A Fast Elitism Gaussian Estimation of Distribution Algorithm and Application for PID Optimization

<table class="table-group" width="800px"><tr class="fig-image" id="a"><td><img alt="597278.fig.006a" src="https://static.hindawi.com/articles/tswj/volume-2014/597278/figures/597278.fig.006a.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(a) </b><svg height="11.3125" id="M124" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 59.75 11.3125" width="59.75" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62
q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" id="x1D435"></path></g><g transform="matrix(.017,-0,0,-.017,10.551,11.113)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g><g transform="matrix(.017,-0,0,-.017,30.116,11.113)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,44.82,11.113)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g>
</svg>, using elitism strategy</td></tr></table></td></tr><tr class="fig-image" id="b"><td><img alt="597278.fig.006b" src="https://static.hindawi.com/articles/tswj/volume-2014/597278/figures/597278.fig.006b.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(b) </b><svg height="11.3125" id="M125" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 59.75 11.3125" width="59.75" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62
q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" id="x1D435"></path></g><g transform="matrix(.017,-0,0,-.017,10.551,11.113)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g><g transform="matrix(.017,-0,0,-.017,30.116,11.113)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,44.82,11.113)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g>
</svg>, without elitism strategy</td></tr></table></td></tr><tr class="fig-image" id="c"><td><img alt="597278.fig.006c" src="https://static.hindawi.com/articles/tswj/volume-2014/597278/figures/597278.fig.006c.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(c) </b><svg height="14.85" id="M126" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 74.912498 14.85" width="74.912498" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62
q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" id="x1D435"></path></g><g transform="matrix(.017,-0,0,-.017,10.551,12.012)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g><g transform="matrix(.017,-0,0,-.017,30.116,12.012)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,44.82,12.012)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g><g transform="matrix(.017,-0,0,-.017,59.677,12.012)"><path d="M368 703l-264 -866h-60l265 866h59z" id="x2F"></path></g><g transform="matrix(.017,-0,0,-.017,66.681,12.012)"><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>, using elitism strategy</td></tr></table></td></tr><tr class="fig-image" id="d"><td><img alt="597278.fig.006d" src="https://static.hindawi.com/articles/tswj/volume-2014/597278/figures/597278.fig.006d.jpg"/></td></tr><tr class="fig-caption"><td align="center"><table><tr><td><b>(d) </b><svg height="14.85" id="M127" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 74.912498 14.85" width="74.912498" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62
q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" id="x1D435"></path></g><g transform="matrix(.017,-0,0,-.017,10.551,12.012)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g><g transform="matrix(.017,-0,0,-.017,30.116,12.012)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,44.82,12.012)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g><g transform="matrix(.017,-0,0,-.017,59.677,12.012)"><path d="M368 703l-264 -866h-60l265 866h59z" id="x2F"></path></g><g transform="matrix(.017,-0,0,-.017,66.681,12.012)"><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>, without elitism strategy</td></tr></table></td></tr></table>

<table>The boxplot of population for different iterations.</table>

The Scientific World Journal

fig6

Figure 6

Figure 6: A Fast Elitism Gaussian Estimation of Distribution Algorithm and Application for PID Optimization 