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<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/624fe37a6caf596192f79c91/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Ne<span class="_ _0"></span>w<span class="_ _1"> </span>variable<span class="_ _1"> </span>gain<span class="_ _1"> </span>super<span class="_ _2"></span>-twisting<span class="_ _1"> </span>sliding<span class="_ _1"> </span>mode<span class="_ _1"> </span>observer<span class="_ _1"> </span>for<span class="_ _1"> </span>sensorless</div><div class="t m0 x1 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">vector<span class="_ _1"> </span>control<span class="_ _3"> </span>of<span class="_ _3"> </span>nonsinusoidal<span class="_ _3"> </span>back<span class="_ _2"></span>-EMF<span class="_ _3"> </span>PMSM</div><div class="t m1 x2 h3 y3 ff2 fs1 fc0 sc0 ls0 ws0">$</div><div class="t m2 x1 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">Cassio<span class="_ _4"> </span>Luciano<span class="_ _4"> </span>Baratieri</div><div class="t m3 x3 h5 y5 ff3 fs3 fc1 sc0 ls0 ws0">n</div><div class="t m2 x4 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">,<span class="_ _4"> </span>Humberto<span class="_ _4"> </span>Pinheiro</div><div class="t m4 x1 h6 y6 ff4 fs4 fc0 sc0 ls0 ws0">T<span class="_ _0"></span>echnology<span class="_ _5"> </span>Center<span class="_ _0"></span>,<span class="_ _5"> </span>Federal<span class="_ _5"> </span>University<span class="_ _5"> </span>of<span class="_ _5"> </span>Santa<span class="_ _5"> </span>Maria,<span class="_ _5"> </span>Santa<span class="_ _5"> </span>Maria,<span class="_ _6"> </span>1<span class="_ _0"></span>000<span class="_ _5"> </span>Roraima<span class="_ _5"> </span>Street,<span class="_ _6"> </span>97105-900<span class="_ _6"> </span>RS,<span class="_ _7"> </span>Brazil</div><div class="t m5 x1 h7 y7 ff5 fs5 fc0 sc0 ls1 ws0">article<span class="_ _8"> </span>info</div><div class="t m4 x1 h6 y8 ff4 fs4 fc0 sc0 ls0 ws0">Article<span class="_ _5"> </span>history:</div><div class="t m4 x1 h8 y9 ff1 fs4 fc0 sc0 ls0 ws0">Received<span class="_ _6"> </span>1<span class="_ _0"></span>7<span class="_ _6"> </span>August<span class="_ _5"> </span>201<span class="_ _0"></span>5</div><div class="t m4 x1 h8 ya ff1 fs4 fc0 sc0 ls0 ws0">Received<span class="_ _6"> </span>in<span class="_ _5"> </span>revised<span class="_ _5"> </span>form</div><div class="t m4 x1 h8 yb ff1 fs4 fc0 sc0 ls0 ws0">3<span class="_ _0"></span>1<span class="_ _5"> </span>December<span class="_ _5"> </span>201<span class="_ _0"></span>5</div><div class="t m4 x1 h8 yc ff1 fs4 fc0 sc0 ls0 ws0">Accepted<span class="_ _6"> </span>3<span class="_ _5"> </span>April<span class="_ _7"> </span>201<span class="_ _9"></span>6</div><div class="t m4 x1 h6 yd ff4 fs4 fc0 sc0 ls0 ws0">Keywords:</div><div class="t m4 x1 h8 ye ff1 fs4 fc0 sc0 ls0 ws0">Discrete-time<span class="_ _6"> </span>sliding<span class="_ _7"> </span>mode</div><div class="t m4 x1 h8 yf ff1 fs4 fc0 sc0 ls0 ws0">Nonsinusoidal<span class="_ _5"> </span>back<span class="_ _9"></span>-EMF</div><div class="t m4 x1 h8 y10 ff1 fs4 fc0 sc0 ls0 ws0">Permanent<span class="_ _5"> </span>magnet<span class="_ _5"> </span>synchronous<span class="_ _5"> </span>motor</div><div class="t m4 x1 h8 y11 ff1 fs4 fc0 sc0 ls0 ws0">Sliding<span class="_ _5"> </span>mode<span class="_ _5"> </span>observer</div><div class="t m4 x1 h8 y12 ff1 fs4 fc0 sc0 ls0 ws0">Sensorless<span class="_ _5"> </span>vector<span class="_ _6"> </span>control</div><div class="t m4 x1 h8 y13 ff1 fs4 fc0 sc0 ls0 ws0">Super<span class="_ _9"></span>-twisting<span class="_ _7"> </span>algorithm</div><div class="t m5 x5 h7 y7 ff5 fs5 fc0 sc0 ls2 ws0">abstract</div><div class="t m5 x5 h9 y14 ff1 fs6 fc0 sc0 ls0 ws0">This<span class="_ _8"> </span>paper<span class="_ _8"> </span>presents<span class="_ _8"> </span>a<span class="_ _7"> </span>new<span class="_ _8"> </span>discrete-time<span class="_ _8"> </span>super<span class="_ _9"></span>-twisting<span class="_ _8"> </span>sliding<span class="_ _8"> </span>mode<span class="_ _8"> </span>observer<span class="_ _8"> </span>with<span class="_ _7"> </span>variable<span class="_ _8"> </span>gains<span class="_ _8"> </span>for</div><div class="t m5 x5 h9 y15 ff1 fs6 fc0 sc0 ls0 ws0">sensorless<span class="_ _4"> </span>nonsinusoidal<span class="_ _a"> </span>vector<span class="_ _a"> </span>control<span class="_ _a"> </span>of<span class="_ _4"> </span>permanent<span class="_ _a"> </span>magnet<span class="_ _4"> </span>synchronous<span class="_ _4"> </span>motors.<span class="_ _a"> </span>This<span class="_ _a"> </span>observer<span class="_ _4"> </span>is</div><div class="t m5 x5 h9 y16 ff1 fs6 fc0 sc0 ls0 ws0">adopted<span class="_ _5"> </span>to<span class="_ _8"> </span>estimate<span class="_ _5"> </span>the<span class="_ _7"> </span>back<span class="_ _7"> </span>electromotive<span class="_ _7"> </span>forces<span class="_ _7"> </span>(back<span class="_ _9"></span>-EMF)<span class="_ _8"> </span>that<span class="_ _7"> </span>are<span class="_ _7"> </span>required<span class="_ _5"> </span>for<span class="_ _7"> </span>the<span class="_ _8"> </span>ro<span class="_ _9"></span>tor<span class="_ _8"> </span>speed<span class="_ _5"> </span>es-</div><div class="t m5 x5 h9 y17 ff1 fs6 fc0 sc0 ls0 ws0">timation<span class="_ _5"> </span>and<span class="_ _7"> </span>for<span class="_ _7"> </span>the<span class="_ _7"> </span>nonsinusoidal<span class="_ _5"> </span>vector<span class="_ _5"> </span>control.<span class="_ _7"> </span>In<span class="_ _7"> </span>addition,<span class="_ _7"> </span>their<span class="_ _7"> </span>gains<span class="_ _5"> </span>are<span class="_ _7"> </span>time-varying<span class="_ _5"> </span>to<span class="_ _7"> </span>minimize</div><div class="t m5 x5 h9 y18 ff1 fs6 fc0 sc0 ls0 ws0">the<span class="_ _8"> </span>chattering.<span class="_ _5"> </span>So,<span class="_ _8"> </span>they<span class="_ _7"> </span>are<span class="_ _7"> </span>adjusted<span class="_ _8"> </span>based<span class="_ _8"> </span>on<span class="_ _7"> </span>internal<span class="_ _7"> </span>states<span class="_ _8"> </span>of<span class="_ _8"> </span>the<span class="_ _7"> </span>super<span class="_ _9"></span>-twisting<span class="_ _8"> </span>algorithm.<span class="_ _8"> </span>The<span class="_ _7"> </span>sta-</div><div class="t m5 x5 h9 y19 ff1 fs6 fc0 sc0 ls0 ws0">bility<span class="_ _6"> </span>analysis<span class="_ _6"> </span>is<span class="_ _6"> </span>investigated<span class="_ _6"> </span>from<span class="_ _6"> </span>the<span class="_ _6"> </span>L<span class="_ _0"></span>yapunov<span class="_ _6"> </span>theory<span class="_ _6"> </span>for<span class="_ _6"> </span>discrete-time<span class="_ _6"> </span>syst<span class="_ _0"></span>ems.<span class="_ _6"> </span>Finally,<span class="_ _6"> </span>simulation<span class="_ _6"> </span>and</div><div class="t m5 x5 h9 y1a ff1 fs6 fc0 sc0 ls0 ws0">experimental<span class="_ _8"> </span>resul<span class="_ _9"></span>ts<span class="_ _8"> </span>are<span class="_ _8"> </span>present<span class="_ _0"></span>ed<span class="_ _8"> </span>to<span class="_ _7"> </span>demonstrate<span class="_ _8"> </span>the<span class="_ _7"> </span>good<span class="_ _8"> </span>performance<span class="_ _8"> </span>and<span class="_ _7"> </span>the<span class="_ _8"> </span>effectiveness<span class="_ _7"> </span>of<span class="_ _8"> </span>the</div><div class="t m5 x5 h9 y1b ff1 fs6 fc0 sc0 ls0 ws0">proposed<span class="_ _7"> </span>observer<span class="_ _9"></span>.</div><div class="t m5 x6 h9 y1c ff3 fs6 fc0 sc0 ls0 ws0">&<span class="_ _7"> </span><span class="ff1">201<span class="_ _2"></span>6<span class="_ _8"> </span>Elsevier<span class="_ _5"> </span>Ltd.<span class="_ _7"> </span>All<span class="_ _8"> </span>rights<span class="_ _5"> </span>reserved.</span></div><div class="t m6 x1 ha y1d ff6 fs7 fc0 sc0 ls0 ws0">1.<span class="_ _b"> </span>Introduction</div><div class="t m6 x7 ha y1e ff1 fs7 fc0 sc0 ls0 ws0">Sensorless<span class="_ _a"> </span>control<span class="_ _8"> </span>of<span class="_ _a"> </span>permanent<span class="_ _a"> </span>magnet<span class="_ _a"> </span>synchronous<span class="_ _a"> </span>motors</div><div class="t m6 x1 ha y1f ff1 fs7 fc0 sc0 ls0 ws0">(PMSMs)<span class="_ _4"> </span>has<span class="_ _4"> </span>been<span class="_ _4"> </span>a<span class="_ _4"> </span>focus<span class="_ _4"> </span>of<span class="_ _4"> </span>intensive<span class="_ _a"> </span>research<span class="_ _4"> </span>during<span class="_ _4"> </span>the<span class="_ _4"> </span>last</div><div class="t m6 x1 ha y20 ff1 fs7 fc0 sc0 ls0 ws0">decades<span class="_ _5"> </span>(<span class="fc1">Betin<span class="_ _5"> </span>et<span class="_ _5"> </span>al.,<span class="_ _7"> </span>201<span class="_ _2"></span>4<span class="fc0">).<span class="_ _5"> </span>The<span class="_ _7"> </span>main<span class="_ _5"> </span>reason<span class="_ _5"> </span>for<span class="_ _5"> </span>this<span class="_ _5"> </span>interest<span class="_ _5"> </span>is<span class="_ _7"> </span>the</span></span></div><div class="t m6 x1 ha y21 ff1 fs7 fc0 sc0 ls0 ws0">elimination<span class="_ _8"> </span>of<span class="_ _8"> </span>rotor<span class="_ _8"> </span>position<span class="_ _8"> </span>or/and<span class="_ _8"> </span>speed<span class="_ _8"> </span>sensors<span class="_ _a"> </span>that<span class="_ _8"> </span>results<span class="_ _8"> </span>in</div><div class="t m6 x1 ha y22 ff1 fs7 fc0 sc0 ls0 ws0">cost<span class="_ _a"> </span>reduction,<span class="_ _4"> </span>increased<span class="_ _4"> </span>robustness<span class="_ _4"> </span>and<span class="_ _a"> </span>compactness<span class="_ _4"> </span>(<span class="fc1">Briz<span class="_ _4"> </span>and</span></div><div class="t m6 x1 ha y23 ff1 fs7 fc1 sc0 ls0 ws0">Degner<span class="_ _9"></span>,<span class="_ _a"> </span>201<span class="_ _2"></span>1<span class="fc0">).<span class="_ _a"> </span>Other<span class="_ _a"> </span>moti<span class="_ _0"></span>vations<span class="_ _8"> </span>result<span class="_ _a"> </span>from<span class="_ _a"> </span>advances<span class="_ _a"> </span>in<span class="_ _8"> </span>power</span></div><div class="t m6 x1 ha y24 ff1 fs7 fc0 sc0 ls0 ws0">electronic<span class="_ _4"> </span>devices<span class="_ _1"> </span>and<span class="_ _4"> </span>in<span class="_ _4"> </span>digital<span class="_ _1"> </span>signal<span class="_ _4"> </span>processors<span class="_ _1"> </span>that<span class="_ _a"> </span>have<span class="_ _4"> </span>al-</div><div class="t m6 x1 ha y25 ff1 fs7 fc0 sc0 ls0 ws0">lowed<span class="_ _a"> </span>product<span class="_ _a"> </span>developers<span class="_ _a"> </span>to<span class="_ _a"> </span>apply<span class="_ _a"> </span>high-performance<span class="_ _4"> </span>sensorless</div><div class="t m6 x1 ha y26 ff1 fs7 fc0 sc0 ls0 ws0">methods<span class="_ _8"> </span>to<span class="_ _5"> </span>low-cost<span class="_ _8"> </span>PMSM<span class="_ _7"> </span>drives<span class="_ _8"> </span>(<span class="fc1">Bose,<span class="_ _7"> </span>20<span class="_ _c"></span>09</span>;<span class="_ _8"> </span><span class="fc1">Pacas,<span class="_ _7"> </span>20<span class="_ _0"></span>1<span class="_ _2"></span>1<span class="fc0">).<span class="_ _8"> </span>As<span class="_ _7"> </span>a</span></span></div><div class="t m6 x1 ha y27 ff1 fs7 fc0 sc0 ls0 ws0">result,<span class="_ _1"> </span>these<span class="_ _4"> </span>motor<span class="_ _4"> </span>drives<span class="_ _1"> </span>hav<span class="_ _0"></span>e<span class="_ _1"> </span>been<span class="_ _4"> </span>widely<span class="_ _4"> </span>adopted<span class="_ _1"> </span>in<span class="_ _4"> </span>manu-</div><div class="t m6 x1 ha y28 ff1 fs7 fc0 sc0 ls0 ws0">facturing<span class="_ _a"> </span>processes<span class="_ _8"> </span>as<span class="_ _a"> </span>well<span class="_ _a"> </span>as<span class="_ _8"> </span>in<span class="_ _a"> </span>large-scale<span class="_ _a"> </span>applications<span class="_ _a"> </span>such<span class="_ _8"> </span>as</div><div class="t m6 x1 ha y29 ff1 fs7 fc0 sc0 ls0 ws0">compressors,<span class="_ _8"> </span>pumps<span class="_ _7"> </span>and<span class="_ _8"> </span>fans<span class="_ _8"> </span>(<span class="fc1">Rahman,<span class="_ _7"> </span>201<span class="_ _2"></span>3<span class="fc0">).</span></span></div><div class="t m6 x7 ha y2a ff1 fs7 fc0 sc0 ls0 ws0">Now<span class="_ _0"></span>adays,<span class="_ _a"> </span>the<span class="_ _4"> </span>sensorless<span class="_ _4"> </span>control<span class="_ _4"> </span>methods<span class="_ _1"> </span>for<span class="_ _a"> </span>PMSM<span class="_ _4"> </span>can<span class="_ _4"> </span>be</div><div class="t m6 x1 ha y2b ff1 fs7 fc0 sc0 ls0 ws0">summarized<span class="_ _4"> </span>in<span class="_ _1"> </span>two<span class="_ _4"> </span>categories<span class="_ _4"> </span>(<span class="fc1">Betin<span class="_ _1"> </span>et<span class="_ _4"> </span>al.,<span class="_ _4"> </span>201<span class="_ _9"></span>4<span class="fc0">;<span class="_ _4"> </span></span>Acarnley<span class="_ _4"> </span>and</span></div><div class="t m6 x1 ha y2c ff1 fs7 fc1 sc0 ls0 ws0">W<span class="_ _9"></span>atson,<span class="_ _8"> </span>20<span class="_ _c"></span>06<span class="fc0">):<span class="_ _8"> </span>fundamental<span class="_ _8"> </span>ex<span class="_ _9"></span>citation<span class="_ _8"> </span>signal<span class="_ _8"> </span>(FES)<span class="_ _8"> </span>and<span class="_ _8"> </span>high-fre-</span></div><div class="t m6 x1 ha y2d ff1 fs7 fc0 sc0 ls0 ws0">quency<span class="_ _8"> </span>signal<span class="_ _a"> </span>injection<span class="_ _8"> </span>(HFSI).<span class="_ _a"> </span>In<span class="_ _8"> </span>the<span class="_ _a"> </span><span class="ff7">fi</span>rst<span class="_ _8"> </span>category<span class="_ _0"></span>,<span class="_ _a"> </span>the<span class="_ _8"> </span>methods</div><div class="t m6 x1 ha y2e ff1 fs7 fc0 sc0 ls0 ws0">are<span class="_ _8"> </span>based<span class="_ _8"> </span>on<span class="_ _8"> </span>the<span class="_ _8"> </span>motor<span class="_ _8"> </span>model,<span class="_ _8"> </span>and<span class="_ _8"> </span>they<span class="_ _8"> </span>can<span class="_ _8"> </span>use<span class="_ _8"> </span>either<span class="_ _8"> </span>reduced-</div><div class="t m6 x1 ha y2f ff1 fs7 fc0 sc0 ls0 ws0">order<span class="_ _5"> </span>or<span class="_ _5"> </span>full-order<span class="_ _7"> </span>observers<span class="_ _5"> </span>for<span class="_ _7"> </span>the<span class="_ _7"> </span>closed-loop<span class="_ _7"> </span>operation.<span class="_ _5"> </span>On<span class="_ _7"> </span>the</div><div class="t m6 x8 ha y30 ff1 fs7 fc0 sc0 ls0 ws0">other<span class="_ _7"> </span>hand,<span class="_ _7"> </span>in<span class="_ _8"> </span>the<span class="_ _7"> </span>second<span class="_ _7"> </span>category<span class="_ _0"></span>,<span class="_ _7"> </span>the<span class="_ _8"> </span>ro<span class="_ _0"></span>tor<span class="_ _7"> </span>position<span class="_ _7"> </span>detection<span class="_ _8"> </span>is</div><div class="t m6 x8 ha y31 ff1 fs7 fc0 sc0 ls0 ws0">achieved<span class="_ _1"> </span>by<span class="_ _4"> </span>exploiting<span class="_ _1"> </span>the<span class="_ _1"> </span>motor<span class="_ _1"> </span>saliences<span class="_ _1"> </span>(i.e.<span class="_ _1"> </span>magnetic<span class="_ _1"> </span>rotor</div><div class="t m6 x8 ha y32 ff1 fs7 fc0 sc0 ls0 ws0">anisotrop<span class="_ _9"></span>y<span class="_ _7"> </span>and/or<span class="_ _7"> </span>magnetic<span class="_ _7"> </span>saturation).<span class="_ _5"> </span>In<span class="_ _7"> </span>this<span class="_ _7"> </span>method,<span class="_ _7"> </span>the<span class="_ _7"> </span>motor</div><div class="t m6 x8 ha y33 ff1 fs7 fc0 sc0 ls0 ws0">control<span class="_ _4"> </span>signals<span class="_ _1"> </span>are<span class="_ _4"> </span>synthesized<span class="_ _1"> </span>superposing<span class="_ _4"> </span>high-frequency<span class="_ _4"> </span>sig-</div><div class="t m6 x8 ha y34 ff1 fs7 fc0 sc0 ls0 ws0">nals<span class="_ _1"> </span>with<span class="_ _3"> </span>the<span class="_ _1"> </span>fundamental<span class="_ _3"> </span>e<span class="_ _9"></span>xcitation<span class="_ _3"> </span>allowing<span class="_ _1"> </span>to<span class="_ _1"> </span>estimate<span class="_ _1"> </span>the</div><div class="t m6 x8 ha y35 ff1 fs7 fc0 sc0 ls0 ws0">rotor<span class="_ _5"> </span>position<span class="_ _7"> </span>even<span class="_ _7"> </span>at<span class="_ _8"> </span>standstill<span class="_ _5"> </span>(<span class="_ _c"></span><span class="fc1">Briz<span class="_ _7"> </span>and<span class="_ _7"> </span>Degner<span class="_ _0"></span>,<span class="_ _7"> </span>201<span class="_ _2"></span>1<span class="fc0">).<span class="_ _5"> </span>Although</span></span></div><div class="t m6 x8 ha y36 ff1 fs7 fc0 sc0 ls0 ws0">FES<span class="_ _7"> </span>methods<span class="_ _8"> </span>ha<span class="_ _0"></span>ve<span class="_ _7"> </span>limitations<span class="_ _8"> </span>to<span class="_ _5"> </span>operate<span class="_ _7"> </span>close<span class="_ _8"> </span>to<span class="_ _5"> </span>zero<span class="_ _8"> </span>speed,<span class="_ _7"> </span>they</div><div class="t m6 x8 ha y37 ff1 fs7 fc0 sc0 ls0 ws0">hav<span class="_ _0"></span>e<span class="_ _4"> </span>been<span class="_ _1"> </span>expansiv<span class="_ _9"></span>ely<span class="_ _1"> </span>used<span class="_ _4"> </span>in<span class="_ _1"> </span>several<span class="_ _a"> </span>applications<span class="_ _1"> </span>that<span class="_ _4"> </span>do<span class="_ _1"> </span>not</div><div class="t m6 x8 ha y38 ff1 fs7 fc0 sc0 ls0 ws0">req<span class="_ _0"></span>uire<span class="_ _8"> </span>v<span class="_ _9"></span>ery<span class="_ _8"> </span>low-speed<span class="_ _7"> </span>operation<span class="_ _8"> </span>or<span class="_ _8"> </span>when<span class="_ _7"> </span>the<span class="_ _8"> </span>rot<span class="_ _9"></span>or<span class="_ _8"> </span>saliences<span class="_ _8"> </span>are</div><div class="t m6 x8 ha y39 ff1 fs7 fc0 sc0 ls0 ws0">inappropriate<span class="_ _7"> </span>for<span class="_ _8"> </span>HFSI<span class="_ _8"> </span>methods<span class="_ _7"> </span>(<span class="fc1">Pacas,<span class="_ _8"> </span>20<span class="_ _0"></span>1<span class="_ _2"></span>1<span class="fc0">).</span></span></div><div class="t m6 x9 ha y3a ff1 fs7 fc0 sc0 ls0 ws0">In<span class="_ _1"> </span>this<span class="_ _1"> </span>regard,<span class="_ _1"> </span>sliding<span class="_ _1"> </span>mode<span class="_ _1"> </span>(SM)<span class="_ _1"> </span>observers<span class="_ _1"> </span>have<span class="_ _1"> </span>become<span class="_ _1"> </span>a</div><div class="t m6 x8 ha y3b ff1 fs7 fc0 sc0 ls0 ws0">promising<span class="_ _8"> </span>alternative<span class="_ _a"> </span>for<span class="_ _8"> </span>FES<span class="_ _a"> </span>methods<span class="_ _a"> </span>applied<span class="_ _8"> </span>to<span class="_ _a"> </span>PMSMs<span class="_ _a"> </span>due<span class="_ _8"> </span>to</div><div class="t m6 x8 ha y3c ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _d"> </span>observation<span class="_ _d"> </span>accuracy<span class="_ _3"> </span>and<span class="_ _d"> </span>the<span class="_ _d"> </span>robustness<span class="_ _d"> </span>with<span class="_ _d"> </span>respect<span class="_ _d"> </span>to</div><div class="t m6 x8 ha y3d ff1 fs7 fc0 sc0 ls0 ws0">bounded<span class="_ _3"> </span>disturbances<span class="_ _3"> </span>and<span class="_ _3"> </span>parameter<span class="_ _3"> </span>uncertainties<span class="_ _3"> </span>(<span class="fc1">Bernardes,</span></div><div class="t m6 x8 ha y3e ff1 fs7 fc1 sc0 ls0 ws0">Foletto<span class="_ _a"> </span>Montagner<span class="_ _9"></span>,<span class="_ _a"> </span>Grundling,<span class="_ _4"> </span>and<span class="_ _a"> </span>Pinheiro,<span class="_ _a"> </span>201<span class="_ _2"></span>4<span class="fc0">;<span class="_ _4"> </span></span>Hamida,<span class="_ _a"> </span>Glu-</div><div class="t m6 x8 ha y3f ff1 fs7 fc1 sc0 ls0 ws0">mineau,<span class="_ _3"> </span>and<span class="_ _3"> </span>De<span class="_ _3"> </span>Leon,<span class="_ _3"> </span>20<span class="_ _9"></span>1<span class="_ _9"></span>3<span class="fc0">;<span class="_ _3"> </span></span>Salgado,<span class="_ _3"> </span>Chairez,<span class="_ _3"> </span>Bandyopadh<span class="_ _9"></span>yay<span class="_ _9"></span>,</div><div class="t m6 x8 ha y40 ff1 fs7 fc1 sc0 ls0 ws0">Fridman,<span class="_ _7"> </span>and<span class="_ _7"> </span>Camacho,<span class="_ _8"> </span>20<span class="_ _9"></span>1<span class="_ _9"></span>4<span class="fc0">;<span class="_ _8"> </span></span>Slotine,<span class="_ _5"> </span>Hedrick,<span class="_ _8"> </span>and<span class="_ _7"> </span>Misaw<span class="_ _9"></span>a,<span class="_ _5"> </span>1<span class="_ _9"></span>986<span class="fc0">;</span></div><div class="t m6 x8 ha y41 ff1 fs7 fc1 sc0 ls0 ws0">Y<span class="_ _2"></span>an<span class="_ _a"> </span>and<span class="_ _a"> </span>Utkin,<span class="_ _a"> </span>20<span class="_ _c"></span>02<span class="fc0">).<span class="_ _a"> </span>Howev<span class="_ _9"></span>er,<span class="_ _8"> </span>it<span class="_ _a"> </span>is<span class="_ _a"> </span>well<span class="_ _a"> </span>known<span class="_ _8"> </span>that<span class="_ _a"> </span>the<span class="_ _a"> </span>major</span></div><div class="t m6 x8 ha y42 ff1 fs7 fc0 sc0 ls0 ws0">dra<span class="_ _0"></span>wback<span class="_ _8"> </span>in<span class="_ _7"> </span>SM<span class="_ _8"> </span>approach<span class="_ _7"> </span>is<span class="_ _8"> </span>the<span class="_ _8"> </span>chattering<span class="_ _7"> </span>phenomenon<span class="_ _8"> </span>(<span class="fc1">Levant,</span></div><div class="t m6 x8 ha y43 ff1 fs7 fc1 sc0 ls0 ws0">201<span class="_ _2"></span>0<span class="fc0">).<span class="_ _a"> </span>Thus,<span class="_ _8"> </span><span class="ff7">fi</span>lters<span class="_ _a"> </span>and<span class="_ _8"> </span>continuous<span class="_ _a"> </span>switching<span class="_ _a"> </span>functions<span class="_ _a"> </span>ar<span class="_ _0"></span>e<span class="_ _a"> </span>often</span></div><div class="t m6 x8 ha y44 ff1 fs7 fc0 sc0 ls0 ws0">used<span class="_ _7"> </span>to<span class="_ _8"> </span>mitigate<span class="_ _5"> </span>the<span class="_ _8"> </span>chattering<span class="_ _7"> </span>(<span class="fc1">Hung,<span class="_ _8"> </span>Gao,<span class="_ _7"> </span>and<span class="_ _7"> </span>Hung,<span class="_ _5"> </span>1<span class="_ _9"></span>993<span class="fc0">;<span class="_ _8"> </span></span>Kim,</span></div><div class="t m6 x8 ha y45 ff1 fs7 fc1 sc0 ls0 ws0">Son,<span class="_ _6"> </span>and<span class="_ _5"> </span>Lee,<span class="_ _6"> </span>201<span class="_ _2"></span>1<span class="fc0">;<span class="_ _6"> </span></span>Yuan<span class="_ _6"> </span>et<span class="_ _6"> </span>al.,<span class="_ _5"> </span>20<span class="_ _0"></span>1<span class="_ _9"></span>3<span class="fc0">).<span class="_ _5"> </span>On<span class="_ _6"> </span>the<span class="_ _5"> </span>other<span class="_ _6"> </span>hand,<span class="_ _6"> </span>a<span class="_ _5"> </span>solution</span></div><div class="t m6 x8 ha y46 ff1 fs7 fc0 sc0 ls0 ws0">to<span class="_ _a"> </span>overcome<span class="_ _a"> </span>this<span class="_ _a"> </span>challenge,<span class="_ _a"> </span>without<span class="_ _a"> </span>compromising<span class="_ _a"> </span>the<span class="_ _a"> </span>observa-</div><div class="t m6 x8 ha y47 ff1 fs7 fc0 sc0 ls0 ws0">tion<span class="_ _8"> </span>performance,<span class="_ _a"> </span>is<span class="_ _8"> </span>to<span class="_ _a"> </span>use<span class="_ _8"> </span>high-order<span class="_ _8"> </span>sliding<span class="_ _a"> </span>mode<span class="_ _8"> </span>(HOSM)<span class="_ _a"> </span>ob-</div><div class="t m6 x8 ha y48 ff1 fs7 fc0 sc0 ls0 ws0">servers<span class="_ _a"> </span>(<span class="fc1">Lev<span class="_ _0"></span>ant,<span class="_ _8"> </span>1<span class="_ _2"></span>993<span class="fc0">,<span class="_ _a"> </span></span>20<span class="_ _c"></span>03<span class="fc0">).<span class="_ _a"> </span>HOSM<span class="_ _a"> </span>can<span class="_ _a"> </span>considerably<span class="_ _8"> </span>reduce<span class="_ _a"> </span>the</span></span></div><div class="t m6 x8 ha y49 ff1 fs7 fc0 sc0 ls0 ws0">chattering<span class="_ _4"> </span>keeping<span class="_ _1"> </span>the<span class="_ _1"> </span><span class="ff7">fi</span>rst<span class="_ _4"> </span>order<span class="_ _4"> </span>SM<span class="_ _1"> </span>properties.<span class="_ _1"> </span>How<span class="_ _9"></span>ever<span class="_ _0"></span>,<span class="_ _1"> </span>this</div><div class="t m6 x8 ha y4a ff1 fs7 fc0 sc0 ls0 ws0">approach<span class="_ _8"> </span>r<span class="_ _9"></span>equires<span class="_ _8"> </span>to<span class="_ _7"> </span>determine<span class="_ _8"> </span>the<span class="_ _8"> </span>switching<span class="_ _8"> </span>function<span class="_ _7"> </span>derivative</div><div class="t m6 xa ha y4b ff1 fs7 fc0 sc0 ls0 ws0">Contents<span class="_ _8"> </span>lists<span class="_ _7"> </span>available<span class="_ _7"> </span>at<span class="_ _8"> </span><span class="fc1">ScienceDirect</span></div><div class="t m6 xb hb y4c ff8 fs7 fc0 sc0 ls3 ws0">journal<span class="_ _8"> </span>homepage:<span class="_ _7"> </span><span class="fc1 ls4">www.elsevier.com/locate/conengprac</span></div><div class="t m7 xc hc y4d ff1 fs8 fc0 sc0 ls0 ws0">Control<span class="_ _3"> </span>Engineering<span class="_ _d"> </span>Pr<span class="_ _9"></span>actice</div><div class="t m4 x1 h8 y4e ff1 fs4 fc1 sc0 ls0 ws0">http://dx.doi.org/10.<span class="_ _2"></span>10<span class="_ _9"></span>1<span class="_ _0"></span>6/j.conengprac.201<span class="_ _2"></span>6.0<span class="_ _c"></span>4.003</div><div class="t m4 x1 h8 y4f ff1 fs4 fc0 sc0 ls0 ws0">0967-0661/<span class="ff3">&<span class="_ _5"> </span></span>201<span class="_ _2"></span>6<span class="_ _7"> </span>Elsevier<span class="_ _5"> </span>Ltd.<span class="_ _5"> </span>All<span class="_ _5"> </span>rights<span class="_ _7"> </span>reserved.</div><div class="t m8 xd hd y50 ff9 fs9 fc0 sc0 ls0 ws0">☆</div><div class="t m4 xe h8 y51 ff1 fs4 fc0 sc0 ls0 ws0">This<span class="_ _8"> </span>wor<span class="_ _0"></span>k<span class="_ _8"> </span>w<span class="_ _0"></span>as<span class="_ _8"> </span>supported<span class="_ _7"> </span>in<span class="_ _8"> </span>part<span class="_ _8"> </span>by<span class="_ _7"> </span>Coordination<span class="_ _8"> </span>of<span class="_ _8"> </span>Impro<span class="_ _9"></span>vement<span class="_ _8"> </span>of<span class="_ _8"> </span>Higher</div><div class="t m4 x1 h8 y52 ff1 fs4 fc0 sc0 ls0 ws0">Education<span class="_ _5"> </span>Personnel<span class="_ _5"> </span>(CAPES)<span class="_ _7"> </span>and<span class="_ _5"> </span>by<span class="_ _7"> </span>National<span class="_ _5"> </span>Council<span class="_ _5"> </span>for<span class="_ _7"> </span>Scienti<span class="ff7">fi</span>c<span class="_ _5"> </span>and<span class="_ _7"> </span>T<span class="_ _9"></span>echnolo-</div><div class="t m4 x1 h8 y53 ff1 fs4 fc0 sc0 ls0 ws0">gical<span class="_ _5"> </span>Development<span class="_ _5"> </span>(CNPq).</div><div class="t m9 xd he y54 ff3 fsa fc0 sc0 ls0 ws0">n</div><div class="t m4 xe h8 y55 ff1 fs4 fc0 sc0 ls0 ws0">Corresponding<span class="_ _5"> </span>author<span class="_ _0"></span>.</div><div class="t m4 x7 h6 y56 ff4 fs4 fc0 sc0 ls0 ws0">E-mail<span class="_ _5"> </span>addresses:<span class="_ _5"> </span><span class="ff1 fc1">cassio.baratieri@gmail.com<span class="_ _5"> </span><span class="fc0">(C.L.<span class="_ _7"> </span>Baratieri),</span></span></div><div class="t m4 x1 h8 y57 ff1 fs4 fc1 sc0 ls0 ws0">humberto.ctlab.ufsm.br@gmail.com<span class="_ _5"> </span><span class="fc0">(H.<span class="_ _5"> </span>Pinheiro).</span></div><div class="t m4 xf h8 y58 ff1 fs4 fc1 sc0 ls0 ws0">Control<span class="_ _5"> </span>Engineering<span class="_ _5"> </span>Practice<span class="_ _5"> </span>52<span class="_ _5"> </span>(201<span class="_ _9"></span>6)<span class="_ _7"> </span>59<span class="_"> </span><span class="ffa">–<span class="_ _c"> </span></span>69</div><a class="l" rel='nofollow' onclick='return false;'><div class="d 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class="l" rel='nofollow' onclick='return false;'><div class="d ma"></div></a></div><div class="pi" data-data='{"ctm":[1.612697,0.000000,0.000000,1.612697,0.000000,0.000000]}'></div></div>
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<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/624fe37a6caf596192f79c91/bg2.jpg"><div class="t m6 x10 ha y59 ff1 fs7 fc0 sc0 ls0 ws0">(s)<span class="_ _a"> </span>which<span class="_ _a"> </span>makes<span class="_ _a"> </span>the<span class="_ _a"> </span>implementation<span class="_ _a"> </span>more<span class="_ _a"> </span>comple<span class="_ _0"></span>x<span class="_ _a"> </span>when<span class="_ _a"> </span>com-</div><div class="t m6 x10 ha y5a ff1 fs7 fc0 sc0 ls0 ws0">pared<span class="_ _7"> </span>to<span class="_ _7"> </span>the<span class="_ _7"> </span><span class="ff7">fi</span>rst-order<span class="_ _8"> </span>SM.<span class="_ _5"> </span>The<span class="_ _8"> </span>e<span class="_ _0"></span>xcepti<span class="_ _0"></span>on<span class="_ _7"> </span>is<span class="_ _8"> </span>the<span class="_ _5"> </span>second<span class="_ _8"> </span>order<span class="_ _7"> </span>SM</div><div class="t m6 x10 ha y5b ff1 fs7 fc0 sc0 ls0 ws0">approach<span class="_ _8"> </span>which<span class="_ _7"> </span>is<span class="_ _8"> </span>known<span class="_ _7"> </span>as<span class="_ _8"> </span>super<span class="_ _9"></span>-twisting<span class="_ _8"> </span>algorithm<span class="_ _7"> </span>(STA).</div><div class="t m6 x11 ha y5c ff1 fs7 fc0 sc0 ls0 ws0">ST<span class="_ _0"></span>A<span class="_ _7"> </span>was<span class="_ _8"> </span>proposed<span class="_ _8"> </span>by<span class="_ _7"> </span><span class="fc1">Levant<span class="_ _8"> </span>(1<span class="_ _9"></span>993)<span class="fc0">.<span class="_ _8"> </span>Since<span class="_ _8"> </span>then,<span class="_ _8"> </span>this<span class="_ _8"> </span>approach</span></span></div><div class="t m6 x10 ha y5d ff1 fs7 fc0 sc0 ls0 ws0">and<span class="_ _5"> </span>similar<span class="_ _7"> </span>ones<span class="_ _5"> </span>have<span class="_ _5"> </span>been<span class="_ _5"> </span>widely<span class="_ _5"> </span>applied<span class="_ _7"> </span>to<span class="_ _5"> </span>state<span class="_ _7"> </span>estimation<span class="_ _7"> </span>and</div><div class="t m6 x10 ha y5e ff1 fs7 fc0 sc0 ls0 ws0">control<span class="_ _a"> </span>(<span class="fc1">Alwi<span class="_ _a"> </span>and<span class="_ _4"> </span>Edwards,<span class="_ _a"> </span>201<span class="_ _2"></span>4<span class="fc0">;<span class="_ _a"> </span></span>Basin<span class="_ _4"> </span>and<span class="_ _4"> </span>Rodriguez<span class="_ _a"> </span>Ramirez,</span></div><div class="t m6 x10 ha y5f ff1 fs7 fc1 sc0 ls0 ws0">201<span class="_ _2"></span>4<span class="fc0">;<span class="_ _a"> </span></span>Di<span class="_ _a"> </span>Gennaro,<span class="_ _a"> </span>Riv<span class="_ _9"></span>era,<span class="_ _a"> </span>and<span class="_ _a"> </span>Castillo-<span class="_ _9"></span>Toledo,<span class="_ _8"> </span>201<span class="_ _9"></span>0<span class="fc0">,<span class="_ _a"> </span></span>20<span class="_ _0"></span>1<span class="_ _9"></span>4<span class="fc0">;<span class="_ _a"> </span></span>Ezzat,</div><div class="t m6 x10 ha y60 ff1 fs7 fc1 sc0 ls0 ws0">de<span class="_ _4"> </span>Leon,<span class="_ _4"> </span>Gonzalez,<span class="_ _4"> </span>and<span class="_ _4"> </span>Glumineau,<span class="_ _4"> </span>201<span class="_ _9"></span>0<span class="fc0">;<span class="_ _a"> </span></span>Kuntanapreeda,<span class="_ _4"> </span>201<span class="_ _2"></span>5<span class="fc0">;</span></div><div class="t m6 x10 ha y61 ff1 fs7 fc1 sc0 ls0 ws0">Kunusch,<span class="_ _4"> </span>Puleston,<span class="_ _1"> </span>Mayosky<span class="_ _9"></span>,<span class="_ _1"> </span>and<span class="_ _4"> </span>Fridman,<span class="_ _1"> </span>20<span class="_ _0"></span>1<span class="_ _9"></span>3<span class="fc0">;<span class="_ _1"> </span></span>Lascu,<span class="_ _1"> </span>Boldea,</div><div class="t m6 x10 ha y62 ff1 fs7 fc1 sc0 ls0 ws0">and<span class="_ _d"> </span>Blaabjerg,<span class="_ _d"> </span>201<span class="_ _2"></span>3<span class="fc0">;<span class="_ _d"> </span></span>Liu,<span class="_ _d"> </span>Laghrouche,<span class="_ _d"> </span>Harmouche,<span class="_ _d"> </span>and<span class="_ _d"> </span>Wack,</div><div class="t m6 x10 ha y63 ff1 fs7 fc1 sc0 ls0 ws0">201<span class="_ _2"></span>4<span class="fc0">;<span class="_ _4"> </span></span>Moreno,<span class="_ _4"> </span>20<span class="_ _c"></span>09<span class="fc0">;<span class="_ _4"> </span></span>Utkin,<span class="_ _a"> </span>201<span class="_ _9"></span>3<span class="fc0">;<span class="_ _4"> </span></span>Utkin<span class="_ _4"> </span>and<span class="_ _4"> </span>Poznyak,<span class="_ _a"> </span>201<span class="_ _2"></span>3<span class="fc0">).<span class="_ _4"> </span>In</span></div><div class="t m6 x10 ha y64 ff1 fs7 fc0 sc0 ls0 ws0">addition,<span class="_ _5"> </span>this<span class="_ _6"> </span>algorithm<span class="_ _5"> </span>has<span class="_ _5"> </span>been<span class="_ _5"> </span>enhanced<span class="_ _5"> </span>by<span class="_ _6"> </span>using<span class="_ _5"> </span>variable<span class="_ _6"> </span>gains</div><div class="t m6 x10 ha y65 ff1 fs7 fc0 sc0 ls0 ws0">(<span class="fc1">Davila,<span class="_ _7"> </span>Moreno,<span class="_ _8"> </span>and<span class="_ _7"> </span>Fridman,<span class="_ _8"> </span>20<span class="_ _0"></span>1<span class="_ _9"></span>0<span class="fc0">;<span class="_ _8"> </span></span>Gonzalez,<span class="_ _7"> </span>Moreno,<span class="_ _8"> </span>and<span class="_ _8"> </span>F<span class="_ _9"></span>rid-</span></div><div class="t m6 x10 ha y66 ff1 fs7 fc1 sc0 ls0 ws0">man,<span class="_ _6"> </span>201<span class="_ _9"></span>0<span class="fc0">).<span class="_ _5"> </span>It<span class="_ _5"> </span>is<span class="_ _5"> </span>important<span class="_ _5"> </span>to<span class="_ _5"> </span>point<span class="_ _6"> </span>out<span class="_ _5"> </span>that<span class="_ _5"> </span>variable<span class="_ _5"> </span>gains<span class="_ _5"> </span>allow<span class="_ _6"> </span>to</span></div><div class="t m6 x10 ha y67 ff1 fs7 fc0 sc0 ls0 ws0">attenuate<span class="_ _e"> </span>the<span class="_ _e"> </span>chattering,<span class="_ _e"> </span>compensating<span class="_ _f"> </span>perturbations<span class="_ _e"> </span>whose</div><div class="t m6 x10 ha y68 ff1 fs7 fc0 sc0 ls0 ws0">bounds<span class="_ _1"> </span>are<span class="_ _1"> </span>time-v<span class="_ _0"></span>ariant.<span class="_ _1"> </span>Recently<span class="_ _2"></span>,<span class="_ _1"> </span>variable<span class="_ _1"> </span>gains<span class="_ _1"> </span>ST<span class="_ _9"></span>A<span class="_ _1"> </span>has<span class="_ _1"> </span>been</div><div class="t m6 x10 ha y69 ff1 fs7 fc0 sc0 ls0 ws0">found<span class="_ _8"> </span>in<span class="_ _8"> </span>several<span class="_ _7"> </span>applications<span class="_ _8"> </span>(<span class="fc1">Davila<span class="_ _8"> </span>et<span class="_ _8"> </span>al.,<span class="_ _8"> </span>20<span class="_ _9"></span>1<span class="_ _0"></span>0<span class="fc0">;<span class="_ _8"> </span></span>Evangelista,<span class="_ _7"> </span>Pu-</span></div><div class="t m6 x10 ha y6a ff1 fs7 fc1 sc0 ls0 ws0">leston,<span class="_ _5"> </span>Valenciaga,<span class="_ _8"> </span>and<span class="_ _5"> </span>Fridman,<span class="_ _7"> </span>201<span class="_ _2"></span>3<span class="fc0">,<span class="_ _7"> </span></span>201<span class="_ _9"></span>4<span class="fc0">;<span class="_ _7"> </span></span>Gonzalez<span class="_ _7"> </span>et<span class="_ _7"> </span>al.,<span class="_ _7"> </span>201<span class="_ _9"></span>0<span class="fc0">,</span></div><div class="t m6 x10 ha y6b ff1 fs7 fc1 sc0 ls0 ws0">Gonzalez,<span class="_ _8"> </span>Moreno,<span class="_ _8"> </span>and<span class="_ _8"> </span>Fridman,<span class="_ _7"> </span>201<span class="_ _9"></span>2<span class="fc0">;<span class="_ _8"> </span></span>Mishra<span class="_ _8"> </span>and<span class="_ _8"> </span>Kurode,<span class="_ _8"> </span>20<span class="_ _9"></span>1<span class="_ _0"></span>4<span class="fc0">;</span></div><div class="t m6 x10 ha y6c ff1 fs7 fc1 sc0 ls0 ws0">Zhan,<span class="_ _a"> </span>Guo,<span class="_ _a"> </span>and<span class="_ _4"> </span>Zhu,<span class="_ _a"> </span>201<span class="_ _2"></span>3<span class="fc0">).<span class="_ _a"> </span>Even<span class="_ _a"> </span>though<span class="_ _4"> </span>these<span class="_ _a"> </span>methods<span class="_ _4"> </span>present</span></div><div class="t m6 x10 ha y6d ff1 fs7 fc0 sc0 ls0 ws0">successfully<span class="_ _8"> </span>the<span class="_ _8"> </span>use<span class="_ _8"> </span>of<span class="_ _8"> </span>STA<span class="_ _7"> </span>with<span class="_ _a"> </span>v<span class="_ _9"></span>ariable<span class="_ _a"> </span>gains<span class="_ _8"> </span>in<span class="_ _8"> </span>different<span class="_ _8"> </span>appli-</div><div class="t m6 x10 ha y6e ff1 fs7 fc0 sc0 ls0 ws0">cations,<span class="_ _7"> </span>they<span class="_ _7"> </span>are<span class="_ _7"> </span>developed<span class="_ _7"> </span>in<span class="_ _8"> </span>continuous-time<span class="_ _7"> </span>domain.<span class="_ _7"> </span>However<span class="_ _9"></span>,</div><div class="t m6 x10 ha y6f ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _8"> </span>control<span class="_ _a"> </span>and<span class="_ _8"> </span>state<span class="_ _8"> </span>estimation<span class="_ _a"> </span>strategies<span class="_ _8"> </span>for<span class="_ _a"> </span>industrial<span class="_ _8"> </span>applica-</div><div class="t m6 x10 ha y70 ff1 fs7 fc0 sc0 ls0 ws0">tions,<span class="_ _5"> </span>especially<span class="_ _5"> </span>for<span class="_ _5"> </span>electrical<span class="_ _5"> </span>motor<span class="_ _6"> </span>drives,<span class="_ _5"> </span>are<span class="_ _5"> </span>implemented<span class="_ _5"> </span>using</div><div class="t m6 x10 ha y71 ff1 fs7 fc0 sc0 ls0 ws0">microcontrollers<span class="_ _1"> </span>and<span class="_ _3"> </span>digital<span class="_ _3"> </span>signal<span class="_ _1"> </span>processors<span class="_ _1"> </span>(DSP)<span class="_ _3"> </span>which<span class="_ _3"> </span>de-</div><div class="t m6 x10 ha y72 ff1 fs7 fc0 sc0 ls0 ws0">mand<span class="_ _7"> </span>discrete-time<span class="_ _5"> </span>algorithms.<span class="_ _7"> </span>As<span class="_ _7"> </span>a<span class="_ _7"> </span>consequence,<span class="_ _7"> </span>SM<span class="_ _7"> </span>approaches</div><div class="t m6 x10 ha y73 ff1 fs7 fc0 sc0 ls0 ws0">in<span class="_ _3"> </span>discrete-time<span class="_ _3"> </span>domain<span class="_ _3"> </span>ha<span class="_ _9"></span>ve<span class="_ _3"> </span>gained<span class="_ _3"> </span>more<span class="_ _3"> </span>attention<span class="_ _3"> </span>from<span class="_ _3"> </span>re-</div><div class="t m6 x10 ha y74 ff1 fs7 fc0 sc0 ls0 ws0">searchers<span class="_ _8"> </span>in<span class="_ _8"> </span>the<span class="_ _8"> </span>last<span class="_ _8"> </span>few<span class="_ _8"> </span>decades<span class="_ _8"> </span>(<span class="fc1">Salgado<span class="_ _8"> </span>et<span class="_ _8"> </span>al.,<span class="_ _8"> </span>20<span class="_ _9"></span>1<span class="_ _0"></span>4<span class="fc0">).<span class="_ _8"> </span>Since<span class="_ _8"> </span>the</span></span></div><div class="t m6 x10 ha y75 ff1 fs7 fc0 sc0 ls0 ws0">discrete-time<span class="_ _3"> </span>approach<span class="_ _3"> </span>differs<span class="_ _3"> </span>from<span class="_ _d"> </span>continuous-time<span class="_ _3"> </span>one<span class="_ _3"> </span>with</div><div class="t m6 x10 ha y76 ff1 fs7 fc0 sc0 ls0 ws0">respect<span class="_ _8"> </span>to<span class="_ _7"> </span>the<span class="_ _8"> </span>sliding<span class="_ _8"> </span>motion<span class="_ _8"> </span>in<span class="_ _8"> </span>the<span class="_ _8"> </span>vicinity<span class="_ _8"> </span>of<span class="_ _7"> </span>sliding<span class="_ _8"> </span>surface<span class="_ _8"> </span>(i.e</div><div class="t m6 x10 ha y77 ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _d"> </span>motion<span class="_ _d"> </span>is<span class="_ _3"> </span>of<span class="_ _c"></span>ten<span class="_ _d"> </span>called<span class="_ _d"> </span>q<span class="_ _9"></span>uasi-sliding<span class="_ _d"> </span>mode)<span class="_ _d"> </span>(<span class="fc1">Bartoszewicz,</span></div><div class="t m6 x10 ha y78 ff1 fs7 fc1 sc0 ls0 ws0">1<span class="_ _9"></span>998<span class="fc0">).<span class="_ _a"> </span>In<span class="_ _8"> </span>regard<span class="_ _8"> </span>to<span class="_ _8"> </span>STA,<span class="_ _8"> </span></span>Dominguez,<span class="_ _a"> </span>Nav<span class="_ _9"></span>arrete,<span class="_ _8"> </span>Meza,<span class="_ _a"> </span>Loukianov<span class="_ _9"></span>,</div><div class="t m6 x10 ha y79 ff1 fs7 fc1 sc0 ls0 ws0">and<span class="_ _8"> </span>Canedo<span class="_ _7"> </span>(201<span class="_ _9"></span>4)<span class="_ _8"> </span><span class="fc0">present<span class="_ _7"> </span>an<span class="_ _8"> </span>algorithm<span class="_ _8"> </span>based<span class="_ _7"> </span>on<span class="_ _8"> </span>(</span>Salgado<span class="_ _8"> </span>et<span class="_ _7"> </span>al.,</div><div class="t m6 x10 ha y7a ff1 fs7 fc1 sc0 ls0 ws0">201<span class="_ _2"></span>4<span class="fc0">).<span class="_ _8"> </span>The<span class="_ _7"> </span>proposed<span class="_ _7"> </span>approach<span class="_ _8"> </span>is<span class="_ _7"> </span>used<span class="_ _7"> </span>to<span class="_ _8"> </span>estimate<span class="_ _7"> </span>sinusoidal<span class="_ _8"> </span>back</span></div><div class="t m6 x10 ha y7b ff1 fs7 fc0 sc0 ls0 ws0">electromotiv<span class="_ _0"></span>e<span class="_ _1"> </span>forces<span class="_ _1"> </span>(EMFs)<span class="_ _1"> </span>for<span class="_ _1"> </span>classical<span class="_ _1"> </span>PMSMs.<span class="_ _3"> </span>Although,<span class="_ _1"> </span>the</div><div class="t m6 x10 ha y7c ff1 fs7 fc0 sc0 ls0 ws0">sinusoidal<span class="_ _7"> </span>back<span class="_ _9"></span>-EMF<span class="_ _7"> </span>is<span class="_ _7"> </span>properly<span class="_ _5"> </span>obtained,<span class="_ _8"> </span>the<span class="_ _5"> </span>gains<span class="_ _7"> </span>are<span class="_ _7"> </span>tuned<span class="_ _7"> </span>just</div><div class="t m6 x10 ha y7d ff1 fs7 fc0 sc0 ls0 ws0">for<span class="_ _7"> </span>the<span class="_ _7"> </span>fundamental<span class="_ _8"> </span>harmonic<span class="_ _7"> </span>component,<span class="_ _7"> </span>which<span class="_ _7"> </span>in<span class="_ _8"> </span>turn<span class="_ _7"> </span>req<span class="_ _0"></span>uires</div><div class="t m6 x10 ha y7e ff1 fs7 fc0 sc0 ls0 ws0">low<span class="_ _7"> </span>constant<span class="_ _8"> </span>gains<span class="_ _7"> </span>(limited<span class="_ _8"> </span>rotor<span class="_ _7"> </span>speed<span class="_ _8"> </span>range).<span class="_ _7"> </span>This<span class="_ _8"> </span>method<span class="_ _8"> </span>with</div><div class="t m6 x10 ha y7f ff1 fs7 fc0 sc0 ls0 ws0">constant<span class="_ _4"> </span>gains<span class="_ _1"> </span>is<span class="_ _a"> </span>inadequate<span class="_ _4"> </span>for<span class="_ _4"> </span>estimating<span class="_ _4"> </span>nonsinusoidal<span class="_ _1"> </span>back<span class="_ _2"></span>-</div><div class="t m6 x10 ha y80 ff1 fs7 fc0 sc0 ls0 ws0">EMFs<span class="_ _4"> </span>in<span class="_ _4"> </span>variable<span class="_ _4"> </span>rot<span class="_ _0"></span>or<span class="_ _4"> </span>speed<span class="_ _4"> </span>applications<span class="_ _1"> </span>due<span class="_ _a"> </span>to<span class="_ _4"> </span>the<span class="_ _4"> </span>chattering</div><div class="t m6 x10 ha y81 ff1 fs7 fc0 sc0 ls0 ws0">increase.</div><div class="t m6 x11 ha y82 ff1 fs7 fc0 sc0 ls0 ws0">Thus,<span class="_ _5"> </span>differently<span class="_ _5"> </span>from<span class="_ _5"> </span>the<span class="_ _7"> </span>above-mentioned<span class="_ _6"> </span>methods<span class="_ _7"> </span>that<span class="_ _5"> </span>have</div><div class="t m6 x10 ha y83 ff1 fs7 fc0 sc0 ls0 ws0">been<span class="_ _6"> </span>proposed<span class="_ _6"> </span>in<span class="_ _5"> </span>continuous-time<span class="_ _5"> </span>domain<span class="_ _5"> </span>or<span class="_ _5"> </span>have<span class="_ _6"> </span>been<span class="_ _6"> </span>presented</div><div class="t m6 x10 ha y84 ff1 fs7 fc0 sc0 ls0 ws0">for<span class="_ _d"> </span>sinusoidal<span class="_ _10"> </span>back<span class="_ _9"></span>-EMF<span class="_ _d"> </span>estimation,<span class="_ _10"> </span>a<span class="_ _d"> </span>new<span class="_ _10"> </span>discrete-time<span class="_ _d"> </span>ST<span class="_ _9"></span>A</div><div class="t m6 x10 ha y85 ff1 fs7 fc0 sc0 ls0 ws0">(DTST<span class="_ _9"></span>A)<span class="_ _d"> </span>sliding<span class="_ _3"> </span>mode<span class="_ _3"> </span>observer<span class="_ _3"> </span>with<span class="_ _11"> </span>variable<span class="_ _3"> </span>gains<span class="_ _3"> </span>for<span class="_ _3"> </span>PMSM</div><div class="t m6 x10 ha y86 ff1 fs7 fc0 sc0 ls0 ws0">drives<span class="_ _8"> </span>is<span class="_ _8"> </span>proposed<span class="_ _7"> </span>in<span class="_ _8"> </span>this<span class="_ _8"> </span>paper<span class="_ _0"></span>.<span class="_ _8"> </span>This<span class="_ _8"> </span>observer<span class="_ _7"> </span>allows<span class="_ _8"> </span>to<span class="_ _8"> </span>estimate</div><div class="t m6 x10 ha y87 ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _3"> </span>nonsinusoidal<span class="_ _3"> </span>back<span class="_ _2"></span>-EMFs<span class="_ _3"> </span>which<span class="_ _3"> </span>are<span class="_ _3"> </span>r<span class="_ _0"></span>equired<span class="_ _1"> </span>for<span class="_ _3"> </span>the<span class="_ _3"> </span>non-</div><div class="t m6 x10 ha y88 ff1 fs7 fc0 sc0 ls0 ws0">sinusoidal<span class="_ _5"> </span>sensorless<span class="_ _5"> </span>vector<span class="_ _6"> </span>control<span class="_ _5"> </span>in<span class="_ _5"> </span>a<span class="_ _5"> </span>wide<span class="_ _5"> </span>rotor<span class="_ _6"> </span>speed<span class="_ _7"> </span>range.<span class="_ _6"> </span>In</div><div class="t m6 x10 ha y89 ff1 fs7 fc0 sc0 ls0 ws0">other<span class="_ _5"> </span>words,<span class="_ _5"> </span>the<span class="_ _5"> </span>estimated<span class="_ _7"> </span>back<span class="_ _9"></span>-EMFs<span class="_ _7"> </span>are<span class="_ _5"> </span>used<span class="_ _5"> </span>to<span class="_ _7"> </span>derive<span class="_ _5"> </span>the<span class="_ _5"> </span>rotor</div><div class="t m6 x10 ha y8a ff1 fs7 fc0 sc0 ls0 ws0">speed,<span class="_ _6"> </span>the<span class="_ _5"> </span>synchronous<span class="_ _5"> </span>reference<span class="_ _6"> </span>frame<span class="_ _5"> </span>and<span class="_ _6"> </span>the<span class="_ _5"> </span>decoupling<span class="_ _5"> </span>terms.</div><div class="t m6 x12 ha y8b ff1 fs7 fc1 sc0 ls0 ws0">Fig.<span class="_ _5"> </span>1<span class="_ _8"> </span><span class="fc0">shows<span class="_ _7"> </span>the<span class="_ _8"> </span>digital<span class="_ _8"> </span>implementation<span class="_ _8"> </span>o<span class="_ _0"></span>verview<span class="_ _7"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span>observer</span></div><div class="t m6 x12 ha y8c ff1 fs7 fc0 sc0 ls0 ws0">and<span class="_ _8"> </span>the<span class="_ _7"> </span>sensorless<span class="_ _8"> </span>vector<span class="_ _7"> </span>control.<span class="_ _7"> </span>In<span class="_ _8"> </span>this<span class="_ _8"> </span>observer<span class="_ _9"></span>,<span class="_ _8"> </span>the<span class="_ _7"> </span>back<span class="_ _9"></span>-EMFs</div><div class="t m6 x12 ha y8d ff1 fs7 fc0 sc0 ls0 ws0">are<span class="_ _5"> </span>assumed<span class="_ _7"> </span>as<span class="_ _7"> </span>bounded<span class="_ _8"> </span>and<span class="_ _5"> </span>unmatched<span class="_ _7"> </span>disturbances.<span class="_ _7"> </span>Hence,<span class="_ _7"> </span>the</div><div class="t m6 x12 ha y8e ff1 fs7 fc0 sc0 ls0 ws0">ST<span class="_ _9"></span>A<span class="_ _a"> </span>gains<span class="_ _a"> </span>are<span class="_ _8"> </span>adjusted<span class="_ _a"> </span>online<span class="_ _a"> </span>according<span class="_ _8"> </span>to<span class="_ _a"> </span>the<span class="_ _a"> </span>ro<span class="_ _9"></span>tor<span class="_ _a"> </span>speed.<span class="_ _a"> </span>As<span class="_ _8"> </span>a</div><div class="t m6 x12 ha y8f ff1 fs7 fc0 sc0 ls0 ws0">result,<span class="_ _7"> </span>the<span class="_ _7"> </span>chattering<span class="_ _7"> </span>that<span class="_ _7"> </span>is<span class="_ _8"> </span>caused<span class="_ _5"> </span>by<span class="_ _7"> </span>the<span class="_ _8"> </span>digital<span class="_ _5"> </span>implementation</div><div class="t m6 x12 ha y90 ff1 fs7 fc0 sc0 ls0 ws0">and<span class="_ _3"> </span>the<span class="_ _1"> </span>parameter<span class="_ _3"> </span>uncertainties<span class="_ _3"> </span>is<span class="_ _3"> </span>signi<span class="ff7">fi</span>cantly<span class="_ _1"> </span>reduced<span class="_ _3"> </span>in<span class="_ _3"> </span>all</div><div class="t m6 x12 ha y91 ff1 fs7 fc0 sc0 ls0 ws0">operational<span class="_ _4"> </span>speed<span class="_ _1"> </span>range.<span class="_ _1"> </span>Moreo<span class="_ _9"></span>ver<span class="_ _0"></span>,<span class="_ _1"> </span>the<span class="_ _4"> </span>stability<span class="_ _1"> </span>analysis<span class="_ _4"> </span>of<span class="_ _1"> </span>the</div><div class="t m6 x12 ha y92 ff1 fs7 fc0 sc0 ls0 ws0">observer<span class="_ _5"> </span>is<span class="_ _5"> </span>investigated<span class="_ _6"> </span>by<span class="_ _5"> </span>means<span class="_ _7"> </span>of<span class="_ _5"> </span>Ly<span class="_ _9"></span>apunov<span class="_ _5"> </span>theory<span class="_ _7"> </span>for<span class="_ _5"> </span>discrete-</div><div class="t m6 x12 ha y93 ff1 fs7 fc0 sc0 ls0 ws0">time<span class="_ _a"> </span>systems.<span class="_ _a"> </span>This<span class="_ _a"> </span>analysis<span class="_ _a"> </span>pro<span class="_ _9"></span>vides<span class="_ _4"> </span>numerical<span class="_ _a"> </span>tools<span class="_ _a"> </span>to<span class="_ _a"> </span>evaluate</div><div class="t m6 x12 ha y94 ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _a"> </span><span class="ff7">fi</span>nite-time<span class="_ _4"> </span>converg<span class="_ _0"></span>ence<span class="_ _a"> </span>and<span class="_ _4"> </span>the<span class="_ _a"> </span>observer<span class="_ _a"> </span>design.<span class="_ _4"> </span>Finally,<span class="_ _a"> </span>si-</div><div class="t m6 x12 ha y95 ff1 fs7 fc0 sc0 ls0 ws0">mulation<span class="_ _3"> </span>and<span class="_ _1"> </span>experimental<span class="_ _3"> </span>r<span class="_ _0"></span>esults<span class="_ _3"> </span>rev<span class="_ _0"></span>eal<span class="_ _3"> </span>the<span class="_ _1"> </span>excellent<span class="_ _1"> </span>perfor<span class="_ _0"></span>-</div><div class="t m6 x12 ha y96 ff1 fs7 fc0 sc0 ls0 ws0">mance<span class="_ _8"> </span>and<span class="_ _a"> </span>effectiveness<span class="_ _8"> </span>of<span class="_ _8"> </span>the<span class="_ _a"> </span>proposed<span class="_ _8"> </span>observer<span class="_ _8"> </span>as<span class="_ _8"> </span>well<span class="_ _8"> </span>as<span class="_ _a"> </span>the</div><div class="t m6 x12 ha y97 ff1 fs7 fc0 sc0 ls0 ws0">sensorless<span class="_ _7"> </span>vector<span class="_ _8"> </span>control.</div><div class="t m6 x13 ha y98 ff1 fs7 fc0 sc0 ls0 ws0">The<span class="_ _8"> </span>rest<span class="_ _7"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span>paper<span class="_ _8"> </span>is<span class="_ _8"> </span>organized<span class="_ _8"> </span>as<span class="_ _8"> </span>follow<span class="_ _0"></span>s.<span class="_ _8"> </span>In<span class="_ _8"> </span><span class="fc1">Section<span class="_ _8"> </span>2</span>,<span class="_ _7"> </span>the</div><div class="t m6 x12 ha y99 ff1 fs7 fc0 sc0 ls0 ws0">proposed<span class="_ _4"> </span>observer<span class="_ _1"> </span>is<span class="_ _1"> </span>described<span class="_ _4"> </span>in<span class="_ _1"> </span>details.<span class="_ _1"> </span>In<span class="_ _4"> </span>the<span class="_ _1"> </span>following,<span class="_ _4"> </span>the</div><div class="t m6 x12 ha y9a ff1 fs7 fc0 sc0 ls0 ws0">sensorless<span class="_ _a"> </span>v<span class="_ _0"></span>ector<span class="_ _8"> </span>control<span class="_ _a"> </span>with<span class="_ _a"> </span>h<span class="_ _9"></span>ybrid<span class="_ _a"> </span>orientation<span class="_ _a"> </span>is<span class="_ _a"> </span>presented<span class="_ _8"> </span>in</div><div class="t m6 x12 ha y9b ff1 fs7 fc1 sc0 ls0 ws0">Section<span class="_ _a"> </span>3<span class="_ _c"></span><span class="fc0 ls5">.I<span class="_ _12"></span>n<span class="fc1 ls0">Section<span class="_ _4"> </span>4<span class="fc0">,<span class="_ _a"> </span>simulation<span class="_ _4"> </span>and<span class="_ _4"> </span>experimental<span class="_ _4"> </span>results<span class="_ _a"> </span>are</span></span></span></div><div class="t m6 x12 ha y9c ff1 fs7 fc0 sc0 ls0 ws0">shown<span class="_ _4"> </span>to<span class="_ _4"> </span>demonstrate<span class="_ _4"> </span>the<span class="_ _1"> </span>effectiveness<span class="_ _a"> </span>and<span class="_ _1"> </span>the<span class="_ _4"> </span>estimation<span class="_ _4"> </span>ac-</div><div class="t m6 x12 ha y9d ff1 fs7 fc0 sc0 ls0 ws0">curacy<span class="_ _6"> </span>of<span class="_ _6"> </span>the<span class="_ _5"> </span>proposed<span class="_ _6"> </span>method.<span class="_ _5"> </span>Then,<span class="_ _6"> </span>conclusions<span class="_ _5"> </span>are<span class="_ _6"> </span>summarized</div><div class="t m6 x12 ha y9e ff1 fs7 fc0 sc0 ls0 ws0">in<span class="_ _5"> </span><span class="fc1">Section<span class="_ _5"> </span>5</span>.<span class="_ _5"> </span>Finally,<span class="_ _6"> </span>the<span class="_ _7"> </span>stability<span class="_ _6"> </span>analysis<span class="_ _5"> </span>of<span class="_ _7"> </span>the<span class="_ _5"> </span>proposed<span class="_ _5"> </span>observer</div><div class="t m6 x12 ha y9f ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _8"> </span>inv<span class="_ _9"></span>estigated<span class="_ _8"> </span>in<span class="_ _8"> </span><span class="fc1">Appendix<span class="_ _7"> </span>A</span>.</div><div class="t m6 x12 ha ya0 ff6 fs7 fc0 sc0 ls0 ws0">2.<span class="_ _10"> </span>DTSTA<span class="_ _7"> </span>SM<span class="_ _8"> </span>observer<span class="_ _8"> </span>with<span class="_ _7"> </span>variable<span class="_ _8"> </span>gains</div><div class="t m6 x13 ha ya1 ff1 fs7 fc0 sc0 ls0 ws0">The<span class="_ _1"> </span>discrete-time<span class="_ _1"> </span>linear<span class="_ _3"> </span>time-inv<span class="_ _9"></span>ariant<span class="_ _3"> </span>model<span class="_ _1"> </span>of<span class="_ _1"> </span>the<span class="_ _3"> </span>PMSM</div><div class="t m6 x12 ha ya2 ff1 fs7 fc0 sc0 ls0 ws0">that<span class="_ _a"> </span>is<span class="_ _a"> </span>shown<span class="_ _a"> </span>in<span class="_ _4"> </span><span class="fc1">Fig.<span class="_ _8"> </span>1<span class="_ _a"> </span></span>can<span class="_ _4"> </span>be<span class="_ _a"> </span>expressed<span class="_ _a"> </span>in<span class="_ _a"> </span>stationary<span class="_ _4"> </span>reference</div><div class="t m6 x12 ha ya3 ff1 fs7 fc0 sc0 ls0 ws0">frame<span class="_ _7"> </span>by</div><div class="t mb x6 hf ya4 ffb fsb fc0 sc0 ls6 ws0">()</div><div class="t mc x8 h10 ya5 ffb fsc fc0 sc0 ls7 ws0">(<span class="_ _1"> </span>+<span class="_"> </span>)=<span class="_ _13"> </span>−<span class="_ _14"> </span>(<span class="_ _6"> </span>)+<span class="_ _15"> </span>(<span class="_ _6"> </span>)−<span class="_ _16"> </span>(<span class="_ _6"> </span>)</div><div class="t mc x14 h10 ya6 ffb fsc fc0 sc0 ls8 ws0">()</div><div class="t mc x15 h11 ya7 ffc fsd fc0 sc0 ls9 ws0">αβ<span class="_ _17"> </span>αβ<span class="_ _18"> </span>αβ<span class="_ _19"> </span>αβ</div><div class="t mc x16 h12 ya8 ffd fsc fc0 sc0 ls0 ws0">⎛</div><div class="t mc x16 h12 ya9 ffd fsc fc0 sc0 ls0 ws0">⎝</div><div class="t mc x16 h12 yaa ffd fsc fc0 sc0 ls0 ws0">⎜</div><div class="t mc x17 h12 yab ffd fsc fc0 sc0 ls0 ws0">⎞</div><div class="t mc x17 h12 yac ffd fsc fc0 sc0 ls0 ws0">⎠</div><div class="t mc x17 h12 yad ffd fsc fc0 sc0 ls0 ws0">⎟</div><div class="t mc x18 h13 yae ffe fsc fc0 sc0 ls0 ws0">k</div><div class="t mc x19 h13 yaf ffe fsc fc0 sc0 lsa ws0">TR</div><div class="t mc x1a h13 yb0 ffe fsc fc0 sc0 ls0 ws0">L</div><div class="t mc x1b h13 yae ffe fsc fc0 sc0 ls0 ws0">k</div><div class="t mc x1c h13 yaf ffe fsc fc0 sc0 ls0 ws0">T</div><div class="t mc x1c h13 yb0 ffe fsc fc0 sc0 ls0 ws0">L</div><div class="t mc x1d h13 yae ffe fsc fc0 sc0 lsb ws0">kk<span class="_ _1a"></span><span class="fff lsc">ii<span class="_ _1b"></span>v<span class="_ _1c"></span>e<span class="_ _1d"></span><span class="ff10 lsd">11</span></span></div><div class="t mc x1e h13 yb1 ff10 fsc fc0 sc0 ls0 ws0">1</div><div class="t mc x1a h14 yb2 ffe fsd fc0 sc0 lse ws0">ss</div><div class="t mc x1f h14 yb3 ffe fsd fc0 sc0 ls0 ws0">s</div><div class="t mc x20 h14 yb2 ffe fsd fc0 sc0 ls0 ws0">s</div><div class="t mc x20 h14 yb3 ffe fsd fc0 sc0 ls0 ws0">s</div><div class="t m6 x12 ha yb4 ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="t mc x21 h15 yb5 ffb fse fc0 sc0 lsf ws0">=[<span class="_ _13"> </span>]</div><div class="t mc x22 h16 yb6 ffc fsf fc0 sc0 ls10 ws0">αβ<span class="_ _1e"> </span>α<span class="_ _10"> </span>β</div><div class="t mc x23 h17 yb5 ffe fse fc0 sc0 ls11 ws0">ii</div><div class="c x24 yb7 w2 h18"><div class="t mc x0 h17 yb8 fff fse fc0 sc0 ls0 ws0">i</div><div class="t mc x25 h19 yb9 ffe fsf fc0 sc0 ls0 ws0">T</div></div><div class="t m6 x26 h1a yb4 ff1 fs7 fc0 sc0 ls0 ws0">are<span class="_ _1"> </span>the<span class="_ _3"> </span>stator<span class="_ _1"> </span>curr<span class="_ _0"></span>ents,<span class="_ _1"> </span><span class="ff4">R</span></div><div class="t md x27 h1b yba ff4 fs10 fc0 sc0 ls0 ws0">s</div><div class="t m6 x28 ha yb4 ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _1"> </span>the<span class="_ _3"> </span>stator<span class="_ _1"> </span>re-</div><div class="t m6 x12 h1a ybb ff1 fs7 fc0 sc0 ls0 ws0">sistance,<span class="_ _7"> </span><span class="ff4">L</span></div><div class="t md x29 h1b ybc ff4 fs10 fc0 sc0 ls0 ws0">s</div><div class="t m6 x2a h1a ybd ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _7"> </span>the<span class="_ _5"> </span>stator<span class="_ _7"> </span>inductance,<span class="_ _7"> </span><span class="ff4">T</span></div><div class="t md x2b h1b ybc ff4 fs10 fc0 sc0 ls0 ws0">s</div><div class="t m6 x2c ha ybd ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _5"> </span>the<span class="_ _7"> </span>sampling<span class="_ _8"> </span>time<span class="_ _5"> </span>period,</div><div class="t mc x13 h15 ybe ffb fse fc0 sc0 lsf ws0">=[<span class="_ _1f"> </span>]</div><div class="t mc x15 h16 ybf ffc fsf fc0 sc0 ls10 ws0">αβ<span class="_ _20"> </span>α<span class="_ _f"> </span>β</div><div class="t mc x2d h17 ybe ffe fse fc0 sc0 ls12 ws0">vv<span class="_ _21"></span><span class="fff ls0">v</span></div><div class="c x12 yc0 w3 h18"><div class="t mc x2e h19 yc1 ffe fsf fc0 sc0 ls0 ws0">T</div></div><div class="t m6 x2f ha yc2 ff1 fs7 fc0 sc0 ls0 ws0">are<span class="_ _6"> </span>the<span class="_ _5"> </span>stator<span class="_ _5"> </span>phase<span class="_ _5"> </span>voltages,<span class="_ _5"> </span>and</div><div class="t mc x30 h15 ybe ffb fse fc0 sc0 lsf ws0">=[<span class="_ _22"> </span>]</div><div class="t mc x27 h16 ybf ffc fsf fc0 sc0 ls10 ws0">αβ<span class="_ _23"> </span>α<span class="_ _f"> </span>β</div><div class="t mc x31 h17 ybe ffe fse fc0 sc0 ls13 ws0">ee</div><div class="c x32 yc0 w4 h18"><div class="t mc x0 h17 yc3 fff fse fc0 sc0 ls0 ws0">e</div><div class="t mc x1 h19 yc1 ffe fsf fc0 sc0 ls0 ws0">T</div></div><div class="t m6 x33 ha yc2 ff1 fs7 fc0 sc0 ls0 ws0">are<span class="_ _6"> </span>the</div><div class="t m6 x12 ha yc4 ff1 fs7 fc0 sc0 ls0 ws0">phase<span class="_ _8"> </span>back<span class="_ _2"></span>-EMFs.<span class="_ _8"> </span>Here,<span class="_ _8"> </span>it<span class="_ _7"> </span>is<span class="_ _8"> </span>assumed<span class="_ _8"> </span>that<span class="_ _7"> </span>the<span class="_ _8"> </span>motor<span class="_ _7"> </span>has<span class="_ _8"> </span>surface-</div><div class="t m6 x12 ha yc5 ff1 fs7 fc0 sc0 ls0 ws0">mounted<span class="_ _7"> </span>permanent<span class="_ _8"> </span>magnets<span class="_ _8"> </span>(PMs).</div><div class="t m6 x13 ha yc6 ff1 fs7 fc0 sc0 ls0 ws0">Now<span class="_ _9"></span>,<span class="_ _a"> </span>assuming<span class="_ _a"> </span>that<span class="_ _a"> </span>the<span class="_ _a"> </span>phase<span class="_ _a"> </span>back<span class="_ _2"></span>-EMFs<span class="_ _a"> </span>in<span class="_ _a"> </span><span class="fc1">(1)<span class="_ _a"> </span></span>are<span class="_ _8"> </span>bounded</div><div class="t m6 x12 ha yc7 ff1 fs7 fc0 sc0 ls0 ws0">and<span class="_ _8"> </span>unmodelled<span class="_ _8"> </span>disturbances,<span class="_ _8"> </span>a<span class="_ _8"> </span>stator<span class="_ _8"> </span>current<span class="_ _8"> </span>state<span class="_ _8"> </span>observer<span class="_ _8"> </span>de-</div><div class="t m6 x12 ha yc8 ff1 fs7 fc0 sc0 ls0 ws0">rived<span class="_ _7"> </span>from<span class="_ _8"> </span><span class="fc1">(1)<span class="_ _7"> </span></span>can<span class="_ _8"> </span>be<span class="_ _8"> </span>written<span class="_ _7"> </span>as</div><div class="t mc x12 h10 yc9 ffb fsc fc0 sc0 ls0 ws0">^</div><div class="t mc x34 h10 yca ffb fsc fc0 sc0 ls14 ws0">(+<span class="_ _c"></span>)<span class="_ _24"></span>=<span class="_ _25"> </span>−</div><div class="t mc x35 h10 yc9 ffb fsc fc0 sc0 ls0 ws0">^</div><div class="t mc x36 h10 yca ffb fsc fc0 sc0 ls15 ws0">()<span class="_ _26"></span>+<span class="_ _27"> </span>()<span class="_ _26"></span>−<span class="_ _28"> </span>()</div><div class="t mc x14 h10 ycb ffb fsc fc0 sc0 ls8 ws0">()</div><div class="t mc x15 h11 ycc ffc fsd fc0 sc0 ls9 ws0">αβ<span class="_ _29"> </span>αβ<span class="_ _2a"> </span>αβ<span class="_ _2b"> </span>αβ</div><div class="t mc x16 h12 ycd ffd fsc fc0 sc0 ls0 ws0">⎛</div><div class="t mc x16 h12 yce ffd fsc fc0 sc0 ls0 ws0">⎝</div><div class="t mc x16 h12 ycf ffd fsc fc0 sc0 ls0 ws0">⎜</div><div class="t mc x17 h12 yd0 ffd fsc fc0 sc0 ls0 ws0">⎞</div><div class="t mc x17 h12 yd1 ffd fsc fc0 sc0 ls0 ws0">⎠</div><div class="t mc x17 h12 yd2 ffd fsc fc0 sc0 ls0 ws0">⎟</div><div class="t mc x18 h13 yd3 ffe fsc fc0 sc0 ls0 ws0">k</div><div class="t mc x19 h13 yd4 ffe fsc fc0 sc0 lsa ws0">TR</div><div class="t mc x1a h13 yd5 ffe fsc fc0 sc0 ls0 ws0">L</div><div class="t mc x37 h13 yd6 ffe fsc fc0 sc0 ls0 ws0">k</div><div class="t mc x38 h13 yd7 ffe fsc fc0 sc0 ls0 ws0">T</div><div class="t mc x38 h13 yd8 ffe fsc fc0 sc0 ls0 ws0">L</div><div class="t mc x39 h13 yd9 ffe fsc fc0 sc0 ls16 ws0">kk<span class="_ _2c"></span><span class="fff ls17">ii<span class="_ _2d"></span>v<span class="_ _2e"></span>u<span class="_ _2f"></span><span class="ff10 lsd">11</span></span></div><div class="t mc x1e h13 yda ff10 fsc fc0 sc0 ls0 ws0">2</div><div class="t mc x1a h14 ydb ffe fsd fc0 sc0 lse ws0">ss</div><div class="t mc x1f h14 ydc ffe fsd fc0 sc0 ls0 ws0">s</div><div class="t mc x3a h14 ydb ffe fsd fc0 sc0 ls0 ws0">s</div><div class="t mc x3a h14 ydc ffe fsd fc0 sc0 ls0 ws0">s</div><div class="t m6 x12 ha ydd ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="t mc x3b h15 yde ffb fse fc0 sc0 ls0 ws0">^</div><div class="t mc x3c h15 ydf ffb fse fc0 sc0 lsf ws0">=[</div><div class="t mc x3d h15 ye0 ffb fse fc0 sc0 ls18 ws0">^^</div><div class="t mc x1f h15 ydf ffb fse fc0 sc0 ls0 ws0">]</div><div class="t mc x3e h16 ye1 ffc fsf fc0 sc0 ls10 ws0">αβ<span class="_ _1e"> </span>α<span class="_ _b"> </span>β</div><div class="t mc x3d h17 ye2 ffe fse fc0 sc0 ls19 ws0">ii<span class="_ _30"></span><span class="fff ls0">i</span></div><div class="t mc x3f h19 ye3 ffe fsf fc0 sc0 ls0 ws0">T</div><div class="t m6 x35 ha ydd ff1 fs7 fc0 sc0 ls0 ws0">are<span class="_ _f"> </span>the<span class="_ _31"> </span>estimated<span class="_ _f"> </span>stator<span class="_ _f"> </span>currents,<span class="_ _31"> </span>and</div><div class="t mc x40 h15 ye4 ffb fse fc0 sc0 lsf ws0">=[<span class="_ _20"> </span>]</div><div class="t mc x41 h16 ye5 ffc fsf fc0 sc0 ls10 ws0">αβ<span class="_ _32"> </span>α<span class="_ _25"> </span>β</div><div class="t mc x42 h17 ye4 ffe fse fc0 sc0 ls1a ws0">uu<span class="_ _33"></span><span class="fff ls0">u</span></div><div class="c x12 ye6 w5 h18"><div class="t mc x43 h19 ye7 ffe fsf fc0 sc0 ls0 ws0">T</div></div><div class="t m6 x2 ha ye8 ff1 fs7 fc0 sc0 ls0 ws0">are<span class="_ _4"> </span>the<span class="_ _1"> </span>forcing<span class="_ _1"> </span>terms.<span class="_ _4"> </span>Here,<span class="_ _1"> </span>the<span class="_ _4"> </span>observer<span class="_ _1"> </span>aims<span class="_ _4"> </span>to</div><div class="t m6 x12 ha ye9 ff1 fs7 fc0 sc0 ls0 ws0">obtain<span class="_ _7"> </span>the<span class="_ _7"> </span>forcing<span class="_ _7"> </span>terms<span class="_ _5"> </span>that<span class="_ _8"> </span>minimize<span class="_ _5"> </span>the<span class="_ _7"> </span>observation<span class="_ _7"> </span>error<span class="_ _9"></span>.<span class="_ _7"> </span>As<span class="_ _7"> </span>a</div><div class="t m6 x12 ha yea ff1 fs7 fc0 sc0 ls0 ws0">result,<span class="_ _7"> </span>a<span class="_ _8"> </span>back<span class="_ _9"></span>-EMF<span class="_ _8"> </span>estimation<span class="_ _7"> </span>can<span class="_ _8"> </span>be<span class="_ _8"> </span>achieved.</div><div class="t m6 x13 ha yeb ff1 fs7 fc0 sc0 ls0 ws0">In<span class="_ _a"> </span>this<span class="_ _a"> </span>paper<span class="_ _9"></span>,<span class="_ _a"> </span>the<span class="_ _4"> </span>forcing<span class="_ _a"> </span>terms<span class="_ _a"> </span>are<span class="_ _8"> </span>determined<span class="_ _a"> </span>from<span class="_ _a"> </span>a<span class="_ _a"> </span>novel</div><div class="t m6 x12 ha yec ff1 fs7 fc0 sc0 ls0 ws0">DTST<span class="_ _9"></span>A<span class="_ _8"> </span>with<span class="_ _7"> </span>variable<span class="_ _8"> </span>gains<span class="_ _7"> </span>that<span class="_ _8"> </span>can<span class="_ _8"> </span>be<span class="_ _7"> </span>expressed<span class="_ _7"> </span>as</div><div class="t m4 x44 h8 yed ff6 fs4 fc0 sc0 ls0 ws0">Fig.<span class="_ _6"> </span>1.<span class="_ _8"> </span><span class="ff1">Block<span class="_ _5"> </span>diagram<span class="_ _7"> </span>of<span class="_ _5"> </span>the<span class="_ _5"> </span>sensorless<span class="_ _5"> </span>nonsinusoidal<span class="_ _5"> </span>vector<span class="_ _5"> </span>control<span class="_ _5"> </span>using<span class="_ _5"> </span>the<span class="_ _5"> </span>proposed<span class="_ _7"> </span>observer<span class="_ _9"></span>.</span></div><div class="t m4 x45 h6 yee ff4 fs4 fc0 sc0 ls0 ws0">C.L.<span class="_ _5"> </span>Baratieri,<span class="_ _5"> </span>H.<span class="_ _5"> </span>Pinheiro<span class="_ _5"> </span>/<span class="_ _5"> </span>Control<span class="_ _5"> </span>Engineering<span class="_ _7"> </span>Practice<span class="_ _6"> </span>52<span class="_ _7"> </span>(201<span class="_ _2"></span>6)<span class="_ _7"> </span>59<span class="_"> </span><span class="ff11">–<span class="_"> </span></span>69<span class="_ _34"></span><span class="ff1">60</span></div><a class="l" rel='nofollow' onclick='return false;'><div class="d ma"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d ma"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d ma"></div></a><a class="l" rel='nofollow' onclick='return false;'><div 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<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/624fe37a6caf596192f79c91/bg3.jpg"><div class="t mc x46 h1c yef ff12 fs11 fc0 sc0 ls0 ws0">ν<span class="_ _35"></span><span class="ffb ls1b">()<span class="_ _26"></span>=<span class="_ _25"> </span>()<span class="_ _26"></span>−<span class="_ _11"> </span>()<span class="_ _24"></span>|<span class="_"> </span>()<span class="_ _24"></span>|<span class="_ _1e"> </span>(<span class="_ _b"> </span>()<span class="_ _24"></span>)</span></div><div class="t mc x47 h1c yf0 ffb fs11 fc0 sc0 ls1c ws0">()</div><div class="t mc x48 h1d yf1 ffc fs12 fc0 sc0 ls1d ws0">αβ<span class="_ _36"> </span>αβ<span class="_ _37"> </span>αβ<span class="_ _38"> </span>αβ</div><div class="t mc x49 h1e yef ffe fs11 fc0 sc0 ls1e ws0">kk<span class="_ _39"></span>k<span class="_ _3a"></span>k<span class="_ _3b"></span>k<span class="_ _3c"> </span>k<span class="_ _3d"></span><span class="fff ls1f">us<span class="_ _3e"></span>s<span class="_ _3f"></span><span class="ff10 ls20">sign</span></span></div><div class="t mc x4a h1e yf0 ff10 fs11 fc0 sc0 ls0 ws0">3</div><div class="t mc x4b h1f yf1 ff10 fs12 fc0 sc0 ls0 ws0">1</div><div class="t mc x4c h1f yf2 ff10 fs12 fc0 sc0 ls0 ws0">1</div><div class="t mc x4c h1f yf3 ff10 fs12 fc0 sc0 ls0 ws0">2</div><div class="t mc x4d h20 yf4 ff12 fs13 fc0 sc0 ls21 ws0">νν<span class="_ _40"></span><span class="ffb ls22">(<span class="_ _5"> </span>+<span class="_ _8"> </span>)<span class="_ _26"></span>=<span class="_ _20"> </span>()<span class="_ _26"></span>−<span class="_ _41"> </span>()<span class="_ _42"> </span>(<span class="_ _b"> </span>()<span class="_ _24"></span>)</span></div><div class="t mc x47 h20 yf5 ffb fs13 fc0 sc0 ls23 ws0">()</div><div class="t mc x43 h1d yf6 ffc fs12 fc0 sc0 ls1d ws0">αβ<span class="_ _43"> </span>ν<span class="_ _a"> </span>αβ<span class="_ _44"> </span>αβ</div><div class="t mc x4e h21 yf4 ffe fs13 fc0 sc0 ls24 ws0">kK<span class="_ _45"></span>k<span class="_ _46"></span>T<span class="_ _47"></span>k<span class="_ _48"></span>k<span class="_ _25"> </span>k<span class="_ _49"></span><span class="fff ls0">s<span class="_ _4a"></span><span class="ff10 ls25">1s<span class="_ _4b"></span>i<span class="_ _4b"></span>g<span class="_ _4b"></span>n</span></span></div><div class="t mc x4a h21 yf5 ff10 fs13 fc0 sc0 ls0 ws0">4</div><div class="t mc x4f h1f yf6 ffe fs12 fc0 sc0 ls0 ws0">s</div><div class="t mc x50 h1f yf7 ff10 fs12 fc0 sc0 ls0 ws0">2</div><div class="t m6 x1 ha yf8 ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="c x51 yf9 w6 h18"><div class="t mc x0 h22 yfa ff12 fse fc0 sc0 ls0 ws0">ν</div></div><div class="t mc x52 h15 yfb ffc fse fc0 sc0 ls26 ws0">νν<span class="_ _4c"></span><span class="ffb lsf">=[<span class="_ _42"> </span>]</span></div><div class="t mc x53 h16 yfc ffc fsf fc0 sc0 ls10 ws0">αβ<span class="_ _27"> </span>α<span class="_ _e"> </span>β</div><div class="c x51 yf9 w6 h18"><div class="t mc x54 h19 ye7 ffe fsf fc0 sc0 ls0 ws0">T</div></div><div class="t m6 x55 h1a yf8 ff1 fs7 fc0 sc0 ls0 ws0">are<span class="_ _5"> </span>the<span class="_ _7"> </span>auxiliary<span class="_ _7"> </span>variables<span class="_ _5"> </span>of<span class="_ _7"> </span>the<span class="_ _5"> </span>DTSTA,<span class="_ _5"> </span><span class="ff4">K</span></div><div class="t m9 x56 h23 yfd ff13 fs14 fc0 sc0 ls0 ws0">ν</div><div class="t m6 x57 ha yf8 ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _5"> </span>a</div><div class="t m6 x1 ha yfe ff1 fs7 fc0 sc0 ls0 ws0">constant<span class="_ _1"> </span>gain<span class="_ _a"> </span>with</div><div class="t me x58 h24 yff ffb fs15 fc0 sc0 ls27 ws0"><<</div><div class="t mc x59 h25 y100 ffc fs16 fc0 sc0 ls0 ws0">ν</div><div class="t me x50 h26 yff ffe fs15 fc0 sc0 ls0 ws0">K</div><div class="c x5a y101 w7 h27"><div class="t me x0 h26 y102 ff10 fs15 fc0 sc0 ls0 ws0">0</div></div><div class="t me x5b h26 yff ff10 fs15 fc0 sc0 ls0 ws0">1</div><div class="t m6 x5c ha y103 ff1 fs7 fc0 sc0 ls0 ws0">,<span class="_ _1"> </span>and<span class="_ _4"> </span>the<span class="_ _1"> </span>switching<span class="_ _1"> </span>functions<span class="_ _1"> </span>ar<span class="_ _0"></span>e</div><div class="t m6 x1 ha y104 ff1 fs7 fc0 sc0 ls0 ws0">de<span class="ff7">fi</span>ned<span class="_ _8"> </span>as</div><div class="t mf xe h28 y105 ffb fs17 fc0 sc0 ls28 ws0">()<span class="_ _4d"></span>≜<span class="_ _25"> </span>()<span class="_ _26"></span>−</div><div class="t mf x5d h28 y106 ffb fs17 fc0 sc0 ls0 ws0">^</div><div class="t mf x5e h28 y105 ffb fs17 fc0 sc0 ls28 ws0">()</div><div class="t mf x47 h28 y107 ffb fs17 fc0 sc0 ls29 ws0">()</div><div class="t m10 x43 h29 y108 ffc fs18 fc0 sc0 ls2a ws0">αβ<span class="_ _4e"> </span>αβ<span class="_ _4e"> </span>αβ</div><div class="t mf x4e h2a y105 ffe fs17 fc0 sc0 ls2b ws0">kkk<span class="_ _4a"></span><span class="fff ls2c">sii<span class="_ _4f"></span><span class="ff10 ls0">.</span></span></div><div class="t mf x4a h2a y107 ff10 fs17 fc0 sc0 ls0 ws0">5</div><div class="t m6 x7 ha y109 ff1 fs7 fc0 sc0 ls0 ws0">In<span class="_ _8"> </span>addition,</div><div class="t m11 x5f h2b y10a ffb fs19 fc0 sc0 ls0 ws0">(</div><div class="c x60 y10b w8 h2c"><div class="t m11 x61 h2b y10c ffb fs19 fc0 sc0 ls0 ws0">)<span class="_ _50"></span><span class="ffe">k</span></div></div><div class="t m11 x62 h2d y10a ffe fs19 fc0 sc0 ls0 ws0">k</div><div class="t m12 x5d h2e y10d ff10 fs1a fc0 sc0 ls0 ws0">1</div><div class="t m6 x63 ha y109 ff1 fs7 fc0 sc0 ls0 ws0">and</div><div class="t m13 x64 h2f y10e ffb fs1b fc0 sc0 ls0 ws0">(</div><div class="c x65 y10b w9 h2c"><div class="t m13 x66 h2f y10f ffb fs1b fc0 sc0 ls0 ws0">)<span class="_ _51"></span><span class="ffe">k</span></div></div><div class="t m13 x67 h30 y10e ffe fs1b fc0 sc0 ls0 ws0">k</div><div class="t m14 x59 h31 y110 ff10 fs1c fc0 sc0 ls0 ws0">2</div><div class="t m6 x68 ha y109 ff1 fs7 fc0 sc0 ls0 ws0">gains<span class="_ _8"> </span>are<span class="_ _7"> </span>de<span class="ff7">fi</span>ned<span class="_ _8"> </span>as<span class="_ _7"> </span>follows:</div><div class="t mc x69 h10 y111 ffb fsc fc0 sc0 ls15 ws0">()<span class="_ _26"></span>=<span class="_ _52"> </span>()</div><div class="t mc x6a h10 y112 ffb fsc fc0 sc0 ls15 ws0">()<span class="_ _26"></span>=<span class="_ _27"> </span>()</div><div class="t mc x47 h10 y113 ffb fsc fc0 sc0 ls8 ws0">()</div><div class="t m10 x46 h29 y114 ffc fs18 fc0 sc0 ls0 ws0">η</div><div class="t m10 x6b h29 y115 ffc fs18 fc0 sc0 ls0 ws0">σ</div><div class="t m10 x6c h29 y116 ffc fs18 fc0 sc0 ls0 ws0">η</div><div class="t m10 x6d h29 y117 ffc fs18 fc0 sc0 ls0 ws0">σ</div><div class="t mc x1 h13 y111 ffe fsc fc0 sc0 ls2d ws0">kk<span class="_ _53"> </span>K<span class="_ _f"> </span>f<span class="_ _6"> </span>k</div><div class="t mc x1 h13 y112 ffe fsc fc0 sc0 ls2e ws0">kk<span class="_ _54"> </span>K<span class="_ _c"></span>f<span class="_ _c"></span>k</div><div class="t mc x4a h13 y113 ff10 fsc fc0 sc0 ls0 ws0">6</div><div class="t m10 x43 h32 y114 ff10 fs18 fc0 sc0 ls2f ws0">11</div><div class="t m10 x6e h32 y116 ff10 fs18 fc0 sc0 ls30 ws0">22</div><div class="t m6 x1 ha y118 ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="t mc x6f h16 y119 ffc fsf fc0 sc0 ls0 ws0">η</div><div class="t mc x70 h17 y11a ffe fse fc0 sc0 ls0 ws0">K</div><div class="t mc x71 h19 y119 ff10 fsf fc0 sc0 ls0 ws0">1</div><div class="t m6 x72 ha y118 ff1 fs7 fc0 sc0 ls0 ws0">,</div><div class="t mc x73 h16 y119 ffc fsf fc0 sc0 ls0 ws0">η</div><div class="t mc x52 h17 y11a ffe fse fc0 sc0 ls0 ws0">K</div><div class="t mc x74 h19 y119 ff10 fsf fc0 sc0 ls0 ws0">2</div><div class="t m6 x75 ha y118 ff1 fs7 fc0 sc0 ls0 ws0">,</div><div class="t m12 x55 h33 y11b ffc fs1d fc0 sc0 ls0 ws0">γ</div><div class="t mc x62 h34 y11c ffe fs1e fc0 sc0 ls0 ws0">K</div><div class="t m12 x76 h35 y11b ff10 fs1d fc0 sc0 ls0 ws0">1</div><div class="t m6 x63 ha y118 ff1 fs7 fc0 sc0 ls0 ws0">,</div><div class="t m12 x77 h33 y11b ffc fs1d fc0 sc0 ls0 ws0">γ</div><div class="t mc x58 h34 y11c ffe fs1e fc0 sc0 ls0 ws0">K</div><div class="c x58 y11d wa h36"><div class="t m12 x78 h35 y11e ff10 fs1d fc0 sc0 ls0 ws0">2</div></div><div class="t m6 x79 ha y118 ff1 fs7 fc0 sc0 ls0 ws0">are<span class="_ _1"> </span>positiv<span class="_ _0"></span>e<span class="_ _1"> </span>constants,<span class="_ _4"> </span>and</div><div class="t mc x7a h15 y11f ffb fse fc0 sc0 ls0 ws0">(</div><div class="c x7b y120 wb h37"><div class="t mc x66 h15 y121 ffb fse fc0 sc0 ls0 ws0">)</div></div><div class="t mc x7c h16 y122 ffc fsf fc0 sc0 ls0 ws0">σ</div><div class="t mc x7d h17 y11f ffe fse fc0 sc0 ls31 ws0">fk</div><div class="t m6 x7e ha y118 ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _1"> </span>a</div><div class="t m6 x1 ha y123 ff1 fs7 fc0 sc0 ls0 ws0">bounded<span class="_ _8"> </span>function<span class="_ _7"> </span>de<span class="ff7">fi</span>ned<span class="_ _8"> </span>by</div><div class="t mc x7f h38 y124 ffc fsc fc0 sc0 ls32 ws0">σσ<span class="_ _55"></span>σ</div><div class="t mc x7f h38 y125 ffc fsc fc0 sc0 ls33 ws0">σσ<span class="_ _24"></span>σ<span class="_ _26"></span>σ</div><div class="t mc x7f h38 y126 ffc fsc fc0 sc0 ls33 ws0">σσ<span class="_ _26"></span>σ</div><div class="t mc x6a h10 y127 ffb fsc fc0 sc0 ls15 ws0">()<span class="_ _4d"></span>≜</div><div class="t mc x5e h10 y124 ffb fsc fc0 sc0 ls34 ws0">≤(<span class="_ _12"></span>)<span class="_ _56"></span><</div><div class="t mc x80 h10 y125 ffb fsc fc0 sc0 ls15 ws0">()<span class="_ _4e"> </span>≤<span class="_ _4"> </span>()<span class="_ _4d"></span><</div><div class="t mc x81 h10 y126 ffb fsc fc0 sc0 ls15 ws0">()<span class="_ _4d"></span>></div><div class="t mc x47 h10 y128 ffb fsc fc0 sc0 ls8 ws0">()</div><div class="t mc x43 h11 y129 ffc fsd fc0 sc0 ls0 ws0">σ</div><div class="t mc x53 h12 y12a ffd fsc fc0 sc0 ls0 ws0">⎧</div><div class="t mc x53 h12 y12b ffd fsc fc0 sc0 ls0 ws0">⎨</div><div class="t mc x53 h12 y12c ffd fsc fc0 sc0 ls0 ws0">⎪</div><div class="t mc x53 h12 y12d ffd fsc fc0 sc0 ls0 ws0">⎩</div><div class="t mc x53 h12 y12e ffd fsc fc0 sc0 ls0 ws0">⎪</div><div class="t mc x82 h13 y12f ffe fsc fc0 sc0 ls35 ws0">fk</div><div class="t mc x4f h13 y130 ffe fsc fc0 sc0 ls0 ws0">k</div><div class="t mc x83 h13 y131 ffe fsc fc0 sc0 ls36 ws0">kk</div><div class="t mc x55 h13 y132 ffe fsc fc0 sc0 ls0 ws0">k</div><div class="t mc x74 h13 y130 ff10 fsc fc0 sc0 ls37 ws0">if<span class="_ _7"> </span>0</div><div class="t mc x74 h13 y131 ff10 fsc fc0 sc0 ls37 ws0">if</div><div class="t mc x74 h13 y132 ff10 fsc fc0 sc0 ls37 ws0">if</div><div class="t mc x4a h13 y133 ff10 fsc fc0 sc0 ls0 ws0">7</div><div class="t mc x72 h14 y134 ffe fsd fc0 sc0 ls38 ws0">min<span class="_ _57"> </span>min</div><div class="t mc x62 h14 y135 ffe fsd fc0 sc0 ls39 ws0">min<span class="_ _43"> </span>max</div><div class="t mc x72 h14 y136 ffe fsd fc0 sc0 ls3a ws0">max<span class="_ _37"> </span>max</div><div class="t m6 x1 ha y137 ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="t m15 x51 h39 y138 ffc fs1f fc0 sc0 ls0 ws0">σ<span class="_ _58"> </span><span class="ffb ls3b">()<span class="_ _4d"></span>≜<span class="_ _4d"></span>(<span class="_ _8"> </span>−<span class="_"> </span>)<span class="_ _24"></span>|<span class="_ _8"> </span>()<span class="_ _24"></span>|<span class="_ _59"></span><span class="ffe ls3c">kK<span class="_ _5a"></span>x<span class="_ _5b"></span>k<span class="_ _5c"></span><span class="ff10 ls0">1</span></span></span></div><div class="t m16 x84 h3a y139 ffe fs20 fc0 sc0 ls3d ws0">ff</div><div class="t m6 x64 h1a y137 ff1 fs7 fc0 sc0 ls0 ws0">,<span class="_ _5"> </span><span class="ff4">x</span></div><div class="t md x85 h1b y13a ff4 fs10 fc0 sc0 ls0 ws0">f</div><div class="t m6 x68 h1a y137 ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _5"> </span>the<span class="_ _7"> </span>low-pass<span class="_ _5"> </span><span class="ff7">fi</span>lter<span class="_ _5"> </span>state,<span class="_ _7"> </span>and<span class="_ _5"> </span><span class="ff4">K</span></div><div class="t md x47 h1b y13a ff4 fs10 fc0 sc0 ls0 ws0">f</div><div class="t m6 x86 ha y137 ff1 fs7 fc0 sc0 ls0 ws0">is</div><div class="t m6 x1 ha y13b ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _8"> </span>low<span class="_ _0"></span>-pass<span class="_ _8"> </span><span class="ff7">fi</span>lte<span class="_ _0"></span>r<span class="_ _8"> </span>gain<span class="_ _7"> </span>that<span class="_ _8"> </span>is<span class="_ _8"> </span>giv<span class="_ _9"></span>en<span class="_ _8"> </span>by</div><div class="t mc xe h10 y13c ffb fsc fc0 sc0 ls0 ws0">=</div><div class="t mc x47 h10 y13d ffb fsc fc0 sc0 ls8 ws0">()</div><div class="t mc x51 h3b y13e ffc fsd fc0 sc0 ls0 ws0">ω<span class="_ _3b"></span><span class="ffb">−</span></div><div class="t mc x1 h13 y13c ffe fsc fc0 sc0 ls3e ws0">Ke</div><div class="t mc x4a h13 y13d ff10 fsc fc0 sc0 ls0 ws0">8</div><div class="t mc x48 h14 y13f ffe fsd fc0 sc0 ls0 ws0">f</div><div class="t mc x46 h14 y140 ffe fsd fc0 sc0 ls0 ws0">T</div><div class="t mc x70 h3c y141 ffe fs21 fc0 sc0 ls0 ws0">f</div><div class="t mc x6f h3c y142 ffe fs21 fc0 sc0 ls0 ws0">s</div><div class="t m6 x1 ha y143 ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="t m17 x51 h3d y143 ff13 fs22 fc0 sc0 ls0 ws0">ω</div><div class="t md x46 h1b y144 ff4 fs10 fc0 sc0 ls0 ws0">f</div><div class="t m6 x71 ha y143 ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _8"> </span>the<span class="_ _7"> </span><span class="ff7">fi</span>lter<span class="_ _8"> </span>cutof<span class="_ _0"></span>f<span class="_ _8"> </span>freq<span class="_ _9"></span>uency.</div><div class="t m6 x7 ha y145 ff1 fs7 fc0 sc0 ls0 ws0">The<span class="_ _8"> </span>upper<span class="_ _7"> </span>limit<span class="_ _8"> </span><span class="ff14">s</span></div><div class="t md x63 h1b y146 ff4 fs10 fc0 sc0 ls0 ws0">max</div><div class="t m6 x87 ha y147 ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _8"> </span>de<span class="ff7">fi</span>ned<span class="_ _7"> </span>as<span class="_ _8"> </span>follows:</div><div class="t mc x1 h3e y148 ffc fs23 fc0 sc0 ls3f ws0">σλ<span class="_ _5d"></span>ω<span class="_ _5e"></span><span class="ffb ls0">≜</span></div><div class="t mc x47 h3e y149 ffb fs23 fc0 sc0 ls40 ws0">()</div><div class="t mc x88 h3f y14a ffe fs23 fc0 sc0 ls0 ws0">T</div><div class="t mc x88 h3f y14b ffe fs23 fc0 sc0 ls0 ws0">L</div><div class="t mc x4a h3f y149 ff10 fs23 fc0 sc0 ls0 ws0">9</div><div class="t mc x6e h40 y14c ffe fs24 fc0 sc0 ls41 ws0">max</div><div class="t mc x89 h40 y14d ffe fs24 fc0 sc0 ls0 ws0">s</div><div class="t mc x89 h40 y14e ffe fs24 fc0 sc0 ls0 ws0">s</div><div class="t mc x8a h40 y14c ffe fs24 fc0 sc0 ls42 ws0">pm<span class="_ _f"> </span>max</div><div class="t m6 x1 ha y14f ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="t m17 x8b h3d y14f ff13 fs22 fc0 sc0 ls0 ws0">λ</div><div class="t md x89 h1b y150 ff4 fs10 fc0 sc0 ls0 ws0">pm</div><div class="t m6 x83 ha y14f ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _4"> </span>the<span class="_ _4"> </span><span class="ff7">fl<span class="_"> </span></span>ux<span class="_ _1"> </span>linkage<span class="_ _4"> </span>due<span class="_ _4"> </span>to<span class="_ _4"> </span>the<span class="_ _4"> </span>PMs,<span class="_ _1"> </span>and</div><div class="t m17 x8c h3d y14f ff13 fs22 fc0 sc0 ls0 ws0">ω</div><div class="t md x8d h1b y150 ff4 fs10 fc0 sc0 ls0 ws0">max</div><div class="t m6 x8e ha y14f ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _4"> </span>the</div><div class="t m6 x1 ha y151 ff1 fs7 fc0 sc0 ls0 ws0">maximum<span class="_ _8"> </span>electrical<span class="_ _8"> </span>angular<span class="_ _8"> </span>speed.<span class="_ _8"> </span>On<span class="_ _8"> </span>the<span class="_ _8"> </span>other<span class="_ _8"> </span>hand,<span class="_ _8"> </span>the<span class="_ _8"> </span>low<span class="_ _0"></span>er</div><div class="t m6 x1 ha y152 ff1 fs7 fc0 sc0 ls0 ws0">limit<span class="_ _8"> </span><span class="ff14">s</span></div><div class="t md x8f h1b y153 ff4 fs10 fc0 sc0 ls0 ws0">min</div><div class="t m6 x71 ha y154 ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _8"> </span>de<span class="ff7">fi</span>ned<span class="_ _7"> </span>by:</div><div class="t mc x1 h3e y155 ffc fs23 fc0 sc0 ls43 ws0">σλ<span class="_ _5f"></span>ω<span class="_ _60"></span><span class="ffb ls0">≜</span></div><div class="t mc x7e h3e y156 ffb fs23 fc0 sc0 ls44 ws0">()</div><div class="t mc x8b h3f y157 ffe fs23 fc0 sc0 ls0 ws0">T</div><div class="t mc x8b h3f y158 ffe fs23 fc0 sc0 ls0 ws0">L</div><div class="t mc x90 h3f y156 ff10 fs23 fc0 sc0 ls45 ws0">10</div><div class="t mc x6e h40 y159 ffe fs24 fc0 sc0 ls46 ws0">min</div><div class="t mc x91 h40 y15a ffe fs24 fc0 sc0 ls0 ws0">s</div><div class="t mc x91 h40 y15b ffe fs24 fc0 sc0 ls0 ws0">s</div><div class="t mc x92 h40 y159 ffe fs24 fc0 sc0 ls42 ws0">pm<span class="_ _f"> </span>min</div><div class="t m6 x1 ha y15c ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="t m17 x51 h3d y15c ff13 fs22 fc0 sc0 ls0 ws0">ω</div><div class="t md x6c h1b y15d ff4 fs10 fc0 sc0 ls0 ws0">min</div><div class="t m6 x93 ha y15c ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _8"> </span>the<span class="_ _a"> </span>minimum<span class="_ _8"> </span>electrical<span class="_ _a"> </span>angular<span class="_ _a"> </span>speed<span class="_ _8"> </span>that<span class="_ _a"> </span>allow<span class="_ _0"></span>s</div><div class="t m6 x1 ha y15e ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _8"> </span>sensorless<span class="_ _7"> </span>control<span class="_ _8"> </span>(<span class="fc1">Zaltni<span class="_ _7"> </span>and<span class="_ _8"> </span>Ghanes,<span class="_ _8"> </span>20<span class="_ _9"></span>1<span class="_ _0"></span>3<span class="fc0">).</span></span></div><div class="t m6 x7 ha y15f ff1 fs7 fc0 sc0 ls0 ws0">A<span class="_ _7"> </span><span class="ff7">fi</span>rst<span class="_ _5"> </span>order<span class="_ _7"> </span>low-pass<span class="_ _5"> </span><span class="ff7">fi</span>lter<span class="_ _7"> </span>is<span class="_ _7"> </span>included<span class="_ _7"> </span>to<span class="_ _5"> </span>the<span class="_ _7"> </span>observer<span class="_ _7"> </span>in<span class="_ _7"> </span>order</div><div class="t m6 x1 ha y160 ff1 fs7 fc0 sc0 ls0 ws0">to<span class="_ _5"> </span>smooth<span class="_ _7"> </span>the<span class="_ _7"> </span>gain<span class="_ _7"> </span>variations.<span class="_ _5"> </span>In<span class="_ _7"> </span>other<span class="_ _7"> </span>words,<span class="_ _5"> </span>the<span class="_ _7"> </span><span class="ff7">fi</span>lter<span class="_ _7"> </span>is<span class="_ _7"> </span>adopted</div><div class="t m6 x1 ha y161 ff1 fs7 fc0 sc0 ls0 ws0">just<span class="_ _5"> </span>for<span class="_ _6"> </span>increasing<span class="_ _5"> </span>the<span class="_ _5"> </span>observer<span class="_ _5"> </span>robustness<span class="_ _6"> </span>during<span class="_ _5"> </span>transients<span class="_ _5"> </span>in<span class="_ _5"> </span>the</div><div class="t m6 x1 ha y162 ff1 fs7 fc0 sc0 ls0 ws0">electrical<span class="_ _a"> </span>syst<span class="_ _0"></span>em.<span class="_ _a"> </span>Its<span class="_ _8"> </span>important<span class="_ _a"> </span>to<span class="_ _a"> </span>point<span class="_ _8"> </span>out<span class="_ _a"> </span>that<span class="_ _a"> </span>the<span class="_ _a"> </span><span class="ff7">fi</span>lter<span class="_ _8"> </span>is<span class="_ _a"> </span>not</div><div class="t m6 x1 ha y163 ff1 fs7 fc0 sc0 ls0 ws0">related<span class="_ _8"> </span>to<span class="_ _8"> </span>the<span class="_ _a"> </span>chattering<span class="_ _8"> </span>reduction.<span class="_ _8"> </span>The<span class="_ _a"> </span>dynamic<span class="_ _8"> </span>behavior<span class="_ _8"> </span>of<span class="_ _a"> </span>the</div><div class="t m6 x1 ha y164 ff7 fs7 fc0 sc0 ls0 ws0">fi<span class="ff1">lter<span class="_ _8"> </span>is<span class="_ _7"> </span>described<span class="_ _8"> </span>by</span></div><div class="t mc x6a h20 y165 ffb fs13 fc0 sc0 ls47 ws0">(+<span class="_ _c"></span>)<span class="_ _24"></span>=<span class="_ _61"> </span>(<span class="_ _62"></span>)<span class="_ _24"></span>+<span class="_ _6"> </span>(<span class="_ _62"></span>)</div><div class="t mc x7e h20 y166 ffb fs13 fc0 sc0 ls48 ws0">()</div><div class="t mc x94 h1d y167 ffc fs12 fc0 sc0 ls0 ws0">ν</div><div class="t mc x4d h21 y165 ffe fs13 fc0 sc0 ls49 ws0">xk<span class="_ _63"> </span>K<span class="_ _64"></span>xk<span class="_ _65"> </span>f<span class="_ _c"></span>k<span class="_ _66"></span><span class="ff10 ls0">1</span></div><div class="t mc x90 h21 y166 ff10 fs13 fc0 sc0 ls4a ws0">11</div><div class="t mc x6e h1f y168 ffe fs12 fc0 sc0 ls4b ws0">ff<span class="_ _67"></span>f</div><div class="t m6 x8 ha y59 ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="t m18 x16 h15 y169 ffb fse fc0 sc0 ls0 ws0">(</div><div class="c x95 y16a wc h41"><div class="t m18 x66 h15 y16b ffb fse fc0 sc0 ls0 ws0">)</div></div><div class="t m19 x96 h16 y16c ffc fsf fc0 sc0 ls0 ws0">ν</div><div class="t m18 x29 h17 y169 ffe fse fc0 sc0 ls4c ws0">fk</div><div class="t m6 x97 ha y59 ff1 fs7 fc0 sc0 ls0 ws0">is<span class="_ _8"> </span>de<span class="ff7">fi</span>ned<span class="_ _7"> </span>as<span class="_ _8"> </span>follows:</div><div class="t me x98 h42 y16d ff12 fs25 fc0 sc0 ls4d ws0">νν</div><div class="t me x36 h42 y16e ff12 fs25 fc0 sc0 ls0 ws0">ν</div><div class="t me x99 h43 y16d ffc fs25 fc0 sc0 ls0 ws0">ν</div><div class="t me x16 h43 y16e ffc fs25 fc0 sc0 ls4e ws0">νν</div><div class="t me x9a h44 y16f ffb fs25 fc0 sc0 ls4f ws0">()<span class="_ _4d"></span>≜</div><div class="t me x16 h44 y170 ffb fs25 fc0 sc0 ls50 ws0">∥(<span class="_ _3b"></span>)<span class="_ _68"></span>∥<span class="_ _25"> </span>≤<span class="_ _69"></span>∥(<span class="_ _3b"></span>)<span class="_ _68"></span>∥<span class="_ _69"></span><</div><div class="t me x9b h44 y171 ffb fs25 fc0 sc0 ls50 ws0">∥(<span class="_ _3b"></span>)<span class="_ _68"></span>∥<span class="_ _69"></span>></div><div class="t me x9c h44 y172 ffb fs25 fc0 sc0 ls51 ws0">()</div><div class="t m10 x40 h29 y173 ffc fs18 fc0 sc0 ls0 ws0">ν</div><div class="t m10 x2 h29 y174 ffc fs18 fc0 sc0 ls2a ws0">αβ<span class="_ _6a"> </span>αβ</div><div class="t m10 x37 h29 y175 ffc fs18 fc0 sc0 ls2a ws0">αβ</div><div class="t m1a x3e h45 y176 ffd fs26 fc0 sc0 ls0 ws0">⎪</div><div class="t m1b x3e h46 y177 ffd fs27 fc0 sc0 ls0 ws0">⎪</div><div class="t me x3e h47 y178 ffd fs25 fc0 sc0 ls0 ws0">⎧</div><div class="t me x3e h47 y179 ffd fs25 fc0 sc0 ls0 ws0">⎨</div><div class="t me x3e h47 y17a ffd fs25 fc0 sc0 ls0 ws0">⎩</div><div class="t me x13 h48 y17b ffe fs25 fc0 sc0 ls52 ws0">fk</div><div class="t me x9d h48 y17c ffe fs25 fc0 sc0 ls53 ws0">kk</div><div class="t me x9e h48 y17d ffe fs25 fc0 sc0 ls0 ws0">k</div><div class="t me x9f h48 y17c ff10 fs25 fc0 sc0 ls37 ws0">if<span class="_ _7"> </span>0</div><div class="t me x9f h48 y17d ff10 fs25 fc0 sc0 ls37 ws0">if</div><div class="t me xa0 h48 y17e ff10 fs25 fc0 sc0 ls54 ws0">12</div><div class="t m10 xa1 h32 y17f ffe fs18 fc0 sc0 ls39 ws0">max</div><div class="t m10 xa2 h32 y180 ffe fs18 fc0 sc0 ls3a ws0">max<span class="_ _6b"> </span>max</div><div class="t m6 x8 ha y181 ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="c x95 y182 wd h49"><div class="t mc x0 h4a y183 ffc fs28 fc0 sc0 ls0 ws0">ν</div></div><div class="t mc x9d h4b y184 ffc fs28 fc0 sc0 ls0 ws0">σ<span class="_ _6c"></span><span class="ffb">≜</span></div><div class="t mc x96 h4c y185 ffe fs29 fc0 sc0 ls55 ws0">max<span class="_ _20"> </span>ma</div><div class="c x95 y182 wd h49"><div class="t mc xa3 h4c y186 ffe fs29 fc0 sc0 ls0 ws0">x</div></div><div class="t m6 xa4 ha y181 ff1 fs7 fc0 sc0 ls0 ws0">.<span class="_ _8"> </span>The<span class="_ _7"> </span>observer<span class="_ _8"> </span>is<span class="_ _7"> </span>summarized<span class="_ _8"> </span>in<span class="_ _8"> </span><span class="fc1">Fig.<span class="_ _7"> </span>2</span>.</div><div class="t m6 x8 h1a y187 ff6 fs7 fc0 sc0 ls0 ws0">Lemma<span class="_ _a"> </span>1.<span class="_ _1"> </span><span class="ff4">The</span></div><div class="t m18 xa4 h15 y188 ffb fse fc0 sc0 ls0 ws0">(</div><div class="c xa5 y189 wc h41"><div class="t m18 x66 h15 y18a ffb fse fc0 sc0 ls0 ws0">)</div></div><div class="t m19 xa6 h16 y18b ffc fsf fc0 sc0 ls0 ws0">ν</div><div class="t m18 xa7 h17 y188 ffe fse fc0 sc0 ls4c ws0">fk</div><div class="t m6 xa8 h1a y18c ff4 fs7 fc0 sc0 ls0 ws0">function<span class="_ _a"> </span>is<span class="_ _4"> </span>proportional<span class="_ _4"> </span>to<span class="_ _a"> </span>the<span class="_ _4"> </span>rotor<span class="_ _a"> </span>angular</div><div class="t m6 x8 h1a y18d ff4 fs7 fc0 sc0 ls0 ws0">frequency<span class="_ _a"> </span>when<span class="_ _4"> </span>the<span class="_ _4"> </span>switching<span class="_ _4"> </span>functions<span class="_ _4"> </span>and<span class="_ _4"> </span>their<span class="_ _4"> </span>derivatives<span class="_ _a"> </span>con-</div><div class="t m6 x8 h1a y18e ff4 fs7 fc0 sc0 ls0 ws0">verge<span class="_ _7"> </span>to<span class="_ _8"> </span>the<span class="_ _7"> </span>sliding<span class="_ _8"> </span>surface.</div><div class="t m6 x8 ha y18f ff6 fs7 fc0 sc0 ls0 ws0">Proof.<span class="_ _1"> </span><span class="ff1">F<span class="_ _9"></span>rom<span class="_ _8"> </span><span class="fc1">(5)</span>,<span class="_ _8"> </span>it<span class="_ _7"> </span>is<span class="_ _8"> </span>possible<span class="_ _8"> </span>to<span class="_ _7"> </span>assume<span class="_ _8"> </span>that</span></div><div class="t mf xa9 h28 y190 ffb fs17 fc0 sc0 ls56 ws0">(+<span class="_ _c"></span>)<span class="_ _24"></span>=<span class="_ _10"> </span>(+<span class="_ _6d"></span>)<span class="_ _24"></span>−</div><div class="t mf xaa h28 y191 ffb fs17 fc0 sc0 ls0 ws0">^</div><div class="t mf xab h28 y190 ffb fs17 fc0 sc0 ls56 ws0">(+<span class="_ _c"></span>)</div><div class="t mf x9c h28 y192 ffb fs17 fc0 sc0 ls57 ws0">()</div><div class="t m10 x40 h29 y193 ffc fs18 fc0 sc0 ls2a ws0">αβ<span class="_ _38"> </span>αβ<span class="_ _38"> </span>αβ</div><div class="t mf x2d h2a y190 ffe fs17 fc0 sc0 ls58 ws0">kk<span class="_ _c"></span>k<span class="_ _6e"></span><span class="fff ls59">si<span class="_ _6d"></span>i<span class="_ _6f"></span><span class="ff10 ls5a">11<span class="_ _c"></span>1<span class="_ _70"></span>.</span></span></div><div class="t mf xa0 h2a y192 ff10 fs17 fc0 sc0 ls54 ws0">13</div><div class="t m6 x8 ha y194 ff1 fs7 fc0 sc0 ls0 ws0">Thus,<span class="_ _5"> </span>substituting<span class="_ _5"> </span><span class="fc1">(1)<span class="_ _5"> </span></span>and<span class="_ _5"> </span><span class="fc1">(2)<span class="_ _5"> </span></span>in<span class="_ _5"> </span><span class="fc1">(1<span class="_ _0"></span>3)<span class="fc0">,<span class="_ _5"> </span>the<span class="_ _5"> </span>dynamic<span class="_ _5"> </span>of<span class="_ _5"> </span>the<span class="_ _5"> </span>switching</span></span></div><div class="t m6 x8 ha y195 ff1 fs7 fc0 sc0 ls0 ws0">functions<span class="_ _8"> </span>can<span class="_ _7"> </span>be<span class="_ _8"> </span>described<span class="_ _8"> </span>by</div><div class="t mc xa9 h20 y196 ffb fs13 fc0 sc0 ls5b ws0">(<span class="_ _1"> </span>+<span class="_"> </span>)=<span class="_ _71"> </span>(<span class="_ _6"> </span>)−<span class="_ _72"> </span>(<span class="_ _6"> </span>)+<span class="_ _73"> </span>(<span class="_ _6"> </span>)</div><div class="t mc x9c h20 y197 ffb fs13 fc0 sc0 ls48 ws0">()</div><div class="t mc x40 h1d y198 ffc fs12 fc0 sc0 ls1d ws0">αβ<span class="_ _74"> </span>αβ<span class="_ _75"> </span>αβ<span class="_ _2b"> </span>αβ</div><div class="t mc x2d h21 y196 ffe fs13 fc0 sc0 ls24 ws0">kK<span class="_ _76"></span>k<span class="_ _46"></span>K<span class="_ _68"></span>kk<span class="_ _77"></span><span class="fff ls5c">ss<span class="_ _46"></span>e<span class="_ _35"></span>u<span class="_ _78"></span><span class="ff10 ls0">1</span></span></div><div class="t mc xa0 h21 y197 ff10 fs13 fc0 sc0 ls4a ws0">14</div><div class="t mc xac h1f y198 ffe fs12 fc0 sc0 ls5d ws0">ab</div><div class="t m6 x8 ha y199 ff1 fs7 fc0 sc0 ls0 ws0">where</div><div class="t m18 xad h4d y19a ffb fs2a fc0 sc0 ls5e ws0">≜−<span class="_ _79"></span>(</div><div class="c x95 y19b we h4e"><div class="t m18 xae h4d y19c ffb fs2a fc0 sc0 ls0 ws0">)</div></div><div class="t m18 x95 h4f y19a ffe fs2a fc0 sc0 ls5f ws0">KT<span class="_ _7a"></span>R<span class="_ _5b"></span>L<span class="_ _7b"></span><span class="ff10 ls60">1/</span></div><div class="t mc x21 h50 y19d ffe fs2b fc0 sc0 ls61 ws0">as<span class="_ _7c"></span>s<span class="_ _47"></span>s</div><div class="t m6 xaf ha y199 ff1 fs7 fc0 sc0 ls0 ws0">and</div><div class="t m15 xb0 h51 y19e ffb fs2c fc0 sc0 ls62 ws0">≜(</div><div class="c xb1 y19b wf h4e"><div class="t m15 xa3 h51 y19f ffb fs2c fc0 sc0 ls0 ws0">)</div></div><div class="t m15 x3a h52 y19e ffe fs2c fc0 sc0 ls63 ws0">KT<span class="_ _45"></span>L<span class="_ _7d"></span><span class="ff10 ls0">/</span></div><div class="t m16 xb2 h53 y1a0 ffe fs2d fc0 sc0 ls64 ws0">bs<span class="_ _68"></span>s</div><div class="t m6 xb3 ha y199 ff1 fs7 fc0 sc0 ls0 ws0">.</div><div class="t m6 x9 ha y1a1 ff1 fs7 fc0 sc0 ls0 ws0">Now<span class="_ _9"></span>,<span class="_ _6"> </span>let<span class="_ _5"> </span>us<span class="_ _5"> </span>assume<span class="_ _6"> </span>the<span class="_ _5"> </span>hypothetical<span class="_ _6"> </span>existence<span class="_ _6"> </span>of<span class="_ _6"> </span>an<span class="_ _5"> </span>ideal<span class="_ _6"> </span>quasi-</div><div class="t m6 x8 ha y1a2 ff1 fs7 fc0 sc0 ls0 ws0">sliding<span class="_ _8"> </span>mode<span class="_ _7"> </span>represented<span class="_ _8"> </span>by</div><div class="t mc xa9 h20 y1a3 ffb fs13 fc0 sc0 ls47 ws0">(+<span class="_ _c"></span>)<span class="_ _24"></span>=<span class="_ _e"> </span>(<span class="_ _62"></span>)<span class="_ _24"></span>=</div><div class="t mc x9c h20 y1a4 ffb fs13 fc0 sc0 ls48 ws0">()</div><div class="t mc x40 h1d y1a5 ffc fs12 fc0 sc0 ls1d ws0">αβ<span class="_ _7e"> </span>αβ</div><div class="t mc x2d h21 y1a3 ffe fs13 fc0 sc0 ls65 ws0">kk<span class="_ _7f"></span><span class="fff ls66">ss<span class="_ _80"></span><span class="ff10 ls67">10<span class="_ _81"></span>.</span></span></div><div class="t mc xa0 h21 y1a4 ff10 fs13 fc0 sc0 ls4a ws0">15</div><div class="t m6 x8 ha y1a6 ff1 fs7 fc0 sc0 ls0 ws0">From<span class="_ _5"> </span><span class="fc1">(1<span class="_ _9"></span>5)<span class="_ _8"> </span><span class="fc0">in<span class="_ _5"> </span></span>(1<span class="_ _0"></span>4)<span class="fc0">,<span class="_ _7"> </span>the<span class="_ _7"> </span>discontinuous<span class="_ _8"> </span>forcing<span class="_ _5"> </span>terms<span class="_ _8"> </span>can<span class="_ _5"> </span>be<span class="_ _8"> </span>giv<span class="_ _9"></span>en<span class="_ _8"> </span>by</span></span></div><div class="t mc x2d h20 y1a7 ffb fs13 fc0 sc0 ls22 ws0">()<span class="_ _26"></span>=<span class="_ _82"> </span>()</div><div class="t mc x9c h20 y1a8 ffb fs13 fc0 sc0 ls48 ws0">()</div><div class="t mc xb4 h1d y1a9 ffc fs12 fc0 sc0 ls1d ws0">αβ<span class="_ _75"> </span>αβ</div><div class="t mc x42 h21 y1a7 ffe fs13 fc0 sc0 ls68 ws0">kK<span class="_ _4"> </span>k<span class="_ _83"></span><span class="fff ls69">ue<span class="_ _84"></span><span class="ff10 ls0">.</span></span></div><div class="t mc xa0 h21 y1a8 ff10 fs13 fc0 sc0 ls4a ws0">16</div><div class="t mc x2f h1f y1a9 ffe fs12 fc0 sc0 ls0 ws0">b</div><div class="t m6 x8 ha y1aa ff1 fs7 fc0 sc0 ls0 ws0">In<span class="_ _8"> </span>addition,<span class="_ _7"> </span>substituting<span class="_ _8"> </span><span class="fc1">(3)<span class="_ _8"> </span></span>in<span class="_ _7"> </span><span class="fc1">(1<span class="_ _9"></span>6)<span class="_ _c"></span><span class="fc0">,<span class="_ _7"> </span>it<span class="_ _8"> </span>is<span class="_ _8"> </span>possible<span class="_ _8"> </span>t<span class="_ _9"></span>o<span class="_ _8"> </span>express</span></span></div><div class="t mc x34 h20 y1ab ff12 fs13 fc0 sc0 ls0 ws0">ν<span class="_ _b"> </span><span class="ffb ls22">()<span class="_ _26"></span>=<span class="_ _82"> </span>()</span></div><div class="t mc x9c h20 y1ac ffb fs13 fc0 sc0 ls48 ws0">()</div><div class="t mc x18 h1d y1ad ffc fs12 fc0 sc0 ls1d ws0">αβ<span class="_ _75"> </span>αβ</div><div class="t mc x2d h21 y1ab ffe fs13 fc0 sc0 ls68 ws0">kK<span class="_ _4"> </span>k<span class="_ _85"></span><span class="fff ls0">e<span class="_ _20"> </span><span class="ff10">.</span></span></div><div class="t mc xa0 h21 y1ac ff10 fs13 fc0 sc0 ls4a ws0">17</div><div class="t mc x16 h1f y1ad ffe fs12 fc0 sc0 ls0 ws0">b</div><div class="t m6 x8 ha y1ae ff1 fs7 fc0 sc0 ls0 ws0">It<span class="_ _8"> </span>is<span class="_ _7"> </span>well<span class="_ _8"> </span>known<span class="_ _7"> </span>that<span class="_ _8"> </span>the<span class="_ _8"> </span>back<span class="_ _2"></span>-EMFs<span class="_ _8"> </span>are<span class="_ _8"> </span>proportional<span class="_ _7"> </span>to<span class="_ _8"> </span>the<span class="_ _7"> </span>rotor</div><div class="t m6 x8 ha y1af ff1 fs7 fc0 sc0 ls0 ws0">speed.<span class="_ _8"> </span>Hence,<span class="_ _7"> </span>it<span class="_ _8"> </span>can<span class="_ _7"> </span>be<span class="_ _8"> </span>concluded<span class="_ _8"> </span>from<span class="_ _5"> </span><span class="fc1">(12)<span class="_ _7"> </span></span>and<span class="_ _8"> </span><span class="fc1">(1<span class="_ _2"></span>7)<span class="_ _8"> </span><span class="fc0">that</span></span></div><div class="t m18 x14 h15 y1b0 ffb fse fc0 sc0 ls0 ws0">(</div><div class="c xb5 y1b1 wc h41"><div class="t m18 x66 h15 y1b2 ffb fse fc0 sc0 ls0 ws0">)</div></div><div class="t m19 xb6 h16 y1b3 ffc fsf fc0 sc0 ls0 ws0">ν</div><div class="t m18 xb7 h17 y1b0 ffe fse fc0 sc0 ls4c ws0">fk</div><div class="t m6 xb8 ha y1b4 ff1 fs7 fc0 sc0 ls0 ws0">is</div><div class="t m6 x8 ha y1b5 ff1 fs7 fc0 sc0 ls0 ws0">also<span class="_ _4"> </span>proportional<span class="_ _4"> </span>to<span class="_ _4"> </span>this<span class="_ _4"> </span>speed.<span class="_ _1"> </span>Therefor<span class="_ _0"></span>e,</div><div class="t m18 x30 h15 y1b6 ffb fse fc0 sc0 ls0 ws0">(</div><div class="c xb9 y1b7 wc h41"><div class="t m18 x66 h15 y1b8 ffb fse fc0 sc0 ls0 ws0">)</div></div><div class="t m19 xba h16 y1b9 ffc fsf fc0 sc0 ls0 ws0">ν</div><div class="t m18 xbb h17 y1b6 ffe fse fc0 sc0 ls4c ws0">fk</div><div class="t m6 xbc ha y1ba ff1 fs7 fc0 sc0 ls0 ws0">has<span class="_ _4"> </span>the<span class="_ _4"> </span>proper</div><div class="t m6 x8 ha y1bb ff1 fs7 fc0 sc0 ls0 ws0">information<span class="_ _3"> </span>to<span class="_ _3"> </span>adjust<span class="_ _3"> </span>the<span class="_ _11"> </span>DTST<span class="_ _9"></span>A<span class="_ _11"> </span>gains<span class="_ _3"> </span>according<span class="_ _3"> </span>to<span class="_ _3"> </span>the<span class="_ _3"> </span>rotor</div><div class="t m6 x8 ha y1bc ff1 fs7 fc0 sc0 ls0 ws0">speed.<span class="ff15">□</span></div><div class="t m6 x9 ha y1bd ff1 fs7 fc0 sc0 ls0 ws0">It<span class="_ _8"> </span>is<span class="_ _7"> </span>straightforward<span class="_ _7"> </span>from<span class="_ _8"> </span><span class="fc1">(1<span class="_ _9"></span>6)<span class="_ _8"> </span><span class="fc0">that<span class="_ _8"> </span>the<span class="_ _7"> </span>back<span class="_ _9"></span>-EMFs<span class="_ _8"> </span>can<span class="_ _8"> </span>be<span class="_ _8"> </span>esti-</span></span></div><div class="t m6 x8 ha y1be ff1 fs7 fc0 sc0 ls0 ws0">mated<span class="_ _7"> </span>from</div><div class="t mc x34 h10 y1bf ffb fsc fc0 sc0 ls0 ws0">^</div><div class="t mc x9 h10 y1c0 ffb fsc fc0 sc0 ls15 ws0">()<span class="_ _26"></span>≈<span class="_ _15"> </span>()</div><div class="t mc x9c h10 y1c1 ffb fsc fc0 sc0 ls6a ws0">()</div><div class="t mc x40 h11 y1c2 ffc fsd fc0 sc0 ls9 ws0">αβ<span class="_ _86"> </span>αβ</div><div class="t mc x2f h3b y1c3 ffb fsd fc0 sc0 ls0 ws0">−</div><div class="t mc xbd h13 y1c4 ffe fsc fc0 sc0 ls6b ws0">kK<span class="_ _87"> </span>k<span class="_ _88"></span><span class="fff ls6c">eu<span class="_ _5d"></span><span class="ff10 ls0">.</span></span></div><div class="t mc xa0 h13 y1c5 ff10 fsc fc0 sc0 ls54 ws0">18</div><div class="t mc xbe h14 y1c6 ffe fsd fc0 sc0 ls0 ws0">b</div><div class="t mc xbf h14 y1c7 ff10 fsd fc0 sc0 ls0 ws0">1</div><div class="t m6 x9 ha y1c8 ff1 fs7 fc0 sc0 ls0 ws0">So,<span class="_ _8"> </span>substituting<span class="_ _7"> </span><span class="fc1">(3)<span class="_ _8"> </span></span>in<span class="_ _8"> </span><span class="fc1">(1<span class="_ _2"></span>4)<span class="fc0">,<span class="_ _8"> </span>it<span class="_ _8"> </span>is<span class="_ _7"> </span>possible<span class="_ _8"> </span>to<span class="_ _7"> </span>obtain</span></span></div><div class="t me xc0 h42 y1c9 ff12 fs25 fc0 sc0 ls0 ws0">ν</div><div class="t me xa9 h44 y1ca ffb fs25 fc0 sc0 ls6d ws0">(<span class="_ _1"> </span>+<span class="_"> </span>)=<span class="_ _71"> </span>(<span class="_ _6"> </span>)−<span class="_ _72"> </span>(<span class="_ _6"> </span>)+<span class="_ _89"> </span>(<span class="_ _6"> </span>)</div><div class="t me xc1 h44 y1cb ffb fs25 fc0 sc0 ls4f ws0">−<span class="_ _d"> </span>()<span class="_ _24"></span>|<span class="_ _e"> </span>()<span class="_ _24"></span>|<span class="_ _1e"> </span>(<span class="_ _e"> </span>()<span class="_ _24"></span>)</div><div class="t me x9c h44 y1cc ffb fs25 fc0 sc0 ls51 ws0">()</div><div class="t m10 x40 h29 y1cd ffc fs18 fc0 sc0 ls2a ws0">αβ<span class="_ _74"> </span>αβ<span class="_ _75"> </span>αβ<span class="_ _36"> </span>αβ</div><div class="t m10 x35 h29 y1ce ffc fs18 fc0 sc0 ls2a ws0">αβ<span class="_ _38"> </span>αβ</div><div class="t mc x2d h13 y1c9 ffe fsc fc0 sc0 ls16 ws0">kK<span class="_ _76"></span>k<span class="_ _46"></span>K<span class="_ _68"></span>k<span class="_ _64"></span>k</div><div class="t mc x9d h13 y1cf ffe fsc fc0 sc0 ls2d ws0">kk<span class="_ _41"> </span>k</div><div class="t mc x8 h13 y1c9 fff fsc fc0 sc0 ls6e ws0">ss<span class="_ _46"></span>e</div><div class="t mc xc2 h13 y1cf fff fsc fc0 sc0 ls6f ws0">ss</div><div class="t mc x29 h13 y1c9 ff10 fsc fc0 sc0 ls0 ws0">1</div><div class="t mc xc3 h13 y1cf ff10 fsc fc0 sc0 ls70 ws0">sign<span class="_ _20"> </span>k<span class="_ _e"> </span>.</div><div class="t mc xa0 h13 y1d0 ff10 fsc fc0 sc0 ls54 ws0">19</div><div class="t m10 xac h32 y1cd ffe fs18 fc0 sc0 ls71 ws0">ab</div><div class="t m10 xc4 h32 y1ce ff10 fs18 fc0 sc0 ls0 ws0">1</div><div class="t m10 xc5 h32 y1d1 ff10 fs18 fc0 sc0 ls0 ws0">1</div><div class="t m10 xc5 h32 y1d2 ff10 fs18 fc0 sc0 ls0 ws0">2</div><div class="t m6 x9 ha y1d3 ff1 fs7 fc0 sc0 ls0 ws0">The<span class="_ _6"> </span>behavior<span class="_ _5"> </span>of<span class="_ _5"> </span>the<span class="_ _5"> </span>DTST<span class="_ _0"></span>A<span class="_ _6"> </span>observer<span class="_ _5"> </span>is<span class="_ _5"> </span>described<span class="_ _5"> </span>by<span class="_ _6"> </span><span class="fc1">(3)</span>,<span class="_ _7"> </span><span class="fc1">(4)</span>,<span class="_ _6"> </span><span class="fc1">(1<span class="_ _2"></span>1)</span></div><div class="t m6 x8 ha y1d4 ff1 fs7 fc0 sc0 ls0 ws0">and<span class="_ _1"> </span><span class="fc1">(1<span class="_ _2"></span>9)<span class="fc0">.<span class="_ _1"> </span>The<span class="_ _1"> </span>stability<span class="_ _4"> </span>analysis<span class="_ _1"> </span>of<span class="_ _4"> </span>this<span class="_ _1"> </span>observer<span class="_ _1"> </span>is<span class="_ _4"> </span>presented<span class="_ _1"> </span>in</span></span></div><div class="t m6 x8 ha y1d5 ff1 fs7 fc1 sc0 ls0 ws0">Appendix<span class="_ _8"> </span>A<span class="fc0">.</span></div><div class="t m6 x8 ha y1d6 ff6 fs7 fc0 sc0 ls0 ws0">Remark<span class="_ _8"> </span>1.<span class="_ _1"> </span><span class="ff1 fc1">Lemma<span class="_ _a"> </span>1<span class="_ _a"> </span><span class="fc0">shows<span class="_ _8"> </span>that<span class="_ _a"> </span>the<span class="_ _a"> </span>back<span class="_ _9"></span>-EMFs<span class="_ _a"> </span>can<span class="_ _a"> </span>be<span class="_ _a"> </span>proper<span class="_ _0"></span>ly</span></span></div><div class="t m6 x8 ha y1d7 ff1 fs7 fc0 sc0 ls0 ws0">estimated<span class="_ _a"> </span>from<span class="_ _8"> </span>the<span class="_ _a"> </span>discontinuous<span class="_ _a"> </span>forcing<span class="_ _8"> </span>terms<span class="_ _a"> </span>of<span class="_ _a"> </span>the<span class="_ _a"> </span>proposed</div><div class="t m6 x8 ha y1d8 ff1 fs7 fc0 sc0 ls0 ws0">DTST<span class="_ _9"></span>A<span class="_ _c"></span>.<span class="_ _6"> </span>In<span class="_ _5"> </span>addition,<span class="_ _6"> </span>it<span class="_ _5"> </span>is<span class="_ _5"> </span>important<span class="_ _6"> </span>to<span class="_ _5"> </span>point<span class="_ _6"> </span>out<span class="_ _5"> </span>that<span class="_ _6"> </span>the<span class="_ _5"> </span>ideal<span class="_ _6"> </span>quasi-</div><div class="t m6 x8 ha y1d9 ff1 fs7 fc0 sc0 ls0 ws0">sliding<span class="_ _8"> </span>mode<span class="_ _7"> </span>is<span class="_ _8"> </span>just<span class="_ _8"> </span>a<span class="_ _7"> </span>theoretical<span class="_ _8"> </span>condition.<span class="_ _8"> </span>In<span class="_ _7"> </span>a<span class="_ _8"> </span>real<span class="_ _7"> </span>system,<span class="_ _8"> </span>spe-</div><div class="t m6 x8 ha y1da ff1 fs7 fc0 sc0 ls0 ws0">ci<span class="ff7">fi</span>cally<span class="_ _4"> </span>in<span class="_ _4"> </span>discrete-time<span class="_ _4"> </span>systems,<span class="_ _4"> </span>the<span class="_ _4"> </span>trajectories<span class="_ _a"> </span>are<span class="_ _4"> </span>not<span class="_ _4"> </span>in<span class="_ _1"> </span>ac-</div><div class="t m6 x8 ha y1db ff1 fs7 fc0 sc0 ls0 ws0">cordance<span class="_ _3"> </span>with<span class="_ _1"> </span><span class="fc1">(1<span class="_ _0"></span>5)<span class="fc0">.<span class="_ _3"> </span>In<span class="_ _3"> </span>this<span class="_ _3"> </span>case,<span class="_ _3"> </span>a<span class="_ _1"> </span>residual<span class="_ _3"> </span>chattering<span class="_ _3"> </span>will<span class="_ _3"> </span>be</span></span></div><div class="t m6 x8 ha y1dc ff1 fs7 fc0 sc0 ls0 ws0">presented<span class="_ _8"> </span>in<span class="_ _a"> </span>the<span class="_ _8"> </span>estimates<span class="_ _8"> </span>of<span class="_ _a"> </span>back<span class="_ _9"></span>-EMF<span class="_ _9"></span>,<span class="_ _a"> </span>because<span class="_ _8"> </span>the<span class="_ _a"> </span>sliding<span class="_ _8"> </span>mo-</div><div class="t m6 x8 ha y1dd ff1 fs7 fc0 sc0 ls0 ws0">tion<span class="_ _8"> </span>occurs<span class="_ _7"> </span>in<span class="_ _8"> </span>the<span class="_ _8"> </span>vicinity<span class="_ _7"> </span>of<span class="_ _8"> </span>the<span class="_ _8"> </span>sliding<span class="_ _7"> </span>hyperplane</div><div class="c xc6 y1de w10 h54"><div class="t mc x0 h15 y1df ffb fse fc0 sc0 ls0 ws0">|</div></div><div class="t mc xc7 h15 y1e0 ffb fse fc0 sc0 ls72 ws0">()<span class="_ _24"></span>|<span class="_ _26"></span>=</div><div class="t mc xc8 h16 y1e1 ffc fsf fc0 sc0 ls10 ws0">αβ</div><div class="t mc x33 h17 y1e0 ffe fse fc0 sc0 ls0 ws0">k<span class="_ _55"></span><span class="fff">s</span></div><div class="c xc6 y1de w10 h54"><div class="t mc xc9 h17 y1df ff10 fse fc0 sc0 ls0 ws0">0</div></div><div class="t m6 xca ha y1e2 ff1 fs7 fc0 sc0 ls0 ws0">.</div><div class="t m6 x8 h1a y1e3 ff6 fs7 fc0 sc0 ls0 ws0">Lemma<span class="_ _4"> </span>2.<span class="_ _1"> </span><span class="ff4">The<span class="_ _4"> </span>switching<span class="_ _a"> </span>function<span class="_ _1"> </span>tr<span class="_ _9"></span>ajectories<span class="_ _4"> </span>of<span class="_ _1"> </span>the<span class="_ _a"> </span>discrete<span class="_ _4"> </span>time</span></div><div class="t m6 x8 h1a y1e4 ff4 fs7 fc0 sc0 ls0 ws0">system<span class="_ _1"> </span>represented<span class="_ _1"> </span>by<span class="_ _1"> </span><span class="fc1">(4)<span class="_ _1"> </span></span>and<span class="_ _1"> </span><span class="fc1">(1<span class="_ _9"></span>9)<span class="_ _3"> </span><span class="fc0">con<span class="_ _9"></span>verge<span class="_ _1"> </span>to<span class="_ _1"> </span>the<span class="_ _1"> </span>quasi-sliding</span></span></div><div class="t m6 x8 h1a y1e5 ff4 fs7 fc0 sc0 ls0 ws0">mode<span class="_ _8"> </span>de<span class="ff16">fi</span>ned<span class="_ _7"> </span>by</div><div class="c x1f y1e6 w11 h55"><div class="t mc xc9 h56 y1e7 ffc fs2e fc0 sc0 ls0 ws0">ε<span class="_ _8a"></span><span class="ffb">|</span></div></div><div class="t mc x35 h56 y1e8 ffb fs2e fc0 sc0 ls73 ws0">()<span class="_ _24"></span>|<span class="_ _26"></span>≤</div><div class="t mc xa4 h57 y1e9 ffc fs2f fc0 sc0 ls74 ws0">αβ</div><div class="t mc xcb h58 y1e8 ffe fs2e fc0 sc0 ls0 ws0">k<span class="_ _55"></span><span class="fff">s</span></div><div class="t m6 xcc h1a y1ea ff4 fs7 fc0 sc0 ls0 ws0">(see<span class="_ _8"> </span><span class="fc1">Bartoszewicz,<span class="_ _5"> </span>1<span class="_ _9"></span>998<span class="fc0">),<span class="_ _8"> </span>when<span class="_ _8"> </span>the<span class="_ _8"> </span>dis-</span></span></div><div class="t m6 x8 h1a y1eb ff4 fs7 fc0 sc0 ls0 ws0">turbance</div><div class="t m18 xa5 h59 y1ec ffb fs30 fc0 sc0 ls0 ws0">(</div><div class="c x98 y1ed w12 h54"><div class="t m18 xcd h59 y1ee ffb fs30 fc0 sc0 ls0 ws0">)</div></div><div class="t mc xce h5a y1ef ffc fs31 fc0 sc0 ls75 ws0">αβ</div><div class="t m18 x98 h5b y1ec ffe fs30 fc0 sc0 ls76 ws0">Kk<span class="_ _8b"></span><span class="fff ls0">e</span></div><div class="t mc xcf h5c y1ef ffe fs31 fc0 sc0 ls0 ws0">b</div><div class="t m6 xd0 h1a y1f0 ff4 fs7 fc0 sc0 ls0 ws0">is<span class="_ _10"> </span>bounded,<span class="_ _e"> </span>K</div><div class="t md xd1 h1b y1f1 ff4 fs10 fc0 sc0 ls0 ws0">a</div><div class="t m6 xd2 h1a y1f0 ff4 fs7 fc0 sc0 ls0 ws0">assumes<span class="_ _b"> </span>a<span class="_ _10"> </span>value<span class="_ _b"> </span>such<span class="_ _b"> </span>that</div><div class="t m1c xd3 h5d y1f2 ffb fs32 fc0 sc0 ls77 ws0"><<<span class="_ _8c"></span><span class="ffe ls0">K</span></div><div class="c x8 y1f3 w13 h5e"><div class="t m1c x0 h5f y1f4 ff10 fs32 fc0 sc0 ls0 ws0">0</div></div><div class="t m1c xd4 h5f y1f2 ff10 fs32 fc0 sc0 ls0 ws0">1</div><div class="t m13 xd5 h60 y1f5 ffe fs33 fc0 sc0 ls0 ws0">a</div><div class="t m6 x2 h1a y1f6 ff4 fs7 fc0 sc0 ls0 ws0">and<span class="_ _e"> </span>the<span class="_ _b"> </span>gains</div><div class="t m11 xd6 h2b y1f7 ffb fs19 fc0 sc0 ls0 ws0">(</div><div class="c x3a y1f8 w8 h61"><div class="t m11 x61 h2b y1f9 ffb fs19 fc0 sc0 ls0 ws0">)<span class="_ _50"></span><span class="ffe">k</span></div></div><div class="t m11 xd7 h2d y1f7 ffe fs19 fc0 sc0 ls0 ws0">k</div><div class="t m12 x6 h2e y1fa ff10 fs1a fc0 sc0 ls0 ws0">1</div><div class="t m6 x39 h1a y1f6 ff4 fs7 fc0 sc0 ls0 ws0">,</div><div class="t m13 xd8 h2f y1fb ffb fs1b fc0 sc0 ls0 ws0">(</div><div class="c xd9 y1f8 w9 h61"><div class="t m13 x66 h2f y1fc ffb fs1b fc0 sc0 ls0 ws0">)<span class="_ _51"></span><span class="ffe">k</span></div></div><div class="t m13 xda h30 y1fb ffe fs1b fc0 sc0 ls0 ws0">k</div><div class="t m14 xdb h31 y1fd ff10 fs1c fc0 sc0 ls0 ws0">2</div><div class="t m6 xdc h1a y1f6 ff4 fs7 fc0 sc0 ls0 ws0">and<span class="_ _e"> </span>K</div><div class="t m9 xdd h23 y1fe ff13 fs14 fc0 sc0 ls0 ws0">ν</div><div class="t m6 xde h1a y1f6 ff4 fs7 fc0 sc0 ls0 ws0">are<span class="_ _b"> </span>given<span class="_ _e"> </span>by</div><div class="t mc xdf h15 y1ff ffc fse fc0 sc0 ls0 ws0">σ<span class="_ _8d"></span><span class="ffb ls78"><(<span class="_ _56"></span>)<span class="_ _79"></span>≤</span></div><div class="t mc xcf h16 y200 ffc fsf fc0 sc0 ls0 ws0">η</div><div class="t mc x42 h17 y1ff ffe fse fc0 sc0 ls79 ws0">kk<span class="_ _53"> </span>K</div><div class="c x8 y201 w14 h55"><div class="t mc x0 h17 y202 ff10 fse fc0 sc0 ls0 ws0">0</div></div><div class="t mc xa7 h19 y200 ffe fsf fc0 sc0 ls7a ws0">max<span class="_ _8e"></span><span class="ff10 ls7b">11</span></div><div class="t m6 xe0 h1a y203 ff4 fs7 fc0 sc0 ls0 ws0">,</div><div class="t m1c x99 h62 y204 ffc fs34 fc0 sc0 ls0 ws0">σ<span class="_ _8f"></span><span class="ffb ls7c"><(<span class="_ _90"></span>)<span class="_ _3b"></span>≤</span></div><div class="t m19 x39 h63 y205 ffc fs35 fc0 sc0 ls0 ws0">η</div><div class="t m1c xe1 h64 y204 ffe fs34 fc0 sc0 ls7d ws0">kk<span class="_ _54"> </span>K</div><div class="c x9b y201 w15 h55"><div class="t m1c x0 h64 y206 ff10 fs34 fc0 sc0 ls0 ws0">0</div></div><div class="t m19 xe2 h65 y205 ffe fs35 fc0 sc0 ls7e ws0">max<span class="_ _91"></span><span class="ff10 ls7f">22</span></div><div class="t m6 xe3 h1a y203 ff4 fs7 fc0 sc0 ls0 ws0">and</div><div class="t me xe4 h24 y207 ffb fs15 fc0 sc0 ls27 ws0"><<</div><div class="t mc xdd h25 y208 ffc fs16 fc0 sc0 ls0 ws0">ν</div><div class="t me xe5 h26 y207 ffe fs15 fc0 sc0 ls0 ws0">K</div><div class="c xb9 y209 w16 h27"><div class="t me x0 h26 y20a ff10 fs15 fc0 sc0 ls0 ws0">0</div></div><div class="t me xe6 h26 y207 ff10 fs15 fc0 sc0 ls0 ws0">1</div><div class="t m6 xe7 h1a y203 ff4 fs7 fc0 sc0 ls0 ws0">.</div><div class="t m6 x8 h1a y20b ff6 fs7 fc0 sc0 ls0 ws0">Proof.<span class="_ _1"> </span><span class="ff1">The<span class="_ _5"> </span>proof<span class="_ _7"> </span>is<span class="_ _7"> </span>based<span class="_ _7"> </span>on<span class="_ _7"> </span><span class="ff4">reductio<span class="_ _5"> </span>ad<span class="_ _7"> </span>absurdum<span class="_ _7"> </span></span>supposing<span class="_ _7"> </span>that</span></div><div class="t m6 x8 ha y20c ff1 fs7 fc0 sc0 ls0 ws0">the<span class="_ _a"> </span>switching<span class="_ _a"> </span>function<span class="_ _a"> </span>tra<span class="_ _9"></span>jectories<span class="_ _a"> </span>diverge<span class="_ _8"> </span>from<span class="_ _a"> </span>sliding<span class="_ _a"> </span>surfaces.</div><div class="t m4 x6f h8 y20d ff6 fs4 fc0 sc0 ls0 ws0">Fig.<span class="_ _5"> </span>2.<span class="_ _a"> </span><span class="ff1">Block<span class="_ _6"> </span>diagram<span class="_ _5"> </span>of<span class="_ _7"> </span>the<span class="_ _5"> </span>DTSTA<span class="_ _6"> </span>sliding<span class="_ _7"> </span>mode<span class="_ _5"> </span>observer.</span></div><div class="t m4 xe8 h6 y20e ff4 fs4 fc0 sc0 ls0 ws0">C.L.<span class="_ _5"> </span>Baratieri,<span class="_ _5"> </span>H.<span class="_ _5"> </span>Pinheiro<span class="_ _5"> </span>/<span class="_ _5"> </span>Control<span class="_ _5"> </span>Engineering<span class="_ _7"> </span>Practice<span class="_ _6"> </span>52<span class="_ _7"> </span>(201<span class="_ _2"></span>6)<span class="_ _7"> </span>59<span class="_"> </span><span 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