Robust Local Stabilization of Discrete-Time Systems with Time-Varying State Delay and Saturating Actuators

<div>Percentage augmentation of the estimate of the region of attraction (<span class="nowrap"><svg height="12.1462pt" id="M325" style="vertical-align:-3.510301pt" version="1.1" viewbox="-0.0498162 -8.6359 15.8935 12.1462" width="15.8935pt" 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="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z" id="g113-83"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.086,3.132)"><path d="M611 626L575 651L170 -9L204 -35L611 626ZM200 635C93 635 50 532 50 449C50 365 94 264 199 264C306 264 349 365 349 450C349 534 304 635 200 635ZM199 601C264 601 273 501 273 450S264 298 200 298C139 298 126 398 126 450S139 601 199 601ZM590 359C485 359 440 256 440 173C440 89 485 -12 589 -12C694 -12 738 89 738 173C738 256 694 359 590 359ZM589 325C653 325 662 225 662 173C662 124 653 22 590 22C529 22 516 122 516 174C516 227 529 325 589 325Z" id="g58-35"></path></g></svg>,</span> see the text) as a function of the maximum time variation of the delay, for <svg height="9.42945pt" id="M326" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -9.22306 27.5889 9.42945" width="27.5889pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><rect height="0.65243" width="6.30254" x="0" y="-8.52081"></rect><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z" id="g113-241"></path></g><g transform="matrix(.013,0,0,-0.013,9.934,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,21.197,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z" id="g113-52"></path></g></svg> (red square), <svg height="9.42945pt" id="M327" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -9.22306 27.5889 9.42945" width="27.5889pt" 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style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -9.22306 27.5889 9.42945" width="27.5889pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><rect height="0.65243" width="6.30254" x="0" y="-8.52081"></rect><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z" id="g113-241"></path></g><g transform="matrix(.013,0,0,-0.013,9.934,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,21.197,0)"><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> (green circle), <svg height="9.42945pt" id="M329" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -9.22306 27.5889 9.42945" width="27.5889pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><rect height="0.65243" width="6.30254" x="0" y="-8.52081"></rect><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z" id="g113-241"></path></g><g transform="matrix(.013,0,0,-0.013,9.934,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,21.197,0)"><path d="M137 343C167 482 260 545 321 574C357 591 397 603 429 609L423 641C382 634 335 622 295 608C189 570 37 457 37 238C37 84 125 -12 242 -12C362 -12 447 89 447 209C447 311 374 393 267 393C247 393 226 386 204 376L137 343ZM227 337C318 337 361 256 361 173C361 105 336 22 258 22C176 22 126 120 126 240C126 266 127 291 132 310C155 323 189 337 227 337Z" id="g113-55"></path></g></svg> (purple star), and <svg height="9.42945pt" id="M330" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -9.22306 27.5889 9.42945" width="27.5889pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><rect height="0.65243" width="6.30254" x="0" y="-8.52081"></rect><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z" id="g113-241"></path></g><g transform="matrix(.013,0,0,-0.013,9.934,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,21.197,0)"><path d="M447 623H65C61 580 56 530 47 475H76C100 541 106 550 172 550H388C308 376 196 170 91 -1L98 -12L172 -2C268 204 360 408 455 611L447 623Z" id="g113-56"></path></g></svg> (▼).</div>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: Robust Local Stabilization of Discrete-Time Systems with Time-Varying State Delay and Saturating Actuators 