An Interval Type-2 Fuzzy System with a Species-Based Hybrid Algorithm for Nonlinear System Control Design

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr><td align="left">Species radius</td><td align="center">16.879<br/>(<svg height="16.450001" id="M272" style="vertical-align:-3.22282pt" version="1.1" viewbox="0 0 32.75 16.450001" width="32.75" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5
h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z" id="x1D451"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.5,16.25)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.012,-0,0,-.012,19.245,16.25)"><path d="M433 39l-85 -51q-27 0 -47 20q-16 16 -24 44q-5 -3 -24 -16.5t-27 -18.5t-23 -13.5t-25 -12t-19 -3.5q-52 0 -86 36.5t-34 85.5q0 71 81 99q128 43 155 65v17q0 54 -23 83.5t-62 29.5q-28 0 -45 -19t-29 -66q-7 -23 -29 -23q-15 0 -29 13t-14 30t31 40q25 19 68.5 40.5
t81.5 29.5q49 0 82 -27q45 -39 45 -123v-185q0 -60 35 -60q18 0 36 11zM275 84v156q-12 -6 -49 -23t-41 -19q-31 -14 -46 -31t-15 -43q0 -35 22.5 -55.5t48.5 -20.5q19 0 42 10.5t38 25.5z" id="x61"></path></g><g transform="matrix(.012,-0,0,-.012,24.469,16.25)"><path d="M474 0h-214v26q32 4 39 12t-5 28l-74 117q-39 -58 -64 -106q-16 -25 -10 -35t44 -16v-26h-186v26q46 6 64 17t48 49t85 120l-87 136q-23 36 -40 47t-58 16v26h199v-26q-31 -4 -36.5 -12.5t9.5 -29.5l59 -93q37 56 58 94q13 23 7 30t-40 11v26h183v-26q-47 -5 -65 -13.5
t-45 -39.5q-24 -29 -78 -110l97 -148q28 -43 46.5 -55.5t63.5 -18.5v-26z" id="x78"></path></g>
</svg>/1.5 )</td><td align="center">12.659<br/>(<svg height="16.450001" id="M273" style="vertical-align:-3.22282pt" version="1.1" viewbox="0 0 32.75 16.450001" width="32.75" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5
h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z" id="x1D451"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.5,16.25)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.012,-0,0,-.012,19.245,16.25)"><path d="M433 39l-85 -51q-27 0 -47 20q-16 16 -24 44q-5 -3 -24 -16.5t-27 -18.5t-23 -13.5t-25 -12t-19 -3.5q-52 0 -86 36.5t-34 85.5q0 71 81 99q128 43 155 65v17q0 54 -23 83.5t-62 29.5q-28 0 -45 -19t-29 -66q-7 -23 -29 -23q-15 0 -29 13t-14 30t31 40q25 19 68.5 40.5
t81.5 29.5q49 0 82 -27q45 -39 45 -123v-185q0 -60 35 -60q18 0 36 11zM275 84v156q-12 -6 -49 -23t-41 -19q-31 -14 -46 -31t-15 -43q0 -35 22.5 -55.5t48.5 -20.5q19 0 42 10.5t38 25.5z" id="x61"></path></g><g transform="matrix(.012,-0,0,-.012,24.469,16.25)"><path d="M474 0h-214v26q32 4 39 12t-5 28l-74 117q-39 -58 -64 -106q-16 -25 -10 -35t44 -16v-26h-186v26q46 6 64 17t48 49t85 120l-87 136q-23 36 -40 47t-58 16v26h199v-26q-31 -4 -36.5 -12.5t9.5 -29.5l59 -93q37 56 58 94q13 23 7 30t-40 11v26h183v-26q-47 -5 -65 -13.5
t-45 -39.5q-24 -29 -78 -110l97 -148q28 -43 46.5 -55.5t63.5 -18.5v-26z" id="x78"></path></g>
</svg>/2 )</td><td align="center">10.127<br/>(<svg height="16.450001" id="M274" style="vertical-align:-3.22282pt" version="1.1" viewbox="0 0 32.75 16.450001" width="32.75" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5
h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z" id="x1D451"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.5,16.25)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.012,-0,0,-.012,19.245,16.25)"><path d="M433 39l-85 -51q-27 0 -47 20q-16 16 -24 44q-5 -3 -24 -16.5t-27 -18.5t-23 -13.5t-25 -12t-19 -3.5q-52 0 -86 36.5t-34 85.5q0 71 81 99q128 43 155 65v17q0 54 -23 83.5t-62 29.5q-28 0 -45 -19t-29 -66q-7 -23 -29 -23q-15 0 -29 13t-14 30t31 40q25 19 68.5 40.5
t81.5 29.5q49 0 82 -27q45 -39 45 -123v-185q0 -60 35 -60q18 0 36 11zM275 84v156q-12 -6 -49 -23t-41 -19q-31 -14 -46 -31t-15 -43q0 -35 22.5 -55.5t48.5 -20.5q19 0 42 10.5t38 25.5z" id="x61"></path></g><g transform="matrix(.012,-0,0,-.012,24.469,16.25)"><path d="M474 0h-214v26q32 4 39 12t-5 28l-74 117q-39 -58 -64 -106q-16 -25 -10 -35t44 -16v-26h-186v26q46 6 64 17t48 49t85 120l-87 136q-23 36 -40 47t-58 16v26h199v-26q-31 -4 -36.5 -12.5t9.5 -29.5l59 -93q37 56 58 94q13 23 7 30t-40 11v26h183v-26q-47 -5 -65 -13.5
t-45 -39.5q-24 -29 -78 -110l97 -148q28 -43 46.5 -55.5t63.5 -18.5v-26z" id="x78"></path></g>
</svg>/2.5 )</td><td align="center">8.439<br/>(<svg height="16.450001" id="M275" style="vertical-align:-3.22282pt" version="1.1" viewbox="0 0 32.75 16.450001" width="32.75" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5
h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z" id="x1D451"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.5,16.25)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.012,-0,0,-.012,19.245,16.25)"><path d="M433 39l-85 -51q-27 0 -47 20q-16 16 -24 44q-5 -3 -24 -16.5t-27 -18.5t-23 -13.5t-25 -12t-19 -3.5q-52 0 -86 36.5t-34 85.5q0 71 81 99q128 43 155 65v17q0 54 -23 83.5t-62 29.5q-28 0 -45 -19t-29 -66q-7 -23 -29 -23q-15 0 -29 13t-14 30t31 40q25 19 68.5 40.5
t81.5 29.5q49 0 82 -27q45 -39 45 -123v-185q0 -60 35 -60q18 0 36 11zM275 84v156q-12 -6 -49 -23t-41 -19q-31 -14 -46 -31t-15 -43q0 -35 22.5 -55.5t48.5 -20.5q19 0 42 10.5t38 25.5z" id="x61"></path></g><g transform="matrix(.012,-0,0,-.012,24.469,16.25)"><path d="M474 0h-214v26q32 4 39 12t-5 28l-74 117q-39 -58 -64 -106q-16 -25 -10 -35t44 -16v-26h-186v26q46 6 64 17t48 49t85 120l-87 136q-23 36 -40 47t-58 16v26h199v-26q-31 -4 -36.5 -12.5t9.5 -29.5l59 -93q37 56 58 94q13 23 7 30t-40 11v26h183v-26q-47 -5 -65 -13.5
t-45 -39.5q-24 -29 -78 -110l97 -148q28 -43 46.5 -55.5t63.5 -18.5v-26z" id="x78"></path></g>
</svg>/3 )</td><td align="center">7.234<br/>(<svg height="16.450001" id="M276" style="vertical-align:-3.22282pt" version="1.1" viewbox="0 0 32.75 16.450001" width="32.75" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5
h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z" id="x1D451"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.5,16.25)"><path d="M797 0h-214v26q46 5 58 17.5t12 63.5v167q0 116 -88 116q-48 0 -112 -52q3 -18 3 -42v-195q0 -46 11 -58t53 -17v-26h-214v26q48 5 59.5 17t11.5 60v172q0 115 -88 115q-47 0 -109 -52v-235q0 -47 10 -59.5t48 -17.5v-26h-211v26q50 5 62.5 17t12.5 60v207q0 48 -9 59.5
t-52 18.5v23q79 14 139 41v-81q3 2 29 20.5t44 29.5q45 28 81 28t66 -22t46 -61q25 20 83 57q39 26 76 26q59 0 93 -40t34 -113v-193q0 -49 10.5 -60.5t54.5 -16.5v-26z" id="x6D"></path></g><g transform="matrix(.012,-0,0,-.012,19.245,16.25)"><path d="M433 39l-85 -51q-27 0 -47 20q-16 16 -24 44q-5 -3 -24 -16.5t-27 -18.5t-23 -13.5t-25 -12t-19 -3.5q-52 0 -86 36.5t-34 85.5q0 71 81 99q128 43 155 65v17q0 54 -23 83.5t-62 29.5q-28 0 -45 -19t-29 -66q-7 -23 -29 -23q-15 0 -29 13t-14 30t31 40q25 19 68.5 40.5
t81.5 29.5q49 0 82 -27q45 -39 45 -123v-185q0 -60 35 -60q18 0 36 11zM275 84v156q-12 -6 -49 -23t-41 -19q-31 -14 -46 -31t-15 -43q0 -35 22.5 -55.5t48.5 -20.5q19 0 42 10.5t38 25.5z" id="x61"></path></g><g transform="matrix(.012,-0,0,-.012,24.469,16.25)"><path d="M474 0h-214v26q32 4 39 12t-5 28l-74 117q-39 -58 -64 -106q-16 -25 -10 -35t44 -16v-26h-186v26q46 6 64 17t48 49t85 120l-87 136q-23 36 -40 47t-58 16v26h199v-26q-31 -4 -36.5 -12.5t9.5 -29.5l59 -93q37 56 58 94q13 23 7 30t-40 11v26h183v-26q-47 -5 -65 -13.5
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</svg>/3.5 )</td></tr><tr><td align="left">Average time</td><td align="center">364.3</td><td align="center">374.4</td><td align="center">419.0</td><td align="center">528.4</td><td align="center">1040.5</td></tr><tr><td align="left">Average MSE</td><td align="center">0.011802</td><td align="center">0.008501</td><td align="center">0.004335</td><td align="center">0.0017631</td><td align="center">0.0017631</td></tr><tr class="table-tr"><td colspan="6"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Comparison of MSEs and computational complexity for Example <a href="https://static.hindawi.com/articles/mpe/volume-2014/735310/figures/#ex1">1</a> when using different radii (<svg height="11.75" id="M269" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 48.174999 11.75" width="48.174999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242
q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" id="x1D43A"></path></g><g transform="matrix(.017,-0,0,-.017,17.078,11.4)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,31.782,11.4)"><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><g transform="matrix(.017,-0,0,-.017,39.942,11.4)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
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</svg>; <svg height="11.375" id="M270" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 57.674999 11.375" width="57.674999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg>; <svg height="15.4" id="M271" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 59.924999 15.4" width="59.924999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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Mathematical Problems in Engineering

tab4

Table 4

Table 4: An Interval Type-2 Fuzzy System with a Species-Based Hybrid Algorithm for Nonlinear System Control Design 