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直流电机的无抖振滑模控制的研究,实现了常规PID控制器,并对其性能进行了评估
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  • Modified Chattering Free Sliding Mode Control of DC motor.pdf
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内容介绍
<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/625fcbbb71c32b15964b5ac5/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625fcbbb71c32b15964b5ac5/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">International Journal of Mod<span class="_ _0"></span>ern Engineering Researc<span class="_ _0"></span>h (IJMER)<span class="_ _0"></span> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">www.ij<span class="_ _0"></span>mer.com<span class="ls1"> </span>Vol.3, Issue.3, May-June. 2013 <span class="_ _0"></span>pp-<span class="ls2">1419</span>-1423 ISSN: 2<span class="_ _0"></span>249-<span class="ls2">6645</span> </div><div class="t m0 x3 h2 y3 ff1 fs0 fc1 sc0 ls0 ws0"> </div><div class="t m0 x4 h2 y4 ff1 fs0 fc0 sc0 ls0 ws0">www.ij<span class="_ _0"></span>mer.com<span class="ls1"> </span> <span class="_ _1"></span>1419 | Page<span class="fc1"> </span></div><div class="t m0 x3 h3 y5 ff2 fs1 fc1 sc0 ls0 ws0"><span class="fc4 sc0"> </span></div><div class="t m0 x3 h3 y6 ff2 fs1 fc1 sc0 ls0 ws0"><span class="fc4 sc0"> </span></div><div class="t m0 x3 h4 y7 ff2 fs0 fc1 sc0 ls0 ws0"> </div><div class="t m0 x5 h5 y8 ff1 fs2 fc2 sc0 ls0 ws0">Arun Prasad K.M,</div><div class="t m0 x6 h6 y9 ff1 fs3 fc2 sc0 ls0 ws0"> 1</div><div class="t m0 x7 h5 y8 ff1 fs2 fc2 sc0 ls0 ws0"> Bindu M. K<span class="_ _2"></span>rishna,</div><div class="t m0 x8 h6 y9 ff1 fs3 fc2 sc0 ls0 ws0"> 2</div><div class="t m0 x9 h5 y8 ff1 fs2 fc2 sc0 ls0 ws0"> Us<span class="_ _2"></span>ha Nair</div><div class="t m0 xa h6 y9 ff1 fs3 fc2 sc0 ls0 ws0">3</div><div class="t m0 xb h5 y8 ff1 fs2 fc2 sc0 ls0 ws0"> </div><div class="t m0 xc h7 ya ff3 fs4 fc1 sc0 ls0 ws0">13</div><div class="t m0 xd h8 yb ff3 fs0 fc1 sc0 ls0 ws0">Division of Electrical E<span class="_ _0"></span>ngg, Schoo<span class="_ _0"></span>l of Engineering, Cochin University o<span class="_ _0"></span>f science <span class="_ _0"></span>&amp; Tec<span class="_ _0"></span>hnology, Koch<span class="_ _0"></span>i, India </div><div class="t m0 xe h7 yc ff3 fs4 fc1 sc0 ls0 ws0">2</div><div class="t m0 xf h8 yd ff3 fs0 fc1 sc0 ls0 ws0">Sophisticated<span class="_ _0"></span> Test and Instrumentatio<span class="_ _0"></span>n centre, <span class="_ _0"></span>Cochin University of S<span class="_ _0"></span>cience and<span class="_ _0"></span> Technology<span class="_ _0"></span>, Kochi, India </div><div class="t m0 x3 h8 ye ff3 fs0 fc1 sc0 ls0 ws0"> </div><div class="t m0 x10 h9 yf ff4 fs5 fc1 sc0 ls0 ws0">Abstract:<span class="fs0"> <span class="_ _3"></span><span class="ff3">This <span class="_ _3"></span>paper <span class="_ _3"></span>is <span class="_ _3"></span>co<span class="_ _0"></span>ncerned<span class="_ _0"></span> <span class="_ _3"></span>with <span class="_ _3"></span>the <span class="_ _3"></span>design <span class="_ _4"></span>of <span class="_ _3"></span>a <span class="_ _4"></span>Modified <span class="_ _4"></span>Sliding <span class="_ _3"></span>Mod<span class="_ _0"></span>e <span class="_ _3"></span>Controller <span class="_ _3"></span>for <span class="_ _4"></span>the <span class="_ _3"></span>position <span class="_ _4"></span>control <span class="_ _3"></span>of <span class="_ _3"></span>DC </span></span></div><div class="t m0 x10 h8 y10 ff3 fs0 fc1 sc0 ls0 ws0">Motor. <span class="_ _4"></span>The <span class="_ _5"></span>DC <span class="_ _4"></span>motor <span class="_ _5"></span>can <span class="_ _4"></span>be <span class="_ _4"></span>mod<span class="_ _0"></span>eled <span class="_ _4"></span>as <span class="_ _4"></span>a <span class="_ _5"></span>linear <span class="_ _4"></span>time <span class="_ _5"></span>invariant <span class="_ _4"></span>single <span class="_ _5"></span>input <span class="_ _3"></span>sing<span class="_ _0"></span>le <span class="_ _4"></span>output <span class="_ _5"></span>(SISO) <span class="_ _4"></span>system. <span class="_ _4"></span>In<span class="_ _0"></span> <span class="_ _4"></span>this <span class="_ _4"></span>pap<span class="_ _0"></span>er<span class="_ _2"></span> </div><div class="t m0 x10 h8 y11 ff3 fs0 fc1 sc0 ls3 ws0">sy<span class="ls0">nthesis a<span class="_ _0"></span>nd a<span class="_ _0"></span>nalysis of <span class="_ _0"></span>positio<span class="_ _0"></span>n co<span class="_ _0"></span>ntrol of <span class="_ _0"></span>a <span class="_ _0"></span>DC motor <span class="_ _0"></span>using <span class="_ _0"></span>chattering<span class="_ _0"></span> free <span class="_ _0"></span>sliding mo<span class="_ _0"></span>de co<span class="_ _0"></span>ntroller an<span class="_ _0"></span>d co<span class="_ _0"></span>nventional PID </span></div><div class="t m0 x10 h8 y12 ff3 fs0 fc1 sc0 ls0 ws0">controllers <span class="_ _5"> </span>are <span class="_ _5"> </span>carried <span class="_ _5"> </span>ou<span class="_ _0"></span>t <span class="_ _5"></span>and <span class="_ _5"> </span>their <span class="_ _5"> </span>performance<span class="_ _0"></span> <span class="_ _5"></span>is <span class="_ _5"></span>evaluated. <span class="_ _5"> </span>The <span class="_ _5"></span>performance <span class="_ _5"> </span>o<span class="_ _0"></span>f <span class="_ _5"></span>modified <span class="_ _5"></span>sliding <span class="_ _5"> </span>mode <span class="_ _5"> </span>contro<span class="_ _0"></span>ller <span class="_ _4"></span>is<span class="_ _0"></span> </div><div class="t m0 x10 h8 y13 ff3 fs0 fc1 sc0 ls0 ws0">superior to con<span class="_ _0"></span>ventional PI<span class="_ _0"></span>D controllers even in the p<span class="_ _0"></span>resence of disturba<span class="_ _0"></span>nces. </div><div class="t m0 x10 h8 y14 ff3 fs0 fc1 sc0 ls0 ws0"> </div><div class="t m0 x10 h9 y15 ff4 fs5 fc1 sc0 ls0 ws0">Keyword: <span class="ff3 fs0">chattering free Sliding mo<span class="_ _0"></span>de controller; DC Motor; PID <span class="_ _0"></span>Controller<span class="_ _0"></span> </span></div><div class="t m0 x10 h8 y16 ff3 fs0 fc1 sc0 ls0 ws0"> </div><div class="t m0 x11 ha y17 ff2 fs5 fc1 sc0 ls4 ws0">I.<span class="ff5 ls0"> <span class="_ _6"> </span><span class="ff2">I<span class="fs3">NTRODUCTION<span class="fs6"> <span class="fs0"> </span></span></span></span></span></div><div class="t m0 xe h2 y18 ff1 fs0 fc1 sc0 ls0 ws0">The <span class="_ _1"></span>proportional-integral-deri<span class="_ _0"></span>vative <span class="_ _1"></span>(PID) <span class="_ _0"></span>co<span class="_ _0"></span>ntroller <span class="_ _1"></span>is <span class="_ _0"></span>extensively <span class="_ _1"></span>used <span class="_ _0"></span>in <span class="_ _1"></span>many <span class="_ _1"></span>industrial <span class="_ _1"></span>control<span class="_ _0"></span> <span class="_ _1"></span>applications <span class="_ _1"></span>[1] </div><div class="t m0 x10 h2 y19 ff1 fs0 fc1 sc0 ls0 ws0">due to <span class="_ _0"></span>is <span class="_ _0"></span>simplicity a<span class="_ _0"></span>nd effect<span class="_ _0"></span>iveness <span class="_ _0"></span>in i<span class="_ _0"></span>mplementation. <span class="_ _0"></span>The <span class="_ _0"></span>three contro<span class="_ _0"></span>ller par<span class="_ _0"></span>ameters, <span class="_ _0"></span>proportional gai<span class="_ _0"></span>n Kp, <span class="_ _0"></span>Integral <span class="_ _0"></span>gai<span class="_ _3"></span><span class="ls5">n </span></div><div class="t m0 x10 h2 y1a ff1 fs0 fc1 sc0 ls0 ws0">Ki, <span class="_ _1"></span>and <span class="_ _1"></span>derivative <span class="_ _1"></span>gai<span class="_ _0"></span>n <span class="_ _1"></span>Kd, <span class="_ _1"></span>are <span class="_ _1"></span>usuall<span class="_ _0"></span>y <span class="_ _1"></span>fixed. <span class="_ _1"></span>T<span class="_ _0"></span>he <span class="_ _1"></span>disadvantage <span class="_ _1"></span>o<span class="_ _0"></span>f <span class="_ _0"></span>P<span class="_ _0"></span>ID <span class="_ _1"></span>controller <span class="_ _1"></span>is<span class="_ _0"></span> <span class="_ _1"></span>poor <span class="_ _1"></span>capabilit<span class="_ _0"></span>y <span class="_ _0"></span>o<span class="_ _0"></span>f <span class="_ _1"></span>dealing<span class="_ _1"></span> <span class="_ _3"></span>with <span class="_ _1"></span>system </div><div class="t m0 x10 h2 y1b ff1 fs0 fc1 sc0 ls0 ws0">uncertainty, <span class="_ _1"></span>i.e., <span class="_ _0"></span>p<span class="_ _0"></span>arameter <span class="_ _1"></span>variations <span class="_ _0"></span>a<span class="_ _0"></span>nd <span class="_ _0"></span>exter<span class="_ _0"></span>nal <span class="_ _1"></span>disturbance. <span class="_ _1"></span>In <span class="_ _0"></span>recent <span class="_ _1"></span>years, <span class="_ _1"></span>there <span class="_ _1"></span>has <span class="_ _1"></span>been <span class="_ _0"></span>exte<span class="_ _0"></span>nsive <span class="_ _0"></span>r<span class="_ _0"></span>esearch <span class="_ _1"></span>interest<span class="_ _1"></span> <span class="_ _1"></span>in </div><div class="t m0 x10 h2 y1c ff1 fs0 fc1 sc0 ls0 ws0">robust control syste<span class="_ _0"></span>ms, where the fuzzy logic, <span class="_ _0"></span>neural network and sliding<span class="_ _1"></span>-mode based controller<span class="_ _0"></span>s [1] [2]. </div><div class="t m0 xe h2 y1d ff1 fs0 fc1 sc0 ls0 ws0">Sliding <span class="_ _3"></span>mode <span class="_ _1"></span>co<span class="_ _0"></span>ntrol <span class="_ _1"></span>(<span class="_ _0"></span>SMC) <span class="_ _1"></span>is <span class="_ _3"></span>one <span class="_ _3"></span>of <span class="_ _1"></span>the <span class="_ _3"></span>popular <span class="_ _3"></span>strategies <span class="_ _3"></span>to <span class="_ _3"></span>deal <span class="_ _3"></span>with <span class="_ _1"></span>uncertain <span class="_ _3"></span>control <span class="_ _3"></span>systems <span class="_ _1"></span>[3<span class="_ _0"></span>]. <span class="_ _1"></span>T<span class="_ _0"></span>he <span class="_ _1"></span>main </div><div class="t m0 x10 h2 y1e ff1 fs0 fc1 sc0 ls0 ws0">feature <span class="_ _3"></span>of <span class="_ _1"></span>SMC <span class="_ _3"></span>is <span class="_ _1"></span>t<span class="_ _0"></span>he <span class="_ _1"></span>r<span class="_ _0"></span>obustness <span class="_ _1"></span>agai<span class="_ _0"></span>nst <span class="_ _3"></span>parameter <span class="_ _3"></span>variations <span class="_ _1"></span>and <span class="_ _3"></span>external <span class="_ _3"></span>disturbances <span class="_ _3"></span>and <span class="_ _3"></span>is <span class="_ _3"></span>widely <span class="_ _1"></span>used<span class="_ _0"></span> <span class="_ _1"></span>to<span class="_ _0"></span> <span class="_ _1"></span>o<span class="_ _0"></span>btain <span class="_ _1"></span>good </div><div class="t m0 x10 h2 y1f ff1 fs0 fc1 sc0 ls0 ws0">dynamic <span class="_ _0"></span>p<span class="_ _0"></span>erformance <span class="_ _1"></span>of <span class="_ _1"></span>contro<span class="_ _0"></span>l <span class="_ _1"></span>systems. <span class="_ _0"></span>Var<span class="_ _0"></span>ious <span class="_ _1"></span>applications <span class="_ _1"></span>of <span class="_ _1"></span>SMC <span class="_ _1"></span>have <span class="_ _1"></span>been <span class="_ _0"></span>co<span class="_ _0"></span>nducted,<span class="_ _0"></span> <span class="_ _0"></span>such <span class="_ _1"></span>as <span class="_ _1"></span>robotic <span class="_ _1"></span>manipulators, </div><div class="t m0 x10 h2 y20 ff1 fs0 fc1 sc0 ls0 ws0">aircrafts, DC motors, chaotic s<span class="_ _0"></span>ystems etc. [4] [5<span class="_ _0"></span>] [6]. <span class="_ _0"></span> </div><div class="t m0 xe h2 y21 ff1 fs0 fc1 sc0 ls0 ws0">The <span class="_ _5"> </span>finite <span class="_ _7"> </span>speed <span class="_ _5"> </span>o<span class="_ _0"></span>f <span class="_ _5"> </span>switch<span class="_ _0"></span>ing <span class="_ _5"> </span>device<span class="_ _0"></span>s <span class="_ _5"> </span>involved <span class="_ _7"> </span>in <span class="_ _7"> </span>SMC <span class="_ _7"> </span>cause <span class="_ _5"> </span>the <span class="_ _7"> </span>phenome<span class="_ _0"></span>non <span class="_ _5"> </span>of <span class="_ _7"> </span>chattering <span class="_ _7"> </span>and <span class="_ _5"> </span>it <span class="_ _7"> </span>affects <span class="_ _7"> </span>the<span class="_ _0"></span> </div><div class="t m0 x10 h2 y22 ff1 fs0 fc1 sc0 ls0 ws0">performance <span class="_ _1"></span>of <span class="_ _1"></span>t<span class="_ _0"></span>he <span class="_ _1"></span>system <span class="_ _1"></span>ad<span class="_ _0"></span>versely. <span class="_ _1"></span>Cha<span class="_ _0"></span>ttering <span class="_ _1"></span>is <span class="_ _1"></span>a <span class="_ _3"></span>phenomenon <span class="_ _1"></span>o<span class="_ _0"></span>f <span class="_ _1"></span>high <span class="_ _1"></span>freque<span class="_ _0"></span>ncy <span class="_ _1"></span>switching <span class="_ _1"></span>o<span class="_ _0"></span>f <span class="_ _1"></span>the <span class="_ _1"></span>co<span class="_ _0"></span>ntrol <span class="_ _1"></span>action <span class="_ _3"></span>which </div><div class="t m0 x10 h2 y23 ff1 fs0 fc1 sc0 ls0 ws0">causes high freque<span class="_ _0"></span>ncy oscillat<span class="_ _0"></span>ions in t<span class="_ _0"></span>he output, heati<span class="_ _0"></span>ng up <span class="_ _0"></span>of electrical circui<span class="_ _0"></span>ts and pre<span class="_ _0"></span>mature wear in <span class="_ _0"></span>actuators. <span class="_ _0"></span>A modifi<span class="_ _1"></span>ed </div><div class="t m0 x10 h2 y24 ff1 fs0 fc1 sc0 ls0 ws0">sliding mode control tech<span class="_ _0"></span>nique can red<span class="_ _0"></span>uce the chattering without sacri<span class="_ _0"></span>ficing the robustness<span class="_ _0"></span> of the system.<span class="_ _1"></span> </div><div class="t m0 xe h2 y25 ff1 fs0 fc1 sc0 ls0 ws0">DC <span class="_ _7"> </span>motor <span class="_ _7"> </span>are <span class="_ _7"> </span>generally <span class="_ _7"> </span>cont<span class="_ _0"></span>rolled <span class="_ _7"> </span>by <span class="_ _7"> </span>conventional <span class="_ _7"> </span>Propo<span class="_ _0"></span>rtional <span class="_ _8"> </span><span class="ff6">&#8211;</span> <span class="_ _7"> </span>Integral <span class="_ _7"> </span><span class="ff6">&#8211;</span> <span class="_ _7"> </span>Der<span class="_ _0"></span>ivative <span class="_ _7"> </span>(PID) <span class="_ _7"> </span>controllers. <span class="_ _7"> </span>For </div><div class="t m0 x10 h2 y26 ff1 fs0 fc1 sc0 ls0 ws0">effective <span class="_ _0"></span>i<span class="_ _0"></span>mplementation <span class="_ _0"></span>of <span class="_ _0"></span>P<span class="_ _0"></span>ID <span class="_ _0"></span>co<span class="_ _0"></span>ntroller <span class="_ _0"></span>it <span class="_ _0"></span>is<span class="_ _0"></span> <span class="_ _1"></span><span class="ff6">necessary <span class="_ _0"></span>to <span class="_ _1"></span>know <span class="_ _0"></span>system&#8217;s <span class="_ _1"></span>mathematical <span class="_ _1"></span>model <span class="_ _0"></span>for <span class="_ _0"></span>tu<span class="_ _0"></span>ning <span class="_ _0"></span>PID <span class="_ _0"></span>p<span class="_ _0"></span>arameters. </span></div><div class="t m0 x10 h2 y27 ff1 fs0 fc1 sc0 ls0 ws0">However, <span class="_ _9"> </span>it <span class="_ _9"> </span>has <span class="_ _a"> </span>been <span class="_ _a"> </span>known <span class="_ _9"> </span>that <span class="_ _a"> </span>co<span class="_ _0"></span>nventional <span class="_ _a"> </span>P<span class="_ _0"></span>ID <span class="_ _9"> </span>controllers <span class="_ _a"> </span>generall<span class="_ _0"></span>y <span class="_ _a"> </span>do <span class="_ _9"> </span>not <span class="_ _9"> </span>work <span class="_ _9"> </span>well <span class="_ _a"> </span>for <span class="_ _a"> </span>non<span class="_ _1"></span>-linear <span class="_ _9"> </span>systems, </div><div class="t m0 x10 h2 y28 ff1 fs0 fc1 sc0 ls0 ws0">particularly <span class="_ _1"></span>complex <span class="_ _1"></span>and<span class="_ _0"></span> <span class="_ _1"></span>vague <span class="_ _1"></span>syste<span class="_ _0"></span>ms <span class="_ _1"></span>that <span class="_ _1"></span>have <span class="_ _3"></span>no <span class="_ _1"></span>precise <span class="_ _3"></span>mathematic<span class="_ _0"></span>al <span class="_ _1"></span>mod<span class="_ _0"></span>els. <span class="_ _1"></span>To <span class="_ _1"></span>overco<span class="_ _0"></span>me <span class="_ _1"></span>these <span class="_ _1"></span>difficultie<span class="_ _0"></span>s, <span class="_ _1"></span>various </div><div class="t m0 x10 h2 y29 ff1 fs0 fc1 sc0 ls0 ws0">types of modified conventio<span class="_ _0"></span>nal PID co<span class="_ _0"></span>ntrollers such as auto<span class="_ _0"></span>-tuning and adap<span class="_ _0"></span>tive PID controllers <span class="_ _0"></span>were developed lately [7<span class="_ _0"></span>]. <span class="_ _0"></span> </div><div class="t m0 xe h2 y2a ff1 fs0 fc1 sc0 ls0 ws0">This <span class="_ _5"> </span>paper <span class="_ _5"> </span>is <span class="_ _5"> </span>focused <span class="_ _5"> </span>on <span class="_ _5"> </span>t<span class="_ _0"></span>he <span class="_ _5"> </span>performance <span class="_ _5"> </span>co<span class="_ _0"></span>mparison <span class="_ _5"></span>of <span class="_ _5"> </span>the <span class="_ _5"> </span>of <span class="_ _5"></span>the <span class="_ _5"> </span>modified <span class="_ _5"> </span>slidi<span class="_ _0"></span>n<span class="_ _0"></span>g <span class="_ _5"> </span>mode <span class="_ _5"> </span>contr<span class="_ _0"></span>oller <span class="_ _5"> </span>with <span class="_ _5"></span>PID </div><div class="t m0 x10 h2 y2b ff1 fs0 fc1 sc0 ls0 ws0">controller for the position cont<span class="_ _0"></span>rol of a DC motor<span class="_ _0"></span> </div><div class="t m0 x10 h2 y2c ff1 fs0 fc1 sc0 ls0 ws0"> </div><div class="t m0 x12 ha y2d ff2 fs5 fc1 sc0 ls4 ws0">II.<span class="ff5 ls0"> <span class="_ _6"> </span><span class="ff2">M<span class="fs3">ATHEMATICAL </span>M<span class="fs3">ODELING </span>O<span class="fs3 ls6">F </span>A<span class="fs3"> </span>D<span class="fs3 ls7">C </span>M<span class="fs3">OTOR <span class="_ _2"></span><span class="fs5"> </span></span></span></span></div><div class="t m0 xe h2 y2e ff1 fs0 fc1 sc0 ls0 ws0">DC <span class="_ _5"></span>motors <span class="_ _4"></span>are <span class="_ _5"> </span>widely <span class="_ _5"></span>used <span class="_ _4"></span>i<span class="_ _0"></span>n <span class="_ _5"></span>various <span class="_ _5"></span>industrial <span class="_ _4"></span>and <span class="_ _5"> </span>electro<span class="_ _0"></span>nic <span class="_ _4"></span>eq<span class="_ _0"></span>uipment <span class="_ _5"></span>where <span class="_ _4"></span>the <span class="_ _5"></span>position <span class="_ _4"></span>co<span class="_ _0"></span>ntrol <span class="_ _5"> </span>with <span class="_ _4"></span>high </div><div class="t m0 x10 h2 y2f ff1 fs0 fc1 sc0 ls0 ws0">accuracy <span class="_ _3"></span>is <span class="_ _4"></span>required. <span class="_ _4"></span>T<span class="_ _0"></span>he <span class="_ _3"></span>elect<span class="_ _0"></span>ric <span class="_ _3"></span>cir<span class="_ _0"></span>cuit <span class="_ _3"></span>of <span class="_ _4"></span>the <span class="_ _4"></span>armature <span class="_ _3"></span>a<span class="_ _0"></span>nd <span class="_ _3"></span>the <span class="_ _4"></span>free <span class="_ _4"></span>bod<span class="_ _0"></span>y <span class="_ _3"></span>diagram <span class="_ _3"></span>o<span class="_ _0"></span>f <span class="_ _3"></span>the <span class="_ _4"></span>rotor <span class="_ _4"></span>are <span class="_ _4"></span>shown <span class="_ _3"></span>in <span class="_ _4"></span>fig. <span class="_ _3"></span>1<span class="_ _0"></span>. <span class="_ _4"></span>The </div><div class="t m0 x10 h2 y30 ff1 fs0 fc1 sc0 ls0 ws0">desired <span class="_ _0"></span>speed <span class="_ _0"></span>is <span class="_ _0"></span>tracked <span class="_ _0"></span>according <span class="_ _0"></span>to <span class="_ _0"></span>the s<span class="_ _0"></span>haft p<span class="_ _0"></span>osition <span class="_ _0"></span>of t<span class="_ _0"></span>he <span class="_ _0"></span>motor. <span class="_ _0"></span>A <span class="_ _0"></span>single co<span class="_ _0"></span>ntroller <span class="_ _0"></span>is <span class="_ _0"></span>required <span class="_ _0"></span>to <span class="_ _0"></span>control <span class="_ _0"></span>the <span class="_ _0"></span>positio<span class="_ _3"></span>n as </div><div class="t m0 x10 h2 y31 ff1 fs0 fc1 sc0 ls0 ws0">well <span class="_ _1"></span>as <span class="_ _3"></span>the <span class="_ _3"></span>spee<span class="_ _0"></span>d <span class="_ _3"></span>of <span class="_ _1"></span>t<span class="_ _0"></span>he <span class="_ _3"></span>motor. <span class="_ _3"></span>Th<span class="_ _0"></span>e <span class="_ _3"></span>controller <span class="_ _3"></span>is <span class="_ _1"></span>selected <span class="_ _4"></span>so <span class="_ _3"></span>that <span class="_ _1"></span>t<span class="_ _0"></span>he <span class="_ _1"></span>er<span class="_ _0"></span>ror <span class="_ _3"></span>between <span class="_ _3"></span>the <span class="_ _3"></span>system <span class="_ _3"></span>output <span class="_ _3"></span>and <span class="_ _3"></span>reference <span class="_ _3"></span>signal </div><div class="t m0 x10 h2 y32 ff1 fs0 fc1 sc0 ls0 ws0">eventually <span class="_ _5"> </span>te<span class="_ _0"></span>nds <span class="_ _5"> </span>to <span class="_ _5"> </span>it<span class="_ _0"></span>s <span class="_ _7"> </span>minimum <span class="_ _5"> </span>value, <span class="_ _5"> </span>id<span class="_ _0"></span>eally <span class="_ _5"> </span>zero. <span class="_ _7"> </span>The <span class="_ _5"> </span>r<span class="_ _0"></span>eference <span class="_ _5"> </span>signal <span class="_ _7"> </span>determines<span class="_ _0"></span> <span class="_ _5"> </span>the <span class="_ _5"> </span>desire<span class="_ _0"></span>d <span class="_ _5"> </span>po<span class="_ _0"></span>sition <span class="_ _5"> </span>and/or <span class="_ _7"> </span>speed. </div><div class="t m0 x10 h2 y33 ff1 fs0 fc1 sc0 ls0 ws0">Depending on type, a DC <span class="_ _0"></span>motor may be controlle<span class="_ _1"></span>d by varying the input voltage or field c<span class="_ _0"></span>urrent. <span class="_ _0"></span> </div><div class="t m0 xe h2 y34 ff1 fs0 fc1 sc0 ls0 ws0">Here the <span class="_ _0"></span>variation <span class="_ _0"></span>of i<span class="_ _0"></span>nput <span class="_ _0"></span>voltage is <span class="_ _0"></span>used <span class="_ _0"></span>as the<span class="_ _0"></span> control <span class="_ _0"></span>para<span class="_ _0"></span>meter for <span class="_ _0"></span>the posi<span class="_ _0"></span>tion co<span class="_ _0"></span>ntrol. A <span class="_ _0"></span>constant d<span class="_ _0"></span>c <span class="_ _0"></span>voltage i<span class="_ _0"></span>s </div><div class="t m0 x10 h2 y35 ff1 fs0 fc1 sc0 ls0 ws0">selected <span class="_ _1"></span>as <span class="_ _3"></span>a <span class="_ _1"></span>re<span class="_ _0"></span>ference <span class="_ _1"></span>signal <span class="_ _3"></span>to <span class="_ _1"></span>ob<span class="_ _0"></span>tain <span class="_ _1"></span>the <span class="_ _1"></span>des<span class="_ _0"></span>ired <span class="_ _1"></span>p<span class="_ _0"></span>osition <span class="_ _1"></span>of <span class="_ _3"></span>the <span class="_ _3"></span>m<span class="_ _2"></span>otor. <span class="_ _3"></span>However, <span class="_ _1"></span>t<span class="_ _1"></span>he <span class="_ _3"></span>method <span class="_ _3"></span>works <span class="_ _1"></span>succe<span class="_ _0"></span>ssfully <span class="_ _1"></span>for <span class="_ _3"></span>any </div><div class="t m0 x10 h2 y36 ff1 fs0 fc1 sc0 ls0 ws0">reference sig<span class="_ _0"></span>nal, particularl<span class="_ _0"></span>y for <span class="_ _0"></span>any stepwise ti<span class="_ _0"></span>me<span class="_ _0"></span>-continuous <span class="_ _0"></span>function. T<span class="_ _0"></span>his signal <span class="_ _0"></span>may b<span class="_ _0"></span>e a <span class="_ _0"></span>periodic <span class="_ _0"></span>signal or an<span class="_ _0"></span>y signal <span class="_ _0"></span>to </div><div class="t m0 x10 h2 y37 ff1 fs0 fc1 sc0 ls0 ws0">get a desired shaft position, i.e<span class="_ _0"></span>. a desired<span class="_ _0"></span> angle between 0 and 3<span class="_ _0"></span>60 degrees from a virtual<span class="_ _1"></span> horizontal line.<span class="_ _0"></span> </div><div class="t m0 x13 h2 y38 ff1 fs0 fc1 sc0 ls0 ws0"> </div><div class="t m0 x10 h2 y39 ff1 fs0 fc1 sc0 ls0 ws0">The dynamics of a DC moto<span class="_ _0"></span>r is given in eqn<span class="_ _0"></span> (1)-eqn (5)<span class="_ _0"></span> </div><div class="c x13 y3a w2 hb"><div class="t m1 x14 hc y3b ff3 fs7 fc1 sc0 ls0 ws0">b</div><div class="t m1 x15 hc y3c ff3 fs7 fc1 sc0 ls0 ws0">a</div><div class="t m1 x16 hc y3b ff3 fs7 fc1 sc0 ls0 ws0">a<span class="_ _b"></span>a<span class="_ _c"></span>a<span class="_ _d"></span>t</div><div class="t m1 x2 hd y3d ff3 fs8 fc1 sc0 ls0 ws0">E</div><div class="t m1 x17 hd y3e ff3 fs8 fc1 sc0 ls8 ws0">dt</div><div class="t m1 xc hd y3f ff3 fs8 fc1 sc0 ls8 ws0">dI</div><div class="t m1 x18 hd y3d ff3 fs8 fc1 sc0 ls0 ws0">L<span class="_ _e"></span>I<span class="_ _f"></span>R<span class="_ _10"></span>V<span class="_ _11"> </span><span class="ff7">&#61483;<span class="_ _12"></span>&#61483;<span class="_ _13"></span>&#61501;</span></div></div><div class="t m0 x19 h2 y40 ff1 fs0 fc1 sc0 ls0 ws0"> <span class="_"> </span> <span class="_ _14"> </span>(1) <span class="_ _15"> </span> <span class="_ _14"> </span> </div><div class="c x1a y3a w3 he"><div class="t m2 x1b hf y41 ff3 fs9 fc1 sc0 ls0 ws0">L</div><div class="t m2 x1c h10 y42 ff3 fsa fc1 sc0 ls0 ws0">T<span class="_ _16"></span>B</div><div class="t m2 x1d h10 y43 ff3 fsa fc1 sc0 ls9 ws0">dt</div><div class="t m2 x1e h10 y44 ff3 fsa fc1 sc0 ls0 ws0">d</div><div class="t m2 x1f h10 y42 ff3 fsa fc1 sc0 ls0 ws0">J<span class="_ _17"></span>T<span class="_ _18"> </span><span class="ff7">&#61485;<span class="_ _19"></span>&#61501;</span></div><div class="t m3 x20 h11 y42 ff7 fsb fc1 sc0 ls0 ws0">&#61559;</div><div class="t m3 x21 h11 y44 ff7 fsb fc1 sc0 ls0 ws0">&#61559;</div></div><div class="t m0 x22 h2 y40 ff1 fs0 fc1 sc0 ls0 ws0"> <span class="_ _1a"> </span> <span class="_ _14"> </span>(2<span class="_ _0"></span>) </div><div class="c x13 y45 w4 h12"><div class="t m4 x23 h13 y46 ff3 fsc fc1 sc0 ls0 ws0">a<span class="_ _1b"></span>t</div><div class="t m4 x24 h14 y47 ff3 fsd fc1 sc0 ls0 ws0">I<span class="_ _1c"></span>K<span class="_ _1d"></span>T<span class="_ _1e"> </span><span class="ff7">&#61501;</span></div></div><div class="t m0 x25 h2 y48 ff1 fs0 fc1 sc0 ls0 ws0"> <span class="_"> </span> <span class="_ _14"> </span> <span class="_ _14"> </span> <span class="_ _1f"> </span>(3) <span class="_ _15"> </span> </div><div class="c x26 y45 w5 h12"><div class="t m5 x27 h15 y49 ff7 fse fc1 sc0 ls0 ws0">&#61559;</div><div class="t m6 x28 h16 y4a ff3 fsf fc1 sc0 ls0 ws0">b<span class="_ _20"></span>b</div><div class="t m6 x29 h17 y49 ff3 fs10 fc1 sc0 ls0 ws0">K<span class="_ _21"></span>E<span class="_ _22"> </span><span class="ff7">&#61501;</span></div></div><div class="t m0 x2a h2 y48 ff1 fs0 fc1 sc0 ls0 ws0"> <span class="_ _23"> </span> <span class="_ _14"> </span> <span class="_ _1f"> </span>(4) </div><div class="c x13 y4b w6 h18"><div class="t m7 x2b h19 y4c ff3 fs11 fc1 sc0 lsa ws0">dt</div><div class="t m7 x2c h19 y4d ff3 fs11 fc1 sc0 ls0 ws0">d</div><div class="t m8 x2d h1a y4d ff7 fs12 fc1 sc0 ls0 ws0">&#61553;</div><div class="t m8 x2e h1a y4e ff7 fs12 fc1 sc0 ls0 ws0">&#61559;</div><div class="t m7 x2f h1b y4e ff7 fs11 fc1 sc0 ls0 ws0">&#61501;</div></div><div class="t m0 x30 h2 y4f ff1 fs0 fc1 sc0 ls0 ws0"> <span class="_ _24"> </span> <span class="_ _14"> </span> <span class="_ _1f"> </span> <span class="_ _14"> </span>(5<span class="_ _0"></span>) </div><div class="t m0 x13 h2 y50 ff1 fs0 fc1 sc0 ls0 ws0">Where </div><div class="c x31 y51 w7 h1c"><div class="t m0 x32 h3 y52 ff2 fs1 fc3 sc0 ls0 ws0">Modified Chattering<span class="_ _0"></span> Free Sliding Mode Control of <span class="_ _0"></span>DC </div><div class="t m0 x33 h3 y53 ff2 fs1 fc3 sc0 ls0 ws0">Motor<span class="ff8 fs5"> </span></div></div></div><div class="pi" data-data='{"ctm":[1.611562,0.000000,0.000000,1.611562,0.000000,0.000000]}'></div></div> </body> </html>
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