" id="IL4" version="1.1" viewbox="0 0 15.1 54.7941" width="15.1pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 21.47)" xlink:href="#B4-3"></use> </svg></span></span>在哪里<我>t</我>≥0,<我>α</我>∈<年代p一个nclass="inline_break"> <svg height="12.3732pt" id="M5" style="vertical-align:-1.576501pt" version="1.1" viewbox="-0.0498162 -10.7967 18.901 12.3732" width="18.901pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g197-29"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,9.711,-5.741)"> <path d="M556 236V289H337V504H275V289H56V236H275V-4H337V236H556Z" id="g54-36"></path> </g> <g transform="matrix(.013,0,0,-0.013,15.79,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span></span>和Γ(·)γ函数(<一个href="#B44">44</一个>]。</p> <p>从关系(<一个href="#EEq2">2</一个>),订单的Riemann-Liouville分数导数的定义<我>α</我>可以被写成以下形式:<年代p一个nclass="equation_break" id="EEq3"><span class="left_break" id="L6"></span><span class="middle_break"><span class="center_break" id="M6"> <svg height="30.6442pt" id="M6-1" version="1.1" viewbox="0 0 44.738 30.6442" width="44.738pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B6-1"> <g transform="matrix(.0091,0,0,-0.0091,0,22.752)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.95,19.934)"> <use xlink:href="#g113-69"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,14.94,12.704)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,14.94,22.752)"> <use xlink:href="#g50-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,20.76,18.968)"> <use xlink:href="#g113-103"></use> </g> <g transform="matrix(.013,0,0,-0.013,31.29,19.151)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,35.8,18.968)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,40.24,19.151)"> <use xlink:href="#g113-42"></use> </g> </g> </svg> <svg height="30.6442pt" id="M6-2" version="1.1" viewbox="0 0 75.479 30.6442" width="75.479pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B6-2"> <g transform="matrix(.013,0,0,-0.013,0,18.968)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,31.95,10.46)"> <use xlink:href="#g113-50"></use> </g> <rect height="0.65243" width="45.2573" x="12.46" y="15.314"></rect> <g transform="matrix(.013,0,0,-0.013,12.46,28.193)"> <use xlink:href="#g113-132"></use> </g> <g transform="matrix(.013,0,0,-0.013,21.23,28.376)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,25.75,28.193)"> <path d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z" id="g113-111"></path> </g> <g transform="matrix(.013,0,0,-0.013,35.18,28.193)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,45.71,28.193)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,53.1,28.376)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,61.84,10.46)"> <use xlink:href="#g113-101"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,69.06,4.719)"> <path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z" id="g50-111"></path> </g> <rect height="0.65243" width="15.8878" x="60.11" y="15.314"></rect> <g transform="matrix(.013,0,0,-0.013,60.11,28.193)"> <use xlink:href="#g190-101"></use> </g> <g transform="matrix(.013,0,0,-0.013,66.81,28.193)"> <path d="M298 36L289 62C276 55 253 45 228 45C202 45 169 60 169 141V397H276C289 405 292 426 282 437H169V574L155 576L90 509V437H45L17 408L21 397H90V107C90 28 125 -12 188 -12C198 -12 213 -8 230 1L298 36Z" id="g190-117"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,70.82,22.178)"> <use xlink:href="#g50-111"></use> </g> </g> </svg> <svg height="30.6442pt" id="M6-3" version="1.1" viewbox="0 0 2.964 30.6442" width="2.964pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B6-3"> <g transform="matrix(.013,0,0,-0.013,0,18.968)"> <path d="M170 255C170 288 146 313 114 313S58 288 58 255C58 221 82 198 114 198S170 221 170 255Z" id="g113-46"></path> </g> </g> </svg> <svg height="30.6442pt" id="M6-4" version="1.1" viewbox="0 0 121.914 30.6442" width="121.914pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B6-4"> <g transform="matrix(.013,0,0,-0.013,0,18.981)"> <use xlink:href="#g119-89"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,8.75,5.43)"> <use xlink:href="#g50-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,8.75,29.578)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,15.88,19.151)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,20.39,18.968)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,27.73,18.968)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,38.27,18.968)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,44.55,19.151)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,49.13,12.704)"> <use xlink:href="#g50-111"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,53.79,12.704)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,59.35,12.704)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,64.65,12.704)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,70.21,12.704)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,77.41,18.968)"> <use xlink:href="#g113-103"></use> </g> <g transform="matrix(.013,0,0,-0.013,87.94,19.151)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,92.46,18.968)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,98.74,19.151)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,105.46,18.968)"> <use xlink:href="#g113-101"></use> </g> <g transform="matrix(.013,0,0,-0.013,112.67,18.968)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,118.95,18.968)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R6"> <svg class="label" height="11.44pt" id="M6-5" version="1.1" viewbox="0 0 14.868 11.44" width="14.868pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B6-5"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <path d="M269 381C298 400 325 421 340 436C360 455 370 476 370 504C370 579 313 635 223 635H222C170 635 123 610 95 579L52 516L72 497C95 533 136 577 193 577C245 577 286 546 286 482C286 408 218 369 127 339L133 314C149 319 174 325 197 325C254 325 324 285 324 192C325 83 266 39 201 39C143 39 103 68 78 91C70 98 62 97 54 91C45 84 33 71 32 58C30 46 34 35 48 22C61 10 102 -12 148 -12C216 -12 410 57 410 225C410 298 357 362 269 379V381Z" id="g190-232"></path> </g> <g transform="matrix(.013,0,0,-0.013,10.37,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>在哪里<我>n</我>−1 <<我>α</我><<我>n</我>,<我>n</我>∈<年代p一个nclass="nowrap"> <svg height="10.8465pt" id="M7" style="vertical-align:-0.04980087pt" version="1.1" viewbox="-0.0498162 -10.7967 15.6651 10.8465" width="15.6651pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <path d="M677 618V662H100V618H469L50 44V0H668V44H252L677 618ZM617 618L200 44H103L523 618H617Z" id="g197-37"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,9.451,-5.741)"> <use xlink:href="#g54-36"></use> </g> </svg>。</年代p一个n>在定义(<一个href="#EEq2">2</一个>)和(<一个href="#EEq3">3</一个>),下标0<我>t</我>是操作的限制和被称为终端部分的集成和分化,分别。为了简单起见,<年代vg height="14.7403pt" id="M8" style="vertical-align:-3.943601pt" version="1.1" viewbox="-0.0498162 -10.7967 26.4365 14.7403" width="26.4365pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.0091,0,0,-0.0091,0,3.784)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.946,.966)"> <use xlink:href="#g113-69"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,14.942,-5.741)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,20.502,-5.741)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,14.942,3.784)"> <use xlink:href="#g50-117"></use> </g> </svg>用<年代vg height="10.8465pt" id="M9" style="vertical-align:-0.04980087pt" version="1.1" viewbox="-0.0498162 -10.7967 23.6653 10.8465" width="23.6653pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-69"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,9.958,-5.741)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,15.518,-5.741)"> <use xlink:href="#g50-223"></use> </g> </svg>和<年代vg height="13.8999pt" id="M10" style="vertical-align:-3.94351pt" version="1.1" viewbox="-0.0498162 -9.95639 20.8556 13.8999" width="20.8556pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.0091,0,0,-0.0091,0,3.784)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.946,.966)"> <use xlink:href="#g113-69"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,14.942,-5.741)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,14.942,3.784)"> <use xlink:href="#g50-117"></use> </g> </svg>通过<年代p一个nclass="nowrap"> <svg height="10.0062pt" id="M11" style="vertical-align:-0.04980946pt" version="1.1" viewbox="-0.0498162 -9.95639 15.9091 10.0062" width="15.9091pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-69"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,9.958,-5.741)"> <use xlink:href="#g50-223"></use> </g> </svg>。</年代p一个n></p> <p>的拉普拉斯变换Riemann-Liouville-based分数导数<年代p一个nclass="equation_break" id="EEq4"><span class="left_break" id="L12"></span><span class="middle_break"><span class="center_break" 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446 563 500 563 583Z" id="g50-224"></path> </g> <g transform="matrix(.0065,0,0,-0.0065,123.19,16.252)"> <use xlink:href="#g176-108"></use> </g> <g transform="matrix(.013,0,0,-0.013,127.57,20.415)"> <path d="M515 96L502 119C471 88 431 62 423 62C416 62 411 70 417 101C440 223 469 341 497 448H486L412 422L380 277C330 188 210 57 152 57C137 57 126 69 139 124L195 366C210 431 205 448 182 448C155 448 89 413 23 350L36 326C73 354 103 376 112 376C118 376 118 365 113 340L61 118C54 90 52 68 52 51C52 0 75 -12 98 -12S151 -3 181 17C242 58 305 119 362 193H364L345 104C323 3 339 -12 359 -12C390 -12 464 35 515 96Z" id="g113-118"></path> </g> <g transform="matrix(.013,0,0,-0.013,136.74,20.598)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,141.25,20.415)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,145.69,20.598)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,152.4,20.415)"> <use xlink:href="#g113-47"></use> </g> </g> </svg></span></span><span class="right_break" id="R14"> <svg class="label" height="11.44pt" id="M14-2" version="1.1" viewbox="0 0 15.428 11.44" width="15.428pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B14-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.49,9.26)"> <path d="M143 344C173 483 265 547 326 576C363 593 402 605 434 611L429 641C389 634 343 623 301 608C195 570 43 456 43 236C43 81 135 -12 245 -12C368 -12 453 88 453 208C453 310 380 393 273 393C253 393 232 386 211 376L143 344ZM234 338C325 338 367 255 367 172C367 102 346 21 264 21C184 21 132 113 132 239C132 267 133 291 138 312C164 327 197 338 234 338Z" id="g190-227"></path> </g> <g transform="matrix(.013,0,0,-0.013,10.93,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span></p> <p>相应的分数阶传递函数不相称的真正的订单有以下形式(<一个href="#B44">44</一个>]:<年代p一个nclass="equation_break" id="EEq7"><span class="left_break" id="L15"></span><span class="middle_break"><span class="center_break" id="M15"> <svg height="36.503pt" id="M15-1" version="1.1" viewbox="0 0 78.559 36.503" width="78.559pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B15-1"> <g transform="matrix(.013,0,0,-0.013,0,23.877)"> <path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path> </g> <g transform="matrix(.013,0,0,-0.013,11.07,24.06)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,15.59,23.877)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.013,0,0,-0.013,20.46,24.06)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,28.63,23.877)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,41.09,12.485)"> <path d="M699 368C699 549 574 666 407 666C186 666 23 488 23 277C23 113 129 -3 288 -13L307 -26C431 -111 501 -139 533 -147C559 -154 613 -163 658 -164L666 -141C597 -111 507 -66 430 -11L416 -1C580 42 699 190 699 368ZM601 371C601 227 518 54 381 22L354 40L278 24C175 47 120 145 120 269C120 451 235 631 398 631C540 631 601 521 601 371Z" id="g113-82"></path> </g> <g transform="matrix(.013,0,0,-0.013,52.65,12.485)"> <path d="M378 -385C239 -253 158 0 158 271S238 795 378 927L356 953C165 798 79 550 79 272V271C79 -10 165 -259 356 -411L378 -385Z" id="g119-133"></path> </g> <g transform="matrix(.013,0,0,-0.013,58.43,12.485)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,63.31,6.744)"> <use xlink:href="#g50-224"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,68.42,8.845)"> <use xlink:href="#g176-108"></use> </g> <g transform="matrix(.013,0,0,-0.013,72.8,12.485)"> <path d="M364 271V272C364 550 278 798 87 953L65 927C205 795 285 542 285 271S204 -253 65 -385L87 -411C278 -259 364 -10 364 271Z" id="g119-134"></path> </g> <rect height="0.65243" width="37.5444" x="41.09" y="20.223"></rect> <g transform="matrix(.013,0,0,-0.013,43.06,33.102)"> <path d="M600 480C600 590 528 650 384 650H143L137 622C222 614 225 607 210 531L130 127C113 41 106 36 23 28L17 0H294L300 28C204 36 195 42 212 127L243 284L314 263C327 263 339 263 352 264C465 271 600 337 600 480ZM508 481C508 351 402 304 329 304C289 304 265 311 250 317L295 559C302 594 310 606 323 611C335 616 350 619 367 619C455 619 508 573 508 481Z" id="g113-81"></path> </g> <g transform="matrix(.013,0,0,-0.013,53.27,33.285)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,57.78,33.102)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,62.66,28.927)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,67.72,31.028)"> <use xlink:href="#g176-108"></use> </g> <g transform="matrix(.013,0,0,-0.013,72.1,33.285)"> <use xlink:href="#g113-42"></use> </g> </g> </svg> <svg height="36.503pt" id="M15-2" version="1.1" viewbox="0 0 202.664 36.503" width="202.664pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B15-2"> <g transform="matrix(.013,0,0,-0.013,0,23.877)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,28.57,14.915)"> <use xlink:href="#g113-99"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,33.56,18.047)"> <use xlink:href="#g50-110"></use> </g> <g transform="matrix(.013,0,0,-0.013,41.41,14.915)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,46.29,9.174)"> <use xlink:href="#g50-224"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,51.4,11.274)"> <path d="M798 93L780 124C747 88 714 75 706 75C699 75 699 79 706 108L752 301C786 442 748 454 723 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transform="matrix(.0091,0,0,-0.0091,89.11,18.047)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,94.1,14.915)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,98.98,9.174)"> <use xlink:href="#g50-224"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,104.09,11.274)"> <use xlink:href="#g176-110"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,109.44,11.274)"> <path d="M578 242V297H59V242H578Z" id="g180-33"></path> </g> <g transform="matrix(.0065,0,0,-0.0065,113.58,11.274)"> <path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z" id="g176-50"></path> </g> <g transform="matrix(.013,0,0,-0.013,120.76,14.915)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,131.3,14.915)"> <use xlink:href="#g113-46"></use> </g> <g transform="matrix(.013,0,0,-0.013,136.44,14.915)"> <use xlink:href="#g113-46"></use> </g> <g transform="matrix(.013,0,0,-0.013,141.58,14.915)"> <use xlink:href="#g113-46"></use> </g> <g transform="matrix(.013,0,0,-0.013,147.49,14.915)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,158.02,14.915)"> <use xlink:href="#g113-99"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,163.02,18.047)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,167.96,14.915)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,172.84,9.174)"> <use xlink:href="#g50-224"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,177.95,11.274)"> <path d="M455 312C455 472 400 635 248 635C95 635 39 458 39 312C39 144 93 -12 248 -12C400 -12 455 166 455 312ZM361 314C361 173 337 30 249 30C163 30 131 174 131 313C131 451 155 594 248 594C334 594 361 454 361 314Z" id="g176-49"></path> </g> <rect height="0.65243" width="186.045" x="12.46" y="20.223"></rect> <g transform="matrix(.013,0,0,-0.013,12.46,33.102)"> <use 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transform="matrix(.013,0,0,-0.013,112.96,33.102)"> <use xlink:href="#g113-46"></use> </g> <g transform="matrix(.013,0,0,-0.013,118.1,33.102)"> <use xlink:href="#g113-46"></use> </g> <g transform="matrix(.013,0,0,-0.013,124,33.102)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,134.54,33.102)"> <use xlink:href="#g113-98"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,140.53,36.234)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,145.48,33.102)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,150.35,28.927)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,155.42,31.028)"> <use xlink:href="#g176-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,162.57,33.102)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,173.1,33.102)"> <use xlink:href="#g113-98"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,179.1,36.234)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,184.04,33.102)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,188.92,28.927)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,193.99,31.028)"> <use xlink:href="#g176-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,199.7,23.877)"> <use xlink:href="#g113-45"></use> </g> </g> </svg> <svg class="alt_equation" height="78.266pt" id="I15" version="1.1" viewbox="0 0 206.33 78.266" width="206.33pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 0)" xlink:href="#B15-1"></use> <use transform="matrix(1 0 0 1 3.63 41.76)" xlink:href="#B15-2"></use> </svg></span></span><span class="right_break" id="R15"> <svg class="label" height="11.44pt" id="M15-3" version="1.1" viewbox="0 0 15.028 11.44" width="15.028pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B15-3"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <path d="M430 623H48C43 580 39 534 30 478H57C80 544 84 550 152 550H369C286 366 187 178 74 -1L81 -12L156 -2C249 201 343 408 438 611L430 623Z" id="g190-223"></path> </g> <g transform="matrix(.013,0,0,-0.013,10.53,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg> <svg class="alt_label" height="78.266pt" id="IL15" version="1.1" viewbox="0 0 15.09 78.266" width="15.09pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 33.2)" xlink:href="#B15-3"></use> </svg></span></span>在P (<年代p一个nclass="nowrap"> <svg height="10.1628pt" id="M16" style="vertical-align:-0.2064095pt" version="1.1" viewbox="-0.0498162 -9.95639 14.4369 10.1628" width="14.4369pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,4.875,-5.741)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,9.944,-3.641)"> <use xlink:href="#g176-108"></use> </g> </svg>)</年代p一个n>和Q (<年代p一个nclass="nowrap"> <svg height="12.4646pt" id="M17" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -12.2582 14.4826 12.4646" width="14.4826pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,4.875,-5.741)"> <use xlink:href="#g50-224"></use> </g> <g transform="matrix(.0065,0,0,-0.0065,9.989,-3.641)"> <use xlink:href="#g176-108"></use> </g> </svg>)</年代p一个n>没有共同的0,<年代vg height="6.1673pt" id="M18" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 6.7023 6.1673" width="6.7023pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-98"></use> </g> </svg><sub><i>k</我></sub>(<我>k</我>= 0…<我>n</我>),<我>b</我><sub>k</年代ub>(<我>k</我>= 0…<我>米</我>)是常数,<年代vg height="6.1673pt" id="M19" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 7.51131 6.1673" width="7.51131pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-223"></use> </g> </svg><sub>k</年代ub>(<我>k</我>= 0…<我>n</我>),<我>ß</我><sub>k</年代ub>(<我>k</我>= 0…<我>米</我>)是任意的真实或有理数,不失一般性,他们可以安排<我>α</我><sub>n</年代ub>><我>α</我><sub>n</年代ub><sub>−</年代ub><sub>1</年代ub>>···><我>α</我><sub>0</年代ub>,<我>β</我><sub>米</年代ub>><我>ß</我><sub><i>米</我></sub><sub><i>−</我></sub><sub><i>1</我></sub>>···><我>ß</我><sub>0</年代ub>。</p> <p>严格的传递函数G (s)由(<一个href="#EEq7">7</一个>)BIBO稳定当且仅当P (s)没有根在{Re (s)≥0}<一个href="#B45">45</一个>]。</p> <p>在特定的情况下,当有一个实数<我>α</我>的最大公约数<我>α</我><sub>k</年代ub>,<我>k</我>= 1,…,<我>n</我>,<我>ß</我><sub>k</年代ub>,<我>k</我>= 1,…,<我>米</我>,它被称为相称的秩序。它认为,<我>α</我><sub>k</年代ub>= k<我>α</我>和<我>β</我><sub>k</年代ub>= k<我>α</我>0 <<我>α</我>< 1,<年代p一个nclass="nowrap"> <svg height="9.49473pt" id="M20" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 33.5135 9.49473" width="33.5135pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <path d="M563 650H502L416 416H157L71 650H10L256 -15H317L563 650ZM397 366L287 68H286L176 366H397Z" id="g117-46"></path> </g> <g transform="matrix(.013,0,0,-0.013,7.449,0)"> <path d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 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xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B21-1"> <g transform="matrix(.013,0,0,-0.013,0,21.763)"> <use xlink:href="#g113-72"></use> </g> <g transform="matrix(.013,0,0,-0.013,11.07,21.946)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,15.58,21.763)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.013,0,0,-0.013,20.46,21.946)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,28.62,21.763)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,41.08,12.263)"> <use xlink:href="#g113-82"></use> </g> <g transform="matrix(.013,0,0,-0.013,52.65,12.263)"> <path d="M332 -241C217 -138 149 60 149 271C149 483 216 680 332 783L312 808C148 686 74 491 74 272V271C74 50 148 -146 312 -266L332 -241Z" id="g119-5"></path> </g> <g transform="matrix(.013,0,0,-0.013,57.67,12.263)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,62.55,6.522)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,68.36,12.263)"> <path d="M311 271V272C311 491 237 686 73 808L53 783C169 680 236 483 236 271C236 60 168 -138 53 -241L73 -266C237 -146 311 50 311 271Z" id="g119-6"></path> </g> <rect height="0.65243" width="32.3553" x="41.08" y="18.109"></rect> <g transform="matrix(.013,0,0,-0.013,42.28,30.988)"> <use xlink:href="#g113-81"></use> </g> <g transform="matrix(.013,0,0,-0.013,52.48,31.171)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,57,30.988)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,61.87,26.813)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,67.69,31.171)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,78.26,21.763)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,91.23,12.265)"> <path d="M697 -46L668 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xlink:href="#g50-223"></use> </g> <rect height="0.65243" width="51.4827" x="90.72" y="18.109"></rect> <g transform="matrix(.013,0,0,-0.013,90.72,31.368)"> <use xlink:href="#g119-65"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,100.51,24.968)"> <use xlink:href="#g50-111"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,100.51,35.146)"> <use xlink:href="#g50-108"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,105.14,35.146)"> <use xlink:href="#g54-34"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,110.7,35.146)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,115.69,31.361)"> <use xlink:href="#g113-98"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,121.68,34.493)"> <use xlink:href="#g50-108"></use> </g> <g transform="matrix(.013,0,0,-0.013,126.83,31.361)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,131.7,27.186)"> <use xlink:href="#g50-108"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,136.33,27.186)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,143.4,21.763)"> <use xlink:href="#g113-47"></use> </g> </g> </svg></span></span><span class="right_break" id="R21"> <svg class="label" height="11.44pt" id="M21-2" version="1.1" viewbox="0 0 15.168 11.44" width="15.168pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B21-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <path d="M246 635C138 635 67 554 67 471C67 401 116 348 178 315C143 296 103 268 87 252C65 231 42 203 42 157C42 50 128 -12 234 -12C319 -12 432 52 432 169C432 256 370 303 299 344C347 376 373 399 381 408C399 430 408 459 408 487C408 569 341 635 246 635ZM235 603C284 603 334 567 334 483C334 420 306 383 273 358C202 393 142 427 142 503C142 552 173 603 235 603ZM245 21C179 21 122 70 122 163C122 219 155 270 204 300C279 263 352 219 352 144C352 66 308 21 245 21Z" id="g190-152"></path> </g> <g transform="matrix(.013,0,0,-0.013,10.67,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span></p> <p>已经证明相应系统G (s)带来了(<一个href="#EEq8">8</一个>)是BIBO稳定的如果所有多项式方程的根<我>P</我>(<我>x</我>)= 0,<我>x</我>=年代<年代up><i>α</我></sup>定位的部门| arg (<我>x</我>)|≤<我>α</我><i>π</我>/ 2 (<一个href="#B45">45</一个>]。</p> <p>考虑<我>n</我>><我>米</我>函数G (s)成为一个适当的有理函数的复杂变量s<年代up><i>α</我></sup>,如果它认为的根源<我>P</我>(<我>x</我>)= 0是截然不同的,传递函数的部分分式扩张(<一个href="#EEq8">8</一个>)可以用下面的一般形式:<年代p一个nclass="equation_break" id="EEq9"><span class="left_break" id="L22"></span><span class="middle_break"><span class="center_break" id="M22"> <svg height="35.6587pt" id="M22-1" version="1.1" viewbox="0 0 100.264 35.6587" width="100.264pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B22-1"> <g transform="matrix(.013,0,0,-0.013,0,20.415)"> <use xlink:href="#g113-72"></use> </g> <g transform="matrix(.013,0,0,-0.013,11.07,20.415)"> <use 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transform="matrix(.0091,0,0,-0.0091,49.56,34.064)"> <use xlink:href="#g54-34"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,55.12,34.064)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,74.61,11.563)"> <path d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z" id="g113-115"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,79.51,14.694)"> <use xlink:href="#g50-106"></use> </g> <rect height="0.65243" width="35.0238" x="61.09" y="16.761"></rect> <g transform="matrix(.013,0,0,-0.013,61.09,29.64)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,65.96,25.465)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,74.68,29.64)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,85.22,29.64)"> <path d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z" id="g113-233"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,92.98,32.772)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.013,0,0,-0.013,97.3,20.415)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R22"> <svg class="label" height="11.44pt" id="M22-2" version="1.1" viewbox="0 0 15.338 11.44" width="15.338pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B22-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <path d="M247 635C117 635 41 518 41 422C41 317 114 239 220 239C239 239 256 244 278 255L348 290C315 139 207 37 63 13L67 -15C93 -15 155 -5 207 17C343 71 443 203 443 385C443 520 371 635 247 635ZM231 602C331 602 355 476 355 390C355 371 354 349 351 325C334 310 298 294 261 294C177 294 128 369 128 457C128 516 155 602 231 602Z" id="g190-193"></path> </g> <g transform="matrix(.013,0,0,-0.013,10.84,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>在哪里<年代vg height="9.49473pt" id="M23" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-233"></use> </g> </svg><sub>我</年代ub>,<我>我</我>= 1,…,<我>n</我>的根<我>P</我>(<我>x</我>)= 0和<我>r</我><sub>我</年代ub>,<我>我</我>= 1,…,<我>n</我>是相应的残留物。从(拉普拉斯逆变换<一个href="#EEq9">9</一个>)脉冲响应的结果<我>G</我>(年代<年代up><i>α</我></sup>(提供)<一个href="#B44">44</一个>]。<年代p一个nclass="equation_break" id="EEq10"><span class="left_break" id="L24"></span><span class="middle_break"><span class="center_break" id="M24"> <svg height="37.2364pt" id="M24-1" version="1.1" viewbox="0 0 126.286 37.2364" width="126.286pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B24-1"> <g transform="matrix(.013,0,0,-0.013,0,21.946)"> <path d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z" id="g113-105"></path> </g> <g 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xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B28-3"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.49,9.26)"> <use xlink:href="#g190-213"></use> </g> <g transform="matrix(.013,0,0,-0.013,8.89,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.76,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>在初始条件通常是作为零获取标准FFOPDT传递函数模型:<年代p一个nclass="equation_break" id="EEq14"><span class="left_break" id="L29"></span><span class="middle_break"><span class="center_break" id="M29"> <svg height="29.8573pt" id="M29-1" version="1.1" viewbox="0 0 82.484 29.8573" width="82.484pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B29-1"> <g transform="matrix(.013,0,0,-0.013,0,20.282)"> <use xlink:href="#g113-81"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.21,20.465)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.72,20.282)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.013,0,0,-0.013,19.6,20.465)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,27.76,20.282)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,44.3,11.774)"> <use xlink:href="#g113-76"></use> </g> <g transform="matrix(.013,0,0,-0.013,54.12,11.774)"> <path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" id="g113-102"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,59.5,6.033)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,65.06,6.033)"> <path d="M567 167L534 181C505 122 482 94 454 70C428 46 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</svg></span></span>在哪里<我>K</我>是一个过程,<我>T</我>> 0是明显的时间常数,<我>l</我>≥0是明显的空时,<我>α</我>的分数阶模型。</p> <p>参数的设置<年代p一个nclass="equation_break" id="EEq15"><span class="left_break" id="L30"></span><span class="middle_break"><span class="center_break" id="M30"> <svg height="13.9183pt" id="M30-1" version="1.1" viewbox="0 0 95.684 13.9183" width="95.684pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B30-1"> <g transform="matrix(.013,0,0,-0.013,0,9.288)"> <path d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z" id="g113-230"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,6.15,12.42)"> <path d="M622 479C622 589 545 650 400 650H146L140 618C229 609 230 602 215 527L135 132C117 43 112 41 27 32L18 0H305L313 32C213 40 205 44 222 132L251 284L324 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11.44" width="19.288pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B30-2"> <g transform="matrix(.013,0,0,-0.013,0,9.25)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.25)"> <use xlink:href="#g190-213"></use> </g> <g transform="matrix(.013,0,0,-0.013,8.89,9.25)"> <use xlink:href="#g190-161"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.79,9.25)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>代表了FFOPDT模型参数,确定本文使用信息从过程反应曲线。</p> <p>标准FOPDT<年代p一个nclass="equation_break" id="EEq16"><span class="left_break" id="L31"></span><span class="middle_break"><span class="center_break" id="M31"> <svg height="29.8573pt" id="M31-1" version="1.1" viewbox="0 0 76.664 29.8573" width="76.664pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B31-1"> <g transform="matrix(.013,0,0,-0.013,0,20.282)"> <use xlink:href="#g113-81"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.2,20.465)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.72,20.282)"> <use xlink:href="#g113-116"></use> </g> <g transform="matrix(.013,0,0,-0.013,19.59,20.465)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,27.76,20.282)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,41.39,11.774)"> <use xlink:href="#g113-76"></use> </g> <g transform="matrix(.013,0,0,-0.013,51.21,11.774)"> <use xlink:href="#g113-102"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,56.59,6.033)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,62.15,6.033)"> <use xlink:href="#g50-77"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,67.25,6.033)"> <use xlink:href="#g50-116"></use> </g> <rect height="0.65243" width="32.2869" x="40.22" y="16.628"></rect> <g transform="matrix(.013,0,0,-0.013,40.22,29.507)"> <use xlink:href="#g113-50"></use> </g> <g 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</svg></span></span>可以看作一种特殊情况下的FFOPDT模型(<一个href="#EEq14">14</一个>),<我>α</我>= 1。FFOPDT模型的一步反应增加的值<我>α</我>,从<我>α</我>= 0.2 - 1.8,如图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig1/" target="_blank">1</一个>。考虑系统的阶跃响应<我>α</我>= 1中虚线表示。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-686cfc124d7c959f2b68a4ce4d95e838"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.001.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-686cfc124d7c959f2b68a4ce4d95e838 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.001_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-686cfc124d7c959f2b68a4ce4d95e838 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图1</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig1/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 认为分数阶系统的一步反应<我>α</我>∈[0.2,1.8]。的阶跃响应<我>α</我>= 1的红色虚线。</d我v> </div> </div> </div> <p>FOPDT模型(<一个href="#EEq16">16</一个>)已经被广泛应用在实践中捕获的基本动态响应工业过程控制为目的的设计(<一个href="#B3">3</一个>]。在这篇文章中,参数的设置(<一个href="#EEq15">15</一个>)被认为是控制过程的动态行为的特点。FFOPDT模型(<一个href="#EEq14">14</一个>)可以被视为一种泛化的古典FOPDT和泛化的相关性有很大影响的动态过程的识别以及动态反馈回路的控制器参数设计(<一个href="#B46">46</一个>]。</p> <p>也有用的考虑一些参数来描述过程动力学。<我>T</我><sub>基于“增大化现实”技术</年代ub>平均停留时间,可以在分数阶模型的上下文中定义为<年代p一个nclass="equation_break" id="EEq17"><span class="left_break" id="L32"></span><span class="middle_break"><span class="center_break" id="M32"> <svg height="41.3388pt" id="M32-1" version="1.1" viewbox="0 0 144.824 41.3388" width="144.824pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B32-1"> <g transform="matrix(.013,0,0,-0.013,0,23.997)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,7.18,27.129)"> <use xlink:href="#g50-98"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,11.86,27.129)"> <use xlink:href="#g50-115"></use> </g> <g transform="matrix(.013,0,0,-0.013,22.87,23.997)"> <path d="M-161 571C-161 600 -183 623 -212 623C-238 623 -262 600 -262 571C-262 544 -238 520 -212 520C-183 520 -161 544 -161 571Z" id="g117-11"></path> </g> <g transform="matrix(.013,0,0,-0.013,16.27,23.997)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,25.13,12.716)"> <path d="M480 853C480 879 456 900 424 900C303 900 237 760 239 491L245 -163C246 -299 234 -327 208 -327C187 -327 171 -289 137 -289C112 -289 90 -309 90 -335C90 -363 116 -389 155 -389C277 -389 342 -245 338 99L332 673C330 820 346 844 370 844C396 844 396 808 432 808C460 808 480 826 480 853Z" id="g119-81"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,32.56,4.215)"> <use xlink:href="#g50-30"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,32.56,18.276)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,42.54,12.71)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,46.06,12.71)"> <use xlink:href="#g113-104"></use> </g> <g transform="matrix(.013,0,0,-0.013,55.64,12.892)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,60.16,12.71)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,64.59,12.892)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,71.3,12.71)"> <use xlink:href="#g190-101"></use> </g> <g transform="matrix(.013,0,0,-0.013,78.19,12.71)"> <use xlink:href="#g113-117"></use> </g> <rect height="0.65243" width="57.555" x="25.13" y="20.343"></rect> <g transform="matrix(.013,0,0,-0.013,26.89,35.619)"> <use xlink:href="#g119-81"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,34.33,27.118)"> <use xlink:href="#g50-30"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,34.33,41.179)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,44.31,35.613)"> <use xlink:href="#g113-104"></use> </g> <g transform="matrix(.013,0,0,-0.013,53.89,35.795)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,58.4,35.613)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,62.84,35.795)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,69.55,35.613)"> <use xlink:href="#g190-101"></use> </g> <g transform="matrix(.013,0,0,-0.013,76.44,35.613)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,87.51,23.997)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,98.77,23.997)"> <use xlink:href="#g113-77"></use> </g> <g transform="matrix(.013,0,0,-0.013,108.86,23.997)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,119.4,23.997)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,127.68,17.733)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,132.12,17.733)"> <path d="M381 704H317L48 -163H112L381 704Z" id="g50-48"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,136.02,17.733)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,141.86,23.997)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R32"> <svg class="label" height="11.44pt" id="M32-2" version="1.1" viewbox="0 0 19.338 11.44" width="19.338pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B32-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-213"></use> </g> <g transform="matrix(.013,0,0,-0.013,8.81,9.26)"> <use xlink:href="#g190-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.84,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>在哪里<年代vg height="9.39034pt" id="M33" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -5.96091 7.52435 9.39034" width="7.52435pt" 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transform="matrix(.0091,0,0,-0.0091,82.07,22.567)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,85.97,22.567)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,93.1,17.144)"> <use xlink:href="#g113-47"></use> </g> </g> </svg></span></span><span class="right_break" id="R34"> <svg class="label" height="11.44pt" id="M34-2" version="1.1" viewbox="0 0 19.498 11.44" width="19.498pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B34-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-213"></use> </g> <g transform="matrix(.013,0,0,-0.013,8.83,9.26)"> <use xlink:href="#g190-152"></use> </g> <g transform="matrix(.013,0,0,-0.013,15,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span></p> <p>这个参数可以用来描述一个过程控制的难度。大致说来,过程与小<我>τ</我>容易控制,在控制系统中增加困难吗<我>τ</我>增加。的具体情况<我>α</我>T = 1,<年代ub>基于“增大化现实”技术</年代ub>和<我>τ</我>参数对应的标准定义(FOPDT模型<一个href="#B3">3</一个>]。</p> <p>虽然被认为是部分的阶跃响应模型可以方便地描述单调或非行为根据分数阶<我>α</我>,这个识别过程只适用于一个有一个s形阶跃响应过程,因为系统本质上是单步的反应是非常普遍的在过程控制<一个href="#B3">3</一个>]。此外,建议在<一个href="#B4">4</一个>)这类的过程PID是合适的基本上可以概括为拥有单调一步反应。描述这些过程的一种方法是引入的单调性指数:<年代p一个nclass="equation_break" id="EEq19"><span class="left_break" id="L35"></span><span class="middle_break"><span class="center_break" id="M35"> <svg height="41.3388pt" id="M35-1" version="1.1" viewbox="0 0 90.984 41.3388" width="90.984pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B35-1"> <g transform="matrix(.013,0,0,-0.013,0,23.997)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,7.06,27.129)"> <use xlink:href="#g50-110"></use> </g> <g transform="matrix(.013,0,0,-0.013,21.51,23.997)"> <use xlink:href="#g117-11"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.91,23.997)"> <use xlink:href="#g117-34"></use> </g> <g 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transform="matrix(.0091,0,0,-0.0091,31.2,27.118)"> <use xlink:href="#g50-30"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,31.2,41.179)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,43.36,28.945)"> <path d="M162 0V316H101V0H162Z" id="g119-34"></path> </g> <g transform="matrix(.013,0,0,-0.013,43.36,32.546)"> <use xlink:href="#g119-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,43.36,36.148)"> <use xlink:href="#g119-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,43.36,39.749)"> <use xlink:href="#g119-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,46.79,35.613)"> <use xlink:href="#g113-104"></use> </g> <g transform="matrix(.013,0,0,-0.013,56.37,35.795)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,60.89,35.613)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,65.32,35.795)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,69.85,28.945)"> <use 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xlink:href="#g190-213"></use> </g> <g transform="matrix(.013,0,0,-0.013,8.76,9.26)"> <use xlink:href="#g190-193"></use> </g> <g transform="matrix(.013,0,0,-0.013,15.1,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>在哪里<年代vg height="9.39034pt" id="M36" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -5.96091 7.52435 9.39034" width="7.52435pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-104"></use> </g> </svg>(<我>t</我>)是系统的脉冲响应。系统与<我>α</我>= 1单调一步反应和系统<我>α</我>> 0.8被认为是单调。</p> <p>图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig2/" target="_blank">2</一个>显示FFOPDT模型的规范化步骤反应(<一个href="#EEq14">14</一个>针对不同的分数阶值)<我>α</我>和不同的归一化值死时间<我>τ</我>。请注意,所有曲线相交于一点<我>t</我>=<我>T</我><sub><i>基于“增大化现实”技术</我></sub>因为正常化。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div 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class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图2</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig2/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 归一化一步反应<我>y</我> <sub><i>α</我></sub>/ K FFOPDT模型(<一个href="#EEq14">14</一个>针对不同的分数阶值)<我>α</我>和不同的归一化值死时间<我>τ</我>。规范化的死亡时间<我>τ</我>= 0(红色),0.25,0.5,0.75,1(蓝色)。注意,时间归一化平均停留时间<我>T</我> <sub><i>基于“增大化现实”技术</我></sub>。</d我v> </div> </div> </div> <h5 id="sec3.1">3.1。使用控制模型识别过程反应曲线</h5> <p>步信号u (<我>t</我>)和振幅Δu将被视为输入和一个信号<我>y</我><sub><i>α</我></sub>(<我>t</我>)和一个振幅Δy作为系统的响应,如图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig3/" target="_blank">3</一个>。FFOPDT模型(<一个href="#EEq14">14</一个>)应对∆u阶跃输入变化<年代p一个nclass="equation_break" id="EEq20"><span class="left_break" id="L37"></span><span class="middle_break"><span class="center_break" id="M37"> <svg height="47.1109pt" id="M37-1" version="1.1" 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transform="matrix(.013,0,0,-0.013,174.44,37.496)"> <use xlink:href="#g113-118"></use> </g> <g transform="matrix(.013,0,0,-0.013,181.43,37.496)"> <use xlink:href="#g113-45"></use> </g> <g transform="matrix(.013,0,0,-0.013,197.56,37.496)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,205.62,37.496)"> <path d="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM531 -40V10H57V-40H531Z" id="g117-94"></path> </g> <g transform="matrix(.013,0,0,-0.013,216.88,37.496)"> <use xlink:href="#g113-77"></use> </g> <g transform="matrix(.013,0,0,-0.013,224.33,37.496)"> <use xlink:href="#g113-45"></use> </g> </g> </svg> <svg class="alt_equation" height="69.1261pt" id="I37" version="1.1" viewbox="0 0 232.24 69.1261" width="232.24pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 -17.7)" xlink:href="#B37-1"></use> <use transform="matrix(1 0 0 1 3.64 22.02)" xlink:href="#B37-2"></use> </svg></span></span><span class="right_break" id="R37"> <svg class="label" height="11.44pt" id="M37-3" version="1.1" viewbox="0 0 21.518 11.44" width="21.518pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B37-3"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-243"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.61,9.26)"> <use xlink:href="#g190-254"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.02,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg> <svg class="alt_label" height="69.1261pt" id="IL37" version="1.1" viewbox="0 0 21.6 69.1261" width="21.6pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 28.63)" xlink:href="#B37-3"></use> </svg></span></span>在哪里<我>E</我><sub><i>α,β</我></sub>是一个米塔格-莱弗勒函数中定义(<一个href="#EEq11">11</一个>)。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-01c178ace7db11db9fdc61369629c46a"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.003.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-01c178ace7db11db9fdc61369629c46a partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.003_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-01c178ace7db11db9fdc61369629c46a partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图3</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig3/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 阶跃输入信号和任意代表点过程的反应曲线。</d我v> </div> </div> </div> <p>从过程反应曲线(<一个href="#EEq20">20.</一个>),是由<年代p一个nclass="equation_break" id="EEq21"><span class="left_break" id="L38"></span><span class="middle_break"><span class="center_break" id="M38"> <svg height="27.0583pt" id="M38-1" version="1.1" viewbox="0 0 45.984 27.0583" width="45.984pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B38-1"> <g transform="matrix(.013,0,0,-0.013,0,17.626)"> <use xlink:href="#g113-76"></use> </g> <g transform="matrix(.013,0,0,-0.013,13.45,17.626)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,25.91,8.636)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,34.24,8.636)"> <use xlink:href="#g113-122"></use> </g> <rect height="0.65243" width="15.9195" x="25.91" y="13.973"></rect> <g transform="matrix(.013,0,0,-0.013,26.18,26.852)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,34.5,26.852)"> <use xlink:href="#g113-118"></use> </g> <g transform="matrix(.013,0,0,-0.013,43.02,17.626)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R38"> <svg class="label" height="11.44pt" id="M38-2" version="1.1" viewbox="0 0 19.468 11.44" width="19.468pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B38-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-243"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.58,9.26)"> <use xlink:href="#g190-213"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.97,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>∆u在哪里输入信号的振幅和∆y是总流程输出变化,如图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig3/" target="_blank">3</一个>。</p> <p>流程输出<我>y</我><sub><i>α</我></sub>(<我>t</我>∆)可以标准化的最终价值<我>y</我>=<我>K</我>·∆u和使用转移和规范化<年代p一个nclass="inline_break"> <svg height="12.2245pt" id="M39" style="vertical-align:-2.268109pt" version="1.1" viewbox="-0.0498162 -9.95639 17.545 12.2245" width="17.545pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,9.914,0)"> <use xlink:href="#g117-34"></use> </g> </svg><span class="irelop"></span> <svg height="12.2245pt" style="vertical-align:-2.268109pt" version="1.1" viewbox="23.3021838 -9.95639 16.094 12.2245" width="16.094pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,23.352,0)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,27.85,0)"> <use xlink:href="#g113-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,34.09,0)"> <path d="M368 703H309L44 -163H104L368 703Z" id="g113-48"></path> </g> </svg><span class="isep"></span><span class="nowrap"> <svg height="12.2245pt" style="vertical-align:-2.268109pt" version="1.1" viewbox="39.4011838 -9.95639 53.011 12.2245" width="53.011pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,39.451,0)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.013,0,0,-0.013,47.736,0)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,52.234,0)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,56.732,0)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.013,0,0,-0.013,64.071,0)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,74.608,0)"> <use xlink:href="#g113-77"></use> </g> <g transform="matrix(.013,0,0,-0.013,81.792,0)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,86.29,-5.741)"> <use xlink:href="#g50-223"></use> </g> </svg>,</年代p一个n></span>与方程(<一个href="#EEq20">20.</一个>)减少到以下表达式:<年代p一个nclass="equation_break" id="EEq22"><span class="left_break" id="L40"></span><span class="middle_break"><span class="center_break" id="M40"> <svg height="14.0918pt" id="M40-1" version="1.1" viewbox="0 0 115.554 14.0918" width="115.554pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B40-1"> <g transform="matrix(.013,0,0,-0.013,1.82,9.027)"> <path d="M470 619L451 641C422 610 391 595 348 595C316 595 288 601 248 612C197 628 172 633 138 633C97 633 43 615 0 555L19 532C54 562 86 578 130 578C166 578 189 571 222 562C267 549 301 540 337 540C381 540 434 557 470 619Z" id="g119-25"></path> </g> <g transform="matrix(.013,0,0,-0.013,0,9.249)"> <use xlink:href="#g113-122"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.94,12.381)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.96,9.432)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,19.48,9.249)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,25.76,9.432)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,33.93,9.249)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,45.19,9.249)"> <use xlink:href="#g113-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,54.34,9.249)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,64.87,9.249)"> <use xlink:href="#g113-70"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,72.82,12.381)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,78.11,12.381)"> <use xlink:href="#g50-45"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,80.27,12.381)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,87.42,9.432)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,91.94,9.249)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,99.57,9.249)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,105.85,9.432)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,112.59,9.249)"> <use xlink:href="#g113-45"></use> </g> </g> </svg> <svg height="14.0918pt" id="M40-2" version="1.1" viewbox="0 0 30.374 14.0918" width="30.374pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B40-2"> <g transform="matrix(.013,0,0,-0.013,0,9.249)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,9.91,9.249)"> <use xlink:href="#g117-94"></use> </g> <g transform="matrix(.013,0,0,-0.013,21.17,9.249)"> <use xlink:href="#g113-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,27.41,9.249)"> <use xlink:href="#g113-47"></use> </g> </g> </svg> <svg class="alt_equation" height="14.0918pt" id="I40" version="1.1" viewbox="0 0 155.91 14.0918" width="155.91pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 0)" xlink:href="#B40-1"></use> <use transform="matrix(1 0 0 1 125.53 0)" xlink:href="#B40-2"></use> </svg></span></span><span class="right_break" id="R40"> <svg class="label" height="11.44pt" id="M40-3" version="1.1" viewbox="0 0 21.158 11.44" width="21.158pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B40-3"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-243"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.61,9.26)"> <use xlink:href="#g190-243"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.66,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg> <svg class="alt_label" height="14.0918pt" id="IL40" version="1.1" viewbox="0 0 21.23 14.0918" width="21.23pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 0.87)" xlink:href="#B40-3"></use> </svg></span></span></p> <p>如果<年代vg height="12.7178pt" id="M41" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 28.06 12.7178" width="28.06pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,1.82,-.222)"> <use xlink:href="#g119-25"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-122"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.942,3.132)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,12.782,0)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.28,0)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.013,0,0,-0.013,23.562,0)"> <use xlink:href="#g113-42"></use> </g> </svg>是规范化的过程输出,<年代vg height="9.25202pt" id="M42" style="vertical-align:-3.29111pt" version="1.1" viewbox="-0.0498162 -5.96091 11.2494 9.25202" width="11.2494pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,5.434,3.132)"> <path d="M545 403C545 425 527 451 498 451C448 451 400 404 312 286L290 341C262 412 246 451 219 451C182 451 138 404 92 345L112 323C152 368 169 378 181 378C192 378 201 359 216 321L257 214C184 115 142 69 113 69C102 69 93 73 88 79S78 89 68 89C47 89 24 61 24 40C24 8 49 -12 75 -12C125 -12 175 31 271 175L312 64C330 15 357 -12 382 -12C421 -12 475 35 515 98L496 121C466 88 439 62 420 62C402 62 384 92 363 147L325 247C349 280 374 309 397 333C422 361 446 381 462 381C471 381 481 374 488 366C493 360 499 357 505 357C521 357 545 380 545 403Z" id="g50-121"></path> </g> </svg>归一化所需的时间达到一些特定的输出归一化值<年代vg height="12.7178pt" id="M43" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 32.908 12.7178" width="32.908pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,1.82,-.222)"> <use xlink:href="#g119-25"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-122"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.942,3.132)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,12.782,0)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.28,0)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,22.714,3.132)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,28.41,0)"> <use xlink:href="#g113-42"></use> </g> </svg>这是在0和1之间,或流程输出总量的0%和100%的变化。的价值<我>τ</我><sub><i>x</我></sub>可以很容易地获得使用(<一个href="#EEq22">22</一个>)。</p> <p>相应地,时间<我>t</我><sub><i>x</我></sub>过程所需的输出<我>x</我>%流程输出总量的变化<年代p一个nclass="equation_break" id="EEq23"><span class="left_break" id="L44"></span><span class="middle_break"><span class="center_break" id="M44"> <svg height="17.782pt" id="M44-1" version="1.1" viewbox="0 0 94.424 17.782" width="94.424pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B44-1"> <g transform="matrix(.013,0,0,-0.013,0,13.818)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,4.37,16.949)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,13.7,13.818)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,24.96,13.818)"> <use xlink:href="#g113-77"></use> </g> <g transform="matrix(.013,0,0,-0.013,35.05,13.818)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,45.59,13.818)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,50.61,13.818)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,56.05,16.949)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,61.74,13.818)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.013,0,0,-0.013,70.03,13.818)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,75.1,6.517)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,79.53,6.517)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,83.44,6.517)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,91.46,13.818)"> <use xlink:href="#g113-47"></use> </g> </g> </svg></span></span><span class="right_break" id="R44"> <svg class="label" height="11.44pt" id="M44-2" version="1.1" viewbox="0 0 21.018 11.44" width="21.018pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B44-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.49,9.26)"> <use xlink:href="#g190-243"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.64,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.52,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span></p> <p>随着剩余FFOPDT模型参数{<我>T</我>,<我>l</我>,<我>α</我>必须获得},有必要确定* {<我>t</我><sub><i>x</我>1</年代ub>,<我>t</我><sub><i>x</我>2</年代ub>,<我>t</我><sub><i>x</我>3</年代ub>}达到三分{<我>y</我><sub><i>α</我></sub>(<我>t</我><sub><i>x</我>1</年代ub>),<我>y</我><sub><i>α</我></sub>(<我>t</我><sub><i>x</我>2</年代ub>),<我>y</我><sub><i>α</我></sub>(<我>t</我><sub><i>x</我>3</年代ub>)}过程反应曲线。考虑到方程(<一个href="#EEq23">23</一个>),下列方程组的定义:<年代p一个nclass="equation_break" id="EEq24"><span 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<p>所谓的比率指数Δ[<一个href="#B18">18</一个>可以适应这部分相应的框架,可以用来确定分数阶<我>α</我>:<年代p一个nclass="equation_break" id="EEq25"><span class="left_break" id="L46"></span><span class="middle_break"><span class="center_break" id="M46"> <svg height="37.6354pt" id="M46-1" version="1.1" viewbox="0 0 137.764 37.6354" width="137.764pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B46-1"> <g transform="matrix(.013,0,0,-0.013,0,22.145)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.1,22.145)"> <use xlink:href="#g117-11"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.5,22.145)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,19.36,13.293)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,23.73,16.425)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,28.9,16.425)"> <use xlink:href="#g50-52"></use> </g> <g 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<use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,102.54,33.618)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,113.08,33.618)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,119.36,27.568)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,123.79,27.568)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,127.7,27.568)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,118.51,37.476)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,123.69,37.476)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,134.8,22.145)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R46"> <svg class="label" height="11.44pt" id="M46-2" version="1.1" viewbox="0 0 21.078 11.44" width="21.078pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B46-2"> <g transform="matrix(.013,0,0,-0.013,0,9.25)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.25)"> <use xlink:href="#g190-243"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.68,9.25)"> <use xlink:href="#g190-161"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.58,9.25)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>在哪里<我>τ</我><sub>x1</年代ub>,<我>τ</我><sub>x2</年代ub>,<我>τ</我><sub>x3</年代ub>归一化倍,可以获得使用方程(<一个href="#EEq22">22</一个>)。方程(<一个href="#EEq25">25</一个>)可以表示为一个函数相关的分数阶<我>α</我>和比率指数∆,<我>α</我>=<我>f</我><sub>1</年代ub>(Δ)。</p> <p>基于时间的参数{<我>T</我>,<我>l</我>}可以解决的<一个href="#EEq24">24</一个>)通过考虑两个点,{<我>y</我><sub><i>α</我></sub>(<我>t</我><sub><i>x1</我></sub>),<我>t</我><sub><i>x1</我></sub>)}和{<我>y</我><sub><i>α</我></sub>(<我>t</我><sub><i>x3</我></sub>),<我>t</我><sub><i>x3</我></sub>)},过程反应曲线,相当于规范化点{<我>ỹ</我><sub><i>α</我></sub>(<我>τ</我><sub><i>x</我>1</年代ub>),<我>τ</我><sub><i>x</我>1</年代ub>},{<我>ỹ</我><sub><i>α</我></sub>(<我>τ</我><sub><i>x</我>3</年代ub>),<我>τ</我><sub><i>x</我>3</年代ub>}。然后,这些参数的表达式<年代p一个nclass="equation_break" id="EEq26"><span class="left_break" id="L47"></span><span class="middle_break"><span class="center_break" id="M47"> <svg height="41.0916pt" id="M47-1" version="1.1" viewbox="0 0 101.284 41.0916" width="101.284pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B47-1"> <g transform="matrix(.013,0,0,-0.013,0,11.516)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.013,0,0,-0.013,11.91,11.516)"> <use xlink:href="#g117-34"></use> </g> <g 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xlink:href="#g119-6"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,90.32,4.215)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,98.32,11.516)"> <use xlink:href="#g113-45"></use> </g> </g> </svg> <svg height="41.0916pt" id="M47-3" version="1.1" viewbox="0 0 95.924 41.0916" width="95.924pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B47-3"> <g transform="matrix(.013,0,0,-0.013,0,37.127)"> <use xlink:href="#g113-77"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.81,37.127)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,22.08,37.127)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,26.44,40.259)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,31.62,40.259)"> <use xlink:href="#g50-52"></use> </g> <g transform="matrix(.013,0,0,-0.013,39.49,37.127)"> <use xlink:href="#g117-33"></use> </g> <g 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xlink:href="#g190-243"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.63,9.26)"> <use xlink:href="#g190-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.66,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>在哪里<年代p一个nclass="equation_break" id="EEq28"><span class="left_break" id="L48"></span><span class="middle_break"><span class="center_break" id="M48"> <svg height="32.3469pt" id="M48-1" version="1.1" viewbox="0 0 81.264 32.3469" width="81.264pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B48-1"> <g transform="matrix(.013,0,0,-0.013,0,16.857)"> <use xlink:href="#g113-98"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.21,16.857)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,46.75,8.349)"> <use xlink:href="#g113-50"></use> </g> <rect height="0.65243" width="54.4376" x="22.67" y="13.203"></rect> <g transform="matrix(.013,0,0,-0.013,22.67,28.329)"> <use 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<p>最后,一组方程(<一个href="#EEq29">29日</一个>)包含的一般表达式确定FFOPDT模型的参数,<我>θ</我><sub><i>P</我></sub>= {<我>K</我>,<我>T</我>,<我>l</我>,<我>α</我>},使用时间所需的响应达到反应曲线上的任何三个点。<年代p一个nclass="equation_break" id="EEq29"><span class="left_break" id="L49"></span><span class="middle_break"><span class="center_break" id="M49"> <svg height="87.3558pt" id="M49-1" version="1.1" viewbox="0 0 169.964 87.3558" width="169.964pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B49-1"> <g transform="matrix(.013,0,0,-0.013,0,17.45)"> <use xlink:href="#g119-47"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,21.234)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,25.018)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,28.802)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,32.586)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,36.371)"> <use 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xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,139.93,75.784)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,145.77,81.525)"> <use xlink:href="#g113-45"></use> </g> <g transform="matrix(.013,0,0,-0.013,153.09,81.525)"> <use xlink:href="#g113-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,159.33,81.525)"> <path d="M273 -404V944H76V909L99 908C205 900 208 895 208 751V-249C208 -355 206 -360 99 -368L76 -369V-404H273Z" id="g119-130"></path> </g> <g transform="matrix(.013,0,0,-0.013,167,81.525)"> <use xlink:href="#g113-45"></use> </g> </g> </svg> <svg class="alt_equation" height="87.3558pt" id="I49" version="1.1" viewbox="0 0 184.28 87.3558" width="184.28pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 0)" xlink:href="#B49-1"></use> </svg></span></span><span class="right_break" id="R49"> <svg class="label" height="11.44pt" id="M49-2" version="1.1" viewbox="0 0 21.388 11.44" width="21.388pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B49-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-243"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.54,9.26)"> <use xlink:href="#g190-193"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.89,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg> <svg class="alt_label" height="87.3558pt" id="IL49" version="1.1" viewbox="0 0 21.46 87.3558" width="21.46pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 37.75)" xlink:href="#B49-2"></use> </svg></span></span>在哪里<我>f</我><sub>1</年代ub>取决于∆,定义为一个函数的通用三分(<一个href="#EEq25">25</一个>),<我>f</我><sub>2</年代ub>(<我>α</我>)=一个<年代up><i>α</我></sup>和<我>f</我><sub>3</年代ub>(<我>α</我>)=<我>τ</我><sub>x3</年代ub><sup>1 /<我>α</我></sup>函数依赖是规范化的时代吗<我>τ</我><sub><i>x</我>1</年代ub>和<我>τ</我><sub><i>x</我>3</年代ub>,<我>τ</我><sub><i>x</我>3</年代ub>分别为,<我>一个</我>参数定义在(<一个href="#EEq28">28</一个>)。注意,在(<一个href="#EEq29">29日</一个>)的值<我>T</我>和<我>l</我>有高度依赖的价值吗<我>α</我>,有重大影响的形状的反应。这一事实的强调的重要性决定的价值<我>α</我>参数准确,将在稍后讨论。</p> <p>在(<一个href="#EEq29">29日</一个>),<我>α</我>> 0,<我>T</我>> 0是一种自然的方式实现,<我>τ</我><sub><i>x</我>1</年代ub><<我>τ</我><sub><i>x</我>2</年代ub><<我>τ</我><sub><i>x</我>3</年代ub>和<我>t</我><sub><i>x</我>1</年代ub><<我>t</我><sub><i>x</我>2</年代ub><<我>t</我><sub><i>x</我>3</年代ub>。另一个条件是必须满足以满足<我>l</我>≥0<年代p一个nclass="equation_break" id="EEq30"><span class="left_break" id="L50"></span><span class="middle_break"><span class="center_break" id="M50"> <svg height="17.782pt" id="M50-1" version="1.1" viewbox="0 0 82.684 17.782" width="82.684pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B50-1"> <g transform="matrix(.013,0,0,-0.013,0,13.818)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,4.37,16.949)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,9.55,16.949)"> <use xlink:href="#g50-52"></use> </g> <g transform="matrix(.013,0,0,-0.013,18.14,13.818)"> <use xlink:href="#g117-94"></use> </g> <g transform="matrix(.013,0,0,-0.013,29.41,13.818)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,34.43,13.818)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,39.86,16.949)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,45.04,16.949)"> <use xlink:href="#g50-52"></use> </g> <g transform="matrix(.013,0,0,-0.013,50.01,13.818)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.013,0,0,-0.013,58.29,13.818)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,63.36,6.517)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,67.8,6.517)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,71.7,6.517)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,79.72,13.818)"> <use xlink:href="#g113-47"></use> </g> </g> </svg></span></span><span class="right_break" id="R50"> <svg class="label" height="11.44pt" id="M50-2" version="1.1" viewbox="0 0 21.438 11.44" width="21.438pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B50-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.53,9.26)"> <use xlink:href="#g190-254"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.94,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span></p> <h4 id="symmetrical-method">4所示。对称的方法</h4> <p>识别的一般开发过程中提出了在前面的小节中,它被假定所需的三个点的反应曲线可以是任何值。在本节中,一个简化的一般方法,这些点可以任意选择而位于对称响应曲线。</p> <p>注意,如图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig4/" target="_blank">4</一个>,中央点位于中间的范围,将对应于所需的时间达到50%流程输出总量的变化对反应曲线<我>y</我><sub><i>α</我></sub>(<我>t</我><sub>50</年代ub>)(<我>t</我><sub><i>x</我>2</年代ub>=<我>t</我><sub>50</年代ub>)。剩下的点可能位于任意但对称放置的中心点。将表示为一个对称点<我>x</我>,另一个是100 - x。这意味着现在将确定的时间<我>t</我><sub><i>x</我>1</年代ub>=<我>t</我><sub><i>x</我></sub>和<我>t</我><sub><i>x</我>3</年代ub>=<我>t</我><sub>100 -<我>x</我></sub>,在那里<我>t</我><sub><i>x</我></sub>和<我>t</我><sub>100 -<我>x</我></sub>表示达到所需的时间<我>x</我>% (100 -<我>x</我>)%流程输出总量的变化,分别在0 <<我>x</我>< 50岁。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-a32eaf72884d0d756353de0999e567d8"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.004.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-a32eaf72884d0d756353de0999e567d8 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.004_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-a32eaf72884d0d756353de0999e567d8 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图4</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig4/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 阶跃输入信号和对称的代表点过程反应曲线。</d我v> </div> </div> </div> <p>采用方程(<一个href="#EEq29">29日</一个>与这些对称性的考虑),获得以下表达式:<年代p一个nclass="equation_break" id="EEq31"><span class="left_break" id="L51"></span><span class="middle_break"><span class="center_break" id="M51"> <svg height="87.3558pt" id="M51-1" version="1.1" viewbox="0 0 194.494 87.3558" width="194.494pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B51-1"> <g transform="matrix(.013,0,0,-0.013,0,17.45)"> <use xlink:href="#g119-47"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,21.234)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,25.018)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,28.802)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,32.586)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,36.371)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,51.507)"> <use xlink:href="#g119-63"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,55.291)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,59.075)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,62.86)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,66.644)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,70.428)"> <use xlink:href="#g119-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,0,84.951)"> <use xlink:href="#g119-71"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.73,17.626)"> <use xlink:href="#g113-76"></use> </g> <g transform="matrix(.013,0,0,-0.013,24.17,17.626)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,36.63,8.636)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,44.96,8.636)"> <use xlink:href="#g113-122"></use> </g> <rect height="0.65243" width="15.9195" x="36.63" y="13.973"></rect> <g transform="matrix(.013,0,0,-0.013,36.9,26.852)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,45.23,26.852)"> <use xlink:href="#g113-118"></use> </g> <g transform="matrix(.013,0,0,-0.013,53.75,17.626)"> <use xlink:href="#g113-45"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.73,38.266)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,21.74,38.266)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,33,38.266)"> <use xlink:href="#g113-103"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,39.73,41.398)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,46.85,38.449)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,51.37,38.266)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,59.69,38.449)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,66.42,38.266)"> <use xlink:href="#g113-45"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.73,59.411)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.013,0,0,-0.013,22.64,59.411)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,33.9,59.411)"> <use xlink:href="#g113-103"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,40.63,62.543)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,47.75,59.594)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,52.27,59.411)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,59.65,59.594)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,66.37,59.411)"> <path d="M308 -259V-228H299C192 -219 190 -215 190 -109V616C190 755 193 759 299 768H308V799H127V-259H308Z" id="g119-1"></path> </g> <g transform="matrix(.013,0,0,-0.013,71.23,59.411)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,76.25,59.411)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,80.62,62.543)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,85.05,62.543)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,89.48,62.543)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,93.91,62.543)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,99.47,62.543)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,105.24,59.411)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.013,0,0,-0.013,113.18,59.411)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,123.72,59.411)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,128.09,62.543)"> <use xlink:href="#g50-121"></use> </g> <g 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transform="matrix(.013,0,0,-0.013,129.22,81.708)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,133.74,81.525)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,141.12,81.708)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,147.84,81.525)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,156.13,75.784)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,160.56,75.784)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,164.47,75.784)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,170.31,81.525)"> <use xlink:href="#g113-45"></use> </g> <g transform="matrix(.013,0,0,-0.013,177.63,81.525)"> <use xlink:href="#g113-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,183.87,81.525)"> <use xlink:href="#g119-130"></use> </g> <g transform="matrix(.013,0,0,-0.013,191.53,81.525)"> <use xlink:href="#g113-45"></use> </g> </g> </svg> <svg class="alt_equation" height="87.3558pt" id="I51" version="1.1" viewbox="0 0 208.81 87.3558" width="208.81pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 0)" xlink:href="#B51-1"></use> </svg></span></span><span class="right_break" id="R51"> <svg class="label" height="11.44pt" id="M51-2" version="1.1" viewbox="0 0 19.078 11.44" width="19.078pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B51-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.19,9.26)"> <use xlink:href="#g190-213"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.58,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg> <svg class="alt_label" height="87.3558pt" id="IL51" version="1.1" viewbox="0 0 19.16 87.3558" width="19.16pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 37.75)" xlink:href="#B51-2"></use> </svg></span></span>在哪里<我>f</我><sub>1</年代ub>分数阶相关的功能吗<我>α</我>与时代∆比例。</p> <p>此外,Δ被定义为一个函数的对称三分<年代p一个nclass="equation_break" id="EEq32"><span class="left_break" id="L52"></span><span class="middle_break"><span class="center_break" id="M52"> <svg height="39.9792pt" id="M52-1" version="1.1" viewbox="0 0 81.398 39.9792" width="81.398pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B52-1"> <g transform="matrix(.013,0,0,-0.013,0,23.317)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.1,23.317)"> <use xlink:href="#g117-11"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.51,23.317)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,19.36,13.818)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,24.38,13.818)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,28.75,16.949)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,33.18,16.949)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,37.62,16.949)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,42.05,16.949)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,47.61,16.949)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,53.37,13.818)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.013,0,0,-0.013,61.32,13.818)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,71.86,13.818)"> <use xlink:href="#g113-117"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,76.22,16.949)"> <use xlink:href="#g50-121"></use> </g> <rect height="0.65243" width="62.6036" x="19.36" y="19.663"></rect> <g 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xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B52-2"> <g transform="matrix(.013,0,0,-0.013,0,23.317)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,12.46,13.818)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.48,13.818)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,22.92,16.949)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,27.35,16.949)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,31.78,16.949)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,36.21,16.949)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,41.77,16.949)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,47.54,13.818)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,52.58,6.517)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,57.02,6.517)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,60.92,6.517)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,69.67,13.818)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,80.21,13.818)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,85.23,13.818)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,90.66,16.949)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,96.36,13.818)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,101.4,6.517)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,105.84,6.517)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,109.74,6.517)"> <use xlink:href="#g50-223"></use> </g> <rect height="0.65243" width="103.153" x="12.46" y="19.663"></rect> <g transform="matrix(.013,0,0,-0.013,20.07,36.041)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,25.1,36.041)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,30.53,39.173)"> <use xlink:href="#g50-54"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,34.96,39.173)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.013,0,0,-0.013,39.93,36.041)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,44.97,28.74)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,49.4,28.74)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,53.31,28.74)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,62.06,36.041)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,72.59,36.041)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,77.62,36.041)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,83.05,39.173)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,88.75,36.041)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,93.79,28.74)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,98.22,28.74)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,102.13,28.74)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,116.81,23.317)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R52"> <svg class="label" height="11.44pt" id="M52-3" version="1.1" viewbox="0 0 20.708 11.44" width="20.708pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B52-3"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.16,9.26)"> <use xlink:href="#g190-243"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.21,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>它依赖于标准化的时代<我>τ</我><sub><i>x</我></sub>,<我>τ</我><sub>50</年代ub>,<我>τ</我><sub>100 -<我>x</我></sub>。<我>f</我><sub><i>2</我></sub>(<我>α</我>)=一个<年代up><i>α</我></sup>,在那里<年代p一个nclass="equation_break" id="EEq33"><span class="left_break" id="L53"></span><span class="middle_break"><span class="center_break" id="M53"> <svg height="33.5188pt" id="M53-1" version="1.1" viewbox="0 0 129.984 33.5188" width="129.984pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B53-1"> <g transform="matrix(.013,0,0,-0.013,0,16.857)"> <use xlink:href="#g113-98"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.21,16.857)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,71.11,8.349)"> <use xlink:href="#g113-50"></use> </g> <rect height="0.65243" width="103.153" x="22.67" y="13.203"></rect> <g transform="matrix(.013,0,0,-0.013,22.67,29.58)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,27.69,29.58)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,33.13,32.712)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,37.56,32.712)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,41.99,32.712)"> <use xlink:href="#g50-49"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,46.42,32.712)"> <use xlink:href="#g54-33"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,51.98,32.712)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,57.75,29.58)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,62.79,22.28)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,67.22,22.28)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,71.13,22.28)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,79.88,29.58)"> <use xlink:href="#g117-33"></use> </g> <g transform="matrix(.013,0,0,-0.013,90.42,29.58)"> <use xlink:href="#g119-5"></use> </g> <g transform="matrix(.013,0,0,-0.013,95.44,29.58)"> <use xlink:href="#g113-241"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,100.87,32.712)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,106.57,29.58)"> <use xlink:href="#g119-6"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,111.61,22.28)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,116.05,22.28)"> <use xlink:href="#g50-48"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,119.95,22.28)"> <use xlink:href="#g50-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,127.02,16.857)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R53"> <svg class="label" height="11.44pt" id="M53-2" version="1.1" viewbox="0 0 20.558 11.44" width="20.558pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B53-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.18,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.06,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>和<我>f</我><sub>3</年代ub>(<我>α</我>)= (<我>τ</我><sub>100 -<我>x</我></sub>)<年代up>1 /<我>α</我></sup>函数依赖是规范化的时代吗<我>τ</我><sub><i>x</我></sub>和<我>τ</我><sub>100 -<我>x</我></sub>,<我>τ</我><sub>100 -<我>x</我></sub>,分别。</p> <p>本文以下组对称点会考虑:(5 - 50 - 95%),(10 - 50 - 90%),(15 - 50 - 85%),(20 - 50 - 80%),(25 - 50 - 75%)和(30 - 50 - 70%),对应下面的值<我>x</我>= 5、10、15、20、25和30。目标是获得洞察的影响选择代表点的准确性得到分数阶模型,如前面提到的。</p> <p>表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab1/" target="_blank">1</一个>- - - - - -<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab6/" target="_blank">6</一个>包含的值归一化<我>τ</我><sub><i>x</我></sub>,<我>τ</我><sub>50</年代ub>,<我>τ</我><sub>100 -<我>x</我></sub>,值<我>x</我>= 5、10、15、20、25和30。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab1"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表1</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab1/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 归一化时代的数值<我>τ</我> <sub>5</年代ub>,<我>τ</我> <sub>50</年代ub>,<我>τ</我> <sub>95年</年代ub>为不同的值<我>α</我>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab1" target="_blank" aria-label="View full page"></a> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab2"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表2</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab2/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 归一化时代的数值<我>τ</我> <sub>10</年代ub>,<我>τ</我> <sub>50</年代ub>,<我>τ</我> <sub>90年</年代ub>为不同的值<我>α</我>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab2" target="_blank" aria-label="View full page"></a> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab3"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表3</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab3/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 归一化时代的数值<我>τ</我> <sub>15</年代ub>,<我>τ</我> <sub>50</年代ub>,<我>τ</我> <sub>85年</年代ub>为不同的值<我>α</我>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab3" target="_blank" aria-label="View full page"></a> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab4"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表4</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab4/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 归一化时代的数值<我>τ</我> <sub>20.</年代ub>,<我>τ</我> <sub>50</年代ub>,<我>τ</我> <sub>80年</年代ub>为不同的值<我>α</我>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab4" target="_blank" aria-label="View full page"></a> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab5"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表5</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab5/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 归一化时代的数值<我>τ</我> <sub>25</年代ub>,<我>τ</我> <sub>50</年代ub>,<我>τ</我> <sub>75年</年代ub>为不同的值<我>α</我>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab5" target="_blank" aria-label="View full page"></a> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab6"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表6</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab6/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 归一化时代的数值<我>τ</我> <sub>30.</年代ub>,<我>τ</我> <sub>50</年代ub>,<我>τ</我> <sub>70年</年代ub>为不同的值<我>α</我>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab6" target="_blank" aria-label="View full page"></a> </div> </div> <p>以后,考虑值相应的归一化时代,数据集{Δ,<我>α</我>},{<我>α</我>,一个<年代up><i>α</我></sup>},{<我>α</我>,(<我>τ</我><sub>100 -<我>x</我></sub>)<年代up>1 /<我>α</我></sup>}获得了每一个被认为是套对称点。从这些数据,分析表达式<我>f</我><sub>1</年代ub>(Δ),<我>f</我><sub>2</年代ub>(<我>α</我>),<我>f</我><sub>3</年代ub>(<我>α</我>使用曲线拟合)估计。最后,分数阶模型参数{<我>T</我>,<我>l</我>,<我>α</我>}表达式可以确定(<一个href="#EEq31">31日</一个>)和实验值<我>t</我><sub><i>x</我></sub>,<我>t</我><sub>50</年代ub>,<我>t</我><sub>100 -<我>x</我></sub>收集到的反应曲线。</p> <p>Δ值不同的值<我>α</我>0.5≤<我>α</我>≤1.1,取得了在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig5/" target="_blank">5</一个>根据表达式(32)和使用标准化的时间<我>τ</我><sub>x</年代ub>,<我>τ</我><sub>50</年代ub>,<我>τ</我><sub>100 -<我>x</我></sub>,因为<我>x</我>= 5、10、15、20、25、30、从表中获得<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab1/" target="_blank">1</一个>- - - - - -<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab6/" target="_blank">6</一个>。注意,Δ指数代表的到达时差之间的比率<我>x</我>(100 -<我>x</我>)%,从<我>x</我>50%的总变异过程的输出,显示在(<一个href="#EEq32">32</一个>)。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-352e9ab449d5191bf49898b649bbcb83"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.005.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-352e9ab449d5191bf49898b649bbcb83 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.005_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-352e9ab449d5191bf49898b649bbcb83 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图5</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig5/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 数据集{Δ,<我>α</我>},Δ定义在(<一个href="#EEq32">32</一个>),<我>x</我>= 5、10、15、20、25、30和曲线拟合的结果<我>f</我> <sub>1</年代ub>(∆)。</d我v> </div> </div> </div> <p>数据集{Δ,<我>α</我>}允许建立一个关系<我>α</我>Δ解析函数的形式,<我>α</我>=<我>f</我><sub>1</年代ub>(Δ)。</p> <p>图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig5/" target="_blank">5</一个>显示了数据集{∆,<我>α</我>从方程(获得}<一个href="#EEq32">32</一个>为不同的值)<我>α</我>≤0.5,<我>α</我>≤1.1,相应的最小二乘曲线拟合的结果不同的对称点。注意,Δ取决于标准化* {<我>τ</我><sub><i>x</我></sub>,<我>τ</我><sub>50</年代ub>,<我>τ</我><sub>100 -<我>x</我></sub>},表示(<一个href="#EEq32">32</一个>),有很强的依赖<我>α</我>参数。</p> <p>使用Levenberg-Marquardt数据拟合最小二乘曲线拟合的算法在所有图在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig5/" target="_blank">5</一个>,以下有理函数被认为是在所有情况下:<年代p一个nclass="equation_break" id="EEq34"><span class="left_break" id="L54"></span><span class="middle_break"><span class="center_break" id="M54"> <svg height="33.2369pt" id="M54-1" version="1.1" viewbox="0 0 159.464 33.2369" width="159.464pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B54-1"> <g transform="matrix(.013,0,0,-0.013,0,20.582)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,11.02,20.582)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,22.28,20.582)"> <use xlink:href="#g113-103"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,29,23.714)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,36.13,20.765)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,40.64,20.582)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,48.97,20.765)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,57.15,20.582)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,69.61,11.591)"> <path d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z" id="g113-113"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,77,14.723)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,81.94,11.591)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,90.27,5.85)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,98.12,11.591)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,108.66,11.591)"> <use xlink:href="#g113-113"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,116.04,14.723)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,120.99,11.591)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,132.22,11.591)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,142.76,11.591)"> <use xlink:href="#g113-113"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,150.14,14.723)"> <use xlink:href="#g50-52"></use> </g> <rect height="0.65243" width="85.6905" x="69.61" y="16.928"></rect> <g transform="matrix(.013,0,0,-0.013,76.86,29.807)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,85.19,25.632)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,93.04,29.807)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,103.58,29.807)"> <path d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z" id="g113-114"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,109.89,32.939)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,114.84,29.807)"> <use xlink:href="#g113-133"></use> </g> <g transform="matrix(.013,0,0,-0.013,126.07,29.807)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,136.61,29.807)"> <use xlink:href="#g113-114"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,142.93,32.939)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,156.5,20.582)"> <use xlink:href="#g113-45"></use> </g> </g> </svg> <svg class="alt_equation" height="33.2369pt" id="I54" version="1.1" viewbox="0 0 159.46 33.2369" width="159.46pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 0)" xlink:href="#B54-1"></use> </svg></span></span><span class="right_break" id="R54"> <svg class="label" height="11.44pt" id="M54-2" version="1.1" viewbox="0 0 21.488 11.44" width="21.488pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B54-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.49,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.53,9.26)"> <use xlink:href="#g190-166"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.99,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg> <svg class="alt_label" height="33.2369pt" id="IL54" version="1.1" viewbox="0 0 21.57 33.2369" width="21.57pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 10.69)" xlink:href="#B54-2"></use> </svg></span></span>在表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab7/" target="_blank">7</一个>显示的参数值){<我>p</我><sub><i>我</我></sub>,<我>问</我><sub><i>我</我></sub>}为函数<我>f</我><sub>1</年代ub>(Δ)和每一个所选组对称点。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab7"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表7</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab7/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 参数{<我>p</我> <sub>我</年代ub>,<我>问</我> <sub>我</年代ub>有理函数的}<我>f</我> <sub>1</年代ub>(Δ)不同的对称点。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab7" target="_blank" aria-label="View full page"></a> </div> </div> <p>重要的是要注意,对于增加的值<我>x</我>Δ范围变得更短。<我>f</我><sub>1</年代ub>(Δ)是一个非线性函数的斜率是陡增加的值<我>x</我>,这表明更大的敏感性。</p> <p>图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig6/" target="_blank">6</一个>显示了数据集{<我>α</我>,一个<年代up><i>α</我></sup>},<我>一个</我>参数是得到方程(<一个href="#EEq33">33</一个>),为不同的值<我>α</我>0.5≤<我>α</我>≤1.1,相应的最小二乘曲线拟合的结果不同的对称点。请注意,<我>f</我><sub>2</年代ub>(<我>α</我>)=一个<年代up><i>α</我></sup>取决于<我>α</我>和规范化的<我>τ</我><sub><i>x</我></sub>和<我>τ</我><sub>100 -<我>x</我></sub>。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-443d620713988d2a5ac34b97326e39d1"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.006.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-443d620713988d2a5ac34b97326e39d1 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.006_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-443d620713988d2a5ac34b97326e39d1 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图6</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig6/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 数据集{<我>α</我>,一个<年代up><i>α</我></sup>},<我>一个</我>定义在(<一个href="#EEq33">33</一个>),<我>x</我>= 5、10、15、20、25、30和曲线拟合的结果<我>f</我> <sub>2</年代ub>(<我>α</我>)。</d我v> </div> </div> </div> <p>在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig6/" target="_blank">6</一个>可以看到,它的振幅<我>f</我><sub>2</年代ub>增加增加的值<我>x</我>。另一方面,对于减少值<我>x</我>、功能<我>f</我><sub>2</年代ub>呈现出更明显的曲率小的值<我>α</我>。</p> <p>使用Levenberg-Marquardt数据拟合最小二乘曲线拟合的算法在所有图在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig6/" target="_blank">6</一个>,下面的有理函数用于所有情况下:<年代p一个nclass="equation_break" id="EEq35"><span class="left_break" id="L55"></span><span class="middle_break"><span class="center_break" id="M55"> <svg height="28.6503pt" id="M55-1" version="1.1" viewbox="0 0 119.824 28.6503" width="119.824pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B55-1"> <g transform="matrix(.013,0,0,-0.013,0,15.995)"> <use xlink:href="#g113-103"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.72,19.127)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,13.84,16.178)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,18.36,15.995)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,25.74,16.178)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,33.92,15.995)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,58.22,7.005)"> <use xlink:href="#g113-113"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,65.6,10.137)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,70.55,7.005)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,80.84,7.005)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,91.38,7.005)"> <use xlink:href="#g113-113"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,98.76,10.137)"> <use xlink:href="#g50-51"></use> </g> <rect height="0.65243" width="69.2872" x="46.38" y="12.342"></rect> <g transform="matrix(.013,0,0,-0.013,46.38,25.221)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,53.76,21.045)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,61.61,25.221)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,72.15,25.221)"> <use xlink:href="#g113-114"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,78.47,28.352)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,83.41,25.221)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,93.7,25.221)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,104.24,25.221)"> <use xlink:href="#g113-114"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,110.56,28.352)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,116.86,15.995)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R55"> <svg class="label" height="11.44pt" id="M55-2" version="1.1" viewbox="0 0 20.708 11.44" width="20.708pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B55-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.31,9.26)"> <use xlink:href="#g190-161"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.21,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>值的参数{<我>p</我><sub>我</年代ub>,<我>问</我><sub>我</年代ub>}为函数<我>f</我><sub>2</年代ub>(<我>α</我>),每一个对称点的选择集如表所示<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab8/" target="_blank">8</一个>。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab8"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表8</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab8/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 参数{<我>p</我> <sub>我</年代ub>,<我>问</我> <sub>我</年代ub>有理函数的}<我>f</我> <sub>2</年代ub>(<我>α</我>)不同的对称点。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab8" target="_blank" aria-label="View full page"></a> </div> </div> <p>图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig7/" target="_blank">7</一个>显示数据集{<我>α</我>,(<我>τ</我><sub>100−<我>x</我></sub>)<年代up>1 /<我>α</我></sup>},规范化<我>τ</我><sub>100 - x</年代ub>从表可以获得吗<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab1/" target="_blank">1</一个>−<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab6/" target="_blank">6</一个>为不同的值<我>α</我>0.5≤<我>α</我>≤1.1,相应的最小二乘曲线拟合的结果不同的对称点。请注意,<我>f</我><sub>3</年代ub>(<我>α</我>)= (<我>τ</我><sub>100−<我>x</我></sub>)<年代up>1 /<我>α</我></sup>取决于<我>α</我>和规范化的<我>τ</我><sub>100 -<我>x</我></sub>。振幅的范围<我>f</我><sub>3</年代ub>值的增加减少了吗<我>x</我>,如可以如图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig7/" target="_blank">7</一个>。使用Levenberg-Marquardt数据拟合最小二乘曲线拟合的算法在所有图在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig7/" target="_blank">7</一个>,下面的有理函数用于所有情况下:<年代p一个nclass="equation_break" id="EEq36"><span class="left_break" id="L56"></span><span class="middle_break"><span class="center_break" id="M56"> <svg height="33.2369pt" id="M56-1" version="1.1" viewbox="0 0 134.324 33.2369" width="134.324pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B56-1"> <g transform="matrix(.013,0,0,-0.013,0,20.582)"> <use xlink:href="#g113-103"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.72,23.714)"> <use xlink:href="#g50-52"></use> </g> <g transform="matrix(.013,0,0,-0.013,13.84,20.765)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,18.36,20.582)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,25.74,20.765)"> <use xlink:href="#g113-42"></use> </g> <g transform="matrix(.013,0,0,-0.013,33.92,20.582)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,46.38,11.591)"> <use xlink:href="#g113-113"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,53.76,14.723)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,58.71,11.591)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,66.09,5.85)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,73.94,11.591)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,84.48,11.591)"> <use xlink:href="#g113-113"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,91.86,14.723)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,96.81,11.591)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,107.1,11.591)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,117.64,11.591)"> <use xlink:href="#g113-113"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,125.02,14.723)"> <use xlink:href="#g50-52"></use> </g> <rect height="0.65243" width="83.7853" x="46.38" y="16.928"></rect> <g transform="matrix(.013,0,0,-0.013,53.63,29.807)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,61.01,25.632)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,68.86,29.807)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,79.4,29.807)"> <use xlink:href="#g113-114"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,85.72,32.939)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,90.66,29.807)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.013,0,0,-0.013,100.95,29.807)"> <use xlink:href="#g117-36"></use> </g> <g transform="matrix(.013,0,0,-0.013,111.49,29.807)"> <use xlink:href="#g113-114"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,117.81,32.939)"> <use xlink:href="#g50-51"></use> </g> <g transform="matrix(.013,0,0,-0.013,131.36,20.582)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R56"> <svg class="label" height="11.44pt" id="M56-2" version="1.1" viewbox="0 0 21.438 11.44" width="21.438pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B56-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-232"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.5,9.26)"> <use xlink:href="#g190-227"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.94,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>值的参数{<我>p</我><sub>我</年代ub>,<我>问</我><sub>我</年代ub>}为函数<我>f</我><sub>3</年代ub>(<我>α</我>),每一个对称点的选择集如表所示<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab9/" target="_blank">9</一个>。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-994ce39713ab2dbd9d86a29ead44b235"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.007.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-994ce39713ab2dbd9d86a29ead44b235 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.007_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-994ce39713ab2dbd9d86a29ead44b235 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图7</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig7/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 数据集{<我>α</我>,(<我>τ</我> <sub>100 -<我>x</我></sub>)<年代up>1 /<我>α</我></sup>},<我>x</我>= 5、10、15、20、25、30,曲线拟合<我>f</我> <sub>3</年代ub>(<我>α</我>)。</d我v> </div> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab9"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表9</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab9/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 参数{<我>p</我> <sub>我</年代ub>,<我>问</我> <sub>我</年代ub>有理函数的}<我>f</我> <sub>3</年代ub>(<我>α</我>)不同的对称点。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab9" target="_blank" aria-label="View full page"></a> </div> </div> <p>重要的是要强调,估计的值<我>α</我>,<我>T</我>,<我>l</我>从表达式(<一个href="#EEq31">31日</一个>)依赖于函数<我>f</我><sub>1</年代ub>(Δ),<我>f</我><sub>2</年代ub>(<我>α</我>),<我>f</我><sub>3</年代ub>(<我>α</我>)。因此,这些函数在这个过程中扮演了一个重要的角色识别过程的特点规范化步骤反应(<一个href="#EEq22">22</一个>由于他们的贡献)可以为特征。注意,对于任何一个不同的选择次集{<我>t</我><sub><i>x</我>1</年代ub>,<我>t</我><sub><i>x</我>2</年代ub>,<我>t</我><sub><i>x</我>3</年代ub>}达到三分{<我>y</我><sub><i>α</我></sub>(<我>t</我><sub><i>x</我>1</年代ub>),<我>y</我><sub><i>α</我></sub>(<我>t</我><sub><i>x</我>2</年代ub>),<我>y</我><sub><i>α</我></sub>(<我>t</我><sub><i>x</我>3</年代ub>)}反应曲线,识别结果的准确性取决于拟合精度。在这种背景下,一个精确的测定<我>α</我>价值是最重要的,因为,随后,功能<我>f</我><sub>2</年代ub>和<我>f</我><sub>3</年代ub>——因此<我>T</我>和L-depend估计的价值<我>α</我>。</p> <p>在曲线拟合过程中,已经发现,一般来说,功能<我>f</我><sub>1</年代ub>(Δ),<我>f</我><sub>2</年代ub>(<我>α</我>),<我>f</我><sub>3</年代ub>(<我>α</我>低的值<我>x</我>需要更多的参数比更高<我>x</我>值能够适应数据与有理函数。虽然一些数据集可以用更少的参数,调整为每一个函数(<我>f</我><sub>1</年代ub>(Δ),<我>f</我><sub>2</年代ub>(<我>α</我>),<我>f</我><sub>3</年代ub>(<我>α</我>),同样的表情有相同数量的参数曲线拟合,分别。通过这种方式,比较估计的值<我>α</我>,<我>T</我>,<我>l</我>,它直接依赖于函数<我>f</我><sub>1</年代ub>,<我>f</我><sub>2</年代ub>,<我>f</我><sub>3</年代ub>对于不同的对称点,可以在相同的条件下。在本节中,通过验证,理性表达适合数据集用于数字<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig5/" target="_blank">5</一个>- - - - - -<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig7/" target="_blank">7</一个>。</p> <p><span class="statement" id="rem1"><span class="statement-title"><i class="statement-title">备注1。</我></span>表达式(<一个href="#EEq34">34</一个>)- (<一个href="#EEq36">36</一个>),参数表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab7/" target="_blank">7</一个>- - - - - -<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab9/" target="_blank">9</一个>为函数<我>f</我><sub>1</年代ub>,<我>f</我><sub>2</年代ub>,<我>f</我><sub>3</年代ub>已经获得了0.5≤<我>α</我>≤1.1。这个范围包括遇到的多数代表过程的动力学过程控制,基本上可以概括为拥有单调一步反应;见,例如,测试批考虑(<一个href="#B4">4</一个>]。的值<我>α</我>在这个范围之外,考虑曲线配件是无效的。一个显著的事实观察是FFOPDT系统的阶跃响应<我>α</我>值小于0.5变得极慢,可以看到在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig1/" target="_blank">1</一个>。注意的归一化值<我>τ</我><sub>100 -<我>x</我></sub>增加<我>α</我>减少,尤其是以一种非常重大的方式对小的值<我>x</我>。<br>尽管FFOPDT模型的阶跃响应只过阻尼<我>α</我>值在0和1之间,1.1还包括价值因为它已经在许多实验,观察到一些过程,一般lag-dominated,与高阶integer-order传输函数的值<我>α</我>略大于1。通过这种方式,一个更好的适合模型的瞬态响应的获得与反应过程的曲线。具体来说,FFOPDT系统的阶跃响应的行为<我>α</我>可以观察到图= 1.1<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig2/" target="_blank">2</一个>。</年代p一个n></p> <h5 id="sec4.1">4.1。算法确定FFOPDT模型参数</h5> <p>促进软件实现所提出的识别方法,开发一个算法在桌子上<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab10/" target="_blank">10</一个>。在这种方法中,输入信号的变化ΔuΔ和过程输出<我>y</我>必要和时间到达<我>x</我>% (<我>t</我><sub>x</年代ub>),50% (<我>t</我><sub>50</年代ub>),(100 -<我>x</我>)% (<我>t</我><sub>100 - x</年代ub>)的总变化对反应过程的输出曲线必须收集为了确定FFOPDT模型参数<我>θ</我><sub>P</年代ub>= {<我>K</我>,<我>T</我>,<我>l</我>,<我>α</我>}。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab10"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表10</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab10/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 算法确定FFOPDT模型参数<我>θ</我> <sub>P</年代ub>= {<我>K</我>,<我>T</我>,<我>l</我>,<我>α</我>}当三个对称点{<我>t</我> <sub><i>x</我></sub>,<我>t</我> <sub>50</年代ub>,<我>t</我> <sub>100 -<我>x</我></sub>}指定反应曲线。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab10" target="_blank" aria-label="View full page"></a> </div> </div> <h4 id="illustrative-examples">5。说明性的例子</h4> <p>在本节中,提出的识别方法为FFOPDT模型测试了好几套对称点的反应曲线和相对于其他识别方法为整数和分数阶模型。</p> <p>以下流程模型被选为了显示的有效性提出程序找到一个近似的四个参数模型(<一个href="#EEq14">14</一个>)和洞察的影响选择的一组对称点识别模型的准确性:<年代p一个nclass="equation_break" id="EEq37"><span class="left_break" id="L57"></span><span class="middle_break"><span class="center_break" id="M57"> <svg height="74.1778pt" id="M57-1" version="1.1" viewbox="0 0 137.504 74.1778" width="137.504pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B57-1"> <g transform="matrix(.013,0,0,-0.013,0,17.379)"> <use xlink:href="#g113-81"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.65,20.51)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,11.08,20.51)"> <use xlink:href="#g50-45"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,13.24,20.51)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.013,0,0,-0.013,18.5,17.561)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,23.02,17.379)"> <use xlink:href="#g113-116"></use> </g> <g 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transform="matrix(.013,0,0,-0.013,16.69,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span></p> <p>一方面,流程<我>P</我><sub>1,我</年代ub>用于比较提出了多个对称点识别过程有两,三点FOPDT识别方法,DPPDT, SOPDT模型和获得洞察对称点的位置的影响确定的分数阶模型的准确性。</p> <p>另一方面,过程<我>P</我><sub>2</年代ub>是用来比较的模型性能方法提出了与其他整数和分数阶的方法。</p> <p>改变输入信号应用于这些过程以注册过程反应曲线。输出响应的过程被用来计算相应的参数模型。请注意,在所有的实验中使用的示例时期成立<我>T</我><sub>年代</年代ub>= 10 ms。</p> <p>考虑到实验数据的识别测试中,需要验证的有效性模型结构采用拟合和相应的模型参数的准确性。拟合目标函数和模型验证方法使用一步激发信号系统识别已经提出了在文献[<一个href="#B8">8</一个>]。</p> <p>不失一般性,下面的时域拟合准则是用来评估模型的性能标识:<年代p一个nclass="equation_break" id="EEq39"><span class="left_break" id="L58"></span><span class="middle_break"><span class="center_break" id="M58"> <svg height="38.9456pt" id="M58-1" version="1.1" viewbox="0 0 138.292 38.9456" width="138.292pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B58-1"> <g transform="matrix(.013,0,0,-0.013,0,23.218)"> <path d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 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xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,12.027,0)"> <use xlink:href="#g113-108"></use> </g> <g transform="matrix(.013,0,0,-0.013,18.566,0)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,25.742,3.132)"> <use xlink:href="#g50-116"></use> </g> <g transform="matrix(.013,0,0,-0.013,29.721,0)"> <use xlink:href="#g113-42"></use> </g> </svg>和<年代p一个nclass="inline_break"> <svg height="15.9799pt" id="M62" style="vertical-align:-3.4294pt" version="1.1" viewbox="-0.0498162 -12.5505 39.949 15.9799" width="39.949pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-122"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.942,3.132)"> <use xlink:href="#g50-110"></use> </g> <g transform="matrix(.013,0,0,-0.013,14.793,0)"> <use xlink:href="#g113-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,19.291,0)"> <use xlink:href="#g113-108"></use> </g> <g transform="matrix(.013,0,0,-0.013,25.83,0)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,33.006,3.132)"> <use xlink:href="#g50-116"></use> </g> <g transform="matrix(.013,0,0,-0.013,36.985,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span><span class="nowrap"> <svg height="15.9799pt" style="vertical-align:-3.4294pt" version="1.1" viewbox="42.077883799999995 -12.5505 11.218 15.9799" width="11.218pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="42.1277" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,42.128,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.013,0,0,-0.013,48.626,0)"> <use xlink:href="#g113-42"></use> </g> </svg>,</年代p一个n></span>分别<我>T</我><sub><i>年代</我></sub>是采样周期,<我>N</我><sub><i>年代</我></sub><i>T</我><sub><i>年代</我></sub>是动态的时间长度(瞬态)的反应。虽然在阶跃响应测试的背景下,<我>N</我><sub><i>年代</我></sub><i>T</我><sub><i>年代</我></sub>可能会被认为是通常定义为时间的沉淀时间进入一个错误的2%或5%对最终的稳态输出偏差在回应一个阶跃变化,时间间隔的上限的即时反应已达到其最终价值。</p> <p>这个时域准则均方误差(MSE)和方块的,可以测量的平均错误。如何准确地繁殖过程反应曲线模型评估指数。</p> <p>在这种背景下,相对性能指标比善以来的绝对值更重要的识别方法对另一个可以量化的。</p> <p>在这篇文章中,使用相对性能指标如下:<年代p一个nclass="equation_break" id="EEq40"><span class="left_break" id="L63"></span><span class="middle_break"><span class="center_break" id="M63"> <svg height="41.2292pt" id="M63-1" version="1.1" viewbox="0 0 121.044 41.2292" width="121.044pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B63-1"> <g transform="matrix(.013,0,0,-0.013,0,23.942)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.07,27.074)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,11.25,27.074)"> <use xlink:href="#g50-45"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,13.41,27.074)"> <path d="M563 395C563 428 543 451 519 451C500 451 480 438 472 423C468 415 465 403 470 394C477 381 482 368 482 347C482 272 386 136 342 74H340C334 163 323 257 312 337C302 410 290 451 257 451C214 451 160 385 127 338L145 310C169 341 203 373 209 373C219 373 225 369 232 325C249 215 269 62 271 -15C220 -80 135 -165 18 -231L26 -257L142 -230C254 -105 279 -74 342 13C394 85 484 211 526 286C553 334 563 365 563 395Z" id="g50-122"></path> </g> <g transform="matrix(.013,0,0,-0.013,21.47,23.942)"> <use xlink:href="#g119-133"></use> </g> <rect height="0.65243" width="6.49826" x="27.25" y="12.094"></rect> <g transform="matrix(.013,0,0,-0.013,27.25,23.942)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,33.75,27.074)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,39.45,23.942)"> <use xlink:href="#g113-45"></use> </g> <rect height="0.65243" width="6.49826" x="44.59" y="12.094"></rect> <g transform="matrix(.013,0,0,-0.013,44.59,23.942)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,51.09,27.074)"> <use xlink:href="#g50-122"></use> </g> <g transform="matrix(.013,0,0,-0.013,56.95,23.942)"> <use xlink:href="#g119-134"></use> </g> <g transform="matrix(.013,0,0,-0.013,66.37,23.942)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,78.99,12.55)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,85.07,15.682)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,92.94,12.55)"> <use xlink:href="#g119-133"></use> </g> <rect height="0.65243" width="6.49826" x="98.72" y=".702"></rect> <g transform="matrix(.013,0,0,-0.013,98.72,12.55)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,105.22,15.682)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,110.91,12.55)"> <use xlink:href="#g119-134"></use> </g> <rect height="0.65243" width="38.052" x="78.83" y="20.288"></rect> <g transform="matrix(.013,0,0,-0.013,78.83,35.399)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,84.9,38.53)"> <use xlink:href="#g50-122"></use> </g> <g transform="matrix(.013,0,0,-0.013,92.94,35.399)"> <use xlink:href="#g119-133"></use> </g> <rect height="0.65243" width="6.49826" x="98.72" y="23.55"></rect> <g transform="matrix(.013,0,0,-0.013,98.72,35.399)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,105.22,38.53)"> <use xlink:href="#g50-122"></use> </g> <g transform="matrix(.013,0,0,-0.013,111.08,35.399)"> <use xlink:href="#g119-134"></use> </g> <g transform="matrix(.013,0,0,-0.013,118.08,23.942)"> <use xlink:href="#g113-45"></use> </g> </g> </svg></span></span><span class="right_break" id="R63"> <svg class="label" height="11.44pt" id="M63-2" version="1.1" viewbox="0 0 22.008 11.44" width="22.008pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B63-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-166"></use> </g> <g transform="matrix(.013,0,0,-0.013,11.11,9.26)"> <use xlink:href="#g190-254"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.51,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg></span></span>在哪里<年代vg height="15.8416pt" id="M64" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -12.5505 33.1122 15.8416" width="33.1122pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.071,3.132)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,11.766,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="16.2644" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,16.264,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,22.763,3.132)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,28.458,0)"> <use xlink:href="#g113-42"></use> </g> </svg>和<年代vg height="18.0794pt" id="M65" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 33.4411 18.0794" width="33.4411pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.071,3.132)"> <use xlink:href="#g50-122"></use> </g> <g transform="matrix(.013,0,0,-0.013,11.931,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="16.4288" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,16.429,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,22.927,3.132)"> <use xlink:href="#g50-122"></use> </g> <g transform="matrix(.013,0,0,-0.013,28.787,0)"> <use xlink:href="#g113-42"></use> </g> </svg>均方误差值(<一个href="#EEq39">39</一个>),<年代vg height="15.8416pt" id="M66" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -12.5505 12.2933 15.8416" width="12.2933pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="0" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.498,3.132)"> <use xlink:href="#g50-121"></use> </g> </svg>和<年代vg height="18.0794pt" id="M67" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 12.4577 18.0794" width="12.4577pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="0" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.498,3.132)"> <use xlink:href="#g50-122"></use> </g> </svg>流程模型参数向量的吗<我>x</我>和<我>y</我>模型,分别。</p> <p>这些说明性的例子的仿真已经使用MATLAB实现,FOTF工具箱(分数阶传递函数)。FOTF是分数阶系统的控制工具箱由雪et al。<一个href="#B47">47</一个>)扩展了许多MATLAB内置函数解决分数阶模型。的参考文本FOTF工具箱(<一个href="#B48">48</一个>彻底),作者解释了所有的命令和应用程序。</p> <h5 id="sec5.1">5.1。示例1</h5> <p>分数阶过程模型(37),<我>K</我><sub>1</年代ub>= 1,<我>T</我><sub>1</年代ub>= 1,<我>l</我><sub>1</年代ub>= 0.1 s<我>λ<年代ub>我</年代ub></i>1.00 = 0.60,0.65,…,,被认为是在这个例子。模型(<一个href="#EEq37">37</一个>)是一个分数二阶过程与死亡时间(FSOPDT)提供了一些从FFOPDT动力学建模错误或偏差,模型结构的选择提出的识别方法。</p> <p>之后的过程这个例子如下:<年代p一个nclass="list"><span class="list-item"><span class="list-label">(1)</年代p一个n><span class="list-content">9个过程植物<我>P</我><sub>1,我</年代ub>,在那里<我>我</我>= 1,…,9<我>λ</我><sub>我</年代ub>1.00 = 0.60,0.65,…,,。</年代p一个n></span><span class="list-item"><span class="list-label">(2)</年代p一个n><span class="list-content">FFOPDT模型参数<年代vg height="18.0794pt" id="M68" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 12.9144 18.0794" width="12.9144pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="0" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.498,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,8.38,3.132)"> <path d="M400 606C400 634 383 656 353 656C316 656 294 620 294 593C294 564 317 545 343 545C375 545 400 573 400 606ZM366 351C379 413 381 451 356 451C323 451 251 408 183 341L199 313C223 335 267 365 277 365C285 365 284 354 277 312C245 132 222 27 193 -100C182 -148 160 -188 131 -188C113 -188 90 -178 75 -170C64 -164 55 -168 48 -175C38 -185 24 -203 24 -222S48 -257 71 -257C89 -257 131 -241 186 -192C243 -141 286 -46 310 74L366 351Z" id="g50-107"></path> </g> </svg>为每个流程<我>P</我><sub>1,我</年代ub>获得使用该识别方法为每个设置点。<年代vg height="18.0794pt" id="M69" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 12.9144 18.0794" width="12.9144pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="0" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.498,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,8.38,3.132)"> <use xlink:href="#g50-107"></use> </g> </svg>表示如下:<年代p一个nclass="equation_break" id="EEq41"><span class="left_break" id="L70"></span><span class="middle_break"><span class="center_break" id="M70"> <svg height="18.3809pt" id="M70-1" version="1.1" viewbox="0 0 116.884 18.3809" width="116.884pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B70-1"> <rect height="0.65243" width="6.49826" x="0" y=".702"></rect> <g transform="matrix(.013,0,0,-0.013,0,12.55)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.5,15.682)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,8.38,15.682)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,16.44,12.55)"> <use xlink:href="#g117-34"></use> </g> <g transform="matrix(.013,0,0,-0.013,27.71,12.551)"> <path d="M363 -410V-380C268 -368 256 -319 256 -264C256 -189 277 -5 277 130C277 195 252 252 189 269V273C253 288 277 346 277 410C277 542 256 679 256 809C256 861 268 908 363 920V950C295 950 197 933 197 793C197 667 208 520 208 391C208 349 204 294 87 287V254C204 246 208 192 208 150C208 26 197 -118 197 -257S295 -410 363 -410Z" id="g119-135"></path> </g> <g transform="matrix(.013,0,0,-0.013,33.42,12.55)"> <use xlink:href="#g113-76"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,42.65,15.682)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,44.53,15.682)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,48.97,12.55)"> <use xlink:href="#g113-45"></use> </g> <g transform="matrix(.013,0,0,-0.013,54.11,12.55)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,61.29,15.682)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,63.17,15.682)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,67.6,12.55)"> <use xlink:href="#g113-45"></use> </g> <g transform="matrix(.013,0,0,-0.013,72.75,12.55)"> <use xlink:href="#g113-77"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,81.04,15.682)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,82.92,15.682)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,87.36,12.55)"> <use xlink:href="#g113-45"></use> </g> <g transform="matrix(.013,0,0,-0.013,92.5,12.55)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,99.56,15.682)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,101.44,15.682)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,105.88,12.551)"> <path d="M351 254V287C234 294 230 349 230 391C230 520 241 667 241 793C241 933 143 950 75 950V920C170 908 182 861 182 809C182 679 161 542 161 410C161 346 185 288 249 273V269C186 252 161 195 161 130C161 -5 182 -189 182 -264C182 -319 170 -368 75 -380V-410C143 -410 241 -396 241 -257S230 26 230 150C230 192 234 246 351 254Z" id="g119-136"></path> </g> <g transform="matrix(.013,0,0,-0.013,113.92,12.55)"> <use xlink:href="#g113-45"></use> </g> </g> </svg> <svg class="alt_equation" height="18.3809pt" id="I70" version="1.1" viewbox="0 0 116.89 18.3809" width="116.89pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 0)" xlink:href="#B70-1"></use> </svg></span></span><span class="right_break" id="R70"> <svg class="label" height="11.44pt" id="M70-2" version="1.1" viewbox="0 0 19.848 11.44" width="19.848pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g id="B70-2"> <g transform="matrix(.013,0,0,-0.013,0,9.26)"> <use xlink:href="#g190-41"></use> </g> <g transform="matrix(.013,0,0,-0.013,4.5,9.26)"> <use xlink:href="#g190-166"></use> </g> <g transform="matrix(.013,0,0,-0.013,10.96,9.26)"> <use xlink:href="#g190-213"></use> </g> <g transform="matrix(.013,0,0,-0.013,15.35,9.26)"> <use xlink:href="#g190-42"></use> </g> </g> </svg> <svg class="alt_label" height="18.3809pt" id="IL70" version="1.1" viewbox="0 0 19.93 18.3809" width="19.93pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <use transform="matrix(1 0 0 1 0 3.26)" xlink:href="#B70-2"></use> </svg></span></span></span></span><span class="list-item"><span class="list-label"></span><span class="list-content">在哪里<我>我</我>= 1,…,9代表的价值<我>λ</我><sub>我</年代ub>过程参数,如表示<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab11/" target="_blank">11</一个>,<我>j</我>= 1,…,6代表使用的识别方法,如表示<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab12/" target="_blank">12</一个>。简而言之,一组54过程模型参数。</年代p一个n></span><span class="list-item"><span class="list-label">(3)</年代p一个n><span class="list-content">54的一步反应为每个模型得到,和他们各自的价值模型的性能指标<年代vg height="18.0794pt" id="M71" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 28.1031 18.0794" width="28.1031pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.013,0,0,-0.013,6.136,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="10.6342" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,10.634,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,17.132,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,19.014,3.132)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,23.449,0)"> <use xlink:href="#g113-42"></use> </g> </svg>确定。</年代p一个n></span></span></p> <div class="floats-partial-section partial-other-wrapper" data-section="tab11"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表11</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab11/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 组过程部分订单、采样时间、样品数量、长度和时间用于FFOPDT一步反应过程<我>P</我> <sub>1,我</年代ub>(<我>我</我>= 1,…,9)。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab11" target="_blank" aria-label="View full page"></a> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab12"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表12</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab12/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 组对称点用于FFOPDT模型识别。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab12" target="_blank" aria-label="View full page"></a> </div> </div> <p>这批使用过程,一方面,验证提出了不同的对称点识别方法,另一方面,能洞察的选择代表点的影响获得的分数阶模型的准确性。</p> <p>注意,表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab11/" target="_blank">11</一个>包含采样周期、样本数量和时间长度被认为是在一步反应的过程。表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab12/" target="_blank">12</一个>列表的集合点考虑FFOPDT的识别模型。图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig8/" target="_blank">8</一个>说明了模型性能指标的值<年代vg height="18.0794pt" id="M72" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 28.1031 18.0794" width="28.1031pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.013,0,0,-0.013,6.136,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="10.6342" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,10.634,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,17.132,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,19.014,3.132)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,23.449,0)"> <use xlink:href="#g113-42"></use> </g> </svg>已获得的提出过程每个进程(<我>我</我>= 1,…,9)和使用不同的方法(<我>j</我>= 1,…,6)。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-5d0925d87f76daa173898f21c1bac3c8"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.008.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-5d0925d87f76daa173898f21c1bac3c8 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.008_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-5d0925d87f76daa173898f21c1bac3c8 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图8</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig8/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 模型性能指标的值<年代vg height="18.0794pt" id="M73" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 28.1031 18.0794" width="28.1031pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.013,0,0,-0.013,6.136,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="10.6342" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,10.634,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,17.132,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,19.014,3.132)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,23.449,0)"> <use xlink:href="#g113-42"></use> </g> </svg>为54 FFOPDT模型与参数<年代p一个nclass="nowrap"> <svg height="18.0794pt" id="M74" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 12.9144 18.0794" width="12.9144pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="0" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.498,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,8.38,3.132)"> <use xlink:href="#g50-107"></use> </g> </svg>,</年代p一个n>获得了每一个过程<我>P</我> <sub>1,我</年代ub>(<我>我</我>= 1,…,9)使用方法<年代up>#</年代up>j (<我>j</我>= 1,…,6)。</d我v> </div> </div> </div> <p>注意,值<年代vg height="18.0794pt" id="M75" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 28.1031 18.0794" width="28.1031pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.013,0,0,-0.013,6.136,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="10.6342" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,10.634,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,17.132,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,19.014,3.132)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,23.449,0)"> <use xlink:href="#g113-42"></use> </g> </svg>在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig8/" target="_blank">8</一个>当他们从方法大幅降低<年代up>#</年代up>6到<年代up>#</年代up>1。这种行为发生的过程,除了那些<我>λ</我><sub><i>我</我></sub>接近1,也就是说,当这个过程方法的一个整数。这种行为已经被观察到在技术文献。123 c的方法(<一个href="#B11">11</一个>使用时间设置(25 - 75%)确定FOPDT DPPDT模型和(25 - 50 - 75%)SOPDT模型。在[<一个href="#B18">18</一个>),点(25 - 50 - 75%)提出了识别RPPDT模型。在这两个文件,这些点任意选择的,而是那些没有优化确定模型参数选择,即。,DPPDT FOPDT SOPDT,分别和RPPDT模型。在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig8/" target="_blank">8</一个>的最小值<年代vg height="18.0794pt" id="M76" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 28.1031 18.0794" width="28.1031pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.013,0,0,-0.013,6.136,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="10.6342" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,10.634,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,17.132,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,19.014,3.132)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,23.449,0)"> <use xlink:href="#g113-42"></use> </g> </svg>为<我>P</我><sub>1、9</年代ub>,这是一个integer-order模型,对应方法<年代up>#</年代up>5 (25 - 50 - 75%)。</p> <p>图8还表明,该识别方法给出了适合相应的识别模型和实际过程之间的反应曲线。这是证实了低的值在时域模型的性能指标<年代vg height="18.0794pt" id="M77" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 28.1031 18.0794" width="28.1031pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.013,0,0,-0.013,6.136,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="10.6342" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,10.634,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,17.132,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,19.014,3.132)"> <use xlink:href="#g50-107"></use> </g> <g transform="matrix(.013,0,0,-0.013,23.449,0)"> <use xlink:href="#g113-42"></use> </g> </svg>所有进程<我>P</我><sub>1,我</年代ub>。</p> <p>从获得的结果在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig8/" target="_blank">8</一个>,它可以观察到的方法<年代up>#</年代up>1是提供最好的结果,因此,点集(5 - 50 - 95%)是建议被选中当使用这个对称的识别过程。在这方面,相对性能指标之间的方法<年代up>#</年代up>6 (<我>j</我>= 6)和方法<年代up>#</年代up>1 (<我>j</我>= 1),<年代p一个nclass="inline_break"> <svg height="16.9011pt" id="M78" style="vertical-align:-4.3506pt" version="1.1" viewbox="-0.0498162 -12.5505 39.095 16.9011" width="39.095pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.071,3.132)"> <path d="M141 347C171 480 264 541 327 570C364 587 403 599 436 605L429 641C389 634 339 622 300 609C191 570 37 458 37 239C37 86 126 -12 245 -12C367 -12 454 89 454 210C454 314 380 397 272 397C252 397 230 390 208 380L141 347ZM231 338C323 338 366 257 366 174C366 108 340 27 263 27C180 27 129 123 129 244C129 268 130 292 135 310C158 323 193 338 231 338Z" id="g50-55"></path> </g> <g transform="matrix(.0091,0,0,-0.0091,10.503,3.132)"> <use xlink:href="#g50-45"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,12.66,3.132)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.63,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="22.1285" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,22.128,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,28.627,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,31.175,3.132)"> <use xlink:href="#g50-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,36.131,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span><span class="nowrap"> <svg height="16.9011pt" style="vertical-align:-4.3506pt" version="1.1" viewbox="41.224083799999995 -12.5505 18.667 16.9011" width="18.667pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="41.2739" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,41.274,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,47.772,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,50.32,3.132)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,55.276,0)"> <use xlink:href="#g113-42"></use> </g> </svg>,</年代p一个n></span>为<我>我</我>= 1,…,9日在图表示<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig9/" target="_blank">9</一个>,它可以注意到方法<年代up>#</年代up>1大约是2.0,2.25,2.5,3.5,4.0,2.75,和1.5倍的年代比获得的一个方法<年代up>#</年代up>6,<我>我</我>= 1,…,分别为7。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-8335e5c16e1ad3936c044b342dff522e"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.009.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-8335e5c16e1ad3936c044b342dff522e partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.009_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-8335e5c16e1ad3936c044b342dff522e partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图9</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig9/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 相对表现索引值<年代p一个nclass="inline_break"> <svg height="16.9011pt" id="M79" style="vertical-align:-4.3506pt" version="1.1" viewbox="-0.0498162 -12.5505 39.095 16.9011" width="39.095pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.071,3.132)"> <use xlink:href="#g50-55"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,10.503,3.132)"> <use xlink:href="#g50-45"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,12.66,3.132)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.63,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="22.1285" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,22.128,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,28.627,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,31.175,3.132)"> <use xlink:href="#g50-55"></use> </g> <g transform="matrix(.013,0,0,-0.013,36.131,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span> <svg height="16.9011pt" style="vertical-align:-4.3506pt" version="1.1" viewbox="43.399183799999996 -12.5505 18.668 16.9011" width="18.668pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="43.4491" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,43.449,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,49.947,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,52.496,3.132)"> <use xlink:href="#g50-50"></use> </g> <g transform="matrix(.013,0,0,-0.013,57.451,0)"> <use xlink:href="#g113-42"></use> </g> </svg></span>为每个值<我>λ</我> <sub><i>我</我></sub>(<我>我</我>9)= 1,…,代表之间的比例模型获得的性能指标与方法<年代up>#</年代up>6和<年代up>#</年代up>1。</d我v> </div> </div> </div> <p>这个案子<我>λ</我><sub>5</年代ub>P = 0.8的过程<年代ub>1、5</年代ub>被认为是在过程<我>P</我><sub>1,我</年代ub>。这个模型的过程反应曲线如图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig10/" target="_blank">10</一个>。从这个数据,总结了表的信息<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab13/" target="_blank">13</一个>。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-5102774399c60ddf74a1dfad41918d44"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0010.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-5102774399c60ddf74a1dfad41918d44 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0010_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-5102774399c60ddf74a1dfad41918d44 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图10</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig10/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 阶跃输入信号和过程反应曲线的过程<我>P</我> <sub>1、5</年代ub>。</d我v> </div> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab13"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表13</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab13/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 处理信息为分数阶模型的识别过程<我>P</我> <sub>1、5</年代ub>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab13" target="_blank" aria-label="View full page"></a> </div> </div> <p>考虑到数据表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab13/" target="_blank">13</一个>,FFOPDT过程参数<年代p一个nclass="inline_break"> <svg height="15.8416pt" id="M80" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -12.5505 25.265 15.8416" width="25.265pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="0" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.498,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,9.046,3.132)"> <use xlink:href="#g50-54"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.634,0)"> <use xlink:href="#g117-34"></use> </g> </svg><span class="irelop"></span> <svg height="15.8416pt" style="vertical-align:-3.291101pt" version="1.1" viewbox="28.8471838 -12.5505 24.209 15.8416" width="24.209pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,28.897,0)"> <use xlink:href="#g113-124"></use> </g> <g transform="matrix(.013,0,0,-0.013,33.408,0)"> <use xlink:href="#g113-76"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,42.638,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,45.186,3.132)"> <use xlink:href="#g50-54"></use> </g> <g transform="matrix(.013,0,0,-0.013,50.142,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span> <svg height="15.8416pt" style="vertical-align:-3.291101pt" version="1.1" viewbox="55.235183799999994 -12.5505 17.644 15.8416" width="17.644pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,55.285,0)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,62.461,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,65.009,3.132)"> <use xlink:href="#g50-54"></use> </g> <g transform="matrix(.013,0,0,-0.013,69.965,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span> <svg height="15.8416pt" style="vertical-align:-3.291101pt" version="1.1" viewbox="75.05818380000001 -12.5505 18.762 15.8416" width="18.762pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,75.108,0)"> <use xlink:href="#g113-77"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,83.402,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,85.95,3.132)"> <use xlink:href="#g50-54"></use> </g> <g transform="matrix(.013,0,0,-0.013,90.906,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span><span class="nowrap"> <svg height="15.8416pt" style="vertical-align:-3.291101pt" version="1.1" viewbox="95.99918380000001 -12.5505 19.389 15.8416" width="19.389pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,96.049,0)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,103.108,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,105.656,3.132)"> <use xlink:href="#g50-54"></use> </g> <g transform="matrix(.013,0,0,-0.013,110.612,0)"> <use xlink:href="#g113-126"></use> </g> </svg>,</年代p一个n></span>在哪里<我>我</我>= 1,…,在表6中,取得了<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab14/" target="_blank">14</一个>为每一个提出的对称点集的表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab12/" target="_blank">12</一个>。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab14"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表14</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab14/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> FFOPDT模型参数代表点的考虑集(5 - 50 - 95%),(10 - 50 - 90%),(15 - 50 - 85%),(20 - 50 - 80%),(25 - 50 - 75%)和(30 - 50 - 70%),分别为。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab14" target="_blank" aria-label="View full page"></a> </div> </div> <p>该识别方法将与各种两,三点FOPDT识别方法,DPPDT和SOPDT模型基于信息也获得反应曲线。</p> <p>的两点方法(<一个href="#B11">11</一个>,<一个href="#B14">14</一个>],FOPDT和DPPDT模型都使用。</p> <p>表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab15/" target="_blank">15</一个>包含FOPDT模型的参数识别使用两点Alfaro Viterkova,提出的识别方法。表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab16/" target="_blank">16</一个>显示的参数DPPDT模型获得使用相同的方法。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab15"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表15</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab15/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> FOPDT模型参数获取使用居多和Viterkova的两点方法,分别。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab15" target="_blank" aria-label="View full page"></a> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab16"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表16</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab16/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> DPPDT模型参数获取使用居多和Viterkova的两点方法,分别。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab16" target="_blank" aria-label="View full page"></a> </div> </div> <p>三分的方法(<一个href="#B15">15</一个>)和斯塔克(<一个href="#B16">16</一个>),SOPDT模型被认为是。</p> <p>表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab17/" target="_blank">17</一个>包含SOPDT模型获得的参数使用三点Jahanmiri提出的识别方法和Fallahi鲜明,分别。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab17"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表17</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab17/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> SOPDT模型参数获取使用鲜明的和Jahanmiri Fallahi三分的方法,分别。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab17" target="_blank" aria-label="View full page"></a> </div> </div> <p>数据<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig11/" target="_blank">11</一个>- - - - - -<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig14/" target="_blank">14</一个>比较过程反应曲线的一步反应FFOPDT模型与该方法获得不同的对称点和FOPDT DPPDT,和SOPDT模型提出的居多,Viterkova,斯塔克,分别和Jahanmiri Fallahi。一步反应在图基于分组的相似点的反应曲线用于识别方法。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-09461e8366838021af89afa8c6199df9"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0011.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-09461e8366838021af89afa8c6199df9 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0011_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-09461e8366838021af89afa8c6199df9 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图11</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig11/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程的反应曲线,一步反应FFOPDT模型方法<年代up>#</年代up>1,<年代up>#</年代up>2和SOPDT方法<年代up>#</年代up>12岁,<年代up>#</年代up>分别为13。</d我v> </div> </div> </div> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-6a5984a66fa59748b70f752e9f48be92"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0012.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-6a5984a66fa59748b70f752e9f48be92 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0012_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-6a5984a66fa59748b70f752e9f48be92 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图12</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig12/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程的反应曲线,一步反应FFOPDT模型方法<年代up>#</年代up>3和<年代up>#</年代up>4和SOPDT方法<年代up>#</年代up>分别为11。</d我v> </div> </div> </div> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-a71d9e624a1d528a3ad9a5579f780b70"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0013.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-a71d9e624a1d528a3ad9a5579f780b70 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0013_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-a71d9e624a1d528a3ad9a5579f780b70 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图13</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig13/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程的反应曲线,一步反应FFOPDT模型方法<年代up>#</年代up>5和FOPDT DPPDT模型方法<年代up>#</年代up>7和<年代up>#</年代up>9,分别。</d我v> </div> </div> </div> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-2dc7d7554cd679699e7c9a5036f66138"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0014.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-2dc7d7554cd679699e7c9a5036f66138 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0014_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-2dc7d7554cd679699e7c9a5036f66138 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图14</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig14/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程的反应曲线,一步反应FFOPDT模型方法<年代up>#</年代up>6和FOPDT DPPDT模型方法<年代up>#</年代up>8和<年代up>#</年代up>分别为10。</d我v> </div> </div> </div> <p>表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab18/" target="_blank">18</一个>显示时域模型性能指标的值<年代vg height="14.8186pt" id="M81" style="vertical-align:-2.2681pt" version="1.1" viewbox="-0.0498162 -12.5505 21.7866 14.8186" width="21.7866pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.013,0,0,-0.013,6.136,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="10.6342" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,10.634,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.132,0)"> <use xlink:href="#g113-42"></use> </g> </svg>为不同的识别方法应用于过程<我>P</我><sub>1、5</年代ub>。相对性能指标的值<年代p一个nclass="inline_break"> <svg height="18.0794pt" id="M82" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 38.949 18.0794" width="38.949pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.071,3.132)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,11.249,3.132)"> <use xlink:href="#g50-45"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,13.406,3.132)"> <use xlink:href="#g50-122"></use> </g> <g transform="matrix(.013,0,0,-0.013,19.293,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="23.7909" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,23.791,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,30.289,3.132)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,35.985,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span><span class="nowrap"> <svg height="18.0794pt" style="vertical-align:-5.528899pt" version="1.1" viewbox="41.077883799999995 -12.5505 17.023 18.0794" width="17.023pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="41.1277" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,41.128,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,47.626,3.132)"> <use xlink:href="#g50-122"></use> </g> <g transform="matrix(.013,0,0,-0.013,53.486,0)"> <use xlink:href="#g113-42"></use> </g> </svg>,</年代p一个n></span>它允许不同的识别方法之间的比较,列出在表吗<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab19/" target="_blank">19</一个>。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab18"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表18</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab18/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 时域模型的性能指标的过程<我>P</我> <sub>1、5</年代ub>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab18" target="_blank" aria-label="View full page"></a> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab19"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表19</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab19/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 相对性能基于索引比较一些考虑识别方法的过程<我>P</我> <sub>1、5</年代ub>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab19" target="_blank" aria-label="View full page"></a> </div> </div> <p>数据<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig11/" target="_blank">11</一个>- - - - - -<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig14/" target="_blank">14</一个>说明模型的阶跃响应与不同的代表点的方法给出一个合适的过程反应曲线,特别是在区间[<我>x</我>−(100 -<我>x</我>)]。</p> <p>对于较小的值<我>x</我>,区间[x−(100 - x))是更大的,因此,这些模型更适合的阶跃响应过程的反应曲线,转化为低价值的模型性能指标<我>年代</我>,如表中可以看到<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab18/" target="_blank">18</一个>。</p> <p>这主要是由于这一事实的价值<我>α</我>确定为较小的值<我>x</我>接近最优值。</p> <p>请注意,<我>α</我>模型的价值有很大的影响对阶跃响应的形状FFOPDT模型,我们可以看到在图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig2/" target="_blank">2</一个>。</p> <p>关于两点方法,数字<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig13/" target="_blank">13</一个>和<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig14/" target="_blank">14</一个>显示一步反应的模型获得的居多(25 - 75%)和Viterkova(33 - 70%)方法,对FOPDT DPPDT,分别比建议的方法(25 - 50 - 75%)和(30 - 50 - 70%)。</p> <p>图11和12显示居多,Viterkova的方法很适合反应曲线范围(25 - 75%)和(33 - 70%),分别为。然而,这些范围以外的两种模型的行为偏离的反应曲线,从而导致更高的<我>年代</我>价值。</p> <p>的S值的方法<年代up>#</年代up>1是7.7和8.1倍低于建议的居多,Viterkova FOPDT模型,分别。反过来,年代为方法的价值<年代up>#</年代up>5和<年代up>#</年代up>6是2.4和2.3倍低于方法提出的居多,Viterkova FOPDT模型,分别。</p> <p>方法<年代up>#</年代up>1是17.8和16.7倍的年代比居多和Viterkova DPPDT方法模型,分别。另一方面,方法<年代up>#</年代up>5和<年代up>#</年代up>6比5.6和4.8倍的居多,Viterkova DPPDT模型。居多的反应DPPDT模型和Viterkova有更高的值在模型中性能指标见表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab18/" target="_blank">18</一个>FOPDT模型相比,相同的方法。</p> <p>关于三分方法,图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig11/" target="_blank">11</一个>展示步骤反应Jahanmiri和Fallahi方法(2 - 70 - 90%)和(5 - 70 - 90%)SOPDT模型,相比,提出的方法(5 - 50 - 95%)和(10 - 50 - 90%)。</p> <p>的S值的方法<年代up>#</年代up>1是11.9和14.1倍,低于一个方法<年代up>#</年代up>12岁,<年代up>#</年代up>分别为13。如果两个三分方法与方法<年代up>#</年代up>2,后者是6.1和7.2倍的方法<年代up>#</年代up>12岁,<年代up>#</年代up>分别为13。</p> <p>图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig12/" target="_blank">12</一个>显示了鲜明的一步反应的方法(15 - 45 - 75%)SOPDT模型相比,提出的方法(15 - 50 - 85%)和(20 - 50 - 80%)。的S值的方法<年代up>#</年代up>1,<年代up>#</年代up>3)低8.6和3.7倍比的方法<年代up>#</年代up>11。</p> <p>有一个改进的S(1.9, 2.3, 2.8, 3.2,和3.4倍的过程<我>P</我><sub>1</年代ub>在使用方法<年代up>#</年代up>1(5 - 50 - 95%)方法<年代up>#</年代up>2,<年代up>#</年代up>3,<年代up>#</年代up>4,<年代up>#</年代up>5,<年代up>#</年代up>6,分别如表中所示<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab19/" target="_blank">19</一个>。</p> <h5 id="sec5.2">5.2。示例2</h5> <p>在这个例子中,该识别方法比较与另一个FFOPDT模型识别方法。</p> <p>流程模型的高阶lag-dominated分数阶(38)<我>K</我><sub>2</年代ub>= 2,<我>T</我><sub>2</年代ub>= 1,<我>λ</我><sub>2</年代ub>= 0.85,<我>n</我>= 5,被认为是在这个例子提出了(<一个href="#B26">26</一个>]。</p> <p>这个模型的过程反应曲线如图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig15/" target="_blank">15</一个>。从这个数据,模型识别所需的处理信息的使用提出的过程在不同的时期,总结了表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab20/" target="_blank">20.</一个>。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-dba95aa405643cc0bcd3dce2de761612"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0015.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-dba95aa405643cc0bcd3dce2de761612 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0015_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-dba95aa405643cc0bcd3dce2de761612 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图15</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig15/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 阶跃输入信号和过程反应曲线的过程<我>P</我> <sub>2</年代ub>。</d我v> </div> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab20"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表20</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab20/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 处理信息为分数阶模型的识别过程<我>P</我> <sub>2</年代ub>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab20" target="_blank" aria-label="View full page"></a> </div> </div> <p>考虑到数据表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab20/" target="_blank">20.</一个>,FFOPDT过程参数<年代p一个nclass="inline_break"> <svg height="15.8416pt" id="M83" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -12.5505 20.817 15.8416" width="20.817pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="0" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.498,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.013,0,0,-0.013,13.186,0)"> <use xlink:href="#g117-34"></use> </g> </svg><span class="irelop"></span> <svg height="15.8416pt" style="vertical-align:-3.291101pt" version="1.1" viewbox="24.399183800000003 -12.5505 19.76 15.8416" width="19.76pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,24.449,0)"> <use xlink:href="#g113-124"></use> </g> <g transform="matrix(.013,0,0,-0.013,28.96,0)"> <use xlink:href="#g113-76"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,38.19,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.013,0,0,-0.013,41.245,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span> <svg height="15.8416pt" style="vertical-align:-3.291101pt" version="1.1" viewbox="48.5141838 -12.5505 13.195 15.8416" width="13.195pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,48.564,0)"> <use xlink:href="#g113-85"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,55.74,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.013,0,0,-0.013,58.795,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span> <svg height="15.8416pt" style="vertical-align:-3.291101pt" version="1.1" viewbox="66.06418380000001 -12.5505 14.313 15.8416" width="14.313pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,66.114,0)"> <use xlink:href="#g113-77"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,74.408,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.013,0,0,-0.013,77.463,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span><span class="nowrap"> <svg height="15.8416pt" style="vertical-align:-3.291101pt" version="1.1" viewbox="84.7321838 -12.5505 14.94 15.8416" width="14.94pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,84.782,0)"> <use xlink:href="#g113-223"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,91.841,3.132)"> <use xlink:href="#g50-106"></use> </g> <g transform="matrix(.013,0,0,-0.013,94.896,0)"> <use xlink:href="#g113-126"></use> </g> </svg>,</年代p一个n></span>在哪里<我>我</我>= 1,…,在表6中,取得了<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab21/" target="_blank">21</一个>提出组的每一个代表点的反应曲线,即。,(5- - - - - -50- - - - - -95%), (10–50–90%), (15–50–85%), (20–50–80%), (25–50–75%), and (30–50–70%), respectively.</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab21"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表21</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab21/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> FFOPDT模型参数代表点的考虑集(5 - 50 - 95%),(10 - 50 - 90%),(15 - 50 - 85%),(20 - 50 - 80%),(25 - 50 - 75%)和(30 - 50 - 70%),分别为。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab21" target="_blank" aria-label="View full page"></a> </div> </div> <p>过程(<一个href="#EEq37">38</一个>)由FFOPDT近似模型三个不同的策略提出了(<一个href="#B26">26</一个>),工艺参数模型<年代vg height="15.8416pt" id="M84" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -12.5505 36.4596 15.8416" width="36.4596pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="0" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.498,3.132)"> <path d="M453 623H65C60 580 56 530 46 472H80C103 538 110 548 175 548H389C310 380 195 168 90 0L98 -12L175 -2C273 206 366 408 462 610L453 623Z" id="g50-56"></path> </g> <g transform="matrix(.013,0,0,-0.013,14.35,0)"> <use xlink:href="#g117-33"></use> </g> <rect height="0.65243" width="6.49826" x="24.8867" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,24.887,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,31.385,3.132)"> <path d="M247 635C116 635 39 518 39 421C39 315 113 237 221 237C239 237 258 241 281 253L349 289C315 140 204 42 60 19L66 -15C90 -15 153 -5 207 16C344 70 447 199 447 385C447 521 373 635 247 635ZM231 598C329 598 356 478 356 390C356 371 355 347 352 325C330 309 297 297 262 297C179 297 127 369 127 456C127 515 156 598 231 598Z" id="g50-58"></path> </g> </svg>给出了在表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab22/" target="_blank">22</一个>。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab22"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表22</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab22/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> FFOPDT模型参数不同的策略提出了(<一个href="#B26">26</一个>]。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab22" target="_blank" aria-label="View full page"></a> </div> </div> <p>过程的反应曲线和相应的步骤的反应被认为是近似模型(5 - 50 - 95%),(10 - 50 - 90%),(15 - 50 - 85%),(20 - 50 - 80%),(25 - 50 - 75%)和(30 - 50 - 70%),分别比较,见图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig16/" target="_blank">16</一个>- - - - - -<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig21/" target="_blank">21</一个>。此外,一步反应FFOPDT模型提出了(<一个href="#B26">26</一个>的过程<我>P</我><sub>2</年代ub>相比,过程反应曲线如图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig22/" target="_blank">22</一个>。</p> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-1c58cd5e741937053e45e11f061d1d53"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0016.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-1c58cd5e741937053e45e11f061d1d53 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0016_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-1c58cd5e741937053e45e11f061d1d53 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图16</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig16/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程反应曲线和FFOPDT模型方法<年代up>#</年代up>1(5 - 50 - 95%)阶跃响应过程<我>P</我> <sub>2</年代ub>。</d我v> </div> </div> </div> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-ffbff5add0b58c74341098a77be0150b"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0017.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-ffbff5add0b58c74341098a77be0150b partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0017_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-ffbff5add0b58c74341098a77be0150b partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图17</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig17/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程反应曲线和FFOPDT模型方法<年代up>#</年代up>2(10 - 50 - 90%)阶跃响应过程<我>P</我> <sub>2</年代ub>。</d我v> </div> </div> </div> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-0e2857b0932c65bbb5490c8927098eae"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0018.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-0e2857b0932c65bbb5490c8927098eae partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0018_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-0e2857b0932c65bbb5490c8927098eae partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图18</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig18/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程反应曲线和FFOPDT模型方法<年代up>#</年代up>3(15 - 50 - 85%)阶跃响应过程<我>P</我> <sub>2</年代ub>。</d我v> </div> </div> </div> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-e56eb8c6c57ec357178803889bba4e3a"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0019.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-e56eb8c6c57ec357178803889bba4e3a partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0019_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-e56eb8c6c57ec357178803889bba4e3a partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图19</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig19/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程反应曲线和FFOPDT模型方法<年代up>#</年代up>3(20 - 50 - 80%)阶跃响应过程<我>P</我> <sub>2</年代ub>。</d我v> </div> </div> </div> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-bfd6c94c940683edc53c9f1e454f1659"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0020.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-bfd6c94c940683edc53c9f1e454f1659 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0020_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-bfd6c94c940683edc53c9f1e454f1659 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图20</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig20/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程反应曲线和FFOPDT模型方法<年代up>#</年代up>5(25 - 50 - 75%)阶跃响应过程<我>P</我> <sub>2</年代ub>。</d我v> </div> </div> </div> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-806b5c3b1b6e5250bb289ce3a4a39bb3"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0021.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-806b5c3b1b6e5250bb289ce3a4a39bb3 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0021_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-806b5c3b1b6e5250bb289ce3a4a39bb3 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图21</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig21/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程反应曲线和FFOPDT模型方法<年代up>#</年代up>6(30 - 50 - 70%)阶跃响应过程<我>P</我> <sub>2</年代ub>。</d我v> </div> </div> </div> <div class="partial-carousel-wrapper"> <div class="floats-partial-section"> <div class="floats-partial-content"> <div class="partial-carousel--full-image partial-carousel-0e09caedd53158450a1d9d90591ac741"> <div class="partial-carousel--single-image"> <img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/figures/7185131.fig.0022.jpg"> <br> <strong></strong> </div> </div> </div> <div class="floats-partial-middle"> <div class="floats-partial-middle--thumbs"> <div class="partial-carousel-dots-0e09caedd53158450a1d9d90591ac741 partial-carousel-thumbnail"> <button aria-label="Slide to " class="partial-carousel-dot"><img loading="lazy" alt="" src="https://static.hindawi.com/articles/jmath/volume-2022/7185131/thumbnails/7185131.fig.0022_th.jpg"><br><strong></strong></button> </div> </div> <div class="floats-partial-middle--nav"> <div class="partial-carousel-next-prev-0e09caedd53158450a1d9d90591ac741 partial-carousel-next-prev"> <button aria-label="Previous" class="carousel-prev"></button> <button aria-label="Next" class="carousel-next"></button> </div> </div> </div> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">图22</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/fig22/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 过程反应曲线和FFOPDT模型方法<年代up>#</年代up>7 -<年代up>#</年代up>9步反应过程P<年代ub>2</年代ub>。</d我v> </div> </div> </div> <p>表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab23/" target="_blank">23</一个>显示时域模型性能指标的值<年代vg height="14.8186pt" id="M85" style="vertical-align:-2.2681pt" version="1.1" viewbox="-0.0498162 -12.5505 21.7866 14.8186" width="21.7866pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.013,0,0,-0.013,6.136,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="10.6342" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,10.634,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.013,0,0,-0.013,17.132,0)"> <use xlink:href="#g113-42"></use> </g> </svg>为不同的识别方法应用于过程<我>P</我><sub>2</年代ub>。相对性能指标的值<年代p一个nclass="inline_break"> <svg height="18.0794pt" id="M86" style="vertical-align:-5.528899pt" version="1.1" viewbox="-0.0498162 -12.5505 38.949 18.0794" width="38.949pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <g transform="matrix(.013,0,0,-0.013,0,0)"> <use xlink:href="#g113-84"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,6.071,3.132)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,11.249,3.132)"> <use xlink:href="#g50-45"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,13.406,3.132)"> <use xlink:href="#g50-122"></use> </g> <g transform="matrix(.013,0,0,-0.013,19.293,0)"> <use xlink:href="#g113-41"></use> </g> <rect height="0.65243" width="6.49826" x="23.7909" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,23.791,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,30.289,3.132)"> <use xlink:href="#g50-121"></use> </g> <g transform="matrix(.013,0,0,-0.013,35.985,0)"> <use xlink:href="#g113-45"></use> </g> </svg><span class="ipunc"></span><span class="nowrap"> <svg height="18.0794pt" style="vertical-align:-5.528899pt" version="1.1" viewbox="41.077883799999995 -12.5505 17.023 18.0794" width="17.023pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"> <rect height="0.65243" width="6.49826" x="41.1277" y="-11.8482"></rect> <g transform="matrix(.013,0,0,-0.013,41.128,0)"> <use xlink:href="#g113-230"></use> </g> <g transform="matrix(.0091,0,0,-0.0091,47.626,3.132)"> <use xlink:href="#g50-122"></use> </g> <g transform="matrix(.013,0,0,-0.013,53.486,0)"> <use xlink:href="#g113-42"></use> </g> </svg>,</年代p一个n></span>显示不同的识别方法之间的比较,列出在表吗<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab24/" target="_blank">24</一个>。</p> <div class="floats-partial-section partial-other-wrapper" data-section="tab23"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表23</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab23/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 时域模型的性能指标的过程<我>P</我> <sub>2</年代ub>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab23" target="_blank" aria-label="View full page"></a> </div> </div> <div class="floats-partial-section partial-other-wrapper" data-section="tab24"> <div class="floats-partial-footer"> <div class="floats-partial-caption"> <span class="caption-text">表24</年代p一个n> <a aria-label="View full page" class="caption-partial-url" href="//www.newsama.com/journals/jmath/2022/7185131/tab24/" title="" target="_blank"></a> </div> <div class="floats-partial-caption-text"> 相对性能基于索引比较一些认为分数阶模型识别方法的过程<我>P</我> <sub>2</年代ub>。</d我v> <a class="table-thumb-overlay" href="//www.newsama.com/journals/jmath/2022/7185131/tab24" target="_blank" aria-label="View full page"></a> </div> </div> <p>在这个例子中,一个高阶分数阶lag-dominated过程为了说明使用的适用性和有效性提出了识别FFOPDT模型识别过程。</p> <p>数据<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig16/" target="_blank">16</一个>- - - - - -<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig21/" target="_blank">21</一个>表明,该方法给出好的结果在区间[<我>x</我>−(100−<我>x</我>)考虑集的代表点。我们需要记得,方法试图将三个对称点(<我>x</我>−−−50 (100<我>x</我>)%)反应曲线。</p> <p>代表点的位置值更高的反应曲线很好<我>x</我>,有轻微偏差<我>x</我>= 5,我们可以看到数据<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig16/" target="_blank">16</一个>- - - - - -<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig21/" target="_blank">21</一个>。</p> <p>然而,获得的模型更适合反应曲线低的值<我>x</我>,因为<我>α</我>价值获得接近最优值,正如前面已经说。这将导致低的值低的值<我>x</我>考虑集的代表点,如表所示<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab23/" target="_blank">23</一个>。</p> <p>有一个改进的1.4,1.7,2.2,2.6,和2.8倍的使用方法<年代up>#</年代up>1(5 - 50 - 95%)方法<年代up>#</年代up>2,<年代up>#</年代up>3,<年代up>#</年代up>4,<年代up>#</年代up>5,<年代up>#</年代up>6,分别如表中所示<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab24/" target="_blank">24</一个>。</p> <p>图<一个href="//www.newsama.com/journals/jmath/2022/7185131/fig22/" target="_blank">22</一个>表明Tavakoli-Kakhki的方法得出的结果类似的一些提议的方法,这可以量化表<一个href="//www.newsama.com/journals/jmath/2022/7185131/tab23/" target="_blank">23</一个>。</p> <p>方法<年代up>#</年代up>1 = 1.8,3.0,和1.1倍的年代比Tavakoli-Kakhki的方法<年代up>#</年代up>1,<年代up>#</年代up>2,<年代up>#</年代up>3,分别。</p> <p>此外,该方法不仅提供了更好的结果比Tavakoli-Kakhki方法的年代,但是在作者的观点,更容易申请。</p> <p><span class="statement" id="rem2"><span class="statement-title"><i class="statement-title">备注2。</我></span>在第二个例子中,仅限于FFOPDT模型的分析方法比较简单和计算工作相似的提议。然而,没有很多分数阶建模过程的分析技术基于过程的反应曲线。因此,提出了对称的比较方法是完成了方法(<一个href="#B26">26</一个>),这是一个公认的参考,提出了三种策略来确定FFOPDT模型的参数利用阶跃响应数据。它结合了数值计算和图形估计。<br>简介中,有一个评估分数阶模型的识别方法,基于过程反应曲线;然而,他们使用优化得到模型参数。在这些情况下,模型的精度提高的成本增加计算工作量。在作者看来,比较该方法与这些的使用优化不会在平等的条件下。</年代p一个n></p> <p><span class="statement" id="rem3"><span class="statement-title"><i class="statement-title">备注3。</我></span>自过程通常是非线性的,动态特性的FFOPDT model-gain,时间常数,死亡时间,和部分定单变化时控制系统的操作点的变化,由于修改设置点或由于扰动的影响。<br>隐式的不确定性名义模型必须考虑。<br>通常有两种方法在技术文献中考虑参数不确定性时植物的识别方法。<br>第一种方法是考虑不确定性的识别过程中明确。这通常会使识别过程更加复杂。<br>第二个可能需要考虑模型的不确定性和变化的动态特性控制的过程控制系统的设计;见,例如,(<一个href="#B3">3</一个>]integer-order控制器和[<一个href="#B45">45</一个>部分情况下)。通常采用第二种方法的方法是保证一定程度的robustness-relative稳定性的控制系统设计,以确保其稳定的变化过程特征。<br>在本文中,一个过程的识别FFOPDT类型的降维分级模型,确定模型的主要用途是用于控制目的。因此,确定模型的参数的不确定性将被认为是在控制系统的设计。<br>在工业背景下,大型流程工业有成百上千的控制回路。出于这个原因,简洁是一个基本特征识别流程模型时用于控制目的。考虑第二种方法,可以保持简单的识别过程。</年代p一个n></p> <h4 id="conclusions">6。结论</h4> <p>摘要一般程序确定一个FFOPDT工业过程模型有s型一步反应和基于拟合三个任意点过程的反应曲线提出了。简化的一般识别过程也一直认为,只有点对称位于反应曲线已经被选中了。对称的方法提供了一个有效的方法获得的参数模型,通过要求的最优位置的选择只有一个点(<我>x</我>),因为另一个立即建立了对称性的要求。</p> <p>一些数值例子已经使用为了显示这个过程的有效性和适用性为分数阶模型的识别和洞察的影响选择组对称点的识别模型的准确性。</p> <p>提出了对称的过程被应用到以下组代表点:(5 - 50 - 95%),(10 - 50 - 90%),(15 - 50 - 85%),(20 - 50 - 80%),(25 - 50 - 75%)和(30 - 50 - 70%)。本文的结果验证所确定的分数阶模型的精度敏感组对称点的选择反应曲线和讨论,更准确的模型是获得低的值<我>x</我>。洞察这个选择的点集的上下文中提供了提出了对称的过程。</p> <p>这个识别过程给好的结果相比与其他整数和分数阶识别方法。</p> <p>强调另一个方面是,该方法是分析,促进其适用性较低的计算工作比较复杂的识别算法通常基于优化。在工业背景下,大型流程工业通常有成百上千的控制回路。出于这个原因,简单识别流程模型时主要关心的是用于控制目的。</p> <p>作者的意见,这种类型的识别方法,简单的强调,将鼓励他们的工业使用和将帮助在弥合差距理论研究部分模型及其实际应用过程中产业。</p> <p>这种期望是本研究的主要动机。</p> <h4 id="data-availability">数据可用性</h4> <p>数据集用于支持本研究的结果包括在本文中。</p> <h4 id="conflicts-of-interest">的利益冲突</h4> <p>作者宣称没有利益冲突。</p> <h4 id="acknowledgments">确认</h4> <p>作者要感谢巴斯克政府部分资金支持通过TRUSTIND ELKARTEK研发项目(ref KK-2020/00054)。</p> </div> </div> <div class="ArticleReferences_xmlContent__p_40j"> <h4 class="ArticleReferences_references__GH2t_" id="references">引用</h4> <ol class="ArticleReferences_orderedReferences__mJr9M"> <li class="ArticleReferences_articleReference__ouEuh" id="B1"> <div class="referenceContent"> <p class="referenceText">问:g . t . 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href="https://doi.org/10.1515/9783110497977-202" target="_blank" rel="noreferrer">出版商的网站</一个> </div></li> </ol> </div> <h4 class="copyright-title" id="copyright">版权</h4> <div class="copyright-body"> 版权©2022胡安·j·古德,巴勃罗•加西亚Bringas。这是一个开放的分布式下文章<一个rel="license" href="http://creativecommons.org/licenses/by/4.0/">知识共享归属许可</一个>,它允许无限制的使用、分配和复制在任何媒介,提供最初的工作是正确引用。</d我v> <div id="related-articles"></div> </div> </div> <div class="ant-col ThreeColumnLayout_rightColumn__b_MuL"> <div> <div> <div class="ArticleRightSection_articleRightSection__rG8Zh"> <div class="ArticleToolbar_articleToolbar__8fsAH"> <a href="https://downloads.hindawi.com/journals/jmath/2022/7185131.pdf" class="ButtonGroup_buttonGroupWrapper__Rqhy0" role="button" type="primary" aria-label="PDF" tabindex="0" unselectable="on" target="_blank" data-testid="download-pdf"><span class="ButtonGroup_leftGroup__FJAmS"><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg xmlns="http://www.w3.org/2000/svg" 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type="ghost" aria-label="Download Citation" tabindex="0" unselectable="on" data-testid="download-citation"><span class="ButtonGroup_leftGroup__FJAmS"><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg width="1em" height="1em" viewbox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"> <path d="M16 12H8C7.44772 12 7 12.4477 7 13C7 13.5523 7.44772 14 8 14H16C16.5523 14 17 13.5523 17 13C17 12.4477 16.5523 12 16 12Z" fill="currentColor"></path> <path d="M16 16H8C7.44772 16 7 16.4477 7 17C7 17.5523 7.44772 18 8 18H16C16.5523 18 17 17.5523 17 17C17 16.4477 16.5523 16 16 16Z" fill="currentColor"></path> <path d="M10 8H8C7.44772 8 7 8.44772 7 9C7 9.55228 7.44772 10 8 10H10C10.5523 10 11 9.55228 11 9C11 8.44772 10.5523 8 10 8Z" fill="currentColor"></path> <path d="M15 2C15 1.44772 14.5523 1 14 1C13.4477 1 13 1.44772 13 2V8C13 8.55228 13.4477 9 14 9H20C20.5523 9 21 8.55228 21 8C21 7.44772 20.5523 7 20 7H15V2Z" fill="currentColor"></path> <path fill-rule="evenodd" clip-rule="evenodd" d="M14.7071 1.29289C14.5196 1.10536 14.2652 1 14 1H6C4.34315 1 3 2.34315 3 4V20C3 21.6569 4.34315 23 6 23H18C19.6569 23 21 21.6569 21 20V8C21 7.73478 20.8946 7.48043 20.7071 7.29289L14.7071 1.29289ZM6 3H13.5858L19 8.41421V20C19 20.5523 18.5523 21 18 21H6C5.44772 21 5 20.5523 5 20V4C5 3.44772 5.44772 3 6 3Z" fill="currentColor"></path> </svg></span><span>下载引用</年代p一个n></span><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg width="1em" height="1em" viewbox="0 0 20 22"> <defs> <clippath> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </clippath> </defs> <g> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </g> </svg></span></a> <div class="ant-collapse ant-collapse-icon-position-end articleToolbarOtherFormats_otherFormatsCollapse__VHaOL"> <div class="ant-collapse-item"> <div class="ant-collapse-header" aria-expanded="false" aria-disabled="false" role="button" tabindex="0"> <div class="ant-collapse-expand-icon"> <span role="img" class="anticon ant-collapse-arrow"> <svg width="1em" height="1em" viewbox="0 0 32 32" fill="none"> <path fill-rule="evenodd" clip-rule="evenodd" d="M17.773 23.453l11.2-11.265a1.918 1.918 0 000-2.641 1.75 1.75 0 00-2.542 0l-9.929 10.101-9.933-10.1a1.75 1.75 0 00-2.542 0 1.919 1.919 0 000 2.64L15.23 23.453c.352.365.811.547 1.271.547.46 0 .92-.182 1.271-.547z" fill="currentColor"></path> </svg></span> </div> <span class="ant-collapse-header-text"><span><span role="img" class="anticon articleToolbarOtherFormats_atofDownloadIcon__YRM5T"> <svg width="1em" height="1em" viewbox="0 0 20 22"> <defs> <clippath> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </clippath> </defs> <g> <path fill-rule="evenodd" clip-rule="evenodd" d="M3,22a3,3,0,0,1-3-3V16a1,1,0,0,1,2,0v3a1,1,0,0,0,1,1H17a1,1,0,0,0,1-1V16a1,1,0,1,1,2,0v3a3,3,0,0,1-3,3Zm6.288-6.3-4-4a1,1,0,0,1,1.414-1.414L9,12.586V1a1,1,0,0,1,2,0V12.586l2.293-2.293a1,1,0,0,1,1.414,1.414l-4,4a1,1,0,0,1-1.423,0Z" fill="currentColor"></path> </g> </svg></span><span>下载其他格式</年代p一个n></span></span> </div> </div> </div> <a href="//www.newsama.com/reprints/7185131/confirm" class="ButtonGroup_buttonGroupWrapper__Rqhy0" role="button" type="ghost" aria-label="Order printed copies" tabindex="0" unselectable="on" target="_blank"><span class="ButtonGroup_leftGroup__FJAmS"><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg width="1em" height="1em" viewbox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"> <path fill-rule="evenodd" clip-rule="evenodd" d="M7.41667 1H18.6667C19.1269 1 19.5 1.44772 19.5 2V22C19.5 22.5523 19.1269 23 18.6667 23H7.41667C5.80584 23 4.5 21.433 4.5 19.5V4.5C4.5 2.567 5.80584 1 7.41667 1ZM6.16667 16.3368V4.5C6.16667 3.67157 6.72631 3 7.41667 3H17.8333V16H7.41667C6.96933 16 6.54552 16.1208 6.16667 16.3368ZM17.8333 18H7.41667C6.72631 18 6.16667 18.6716 6.16667 19.5C6.16667 20.3284 6.72631 21 7.41667 21H17.8333V18Z" fill="currentColor"></path> </svg></span><span>订单打印副本</年代p一个n></span><span role="img" class="anticon ButtonGroup_buttonGroupWrapperIcon__CFikg"> <svg width="1em" height="1em" viewbox="0 0 18 14"> <defs> <clippath id="a"> <path d="M10.293,13.707a1,1,0,0,1,0-1.414L14.586,8H1A1,1,0,0,1,1,6H14.586L10.293,1.707A1,1,0,0,1,11.707.293l6,6a1,1,0,0,1,.172.231h0l.01.019,0,.005.007.014.006.012,0,.008.008.018v0a1,1,0,0,1-.009.817l0,0L17.9,7.44l0,.009-.005.011-.007.013,0,.008-.008.015,0,.005-.01.016,0,0-.011.017,0,0-.012.018,0,0-.012.017,0,0L17.8,7.6l0,0-.011.014,0,.006-.01.012-.009.011-.006.007-.015.017h0l-.04.042-6,6a1,1,0,0,1-1.414,0Z" transform="translate(3 5)"></path> </clippath> </defs> <g transform="translate(-3 -5)"> <path d="M10.293,13.707a1,1,0,0,1,0-1.414L14.586,8H1A1,1,0,0,1,1,6H14.586L10.293,1.707A1,1,0,0,1,11.707.293l6,6a1,1,0,0,1,.172.231h0l.01.019,0,.005.007.014.006.012,0,.008.008.018v0a1,1,0,0,1-.009.817l0,0L17.9,7.44l0,.009-.005.011-.007.013,0,.008-.008.015,0,.005-.01.016,0,0-.011.017,0,0-.012.018,0,0-.012.017,0,0L17.8,7.6l0,0-.011.014,0,.006-.01.012-.009.011-.006.007-.015.017h0l-.04.042-6,6a1,1,0,0,1-1.414,0Z" transform="translate(3 5)" fill="currentColor"></path> </g> </svg></span></a> <div class="articleToolbarStatistics_articleToolbarStatistics__8_woN"> <div class="articleToolbarStatistics_column__R2GqD"> <div> <div> <span>的观点</年代p一个n> <div class="articleToolbarStatistics_amount__vJIDp"> 499年</d我v> </div> <div> <span>下载</年代p一个n> <div class="articleToolbarStatistics_amount__vJIDp"> 289年</d我v> </div> </div> </div> <div class="articleToolbarStatistics_column__R2GqD articleToolbarStatistics_dimensions__pe7pO"> <div> 引用</d我v> <span class="__dimensions_badge_embed__ articleToolbarStatistics_circle__p2dSc" data-doi="10.1155/2022/7185131" data-style="small_circle"></span> </div> </div> <ul class="articleToolbarSocials_articleToolbarSocials__g02HH"> <li><a href="https://twitter.com/share?url=https://doi.org/10.1155/2022/7185131&via=Hindawi" unselectable="on" title="" rel="noopener noreferrer" target="_blank"><span role="img" class="anticon"> <svg xmlns="http://www.w3.org/2000/svg" width="1em" height="1em" viewbox="126.444 2.281 589 589"> <circle 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