Output-Feedback Control of a Chaotic MEMS Resonator for Oscillation Amplitude Enhancement

<table>Estimated parameter <svg height="9.89089pt" id="M157" style="vertical-align:-0.2129898pt" version="1.1" viewbox="-0.0498162 -9.6779 7.1439 9.89089" width="7.1439pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M501 444C501 643 409 712 321 712C208 712 108 618 108 502C108 412 172 331 293 331C337 331 394 359 420 378C397 154 341 41 258 41C209 41 178 76 178 161C178 232 171 299 114 299C76 299 38 272 23 257L29 232C42 236 54 239 66 239C91 239 106 221 102 149C95 20 158 -12 219 -12C264 -12 314 10 353 47C461 150 501 295 501 444ZM423 443C423 433 422 422 421 412C401 396 370 385 335 385C245 385 180 444 180 537C180 609 212 671 286 671C349 671 423 595 423 443Z" id="g113-248"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="518" vert-adv-y="518"></glyph.data></g></svg> for the nonimmersed orbit: (a) <svg height="12.9915pt" id="M158" style="vertical-align:-3.3136pt" version="1.1" viewbox="-0.0498162 -9.6779 16.5738 12.9915" width="16.5738pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M501 444C501 643 409 712 321 712C208 712 108 618 108 502C108 412 172 331 293 331C337 331 394 359 420 378C397 154 341 41 258 41C209 41 178 76 178 161C178 232 171 299 114 299C76 299 38 272 23 257L29 232C42 236 54 239 66 239C91 239 106 221 102 149C95 20 158 -12 219 -12C264 -12 314 10 353 47C461 150 501 295 501 444ZM423 443C423 433 422 422 421 412C401 396 370 385 335 385C245 385 180 444 180 537C180 609 212 671 286 671C349 671 423 595 423 443Z" id="g113-248"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="518" vert-adv-y="518"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.679,3.264)"><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"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,11.298,3.264)"><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"></glyph.data></g></svg>; (b) <svg height="12.9915pt" id="M159" style="vertical-align:-3.3136pt" version="1.1" viewbox="-0.0498162 -9.6779 16.5738 12.9915" width="16.5738pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M501 444C501 643 409 712 321 712C208 712 108 618 108 502C108 412 172 331 293 331C337 331 394 359 420 378C397 154 341 41 258 41C209 41 178 76 178 161C178 232 171 299 114 299C76 299 38 272 23 257L29 232C42 236 54 239 66 239C91 239 106 221 102 149C95 20 158 -12 219 -12C264 -12 314 10 353 47C461 150 501 295 501 444ZM423 443C423 433 422 422 421 412C401 396 370 385 335 385C245 385 180 444 180 537C180 609 212 671 286 671C349 671 423 595 423 443Z" id="g113-248"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="518" vert-adv-y="518"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.679,3.264)"><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"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,11.298,3.264)"><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"></glyph.data></g></svg>; (c) <svg height="13.1057pt" id="M160" style="vertical-align:-3.427799pt" version="1.1" viewbox="-0.0498162 -9.6779 16.5738 13.1057" width="16.5738pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M501 444C501 643 409 712 321 712C208 712 108 618 108 502C108 412 172 331 293 331C337 331 394 359 420 378C397 154 341 41 258 41C209 41 178 76 178 161C178 232 171 299 114 299C76 299 38 272 23 257L29 232C42 236 54 239 66 239C91 239 106 221 102 149C95 20 158 -12 219 -12C264 -12 314 10 353 47C461 150 501 295 501 444ZM423 443C423 433 422 422 421 412C401 396 370 385 335 385C245 385 180 444 180 537C180 609 212 671 286 671C349 671 423 595 423 443Z" id="g113-248"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="518" vert-adv-y="518"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.679,3.264)"><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"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,11.298,3.264)"><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 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z" id="g50-52"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg>; (d) <svg height="12.9915pt" id="M161" style="vertical-align:-3.3136pt" version="1.1" viewbox="-0.0498162 -9.6779 16.5738 12.9915" width="16.5738pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M501 444C501 643 409 712 321 712C208 712 108 618 108 502C108 412 172 331 293 331C337 331 394 359 420 378C397 154 341 41 258 41C209 41 178 76 178 161C178 232 171 299 114 299C76 299 38 272 23 257L29 232C42 236 54 239 66 239C91 239 106 221 102 149C95 20 158 -12 219 -12C264 -12 314 10 353 47C461 150 501 295 501 444ZM423 443C423 433 422 422 421 412C401 396 370 385 335 385C245 385 180 444 180 537C180 609 212 671 286 671C349 671 423 595 423 443Z" id="g113-248"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="518" vert-adv-y="518"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,6.679,3.264)"><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"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,11.298,3.264)"><path d="M462 177V227H365V632H320C217 496 116 350 21 208V177H284V109C284 43 280 38 190 31V0H451V31C369 38 365 43 365 108V177H462ZM284 227H88C155 336 218 430 282 519H284V227Z" id="g50-53"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg>.</table>

Mathematical Problems in Engineering

fig12

Figure 12

Figure 12: Output-Feedback Control of a Chaotic MEMS Resonator for Oscillation Amplitude Enhancement 