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some mathematical aspects of rna velocity

Some mathematical aspects of RNA velocity Loc D EMEULENAERE Universit - PowerPoint PPT Presentation

Aug 23, 2022 •38 likes •1.05k views

Some mathematical aspects of RNA velocity Loc D EMEULENAERE Universit de Lige GIGA- Genomics Lige, January 9, 2019 rs tt t

  1. t❀❡ ✈❡❧♊❝✐t② ♊❢ t❀❡ ♣❛rt✐❝❧❡ ✭❛t t✐♠❡ ✮ ✐s ❌ ■♥ ✞❉✿ ❧✐❊❡✇✐s❡✊ ■❢ ✐s t❀❡ ♣♊s✐t✐♊♥ ♊❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❀❡ s♣❛❝❡✱ t❀❡♥ ✐ts ✈❡❧♊❝✐t② ✐s ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❱❛r✐❛t✐♊♥s ✇✐t❀ r❡s♣❡❝t t♩ t✐♠❡ ❛✈❡r❛❣❡ ❝❀❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♊❢ t✐♠❡ ï¿œ ￜￜ ï¿œ f ( t ✵ + t ) − f ( t ✵ ) df dt ( t ✵ ) := lim t t → ✵ ï¿œ ￜￜ ï¿œ ■♥st❛♥t❛♥❡♊✉s r❛t❡ ♊❢ ✈❛r❛t✐♊♥ ✇✐t❀ r❡s♣❡❝t t♩ t✐♠❡ ❊①❛♠♣❧❡s ✭P❀②s✐❝s✮ ❌ f ( t ) = x ( t ) ✿ ♠♩t✐♊♥ ✭✶❉✮ ♊❢ ❛ ♣❛rt✐❝❧❡❀

  2. ❌ ■♥ ✞❉✿ ❧✐❊❡✇✐s❡✊ ■❢ ✐s t❀❡ ♣♊s✐t✐♊♥ ♊❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❀❡ s♣❛❝❡✱ t❀❡♥ ✐ts ✈❡❧♊❝✐t② ✐s ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❱❛r✐❛t✐♊♥s ✇✐t❀ r❡s♣❡❝t t♩ t✐♠❡ ❛✈❡r❛❣❡ ❝❀❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♊❢ t✐♠❡ ï¿œ ￜￜ ï¿œ f ( t ✵ + t ) − f ( t ✵ ) df dt ( t ✵ ) := lim t t → ✵ ï¿œ ￜￜ ï¿œ ■♥st❛♥t❛♥❡♊✉s r❛t❡ ♊❢ ✈❛r❛t✐♊♥ ✇✐t❀ r❡s♣❡❝t t♩ t✐♠❡ ❊①❛♠♣❧❡s ✭P❀②s✐❝s✮ ❌ f ( t ) = x ( t ) ✿ ♠♩t✐♊♥ ✭✶❉✮ ♊❢ ❛ ♣❛rt✐❝❧❡❀ t❀❡ ✈❡❧♊❝✐t② ♊❢ t❀❡ ♣❛rt✐❝❧❡ ✭❛t t✐♠❡ t ✮ ✐s v ( t ) := dx dt ( t )

  3. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❱❛r✐❛t✐♊♥s ✇✐t❀ r❡s♣❡❝t t♩ t✐♠❡ ❛✈❡r❛❣❡ ❝❀❛♥❣❡ ❞✉r✐♥❣ t ✉♥✐ts ♊❢ t✐♠❡ ï¿œ ￜￜ ï¿œ f ( t ✵ + t ) − f ( t ✵ ) df dt ( t ✵ ) := lim t t → ✵ ï¿œ ￜￜ ï¿œ ■♥st❛♥t❛♥❡♊✉s r❛t❡ ♊❢ ✈❛r❛t✐♊♥ ✇✐t❀ r❡s♣❡❝t t♩ t✐♠❡ ❊①❛♠♣❧❡s ✭P❀②s✐❝s✮ ❌ f ( t ) = x ( t ) ✿ ♠♩t✐♊♥ ✭✶❉✮ ♊❢ ❛ ♣❛rt✐❝❧❡❀ t❀❡ ✈❡❧♊❝✐t② ♊❢ t❀❡ ♣❛rt✐❝❧❡ ✭❛t t✐♠❡ t ✮ ✐s v ( t ) := dx dt ( t ) ❌ ■♥ ✞❉✿ ❧✐❊❡✇✐s❡✊ ■❢ ï¿œ x ( t ) := ( x ( t ) , y ( t ) , z ( t )) ✐s t❀❡ ♣♊s✐t✐♊♥ ♊❢ ❛ ♣❛rt✐❝❧❡ ✐♥ t❀❡ s♣❛❝❡✱ t❀❡♥ ✐ts ✈❡❧♊❝✐t② ✐s ï¿œ dx ï¿œ v ( t ) := d ï¿œ x dt ( t ) , dy dt ( t ) , dz ï¿œ dt ( t ) := dt ( t ) .

  4. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐①

  5. ❘❛t❡ ◗✉❛♥t✐t② ❘❛t❡ ◗✉❛♥t✐t② ❘❛t❡ ï¿œ ï¿œ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ t❀❡ ✐❞❡❛ ■♥ ♊♥❡ ❝❡❧❧ ✱ ❢♊r ♊♥❡ ❣❡♥❡ ✳✳✳ ❚r❛♥s❝r✐♣t✐♊♥ ï¿œ ❯♥s♣❧✐❝❡❞ ❘◆❆ ❉◆❆ ❙♣❧✐❝✐♥❣ ∅ ❙♣❧✐❝❡❞ ❘◆❆ ❉❡❣r❛❞❛t✐♊♥

  6. ◗✉❛♥t✐t② ❘❛t❡ ◗✉❛♥t✐t② ❘❛t❡ ï¿œ ï¿œ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ t❀❡ ✐❞❡❛ ■♥ ♊♥❡ ❝❡❧❧ ✱ ❢♊r ♊♥❡ ❣❡♥❡ ✳✳✳ ❚r❛♥s❝r✐♣t✐♊♥ ❘❛t❡ α ( t ) ï¿œ ❯♥s♣❧✐❝❡❞ ❘◆❆ ❉◆❆ ❙♣❧✐❝✐♥❣ ∅ ❙♣❧✐❝❡❞ ❘◆❆ ❉❡❣r❛❞❛t✐♊♥

  7. ◗✉❛♥t✐t② ◗✉❛♥t✐t② ❘❛t❡ ï¿œ ï¿œ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ t❀❡ ✐❞❡❛ ■♥ ♊♥❡ ❝❡❧❧ ✱ ❢♊r ♊♥❡ ❣❡♥❡ ✳✳✳ ❚r❛♥s❝r✐♣t✐♊♥ ❘❛t❡ α ( t ) ï¿œ ❯♥s♣❧✐❝❡❞ ❘◆❆ ❉◆❆ ❙♣❧✐❝✐♥❣ ❘❛t❡ β ( t ) ∅ ❙♣❧✐❝❡❞ ❘◆❆ ❉❡❣r❛❞❛t✐♊♥

  8. ◗✉❛♥t✐t② ◗✉❛♥t✐t② ï¿œ ï¿œ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ t❀❡ ✐❞❡❛ ■♥ ♊♥❡ ❝❡❧❧ ✱ ❢♊r ♊♥❡ ❣❡♥❡ ✳✳✳ ❚r❛♥s❝r✐♣t✐♊♥ ❘❛t❡ α ( t ) ï¿œ ❯♥s♣❧✐❝❡❞ ❘◆❆ ❉◆❆ ❙♣❧✐❝✐♥❣ ❘❛t❡ β ( t ) ∅ ❙♣❧✐❝❡❞ ❘◆❆ ❉❡❣r❛❞❛t✐♊♥ ❘❛t❡ γ ( t )

  9. ◗✉❛♥t✐t② ï¿œ ï¿œ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ t❀❡ ✐❞❡❛ ■♥ ♊♥❡ ❝❡❧❧ ✱ ❢♊r ♊♥❡ ❣❡♥❡ ✳✳✳ ❚r❛♥s❝r✐♣t✐♊♥ ❘❛t❡ α ( t ) ï¿œ ❯♥s♣❧✐❝❡❞ ❘◆❆ ❉◆❆ ◗✉❛♥t✐t② u ( t ) ❙♣❧✐❝✐♥❣ ❘❛t❡ β ( t ) ∅ ❙♣❧✐❝❡❞ ❘◆❆ ❉❡❣r❛❞❛t✐♊♥ ❘❛t❡ γ ( t )

  10. ï¿œ ï¿œ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ t❀❡ ✐❞❡❛ ■♥ ♊♥❡ ❝❡❧❧ ✱ ❢♊r ♊♥❡ ❣❡♥❡ ✳✳✳ ❚r❛♥s❝r✐♣t✐♊♥ ❘❛t❡ α ( t ) ï¿œ ❯♥s♣❧✐❝❡❞ ❘◆❆ ❉◆❆ ◗✉❛♥t✐t② u ( t ) ❙♣❧✐❝✐♥❣ ❘❛t❡ β ( t ) ∅ ❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② s ( t ) ❉❡❣r❛❞❛t✐♊♥ ❘❛t❡ γ ( t )

  11. ï¿œ ï¿œ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ t❀❡ ✐❞❡❛ ■♥ ♊♥❡ ❝❡❧❧ ✱ ❢♊r ♊♥❡ ❣❡♥❡ ✳✳✳ ❚r❛♥s❝r✐♣t✐♊♥ ❘❛t❡ α ( t ) ï¿œ ❯♥s♣❧✐❝❡❞ ❘◆❆ ❉◆❆ ◗✉❛♥t✐t② u ( t )  du  α ( t ) − β ( t ) u ( t ) dt ( t ) =   ❙♣❧✐❝✐♥❣ ❘❛t❡ β ( t )    ∅ ❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② s ( t ) ❉❡❣r❛❞❛t✐♊♥ ❘❛t❡ γ ( t )

  12. ï¿œ ï¿œ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ t❀❡ ✐❞❡❛ ■♥ ♊♥❡ ❝❡❧❧ ✱ ❢♊r ♊♥❡ ❣❡♥❡ ✳✳✳ ❚r❛♥s❝r✐♣t✐♊♥ ❘❛t❡ α ( t ) ï¿œ ❯♥s♣❧✐❝❡❞ ❘◆❆ ❉◆❆ ◗✉❛♥t✐t② u ( t )  du  α ( t ) − β ( t ) u ( t ) dt ( t ) =   ❙♣❧✐❝✐♥❣ ❘❛t❡ β ( t ) ds   dt ( t ) = β ( t ) u ( t ) − γ ( t ) s ( t )  ∅ ❙♣❧✐❝❡❞ ❘◆❆ ◗✉❛♥t✐t② s ( t ) ❉❡❣r❛❞❛t✐♊♥ ❘❛t❡ γ ( t )

  13. ❌ ❘❡❛❧ ♥✉♠❜❡rs ♊❢ ♠♊❧❡❝✉❧❡s ✭❛t t✐♠❡ ✮ ❀❛✈❡ ❛ ❜✐✈❛r✐❛t❡ P♩✐ss♊♥ ❞✐str✐❜✉t✐♊♥ ✇✐t❀ ♣❛r❛♠❡t❡rs ✭❡①♣❡❝t❡❞ ✈❛❧✉❡s✮ ❛♥❞ ✳ ❖✉r ❡q✉❛t✐♊♥s ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s ■♥ t❀✐s ❝♊♥t❡①t ❌ u ( t ) ❛♥❞ s ( t ) ❛r❡ t❀❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ♊❢ t❀❡ ♥✉♠❜❡rs ♊❢ ♠♊❧❡❝✉❧❡s ♊❢ ✉♥s♣❧✐❝❡❞ ❛♥❞ s♣❧✐❝❡❞ ❘◆❆ ✭❛t t✐♠❡ t ✮

  14. ❖✉r ❡q✉❛t✐♊♥s ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s ■♥ t❀✐s ❝♊♥t❡①t ❌ u ( t ) ❛♥❞ s ( t ) ❛r❡ t❀❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ♊❢ t❀❡ ♥✉♠❜❡rs ♊❢ ♠♊❧❡❝✉❧❡s ♊❢ ✉♥s♣❧✐❝❡❞ ❛♥❞ s♣❧✐❝❡❞ ❘◆❆ ✭❛t t✐♠❡ t ✮ ❌ ❘❡❛❧ ♥✉♠❜❡rs ♊❢ ♠♊❧❡❝✉❧❡s ✭❛t t✐♠❡ t ✮ ❀❛✈❡ ❛ ❜✐✈❛r✐❛t❡ P♩✐ss♊♥ ❞✐str✐❜✉t✐♊♥ ✇✐t❀ ♣❛r❛♠❡t❡rs ✭❡①♣❡❝t❡❞ ✈❛❧✉❡s✮ u ( t ) ❛♥❞ s ( t ) ✳

  15. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s ■♥ t❀✐s ❝♊♥t❡①t ❌ u ( t ) ❛♥❞ s ( t ) ❛r❡ t❀❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ♊❢ t❀❡ ♥✉♠❜❡rs ♊❢ ♠♊❧❡❝✉❧❡s ♊❢ ✉♥s♣❧✐❝❡❞ ❛♥❞ s♣❧✐❝❡❞ ❘◆❆ ✭❛t t✐♠❡ t ✮ ❌ ❘❡❛❧ ♥✉♠❜❡rs ♊❢ ♠♊❧❡❝✉❧❡s ✭❛t t✐♠❡ t ✮ ❀❛✈❡ ❛ ❜✐✈❛r✐❛t❡ P♩✐ss♊♥ ❞✐str✐❜✉t✐♊♥ ✇✐t❀ ♣❛r❛♠❡t❡rs ✭❡①♣❡❝t❡❞ ✈❛❧✉❡s✮ u ( t ) ❛♥❞ s ( t ) ✳ ❖✉r ❡q✉❛t✐♊♥s  du  dt ( t ) = α ( t ) − β ( t ) u ( t )   ds   β ( t ) u ( t ) − γ ( t ) s ( t ) dt ( t ) = 

  16. ✶ ✭❛❧❧ ✉♥✐ts ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♊❢ ✱ ✐✳❡✳ ❡✈❡r②t❀✐♥❣ ❞✐✈✐❞❡❞ ❌ ❜② ✮✳ ❋✐♥❛❧ ❡q✉❛t✐♊♥s ✏✭▲✐♥❡❛r✮ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♊♥s✑ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s ❆ss✉♠♣t✐♊♥s ❌ ❚❀❡ r❛t❡s α ✱ β ✱ γ ❛r❡ ❝♊♥st❛♥t✿ α ≥ ✵✱ β > ✵✱ γ > ✵✳

  17. ❋✐♥❛❧ ❡q✉❛t✐♊♥s ✏✭▲✐♥❡❛r✮ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♊♥s✑ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s ❆ss✉♠♣t✐♊♥s ❌ ❚❀❡ r❛t❡s α ✱ β ✱ γ ❛r❡ ❝♊♥st❛♥t✿ α ≥ ✵✱ β > ✵✱ γ > ✵✳ ❌ β = ✶ ✭❛❧❧ ✉♥✐ts ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♊❢ β ✱ ✐✳❡✳ ❡✈❡r②t❀✐♥❣ ❞✐✈✐❞❡❞ ❜② β ✮✳

  18. ✏✭▲✐♥❡❛r✮ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♊♥s✑ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s ❆ss✉♠♣t✐♊♥s ❌ ❚❀❡ r❛t❡s α ✱ β ✱ γ ❛r❡ ❝♊♥st❛♥t✿ α ≥ ✵✱ β > ✵✱ γ > ✵✳ ❌ β = ✶ ✭❛❧❧ ✉♥✐ts ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♊❢ β ✱ ✐✳❡✳ ❡✈❡r②t❀✐♥❣ ❞✐✈✐❞❡❞ ❜② β ✮✳ ❋✐♥❛❧ ❡q✉❛t✐♊♥s  du  dt ( t ) = α − u ( t )   ds   dt ( t ) = u ( t ) − γ s ( t ) 

  19. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s ❆ss✉♠♣t✐♊♥s ❌ ❚❀❡ r❛t❡s α ✱ β ✱ γ ❛r❡ ❝♊♥st❛♥t✿ α ≥ ✵✱ β > ✵✱ γ > ✵✳ ❌ β = ✶ ✭❛❧❧ ✉♥✐ts ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♊❢ β ✱ ✐✳❡✳ ❡✈❡r②t❀✐♥❣ ❞✐✈✐❞❡❞ ❜② β ✮✳ ❋✐♥❛❧ ❡q✉❛t✐♊♥s  du  dt ( t ) = α − u ( t )   ds   dt ( t ) = u ( t ) − γ s ( t )  ✏✭▲✐♥❡❛r✮ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♊♥s✑

  20. ✵ ✵ ■♥ ❛❧❧ ❝❛s❡s✱ ✱ ✐✳❡✳ ✐❢ ✵✳ ❙♩❧✉t✐♊♥ ■❢ ✵ ✱ ✵ ✵ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ ✜rst ❡q✉❛t✐♊♥ du dt ( t ) = α − u ( t )

  21. ✵ ✵ ■♥ ❛❧❧ ❝❛s❡s✱ ✱ ✐✳❡✳ ✐❢ ✵✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ ✜rst ❡q✉❛t✐♊♥ du dt ( t ) = α − u ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ := u ( ✵ ) ✱ u ( t ) = α + ( u ✵ − α ) e − t .

  22. ✵ ✵ ■♥ ❛❧❧ ❝❛s❡s✱ ✱ ✐✳❡✳ ✐❢ ✵✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ ✜rst ❡q✉❛t✐♊♥ du dt ( t ) = α − u ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ := u ( ✵ ) ✱ u ( t ) = α + ( u ✵ − α ) e − t .

  23. ■♥ ❛❧❧ ❝❛s❡s✱ ✱ ✐✳❡✳ ✐❢ ✵✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ ✜rst ❡q✉❛t✐♊♥ du dt ( t ) = α − u ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ := u ( ✵ ) ✱ u ( t ) = α + ( u ✵ − α ) e − t . u ✵ < α u ✵ > α 1.0 1.0 0.8 0.8 0.6 0.6 α α u(t) u(t) 0.4 0.4 0.2 0.2 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 t t

  24. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ ✜rst ❡q✉❛t✐♊♥ du dt ( t ) = α − u ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ := u ( ✵ ) ✱ u ( t ) = α + ( u ✵ − α ) e − t . u ✵ < α u ✵ > α 1.0 1.0 0.8 0.8 0.6 0.6 α α u(t) u(t) 0.4 0.4 0.2 0.2 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 t t ■♥ ❛❧❧ ❝❛s❡s✱ lim t →∞ u ( t ) = α ✱ ✐✳❡✳ u ( t ) ≈ α ✐❢ t ≫ ✵✳

  25. ❙♩❧✉t✐♊♥ ■❢ ✵ ❛♥❞ ✵ ✱ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ▌❛♥② ❣r❛♣❀✐❝❛❧ ♣♊ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❀❛✈❡ ✐✳❡✳ ✐❢ ✵✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ s❡❝♊♥❞ ❡q✉❛t✐♊♥ ds dt ( t ) = u ( t ) − γ s ( t )

  26. ▌❛♥② ❣r❛♣❀✐❝❛❧ ♣♊ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❀❛✈❡ ✐✳❡✳ ✐❢ ✵✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ s❡❝♊♥❞ ❡q✉❛t✐♊♥ ds dt ( t ) = u ( t ) − γ s ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ ï¿œ ï¿œ γ + u ✵ − α s ✵ + α − u ✵ s ( t ) = α γ − ✶ − α γ − ✶ e − t + e − γ t . γ

  27. ▌❛♥② ❣r❛♣❀✐❝❛❧ ♣♊ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❀❛✈❡ ✐✳❡✳ ✐❢ ✵✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ s❡❝♊♥❞ ❡q✉❛t✐♊♥ ds dt ( t ) = u ( t ) − γ s ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ ï¿œ ï¿œ γ + u ✵ − α s ✵ + α − u ✵ s ( t ) = α γ − ✶ − α γ − ✶ e − t + e − γ t . γ

  28. ▌❛♥② ❣r❛♣❀✐❝❛❧ ♣♊ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❀❛✈❡ ✐✳❡✳ ✐❢ ✵✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ s❡❝♊♥❞ ❡q✉❛t✐♊♥ ds dt ( t ) = u ( t ) − γ s ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ ï¿œ ï¿œ  γ + u ✵ − α s ✵ + α − u ✵ α γ − ✶ − α γ − ✶ e − t + e − γ t  ✐❢ γ ï¿œ = ✶  γ s ( t ) =   α + [( u ✵ − α ) t + s ✵ − α ] e − t ✐❢ γ = ✶ (= β ) .

  29. ▌❛♥② ❣r❛♣❀✐❝❛❧ ♣♊ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❀❛✈❡ ✐✳❡✳ ✐❢ ✵✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ s❡❝♊♥❞ ❡q✉❛t✐♊♥ ds dt ( t ) = u ( t ) − γ s ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ ï¿œ ï¿œ  γ + u ✵ − α s ✵ + α − u ✵ α γ − ✶ − α γ − ✶ e − t + e − γ t  ✐❢ γ ï¿œ = ✶  γ s ( t ) =   α + [( u ✵ − α ) t + s ✵ − α ] e − t ✐❢ γ = ✶ (= β ) .

  30. ❇✉t ✇❡ ❛❧✇❛②s ❀❛✈❡ ✐✳❡✳ ✐❢ ✵✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ s❡❝♊♥❞ ❡q✉❛t✐♊♥ ds dt ( t ) = u ( t ) − γ s ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ ï¿œ ï¿œ  γ + u ✵ − α s ✵ + α − u ✵ α γ − ✶ − α γ − ✶ e − t + e − γ t  ✐❢ γ ï¿œ = ✶  γ s ( t ) =   α + [( u ✵ − α ) t + s ✵ − α ] e − t ✐❢ γ = ✶ (= β ) . ▌❛♥② ❣r❛♣❀✐❝❛❧ ♣♊ss✐❜✐❧✐t✐❡s✳✳✳

  31. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ s❡❝♊♥❞ ❡q✉❛t✐♊♥ ds dt ( t ) = u ( t ) − γ s ( t ) ❙♩❧✉t✐♊♥ ■❢ u ✵ = u ( ✵ ) ❛♥❞ s ✵ = s ( ✵ ) ✱ ï¿œ ï¿œ  γ + u ✵ − α s ✵ + α − u ✵ α γ − ✶ − α γ − ✶ e − t + e − γ t  ✐❢ γ ï¿œ = ✶  γ s ( t ) =   α + [( u ✵ − α ) t + s ✵ − α ] e − t ✐❢ γ = ✶ (= β ) . ▌❛♥② ❣r❛♣❀✐❝❛❧ ♣♊ss✐❜✐❧✐t✐❡s✳✳✳ ❇✉t ✇❡ ❛❧✇❛②s ❀❛✈❡ t →∞ s ( t ) = α lim γ , ✐✳❡✳ s ( t ) ≈ α γ ✐❢ t ≫ ✵✳

  32. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥ ♊❢ t❀❡ s❡❝♊♥❞ ❡q✉❛t✐♊♥✿ ❣r❛♣❀✐❝❛❧ ❡①❛♠♣❧❡s u ✵ = s ✵ = ✵ , α = ✵ . ✷✺ , γ = ✵ . ✌✺ u ✵ = ✾ , s ✵ = ✶ , α = ✵ . ✷✺ , γ = ✵ . ✌✺ α/γ 0.30 1.5 0.25 0.20 s(t) s(t) 0.15 1.0 0.10 0.05 0.5 α/γ 0.00 0 2 4 6 8 10 0 2 4 6 8 10 t t u ✵ = ✾ , s ✵ = ✹ , α = ✞✵ , γ = ✺ u ✵ = ✹ , s ✵ = ✻ , α = ✶ , γ = ✶ 6 α/γ 6 5 5 4 s(t) s(t) 4 3 2 3 1 α/γ 0 2 0 2 4 6 8 10 0 2 4 6 8 10 t t

  33. ■♥ ♣❛rt✐❝✉❧❛r✱ ❙t❡❛❞② st❛t❡ ❲❀❡♥ ✵✱ t❀❡ s②st❡♠ r❡❛❝❀❡s ❛ st❡❛❞② st❛t❡✱ ✇✐t❀ ❛♥❞ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ s✉♠♠❛r② ❚❀❡r❡ ❡①✐st s♩❧✉t✐♊♥s u ✱ s ✱ ❞❡♣❡♥❞✐♥❣ ♊♥ u ✵ ✱ s ✵ ✭✐♥✐t✐❛❧ ❝♊♥❞✐t✐♊♥s✮ ❛♥❞ ♊♥ α ✱ γ ✭♣❛r❛♠❡t❡rs✮✱ ✇✐t❀ t →∞ s ( t ) = α t →∞ u ( t ) = α lim ❛♥❞ lim γ .

  34. ❙t❡❛❞② st❛t❡ ❲❀❡♥ ✵✱ t❀❡ s②st❡♠ r❡❛❝❀❡s ❛ st❡❛❞② st❛t❡✱ ✇✐t❀ ❛♥❞ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ s✉♠♠❛r② ❚❀❡r❡ ❡①✐st s♩❧✉t✐♊♥s u ✱ s ✱ ❞❡♣❡♥❞✐♥❣ ♊♥ u ✵ ✱ s ✵ ✭✐♥✐t✐❛❧ ❝♊♥❞✐t✐♊♥s✮ ❛♥❞ ♊♥ α ✱ γ ✭♣❛r❛♠❡t❡rs✮✱ ✇✐t❀ t →∞ s ( t ) = α t →∞ u ( t ) = α lim ❛♥❞ lim γ . ■♥ ♣❛rt✐❝✉❧❛r✱ u ( t ) lim s ( t ) = γ. t →∞

  35. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ❞②♥❛♠✐❝s✿ s✉♠♠❛r② ❚❀❡r❡ ❡①✐st s♩❧✉t✐♊♥s u ✱ s ✱ ❞❡♣❡♥❞✐♥❣ ♊♥ u ✵ ✱ s ✵ ✭✐♥✐t✐❛❧ ❝♊♥❞✐t✐♊♥s✮ ❛♥❞ ♊♥ α ✱ γ ✭♣❛r❛♠❡t❡rs✮✱ ✇✐t❀ t →∞ s ( t ) = α t →∞ u ( t ) = α lim ❛♥❞ lim γ . ■♥ ♣❛rt✐❝✉❧❛r✱ u ( t ) lim s ( t ) = γ. t →∞ ❙t❡❛❞② st❛t❡ ❲❀❡♥ t ≫ ✵✱ t❀❡ s②st❡♠ r❡❛❝❀❡s ❛ st❡❛❞② st❛t❡✱ ✇✐t❀ s ( t ) ≈ α u ( t ) ≈ α, u ( t ) ≈ γ s ( t ) . γ , ❛♥❞

  36. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① P❀❛s❡ ♣♊rtr❛✐t ●r❛♣❀✐❝ ✧s♣❧✐❝❡❞ ✈s✳ ✉♥s♣❧✐❝❡❞✧ u ✵ = ✵ , s ✵ = ✵ , α = ✾ , γ = ✵ . ✌✺ u ✵ = ✾ , s ✵ = ✹ , α = ✵ , γ = ✵ . ✌✺ 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 u u 1.0 1.0 0.5 0.5 0.0 0.0 0 1 2 3 4 0 1 2 3 4 s s u ≥ γ s u ≀ γ s ❚❀❡ s②st❡♠ r❡❛❝❀❡s t❀❡ st❡❛❞② st❛t❡✱ ✐✳❡✳ t❀❡ str❛✐❣❀t ❧✐♥❡ u = γ s ✳

  37. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① P❀❛s❡ ♣♊rtr❛✐t ●r❛♣❀✐❝ ✧s♣❧✐❝❡❞ ✈s✳ ✉♥s♣❧✐❝❡❞✧ u ✵ = ✵ , s ✵ = ✵ , α = ✾ , γ = ✵ . ✌✺ u ✵ = ✾ , s ✵ = ✹ , α = ✵ , γ = ✵ . ✌✺ 3.0 3.0 2.5 2.5 2.0 2.0 1.5 = ⇒ 1.5 u u 1.0 1.0 0.5 0.5 0.0 0.0 0 1 2 3 4 0 1 2 3 4 s s u ≥ γ s u ≀ γ s ❚❀❡ s②st❡♠ r❡❛❝❀❡s t❀❡ st❡❛❞② st❛t❡✱ ✐✳❡✳ t❀❡ str❛✐❣❀t ❧✐♥❡ u = γ s ✳ 3.0 2.5 2.0 u 1.5 1.0 0.5 0.0 0 1 2 3 4 s

  38. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐①

  39. ❌ ▲❡t ❜❡ t❀❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♊❢ t❀❡✮ q✉❛♥t✐t② ♊❢ s♣❧✐❝❡❞ ❘◆❆ ❛ss♩❝✐❛t❡❞ t♩ t❀❡ t❀ ❣❡♥❡ ✭❛t t✐♠❡ ✮✳ ❌ ❊❛❝❀ ✈❡r✐✜❡s t❀❡ ♣r❡✈✐♊✉s ❡q✉❛t✐♊♥s✱ ✇✐t❀ ✐ts ♊✇♥ ♣❛r❛♠❡t❡rs ✵✱ ✶✱ ❛♥❞ ✵✳ ❲❛r♥✐♥❣✊ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♊♥✊ ✶ ❢♊r ❛❧❧ ✿ t❀❡ r❛t❡s ♊❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♊r ❛❧❧ ❣❡♥❡s✩ ❉❡✜♥✐t✐♊♥ ❚❀❡ ❘◆❆ ✈❡❧♊❝✐t② ♊❢ t❀❡ ❝❡❧❧ ✭❛t t✐♠❡ ✮ ✐s ✶ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ✈❡❧♊❝✐t② ❈♊♥t❡①t ❌ ❍❡r❡✱ ✇❡ ❝♊♥s✐❞❡r ♊♥❡ ❝❡❧❧✱ ✇✐t❀ p ❣❡♥❡s✳

  40. ❌ ❊❛❝❀ ✈❡r✐✜❡s t❀❡ ♣r❡✈✐♊✉s ❡q✉❛t✐♊♥s✱ ✇✐t❀ ✐ts ♊✇♥ ♣❛r❛♠❡t❡rs ✵✱ ✶✱ ❛♥❞ ✵✳ ❲❛r♥✐♥❣✊ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♊♥✊ ✶ ❢♊r ❛❧❧ ✿ t❀❡ r❛t❡s ♊❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♊r ❛❧❧ ❣❡♥❡s✩ ❉❡✜♥✐t✐♊♥ ❚❀❡ ❘◆❆ ✈❡❧♊❝✐t② ♊❢ t❀❡ ❝❡❧❧ ✭❛t t✐♠❡ ✮ ✐s ✶ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ✈❡❧♊❝✐t② ❈♊♥t❡①t ❌ ❍❡r❡✱ ✇❡ ❝♊♥s✐❞❡r ♊♥❡ ❝❡❧❧✱ ✇✐t❀ p ❣❡♥❡s✳ ❌ ▲❡t s j ( t ) ❜❡ t❀❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♊❢ t❀❡✮ q✉❛♥t✐t② ♊❢ s♣❧✐❝❡❞ ❘◆❆ ❛ss♩❝✐❛t❡❞ t♩ t❀❡ j t❀ ❣❡♥❡ ✭❛t t✐♠❡ t ✮✳

  41. ❲❛r♥✐♥❣✊ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♊♥✊ ✶ ❢♊r ❛❧❧ ✿ t❀❡ r❛t❡s ♊❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♊r ❛❧❧ ❣❡♥❡s✩ ❉❡✜♥✐t✐♊♥ ❚❀❡ ❘◆❆ ✈❡❧♊❝✐t② ♊❢ t❀❡ ❝❡❧❧ ✭❛t t✐♠❡ ✮ ✐s ✶ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ✈❡❧♊❝✐t② ❈♊♥t❡①t ❌ ❍❡r❡✱ ✇❡ ❝♊♥s✐❞❡r ♊♥❡ ❝❡❧❧✱ ✇✐t❀ p ❣❡♥❡s✳ ❌ ▲❡t s j ( t ) ❜❡ t❀❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♊❢ t❀❡✮ q✉❛♥t✐t② ♊❢ s♣❧✐❝❡❞ ❘◆❆ ❛ss♩❝✐❛t❡❞ t♩ t❀❡ j t❀ ❣❡♥❡ ✭❛t t✐♠❡ t ✮✳ ❌ ❊❛❝❀ s j ( t ) ✈❡r✐✜❡s t❀❡ ♣r❡✈✐♊✉s ❡q✉❛t✐♊♥s✱ ✇✐t❀ ✐ts ♊✇♥ ♣❛r❛♠❡t❡rs α j ≥ ✵✱ β j = ✶✱ ❛♥❞ γ j > ✵✳

  42. ❉❡✜♥✐t✐♊♥ ❚❀❡ ❘◆❆ ✈❡❧♊❝✐t② ♊❢ t❀❡ ❝❡❧❧ ✭❛t t✐♠❡ ✮ ✐s ✶ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ✈❡❧♊❝✐t② ❈♊♥t❡①t ❌ ❍❡r❡✱ ✇❡ ❝♊♥s✐❞❡r ♊♥❡ ❝❡❧❧✱ ✇✐t❀ p ❣❡♥❡s✳ ❌ ▲❡t s j ( t ) ❜❡ t❀❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♊❢ t❀❡✮ q✉❛♥t✐t② ♊❢ s♣❧✐❝❡❞ ❘◆❆ ❛ss♩❝✐❛t❡❞ t♩ t❀❡ j t❀ ❣❡♥❡ ✭❛t t✐♠❡ t ✮✳ ❌ ❊❛❝❀ s j ( t ) ✈❡r✐✜❡s t❀❡ ♣r❡✈✐♊✉s ❡q✉❛t✐♊♥s✱ ✇✐t❀ ✐ts ♊✇♥ ♣❛r❛♠❡t❡rs α j ≥ ✵✱ β j = ✶✱ ❛♥❞ γ j > ✵✳ ❲❛r♥✐♥❣✊ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♊♥✊ β j = ✶ ❢♊r ❛❧❧ j ✿ t❀❡ r❛t❡s ♊❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♊r ❛❧❧ ❣❡♥❡s✩

  43. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘◆❆ ✈❡❧♊❝✐t② ❈♊♥t❡①t ❌ ❍❡r❡✱ ✇❡ ❝♊♥s✐❞❡r ♊♥❡ ❝❡❧❧✱ ✇✐t❀ p ❣❡♥❡s✳ ❌ ▲❡t s j ( t ) ❜❡ t❀❡ ✭❡①♣❡❝t❡❞ ✈❛❧✉❡ ♊❢ t❀❡✮ q✉❛♥t✐t② ♊❢ s♣❧✐❝❡❞ ❘◆❆ ❛ss♩❝✐❛t❡❞ t♩ t❀❡ j t❀ ❣❡♥❡ ✭❛t t✐♠❡ t ✮✳ ❌ ❊❛❝❀ s j ( t ) ✈❡r✐✜❡s t❀❡ ♣r❡✈✐♊✉s ❡q✉❛t✐♊♥s✱ ✇✐t❀ ✐ts ♊✇♥ ♣❛r❛♠❡t❡rs α j ≥ ✵✱ β j = ✶✱ ❛♥❞ γ j > ✵✳ ❲❛r♥✐♥❣✊ ■♠♣❧✐❝✐t ❛ss✉♠♣t✐♊♥✊ β j = ✶ ❢♊r ❛❧❧ j ✿ t❀❡ r❛t❡s ♊❢ s♣❧✐❝✐♥❣ ❛r❡ ❡q✉❛❧ ❢♊r ❛❧❧ ❣❡♥❡s✩ ❉❡✜♥✐t✐♊♥ ❚❀❡ ❘◆❆ ✈❡❧♊❝✐t② ♊❢ t❀❡ ❝❡❧❧ ✭❛t t✐♠❡ t ✮ ✐s ï¿œ ds ✶ ï¿œ d ï¿œ s dt ( t ) , ..., ds p dt ( t ) := dt ( t ) .

  44. ❌ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♩r② ♊❢ t❀❡ ❝❡❧❧ ✭ ✮✳ ✶ ✷ ❆rr♩✇s✿ ❘◆❆ ✈❡❧♊❝✐t② ❘❡❞ ♣♊✐♥t✿ st❡❛❞② st❛t❡ ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶

  45. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ❌ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♩r② ♊❢ 2.5 t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ • ❆rr♩✇s✿ s2 1.5 ❘◆❆ 1.0 ✈❡❧♊❝✐t② 0.5 • ❘❡❞ ♣♊✐♥t✿ st❡❛❞② st❛t❡ 0.0 0 1 2 3 4 s1 ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊

  46. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ❌ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♩r② ♊❢ 2.5 t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ❌ ❆rr♩✇s✿ s2 1.5 ❘◆❆ 1.0 ✈❡❧♊❝✐t② 0.5 • ❘❡❞ ♣♊✐♥t✿ st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊

  47. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ❌ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♩r② ♊❢ 2.5 t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ❌ ❆rr♩✇s✿ s2 1.5 ❘◆❆ 1.0 ✈❡❧♊❝✐t② 0.5 • ❘❡❞ ♣♊✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊

  48. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ❌ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♩r② ♊❢ 2.5 t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ❌ ❆rr♩✇s✿ s2 1.5 ❘◆❆ 1.0 ✈❡❧♊❝✐t② ● 0.5 • ❘❡❞ ♣♊✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊

  49. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ❌ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♩r② ♊❢ 2.5 t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ❌ ❆rr♩✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♊❝✐t② ● 0.5 • ❘❡❞ ♣♊✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊

  50. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ❌ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♩r② ♊❢ 2.5 t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❌ ❆rr♩✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♊❝✐t② ● 0.5 • ❘❡❞ ♣♊✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊

  51. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ❌ ●r❡② ❝✉r✈❡✿ tr❛❥❡❝t♩r② ♊❢ 2.5 ● t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❌ ❆rr♩✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♊❝✐t② ● 0.5 • ❘❡❞ ♣♊✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊

  52. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ❌ ●r❡② ❝✉r✈❡✿ ● tr❛❥❡❝t♩r② ♊❢ 2.5 ● t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❌ ❆rr♩✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♊❝✐t② ● 0.5 • ❘❡❞ ♣♊✐♥t✿ ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊

  53. ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ● ❌ ●r❡② ❝✉r✈❡✿ ● tr❛❥❡❝t♩r② ♊❢ 2.5 ● t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❌ ❆rr♩✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♊❝✐t② ● ❌ ❘❡❞ ♣♊✐♥t✿ 0.5 ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1

  54. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥✿ ❛♥ ✉♥r❡❛❧ ✇♩r❧❞✳✳✳ ❌ ❆ ❝❡❧❧ ✇✐t❀ ✷ ❣❡♥❡s✳✳✳ ❌ α ✶ = ✷✱ γ ✶ = ✵ . ✺❀ α ✷ = ✞✱ γ ✷ = ✶ 3.0 ● ❌ ●r❡② ❝✉r✈❡✿ ● tr❛❥❡❝t♩r② ♊❢ 2.5 ● t❀❡ ❝❡❧❧ 2.0 ✭ ( s ✶ ( t ) , s ✷ ( t )) ✮✳ ● ❌ ❆rr♩✇s✿ s2 1.5 ● ❘◆❆ 1.0 ✈❡❧♊❝✐t② ● ❌ ❘❡❞ ♣♊✐♥t✿ 0.5 ● st❡❛❞② st❛t❡ 0.0 ● 0 1 2 3 4 s1 ✏P❀②s✐❝❛❧ ✈❡❧♊❝✐t②✑ ✐♥ ❘◆❆✬s s♣❛❝❡✊

  55. ❌ Pr✐♥❝✐♣❧❡ ❝♊♠♣♊♥❡♥t ❛♥❛❧②s✐s✿ q✉✐t❡ ♥❛t✉r❛❧✱ ♣r♊❥❡❝t✐♊♥ ♊♥ P✳❈✳❀ ❌ t✲❙◆❊❄ P♩ss✐❜❧❡✱ ❜✉t ♠♩r❡ tr✐❝❊②✳✳✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥ ♊❢ ❘◆❆ ✈❡❧♊❝✐t② ❆♥❞ ✐❢ p > ✾❄

  56. ❌ t✲❙◆❊❄ P♩ss✐❜❧❡✱ ❜✉t ♠♩r❡ tr✐❝❊②✳✳✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥ ♊❢ ❘◆❆ ✈❡❧♊❝✐t② ❆♥❞ ✐❢ p > ✾❄ ❌ Pr✐♥❝✐♣❧❡ ❝♊♠♣♊♥❡♥t ❛♥❛❧②s✐s✿ q✉✐t❡ ♥❛t✉r❛❧✱ ♣r♊❥❡❝t✐♊♥ ♊♥ P✳❈✳❀

  57. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥ ♊❢ ❘◆❆ ✈❡❧♊❝✐t② ❆♥❞ ✐❢ p > ✾❄ ❌ Pr✐♥❝✐♣❧❡ ❝♊♠♣♊♥❡♥t ❛♥❛❧②s✐s✿ q✉✐t❡ ♥❛t✉r❛❧✱ ♣r♊❥❡❝t✐♊♥ ♊♥ P✳❈✳❀ ❌ t✲❙◆❊❄ P♩ss✐❜❧❡✱ ❜✉t ♠♩r❡ tr✐❝❊②✳✳✳

  58. P❈❆ t✲❙◆❊ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥ ♊❢ ❘◆❆ ✈❡❧♊❝✐t② ❊①❛♠♣❧❡✿ ❙❝❀✇❛♥♥ ❝❡❧❧ ♣r❡❝✉rs♩rs ✭❝♊♠✐♥❣ ❢r♩♠ ❬✶❪✮

  59. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❘❡♣r❡s❡♥t❛t✐♊♥ ♊❢ ❘◆❆ ✈❡❧♊❝✐t② ❊①❛♠♣❧❡✿ ❙❝❀✇❛♥♥ ❝❡❧❧ ♣r❡❝✉rs♩rs ✭❝♊♠✐♥❣ ❢r♩♠ ❬✶❪✮ P❈❆ t✲❙◆❊

  60. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐①

  61. ❆ss✉♠♣t✐♊♥s ❌ ❚❀❡ s❛♠♣❧❡ ♊❢ ❝❡❧❧s ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡ t♩ ❝♊✈❡r ❛❧❧ t❀❡ ✏❘◆❆ ❝②❝❧❡✑ ✭❢r♩♠ ❜❡❣✐♥♥✐♥❣ ♊❢ ♣r♩❞✉❝t✐♊♥ t♩ st❡❛❞② st❛t❡✮✳ ❌ ❚❀❡ r❛t❡ ♊❢ ❞❡❣r❛❞❛t✐♊♥ ♊❢ t❀❡ ❣❡♥❡ ✐s t❀❡ s❛♠❡ ✐♥ ❛❧❧ ❝❡❧❧s ✳ ❊st✐♠❛t✐♊♥ ♊❢ ✇✐t❀ ♣❀❛s❡ ♣♊rtr❛✐ts✳✳✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❊st✐♠❛t✐♊♥ ♊❢ γ ❲❡ st✉❞② ♊♥❡ ❣❡♥❡ ✭✐✳❡✳ ✐ts ♣❛r❛♠❡t❡rs✮ t❀r♊✉❣❀ ❛ s❛♠♣❧❡ ♊❢ s❡✈❡r❛❧ ❝❡❧❧s✳

  62. ❊st✐♠❛t✐♊♥ ♊❢ ✇✐t❀ ♣❀❛s❡ ♣♊rtr❛✐ts✳✳✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❊st✐♠❛t✐♊♥ ♊❢ γ ❲❡ st✉❞② ♊♥❡ ❣❡♥❡ ✭✐✳❡✳ ✐ts ♣❛r❛♠❡t❡rs✮ t❀r♊✉❣❀ ❛ s❛♠♣❧❡ ♊❢ s❡✈❡r❛❧ ❝❡❧❧s✳ ❆ss✉♠♣t✐♊♥s ❌ ❚❀❡ s❛♠♣❧❡ ♊❢ ❝❡❧❧s ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡ t♩ ❝♊✈❡r ❛❧❧ t❀❡ ✏❘◆❆ ❝②❝❧❡✑ ✭❢r♩♠ ❜❡❣✐♥♥✐♥❣ ♊❢ ♣r♩❞✉❝t✐♊♥ t♩ st❡❛❞② st❛t❡✮✳ ❌ ❚❀❡ r❛t❡ ♊❢ ❞❡❣r❛❞❛t✐♊♥ γ ♊❢ t❀❡ ❣❡♥❡ ✐s t❀❡ s❛♠❡ ✐♥ ❛❧❧ ❝❡❧❧s ✳

  63. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❊st✐♠❛t✐♊♥ ♊❢ γ ❲❡ st✉❞② ♊♥❡ ❣❡♥❡ ✭✐✳❡✳ ✐ts ♣❛r❛♠❡t❡rs✮ t❀r♊✉❣❀ ❛ s❛♠♣❧❡ ♊❢ s❡✈❡r❛❧ ❝❡❧❧s✳ ❆ss✉♠♣t✐♊♥s ❌ ❚❀❡ s❛♠♣❧❡ ♊❢ ❝❡❧❧s ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡ t♩ ❝♊✈❡r ❛❧❧ t❀❡ ✏❘◆❆ ❝②❝❧❡✑ ✭❢r♩♠ ❜❡❣✐♥♥✐♥❣ ♊❢ ♣r♩❞✉❝t✐♊♥ t♩ st❡❛❞② st❛t❡✮✳ ❌ ❚❀❡ r❛t❡ ♊❢ ❞❡❣r❛❞❛t✐♊♥ γ ♊❢ t❀❡ ❣❡♥❡ ✐s t❀❡ s❛♠❡ ✐♥ ❛❧❧ ❝❡❧❧s ✳ ï¿œ ❊st✐♠❛t✐♊♥ ♊❢ γ ✇✐t❀ ♣❀❛s❡ ♣♊rtr❛✐ts✳✳✳

  64. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❚❀❡♊r✐t✐❝❛❧ ❡st✐♠❛t✐♊♥ ♊❢ γ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❌ α = ✾ , γ = ✵ . ✌✺ ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❌ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● u 1.5 ● ❌ ✹✵✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♊r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● 0.0 ● 0 1 2 3 4 s Pr♊❝❡ss ❙❡❧❡❝t✐♊♥ ♊❢ t❀❡ ❡①tr❡♠❡ ❝❡❧❧s ✭❀❡r❡ s♠❛❧❧❡st ❛♥❞ ❣r❡❛t❡st ✶✪✬s✮ ▲✐♥❡❛r r❡❣r❡ss✐♊♥ ♊♥ t❀❡ ❡①tr❡♠❡ ❝❡❧❧s ❍❡r❡✱ ❡st✐♠❛t✐♊♥ ♊❢ γ ✭s❧♊♣❡✮✿ ✵✳✌✞✹✟✞

  65. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❚❀❡♊r✐t✐❝❛❧ ❡st✐♠❛t✐♊♥ ♊❢ γ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❌ α = ✾ , γ = ✵ . ✌✺ ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❌ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● u 1.5 ● ❌ ✹✵✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♊r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s Pr♊❝❡ss ✶✳ ❙❡❧❡❝t✐♊♥ ♊❢ t❀❡ ❡①tr❡♠❡ ❝❡❧❧s ✭❀❡r❡ s♠❛❧❧❡st ❛♥❞ ❣r❡❛t❡st ✶✪✬s✮ ▲✐♥❡❛r r❡❣r❡ss✐♊♥ ♊♥ t❀❡ ❡①tr❡♠❡ ❝❡❧❧s ❍❡r❡✱ ❡st✐♠❛t✐♊♥ ♊❢ γ ✭s❧♊♣❡✮✿ ✵✳✌✞✹✟✞

  66. ❍❡r❡✱ ❡st✐♠❛t✐♊♥ ♊❢ ✭s❧♊♣❡✮✿ ✵✳✌✞✹✟✞ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❚❀❡♊r✐t✐❝❛❧ ❡st✐♠❛t✐♊♥ ♊❢ γ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❌ α = ✾ , γ = ✵ . ✌✺ ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❌ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● u 1.5 ● ❌ ✹✵✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♊r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s Pr♊❝❡ss ✶✳ ❙❡❧❡❝t✐♊♥ ♊❢ t❀❡ ❡①tr❡♠❡ ❝❡❧❧s ✭❀❡r❡ s♠❛❧❧❡st ❛♥❞ ❣r❡❛t❡st ✶✪✬s✮ ✷✳ ▲✐♥❡❛r r❡❣r❡ss✐♊♥ ♊♥ t❀❡ ❡①tr❡♠❡ ❝❡❧❧s

  67. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❚❀❡♊r✐t✐❝❛❧ ❡st✐♠❛t✐♊♥ ♊❢ γ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❌ α = ✾ , γ = ✵ . ✌✺ ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❌ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● u 1.5 ● ❌ ✹✵✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♊r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s Pr♊❝❡ss ✶✳ ❙❡❧❡❝t✐♊♥ ♊❢ t❀❡ ❡①tr❡♠❡ ❝❡❧❧s ✭❀❡r❡ s♠❛❧❧❡st ❛♥❞ ❣r❡❛t❡st ✶✪✬s✮ ✷✳ ▲✐♥❡❛r r❡❣r❡ss✐♊♥ ♊♥ t❀❡ ❡①tr❡♠❡ ❝❡❧❧s ❍❡r❡✱ ❡st✐♠❛t✐♊♥ ♊❢ γ ✭s❧♊♣❡✮✿ ✵✳✌✞✹✟✞

  68. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❉✐✣❝✉❧t✐❡s ♊❢ ❡st✐♠❛t✐♊♥ ❢♊r γ ✳✳✳ ❆ss✉♠♣t✐♊♥ ✶ ♥♊t r❡s♣❡❝t❡❞✳✳✳ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❌ α = ✾ , γ = ✵ . ✌✺ ● ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❌ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● ● ● u 1.5 ● ❌ ✶✺✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♊r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● 0.0 ● 0 1 2 3 4 s ❊st✐♠❛t✐♊♥ ♊❢ γ ✿ ✵✳✟✌✹✞✞✳✳✳ ❈♩rr❡❝t✐♊♥s❄ ❊st✐♠❛t✐♊♥ ♊♥ ✈❡r② ❝♩rr❡❧❛t❡❞ ❣❡♥❡s✱ ❝❧✉st❡rs ♊❢ ✈❡r② ❝♩rr❡❧❛t❡❞ ❝❡❧❧s✳✳✳

  69. ❊st✐♠❛t✐♊♥ ♊❢ ✿ ✵✳✟✌✹✞✞✳✳✳ ❈♩rr❡❝t✐♊♥s❄ ❊st✐♠❛t✐♊♥ ♊♥ ✈❡r② ❝♩rr❡❧❛t❡❞ ❣❡♥❡s✱ ✜❧t❡r✐♥❣ s♊♠❡ ❝❡❧❧s✳✳✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❉✐✣❝✉❧t✐❡s ♊❢ ❡st✐♠❛t✐♊♥ ❢♊r γ ✳✳✳ ❆ss✉♠♣t✐♊♥ ✶ ♥♊t r❡s♣❡❝t❡❞✳✳✳ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❌ α = ✾ , γ = ✵ . ✌✺ ● ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❌ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● ● ● u 1.5 ● ❌ ✶✺✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♊r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s

  70. ❈♩rr❡❝t✐♊♥s❄ ❊st✐♠❛t✐♊♥ ♊♥ ✈❡r② ❝♩rr❡❧❛t❡❞ ❣❡♥❡s✱ ✜❧t❡r✐♥❣ s♊♠❡ ❝❡❧❧s✳✳✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❉✐✣❝✉❧t✐❡s ♊❢ ❡st✐♠❛t✐♊♥ ❢♊r γ ✳✳✳ ❆ss✉♠♣t✐♊♥ ✶ ♥♊t r❡s♣❡❝t❡❞✳✳✳ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❌ α = ✾ , γ = ✵ . ✌✺ ● ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❌ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● ● ● u 1.5 ● ❌ ✶✺✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♊r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s ❊st✐♠❛t✐♊♥ ♊❢ γ ✿ ✵✳✟✌✹✞✞✳✳✳

  71. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❉✐✣❝✉❧t✐❡s ♊❢ ❡st✐♠❛t✐♊♥ ❢♊r γ ✳✳✳ ❆ss✉♠♣t✐♊♥ ✶ ♥♊t r❡s♣❡❝t❡❞✳✳✳ 3.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ❌ α = ✾ , γ = ✵ . ✌✺ ● ● ● 2.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ❌ ❙t❡❛❞② st❛t❡✿ 2.0 u = γ s ● ● ● ● u 1.5 ● ❌ ✶✺✵ ❝❡❧❧s✱ ● ● ● ● ● ● ● ● ● ● ● ✉♥✐❢♊r♠❧② ● ● 1.0 ● ● ● ● ● ● ❣❡♥❡r❛t❡❞ ✐♥ ● ● ● ● ● 0.5 ● ● t✐♠❡✳ ● ● ● ● ● ● ● ● ● 0.0 ● ● 0 1 2 3 4 s ❊st✐♠❛t✐♊♥ ♊❢ γ ✿ ✵✳✟✌✹✞✞✳✳✳ ❈♩rr❡❝t✐♊♥s❄ ❊st✐♠❛t✐♊♥ ♊♥ ✈❡r② ❝♩rr❡❧❛t❡❞ ❣❡♥❡s✱ ✜❧t❡r✐♥❣ s♊♠❡ ❝❡❧❧s✳✳✳

  72. ❜✉t ✶✶ ✪ s❀♊✇❡❞ s❡✈❡r❛❧ ❞❡❣r❛❞❛t✐♊♥ r❛t❡s✩ ❌ ❊①❛♠♣❧❡ ❢r♩♠ ❬✶❪✿ ◆tr❊✷ ❌ ❚❀❡♥ t❀❡ ♠♊❞❡❧ ❢❛✐❧s✳✳✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ▌✉❧t✐♣❧❡ s♣❧✐❝✐♥❣ ❌ ■♥ ❬✶❪✱ ± ✜✟ ✪ ♊❢ st✉❞✐❡❞ ❣❡♥❡s s❀♊✇❡❞ ❛ ✉♥✐q✉❡ ❞❡❣r❛❞❛t✐♊♥ r❛t❡ γ ✳✳✳

  73. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ▌✉❧t✐♣❧❡ s♣❧✐❝✐♥❣ ❌ ■♥ ❬✶❪✱ ± ✜✟ ✪ ♊❢ st✉❞✐❡❞ ❣❡♥❡s s❀♊✇❡❞ ❛ ✉♥✐q✉❡ ❞❡❣r❛❞❛t✐♊♥ r❛t❡ γ ✳✳✳ ❜✉t ✶✶ ✪ s❀♊✇❡❞ s❡✈❡r❛❧ ❞❡❣r❛❞❛t✐♊♥ r❛t❡s✩ ❌ ❊①❛♠♣❧❡ ❢r♩♠ ❬✶❪✿ ◆tr❊✷ ❌ ❚❀❡♥ t❀❡ ♠♊❞❡❧ ❢❛✐❧s✳✳✳

  74. ❌ ▌♊❞❡❧ ■ ✿ ✐s ❛ss✉♠❡❞ t♩ ❜❡ ❝♊♥st❛♥t❀ t❀❡♥✱ ✵ ✇✐t❀ ✵ ✳ ✵ ❌ ▌♊❞❡❧ ■■ ✿ ✐s ❛ss✉♠❡❞ t♩ ❜❡ ❝♊♥st❛♥t❀ t❀❡♥✱ ✵ ✵ ✵ ❚❀❡s❡ t✇♩ ♠♊❞❡❧s ❛r❡ ❝♩rr❡❝t ✐♥ t❀❡ s❀♊rt t❡r♠❀ t❀❡② ❀❛✈❡ t♩ ❜❡ ✉s❡❞ ✏st❡♣ ❜② st❡♣✑ t♩ ♣r❡❞✐❝t t❀❡ ❢✉t✉r❡ ✭▌❛r❩♩✈ ♣r♊❝❡ss✮✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❊st✐♠❛t✐♊♥ ♊❢ α ❆❝❝♩r❞✐♥❣ t♩ ❬✶❪✱ ✐t ✐s ✈❡r② ❞✐✣❝✉❧t t♩ ❡st✐♠❛t❡ α ✳✳✳ ❚✇♩ ❛♣♣r♩①✐♠❛t✐♊♥s ❛r❡ ❝♊♥s✐❞❡r❡❞✿

  75. ❌ ▌♊❞❡❧ ■■ ✿ ✐s ❛ss✉♠❡❞ t♩ ❜❡ ❝♊♥st❛♥t❀ t❀❡♥✱ ✵ ✵ ✵ ❚❀❡s❡ t✇♩ ♠♊❞❡❧s ❛r❡ ❝♩rr❡❝t ✐♥ t❀❡ s❀♊rt t❡r♠❀ t❀❡② ❀❛✈❡ t♩ ❜❡ ✉s❡❞ ✏st❡♣ ❜② st❡♣✑ t♩ ♣r❡❞✐❝t t❀❡ ❢✉t✉r❡ ✭▌❛r❩♩✈ ♣r♊❝❡ss✮✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❊st✐♠❛t✐♊♥ ♊❢ α ❆❝❝♩r❞✐♥❣ t♩ ❬✶❪✱ ✐t ✐s ✈❡r② ❞✐✣❝✉❧t t♩ ❡st✐♠❛t❡ α ✳✳✳ ❚✇♩ ❛♣♣r♩①✐♠❛t✐♊♥s ❛r❡ ❝♊♥s✐❞❡r❡❞✿ ❌ ▌♊❞❡❧ ■ ✿ v := ds dt ✐s ❛ss✉♠❡❞ t♩ ❜❡ ❝♊♥st❛♥t❀ t❀❡♥✱ s ( t ) = vt + s ✵ , ✇✐t❀ v := u ✵ − γ s ✵ ✳

  76. ❚❀❡s❡ t✇♩ ♠♊❞❡❧s ❛r❡ ❝♩rr❡❝t ✐♥ t❀❡ s❀♊rt t❡r♠❀ t❀❡② ❀❛✈❡ t♩ ❜❡ ✉s❡❞ ✏st❡♣ ❜② st❡♣✑ t♩ ♣r❡❞✐❝t t❀❡ ❢✉t✉r❡ ✭▌❛r❩♩✈ ♣r♊❝❡ss✮✳ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❊st✐♠❛t✐♊♥ ♊❢ α ❆❝❝♩r❞✐♥❣ t♩ ❬✶❪✱ ✐t ✐s ✈❡r② ❞✐✣❝✉❧t t♩ ❡st✐♠❛t❡ α ✳✳✳ ❚✇♩ ❛♣♣r♩①✐♠❛t✐♊♥s ❛r❡ ❝♊♥s✐❞❡r❡❞✿ ❌ ▌♊❞❡❧ ■ ✿ v := ds dt ✐s ❛ss✉♠❡❞ t♩ ❜❡ ❝♊♥st❛♥t❀ t❀❡♥✱ s ( t ) = vt + s ✵ , ✇✐t❀ v := u ✵ − γ s ✵ ✳ ❌ ▌♊❞❡❧ ■■ ✿ u ✐s ❛ss✉♠❡❞ t♩ ❜❡ ❝♊♥st❛♥t❀ t❀❡♥✱ ï¿œ ï¿œ s ( t ) = u ✵ s ✵ − u ✵ e − γ t . γ + γ

  77. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❊st✐♠❛t✐♊♥ ♊❢ α ❆❝❝♩r❞✐♥❣ t♩ ❬✶❪✱ ✐t ✐s ✈❡r② ❞✐✣❝✉❧t t♩ ❡st✐♠❛t❡ α ✳✳✳ ❚✇♩ ❛♣♣r♩①✐♠❛t✐♊♥s ❛r❡ ❝♊♥s✐❞❡r❡❞✿ ❌ ▌♊❞❡❧ ■ ✿ v := ds dt ✐s ❛ss✉♠❡❞ t♩ ❜❡ ❝♊♥st❛♥t❀ t❀❡♥✱ s ( t ) = vt + s ✵ , ✇✐t❀ v := u ✵ − γ s ✵ ✳ ❌ ▌♊❞❡❧ ■■ ✿ u ✐s ❛ss✉♠❡❞ t♩ ❜❡ ❝♊♥st❛♥t❀ t❀❡♥✱ ï¿œ ï¿œ s ( t ) = u ✵ s ✵ − u ✵ e − γ t . γ + γ ❚❀❡s❡ t✇♩ ♠♊❞❡❧s ❛r❡ ❝♩rr❡❝t ✐♥ t❀❡ s❀♊rt t❡r♠❀ t❀❡② ❀❛✈❡ t♩ ❜❡ ✉s❡❞ ✏st❡♣ ❜② st❡♣✑ t♩ ♣r❡❞✐❝t t❀❡ ❢✉t✉r❡ ✭▌❛r❩♩✈ ♣r♊❝❡ss✮✳

  78. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❚❀❛♥❊ ②♊✉ ❢♊r ②♊✉r ❛tt❡♥t✐♊♥✊

  79. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐①

  80. ❘◆❆ ❡q✉❛t✐♊♥s ✶ ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❆♥❞ ✐❢ t❀❡ ♣❛r❛♠❡t❡rs ❛r❡ ♥♊♥✲❝♊♥st❛♥t❄ ▌✉❝❀ ♠♩r❡ ❝♊♠♣❧❡①✳✳✳ ❊①❛♠♣❧❡ ❆ss✉♠❡ t❀❛t α ( t ) = ✶ − cos( t ) ✱ β = ✶✱ ❛♥❞ γ > ✵ ✐s ❝♊♥st❛♥t✳ 2.0 1.5 1 − cos(t) 1.0 0.5 0.0 0 5 10 15 20 25 t

  81. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❆♥❞ ✐❢ t❀❡ ♣❛r❛♠❡t❡rs ❛r❡ ♥♊♥✲❝♊♥st❛♥t❄ ▌✉❝❀ ♠♩r❡ ❝♊♠♣❧❡①✳✳✳ ❊①❛♠♣❧❡ ❆ss✉♠❡ t❀❛t α ( t ) = ✶ − cos( t ) ✱ β = ✶✱ ❛♥❞ γ > ✵ ✐s ❝♊♥st❛♥t✳ 2.0 1.5 1 − cos(t) 1.0 0.5 0.0 0 5 10 15 20 25 t ❘◆❆ ❡q✉❛t✐♊♥s  du  dt ( t ) = [ ✶ − cos( t )] − u ( t )   ds   u ( t ) − γ s ( t ) dt ( t ) = 

  82. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥s ❯♥s♣❧✐❝❡❞ ❘◆❆ ï¿œ ï¿œ u ( t ) = ✶ − ✶ u ✵ − ✶ e − t . ✷ (cos( t ) + sin( t )) + ✷ u ✵ = ✵ u ✵ = ✾ 3.0 1.5 2.5 2.0 1.0 u(t) u(t) 1.5 0.5 1.0 0.5 0.0 0 5 10 15 20 25 0 5 10 15 20 25 t t

  83. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥s ❙♣❧✐❝❡❞ ❘◆❆ ■❢ γ ï¿œ = ✶✱ s ( t ) = ✶ ✶ γ − ✷ ( ✶ + γ ✷ ) (( γ − ✶ ) cos( t ) + ( γ + ✶ ) sin( t )) ï¿œ ï¿œ + u ✵ − ✶ / ✷ s ✵ − ✶ γ + ✶ / ✷ − u ✵ γ − ✶ γ − ✶ e − t + e − γ t + ✷ ( ✶ + γ ✷ ) γ − ✶ ❛♥❞✱ ✐❢ γ = ✶✱ ￜￜ ï¿œ ï¿œ s ( t ) = ✶ − ✶ u ✵ − ✶ e − t . ✷ sin( t ) + t + s ✵ − ✶ ✷

  84. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① ❙♩❧✉t✐♊♥s u ✵ = s ✵ = ✵ , γ = ✵ . ✷ u ✵ = ✌ , s ✵ = ✺ , γ = ✵ . ✷ 5 8 4 7 3 s(t) s(t) 6 2 1 5 0 0 5 10 15 20 25 0 5 10 15 20 25 t t u ✵ = ✶ , s ✵ = ✷ , γ = ✷ . ✺ u ✵ = ✷✵ , s ✵ = ✺ , γ = ✶ 2.0 8 1.5 6 s(t) s(t) 1.0 4 0.5 2 0 5 10 15 20 25 0 5 10 15 20 25 t t

  85. ❙♊♠❡ r❡❝❛❧❧s ❆ ♠❛t❀❡♠❛t✐❝❛❧ ♠♊❞❡❧ ❘◆❆ ✈❡❧♊❝✐t② ❊st✐♠❛t✐♊♥ ♊❢ ♣❛r❛♠❡t❡rs ❛♥❞ ♣r❡❞✐❝t✐♊♥ ❆♣♣❡♥❞✐① P❀❛s❡ ♣♊rtr❛✐ts u ✵ = s ✵ = ✵ , γ = ✵ . ✷ u ✵ = ✌ , s ✵ = ✺ , γ = ✵ . ✷ 7 1.5 6 5 1.0 4 u(t) u(t) 3 0.5 2 1 0.0 5 6 7 8 0 1 2 3 4 5 s(t) s(t) u ✵ = ✶ , s ✵ = ✷ , γ = ✷ . ✺ u ✵ = ✷✵ , s ✵ = ✺ , γ = ✶ 20 1.6 1.4 15 1.2 u(t) u(t) 1.0 10 0.8 5 0.6 0.4 0 0.5 1.0 1.5 2.0 2 4 6 8 s(t) s(t)

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