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.006a" src="https://static.hindawi.com/articles/mpe/volume-2014/267307/figures/267307.fig.006a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>The finite-time observer estimation error <svg height="16.3125" id="M232" style="vertical-align:-3.22282pt" 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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</svg></td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="267307.fig.006b" src="https://static.hindawi.com/articles/mpe/volume-2014/267307/figures/267307.fig.006b.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="M233" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 42.599998 16.3125" width="42.599998" 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
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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><tr class="fig-image" id="c"><td><img alt="267307.fig.006c" src="https://static.hindawi.com/articles/mpe/volume-2014/267307/figures/267307.fig.006c.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(c) </b>The infinite-time observer estimation error <svg height="16.3125" id="M234" style="vertical-align:-3.22282pt" 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="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>
</svg></td></tr></table></td></tr><tr class="fig-image" id="d"><td><img alt="267307.fig.006d" src="https://static.hindawi.com/articles/mpe/volume-2014/267307/figures/267307.fig.006d.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="M235" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 42.599998 16.3125" width="42.599998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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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

fig6

Figure 6

Figure 6: Neural Network Observer-Based Finite-Time Formation Control of Mobile Robots 