Predefined-Time Consensus of Nonlinear First-Order Systems Using a Time Base Generator

<div>Comparison of the TBG-tracking controller (<a href="https://static.hindawi.com/articles/mpe/volume-2018/1957070/figures/#EEq13">16</a>) with other approaches for <span class="nowrap"><svg height="12.3524pt" id="M339" style="vertical-align:-3.181499pt" version="1.1" viewbox="-0.0498162 -9.1709 15.9287 12.3524" width="15.9287pt" 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="M422 682C387 695 357 699 317 699C181 699 82 633 82 548C82 472 149 424 253 401C233 362 226 321 226 296C226 212 296 162 389 162C421 162 463 169 491 181L493 179C436 80 364 34 264 34C219 34 189 50 178 65V66C218 73 236 105 236 134C236 169 211 201 167 201C120 201 84 158 84 108C84 32 155 -21 272 -21C459 -21 585 56 641 209C664 269 678 309 707 336L668 368C537 250 498 217 406 217C371 217 360 244 360 279C360 318 375 364 390 391C485 394 576 415 649 448C738 488 799 543 799 602C799 662 746 687 665 687C543 687 382 600 284 457C220 472 173 496 173 548C173 609 232 644 324 644C347 644 380 642 408 634L422 682ZM424 447C488 548 589 632 665 632C696 632 708 617 708 593C708 519 576 454 424 446V447Z" id="g199-8"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.842,3.132)"><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></g></svg>.</span> Top: settling time as a function of the initial condition <span class="nowrap"><svg height="9.35641pt" id="M340" style="vertical-align:-3.29111pt" version="1.1" viewbox="-0.0498162 -6.0653 12.523 9.35641" width="12.523pt" 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="M552 399C552 425 539 454 501 454C451 454 411 416 332 310L314 361C295 415 272 454 241 454C202 454 143 409 83 325L106 298C141 343 161 356 170 356C182 356 194 353 216 283L240 205C186 131 142 82 117 82C108 82 103 85 92 94C86 99 78 104 71 104C47 104 21 82 21 48S46 -12 78 -12C135 -12 174 38 256 161L281 89C306 16 325 -12 364 -12S468 27 529 120L509 149C465 97 443 83 430 83C416 83 404 107 392 140L346 271C370 300 392 326 410 343C430 360 443 367 455 367C465 367 474 363 480 356S492 345 500 345C522 345 552 365 552 399Z" id="g5-121"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.477,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z" id="g50-49"></path></g></svg>.</span> Bottom: maximum value of the auxiliary control input as a function of the initial condition <span class="nowrap"><svg height="9.35641pt" id="M341" style="vertical-align:-3.29111pt" version="1.1" viewbox="-0.0498162 -6.0653 12.523 9.35641" width="12.523pt" 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="M552 399C552 425 539 454 501 454C451 454 411 416 332 310L314 361C295 415 272 454 241 454C202 454 143 409 83 325L106 298C141 343 161 356 170 356C182 356 194 353 216 283L240 205C186 131 142 82 117 82C108 82 103 85 92 94C86 99 78 104 71 104C47 104 21 82 21 48S46 -12 78 -12C135 -12 174 38 256 161L281 89C306 16 325 -12 364 -12S468 27 529 120L509 149C465 97 443 83 430 83C416 83 404 107 392 140L346 271C370 300 392 326 410 343C430 360 443 367 455 367C465 367 474 363 480 356S492 345 500 345C522 345 552 365 552 399Z" id="g5-121"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.477,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z" id="g50-49"></path></g></svg>.</span></div>

Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: Predefined-Time Consensus of Nonlinear First-Order Systems Using a Time Base Generator 