<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/626773634c65f412596c8308/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/626773634c65f412596c8308/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Oscillation Mitig<span class="_ _0"></span>ation for Sliding<span class="_ _0"></span>-Mode Observers in </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws1">Sensorless Control of IPMSMs</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls2 ws2">Yue Zhao, Wei<span class="_ _0"></span> Qiao </div><div class="t m0 x4 h4 y4 ff1 fs2 fc0 sc0 ls3 ws3">Department<span class="_ _0"></span> of Electrical Engineering </div><div class="t m0 x5 h4 y5 ff1 fs2 fc0 sc0 ls4 ws4">Un<span class="_ _0"></span>ive<span class="_ _0"></span>rsi<span class="_ _0"></span>ty<span class="_ _0"></span> of N<span class="_ _0"></span>ebr<span class="_ _0"></span>ask<span class="_ _0"></span>a-L<span class="_ _0"></span>in<span class="_ _0"></span>coln </div><div class="t m0 x6 h4 y6 ff1 fs2 fc0 sc0 ls5 ws5">Lincol<span class="_ _0"></span>n, NE 68588-<span class="_ _0"></span>0511 US<span class="_ _0"></span>A </div><div class="t m0 x7 h4 y7 ff1 fs2 fc0 sc0 ls6 ws6">yue.zh<span class="_ _0"></span>ao@husk<span class="_ _0"></span>ers.unl<span class="_ _0"></span>.edu; wqi<span class="_ _0"></span>ao@eng<span class="_ _0"></span>r.unl.edu </div><div class="t m0 x8 h3 y3 ff1 fs1 fc0 sc0 ls7 ws7">Long Wu </div><div class="t m0 x9 h4 y4 ff1 fs2 fc0 sc0 ls8 ws8">Pho<span class="_ _1"></span>enix In<span class="_ _1"></span>ter<span class="_ _1"></span>natio<span class="_ _1"></span>nal<span class="_ _1"></span> – A Jo<span class="_ _1"></span>hn De<span class="_ _1"></span>ere<span class="_ _1"></span> Compa<span class="_ _1"></span>n<span class="_ _1"></span>y </div><div class="t m0 xa h4 y5 ff1 fs2 fc0 sc0 ls9 ws9">1750 NDSU Res<span class="_ _0"></span>earch Park<span class="_ _0"></span> Dr. </div><div class="t m0 xb h4 y6 ff1 fs2 fc0 sc0 lsa wsa">Fargo, ND<span class="_ _0"></span> 58102 USA<span class="_ _0"></span> </div><div class="t m0 xc h4 y7 ff1 fs2 fc0 sc0 lsb ws7">WuLong@JohnDeere.com<span class="fc1 ls7"> </span></div><div class="t m0 xd h4 y8 ff1 fs2 fc0 sc0 ls7 ws7"> </div><div class="t m0 xe h5 y9 ff2 fs3 fc0 sc0 lsc ws7">Abstract<span class="ff3 lsd wsb">—Bac<span class="_ _1"></span>k electro<span class="_ _1"></span>magnetic force (EMF)-base<span class="_ _1"></span>d methods are<span class="_ _1"></span> </span></div><div class="t m0 xe h5 ya ff3 fs3 fc0 sc0 lse wsc">commonly use<span class="_ _1"></span>d for sens<span class="_ _1"></span>orless co<span class="_ _1"></span>ntrol of inte<span class="_ _1"></span>rior per<span class="_ _1"></span>manent </div><div class="t m0 xe h5 yb ff3 fs3 fc0 sc0 lsf wsd">magne<span class="_ _1"></span>t sy<span class="_ _1"></span>nchr<span class="_ _1"></span>ono<span class="_ _1"></span>us <span class="_ _1"></span>machi<span class="_ _1"></span>nes<span class="_ _1"></span> (IPM<span class="_ _1"></span>SM<span class="_ _1"></span>s) in medi<span class="_ _1"></span>um a<span class="_ _1"></span>nd hi<span class="_ _1"></span>gh </div><div class="t m0 xe h5 yc ff3 fs3 fc0 sc0 lsd wsb">speed range.<span class="_ _1"></span> The featur<span class="_ _1"></span>e of hig<span class="_ _1"></span>h robustness to sy<span class="_ _1"></span>stem struct<span class="_ _1"></span>ure </div><div class="t m0 xe h5 yd ff3 fs3 fc0 sc0 ls10 wse">and pa<span class="_ _1"></span>ra<span class="_ _1"></span>mete<span class="_ _1"></span>r unc<span class="_ _1"></span>ert<span class="_ _1"></span>ainti<span class="_ _1"></span>es<span class="_ _1"></span> ma<span class="_ _1"></span>kes t<span class="_ _1"></span>he sl<span class="_ _1"></span>idi<span class="_ _1"></span>ng-<span class="_ _1"></span>mode <span class="_ _1"></span>obse<span class="_ _1"></span>rver<span class="_ _1"></span> </div><div class="t m0 xe h5 ye ff3 fs3 fc0 sc0 ls11 wsf">(SMO) a promising candidate for rotor <span class="_ _1"></span>position estimation. In a </div><div class="t m0 xe h5 yf ff3 fs3 fc0 sc0 ls12 ws10">practical dri<span class="_ _1"></span>ve system, because of phy<span class="_ _1"></span>sical li<span class="_ _1"></span>m<span class="_ _0"></span>itatio<span class="_ _1"></span>ns, e.g., </div><div class="t m0 xe h5 y10 ff3 fs3 fc0 sc0 ls13 ws11">samplin<span class="_ _0"></span>g frequenc<span class="_ _0"></span>y and com<span class="_ _0"></span>putational resou<span class="_ _0"></span>rce, it is cha<span class="_ _0"></span>llenging </div><div class="t m0 xe h5 y11 ff3 fs3 fc0 sc0 ls14 ws12">to obtain a <span class="_ _1"></span>perfect sinus<span class="_ _1"></span>oidal <span class="_ _1"></span>w<span class="_ _1"></span>aveform for the bac<span class="_ _1"></span>k EMF by<span class="_ _1"></span> a </div><div class="t m0 xe h5 y12 ff3 fs3 fc0 sc0 ls15 ws13">SMO, especiall<span class="_ _1"></span>y in high s<span class="_ _1"></span>peed range.<span class="_ _1"></span> As a result, the r<span class="_ _1"></span>otor </div><div class="t m0 xe h5 y13 ff3 fs3 fc0 sc0 ls16 ws14">position obtained<span class="_ _0"></span> from the estimated back EMF by usin<span class="_ _0"></span>g the </div><div class="t m0 xe h5 y14 ff3 fs3 fc0 sc0 ls17 ws15">trad<span class="_ _0"></span>itional invers<span class="_ _0"></span>e tangen<span class="_ _0"></span>t method<span class="_ _0"></span> will ha<span class="_ _0"></span>ve nonn<span class="_ _0"></span>egligible </div><div class="t m0 xe h5 y15 ff3 fs3 fc0 sc0 ls18 ws16">oscilla<span class="_ _0"></span>tions. This pap<span class="_ _0"></span>er propo<span class="_ _0"></span>ses a nov<span class="_ _0"></span>el algor<span class="_ _0"></span>ithm, which<span class="_ _0"></span> uses </div><div class="t m0 xe h5 y16 ff3 fs3 fc0 sc0 ls19 ws17">the estimated rotor speed as a feedbac<span class="_ _1"></span>k signal wi<span class="_ _1"></span>th th<span class="_ _0"></span>e </div><div class="t m0 xe h5 y17 ff3 fs3 fc0 sc0 ls12 ws18">convention<span class="_ _1"></span>al back EM<span class="_ _1"></span>F-based inve<span class="_ _1"></span>rse ta<span class="_ _1"></span>ngent method t<span class="_ _1"></span>o extract<span class="_ _1"></span> </div><div class="t m0 xe h5 y18 ff3 fs3 fc0 sc0 ls1a ws19">the rotor positio<span class="_ _1"></span>n. The pr<span class="_ _1"></span>oposed alg<span class="_ _1"></span>orith<span class="_ _1"></span>m can effectivel<span class="_ _1"></span>y </div><div class="t m0 xe h5 y19 ff3 fs3 fc0 sc0 ls1b ws1a">mitiga<span class="_ _0"></span>te the o<span class="_ _0"></span>scillation<span class="_ _0"></span> and <span class="_ _0"></span>improve <span class="_ _0"></span>the d<span class="_ _0"></span>ynamic<span class="_ _0"></span> performan<span class="_ _0"></span>ce of </div><div class="t m0 xe h5 y1a ff3 fs3 fc0 sc0 ls15 ws1b">the SMO for rotor position<span class="_ _1"></span> estimation. The prop<span class="_ _1"></span>osed algorit<span class="_ _1"></span>hm is </div><div class="t m0 xe h5 y1b ff3 fs3 fc0 sc0 ls1c ws1c">validat<span class="_ _0"></span>ed by simulation<span class="_ _0"></span>s in MATLAB Sim<span class="_ _0"></span>ulink as <span class="_ _0"></span>well as </div><div class="t m0 xe h5 y1c ff3 fs3 fc0 sc0 ls1d ws1d">experi<span class="_ _1"></span>ments on a high-<span class="_ _1"></span>pow<span class="_ _1"></span>er IPMSM drive syste<span class="_ _1"></span>m. </div><div class="t m0 xe h5 y1d ff2 fs3 fc0 sc0 ls1e ws7">Key<span class="_ _0"></span>words—<span class="_ _0"></span><span class="ff3 ls1f ws1e">Interi<span class="_ _1"></span>or p<span class="_ _1"></span>erma<span class="_ _1"></span>ne<span class="_ _1"></span>nt <span class="_ _1"></span>magnet s<span class="_ _1"></span>ync<span class="_ _1"></span>hrno<span class="_ _1"></span>us ma<span class="_ _1"></span>chi<span class="_ _1"></span>ne </span></div><div class="t m0 xe h5 y1e ff3 fs3 fc0 sc0 ls20 ws1f">(IPMSM); oscillatio<span class="_ _0"></span>n mitiga<span class="_ _0"></span>tion; sensor<span class="_ _0"></span>less control; slidin<span class="_ _0"></span>g-mode </div><div class="t m0 xe h5 y1f ff3 fs3 fc0 sc0 ls21 ws20">observer (SM<span class="_ _1"></span>O); speed feedback<span class="ff2 ls22 ws7"> </span></div><div class="t m0 xf h4 y20 ff1 fs2 fc0 sc0 ls23 ws7">I.<span class="ff4 ls7"> <span class="_ _2"> </span><span class="ff1">I<span class="fs4 ls24">NTRODUCTION <span class="_ _1"></span></span> </span></span></div><div class="t m0 x10 h4 y21 ff1 fs2 fc0 sc0 ls25 ws21">Elect<span class="_ _0"></span>romech<span class="_ _0"></span>anical<span class="_ _0"></span> senso<span class="_ _0"></span>rs, e.g<span class="_ _0"></span>., resolv<span class="_ _0"></span>ers<span class="_ _0"></span>, optical<span class="_ _0"></span> enco<span class="_ _0"></span>ders<span class="_ _0"></span>, </div><div class="t m0 xe h4 y22 ff1 fs2 fc0 sc0 ls26 ws22">and hall-<span class="_ _0"></span>effe<span class="_ _0"></span>ct sens<span class="_ _0"></span>ors,<span class="_ _0"></span> are <span class="_ _0"></span>comm<span class="_ _0"></span>only u<span class="_ _0"></span>sed to <span class="_ _0"></span>obtain<span class="_ _0"></span> roto<span class="_ _0"></span>r </div><div class="t m0 xe h4 y23 ff1 fs2 fc0 sc0 ls23 ws23">posit<span class="_ _0"></span>ion<span class="_ _0"></span>s/s<span class="_ _0"></span>peeds<span class="_ _0"></span>, w<span class="_ _0"></span>h<span class="_ _0"></span>ich are<span class="_ _0"></span> in<span class="_ _0"></span>dis<span class="_ _0"></span>pensa<span class="_ _0"></span>bl<span class="_ _0"></span>e for h<span class="_ _0"></span>igh<span class="_ _0"></span>-p<span class="_ _0"></span>erf<span class="_ _0"></span>orma<span class="_ _0"></span>nce </div><div class="t m0 xe h4 y24 ff1 fs2 fc0 sc0 ls27 ws24">control<span class="_ _1"></span> of interior per<span class="_ _1"></span>manent magnet synchro<span class="_ _1"></span>nous machines </div><div class="t m0 xe h4 y25 ff1 fs2 fc0 sc0 ls28 ws25">(IPM<span class="_ _1"></span>SMs)<span class="_ _1"></span>. The <span class="_ _1"></span>use of t<span class="_ _1"></span>hese se<span class="_ _1"></span>nsors<span class="_ _1"></span> incre<span class="_ _1"></span>ases co<span class="_ _1"></span>st, s<span class="_ _1"></span>ize and </div><div class="t m0 xe h4 y26 ff1 fs2 fc0 sc0 ls29 ws26">wiring c<span class="_ _1"></span>omplexity of IP<span class="_ _1"></span>MSM dr<span class="_ _1"></span>ive systems. Mo<span class="_ _1"></span>reover, </div><div class="t m0 xe h4 y27 ff1 fs2 fc0 sc0 ls2a ws27">sensors are subj<span class="_ _1"></span>ected to high failure ra<span class="_ _1"></span>tes in harsh </div><div class="t m0 xe h4 y28 ff1 fs2 fc0 sc0 ls2b ws28">enviro<span class="_ _1"></span>nments, such a<span class="_ _1"></span>s high environment te<span class="_ _1"></span>mperature<span class="_ _1"></span>, high-</div><div class="t m0 xe h4 y29 ff1 fs2 fc0 sc0 ls2c ws29">speed op<span class="_ _1"></span>eratio<span class="_ _1"></span>n, and adve<span class="_ _1"></span>rse or hea<span class="_ _1"></span>vy load<span class="_ _1"></span>ing condi<span class="_ _1"></span>tions [1]. </div><div class="t m0 xe h4 y2a ff1 fs2 fc0 sc0 ls2d ws2a">To ove<span class="_ _0"></span>rcom<span class="_ _0"></span>e thes<span class="_ _0"></span>e draw<span class="_ _0"></span>back<span class="_ _0"></span>s, m<span class="_ _0"></span>uch<span class="_ _0"></span> research<span class="_ _0"></span> ef<span class="_ _0"></span>fort h<span class="_ _0"></span>as g<span class="_ _0"></span>one </div><div class="t m0 xe h4 y2b ff1 fs2 fc0 sc0 ls2e ws2b">into the<span class="_ _0"></span> develo<span class="_ _0"></span>pment of<span class="_ _0"></span> sens<span class="_ _0"></span>orless drives<span class="_ _0"></span> that h<span class="_ _0"></span>ave <span class="_ _0"></span>comparab<span class="_ _0"></span>le </div><div class="t m0 xe h4 y2c ff1 fs2 fc0 sc0 ls2f ws2c">or s<span class="_ _0"></span>im<span class="_ _0"></span>ilar<span class="_ _0"></span> dy<span class="_ _0"></span>nam<span class="_ _0"></span>ic p<span class="_ _0"></span>erf<span class="_ _0"></span>orma<span class="_ _0"></span>nce<span class="_ _0"></span> to<span class="_ _0"></span> sens<span class="_ _0"></span>or-<span class="_ _0"></span>base<span class="_ _0"></span>d <span class="_ _0"></span>driv<span class="_ _0"></span>es <span class="_ _0"></span>duri<span class="_ _0"></span>ng<span class="_ _0"></span> </div><div class="t m0 xe h4 y2d ff1 fs2 fc0 sc0 ls30 ws2d">last<span class="_ _0"></span> deca<span class="_ _0"></span>des. A<span class="_ _0"></span>mon<span class="_ _0"></span>g diff<span class="_ _0"></span>eren<span class="_ _0"></span>t r<span class="_ _0"></span>otor<span class="_ _0"></span> posi<span class="_ _0"></span>tion<span class="_ _0"></span>/s<span class="_ _0"></span>peed<span class="_ _0"></span> obse<span class="_ _0"></span>rv<span class="_ _0"></span>ers </div><div class="t m0 xe h4 y2e ff1 fs2 fc0 sc0 ls8 ws2e">used in sensorl<span class="_ _1"></span>ess control schemes, t<span class="_ _1"></span>he sliding-mode<span class="_ _1"></span> observer </div><div class="t m0 xe h4 y2f ff1 fs2 fc0 sc0 ls2a ws2f">(SMO)<span class="_ _1"></span> is a promising ca<span class="_ _1"></span>ndidat<span class="_ _1"></span>e. Gener<span class="_ _1"></span>ally spea<span class="_ _1"></span>king, a SMO is </div><div class="t m0 xe h4 y30 ff1 fs2 fc0 sc0 ls31 ws30">an observ<span class="_ _0"></span>er w<span class="_ _0"></span>hose in<span class="_ _0"></span>put is a <span class="_ _0"></span>discontin<span class="_ _0"></span>uous<span class="_ _0"></span> functi<span class="_ _0"></span>on of<span class="_ _0"></span> the er<span class="_ _0"></span>ror<span class="_ _0"></span> </div><div class="t m0 xe h4 y31 ff1 fs2 fc0 sc0 ls32 ws31">between<span class="_ _0"></span> the<span class="_ _0"></span> estim<span class="_ _0"></span>ated <span class="_ _0"></span>and m<span class="_ _0"></span>easur<span class="_ _0"></span>ed ou<span class="_ _0"></span>tputs<span class="_ _0"></span> [2]<span class="_ _0"></span>. If<span class="_ _0"></span> a s<span class="_ _0"></span>lidi<span class="_ _0"></span>ng-</div><div class="t m0 xe h4 y32 ff1 fs2 fc0 sc0 ls33 ws32">mode mani<span class="_ _1"></span>fold is well de<span class="_ _1"></span>signed and<span class="_ _1"></span> when the state traj<span class="_ _1"></span>ector<span class="_ _1"></span>y </div><div class="t m0 xe h4 y33 ff1 fs2 fc0 sc0 ls34 ws33">reaches<span class="_ _0"></span> the m<span class="_ _0"></span>anif<span class="_ _0"></span>old, the<span class="_ _0"></span> slid<span class="_ _0"></span>ing m<span class="_ _0"></span>ode will b<span class="_ _0"></span>e enf<span class="_ _0"></span>orced. <span class="_ _0"></span>The </div><div class="t m0 xe h4 y34 ff1 fs2 fc0 sc0 ls2b ws34">dynamic be<span class="_ _1"></span>havior o<span class="_ _1"></span>f state trajector<span class="_ _1"></span>y under slidi<span class="_ _1"></span>ng mode only </div><div class="t m0 x11 h4 y35 ff1 fs2 fc0 sc0 ls35 ws35">depen<span class="_ _0"></span>ds<span class="_ _0"></span> on th<span class="_ _0"></span>e sli<span class="_ _0"></span>ding<span class="_ _0"></span> surf<span class="_ _0"></span>ac<span class="_ _0"></span>e and<span class="_ _0"></span> is<span class="_ _0"></span> not af<span class="_ _0"></span>fect<span class="_ _0"></span>ed<span class="_ _0"></span> by sy<span class="_ _0"></span>stem </div><div class="t m0 x11 h4 y36 ff1 fs2 fc0 sc0 ls36 ws36">structu<span class="_ _0"></span>re an<span class="_ _0"></span>d param<span class="_ _0"></span>eter un<span class="_ _0"></span>cert<span class="_ _0"></span>ainty<span class="_ _0"></span>. </div><div class="t m0 x12 h4 y37 ff1 fs2 fc0 sc0 ls37 ws37">Contro<span class="_ _1"></span>l systems with sl<span class="_ _1"></span>iding mode<span class="_ _1"></span> realizat<span class="_ _1"></span>ion are be<span class="_ _1"></span>coming </div><div class="t m0 x11 h4 y38 ff1 fs2 fc0 sc0 ls38 ws38">more<span class="_ _1"></span> and more trac<span class="_ _1"></span>tabl<span class="_ _1"></span>e due to the wide<span class="_ _1"></span>spread us<span class="_ _1"></span>e of digita<span class="_ _1"></span>l </div><div class="t m0 x11 h4 y39 ff1 fs2 fc0 sc0 ls39 ws39">contr<span class="_ _0"></span>ollers<span class="_ _0"></span> ov<span class="_ _0"></span>er the<span class="_ _0"></span> last f<span class="_ _0"></span>ew<span class="_ _0"></span> years<span class="_ _0"></span>. In <span class="_ _0"></span>a dis<span class="_ _0"></span>crete-<span class="_ _0"></span>tim<span class="_ _0"></span>e slidi<span class="_ _0"></span>ng<span class="_ _0"></span>-</div><div class="t m0 x11 h4 y3a ff1 fs2 fc0 sc0 lsa ws3a">mode <span class="_ _0"></span>con<span class="_ _0"></span>tr<span class="_ _0"></span>olle<span class="_ _0"></span>r o<span class="_ _0"></span>r <span class="_ _0"></span>obs<span class="_ _0"></span>erv<span class="_ _0"></span>er,<span class="_ _0"></span> in<span class="_ _0"></span> or<span class="_ _0"></span>der<span class="_ _0"></span> t<span class="_ _0"></span>o fa<span class="_ _0"></span>cili<span class="_ _0"></span>ta<span class="_ _0"></span>te<span class="_ _0"></span> com<span class="_ _0"></span>pu<span class="_ _0"></span>ter-</div><div class="t m0 x11 h4 y3b ff1 fs2 fc0 sc0 ls3a ws3b">based im<span class="_ _0"></span>plem<span class="_ _0"></span>entation,<span class="_ _0"></span> the contro<span class="_ _0"></span>l input<span class="_ _0"></span> is calcul<span class="_ _0"></span>ated<span class="_ _0"></span> once in<span class="_ _0"></span> </div><div class="t m0 x11 h4 y3c ff1 fs2 fc0 sc0 ls3b ws3c">ever<span class="_ _1"></span>y sampling period and is held co<span class="_ _1"></span>nstant during this interva<span class="_ _1"></span>l. </div><div class="t m0 x11 h4 y3d ff1 fs2 fc0 sc0 ls3c ws3d">Beca<span class="_ _1"></span>use of a fini<span class="_ _1"></span>te samp<span class="_ _1"></span>ling freq<span class="_ _1"></span>uency, t<span class="_ _1"></span>he stat<span class="_ _1"></span>e traje<span class="_ _1"></span>ctory is </div><div class="t m0 x11 h4 y3e ff1 fs2 fc0 sc0 ls3d ws3e">unable t<span class="_ _0"></span>o exactly<span class="_ _0"></span> m<span class="_ _0"></span>ove along<span class="_ _0"></span> the sli<span class="_ _0"></span>ding su<span class="_ _0"></span>rfac<span class="_ _0"></span>e, w<span class="_ _0"></span>hich will<span class="_ _0"></span> </div><div class="t m0 x11 h4 y3f ff1 fs2 fc0 sc0 ls3e ws3f">give <span class="_ _0"></span>only a sli<span class="_ _0"></span>ding-lik<span class="_ _0"></span>e <span class="_ _0"></span>or quas<span class="_ _0"></span>i-sliding-<span class="_ _0"></span>mode<span class="_ _0"></span> motion<span class="_ _0"></span> [3],<span class="_ _0"></span> [4]<span class="_ _0"></span>. </div><div class="t m0 x12 h4 y40 ff1 fs2 fc0 sc0 ls3f ws40"> <span class="_ _3"></span>In s<span class="_ _0"></span>everal<span class="_ _0"></span> <span class="_ _0"></span>previou<span class="_ _0"></span>s w<span class="_ _0"></span>ork<span class="_ _0"></span>s, th<span class="_ _0"></span>e SMO h<span class="_ _0"></span>as b<span class="_ _0"></span>een <span class="_ _0"></span>applie<span class="_ _0"></span>d f<span class="_ _0"></span>or<span class="_ _0"></span> </div><div class="t m0 x11 h4 y41 ff1 fs2 fc0 sc0 ls40 ws41">sensorl<span class="_ _1"></span>ess control of PM<span class="_ _1"></span>SM drives [5]-[8]. T<span class="_ _1"></span>he SMO uses a </div><div class="t m0 x11 h4 y42 ff1 fs2 fc0 sc0 ls3e ws3f">discontinu<span class="_ _0"></span>ous c<span class="_ _0"></span>ontrol <span class="_ _0"></span>(i.e<span class="_ _0"></span>., a s<span class="_ _0"></span>witch<span class="_ _0"></span>ing f<span class="_ _0"></span>unction) t<span class="_ _0"></span>o estim<span class="_ _0"></span>ate <span class="_ _0"></span>the </div><div class="t m0 x11 h4 y43 ff1 fs2 fc0 sc0 ls41 ws42">back <span class="_ _0"></span>EMF b<span class="_ _0"></span>ased<span class="_ _0"></span> on<span class="_ _0"></span> the e<span class="_ _0"></span>rro<span class="_ _0"></span>rs of<span class="_ _0"></span> the<span class="_ _0"></span> sta<span class="_ _0"></span>tor cu<span class="_ _0"></span>rr<span class="_ _0"></span>ent es<span class="_ _0"></span>timati<span class="_ _0"></span>on<span class="_ _0"></span>. </div><div class="t m0 x11 h4 y44 ff1 fs2 fc0 sc0 ls42 ws43">However<span class="_ _1"></span>, in these works a<span class="_ _1"></span> high sampling fre<span class="_ _1"></span>quency, e.<span class="_ _1"></span>g., 20 </div><div class="t m0 x11 h4 y45 ff1 fs2 fc0 sc0 ls43 ws44">kHz, is commo<span class="_ _1"></span>nly used, and the spe<span class="_ _1"></span>ed range is not wide </div><div class="t m0 x11 h4 y46 ff1 fs2 fc0 sc0 ls44 ws45">enough.<span class="_ _1"></span> In some pr<span class="_ _1"></span>actica<span class="_ _1"></span>l applica<span class="_ _1"></span>tions, e.g.,<span class="_ _1"></span> the gener<span class="_ _1"></span>ators in </div><div class="t m0 x11 h4 y47 ff1 fs2 fc0 sc0 ls45 ws46">electric<span class="_ _0"></span> vehicles<span class="_ _0"></span>, w<span class="_ _0"></span>hen consi<span class="_ _0"></span>dering<span class="_ _0"></span> the<span class="_ _0"></span> sw<span class="_ _0"></span>itching losses<span class="_ _0"></span>, dri<span class="_ _0"></span>ve </div><div class="t m0 x11 h4 y48 ff1 fs2 fc0 sc0 ls7 ws47">size, an<span class="_ _0"></span>d EMI <span class="_ _0"></span>issues, the<span class="_ _0"></span> PWM f<span class="_ _0"></span>requency<span class="_ _0"></span> w<span class="_ _0"></span>ill be relati<span class="_ _0"></span>vely </div><div class="t m0 x11 h4 y49 ff1 fs2 fc0 sc0 ls26 ws48">lower, n<span class="_ _0"></span>ormally<span class="_ _0"></span> less than<span class="_ _0"></span> 10 k<span class="_ _0"></span>Hz. Low<span class="_ _0"></span> sapling fr<span class="_ _0"></span>equency<span class="_ _0"></span> and<span class="_ _0"></span> </div><div class="t m0 x11 h4 y4a ff1 fs2 fc0 sc0 ls46 ws49">high spee<span class="_ _0"></span>d w<span class="_ _0"></span>ill m<span class="_ _0"></span>ake the<span class="_ _0"></span> app<span class="_ _0"></span>licati<span class="_ _0"></span>on of<span class="_ _0"></span> a dis<span class="_ _0"></span>crete-t<span class="_ _0"></span>ime S<span class="_ _0"></span>MO </div><div class="t m0 x11 h4 y4b ff1 fs2 fc0 sc0 ls3b ws4a">more c<span class="_ _1"></span>hallenging. </div><div class="t m0 x12 h4 y4c ff1 fs2 fc0 sc0 ls47 ws4b">In the app<span class="_ _1"></span>lica<span class="_ _1"></span>tio<span class="_ _1"></span>ns where the bac<span class="_ _1"></span>k EMF i<span class="_ _1"></span>s estimat<span class="_ _1"></span>ed from a </div><div class="t m0 x11 h4 y4d ff1 fs2 fc0 sc0 ls33 ws4c">discre<span class="_ _1"></span>te-ti<span class="_ _1"></span>me SMO using a low sam<span class="_ _0"></span>pli<span class="_ _1"></span>ng freque<span class="_ _1"></span>ncy, the </div><div class="t m0 x11 h4 y4e ff1 fs2 fc0 sc0 ls3e ws4d">waveform<span class="_ _0"></span> of<span class="_ _0"></span> the <span class="_ _0"></span>estimate<span class="_ _0"></span>d <span class="_ _0"></span>back<span class="_ _0"></span> EMF w<span class="_ _0"></span>ill h<span class="_ _0"></span>ave <span class="_ _0"></span>distorti<span class="_ _0"></span>ons, </div><div class="t m0 x11 h4 y4f ff1 fs2 fc0 sc0 ls48 ws4e">which incl<span class="_ _1"></span>ude both pha<span class="_ _1"></span>se shift and magnitud<span class="_ _1"></span>e variat<span class="_ _1"></span>ion. The </div><div class="t m0 x11 h4 y50 ff1 fs2 fc0 sc0 ls45 ws4f">degree <span class="_ _0"></span>of the <span class="_ _0"></span>distorti<span class="_ _0"></span>on w<span class="_ _0"></span>ill becom<span class="_ _0"></span>e larger w<span class="_ _0"></span>hen<span class="_ _0"></span> the sam<span class="_ _0"></span>pling </div><div class="t m0 x11 h4 y51 ff1 fs2 fc0 sc0 ls49 ws50">freq<span class="_ _1"></span>uency decr<span class="_ _1"></span>ea<span class="_ _1"></span>ses. As a result, conve<span class="_ _1"></span>ntiona<span class="_ _1"></span>l angle ex<span class="_ _1"></span>traction </div><div class="t m0 x11 h4 y52 ff1 fs2 fc0 sc0 ls3e ws51">methods,<span class="_ _0"></span> e.g.<span class="_ _0"></span>, the<span class="_ _0"></span> inve<span class="_ _0"></span>rse tan<span class="_ _0"></span>gent m<span class="_ _0"></span>ethod <span class="_ _0"></span>and an<span class="_ _0"></span>gle <span class="_ _0"></span>tracki<span class="_ _0"></span>ng </div><div class="t m0 x11 h4 y53 ff1 fs2 fc0 sc0 ls46 ws52">observer, w<span class="_ _0"></span>ill hav<span class="_ _0"></span>e an <span class="_ _0"></span>oscillati<span class="_ _0"></span>on pro<span class="_ _0"></span>blem, w<span class="_ _0"></span>hich resu<span class="_ _0"></span>lts in<span class="_ _0"></span> </div><div class="t m0 x11 h4 y54 ff1 fs2 fc0 sc0 ls4a ws53">degra<span class="_ _0"></span>dation<span class="_ _0"></span> of the<span class="_ _0"></span> dy<span class="_ _0"></span>nami<span class="_ _0"></span>c perfo<span class="_ _0"></span>rm<span class="_ _0"></span>ance of th<span class="_ _0"></span>e SMO an<span class="_ _0"></span>d lar<span class="_ _0"></span>ge </div><div class="t m0 x11 h4 y55 ff1 fs2 fc0 sc0 ls34 ws54">noise an<span class="_ _0"></span>d err<span class="_ _0"></span>ors in<span class="_ _0"></span> the<span class="_ _0"></span> estim<span class="_ _0"></span>ated rot<span class="_ _0"></span>or <span class="_ _0"></span>position<span class="_ _0"></span>. <span class="_ _0"></span> </div><div class="t m0 x12 h4 y56 ff1 fs2 fc0 sc0 ls2a ws55">This p<span class="_ _1"></span>aper propo<span class="_ _1"></span>ses a <span class="_ _1"></span>novel e<span class="_ _1"></span>stimated <span class="_ _1"></span>speed fe<span class="_ _1"></span>edback </div><div class="t m0 x11 h4 y57 ff1 fs2 fc0 sc0 ls3d ws56">algorithm<span class="_ _0"></span>, w<span class="_ _0"></span>hich will w<span class="_ _0"></span>ork togeth<span class="_ _0"></span>er w<span class="_ _0"></span>ith th<span class="_ _0"></span>e conventi<span class="_ _0"></span>onal<span class="_ _0"></span> </div><div class="t m0 x11 h4 y58 ff1 fs2 fc0 sc0 ls33 ws57">inver<span class="_ _1"></span>se tangent method<span class="_ _1"></span> for rotor po<span class="_ _1"></span>sition extrac<span class="_ _1"></span>tion. This </div><div class="t m0 x11 h4 y59 ff1 fs2 fc0 sc0 ls7 ws47">method w<span class="_ _0"></span>ill h<span class="_ _0"></span>ave a f<span class="_ _0"></span>ilte<span class="_ _0"></span>ring <span class="_ _0"></span>eff<span class="_ _0"></span>ect t<span class="_ _0"></span>o the <span class="_ _0"></span>estim<span class="_ _0"></span>ated <span class="_ _0"></span>position,<span class="_ _0"></span> </div><div class="t m0 x11 h4 y5a ff1 fs2 fc0 sc0 ls4b ws58">and m<span class="_ _0"></span>itigate th<span class="_ _0"></span>e p<span class="_ _0"></span>osition <span class="_ _0"></span>oscillati<span class="_ _0"></span>on cau<span class="_ _0"></span>sed <span class="_ _0"></span>by the<span class="_ _0"></span> low<span class="_ _0"></span> </div><div class="t m0 x11 h4 y5b ff1 fs2 fc0 sc0 ls4c ws59">sampling fre<span class="_ _0"></span>quency. The<span class="_ _0"></span> proposed os<span class="_ _0"></span>cillati<span class="_ _0"></span>on mitig<span class="_ _0"></span>ation </div><div class="t m0 x11 h4 y5c ff1 fs2 fc0 sc0 ls2e ws2b">algorithm<span class="_ _0"></span> is v<span class="_ _0"></span>alidated<span class="_ _0"></span> by<span class="_ _0"></span> simulations<span class="_ _0"></span> in <span class="_ _0"></span>MA<span class="_ _0"></span>TLAB Simulin<span class="_ _0"></span>k as </div><div class="t m0 x11 h4 y5d ff1 fs2 fc0 sc0 ls48 ws5a">well as exper<span class="_ _1"></span>iments on a high-po<span class="_ _1"></span>wer (155 kW) IPMSM drive </div><div class="t m0 x11 h4 y5e ff1 fs2 fc0 sc0 ls4d ws5b">system for<span class="_ _1"></span> off-road hyb<span class="_ _1"></span>rid electri<span class="_ _1"></span>c vehicles<span class="_ _1"></span>. </div><div class="t m0 x10 h6 y5f ff5 fs5 fc0 sc0 ls7 ws7">978-1-4673-1408-4/12/$31.00 ©2012 IEEE</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>
</body>
</html>
<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/626773634c65f412596c8308/bg2.jpg"><div class="t m0 x10 h4 y60 ff1 fs2 fc0 sc0 ls23 ws7">II.<span class="ff4 ls7"> <span class="_ _4"> </span><span class="ff1">D<span class="fs4 ls4e">IS<span class="_ _1"></span>C<span class="_ _1"></span>R<span class="_ _1"></span>E<span class="_ _1"></span>T<span class="_ _1"></span>E<span class="_ _1"></span></span><span class="ls4f">-T<span class="_ _1"></span><span class="fs4 ls4e">IM<span class="_ _1"></span>E<span class="_ _1"></span> <span class="_ _1"></span></span></span>S<span class="_ _1"></span><span class="fs4 ls50">LID<span class="_ _1"></span>ING</span><span class="ls4f">-M<span class="_ _1"></span><span class="fs4 ls51">ODE <span class="_ _1"></span></span></span>P<span class="fs4 ls52">OSIITON </span>O<span class="fs4 ls53">BSERVER <span class="_ _0"></span><span class="fs2 ls7"> </span></span></span></span></div><div class="t m0 x10 h4 y61 ff1 fs2 fc0 sc0 ls25 ws5c">Due to<span class="_ _0"></span> the s<span class="_ _0"></span>ali<span class="_ _0"></span>ency<span class="_ _0"></span> of<span class="_ _0"></span> IPMS<span class="_ _0"></span>M (i<span class="_ _0"></span>.e.,<span class="_ _0"></span> <span class="ff6 ls7 ws7">L</span></div><div class="t m0 x13 h7 y62 ff6 fs6 fc0 sc0 ls7 ws7">d</div><div class="t m0 x14 h4 y63 ff6 fs2 fc0 sc0 ls7 ws7"> <span class="_ _5"></span><span class="ff7">≠<span class="ff1"> <span class="_ _5"></span></span></span>L</div><div class="t m0 x15 h7 y62 ff6 fs6 fc0 sc0 ls7 ws7">q</div><div class="t m0 x16 h4 y63 ff1 fs2 fc0 sc0 ls34 ws5d">), both th<span class="_ _0"></span>e back </div><div class="t m0 xe h4 y64 ff1 fs2 fc0 sc0 ls26 ws5e">EMF and<span class="_ _0"></span> the in<span class="_ _0"></span>ductan<span class="_ _0"></span>ce m<span class="_ _0"></span>atrix c<span class="_ _0"></span>ontain<span class="_ _0"></span> the in<span class="_ _0"></span>form<span class="_ _0"></span>ation of<span class="_ _0"></span> th<span class="_ _0"></span>e </div><div class="t m0 xe h4 y65 ff1 fs2 fc0 sc0 ls54 ws5f">rotor <span class="_ _0"></span>positi<span class="_ _0"></span>on ang<span class="_ _0"></span>le. <span class="_ _0"></span>Moreov<span class="_ _0"></span>er, s<span class="_ _0"></span>ince<span class="_ _0"></span> the in<span class="_ _0"></span>duct<span class="_ _0"></span>ance m<span class="_ _0"></span>atrix<span class="_ _0"></span> </div><div class="t m0 xe h4 y66 ff1 fs2 fc0 sc0 ls55 ws60">contains bo<span class="_ _1"></span>th <span class="ff6 ls7 ws7">2<span class="ff8">θ</span></span></div><div class="t m0 x17 h7 y67 ff6 fs6 fc0 sc0 ls56 ws7">re</div><div class="t m0 x18 h4 y68 ff1 fs2 fc0 sc0 ls27 ws61"> and <span class="ff8 ls7 ws7">θ</span></div><div class="t m0 x19 h7 y67 ff6 fs6 fc0 sc0 ls56 ws7">re</div><div class="t m0 x1a h4 y68 ff1 fs2 fc0 sc0 ls4c ws62"> term<span class="_ _0"></span>s, it is<span class="_ _0"></span> not easy<span class="_ _0"></span> t<span class="_ _0"></span>o obtain <span class="_ _0"></span>th<span class="_ _0"></span>e </div><div class="t m0 xe h4 y69 ff1 fs2 fc0 sc0 ls34 ws63">rotor p<span class="_ _0"></span>ositi<span class="_ _0"></span>on from<span class="_ _0"></span> the <span class="_ _0"></span>back EMF<span class="_ _0"></span> dire<span class="_ _0"></span>ctly<span class="_ _0"></span>. To f<span class="_ _0"></span>acilit<span class="_ _0"></span>ate <span class="_ _0"></span>the </div><div class="t m0 xe h4 y6a ff1 fs2 fc0 sc0 ls35 ws64">rotor<span class="_ _0"></span> pos<span class="_ _0"></span>it<span class="_ _0"></span>ion<span class="_ _0"></span> obs<span class="_ _0"></span>erv<span class="_ _0"></span>ati<span class="_ _0"></span>on,<span class="_ _0"></span> an <span class="_ _0"></span>ext<span class="_ _0"></span>ende<span class="_ _0"></span>d b<span class="_ _0"></span>ack<span class="_ _0"></span> EMF-<span class="_ _0"></span>based<span class="_ _0"></span> m<span class="_ _0"></span>odel<span class="_ _0"></span> </div><div class="t m0 xe h4 y6b ff1 fs2 fc0 sc0 ls31 ws30">for IPMSMs<span class="_ _0"></span> is <span class="_ _0"></span>propos<span class="_ _0"></span>ed in <span class="_ _0"></span>[9]<span class="_ _0"></span>, wh<span class="_ _0"></span>ich can <span class="_ _0"></span>be fu<span class="_ _0"></span>rther<span class="_ _0"></span> written<span class="_ _0"></span> as: <span class="_ _0"></span> </div><div class="t m0 x1a h8 y6c ff1 fs7 fc0 sc0 ls7 ws7">+<span class="_ _6"> </span><span class="fs8 ls57">sin</span></div><div class="t m0 x1b h8 y6d ff1 fs8 fc0 sc0 ls58 ws7">cos</div><div class="t m0 x1c h9 y6e ff6 fs9 fc0 sc0 ls59 ws7">qd</div><div class="t m0 x1d h9 y6f ff6 fs9 fc0 sc0 ls5a ws7">re<span class="_ _7"> </span>re</div><div class="t m0 x1e h9 y70 ff6 fs9 fc0 sc0 ls5b ws7">dd<span class="_ _8"> </span>d<span class="_ _9"> </span>d</div><div class="t m0 x1f h9 y71 ff6 fs9 fc0 sc0 ls59 ws7">qd</div><div class="t m0 x20 h9 y72 ff6 fs9 fc0 sc0 ls5a ws7">re<span class="_ _a"> </span>re</div><div class="t m0 x21 h9 y73 ff6 fs9 fc0 sc0 ls5c ws7">dd<span class="_ _b"> </span>d<span class="_ _9"> </span>d</div><div class="t m0 x22 ha y74 ff6 fs8 fc0 sc0 ls5d ws7">LL</div><div class="t m0 x23 ha y75 ff6 fs8 fc0 sc0 ls5e ws7">di<span class="_ _c"> </span>v</div><div class="t m0 x24 ha y76 ff6 fs8 fc0 sc0 ls7 ws7">R</div><div class="t m0 x25 ha y77 ff6 fs8 fc0 sc0 ls5f ws7">ii</div><div class="t m0 x26 ha y78 ff6 fs8 fc0 sc0 ls60 ws7">dt<span class="_ _d"> </span>L<span class="_ _e"> </span>L<span class="_ _f"> </span>L<span class="_ _10"> </span>L</div><div class="t m0 x27 ha y79 ff6 fs8 fc0 sc0 ls5d ws7">LL</div><div class="t m0 x28 ha y7a ff6 fs8 fc0 sc0 ls5e ws7">di<span class="_ _c"> </span>v</div><div class="t m0 x29 ha y7b ff6 fs8 fc0 sc0 ls7 ws7">R</div><div class="t m0 x2a ha y7c ff6 fs8 fc0 sc0 ls61 ws7">ii</div><div class="t m0 x23 ha y7d ff6 fs8 fc0 sc0 ls60 ws7">dt<span class="_ _d"> </span>L<span class="_ _e"> </span>L<span class="_ _11"> </span>L<span class="_ _10"> </span>L</div><div class="t m1 x2b hb y7e ff9 fsa fc0 sc0 ls62 ws7">αα</div><div class="t m1 x2c hb y7f ff9 fsa fc0 sc0 ls63 ws7">αβ</div><div class="t m1 x2d hb y80 ff9 fsa fc0 sc0 ls64 ws7">ββ</div><div class="t m1 x25 hb y81 ff9 fsa fc0 sc0 ls65 ws7">βα</div><div class="t m2 x2e hc y82 ff9 fsb fc0 sc0 ls7 ws7">η</div><div class="t m2 x2 hc y83 ff9 fsb fc0 sc0 ls66 ws7">ωθ</div><div class="t m2 x2f hc y84 ff9 fsb fc0 sc0 ls7 ws7">η</div><div class="t m2 x30 hc y85 ff9 fsb fc0 sc0 ls67 ws7">ωθ</div><div class="t m0 x31 hd y74 ff9 fs8 fc0 sc0 ls7 ws7">−</div><div class="t m0 x32 he y86 ffa fs8 fc0 sc0 ls7 ws7">⎧</div><div class="t m0 x33 hd y77 ff9 fs8 fc0 sc0 ls68 ws7">=−<span class="_ _12"> </span>+</div><div class="t m0 x32 he y87 ffa fs8 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x32 he y88 ffa fs8 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x32 he y89 ffa fs8 fc0 sc0 ls7 ws7">⎨</div><div class="t m0 x34 hd y8a ff9 fs8 fc0 sc0 ls7 ws7">−</div><div class="t m0 x32 he y8b ffa fs8 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x35 hd y8c ff9 fs8 fc0 sc0 ls69 ws7">=−<span class="_ _13"> </span>−<span class="_ _14"> </span>−</div><div class="t m0 x32 he y8d ffa fs8 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x32 he y8e ffa fs8 fc0 sc0 ls7 ws7">⎩</div><div class="t m0 x36 h4 y6d ff1 fs2 fc0 sc0 ls3f ws7"> </div><div class="t m0 x37 h4 y8f ff1 fs2 fc0 sc0 ls6a ws7">(1) </div><div class="t m0 xe h4 y90 ff1 fs2 fc0 sc0 ls2d ws7">where <span class="_ _1"></span><span class="ff6 ls7">v</span></div><div class="t m0 x38 h7 y91 ff8 fs6 fc0 sc0 ls7 ws7">α</div><div class="t m0 x39 h4 y92 ff1 fs2 fc0 sc0 ls7 ws7">,<span class="ff6 ls6b"> v</span></div><div class="t m0 x2d h7 y91 ff8 fs6 fc0 sc0 ls7 ws7">β</div><div class="t m0 x4 h4 y92 ff1 fs2 fc0 sc0 ls3f ws7">, <span class="_ _1"></span><span class="ff6 ls7">i</span></div><div class="t m0 x33 h7 y91 ff8 fs6 fc0 sc0 ls7 ws7">α</div><div class="t m0 x5 h4 y92 ff1 fs2 fc0 sc0 ls6c ws65"> and<span class="_ _0"></span> <span class="ff6 ls7 ws7">i</span></div><div class="t m0 x3a h7 y91 ff8 fs6 fc0 sc0 ls7 ws7">β</div><div class="t m0 x29 h4 y92 ff1 fs2 fc0 sc0 ls2d ws66"> are stator voltages and cu<span class="_ _0"></span>rrents in the <span class="ff8 ls7 ws7">α<span class="ff6">-</span></span></div><div class="t m0 xe h4 y93 ff8 fs2 fc0 sc0 ls7 ws7">β<span class="ff1 ls6d ws67"> st<span class="_ _0"></span>at<span class="_ _0"></span>ion<span class="_ _0"></span>a<span class="_ _0"></span>ry<span class="_ _0"></span> ref<span class="_ _0"></span>e<span class="_ _0"></span>re<span class="_ _0"></span>nc<span class="_ _0"></span>e f<span class="_ _0"></span>ram<span class="_ _0"></span>e; <span class="_ _0"></span><span class="ff8 ls7 ws7">ω</span></span></div><div class="t m0 x3b hf y94 ff6 fs6 fc0 sc0 ls56 ws7">re<span class="ff1 ls7"> </span></div><div class="t m0 x3c h4 y95 ff1 fs2 fc0 sc0 ls6e ws68">is rotor electrical angular </div><div class="t m0 xe h4 y96 ff1 fs2 fc0 sc0 ls39 ws7">speed; <span class="_ _15"> </span><span class="ff6 ls7">L</span></div><div class="t m0 x23 h7 y97 ff6 fs6 fc0 sc0 ls7 ws7">d </div><div class="t m0 x3d h4 y98 ff1 fs2 fc0 sc0 ls30 ws7">and <span class="_ _15"> </span><span class="ff6 ls7">L</span></div><div class="t m0 x3e h7 y97 ff6 fs6 fc0 sc0 ls7 ws7">q</div><div class="t m0 x3a h4 y98 ff1 fs2 fc0 sc0 ls2d ws69"> are <span class="ff6 ls7 ws7">d</span><span class="ls6f ws6a">-axis and<span class="_ _1"></span> <span class="ff6 ls7 ws7">q</span><span class="ls70 ws6b">-ax<span class="_ _0"></span>is<span class="_ _0"></span> in<span class="_ _0"></span>du<span class="_ _0"></span>ct<span class="_ _0"></span>ance<span class="_ _0"></span>s,<span class="_ _0"></span> </span></span></div><div class="t m0 xe h4 y99 ff1 fs2 fc0 sc0 ls71 ws7">respectively; <span class="_ _16"></span><span class="ff6 ls7">R</span><span class="ls72 ws6c"> is the stator resist<span class="_ _0"></span>ance; and <span class="ff8 ls7 ws7">η</span><span class="ls45 ws6d"> is the magnitude </span></span></div><div class="t m0 xe h4 y9a ff1 fs2 fc0 sc0 ls6e ws6e">of the ex<span class="_ _0"></span>tended back EMF term, wh<span class="_ _0"></span>ich equals to (<span class="ff6 ls7 ws7">L</span></div><div class="t m0 x3f h7 y9b ff6 fs6 fc0 sc0 ls7 ws7">d</div><div class="t m0 x40 h4 y9c ff1 fs2 fc0 sc0 ls27 ws6f"> – <span class="ff6 ls7 ws7">L</span></div><div class="t m0 x41 h7 y9b ff6 fs6 fc0 sc0 ls7 ws7">q</div><div class="t m0 x42 h4 y9c ff1 fs2 fc0 sc0 ls23 ws7">) </div><div class="t m0 xe h4 y9d ff1 fs2 fc0 sc0 ls7 ws7">(<span class="ff8">ω</span></div><div class="t m0 x43 h7 y9e ff6 fs6 fc0 sc0 ls56 ws7">re</div><div class="t m0 x44 h10 y9f ff6 fs2 fc0 sc0 ls7 ws7">i</div><div class="t m0 x45 h7 y9e ff6 fs6 fc0 sc0 ls7 ws7">d</div><div class="t m0 x46 h4 y9f ff6 fs2 fc0 sc0 ls7 ws7"> <span class="_ _1"></span><span class="ff1">–<span class="_ _1"></span></span><span class="ls73 ws70"> pi</span></div><div class="t m0 x47 h7 y9e ff6 fs6 fc0 sc0 ls7 ws7">q</div><div class="t m0 x48 h4 y9f ff1 fs2 fc0 sc0 ls4f ws71">) +<span class="ff6 ls7 ws7"> <span class="_ _16"></span><span class="ff8">ω</span></span></div><div class="t m0 x49 h7 y9e ff6 fs6 fc0 sc0 ls56 ws7">re</div><div class="t m0 x4a h10 y9f ff8 fs2 fc0 sc0 ls7 ws7">ψ</div><div class="t m0 x3 h7 y9e ff6 fs6 fc0 sc0 ls7 ws7">m</div><div class="t m0 x3a h4 y9f ff1 fs2 fc0 sc0 ls72 ws72">. In (1<span class="_ _0"></span>) only<span class="_ _0"></span> the extended back EMF term </div><div class="t m0 xe h4 ya0 ff1 fs2 fc0 sc0 ls74 ws73">contains the information of rotor position. If the extended back </div><div class="t m0 xe h4 ya1 ff1 fs2 fc0 sc0 ls34 ws74">EMF can be estimated, t<span class="_ _1"></span>he rotor po<span class="_ _1"></span>sition can be obtained </div><div class="t m0 xe h4 ya2 ff1 fs2 fc0 sc0 ls75 ws75">directly. The s<span class="_ _0"></span>liding-m<span class="_ _0"></span>ode current observ<span class="_ _0"></span>er is desi<span class="_ _0"></span>gned with </div><div class="t m0 xe h4 ya3 ff1 fs2 fc0 sc0 ls6e ws76">the same structure as (1): </div><div class="t m0 x10 h4 ya4 ff1 fs2 fc0 sc0 ls3f ws7"> </div><div class="t m0 x4b h11 ya5 ff1 fsc fc0 sc0 ls7 ws7">*</div><div class="t m0 x49 h11 ya6 ff1 fsc fc0 sc0 ls7 ws7">*</div><div class="t m0 x48 h12 ya7 ff1 fsd fc0 sc0 ls7 ws7">ˆ</div><div class="t m0 x2a h12 ya8 ff1 fsd fc0 sc0 ls7 ws7">ˆ</div><div class="t m0 x4c h12 ya9 ff1 fsd fc0 sc0 ls7 ws7">+</div><div class="t m0 x26 h12 yaa ff1 fsd fc0 sc0 ls7 ws7">ˆ</div><div class="t m0 x4d h12 yab ff1 fsd fc0 sc0 ls7 ws7">ˆ</div><div class="t m0 x27 h13 yac ff6 fsc fc0 sc0 ls76 ws7">qd</div><div class="t m0 x4e h13 yad ff6 fsc fc0 sc0 ls77 ws7">re</div><div class="t m0 x4b h13 yae ff6 fsc fc0 sc0 ls78 ws7">dd<span class="_ _17"> </span>d</div><div class="t m0 x4f h13 yaf ff6 fsc fc0 sc0 ls76 ws7">qd</div><div class="t m0 x50 h13 yb0 ff6 fsc fc0 sc0 ls77 ws7">re</div><div class="t m0 x4b h13 yb1 ff6 fsc fc0 sc0 ls78 ws7">dd<span class="_ _18"> </span>d</div><div class="t m0 x51 h14 yb2 ff6 fsd fc0 sc0 ls79 ws7">LL</div><div class="t m0 x28 h14 yb3 ff6 fsd fc0 sc0 ls7a ws7">di<span class="_ _e"> </span>v</div><div class="t m0 x52 h14 yb4 ff6 fsd fc0 sc0 ls7 ws7">R</div><div class="t m0 x4d h14 yb5 ff6 fsd fc0 sc0 ls7b ws7">ii<span class="_ _19"></span>l<span class="_ _1a"></span>Z</div><div class="t m0 x23 h14 yb6 ff6 fsd fc0 sc0 ls7c ws7">dt<span class="_ _d"> </span>L<span class="_ _d"> </span>L<span class="_ _1b"> </span>L</div><div class="t m0 x53 h14 yb7 ff6 fsd fc0 sc0 ls79 ws7">LL</div><div class="t m0 x39 h14 yb8 ff6 fsd fc0 sc0 ls7d ws7">di<span class="_ _e"> </span>v</div><div class="t m0 x54 h14 yb9 ff6 fsd fc0 sc0 ls7 ws7">R</div><div class="t m0 x55 h14 yba ff6 fsd fc0 sc0 ls7e ws7">ii<span class="_ _1c"></span>l<span class="_ _1d"></span>Z</div><div class="t m0 x56 h14 ybb ff6 fsd fc0 sc0 ls7c ws7">dt<span class="_ _d"> </span>L<span class="_ _d"> </span>L<span class="_ _1e"> </span>L</div><div class="t m3 x2d h15 ybc ff9 fse fc0 sc0 ls7f ws7">αα</div><div class="t m3 x57 h15 ybd ff9 fse fc0 sc0 ls7 ws7">α</div><div class="c x57 ybe w2 h16"><div class="t m3 x58 h15 ybf ff9 fse fc0 sc0 ls7 ws7">β</div></div><div class="t m3 x59 h15 yc0 ff9 fse fc0 sc0 ls7 ws7">α</div><div class="t m3 x48 h15 yc1 ff9 fse fc0 sc0 ls80 ws7">ββ</div><div class="c x2a yc2 w3 h16"><div class="t m3 x0 h15 ybf ff9 fse fc0 sc0 ls7 ws7">β</div></div><div class="t m3 x5a h15 yc3 ff9 fse fc0 sc0 ls7 ws7">α</div><div class="c x2a yc2 w3 h16"><div class="t m3 x2b h15 ybf ff9 fse fc0 sc0 ls7 ws7">β</div></div><div class="t m4 x5b h17 yc4 ff9 fsf fc0 sc0 ls7 ws7">ω</div><div class="t m4 x5c h17 yc5 ff9 fsf fc0 sc0 ls7 ws7">ω</div><div class="t m5 x5d h18 yb2 ffa fsd fc0 sc0 ls7 ws7">⎧<span class="_ _1f"> </span><span class="ff9">−</span></div><div class="t m5 x35 h19 yb5 ff9 fsd fc0 sc0 ls81 ws7">=−<span class="_ _20"> </span>+<span class="_ _21"></span>×</div><div class="t m5 x5d h18 yc6 ffa fsd fc0 sc0 ls7 ws7">⎪</div><div class="t m5 x5d h18 yc7 ffa fsd fc0 sc0 ls7 ws7">⎪</div><div class="t m5 x5d h18 yc8 ffa fsd fc0 sc0 ls7 ws7">⎨</div><div class="t m5 x5e h19 yb7 ff9 fsd fc0 sc0 ls7 ws7">−</div><div class="t m5 x5d h18 yc9 ffa fsd fc0 sc0 ls7 ws7">⎪</div><div class="t m5 x5f h19 yba ff9 fsd fc0 sc0 ls81 ws7">=−<span class="_ _22"> </span>−<span class="_ _23"> </span>−<span class="_ _21"></span>×</div><div class="t m5 x5d h18 yca ffa fsd fc0 sc0 ls7 ws7">⎪</div><div class="t m5 x5d h18 ycb ffa fsd fc0 sc0 ls7 ws7">⎩</div><div class="t m5 x60 h4 ycc ff1 fs2 fc0 sc0 ls6b ws7"> <span class="_ _0"></span> <span class="_ _0"></span> <span class="_ _0"></span> </div><div class="t m5 x37 h4 ycd ff1 fs2 fc0 sc0 ls23 ws7">(2) </div><div class="t m5 xe h4 yce ff1 fs2 fc0 sc0 ls2d ws7">where <span class="_ _5"></span><span class="ff6 ls7">l</span><span class="ls3d ws77"> is the SMO gain <span class="_ _0"></span>of the sw<span class="_ _0"></span>itch<span class="_ _0"></span>ing control vector <span class="ff6 ls7 ws7">Z</span></span></div><div class="t m5 x61 h7 ycf ff8 fs6 fc0 sc0 ls82 ws7">αβ</div><div class="t m5 x42 h4 yd0 ff1 fs2 fc0 sc0 ls4 ws7">; </div><div class="t m5 xe h10 yd1 ff6 fs2 fc0 sc0 ls7 ws7">v</div><div class="t m5 x1 h7 yd2 ff6 fs6 fc0 sc0 ls7 ws7">*</div><div class="t m5 x62 h7 yd3 ff8 fs6 fc0 sc0 ls7 ws7">α</div><div class="t m5 x43 h4 yd4 ff1 fs2 fc0 sc0 ls30 ws78"> and <span class="ff6 ls7 ws7">v</span></div><div class="t m5 x63 h7 yd2 ff6 fs6 fc0 sc0 ls7 ws7">*</div><div class="t m5 x56 h7 yd3 ff8 fs6 fc0 sc0 ls7 ws7">β</div><div class="t m5 x26 h4 yd4 ff1 fs2 fc0 sc0 ls2d ws79"> are commanded voltag<span class="_ _0"></span>es obtained from th<span class="_ _0"></span>e current </div><div class="t m5 xe h4 yd5 ff1 fs2 fc0 sc0 ls4a ws7a">regulated vector con<span class="_ _0"></span>trol of the <span class="ls41 ws7b">IPMSM. If the IGBT<span class="_ _1"></span> dead-time </span></div><div class="t m5 xe h4 yd6 ff1 fs2 fc0 sc0 ls39 ws7c">effect is w<span class="_ _0"></span>ell compensated, <span class="ff6 ls7 ws7">v</span></div><div class="t m5 x22 h7 yd7 ff6 fs6 fc0 sc0 ls7 ws7">*</div><div class="t m5 x4f h7 yd8 ff8 fs6 fc0 sc0 ls82 ws7">αβ</div><div class="t m5 x5e h4 yd9 ff1 fs2 fc0 sc0 ls30 ws7d"> should be i<span class="_ _0"></span>dentical t<span class="_ _0"></span>o <span class="ff6 ls7 ws7">v</span></div><div class="t m5 x41 h7 yda ff8 fs6 fc0 sc0 ls82 ws7">αβ</div><div class="t m5 x64 h4 yd9 ff1 fs2 fc0 sc0 ls7 ws7"> </div><div class="t m5 xe h4 ydb ff1 fs2 fc0 sc0 ls41 ws7e">measured from the IPMSM stator term<span class="_ _0"></span>inals. Subtracting (2) </div><div class="t m5 xe h4 ydc ff1 fs2 fc0 sc0 ls35 ws7f">from (1) t<span class="_ _0"></span>he foll<span class="_ _0"></span>owing equati<span class="_ _0"></span>ons can be obtain<span class="_ _0"></span>ed. </div><div class="t m5 x46 h4 ydd ff1 fs2 fc0 sc0 ls3f ws7"> </div><div class="t m5 x65 h1a yde ff1 fs10 fc0 sc0 ls83 ws7">re</div><div class="t m5 x65 h1a ydf ff1 fs10 fc0 sc0 ls83 ws7">re</div><div class="t m5 x3e h1b ye0 ff1 fs11 fc0 sc0 ls7 ws7">)</div><div class="t m5 x5e h1b ye1 ff1 fs11 fc0 sc0 ls7 ws7">)</div><div class="t m5 x3e h1b ye2 ff1 fs11 fc0 sc0 ls7 ws7">)</div><div class="t m5 x3b h1b ye3 ff1 fs11 fc0 sc0 ls7 ws7">)</div><div class="t m5 x66 h1c ye4 ff1 fs12 fc0 sc0 ls7 ws7">ˆ</div><div class="t m5 x33 h1c ye5 ff1 fs12 fc0 sc0 ls7 ws7">(</div><div class="t m5 x27 h1c ye6 ff1 fs12 fc0 sc0 ls7 ws7">ˆ</div><div class="t m5 x2 h1c ye7 ff1 fs12 fc0 sc0 ls84 ws7">(s<span class="_ _24"></span>i<span class="_ _24"></span>n</div><div class="t m5 x67 h1c ye8 ff1 fs12 fc0 sc0 ls7 ws7">ˆ</div><div class="t m5 x68 h1c ye9 ff1 fs12 fc0 sc0 ls7 ws7">(</div><div class="t m5 x27 h1c yea ff1 fs12 fc0 sc0 ls7 ws7">ˆ</div><div class="t m5 x2 h1c yeb ff1 fs12 fc0 sc0 ls84 ws7">(c<span class="_ _24"></span>o<span class="_ _24"></span>s</div><div class="t m5 x69 h1d yec ff6 fs10 fc0 sc0 ls85 ws7">dd</div><div class="t m5 x69 h1d yed ff6 fs10 fc0 sc0 ls86 ws7">dd</div><div class="t m5 x3d h1e ye0 ff6 fs12 fc0 sc0 ls87 ws7">di<span class="_ _25"> </span>i</div><div class="t m5 x19 h1e yee ff6 fs12 fc0 sc0 ls7 ws7">R</div><div class="t m5 x4e h1e yef ff6 fs12 fc0 sc0 ls88 ws7">ii<span class="_ _1e"> </span>l<span class="_ _26"></span>Z</div><div class="t m5 x6 h1e yf0 ff6 fs12 fc0 sc0 ls89 ws7">dt<span class="_ _27"> </span>L<span class="_ _28"> </span>L</div><div class="t m5 x6a h1e yf1 ff6 fs12 fc0 sc0 ls87 ws7">di<span class="_ _25"> </span>i</div><div class="t m5 x19 h1e yf2 ff6 fs12 fc0 sc0 ls7 ws7">R</div><div class="t m5 x6b h1e yf3 ff6 fs12 fc0 sc0 ls88 ws7">ii<span class="_ _29"> </span>l<span class="_ _26"></span>Z</div><div class="t m5 x6 h1e yf4 ff6 fs12 fc0 sc0 ls8a ws7">dt<span class="_ _2a"> </span>L<span class="_ _28"> </span>L</div><div class="t m6 x6c h1f yf5 ff9 fs13 fc0 sc0 ls8b ws7">αα</div><div class="t m6 x50 h1f yf6 ff9 fs13 fc0 sc0 ls8b ws7">αα<span class="_ _2b"> </span>α</div><div class="t m6 x6c h1f yf7 ff9 fs13 fc0 sc0 ls8c ws7">ββ</div><div class="t m6 x50 h1f yf8 ff9 fs13 fc0 sc0 ls8b ws7">αα<span class="_ _2c"> </span>β</div><div class="t m7 x6d h20 yf9 ff9 fs14 fc0 sc0 ls7 ws7">η</div><div class="t m7 x6e h20 yfa ff9 fs14 fc0 sc0 ls7 ws7">θ</div><div class="t m7 x6d h20 yfb ff9 fs14 fc0 sc0 ls7 ws7">η</div><div class="t m7 x6f h20 yfc ff9 fs14 fc0 sc0 ls7 ws7">θ</div><div class="t m5 x4b h21 ye0 ff9 fs11 fc0 sc0 ls7 ws7">−</div><div class="t m5 x51 h21 ye1 ff9 fs11 fc0 sc0 ls7 ws7">−</div><div class="t m5 x4b h21 ye2 ff9 fs11 fc0 sc0 ls7 ws7">−</div><div class="t m5 x70 h21 ye3 ff9 fs11 fc0 sc0 ls7 ws7">−</div><div class="t m5 x48 h22 yfd ffa fs12 fc0 sc0 ls7 ws7">⎧</div><div class="t m5 x52 h23 yfe ff9 fs12 fc0 sc0 ls8d ws7">=−<span class="_ _23"> </span>+<span class="_ _2d"> </span>+<span class="_ _4"> </span>×</div><div class="t m5 x48 h22 yff ffa fs12 fc0 sc0 ls7 ws7">⎪</div><div class="t m5 x48 h22 y100 ffa fs12 fc0 sc0 ls7 ws7">⎪</div><div class="t m5 x48 h22 y101 ffa fs12 fc0 sc0 ls7 ws7">⎨</div><div class="t m5 x48 h22 y102 ffa fs12 fc0 sc0 ls7 ws7">⎪</div><div class="t m5 x52 h23 y103 ff9 fs12 fc0 sc0 ls8d ws7">=−<span class="_ _23"> </span>−<span class="_ _2e"> </span>+<span class="_ _4"> </span>×</div><div class="t m5 x48 h22 y104 ffa fs12 fc0 sc0 ls7 ws7">⎪</div><div class="t m5 x48 h22 y105 ffa fs12 fc0 sc0 ls7 ws7">⎩</div><div class="t m5 x71 h4 y106 ff1 fs2 fc0 sc0 ls3f ws7"> </div><div class="t m5 x37 h4 y107 ff1 fs2 fc0 sc0 ls4f ws7">(3)</div><div class="t m5 x64 h4 y108 ff1 fs2 fc0 sc0 ls7 ws7"> </div><div class="t m5 xe h4 y109 ff1 fs2 fc0 sc0 ls8e ws7">Let</div><div class="t m5 x56 h24 y10a ff1 fs15 fc0 sc0 ls8f ws7">[s<span class="_ _2f"></span>i<span class="_ _2f"></span>n<span class="_ _30"> </span>c<span class="_ _2f"></span>o<span class="_ _31"></span>s<span class="_ _32"></span>]</div><div class="t m5 x72 h25 y10b ff6 fs16 fc0 sc0 ls7 ws7">T</div><div class="t m5 x33 h25 y10c ff6 fs16 fc0 sc0 ls90 ws7">dd</div><div class="t m5 x73 h25 y10d ff6 fs16 fc0 sc0 ls91 ws7">re<span class="_ _33"> </span>re</div><div class="t m5 x44 h26 y10a ff6 fs15 fc0 sc0 ls92 ws7">eL<span class="_ _34"> </span>L</div><div class="t m8 x72 h27 y10e ff9 fs17 fc0 sc0 ls93 ws7">αβ</div><div class="c x47 y10f w4 h28"><div class="t m9 x0 h29 y110 ff9 fs18 fc0 sc0 ls7 ws7">η</div></div><div class="t m9 x18 h29 y10a ff9 fs18 fc0 sc0 ls94 ws7">θη<span class="_ _35"> </span>θ</div><div class="t m0 x32 h2a y10a ff9 fs15 fc0 sc0 ls95 ws7">=−</div><div class="t m0 x74 h4 y111 ff1 fs2 fc0 sc0 ls7 ws7">,</div><div class="t m0 x75 h24 y10a ff1 fs15 fc0 sc0 ls96 ws7">[]</div><div class="t m0 x76 h25 y10b ff6 fs16 fc0 sc0 ls7 ws7">T</div><div class="c x77 y112 w5 h2b"><div class="t m0 x0 h26 y113 ff6 fs15 fc0 sc0 ls7 ws7">Z</div></div><div class="t m0 x59 h26 y10a ff6 fs15 fc0 sc0 ls97 ws7">ZZ</div><div class="c x78 y114 w6 h2c"><div class="t m8 x0 h27 y115 ff9 fs17 fc0 sc0 ls7 ws7">β</div></div><div class="t m8 x13 h27 y10e ff9 fs17 fc0 sc0 ls98 ws7">αβ<span class="_ _36"> </span>α</div><div class="t m0 x65 h2a y10a ff9 fs15 fc0 sc0 ls99 ws7">=−</div><div class="t m0 x79 h4 y111 ff1 fs2 fc0 sc0 ls9a ws80">, and </div><div class="t m0 x47 h24 y116 ff1 fs15 fc0 sc0 ls9b ws7">ˆˆ</div><div class="t m0 x7a h24 y117 ff1 fs15 fc0 sc0 ls9c ws7">[]</div><div class="t m0 x62 h25 y118 ff6 fs16 fc0 sc0 ls7 ws7">T</div><div class="t m0 x7b h26 y119 ff6 fs15 fc0 sc0 ls9d ws7">Si<span class="_ _37"></span>i<span class="_ _38"></span>i<span class="_ _37"></span>i</div><div class="t m8 x7c h27 y11a ff9 fs17 fc0 sc0 ls9e ws7">αα<span class="_ _4"> </span>β<span class="_ _16"></span>β</div><div class="t m0 x39 h2a y119 ff9 fs16 fc0 sc0 ls9f ws7">−−<span class="_ _39"></span><span class="fs15 ls7">=</span></div><div class="t m0 x54 h4 y11b ff1 fs2 fc0 sc0 lsa0 ws81">, (3) can be expres<span class="_ _0"></span>sed as:</div><div class="t m0 x15 h4 y11c ff1 fs2 fc0 sc0 ls7 ws7"> </div><div class="t m0 x63 h4 y11d ff1 fs2 fc0 sc0 ls3f ws7"> </div><div class="t m0 x17 h2d y11e ff6 fs19 fc0 sc0 lsa1 ws7">TT<span class="_ _3a"></span>T<span class="_ _3b"></span>T</div><div class="t m0 x2a h2d y11f ff6 fs19 fc0 sc0 ls7 ws7">d</div><div class="t m0 x7d h2e y120 ff6 fs1a fc0 sc0 ls7 ws7">R</div><div class="t m0 x7e h2e y121 ff6 fs1a fc0 sc0 lsa2 ws7">SS<span class="_ _31"></span>e<span class="_ _3c"></span>l<span class="_ _3d"></span>Z</div><div class="t m0 x7f h2e y122 ff6 fs1a fc0 sc0 ls7 ws7">L</div><div class="t ma x22 h2f y123 ff9 fs1b fc0 sc0 ls7 ws7">α</div><div class="c x22 y124 w7 h16"><div class="t ma x80 h2f ybf ff9 fs1b fc0 sc0 ls7 ws7">β</div></div><div class="t ma x81 h2f y123 ff9 fs1b fc0 sc0 ls7 ws7">α</div><div class="c x22 y124 w7 h16"><div class="t ma x82 h2f ybf ff9 fs1b fc0 sc0 ls7 ws7">β</div></div><div class="t m0 x83 h30 y125 ff9 fs1c fc0 sc0 ls7 ws7">=<span class="_ _3e"> </span><span class="fs1a lsa3">−+<span class="_ _3f"></span>−<span class="_ _3a"></span>×</span></div><div class="t m0 x49 h31 y126 ffb fs1a fc0 sc0 ls7 ws7"></div><div class="t m0 x6e h4 y127 ff1 fs2 fc0 sc0 lsa4 ws82"> <span class="_ _25"> </span> <span class="_ _40"> </span> (4)</div><div class="t m0 x84 h4 y128 ff1 fs2 fc0 sc0 ls7 ws7"> </div><div class="t m0 x10 h4 y129 ff1 fs2 fc0 sc0 ls6e ws83">Since the extended back EMF are boun<span class="_ _0"></span>ded, they can be </div><div class="t m0 xe h4 y12a ff1 fs2 fc0 sc0 ls6a ws84">suppres<span class="_ _0"></span>sed by th<span class="_ _0"></span>e discon<span class="_ _0"></span>tinuou<span class="_ _0"></span>s inpu<span class="_ _0"></span>t with <span class="ff6 ls7 ws85">l > max<span class="ff1 ws7">{<span class="ff6 ls25">|e</span></span></span></div><div class="t m0 x78 h7 y12b ff8 fs6 fc0 sc0 ls7 ws7">α</div><div class="t m0 x3f h10 y12a ff6 fs2 fc0 sc0 ls41 ws86">|, |e</div><div class="t m0 x85 h7 y12b ff8 fs6 fc0 sc0 ls7 ws7">β</div><div class="t m0 x61 h4 y12a ff6 fs2 fc0 sc0 ls7 ws7">|<span class="ff1">}</span>.<span class="ff1"> </span></div><div class="t m0 xe h4 y12c ff1 fs2 fc0 sc0 lsa5 ws87">When the state trajectory is en<span class="lsa6 ws88">forced to the <span class="_ _1"></span>sliding <span class="_ _1"></span>mode,</span></div><div class="t m0 x64 h4 y12d ff1 fs2 fc0 sc0 ls7 ws7"> </div><div class="t m0 x86 h32 y12e ff1 fs1d fc0 sc0 ls7 ws7">0</div><div class="t m0 x62 h33 y12f ff6 fs1e fc0 sc0 lsa7 ws7">TT</div><div class="t m0 x7b h34 y12e ff6 fs1d fc0 sc0 lsa8 ws7">SS<span class="_ _41"></span><span class="ff9 lsa9">==</span></div><div class="t m0 x1 h35 y130 ffb fs1d fc0 sc0 ls7 ws7"></div><div class="t m0 x6a h32 y12e ff1 fs1d fc0 sc0 lsaa ws89"> and </div><div class="t m0 x18 h25 y131 ff6 fs16 fc0 sc0 lsab ws7">TT</div><div class="t m0 x1e h26 y132 ff6 fs15 fc0 sc0 lsac ws7">el<span class="_ _42"></span>Z</div><div class="t m8 x18 h27 y133 ff9 fs17 fc0 sc0 lsad ws7">αβ<span class="_ _43"> </span>αβ</div><div class="t m0 x52 h2a y132 ff9 fs15 fc0 sc0 lsae ws7">=×</div><div class="t m0 x1d h32 y12e ff1 fs1d fc0 sc0 lsaf ws7"> . </div><div class="t m0 x10 h32 y134 ff1 fs1d fc0 sc0 lsb0 ws8a">In dig<span class="_ _0"></span>ital<span class="_ _0"></span> cont<span class="_ _0"></span>rol<span class="_ _0"></span> appl<span class="_ _0"></span>icat<span class="_ _0"></span>ion<span class="_ _0"></span>s, a dis<span class="_ _0"></span>cret<span class="_ _0"></span>e-t<span class="_ _0"></span>im<span class="_ _0"></span>e model<span class="_ _0"></span> of th<span class="_ _0"></span>e </div><div class="t m0 xe h32 y135 ff1 fs1d fc0 sc0 lsb1 ws8b">SMO is<span class="_ _0"></span> needed<span class="_ _0"></span>. Due<span class="_ _0"></span> to f<span class="_ _0"></span>ast calcu<span class="_ _0"></span>lati<span class="_ _0"></span>on and<span class="_ _0"></span> im<span class="_ _0"></span>plemen<span class="_ _0"></span>tation <span class="_ _0"></span>of </div><div class="t m0 xe h32 y136 ff1 fs1d fc0 sc0 lsb2 ws8c">switchin<span class="_ _0"></span>g fu<span class="_ _0"></span>nction in<span class="_ _0"></span> the SM<span class="_ _0"></span>O, the <span class="_ _0"></span>Eule<span class="_ _0"></span>r meth<span class="_ _0"></span>od is us<span class="_ _0"></span>ed <span class="_ _0"></span>to </div><div class="t m0 xe h32 y137 ff1 fs1d fc0 sc0 lsb3 ws8d">transfo<span class="_ _0"></span>rm th<span class="_ _0"></span>e cont<span class="_ _0"></span>inu<span class="_ _0"></span>ous-tim<span class="_ _0"></span>e SMO t<span class="_ _0"></span>o a dis<span class="_ _0"></span>cret<span class="_ _0"></span>e-tim<span class="_ _0"></span>e obse<span class="_ _0"></span>rv<span class="_ _0"></span>er, </div><div class="t m0 xe h32 y138 ff1 fs1d fc0 sc0 lsb4 ws8e">which can be e<span class="_ _1"></span>xpressed as following: </div><div class="t m0 x87 h36 y139 ff1 fs1f fc0 sc0 ls7 ws7">*</div><div class="t m0 x88 h37 y13a ff1 fs20 fc0 sc0 lsb5 ws7">[1<span class="_ _3f"></span>]<span class="_ _44"> </span>[<span class="_ _45"></span>]<span class="_ _46"> </span>[<span class="_ _45"></span>]<span class="_ _47"> </span>[<span class="_ _45"></span>]</div><div class="t m0 x89 h37 y13b ff1 fs20 fc0 sc0 lsb6 ws7">[1<span class="_ _48"></span>]<span class="_ _44"> </span>[<span class="_ _45"></span>]<span class="_ _46"> </span>[<span class="_ _45"></span>]<span class="_ _47"> </span>[<span class="_ _45"></span>]</div><div class="t m0 x8a h38 y13c ff1 fs21 fc0 sc0 lsb7 ws7">ˆˆ</div><div class="t m0 x8b h38 y13d ff1 fs21 fc0 sc0 ls7 ws7">1</div><div class="t m0 x87 h38 y13e ff1 fs21 fc0 sc0 ls7 ws7">*</div><div class="t m0 x8a h38 y13f ff1 fs21 fc0 sc0 lsb8 ws7">ˆˆ</div><div class="t m0 x8b h38 y140 ff1 fs21 fc0 sc0 ls7 ws7">1</div><div class="t m0 x8c h39 y141 ff6 fs1f fc0 sc0 lsb9 ws7">qd</div><div class="t m0 x8d h39 y142 ff6 fs1f fc0 sc0 ls7 ws7">s</div><div class="t m0 x8e h39 y143 ff6 fs1f fc0 sc0 lsba ws7">sr<span class="_ _49"></span>e</div><div class="t m0 x87 h39 y144 ff6 fs1f fc0 sc0 lsbb ws7">dd<span class="_ _4a"> </span>d</div><div class="t m0 x8c h39 y145 ff6 fs1f fc0 sc0 lsb9 ws7">qd</div><div class="t m0 x8d h39 y146 ff6 fs1f fc0 sc0 ls7 ws7">s</div><div class="t m0 x8f h39 y147 ff6 fs1f fc0 sc0 lsbc ws7">sr<span class="_ _49"></span>e</div><div class="t m0 x90 h39 y148 ff6 fs1f fc0 sc0 lsbd ws7">dd<span class="_ _4a"> </span>d</div><div class="t m0 x91 h3a y13a ff6 fs20 fc0 sc0 lsbe ws7">kk<span class="_ _39"></span>k<span class="_ _4b"></span>k</div><div class="t m0 x91 h3a y13b ff6 fs20 fc0 sc0 lsbf ws7">kk<span class="_ _4c"></span>k<span class="_ _4b"></span>k</div><div class="t m0 x92 h3b y149 ff6 fs21 fc0 sc0 lsc0 ws7">LL</div><div class="t m0 x93 h3b y14a ff6 fs21 fc0 sc0 lsc1 ws7">vT<span class="_ _4d"></span>R</div><div class="t m0 x94 h3b y14b ff6 fs21 fc0 sc0 lsc2 ws7">iT<span class="_ _4e"> </span>i<span class="_ _4f"></span>l<span class="_ _50"></span>Z<span class="_ _2a"> </span>i</div><div class="t m0 x93 h3b y14c ff6 fs21 fc0 sc0 lsc3 ws7">LL<span class="_ _4a"> </span>L</div><div class="t m0 x92 h3b y14d ff6 fs21 fc0 sc0 lsc0 ws7">LL<span class="_ _51"></span><span class="ls7">v</span></div><div class="t m0 x95 h3b y14e ff6 fs21 fc0 sc0 lsc4 ws7">TR</div><div class="t m0 x94 h3b y14f ff6 fs21 fc0 sc0 lsc5 ws7">iT<span class="_ _4e"> </span>i<span class="_ _48"></span>l<span class="_ _50"></span>Z<span class="_ _2a"> </span>i</div><div class="t m0 x96 h3b y150 ff6 fs21 fc0 sc0 lsc6 ws7">LL<span class="_ _4a"> </span>L</div><div class="t mb x87 h3c y151 ff9 fs22 fc0 sc0 ls7 ws7">α</div><div class="t mb x12 h3c y152 ff9 fs22 fc0 sc0 lsc7 ws7">αβ<span class="_ _52"></span>α<span class="_ _1d"></span>α</div><div class="t mb x87 h3c y153 ff9 fs22 fc0 sc0 ls7 ws7">β</div><div class="t mb x97 h3c y154 ff9 fs22 fc0 sc0 lsc8 ws7">βα<span class="_ _53"></span>β<span class="_ _54"></span>β</div><div class="t mc x98 h3d y13a ff9 fs23 fc0 sc0 ls7 ws7">ω</div><div class="t mc x98 h3d y155 ff9 fs23 fc0 sc0 ls7 ws7">ω</div><div class="t m0 x99 h3e y13a ff9 fs20 fc0 sc0 ls7 ws7">+</div><div class="t m0 x99 h3e y13b ff9 fs20 fc0 sc0 ls7 ws7">+</div><div class="t m0 x9a h3f y156 ffa fs21 fc0 sc0 ls7 ws7">⎧</div><div class="t m0 x9b h40 y157 ff9 fs21 fc0 sc0 ls7 ws7">−</div><div class="t m0 x9c h3f y158 ffa fs21 fc0 sc0 lsc9 ws7">⎛⎞</div><div class="t m0 x9d h3f y159 ffa fs21 fc0 sc0 lsca ws7">⎛⎞</div><div class="t m0 x9e h40 y15a ff9 fs21 fc0 sc0 lscb ws7">=+<span class="_ _40"> </span>+<span class="_ _16"></span>+<span class="_ _55"></span>−</div><div class="t m0 x9a h3f y15b ffa fs21 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x9c h3f y15c ffa fs21 fc0 sc0 lsc9 ws7">⎜⎟</div><div class="t m0 x9d h3f y15d ffa fs21 fc0 sc0 lsca ws7">⎜⎟</div><div class="t m0 x9c h3f y15e ffa fs21 fc0 sc0 lsc9 ws7">⎜⎟</div><div class="t m0 x9a h3f y15f ffa fs21 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x9d h3f y160 ffa fs21 fc0 sc0 lsca ws7">⎝⎠</div><div class="t m0 x9c h3f y161 ffa fs21 fc0 sc0 lsc9 ws7">⎝⎠</div><div class="t m0 x9a h3f y162 ffa fs21 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x9a h3f y163 ffa fs21 fc0 sc0 ls7 ws7">⎨</div><div class="t m0 x9c h3f y164 ffa fs21 fc0 sc0 lscc ws7">⎛⎞</div><div class="t m0 x9b h40 y165 ff9 fs21 fc0 sc0 ls7 ws7">−</div><div class="t m0 x9d h3f y166 ffa fs21 fc0 sc0 lsca ws7">⎛⎞</div><div class="t m0 x9a h3f y167 ffa fs21 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x9c h3f y168 ffa fs21 fc0 sc0 lscc ws7">⎜⎟</div><div class="t m0 x9e h40 y169 ff9 fs21 fc0 sc0 lscb ws7">=−<span class="_ _40"> </span>−<span class="_ _16"></span>+<span class="_ _3c"></span>−</div><div class="t m0 x9d h3f y16a ffa fs21 fc0 sc0 lsca ws7">⎜⎟</div><div class="t m0 x9a h3f y16b ffa fs21 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x9c h3f y16c ffa fs21 fc0 sc0 lscc ws7">⎜⎟</div><div class="t m0 x9d h3f y16d ffa fs21 fc0 sc0 lsca ws7">⎝⎠</div><div class="t m0 x9a h3f y16e ffa fs21 fc0 sc0 ls7 ws7">⎪</div><div class="t m0 x9c h3f y16f ffa fs21 fc0 sc0 lscc ws7">⎝⎠</div><div class="t m0 x9a h3f y170 ffa fs21 fc0 sc0 ls7 ws7">⎩</div><div class="t m0 x9f h32 y171 ff1 fs1d fc0 sc0 lsaf ws7"> </div><div class="t m0 xa0 h32 y172 ff1 fs1d fc0 sc0 lscd ws7">(5)</div><div class="t m0 xa1 h32 y173 ff1 fs1d fc0 sc0 ls7 ws7"> </div><div class="t m0 x11 h32 y174 ff1 fs1d fc0 sc0 lsce ws7">wher<span class="_ _1"></span>e <span class="_ _1"></span><span class="ff6 ls7">T</span></div><div class="t m0 x9 h7 y175 ff6 fs6 fc0 sc0 ls7 ws7">s</div><div class="t m0 xa2 h32 y176 ff1 fs1d fc0 sc0 lscf ws8f"> is <span class="_ _0"></span>the sam<span class="_ _0"></span>pling<span class="_ _0"></span> peri<span class="_ _0"></span>od <span class="_ _0"></span>of th<span class="_ _0"></span>e SMO<span class="_ _0"></span>, w<span class="_ _0"></span>hich is<span class="_ _0"></span> no<span class="_ _0"></span>rmally </div><div class="t m0 x11 h32 y177 ff1 fs1d fc0 sc0 lsd0 ws90">equal to<span class="_ _0"></span> one PWM<span class="_ _0"></span> cy<span class="_ _0"></span>cle; </div><div class="t m0 xa3 h41 y178 ff1 fs24 fc0 sc0 ls7 ws7">ˆ</div><div class="t m0 xa4 h41 y179 ff1 fs24 fc0 sc0 lsd1 ws7">S<span class="_ _3b"></span>a<span class="_ _56"></span>t<span class="_ _3b"></span>(<span class="_ _57"> </span>[]<span class="_ _46"> </span>[]<span class="_ _56"></span>)</div><div class="c xa5 y17a w8 h42"><div class="t m0 x0 h43 y17b ff6 fs24 fc0 sc0 ls7 ws7">Z</div></div><div class="t m0 x9d h43 y4 ff6 fs24 fc0 sc0 lsd2 ws7">ik<span class="_ _58"> </span>ik</div><div class="t m2 xa6 h44 y17c ff9 fs25 fc0 sc0 lsd3 ws7">αα<span class="_ _59"></span>α</div><div class="t m5 xa7 h45 y4 ff9 fs24 fc0 sc0 lsd4 ws7">=−</div><div class="t m5 xa8 h32 y17d ff1 fs1d fc0 sc0 lsaa ws7">and </div><div class="t m5 xa9 h24 y17e ff1 fs15 fc0 sc0 ls7 ws7">ˆ</div><div class="t m5 x99 h24 y17f ff1 fs15 fc0 sc0 lsd5 ws7">S<span class="_ _3b"></span>a<span class="_ _56"></span>t<span class="_ _3b"></span>(<span class="_ _57"> </span>[]<span class="_ _5a"> </span>[]<span class="_ _3b"></span>)</div><div class="c x9a y180 w9 h2b"><div class="t m5 x0 h26 y113 ff6 fs15 fc0 sc0 ls7 ws7">Z</div></div><div class="t m5 xaa h26 y181 ff6 fs15 fc0 sc0 lsd6 ws7">ik<span class="_ _58"> </span>ik</div><div class="t m8 x94 h27 y182 ff9 fs17 fc0 sc0 lsd7 ws7">ββ<span class="_ _45"></span>β</div><div class="t m0 x89 h2a y181 ff9 fs15 fc0 sc0 lsd8 ws7">=−</div><div class="t m0 xab h32 y183 ff1 fs1d fc0 sc0 lsd9 ws91">, w<span class="_ _0"></span>here a<span class="_ _0"></span> sat<span class="_ _0"></span>urat<span class="_ _0"></span>ion<span class="_ _0"></span> fu<span class="_ _0"></span>ncti<span class="_ _0"></span>on i<span class="_ _0"></span>ns<span class="_ _0"></span>tead <span class="_ _0"></span>of<span class="_ _0"></span> </div><div class="t m0 x11 h32 y184 ff1 fs1d fc0 sc0 lsda ws92">the co<span class="_ _1"></span>nventional si<span class="_ _1"></span>gn function is used as the switchi<span class="_ _1"></span>ng function. </div><div class="t m0 x11 h32 y185 ff1 fs1d fc0 sc0 lsdb ws93">Since th<span class="_ _0"></span>e Eule<span class="_ _0"></span>r m<span class="_ _0"></span>ethod is<span class="_ _0"></span> used<span class="_ _0"></span>, the <span class="_ _0"></span>discre<span class="_ _0"></span>te fo<span class="_ _0"></span>rm<span class="_ _0"></span> of curr<span class="_ _0"></span>ent<span class="_ _0"></span> </div><div class="t m0 x11 h32 y186 ff1 fs1d fc0 sc0 lsaf ws7">iterat<span class="_ _0"></span>ion<span class="_ _0"></span> can<span class="_ _0"></span> be<span class="_ _0"></span> ex<span class="_ _0"></span>press<span class="_ _0"></span>ed <span class="_ _0"></span>as: </div><div class="t m0 xac h46 y187 ff1 fs26 fc0 sc0 lsdc ws7">[1<span class="_ _5b"></span>]<span class="_ _5c"></span>[<span class="_ _32"></span>]</div><div class="t m0 xad h47 y188 ff6 fs27 fc0 sc0 ls7 ws7">s</div><div class="t m0 xae h48 y189 ff6 fs26 fc0 sc0 ls7 ws7">i</div><div class="t m0 xaf h48 y18a ff6 fs26 fc0 sc0 lsdd ws7">ik<span class="_ _5d"> </span>ik<span class="_ _13"> </span>T</div><div class="t m0 xae h48 y18b ff6 fs26 fc0 sc0 ls7 ws7">t</div><div class="t m0 xb0 h49 y189 ff9 fs26 fc0 sc0 ls7 ws7">Δ</div><div class="t m0 xb1 h49 y18a ff9 fs26 fc0 sc0 lsde ws7">+=<span class="_ _5e"> </span>+</div><div class="t m0 xb0 h49 y18b ff9 fs26 fc0 sc0 ls7 ws7">Δ</div><div class="t m0 xb2 h32 y18c ff1 fs1d fc0 sc0 lsdf ws7"> <span class="_ _1"></span> <span class="_ _1"></span> <span class="_ _1"></span> <span class="_ _1"></span> <span class="_ _1"></span> </div><div class="t m0 xa0 h32 y18d ff1 fs1d fc0 sc0 lse0 ws7">(6) </div><div class="t m0 x11 h32 y18e ff1 fs1d fc0 sc0 lse1 ws94">The<span class="_ _1"></span> curr<span class="_ _1"></span>ent d<span class="_ _1"></span>eri<span class="_ _1"></span>vati<span class="_ _1"></span>ve <span class="ff7 ls7 ws7">Δ<span class="ff6 lse2">i/</span>Δ<span class="ff6">t</span></span><span class="lsd9 ws95"> c<span class="_ _1"></span>an be obtain<span class="_ _0"></span>ed from th<span class="_ _0"></span>e current </span></div><div class="t m0 x11 h32 y18f ff1 fs1d fc0 sc0 lse3 ws96">observer equ<span class="_ _0"></span>ation<span class="_ _0"></span> (3). Fig. 1 sh<span class="_ _0"></span>ows a block<span class="_ _0"></span> diagram for th<span class="_ _0"></span>e </div><div class="t m0 x11 h32 y190 ff1 fs1d fc0 sc0 lse4 ws97">current obs<span class="_ _0"></span>erver in the dis<span class="_ _0"></span>crete-time SMO. </div><div class="t m0 x12 h32 y191 ff1 fs1d fc0 sc0 lsaf ws7"> </div><div class="t m0 xa1 h32 y192 ff1 fs1d fc0 sc0 ls7 ws7"> </div><div class="t m0 x11 h4a y193 ff1 fs4 fc0 sc0 lse5 ws1b">Fig<span class="_ _1"></span>. 1<span class="_ _1"></span>.<span class="_ _1"></span> <span class="ff4 ls7 ws7"> <span class="_"> </span></span><span class="lse6 ws98">Bl<span class="_ _0"></span>ock di<span class="_ _0"></span>agr<span class="_ _0"></span>am of <span class="ff7 ls7 ws7">α</span><span class="lse7 ws99">-loop current ob<span class="_ _1"></span>server for discret<span class="_ _1"></span>e-time SMO. </span></span></div><div class="t m0 xb3 h32 y194 ff1 fs1d fc0 sc0 lse8 ws7">III.<span class="ff4 fs2 ls7"> <span class="_ _57"> </span></span><span class="ls7">O<span class="fs4 lse9">SCIL<span class="_ _0"></span>LATION <span class="fs1d ls7">M</span><span class="lsea ws9a">IT<span class="_ _1"></span>IGAT<span class="_ _1"></span>IO<span class="_ _1"></span>N FO<span class="_ _1"></span>R<span class="_ _1"></span> </span><span class="fs1d ls7">D</span><span class="ls4e">IS<span class="_ _1"></span>C<span class="_ _1"></span>R<span class="_ _1"></span>E<span class="_ _1"></span>TE<span class="_ _1"></span><span class="fs1d lscd">-T<span class="_ _16"></span></span>IM<span class="_ _1"></span>E<span class="_ _1"></span> <span class="_ _1"></span><span class="fs1d lseb">SMO</span><span class="ls7"> <span class="fs1d">U</span><span class="lsec">SI<span class="_ _0"></span>NG </span></span></span></span></span></div><div class="t m0 xb4 h32 y195 ff1 fs1d fc0 sc0 ls7 ws7">E<span class="fs4 lsed">STIMATED </span>R<span class="fs4 lsee">OT<span class="_ _0"></span>OR <span class="fs1d ls7">S<span class="_ _0"></span><span class="fs4 lsef">PEED <span class="fs1d ls7">F</span><span class="lsf0">EEDBA<span class="_ _0"></span>CK<span class="_ _0"></span> <span class="fs1d ls7">A</span><span class="ls50">LGOR<span class="_ _1"></span>IT<span class="_ _1"></span>HM<span class="fs1d ls7"> </span></span></span></span></span></span></div><div class="t m0 x11 h4b y196 ff6 fs1d fc0 sc0 lsf1 ws7">A.</div><div class="t m0 x8a h4b y197 ffc fs2 fc0 sc0 ls7 ws7"> <span class="_"> </span><span class="ff6 fs1d lsf2 ws9b">P<span class="_ _1"></span>roblem Discrip<span class="_ _1"></span>tion </span></div><div class="t m0 x12 h32 y198 ff1 fs1d fc0 sc0 lsf3 ws9c">In a c<span class="_ _0"></span>ontinu<span class="_ _0"></span>ous-tim<span class="_ _0"></span>e sy<span class="_ _0"></span>stem<span class="_ _0"></span>, the stat<span class="_ _0"></span>e tra<span class="_ _0"></span>ject<span class="_ _0"></span>ory can<span class="_ _0"></span> exact<span class="_ _0"></span>ly<span class="_ _0"></span> </div><div class="t m0 x11 h32 y199 ff1 fs1d fc0 sc0 lsf4 ws9d">move along<span class="_ _0"></span> the sli<span class="_ _0"></span>ding su<span class="_ _0"></span>rfac<span class="_ _0"></span>e; w<span class="_ _0"></span>hile in<span class="_ _0"></span> a discre<span class="_ _0"></span>te-tim<span class="_ _0"></span>e sy<span class="_ _0"></span>stem, </div><div class="t m0 x11 h32 y19a ff1 fs1d fc0 sc0 lsd0 ws9e">due to<span class="_ _0"></span> the u<span class="_ _0"></span>se of<span class="_ _0"></span> a fi<span class="_ _0"></span>nite<span class="_ _0"></span> sam<span class="_ _0"></span>pling f<span class="_ _0"></span>requen<span class="_ _0"></span>cy<span class="_ _0"></span>, the s<span class="_ _0"></span>tate<span class="_ _0"></span> </div><div class="t m0 x11 h32 y19b ff1 fs1d fc0 sc0 lsf5 ws9f">traj<span class="_ _1"></span>ecto<span class="_ _1"></span>ry canno<span class="_ _1"></span>t exac<span class="_ _1"></span>tly move alo<span class="_ _1"></span>ng the slid<span class="_ _1"></span>ing sur<span class="_ _1"></span>face. </div><div class="t m0 x11 h32 y19c ff1 fs1d fc0 sc0 lsf6 wsa0">Bec<span class="_ _1"></span>ause of the e<span class="_ _1"></span>xiste<span class="_ _1"></span>nce of a trac<span class="_ _1"></span>king err<span class="_ _1"></span>or, the re<span class="_ _1"></span>aching la<span class="_ _1"></span>w </div><div class="t m0 x11 h32 y19d ff1 fs1d fc0 sc0 lsf7 wsa1">will sti<span class="_ _0"></span>ll fo<span class="_ _0"></span>rce th<span class="_ _0"></span>e s<span class="_ _0"></span>tate<span class="_ _0"></span> traject<span class="_ _0"></span>ory m<span class="_ _0"></span>ovin<span class="_ _0"></span>g tow<span class="_ _0"></span>ards th<span class="_ _0"></span>e <span class="_ _0"></span>desig<span class="_ _0"></span>ned </div><div class="t m0 x11 h32 y19e ff1 fs1d fc0 sc0 lscd wsa2">sliding sur<span class="_ _1"></span>face. However<span class="_ _1"></span>, the trac<span class="_ _1"></span>king error ca<span class="_ _1"></span>nnot be fully </div><div class="t m0 x11 h32 y19f ff1 fs1d fc0 sc0 lsf8 wsa3">elimin<span class="_ _0"></span>ated bec<span class="_ _0"></span>ause of th<span class="_ _0"></span>e fini<span class="_ _0"></span>te sam<span class="_ _0"></span>pling fr<span class="_ _0"></span>equency<span class="_ _0"></span>. <span class="_ _0"></span>This will </div><div class="t m0 x11 h32 y1a0 ff1 fs1d fc0 sc0 lsf9 wsa4">make the sta<span class="_ _1"></span>te trajecto<span class="_ _1"></span>ry have a bounded motio<span class="_ _1"></span>n around the </div><div class="t m0 x11 h32 y1a1 ff1 fs1d fc0 sc0 lse0 wsa5">sliding sur<span class="_ _1"></span>face,<span class="_ _1"></span> causing a chat<span class="_ _1"></span>tering prob<span class="_ _1"></span>lem, which in tur<span class="_ _1"></span>n </div><div class="t m0 x11 h32 y1a2 ff1 fs1d fc0 sc0 lsf4 ws9d">will cause os<span class="_ _0"></span>cillati<span class="_ _0"></span>ons in th<span class="_ _0"></span>e output of<span class="_ _0"></span> the SMO. The<span class="_ _0"></span> amplit<span class="_ _0"></span>ude </div><div class="t m0 x11 h32 y1a3 ff1 fs1d fc0 sc0 lsfa wsa6">of the chat<span class="_ _1"></span>teri<span class="_ _1"></span>ng can be decrea<span class="_ _1"></span>sed by increa<span class="_ _1"></span>sing the s<span class="_ _1"></span>ampling </div><div class="t m0 x11 h32 y1a4 ff1 fs1d fc0 sc0 ls7 wsa7">rate, <span class="_ _0"></span>but<span class="_ _0"></span> cann<span class="_ _0"></span>ot be<span class="_ _0"></span> elim<span class="_ _0"></span>in<span class="_ _0"></span>ated u<span class="_ _0"></span>nless th<span class="_ _0"></span>e sam<span class="_ _0"></span>ple<span class="_ _0"></span> tim<span class="_ _0"></span>e <span class="ff6 ws7">T</span></div><div class="t m0 xb5 h7 y1a5 ff6 fs6 fc0 sc0 ls7 ws7">s</div><div class="t m0 xb6 h32 y1a6 ff7 fs1d fc0 sc0 ls7 ws7">→<span class="ff1 lsf9 wsa8">0 [2]. </span></div><div class="t m0 x11 h32 y1a7 ff1 fs1d fc0 sc0 lsf7 wsa9">In an<span class="_ _0"></span> electr<span class="_ _0"></span>ic driv<span class="_ _0"></span>e sy<span class="_ _0"></span>stem, du<span class="_ _0"></span>e to<span class="_ _0"></span> the phy<span class="_ _0"></span>sical<span class="_ _0"></span> lim<span class="_ _0"></span>itation<span class="_ _0"></span> of t<span class="_ _0"></span>he </div><div class="t m0 x11 h32 y1a8 ff1 fs1d fc0 sc0 lsfb wsaa">comput<span class="_ _1"></span>atio<span class="_ _1"></span>nal resour<span class="_ _1"></span>ce, switching no<span class="_ _1"></span>ise<span class="_ _1"></span>, losses, a<span class="_ _1"></span>nd therma<span class="_ _1"></span>l </div><div class="t m0 x11 h32 y1a9 ff1 fs1d fc0 sc0 lsfc wsab">issue in th<span class="_ _0"></span>e <span class="_ _0"></span>inverte<span class="_ _0"></span>r, the s<span class="_ _0"></span>am<span class="_ _0"></span>pling ra<span class="_ _0"></span>te shoul<span class="_ _0"></span>d <span class="_ _0"></span>be select<span class="_ _0"></span>ed </div><div class="t m0 x11 h32 y1aa ff1 fs1d fc0 sc0 lsfd wsac">appropr<span class="_ _1"></span>iately accord<span class="_ _1"></span>ing to the system dyna<span class="_ _1"></span>mics to gua<span class="_ _1"></span>rantee </div><div class="t m0 x11 h32 y1ab ff1 fs1d fc0 sc0 lsf9 wsa8">fast resp<span class="_ _1"></span>onse, in<span class="_ _1"></span>stead of for t<span class="_ _1"></span>he sake of contro<span class="_ _1"></span>l algorit<span class="_ _1"></span>hms. </div><div class="t m0 x12 h32 y1ac ff1 fs1d fc0 sc0 lsf7 wsad">In rotor<span class="_ _0"></span> position estim<span class="_ _0"></span>ati<span class="_ _0"></span>on, the ext<span class="_ _0"></span>ended ba<span class="_ _0"></span>ck EMF is </div><div class="t m0 x11 h32 y1ad ff1 fs1d fc0 sc0 lsfe wsae">obtained <span class="_ _0"></span>by a SMO<span class="_ _0"></span>. If th<span class="_ _0"></span>e posit<span class="_ _0"></span>ion inf<span class="_ _0"></span>ormati<span class="_ _0"></span>on is ext<span class="_ _0"></span>racted<span class="_ _0"></span> </div><div class="t m0 x11 h32 y1ae ff1 fs1d fc0 sc0 lsfe wsaf">from th<span class="_ _0"></span>e estimate<span class="_ _0"></span>d back<span class="_ _0"></span> EMF <span class="_ _0"></span>with oscil<span class="_ _0"></span>lations<span class="_ _0"></span>, the<span class="_ _0"></span>re w<span class="_ _0"></span>ill be an<span class="_ _0"></span> </div><div class="t md x12 h4c y1af ff1 fs28 fc0 sc0 lsff ws7">[]<span class="_ _5f"></span><span class="ff6 ls100">ik</span></div><div class="t me xb7 h4d y1b0 ff9 fs29 fc0 sc0 ls7 ws7">α</div><div class="c xa4 y1b1 wa h4e"><div class="t mf x0 h4f y1b2 ff6 fs2a fc0 sc0 ls7 ws7">s</div></div><div class="t m10 xb8 h50 y1b3 ff6 fs2b fc0 sc0 ls7 ws7">T</div><div class="t m11 xa9 h51 y1b4 ff1 fs2c fc0 sc0 ls7 ws7">ˆ</div><div class="t m11 xb9 h51 y1b5 ff1 fs2c fc0 sc0 ls101 ws7">[]<span class="_ _31"></span><span class="ff6 ls102">ik</span></div><div class="t m12 xba h52 y1b6 ff9 fs2d fc0 sc0 ls7 ws7">α</div><div class="c xbb y1b7 wb h53"><div class="t m13 x0 h54 ybf ff6 fs2e fc0 sc0 ls7 ws7">s</div></div><div class="t m13 xbc h54 y1b8 ff6 fs2e fc0 sc0 ls7 ws7">d</div><div class="t m14 xaf h55 y1b9 ff6 fs2f fc0 sc0 ls7 ws7">R</div><div class="c xbd y1ba wc h56"><div class="t m14 x0 h55 y1bb ff6 fs2f fc0 sc0 ls7 ws7">L</div></div><div class="t m15 x8e h57 y1bc ff1 fs30 fc0 sc0 ls7 ws7">*</div><div class="t m15 x87 h58 y1bd ff6 fs30 fc0 sc0 ls103 ws7">dq</div><div class="t m15 xb h58 y1be ff6 fs30 fc0 sc0 ls7 ws7">r</div><div class="t m15 x8f h58 y1bf ff6 fs30 fc0 sc0 ls104 ws7">dd</div><div class="t m16 x96 h59 y1c0 ff6 fs31 fc0 sc0 ls105 ws7">LL</div><div class="t m16 xbe h59 y1c1 ff6 fs31 fc0 sc0 ls7 ws7">V</div><div class="t m16 xac h59 y1c2 ff6 fs31 fc0 sc0 ls7 ws7">i</div><div class="t m16 xbf h59 y1c3 ff6 fs31 fc0 sc0 ls106 ws7">LL</div><div class="t m17 xc0 h5a y1c4 ff9 fs32 fc0 sc0 ls7 ws7">α</div><div class="c xbb y1c5 wd h5b"><div class="t m17 x0 h5a y1c6 ff9 fs32 fc0 sc0 ls7 ws7">β</div></div><div class="t m18 xc1 h5c y1c7 ff9 fs33 fc0 sc0 ls7 ws7">ω</div><div class="t m16 xc2 h5d y1c0 ff9 fs31 fc0 sc0 ls7 ws7">−</div><div class="t m16 xc3 h5d y1c2 ff9 fs31 fc0 sc0 ls7 ws7">−</div><div class="t m19 xc4 h5e y1c8 ff1 fs34 fc0 sc0 ls7 ws7">1</div><div class="t m19 xc4 h5e y1c9 ff1 fs34 fc0 sc0 ls7 ws7">Z</div><div class="t m1a xc5 h5f y1ca ff1 fs35 fc0 sc0 ls7 ws7">ˆ</div><div class="t m1a xc6 h5f y1cb ff1 fs35 fc0 sc0 ls107 ws7">[]<span class="_ _60"></span><span class="ff6 ls108">ik</span></div><div class="t m1b xc7 h60 y1cc ff9 fs36 fc0 sc0 ls7 ws7">α</div><div class="t m1c xc8 h61 y1cd ff1 fs37 fc0 sc0 ls7 ws7">ˆ</div><div class="t m1c xc9 h61 y1ce ff1 fs37 fc0 sc0 ls109 ws7">[1<span class="_ _61"></span>]<span class="_ _62"></span><span class="ff6 ls10a">Ik</span></div><div class="t m1d x8d h62 y1cf ff9 fs38 fc0 sc0 ls7 ws7">α</div><div class="t m1c xca h63 y1d0 ff9 fs37 fc0 sc0 ls7 ws7">+</div><div class="c xcb y1d1 we h64"><div class="t m1e x0 h65 y110 ff6 fs39 fc0 sc0 ls7 ws7">Z</div></div><div class="t m1f xcc h66 y1d2 ff9 fs3a fc0 sc0 ls7 ws7">α</div><div class="t m20 xb2 h67 y1d3 ff6 fs3b fc0 sc0 ls7 ws7">i</div><div class="t m20 xb2 h67 y1d4 ff6 fs3b fc0 sc0 ls7 ws7">t</div><div class="t m20 xcd h68 y1d3 ff9 fs3b fc0 sc0 ls7 ws7">Δ</div><div class="t m20 xcd h68 y1d4 ff9 fs3b fc0 sc0 ls7 ws7">Δ</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>
