Neural Network Observer-Based Finite-Time Formation Control of Mobile Robots

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="267307.fig.007a" src="https://static-preview.hindawi.com/articles/mpe/volume-2014/267307/figures/267307.fig.007a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>The finite-time observer estimation error <svg height="17.5" id="M236" style="vertical-align:-3.22282pt" version="1.1" viewbox="0 0 46.0625 17.5" width="46.0625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,10.138,17.3)"><path d="M238 681l-124 -585q-7 -31 4 -31q10 0 37.5 18.5t49.5 41.5l16 -22q-40 -48 -89.5 -81.5t-76.5 -33.5q-42 0 -16 122l105 488q7 32 0 41t-39 9h-35l5 26q35 3 71 13t58 17.5t26 7.5q14 0 8 -31z" id="x1D459"></path></g><g transform="matrix(.012,-0,0,-.012,13.338,17.3)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
<g transform="matrix(.017,-0,0,-.017,19.675,12.337)"><path d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" id="x2C"></path></g><g transform="matrix(.017,-0,0,-.017,27.438,10.962)"><path d="M470 619q-46 -79 -133 -79q-39 0 -115 22q-59 16 -92 16q-57 0 -111 -46l-19 23q56 78 138 78q43 0 110 -21q62 -17 100 -17q60 0 103 46z" id="xF5C11"></path></g><g transform="matrix(.017,-0,0,-.017,26.387,12.337)"><path d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36
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<g transform="matrix(.012,-0,0,-.012,36.463,17.3)"><path d="M238 681l-124 -585q-7 -31 4 -31q10 0 37.5 18.5t49.5 41.5l16 -22q-40 -48 -89.5 -81.5t-76.5 -33.5q-42 0 -16 122l105 488q7 32 0 41t-39 9h-35l5 26q35 3 71 13t58 17.5t26 7.5q14 0 8 -31z" id="x1D459"></path></g><g transform="matrix(.012,-0,0,-.012,39.663,17.3)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg></td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="267307.fig.007b" src="https://static-preview.hindawi.com/articles/mpe/volume-2014/267307/figures/267307.fig.007b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b>The finite-time observer estimation error <svg height="16.3125" id="M237" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 18 16.3125" width="18" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.237,10.963)"><path d="M470 619q-46 -79 -133 -79q-39 0 -115 22q-59 16 -92 16q-57 0 -111 -46l-19 23q56 78 138 78q43 0 110 -21q62 -17 100 -17q60 0 103 46z" id="xF5C11"></path></g><g transform="matrix(.017,-0,0,-.017,.062,11.15)"><path d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387
q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z" id="x1D45E"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.413,16.113)"><path d="M238 681l-124 -585q-7 -31 4 -31q10 0 37.5 18.5t49.5 41.5l16 -22q-40 -48 -89.5 -81.5t-76.5 -33.5q-42 0 -16 122l105 488q7 32 0 41t-39 9h-35l5 26q35 3 71 13t58 17.5t26 7.5q14 0 8 -31z" id="x1D459"></path></g><g transform="matrix(.012,-0,0,-.012,11.613,16.113)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
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</svg></td></tr></table></td></tr><tr class="fig-image" id="c"><td><img alt="267307.fig.007c" src="https://static-preview.hindawi.com/articles/mpe/volume-2014/267307/figures/267307.fig.007c.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(c) </b>The infinite-time observer estimation error <svg height="17.5" id="M238" style="vertical-align:-3.22282pt" version="1.1" viewbox="0 0 46.0625 17.5" width="46.0625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,1.113,10.963)"><path d="M470 619q-46 -79 -133 -79q-39 0 -115 22q-59 16 -92 16q-57 0 -111 -46l-19 23q56 78 138 78q43 0 110 -21q62 -17 100 -17q60 0 103 46z" id="xF5C11"></path></g><g transform="matrix(.017,-0,0,-.017,.063,12.338)"><path d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36
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</svg></td></tr></table></td></tr><tr class="fig-image" id="d"><td><img alt="267307.fig.007d" src="https://static-preview.hindawi.com/articles/mpe/volume-2014/267307/figures/267307.fig.007d.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(d) </b>The infinite-time observer estimation error <svg height="16.3125" id="M239" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 18 16.3125" width="18" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z" id="x1D45E"></path></g>
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</svg></td></tr></table></td></tr></table>

<table>The infinite-time observer and finite-time observer estimation errors.</table>

Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: Neural Network Observer-Based Finite-Time Formation Control of Mobile Robots 