3 T. Terano, K Asai, M. Sugeno Fuzzy Systems A CONTROLLER PERFORMANCE RESULTS Theory and Its App lications. Academic Press Boston, MA, USA 4 W.Pedrycz. Fuzzy Control and Fuzzy s 2nd extended ed John wiley New York, NY,USA H. B i,“ Fuzzy Logic Controllers”,in An Introduction to Fuzzy Logic Applications in Intelligent Systems, R. R. Yager and L. A. Zadeh. eds. Kluwer Academic Publish ers, Boston, MA, USA, 1992, Pp 69-96 6 S. Chiu, S Chand, D. Moore, A Chaudhary Fuzzy Logic for Control of Roll and Moment for a Flexible Wing Aircraft", IEEE Control Systems Magazine, pp 42-48, Vol ll, No 4 une 7 A.L. Schwartz, Comments on Fuzzy L Control of roll and moment for a flexible Wing Aircraft, IEEE Control Systems Mag azine, pp 61-62, Vol 12, No 1, February 1992 S.Chiu,“ Author' s Reply”, TEEE Control Sys tems Magazine, pp 62-63, Vol 12, No 1, Feb ary 1992 “ Adaptive Fuzzy Systems”,IEEE Spectrum, pp 27-31, Vol 30, No 2, February Figure 3: Closed Loop Step Response of nominal plant 10]C.J. Herget,Ed.,“ Reader' s Forun”,IEEE Control Systems Magazine, pp 5-7, Vol 13, No June 1993 [11 M. Athans, Control-The Adventure Cont in ues", Bode Lecture, 3end IEEE Conference on Decision and Control. San Antonio. TX USA December 15-17 1993 12]E. H. Mamdani, "Twenty Years of Fuzz Learnt P.可r9 nd Ieee Intl. Conf on i" Fuzzy Systems, pp. 339-344 13 M. Tomizuka," Fuzzy Control in Control Lecture, ARO/NASA Workshop on Formal Models for Intelligent Control, MIT, Cambridge, MA 02139, USA September 30 ber2,1993 [14 C. W. Tao, R. Mamlook, W.E. Thompson, Reduction of Complexity for a Robust Fuzzy Controller, Proc. of 2nd IEEE Intl. Conf.on 1346-134 [15H. Ying, W. Siler, JJ Buckley, "Fuzzy Con trol Theory: A Nonlinear Case", Automatica Vol.26.No.3.pp513-420.1990 Control. prentice hall. 2nd ed.19 91 Figure 4: Closed Loop Step Response of the per turbed model G2(s)þ❂✁ æ ✝ æ✌è✉ö❾÷rô❉ï❅✲✁￾ ✝ ★é➏÷➁ó✗✲☛☞ ✝ ✯ ê❉ò❢è✉ô❉ï ✝ ♥❂✈✎✖✜✖✘✓P✟✞✓❅ß❾Û②Ö✉ã❝ß Þ①✺➡Ö❾ÙrÜ✘✓✤✢rÚ❅✥✞✕❾Û▲ß Ó✡☛✡❭Ý➈ä✆❥✣✢❅Û❤ä❇Ù➁Ú◆ß ✝ ★ë✉÷➍ú◆è✮✡❳ó①ë❵❺✌ö②èsé②é✜✲ ❙ï❢é②õ②ï❢ô❲✲❴☞★ ✲✄✂ ✯✠★ ✝ þ✍✁ ✇✝ ❺❬è❫ú◆ö②ñ❉ë❾ð ✝ ♥❂✈ ✖✗✖✘✓✜Ø✐Ù➁Ú◆Û➶Ü❾ÙrÝ✤✢rÚ✦✥ ♥❂✈✎✖✗✖✶✓ ✟✞✓❅ß✣✶ Û➔Ösã✿ß ✲✯✻rô❑ú✼è❾ø❉õ➃è✉ô❑ú◆è❫ú❖è❫ú ✝✆☎ ï❢ç❉ô ✇ó↕î➪è✉ñ✌☞ ✯ ï❢ô❉é✜✲ ✕è✉ü✞✝✐ï❢ö❈❀❂✲ ✕✝✤✲✟✂✯✠★ ✝ þ✶ ✁ ✰☞✝✁❙è✉ö➃è✉ô✔✓❾ó✗✲ ✙❚s❃ê❉ð✉ð✉ñ ❤qï➍ò❢ó①ë ✓■ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö➃é✭✢✦✲âó➪ô Ó❥Ú ✕✉Ú◆Û❤Ü➏Ù✏✥✓✈②❥ìÛ➶ä▲ÙrÚ❴Û②Ù✵♥❂✈✎✖✜✖✘✓✵❧❨Ù✣✸rä✆❥➲Ó✦✡❜✡❑Ý➈ä✻❥❚✢❱Û➶ä▲ÙrÚ◆ß ä①Ú ✕✉Ú◆Û②ÖsÝ①Ý➈ä✄✸➍ÖsÚ◆Û▲✟✞✓❅ß❾Û➔Ösã❝ß ✲ ❥✝ ❥ ✝ ✝÷rò❢è✉ö✫÷➁ô❑ú ❤ ✝ ★ ✝ ❚ ÷➍ú➡è✉ç❲✲ ✥ú◆é ✝ ✲✠￾❍î➪ê◆ü✐è✉ö ★ës÷➍ú◆è✞✡➲ó①ë❵❺✌ê②✑❉î➪ó➪é②ç✠✟ è✉ö②é✏✲ ❙ï➍é➃õ②ï❢ô❲✲❴☞★ ✲✟✂✯✠★ ✲✲✍ ❄ ❄☛✻ ✲➍ù❃ù ✫ ❄✂❄ ✫ ✝ þ✫✁ ✯ ✝ ✓■ç❉ó➪ê❲✲ ✯ ✝ ✓■ç❑÷rô❭ú❂✲✦❻ ✝ ☞➐ï❵ï❢ö②è☛✲ ★ ✝ ✓■ç❑÷➁ê❑ú◆ç❑÷➁ö②ñ✠✲ ✙❚s❃ê❉ð✉ð✉ñ ❤ ï➍ò➍ó①ë í❇ï➍ö✺✓■ï❢ô◆õ②ö②ï❢î❑ï➁í ❥ï➍î➪î❑÷rô❭ú ☞❚ï✬✡❳è✉ô◆õ í❇ï➍ö✇÷④s❬î➪è❾ø❉ó✄✑❉î➪è ✇ó➪ô❉ò ★ó↕ö❾ë❾ö➏÷❱í❇õ✭✢❅✲ ✕ìá✹á✹á✒Ø✐ÙrÚ◆Û❤Ü➏Ù➁Ý ✟✞✓❅ß❾Û➔Ösã✿ß✙✝✪✢ ✸✔✢ ✖✉ä①Ú❑Ö ✲➹ù❉ù ✍ ✻✂✍ ❃ ✲✤❆✹ï❢î✞✍☛✍✬✲ ✕ï ✍✦✲ ☎ ê❉ô❉è ✍✔❄ ❄❣✍ ✝ þ✳✁ ★ ✝ ❤ ✝ ✯ ë②ç◆ü➹÷rö➃õ②ð☛✲✟✙❃✓■ï✬✡✘✡❋è✉ô◆õ➃é➬ï❢ô s❃ê❉ð✉ð✉ñ❁❤qï❢ò➍ó①ë ✓■ï➍ô◆õ②ö➃ï➍î✿ï➁í ❥ï➍î➪î➬÷➁ô❑ú▲☞❚ï☛✡➲è✉ô◆õ❋í❇ï❢ö✙÷ s❬î➪è❾ø❉ó✄✑❉î➪è ✇ó➪ô❉ò ★ó↕ö❾ë❾ö➏÷❱í❇õ✭✢❅✲ ✕❾á✹á✹á Ø✐ÙrÚ➡Û➶Ü❾ÙrÝ❛✟✞✓❅ß❾Û➔Ösã❝ß✞✝③✢✏✸✏✶ ✢✣✖✉ä①Ú❑Ö ✲❉ù❉ù✗✫❣✍✣✟ ✫☛✻②✲❜❆✹ï❢î✤✍❇✻②✲ ✕ï✔✍☛✲②s❃è✞✑❉ö➃ê❑÷rö➃ñ✁✍✔❄ ❄☛✻ ✝ þ❃✁ ✯ ✝ ✓■ç❉ó➪ê❲✲✮✙★ê❃õ②ç❉ï❢ö❇❅ é✪❥è✉ù❉î➪ñ✢✦✲ ✕ìá✹á✹á✣Ø✐Ù➁Ú◆Û➶Ü❾ÙrÝ ✟✞✓❅ß✣✶ Û➔Ösã✿ß ✝✪✢ ✸✔✢ ✖✉ä①Ú❑Ö ✲rù❉ù ✫❜✻✂✫❜❂②✲ ❆✹ï➍î✆✍✾✻②✲ ✕ï✞✍✬✲ s❃è✞✑❉ö➃ê✠✟ ÷rö②ñ ✍✔❄ ❄☛✻ ✝ þ❄✁ ✥❵✝ ✓■ï❅ø❶✲ ✙★ú❉÷rù❃õ②ó✄✎➍è❼s❃ê❉ð✉ðsñ ✯ ñ◆é②õ➃è✞✡❳é✱✢✦✲ ✕❾á✹á✹á ✟☛✡➡Ö✣❥ìÛ➶Ü✮✈❵ã✲✌ù❉ù❁✻ ✳✂❂✆✍✬✲■❆✹ï➍î✦❂➍ÿ ✲ ✕ï✁✻②✲✖s❃è✞✑❉ö➃ê❑÷rö➃ñ ✍✔❄ ❄❜❂ ✝ þ ✍❫ÿ ✁ ✓ ✝✡☎✦✝✏✰è✉ö②ò❢è✉õ✜✲ ✥ú ✝ ✲ ✙✽❥➹è❫÷❢ú◆è✉ö❇❅ é✘s❃ï➍ö➃ê②✡✳✢❅✲ ✕❾á✹á✹á Ø✐ÙrÚ◆Û❤Ü➏Ù➁Ý☛✟✮✓❅ß➏Û②Ösã✿ß✦✝③✢✏✸☛✢✣✖✉ä①Ú❭Ö ✲rù❃ù ✶✂✳②✲✾❆✹ï❢î❴✍✾❂②✲ ✕ï ❂②✲ ☎ ê❉ô❉è✏✍✔❄ ❄☛❂ ✝ þ ✍☛✍ ✁ ☞ ✝ ★õ②ç❭÷rô❉é✏✲✫✙❃✓■ï❢ô❵õ➃ö②ï❢î☛✟qæ➹ç❉è ★ú✠✎❢è✉ô◆õ②ê❉ö➃è✩✓■ï➍ô◆õ②ó➪ô✠✟ ê❉è✉é✭✢❅✲ ❙ï➡ú➡è✺❤qè❫ëìõ②ê❉ö➃è☛✲✟☛✲❢Ú✦✥✦✕ìá✹á✹á❴Ø✐ÙrÚ✞✹❫ÖsÜ➏ÖsÚ✦❥❾Ö➹ÙrÚ ✚❞Ö✣❥✉ä ßìä▲ÙrÚ✘✢rÚ❅✥④Ø✐ÙrÚ◆Û❤Ü➏Ù➁Ý✲ ✯ ÷➁ô ★ô◆õ②ï❢ô❉ó➪ï✦✲❅æ✌☞❦✲✍✂ ✯✠★ ✲ ❻➳è❅ë❾è✞✡✤✑❭è✉ö ✍ ✶✂ ✍ ✳②✲❵✍✔❄ ❄☛❂ ✝ þ ✍❇✻ ✁ ✥❵✝ ✰t✝ ☞✙÷✔✡➮ú❉÷➁ô❉ó✗✲☎✙②æ➹ü■è✉ô◆õ❤ñ✎✝✐è❫÷➁ö②é❙ïrí s❉ê❉ðsð✉ñ ✓■ï➍ô◆õ②ö➃ï➍î✵✑ ✥ø❉ù❑è✉ö➃ó➪è✉ô❑ë❾è✉é ❯❝÷ró➪ô❉è❫ú✏÷➁ô❑ú ❤qè✉é➃é②ï❢ô❉é ❤qè❫÷➁ö②ô◆õ✭✢❅✲ ✐✐Ü❾Ù✜❥sÔ❳Ù✗✹✌✲❢Ú✦✥ ✕ìá✹á✹á ✕✉Ú◆Û❤Ý➈Ô❚Ø✐Ù➁Ú✞✹✉Ô❳ÙrÚ ♥✈✎✖✗✖✘✓✏✟✞✓❅ß❾Û②Ö✉ã❝ß ✲◆ù❉ù ✝ ❂❜❂ ❄✂❂✯✍✍ ✝ þ ✍❇❂ ✁ ☞ ✝ æ➘ï✬✡❳ó➪ð✉ê②❀➍÷ ✲ ✙❚s❃ê❉ð✉ð✉ñ ✓■ï❢ô◆õ②ö②ï❢î☛ó➪ô ✓■ï➍ô◆õ②ö➃ï➍î ✥ô❉ò➍ó➪ô❉èsè✉ö❇❅ é❳æ✌ï◆ï➍î ❙ï❅ø✢❅✲✺❤ è❫ë❾õ②ê❃ö②è☛✲ Ó❯✉✱✘✆✏❉Õ➹Ó✱✟❵Ó ✴➮ÙrÜ✒✑❫ß✣✺➡Ù❉✡✫ÙrÚ ♥Ù➁Ü✉ã ✢➁Ý✱✝✍Ù✜✥❢Ö✉Ý➤ß✘✹❫ÙrÜ✔✕sÚ◆Û②Ö✉Ý①Ý➟ä✄✸➍Ö✉Ú◆Û Ø✐ÙrÚ◆Û❤Ü➏Ù➁Ý✲✤☞④✼❤æ☞✲✤✓➹÷✔✡✤✑❉ö②ó①ú◆ò❢è☛✲✪☞★ ÿ❜✻✆✍❇❂ ❄②✲✡✂ ✯✠★ ✲ ✯ è✉ù❉õ➃è✞✡✤✑❑èsö✛❂➍ÿ ✂ ✾ë❾õ②ï✬✑❑èsö✛✻②✲■✍ ❄ ❄❜❂ ✝ þ ✍✺✍✁ ✓ ✝ ✟✇✝ æ✹÷rï❅✲✤❥✝ ☞✙÷✔✡❳î➪ï◆ï☛❀❂✲ ✇✝✤✥❵✝ æ➹ç❃ï☛✡❳ù❃é②ï➍ô①✲ ✙❃❥➹è❅ú◆ê❑ë❾õ➃ó↕ï❢ô❋ïrí✲✓■ï☛✡➲ù❉î➪è❾ø❉ó↕õ❤ñ❍í❇ï❢ö✐÷✱❥➹ï✬✑❉ê❉é②õ✽s❉ê❉ðsð✉ñ ✓■ï➍ô◆õ②ö➃ï➍î➪î➪è✉ö✭✢❅✲ ✐✐Ü❾Ù✜❥sÔ➘Ùr✹ ✲❢Ú✦✥✞✕ìá✹á✹á ✕✉Ú◆Û❤Ý➈Ô➋Ø✐ÙrÚ✞✹sÔ➘ÙrÚ ♥✈✎✖✗✖✘✓✏✟✞✓❅ß❾Û②Ö✉ã❝ß ✲◆ù❉ù ✝ ✍✾❂✯✍✼✫✂ ✍✾❂✯✍✼❄ ✝ þ ✍ ✶ ✁ ✰☞✝ ✝➬ó➪ô❉ò❅✲ ✇✝ ✯ ó➪î↕èsö✜✲ ☎✦✝✓☎❅✝✆❙ê❭ë❈❀◆î➪è✉ñ✠✲ ✙❚s❃ê❉ð✉ð✉ñ❱✓■ï➍ô✠✟ õ②ö②ï❢î❨æ➹ç❉èsï➍ö②ñ■✑ ★ ✕ï➍ô❃î↕ó➪ô❉è❅÷rö✦✓➹÷ré②è ✢✦✲ Ó❯✈❢Û②Ùrã✤✢❅Û❤ä✆❥✣✢ ✲ ❆✹ï➍î ✝ ✻ ✫ ✲ ✕ï ✝ ❂②✲◆ù❉ù ✶ ✍✾❂✂✍ ✻➍ÿ ✲❴✍ ❄ ❄❢ÿ ✝ þ ✍✔✫ ✁✔☎✦✝ ✟ ☎❅✝❴✥❵✝ ✯ î➪ï➍õ➃ó↕ô❃è➬÷rô❭ú ✇✝ ❤qó✗✲ Ó✦✡❜✡❑Ý➈ä▲Ö❚✥✈Õ❞Ù➁Ú❉Ý➈ä①Ú❭Ö❚✢➁Ü Ø✐ÙrÚ◆Û❤Ü➏Ù➁Ý✝ ❺✌ö➃è✉ô◆õ②ó①ë❾è ✰ ÷rî➪î✗✲ ✻➁ô❑ú➲è❫ú ✝ ✲■✍✔❄ ❄✆✍ ✝ ● ⑤➬❰ ❐q❒q❮❰❶⑩☛⑩✬⑦❉❮ ￾✒⑦❉❮❂⑧◆❰q❮■☎✩❍q❐❨Ñ❅⑦❁❑✿⑦✦❸✉Ð❂⑩s❒❲❸ 0 5 10 Closed loop Step Response of System 1 1 0.8 0.6 0.4 0.2 0 1.2 Time 15 20 25 30 Sliding Mode Controller PID Controller Fuzzy Logic Controller s❬ó➪ò➍ê❉ö➃è✁❂✆✑✔✓■î➪ï➍é➃è❫ú❁❤qï◆ï➍ù ✯ õ②èsù✹❥➹è✉é➃ù❑ï❢ô❉é②è➙ïrí➬ô❃ï☛✡❳ó➪ô❑÷➁î ù❉î①÷rô◆õ 0 5 10 Closed loop Step Response of System 2 1 0.8 0.6 0.4 0.2 0 1.2 Time 15 20 25 30 Sliding Mode Controller PID Controller Fuzzy Logic Controller s❬ó➪ò➍ê❉ö➃è ✍❵✑✐✓■î➪ï❢é②è❫ú❁❤qï◆ï❢ù ✯ õ➃è✉ù ❥➹è✉é➃ù❑ï❢ô❉é②è✙ï➁í➬õ➃ç❉è❚ù❑è✉ör✟ õ②ê❃ö❈✑❭è❫ú✾✡➲ï❉ú◆è✉î ✳✿ ý✷✴ ☎