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PI controller design and Observer Design for magnetic ball suspended in the air
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<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/626fad28d973ef42a41f38c3/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/626fad28d973ef42a41f38c3/bg1.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x1 h3 y2 ff1 fs0 fc0 sc0 ls0 ws0">EXPERIMENT NO. 3</div><div class="t m0 x2 h4 y3 ff1 fs1 fc0 sc0 ls0 ws0">EXPERIMENT: <span class="ff2">To simulate pole placement control of a magnetic ball suspended in air</span></div><div class="t m0 x2 h4 y4 ff1 fs1 fc0 sc0 ls0 ws0">OBJECT:</div><div class="t m0 x2 h4 y5 ff2 fs1 fc0 sc0 ls0 ws0">(A): To simulate pole placement control of a magnetic ball system suspended in air.</div><div class="t m0 x2 h4 y6 ff2 fs1 fc0 sc0 ls0 ws0">(B): To design an observer for the estimation of unobservable states of the magnetic ball system.</div><div class="t m0 x2 h4 y7 ff1 fs1 fc0 sc0 ls0 ws0">(A): To simulate pole placement control of a magnetic ball system suspended in air</div><div class="t m0 x2 h4 y8 ff2 fs1 fc0 sc0 ls0 ws0">When<span class="_ _0"></span> <span class="_ _0"></span>we<span class="_ _0"></span> <span class="_ _0"></span>can't<span class="_ _0"></span> <span class="_ _0"></span>measure<span class="_ _0"></span> <span class="_ _0"></span>all<span class="_ _0"></span> <span class="_ _0"></span>the<span class="_ _0"></span> <span class="_ _0"></span>states<span class="_ _0"></span> <span class="_ _0"></span>x<span class="_ _0"></span> <span class="_ _0"></span>(often<span class="_ _0"></span> <span class="_ _0"></span>the<span class="_ _0"></span> <span class="_ _0"></span>case<span class="_ _0"></span> <span class="_ _0"></span>i<span class="_ _0"></span>n <span class="_ _1"></span>practice),<span class="_ _0"></span> <span class="_ _0"></span>we<span class="_ _0"></span> <span class="_ _0"></span>can<span class="_ _0"></span> <span class="_ _0"></span>build<span class="_ _0"></span> <span class="_ _0"></span>an<span class="_ _0"></span> <span class="_ _0"></span>observer<span class="_ _0"></span> <span class="_ _0"></span>to<span class="_ _0"></span> <span class="_ _0"></span>estimate</div><div class="t m0 x2 h4 y9 ff2 fs1 fc0 sc0 ls0 ws0">them, while measuring only the output y = C x. </div><div class="t m0 x2 h5 ya ff2 fs1 fc0 sc0 ls0 ws0">Consider<span class="_ _0"></span> <span class="_ _0"></span>the<span class="_ _0"></span> <span class="_ _0"></span>magnetic<span class="_ _0"></span> <span class="_ _0"></span>ball<span class="_ _0"></span> <span class="_ _0"></span>example,<span class="_ _0"></span><span class="ff3">.<span class="_ _0"></span> <span class="_ _0"></span>The<span class="_ _0"></span> <span class="_ _0"></span>c<span class="_ _0"></span>urrent <span class="_ _0"></span>t<span class="_ _0"></span>hrough<span class="_ _0"></span> <span class="_ _0"></span>the <span class="_ _1"></span>coils<span class="_ _0"></span> <span class="_ _0"></span>induces<span class="_ _0"></span> <span class="_ _0"></span>a<span class="_ _0"></span> <span class="_ _0"></span>magne&#58899;c<span class="_ _0"></span> <span class="_ _0"></span>force<span class="_ _0"></span> <span class="_ _0"></span>which<span class="_ _0"></span> <span class="_ _0"></span>can</span></div><div class="t m0 x2 h5 yb ff3 fs1 fc0 sc0 ls0 ws0">balance <span class="_ _0"></span>the<span class="_ _0"></span> <span class="_ _0"></span>force<span class="_ _0"></span> <span class="_ _0"></span>of <span class="_ _0"></span>g<span class="_ _0"></span>ravity <span class="_ _0"></span>and<span class="_ _0"></span> <span class="_ _0"></span>cause<span class="_ _0"></span> <span class="_ _0"></span>the<span class="_ _0"></span> ball<span class="_ _0"></span> <span class="_ _0"></span>(which<span class="_ _0"></span> <span class="_ _0"></span>is<span class="_ _0"></span> <span class="_ _0"></span>made <span class="_ _0"></span>of<span class="_ _0"></span> <span class="_ _0"></span>a<span class="_ _0"></span> m<span class="_ _0"></span>agne&#58899;c <span class="_ _0"></span>material)<span class="_ _0"></span> <span class="_ _0"></span>to<span class="_ _0"></span> <span class="_ _0"></span>be<span class="_ _0"></span> <span class="_ _0"></span>suspended</div><div class="t m0 x2 h5 yc ff3 fs1 fc0 sc0 ls0 ws0">in<span class="_ _2"></span> <span class="_ _2"></span>midair.<span class="_ _2"></span> <span class="_ _2"></span>The<span class="_ _2"></span> <span class="_ _2"></span>modeling<span class="_ _2"></span> <span class="_ _2"></span>of<span class="_ _2"></span> <span class="_ _2"></span>this<span class="_ _2"></span> <span class="_ _2"></span>system<span class="_ _2"></span> <span class="_ _2"></span>has<span class="_ _2"></span> <span class="_ _2"></span>been<span class="_ _2"></span> <span class="_ _2"></span>established<span class="_ _2"></span> <span class="_ _2"></span>in<span class="_ _2"></span> <span class="_ _2"></span>many<span class="_ _2"></span> <span class="_ _2"></span>control<span class="_ _2"></span> <span class="_ _2"></span>text<span class="_ _2"></span> <span class="_ _2"></span>books<span class="_ _2"></span> <span class="_ _2"></span>(including</div><div class="t m0 x2 h5 yd ff4 fs1 fc0 sc0 ls0 ws0">Automa&#58887;c Control Systems<span class="ff3"> by B. C. Kuo, the seventh edi&#58899;on). </span></div><div class="t m0 x2 h3 ye ff2 fs0 fc0 sc0 ls0 ws0">The equations for the system are given by:</div></div><div class="c x3 yf w3 h6"><div class="t m0 x4 h7 y10 ff5 fs2 fc0 sc0 ls0 ws0">M</div><div class="t m0 x5 h7 y11 ff5 fs2 fc0 sc0 ls0 ws0">d</div><div class="t m0 x6 h8 y12 ff2 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 x7 h7 y11 ff5 fs2 fc0 sc0 ls0 ws0">h</div><div class="t m0 x8 h7 y13 ff5 fs2 fc0 sc0 ls0 ws0">dt</div><div class="t m0 x9 h8 y14 ff2 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa h9 y10 ff6 fs2 fc0 sc0 ls0 ws0">=<span class="_ _2"></span><span class="ff5">Mg<span class="_ _0"></span></span>&#8722;</div><div class="t m0 x2 h7 y11 ff5 fs2 fc0 sc0 ls0 ws0">K<span class="_ _3"> </span>i</div><div class="t m0 xb h8 y12 ff2 fs3 fc0 sc0 ls0 ws0">2</div><div class="t m0 xc h7 y15 ff5 fs2 fc0 sc0 ls0 ws0">h</div></div><div class="c xd y16 w4 ha"><div class="t m0 x0 h9 y17 ff5 fs2 fc0 sc0 ls0 ws0">V<span class="_ _4"> </span><span class="ff6">=<span class="_ _2"></span></span>L</div><div class="t m0 xe h7 y18 ff5 fs2 fc0 sc0 ls0 ws0">di</div><div class="t m0 xe h7 y19 ff5 fs2 fc0 sc0 ls0 ws0">dt</div><div class="t m0 xf h9 y17 ff6 fs2 fc0 sc0 ls0 ws0">+<span class="_ _1"></span><span class="ff5">iR</span></div></div><div class="c x0 y1 w2 h2"><div class="t m0 x2 h3 y1a ff2 fs0 fc0 sc0 ls0 ws0">where<span class="_ _2"></span> <span class="ff7 fs4">h</span> <span class="_ _2"></span>is<span class="_ _1"></span> <span class="_ _1"></span>the<span class="_ _1"></span> <span class="_ _1"></span>vertical<span class="_ _1"></span> <span class="_ _1"></span>position<span class="_ _1"></span> <span class="_ _1"></span>of<span class="_ _1"></span> <span class="_ _1"></span>the<span class="_ _1"></span> <span class="_ _1"></span>ball,<span class="_ _2"></span> <span class="_ _1"></span><span class="ff7 fs4">i</span> <span class="_ _2"></span>is<span class="_ _1"></span> <span class="_ _1"></span>the<span class="_ _1"></span> <span class="_ _1"></span>current<span class="_ _1"></span> <span class="_ _1"></span>through<span class="_ _1"></span> <span class="_ _1"></span>the<span class="_ _1"></span> <span class="_ _1"></span>electromagnet,<span class="_ _2"></span> <span class="_ _0"></span><span class="ff7 fs4">V</span> <span class="_ _5"> </span>is<span class="_ _0"></span> <span class="_ _1"></span>the</div><div class="t m0 x2 h3 y1b ff2 fs0 fc0 sc0 ls0 ws0">applied <span class="_ _0"></span>voltage,<span class="_ _0"></span> <span class="ff7 fs4">M</span> is<span class="_ _0"></span> the <span class="_ _0"></span>mass <span class="_ _0"></span>of <span class="_ _0"></span>the ball,<span class="_ _0"></span> <span class="_ _0"></span><span class="ff7 fs4">g</span> <span class="_ _0"></span>is gravity,<span class="_ _0"></span> <span class="ff7 fs4">L</span> <span class="_ _0"></span>is<span class="_ _0"></span> the <span class="_ _0"></span>inductance,<span class="_ _0"></span> <span class="ff7 fs4">R</span> <span class="_ _0"></span>is <span class="_ _0"></span>the resistance,<span class="_ _0"></span> and</div><div class="t m0 x2 h3 y1c ff7 fs4 fc0 sc0 ls0 ws0">K<span class="ff2 fs0"> <span class="_ _1"></span>is<span class="_ _0"></span> <span class="_ _1"></span>a<span class="_ _0"></span> <span class="_ _0"></span>coefficient<span class="_ _1"></span> <span class="_ _0"></span>that<span class="_ _0"></span> <span class="_ _0"></span>determines<span class="_ _1"></span> <span class="_ _0"></span>the<span class="_ _0"></span> <span class="_ _0"></span>magnetic<span class="_ _1"></span> <span class="_ _0"></span>force<span class="_ _0"></span> <span class="_ _0"></span>exerted<span class="_ _1"></span> <span class="_ _0"></span>on<span class="_ _0"></span> <span class="_ _0"></span>the<span class="_ _0"></span> <span class="_ _1"></span>ball.<span class="_ _0"></span> <span class="_ _0"></span>For<span class="_ _0"></span> <span class="_ _1"></span>simplicity,<span class="_ _0"></span> <span class="_ _0"></span>we<span class="_ _0"></span> <span class="_ _1"></span>will</span></div><div class="t m0 x2 h3 y1d ff2 fs0 fc0 sc0 ls0 ws0">choose<span class="_ _0"></span> <span class="_ _0"></span>values<span class="_ _1"></span> <span class="ff7 fs4">M<span class="_ _0"></span> <span class="_ _0"></span>=<span class="_ _0"></span> <span class="_ _0"></span>0.05<span class="_ _0"></span> <span class="_ _0"></span>Kg</span>,<span class="_ _1"></span> <span class="ff7 fs4">K<span class="_ _0"></span> <span class="_ _0"></span>=<span class="_ _0"></span> <span class="_ _0"></span>0.0001</span>,<span class="_ _1"></span> <span class="ff7 fs4">L<span class="_ _0"></span> <span class="_ _0"></span>=<span class="_ _0"></span> <span class="_ _0"></span>0.01<span class="_ _0"></span> <span class="_ _0"></span>H<span class="_ _0"></span></span>,<span class="_ _1"></span> <span class="ff7 fs4">R<span class="_ _0"></span> <span class="_ _0"></span>= <span class="_ _0"></span>1<span class="_ _0"></span> <span class="_ _0"></span>Ohm<span class="_ _0"></span></span>,<span class="_ _1"></span> <span class="ff7 fs4">g<span class="_ _0"></span> <span class="_ _0"></span>=<span class="_ _0"></span> <span class="_ _0"></span>9.81<span class="_ _0"></span> <span class="_ _0"></span>m/sec^2</span>.<span class="_ _0"></span> <span class="_ _0"></span>The</div><div class="t m0 x2 h3 y1e ff2 fs0 fc0 sc0 ls0 ws0">system<span class="_ _1"></span> <span class="_ _2"></span>is<span class="_ _0"></span> <span class="_ _2"></span>at<span class="_ _0"></span> <span class="_ _1"></span>equilibrium<span class="_ _2"></span> <span class="_ _0"></span>(the<span class="_ _2"></span> <span class="_ _1"></span>ball<span class="_ _1"></span> <span class="_ _6"></span>is<span class="_ _1"></span> <span class="_ _1"></span>suspended<span class="_ _1"></span> <span class="_ _6"></span>in<span class="_ _6"></span> <span class="_ _1"></span>midair)<span class="_ _1"></span> <span class="_ _6"></span>whenever<span class="_ _5"> </span> <span class="_ _6"></span><span class="ff7 fs4">h<span class="_ _1"></span> <span class="_ _1"></span>=<span class="_ _6"></span> <span class="_ _1"></span>K<span class="_ _6"></span> <span class="_ _1"></span>i^2/Mg<span class="_ _0"></span></span> <span class="_ _5"> </span>(at<span class="_ _1"></span> <span class="_ _1"></span>w<span class="_ _0"></span>hich</div><div class="t m0 x2 h3 y1f ff2 fs0 fc0 sc0 ls0 ws0">point<span class="_ _6"></span> <span class="ff7 fs4">dh/dt<span class="_ _0"></span> <span class="_ _1"></span>=<span class="_ _0"></span> <span class="_ _0"></span>0</span>).<span class="_ _1"></span> <span class="_ _0"></span>We<span class="_ _0"></span> <span class="_ _1"></span>linearize<span class="_ _1"></span> <span class="_ _0"></span>the<span class="_ _1"></span> <span class="_ _0"></span>equations<span class="_ _0"></span> <span class="_ _1"></span>about<span class="_ _0"></span> <span class="_ _1"></span>the<span class="_ _0"></span> <span class="_ _1"></span>point<span class="_ _6"></span> <span class="ff7 fs4">h<span class="_ _0"></span> <span class="_ _1"></span>=<span class="_ _0"></span> <span class="_ _1"></span>0.01<span class="_ _0"></span> <span class="_ _1"></span>m</span> <span class="_ _6"></span>(where<span class="_ _0"></span> <span class="_ _1"></span>the<span class="_ _0"></span> <span class="_ _0"></span>nominal</div><div class="t m0 x2 h3 y20 ff2 fs0 fc0 sc0 ls0 ws0">current is about 7 amp) </div><div class="t m0 x10 h5 y21 ff3 fs1 fc0 sc0 ls0 ws0">1 | <span class="fc1">P<span class="_ _7"> </span>a<span class="_ _7"> </span>g<span class="_ _7"> </span>e</span></div></div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>
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