Proportional Retarded Controller to Stabilize Underactuated Systems with Measurement Delays: Furuta Pendulum Case Study

<div>Control actions (<svg height="8.8423pt" id="M201" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 11.0475 8.8423" width="11.0475pt" 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="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z" id="g113-79"></path></g></svg>), PR controller (<svg height="11.2615pt" id="M202" style="vertical-align:-5.300591pt" version="1.1" viewbox="-0.0498162 -5.96091 15.7917 11.2615" width="15.7917pt" 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="M515 96L502 119C471 88 431 62 423 62C416 62 411 70 417 101C440 223 469 341 497 448H486L412 422L380 277C330 188 210 57 152 57C137 57 126 69 139 124L195 366C210 431 205 448 182 448C155 448 89 413 23 350L36 326C73 354 103 376 112 376C118 376 118 365 113 340L61 118C54 90 52 68 52 51C52 0 75 -12 98 -12S151 -3 181 17C242 58 305 119 362 193H364L345 104C323 3 339 -12 359 -12C390 -12 464 35 515 96Z" id="g113-118"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.929,3.132)"><path d="M173 382V461C125 442 68 425 25 418V390C88 382 92 382 92 317V-127C92 -191 84 -199 18 -204V-235H259V-204C180 -196 173 -192 173 -120V7C188 -1 212 -11 243 -12C375 12 495 109 495 261C495 361 427 451 316 451C303 451 283 446 266 435L173 382ZM173 345C200 365 243 388 274 388C346 388 409 329 409 222C409 111 353 40 266 40C231 40 196 56 173 79V345Z" id="g58-110"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.679,3.132)"><path d="M185 344V453C137 433 92 421 41 413V386C100 378 104 375 104 310V106C104 41 97 35 33 30V0H268V30C190 35 185 41 185 107V283C206 340 240 371 267 371C282 371 296 364 310 351C316 344 325 343 336 348C357 357 369 378 369 399C369 424 345 451 310 451C262 451 219 397 187 344H185Z" id="g58-112"></path></g></svg>), and state feedback controller (<svg height="9.25658pt" id="M203" style="vertical-align:-3.295671pt" version="1.1" viewbox="-0.0498162 -5.96091 14.5677 9.25658" width="14.5677pt" 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="M515 96L502 119C471 88 431 62 423 62C416 62 411 70 417 101C440 223 469 341 497 448H486L412 422L380 277C330 188 210 57 152 57C137 57 126 69 139 124L195 366C210 431 205 448 182 448C155 448 89 413 23 350L36 326C73 354 103 376 112 376C118 376 118 365 113 340L61 118C54 90 52 68 52 51C52 0 75 -12 98 -12S151 -3 181 17C242 58 305 119 362 193H364L345 104C323 3 339 -12 359 -12C390 -12 464 35 515 96Z" id="g113-118"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.929,3.132)"><path d="M327 325C324 350 312 408 304 433C283 442 256 451 213 451C120 451 59 391 59 320C59 243 125 210 188 181C237 158 266 136 266 93C266 52 231 24 195 24C134 24 93 81 74 148L42 143C42 106 51 37 59 23C78 8 125 -12 177 -12C257 -12 345 35 345 127C345 198 292 234 216 264C171 283 131 305 131 348C131 387 156 416 195 416C243 416 277 378 297 317L327 325Z" id="g58-113"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.351,3.132)"><path d="M56 439L27 407L32 396H103V106C103 41 96 35 31 30V0H271V29C191 37 184 40 184 110V396H293C304 404 309 428 303 439H184V473C183 558 195 604 207 623S234 653 261 653C293 653 322 635 341 616C350 607 360 606 370 614C381 622 386 631 389 638C394 652 393 664 384 675C366 696 339 708 303 710C263 705 225 686 190 655C137 608 121 561 114 538S103 487 103 456V439H56Z" id="g58-100"></path></g></svg>).</div>

Mathematical Problems in Engineering

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Figure 9

Figure 9: Proportional Retarded Controller to Stabilize Underactuated Systems with Measurement Delays: Furuta Pendulum Case Study 