Introduction to Modern Control Theory (1)Classical Control Techniques 1.Root Locus 2.Nyquist Plot 3.Bode Plot 4.PID(Proportional-Integral-Derivative )Control Technique Features 1.Limited to Linear Time Invariant Systems 2.Frequency Domain Design Technique 3.Limited to Single-Input Single-Output System (2)Modern control techniques 1.Optimal Control√ 2.Kalman Filter√ 3.Adaptive Control√ 4.Nonlinear Control√ 5.Robust Control 6.Intelligent Control√ (Neural Network,Fuzzy Control,Genetic Algorithm Features 1
✂✁☎✄✝✆✝✞✠✟☛✡☛☞✌✄✝✍✎✞✏✁✑✄✒✞✔✓✞✠✟✖✕✗✆✘✁✚✙✛✞✏✁☎✄✝✆✝✞✏✜✣✢✥✤✖✕✦✞✏✆✒✧ ★✪✩✒✫✭✬✯✮✱✰✝✲✳✲✵✴✱✶✷✰✘✮✠✬✹✸✻✺✻✼✵✽✾✸✻✮❀✿✏❁✂✶❃❂✗✺✗✴✱❄❆❅✦❁✂✲ ❇✪❈❊❉❊❋●❋■❍❆❏✷❋●❑✎▲✵▼ ◆●❈❊❖❊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 i odeling based on State ariables 2. Time Domain Design Technique Applicable to I ulti-Input I ulti-Output(IIIO) System 4. arious System Specifications (3) dustec di ecifications for Controller Mesien Stabilit a) Absolute Stability i. All system poles must be in cC ThEC(Re h) Routh-Hurwitz criterion (a)relative Stability vershoot(a5s) dIpr Tracking Capability a)Steady-state t rror r OfTpossible (b) System Type r Optimization (a)Control t nergy minimization (b)Obeect 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) r(t)=u(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 Pl
✂✁✰✾☞✍✗✮✱❁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❃❬❭❱✝❫✉❈ ❳
*I e can try n+rts( PD controller )or gro( Lead or Lag kts term has the problem of amplifying noise and kt=o is the better choice o Next, consider the canonical or relative stability spec it cation and try to design a controller M) that satisfy the following system specit cations In this case, overshoot <5 % ≥ 0≤6 fi 65 %)settling time s2 (sec)e In2 yy 1.65 Here, the roots of the above canonical system is SIn alnvI-sni 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 requirements absolute stabilit The shaded region where the closed loop poles must be Now, as an example, consider the plant 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 ❪
C(s)=l as a simple controller Then the closed loop TF becomes d loop poles the above specs are not met Next, if we apply C(s)=4(s+4) Ge(s C(sG(s) 1+C(sG(s)1+x4 (8+4)832+4s+4 closed loop Poles are S1.2= the above specs are met Next, letr(t)=3 and r(s)=3.Then y(s)=G(s)r(s) lim y(t)=lim sy(s)=lim 0s2+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
