Generalized Fractional-Order Discrete-Time Integrator

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/complexity/volume-2017/3452409/figures/3452409.fig.009a.svgz" name="3452409.fig.009a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>Sinusoidal order function</td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/complexity/volume-2017/3452409/figures/3452409.fig.009b.svgz" name="3452409.fig.009b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>The VFOI homogenous response to <svg height="15.6446pt" id="M224" style="vertical-align:-4.0532pt" version="1.1" viewbox="-0.0498162 -11.5914 16.6947 15.6446" width="16.6947pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M483 442H303V410C352 401 362 388 351 355C333 298 299 198 276 142L201 363C190 396 197 401 240 410V442H-7V410C40 403 52 391 70 348C89 304 144 162 196 18C202 1 204 -14 199 -30C196 -41 190 -53 180 -71C161 -103 138 -122 99 -143C78 -156 71 -165 71 -181C71 -217 114 -245 142 -245C158 -245 172 -240 179 -218C196 -162 216 -99 284 63C342 202 375 285 400 338C425 392 435 402 483 410V442Z" id="g13-119"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="465" vert-adv-y="465" xpluscharwidth="6.045"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.28,-5.741)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487" xpluscharwidth="10.71168539"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.045,3.784)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611" xpluscharwidth="11.60508167"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,11.605,3.784)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487" xpluscharwidth="16.036685390000002"></glyph.data></g></svg></td></tr><tr class="fig-image" id="c"><td><object data="https://static.hindawi.com/articles/complexity/volume-2017/3452409/figures/3452409.fig.009c.svgz" name="3452409.fig.009c" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(c) </b>The VFOI homogenous response to <svg height="15.6446pt" id="M225" style="vertical-align:-4.0532pt" version="1.1" viewbox="-0.0498162 -11.5914 16.6947 15.6446" width="16.6947pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M483 442H303V410C352 401 362 388 351 355C333 298 299 198 276 142L201 363C190 396 197 401 240 410V442H-7V410C40 403 52 391 70 348C89 304 144 162 196 18C202 1 204 -14 199 -30C196 -41 190 -53 180 -71C161 -103 138 -122 99 -143C78 -156 71 -165 71 -181C71 -217 114 -245 142 -245C158 -245 172 -240 179 -218C196 -162 216 -99 284 63C342 202 375 285 400 338C425 392 435 402 483 410V442Z" id="g13-119"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="465" vert-adv-y="465" xpluscharwidth="6.045"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.28,-5.741)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487" xpluscharwidth="10.71168539"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.045,3.784)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611" xpluscharwidth="11.60508167"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,11.605,3.784)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487" xpluscharwidth="16.036685390000002"></glyph.data></g></svg></td></tr></table>

<div>Plots of sinusoidal order function <svg height="12.5794pt" id="M220" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 117.852 12.5794" width="117.852pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z" id="g185-47"></path><glyph.data ascent="989" descent="-303" horiz-adv-x="454" vert-adv-y="454" xpluscharwidth="5.9241552"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,5.902,3.132)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 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style="vertical-align:-4.0532pt" version="1.1" viewbox="-0.0498162 -11.5914 16.6947 15.6446" width="16.6947pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M483 442H303V410C352 401 362 388 351 355C333 298 299 198 276 142L201 363C190 396 197 401 240 410V442H-7V410C40 403 52 391 70 348C89 304 144 162 196 18C202 1 204 -14 199 -30C196 -41 190 -53 180 -71C161 -103 138 -122 99 -143C78 -156 71 -165 71 -181C71 -217 114 -245 142 -245C158 -245 172 -240 179 -218C196 -162 216 -99 284 63C342 202 375 285 400 338C425 392 435 402 483 410V442Z" id="g13-119"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="465" vert-adv-y="465" xpluscharwidth="6.045"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.28,-5.741)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487" xpluscharwidth="10.71168539"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.045,3.784)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611" xpluscharwidth="11.60508167"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,11.605,3.784)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487" xpluscharwidth="16.036685390000002"></glyph.data></g></svg> and <svg height="15.6446pt" id="M222" style="vertical-align:-4.0532pt" version="1.1" viewbox="-0.0498162 -11.5914 16.6947 15.6446" width="16.6947pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M483 442H303V410C352 401 362 388 351 355C333 298 299 198 276 142L201 363C190 396 197 401 240 410V442H-7V410C40 403 52 391 70 348C89 304 144 162 196 18C202 1 204 -14 199 -30C196 -41 190 -53 180 -71C161 -103 138 -122 99 -143C78 -156 71 -165 71 -181C71 -217 114 -245 142 -245C158 -245 172 -240 179 -218C196 -162 216 -99 284 63C342 202 375 285 400 338C425 392 435 402 483 410V442Z" id="g13-119"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="465" vert-adv-y="465" xpluscharwidth="6.045"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.28,-5.741)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487" xpluscharwidth="10.71168539"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.045,3.784)"><path d="M556 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Complexity

Generalized Fractional-Order Discrete-Time Integrator

Figure 9