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❵✻✯✍ ❄✂ ✻ ✫ ❄ ✝