Introduction to Modern Control Theory (1) Classical Control Technique quist Plot 4. PID( Proportional-Integral-Derivative) Control Technique Time Invariant Systems 3. Limited to Single-Input Single-Output System 2)Modern control techniques 1. Optimal Control 2. Kalman filter Features
✂✁☎✄✝✆✝✞✠✟☛✡☛☞✌✄✝✍✎✞✏✁✑✄✒✞✔✓✞✠✟✖✕✗✆✘✁✚✙✛✞✏✁☎✄✝✆✝✞✏✜✣✢✥✤✖✕✦✞✏✆✒✧ ★✪✩✒✫✭✬✯✮✱✰✝✲✳✲✵✴✱✶✷✰✘✮✠✬✹✸✻✺✻✼✵✽✾✸✻✮❀✿✏❁✂✶❃❂✗✺✗✴✱❄❆❅✦❁✂✲ ❇✪❈❊❉❊❋●❋■❍❆❏✷❋●❑✎▲✵▼ ◆●❈❊❖❊P●◗❘▲✵❙❚▼❯❍✦❱✝❲❚❋■❍ ❳✳❈❊❨✻❋●❩✵❬❭❱✝❲❚❋■❍ ❪ ❈❊❱✝❫✱❴❛❵✝❱✝❜❝❋✪❞❃❋✪❜❯❍❡❙❚❋✪❢✾❣■❲✐❤❥❫❥❢❦❍❡❬✎❧✪❜❡❣■❲✐❤✱❴✦❬✎❜❝❙♥♠■❣♦❍❡❙♥♠✪❬❆♣❊q✻❋✪❢❦❍❡❜❝❋✪❲✷rs❬✎❑✉t✵❢✵❙❚◗❘▲✵❬ ✈❁✂✰✇✼✾❅✦✽✾❁✂✲ ① ❇✪❈❊❏✷❙❚②✏❙♥❍❡❬✎❩☎❍❡❋✏❏✷❙❚❢✵❬③❣■❜✌r✌❙❚②✏❬✦❫❥❢❦♠■❣■❜❝❙④❣■❢❦❍✦⑤❘P●▼❯❍❡❬✎②✏▼ ◆●❈❊⑥✾❜❝❬✎◗❘▲✵❬✎❢✵❑⑦P⑧❴✦❋✪②❀❣■❙❚❢⑨❴✦❬✎▼❝❙❚❧✪❢✠rs❬✎❑✉t✵❢✵❙❚◗❘▲✵❬ ❳✳❈❊❏✷❙❚②✏❙♥❍❡❬✎❩☎❍❡❋❀⑤●❙❚❢✵❧✪❲❚❬⑩❤❥❫❥❢✵❞✵▲✳❍❆⑤●❙❚❢✵❧✪❲❚❬⑩❤❯❶✦▲✳❍❡❞✵▲✳❍❆⑤❘P●▼❯❍❡❬✎② ★❸❷❹✫✯❺❻✸❊❼✗❁✂✽❽✺❾✶✷✸✻✺✻✼✵✽✾✸✻✮❿✼✵❁✂✶❃❂✗✺✗✴✱❄❆❅✦❁✂✲ ❇✪❈❆❶✦❞✳❍❡❙❚②❀❣■❲✷q✻❋✪❢❦❍❡❜❝❋✪❲✝➀ ◆●❈❊➁➂❣■❲❚②❀❣■❢✠⑥✇❙❚❲♥❍❡❬✎❜➃➀ ❳✳❈✌➄➃❩✾❣■❞✳❍❡❙♥♠✪❬➅q✻❋✪❢❦❍❡❜❝❋✪❲✝➀ ❪ ❈❊❖❆❋✪❢✵❲❚❙❚❢✵❬③❣■❜❊q✻❋✪❢❦❍❡❜❝❋✪❲✝➀ ➆●❈❊❉❊❋✪➇✵▲✵▼❯❍✦q✻❋✪❢❦❍❡❜❝❋✪❲ ➈✳❈✌❫❥❢❦❍❡❬✎❲❚❲❚❙❚❧✪❬✎❢❦❍➃q✻❋✪❢❦❍❡❜❝❋✪❲✝➀ ❵✝❖❆❬✎▲✵❜❡❣■❲✂❖❆❬⑦❍❥➉✻❋✪❜❝➊❽➋❃⑥✾▲✵➌✎➌⑦P➍q✻❋✪❢❦❍❡❜❝❋✪❲➎➋❽➏✗❬✎❢✵❬⑦❍❡❙❚❑➂➄➃❲❚❧✪❋✪❜❝❙♥❍❡t✵②➐♣ ✈❁✂✰✇✼✾❅✦✽✾❁✂✲ ① ❇
System Modeling based on State Variables 2. Time Domain Design Technique 3. Applicable to Multi-Input Multi-Output( MIMO )System 4. Various System Specifications (3)System Specifications for Controller Design Stabilit Absolute Stability i. All system poles must be in C-=SECReS<01 Routh-Hurwitz criterion (a)relative Stability t(≤5%) Rise time(≤0.5scc iii. Settling Time( <2 sec d/ or Tracking Capabili a)Steady-state Error: 0 if possible or as small as possible (b) System Type Optimization (a)Control Energy minimization (b)Object Accomplishment Time minimization
❇✪❈❆⑤❘P●▼❯❍❡❬✎② ❋●❩✵❬✎❲❚❙❚❢✵❧❛➇✾❣■▼❝❬✎❩➍❋✪❢✖⑤❘❍✉❣♦❍❡❬✂✁❹❣■❜❝❙④❣■➇✵❲❚❬✎▼ ◆●❈✌r✌❙❚②✏❬➂❴✦❋✪②❀❣■❙❚❢⑨❴✦❬✎▼❝❙❚❧✪❢✠rs❬✎❑✉t✵❢✵❙❚◗❘▲✵❬ ❳✳❈✌➄➃❞✵❞✵❲❚❙❚❑③❣■➇✵❲❚❬➃❍❡❋ ▲✵❲♥❍❡❙✐❤❥❫❥❢✵❞✵▲✳❍ ▲✵❲♥❍❡❙✐❤❯❶✦▲✳❍❡❞✵▲✳❍❸❵ ❫❶ ♣❊⑤❘P●▼❯❍❡❬✎② ❪ ❈✄✁❹❣■❜❝❙❚❋✪▲✵▼❆⑤❘P●▼❯❍❡❬✎② ⑤●❞❃❬✎❑✎❙✆☎✾❑③❣♦❍❡❙❚❋✪❢✵▼ ★✞✝✻✫✠✟☛✡❆✲❘✼✵❁✌☞ ✟✎✍➅❁✂✶✂✴✑✏➂✶✷✰✇✼✾✴✱✸✻✺✦✲✠✒⑩✸❹✽ ✬✹✸✻✺✻✼✵✽✾✸✻✮❯✮✱❁✂✽✔✓❁✂✲✵✴✖✕✻✺ ❇✪❈❆⑤❘❍✉❣■➇✵❙❚❲❚❙♥❍❥P ❵ ❣❦♣⑨➄➃➇✵▼❝❋✪❲❚▲✳❍❡❬➅⑤❘❍✉❣■➇✵❙❚❲❚❙♥❍❥P ❙➎❈✌➄➃❲❚❲ ▼❯P●▼❯❍❡❬✎② ❞❃❋✪❲❚❬✎▼✌②▲✵▼❯❍❊➇❃❬➂❙❚❢✘✗✚✙✜✛✣✢✥✤✔✦✧✗✩★●❉❊❬✪✤✩✫✭✬✘✮ ❙❚❙➎❈✌❉❊❋✪▲✳❍❡t●❤✖✯❆▲✵❜❯➉❊❙♥❍❡➌❭q✻❜❝❙♥❍❡❬✎❜❝❙❚❋✪❢ ❵ ❣❦♣✠❉❊❬✎❲④❣♦❍❡❙♥♠✪❬❭⑤❘❍✉❣■➇✵❙❚❲❚❙♥❍❥P ❙➎❈❆❶➃♠✪❬✎❜❝▼❝t✵❋●❋■❍❿❵✱✰ ➆✳✲➐♣ ❙❚❙➎❈✌❉❊❙❚▼❝❬➂r✌❙❚②✏❬❛❵✱✰✩✬✵✴ ➆✳✶✞✷✞✸✗♣ ❙❚❙❚❙➎❈✌⑤●❬⑦❍❝❍❡❲❚❙❚❢✵❧❿r✌❙❚②✏❬✏❵✱✰ ◆✹✶✞✷✞✸✗♣ ◆●❈❊❉❊❬✎❧✪▲✵❲④❣♦❍❡❙❚❋✪❢✠❣■❢✵❩✻✺♦❋✪❜✌rs❜❡❣■❑✉➊❘❙❚❢✵❧☎q ❣■❞✾❣■➇✵❙❚❲❚❙♥❍❥P ❵ ❣❦♣➍⑤❘❍❡❬③❣■❩✳P❦❤❥▼❯❍✉❣♦❍❡❬✥✼✝❜❝❜❝❋✪❜ ✽✾✬❛❙✆✿s❞❃❋✪▼❝▼❝❙❚➇✵❲❚❬✗❋✪❜❊❣■▼❊▼❝②❀❣■❲❚❲ ❣■▼❊❞❃❋✪▼❝▼❝❙❚➇✵❲❚❬■❈ ❵ ➇❽♣➍⑤❘P●▼❯❍❡❬✎② r✘P●❞❃❬ ✽ ❳✳❈❆❶✦❞✳❍❡❙❚②✏❙❚➌③❣♦❍❡❙❚❋✪❢ ❵ ❣❦♣➍q✻❋✪❢❦❍❡❜❝❋✪❲❀✼✝❢✵❬✎❜❝❧■P⑨②✏❙❚❢✵❙❚②✏❙❚➌③❣♦❍❡❙❚❋✪❢ ❵ ➇❽♣➍❶✦➇❂❁❝❬✎❑⑦❍❆➄➃❑✎❑✎❋✪②✏❞✵❲❚❙❚▼❝t✵②✏❬✎❢❦❍❊r✌❙❚②✏❬✗②✏❙❚❢✵❙❚②✏❙❚➌③❣♦❍❡❙❚❋✪❢ ❵ ❑ ♣⑨❋■❍❡t✵❬✎❜❝▼ ◆
Example (s)=1/ G(s) 1t≥0 (t)=(t) 0t<0 System Type r(t)e(t) Controlle] u(t)plant (t Type: 1 1 setpoint tracking possible K PI controller Since exists in the plant, the controller does not have to be Pi
✂✁✰✾☞✍✗✮✱❁❹① G(s) r(s) = 1/s y(s) ✄ ❵✆☎❝♣ ✛✞✝✇❵✆☎❝♣ ✛ ✟✠ ✡ ✠☛ ❇☞☎✞✌ ✬ ✬✍☎✩✫✭✬ ❋☎⑤❘P●▼❯❍❡❬✎② r✘P●❞❃❬ ✛✏✎ Controller 1 s(s+1) r(t) e(t) u(t) plant Type:1 y(t) 1 s setpoint tracking possible ✑✓✒✕✔✖✑✘✗ ✶ ✶ ✛ ✑✘✗✙✔ ✑✓✒ ✶ ❵✣❱✝❫ ❑✎❋✪❢❦❍❡❜❝❋✪❲❚❲❚❬✎❜ ♣ ✚ ⑤●❙❚❢✵❑✎❬✜✛✢ ❬✤✣✳❙❚▼❯❍❡▼❊❙❚❢⑨❍❡t✵❬➂❞✵❲④❣■❢❦❍③➋✳❍❡t✵❬➂❑✎❋✪❢❦❍❡❜❝❋✪❲❚❲❚❬✎❜✌❩✵❋●❬✎▼❊❢✵❋■❍❊t✾❣③♠✪❬➂❍❡❋❛➇❃❬❭❱✝❫✉❈ ❳
Me can try Vp+ Vps( Pd controller )or i2( Lead or Lag Ds term has the problem of amplifying noise and O Next, consider the canonical system 32+Ii+= for relative stability spec- nn ification and try to design a controller d) that satisfy the following system In this case. overshoot <5 ≤ (5%)settling time <2(sec) g Here. the roots of the above canonical system is and the above requirements are mapped as follows in the complex plane In other words, the closed loop poles must be in the shaded region absolute stabilit The shaded region where the closed loop poles must be Now, as an example, consider the plant s(s+'s motor system )and apply
✚ ❬➂❑③❣■❢⑨❍❡❜❯P✂✁✗✙✔ ✁☎✄ ✶ ❵✒❱✘❴ ❑✎❋✪❢❦❍❡❜❝❋✪❲❚❲❚❬✎❜❆♣✻❋✪❜ ✢✝✆✟✞ ✢✝✆✡✠ ❵✒❏✷❬③❣■❩➍❋✪❜❊❏✂❣■❧❀♣⑩❈ ✚ ✑✄ ✶✗❍❡❬✎❜❝②➐t✾❣■▼✌❍❡t✵❬➂❞✵❜❝❋✪➇✵❲❚❬✎② ❋✿✝❣■②✏❞✵❲❚❙✆✿P●❙❚❢✵❧➅❢✵❋✪❙❚▼❝❬➂❣■❢✵❩ ✑✄ ✢ ✢✝✆✡✠ ❙❚▼ ❍❡t✵❬➂➇❃❬⑦❍❝❍❡❬✎❜❊❑✉t✵❋✪❙❚❑✎❬■❈ ☛ ❖❆❬✤✣●❍③➋✳❑✎❋✪❢✵▼❝❙❚❩✵❬✎❜✘❍❡t✵❬➃❑③❣■❢✵❋✪❢✵❙❚❑③❣■❲❽▼❯P●▼❯❍❡❬✎② ☞✍✌✎ ✢ ✌ ✆✡✏✒✑☞✎ ✢✝✆☞✎✌ ✿❋✪❜❹❜❝❬✎❲④❣♦❍❡❙♥♠✪❬❊▼❯❍✉❣■➇✵❙❚❲❚❙♥❍❥P❿▼❝❞❃❬✎❑⑩❤ ❙✆☎✾❑③❣♦❍❡❙❚❋✪❢❀❣■❢✵❩✏❍❡❜❯P❛❍❡❋➅❩✵❬✎▼❝❙❚❧✪❢☎❣❭❑✎❋✪❢❦❍❡❜❝❋✪❲❚❲❚❬✎❜✔✓❀❵ ✶❸♣✒❍❡t✾❣♦❍ ▼❡❣♦❍❡❙❚▼ ✿P❿❍❡t✵❬ ✿❋✪❲❚❲❚❋➉❊❙❚❢✵❧✗▼❯P●▼❯❍❡❬✎② ▼❝❞❃❬✎❑✎❙✆☎✾❑③❣♦❍❡❙❚❋✪❢✵▼✎❈ ❫❥❢✠❍❡t✵❙❚▼✌❑③❣■▼❝❬■➋❽❋♠✪❬✎❜❝▼❝t✵❋●❋■❍ ✰ ➆✳✲ ✕ ✖✓✌ ✛ ✗ ✏ ✕ ✘ ✰✚✙✛ ❵➎➆✳✲❛♣ ▼❝❬⑦❍❝❍❡❲❚❙❚❢✵❧ ❍❡❙❚②✏❬ ✰ ◆✠❵ ✶✞✷✞✸✎♣ ✕ ✖✢✜✤✣ ✌✦✥★✧ ✥ ✩✫✪ ✛ ❇ ✴ ➈❦➆ ✯❆❬✎❜❝❬■➋✾❍❡t✵❬➂❜❝❋●❋■❍❡▼✌❋✿s❍❡t✵❬❭❣■➇❃❋♠✪❬➂❑③❣■❢✵❋✪❢✵❙❚❑③❣■❲✷▼❯P●▼❯❍❡❬✎② ❙❚▼ ✶ ✛✒✬ ✏ ✛✮✭✯✖✢✜✤✣✱✰✲✜✤✣✴✳❇✱✭✲✖ ✏ ❣■❢✵❩✠❍❡t✵❬❭❣■➇❃❋♠✪❬➂❜❝❬✎◗❘▲✵❙❚❜❝❬✎②✏❬✎❢❦❍❡▼➃❣■❜❝❬✗②❀❣■❞✵❞❃❬✎❩⑧❣■▼✄✿❋✪❲❚❲❚❋➉❊▼✌❙❚❢⑨❍❡t✵❬➂❑✎❋✪②✏❞✵❲❚❬✤✣⑨❞✵❲④❣■❢✵❬■❈ ❫❥❢➍❋■❍❡t✵❬✎❜✌➉✻❋✪❜❝❩✵▼✎➋✾❍❡t✵❬➂❑✎❲❚❋✪▼❝❬✎❩✠❲❚❋●❋✪❞✠❞❃❋✪❲❚❬✎▼✌②▲✵▼❯❍❊➇❃❬➂❙❚❢⑨❍❡t✵❬➂▼❝t✾❣■❩✵❬✎❩⑧❜❝❬✎❧✪❙❚❋✪❢ ❈ overshoot requirements absolute stability boundary settling time requirements Im s Re s r✌t✵❬➂▼❝t✾❣■❩✵❬✎❩⑧❜❝❬✎❧✪❙❚❋✪❢⑨➉❊t✵❬✎❜❝❬➂❍❡t✵❬➂❑✎❲❚❋✪▼❝❬✎❩✠❲❚❋●❋✪❞⑨❞❃❋✪❲❚❬✎▼❊②▲✵▼❯❍❊➇❃❬■❈ ❖❆❋➉➂➋✒❣■▼➅❣■❢ ❬✤✣✵❣■②✏❞✵❲❚❬■➋s❑✎❋✪❢✵▼❝❙❚❩✵❬✎❜➂❍❡t✵❬❀❞✵❲④❣■❢❦❍ ✛ ✢✒✵ ✢✝✆ ✛✝✶ ❵✌②✏❋■❍❡❋✪❜✗▼❯P●▼❯❍❡❬✎② ♣➂❣■❢✵❩ ❣■❞✵❞✵❲♥P ❪
Ds=l as a simple controller Then the closed loop TF becomes Irs) (s+ t the closed loop poles are the above specs are not met Next, if we apply ds)=4(5 Im)=DsⅠo)-=可 sfi+ 48+4 t closed loop Poles are R2d2 t the above specs are Next, let r(t)=3 and r(s) Then Ins)r(s) s(sf+4s+4) →00 ut)=lim sus=lim 0 sfi+ 4s+ final value thm This Cl system is able to track the given set-point
✓❀❵ ✶❸♣✾✛ ❇➅❣■▼❊❣❛▼❝❙❚②✏❞✵❲❚❬✦❑✎❋✪❢❦❍❡❜❝❋✪❲❚❲❚❬✎❜✎❈ r✌t✵❬✎❢➍❍❡t✵❬➂❑✎❲❚❋✪▼❝❬✎❩➍❲❚❋●❋✪❞⑨r❊⑥ ➇❃❬✎❑✎❋✪②✏❬✎▼ ✂✁ ❵ ✶❸♣✾✛ ✛ ✢✒✵ ✢✝✆ ✛✝✶ ❇ ✔ ✛ ✢✒✵ ✢✝✆ ✛✝✶ ✛ ❇ ✶ ✏ ✔ ✶ ✔ ❇ ✄ ❍❡t✵❬➂❑✎❲❚❋✪▼❝❬✎❩➍❲❚❋●❋✪❞⑨❞❃❋✪❲❚❬✎▼❊❣■❜❝❬ ✶ ✛✒✬ ✏ ✛ ✭ ❇ ◆ ✰✆☎❳ ◆✞✝ ✄ ❍❡t✵❬ ❣■➇❃❋♠✪❬ ▼❝❞❃❬✎❑✎▼ ❣■❜❝❬ ❢✵❋■❍ ②✏❬⑦❍ ❖❆❬✤✣●❍③➋❽❙✆✿✂➉✻❬➅❣■❞✵❞✵❲♥P ✓❀❵ ✶❸♣✾✛ ❪ ❵ ✢✝✆ ✛ ✢✝✆■♣⑩➋ ✂✁ ❵ ✶❸♣✾✛ ✓❀❵ ✶❸♣ ❵ ✶❸♣ ❇ ✔ ✓❀❵ ✶❸♣ ❵ ✶❸♣ ✛ ✛ ✢✒✵ ✢✝✆✛ ✶ ❇ ✔ ✛ ✢✒✵ ✢✝✆✛ ✶ ✛ ❪ ✶ ✏ ✔ ❪ ✶ ✔ ❪ ✄ ❑✎❲❚❋✪▼❝❬✎❩➍❲❚❋●❋✪❞✠❱s❋✪❲❚❬✎▼❆❣■❜❝❬ ✶ ✛✒✬ ✏ ✛ ✭✦◆✠✟ ✭✦◆ ✄ ❍❡t✵❬ ❣■➇❃❋♠✪❬ ▼❝❞❃❬✎❑✎▼ ❣■❜❝❬ ②✏❬⑦❍ ❖❆❬✤✣●❍③➋❽❲❚❬⑦❍ ✄ ❵✆☎❝♣ ✛ ❳❛❣■❢✵❩ ✄ ❵ ✶❸♣ ✛ ✥ ✢ ❈✝r✌t✵❬✎❢ ➋ ✡ ❵ ✶❸♣ ✛ ✂✁ ❵ ✶❸♣ ✄ ❵ ✶❸♣ ✛ ❇❸◆ ✶❘❵ ✶ ✏ ✔ ❪ ✶ ✔ ❪ ♣ ✄ ❲❚❙❚② ✩☞☛✍✌ ✡ ❵✆☎❝♣ ✛ ❲❚❙❚② ✢☛✏✎ ✶ ✡ ❵ ✶❸♣ ✛ ❲❚❙❚② ✢☛✏✎ ❇❸◆ ✶ ✏ ✔ ❪ ✶ ✔ ❪ ✛ ❳ ❵✚☎✾❢✾❣■❲ ♠■❣■❲❚▲✵❬✦❍❡t✵② ♣ ✄ r✌t✵❙❚▼➃q ❏ ▼❯P●▼❯❍❡❬✎② ❙❚▼❊❣■➇✵❲❚❬✦❍❡❋❿❍❡❜❡❣■❑✉➊❀❍❡t✵❬➂❧✪❙♥♠✪❬✎❢➍▼❝❬⑦❍❯❤❥❞❃❋✪❙❚❢❦❍③❈ ➆
