Design of Fuzzy Fractional PD + I Controllers Tuned by a Genetic Algorithm

<table>Parameters <svg height="14.4125" id="M183" style="vertical-align:-3.2316pt" version="1.1" viewbox="0 0 11 14.4125" width="11" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg>, <svg height="14.4125" id="M184" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 11.5625 14.4125" width="11.5625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg>, <svg height="17.325001" id="M185" style="vertical-align:-4.74141pt" version="1.1" viewbox="0 0 13.6 17.325001" width="13.6" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.125)"><path d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" id="x1D461"></path></g>
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</svg>, <svg height="7.9499998" id="M186" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 15.4375 7.9499998" width="15.4375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path d="M445 282q0 -64 -27.5 -125t-80.5 -104q-80 -65 -168 -65q-73 0 -109.5 46.5t-36.5 116.5q0 92 46.5 163t119.5 104q66 30 108 30q67 0 107.5 -43.5t40.5 -122.5zM355 280q0 62 -26.5 94.5t-64.5 32.5q-26 0 -44 -8q-39 -18 -74 -82t-35 -163q0 -57 25.5 -90.5t70.5 -33.5
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</svg>(%) versus <svg height="16.6625" id="M187" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 10 16.6625" width="10" 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="M558 587q0 -32 -14 -61t-40 -53.5t-48.5 -41t-54.5 -36.5q144 -51 144 -174q0 -55 -43.5 -108t-104.5 -87q-77 -42 -131 -42q-31 0 -54 20t-31 47l11 18q48 -29 108 -29q79 0 119.5 43t40.5 109t-44.5 107.5t-119.5 50.5l22 47q34 1 65 21q96 61 96 157q0 42 -24 67.5
t-62 25.5q-24 0 -43.5 -9t-35 -29.5t-27 -44t-22.5 -63t-19.5 -75.5t-18.5 -91q-57 -294 -68 -380q-26 -190 -35 -200q-26 -31 -97 -37l-4 26q19 9 48 170l77 413q23 121 52.5 187.5t83.5 114.5q70 62 148 62q51 0 88.5 -34t37.5 -91z" id="x1D6FD"></path></g>
</svg> of the step responses of the closed-loop system with <svg height="16.275" id="M188" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 36.900002 16.275" width="36.900002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.137)"><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
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<g transform="matrix(.012,-0,0,-.012,12.375,16.212)"><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,-.017,18.7,12.137)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,24.582,12.137)"><path d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81
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</svg> and with a fuzzy <svg height="16.8125" id="M189" style="vertical-align:-0.51414pt" version="1.1" viewbox="0 0 52.974998 16.8125" width="52.974998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><path d="M46 650h247q112 0 169 -41q64 -49 64 -141q0 -91 -56.5 -143t-138.5 -62q-23 -3 -45 -1l-74 19v-157q0 -63 15.5 -77t83.5 -19v-28h-271v28q60 5 74 20t14 77v400q0 65 -12.5 79t-69.5 18v28zM212 559v-245q31 -12 73 -12q59 0 102 38t43 124q0 77 -42.5 115.5
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<g transform="matrix(.012,-0,0,-.012,22.038,8.537)"><path d="M558 587q0 -32 -14 -61t-40 -53.5t-48.5 -41t-54.5 -36.5q144 -51 144 -174q0 -55 -43.5 -108t-104.5 -87q-77 -42 -131 -42q-31 0 -54 20t-31 47l11 18q48 -29 108 -29q79 0 119.5 43t40.5 109t-44.5 107.5t-119.5 50.5l22 47q34 1 65 21q96 61 96 157q0 42 -24 67.5
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<g transform="matrix(.017,-0,0,-.017,33.35,16.7)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g><g transform="matrix(.017,-0,0,-.017,47.102,16.7)"><path d="M303 0h-265v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28h265v-28q-62 -5 -76 -19.5t-14 -77.5v-400q0 -64 14 -78t76 -19v-28z" id="x49"></path></g>
</svg> controller.</table>

Mathematical Problems in Engineering

fig10

Figure 10

Figure 10: Design of Fuzzy Fractional PD + I Controllers Tuned by a Genetic Algorithm 