Leader-Following Consensus in Networks of Agents with Nonuniform Time-Varying Delays

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="848942.fig.002a" src="https://static-preview.hindawi.com/articles/mpe/volume-2012/848942/figures/848942.fig.002a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b>Subgraph <svg height="20.137501" id="M132" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 25.3125 20.137501" width="25.3125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(1.25,0,0,-1.25,90,90)">
<path d="m -71.95,70.93 10.51,0" style="fill:none;stroke:#000000;stroke-width:0.68000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
</g>
<g transform="matrix(.017,-0,0,-.017,.062,15.987)"><use xlink:href="#x1D4A2"></use></g>
<g transform="matrix(.012,-0,0,-.012,13.2,20.075)"><use xlink:href="#x31"></use></g><g transform="matrix(.012,-0,0,-.012,18.911,20.075)"><use xlink:href="#x31"></use></g>
</svg></td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="848942.fig.002b" src="https://static-preview.hindawi.com/articles/mpe/volume-2012/848942/figures/848942.fig.002b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b>Subgraph <svg height="20.137501" id="M133" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 25.3125 20.137501" width="25.3125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(1.25,0,0,-1.25,90,90)">
<path d="m -71.95,70.93 10.51,0" style="fill:none;stroke:#000000;stroke-width:0.68000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
</g>
<g transform="matrix(.017,-0,0,-.017,.062,15.987)"><use xlink:href="#x1D4A2"></use></g>
<g transform="matrix(.012,-0,0,-.012,13.2,20.075)"><use xlink:href="#x32"></use></g><g transform="matrix(.012,-0,0,-.012,18.911,20.075)"><use xlink:href="#x31"></use></g>
</svg></td></tr></table></td></tr><tr class="fig-image" id="c"><td><img alt="848942.fig.002c" src="https://static-preview.hindawi.com/articles/mpe/volume-2012/848942/figures/848942.fig.002c.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(c) </b>Subgraph <svg height="20.275" id="M134" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 25.3125 20.275" width="25.3125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(1.25,0,0,-1.25,90,90)">
<path d="m -71.95,70.93 10.51,0" style="fill:none;stroke:#000000;stroke-width:0.68000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
</g>
<g transform="matrix(.017,-0,0,-.017,.062,15.987)"><use xlink:href="#x1D4A2"></use></g>
<g transform="matrix(.012,-0,0,-.012,13.2,20.075)"><use xlink:href="#x33"></use></g><g transform="matrix(.012,-0,0,-.012,18.911,20.075)"><use xlink:href="#x31"></use></g>
</svg></td></tr></table></td></tr></table>

<table>The subgraphs of <svg height="20.137501" id="M128" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 19.6 20.137501" width="19.6" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(1.25,0,0,-1.25,90,90)">
<path d="m -71.95,70.93 10.51,0" style="fill:none;stroke:#000000;stroke-width:0.68000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
</g>
<g transform="matrix(.017,-0,0,-.017,.062,15.987)"><use xlink:href="#x1D4A2"></use></g>
<g transform="matrix(.012,-0,0,-.012,13.2,20.075)"><use xlink:href="#x31"></use></g>
</svg> associated to delays <svg height="12.0125" id="M129" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 14.675 12.0125" width="14.675" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.863)"><path d="M471 456q-22 -60 -37 -85q-20 -3 -84 -3q-9 0 -29 2t-29 2q-38 -145 -54 -234q-15 -80 14 -80q23 0 68 38l14 -25q-78 -83 -137 -83q-50 0 -50 64q0 25 7 57l67 266q-57 4 -98 -9t-80 -49l-20 21q14 18 27 31t36.5 30t56.5 26t73 9q26 0 82.5 -2t89.5 -2q21 0 32 5t24 24
z" id="x1D70F"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.275,11.95)"><use xlink:href="#x31"></use></g>
</svg>, <svg height="12.0125" id="M130" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 14.675 12.0125" width="14.675" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.863)"><use xlink:href="#x1D70F"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.275,11.95)"><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>, and <svg height="12.15" id="M131" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 14.675 12.15" width="14.675" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.863)"><use xlink:href="#x1D70F"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.275,11.95)"><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
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" id="x33"></path></g>
</svg>.</table>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: Leader-Following Consensus in Networks of Agents with Nonuniform Time-Varying Delays 