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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/622bae3615da9b288b763120/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Ch.II – Perform<span class="_ _0"></span>ances des systèmes ass<span class="_ _0"></span>ervis - p1</div><div class="t m1 x2 h3 y2 ff2 fs1 fc0 sc0 ls0 ws0">PERFORMA<span class="_ _1"></span>NCES DES SYSTEMES</div><div class="t m1 x3 h3 y3 ff2 fs1 fc0 sc0 ls0 ws0">A<span class="_ _1"></span>UTOMA<span class="_ _1"></span>TISES</div><div class="t m1 x4 h4 y4 ff1 fs1 fc0 sc0 ls0 ws0">I – Analyse d'un système – Démarches <span class="_ _0"></span>d'étude</div><div class="t m2 x5 h5 y5 ff3 fs2 fc0 sc0 ls0 ws0">Le <span class="_ _2"> </span>chapitre <span class="_ _2"> </span>précédent <span class="_ _2"> </span>a <span class="_ _2"> </span>mis <span class="_ _2"> </span>en <span class="_ _2"> </span>évidence <span class="_ _2"> </span>la <span class="_ _2"> </span>schématisation <span class="_ _2"> </span>structurelle <span class="_ _3"> </span>de <span class="_ _2"> </span>la <span class="_ _2"> </span>commande <span class="_ _2"> </span>d'un</div><div class="t m2 x6 h5 y6 ff3 fs2 fc0 sc0 ls0 ws0">sy<span class="_ _1"></span>stème <span class="_ _4"></span>asservi, <span class="_ _4"></span>le <span class="_ _5"></span>schéma-blocs <span class="_ _4"></span>ou <span class="_ _5"></span>diagramm<span class="_ _1"></span>e <span class="_ _5"></span>fonctionnel. <span class="_ _4"></span>Cette <span class="_ _5"></span>étape <span class="_ _4"></span>est <span class="_ _5"></span>le <span class="_ _4"></span>point <span class="_ _5"></span>de <span class="_ _4"></span>départ <span class="_ _5"></span>de</div><div class="t m2 x6 h5 y7 ff3 fs2 fc0 sc0 ls0 ws0">l'analy<span class="_ _1"></span>se d'un sy<span class="_ _1"></span>stème asservi.</div><div class="t m2 x5 h5 y8 ff3 fs2 fc0 sc0 ls0 ws0">Les <span class="_ _6"></span>sy<span class="_ _1"></span>stèmes <span class="_ _6"></span>asservis <span class="_ _6"></span>sont <span class="_ _6"></span>des <span class="_ _6"></span>sy<span class="_ _1"></span>stèmes <span class="_ _6"></span>com<span class="_ _1"></span>mandés, <span class="_ _6"></span>électromécaniques, <span class="_ _6"></span>régis <span class="_ _6"></span>par <span class="_ _6"></span>les <span class="_ _6"></span>lois <span class="_ _6"></span>de <span class="_ _6"></span>la</div><div class="t m2 x6 h5 y9 ff3 fs2 fc0 sc0 ls0 ws0">dy<span class="_ _1"></span>namique, <span class="_ _6"></span>et <span class="_ _4"></span>de<span class="_ _0"></span> <span class="_ _4"></span>l'électricité. <span class="_ _4"></span>La <span class="_ _4"></span>mise <span class="_ _4"></span>en <span class="_ _5"></span>équation <span class="_ _6"></span>d'un <span class="_ _4"></span>système <span class="_ _6"></span>conduit <span class="_ _4"></span>à <span class="_ _5"></span>un <span class="_ _6"></span>système <span class="_ _6"></span>d'équations</div><div class="t m2 x6 h5 ya ff3 fs2 fc0 sc0 ls0 ws0">diff<span class="_ _0"></span>érentielles. <span class="_"> </span>Le <span class="_ _7"> </span>but <span class="_"> </span>de <span class="_ _7"> </span>la <span class="_"> </span>modélisation <span class="_ _7"> </span>d'un <span class="_ _7"> </span>asservissement, <span class="_"> </span>est <span class="_ _7"> </span>de <span class="_ _7"> </span>parvenir <span class="_ _7"> </span>à <span class="_ _7"> </span>déterminer <span class="_ _7"> </span>la</div><div class="t m2 x6 h5 yb ff3 fs2 fc0 sc0 ls0 ws0">commande <span class="_ _2"> </span>du <span class="_ _3"> </span>sy<span class="_ _1"></span>stème <span class="_ _2"> </span>qui <span class="_"> </span>remplisse <span class="_ _2"> </span>les <span class="_"> </span>exigences <span class="_"> </span>du <span class="_ _2"> </span>cahier <span class="_"> </span>des <span class="_ _3"> </span>charges <span class="_"> </span>fonctionnel <span class="_"> </span>(<span class="ff4 fc1">Analyse</span></div><div class="t m2 x6 h5 yc ff4 fs2 fc1 sc0 ls0 ws0">fonctionnelle, caractérisation des fonctions de service attendues, et fonctions de se<span class="_ _0"></span>rvice réalisées<span class="ff3 fc0">).</span></div><div class="t m2 x5 h5 yd ff3 fs2 fc0 sc0 ls0 ws0">Le <span class="_ _6"></span>schéma <span class="_ _6"></span>ci-dessous <span class="_ _4"></span>présente <span class="_ _4"></span>la <span class="_ _4"></span>démarche <span class="_ _6"></span>globale <span class="_ _4"></span>d'étude <span class="_ _4"></span>d'un <span class="_ _4"></span>sy<span class="_ _1"></span>stème <span class="_ _6"></span>asservi, <span class="_ _4"></span>avec <span class="_ _4"></span>les <span class="_ _4"></span>deux</div><div class="t m2 x6 h5 ye ff3 fs2 fc0 sc0 ls0 ws0">méthodes <span class="_ _2"> </span>conduisant <span class="_ _8"> </span>au <span class="_ _8"> </span>diagramme <span class="_ _8"> </span>f<span class="_ _0"></span>onctionnel <span class="_ _8"> </span>(la <span class="_ _2"> </span>méthode <span class="_ _8"> </span>théorique <span class="_ _2"> </span>qui <span class="_ _2"> </span>passe <span class="_ _8"> </span>par <span class="_ _2"> </span>la <span class="_ _2"> </span>mise <span class="_ _8"> </span>en</div><div class="t m2 x6 h5 yf ff3 fs2 fc0 sc0 ls0 ws0">équations, et la méthode "pratique" qui consiste à eff<span class="_ _0"></span>ectuer des essais).</div><div class="t m3 x7 h6 y10 ff5 fs3 fc0 sc0 ls0 ws0">SY<span class="_ _0"></span>ST<span class="_ _0"></span>EM<span class="_ _0"></span>E</div><div class="t m3 x8 h6 y11 ff5 fs3 fc0 sc0 ls0 ws0">MA<span class="_ _0"></span>TE<span class="_ _0"></span>RI<span class="_ _0"></span>EL</div><div class="t m3 x9 h6 y12 ff5 fs3 fc0 sc0 ls0 ws0">AS<span class="_ _0"></span>SER<span class="_ _0"></span>VI</div><div class="t m3 xa h6 y13 ff5 fs3 fc0 sc0 ls0 ws0">Mis<span class="_ _0"></span>e <span class="_ _0"></span>en <span class="_ _0"></span>équa<span class="_ _0"></span>ti<span class="_ _0"></span>on<span class="_ _0"></span>s</div><div class="t m4 xa h7 y14 ff5 fs4 fc0 sc0 ls0 ws0">Sy<span class="_ _0"></span>stèm<span class="_ _0"></span>e d<span class="_ _0"></span>'équ<span class="_ _0"></span>ati<span class="_ _0"></span>ons</div><div class="t m3 xb h6 y15 ff5 fs3 fc0 sc0 ls0 ws0">Exp<span class="_ _0"></span>éri<span class="_ _0"></span>me<span class="_ _0"></span>nt<span class="_ _0"></span>atio<span class="_ _0"></span>n</div><div class="t m4 xc h7 y16 ff5 fs4 fc0 sc0 ls0 ws0">So<span class="_ _0"></span>llici<span class="_ _0"></span>tatio<span class="_ _0"></span>n d<span class="_ _0"></span>u sy<span class="_ _0"></span>st<span class="_ _0"></span>ème</div><div class="t m4 xd h7 y17 ff5 fs4 fc0 sc0 ls0 ws0">par<span class="_ _0"></span> un<span class="_ _0"></span> sig<span class="_ _0"></span>nal</div><div class="t m3 xe h6 y18 ff5 fs3 fc0 sc0 ls0 ws0">Sc<span class="_ _0"></span>hém<span class="_ _0"></span>ati<span class="_ _0"></span>sat<span class="_ _0"></span>ion<span class="_ _0"></span> f<span class="_ _0"></span>on<span class="_ _0"></span>ctio<span class="_ _0"></span>nn<span class="_ _0"></span>ell<span class="_ _0"></span>e</div><div class="t m4 xf h7 y19 ff5 fs4 fc0 sc0 ls0 ws0">(sc<span class="_ _0"></span>héma<span class="_ _0"></span>-b<span class="_ _0"></span>locs)</div><div class="t m3 x10 h6 y1a ff5 fs3 fc0 sc0 ls0 ws0">Rég<span class="_ _0"></span>la<span class="_ _0"></span>ge e<span class="_ _0"></span>t c<span class="_ _0"></span>or<span class="_ _0"></span>rec<span class="_ _0"></span>tion<span class="_ _0"></span>, <span class="_ _0"></span>en</div><div class="t m3 x11 h6 y1b ff5 fs3 fc0 sc0 ls0 ws0">fon<span class="_ _0"></span>ct<span class="_ _0"></span>ion<span class="_ _0"></span> de<span class="_ _0"></span>s <span class="_ _0"></span>per<span class="_ _0"></span>for<span class="_ _0"></span>m<span class="_ _0"></span>ance<span class="_ _0"></span>s</div><div class="t m3 x9 h6 y1c ff5 fs3 fc0 sc0 ls0 ws0">sou<span class="_ _0"></span>hai<span class="_ _0"></span>tée<span class="_ _0"></span>s</div><div class="t m5 x12 h8 y1d ff6 fs5 fc0 sc0 ls0 ws0">Si l<span class="_ _0"></span>e sy<span class="_ _0"></span>stè<span class="_ _0"></span>me</div><div class="t m5 x13 h8 y1e ff6 fs5 fc0 sc0 ls0 ws0">est<span class="_ _0"></span> sim<span class="_ _0"></span>ple</div><div class="t m5 x14 h8 y1d ff6 fs5 fc0 sc0 ls0 ws0">Si<span class="_ _0"></span> le <span class="_ _0"></span>systè<span class="_ _0"></span>me</div><div class="t m5 x14 h8 y1e ff6 fs5 fc0 sc0 ls0 ws0">est<span class="_ _0"></span> com<span class="_ _0"></span>ple<span class="_ _0"></span>xe</div><div class="t m5 x15 h8 y1f ff6 fs5 fc0 sc0 ls0 ws0">Id<span class="_ _0"></span>entif<span class="_ _0"></span>ica<span class="_ _0"></span>tion<span class="_ _9"></span>Mod<span class="_ _0"></span>éli<span class="_ _0"></span>sati<span class="_ _0"></span>on</div><div class="t m5 x16 h8 y20 ff6 fs5 fc0 sc0 ls0 ws0">Sol<span class="_ _0"></span>licit<span class="_ _0"></span>atio<span class="_ _0"></span>n p<span class="_ _0"></span>ar <span class="_ _0"></span>un</div><div class="t m5 x17 h8 y21 ff6 fs5 fc0 sc0 ls0 ws0">sig<span class="_ _0"></span>nal<span class="_ _0"></span> typ<span class="_ _0"></span>iqu<span class="_ _0"></span>e</div><div class="t m3 x11 h6 y22 ff5 fs3 fc0 sc0 ls0 ws0">Co<span class="_ _0"></span>mp<span class="_ _0"></span>or<span class="_ _0"></span>tem<span class="_ _0"></span>en<span class="_ _0"></span>t <span class="_ _0"></span>du s<span class="_ _0"></span>ys<span class="_ _0"></span>tè<span class="_ _0"></span>me</div><div class="t m4 x18 h7 y23 ff5 fs4 fc0 sc0 ls0 ws0">Ré<span class="_ _0"></span>pons<span class="_ _0"></span>e d<span class="_ _0"></span>u sy<span class="_ _0"></span>stè<span class="_ _0"></span>me</div><div class="t m5 x19 h8 y24 ff6 fs5 fc0 sc0 ls0 ws0">An<span class="_ _0"></span>alys<span class="_ _0"></span>e d<span class="_ _0"></span>es</div><div class="t m5 x1a h8 y25 ff6 fs5 fc0 sc0 ls0 ws0">per<span class="_ _0"></span>for<span class="_ _0"></span>man<span class="_ _0"></span>ces</div><div class="t m5 x1b h8 y26 ff6 fs5 fc0 sc0 ls0 ws0">Sim<span class="_ _0"></span>ulati<span class="_ _0"></span>on</div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></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/622bae3615da9b288b763120/bg2.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Ch.II – Perform<span class="_ _0"></span>ances des systèmes ass<span class="_ _0"></span>ervis - p2</div><div class="t m1 x4 h4 y27 ff1 fs1 fc0 sc0 ls0 ws0">II – Les<span class="_ _0"></span> signaux d'entrée typiques</div><div class="t m2 x4 h9 y28 ff1 fs2 fc0 sc0 ls0 ws0">1. Introduction</div><div class="t m2 x5 h5 y29 ff3 fs2 fc0 sc0 ls0 ws0">Dans <span class="_ _0"></span>le <span class="_ _a"></span>cas <span class="_ _0"></span>général <span class="_ _a"></span>les <span class="_ _0"></span>signaux <span class="_ _a"></span>d'entrée <span class="_ _0"></span>ont <span class="_ _a"></span>une <span class="_ _a"></span>forme <span class="_ _a"></span>quelconque <span class="_ _0"></span>et <span class="_ _a"></span>inconnue. <span class="_ _a"></span>Néanmoins <span class="_ _a"></span>pour</div><div class="t m2 x6 h5 y2a ff3 fs2 fc0 sc0 ls0 ws0">des <span class="_ _6"></span>besoins <span class="_ _6"></span>d'<span class="_ _1"></span>analyse <span class="_ _a"></span>on <span class="_ _6"></span>définit <span class="_ _6"></span>des <span class="_ _6"></span>signaux <span class="_ _6"></span>de <span class="_ _a"></span>f<span class="_ _0"></span>orme <span class="_ _6"></span>sim<span class="_ _1"></span>ple <span class="_ _6"></span>dont <span class="_ _6"></span>on <span class="_ _a"></span>pourra <span class="_ _6"></span>calculer <span class="_ _6"></span>les <span class="_ _6"></span>ef<span class="_ _0"></span>fets <span class="_ _6"></span>en</div><div class="t m2 x6 h5 y2b ff3 fs2 fc0 sc0 ls0 ws0">sortie d'un sy<span class="_ _1"></span>stème. Parmi ces signaux, les plus courants sont l'échelon et la sinusoïde.</div><div class="t m2 x5 ha y2c ff3 fs2 fc0 sc0 ls0 ws0">Pour <span class="_ _b"> </span>l'étude <span class="_ _b"> </span>des <span class="_ _b"> </span>sy<span class="_ _1"></span>stèmes <span class="_ _b"> </span>sollicités <span class="_ _b"> </span>par <span class="_ _b"> </span>un <span class="_ _b"> </span>signal, <span class="_ _b"> </span>on <span class="_ _b"> </span>distingue <span class="_ _b"> </span>deux <span class="_ _b"> </span>phases <span class="_ _b"> </span>: <span class="_ _b"> </span>le <span class="_ _b"> </span><span class="ff7">régime</span></div><div class="t m2 x6 ha y2d ff7 fs2 fc0 sc0 ls0 ws0">transitoire<span class="ff3">, <span class="_ _8"> </span>pha<span class="_ _0"></span>se <span class="_ _8"> </span>durant <span class="_ _2"> </span>laquelle <span class="_ _2"> </span>le <span class="_ _2"> </span>systèm<span class="_ _1"></span>e <span class="_ _2"> </span>"réagit" <span class="_ _2"> </span>au <span class="_ _2"> </span>signal, <span class="_ _2"> </span>puis <span class="_ _2"> </span>le <span class="_ _2"> </span><span class="ff7">régime <span class="_ _2"> </span>permanent <span class="_ _8"> </span>ou</span></span></div><div class="t m2 x6 hb y2e ff7 fs2 fc0 sc0 ls0 ws0">établi<span class="ff3">, qui correspond au comportement lorsque <span class="ff1">t <span class="ff8">→</span> +<span class="ff8">∞</span></span>.</span></div><div class="t m2 x6 hc y2f ff2 fs2 fc0 sc0 ls0 ws0">Remarque :<span class="ff3"> Tous les signaux seront consid<span class="_ _0"></span>érés nuls pour </span>t < 0<span class="ff3">.</span></div><div class="t m2 x4 hb y30 ff1 fs2 fc0 sc0 ls0 ws0">2. Signal d'entrée Dirac <span class="ff8">δ</span>(t) (Cf. tableau des transformées de Laplace)</div><div class="t m2 x5 h5 y31 ff3 fs2 fc0 sc0 ls0 ws0">Mathématiquement, le Dirac est <span class="_ _0"></span>défini <span class="_ _0"></span>comme étant</div><div class="t m2 x6 h5 y32 ff3 fs2 fc0 sc0 ls0 ws0">le <span class="_ _8"> </span>signal <span class="_ _8"> </span>d'amplitude <span class="_ _8"> </span>infinie, <span class="_ _2"> </span>pour <span class="_ _8"> </span>une <span class="_ _2"> </span>durée <span class="_ _8"> </span>nulle <span class="_ _2"> </span>:</div><div class="t m2 x6 hc y33 ff2 fs2 fc2 sc0 ls0 ws0">e(t) = </div><div class="t m6 x1c hd y33 ff8 fs6 fc2 sc0 ls0 ws0">δ</div><div class="t m6 x1c hd y34 ff8 fs6 fc2 sc0 ls0 ws0">δ<span class="_ _c"></span>δ</div><div class="t m6 x1c hd y33 ff8 fs6 fc2 sc0 ls0 ws0">δ</div><div class="t m2 x1d hc y33 ff2 fs2 fc2 sc0 ls0 ws0">(t) = <span class="_ _0"></span>0<span class="ff3 fc0"> </span><span class="ff7">pour </span>t <span class="_ _0"></span>< 0<span class="ff7"> e<span class="_ _0"></span>t <span class="_ _0"></span></span>t > <span class="_ _0"></span>0<span class="ff3 fc0">. <span class="_ _0"></span>En <span class="_ _0"></span>pratique <span class="_ _0"></span>on ne</span></div><div class="t m2 x6 h5 y35 ff3 fs2 fc0 sc0 ls0 ws0">peut <span class="_ _6"></span>que <span class="_ _6"></span>gé<span class="_ _0"></span>nérer <span class="_ _6"></span>une <span class="_ _4"></span>impulsion <span class="_ _6"></span>proche <span class="_ _4"></span>du <span class="_ _4"></span>Dirac, <span class="_ _4"></span>qui</div><div class="t m2 x6 h5 y36 ff3 fs2 fc0 sc0 ls0 ws0">modélise <span class="_ _d"> </span>alors <span class="_ _d"> </span>une <span class="_ _b"> </span>action <span class="_ _d"> </span>qui <span class="_ _b"> </span>s'exerce <span class="_ _d"> </span>pendant <span class="_ _b"> </span>un</div><div class="t m2 x6 h5 y37 ff3 fs2 fc0 sc0 ls0 ws0">temps très court (Choc, secousse…).</div><div class="t m7 x1e he y38 ff1 fs7 fc0 sc0 ls0 ws0">e(t)</div><div class="t m7 x1f he y39 ff1 fs7 fc0 sc0 ls0 ws0">t</div><div class="t m8 x20 hf y3a ff1 fs8 fc0 sc0 ls0 ws0">t = 0</div><div class="t m2 x6 hc y3b ff3 fs2 fc0 sc0 ls0 ws0">La <span class="_ _6"></span>réponse <span class="_ _6"></span>à <span class="_ _a"></span>une <span class="_ _6"></span>impulsion <span class="_ _6"></span>de <span class="_ _a"></span>D<span class="_ _0"></span>irac, <span class="_ _6"></span><span class="ff2 fc2">réponse <span class="_ _a"></span>impu<span class="_ _0"></span>lsionnelle</span>, <span class="_ _6"></span>est <span class="_ _6"></span>très <span class="_ _6"></span>intéressante <span class="_ _6"></span>en <span class="_ _6"></span>théorie,</div><div class="t m2 x4 h5 y3c ff3 fs2 fc0 sc0 ls0 ws0">permettant de caractériser parfaitement le sy<span class="_ _1"></span>stème. En pratique il est diff<span class="_ _0"></span>icile de procéder à cet essai.</div><div class="t m2 x4 h9 y3d ff1 fs2 fc0 sc0 ls0 ws0">3. Signal d'entrée constant e(t) = A : échelon</div><div class="t m2 x5 h5 y3e ff3 fs2 fc0 sc0 ls0 ws0">Soit le signal échelon défini ci-contr<span class="_ _0"></span>e :</div><div class="t m2 x5 hc y3f ff7 fs2 fc2 sc0 ls0 ws0">pour <span class="ff2">t < 0 </span></div><div class="t m1 x21 h3 y3f ff2 fs1 fc2 sc0 ls0 ws0">e(t) = 0</div><div class="t m2 x5 hc y40 ff7 fs2 fc2 sc0 ls0 ws0">pour <span class="ff2">t > 0 </span></div><div class="t m1 x21 h3 y40 ff2 fs1 fc2 sc0 ls0 ws0">e(t) = A</div><div class="t m2 x5 h5 y41 ff3 fs2 fc0 sc0 ls0 ws0">Fonction non définie en <span class="ff1">t = 0</span></div><div class="t m7 x22 he y42 ff1 fs7 fc0 sc0 ls0 ws0">e(t)</div><div class="t m7 x23 he y43 ff1 fs7 fc0 sc0 ls0 ws0">t</div><div class="t m7 x24 he y44 ff1 fs7 fc0 sc0 ls0 ws0">A</div><div class="t m8 x25 hf y45 ff1 fs8 fc0 sc0 ls0 ws0">t = 0</div><div class="t m2 x5 h5 y46 ff3 fs2 fc0 sc0 ls0 ws0">Lorsqu'on applique<span class="_ _0"></span> un <span class="_ _0"></span>échelon <span class="_ _0"></span>à <span class="_ _0"></span>l'entrée <span class="_ _0"></span>d'un systèm<span class="_ _1"></span>e, <span class="_ _0"></span>il s'agit <span class="_ _0"></span>d'une <span class="_ _0"></span>brutale <span class="_ _0"></span>variation, <span class="_ _0"></span>passage de</div><div class="t m2 x6 h5 y47 ff3 fs2 fc0 sc0 ls0 ws0">zéro <span class="_ _0"></span>à <span class="_ _0"></span>une <span class="_ _0"></span>amplitude <span class="_ _0"></span><span class="ff1">A</span>. <span class="_ _0"></span>La <span class="_ _a"></span>sortie <span class="_ _0"></span>du <span class="_ _a"></span>sy<span class="_ _1"></span>stème <span class="_ _0"></span>ne <span class="_ _0"></span>peut <span class="_ _a"></span>suivre <span class="_ _0"></span>instantanément <span class="_ _a"></span>cette <span class="_ _0"></span>brusque <span class="_ _a"></span>variation,</div><div class="t m2 x6 h5 y48 ff3 fs2 fc0 sc0 ls0 ws0">et <span class="_ _5"></span>on <span class="_ _8"> </span>observe <span class="_ _5"></span>une <span class="_ _5"></span><span class="ff4">phase <span class="_ _8"> </span>transitoire</span>, <span class="_ _5"></span>qui <span class="_ _5"></span>met <span class="_ _5"></span>en <span class="_ _5"> </span>é<span class="_ _0"></span>vidence <span class="_ _5"></span>certaines <span class="_ _8"> </span>caractéristiques <span class="_ _5"> </span>de <span class="_ _8"> </span>la <span class="_ _5"> </span>c<span class="_ _0"></span>haîne</div><div class="t m2 x6 h5 y49 ff3 fs2 fc0 sc0 ls0 ws0">fonctionnelle.</div><div class="t m2 x5 h5 y4a ff3 fs2 fc0 sc0 ls0 ws0">Par ailleurs, on peut observer le comportement en régime établi, stabilité ou divergence, et définir</div><div class="t m2 x6 hc y4b ff3 fs2 fc0 sc0 ls0 ws0">ainsi d'autres caractéristiques du sy<span class="_ _1"></span>stème. La réponse à un échelon est appelée <span class="ff2 fc2">réponse indicielle</span>.</div><div class="t m2 x4 hc y4c ff1 fs2 fc3 sc0 ls0 ws0">Remarque <span class="_ _0"></span>:<span class="ff3 fc0"> <span class="_ _0"></span><span class="ff2 fc2">fonction <span class="_ _0"></span>existence <span class="_ _0"></span>u(<span class="_ _0"></span>t)</span>, <span class="_ _0"></span>définie <span class="_ _0"></span>par <span class="_ _a"></span><span class="ff1">u(t) <span class="_ _0"></span>= <span class="_ _0"></span>0</span> <span class="_ _0"></span>pour <span class="_ _a"></span><span class="ff1">t <span class="_ _0"></span>< <span class="_ _a"></span>0</span> <span class="_ _0"></span> <span class="_ _a"></span>et <span class="_ _0"></span> <span class="_ _a"></span><span class="ff1">u(t) <span class="_ _0"></span>= <span class="_ _0"></span>1</span> <span class="_ _a"></span>pour <span class="_ _0"></span><span class="ff1">t <span class="_ _a"></span>> <span class="_ _0"></span>0</span>. <span class="_ _a"></span>Cette</span></div><div class="t m2 x4 h5 y4d ff3 fs2 fc0 sc0 ls0 ws0">fonction est utile si le signal <span class="ff1">e(t)</span> n'est pas nul pour <span class="ff1">t < 0</span>, on utilisera<span class="_ _0"></span> alors <span class="ff1">[e(t) <span class="fs9">x</span> u(t)]</span>.</div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>
<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/622bae3615da9b288b763120/bg3.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Ch.II – Perform<span class="_ _0"></span>ances des systèmes ass<span class="_ _0"></span>ervis - p3</div><div class="t m2 x4 h9 y4e ff1 fs2 fc0 sc0 ls0 ws0">4. Signal rampe</div><div class="t m2 x5 h5 y4f ff3 fs2 fc0 sc0 ls0 ws0">Soit le signal rampe défini ci-contre :</div><div class="t m2 x5 hc y50 ff7 fs2 fc2 sc0 ls0 ws0">pour <span class="ff2">t < 0 </span></div><div class="t m1 x21 h3 y50 ff2 fs1 fc2 sc0 ls0 ws0">e(t) = 0</div><div class="t m2 x5 hc y51 ff7 fs2 fc2 sc0 ls0 ws0">pour <span class="ff2">t > 0 </span></div><div class="t m1 x21 h3 y51 ff2 fs1 fc2 sc0 ls0 ws0">e(t) = A<span class="_ _1"></span>t</div><div class="t m2 x5 h5 y52 ff3 fs2 fc0 sc0 ls0 ws0">Fonction non définie en <span class="ff1">t = 0</span></div><div class="t m9 x1a h10 y53 ff1 fsa fc0 sc0 ls0 ws0">e(t)</div><div class="t m9 x26 h10 y54 ff1 fsa fc0 sc0 ls0 ws0">t</div><div class="t ma x27 h11 y55 ff1 fsb fc0 sc0 ls0 ws0">t = 0</div><div class="t m2 x5 h5 y56 ff3 fs2 fc0 sc0 ls0 ws0">Ce <span class="_ _6"></span>signal <span class="_ _a"></span>va <span class="_ _6"></span>perm<span class="_ _1"></span>ettre <span class="_ _6"></span>d'observer <span class="_ _6"></span>la <span class="_ _a"></span>f<span class="_ _0"></span>açon <span class="_ _6"></span>dont <span class="_ _6"></span>le <span class="_ _a"></span>système <span class="_ _a"></span>suit <span class="_ _6"></span>l'évolution <span class="_ _6"></span>du <span class="_ _e"></span>signal <span class="_ _e"></span>d'entrée, <span class="_ _6"></span>et</div><div class="t m2 x6 h5 y57 ff3 fs2 fc0 sc0 ls0 ws0">mettre <span class="_ _5"></span>ainsi <span class="_ _5"></span>en <span class="_ _8"> </span>évidence <span class="_ _5"></span>le <span class="_ _5"> </span>phénomène <span class="_ _5"> </span>de <span class="_ _8"> </span>vitesse. <span class="_ _5"> </span>C'est <span class="_ _8"> </span>très <span class="_ _5"> </span>utile <span class="_ _8"> </span>pour <span class="_ _8"> </span>caractériser <span class="_ _5"> </span>les <span class="_ _8"> </span>sy<span class="_ _1"></span>stèmes</div><div class="t m2 x6 h5 y58 ff3 fs2 fc0 sc0 ls0 ws0">suiveurs.</div><div class="t m2 x4 h9 y59 ff1 fs2 fc0 sc0 ls0 ws0">5. Signal d'entrée sinusoïdal</div><div class="t m2 x5 h5 y5a ff3 fs2 fc0 sc0 ls0 ws0">Soit le signal sinusoïdal défini ci-contre :</div><div class="t m2 x5 hc y5b ff7 fs2 fc2 sc0 ls0 ws0">pour <span class="ff2">t < 0 </span></div><div class="t m1 x21 h3 y5b ff2 fs1 fc2 sc0 ls0 ws0">e(t) = 0</div><div class="t m2 x5 hc y5c ff7 fs2 fc2 sc0 ls0 ws0">pour <span class="ff2">t > 0 </span></div><div class="t m1 x21 h12 y5c ff2 fs1 fc2 sc0 ls0 ws0">e(t) = A<span class="_ _1"></span> sin (<span class="_ _0"></span><span class="ff8">ω</span></div><div class="t m1 x28 h12 y5d ff8 fs1 fc2 sc0 ls0 ws0">ω<span class="_ _f"></span>ω</div><div class="t m1 x28 h12 y5c ff8 fs1 fc2 sc0 ls0 ws0">ω</div><div class="t mb x29 h13 y5e ff2 fs9 fc2 sc0 ls0 ws0">0 </div><div class="t m1 x2a h3 y5c ff2 fs1 fc2 sc0 ls0 ws0">t)</div><div class="t m2 x5 h5 y5f ff3 fs2 fc0 sc0 ls0 ws0">Fonction non définie en <span class="ff1">t = 0</span></div><div class="c x16 y60 w2 h14"><div class="t m1 x0 h15 y61 ff1 fsc fc0 sc0 ls0 ws0">e(t)</div></div><div class="t m1 x2b h15 y62 ff1 fsc fc0 sc0 ls0 ws0">t</div><div class="t mc x2c h16 y63 ff1 fsd fc0 sc0 ls0 ws0">t = 0</div><div class="t m2 x5 h5 y64 ff3 fs2 fc0 sc0 ls0 ws0">L'hy<span class="_ _1"></span>pothèse <span class="_ _7"> </span>de <span class="_ _7"> </span>système <span class="_"> </span>linéaire, <span class="_ _d"> </span>assure <span class="_ _7"> </span>que <span class="_ _d"> </span>la <span class="_ _7"> </span>sortie <span class="_ _d"> </span>d'un <span class="_ _7"> </span>systèm<span class="_ _1"></span>e <span class="_ _7"> </span>sollicité <span class="_ _d"> </span>par <span class="_ _d"> </span>une <span class="_ _7"> </span>e<span class="_ _0"></span>ntrée</div><div class="t m2 x6 h5 y65 ff3 fs2 fc0 sc0 ls0 ws0">sinusoïdale, <span class="_ _a"></span>est <span class="_ _e"></span>également <span class="_ _a"></span>sinusoïdale. <span class="_ _e"></span>La <span class="_ _e"></span>sortie <span class="_ _e"></span>e<span class="_ _0"></span>st <span class="_ _a"></span>de <span class="_ _e"></span>même <span class="_ _e"></span>fréquence <span class="_ _e"></span>que <span class="_ _e"></span>l'entrée, <span class="_ _e"></span>mais <span class="_ _e"></span>possède</div><div class="t m2 x6 h5 y66 ff3 fs2 fc0 sc0 ls0 ws0">une <span class="_ _4"></span><span class="ff4">am<span class="_ _0"></span>plitude <span class="_ _5"></span>différente</span>, <span class="_ _5"></span>et <span class="_ _5"></span>présente <span class="_ _4"></span>un <span class="_ _5"></span><span class="ff4">déphasage</span> <span class="_ _5"></span>par <span class="_ _5"></span>rapport <span class="_ _5"></span>au <span class="_ _4"></span>signal <span class="_ _5"></span>d'entrée. <span class="_ _5"></span>Il <span class="_ _4"></span>s'agit <span class="_ _5"></span>de <span class="_ _5"></span>la</div><div class="t m2 x6 hc y67 ff2 fs2 fc2 sc0 ls0 ws0">réponse fréquentielle ou harmonique<span class="ff3 fc0">.</span></div><div class="t m2 x5 h5 y68 ff9 fs2 fc0 sc0 ls0 ws0">L'étude portera sur l'analyse du système en régime établi. <span class="ff3">Elle sera f<span class="_ _0"></span>aite à partir de la variation de</span></div><div class="t m2 x6 h5 y69 ff3 fs2 fc0 sc0 ls0 ws0">la fréquence du signal d'entrée (variation<span class="_ _0"></span> de fréquence de zéro à l'inf<span class="_ _0"></span>ini).</div><div class="t m2 x4 h9 y6a ff1 fs2 fc0 sc0 ls0 ws0">6. Signaux complexes</div><div class="t m2 x5 h5 y6b ff3 fs2 fc0 sc0 ls0 ws0">Tout <span class="_ _10"> </span>signal <span class="_ _10"> </span>peut <span class="_ _10"> </span>s'écrire <span class="_ _10"> </span>comme <span class="_ _b"> </span>la <span class="_ _10"> </span>somme <span class="_ _10"> </span>de <span class="_ _10"> </span>signaux <span class="_ _11"> </span>simples. <span class="_ _10"> </span>Le <span class="_ _10"> </span>principe <span class="_ _11"> </span>réside <span class="_ _10"> </span>dans</div><div class="t m2 x6 h5 y6c ff3 fs2 fc0 sc0 ls0 ws0">l'utilisation <span class="_ _5"></span>d'un <span class="_ _5"></span>signal <span class="_ _5"></span>élémentaire <span class="_ _5"></span>(échelon, <span class="_ _5"></span>rampe <span class="_ _5"> </span>ou <span class="_ _8"> </span>sinusoïde) <span class="_ _5"></span>et <span class="_ _5"> </span>dans <span class="_ _8"> </span>la <span class="_ _5"></span>prise <span class="_ _5"></span>en <span class="_ _8"> </span>compte <span class="_ _5"></span>du</div><div class="t m2 x6 hc y6d ff3 fs2 fc0 sc0 ls0 ws0">retard (<span class="ff4 fc1">voir exercice <span class="ff2">1</span> en fin de chapitre, et chapitre "Transform<span class="_ _0"></span>ée de Laplace"</span>)<span class="_ _0"></span>.</div><div class="t m1 x4 h4 y6e ff1 fs1 fc0 sc0 ls0 ws0">III – <span class="_ _0"></span>Performances des <span class="_ _0"></span>systèmes</div><div class="t m2 x5 h5 y6f ff3 fs2 fc0 sc0 ls0 ws0">Il <span class="_"> </span>s<span class="_ _0"></span>'agit <span class="_"> </span>maintenant <span class="_ _7"> </span>d'analy<span class="_ _1"></span>ser <span class="_ _7"> </span>la <span class="_ _d"> </span>réponse <span class="_ _7"> </span>d'un <span class="_ _7"> </span>système <span class="_"> </span>à <span class="_ _d"> </span>un <span class="_ _7"> </span>signal, <span class="_ _d"> </span>que <span class="_ _7"> </span>ce <span class="_ _d"> </span>soit <span class="_ _7"> </span>lors <span class="_ _7"> </span>d'une</div><div class="t m2 x6 hc y70 ff2 fs2 fc2 sc0 ls0 ws0">expérimentation<span class="ff3 fc0"> ou d'une </span>simulation<span class="ff3 fc0">.</span></div><div class="t m2 x4 h9 y71 ff1 fs2 fc0 sc0 ls0 ws0">1. Introduction : critères de performance</div><div class="t m2 x5 h5 y72 ff3 fs2 fc0 sc0 ls0 ws0">Une <span class="_"> </span>simulation <span class="_"> </span>(ou <span class="_"> </span>expérimentation) <span class="_"> </span>ay<span class="_ _1"></span>ant <span class="_"> </span>été <span class="_ _7"> </span>réalisée, <span class="_"> </span>il <span class="_ _7"> </span>est <span class="_"> </span>alo<span class="_ _0"></span>rs <span class="_"> </span>nécessaire <span class="_ _7"> </span>d'analy<span class="_ _1"></span>ser <span class="_"> </span>la</div><div class="t m2 x6 h5 y73 ff3 fs2 fc0 sc0 ls0 ws0">réponse <span class="_ _8"> </span>du <span class="_ _8"> </span>sy<span class="_ _1"></span>stème <span class="_ _5"> </span>obtenue. <span class="_ _8"> </span>Si <span class="_ _8"> </span>une <span class="_ _8"> </span>expérience <span class="_ _8"> </span>a <span class="_ _8"> </span>été <span class="_ _8"> </span>menée <span class="_ _8"> </span>en <span class="_ _8"> </span>parallèle, <span class="_ _2"> </span>on <span class="_ _8"> </span>peut <span class="_ _8"> </span>conf<span class="_ _0"></span>ronter <span class="_ _8"> </span>la</div><div class="t m2 x6 h5 y74 ff3 fs2 fc0 sc0 ls0 ws0">réponse expérimentale à <span class="_ _0"></span>la réponse <span class="_ _0"></span>simulée. La seule <span class="_ _0"></span>comparaison visuelle <span class="_ _0"></span>des <span class="_ _0"></span>courbes n<span class="_ _0"></span>e suf<span class="_ _0"></span>fit <span class="_ _0"></span>pas</div><div class="t m2 x6 h5 y75 ff3 fs2 fc0 sc0 ls0 ws0">(<span class="ff4 fc1">voir identification</span>).</div></div><div class="pi" data-data='{"ctm":[1.611639,0.000000,0.000000,1.611639,0.000000,0.000000]}'></div></div>
