a Comparison of robustness Fuzzy logic, PID, Sliding Mode Control Charles p coleman and datta godbole Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, CA 94720, USA abstract-Th erformed in order to D. A Call for Peace or at Least a Cease Fire! and foster and promote Negotiations son of fuzzy logic control and classical cont Fortunately ome nave called for control engineers u ratiisits fof ty ow cy pise. coat rsldi given de to) n- control enginering toolboxes, and for the utilization trollers are designed to control the speed of a nominal of both fuzzy logic control and classical control when The step response performance of each controller, ap- and where appropriate ([O[OD. This appears to be plied to the nominal and two perturbed motor plants reasonable request. At the very least, the fuzzy logic tool can be compared to classical control tools e conclude zy logic contro can be a use too Should the fuzzy logic control tool prove useful and control techniques for the benefit of the practicical acceptable to the control engineer, it should become ng welcome addition to her control toolbox L INTRODUCTION The control engineer is concerned about such prop A. New control Tools, New enemies erties as the robustness of a controller to plant per turbation and uncertainty. We take the perspective The successful use of fuzzy logic controllers h as greatly of the control engineer who has several control tools expanded in the last twenty years([0][O). This e available for the synthesis of robust cont rollers, and pansion has prompted much compar ison to classic ntrol techniques. The ensuing discussions have who is interested in determining the capabilities of each control tool to synthesize a robust controller not always been amicable. Nor have they necessar- To assess the usefulness of each tool in designing ily lead to manifest results and conclusions for the practicing control engineer a robust controller, we choose a benchmark prob lem, and then engage in a comp arison of the control B. Debate and dilemma designs which result from the application of each At times the debate over the use of fuzzy logic con- control design method trol techniques versus classical control techniques has become quite heated with ardent detractors on Fuzzy logic control, PID control, and slidin both sides([O-[o) control tools are chosen for our investig ation. Fuzzy Some control engineers have been placed in a logic control is chosen because of its empirically demon lemma by this debate, and have been left wit out an appeasing study or comparison upon which strated robustness properties shown in[0]. PID con trol is chosen because it is com monly trolle it C. Well-Being of the Control Engineer has plant perturbation robustness properties which In the process of this raging and seemingly grow- can be mathematically analyzed. We choose a slid- ing battle, it appears that the best interest of the g mode controller because it is a robust non-linear control engineer has been occluded. The practicing control method whose robustness properties can also control engineer usually has little interest in ideolog be mathematically analyzed I deb ates. It is her interest to produce a working ntroller. with eff aking use of any and all robust control technique. Indeed, the popular and techniques at her disposa successful Hoo and adapti ive cont rol techniques noticeably absent from this study. We felt it was DAAL-91-G-091 and an AASERT sup and two other control methods, in order to keep our
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➛❾➌②➊r➧➟➧➈➒r➌➃➇❨➜➍➌➃➒✌➣❉➒❫➇➃➆➈➨➍↔◆➒❅➣❥➛②➊■➠✉➊➍↔❢➛➏➌➃➊r➧➍➛❾➅◆➒✌➇➏➑❃➒❫➒❅➣➳➊❅➓❉➜➳↔◆➊➍→④➆➪↔◆➜r➧ ➛❾➅◆➆↕➌➃➣✼➊➍➌②➣❃➒❱➌❥➧➟➆↕↔◆➒❅➜➍➌❥➛➃➆↕→④➒s➢➶➆➪↔➍➭❅➜➍➌➃➆➈➜➍↔❢➛➳→④➊➍➣❃➒❫➧✌➊❅➓➹➜➐→④➊r➛➃➊➍➌s➩ ➄✐➅➡➒✹➇②➛➃➒❱➑❍➌②➒❅➇➏➑❃➊➍↔◆➇➃➒❁➑❃➒r➌❤➓①➊➍➌❾→④➜➍↔◆➠✉➒❬➊❅➓❉➒❫➜r➠✉➅➹➠s➊➍↔➍➛❾➌➃➊r➧➈➧➈➒r➌s➼r➜➍➑❢➢ ➑◆➧➟➆➈➒❅➣❥➛②➊➹➛➏➅➡➒✐↔◆➊➍→④➆↕↔➡➜r➧❫➜➍↔◆➣❥➛❤➉✹➊➹➑❃➒❱➌➃➛➏➝❃➌➏➞❃➒❅➣➳→④➊r➛②➊➍➌❨➑➡➧➈➜➍↔❢➛②➇s➼ ➆➟➇➳➑❉➌➃➒❫➇➃➒❱↔❢➛②➒❅➣q➩ ➫✼➒➘➠s➊➍↔◆➠✉➧↕➝➡➣❉➒✌➓❇➝◆➥✉➥✉➦❍➧➈➊r➨r➆➟➠✐➠✉➊➍↔❢➛❾➌②➊r➧❢➠✉➜➍↔➳➞❃➒✹➜➳➝◆➇➃➒✉➓❇➝◆➧r➛➃➊➍➊r➧ ➓①➊➍➌➹➛❾➅◆➒➳➠s➊➍↔➍➛❾➌➃➊r➧q➒❱↔◆➨r➆➪↔◆➒❅➒❱➌s➩q➫✍➒❥➒r↔◆➠✉➊➍➝❉➌➃➜r➨r➒❥→④➊➍➌②➒➳➞❃➒r↔◆➠✉➅❢➢ →④➜➍➌❾➎■➠✉➊➍→❋➑◆➜➍➌➃➆➟➇②➊➍↔◆➇q➊❅➓❉➓❇➝◆➥s➥✉➦➳➧➟➊r➨r➆➈➠➹➠✉➊➍↔❢➛➏➌➃➊r➧❢➉✐➆➟➛➏➅❥➠✉➧➟➜r➇②➇➃➆➈➠✉➜r➧ ➠✉➊➍↔❢➛❾➌②➊r➧■➛②➒❅➠✉➅❉↔◆➆➟➴❵➝◆➒❅➇❥➓①➊➍➌④➛➏➅◆➒❚➞❉➒r↔◆➒s➷❑➛❥➊❅➓➬➛❾➅◆➒➮➑❉➌➃➜r➠✉➛➃➆➈➠✉➆➪↔◆➨ ➠✉➊➍↔❢➛❾➌②➊r➧q➒r↔◆➨r➆➪↔◆➒❫➒r➌s➩ ➱❫✃✈➱➶❐q❒q❮q❰qÏ❬Ð❨Ñ❭❒qÒ❇❰q❐ Ó❝Ô✿Õ✿Ö✉×❙Ø✐ÙrÚ➡Û➶Ü❾ÙrÝ✐ÞqÙ❅ÙrÝ➤ß➏à✌Õ✿Ö✉×âá✐Ú❑Ösã④ä❇Ö❾ß◆å æ➹ç❉è➋é②ê❑ë✉ëìè✉é②é➔í❇ê❉î❉ê❉é➃è✹ïrí◆í❇ê❃ð✉ð✉ñ❥î➪ï➍ò➍ó①ë❬ë❾ï➍ô◆õ②ö➃ï➍î➪î➪è✉ö②éqç❭÷ré❨ò❢ö②è❫÷➁õ②î➪ñ è❾ø❉ù❑÷➁ô❑ú◆è❫úûó➪ô❋õ②ç❉è➹î①÷ré➃õ✹õ❤ü■è✉ô◆õ❤ñ✇ñ❢è❫÷rö➃é➹ý➏þÿ✁✄✂ þÿ✁✆☎✞✝ æ➹ç❉ó➪é✹è❾ø✠✟ ù❑÷➁ô❉é②ó➪ï❢ô❋ç❑÷ré✹ù❃ö②ï☛✡➲ù❉õ②è❅ú☞✡❋ê❑ë➃ç❋ë❾ï☛✡➲ù❑÷rö➃ó➪é②ï➍ô④õ②ï④ë❾î①÷ré②é➃ó①ë✉÷rî ë❾ï❢ô◆õ②ö②ï❢î❥õ②è❫ë➃ç❉ô❉ó✍✌❢ê❉è✉é ✝ æ➹ç❉è✼è✉ô❉é②ê❃ó↕ô❃ò❖ú➡ó↕é❾ë❾ê❉é➃é②ó➪ï➍ô❉é❚ç❑÷✏✎❢è ô❉ï❢õ➬÷➁î↕ü➹÷❅ñ◆é✒✑❭è✉è✉ô✼÷✓✡❳ó①ë✉÷✔✑❉î➪è ✝✖✕ï❢ö➹ç❑÷✏✎➍è❝õ②ç❉è✉ñ➲ô❉è❫ë❾èsé②é➏÷➁ö✗✟ ó➪î➪ñ❚î↕è❅÷➍ú➙õ②ï✘✡❚÷rô❉ó➟í❇èsé②õ➳ö➃è✉é②ê❃î↕õ➃é✇÷rô❭ú✙ë❾ï➍ô❭ë❾î➪ê❉é②ó➪ï➍ô❃é➳í▲ï➍ö❥õ②ç❉è ù❉ö❾÷➍ë❾õ➃ó①ë❾ó➪ô❉ò❳ë❾ï➍ô◆õ➃ö②ï➍î❬è✉ô❉ò❢ó↕ô❃è✉è✉ö ✝ ✙Ô☞✚✿Ö✜✛✣✢❱Û➔Ö✤✢➁Ú✦✥✧✚❥ä①Ý↕Ösã✇ã✤✢ ★õ➹õ➃ó✩✡➲è✉é✹õ➃ç❉è✿ú◆è✞✑❑÷➁õ②è❥ï✓✎➍è✉ö■õ②ç❃è➬ê❃é②è✿ï➁íqí❇ê❉ð✉ð✉ñ➲î➪ï➍ò➍ó①ë➹ë❾ï❢ô✠✟ õ➃ö②ï➍î❝õ②è❫ë➃ç❉ô❉ó✍✌❢ê❉è✉é✪✎➍è✉ö➃é②ê❉é✚ë❾î①÷ré➃é②ó①ë✉÷rî✿ë❾ï❢ô◆õ②ö②ï❢î✿õ②è❅ë②ç❃ô❉ó✍✌➍ê❃è✉é ç❑÷➁é✫✑❭è❫ë❾ï✬✡❳è☞✌❢ê❉ó➪õ②è✿ç❉è❅÷rõ②è❅ú❚ü➹ó➪õ②ç✍÷➁ö➏ú◆è✉ô◆õ❥ú◆è✉õ②ö❾÷➍ë❾õ➃ï➍ö➃é➬ï❢ô ✑❭ï➍õ➃ç➮é➃ó①ú◆è✉é✿ý❾þÿ✁✩✂ þÿ✁✭☎✮✝ ✯ ï✬✡❳è☛ë❾ï❢ô❵õ➃ö②ï❢î❋è✉ô❉ò➍ó➪ô❉èsè✉ö②éâç❑÷✏✎➍è✰✑❑è✉èsô✮ù❉î①÷➍ë❾è❅ú ó➪ô ÷ ú◆ó➪î➪è✞✡✱✡➮÷✤✑◆ñ✚õ➃ç❉ó➪é❳ú◆è✞✑❑÷➁õ②è☛✲➹÷rô❭ú✚ç❑÷✏✎❢è✘✑❭è✉è✉ô✣î➪è❾í❇õ④ü➹ó➪õ②ç✠✟ ï❢ê❉õ✿÷rô✍÷➁ù❉ù❑è❅÷ré②ó➪ô❉ò❚é➃õ②ê❑ú◆ñ➙ï➍ö✿ë❾ï☛✡➲ù❑÷rö➃ó➪é②ï➍ô❚ê❉ù❭ï➍ô✼ü➹ç❉ó①ë➃ç õ➃ï④í❇ï➍ö✳✡❋ê❉î①÷rõ②è➳÷④ê❃é②è❾í❇ê❃î❨ï➍ù❃ó↕ô❃ó↕ï❢ô ✝ Ø✹Ô✵✴➮ÖsÝ①Ý✷✶ ✙ Ösä①Ú✠✸❳Ù✗✹❥Û✻✺➡ÖûØ✐ÙrÚ◆Û❤Ü➏Ù➁Ý❭á✐Ú✠✸rä①Ú❑Ö❾ÖsÜ ✼❤ô❴õ②ç❃è➮ù❃ö②ï❉ë❾è✉é➃é❳ïrí❥õ②ç❉ó➪é④ö➏÷➁ò➍ó➪ô❉ò✚÷rô❑ú é②è✉è✞✡➲ó➪ô❉ò➍î➪ñ✼ò➍ö②ïrü✽✟ ó➪ô❉ò✾✑❭÷rõ②õ➃î➪è☛✲❨ó➪õ✇÷➁ù❉ù❭è❫÷rö➃é✇õ➃ç❑÷rõ④õ②ç❉è✤✑❭è✉é➃õ❋ó↕ô◆õ➃è✉ö②è✉é➃õ✇ï➁í➳õ②ç❉è ë❾ï❢ô◆õ②ö②ï❢î❨è✉ô❃ò➍ó➪ô❉è✉è✉ö❍ç❑÷ré✿✑❑èsè✉ô✙ï❉ë✉ë❾î➪ê❑ú◆è❅ú ✝ æ➹ç❉è❥ù❉ö➏÷❢ë❾õ②ó①ë❾ó➪ô❉ò ë❾ï❢ô◆õ②ö②ï❢î◆è✉ô❉ò➍ó➪ô❉èsè✉ö✌ê❉é②ê❑÷➁î➪î↕ñ❥ç❑÷➁é❨î➪ó➪õ②õ➃î↕è➋ó↕ô◆õ➃è✉ö②è✉é➃õ✌ó↕ô❝ó ú➡è✉ï➍î➪ï➍ò✔✟ ó①ë✉÷➁î❨ú◆è✞✑❑÷➁õ②è✉é ✝ ✼❤õ■ó↕é■ç❉è✉ö➹ó➪ô◆õ②èsö②è✉é➃õ➳õ②ï④ù❉ö➃ï➡ú➡ê❑ë❾è✿÷✿ü■ï➍ö✳❀◆ó↕ô❃ò ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö✜✲qü➹ó➪õ②ç✼è✣❁➮ë✉÷❢ë❾ñ✠✲❂✡➮÷✔❀❵ó➪ô❉ò④ê❉é➃è❋ïrí ÷➁ô❵ñ✙÷➁ô❑ú✍÷➁î➪î õ➃è❫ë➃ç❉ô❉ó✍✌❢ê❉è✉é➬÷➁õ➹ç❉è✉ö❥ú◆ó➪é②ù❭ï➍é➏÷➁î ✝ ❃❅❄✗❆✍❄❈❇✳❉✍❊✻❋✾❆✍●✞❍✮❍✜■✣❉✍❏✍❄✗❑✵▲ ▼✾❍✜❇✳❉✍❏☞◆✣❖✾P❂❃✦◗❘●✮▼✞❑✞❄✗❉✧❙❚❉✻❇❈▼✣❏❯▼✣●✞❱❲◆✏❄✗❉ ❳❅P❂P❂❨✓❩✍❬✞❭✻❩✍❪❅❩❴❫✣❬✮❭✦❇✳▼✞❑❵❇❈▼❵P❂P❲❛❝❜❞❃✬❡✤❆✍●✞❍✞❍✮❢ ❄✗❱❣❄✗▼✣❏✗❤ ✚➬Ô✿Ó✒Ø✒✢➁Ý①Ý☛✹❅ÙrÜ✿✐Ö✣✢☛❥❾Ö④ÙrÜ❦✢❱Û✒❧❨Ö✣✢❅ß❾Û♠✢❚Ø✐Ö✣✢❱ßsÖ♦♥✹ä①Ü➏Ö✮♣q✢➁Ú✦✥ Õ✿Ör✸➁Ù❱Û➶ä✆✢❱Û➶ä▲ÙrÚ◆ß s❃ï➍ö②õ➃ê❉ô❑÷➁õ②è✉î➪ñ✠✲❑é➃ï☛✡➲è✿ç❑÷✏✎➍è④ë✉÷rî➪î➪è❫ú➐í❇ï➍ö❥ë❾ï➍ô◆õ②ö➃ï➍î✌è✉ô❉ò❢ó➪ô❉è✉è✉ö➃é õ②ï④ë❾ï➍ô❉é➃ó①ú◆è✉ö■ó↕ô❭ë❾î➪ê❉é②ó➪ï➍ô④ïrí❭í❇ê❉ðsð✉ñ❋î➪ï➍ò➍ó①ë■ë❾ï➍ô◆õ➃ö②ï➍î❃ó↕ô◆õ➃ï➬õ➃ç❉è✉ó➪ö ë❾ï❢ô❵õ➃ö②ï❢îqè✉ô❉ò➍ó➪ô❉èsö②ó➪ô❉ò④õ②ï◆ï➍î✄✑❭ï❅ø➡èsé✜✲◆÷rô❑ú✈í❇ï➍ö■õ②ç❉è❍ê❉õ②ó➪î➪ó➪ð❫÷rõ➃ó➪ï➍ô ïrí❅✑❑ï❢õ②ç❍í❇ê❉ð✉ðsñ✿î➪ï➍ò➍ó①ë❬ë❾ï➍ô◆õ②ö➃ï➍î❉÷rô❭ú➬ëìî ÷➁é②é➃ó ës÷rî❉ë❾ï➍ô◆õ➃ö②ï➍î❢ü➹ç❉è✉ô ÷rô❭ú❳ü➹ç❃è✉ö②è✇÷➁ù❉ù❉ö➃ï➍ù❉ö➃ó ÷➁õ②è④ý➏þÿ ✁ þÿ ✁✆☎✞✝ æ➹ç❉ó➪é➹÷rù❉ù❭è❫÷rö➃é➳õ➃ï✤✑❑è ÷➙ö②è❫÷➁é②ï➍ô❭÷✓✑❉î➪è❋ö②è✏✌➍ê❃è✉é②õ ✝ ★õ④õ②ç❃è✘✎➍èsö②ñ✼î↕è❅÷ré②õ✏✲✌õ②ç❉èûí❇ê❉ð✉ðsñ î➪ï➍ò❢ó ë➋õ②ï◆ï❢î❑ë✉÷rô✧✑❭è➳ë❾ï☛✡➲ù❑÷rö➃è❫ú✿õ➃ï✿ë❾î①÷ré②é➃ó①ë✉÷rî❑ëìï➍ô◆õ②ö➃ï➍î❉õ➃ï❵ï❢î➪é ✝ ✯ ç❉ï❢ê❉î①ú❋õ➃ç❉è❍í❇ê❉ð✉ð✉ñ➲î➪ï➍ò➍ó①ë➹ë❾ï❢ô◆õ②ö②ï❢îqõ②ï◆ï➍î❭ù❉ö➃ï✓✎➍è❥ê❉é②èìí❇ê❉î❨÷rô❭ú ÷➍ësë❾è✉ù❉õ❾÷✓✑❉î➪è➳õ➃ï❥õ②ç❉è➳ë❾ï❢ô◆õ②ö②ï❢î❉è✉ô❉ò❢ó➪ô❉è✉è✉ö✏✲◆ó↕õ✌é➃ç❉ï➍ê❉î①út✑❑è❅ë❾ï☛✡➲è ü■è✉î①ë❾ï☛✡➲è➬÷➍ú❃ú◆ó➪õ②ó➪ï➍ô➲õ②ï❋ç❉è✉ö➳ëìï➍ô◆õ②ö➃ï➍î❨õ➃ï◆ï➍î✄✑❑ï❅ø ✝ á✐Ô✧✉Ù✬✛✞✈❢ß❾Û❥Ø✐ÙrÚ◆Û❤Ü➏Ù➁Ý①Ý↕ÖsÜ❾ß æ➹ç❉è✿ë❾ï❢ô❵õ➃ö②ï❢î❨è✉ô❉ò❢ó➪ô❉è✉è✉ö➹ó➪é➳ë❾ï❢ô❑ë❾è✉ö➃ô❉è❫ú✙÷✔✑❑ï❢ê❉õ✐é➃ê❑ë➃ç✙ù❉ö➃ï➍ù✠✟ è✉ö➃õ②ó➪è✉é➬÷➁é✐õ➃ç❉è❥ö②ï✬✑❉ê❉é②õ➃ô❉è✉é➃é➬ï➁í❁÷④ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö■õ②ï④ù❉î①÷rô◆õ■ù❑è✉ör✟ õ②ê❃ö❈✑❑÷➁õ②ó➪ï➍ô❉é✹÷➁ô❑ú❥ê❉ô❑ëìè✉ö②õ❾÷ró➪ô◆õ❤ñ✝①✇è✌õ➏÷✓❀❢è✐õ➃ç❉è✹ù❭è✉ö②é➃ù❑è❅ë❾õ②ó✄✎➍è ïrí❨õ➃ç❉è➳ë❾ï❢ô❵õ➃ö②ï❢î❑è✉ô❃ò➍ó➪ô❉è✉è✉ö■ü➹ç❉ï✿ç❭÷ré✹é➃è✞✎➍è✉ö❾÷rîqë❾ï➍ô◆õ➃ö②ï➍î❭õ②ï◆ï❢î↕é ÷✏✎➍÷ró➪î①÷✓✑❃î↕è➋í❇ï❢ö✹õ②ç❉è❥é➃ñ❵ô◆õ➃ç❉è✉é②ó➪é➹ï➁í❨ö②ï☛✑❃ê❉é②õ➹ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö➃é✜✲❉÷rô❭ú ü➹ç❉ï❚ó➪é➳ó➪ô◆õ②è✉ö➃è✉é②õ➃è❫ú✼ó➪ô✍ú◆è✉õ➃è✉ö❈✡➲ó➪ô❉ó➪ô❉ò❋õ②ç❉è❋ë✉÷➁ù❑÷✓✑❉ó➪î➪ó➪õ②ó➪è✉é➹ï➁í è❫÷❢ë②ç✼ë❾ï➍ô◆õ②ö➃ï➍î❬õ②ï◆ï➍îqõ➃ï✇é➃ñ❵ô◆õ➃ç❉è✉é②ó➪ð✉è❋÷④ö②ï✬✑❉ê❉é➃õ➳ë❾ï➍ô◆õ②ö➃ï➍î➪î➪è✉ö ✝ æ✌ï❳÷ré②é➃è✉é②é❥õ➃ç❉è❥ê❉é②è❾í▲ê❉î➪ô❉è✉é②é❥ï➁í❁è❅÷➍ë➃ç✙õ②ï◆ï❢îqó↕ô✙ú◆èsé②ó➪ò➍ô❉ó➪ô❉ò ÷➙ö②ï☛✑❃ê❉é②õ❳ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö✜✲✹ü■è✙ë➃ç❉ï◆ï➍é②è✙÷✪✑❑èsô❑ë➃ç②✡➮÷➁ö❈❀✚ù❉ö➃ï☛✑✠✟ î➪è✞✡③✲❢÷rô❑ú④õ➃ç❉è✉ô❋è✉ô❉ò◆÷rò❢è➹ó↕ô❳÷❝ë❾ï☛✡➲ù❑÷rö➃ó↕é➃ï➍ô✿ï➁í õ➃ç❉è➬ëìï➍ô◆õ②ö➃ï➍î ú◆è✉é➃ó➪ò➍ô❉é❚ü➹ç❉ó①ë➃ç✣ö②èsé②ê❉î➪õ➲í❇ö②ï✬✡ õ➃ç❉è✍÷➁ù❉ù❉î➪ó①ë✉÷rõ➃ó↕ï❢ô❴ïrí✿è❫÷➍ë➃ç ë❾ï❢ô❵õ➃ö②ï❢î➘ú◆è✉é➃ó➪ò➍ô④✡❳è✉õ➃ç❉ï❉ú ✝ ➱✉➱❫✃✧⑤✫⑥❨❰ Ò❇Ñ❅⑦✚❰❂⑧⑨⑤➬❰ ❐q❒q❮❰❶⑩✵❷➬❰❨❰❶⑩❞❸ s❃ê❉ð✉ð✉ñ✼î➪ï➍ò➍ó①ë❥ë❾ï❢ô❵õ➃ö②ï❢î❹✲①❺❣✼❈❻ ë❾ï➍ô◆õ②ö➃ï➍î✗✲❨÷➁ô❑ú❚é②î➪ó①ú◆ó➪ô❉ò✱✡❳ï❉ú◆è ë❾ï❢ô❵õ➃ö②ï❢î❑õ➃ï❵ï❢î➪é❁÷➁ö②è➳ë➃ç❉ï❢é②è✉ôûí❇ï➍ö✌ï➍ê❃ö✹ó➪ô❞✎❢è✉é②õ➃ó➪ò❵÷rõ➃ó➪ï➍ô ✝ s❃ê❉ð✉ðsñ î➪ï➍ò❢ó ë✌ë❾ï❢ô◆õ②ö②ï❢î➍ó➪é❨ë➃ç❉ï➍é➃è✉ô☞✑❭è❫ë✉÷➁ê❉é②è■ïrí❉ó➪õ②éqè✞✡➲ù❉ó➪ö②ó①ë✉÷➁î↕î➪ñ➹ú◆è✞✡➲ï➍ô✠✟ é②õ➃ö➏÷➁õ②è❫ú④ö➃ï☛✑❉ê❉é➃õ②ô❉èsé②é✹ù❉ö➃ï➍ù❭è✉ö②õ➃ó➪è✉é✹é②ç❃ïrü➹ô✇ó➪ô❳þÿ✁❈✝ ❺❣✼❈❻✣ëìï➍ô✠✟ õ②ö➃ï➍î❥ó➪é❳ë➃ç❉ï➍é➃è✉ô❼✑❭è❫ë✉÷➁ê❉é②è✼ó➪õ❋ó➪é❳ï➍ô❃è➮ï➁í✿õ②ç❉è✵✡❳ï❢é②õ❋ë❾ï☛✡❽✟ ✡❳ï❢ô❉î➪ñ✙ê❉é➃è❫ú☛ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö②é✿ó➪ô✚ó➪ô❑ú◆ê❉é➃õ②ö➃ñ❞✲✹÷➁ô❑ú✵✑❑è❅ë✉÷rê❉é➃è❚ó↕õ ç❑÷➁é➳ù❉î①÷rô◆õ➹ù❭è✉ö②õ➃ê❉ö❈✑❭÷rõ②ó➪ï❢ô✙ö②ï✬✑❉ê❉é②õ➃ô❉è✉é➃é➬ù❃ö②ï➍ù❭è✉ö➃õ②ó➪è✉é❥ü➹ç❉ó①ë➃ç ë✉÷➁ô✾✑❑èt✡➮÷rõ➃ç❉è✞✡❚÷rõ②ó①ë✉÷➁î➈î➪ñ✇÷➁ô❑÷rî➪ñ◆ð✉è❫ú ✝❾✇è✿ë②ç❃ï❵ï❢é②è✿÷④é②î➪ó①ú✬✟ ó➪ô❉ò❯✡❳ï❉ú◆è➹ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö❣✑❭è❫ë✉÷rê❃é②è❥ó➪õ➘ó➪é✹÷❍ö②ï✬✑❉ê❉é②õ➋ô❉ï➍ô✠✟❤î➪ó➪ô❉è❫÷rö ë❾ï❢ô❵õ➃ö②ï❢î❞✡➲è✉õ②ç❃ï➡ú ü➹ç❉ï➍é➃è✹ö②ï✬✑❉ê❉é➃õ②ô❉è✉é➃é✹ù❉ö➃ï➍ù❭è✉ö②õ➃ó↕èsé❁ës÷rô✿÷rî➪é②ï ✑❭è☞✡➮÷➁õ②ç❉è✞✡❚÷rõ➃ó ës÷rî➪î➈ñ✇÷➁ô❑÷rî➪ñ◆ð✉è❫ú ✝ ✇è➬ú◆ï❋ô❃ï➍õ➹ë❾î①÷ró✄✡ õ②ï❋ó➪ô✠✎➍è✉é➃õ②ó➪ò❵÷➁õ②è➬÷➁ô❑ú➲ù❉ö②èsé②è✉ô◆õ➳è✮✎➍è✉ö➃ñ ö②ï✬✑❉ê❉é➃õ✈ë❾ï❢ô◆õ②ö②ï❢î✹õ②è❅ë②ç❃ô❉ó✍✌➍ê❃è ✝ ✼❤ô❑ú◆èsè❫ú❂✲✌õ②ç❃è❋ù❑ï❢ù❉ê❉î①÷rö✿÷rô❭ú é②ê❭ë✉ë❾è✉é➃é❤í❇ê❉î✫❿✤➀ ÷rô❑ú➙÷➍ú❉÷➁ù❉õ②ó✄✎➍è❋ë❾ï❢ô❵õ➃ö②ï❢î➘õ➃è❫ë➃ç❉ô❉ó✍✌❢ê❉è✉é✇÷➁ö②è ô❉ï❢õ②ó①ë❾è❫÷✔✑❉î➪ñ☛÷✓✑❉é➃è✉ô◆õ④í❇ö②ï✬✡⑨õ②ç❉ó➪é④é②õ➃ê❑ú◆ñ✝⑨✇è❋í❇èsî↕õ④ó➪õ✇ü➹÷ré ô❉è❫ëìè✉é②é❾÷rö②ñ➙õ②ï④ö②è✉é➃õ②ö②ó①ë❾õ❍ï➍ê❉ö➃é②è✉î✄✎❢è✉é➹õ②ï❝í❇ê❉ð✉ðsñ❳î➪ï➍ò❢ó ë➹ëìï➍ô◆õ②ö➃ï➍î ÷rô❭ú④õ❤ü✐ï✿ï❢õ②ç❉èsö✐ë❾ï➍ô◆õ➃ö②ï➍î❅✡❳èsõ②ç❉ï❉ú◆é✏✲ró➪ô❋ï➍ö➏ú➡è✉ö✹õ②ït❀➍è✉è✉ù➲ï➍ê❉ö
study brief, intelligible, and loop unit step responses of the nominal plant and I CONTROL PROBLEM the two perturbed plants. To be acceptable, th closed loop step resp onses must simultaneously have A. Motor Speed contro short rise time an Motivated by the work in [o] and [ol, and encour- F. Design approach aged by the robustness results presented for fuzzy logic control, we consider the robust speed control We design the fuzzy, PID, and sliding mode con trollers based on the nominal plant given in equa- time-invariant transfer function. The general con- tion(1 ) Each designed controller is then applied figuration for the motor speed control problem is to all three transfer functions and numerical sim given in Figure 1 ulations are performed to analy ze the controller The design of the fuzzy logic controller is given Controller- in Section IV. The design of the PID controller is given in Section V. The design of the sliding mode controller is given in Section VI. Plots of each con- troller's closed e and control effor are shown in Appendix A Figure 1: Motor Speed Control Problem IV, ANALYSIS AND RECONSTRUCTION OF A Fuzzy RoI We use the nominal plant, perturbed motor plants, We implement the robust fuzzy controller given in and the performance criteria given in [] to design [o] and [0]. The inputs to the fuzzy controller are e and compare robust fuzzy, PID, and sliding mode and we. The output of the fuzzy controller controllers universes of discourse of e, wc, and u are partitioned B.Restrictions on Controller Inputs and Outputs Into seven As shown in Figure 1, the reference step input speed NB- negative big nd the out put motor speed wc are available fo ·NM- negative mediun parison. Each controller only has access to the Ns-negative small motor speed error e =Wr-wc and the motor angular ZE -zero acceleration wc as inputs. Each controller produces PS- positive small only one output u ●PM- positive mediun C. Nominal plant The nominal motor plant is modeled by the follow Each fuzzy set is represented by a Gaussian men ing transfer function bership function. The rule base of the fuzzy logi controller contains forty-nine rules which are tabu- lated in Figure 2. The output of each rule is de- (1) termined by min-inference. The crisp output u of the fuzzy logic controller is generated by centroid D. Perturbed plants The two perturbed motor plants are given by the fo llowing transfer functions ZE NB NM NS PS PM PB (s) (2) NS ZE PM PB PB PB PB PB NB ZE PS PM PBPBPBPB G3(s) Figure 2: Fuzzy Logic Controller Rule Base E. Perform ane The designed controllers must be robust to varia- The step response of this fuzzy logic controller ap tions of system parameters. Specifically, for the plied to the nominal and perturbed plants is shown fuzzy logic, PID, and sliding control designs, one in Appendix A. The step responses for all three single controller must render acceptable the closed plants have short rise times and no overshoot. Thus
é➃õ②ê❑ú◆ñ④✑❉ö➃ó➪è❾í❚✲◆ó➪ô◆õ②è✉î➪î➪ó➪ò➍ó✄✑❉î➪è☛✲➍÷➁ô❑ú✱✡❳è❫÷➁ô❉ó➪ô❉òrí❇ê❃î ✝ ➱s➱✉➱❫✃✧⑤➬❰ ❐q❒q❮❰❶⑩✁■❮❰✄✂✦⑩✬⑦✆☎ Ó❝Ô✞✝✍Ù❅Û②ÙrÜ✠✟☛✡➡Ö❾Ö✣✥❚Ø✐Ù➁Ú◆Û❤Ü➏ÙrÝ ☞❚ï❢õ②ó✄✎➍÷rõ➃è❫ú✵✑◆ñ✍õ➃ç❉è❳ü■ï➍ö✳❀✍ó➪ô✣þÿ✁ ÷rô❑ú❴þÿ✁ ✲➘÷➁ô❑ú✚èsô❑ë❾ï➍ê❃ö✗✟ ÷➁ò➍è❫ú ✑❵ñ✼õ②ç❃è❳ö②ï✬✑❉ê❉é➃õ②ô❉è✉é➃é❳ö➃è✉é②ê❉î➪õ➃é✇ù❉ö➃è✉é②èsô❵õ➃è❫úâí❇ï❢ö❥í❇ê❉ðsð✉ñ î➪ï➍ò❢ó①ë✇ë❾ï❢ô◆õ②ö②ï❢î✗✲qü✐è❳ëìï➍ô❉é➃ó ú➡è✉ö✿õ②ç❃è❳ö②ï✬✑❉ê❉é➃õ✿é②ù❭è✉è❫úâë❾ï➍ô◆õ②ö➃ï➍î ï➁í➘÷❥ö②ï❢õ➏÷rõ➃ó➪ô❉ò❯✡❳ï➍õ➃ï➍ö✏✲✓✡❳ï❉ú◆è✉î➪è❫út✑◆ñ✈÷❞õ②ç❉ó➪ö➏ú④ï❢ö➏ú◆èsö✐î➪ó➪ô❉è❫÷➁ö õ➃ó✩✡➲è✣✟❤ó↕ô✠✎➍÷➁ö②ó①÷rô◆õ➳õ➃ö➏÷➁ô❉é❤í❇èsö✿í❇ê❃ô❑ë❾õ②ó➪ï❢ô ✝ æ➹ç❉è➲ò➍è✉ô❃è✉ö➏÷➁î ë❾ï❢ô✠✟ ✌ò❢ê❉ö➏÷➁õ②ó➪ï➍ôâí❇ï❢ö❳õ➃ç❉è✾✡❳ï❢õ②ï❢ö✇é➃ù❑è✉è❅ú✜ë❾ï➍ô◆õ②ö➃ï➍î❍ù❉ö②ï✬✑❉î➪è✞✡✞ó➪é ò❢ó✩✎❢è✉ô❚ó➪ô s❬ó➪ò➍ê❉ö➃è✎✍ ✝ . r c c e y ω ω ω - + s Plant Motor Controller s❬ó➪ò➍ê❉ö➃è✏✍☛✑✒☞❚ï➍õ➃ï➍ö ✯ ù❭è✉è❫ú✔✓■ï❢ô◆õ②ö②ï❢î❲❺✌ö②ï✬✑❉î➪è✞✡ ✇è✌ê❉é②è➋õ②ç❉è✹ô❃ï☛✡❳ó➪ô❑÷➁î✉ù❉î①÷rô◆õ✏✲✉ù❭è✉ö②õ➃ê❉ö❈✑❭è❫ú✧✡❳ï❢õ②ï❢ö❉ù❉î①÷rô◆õ②é✏✲ ÷➁ô❑ú✙õ➃ç❉è❋ù❭è✉ö❤í❇ï❢ö❈✡❚÷rô❑ë❾è❳ëìö②ó➪õ②è✉ö➃ó①÷❳ò❢ó✄✎➍è✉ô✼ó➪ô þÿ✁ õ➃ï➐ú➡è✉é②ó➪ò➍ô ÷➁ô❑ú☛ë❾ï✬✡❳ù❑÷➁ö②è❋ö➃ï☛✑❉ê❉é➃õ✿í▲ê❉ð✉ð✉ñ✠✲✖❺❣✼❈❻✧✲➘÷➁ô❑ú✚é➃î➪ó ú➡ó↕ô❃ò ✡➲ï❉ú◆è ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö②é ✝ ✙Ô☞✉➹Ö❾ß❾Û❤Ü✉ä✆❥ìÛ➶ä❇Ù➁Ú◆ß✇ÙrÚ✚Ø✐Ù➁Ú◆Û❤Ü➏ÙrÝ Ý↕ÖsÜ✖✕sÚ✗✡✦✈❢Û▲ß❦✢rÚ✦✥✙✘❵✈❢Û✚✡❅✈➍Û▲ß ★é✹é②ç❉ïrü➹ô④ó➪ô❦s❬ó➪ò➍ê❉ö➃è✛✍✬✲rõ②ç❉è■ö②èìí❇è✉ö②èsô❑ë❾è➳é➃õ②è✉ù❋ó➪ô❉ù❃ê❉õ✌é②ù❭è✉è❫ú ✜✒✢ ÷rô❑úûõ②ç❉è❝ï➍ê❉õ➃ù❉ê❉õ❵✡❳ï❢õ②ï❢ö✐é➃ù❑è✉è❅ú ✜✤✣ ÷rö②è④÷✏✎➍÷ró➪î①÷✓✑❉î➪è■í❇ï❢ö ë❾ï✬✡❳ù❑÷➁ö②ó➪é②ï❢ô ✝✦✥÷➍ë➃ç✚ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö❍ï➍ô❉î➪ñ❳ç❭÷ré➬÷❢ë✉ë❾è✉é➃é✿õ②ï➲õ②ç❉è ✡➲ï➍õ②ï❢öqé②ù❭è✉è❫ú❝è✉ö②ö➃ï➍ö✤✧✩★ ✜✪✢✬✫✦✜✤✣ ÷➁ô❑ú❥õ➃ç❉è✖✡❳ï❢õ②ï❢ö ÷➁ô❉ò➍ê❉î①÷➁ö ÷❢ë✉ë❾è✉î➪è✉ö❾÷rõ②ó➪ï❢ô✮✭ ✜ ✣ ÷ré➹ó➪ô❉ù❉ê❉õ➃é ✝✯✥÷➍ë➃ç✍ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö➹ù❉ö②ï❉ú◆ê❑ëìè✉é ï❢ô❉î➪ñ❋ï➍ô❉è❥ï➍ê❃õ②ù❉ê❉õ✱✰ ✝ Ø✹Ô❝Õ❞Ù➁ã✇ä①Ú❅✢rÝ❂✐■Ý✩✢rÚ➡Û æ➹ç❉è❝ô❉ï☛✡➲ó↕ô❭÷rî②✡➲ï➍õ②ï❢ö✐ù❃î ÷➁ô◆õ✐ó➪é❵✡❳ï❉ú◆èsî↕è❅ú✤✑◆ñ❳õ➃ç❉è❥í❇ï❢î↕î➪ïrü✽✟ ó➪ô❉ò④õ②ö❾÷rô❉é➔í❇è✉ö➹í▲ê❉ô❑ë❾õ➃ó↕ï❢ô✲✑ ✳ ❭ ý✵✴ ☎ ★ ✶ ✴➁ý✷✴✯✸✹✍ ☎ ý✵✴✺✸✼✻ ☎ ý✽✍ ☎ ✚❥Ô✧✐Ö✉ÜìÛ❹✈◆Ü✜✛❾Ö✣✥t✐✐Ý✩✢➁Ú◆Û▲ß æ➹ç❉è❚õ❤ü■ï✍ù❭è✉ö➃õ②ê❉ö✳✑❑è❫ú ✡❳ï➍õ➃ï➍ö✿ù❃î ÷➁ô◆õ②é❋÷rö②è➲ò➍ó✄✎➍èsô ✑◆ñ✚õ②ç❉è í❇ï❢î➪î↕ïrü➹ó➪ô❉ò❝õ②ö➏÷➁ô❉é❤í▲è✉ö■í❇ê❉ô❑ëìõ②ó➪ï➍ô❉é✾✑ ✳✞✿ ý✵✴ ☎ ★ ✍ ✶ ✴➁ý✷✴✯✸✹✍ ☎ ý✵✴✺✸✼✻ ☎ ý✵✻ ☎ ✳✱❀ ý✵✴ ☎ ★ ✶ ✴ ✿ ý✵✴✺✸❁✻ ☎ ý✵❂ ☎ á✐Ôt✐ÖsÜ✍✹❫Ù➁Ü✉ã ✢➁Ú✦❥❾Ö❋Ø✹Üsä➪Û➔ÖsÜ✉ä✆✢ æ➹ç❉è✙ú◆èsé②ó➪ò➍ô❉è❅ú✣ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö➃é❦✡❋ê❉é➃õ❦✑❭è❚ö②ï✬✑❉ê❉é②õ❋õ➃ï⑨✎➍÷rö➃ó ÷✟ õ➃ó↕ï❢ô❉é✙ï➁í❳é②ñ◆é②õ➃è✞✡✴ù❑÷rö❾÷✓✡❳èsõ②è✉ö➃é ✝ ✯ ù❭è❫ë❾ó✌ë✉÷➁î➪î↕ñ✠✲❍í❇ï❢ö✙õ②ç❉è í❇ê❃ð✉ð✉ñ✣î➪ï➍ò❢ó①ë✓✲❵❺❣✼❈❻✧✲✐÷rô❭ú❴é②î➪ó①ú◆ó➪ô❉ò✚ë❾ï➍ô◆õ②ö➃ï➍î✿ú◆è✉é➃ó➪ò➍ô❉é✏✲➹ï➍ô❉è é➃ó↕ô❃ò➍î➪è✈ëìï➍ô◆õ②ö➃ï➍î➪î➪è✉ö☞✡❋ê❉é➃õ➳ö②èsô❑ú◆è✉ö❋÷➍ë✉ëìè✉ù❉õ➏÷✔✑❉î➪è❋õ②ç❉è❳ëìî↕ï❢é②è❫ú î➪ï◆ï➍ù✚ê❉ô❉ó➪õ✿é➃õ②è✉ù❴ö➃è✉é②ù❭ï➍ô❉é➃è✉é❋ïrí➳õ➃ç❉è❳ô❃ï☛✡❳ó➪ô❑÷➁î✌ù❉î①÷rô◆õ✇÷rô❭ú õ②ç❃è✍õ❤ü■ï☛ù❭è✉ö②õ➃ê❉ö❈✑❭è❫ú❙ù❉î①÷➁ô❵õ➃é ✝ æ✌ï⑨✑❑è✚÷➍ë✉ëìè✉ù❉õ➏÷✔✑❉î➪è☛✲❍õ②ç❉è ë❾î➪ï➍é➃è❫ú❝î↕ï◆ï❢ù✿é②õ➃è✉ù✇ö➃è✉é②ù❭ï➍ô❃é②è✉é✽✡❋ê❉é②õqé➃ó✄✡❋ê❉î➪õ➏÷rô❃è✉ï➍ê❉é➃î➪ñ➳ç❑÷✏✎❢è é②ç❃ï➍ö②õ❍ö②ó➪é②è❥õ➃ó✩✡➲è➬÷➁ô❑ú➲ô❉ï✇ï✓✎❢è✉ö②é➃ç❉ï◆ï➍õ ✝ ♥✹Ô✧✚❞Öìß✉ä✄✸❱Ú➲Ó✦✡☛✡❭Ü➏Ù✏✢☛❥❝✺ ✇è✙ú◆è✉é➃ó↕ò❢ô✣õ②ç❃è❚í❇ê❉ð✉ðsñ❞✲❵❺❣✼❈❻✧✲✹÷rô❭ú☛é➃î➪ó ú➡ó↕ô❃ò③✡➲ï➡ú➡è➐ëìï➍ô✠✟ õ②ö➃ï➍î➪î➪è✉ö②é✧✑❑÷➁é②è❫ú✼ï❢ô✚õ②ç❉è❋ô❉ï✬✡❳ó➪ô❑÷➁î➘ù❃î ÷➁ô◆õ➬ò❢ó✄✎➍è✉ô✼ó➪ô✚è✜✌❢ê❑÷✏✟ õ②ó➪ï❢ô✣ý❃✍ ☎✞✝❄✥÷❢ë②ç✣ú◆èsé②ó➪ò➍ô❉è❅ú ëìï➍ô◆õ②ö➃ï➍î➪î➪è✉ö✇ó➪é④õ②ç❉èsô ÷rù❃ù❉î➪ó↕è❅ú õ②ï✚÷rî➪î✹õ➃ç❉ö②èsè❳õ②ö❾÷rô❉é➔í❇è✉ö④í❇ê❃ô❑ë❾õ②ó➪ï❢ô❉é✜✲✌÷rô❑ú ô◆ê②✡❳è✉ö➃ó①ë✉÷rî✌é②ó✄✡❽✟ ê❉î①÷rõ➃ó➪ï➍ô❉é❳÷➁ö②è✼ù❑è✉ö➔í❇ï➍ö✳✡❳è❫úâõ➃ï ÷rô❭÷rî➪ñ◆ð✉è❚õ②ç❉è✚ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö✾❅ é ö②ï✬✑❉ê❉é➃õ②ô❉è✉é➃é ✝ æ➹ç❉è➮ú➡è✉é②ó➪ò➍ôâïrí➳õ➃ç❉è➲í❇ê❉ð✉ð✉ñ✚î➪ï❢ò➍ó①ë❋ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö④ó➪é✿ò➍ó✄✎➍èsô ó➪ô ✯ è❫ë❾õ②ó➪ï❢ô⑨✼✵❆ ✝ æ➹ç❉è❳ú➡è✉é②ó➪ò➍ô✚ï➁í✐õ②ç❃è✘❺❣✼❈❻✮ë❾ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö❝ó↕é ò➍ó✄✎❢è✉ô✙ó➪ô ✯ è❫ë❾õ➃ó↕ï❢ô✔❆ ✝ æ➹ç❉è✇ú◆èsé②ó➪ò➍ô✙ï➁í✹õ②ç❉è✿é➃î➪ó ú➡ó↕ô❃ò✤✡❳ï❉ú◆è ë❾ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö❍ó➪é➹ò➍ó✄✎➍è✉ô➲ó➪ô ✯ è❫ë❾õ②ó➪ï❢ô✔❆✫✼ ✝ ❺✌î➪ï➍õ➃é➹ïrí✹è❫÷➍ë➃ç✍ëìï➍ô✠✟ õ②ö➃ï➍î➪î➪è✉ö❇❅ é➹ë❾î➪ï➍é➃è❫ú➲î↕ï◆ï❢ù❋é②õ②èsù➮ö➃è✉é②ù❭ï➍ô❃é②è✿÷rô❑ú➲ë❾ï➍ô◆õ➃ö②ï➍îqè❉❈ï❢ö②õ ÷rö➃è✿é②ç❃ïrü➹ô➮ó➪ô ★ù❉ù❭è✉ô❑ú◆ó➟ø ★ ✝ ➱❋❊➮✃✱●❝❐■❍❂⑩✽❏❸sÒ✆❸✏❍q❐qÏ▲❑✿⑦❑Ñq❰q❐❲❸s❒q❮❑Ð❨Ñ❭❒qÒ▲❰q❐ ❰❶⑧✔❍ ▼✹Ð✲◆☛◆❖❏✹❑➳❰■✂❭Ð❲❸✉❒ ⑤➬❰q❐q❒q❮q❰❂⑩✬⑩☛⑦❉❮ ✇è④ó✩✡➲ù❉î➪è✞✡❳èsô❵õ➹õ➃ç❉è❋ö②ï✬✑❉ê❉é➃õ❥í❇ê❉ð✉ðsñ ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö➬ò❢ó✄✎➍è✉ô✼ó➪ô þÿ✁ ÷➁ô❑ú✼þÿ✁❈✝ æ➹ç❉è❝ó↕ô❃ù❉ê❉õ②é➹õ➃ï❋õ②ç❉è❍í❇ê❃ð✉ð✉ñ✙ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö➳÷➁ö②è✞✧ ÷rô❭ú✹✭ ✜ ✣ ✝ æ➹ç❉è➹ï❢ê❉õ②ù❉ê❃õ➘ï➁í õ➃ç❉è■í❇ê❉ðsð✉ñ✇ë❾ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö✌ó➪é✤✰ ✝ æ➹ç❉è ê❉ô❉ó✄✎❢è✉ö②é➃è✉é✐ï➁í ú➡ó↕é❾ë❾ï➍ê❃ö②é②è❍ïrí✲✧☛✲P✭ ✜ ✣ ✲❵÷➁ô❑ú◗✰❋÷rö②è➹ù❭÷rö②õ➃ó➪õ②ó➪ï➍ô❉è❅ú ó➪ô◆õ②ï❋é②è✮✎➍è✉ô❚í▲ê❉ð✉ð✉ñ➲é②è✉õ➃é❇✑ ❘ ✕✩❙ ✟✐ô❃è✉ò❵÷➁õ②ó✄✎➍è✿✑❉ó➪ò ❘ ✕☞ ✟✹ô❉è✉ò❵÷➁õ②ó✄✎➍è❯✡➲è❫ú◆ó➪ê②✡ ❘ ✕✯ ✟✹ô❉è✉ò❵÷➁õ②ó✄✎➍è❥é✳✡➮÷rî➪î ❘❁❚ ✥ ✟■ð✉è✉ö②ï ❘ ❺ ✯ ✟❁ù❭ï➍é➃ó➪õ②ó✄✎➍è❥é❈✡❚÷rî➪î ❘ ❺✤☞ ✟❁ù❭ï➍é➃ó➪õ②ó✄✎➍è❯✡❳è❅ú◆ó➪ê②✡ ❘ ❺ ❙ ✟✐ù❭ï➍é➃ó↕õ➃ó✄✎➍è✫✑❃ó↕ò ✥÷❢ë②ç✼í▲ê❉ð✉ð✉ñ✼é②è✉õ④ó➪é❥ö②è✉ù❉ö➃è✉é②èsô❵õ➃è❫ú✰✑◆ñ✍÷❱❯✿÷rê❉é➃é②ó①÷rô✵✡❳è✮✡✤✟ ✑❭è✉ö②é➃ç❉ó➪ù☛í▲ê❉ô❑ë❾õ➃ó↕ï❢ô ✝ æ➹ç❉è➲ö②ê❉î➪è✤✑❑÷➁é②è❳ï➁í➳õ②ç❉èûí❇ê❉ð✉ðsñ✚î↕ï❢ò➍ó①ë ë❾ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö✿ë❾ï❢ô◆õ➏÷ró➪ô❉é í❇ï➍ö➃õ❤ñ✬✟➶ô❉ó➪ô❉è❥ö➃ê❉î➪è✉é❥ü➹ç❉ó①ë②ç✼÷rö②è④õ➏÷✔✑❉ê✠✟ î①÷rõ➃è❫úâó↕ô s❬ó↕ò❢ê❉ö②è❱✻ ✝ æ➹ç❉è❚ï➍ê❉õ➃ù❉ê❉õ④ïrí❥è❫÷➍ë➃ç❴ö②ê❉î➪è❚ó➪é✈ú➡è✣✟ õ②èsö❈✡❳ó➪ô❉è❅ú⑨✑◆ñ③✡➲ó➪ô✠✟➶ó➪ô◆í▲è✉ö②è✉ô❭ë❾è ✝ æ➹ç❉è❚ë❾ö②ó➪é②ùâï➍ê❉õ➃ù❉ê❉õ✏✰✚ï➁í õ②ç❃è❚í❇ê❉ð✉ðsñ☛î➪ï➍ò❢ó ë➲ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö④ó➪é❋ò➍è✉ô❉èsö➏÷rõ➃è❫ú ✑◆ñ☛ë❾è✉ô◆õ➃ö②ï➍ó①ú ú◆è❾í▲ê❉ð✉ð✉ó✌ë✉÷➁õ②ó➪ï➍ô ✝ ❲ ❳✲❨ ❳✲❩ ❳✪❬ ❭☛❪ ❫■❬ ❫❴❩ ❫❴❨ ❫❵❨ ❳✲❨ ❳✲❨ ❳✲❨ ❳❛❨ ❳❛❩ ❳✪❬ ❭❜❪ ❫❵❩ ❳✲❨ ❳✲❨ ❳✲❨ ❳✲❩ ❳✪❬ ❭❜❪ ❫❝❬ ❫■❬ ❳✲❨ ❳✲❨ ❳✲❩ ❳✪❬ ❭❜❪ ❫■❬ ❫❵❩ ❡✆❢ ❞ ❭❜❪ ❳✲❨ ❳✲❩ ❳✪❬ ❭☛❪ ❫■❬ ❫❴❩ ❫❴❨ ❳✒❬ ❳✲❩ ❳✪❬ ❭❜❪ ❫■❬ ❫❴❩ ❫❵❨ ❫❴❨ ❳❛❩ ❳✒❬ ❭❜❪ ❫■❬ ❫❴❩ ❫❵❨ ❫❵❨ ❫❴❨ ❳✲❨ ❭❜❪ ❫■❬ ❫❵❩ ❫❴❨ ❫❵❨ ❫❵❨ ❫❴❨ s❬ó➪ò➍ê❉ö➃è✠✻❣✑✖s❉ê❃ð✉ð✉ñ✐❤ ï➍ò❢ó ëP✓■ï➍ô◆õ②ö➃ï➍î➪î➪è✉ö✩❥➹ê❃î↕è ❙ ÷➁é②è æ➹ç❉è■é②õ②èsù✇ö➃è✉é②ù❭ï➍ô❉é➃è➹ïrí❑õ➃ç❉ó➪é❬í❇ê❉ð✉ð✉ñ❥î➪ï➍ò❢ó①ë✹ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö✌÷rù✠✟ ù❉î➪ó➪è❫ú➲õ②ï④õ②ç❉è❥ô❉ï✬✡❳ó➪ô❑÷➁î➡÷➁ô❑ú➲ù❑èsö②õ②ê❃ö❈✑❭è❫ú❚ù❉î①÷rô◆õ➃é✐ó➪é➹é②ç❃ïrü➹ô ó➪ô ★ù❃ù❑è✉ô❭ú◆ó➟ø ★ ✝ æ➹ç❃è✙é②õ➃è✉ù✦ö➃è✉é②ù❭ï➍ô❃é②è✉é❚í▲ï➍ö➮÷➁î➪î➹õ②ç❉ö➃è✉è ù❉î①÷rô◆õ➃é❨ç❑÷✏✎❢è✐é➃ç❉ï➍ö➃õ❨ö②ó➪é➃è✐õ➃ó✩✡➲è✉é➘÷➁ô❑ú❥ô❉ï❍ï✓✎➍è✉ö➃é②ç❉ï◆ï❢õ ✝ æ➹ç❵ê❃é✜✲
they meet the specified robust performance criteria. We consider the following sliding surface The control effort generated by the fuzzy logic con- design troller is also shown in Appendix a V. ANALYSIS AND SYNTHESIS OF A ROBUST PID S=(-ja)+h1(y-ya)+A2(y-y)(7) CONTrOLleR The choice of sliding surface is based on the follow The fuzzy logic controller designed in [0, 0] is similar ng considerations to a nonlinear PD controller [O. We design a linear The relative degree of the system(6)with S as PID controller to obtain satisfactory step response output should be 1. This ensures that the input for all three transfer funct u appears explicitly on the right hand side of s be characterized b the following transfer function System dynamics on the sliding surface should be stable.(This requires A1, A2>0) C(s)=Kp+Kd (4)ya(t) by output. If one knows the entire trajectory of ye Because, it is very difficult to realize a pure differ- then the derivative information ja(t), ya(t)can be entiator, we use the following transfer function for extracted off line and used in the design to improve implementation the performance. If the information is not available nis case, ya(t) being unit step, is nondif- C(s)=k (5) ferentiable) we can assume yd and ya to be zero and let the robustness property of the sliding mode con- To have a fair comparison, we use a small value for troller take care of the mismatch Ki=0.001, just enough to keep stea dy state error zero. We use a small value for T=0.01. so that its With the definition of sliding surface as above, we effect on the dynamics is minimal, but the derivative have reduced the design requirement from tracking ya(t) to being on the surface S=0. Once on the The selection of Kp and Kd is based on the root slidin locus of C(s)GI(s). The root locus gives us the nentially stable and asymptotic trajectory tracking locations of closed loop poles. The proportional is achieved. The control u is designed to make the nd derivative gain values are chosen such that the surface S=0 at tractive and to reach the surface in closed loop pole locations are in the left half com- finite time plex plain for all three plants. Another criterion for selection of these gains is the step response of the Consider the following lyapunov function closed loop system We select the following gains V= Kp=2, Kd=5 With these gains, the closed loop system is stable for Its time derivative is given by the open loop gain up to 51. Thus in particular, the above PID controller stabilizes the systems given b the transfer functions(1, 2). For the model of(3), As the system has relative degree 1 with S as output we have closed loop stability for the open loop gain upto 30. The step responses have no overshoot and we can solve for u from the equation are critically dampe S=-K sgn(S) VI. ANALYSIS AND SYNTHESIS OF ROBUST SLIDING MODE CONTROLLER This results in a negative definite V and also guar entees finite time convergence to the sliding surface To design a sliding mode controller, we convert the But, the controller will result in high frequ transfer function model into state space format. The tering near sliding surface. To avoid this, we use the controllable canonical form realization of the model following expression to solve for u 2 (1) →V )msm基 su face in finite time ory s ot gu
õ➃ç❉è✉ñ✤✡❳èsè✉õ✹õ②ç❃è➹é②ù❭è❫ë❾ó✌è❫ú➲ö②ï✬✑❉ê❉é➃õ✐ù❭è✉ö❤í▲ï➍ö❈✡❚÷rô❑ëìè➳ë❾ö②ó➪õ②èsö②ó①÷ ✝ æ➹ç❉è✿ë❾ï❢ô◆õ②ö②ï❢îqè✽❈qï➍ö②õ❍ò➍è✉ô❃è✉ö➏÷➁õ②è❫ú④✑◆ñ❋õ➃ç❉è❍í❇ê❉ð✉ð✉ñ➲î➪ï➍ò➍ó①ë➹ë❾ï❢ô✠✟ õ➃ö②ï➍î➪î➪è✉ö➹ó➪é➳÷rî➪é➃ï✇é➃ç❉ïrü➹ô❚ó↕ô ★ù❉ù❭è✉ô❑ú➡ó➈ø ★ ✝ ❊❚✃✞●❝❐■❍❂⑩✽❏❸✉Ò✻❸✏❍q❐qÏ ❏❨❐q❒❂⑥❂⑦✦❸sÒ✆❸❳❰❶⑧✔❍ ❑❰✄✂❑Ð❲❸s❒ ❍➱✂✁ ⑤➬❰q❐q❒q❮q❰❂⑩✬⑩☛⑦❉❮ æ➹ç❉è➋í❇ê❉ðsð✉ñ✿î➪ï➍ò❢ó ë➋ë❾ï➍ô◆õ②ö➃ï➍î➪î➪è✉ö➘ú➡è✉é②ó➪ò➍ô❃è❫ú✿ó➪ô❚þÿ✠✲❱ÿ ✁ ó↕é❬é②ó✄✡❳ó➪î①÷rö õ➃ï✈÷④ô❉ï❢ô❉î➪ó➪ô❉è❫÷rö✽❺✖❻✜ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö✿þÿ✁❈✝❲✇è❥ú◆è✉é➃ó↕ò❢ô✙÷✿î➪ó➪ô❉è❫÷➁ö ❺❣✼❈❻✠ë❾ï➍ô◆õ②ö➃ï➍î➪î➪è✉ö❥õ②ï❚ï☛✑❃õ➏÷ró➪ô✙é❾÷rõ➃ó↕é➔í➶÷➍ëìõ②ï➍ö➃ñ✍é➃õ②è✉ù✚ö➃è✉é②ù❭ï➍ô❃é②è í❇ï❢ö➳÷rî➪î❭õ②ç❉ö➃è✉è✇õ➃ö➏÷➁ô❉é❤í❇èsö➹í❇ê❉ô❭ë❾õ②ó➪ï➍ô❃é ✝ ★ î➪ó➪ô❉è❫÷➁ö❦❺❣✼❈❻✏ë❾ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö④ë✉÷rô✰✑❑è❚ë②ç❭÷rö➏÷❢ë❾õ②èsö②ó➪ð✉è❫ú ✑◆ñ õ➃ç❉è❥í❇ï❢î↕î➪ïrü➹ó➪ô❉ò❥õ②ö❾÷rô❉é➔í❇è✉ö í❇ê❉ô❑ë❾õ➃ó➪ï➍ô✲✑ ☎✄ ý✷✴ ☎ ★✝✆✟✞✛✸✠✆☛✡✩✴✯✸ ✆✌☞ ✴ ý✎✍ ☎ ❙è❫ë✉÷➁ê❉é②è☛✲✌ó➪õ❍ó↕é❯✎❢è✉ö②ñ✍ú➡ó❁➮ëìê❉î➪õ➳õ②ï❚ö➃è❫÷rî➪ó➪ð✉è❋÷❳ù❃ê❉ö②è❋ú◆ó❈èsö✗✟ è✉ô◆õ➃ó ÷➁õ②ï❢ö✜✲❬ü✐è④ê❉é②è➲õ②ç❉è④í▲ï➍î➪î➪ïrü➹ó↕ô❃ò✇õ➃ö➏÷rô❃é❤í❇è✉ö❝í❇ê❉ô❭ë❾õ②ó➪ï➍ô➙í❇ï❢ö ó✄✡❳ù❃î↕è✮✡❳è✉ô◆õ➏÷➁õ②ó➪ï➍ô ✝ ☎ý✷✴ ☎ ★✝✆✟✞✛✸ ✆☛✡✩✴ ✏ ✴✤✸ ✍ ✸ ✆✌☞ ✴ ý ✶ ☎ æ✌ï④ç❑÷✏✎➍è✿÷❝í➶÷➁ó↕ö➹ë❾ï✬✡❳ù❑÷➁ö②ó➪é②ï❢ô❲✲➍ü■è❥ê❉é②è④÷④é❈✡➮÷➁î➪î②✎➍÷rî➪ê❉è❍í❇ï❢ö ✆☞ ★ ÿ✒✑ ÿ❢ÿ✆✍✬✲✔✓❾ê❉é➃õ➳è✉ô❃ï➍ê❉ò❢ç✍õ➃ï✘❀➍èsè✉ù✍é➃õ②è❫÷❢ú◆ñ✙é➃õ➏÷rõ➃è❋è✉ö②ö➃ï➍ö ð✉èsö②ï ✝ ✇è➬ê❃é②è✇÷④é✳✡➮÷rî➪î❅✎➍÷rî➪ê❉è❍í❇ï❢ö ✏ ★✜ÿ✕✑ ÿ✆✍✬✲◆é②ï❋õ②ç❭÷rõ➹ó➪õ②é è✽❈qè❫ë❾õ➋ï➍ô✿õ➃ç❉è✐ú◆ñ◆ô❑÷✔✡❳ó①ë❾éqó➪é❲✡➲ó↕ô❃ó✩✡❳÷➁î✗✲❈✑❉ê❃õ❨õ②ç❃è ú◆èsö②ó✄✎➍÷rõ➃ó✩✎❢è ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö➹ó↕é➹é➃õ②ó➪î➪îqö②è❫÷➁î➪ó↕ð❅÷✓✑❉î➪è ✝ æ➹ç❉è✿é➃è✉î➪è❫ë❾õ➃ó↕ï❢ô✍ï➁í✖✆✞ ÷➁ô❑ú✗✆✡ ó↕é✿✑❑÷ré➃è❫ú❚ï❢ô✙õ②ç❉è❝ö②ï◆ï➍õ î➪ï❉ë❾ê❉é❚ïrí ☎ý✵✴ ☎ ✳ ❭ ý✷✴ ☎✞✝ æ➹ç❉è✼ö②ï◆ï❢õ➮î➪ï❉ë❾ê❉é❚ò❢ó✩✎❢è✉é❚ê❉é❚õ②ç❉è î➪ï❉ë✉÷rõ➃ó➪ï➍ô❉é✙ï➁íûëìî↕ï❢é②è❫ú❙î➪ï◆ï➍ù✜ù❭ï➍î➪è✉é ✝ æ➹ç❃è✚ù❉ö②ï❢ù❑ï❢ö②õ②ó➪ï❢ô❑÷rî ÷➁ô❑ú❚ú◆è✉ö②ó✄✎➍÷➁õ②ó✄✎➍è❥ò❵÷➁ó➪ô ✎➍÷➁î➪ê❉è✉é➳÷rö➃è✿ë②ç❃ï➍é②èsô✙é②ê❑ë➃ç✼õ②ç❑÷➁õ➹õ②ç❉è ë❾î➪ï❢é②è❫ú✼î➪ï◆ï➍ù✼ù❑ï❢î➪è✇î➪ï❉ë✉÷rõ➃ó➪ï➍ô❉é✿÷rö➃è❋ó➪ô✍õ➃ç❉è❋î➪è❾í❇õ❥ç❑÷rî➟í➳ë❾ï☛✡❽✟ ù❉î➪è❾ø❳ù❃î ÷➁ó➪ô④í❇ï➍ö✐÷➁î➪î❑õ➃ç❉ö②è✉è❥ù❉î①÷➁ô❵õ➃é ✝ ★ô❉ï➍õ➃ç❉è✉ö➳ë❾ö➃ó↕õ➃è✉ö②ó➪ï❢ô❋í❇ï❢ö é➃è✉î➪è❫ë❾õ②ó➪ï❢ô☛ïrí➳õ➃ç❉è✉é➃è❳ò❵÷➁ó➪ô❉é✿ó➪é✿õ②ç❃è❋é②õ②èsù☛ö②èsé②ù❭ï➍ô❉é➃è➮ï➁í➳õ②ç❉è ë❾î➪ï❢é②è❫ú❚î➪ï◆ï➍ù➲é②ñ◆é②õ➃è✞✡✝ ✇è➳é➃è✉î➪è❫ë❾õ❥õ②ç❃è➳í▲ï➍î➪î➪ïrü➹ó↕ô❃ò➬ò◆÷ró➪ô❉é✾✑ ✆✟✞◗★✹✻✒✘✙✆☛✡✦★ ✇ ✶ ó➪õ➃ç✿õ②ç❉èsé②è■ò❵÷ró➪ô❉é✏✲❅õ②ç❉è✐ëìî↕ï❢é②è❫ú❝î➪ï◆ï➍ù❥é②ñ◆é②õ➃è✞✡ ó➪é❬é②õ➏÷✔✑❉î➪è✌í❇ï❢ö õ➃ç❉è➳ï❢ù❑è✉ô❋î➪ï◆ï➍ù④ò❵÷➁ó➪ô✇ê❉ù❋õ➃ï ✶ ✍ ✝ æ➹ç◆ê❉é➋ó↕ô④ù❑÷➁ö②õ②ó①ë❾ê❃î ÷➁ö✜✲❢õ②ç❉è ÷✔✑❑ï✓✎❢è✫❺❣✼❈❻✣ë❾ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö✌é➃õ➏÷✓✑❃ó↕î➪ó➪ð✉è✉é➋õ②ç❉è➹é➃ñ❵é➃õ②è✞✡➲é✹ò➍ó✄✎❢è✉ô❦✑◆ñ õ➃ç❉è❳õ➃ö➏÷rô❃é❤í❇è✉ö❝í❇ê❉ô❭ë❾õ②ó➪ï➍ô❃é❋ý❃✍✬✲ ✻ ☎✞✝ s❃ï➍ö❝õ②ç❉è✤✡➲ï➡ú➡è✉î✹ïrí❥ý✷❂ ☎ ✲ ü■è❥ç❑÷✏✎➍è❥ë❾î➪ï➍é②è❅ú❳î➪ï◆ï➍ù➲é②õ❾÷✓✑❉ó➪î➪ó➪õ❤ñ✿í❇ï❢ö➹õ②ç❉è❥ï❢ù❑è✉ô❚î➪ï◆ï➍ù❋ò◆÷ró➪ô ê❉ù❃õ②ï ❂➍ÿ ✝ æ➹ç❃è➬é➃õ②è✉ù➙ö②è✉é➃ù❑ï❢ô❉é②è✉é❥ç❑÷✏✎❢è➳ô❉ï❋ï✓✎❢è✉ö②é➃ç❉ï◆ï➍õ➳÷➁ô❑ú ÷➁ö②è✇ë❾ö➃ó➪õ②ó①ë✉÷rî➪î➪ñ❳ú❉÷✔✡❳ù❭è❫ú ✝ ❊û➱❫✃✞●❝❐■❍❂⑩✽❏❸sÒ✆❸✏❍q❐qÏ ❏❨❐q❒❂⑥❂⑦✦❸sÒ✆❸❳❰❶⑧❄❑➳❰■✂❭Ð❲❸✉❒ ⑩❢Ò①Ï❨Ò①❐✛✚✢✜❰qÏ①⑦ ⑤❥❰q❐q❒q❮q❰❂⑩☛⑩✬⑦❉❮ æ✌ï❳ú◆è✉é➃ó➪ò➍ô✙÷④é②î➪ó①ú◆ó➪ô❉ò✧✡❳ï❉ú◆è➬ëìï➍ô◆õ②ö➃ï➍î➪î➪è✉ö✜✲❃ü✐è✿ë❾ï❢ô❞✎❢è✉ö②õ❍õ②ç❉è õ➃ö➏÷rô❃é❤í❇è✉ö❬í❇ê❃ô❑ë❾õ②ó➪ï❢ôq✡➲ï❉ú◆è✉îró➪ô◆õ②ï➹é②õ❾÷rõ➃è✹é②ù❑÷❢ë❾è✌í❇ï❢ö❈✡❚÷rõ ✝ æ➹ç❉è ë❾ï❢ô◆õ②ö②ï❢î➪î ÷✔✑❉î➪è➳ë✉÷rô❃ï➍ô❉ó①ë✉÷➁î❭í❇ï➍ö✳✡ ö➃è❫÷rî➪ó➪ð❫÷➁õ②ó➪ï➍ô❋ï➁í❨õ②ç❉è❯✡❳ï❉ú◆èsî ï➁í ý✽✍ ☎ ó➪é➹ò➍ó✄✎➍è✉ô④✑◆ñ■✑ ✭ ✣ ★ ✤✥ ÿ ✍ ÿ ÿ ÿ ✍ ÿ ✫ ✻ ✫ ❂ ✦✧ ✣ ✸ ✤✥ ÿ ÿ ✶ ✦✧ ✰ ★ ★ ✩✎✍ ÿ ÿ✠✪ ✣ ý✬✫ ☎ ✇è✚ë❾ï➍ô❉é➃ó①ú◆è✉ö✙õ➃ç❉è✙í▲ï➍î➪î➪ïrü➹ó↕ô❃ò✚é②î➪ó①ú◆ó➪ô❉ò✚é②ê❉ö➔í➶÷➍ëìè✼í❇ï➍ö❚õ②ç❃ó↕é ú◆è✉é➃ó➪ò➍ô✲✑ ❭ ✭ ★✜ý✯✮★✱✫ ✮★✡ ☎ ✸✱✰ ❭ ý❴✭ ★✱✫ ✭ ★✡ ☎ ✸✠✰ ✿ ý ★✠✫✲★✡ ☎ ý✴✳ ☎ æ➹ç❉è✇ë➃ç❉ï❢ó①ë❾è❥ïrí➘é➃î➪ó ú➡ó↕ô❃ò✿é②ê❉ö➔í➶÷➍ëìè✿ó➪é✒✑❑÷➁é②è❫ú➲ï❢ô➮õ➃ç❉è❍í❇ï➍î➪î➪ïrü✽✟ ó➪ô❉ò❳ë❾ï❢ô❉é②ó①ú◆è✉ö❾÷rõ➃ó↕ï❢ô❉é❇✑ ❘ æ➹ç❉è➳ö➃è✉î①÷rõ②ó✄✎❢è➳ú◆è✉ò➍ö➃è✉è❥ïrí❨õ➃ç❉è❥é②ñ◆é②õ➃è✞✡✏ý✴✫ ☎ ü➹ó➪õ➃ç ✭ ÷ré ï➍ê❉õ➃ù❉ê❉õ❨é➃ç❉ï➍ê❃î ú♦✑❑è✱✍ ✝ æ➹ç❉ó➪é❨è✉ô❃é②ê❉ö➃è✉é✹õ②ç❭÷rõ✌õ②ç❉è■ó➪ô❉ù❉ê❉õ ✰❋÷rù❉ù❭è❫÷rö➃é➹è❾ø❉ù❉î➪ó①ë❾ó➪õ②î➪ñ✿ï➍ô❋õ➃ç❉è➳ö➃ó➪ò➍ç◆õ✌ç❑÷rô❑úûé②ó①ú◆è➹ï➁í ✭✭ è✜✌❢ê❑÷rõ➃ó↕ï❢ô ✝ ❘ ✯ ñ◆é②õ②è✮✡☎ú◆ñ◆ô❑÷✔✡❳ó①ë❾é■ï➍ô❚õ②ç❃è✿é②î➪ó①ú◆ó➪ô❉ò④é②ê❉ö➔í➶÷➍ë❾è❝é②ç❉ï❢ê❉î①ú ✑❑è❥é②õ❾÷✓✑❉î➪è ✝ ý▲æ➹ç❉ó➪é➹ö②è✜✌❢ê❉ó➪ö➃è✉é✵✰ ❭ ✘✂✰ ✿✷✶ ÿ ☎ ★✡❢ý✹✸ ☎ é②ù❭è❫ë❾ó✌èsé➹õ②ç❉è❍ö②è❾í❇èsö②è✉ô❑ëìè✿õ②ö❾÷✺✓❾è❫ë❾õ➃ï➍ö➃ñ❳õ②ït✑❑è❍õ②ö➏÷❢ë✳❀➍è❫ú ✑◆ñ❳ï❢ê❉õ②ù❉ê❃õ ✝ ✼▲í➘ï❢ô❉è❯❀◆ô❉ïrü➹é➹õ②ç❉è❥è✉ô◆õ➃ó↕ö➃è✿õ②ö❾÷✺✓❾è❫ëìõ②ï➍ö➃ñ❳ï➁í ★✡✠✲ õ②ç❃è✉ô❴õ②ç❉è✙ú◆è✉ö➃ó✄✎➍÷rõ②ó✄✎❢è❋ó↕ô➡í❇ï➍ö✳✡➮÷rõ➃ó➪ï➍ô✮✭ ★✡❢ý✻✸ ☎ ✘✼✮★✡❢ý✹✸ ☎ ë✉÷rô✰✑❑è è❾ø❉õ②ö❾÷➍ë❾õ➃è❫ú❚ï❈➐î➪ó➪ô❉è❥÷rô❑úûê❉é②è❫ú➲ó➪ô❋õ②ç❃è❞ú➡è✉é②ó➪ò➍ô➲õ②ï④ó✄✡❳ù❉ö➃ï✓✎➍è õ②ç❃è➬ù❭è✉ö➔í❇ï➍ö✳✡➮÷rô❭ë❾è ✝ ✼▲í➘õ➃ç❉è❥ó➪ô◆í❇ï➍ö✳✡➮÷➁õ②ó➪ï➍ô✿ó➪é■ô❉ï➍õ➹÷✏✎➍÷ró➪î①÷✓✑❃î↕è ý❇ï❢ö✿÷ré➳ó➪ô✙õ➃ç❉ó➪é✿ë✉÷ré➃è☛✲ ★✡➁ý✹✸ ☎ ✑❑èsó↕ô❃ò❳ê❉ô❉ó➪õ❍é②õ②èsù❲✲❨ó➪é❍ô❉ï➍ô❑ú➡ó➈í✭✟ í❇è✉ö➃è✉ô◆õ②ó①÷✓✑❃î↕è ☎ ü■è➳ë✉÷rô❳÷➁é②é➃ê②✡❳è❄✭ ★✡ ÷rô❭ú✽✮★✡ õ②ït✑❑è❍ð✉è✉ö②ï❋÷rô❭ú î➪è✉õ■õ②ç❉è❍ö②ï☛✑❃ê❉é②õ➃ô❉è✉é②é➹ù❃ö②ï➍ù❭è✉ö➃õ❤ñ❋ïrí❨õ②ç❃è➳é②î➪ó①ú◆ó➪ô❉ò❯✡❳ï❉ú◆è➳ëìï➍ô✠✟ õ②ö➃ï➍î➪î➪è✉ö➹õ➏÷✔❀➍è✿ë✉÷➁ö②è✿ï➁í➘õ②ç❃è☞✡❳ó➪é❈✡❚÷rõ❾ë②ç ✝ ✇ó➪õ②ç④õ②ç❉è➳ú➡è✌ô❉ó➪õ②ó➪ï❢ô✇ï➁í❑é②î➪ó①ú◆ó➪ô❉ò❥é②ê❉ö➔í➶÷➍ëìè➳÷ré✹÷✓✑❭ï✓✎➍è☛✲➁ü✐è ç❑÷✏✎❢è✿ö②è❫ú➡ê❑ë❾è❫ú✼õ➃ç❉è✇ú◆è✉é➃ó➪ò➍ô✙ö➃è✜✌❢ê❉ó➪ö②è✞✡➲è✉ô◆õ➹í❇ö➃ï☛✡✒õ②ö➏÷❢ë✳❀❵ó➪ô❉ò ★✡ ý✹✸ ☎ õ②ï✾✑❭è✉ó➪ô❉ò✙ï❢ô✚õ②ç❉è❋é➃ê❉ö❤í➶÷❢ë❾è ✭ ★✠ÿ ✝✿✾ô❭ë❾è❳ï❢ô✚õ②ç❉è é②î➪ó①ú◆ó➪ô❉ò➲é②ê❉ö➔í➶÷➍ë❾è✬✲❨õ➃ç❉è❳ú◆ñ◆ô❑÷✓✡➲ó①ë❾é✿ý▲è✜✌❢ê❑÷rõ➃ó↕ï❢ô❀✳ ☎ ó↕é❥è❾ø❉ù❭ï✓✟ ô❉è✉ô◆õ➃ó ÷➁î➪î↕ñ➲é②õ❾÷✓✑❉î➪è✇÷➁ô❑ú✙÷ré➃ñ✠✡❳ù❉õ➃ï➍õ②ó①ë õ②ö➏÷❁✓❾è❫ë❾õ➃ï➍ö②ñ➙õ②ö➏÷❢ë✳❀❵ó➪ô❉ò ó➪é✿÷➍ë➃ç❉ó➪è✞✎➍è❅ú ✝ æ➹ç❉è✇ë❾ï❢ô◆õ②ö②ï❢î✺✰➙ó↕é❥ú◆è✉é➃ó↕ò❢ô❉è❫ú➙õ②ï✱✡➮÷✓❀❢è❥õ②ç❉è é②ê❃ö❤í➶÷❢ë❾è ✭ ★✦ÿ④÷rõ➃õ②ö➏÷❢ë❾õ②ó✄✎❢è❞÷➁ô❑ú➲õ②ï④ö②è❫÷❢ë➃ç➮õ➃ç❉è✿é➃ê❉ö❤í➶÷❢ë❾è❥ó➪ô ✌ô❉ó➪õ➃è✿õ②ó✄✡❳è ✝ ✓■ï➍ô❉é➃ó①ú◆è✉ö➹õ②ç❃è❥í❇ï➍î➪î➪ïrü➹ó➪ô❉ò❥î➪ñ❵÷rù❃ê❉ô❉ï✓✎④í❇ê❉ô❭ë❾õ②ó➪ï➍ô❛✑ ❂✹★ ✍ ✻ ✭ ✿ ý✴❃ ☎ ✼❤õ②é❍õ②ó✄✡❳è➳ú➡è✉ö②ó✄✎➍÷rõ➃ó✄✎➍è❥ó➪é➹ò➍ó✄✎➍è✉ô✱✑◆ñ ❂✹★✭ ✭ ✭✭ ý✴❄ ☎ ★é❬õ②ç❉è➋é②ñ◆é②õ➃è✞✡✜ç❑÷ré❬ö②è✉î①÷➁õ②ó✄✎➍è✹ú◆è✉ò❢ö②è✉èP✍✌ü➹ó↕õ➃ç ✭ ÷réqï❢ê❉õ②ù❉ê❃õ✜✲ ü■è✇ë✉÷rô❚é➃ï➍î✄✎➍è í❇ï➍ö✩✰➲í▲ö②ï☛✡✠õ②ç❉è❥è✜✌❢ê❑÷➁õ②ó➪ï➍ô ✭✭ ★ ✫✆ é➃ò➍ôqý ✭☎ ý✽✍❫ÿ ☎ æ➹ç❉ó➪é➳ö➃è✉é②ê❃î↕õ➃é➳ó➪ô✙÷④ô❉è✉ò◆÷rõ②ó✄✎❢è❞ú➡è✌ô❉ó➪õ②è ❂✭ ÷rô❑ú❚÷➁î↕é➃ï❋ò➍ê❑÷➁ö✗✟ è✉ô◆õ②èsè✉é ✌ô❉ó➪õ②è➹õ➃ó✄✡❳è➹ë❾ï➍ô✠✎➍èsö②ò➍èsô❑ë❾è➳õ➃ï✿õ②ç❃è➹é②î➪ó①ú◆ó➪ô❉ò❥é②ê❉ö➔í➶÷➍ë❾è ✝ ❙ê❉õ✜✲rõ➃ç❉è✐ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö❨ü➹ó➪î➪îrö②è✉é➃ê❉î➪õ❨ó➪ô❥ç❉ó➪ò➍ç❍í❇ö②è✏✌➍ê❃è✉ô❑ë❾ñ✇ë➃ç❑÷➁õ✗✟ õ②èsö②ó➪ô❉ò✿ô❉è❅÷rö✌é②î➪ó①ú◆ó➪ô❉ò➳é➃ê❉ö❤í❤÷➍ë❾è ✝ æ✌ï✿÷✏✎➍ï❢ó①ú➳õ➃ç❉ó➪é✜✲❢ü✐è■ê❉é➃è➹õ②ç❉è í❇ï❢î↕î➪ïrü➹ó➪ô❉ò✿èìø➡ù❃ö②è✉é➃é②ó➪ï➍ô✙õ➃ï❋é②ï❢î✩✎❢è➹í❇ï❢ö✩✰ ✝ ✿ ✭✭ ★ ✫✆ ✭ ý✽✍☛✍ ☎ ❅ ❂✭ ★ ✫✆ ✭ ✿ ý✽✍❇✻ ☎ ❆ ❃❅❄✎❇❄✗❉✦❏✍■❉❈❫❋❊✯❇■✣❉✦❑✮❄✗❏✆❇✳▲ ❢ ❆❂■●❇❞❆✍❢ ▲❑✮▲▼✞❙❵❱❣■✞❑✮❄❲❊✗■✣▼✣❏✍❉✍■✣❢ ❢ ❄✗❉❅❑✞❄✗❆✍▲ ❙❚▼✔❤ ❍✂■▲ ❏✍❋❣❏✍❋✞▲ ❆✠❊✗■❚▼✣❏✍❉✭■❚❢ ❢ ❄✗❉✬❏❈❏✍❋✞❄ ❏✍❉✆❇❋❑✍❄✗❊✗❏✍■✣❉✍❖✦▲ ❆☛▼✮■❚❏✬❙✣●✜❇✳❉✍❄✗▼✣❏✍❄✗❄✗❑❲❏✍■❅❉✍❄❈❇✳❊✻❋ ❏✍❋✞❄❣❆✍❢ ▲ ❑✞▲▼✮❙✒❆✍●✮❉▲❇✩❇✳❊✗❄❲▲ ▼❉▼▼✞▲ ❏✍❄❣❏✍▲❱❣❄✣❤
d xpanding both sides of the equation, we get results. Software tools were also available for the design and simulation of the PID and sliding mode 3akg 5ug A1zk g A222 g A132 g A221-A2yd) Desired A strollers for this problems, and so these two con- oilers were easy to implement The u obtained from this equation will be used for B. Analytical Tools for the Analysis of Fuzzy Logic Control A control engineer adding the fuzzy logic control This controller needs access to all three states, tool to her toolbox already full of classical control whereas the fuzzy logic controller of [0, 0] make use tools, might ask "Why does this method demon of only a1 and a2. To have a fair comparison, we strate robustness ?", and "When will it fail? "Thes will construct an observer to get an estimate of the questions often have mathematically analytical an state variable ak swers for classical control tools. To satisfy this in ll for the continued development of math Standard Luenburger observer is designed to get ematically analytical tools to answer such questions an estimate of ak. We use the model of (1)to design to the satisfaction of the control engineering com the observer as follows b munitY A Aig bug L(y-y (13) C. Fuzzy Logic Control: A Useful Tool y a ci From our brief study, we conclude that fuzzy logic control should have a place in the control engineers where, the matrices A, b and c correspond to the toolbox. Only time, experience, and further anal model in equation(1). The matrix L is chosen such vsis will determine whet her fuzzy logic control be- L)are in the left half plane and the error dynamics comes a prominent tool in the control toolbox used that the eigenvalues of observer error dy namics(A- by devotees of classical control techniques is faster than the dy namics on the sliding surface The closed loop step responses with this controller VIII. FUTURE WORK re shown in Appendix A. The controller is robust We encourage more extensive fair and unbiased bench- to input gain changes as well as to the changes in mark comp arisons of fuzzy logic control with classi system dynamics while keeping the input magnitude cal control techniques for the benet of the practic- ee Noteb With the use of equation(11), the con- ing control engineer, and we hope to become a part of this effort oller is essentially a linear state feedback controller. In the future, we hope to contribute to the e cre- It Ilts in closed loop eigenvalues of-13392 ation of new analytical tools for the analysis of fuzzy 4.33+j 1.91 for model(1). The same controller, logic controllers. We also hope to be able to con- when applied to other plants will give the following tribute to the continuing effort to identify classes of closed loop eigenvalue locations b problems where rule based fuzzy logic control tech Model of(2)→-1,32,-355,-19.12 niques have advantages over and are more appropri Thus, by design, the increase in gain makes the ate than classical control techniques closed loop system more stable Model of(3)→-1.4,-3.79±j2.62 X. ACKNOWLEDG)ENTS ⅤII. CONCLUSIONS The authors would like to thank Professor S. Shankar A. Comment ontroller Desic Sastry for his gracious support and encouragement The aut hors would also like to thank Dr. Shahram The control Icient in PID and sliding M. Shahruz for helpful di mode control techniques can readily sy nt hesize ro- bust controllers to perform the task dictated by this ReFERENCE S xample. As demonstrated in [0] and veried in this [1 H. T Nguyen, C. W. Tao, W.d. Thompson study, the control engineer utilizing fuzzy control An d mpirical Study of c obus tness of Fuzzy techniques can readily achieve the same goal. Thus Systems'", Proc. of 2nd NEE Nitl. Conf.on l three control tools Fuzzy-ystems, pp. 1340-1345 gineer for use in synthesizing an acceptable robust c. M Tong, An Annotated Bibliograph controller for this control problem of Fuzzy Control",, in Nadustrial Amdications with the appropriate software tools available, the g oller for this problem was relatively Holland, Amsterdam, Holland, 1985, pp249 simple to implement, and it provided satisfactory 269
✥ø❉ù❑÷➁ô❑ú◆ó➪ô❉ò✤✑❭ï➍õ➃ç➮é➃ó①ú◆è✉é➹ïrí✹õ➃ç❉è❥è✜✌❢ê❑÷rõ➃ó↕ï❢ô❲✲◆ü■è➬ò❢è✉õ ✭✭ ★ ✫ ✻ ✣ ✿ ✫ ❂ ✣ ❀ ✸ ✶ ✰✱✸ ✰ ❭ ✣ ❀ ✸ ✰ ✿ ✣ ✿ ★ ✫✆✼ý ✣ ❀ ✸✱✰ ❭ ✣ ✿ ✸✠✰ ✿ ✣ ❭ ✫ ✰ ✿ ★✡ ☎ ❻❍è✉é➃ó↕ö➃è❫ú æ➹ç❉è✏✰❚ï✬✑❉õ➏÷➁ó↕ô❃è❫ú➲í❇ö➃ï☛✡✒õ②ç❉ó➪é➳è✏✌➍ê❭÷rõ②ó➪ï❢ô✙ü➹ó➪î↕î❂✑❭è④ê❉é②è❫ú➐í❇ï❢ö õ➃ç❉è✇ë❾ï❢ô❵õ➃ö②ï❢î ✝ æ➹ç❉ó➪é➮ëìï➍ô◆õ②ö➃ï➍î➪î➪è✉ö❳ô❉èsè❫ú◆é✙÷➍ë✉ëìè✉é②é✙õ➃ï ÷➁î↕î➹õ➃ç❉ö②èsè✚é②õ➏÷➁õ②è✉é✏✲ ü➹ç❉èsö②è❫÷➁é➳õ②ç❃è➳í▲ê❉ð✉ð✉ñ❋î➪ï➍ò❢ó①ë➳ë❾ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö■ïrí✐þÿ ✲❵ÿ ✁ ✡➮÷✓❀❢è➹ê❉é②è ï➁í➳ï➍ô❉î➪ñ ✣ ❭ ÷rô❑ú ✣ ✿ ✝ æ✌ï❚ç❑÷✏✎➍è❳÷➮í➶÷ró➪ö✿ë❾ï☛✡➲ù❑÷rö➃ó➪é②ï➍ô①✲qü✐è ü➹ó➪î➪î❨ë❾ï➍ô❃é②õ②ö➃ê❑ë❾õ❥÷rô❚ï☛✑❉é➃è✉ö❈✎❢è✉ö➳õ➃ï❋ò➍è✉õ➳÷➁ô❳è✉é➃õ②ó✄✡➮÷➁õ②è❥ïrí✌õ②ç❉è é➃õ➏÷rõ➃è☞✎➍÷rö➃ó①÷✓✑❉î➪è ✣ ❀ ✝ ✯ õ❾÷rô❑ú❉÷➁ö➏ú✔❤ ê❉è✉ô✠✑❉ê❉ö➃ò➍è✉ö❝ï☛✑❉é➃è✉ö❈✎❢è✉ö❥ó↕é❥ú◆è✉é➃ó↕ò❢ô❉è❫ú➙õ②ï➲ò➍è✉õ ÷➁ô✇è✉é➃õ②ó✄✡➮÷➁õ②è■ïrí ✣ ❀ ✝❲✇è✹ê❃é②è➹õ②ç❃è❵✡❳ï❉ú◆è✉î◆ïrí❬ý❃✍ ☎ õ➃ï❞ú➡è✉é②ó➪ò➍ô õ➃ç❉è✿ï☛✑❃é②è✉ö✳✎➍è✉ö❥÷ré■í❇ï❢î↕î➪ïrü➹é✾✑ ✭✣ ★ ✁✂✣ ✸☎✄ ✰✱✸☎✆■ý ★✱✫ ★ ☎ ý✽✍❇❂ ☎ ★ ★ ✝✞✣ ü➹ç❉èsö②è☛✲■õ②ç❃è ✡❚÷rõ➃ö②ó①ë❾è✉é✟✁❦✲✠✄❳÷rô❭ú✡✝❳ëìï➍ö②ö➃è✉é②ù❭ï➍ô❭ú❴õ②ï✼õ②ç❉è ✡➲ï➡ú➡è✉î❉ó➪ô❳è✜✌❢ê❑÷➁õ②ó➪ï➍ô❚ý✽✍ ☎✮✝ æ➹ç❉è✫✡❚÷rõ②ö➃ó➟ø☛✆✚ó➪é✐ë➃ç❉ï❢é②è✉ô❚é②ê❭ë②ç õ➃ç❑÷rõ➹õ➃ç❉è❥è✉ó➪ò➍è✉ô✠✎➍÷rî➪ê❉èsé✐ï➁í➘ï☛✑❃é②è✉ö✳✎➍è✉ö❍è✉ö②ö➃ï➍ö➳ú◆ñ◆ô❑÷✔✡❳ó①ë❾é➹ý★✟ ❤❛✓ ☎ ÷➁ö②è■ó➪ô✿õ②ç❉è■î➪è❾í❇õ✌ç❑÷➁î➈í❉ù❉î①÷➁ô❉è ÷➁ô❑ú❥õ➃ç❉è✐èsö②ö②ï❢ö✐ú◆ñ◆ô❑÷✔✡❳ó①ë❾é ó➪é■í➶÷➁é②õ②èsö➳õ②ç❭÷rô❳õ➃ç❉è✇ú◆ñ◆ô❑÷✔✡❳ó①ë❾é✹ï❢ô❚õ②ç❉è✿é➃î➪ó ú➡ó↕ô❃ò✿é②ê❉ö➔í➶÷➍ëìè ✝ æ➹ç❉è✐ëìî↕ï❢é②è❫ú❞î↕ï◆ï❢ù➬é➃õ②è✉ù④ö②èsé②ù❭ï➍ô❉é➃è✉é✹ü➹ó➪õ②ç✿õ➃ç❉ó➪é❨ë❾ï➍ô◆õ➃ö②ï➍î➪î➪è✉ö ÷➁ö②è④é②ç❉ïrü➹ô✼ó➪ô ★ù❉ù❭è✉ô❑ú➡ó➈ø ★ ✝ æ➹ç❃è✈ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö➬ó➪é❍ö②ï☛✑❃ê❉é②õ õ➃ï✙ó➪ô❉ù❉ê❉õ✿ò◆÷ró➪ô✚ë②ç❭÷rô❉ò❢è✉é✇÷ré❝ü✐è✉î➪î✐÷➁é✿õ②ï➙õ②ç❉è❳ë➃ç❑÷➁ô❉ò➍è✉é④ó➪ô é➃ñ❵é➃õ②è✞✡ ú◆ñ◆ô❑÷✓✡➲ó①ë❾éqü➹ç❉ó➪î➪è✖❀➍è✉èsù❉ó➪ô❉ò➳õ➃ç❉è✐ó➪ô❉ù❃ê❉õ❲✡❚÷rò➍ô❃ó↕õ➃ê❑ú◆è é✳✡➮÷rî➪î ✝ ✕ï❢õ②è☛✑ ✇ó➪õ②ç✣õ➃ç❉è✍ê❃é②è✼ïrí✿è✜✌❢ê❑÷rõ➃ó➪ï➍ô✜ý✽✍☛✍ ☎ ✲➹õ②ç❃è ë❾ï❢ô✠✟ õ➃ö②ï➍î➪î➪è✉öqó➪é❨èsé②é②èsô❵õ➃ó①÷rî➪î➪ñ➬÷■î➪ó➪ô❉è❫÷röqé➃õ➏÷➁õ②è✌í❇èsè❫ú✠✑❑÷❢ë❈❀✿ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö ✝ ✼❤õ❚ö②èsé②ê❉î➪õ②é➙ó↕ô❙ë❾î➪ï➍é②è❅ú❙î↕ï◆ï❢ù✣è✉ó➪ò➍è✉ô✠✎➍÷➁î↕ê❃è✉é❳ï➁í☞✟❉✍ ✝ ❂☛❂ ❄❜✻✌☞ ✟✬✍ ✝ ❂☛❂✎✍✓✁✍ ✝ ❄✆✍ûí❇ï➍ö✱✡❳ï❉ú◆è✉î❥ý✽✍ ☎✞✝ æ➹ç❉è✙é❾÷✓✡➲è➐ëìï➍ô◆õ②ö➃ï➍î➪î➪è✉ö✜✲ ü➹ç❉èsô➐÷➁ù❉ù❉î➪ó➪è❫ú❋õ➃ï❋ï➍õ➃ç❉è✉ö➹ù❉î①÷rô◆õ➃é➹ü➹ó↕î➪î❭ò➍ó✄✎➍è❍õ②ç❉è❍í❇ï❢î➪î↕ïrü➹ó➪ô❉ò ë❾î➪ï❢é②è❫ú❚î➪ï◆ï➍ù➲è✉ó➪ò➍è✉ô✠✎➍÷➁î↕ê❃è➳î➪ï❉ë✉÷rõ➃ó↕ï❢ô❉é❇✑ ☞❚ï❉ú◆èsî❨ïrí■ý✷✻ ☎ ❅ ✫ ✍ ✑ ❂☛✻✒✘ ✫ ❂✕✑ ✶❜✶ ✘ ✫ ✍ ❄✕✑ ✍❇✻ æ➹ç◆ê❉é✏✲☞✑◆ñ ú➡è✉é②ó➪ò➍ô①✲✿õ②ç❃è✚ó↕ô❭ë❾ö②è❫÷➁é②èâó↕ô✜ò◆÷ró➪ô❘✡➮÷✓❀❢è✉é❚õ②ç❉è ë❾î➪ï❢é②è❫ú❚î➪ï◆ï➍ù➲é②ñ◆é②õ➃è✞✡ ✡❳ï❢ö②è➹é➃õ➏÷✓✑❃î↕è ✝✩✝✄✝ ☞❚ï❉ú◆èsî❨ïrí■ý✷❂ ☎ ❅ ✫ ✍ ✑ ✍ ✘ ✫ ❂✕✑ ✳ ❄✏✍✒✑☛✻✕✑ ✫☛✻ ❊❋➱s➱s✃ ⑤➬❰q❐❨Ñ❅⑩❢Ð❲❸✉Ò▲❰q❐❲❸ Ó❝Ô❳Ø✐Ù➁ã✇ã❋Ö✉Ú➡Û❇ß✇ÙrÚ Ø✐ÙrÚ◆Û❤Ü➏Ù➁Ý①Ý↕ÖsÜ✿✚❞Öìßìä❴✸❱Ú◆ß æ➹ç❉è❚ë❾ï➍ô◆õ➃ö②ï➍î■è✉ô❉ò❢ó➪ô❉è✉è✉ö④ù❉ö②ï✌ë❾ó➪è✉ô◆õ④ó↕ô ❺ ✼❈❻✫÷➁ô❑ú✼é②î➪ó①ú◆ó➪ô❉ò ✡➲ï➡ú➡è✈ë❾ï❢ô◆õ②ö②ï❢î✹õ②è❅ë②ç❃ô❉ó✍✌➍ê❃è✉é✈ës÷rô✚ö②è❅÷➍ú◆ó➪î➪ñ✙é②ñ◆ô◆õ②ç❃è✉é②ó➪ð✉è➲ö②ï✓✟ ✑❉ê❃é②õ➹ë❾ï➍ô◆õ②ö➃ï➍î➪î➪è✉ö②é➋õ②ï❥ù❑èsö❤í❇ï❢ö❈✡ õ②ç❉è➹õ❾÷ré✳❀✈ú◆ó①ë❾õ❾÷rõ➃è❫ú✧✑◆ñ✿õ②ç❉ó➪é è❾ø❑÷✔✡❳ù❉î➪è ✝ ★é✐ú◆è✞✡➲ï➍ô❉é➃õ②ö❾÷rõ②è❅ú④ó↕ô❚þÿ✁ ÷rô❭ú☞✎❢è✉ö②ó✌è❫ú④ó➪ô④õ②ç❉ó➪é é➃õ②ê❑ú◆ñ✠✲➹õ➃ç❉è ëìï➍ô◆õ②ö➃ï➍î❥è✉ô❉ò❢ó↕ô❃è✉è✉ö❚ê❉õ②ó➪î➪ó➪ð✉ó➪ô❉ò➙í❇ê❉ð✉ð✉ñ❙ë❾ï➍ô◆õ②ö➃ï➍î õ➃è❫ë➃ç❉ô❉ó✍✌❢ê❉è✉é➳ë✉÷➁ô❋ö②è❫÷❢ú◆ó➪î↕ñ④÷➍ë➃ç❉ó➪è✞✎➍è❍õ②ç❉è➹é❾÷✓✡➲è✐ò❢ï❵÷rî ✝ æ➹ç◆ê❉é✜✲ ÷➁î↕î◆õ➃ç❉ö②è✉è➹ë❾ï❢ô❵õ➃ö②ï❢î❉õ②ï◆ï➍î➪é✌÷rö②è➹÷✏✎➍÷ró➪î①÷✓✑❃î↕è❬õ②ï❥õ②ç❃è ë❾ï❢ô◆õ②ö②ï❢î◆è✉ô✠✟ ò❢ó↕ô❃è✉è✉ö❥í❇ï❢ö❥ê❉é②è❋ó➪ô✙é➃ñ❵ô◆õ➃ç❉è✉é②ó➪ð✉ó➪ô❉ò✼÷rô✚÷➍ë✉ë❾èsù❉õ➏÷✔✑❉î➪è❋ö②ï☛✑❃ê❉é②õ ë❾ï❢ô◆õ②ö②ï❢î➪î↕èsö■í❇ï➍ö➹õ➃ç❉ó➪é➳ë❾ï➍ô◆õ②ö➃ï➍îqù❉ö➃ï☛✑❉î➪è✞✡ ✝ ✇ó➪õ②ç❝õ②ç❉è➳÷➁ù❉ù❉ö②ï❢ù❉ö②ó①÷➁õ②è■é②ïrí▲õ❤ü÷➁ö②è■õ②ï◆ï➍î➪é✹÷✏✎➍÷ró➪î①÷✓✑❉î➪è☛✲sõ②ç❉è í❇ê❃ð✉ð✉ñ❋î➪ï➍ò❢ó ë■ë❾ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö❁í❇ï➍ö✌õ②ç❃ó↕é■ù❉ö➃ï☛✑❉î➪è✞✡ ü÷➁é✹ö②è✉î①÷➁õ②ó✄✎➍è✉î➪ñ é➃ó✩✡➲ù❉î➪è❳õ➃ï✚ó✩✡➲ù❉î➪è✞✡❳èsô❵õ✏✲✹÷rô❑úâó➪õ❋ù❉ö➃ï✓✎❵ó①ú◆è❅ú☛é❾÷rõ➃ó↕é➔í➶÷➍ëìõ②ï➍ö➃ñ ö②èsé②ê❉î➪õ②é ✝ ✯ ï➁í❇õ❤ü➹÷rö②è➲õ②ï◆ï➍î➪é④ü✐èsö②è✍÷➁î↕é➃ï ÷✏✎➍÷➁ó➪î ÷✔✑❉î➪è✿í▲ï➍ö④õ②ç❉è ú◆è✉é➃ó➪ò➍ô✍÷➁ô❑ú➲é②ó✄✡❋ê❉î①÷rõ➃ó↕ï❢ô❋ïrí✌õ②ç❉è❦❺ ✼❈❻ ÷rô❭ú❚é②î➪ó①ú◆ó➪ô❉ò✧✡❳ï❉ú◆è ë❾ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö➃é➹í❇ï❢ö➹õ②ç❉ó➪é➹ù❉ö➃ï☛✑❉î➪è✞✡➲é✜✲❉÷➁ô❑ú➲é②ï❋õ②ç❃è✉é②è④õ❤ü✐ï❳ëìï➍ô✠✟ õ②ö➃ï➍î➪î➪è✉ö②é❍ü✐èsö②è❥è❫÷ré➃ñ❚õ②ï❋ó✄✡❳ù❉î➪è✞✡❋èsô❵õ ✝ ✙Ô✿Ó❥Ú✦✢rÝ✔✓❅Û➶ä✆❥✣✢➁Ý✹ÞÙ❅ÙrÝ➤ß ✹❫Ù➁Ü❥Û✻✺➡Ö➹Ó❥Ú✦✢➁Ý✕✓❅ß✉ä➪ß✇Ù✗✹♦♥✈✎✖✗✖✘✓✧❧ Ù❝✸❱ä✻❥ Ø✐ÙrÚ◆Û❤Ü➏Ù➁Ý ★ ë❾ï➍ô◆õ➃ö②ï➍î❥è✉ô❃ò➍ó➪ô❉è✉è✉ö✙÷❢ú❉ú◆ó➪ô❉ò✚õ②ç❃è✙í❇ê❃ð✉ð✉ñ✣î➪ï➍ò❢ó①ë➮ëìï➍ô◆õ②ö➃ï➍î õ②ï◆ï❢î✐õ➃ï➮ç❃è✉ö✇õ➃ï◆ï➍î✄✑❑ï❅ø✍÷➁î↕ö➃è❫÷➍ú➡ñ✙í❇ê❃î↕î✌ï➁í ë❾î①÷➁é②é②ó①ë✉÷➁î ëìï➍ô◆õ②ö➃ï➍î õ②ï◆ï❢î↕é✏✲✫✡➲ó➪ò➍ç◆õ✈÷➁é❈❀✚✙✇ç❵ñ✣ú◆ï◆è✉é❚õ➃ç❉ó➪é✘✡❳èsõ②ç❉ï❉ú✣ú◆è✮✡❳ï➍ô✟ é②õ➃ö➏÷➁õ②è❥ö②ï✬✑❉ê❉é②õ➃ô❉è✉é➃é✜✛✣✢✦✲❭÷rô❑ú✒✙✇ç❃è✉ô❳ü➹ó➪î➪î❉ó➪õ✌í➶÷➁ó↕î✤✛✣✢❚æ➹ç❉è✉é➃è ✌❢ê❉è✉é➃õ②ó➪ï➍ô❉é❥ï➁í❇õ②è✉ô✼ç❑÷✏✎❢è☞✡➮÷➁õ②ç❉è✞✡❚÷rõ➃ó ës÷rî➪î➈ñ❋÷rô❑÷➁î➪ñ❵õ➃ó①ë✉÷rî✌÷rô✠✟ é②ü■è✉ö➃é✿í❇ï❢ö➬ë❾î①÷➁é②é②ó①ë✉÷➁î✐ë❾ï❢ô❵õ➃ö②ï❢î✹õ②ï◆ï➍î➪é ✝ æ✌ï❳é❾÷rõ②ó➪é➔í❇ñ✙õ②ç❃ó↕é❥ó➪ô✠✟ ✌❢ê❉ó➪ö②ñ✠✲✉ü■è✐ë✉÷➁î↕î❅í▲ï➍öqõ②ç❃è✐ë❾ï➍ô◆õ➃ó↕ô◆ê❉è❅ú✿ú◆è✞✎❢è✉î➪ï➍ù②✡➲è✉ô◆õ❑ï➁í ✡❚÷rõ➃ç✠✟ è✞✡❚÷rõ②ó①ë✉÷➁î➪î↕ñ④÷rô❑÷➁î↕ñ◆õ➃ó ës÷rî◆õ②ï◆ï➍î➪é✹õ➃ï✈÷➁ô❉é②ü■è✉ö■é②ê❑ë➃ç④✌➍ê❃è✉é②õ➃ó↕ï❢ô❉é õ②ï✼õ②ç❃è❳é➏÷➁õ②ó➪é❤í❤÷➍ë❾õ➃ó↕ï❢ô✚ïrí➳õ➃ç❉è➮ë❾ï❢ô◆õ②ö②ï❢î✐èsô❉ò➍ó➪ô❉è✉èsö②ó➪ô❉ò✚ë❾ï☛✡❽✟ ✡❋ê❉ô❉ó➪õ❤ñ✝ Ø✹Ôt♥✈✎✖✗✖✘✓✧❧ Ù❝✸❱ä✻❥❋Ø✐Ù➁Ú◆Û➶Ü❾ÙrÝ✔✥✐Ó✧✦ß❫Ö✍✹✞✈◆Ý✹ÞqÙ❅ÙrÝ s❃ö②ï☛✡ ï➍ê❉ö❯✑❉ö➃ó➪è❾í✐é②õ➃ê❑ú◆ñ✠✲qü■è❳ë❾ï➍ô❑ëìî↕ê❭ú◆è✇õ➃ç❑÷rõ❍í❇ê❃ð✉ð✉ñ✼î↕ï❢ò➍ó①ë ë❾ï❢ô❵õ➃ö②ï❢î❑é➃ç❉ï➍ê❉î①ú④ç❑÷✏✎❢è➳÷❥ù❉î①÷➍ëìè➹ó↕ô❋õ➃ç❉è➬ëìï➍ô◆õ②ö➃ï➍î❭è✉ô❉ò➍ó➪ô❉èsè✉ö❇❅ é õ②ï◆ï❢î✩✑❭ï❅ø ✝✽✾ô❉î➪ñ✼õ②ó✄✡❳è☛✲❬è❾ø❉ù❭è✉ö②ó➪è✉ô❑ëìè☛✲➹÷rô❑ú➙í❇ê❃ö②õ②ç❃è✉ö❳÷rô❑÷➁î ✟ ñ◆é②ó➪é✿ü➹ó➪î➪î✹ú◆è✉õ②èsö❈✡❳ó➪ô❉è❋ü➹ç❃è✉õ②ç❉èsö❥í❇ê❉ð✉ðsñ✍î➪ï➍ò❢ó①ë✇ë❾ï❢ô◆õ②ö②ï❢î✖✑❭è✣✟ ë❾ï✬✡❳è✉é➳÷④ù❉ö➃ï☛✡➲ó↕ô❃è✉ô◆õ✹õ②ï◆ï➍îqó➪ô❚õ②ç❃è❞ëìï➍ô◆õ②ö➃ï➍î❨õ➃ï◆ï➍î✄✑❑ï❅ø❳ê❃é②è❫ú ✑◆ñ➮ú◆è✮✎➍ï➍õ➃è✉è✉é❍ïrí✐ë❾î①÷ré➃é②ó①ë✉÷rî❨ëìï➍ô◆õ②ö➃ï➍î❨õ➃è❫ë➃ç❉ô❉ó✍✌❢ê❉è✉é ✝ ❊❋➱✉➱s➱s✃✎▼✹Ðq❒qÐq❮❂⑦✩★✜❰ ❮✫✪ ✇è✌è✉ô❑ë❾ï❢ê❉ö➏÷➁ò➍è✽✡❳ï❢ö②è❨èìø➡õ➃è✉ô❉é➃ó✩✎❢è✌í➶÷➁ó↕ö❨÷➁ô❑ú❍ê❉ô✠✑❉ó①÷ré➃è❫ú✿✑❑è✉ô❭ë②ç✟ ✡➮÷➁ö❈❀❋ë❾ï☛✡➲ù❑÷rö➃ó➪é②ï➍ô❃é✹ïríqí❇ê❃ð✉ð✉ñ❋î➪ï➍ò❢ó ë➹ëìï➍ô◆õ②ö➃ï➍î❭ü➹ó↕õ➃ç➮ë❾î①÷➁é②é②ó ✟ ë✉÷➁î❨ë❾ï➍ô◆õ②ö➃ï➍îqõ➃è❫ë➃ç❉ô❉ó✍✌❢ê❉è✉é➹í▲ï➍ö■õ②ç❉è✿✑❑è✉ô❃è✌õ➳ï➁í❨õ②ç❉è❥ù❉ö❾÷➍ë❾õ➃ó①ë✳✟ ó➪ô❉ò❋ë❾ï➍ô◆õ②ö➃ï➍îqè✉ô❃ò➍ó➪ô❉è✉è✉ö✏✲❑÷➁ô❑ú❋ü■è❥ç❉ï➍ù❭è➳õ➃ï❦✑❭è❫ë❾ï✬✡❳è➬÷❝ù❑÷➁ö②õ ïrí✹õ➃ç❉ó➪é➹è✽❈ï❢ö②õ ✝ ✼❤ô✚õ②ç❉èûí❇ê❉õ➃ê❉ö②è☛✲✌ü■è❋ç❉ï❢ù❑è➲õ②ï✍ë❾ï❢ô◆õ②ö②ó✄✑❉ê❃õ②è❳õ➃ï✙õ②ç❃è➮ë❾ö➃è✣✟ ÷rõ➃ó➪ï➍ô✿ï➁í❑ô❉è✉ü❴÷rô❑÷➁î↕ñ◆õ➃ó ës÷rî➍õ➃ï◆ï➍î➪éqí❇ï❢ö❨õ➃ç❉è ÷➁ô❑÷rî➪ñ◆é②ó➪éqï➁í❉í❇ê❉ð✉ðsñ î➪ï➍ò❢ó ë❋ëìï➍ô◆õ②ö➃ï➍î➪î➪è✉ö②é ✝✾✇è❳÷rî➪é➃ï✙ç❉ï➍ù❭è❳õ➃ï✾✑❭è➮÷✓✑❃î↕è❋õ➃ï ëìï➍ô✠✟ õ②ö➃ó✄✑❉ê❉õ②è❥õ➃ï✿õ②ç❉è✿ë❾ï❢ô◆õ②ó➪ô◆ê❉ó➪ô❉ò❥è✽❈ï❢ö②õ➹õ➃ï✿ó①ú◆è✉ô◆õ②ó➟í❇ñ❋ë❾î①÷ré➃é②è✉é➹ï➁í ù❉ö➃ï☛✑❉î➪è✞✡➲é➹ü➹ç❉è✉ö②è❝ö②ê❉î➪è❯✑❑÷ré➃è❫ú➲í❇ê❃ð✉ð✉ñ❳î➪ï❢ò➍ó①ë➳ë❾ï❢ô◆õ②ö②ï❢î❨õ➃è❫ë➃ç✠✟ ô❉ó✍✌❢ê❉è✉é✹ç❭÷✏✎➍è➳÷➍ú✎➍÷rô◆õ➏÷➁ò➍è✉é✌ï✓✎➍èsö❁÷➁ô❑ú❋÷rö➃è❵✡❳ï❢ö②è✐÷➁ù❉ù❉ö②ï❢ù❉ö②ó ✟ ÷rõ➃è✿õ②ç❭÷rô✙ë❾î①÷ré➃é②ó①ë✉÷rî❨ëìï➍ô◆õ②ö➃ï➍î❨õ➃è❫ë➃ç❉ô❉ó✍✌❢ê❉è✉é ✝ ➱✭✬❚✃✞●✿Ñ✮✪❬❐❨❰✮✯✤⑩☛⑦❉Ï ✚✲☎✿⑦❉❐q❒❲❸ æ➹ç❉è✐÷➁ê❉õ②ç❃ï➍ö②é❬ü✐ï❢ê❉î①ú➹î➪ó✄❀➍è✌õ②ï➹õ➃ç❑÷rô❀q❺✌ö➃ïrí❇è✉é➃é②ï❢ö ✯ ✝ ✯ ç❑÷rô❀➍÷rö ✯ ÷➁é②õ②ö➃ñ❋í❇ï❢ö✐ç❃ó↕é■ò➍ö❾÷➍ë❾ó➪ï❢ê❉é✐é➃ê❉ù❉ù❭ï➍ö➃õ➳÷rô❑úûè✉ô❑ë❾ï❢ê❉ö➏÷➁ò➍è✞✡➲è✉ô◆õ ✝ æ➹ç❉è✇÷➁ê❉õ②ç❃ï➍ö②é❍ü✐ï❢ê❉î①ú❚÷rî➪é②ï❋î➪ó✄❀➍è❥õ②ï❋õ➃ç❑÷rô❀ ❻❍ö ✝ ✯ ç❑÷➁ç❉ö➏÷✔✡ ☞ ✝ ✯ ç❑÷rç❃ö②ê❉ð❍í❇ï❢ö➹ç❉è✉î➪ù◆í❇ê❉î❨ú➡ó↕é❾ë❾ê❉é➃é②ó➪ï➍ô❉é ✝ ❑✫⑦②⑧✠⑦❉❮❂⑦❃❐❨Ñ❅⑦✦❸ þ ✍ ✁ ✰☞✝ æ ✝ ✕ò➍ê❉ñ❢è✉ô❲✲■✓ ✝ ✟✇✝ æ✹÷rï❅✲ ✇✝■✥❵✝ æ➹ç❉ï☛✡➲ù❉é②ï❢ô❲✲ ✙★ô ✥✡❳ù❃ó↕ö➃ó①ë✉÷rî ✯ õ➃ê❑ú◆ñ❚ïrí✺❥ï☛✑❉ê❉é➃õ②ô❉èsé②é✿ï➁í❾s❃ê❉ð✉ðsñ ✯ ñ◆é②õ➃è✞✡❳é✱✢✦✲ ✐✐Ü❾Ù✜❥sÔ➬Ùr✹✳✲➍Ú❅✥✔✕❾á➋á✹á ✕✉Ú◆Û❤Ý➈Ô❝Ø✐ÙrÚ✞✹✉Ô❥ÙrÚ ♥✈✎✖✗✖✘✓◗✟✞✓❅ß❾Û➔Ösã❝ß ✲◆ù❃ù ✝ ✍✾❂✯✍❵ÿ✂ ✍✾❂✯✍ ✶ ✝ þ✻✁ ❥✝ ☞ æ✌ï➍ô❉ò❅✲✴✙★ô ★ô❉ô❉ï❢õ➏÷➁õ②è❫ú ❙ó✄✑❉î➪ó➪ï➍ò❢ö➏÷rù❃ç❵ñ ïrí♠s❃ê❉ð✉ðsñ ✓■ï❢ô❵õ➃ö②ï❢î✵✢❅✲qó➪ô ✕sÚ✦✥✓✈❢ß❾Û❤Ü✉ä✆✢rÝ❬Ó✦✡☛✡❭Ý➈ä✆❥✣✢❱Û➶ä❇Ù➁Ú◆ß Ù✗✹ ♥❂✈✎✖✗✖✶✓ Ø✐Ù➁Ú◆Û❤Ü➏ÙrÝ ✲ ☞ ✝ ✯ ê❉ò❢è✉ô❉ï❅✲ ✥ú ✝ ✲ ✕ï❢ö②õ➃ç✠✟ ✰ï❢î↕î①÷➁ô❑ú❂✲ ★✡❳é②õ➃è✉ö➏ú❃÷✓✡③✲ ✰ï➍î➪î①÷rô❭ú❂✲✤✍✔❄ ❃ ✶ ✲❨ù❉ù❵✻✯✍ ❄✂ ✻ ✫ ❄ ✝
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✷✴ ☎
o2.b8 Time Figure 5: Closed Loop Step Resp onse of the per- Figure 7: PID Controller Control Effort tur.ed a odel G3(sC Cont roller ontrol effort Figure R: Sliding Mode Controller Control Effort
0 5 10 Closed loop Step Response of System 3 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❁❤ ï❵ï❢ù ✯ õ➃è✉ù ❥➹è✉é➃ù❑ï❢ô❉é②è➙ïrí➬õ➃ç❉è❚ù❑èsö✗✟ õ➃ê❉ö❈✑❭è❫ú④✡➲ï➡ú➡è✉î ✳✱❀ ý✵✴ ☎ 0 5 10 Control Effort for Fuzzy Logic Controller 1.5 1 0.5 0 -0.5 -1 -1.5 2 Time 15 20 25 30 System3 System2 System1 (nominal) s❬ó➪ò➍ê❃ö②è ✫✆✑❣s❉ê❃ð✉ð✉ñ✔❤qï❢ò➍ó①ë✩✓■ï❢ô❵õ➃ö②ï❢î↕î➪è✉ö✦✓■ï➍ô◆õ➃ö②ï➍î ✥❈qï➍ö②õ 0 5 10 Control Effort for PID Controller 1.5 1 0.5 0 -0.5 -1 -1.5 2 Time 15 20 25 30 System3 System2 System1 (nominal) s❨ó➪ò❢ê❉ö②è ✳✆✑❣❺❣✼❈❻ ✓■ï➍ô◆õ②ö➃ï➍î➪î➪è✉ö✩✓■ï➍ô◆õ②ö➃ï➍î ✥❈qï➍ö➃õ 0 5 10 Control Effort for Sliding Mode Controller 2 1 0 -1 -2 -3 3 Time 15 20 25 30 System3 System2 System1 (nominal) s❬ó➪ò➍ê❃ö②è ❃✆✑ ✯ î➪ó①ú◆ó➪ô❉ò ☞➐ï➡ú➡è ✓■ï➍ô◆õ②ö➃ï➍î➪î➪è✉ö✦✓■ï➍ô◆õ②ö➃ï➍î ✥❈qï➍ö➃õ
