Robust Preview Control for a Class of Uncertain Discrete-Time Lipschitz Nonlinear Systems

<table class="fixed-width table-group" id="tab4"><tr><td><table class="table"><colgroup><col style="width:18.00em"/><col style="width:14.65em"/><col style="width:14.65em"/><col style="width:14.65em"/></colgroup><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td class="align_left">Performance Index</td><td class="align_center">Standard <svg height="11.927pt" id="M248" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 20.1917 11.927" width="20.1917pt" 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="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z" id="g113-73"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.075,3.132)"><path d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z" id="g50-30"></path></g></svg> controller</td><td class="align_center">Preview controller <br/>(<svg height="11.927pt" id="M249" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 37.9506 11.927" width="37.9506pt" 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="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z" id="g113-78"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.22,3.132)"><path d="M397 380C406 395 404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z" id="g50-115"></path></g><g transform="matrix(.013,0,0,-0.013,20.25,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,31.513,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path></g></svg>)</td><td class="align_center">Preview controller <br/>(<svg height="11.927pt" id="M250" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 37.9506 11.927" width="37.9506pt" 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="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z" id="g113-78"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.22,3.132)"><path d="M397 380C406 395 404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z" id="g50-115"></path></g><g transform="matrix(.013,0,0,-0.013,20.25,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,31.513,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z" id="g113-54"></path></g></svg>)</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td class="align_left">IAE</td><td class="align_center">12.6485</td><td class="align_center">7.8533</td><td class="align_center">6.5527</td></tr><tr><td class="align_left">ISE</td><td class="align_center">26.2037</td><td class="align_center">11.1251</td><td class="align_center">5.5557</td></tr><tr><td class="align_left">ITAE</td><td class="align_center">280.9896</td><td class="align_center">170.9029</td><td class="align_center">132.8920</td></tr><tr><td class="align_left">ITSE</td><td class="align_center">558.5424</td><td class="align_center">232.7349</td><td class="align_center">110.0990</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Mathematical Problems in Engineering

tab4

Table 4

Table 4: Robust Preview Control for a Class of Uncertain Discrete-Time Lipschitz Nonlinear Systems 