Controlling the PVTOL Aircraft System with an Inverted Pendular Load by means of Nested Saturation Functions and GPI Controller

<div>Angular velocities’ closed-loop response of the proposed controller. (a) Behaviour of state <span class="nowrap"><svg height="12.0627pt" id="M305" style="vertical-align:-3.309361pt" version="1.1" viewbox="-0.0498162 -8.75334 11.9462 12.0627" width="11.9462pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,7.262,-.222)"><path d="M-161 571C-161 600 -183 623 -212 623C-238 623 -262 600 -262 571C-262 544 -238 520 -212 520C-183 520 -161 544 -161 571Z"></path></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.202,3.132)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"></path></g></svg>.</span> (b) Behaviour of state <span class="nowrap"><svg height="12.1828pt" id="M306" style="vertical-align:-3.429461pt" version="1.1" viewbox="-0.0498162 -8.75334 11.6853 12.1828" width="11.6853pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,7.653,-.222)"><path d="M-161 571C-161 600 -183 623 -212 623C-238 623 -262 600 -262 571C-262 544 -238 520 -212 520C-183 520 -161 544 -161 571Z"></path></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.942,3.132)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"></path></g></svg>.</span> (c) Behaviour of state <span class="nowrap"><svg height="8.95973pt" id="M307" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.75334 7.51131 8.95973" width="7.51131pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,7.059,-.222)"><path d="M-161 571C-161 600 -183 623 -212 623C-238 623 -262 600 -262 571C-262 544 -238 520 -212 520C-183 520 -161 544 -161 571Z"></path></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"></path></g></svg>.</span> (d) Behaviour of state <span class="nowrap"><svg height="12.2872pt" id="M308" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -12.0808 6.86362 12.2872 " width="6.86362pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,6.864,-3.549)"><path d="M-161 571C-161 600 -183 623 -212 623C-238 623 -262 600 -262 571C-262 544 -238 520 -212 520C-183 520 -161 544 -161 571Z"></path></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"></path></g></svg>.</span></div>

Mathematical Problems in Engineering

fig5

Figure 5

Figure 5: Controlling the PVTOL Aircraft System with an Inverted Pendular Load by means of Nested Saturation Functions and GPI Controller 