observer design.rar

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observer design robust
observer design.rar
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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/6259344192dc900e62b7ea50/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/6259344192dc900e62b7ea50/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">www<span class="_ _0"></span>.ietdl.or<span class="_ _1"></span>g</div><div class="t m0 x2 h3 y2 ff2 fs1 fc0 sc0 ls0 ws0">Published<span class="_ _2"> </span>in<span class="_ _2"> </span>IET<span class="_ _2"> </span>Control<span class="_ _2"> </span>Theory<span class="_ _2"> </span>and<span class="_ _2"> </span>Applications</div><div class="t m0 x2 h3 y3 ff2 fs1 fc0 sc0 ls0 ws0">Received<span class="_ _2"> </span>on<span class="_ _2"> </span>30th<span class="_ _2"> </span>January<span class="_ _2"> </span>2013</div><div class="t m0 x2 h3 y4 ff2 fs1 fc0 sc0 ls0 ws0">Revised<span class="_ _2"> </span>on<span class="_ _2"> </span>22nd<span class="_ _2"> </span>April<span class="_ _2"> </span>2013</div><div class="t m0 x2 h3 y5 ff2 fs1 fc0 sc0 ls0 ws0">Accepted<span class="_ _2"> </span>on<span class="_ _2"> </span>2nd<span class="_ _2"> </span>June<span class="_ _2"> </span>2013</div><div class="t m0 x2 h3 y6 ff2 fs1 fc0 sc0 ls0 ws0">doi:<span class="_ _2"> </span>10.1049/iet-cta.2013.0096</div><div class="t m0 x3 h4 y7 ff2 fs2 fc0 sc0 ls0 ws0">ISSN<span class="_ _3"> </span>1751-8644</div><div class="t m0 x4 h5 y8 ff1 fs3 fc0 sc0 ls0 ws0">Observ<span class="_ _4"></span>ability<span class="_ _5"> </span>analysis<span class="_ _5"> </span>and<span class="_ _5"> </span>observ<span class="_ _1"></span>er<span class="_ _5"> </span>design<span class="_ _5"> </span>f<span class="_ _1"></span>or<span class="_ _5"> </span>&#64257;nit<span class="_ _1"></span>e</div><div class="t m0 x4 h5 y9 ff1 fs3 fc0 sc0 ls0 ws0">aut<span class="_ _1"></span>omata<span class="_ _5"> </span>via<span class="_ _5"> </span>matr<span class="_ _4"></span>ix<span class="_ _5"> </span>appr<span class="_ _1"></span>oac<span class="_ _1"></span>h</div><div class="t m0 x4 h6 ya ff3 fs4 fc0 sc0 ls0 ws0">Xu<span class="_"> </span>Xiangru,<span class="_ _6"> </span>Hong<span class="_ _7"></span>Y<span class="_ _1"></span>iguang</div><div class="t m0 x4 h7 yb ff3 fs2 fc0 sc0 ls0 ws0">Key<span class="_"> </span>Laboratory<span class="_"> </span>of<span class="_"> </span>S<span class="_ _1"></span>ystems<span class="_"> </span>and<span class="_"> </span>Control,<span class="_ _2"> </span>Academ<span class="_ _1"></span>y<span class="_"> </span>of<span class="_"> </span>Mathematics<span class="_"> </span>and<span class="_"> </span>Systems<span class="_"> </span>S<span class="_ _1"></span>cience,<span class="_"> </span>Chinese<span class="_ _2"> </span>Academy<span class="_"> </span>of<span class="_"> </span>S<span class="_ _1"></span>ciences,</div><div class="t m0 x4 h7 yc ff3 fs2 fc0 sc0 ls0 ws0">Beijing<span class="_"> </span>1<span class="_ _1"></span>0<span class="_ _8"></span>0<span class="_ _1"></span>1<span class="_ _4"></span>90,<span class="_"> </span>People<span class="_ _1"></span>&#8217;<span class="_ _4"></span>s<span class="_"> </span>Republic<span class="_"> </span>of<span class="_"> </span>China</div><div class="t m0 x4 h7 yd ff3 fs2 fc0 sc0 ls0 ws0">E-mail:<span class="_"> </span>xuxiangru@amss.ac.cn</div><div class="t m0 x4 h8 ye ff4 fs5 fc0 sc0 ls0 ws0">Abstract:<span class="_ _9"> </span><span class="ff5">This<span class="_ _9"> </span>study<span class="_ _9"> </span>investigates<span class="_ _9"> </span>the<span class="_ _9"> </span>observability<span class="_ _9"> </span>problem<span class="_ _9"> </span>and<span class="_ _9"> </span>the<span class="_ _9"> </span>observer<span class="_ _9"> </span>design<span class="_ _9"> </span>of<span class="_ _9"> </span>partially<span class="_ _9"> </span>observed<span class="_ _9"> </span>&#64257;nite<span class="_ _9"> </span>automata<span class="_ _9"> </span>via</span></div><div class="t m0 x4 h8 yf ff5 fs5 fc0 sc0 ls0 ws0">a<span class="_ _9"> </span>matrix<span class="_ _a"> </span>approach.<span class="_ _9"> </span>Using<span class="_ _a"> </span>semi-tensor<span class="_ _9"> </span>product<span class="_ _a"> </span>of<span class="_ _9"> </span>matrices,<span class="_ _a"> </span>&#64257;nite<span class="_ _9"> </span>automata<span class="_ _a"> </span>are<span class="_ _9"> </span>modelled<span class="_ _a"> </span>in<span class="_ _9"> </span>the<span class="_ _a"> </span>form<span class="_ _9"> </span>of<span class="_ _a"> </span>discrete-time<span class="_ _9"> </span>bilinear</div><div class="t m0 x4 h8 y10 ff5 fs5 fc0 sc0 ls0 ws0">systems.<span class="_ _b"> </span>Matrix-form<span class="_ _b"> </span>necessary<span class="_ _b"> </span>and<span class="_ _b"> </span>suf&#64257;cient<span class="_ _b"> </span>conditions<span class="_ _b"> </span>for<span class="_ _b"> </span>both<span class="_ _b"> </span>the<span class="_ _b"> </span>initial<span class="_ _b"> </span>and<span class="_ _b"> </span>current<span class="_ _b"> </span>state<span class="_ _b"> </span>observability,<span class="_ _b"> </span>either<span class="_ _b"> </span>with<span class="_ _b"> </span>or</div><div class="t m0 x4 h8 y11 ff5 fs5 fc0 sc0 ls0 ws0">without<span class="_ _9"> </span>input<span class="_ _a"> </span>information,<span class="_ _9"> </span>are<span class="_ _a"> </span>&#64257;rst<span class="_ _9"> </span>proposed.<span class="_ _a"> </span>Based<span class="_ _9"> </span>on<span class="_ _a"> </span>that,<span class="_ _9"> </span>a<span class="_ _a"> </span>constructive<span class="_ _9"> </span>method<span class="_ _a"> </span>for<span class="_ _9"> </span>the<span class="_ _a"> </span>observer<span class="_ _9"> </span>design<span class="_ _a"> </span>is<span class="_ _9"> </span>provided.</div><div class="t m0 x4 h9 y12 ff1 fs6 fc0 sc0 ls0 ws0">1<span class="_ _c"> </span>Intr<span class="_ _1"></span>oduction</div><div class="t m0 x4 h8 y13 ff5 fs5 fc0 sc0 ls0 ws0">Finite<span class="_ _9"> </span>automata<span class="_ _9"> </span>are<span class="_ _d"> </span>important<span class="_ _d"> </span>models<span class="_ _9"> </span>in<span class="_ _9"> </span>the<span class="_ _d"> </span>study<span class="_ _9"> </span>of<span class="_ _9"> </span>hybrid</div><div class="t m0 x4 h8 y14 ff5 fs5 fc0 sc0 ls0 ws0">systems<span class="_ _3"> </span>and<span class="_ _3"> </span>discrete<span class="_ _d"> </span>event<span class="_ _3"> </span>dynamical<span class="_ _3"> </span>systems<span class="_ _3"> </span>and<span class="_ _d"> </span>also<span class="_ _2"> </span>serve</div><div class="t m0 x4 h8 y15 ff5 fs5 fc0 sc0 ls0 ws0">for<span class="_ _d"> </span>the<span class="_ _3"> </span>&#64257;nite-state<span class="_ _d"> </span>approximation<span class="_ _d"> </span>of<span class="_ _d"> </span>complex<span class="_ _3"> </span>dynamical<span class="_ _d"> </span>sys-</div><div class="t m0 x4 h8 y16 ff5 fs5 fc0 sc0 ls0 ws0">tems<span class="_ _e"> </span>[<span class="fc1">1</span>&#8211;<span class="fc1">5</span>].<span class="_ _e"> </span>They<span class="_ _e"> </span>received<span class="_ _e"> </span>considerable<span class="_ _e"> </span>attention<span class="_ _e"> </span>in<span class="_ _e"> </span>the</div><div class="t m0 x4 h8 y17 ff5 fs5 fc0 sc0 ls0 ws0">control<span class="_ _f"> </span>and<span class="_ _f"> </span>computer<span class="_ _f"> </span>communities<span class="_ _f"> </span>for<span class="_ _f"> </span>their<span class="_ _f"> </span>broad<span class="_ _f"> </span>appli-</div><div class="t m0 x4 h8 y18 ff5 fs5 fc0 sc0 ls0 ws0">cations<span class="_ _f"> </span>to<span class="_ _f"> </span>embedded<span class="_ _f"> </span>systems,<span class="_ _f"> </span>automotive<span class="_ _f"> </span>controllers<span class="_ _f"> </span>and</div><div class="t m0 x4 h8 y19 ff5 fs5 fc0 sc0 ls0 ws0">algorithm<span class="_ _5"> </span>design,<span class="_ _5"> </span>etc.<span class="_ _5"> </span>In<span class="_ _e"> </span>particular,<span class="_ _5"> </span>automata<span class="_ _5"> </span>have<span class="_ _5"> </span>been</div><div class="t m0 x4 h8 y1a ff5 fs5 fc0 sc0 ls0 ws0">widely<span class="_ _2"> </span>used<span class="_ _3"> </span>in<span class="_ _3"> </span>the<span class="_ _3"> </span>control<span class="_ _2"> </span>analysis<span class="_ _3"> </span>and<span class="_ _3"> </span>synthesis<span class="_ _3"> </span>problems<span class="_ _2"> </span>of</div><div class="t m0 x4 h8 y1b ff5 fs5 fc0 sc0 ls0 ws0">discrete<span class="_ _a"> </span>event<span class="_ _b"> </span>system<span class="_ _a"> </span>under<span class="_ _a"> </span>the<span class="_ _b"> </span>Ramadge&#8211;Wonham<span class="_ _a"> </span>frame-</div><div class="t m0 x4 h8 y1c ff5 fs5 fc0 sc0 ls0 ws0">work<span class="_ _b"> </span>[<span class="fc1">6</span>].<span class="_ _b"> </span>Furthermore,<span class="_ _10"> </span>some<span class="_ _b"> </span>concepts<span class="_ _b"> </span>related<span class="_ _10"> </span>to<span class="_ _b"> </span>automata</div><div class="t m0 x4 h8 y1d ff5 fs5 fc0 sc0 ls0 ws0">such<span class="_ _f"> </span>as<span class="_ _f"> </span>simulation/bisimulation<span class="_ _f"> </span>have<span class="_ _11"> </span>been<span class="_ _11"> </span>applied<span class="_ _f"> </span>to<span class="_ _f"> </span>the</div><div class="t m0 x4 h8 y1e ff5 fs5 fc0 sc0 ls0 ws0">control<span class="_ _9"> </span>synthesis<span class="_ _a"> </span>problems<span class="_ _9"> </span>[<span class="fc1">7</span>,<span class="_ _a"> </span><span class="fc1">8</span>].</div><div class="t m0 x5 h8 y1f ff5 fs5 fc0 sc0 ls0 ws0">The<span class="_ _10"> </span>state<span class="_ _b"> </span>estimation<span class="_ _10"> </span>of<span class="_ _10"> </span>&#64257;nite<span class="_ _10"> </span>automata<span class="_ _10"> </span>is<span class="_ _b"> </span>an<span class="_ _10"> </span>interesting</div><div class="t m0 x4 h8 y20 ff5 fs5 fc0 sc0 ls0 ws0">topic<span class="_ _3"> </span>and<span class="_ _d"> </span>has<span class="_ _3"> </span>been<span class="_ _d"> </span>investigated<span class="_ _3"> </span>under<span class="_ _d"> </span>different<span class="_ _3"> </span>formulations.</div><div class="t m0 x4 h8 y21 ff5 fs5 fc0 sc0 ls0 ws0">For<span class="_ _a"> </span>example,<span class="_ _b"> </span>the<span class="_ _a"> </span>authors<span class="_ _b"> </span>in<span class="_ _a"> </span>[<span class="fc1">9</span>&#8211;<span class="fc1">11</span>]<span class="_ _b"> </span>considered<span class="_ _a"> </span>the<span class="_ _a"> </span>current-</div><div class="t m0 x4 h8 y22 ff5 fs5 fc0 sc0 ls0 ws0">location<span class="_ _a"> </span>state<span class="_ _a"> </span>recovery<span class="_ _a"> </span>problems<span class="_ _a"> </span>of<span class="_ _b"> </span>non-deterministic<span class="_ _a"> </span>&#64257;nite</div><div class="t m0 x4 h8 y23 ff5 fs5 fc0 sc0 ls0 ws0">automata<span class="_ _12"> </span>with<span class="_ _12"> </span>different<span class="_ _12"> </span>observability<span class="_ _12"> </span>notions<span class="_ _12"> </span>under<span class="_ _12"> </span>the</div><div class="t m0 x4 h8 y24 ff5 fs5 fc0 sc0 ls0 ws0">supervisory<span class="_ _12"> </span>control<span class="_ _12"> </span>framework,<span class="_ _12"> </span>and<span class="_ _12"> </span>Ozveren<span class="_ _13"> </span>and<span class="_ _12"> </span>Will-</div><div class="t m0 x4 h8 y25 ff5 fs5 fc0 sc0 ls0 ws0">sky<span class="_ _b"> </span>[<span class="fc1">12</span>]<span class="_ _10"> </span>further<span class="_ _b"> </span>studied<span class="_ _10"> </span>the<span class="_ _b"> </span>output<span class="_ _10"> </span>stabilisability<span class="_ _10"> </span>problem;</div><div class="t m0 x4 h8 y26 ff5 fs5 fc0 sc0 ls0 ws0">Caines<span class="_ _11"> </span><span class="ff6">et<span class="_ _11"> </span>al.<span class="_ _11"> </span></span>[<span class="fc1">13</span>,<span class="_ _11"> </span><span class="fc1">14</span>]<span class="_ _11"> </span>considered<span class="_ _11"> </span>the<span class="_ _10"> </span>classical<span class="_ _11"> </span>and<span class="_ _11"> </span>logic-</div><div class="t m0 x4 h8 y27 ff5 fs5 fc0 sc0 ls0 ws0">based<span class="_ _a"> </span>observer<span class="_ _a"> </span>design<span class="_ _b"> </span>of<span class="_ _a"> </span>&#64257;nite<span class="_ _a"> </span>automata,<span class="_ _b"> </span>whereas<span class="_ _a"> </span>Balluchi</div><div class="t m0 x4 h8 y28 ff6 fs5 fc0 sc0 ls0 ws0">et<span class="_ _a"> </span>al.<span class="_ _a"> </span><span class="ff5">[<span class="fc1">15</span>]<span class="_ _a"> </span>investigated<span class="_ _a"> </span>the<span class="_ _a"> </span>design<span class="_ _a"> </span>of<span class="_ _a"> </span>the<span class="_ _a"> </span>location-observers</span></div><div class="t m0 x4 h8 y29 ff5 fs5 fc0 sc0 ls0 ws0">based<span class="_ _3"> </span>on<span class="_ _d"> </span>the<span class="_ _d"> </span>computation<span class="_ _3"> </span>of<span class="_ _d"> </span>state-observer<span class="_ _3"> </span>trees<span class="_ _d"> </span>proposed<span class="_ _3"> </span>in</div><div class="t m0 x4 h8 y2a ff5 fs5 fc0 sc0 ls0 ws0">[<span class="fc1">14</span>];<span class="_ _9"> </span>Geng<span class="_ _a"> </span>[<span class="fc1">16</span>]<span class="_ _9"> </span>studied<span class="_ _a"> </span>the<span class="_ _9"> </span>observer<span class="_ _a"> </span>design<span class="_ _a"> </span>of<span class="_ _9"> </span>input/output</div><div class="t m0 x4 h8 y2b ff5 fs5 fc0 sc0 ls0 ws0">asynchronous<span class="_ _10"> </span>sequential<span class="_ _b"> </span>machines<span class="_ _10"> </span>for<span class="_ _10"> </span>the<span class="_ _b"> </span>model-matching</div><div class="t m0 x4 h8 y2c ff5 fs5 fc0 sc0 ls0 ws0">problem,<span class="_ _2"> </span>and<span class="_ _3"> </span>moreover,<span class="_ _3"> </span>Grosu<span class="_ _3"> </span>[<span class="fc1">17</span>]<span class="_ _3"> </span>treated<span class="_ _3"> </span>&#64257;nite<span class="_ _3"> </span>automata<span class="_ _3"> </span>as</div><div class="t m0 x4 h8 y2d ff5 fs5 fc0 sc0 ls0 ws0">discrete-time<span class="_ _d"> </span>invariant<span class="_ _d"> </span>linear<span class="_ _d"> </span>systems<span class="_ _d"> </span>over<span class="_ _d"> </span>semimodules<span class="_ _d"> </span>and</div><div class="t m0 x4 h8 y2e ff5 fs5 fc0 sc0 ls0 ws0">studied<span class="_ _9"> </span>the<span class="_ _9"> </span>observability<span class="_ _9"> </span>properties<span class="_ _a"> </span>analogy<span class="_ _9"> </span>to<span class="_ _9"> </span>conventional</div><div class="t m0 x4 h8 y2f ff5 fs5 fc0 sc0 ls0 ws0">linear<span class="_ _9"> </span>systems.</div><div class="t m0 x5 h8 y30 ff5 fs5 fc0 sc0 ls0 ws0">Based<span class="_ _10"> </span>on<span class="_ _10"> </span>semi-tensor<span class="_ _11"> </span>product<span class="_ _10"> </span>(STP)<span class="_ _10"> </span>of<span class="_ _10"> </span>matrices<span class="_ _10"> </span>devel-</div><div class="t m0 x4 h8 y31 ff5 fs5 fc0 sc0 ls0 ws0">oped<span class="_ _a"> </span>by<span class="_ _a"> </span>Cheng<span class="_ _b"> </span>[<span class="fc1">18</span>],<span class="_ _a"> </span>the<span class="_ _a"> </span>observability<span class="_ _b"> </span>problem<span class="_ _a"> </span>of<span class="_ _a"> </span>Boolean</div><div class="t m0 x4 h8 y32 ff5 fs5 fc0 sc0 ls0 ws0">networks,<span class="_ _a"> </span>which<span class="_ _a"> </span>can<span class="_ _a"> </span>be<span class="_ _a"> </span>seen<span class="_ _a"> </span>as<span class="_ _a"> </span>a<span class="_ _a"> </span>special<span class="_ _a"> </span>case<span class="_ _b"> </span>of<span class="_ _a"> </span>automata</div><div class="t m0 x4 h8 y33 ff5 fs5 fc0 sc0 ls0 ws0">driven<span class="_ _5"> </span>by<span class="_ _e"> </span>logical<span class="_ _e"> </span>relations<span class="_ _5"> </span>between<span class="_ _e"> </span>states,<span class="_ _5"> </span>were<span class="_ _e"> </span>consid-</div><div class="t m0 x4 h8 y34 ff5 fs5 fc0 sc0 ls0 ws0">ered<span class="_ _9"> </span>in<span class="_ _9"> </span>[<span class="fc1">19</span>&#8211;<span class="fc1">24</span>].<span class="_ _9"> </span>In<span class="_ _a"> </span>spite<span class="_ _9"> </span>of<span class="_ _9"> </span>these<span class="_ _9"> </span>existing<span class="_ _9"> </span>results,<span class="_ _9"> </span>necessary</div><div class="t m0 x4 h8 y35 ff5 fs5 fc0 sc0 ls0 ws0">and<span class="_ _b"> </span>suf&#64257;cient<span class="_ _b"> </span>conditions<span class="_ _a"> </span>that<span class="_ _b"> </span>are<span class="_ _b"> </span>readily<span class="_ _b"> </span>veri&#64257;able<span class="_ _b"> </span>for<span class="_ _b"> </span>dif-</div><div class="t m0 x4 h8 y36 ff5 fs5 fc0 sc0 ls0 ws0">ferent<span class="_ _e"> </span>observability<span class="_ _e"> </span>concepts<span class="_ _e"> </span>of<span class="_ _5"> </span>&#64257;nite<span class="_ _e"> </span>automata<span class="_ _e"> </span>are<span class="_ _e"> </span>still</div><div class="t m0 x6 h8 y37 ff5 fs5 fc0 sc0 ls0 ws0">in<span class="_ _f"> </span>need.<span class="_ _f"> </span>On<span class="_ _5"> </span>the<span class="_ _f"> </span>other<span class="_ _f"> </span>hand,<span class="_ _f"> </span>a<span class="_ _f"> </span>matrix-based<span class="_ _5"> </span>approach<span class="_ _f"> </span>to</div><div class="t m0 x6 h8 y38 ff5 fs5 fc0 sc0 ls0 ws0">model<span class="_ _d"> </span>&#64257;nite<span class="_ _9"> </span>automata<span class="_ _d"> </span>with<span class="_ _d"> </span>the<span class="_ _9"> </span>help<span class="_ _d"> </span>of<span class="_ _9"> </span>STP<span class="_ _d"> </span>was<span class="_ _d"> </span>provided<span class="_ _9"> </span>in</div><div class="t m0 x6 h8 y39 ff5 fs5 fc0 sc0 ls0 ws0">[<span class="fc1">25</span>].<span class="_ _9"> </span>The<span class="_ _9"> </span>essentially<span class="_ _d"> </span>non-linear<span class="_ _9"> </span>dynamics<span class="_ _9"> </span>of<span class="_ _9"> </span>&#64257;nite<span class="_ _9"> </span>automata</div><div class="t m0 x6 h8 y3a ff5 fs5 fc0 sc0 ls0 ws0">can<span class="_ _a"> </span>be<span class="_ _b"> </span>expressed<span class="_ _a"> </span>in<span class="_ _a"> </span>a<span class="_ _a"> </span>discrete-time<span class="_ _b"> </span>bilinear<span class="_ _a"> </span>equation<span class="_ _a"> </span>form,</div><div class="t m0 x6 h8 y3b ff5 fs5 fc0 sc0 ls0 ws0">and<span class="_ _11"> </span>the<span class="_ _11"> </span>run<span class="_ _11"> </span>of<span class="_ _11"> </span>a<span class="_ _f"> </span>string<span class="_ _11"> </span>can<span class="_ _11"> </span>be<span class="_ _11"> </span>considered<span class="_ _11"> </span>as<span class="_ _11"> </span>straightfor-</div><div class="t m0 x6 h8 y3c ff5 fs5 fc0 sc0 ls0 ws0">ward<span class="_ _10"> </span>matrix<span class="_ _10"> </span>products.<span class="_ _10"> </span>This<span class="_ _10"> </span>approach,<span class="_ _11"> </span>therefore,<span class="_ _10"> </span>facilitates</div><div class="t m0 x6 h8 y3d ff5 fs5 fc0 sc0 ls0 ws0">the<span class="_ _5"> </span>analysis<span class="_ _5"> </span>and<span class="_ _5"> </span>design<span class="_ _5"> </span>procedure<span class="_ _5"> </span>because<span class="_ _5"> </span>it<span class="_ _5"> </span>makes<span class="_ _e"> </span>the</div><div class="t m0 x6 h8 y3e ff5 fs5 fc0 sc0 ls0 ws0">conventional<span class="_ _5"> </span>state-space<span class="_ _5"> </span>approach<span class="_ _5"> </span>to<span class="_ _5"> </span>dynamical<span class="_ _e"> </span>(control)</div><div class="t m0 x6 h8 y3f ff5 fs5 fc0 sc0 ls0 ws0">systems<span class="_ _9"> </span>and<span class="_ _9"> </span>the<span class="_ _9"> </span>well-developed<span class="_ _a"> </span>matrix<span class="_ _9"> </span>analysis<span class="_ _9"> </span>tools<span class="_ _9"> </span>appli-</div><div class="t m0 x6 h8 y40 ff5 fs5 fc0 sc0 ls0 ws0">cable.<span class="_ _d"> </span>Following<span class="_ _d"> </span>this<span class="_ _3"> </span>technical<span class="_ _d"> </span>line,<span class="_ _d"> </span>(bi-)simulation<span class="_ _d"> </span>analysis,</div><div class="t m0 x6 h8 y41 ff5 fs5 fc0 sc0 ls0 ws0">decomposition<span class="_ _e"> </span>and<span class="_ _12"> </span>model-matching<span class="_ _e"> </span>problems<span class="_ _14"> </span>have<span class="_ _e"> </span>been</div><div class="t m0 x6 h8 y42 ff5 fs5 fc0 sc0 ls0 ws0">investigated<span class="_ _9"> </span>[<span class="fc1">26</span>&#8211;<span class="fc1">28</span>].</div><div class="t m0 x7 h8 y43 ff5 fs5 fc0 sc0 ls0 ws0">In<span class="_ _10"> </span>this<span class="_ _10"> </span>paper,<span class="_ _b"> </span>we<span class="_ _10"> </span>adopt<span class="_ _10"> </span>the<span class="_ _10"> </span>matrix<span class="_ _10"> </span>expression<span class="_ _10"> </span>for<span class="_ _10"> </span>&#64257;nite</div><div class="t m0 x6 h8 y44 ff5 fs5 fc0 sc0 ls0 ws0">automata<span class="_ _a"> </span>proposed<span class="_ _b"> </span>in<span class="_ _a"> </span>[<span class="fc1">25</span>]<span class="_ _b"> </span>to<span class="_ _b"> </span>study<span class="_ _a"> </span>the<span class="_ _b"> </span>observability<span class="_ _a"> </span>prob-</div><div class="t m0 x6 h8 y45 ff5 fs5 fc0 sc0 ls0 ws0">lem<span class="_ _d"> </span>of<span class="_ _d"> </span>partially<span class="_ _9"> </span>observed<span class="_ _d"> </span>non-deterministic<span class="_ _d"> </span>automata.<span class="_ _9"> </span>Using</div><div class="t m0 x6 h8 y46 ff5 fs5 fc0 sc0 ls0 ws0">the<span class="_ _2"> </span>matrix<span class="_ _3"> </span>expression,<span class="_ _3"> </span>the<span class="_ _3"> </span>initial<span class="_ _2"> </span>and<span class="_ _3"> </span>current<span class="_ _3"> </span>state<span class="_ _3"> </span>observabil-</div><div class="t m0 x6 h8 y47 ff5 fs5 fc0 sc0 ls0 ws0">ity,<span class="_ _a"> </span>either<span class="_ _9"> </span>with<span class="_ _a"> </span>or<span class="_ _a"> </span>without<span class="_ _a"> </span>input<span class="_ _a"> </span>information,<span class="_ _9"> </span>are<span class="_ _a"> </span>dealt<span class="_ _a"> </span>with</div><div class="t m0 x6 h8 y48 ff5 fs5 fc0 sc0 ls0 ws0">in<span class="_ _d"> </span>a<span class="_ _9"> </span>uni&#64257;ed<span class="_ _d"> </span>way,<span class="_ _9"> </span>and<span class="_ _d"> </span>veri&#64257;able<span class="_ _d"> </span>algebraic<span class="_ _9"> </span>necessary<span class="_ _d"> </span>and<span class="_ _9"> </span>suf-</div><div class="t m0 x6 h8 y49 ff5 fs5 fc0 sc0 ls0 ws0">&#64257;cient<span class="_ _b"> </span>conditions<span class="_ _a"> </span>are<span class="_ _b"> </span>provided<span class="_ _b"> </span>for<span class="_ _b"> </span>the<span class="_ _a"> </span>&#64257;rst<span class="_ _b"> </span>time.<span class="_ _b"> </span>Based<span class="_ _a"> </span>on</div><div class="t m0 x6 h8 y4a ff5 fs5 fc0 sc0 ls0 ws0">the<span class="_ _b"> </span>algebraic<span class="_ _10"> </span>expressions,<span class="_ _b"> </span>an<span class="_ _10"> </span>observer<span class="_ _b"> </span>design<span class="_ _b"> </span>procedure<span class="_ _10"> </span>is</div><div class="t m0 x6 h8 y4b ff5 fs5 fc0 sc0 ls0 ws0">also<span class="_ _b"> </span>given.<span class="_ _b"> </span>Compared<span class="_ _a"> </span>with<span class="_ _b"> </span>existing<span class="_ _b"> </span>results,<span class="_ _b"> </span>our<span class="_ _b"> </span>results<span class="_ _b"> </span>are</div><div class="t m0 x6 h8 y4c ff5 fs5 fc0 sc0 ls0 ws0">constructive<span class="_ _9"> </span>and<span class="_ _a"> </span>easy<span class="_ _9"> </span>to<span class="_ _a"> </span>implement.</div><div class="t m0 x7 h8 y4d ff5 fs5 fc0 sc0 ls0 ws0">The<span class="_ _3"> </span>rest<span class="_ _3"> </span>of<span class="_ _3"> </span>this<span class="_ _3"> </span>paper<span class="_ _3"> </span>is<span class="_ _3"> </span>organised<span class="_ _3"> </span>as<span class="_ _3"> </span>follows.<span class="_ _3"> </span>In<span class="_ _3"> </span>Section<span class="_ _3"> </span>2,</div><div class="t m0 x6 h8 y4e ff5 fs5 fc0 sc0 ls0 ws0">important<span class="_ _d"> </span>concepts<span class="_ _d"> </span>and<span class="_ _d"> </span>preliminary<span class="_ _d"> </span>results<span class="_ _d"> </span>of<span class="_ _d"> </span>STP<span class="_ _d"> </span>are<span class="_ _d"> </span>given</div><div class="t m0 x6 h8 y4f ff5 fs5 fc0 sc0 ls0 ws0">along<span class="_ _5"> </span>with<span class="_ _f"> </span>the<span class="_ _5"> </span>matrix<span class="_ _5"> </span>expression<span class="_ _5"> </span>for<span class="_ _5"> </span>&#64257;nite<span class="_ _f"> </span>automata.<span class="_ _5"> </span>In</div><div class="t m0 x6 h8 y50 ff5 fs5 fc0 sc0 ls0 ws0">Section<span class="_ _5"> </span>3,<span class="_ _e"> </span>the<span class="_ _5"> </span>initial<span class="_ _e"> </span>and<span class="_ _5"> </span>current<span class="_ _e"> </span>state<span class="_ _5"> </span>observability<span class="_ _e"> </span>are</div><div class="t m0 x6 h8 y51 ff5 fs5 fc0 sc0 ls0 ws0">de&#64257;ned,<span class="_ _b"> </span>and<span class="_ _b"> </span>the<span class="_ _b"> </span>respective<span class="_ _10"> </span>suf&#64257;cient<span class="_ _b"> </span>and<span class="_ _b"> </span>necessary<span class="_ _b"> </span>condi-</div><div class="t m0 x6 h8 y52 ff5 fs5 fc0 sc0 ls0 ws0">tions<span class="_ _9"> </span>for<span class="_ _d"> </span>each<span class="_ _9"> </span>problem<span class="_ _9"> </span>are<span class="_ _9"> </span>derived.<span class="_ _d"> </span>The<span class="_ _9"> </span>observer<span class="_ _9"> </span>design<span class="_ _9"> </span>are</div><div class="t m0 x6 h8 y53 ff5 fs5 fc0 sc0 ls0 ws0">considered<span class="_ _11"> </span>in<span class="_ _11"> </span>Section<span class="_ _11"> </span>4,<span class="_ _11"> </span>where<span class="_ _11"> </span>an<span class="_ _11"> </span>illustrative<span class="_ _11"> </span>example<span class="_ _11"> </span>is</div><div class="t m0 x6 h8 y54 ff5 fs5 fc0 sc0 ls0 ws0">also<span class="_ _b"> </span>provided.<span class="_ _a"> </span>Finally,<span class="_ _b"> </span>some<span class="_ _b"> </span>concluding<span class="_ _a"> </span>remarks<span class="_ _b"> </span>are<span class="_ _b"> </span>made</div><div class="t m0 x6 h8 y55 ff5 fs5 fc0 sc0 ls0 ws0">in<span class="_ _9"> </span>Section<span class="_ _a"> </span>5.</div><div class="t m0 x6 h9 y56 ff1 fs6 fc0 sc0 ls0 ws0">2<span class="_ _c"> </span>Pr<span class="_ _1"></span>eliminary</div><div class="t m0 x6 h8 y57 ff5 fs5 fc0 sc0 ls0 ws0">In<span class="_ _9"> </span>this<span class="_ _9"> </span>section,<span class="_ _a"> </span>we<span class="_ _9"> </span>introduce<span class="_ _9"> </span>some<span class="_ _9"> </span>useful<span class="_ _a"> </span>concepts<span class="_ _9"> </span>and<span class="_ _9"> </span>pre-</div><div class="t m0 x6 h8 y58 ff5 fs5 fc0 sc0 ls0 ws0">liminaries<span class="_ _d"> </span>about<span class="_ _3"> </span>the<span class="_ _d"> </span>STP<span class="_ _d"> </span>and<span class="_ _3"> </span>the<span class="_ _d"> </span>matrix<span class="_ _d"> </span>expression<span class="_ _d"> </span>for<span class="_ _3"> </span>&#64257;nite</div><div class="t m0 x6 h8 y59 ff5 fs5 fc0 sc0 ls0 ws0">automata.</div><div class="t m0 x4 ha y5a ff3 fs1 fc0 sc0 ls0 ws0">IET<span class="_"> </span>Control<span class="_ _15"> </span>Theory<span class="_ _15"> </span>Appl.,<span class="_"> </span><span class="ff2">2013,<span class="_ _2"> </span>V<span class="_ _1"></span>ol.<span class="_ _2"> </span>7,<span class="_ _2"> </span>Iss.<span class="_ _2"> </span>12,<span class="_ _2"> </span>pp.<span class="_ _2"> </span>1609&#8211;1615<span class="_ _16"> </span>1609</span></div><div class="t m0 x4 h3 y5b ff2 fs1 fc0 sc0 ls0 ws0">doi:<span class="_ _2"> </span>10.1049/iet-cta.2013.0096<span class="_ _17"> </span>&#169;<span class="_ _2"> </span>The<span class="_ _2"> </span>Institution<span class="_ _2"> </span>of<span class="_ _2"> </span>Engineering<span class="_ _2"> </span>and<span class="_ _2"> </span>T<span class="_ _4"></span>echnology<span class="_ _2"> </span>2013</div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div> </body> </html>
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