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-This work is performed in order to D.A Call for Peace or at Least a Cease Fire!and foster and promote unbiased and accurate compari- Negotiations son of fuzzy logic control and classical control design methodologies.We are motivated to execute this Fortunately,some have called for control engineers study by the demonstrated robustness to plant per-to consider inclusion of fuzzy logic control into their turbations of the fuzzy logic controller given in [0]. Robust fuzzy logic,PID,and sliding mode con-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 third order linear time-invariant model of a motor.and where appropriate ([0][0]).This appears to be The step response performance of each controller,ap- plied to the nominal and two perturbed motor plants,a reasonable request.At the very least,the fuzzy is presented. logic tool can be compared to classical control tools. We conclude fuzzy logic control can be a useful tool Should the fuzzy logic control tool prove useful and for the control engineer.We encourage more bench- mark comparisons of fuzzy logic control with classical acceptable to the control engineer,it should become control techniques for the benefit of the practicing welcome addition to her control toolbox. control engineer. E.Robust Controllers I.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- turbations and uncertainty.We take the perspective The successful use of fuzzy logic controllers has greatly of the control engineer who has several control tools expanded in the last twenty years ([0]-[0).This ex-available for the synthesis of robust controllers,and pansion has prompted much comparison to classical who is interested in determining the capabilities of control techniques.The ensuing discussions have 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 a robust controller,we choose a benchmark prob- practicing control engineer. 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 II.CHOICE OF CONTROL TOOLS has become quite heated with ardent detractors on Fuzzy logic control,PID control,and sliding mode both sides ([0]-[0]). control tools are chosen for our investigation.Fuzzy Some control engineers have been placed in a logic control is chosen because of its empirically demon- dilemma by this debate,and have been left with- strated robustness properties shown in [0].PID con- out an appeasing study or comparison upon which trol is chosen because it is one of the most com- to formulate a useful opinion. monly used controllers in industry,and because 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 mode controller because it is a robust non-linear ing battle,it appears that the best interest of the 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. ical debates.It is her interest to produce a working We do not claim to investigate and present every controller,with efficacy,making use of any and all robust control technique.Indeed,the popular and techniques at her disposal. successful Hoo and adaptive control techniques are noticeably absent from this study.We felt it was Research supported in part by ARO under grant number necessary to restrict ourselves to fuzzy logic control DAAL-91-G-091 and an AASERT supplement. and two other control methods,in order to keep our￾ ✁✂☎✄✆✞✝✠✟☛✡✌☞✍✂✏✎ ✂✒✑ ✓✂✒✔✖✕✗☞✙✘✚✎✗✛✜☞✍☞✣✢ ✤✗✕✗✥✦✥✣✧ ★✩✂✒✪✫✡✌✬✮✭✰✯✱✍✲ ✭✴✳ ✵✷✶✌✡✌✸✞✡✹✎✗✪ ✺✂✻✸✞✛ ✁✂✏✎✏✘✼✟☛✂✒✶ ✽✿✾❁❀❃❂❅❄❇❆❉❈❋❊❍●■✽✿❏❑❄▲❆◆▼❖❀❃P✣❀❉P✹◗❙❘❚❀❉❯❅❯❱❀❖❲❳❏❨◗❬❩✹❏❭❄❇❆ ❘❳❆◆❪✹❀❃❂❫❯❅▼❴❆❵P❬❯✿❏❃❛❝❜❞❄❇❆❉❡❢❯❫❂❅❣❤❡❵❀❃❄✐❜❥P✌❦❭❣❇P✌❆◆❆◆❂❫❣▲P✌❦☛❀❉P✹◗❧✽✿❏❑▼❴❪✌♠✌❯❫❆◆❂❚♥❨❡❢❣❇❆◆P✹❡❢❆❉❈ ♦P✌❣▲♣q❆❵❂r❈s❣❇❯✉t✼❏❃❛✈✽✇❀❃❄▲❣①❛②❏❭❂❫P✌❣❤❀✚❀❉❯❚③④❆◆❂❫⑤q❆◆❄▲❆❵t ③④❆◆❂❅⑤q❆◆❄❇❆◆tq⑥❬✽✇⑦⑨⑧❉⑩❬❶❉❷❑❸❬⑥ ♦ ♥❨⑦ ❹❳❺◆❻❢❼❾❽❑❿❨➀➁❼➃➂ ➄■➅◆➆➈➇✿➉➋➊➍➌➏➎➐➆➈➇✿➑❉➒r➌➔➓①➊➍➌➏→④➒❅➣❚➆↕↔➙➊➍➌➃➣❉➒❱➌❍➛➃➊ ➓①➊r➇②➛➃➒❱➌❞➜➍↔◆➣✙➑❉➌➃➊➍→④➊r➛②➒✇➝❉↔◆➞◆➆➟➜r➇②➒❅➣❳➜➍↔◆➣❚➜r➠✉➠❅➝❉➌➃➜r➛②➒✿➠✉➊➍→❋➑➡➜➍➌②➆➤➢ ➇➃➊➍↔❋➊❅➓q➓❇➝◆➥✉➥✉➦④➧➈➊r➨r➆➟➠❥➠✉➊➍↔❢➛➏➌➃➊r➧❉➜➍↔◆➣④➠✉➧➈➜r➇➃➇②➆➟➠✉➜r➧❑➠s➊➍↔➍➛❾➌➃➊r➧❑➣❉➒❅➇②➆➟➨➍↔ →④➒❅➛➏➅◆➊❢➣❉➊r➧➈➊r➨r➆➟➒❫➇s➩✮➫✍➒➙➜➍➌②➒✚→④➊r➛②➆➟➭❫➜r➛➃➒❫➣❴➛➃➊✚➒❫➯❢➒❫➠❫➝➡➛②➒✼➛➏➅◆➆➟➇ ➇➃➛➏➝◆➣❉➦❳➞❢➦➲➛➏➅◆➒❥➣❉➒r→④➊➍↔◆➇②➛❾➌➃➜r➛②➒❅➣❳➌②➊➍➞❃➝◆➇②➛❾↔◆➒❫➇➃➇✐➛➃➊✈➑➡➧➈➜➍↔❢➛➳➑❉➒r➌➔➢ ➛❾➝❉➌❾➞◆➜r➛②➆➟➊➍↔◆➇✐➊❅➓✌➛➏➅◆➒❍➓❇➝◆➥✉➥s➦❋➧➈➊r➨r➆➈➠❝➠✉➊➍↔❢➛➏➌➃➊r➧➈➧➟➒❱➌■➨r➆➟➭r➒❱↔➲➆↕↔➐➵➸❱➺❇➩ ➻➊➍➞❉➝◆➇➃➛④➓❇➝◆➥✉➥✉➦❴➧➈➊r➨r➆➟➠r➼✿➽✌➾②➚❥➼■➜➍↔◆➣☛➇➃➧➈➆➈➣❃➆↕↔◆➨❴→④➊➍➣❉➒❚➠✉➊➍↔❢➢ ➛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